Full text of "Rotorcraft dynamics"

See other formats

NASA SP-352

JL\.\^ir JL \«^r JL\.\_^A\.^ljLJL JL I *P JL J. ^I JT ^JLVJLJL\_aJ'

A conference held at

AMES RESEARCH CENTER

Moffett Field, California

February 13-15, 1974

US. ft-

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

NASA SP-352

ROTORCRAFT DYNAMICS

A conference sponsored by

Ames Research Center and the American Helicopter Society

and held at Ames Research Center, Moffett Field, California

February 13-15, 1974

Prepared at Ames Research Center

Scientific and Technical Information Office 1974

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

Washington, D.C.

For sale by the National Technical Information Service
Springfield, Virginia 22151
Price $8.00

PREFACE

Events of recent years have clearly identified rotorcraft dynamics as one of the most critical technologies
pacing the helicopter industry's efforts to develop new rotor concepts. And as rotary wing program
investments have escalated, the financial stakes in the technical risks have become tremendous. The rotor
dynamicist is suddenly very much in the critical path.

Fortunately Research and Development efforts by industry in rotorcraft dynamics have been greatly
augmented in recent years by stepped up in-house efforts on the part of NASA and the U. S. Army. New
computer tools and more complete experimental data coming from many quarters^are bringing us to the
threshold of a far more complete understanding of the problem. This increased R&D is reflected in the
number of high quality papers that have exceeded the capacity of the dynamics session at recent AHS
Annual Forums.

With these thoughts in mind, the Specialists' Meeting on Rotorcraft Dynamics was organized to provide an
opportunity for the principal investigators in the field to dialogue in greater depth than is possible at the
American Helicopter Society Annual Forum. This is the first such meeting in the U. S. since the
CAL/TRECOM Helicopter and V/STOL Dynamic Loads Symposium in 1963.

This volume contains the formal presentations of the first four sessions of the meeting. Presentations of the
fifth session and transcriptions of questions and panel discussions are contained in the supplement to this
volume.

E. S. Carter, Jr.

Meeting General Chairman

ill

ORGANIZING COMMITTEE

General Chairman

Edward S. Carter, Jr., Sikorsky Aircraft

Technical Chairman

Robert A. Ormiston, U. S. Army Air Mobility R&D Laboratory,
Ames Directorate

Administrative Chairman

James C. Biggers, NASA-Ames Research Center

Chairman, AHS Dynamics Technical Committee
E. Roberts Wood, Lockheed-California Company

Session Chairmen and Co-Chairmen

Session I

Kurt H. Hohenemser, Washington University
James R. Neff , Hughes Helicopters

Session II

E. Roberts Wood, Lockheed-California Company
G. Alvin Pierce, Georgia Institute of Technology

Session HI

Peter J. Arcidiacono, Sikorsky Aircraft
William E. Nettles, U. S. Army Air Mobility R&D Laboratory
Eustis Directorate

Session IV

James J. CLeary, Boeing Vertol Company

William G. Flannelly, Kaman Aerospace Corporation

Session V

Troy M. Gaffey, Bell Helicopter Company

CONTENTS

Paper Page

Preface iii

Organizing Committee iv

SESSION I - ROTOR SYSTEM DYNAMICS

1 Hingeless Rotor Frequency Response with Unsteady Inflow

D.A.Peters 1

2 Dynamic Stall Modeling and Correlation with Experimental Data
on Airfoils and Rotors

R. G. Carlson, R. H. Blackwell, G. L. Commerford and P. H. Mirick 13

3 Computer Experiments on Periodic Systems Identification Using
Rotor Blade Transient Flapping-Torsion Responses at High
Advance Ratio

K. H. Hohenemser and D. A. Prelewicz 25

4 Dynamic Analysis of Multi-Degree-of-Freedom Systems Using
Phasing Matrices

R. L. Bielawa 35

5 Some Approximations to the Flapping Stability of Helicopter Rotors

J. C. Biggers 45

6 Flap-Lag Dynamics of Hingeless Helicopter Blades at Moderate and
High Advance Ratios

P. Friedmann and L. J. Silverthom 55

SESSION II - HELICOPTER VIBRATION AND LOADS - THEORY

7 Correlation of Finite-Element Structural Dynamic Analysis with
Measured Free Vibration Characteristics for a Full-Scale
Helicopter Fuselage

I. J. Kenigsberg, M. W. Dean and R. Malatino 67

8 Coupled Rotor/ Airframe Vibration Prediction Methods

J. A. Staley and J. J. Sciarra 81

9 Helicopter Gust Response Characteristics Including Unsteady Aerodynamic
Stall Effects

P. J. Arcidiacono, R. R. Bergquist and W. T. Alexander, Jr 91

10 Application of Antiresonance Theory to Helicopters

F. D. Bartlett, Jr. and W. G. Flannelly 101

Paper Page

1 1 The Effect of Cyclic Feathering Motions on Dynamic Rotor Loads

K. W. Harvey , 107

12 Control Load Envelope Shaping by live Twist

F. J. Tarzanin, Jr. and P. H. Mirick 115

13 Application to Rotary Wings of a Simplified Aerodynamic Lifting
Surface Theory for Unsteady Compressible Flow

B. M. Rao and W. P. Jones 127

SESSION III - ROTOR/VEHICLE DYNAMICS

14 Rotor Aeroelastic Stability Coupled with Helicopter
Body Motion

Wen-Liu Miao and H. B. Huber 137

15 An Application of Floquet Theory to Prediction of Mechanical
Instability

C. E. Hammond 147

16 Theory and Comparison with Tests of Two Full-Scale Proprotors

W. Johnson 159

17 Experimental and Analytical Studies in Tilt-Rotor Aeroelasticity

R. G. Kvaternik 171

18 Comparison of Flight Data and Analysis for Hingeless Rotor
Regressive Inplane Mode Stability

W. D. Anderson and J. F. Johnston 185

19 Hub Moment Springs on Two-Bladed Teetering Rotors

W. G. O. Sonneborn and J. Yen 199

20 Open and Closed Loop Stability of Hingeless Rotor Helicopter Air
and Ground Resonance

M. I. Young, D.J. Bailey and M.J. Hirschbein 205

SESSION IV -HELICOPTER VIBRATION AND LOADS - APPLICATIONS

21 Vertical-Plane Pendulum Absorbers for Minimizing Helicopter
Vibratory Loads

K. B. Amer and J. R. Neff 219

22 The Evaluation of a Stall-Flutter Spring-Damper Pushrod in
the Rotating Control System of a CH-54B Helicopter

W. E. Nettles, W. F. Paul and D. 0. Adams 223

23 Multicyclic Jet-Flap Control for Alleviation of Helicopter Blade
Stresses and Fuselage Vibration

J.L.McCloudIIIandM.Kretz . 233

Paper Page

24 Identification of Structural Parameters from Helicopter Dynamic
Test Data

N.Giansante and W. G. Flannelly 239

25 Engine/Airframe Interface Dynamics Experience

C. A. Fredrickson 249

26 Hingeless Rotor Theory and Experiment on Vibration Reduction by
Periodic Variation of Conventional Controls

G. J. Sissingh and R. E. Donham 261

SUPPLEMENT

Foreword to Supplement 279

Welcome

C. A. Syvertson 280

Opening Remarks
E. S. Carter . . 280

Dinner Address - What Can the Dynamicist Do for Future Army Aircraft?
P. F. Yaggy 281

Session V — Application of Dynamics Technology to Helicopter Design 283

Panel 1 : Prediction of Rotor and Control System Loads

Panel Members 283

Comparison of Several Methods for Predicting Loads on a Hypothetical Helicopter Rotor
R. A. Ormiston , 284

Discussion, Panel 1 303

Prepared Comments

W. D. Anderson 303

P. J. Arcidiacono 304

R. L. Bennett 306

W. Johnson 307

A. Z. Lemnios 307

R. H. MacNeal • • • 308

F. J. Tarzanin 309

R. P. White 311

Survey of Panelists 313

Questions and Answers, Panel 1 314

Vil

Page

Panel 2 : Control of 1 /Rev Vibration

Panel Members 315

The User's Problem

R. J. van der Harten 316

D..F, Benton 320

Technical Aspects of 1 /Rev Vibration

W. F. Wilson . . , 331

Discussion, Panel 2 . 322

Questions and Answers, Panel 2 326

Panel 3 : Integrating Dynamic Analysis and Helicopter Design

Panel Members 327

Discussion, Panel 3 328

Prepared Comments

R. W. Balke 328

R. Gabel 329

J. F. Johnston 332

J. R. Neff 334

W. F. Paul 335

Questions and Answers, Panel 3 / 338

Questions and Answers — Sessions I— IV

Session I — Rotor System Dynamics 339

Session II — Helicopter Vibration and Loads — Theory 347

Session HI - Rotor/Vehicle Dynamics 355

Session IV — Helicopter Vibration and Loads — Applications 361

List of Attendees 367

Vlll

HIHGELESS ROTOR FREQUENCY RESPONSE WITH UNSTEADY INFLOW

U.S.

David A. Peters
Research Scientist

Ames Directorate
Army Air Mobility R&D Laboratory
Moffett Field, Calif. 94035

Abstract

Hingeless rotor frequency response calcula-
tions are obtained by applying a generalized har-
monic balance to the elastic blade flapping equa-
tions. Nonuniform, unsteady induced flow effects
are included by assuming a simple three-degree-
of-freedom description of the rotor wake. Results
obtained by using various models of elastic blade
bending and induced flow are compared with exper-
imental data obtained from a 7.5-ft diameter wind
tunnel model at advance ratios from 0.0 to 0.6.
It is shown that the blade elasticity and nonuni-
form, unsteady induced flow can have a signifi-
cant effect on the transient response character-
istics of rotor systems. Good correlation be-
tween theory and experiment is obtained by using:
(i) a single rotating mode shape description of
the elastic blade bending, (ii) an empirical form-
ula for the quasi-steady induced flow behavior,
and (iii) the apparent mass terms from potential
flow for the unsteady induced flow characteris-
tics.

Notation

a jn> b jn

c M

e
e pc

{£}
F.G

go>gs»Sc

[I]

two-dimensional lift-curve slope,

rad -1

harmonics of jth flapping mode

number of blades

tip loss factor

blade chord, ft

steady value of thrust coefficient,

steady thrust/pirn 2 R l *

harmonic perturbation of thrust

coefficient

harmonic perturbation of roll moment

coefficient « roll moment/pir$2 2 R 5 ,

positive advancing blade down

harmonic perturbation of pitch

moment coefficient = pitch moment/

pirfi 2 R 5 , positive nose up

dimensionless flapping hinge offset

dimensionless radius of pocket

cutout

rotor blade bending stiffness, .

lb-ft 2

generalized response vector

aerodynamic and inertial forces per

unit blade span, lb/ft

nondimensional harmonics of inertial

forcing function, Eq. (12)

apparent inertia of air, slug-ft 2

identity matrix

index referring to mode number

[K]
[L]
[L E ]

ffi.niyy.myj

M,[M']

dm
dm

m A

[0]

r
R
S(0,i{i)

{u}
Up,U T

U„o,V„„

v(o,<|0

w
[W]
{x}
[Y]

number of flap bending modes
nondimensional apparent mass and in-
ertia of impermeable disk
control feedback matrix
nonuniform induced flow matrix
empirical value for quasi-steady
portion of [L]

rotor blade mass distribution, slug/ft
nondimensional blade parameters
R -R

pacR^

pacR d

m dr,

pacR 1 *

mr^ dr,

mr<j>, dr

rotor response matrix open loop,

closed loop

elemental apparent mass, slugs

elemental mass flow, slugs/sec

apparent mass of air, slugs

index referring to harmonic number

number of azimuthal harmonics

null matrix

first flap frequency divided by Q

generalized coordinates

steady values of q^

rotor blade radius coordinate, ft

rotor blade radius, ft

blade root moment, ft-lb

blade parameter

pacR2

m *j

Presented at the AHS/NASA-Ames Specialists' Meet-
ing on Rotor craft Dynamics, February 13-15, 1974.

generalized control vector

perpendicular, tangential components of

air speed in undef ormed blade coordi-
nate system, ft/sec
freestream airspeed perpendicular and
parallel to rotor shaft (V» positive
down), ft /sec
induced flow parameter «■

[y 2 + A(X + v)]/(y 2 + T 2 ) 1/2

blade root shear, lb

rotor blade flap deflection, ft

frequency transform, Eq. (23)

physical control vector <@ o s c z<f>a>

control coupling matrix

hub plunge deflection divided by R,

positive down

hub pitch angle, positive nose up,

rad

Y
Y*

e o» 6 s» 6 c
X

Lock number, 1/myy

equivalent Lock number, Eq. (35)

blade pitch angle = 5 + (6 + 6 S

sin iji + 6 C cos ^)e ±la ^

steady collective pitch angle

rotor pitch perturbations

total inflow (including induced flow)

= X +

X b f sln

X c f cos

iuf

^o»^s»^c

steady inflow ratio = V^/QR + v
inflow perturbations (including in-
duced flow), Eq. (10)
advance ratio = U<»/fiR
total induced flow =

v +

+ v

S R

sin iji
itoijj

v o» v s» v c

r T ,T S

♦J

u
SI

+ v c — cos i|> I e'

induced flow due to steady rotor
thrust

induced flow perturbations
air density, slug/ft 3
rotor solidity, bc/ffR
induced flow time constants, rad~l
hub roll angle, positive advancing
blade down, rad
orthogonal functions
rotor blade azimuth position, non-
dimensional time, rad
' excitation frequency divided by Si
rotor blade angular velocity,
rad /sec
3/3r
3/3*

The dynamic response characteristics of
hingeless rotors are dependent upon the distrib-
uted structural properties of the rotor blades,
the local aerodynamic properties of the blade
sections, and the detailed description of the
aerodynamic environment. It is generally be-
lieved i however j that reasonable predictions of
rotor thrust and moments at low lift can be ob-
tained by using some appropriately simplified
models for the blade structure, section aero-
dynamics^ and inflow distribution. The develop-
ment of these simplified rotor models is useful
for gaining insight into the basic dynamic mech-
anisms of rotor response. Detailed calculations
of dynamic airloads, necessary for many applica-
tions, are usually too complex for use in basic
dynamic research or preliminary design calcula-
tions.

The formulation of a minimum complexity
rotor response model is the subject of several
recent papers. One area of interest is the ef-
fect of mode shape and mode number on rotor
flapping response. Shupe 1 addresses the effects
of the second flap mode, Ormiston and Peters 2
compare various mode shape models for first and
second flap modes, and Hohenemser and Yin 3 con-
sider the effect of using rotating rather than
nonrotating modes as generalized degrees of
freedom. The fundamental conclusion, as

clarified in Reference 3, is that for u < 0.8 a
single rotating mode shape is adequate for model-
ing the steady rotor flapping response.

A second area of interest is the effect of
induced flow perturbations on rotor flapping
response. In Reference 1, a simple momentum
theory predicts a significant effect of induced
flow on steady rotor response. In Reference 2', a
comparison of steady experimental and theoretical
results indicates that, although there is a sig-
nificant effect due to induced flow, momentum
theory is inadequate for predicting this effect in
forward flight. Alternate induced flow models are
introduced and compared with the data, but no
clear choice for the best model is found. In
Reference 4, an unsteady momentum theory is used
in hover to improve correlations with experimental
frequency response data. '

The work in References 1 through 4 indicates
that a minimum complexity analytic model for
rotor dynamics must include appropriate degrees of
freedom for both structural and induced flow per-
turbations (certain flight dynamics programs
presently include a simplified dynamic treatment
of the induced f low^) . Unfortunately, while some
success has been achieved using simple models of
the rotor induced flow in hover, a completely
satisfactory induced flow model for forward flight
has not been found, not even for the condition of
steady response. In addition, neither the phy-
sical values of the induced flow time constants
nor the frequency range in which they are impor-
tant is known. The unsteady behavior of the in-
duced flow contributes directly to the low fre-
quency rotor control characteristics and to the
coupled rotor/fuselage aeroelastic stability. In
particular, induced flow perturbations contribute
to the rotor damping available in pitch and roll
(which is important for ground and air resonance
calculations) . It is consequently important to
understand the dynamic characteristics of the in-
duced flow.

The purpose of this paper is to provide ad-
ditional insight into the question of rotor
structural and induced flow modeling. To this
end, experimental rotor frequency response data
in hover and in forward flight are compared with
theoretical results that are calculated by using
several different models for the elastic blade
bending and induced flow. The frequency response
data provide a broad base of comparison so 'that
the effects of mode shape and induced flow model
can be clearly determined throughout the fre-
quency range of interest.

Basic Equations

Analysis

The mathematical technique used here is a
further generalization of the harmonic balance
approach of Reference 2. In addition to an arbi-
trary number of bending modes (with an arbitrary
number of azimuthal harmonics for each mode), the
generalized harmonic balance allows for a

rational treatment of reversed flow aerodynamics
and the possibility of harmonically oscillating
control inputs.

The linear equation of motion for the de-
flection of an elastic beam subject to distrib-
uted aerodynamic and inertial loadings F(r,i)0
and G(r,ijO is 6

(EIw") + m« 2 w + n 2 (mrw' - w" f mr dr 1

are obtained for the aj n and bj n . Solution of
these equations, followed by a substitution of Eq.
(5) into Eqs. (2) and (3), results in the phase
and magnitude of all desired harmonics of the
flapping deflections and hub forces and moments.

Blade Loading

The aerodynamic loading of each Blade is
given by

F(r,i|0 + G(r,<J;)

(1)

pac

l u T Kv- u p)

(6)

The associated expressions for bending moment and
shear at the blade root are

S(0,*) » J (F + G - mfl 2 w - mfl 2 w)r dr (2)

where

U = S2r + fiRu sin <fi

(7)

V(0,*)

,R
J (F + G - mC 2 w)dr

(3)

U = Qw + SiRX + fiRuw 1 cos ii

(8)

The blade root bending moment is transformed into
a stationary coordinate system to yield the pitch
and roll moment of the rotor. The solution of
Eq. (1) yields directly the blade deflections,
and substitution into Eqs. (2) and (3) then
yields the forces and moments.

Application of the harmonic balance involves,
first of all, an orthogonal expansion of w:

R = 2v
J-l

qjOlOfjCr)

(4)

For the present analysis, the <f>* are taken to
be the exact mode shapes of the rotating beam
without aerodynamics. Galerkin's method is then
used to transform Eq. (1) into J ordinary dif-
ferential equations (with periodic coefficients)
for the modal coordinates qj.^ When the forcing
terms contain a steady portion superposed onto
periodic functions that are modulated by an ex-
citation frequency u (cycles per revolution) ,
Floquet's theorem implies that the qj have a
solution of the form'

q J = q 3 +

jo + S [ a jn cos(n "' )

n=l

+ b. sin(ni())

•]

*«>*

(5)

where q. are the steady coning displacements
and the aj n and bj n are complex quantities
indicating the magnitude and phase shift of each
modulated harmonic of the perturbation response.
The harmonic balance approach entails substitut-
ing Eq. (5) into the J ordinary differential
equations for qj and setting coefficients of
like harmonics equal. When n is truncated at
the highest harmonic of interest N, then
(2 • N + 1) • J linear algebraic equations

Eq. (8) contains the primary contributions of mode
shape and induced flow to the flapping equations.
The details of blade mode shape become important
as u increases because Up depends upon both
the blade deflection w and its first derivative
w" . The induced flow is important because first
order perturbations to the inflow X create first
order changes to Up and F.

Although the inflow is in general a compli-
cated function of radius and azimuth, as a first
approximation, the total inflow can be represented
by

A +

A + X

f sin

ijj +

c R

cos ifi e

lWTp

(9)

The steady portion of the total inflow X con-
tains contributions from the f reestream velocity
V„/nR and from the steady induced flow due to
rotor thrust v. The unsteady inflow components
X ,X S ,X C contain contributions from harmonic
plunging ze iu *, rolling ()>e iw *, and pitching
ae"' of the shaft, as well as contributions
from the unsteady induced flow components
v ,v s ,v due to perturbations in rotor thrust and
moments :

X = -iwz + v
o o

-ioxji + v
-iuio + v

(10 a-c)

c c

The blade pitch angle 9 is given by

= 6 +

8 + 9 sin ifj + 8 cos ty

iuif.

(11)

where 8 is the steady value of 6 and
8 ,8 S ,8 are control system perturbations. The
inflow perturbations X ,X g ,X are assumed to be

small compared with unity. This implies that the
induced flow perturbations v ,v s ,v c and the con-
trol perturbations 9 ,6 s ,8 c ,z,<j>,a are also small
quantities yielding linear perturbation equa-
tions.

The inertial loading of each blade is given

moments influence the induced flow. The induced
flow, therefore, is a feedback loop of Eq. (15) ,
causing the uj to depend upon the f 4 .

From the standpoint of calculation, it is con-
venient to express the coupling relation (between
the generalized controls, the physical controls,
and the rotor response) in matrix form:

G = -mfi 2 R[g + g s | sin * + g c | cos * e*" r (12)

iuij,

{u> = me*} + [K]{f>

(17)

where

8 o

— g

U) 2 Z

g s

to 2 * +

2iua

S c

-2iu<(>

+ <o 2 a

(13 a-c)

The inertial loading is a result of centrifugal,
Coriolis, and gyroscopic forces which occur in
the rotating reference frame of the blade due to
hub motions z,<(i,a in the inertial reference
frame.

When Eqs. (6) through (12) are combined and
appropriately integrated in Eqs. (1), (2), and
(3), the steady deflections and forces qj.Oj/aa
are obtained as linear functions of the steady
inputs 6, A; and the perturbation blade deflec-
tions and hub forces and moments are obtained as
linear combinations of the generalized control
variables

<u> =(ee e gggHi \

N o s c £> o°s B c o s c S

(14)

Although g ,g s ,g c are simply related to the
shaft motion through Eq. (13), they are retained
as generalized controls so that the generalized
controls can be separated into physical, in-
ertial, and aerodynamic groupings. This will
facilitate the calculation of rotor response when
induced flow is included later.

Interpretation of Results

The results of the harmonic balance can be
expressed in matrix form as

{f} - [M]{u}

(15)

where {f } represents the perturbation harmon-
ics of thrust, moments, and generalized coordi-
nates. The elements of [M] , therefore, have
direct physical significance. They are the par-
tial derivatives of each of the response harmon-
ics taken with respect to each of the generalized
controls uj. The generalized control variables
are in turn functions of the physical controls

<x> "(s 8 6 zAaS

^ O S C r

(16)

Eq. (17) is simply a set of linear equations de-
scribing: (i) the generalized control perturba-
tions due to application of the physical controls
[Y] and (ii) the generalized control perturba-
tions due to the effect that rotor response has
on the induced flow [K] . The matrices [Y] and
[K] will be obtained later by using an appropri-
ate induced flow model. It follows that the par-
tial derivatives of the f j with respect to the
physical controls -x.± can be found (including in-
duced flow effects) from Eqs. (15) and (17). The
derivative matrix is designated [M'] and has the
properties

{f} - [M']{xl

[M'l = [[I] - [M][K]

(18)
(19)

Although the higher harmonics are often necessary
in the harmonic balance calculation of [M] , the
subsequent calculation of [M'] by Eq. (19) may
be performed for only those response and inflow
harmonics of interest. In this paper, five har-
monics are used in the calculation of [M] , but
only first harmonics are retained in Eqs. (18) and
(19), so that the f*. are taken to be

f j

<f> / C T C L C M „ \

<f> = \«M« a JoVw

(20)

Induced Flow

Form of Induced Flow Model

A useful form of the induced flow model is
given by*

v o

. c

Cj/aa
C L /aa
C M /aa

aerodynamic only

(21)

Although not completely general, Eq. (21) can
accommodate a variety of induced flow models.
Only aerodynamic contributions are included on
the right-hand side, because they are the only
loads which produce reaction forces on the rotor
wake. Using Eqs. (2) and (3), these aerodynamic
forces and moments can be expressed in matrix
form as

as evidenced in Eqs. (10) and (13).

The generalized control variables uj axe
also coupled to the fj, because the thrust and

LC m /caJ

1 I

(C /era)

J 1

/C„/aa\
^ M ^aero-
dynamic

where

iC T /aa,
?C L /aa(

0a 1

- [W]

- — m
2 yy

-2 m yy

J 1

-Kj

2 yj.

a ji

(22)

directly in Eq. (19) to obtain the complete rotor
response to physical" control inputs .

Unsteady Momentum Theory

An approximation of the induced flow that is
suitable for Eq. (21) can be obtained as an ex-
tension of the momentum theory used in Reference
2. The differential force on an elemental area of
rotor disk is written as

dF = 2£2Rvdm + fl 2 Rvdm

(26)

where 2QRv is the total change in velocity nor-
mal to the disk, dm is the differential mass flow
through the element, dm is the # apparent mass
associated with the flow, and v is the time
derivative of v in the nonrotating system. The
differential mass flow relation

w=-

<u 2
-2iu)

2i<u

U, 2

(23)

With the induced flow v described by Eqs. (21)
and (22), the inflow relation follows directly
from Eqs. (10), (13), and (21). The matrices
[Y] and [K] of Eq. (17) may then be identified
as

m -

C"j,

°3*3 [t] [W]

Ky
o

(24)

[K]

3*3J

ai,

-[L] [W]

J 1

J J 3*3J

(25)

Eq. (24) represents the control coupling be-
tween the physical controls Xj and the general-
ized controls u-j. The presence of [L] in this
matrix indicates that the X's are indirectly
coupled (through the induced flow), as well as
geometrically coupled [Eqs. (10) and (13)] to the
rotor plunge, pitch, and roll motions. Eq. (25)
represents the induced flow caused by the de-
pendence of X upon the thrust and first harmon-
ic flapping. If a suitable approximation to the
inflow can be modeled in the form of Eq. (21),
then Eqs. (24) and (25) may be substituted

dm = pflR Vu z + A 2 r dr diji

(27)

can be used to integrate the first term of Eq.
(26) over the disk to obtain a quasi-steady in-
duced flow relation for rotors that have combined
conditions of thrust and forward speed. The eval-
uation of the second term in Eq. (26) (the un-
steady effect) requires the additional knowledge
of the apparent mass dm associated with the flow.

An approximation to the apparent mass terms
of a lifting rotor can be made in terms of the
reaction forces (or moments) on an impermeable
disk which is instantaneously accelerated (or ro-
tated) in still air. This approximation was used
in Reference 8, giving good agreement with trans-
ient thrust measurements for an articulated rotor.
The reactions on such an impermeable disk are
given from potential flow theory in terms of el-
liptic integrals which are evaluated in the lit-
erature. 9 They result in apparent mass and in-
ertia values

m A = f pR3 ,

^-if* 8

(28)

(For

v_ r/R, a radial velocity distribution,

m A = pR 3 .) These values represent 64 percent of
the mass and 57 percent of the rotary inertia of
a sphere of air having radius R. It is empha-
sized that they are only approximations to the
actual values for a lifting rotor.

Using this approximation, the steady induced
flow equation and the unsteady induced flow per-
turbation equations can be derived from Eqs. (26)
through (28):

2v \/u 2 + X 2 - C T
Vo + 2vV o " C T
Kl v s +|vv 8 --C L

(29a)

(29b-d)

where

v= u 2 + X(X + v)

vV + X 2

(29e)

and

K = __A_ = JL = 0.8488
m pnR 3 3ir

A 16
^ s — &- = -^2- = 0.1132

(30a-b)

pirR 5 45ir

Eq. (29a) expresses the nonlinear relation be-
tween the steady thrust and the steady induced
flow v. Eqs. (29b-d) are then the linear per-
turbation equations for small changes in thrust,
moments, and induced flow. In order for the per-
turbation equations to be valid, it is assumed
that v ,v s ,v c are much smaller than Cli 2 + X 2 )\

The time constants associated with the induc-
ed flow model in Eq, (29) are

(24) , (25) , and (19) to obtain the rotor response
that includes inflow.

Empirical Model

Experimental data have shown that momentum
theory, although particularly simple to use, is
qualitatively inaccurate for certain steady re-
sponse derivatives in forward flight. Reference
2 introduces an alternate induced flow model for
forward flight in which the elements of [L]
(with u = 0) are chosen to give the best fit of
experimental response data for several configura-
tions at conditions of near zero lift. If this
empirical inflow model, [Lg] , is taken for the
quasi-steady portion of the induced flow law, and
if the theoretical apparent mass terms (from po-
tential flow) are taken as a model for the un-
steady portion of the induced flow law, then a
complete induced flow equation can be expressed as

1_
oa

-K.

I
-K,

(33)

t T = -^ = 0.4244/v (for v q )

2K I

(31a-b)

T = = 0.2264/v (for v ,v )

S v s' c

In Reference 4, the steady induced flow v and
the time constant for v s ,v c are obtained by
correlating experimental hover frequency response
data. Two operating conditions are considered,
and the best fit in these cases is found to be
v - .014, t s = 8 (with 8 = 2°) and v = .028,

l s

4 (with 8=8°). From the

values indi-

cated for these cases, it can be shown that each
t s implies the same value of Kj = 0.112. Thus,
there is some experimental evidence that the po-
tential flow value
valid .

Kj; = 0.113 is approximately

By assuming simple harmonic motion, Eqs.
(29b-d) can be brought into the form of Eq. (21) ,
yielding the components of [L] for unsteady
momentum theory.

<3&

[L]

2v + K iw

-era

v/2 + Rjiio

v/2 + ICj.iu)

(32)

(L22 ana L 33 differ by a factor of 4/3 from Ref-
erence 2, because v s and v c are taken uniform
with r in that reference, whereas they are
taken linear with r here.) The matrix [L]
from Eq. (32) may now be substituted into Eqs.

The assumption that the apparent mass terms
may be superposed on the quasi-steady terms is not
rigorous, but it can be considered analogous to
unsteady wing theory in which the apparent mass
terms are theoretically independent of the free-
stream velocity. Under the superposition assump-
tion, the empirical inflow model modified for the
unsteady case is

[L]

~ K T

-K

I- 1

I 1-1

-1

(34)

Although this particular formulation of [L] is
valuable for predicting the effects of induced
flow, ultimately a more consistent formulation of
[L] should be made, as discussed in Reference 10.

Equivalent Lock Number

Another method of accounting for the unsteady
induced flow is the use of an equivalent Lock num-
ber y*, which can be derived from a single har-
monic balance of the root moment equation:

1 -

1 + 8v/oa + 16K Waa

(35)

Although this approach is not a completely con-
sistent treatment of the induced flow, since it
does not give an exact harmonic balance of the
blade flapping and thrust equations, it yields
results which are nearly the same as those ob-
tained from momentum theory.

The practical use of Eq. (35) is somewhat
limited because of the inaccuracies of momentum
theory in forward flight, but a y* approach is
nevertheless a valuable conceptual tool for under-
standing the effects of induced flow. In particu-
lar, Eq. (35) shows that one effect of induced
flow perturbations is to decrease the effective
Lock number (i. e. , decrease the aerodynamic ef-
fectiveness) . This decrease is most pronounced at
low values of v (i.e., low u and 8 ) and low
values of w. For example, rotor roll moment is
plotted in Figure 1 for two values of and com-
pared with the value from elementary theory
(steady induced flow only, induced flow perturba-
tions neglected, equivalent to lim 8 •*■ °° )'. The
curves for 8 " 0, 0.05 result in values of roll
moment well below the elementary value.

.020

NO INDUCED FLOW
PERTURBATIONS \^

8=oo

.05

.015

.010

/ /

.005

&£^ X 1

.2 .3

Sst rod

Figure 1.

Effect of induced flow on steady rotor
response in hover, u = 0, m = 0,
a ■ 0.1, a = 2ir, p » <*>.

The effect of induced flow is most pro-
nounced in the response derivative (the slope of
the response curve at 8 S • 0) . For p = °°, the
derivative is given by

sfcj/oaj/i

8 =0
1 s

_1_
16

(l + 3/2 u 2 ) (36)

indicating that y*/y < 1 results in a reduction
of the roll moment response (or control power)
from the elementary value. When the rotor is in
hover with no lift (v » 0) , a quasi-steady per-
turbation of 8 S (u » 0) results in no response
because of the zero slope of the curve in Figure
1. The mathematical justification for the van-
ishing response derivative can be seen in Eqs.
(35) and (36). With u - v - 0, y*ly and the
response derivative must equal zero. As 9" in-
creases, however, v and v increase so that
Y*/y approaches unity and the derivative ap-
proaches -1/16, as illustrated in Figure 2. With-
in the practical range of thrust coefficients,
however, the response derivative never recovers
more than about 80 percent of the elementary
value. Eq. (35) also implies that increasing ad-
vance ratio (which increases v) will result in a
partial recovery of y*/y (and of the response
derivative). This recovery is evident in

-.08

NO INDUCED FLOW
•PERTURBATIONS^,^ -_„

-.06

-.04

' ^^-~~~~~

-.02

i i — i 1

.05

.10
I

.12

Figure 2.

C T Ar

Eff ect of induced flow on steady rotor
response derivatives in hover, u = 0,"
to = 0, = 0.1, a = 2ir, p = ».

Figure 3, where the roll response is given versus
u; but no more than 90 percent of the elementary
value is reached in the practical range of thrust
and advance ratio.

-.10

NO INDUCED FLOW -^
PERTURBATIONS \^"^

-.08

F = co —

£ -.06

■ 15 —

£-.04
*>

- .05^^

-.02

i i i i i

.1 .2 .3 .4 .5 .6

Figure 3. Effect of induced flow on steady rotor
response derivatives in forward
flight, w • 0, a = 0.1, a « 2ir, p » ».

The unsteady terms (apparent inertia K_)
also bring y*/y closer to unity, as seen by the
role of Kx in Eq. (35). This recovery with fre-
quency is illustrated in Figure 4, where, as us
becomes large, the response derivative approaches
the elementary value of -1/16. The rate at
which the response approaches -1/16 is dependent
upon the magnitude of the apparent inertia Kj.
Large values of Kx result in a rapid return to
the elementary value, and small values of Kx re-
sult in a slow return. For Kx " 0.1132 and
u < 0.3, the unsteady terms provide only small
contributions to the response. Thus, the quasi-

4 o

Sf 200

.ua

NO INDUCED FLOW
K I = O / PERTURBATIONS

.06
.04

/ .M32_^^ ^ . — '

.02

\. QUASI-STEADY
INDUCED FLOW

1 I i i i i

1.0

1.2

132

-£.

0, oo

1.0

1.2

Figure 4.

Effect of induced flow time constant on
rotor frequency response derivatives,
u = 0, a = 0,1, a = 2ir, p = =°,
v = X = 0.05.

steady theory (with Ki = 0) would be adequate in
this range. In the frequency range 0.3 < u> <
1.0, the unsteady terms have a more significant
effect. Above m = 1.2, the total effect of in-
duced flow diminishes so that the elementary
theory and the unsteady theory give similar re-
sults; but the quasi-steady theory (with Ki = 0)
is in considerable error in this region.

The frequency range in which unsteady in-
duced flow is important_is also dependent upon the
thrust or mean inflow v as shown in Figure 5.
For low values of v, the unsteady effects domi-
nate at low frequencies; and for large values of
v, the unsteady effects are delayed into the
higher frequencies. This effect is implicit in
Eq. (35) and is a direct result of the inverse
dependence of time constant upon v, Eq. (31) .
Thus, a low v implies a slow induced flow re-
sponse; and a high v implies a rapid induced
flow response. Equation (35) shows that advance
ratio (which also increases v) • has a similar ef-
fect on the induced flow behavior. It follows
that the relative importance of the unsteady and
quasi-steady nonuniform induced flow terms de-
pends upon both the rotor operating conditions
and the frequency range of interest.

.08

.06

-, .04

.02 -

f = 00

^N0 INDUCED FLOW PERTURBATIONS

^=r^^^^^

' .05 y

iii.,

Figure 5. Effect of unsteady induced flow on

rotor frequency response derivatives,
u = 0, a = 0.1, a •* 2ir, p = ■»,
Kj. = 0.1132.

NO INDUCED FLOW

QUASI -STEADY INDUCED FLOW

UNSTEADY INDUCED FLOW

HIGH /J.

MODERATE p.
LOW/Lt i

OR HIGH ~T~
C T /<r ,

VERY LOW jJl
AND LOW _!_
C T /o-

O .5

ROTOR /FUSELAGE DYNAMICS

INDUCED

FLOW

UNSTEADY

INDUCED FLOW

1.5 2.0

BLADE DYNAMICS

In Figure 6, the relative importance of
these terms is presented qualitatively through a
chart of the operating regimes in which (for no
induced flow or quasi-steady induced flow) |y*j
differs by less than 10 percent from the unsteady
value. This is a subjective criterion and is
merely intended to illustrate the trends with
thrust, advance ratio, and frequency. Four
regions are defined: (i) at high u and v,
induced flow effects are small and either the
elementary or quasi-steady approximation is ade-
quate; (ii) at high u and low v, although in-
duced flow effects are small (no induced flow
being a good approximation) , the quasi-steady

Figure 6. Regions of validity for steady (no
induced flow perturbations) quasi-
steady (Kj = K = 0) , and unsteady
(K x = 0.1132, \ = 0.8488) induced
flow models based on y*» Eq. (35),
a = 0.1, a = 2ir.

theory alone will be in error; (iii) at low id
and high v, the opposite is true (i.e., the
quasi-steady nonuniform theory is required, where-
as neglecting induced flow results in error) ; and
(iv) for low a) and v, complete unsteady theory
is required.

Comparison of Theory and Experiment

the experimental data used in the following
correlations were obtained with a 7 . 5-f t-diameter
hingeless rotor model tested in the USAAMRDL-Ames
wind tunnel. •"• The model configuration and test
conditions covered a wide range of parameters.
The results included here are for p = 1.15 and
advance ratios from 0.0 to 0.6.

Elastic Blade Bending

In Fig. 7, experimental values of roll and
pitch moments due to 6 S are compared with theo-
retical results which are calculated neglecting
induced flow perturbations. Two sets of theory
are presented. The first theory employs a rigid
centrally-hinged blade with root spring to model
the elastic blade bending, and the second theory
[uses a similar model, except that hinge offset is
llowed. The largest differences between the two
heories occur near resonant frequencies, i.e.,

0.15, 1.15. (The primary effect of mode
ihape is aerodynamic, Eq. (8); it causes domi-
nce at resonance.) A surprising element in
figure 7 is that the centrally hinged model gives
closer agreement with the high frequency response
than does the hinge offset model. This reversal,
however, is not a consistent trend in the data
and may be somewhat coincidental.

'[cmAoJ/W,

Figure 7. Comparison of experimental data with
rigid blade approximations without
induced flow, p •> 1.15, y = 4.25,
B = 0.97, e pc = 0.25, u - 0.60.

Similar frequency response comparisons have
been made when the blade is modeled by one or two
of the rotating elastic mode shapes. When
u < 0.8 and <d is at least once-per-revolution
below the second flap frequency, the one- and two-
mode calculations are within a few percent of the
hinge-offset results. At higher advance ratios
and frequencies, the effects of second-mode bend-
ing can become significant; but in the range of
operating conditions considered here, a single ro-
tating mode is sufficient to model the blade.

Three major types of discrepancies between
theory and experiment which are found in Figure 7

cannot be explained in terms of flapping mode shape
effects. The first is the difference encountered
at frequencies near one and two per revolution.
This difference may be explained by the fact that
the lead-lag frequency of this configuration is
near two per revolution, causing resonance at these
frequencies. The second discrepancy is the irregu-
larity in the pitch response at id = 0.6. Here, a
natural frequency of the rotor support stand is
being excited and contaminates the data.-'-- 1 - The
third discrepancy is found at u < 0.6, and will be
shown to result from unsteady inflow perturbations.

Effect of Induced Flow In Hover

The low-frequency hover data provide some in-
sight into the effects of unsteady induced flow.
In Figure 8, rotor roll and pitch moments versus
9 g are presented. The experimental results are
for 5 = A", v = 0.03. The theoretical results are
calculated using the actual blade rotating mode
shape as a generalized coordinate and using three
different representations of the induced flow. The
first representation is the elementary model, which
completely neglects induced flow perturbations.
The second representation is quasi-steady momentum
theory, which neglects the apparent inertia
(Kj «■ 0) , assuming that nonuniform induced flow
perturbations instantaneously follow the blade dy-
namics. The third representation is unsteady mo-
mentum theory, which gives a time lag on the in-
duced flow perturbations. (The empirical model is
not applicable in hover.)

A comparison of theory and experiment reveals
that the elementary theory is unsatisfactory below
w = 0.6, failing to reproduce even the qualitative
character of the data. On the other hand, the
theories which include induced flow perturbations
account for most of the important features of the
response. The loss of aerodynamic effectiveness,
which is a result of induced flow perturbations,
causes a decrease in the excitation forces and an
overall decrease in the response. But the loss of
aerodynamic effectiveness also lowers the blade
damping, causing a resonant peak effect near the
blade natural frequency (with p = 1.15, w = 0.15).

The effect of the unsteady induced flow terms
is also evidenced in Figure 8. The major contri-
bution of Kx is the determination of how rapidly
with on the aerodynamic effectiveness returns to
the elementary value. Above w = 0.6, the theo-
retical value of Kj gives the proper amplitude
and phase for the hub moments, while the quasi-
steady theory (Kj * 0) fails to return to the con-
ventional value and does not agree with the data.
Below to = 0.6 the comparison is less clear. In
the roll-moment phase and amplitude, a Kj less
than 0.1132 would give better correlation than
does this theoretical value. In the pitch-moment
response, however, a smaller Kj would give worse
correlation than does Kj = 0.1132. Further work
would be necessary to determine if this effect is
due to experimental difficulties (such as recir-
culation) or to an actual deficiency in the in-
duced flow model.

.06

.04

3.02

360

■a

180

d[c L /a-a]/38 s

.06

O DATA REF II

NO INDUCED FLOW

QUASI-STEADY INDUCED FLOW .04

UNSTEADY INDUCED FLOW

(Kl- 0.1132, K„. 0.8488)

a[c„/<ro]/as,

Figure 8. Rotor response to cyclic pitch in

hover, p =1.15, y = 4.25, B - 0.97,
e pc = 0.25, u = 0, era « 0.7294,
v «■ X = 0.03, momentum theory, single
rotating mode.

In Figure 9, rotor roll and pitch moments
versus a are presented for the same test condi-
tions as in Figure 8. Data are presented for
shaft excitations in both roll and pitch, since
in hover the response to these controls is
ideally symmetric. A comparison of the two sets
of data gives an indication of the experimental
error due to test stand dynamics (and possibly
recirculation) . Although the data are question-
able for w > 0.3, the lower frequency data sub-
stantiate three of the observations made from
Figure 8. First, the elementary theory is quali-
tatively inaccurate for amplitude and phase re-
sponse. Second, a major effect of induced flow
is a resonant peak effect near os » 0.15. Third,
Ki < 0.1132 would give better correlation than
the theoretical value at low w. Figure 9 also

.06

i[c,_/<ro]/ia
O DATA REF II
D DATA REF II (ROLL I
NO INDUCED FLOW

.06

ra]/da

Figure 9. Rotor response to hub motions in hover,
p - 1.15, y - 4.25, B - 0.97,
e pc ■ 0.25, u » 0, aa « 0.7294,
v * X ■ 0.03, momentum theory, single
rotating mode.

shows that although induced flow decreases the
blade damping, it can actually increase the rotor
pitch/rate damping = -Re[3(C M /aa)/3ot] and also
increase the rotor pitch/roll coupling
= -Re[3(CL/aa)/3o]. The damping and coupling can
be found by dividing the plotted curves by
-iuj, A = iwa, which is approximately equivalent to
taking the slope of the plotted curves with a 90-
degree shift in phase angle. For this particular
configuration, the damping and coupling are in-
creased by induced flow effects, indicating that
induced flow perturbations can be important in
coupled rotor /fuselage dynamics.

Effect of Induced Flow in Forward Flight

In the next three figures, experimental data
at high-advance ratio (u= 0.51) and very low lift
(6 = 0.5°) are compared with theory using three
induced flow descriptions. The first description/
is an analysis which neglects Induced flow perturf
bations, the second description is the empirical
model of Reference 2 with no time lag (quasi-
steady, Kj_ ■ Kji = 0), and the third description j
is the empirical model of Reference 2 adapted to :
the unsteady case according to Eq. (34) (with the

»[c L /o-o]/a8,

afCM/o-aJ/se,,

Figure 10. Rotor response to collective pitch in
forward flight, p - 1.15, y - 4.25,
B - 0.97,, e pc - 0^25, p - 0.51,
era - 0.7294, v - X ■ 0, single
rotating mode.

theoretical values of Kj_ and K m ). The first
comparison of theory and experiment is shown in
Figure 10 for the roll- and pitch-moment response
due to 6 . The elementary theory predicts a
roll moment of 0.017 at w *• and a near-zero
crossing (amplitude ■ 0, phase angle discontinu-
ous) at co ■= 0.4. The data, however, displays a
much lower steady value and completely avoids the
zero crossing. The unsteady and quasi-steady
empirical models provide a fairly accurate
description of this behavior, showing quantita-
tive agreement with phase and magnitude for
a) < 0.6. For the pitch moment derivative, the
empirical models predict the qualitative (but not
the quantitative) aspects of the reduction in
moment (from the conventional value) due to
induced flow.

Another comparison of theory and experiment
is shown in Figure 11 for the roll- and pitch-
moment response due to S . The empirical models
predict a roll-moment derivative which is less
than the elementary value, exhibiting a near-zero
crossing at a = 0.26. This characteristic is
clearly evident in the magnitude and phase of the
data, but it does not appear in the theory

Figure 11. Rotor response to longitudinal cyclic
pitch in forward flight, p = 1.15,
Y = 4.25, B = 0.97, e pc = 0.25,
u = 0.51, aa = 0.7294, v = X = 0,
single rotating mode.

without induced flow. For the pitch-moment deriva-
tive, the elementary theory agrees with the data
only for u > 1.2; the quasi-steady theory shows
good correlation for < u < 0.6, and the unsteady
theory gives quantitative correlation at all fre-
quencies.

The third comparison is shown in Figure 12 for
the roll- and pitch-moment response due to 8 C .
The data show that the roll-moment derivative is
less than the elementary value at • u «■ 0, display-
ing a resonant peak (hear oj = 0.15) which is
greater than the elementary value and which is ac-
companied by a 10-rdegree phase shift. The empiri-
cal models predict the qualitative character of the
resonant peak and quantitative character of the
phase shift. The empirical models also correlate
well with the pitch-moment response, for which the
experiment shows the derivative to be greater than
the elementary value for « < 0.3 and less than
the elementary value for u > 0.3.

In general, the empirical inflow models show
this same degree of correlation at all advance
ratios considered (u » 0.27, 0.36, 0.51, 0.60).
This substantiates one of the qualitative
conclusions of Figure 6. For moderate advance
ratios and w < 1.0, an appropriate unsteady or
quasi-steady induced flow theory is adequate, but
the theory without induced flow is in consider-
able error. Of course, Figure 6 only implies in
which regions quasi-steady or unsteady terms may
be significant. It does not Imply that any par-
ticular quasi-steady or unsteady model will be

.06
g.04

3.02

360

j[c L /«i]/ae c

O DATA REF II

* NO INDUCED FLOW

QUASI-STEADY EMPIRICAL MODEL

UNSTEADY EMPIRICAL MODEL

(Kj-0.1132, K m »0.84B8)

.06

*fc M /<™]/*>C

.02

^V-^

. . . . *r - Bv r= -?"7 . °. . ,

,-180

1.2
360

180

1.2

.4 .8
u

1.2

.4 .8

Figure 12. Rotor response to lateral cyclic

pitch in forward flight, p = 1.15,
Y = 4.25, B - 0.97, ep C = 0.25,
U = 0.51, aa » 0.7294, v = A = 0,
single rotating mode,
adequate. For example, in Figure 13, pitch mo-
ment derivates (as calculated using the theory
without induced flow, unsteady momentum theory,
and unsteady empirical theory) are compared with
the experimental data. The comparison shows that
unsteady momentum theory can be in qualitative
disagreement with the data even though empirical
theory shows good correlation. Even the empiri-
cal model, however, does not show complete quan-
titative correlation; and further refinements in
the induced flow model may be necessary.

Conclusions

1. On the basis of an equivalent Lock number
relation and p = °°, quasi-steady nonuniform in-
duced flow perturbations can have a significant
effect on rotor response throughout the entire
thrust /advance ratio range; but the time lag of
the induced flow is only important at low lift and
low advance ratio.

2. In hover, unsteady momentum theory with appar-
ent mass terms from potential flow provides a
significant improvement in data correlation over
the theory without induced flow perturbations;

but further work is required to refine the induced
flow model.

3. In forward flight and near-zero lift, the
empirical inflow model of Reference 2, whether
used with the unsteady time-lag effect or with-
out the time-lag effect (quasi-steady) , corre-
lates well with most qualitative and some quanti-
tative aspects of the data, while unsteady momen-
tum theory and the theory without induced flow
provide little agreement with the data.

4. A single rotating mode is sufficient for
flapping response calculations when u < 0.8 and
when the major excitation frequency is at least
once-per-revolution below the second flapping
frequency.

.03

kj .02
o

« .01

a[c L /o-a]/ae

.03

a[c L /o-a]/ae 5

O DATA REF 10

NO INDUCED FLOW

MOMENTUM THEORY, g - 02

UNSTEADY =J

■EMPIRICAL MODEL, g

UNSTEADY S

Ov\ < -o

.03

uj .02

.01

a[c L /o-a]/ae c

.2 .4

360

lj" 180

■ CU3 =^et =:

^^ == ^^ = ~ :=:: =Q~.

1 1 1 1 1 1

Figure 13. Effect of induced flow model on low frequency, roll response, p = 1.15, Y = 4.25, B = 0.97,

e = 0.25, \i = 0.51, oa = 0.7294, V = X - 0, K T
pc I

0.1132, K. = 0.8488, single rotating mode.

References

1. Shupe, N. K. , "A Study of the Dynamic Motions
of Hingeless Rotored Helicopter," PhD. Thesis,
Princeton Univ.

2. Ormiston, R. A. and Peters, D. A., "Hingeless
Rotor Response with Nonuniform Inflow and
Elastic Blade Bending," Journal of Aircraft ,
Vol. 9, No. 10, October 1972, pp. 730-736.

3. Hohenemser, K. H. and Yin, Sheng-Kwang, "On
the Question of Adequate Hingeless Rotor
Modeling in Flight Dynamics," 29th Annual
National Forum of the American Helicopter
Society , Preprint No. 732, May 1973.

4. Crews, S. T., Hohenemser, K. H. , and Ormiston,
R. A., "An Unsteady Wake Model for a Hinge-
less Rotor," Journal of Aircraft , Vol. 10,
No. 12, December 1973.

5. Potthast, A. J., "Lockheed Hingeless Rotor
Technology Summary - Flight Dynamics", Lock-
heed Report LR 259871, June 1973, p. 43.

Bisplinghoff , R. L., Ashley, H. , and Halfman,
R. L. , Aeroelasticity , Addison-Wesley, Read-
ing, Mass., c. 1955.

Peters, D. A. and Hohenemser, K. H. , "Appli-
cation of the Floquet Transition Matrix to
Problems of Lifting Rotor Stability," Journal
of the American Helicopter Society , Vol. 16,
No. 2, April 1971, pp. 25-33,

Carpenter, P. J. and Fridovich, B., "Effect of
Rapid Blade Pitch Increase on the Thrust and
Induced Velocity Response of a Full Scale
Helicopter Rotor," NACA TN 3044, Nov. 1953.

Tuckerman, L. B. , "Inertia Factors of Ellip-
soids for Use in Airship Design," NACA Report
No. 210, 1925.

10. Ormiston, R. A. , "An Actuator Disc Theory for
Rotor Wake Induced Velocities," presented at
AGARD Specialists' Meeting on the Aerodynam-
ics of Rotary Wings, September 1972.

11. Kuczynski, W. A. , "Experimental Hingeless
Rotor Characteristics at Full Scale First
Flap Mode Frequencies," NASA CR 114519,
October 1972.

DYNAMIC STALL MODELING AND CORRELATION WITH
EXPERIMENTAL DATA ON AIRFOILS AND ROTORS

R. G. Carlson, Supervisor

R. H; Blaekwell, Dynamics Engineer

Rotor Dynamics Section

Sikorsky Aircraft Division of United Aircraft Corporation

Stratford, Connecticut

G. L. Commerford, Research Engineer

Aeroelastics Group, Fluid Dynamics Laboratory

United Aircraft Research Laboratories

East Hartford, Connecticut

P. H. Mirick, Aerospace Engineer

U. S. Army Air Mobility Research and Development Laboratory

Fort Eustis, Virginia

Abstract

Two methods for modeling dynamic stall have
been developed at United Aircraft. The a, A, B
Method generates lift and pitching moments as
functions of angle of attack and its first two
time derivatives . The coefficients are derived
from experimental data for oscillating airfoils.
The Time Delay Method generates the coefficients
from steady state airfoil characteristics and an
associated time delay in stall beyond the steady
state stall angle. Correlation with three types
of test data shows that the a, A, B Method is
somewhat better for use in predicting helicopter
rotor response in forward flight . Correlation
with lift and moment hysteresis loops generated
for oscillating airfoils was good for both models.
Correlation with test data in which flexibly
mounted two-dimensional airfoils were oscillated
to simulate the IP pitch variation of a helicopter
rotor blade showed that both methods overpredicted
the response, and neither gave a clear advantage.
The <*, A, B Method gave better correlation of
torsional response of full scale rotors and re-
mains the method in general use. The Time Delay
Method has the potential to be applied more easily
and probably can be improved by consideration of
spanwise propagation of stall effects .

Stall-related phenomena limit the operation-
al capabilities of the helicopter. Power, blade
stress , and control system loads can all increase
substantially due to blade stall. To predict
such phenomena unsteady aerodynamics in stall must
be modeled in blade aeroelastlc analyses. A num-
ber of unsteady aerodynamic models have been
developed. These include methods described in
References 1 and 2. Reference 3 is a recent
general survey article of rotor dynamic stall.
The a, A, B Method and the Time Delay Method are
two methods developed by United Aircraft. The
a, A, B Method was developed to use airfoil test
data obtained for a sinusoidally oscillating

Presented at the AHS /NASA-Ames Specialists'
Meeting on Rotorcraft Dynamics, February 13-15 »
191b.

Based on work performed under U. S. Army Air Mobil-
ity Research and Development Laboratory Contract
No. DAAJ02-72-C-0105 .

two-dimensional model airfoil. The Time Delay
Method was developed to provide an empirical method
that would agree with the lift and pitching moment
hysteresis characteristics measured in oscillating
airfoil tests for a number of airfoils and test
conditions .

Evaluation of unsteady aerodynamic modeling
techniques generally proceeds from correlation with
data obtained in two-dimensional oscillating air-
foil tests to correlation of full scale rotor blade
torsional response. Two-dimensional rigid airfoil
results are compared on the basis of aerodynamic
pitch damping and lift and pitching moment hys-
teresis loops , and full scale correlation is judged
on the basis of agreement in blade torsional mo-
ments or control rod loads . Evaluation of an un-
steady model on the basis of full scale torsional
response is made difficult by uncertainties in
three-dimensional rotor inflow and blade bending
and plunging motion. Correlation of the lift and
pitching moment time histories of rigidly driven
airfoils, on the other hand, is not the best method
of comparison because it does not treat blade dy-
namic response to stall. As an intermediate ap-
proach, model test data were obtained for a flex-
ibly mounted model airfoil which was dynamically
scaled to simulate the dynamics of the first tor-
sional mode of a rotor blade. This paper summa-
rizes unsteady aerodynamic modeling techniques and
includes comparisons based on two-dimensional aero-
dynamic pitch damping, lift and pitching moment
hysteresis loops , two-dimensional flexured airfoil
response, and full scale rotor blade torsional
moments .

Description of the Unsteady Models

a, A, B Method

In the a, A, B method the aerodynamic moment
is assumed to be a function of angle of attack and
its first two time derivatives . Reference 1* demon-
strated that unsteady normal force and moment data
generated during sinusoidal airfoil tests and tabu-
lated as functions of a, A = ba a nd B = b 2 *qj

u u?r

(where b is the airfoil semi-chord and U is the
free stream velocity) could be used to predict the
aerodynamic response of an airfoil executing

a nonsinusoidal motion. In a limited number of
flexured airfoil tests described in Eeferenoe k, '
good correlation was achieved between measured
and predicted airfoil dynamic response. The a,
A, B lift and pitching moment data tabulations
of Heference h were used in the calculation of
torsional response for the dynamically scaled
model airfoil. As applied in this investigation,
two changes were made in the calculation. First,
to consider the pitch axis of the model airfoil as
a variable, provision was made to include pitching
moment due to chordwise offset of the aerodynamic
center from the pitch axis:

c m (a,A,B)=c m (a,A,B)+(Xp A -

7 cA

X PA

.25)c 1 (a,A,B)

The second change involved scaling the un-
steady data tables to account for differences in
wind tunnel characteristics. The steady state
lift and moment data for the present test program
differed from the corresponding steady data
obtained in Heference h because the tests were
conducted in an open jet wind tunnel and because
the airfoil effective aspect ratio was much
higher. The method of scaling used for these
analyses required a shift in the entire data tabu-
lation by constant values of angle of attack, un-
steady lift coefficient, and unsteady moment co-
efficient according to the following relations :

C l (a > A > B) open jet =c l (a+5( V A > B) TAB +

6ci

and

c (a,A,B) . =c (a+6a ,A,BL.„+ 6e
m '. 'open jet m m' ' TAB m

The constants Sa^, 60^, 6c-|_ and S^ were es-
tablished for each airfoil and were equal to the
amount of shift necessary to make the open jet
steady state stall points in lift and moment
match the steady state stall points of the
airfoil of Reference h.

Time Delay Unsteady Model

. Wind tunnel airfoil dynamic response was
also calculated with the Sikorsky Time Delay un-
steady aerodynamic method. This formulation was
developed empirically by generalizing the re-
sults of a set of oscillating airfoil test pro-
grams. It is intended to predict the unsteady
aerodynamic characteristics of arbitrary airfoils.
Its aim is to provide the blade designer with un-
steady lift and pitching moment characteristics
of various airfoils without conducting extensive
oscillating airfoil tests. This model, based on
a hypothesis of the physical separation process ,
does not depend on an assumed harmonic variation
of angle of attack. The basic assumption is that
there exists a maximum quasi-static angle of
attack at which the pressure distribution and the
boundary layer are in equilibrium. During in-
creases in angle of attack beyond this static
stall angle, there are finite time delays before
a redistribution of pressure causes first a moment

break and then a loss of lift corresponding to flow-
separation. The relative phasing of the moment and
lift breaks with angle of attack produces either
positive or negative damping of the motion.

To test the Time Delay hypothesis , harmonic

data from Reference 5 were examined. It was noted

that the onset of stall can occur before, with, or

after maximum amplitude of the oscillation. In

accordance with the Time Delay hypothesis, the

spread between the static moment stall angle and

the dynamic lift break was evaluated in terms of

elapsed time nondimensionalized by free. stream

velocity and chord length, t* =At r ,__(U /c).

olLr o

Typical results show that separation generally
occurs when t* exceeds about 6.

Dynamic pitching moment stall has been
handled similarly. Test data showed, in general,
that the dynamic moment break occurred before the
lift break. This has been noted in Reference 6
and attributed to the shedding of a vortex at the
airfoil leading edge at the beginning of the
separation process . Rearward movement of the vor-
tex over the surface of the airfoil tends to main-
tain lift, but drastically alters the pitching
moment .

To apply the Time Delay Model to a given air-
foil, only static aerodynamic data are required.
First, the airfoil static lift and pitching moment
data are used to define the approximate variation
in center of pressure between the static moment
stall angle ol and an angle of attack a_ above
which the center of pressure is assumed fixed.
Secondly, an approximation is made to the c.
versus a curve for fully separated flow. The se-
quence of events occurring during one stall-unstall
cycle is detailed in Figure 1. Briefly stated,
lift and pitching moment are determined from po-
tential flow theory until the nondimensional time
t_ (which begins counting when the angle of attack

exceeds the static moment stall angle) reaches t .
At this point the pressure distribution begins to
change, leading to rearward movement of the center
of pressure and loss of potential flow pitching
moment. Later, when x_ = t*, the lift breaks from

the static line and decreases gradually with time
to the fully separated value, c. (a). For

•^EP
t_>t* the center of pressure coincides with

C.P.a—pCa) . At the point where a = 0, the rates

at which c. approaches c lqii , p (a) and C.P. approaches

C.P. SEp (a) (if it does not already equal C.P. g _ p {ot))

are increased. When a falls back below the quasi-
static stall angle a-, , the center of pressure
returns to the quarter chord, potential flow
pitching moment effects are reintroduced and a
second time parameter x_ is recorded to govern the
rate at which c 1 returns to c.

1 -T?0T

® o^.

"1 "1F0T
O.P. .= 0.25

°m = C mPOT =

ANGLE OF ATTACK, o

it oe.

IT dT

at a-a_

moment stall time constant t

2.0

. C.P. *begins to move rearward with time toward c, ^ , cep(~)
. g -_- is eliminated

T s <T g <T«

c. remains equal to c r

C.P. continues to shift aft with time, T„
C.P. - 0.25 * Cz ' \ ) [c.P. SEp (tx) - O.25]

e ffl = c l( C.P.

0.25)

T >

. c. begins to decay toward ci_j, F (oi)

1 1P0T
. moves to
variation in a

, C.P. moves to C.P. g __(a) independent- of subsequent

the exponential rate at which c^ approaches ci SEp (o) is

increased by a factor of 3

if t <t„<t* the rate at which C.P. approaches C.P. gEp (ci)

is increased by doubling the time increment

T 2n+1 = T 2n + 24t n(^°n)

t, counting begins t, » I At n (_£cJ

n«o
at cc«ai,a<o

C.P. returns to 0.25

potential flow moment is reintroduced c

c l * C1 P0T
C.P. » 0.25
c_ « c_

T 3A

"mPOT

— ORIGINAL LOOP

o RECONSTRUCTED LOOP
TIME DELAY PREDICTION

«M- "'

f = 75.25 cps

1.80

N
O

1.40

»-

J* /

JS J

1.00

S'T

yyy

0.60

1-
u.

0,20

£~~^

16 20

ui
2

{£ -0.I6P

4 8 12 16 20
ANGLE OF ATTACK , a , DEG

Figure 2. Correlation with NACA 0012 Lift and
Pitching Moment Hysteresis Loops. ■

Figure 1. Time Delay Unsteady Aerodynamic Model.

Although additional correlation studies must
be made to identify the effects of airfoil type on
the time delay constants and although refinements
to the present model may he implemented, this
rather simple model represents well the essential
features of the dynamic stall process. Correlation,
typical of that claimed for other empirical methods
(References 2 and 7) has been found with data from
References k, 5» and 8. Only the o, A, B Method
has produced better correlation (Reference k), but
it suffers from the requirement for extensive
testing and data processing. Figure 2 compares
the NACA 0012 unsteady lift and pitching moment
hysteresis loops measured in Reference k with Time
Delay results. This correlation was achieved by
setting the lift break time constant t* equal to
k.Q instead of 6.0. Three-dimensional effects
encountered in this test apparently reduced the
time interval between static stall and dynamic
lift stall. Also shown are the hysteresis loops

FAIRED CURVE OF REFERENCE 3

A 5 = 6° DATA
Q — T iME DELAY PREDICTION FOR 5«6°

M0.1IZ3

£~- -S3

'AIREO CU

'VE-v^ ^0-

i" 4

=&a^

^— <

STABLE

' tar'

UNSTABLE

ks 0.3375

< s&

k A

<*>

r 1

V\ f

1 !/

-.2

UNSTABLE

i
i /

-.6

Figure 3.

MEAN INCIDENCE ANGLE, a„,DEG

Correlation of Two-Dimensional Aero-
Dynwni c Pitch Damping.

predicted using the a, A, B Method. The a, A, B

Method correlation is with the data from which the

a, A, B coefficients were derived.
*

In addition to predicting the exact form of
lift and moment hysteresis loops j an unsteady
model should represent faithfully aerodynamic
pitch: damping. Accordingly, the Time Delay
Model was used to calculate two-dimensional aero-
dynamic damping for the reduced frequency/mean
angles of attack test points of Reference 9.
Sample results plotted versus airfoil mean in-
cidence angle of attack are shown in Figure 3.
Generally excellent correlation of measured and
predicted damping is noted.

Other correlation of the Time Delay Method
with two-dimensional oscillating airfoil test
data has been good. During development of the
theory, correlation was carried out with forced
oscillating airfoil data for a range of airfoils,
frequencies of f breed oscillation, Mach numbers,
and angles of attack. Typical examples of the
correlation obtained are shown in Figure h where
measured and calculated hysteresis loops are
shown for the V13006-7 airfoil. These test data
taken from Reference 1 show the correlation with
the Boeing Theory of Reference 1 as well. Corre-
lation included hysteresis loops for different
airfoils and covered a Mach number range from 0.2
to 0.6. In all cases, the general character and
magnitude of the hysteresis loops were well match-
ed. In particular, the method provides the sharp
drop in pitching moment that is often found when
stall occurs . The oscillation frequency in

Figure It is constant for the two cases, but Mach
number and mean angle of attack are changed. The
lift break occurs before the angle of attack reach-
es its maximum value. In terms of the non-dimen-
sional time parameter t*, the x* value of 6 at
which lift stall occurs is reached before the maxi-
mum angle of attack is reached. The Time Delay and
Boeing Methods show comparable correlation for
lift. For the Mach number, 0.!* case (Figure kb) the
return to potential flow occurs earlier for de-
creasing angle of attack than the return given by
the Time Delay Method. Pitching moment correlation
is better for the Time Delay Method. The triple
loop characteristic is well duplicated. Similar
correlation obtained with the Time Delay Method for
a wide range of conditions demonstrated its promise
as a practical method for analyzing unsteady aero-
dynamics .

Dynamic Stall Tests

In order to obtain data useful in evaluating
the two unsteady aerodynamic methods dynamic stall
wind tunnel tests were run using a two-dimensional
airfoil model. The model was oscillated at a
frequency simulating the cyclic pitch variation on
a helicopter rotor blade. Torsional frequencies
representative of helicopter blade frequencies were
obtained by varying a torsional stiffness element
between the drive system and the airfoil section.
The airfoil models were made to be as stiff as
possible along their span and light in weight to
approximate scaled helicopter blade mass and iner-
tia properties . Hence the non-dimensional coeffi-
cients in the equation of motion of the' model air-
foil were close to those of the helicopter blade
torsional equation of motion based on the aero-
dynamics of the three-quarter radius on the re-
tracting blade. Two different airfoils were fab-
ricated, an HA.CA 0012 and an SC 1095.

The model airfoils and drive system were
designed to permit investigation of the effects on
torsional response of torsional natural frequency,
chordwise pitch axis location and torsional inertia
over a range of IP frequencies for an NACA 0012
airfoil and a cambered SC 1095 airfoil. The
oscillating mechanism provided an 8-degree ampli-
tude of motion of the model with an adjustable
mean angle of attack. The model has a span of
1.75 feet and a chord of 0.5 feet. The wind tunnel
velocity was 275 fps for all tests . Time histories
of the model angular motion were recorded at
nominal driving frequencies of 8.0, 10.0, and 12.5
eps . Tests were run for the full range of angle of
attack for all the combinations of pitch axis ,
torsional inertia, torsional natural frequency, and
airfoil type. A typical set of time histories
for a basic reference condition (HACA 0012 airfoil,
25 percent pivot axis, nominal blade inertia, and
5P natural frequency ratio ) is shown for four mean
angles of attack in Figure 5. These time histories
represent the time average of ten cycles .

The elastic torsional deflection of the
airfoil (difference between total angular motion
and input angular motion) was obtained for each
test condition by subtracting the input angular

2.4

2.0

LlI

o
o

TEST DATA

BOEING METHOD

TIME DELAY METHOD.

STATIC DATA

1.6

1.4

1.2

1.0

0.8

0.6

0.4

\~~" N

'// — ■

. —

x" \

^
^

0.1

E
o

W -0.1

-0.2 —

5 -0.3

-0.4

i
i

"~~\

4 8 12 16 20 24

ANGLE OF ATTACK , a , DEGREES
A)f= 11.92 Hz, K = 0.l(5, M = 0.2

8 12 16 20 24

ANGLE OF ATTACK , a , DEGREES
8) f= 12,07 Hz, K = 0.057, M = 0.4

Figure k. Correlation of Bynamie Loops for the V130C-6-7 Airfoil in Forced Pitch Oscillation.

s s

4 \

f r

S>s a

Figure 5.

NON-DIMENSIONAL TIME, Slt/Zt

Averaged Time Histories of Angle of
Attack for the Model Airfoil.

position time history from the averaged airfoil
angular position time history:

S(t) ■ o(t) - (a + BBin2irr).

where 0(t) is the difference between the measured
non-dimensional angular time history response «(t)
and the input driving motion. The non-dimensional
time t is given by t/T, where the period, T, was
established from the ten-cycle time-averaging
process for that run. Some statistical variation
in measured response was noted when stall flutter
occurred, but in general the ten cycle time
averaged response was representative of the

I'Iref

I,l.5xI REF

1 1

Li
_l
(9

Z
<

S 3

MAC A 0012
5? PA =0.25

S« = !

\"7

tug

2 -

MACA 001
*PA=0.22

i^^

■as

-Og =

1<"

/ /

f\ *

.'"

— •>

' 4

Ada =7

SC 1095 |
Xp A = 0.25

F S« = 7

SC (095 |
5! P a=0.22

,«■

^mm

(SqsS

//,

_J.

. W

10 14 18 ~6 10 14

MEAN INCIDENCE AN~GLE,a. M , PEG.
Figure 6. Model Airfoil Elastic Deflection.

measured data. Two measures of stall response
amplitude were extracted from each of the Q(t) time
histories . These were A0^ which is one-half of
the initial response to stall and 9 JgpTp which is
one-half of the overall peak-to-peak elastic de-
flection.

It was found that the initial stall response
parameter A0, gave the most consistent indication
of susceptibility to stall flutter. The possible
reduction in flutter amplitude introduced by the
time averaging procedure when there was cycle-to-
cycle variation in phase made it somewhat difficult
to assess the amplitude of flutter response. For-
tunately, the initial stall deflection showed
virtually no cycle-to-cycle variation. Figure 6
compares measured initial deflection angles for an
excitation frequency B of 10 cps for the two air-
foils at all combinations of airfoil natural
frequency ratio (Wq = us Q /&l torsional inertia, and
pitch axis. Certain general trends of deflection
angle can be identified in the test results.

1. Elastic deflection increases with mean
incidence angle.

2. For the same torsional inertia, response
is generally greater for the lower
frequency airfoil section.

3.. The amplitude of response is inversely
related to torsional inertia.

k. Forward movement of the pitch axis leads
to a decrease in deflection.

5. SC 1095 airfoil dynamic stall response
begins to build up at a higher mean
incidence angle than the 0012, but the
two airfoils have comparable responses
once stall is penetrated.

Correlation Study of Two -Dimensional Results

The two-dimensional flexured airfoil test
data were compared with predictions based on
various unsteady aerodynamic methods. The single
torsional degree of freedom differential equation
of motion for the flexibly mounted airfoil section
oscillating in the wind tunnel test section is
given by

I-£.+ coi + KU-Ojh) = M(t) +Ko sin fit

where c ■ equivalent mechanical damping per unit
span
I = airfoil torsional inertia per unit span
K = torsional spring constant
Mfc= applied aerodynamic moment
t = '.time'

-a~=-aix£oil angle of attack

tt «*.. amplitude of angular oscillation

om= mean angle of the oscillation

Q » angular frequency of the applied" torque •

This equation was solved numerically using
the unsteady aerodynamic models to calculate the '-
applied aerodynamic moment M(t). For the a, A, B
Method unsteady data tables obtained from earlier
oscillating airfoil tests, Reference l*,were scaled
for both airfoils . The measured steady state lift

and pitching moment data served as inputs in the
Time Delay calculations. Additionally, the air-
foil mean incidence angle used in the Time Delay
solution was two degrees less than that set in
the wind tunnel. The open jet flow deflection
experienced at high unsteady lift coefficients was
sufficient to decrease actual peak angles of
attack to a value somewhat lower than the geo-
metrically impressed; pitch angle. The two-degree
correction to ay, gave consistently better corre-
lation of the initial stall time.

Correlation between measured wind tunnel
model response and response calculated with the
unsteady models was examined for thirty-six test
conditions. The set of cases studied was suffi-
cient to evaluate the independent effects on air-
foil stall response of mean incidence angle,

TEST

a,A,B

TIME DELAY

g. °

NACA 0012 AIRFOIL
a M .14 e w s /ft"?

■;

CHANGE IN BATUMI.
FEEQUSHCY

I=IrEF Xpa-0.25

l I

J-

\S,

f—

^? J

"""

— "S

■4 /

___

i I i

SC 1095 AIRFOIL

■

■ i

CHAHGE

*■*» *»-«h

\ —

— ^

> N

-*C;

-7

■•>

l i

torsional natural frequency, chordwise pitch axis,
torsional inertia, and airfoil type. Relative to
a baseline case 1*aken to be the NACA 0012 airfoil
at ^ = W 5 , ae/fl = 5, ^, A = 0.25, and I = I-^,,
Figure 7 shows measured and predicted effects of
mean angle, torsional natural frequency, pitch axis,
and airfoil type on time histories of elastic de-
flection. Comparison of the measured and predicted
effects of airfoil mean angle of attack indicates
that deeper penetration into stall results in
sharper initial stall deflection and larger resid-
ual stall flutter response. The two analyses pre-
dict these effects qualitatively, but each -
especially the Time Delay model - overprediets the
amplitude of response. The main effects of an in-
crease in torsional natural frequency are a shift
in response frequency and a decrease in the am-
plitude of elastic defleetion. Figure 7 shows
good correlation of response amplitude, although
both analyses predict an initial stall response
earlier than that measured. Moving the airfoil
pitch axis forward causes delay in initial stall
time and reduction in amplitude of response. The
analytical results do predict the reduction in
response amplitude, but the Time Delay model still
results in overpredicted response. Finally, a
comparison between the NACA 0012 and the SC 1095
airfoils shows a delay in the initial stall time
for the SC 1095 airfoil, which had a static stall
angle measured in this wind tunnel to be about
three degrees higher than that of the NACA 0012.
However , the SC 1095 stall flutter amplitude was
comparable to that experienced by the NACA 0012
at this condition.

The time history correlation was good in
that the initial response and the frequency of the
subsequent oscillations were predicted. The trends
observed in test were well matched by the analysis ,
although. the Time Delay model generally overpre-
dicted stall flutter response. The basic effects
of structural changes on blade response time
histories were well predicted by either analysis .

Although torsional elastic deflection is
important in determining rotor stability and per-
formance, the torsional moments resulting from
stall flutter are the designer's primary concern.
To measure the trends of torsional moment with
parameter changes , the twisting moment experienced
by the flexible connector in the model airfoil
drive system was calculated for each test condition.
The torsion moment Mq was calculated using the
equivalent spring stiffness of the connector:

M„

K
eq

airfoil 9

2
<a„ )

NON0IMENSIONAL TIME,Q</2r

Figure 7, Effect of ParanBters~an rftirfoil Response.

The torsional moments corresponding to the initial
stall deflection angle A0 1 were used to show the
effects of blade parameters on structural moments.
Figure 8 presents typical results for three com-
binations of airfoil type and mean angle of attack.
It was generally found that decreasing torsional
natural frequency reduced stall flutter moments.
Although the stiffer system experienced lower
response amplitudes, the corresponding structural
moments were increased:

TEST

40
20
L
[ 60
40
20

80 -
60 -
40-
20 -

TIME
DELAY

NACA 0012 AIRFOIL, ct M = I

a,A,B

ACA C

012

AIRFO

*M= ,4 °

SC 1095 AIRFOIL, a M = 14°

« H

m O ix

13 |x H

Figure 8.

Effect of Airfoil Parameters on Model
Airfoil Vibratory Torsional Moments.

05p u 05p 5p P 5p

Forward placement of the airfoil pitch axis gener-
ally decreased vibratory torsional moments. The
two analyses predicted this trend with comparable
accuracy. That forward movement .of the airfoil
pitch axis relative to the aerodynamic center
reduces stall flutter moments can he understood
based on lift and pitching moment hysteresis
loops . For an airfoil with pitch axis forward
of the center of pressure, positive lift forces
cause negative moments about the pitch axis.
For positive lift, the lift hysteresis loop is
usually traversed in the clockwise direction,
which contributes a negative pitching moment
loop in the counterclockwise (stabilizing)
direction. A decrease in torsional moment am-
plitude with decreasing torsional inertia was
generally found throughout the testing. This
trend, evident in two of the conditions shown in
Figure 8, is predicted somewhat more correctly
by the Time Delay Analysis. Finally the two
airfoils are compared in Figure 9- For two
different combinations of inertia and pitch axis,
high stall flutter moments are delayed in mean
angle with the SC 1095 airfoil.

I=I REF ,XpA=0.25,Og = 5

TEST

,S H

NACA 0012,

-SC 1

/ /

' 1

I«l.5xI REF ,5?

»«0.22.

"8=5

1
TEST

Ss'—~

-NACA

' 'SC 1095

s .

TIME

3ELAY

S 40

-J
<

..._.

§20

-1

P n

TIME DELAY

•

1
/

a,/

*,B

■-

---••

6 10 14 18 6 10 14

MEAN INCIDENCE ANGLE, a M , DEGREES

Figure 9. Effect of Airfoil Type on Structural
Moments

Flight Test Correlation

Flight test data were correlated with the
Normal Modes Blade Aeroelastic Analysis for both
the CH-53A and CH-5ta aircraft. Both models of
unsteady aerodynamics were used. Information on
the blade analysis used can be found in Refer-
ence 10.

Correlation of CH-53 control system loads ,
blade stresses and required power was studied at
a nominal aircraft gross weight of fe?,000 lb
(Cm/a = 0.083), a tip speed of 710 ft/sec, and a
3000 ft density altitude for airspeeds ranging from
100 knots to 170 knots . Inclusion of variable in^
flow was found to be essential in calculating the
proper levels of blade bending moments . It also
provided some improvement in the correlation of
blade torsional moments .

The a, A, B and Time Delay aerodynamic models
are compared at 137 knots in Figure 10. Figures
10a and 10b shows that the computed blade stresses
are comparable for the two methods. However, the
push rod loads calculated with the Time Delay
Model are much less than values calculated with the
o , A, B Method and measured values . The Time

'(D)

Cc)

8000

Q-_ 4000
in

-4000

-8000

4000

r/R

= 08

•

n as

- TIME DELAt

^^ —

11/
f

\>^

z.^

J(V

%tf\

120 180 240

BLADE AZIMUTH ANGLE , DEGREES

300

360

Figure 10. Correlation .of CH-53A Blade Stresses and Pushrod Loads.

Delay* results generally do not give sufficiently
large oscillations in stall.

That better correlation of stall flutter
moments was possible with the a, A, B Method is
evident in Figure 11a which shows the buildup
of vibratory pushrod load amplitude with airspeed.
The a, A, B model predicts a buildup rate almost
identical with the mean of the' test data. A
discrepancy of no more than 10 knots in the knee
of the control load curve is evident at this thrust
coefficient. Figure lib shows the correlation of
pushrod load amplitude achieved with the a, A, B
Method at three thrust coefficients .

Calculated C&-5^ control loads were also
generally less than measured values . Figure 12
indicates that a definite stall boundary is pre-

dicted by the analysis . Relative to the CH-53A
calculations , a decreased control load stall
speed and an increased rate of buildup with air-
speed are clearly predicted. Again, higher loads
are computed based on the a, A, B Model. The
comparison of measured and predicted push rod
load time histories indicates that the a, A, B
results reflect a buildup in the higher fre-
quency loads much more accurately than do the
Time Delay calculations.

It is not entirely clear why, relative to
the o, A, B method, the Time Delay model under-
prediefcs helicopter control loads while over-
predicting the stall flutter oscillations of the
two-dimensional wind tunnel model. Examination
of several blade section pitching moment /angle
of attack hysteresis loops indicates not so much

(a)
3200

m
«2400

3 '

£1600

S 800

...

1 r ■

— TEST DATA

— a,A,B

— Tl

ME Dt

.LAY

]//

/fi

s"-

<La

80 120

AIRSPEED, KNOTS

160

200

(b)

4000

.3200

2400

1600

800

1 1 1 1 1

C T /cr TEST ANALYSIS

0.083

PUSH

ROD

JOAD=

±220

/ 1

n/i

/m,

1 f

r ■(

i i

/ i

Y^"

120 160 '

AIRSPEED, KNOTS

200

240

Figure 11.

2400

Correlation of CH-53A Vibratory
Pushrod Loads .

1600

800

Figure 12.

40 80

AIRSPEED, KNOTS

Correlation of CH-^B Vibratory
Pushrod Loads.

160

that more negative pitch damping is present in the
a, A, B results. Rather pitching moments along the
blade are more in phase with each other, leading to
larger modal excitation. In the o, A, B formu-
lation, pitching moment coefficients are tabulated
as functions of a, a and 8 values all along the
blade. This formulation leads to similarly phased
pitching moments. In the Time Delay Model, moments
are calculated based on the angles of attack ex-
ceeding the steady state stall angle for a certain
interval of time and are not solely dependent on
the instantaneous angle of attack characteristics .
For small differences in calculated angles of
attack, computed pitching moments for adjacent
blade sections can be different in phase. Because
the two-dimensional wind tunnel airfoil was modeled
as a single panel for the calculation of aerodynamic
forces, the effects of simultaneous spanwise stall-
ing were not a factor in the correlation with that
data.

Comparison of Methods

Because the a, A, B Method has demonstrated
better correlation with flight test data, it con-
tinues to be the method in use for blade design
analysis. However , development of both methods
continues. The ot, A, B Method provides a relative-
ly direct and simple procedure for calculating un-
steady aerodynamic loads . Correlation has been
good with test data but its disadvantage centers
largely on the apparent need for extensive tests
to provide the body of tabulated data required for
each airfoil. Some success has been obtained by
scaling the KACA 0012 unsteady aerodynamic tables
based on steady state differences between airfoils .
Work is also being done on developing analytical
expressions to replace the tabulated data. These
may lead to the ability to synthesize the data
required for a given airfoil , which would make the
method more desirable for general applications .

The Time Delay Method has the great advantage
of requiring only .steady state airfoil data for its
application. The correlation with forced oscil-
lations of two-dimensional airfoils demonstrated
its applicability over a wide range of conditions .
Correlation with the tests described in this paper
showed no clear advantage of the Time Delay Method
over the a, A, B Method, and correlation with
flight test data was definitely poorer with the
Time Delay Method. Further work must be done to
investigate the reasons for the poor flight test
correlation. The problem may result from the
assumption in the analysis that, on a blade, each
radial section acts independently of its neighbor-
ing sections . This causes a more random stalling
along the span with time, which smoothes out the
changes in blade loading. The propagation of stall
along the span for the three-dimensional case of a
helicopter blade must be added to the Time Delay
Method. The a, A, B Method does provide spanwise
correlation in loading by use of torsional mode
acceleration to calculate the B parameter. This
acceleration is in phase for each point along the
blade span. Incorporation of a suitable radial
propagation model in the Time Delay Method may make
this a more versatile, more easily applicable, and

more accurate model of unsteady aerodynamics.
Until this can be shown the a. A, B Method
continues in use in blade design.

Conclusions

1. The a, A, B and Time Delay unsteady aerodynamic
models predict with good accuracy the lift and
pitching moment hysteresis loops and the aero-
dynamic pitch damping of rigidly driven os-
cillating airfoils.

2. Two-dimensional stall flutter tests indicate
that reducing blade torsional stiffness , re-
ducing blade torsional inertia and moving
blade pitch axis forward decrease stall flutter
induced moments. Inception of stall flutter
was delayed with the SC 1095 airfoil relative
to the NACA 0012 airfoil; however, once
initiated, stall flutter loads for the two
airfoils were generally comparable.

3. Stall flutter response of the two-dimensional
model airfoils and the effects of airfoil
structural design parameters on blade torsion-
al moments can be calculated using both un-
steady models. The Time Delay method gives a
high prediction of response amplitude.

k. Good correlation of CH-53A and CH-5^B blade
stresses and control loads was obtained with
a rotor aeroelastic analysis employing vari-
able rotor inflow and unsteady aerodynamics.
Best correlation was achieved using the a,
A, B unsteady model. The Time Delay method
generally underpredicted full scale rotor
stall flutter response.

5. The a, A, B model is in use for blade design
analysis. Refinements to the Time Delay Method
may make it a more versatile and more easily
applied unsteady aerodynamic model.

References

1. Gormont, R. E., A MATHEMATICAL MODEL OF UN-
STEADY AERODYNAMICS AND RADIAL SLOW KIR
APPLICATION TO HELICOPTER ROTORS, USAAMRDL
TR 72-67. U. S. Army Air Mobility Research
and Development Laboratory, Fort Eustis,
Virginia, May 1973.

2. Ericsson, L. E. and Reding, J. P., UNSTEADY
AIRFOIL STALL REVIEW AND EXTENSION, AIAA
Journal of Aircraft, Vol. 8, No. 8,
August 1971-

3. McCroskey, W. J., RECENT DEVELOPMENTS IN
ROTOR BLADE STALL, AGABD Conference Pre-
print No. Ill on Aerodynamics of Rotary
Wings, September 1972.

h. Carta, P. 0., Commerford, G. L.,

Carlson, R. G. , and Blackwell, R. H.,
INVESTIGATION OF AIRFOIL DYNAMIC STALL AND
ITS INFLUENCE ON HELICOPTER CONTROL LOADS,
United Aircraft Research Laboratories;
USAAMRDL TR 72-51, U. S. Army Air Mobility
Research and Development Laboratory,
Fort Eustis, Virginia, September 1972.

5. Gray, L. and Liiva, J., WIND TUNNEL TESTS
OF THIN AIRFOILS OSCILLATING NEAR STALL,
The Boeing Company, Vertol Division;
USAAMRDL TR 68-89A and 68-89B, U. S. Army
Aviation Materiel Laboratories, Fort
Eustis, Virginia, January 1969.

6. Ham, N. D. and Garelick, M. S. , DYNAMIC
STALL CONSIDERATIONS IN HELICOPTER ROTORS,
Journal of the American Helicopter Society,
Vol. 13, No. 2, April 1968.

7. Tarzanin, F. J., PREDICTION OF CONTROL
LOADS DUE TO BLADE STALL, American Heli-
copter Society, 27th Annual National
Forum, May 1971.

8. Arcidiacono, P. J., Carta, F. 0.,
Caselini, L. M., and Elman, H. L.,
INVESTIGATION OF HELICOPTER CONTROL LOADS
INDUCED BY STALL FLUTTER, United Aircraft
Corporation, Sikorsky Aircraft Division;
USAAVLABS TR 70-2, U. S. Army Aviation
Material Laboratories, Fort Eustis
Virginia, March 1970.

9. Carta, F. 0., and Niebanck, C. F.,
PREDICTION OF ROTOR INSTABILITY AT HIGH
FORWARD SPEEDS, Volume III, STALL FLUTTER, ■
USAAVLABS TR 68-18C, U. S. Army Aviation
Materiel Laboratories, Fort Eustis,
Virginia, February 1969.

10. Arcidiacono, P.' J., PREDICTION OF ROTOR
INSTABILITY AT HIGH FORWARD SPEEDS, Vol. I,
Steady Flight Differential Equations of
Motion for a Flexible Helicopter Blade with
Chordwise Mass Unbalance, United Aircraft
Corporation, Sikorsky Aircraft Division;
USAAVLABS TR 68-18A, U. S. Army Aviation
Materiel Laboratories, Fort Eustis, Virginia,
February 1969.

COMPUTER EXPERIMENTS .ON PERIODIC SYSTEMS

IDENTIFICATION USING ROTOR BLADE TRANSIENT

FLAPPING-TORSION RESPONSES AT HIGH ADVANCE RATIO

K. H. Hohenemser and D. A. Prelewicz
Washington University, St. Louis, Missouri 63130

Abstract

Systems identification methods have
recently been applied to rotorcraft to
estimate stability derivatives from
transient flight control response data.
While these applications assumed a
linear constant coefficient representa-
tion of the rotorcraft, the computer
experiments described in this paper used
transient responses in flap-bending and
torsion of a rotor blade at high advance
ratio which is a rapidly time varying
periodic system. It was found that a
simple system identification method ap-
plying a linear sequential estimator
also called equation of motion estimator,
is suitable for this periodic system and
can be used directly if only the accel-
eration data are noise polluted. In the
case of noise being present also in the
state variable data the direct applica-
tion of the estimator gave poor results,
however after pref iltering the data with
a digital Graham filter having a cut-off
frequency above the natural blade torsion
frequency, the linear sequential estima-
tor successfully recovered the parameters
of the periodic coefficient analytical
model.

Notation

B Blade tip loss factor

F = (Ix/lSIfXc/R) 2 First blade tor-
sional inertia
number

State matrix

Process noise modulating
matrix

Fourier transform of
weighting function

Measurement matrix

State matrix = measurement
matrix

F(x,t)
G(t)

H(u)

H(x,t)
H(5,a)

Presented at the AHS/NASA-Ames Special-
ists' Meeting on Rotorcraft Dynamics,
February 13-15, 197t. This work was
sponsored by AMRDL, Ames Directorate,
under Contract No. NAS2-4151.

"Now at the Westinghouse Bettis Atomic
Power Lab. Westmifflin, Pennsylvania.

P(t) or P

Q= (I 1 /itI f )c/R

R
a,b,c

t
v
w(jAt)

Blade flapping moment of
inertia.

Blade feathering moment
of inertia.

Quadratic cost function.

Covariance matrix of
conditional state vector
probability distribution
given measurements.

Blade flapping natural
frequency .

Second blade torsional
inertia number.

Measurement noise co-
variance matrix.

Blade radius

Unknown parameters to be
estimated in flapping-
torsion problem.

Parameter vector.

Blade chord.

Blade torsional natural
frequency.

Non-dimensional time.

Measurement noise vector

Smoothing weights.

Process noise vector

State vector

Measurement vector.

Flapping angle.

Blade Lock number.

Blade torsion deflection

Acceleration vector.

Rate of displacement
vector .

Blade pitch angle.

Rotor inflow ratio, con-
stant over disk.

Rotor advance ratio.

Displacement vector.

T In order to retain the conventional sym-
bols in helicopter aerodynamics (Ref-
erence 7) and in systems analysis (Ref-
erence 9 ) some symbols are used in two
different meanings.

Subscripts

c,t

Superscripts

Notation
(conf)

Standard deviation.
Circular frequency.

Initial or mean value.

Beginning and end of fil-
ter cut-off frequencies.

Time differentiation.

Smoothed data after fil-
tering.

Estimate

Matrix transpose

The question often arises, how to
best select some parameters of a given
analytical model of a dynamic system on
the basis of transient responses to
certain inputs either obtained analyti-
cally with a more complete math model or
obtained experimentally. In rotor craft
flight dynamics one may want to use a
linear constant coefficient math model
and select the state matrix in an opti-
mal way from the measured data ob-
tained in a number of transient flight
maneuvers. One also may have a more
sophisticated non-linear analytical
model of the rotorcraft. The problem
then is how can the simpler linear math
model be selected to best represent the
responses of the more complete analyti-
cal model; or one may have the dynamic
equations of a rotorcraft without the
effects of dynamic inflow and one de-
sires to modify some of the parameters
in such a way that dynamic inflow ef-
fects are best approximated. It is
known from theoretical studies, for
example Reference 1, that a reduction in
blade Lock number can approximately
account for rotor inflow effects in
steady conditions. The question then is
whether changes in parameters can also
account for inflow effects during
transient conditions.

The idea of using transient re-
sponse data to determine parameters of
an analytical model is certainly not
new. Recently, however, considerable
interest in this area has been de-
veloped and a number of approaches have
been studied which are unified under the
title of "system identification". There
is a considerable and rapidly growing
literature in this field. System iden-
tification methods generally fall into

two classes: (1) deterministic methods -
usually some variation of the classical
least squares technique and (2) proba-
bilistic methods which determine the
parameters as maximum likelihood esti-
mates of random variables. Some methods
can also be interpreted either on a
deterministic or on a probabilistic
basis. References 2 and 3 are typical of
recent work using deterministic methods.
Both of these studies illustrate the
feasibility of determining coefficients
in time invariant linear systems from
transient response data. Reference 4 de-
scribes many of the probabilistic tech-
niques. Reference 5 gives a detailed
discussion of the various methods in
their application to V/STOL aircraft and
Reference 6 presents an identification
method suitable for obtaining stability
derivatives for a helicopter from flight
test data in transient maneuvers. The
studies of References 5 and 6 assume a
linear constant coefficient representation
of the system. A rotorcraft blade is,
however, a dynamic system with rapidly
changing periodic coefficients. It ap-
peared, therefore, desirable to try out
methods of system identification for a
periodic dynamic system.

Selection of Identification Method

If one assumes that only the state
variables have been measured but not the
accelerations, one must use a non-linear
estimator since the estimate of a system
parameter and the estimate of a state
variable appear as a product of two un-
knowns. A non-linear sequential esti-
mator was tried on the simplest linear
periodic system described by the Mathieu
Equation. It was found that the non-
linear estimating process diverged in
most cases, unless the initial estimate
and its standard deviation were selected
within rather narrow limits. Reference 6
uses a sequential non-linear estimator
but initializes the process by first ap-
plying a least square estimator, which
needs in addition to the state variable
measurements also measurements of the ac-
celerations. In the case of the problem
of Reference 6 the least square estimator
yielded a rather good set of derivatives
and the improvement from the much more
involved non-linear estimation was not
very pronounced. From this experience
it would appear that one needs to apply
the least square or an equivalent linear
estimator any way and that in some cases
it is doubtful whether or not the sub-
sequent application of a non-linear es-
timator is worth the considerable effort.

After conducting the rather unsatis-
factory computer experiments to identify
a simple periodic system with the

non-linear estimator, all subsequent work
was done with a linear sequential esti-
mator. This estimator is equivalent to
least square estimation but has the ad-
vantage of being usable for "on-line"
system identification. The inversion of
large matrices is avoided and replaced by
numerical integration of a number of
ordinary differential equations. The
computer experiments were conducted with
the system equations of Reference 7 for
the flapping - torsion dynamics of a
rotorblade operating at advance ratio 1.6.
Reference 7 assumes a straight blade
elastically hinged at the rotor center
and stipulates linear elastic blade twist.
The system used here for the computer ex-
periments represents only a relatively
crude approximation, since at 1.6 advance
ratio blade bending flexibility is of im-
portance, see for example Reference 8.
The coefficients in the system equations
are non-analytic periodic functions which
include the effects of reversed flow.

The identification algorithm used in
this report is easily derived using the
extended Kalman filter discussed in the
next section. Although the algorithm
does not provide for noise in the state
variables, one can nevertheless use it
also for noisy data if one interprets the
estimate, which normally is a determinis-
tic variable, as a sample of a random
variable. The effects on system identi-
fication of computer generated noise in
both the acceleration data and in the
state variable data were studied. However,
no errors in modeling were introduced
since their effects can only be evaluated
on a case by case basis.

Extended Kalman Filter

The extended Kalman filter is an
algorithm for obtaining an estimate x of a
state vector x satisfying

x = F(x,t) + G(t)w

Process Equation(l)

given noisy measurements z related to x
via

3F
3x

P + P

(s) 1 * <«* - -(s)v^

3H
3x

x(o)

Covariance Equation (H)

P(o) = P
Initial Conditions (5)

x and P can be interpreted as vector mean
and covariance matrix of a conditional
probability distribution of the state
vector x, given the measurement vector z.

However, since the extended Kalman
filter is a biased estimator (see Refer-
ence 5) and since the correct value of
P is not known, P cannot be used as a
measure of the quality of the estimate.
Rather, the rate of decrease of P is an
indication of the amount of information
being obtained from the data. When P
approaches a constant value then no
further information is being obtained.

The extended Kalman filter may also
be interpreted as an algorithm for ob-
taining a least square estimate recur-
sively. The estimate is such as to
minimize the following quadratic oost
function

J = 1/2 { (x - V^ 1 (x - x )

J o wV" 1 * + |z-H(x,t)) T

,t)W

z - H(x,

Cost Function (6)

where now P , R and Q are arbitrary
weighting matrices, which may be se-
lected for good convergence of the
algorithm. Since 1.) numerical methods
for solving ordinary differential equa-
tions are well developed and 2.) R is
usually a diagonal matrix so that R -1 is
easy to obtain, this algorithm is compu-
tationally very efficient.

H(x,t) + v

Measurement Equation (2) Estimation of Unknown Parameters

In the above equations w represents zero
mean white Gaussian process noise with
covariance matrix Q, v represents zero
mean white Gaussian measurement noise
with covariance matrix R. An optimum es-
timate x of x can be obtained by solving
the extended Kalman filter equations
(see Reference 9)

x = F(x,t) + Pj

mVr

i,t)J

I - H(x,
Filter Equation (3)

If we wish to estimate the vector a
of unknown parameters we substitute a for
x in the Kalman filter Eq. 3. For con-
stant parameters we have

a = o Process Equation (7)

so that F(x,t) = w = o. The system equa-
tion is then used as the measurement
equation

H(c,a) + v

Measurement Equation
System Equation (8)

C is the vector of measured accelerations,
5 is the measured state vector and v can
be interpreted as acceleration measure-
ment noise or as system noise (including
modeling errors). The Kalman filter
equations are then

R _1 U - H(5,a)]

/3H\ T -1

Filter Equation (9)

P = -Pf-rr

3H
3a

Covariance Equation (10)

For P ■*■ o the measurements lose influence
on the estimate and one obtains

a = o Asymptotic Filter Equation (11)

which agrees with the process equation.
Again P and R may be selected for good
convergence. A convenient choice for the
initial estimate is a(o) = o. The ele-
ments of R should be large enough to pre-
vent the elements of P from becoming
negative due to computation errors in the
numerical integration.

Note that £, the state vector, is
also a measured quantity. If measurement
errors are present then this estimation
algorithm is biased by an amount approxi-
mately proportional to the noise to sig-
nal ratio in the state variable measure-
ments, see Reference 5. It is therefore
advantageous to reduce the noise ratio
before using the estimator. Methods for
doing this are discussed in a later sec-
tion on filtering of the response data.

In practice, one can almost always
choose the parameters to be identified in
such a way that H(£,a) is a linear func-
tion of a. The estimator (9), (10) is
then linear and problems of nonuniqueness
and filter divergence are easily avoided.
For this case, we call the algorithm the
linear sequential estimator.

The extended Kalman filter assumes
that the noise processes w and v are
white and Gaussian. This will never be
the case in practice especially if w must
account for the effects of modeling er-
rors. Because the extended Kalman fil-
ter may also be interpreted as yielding a
least squares estimate for a given sample
of the state £ and acceleration c, we can
regard the resulting estimate as a sample
from a random variable. Determination of
this random variable would necessitate a
complete simulation, i.e., mean and
variance determined by averaging over
many runs. Since this approach is
expensive of computing time, efforts here

have been directed toward recovering para-
meters from a single run of computer gen-
erated response data.

The above approach to parameter es-
timation allows the use of high order of
accuracy numerical integration (i.e., pre-
dictor corrector) schemes to solve the
system of ordinary differential equations
provided that the response data are suf-
ficiently smooth. The parameter estima-
tion is rapid and requires little com-
puter time. R and P can be freely se-
lected to obtain good convergence. The
reason for this benign behavior of the
estimation method is the linearity of the
filter equations in the unknowns. If the
accelerations of the system are not
measured, one must estimate state vari-
ables and parameters simultaneously from
a nonlinear filter equation. This non-
linear estimation requires an order of
magnitude more computer effort and it is
very sensitive to the initializations and
to the correct assumptions of process
noise and measurement noise. As
mentioned before, we began by applying
the nonlinear estimator to the identi-
fication of parameters in Mathieu's
equation for a periodic system. The re-
sults were unsatisfactory since filter
divergence occured for many choices of
P and R. However, for the linear
sequential estimator divergence could be
avoided by following simple rules in
selecting x(o), P Q and R.

Identif iability of System Parameters

It is obvious ^from the filter
equation (9) that a will asymptotically
approach a constant value provided that
P -*• o. The covariance equation (10) can
be solved explicitly (see Appendix A) to
yield

fl&

3H
3a

(12)

If the integral is replaced by a sum, this
is the error equation for the standard
least square method. If P Q / o, then
P(t) -»• o whenever the integrand in the
above equation is positive definite for
all t. This is then a sufficient condi-
tion for identif iability. Note that
3H

3a is a function of the system response
and hence also of the excitation, so that
the identifiability depends not only upon
the system but also upon the type of ex-
citation. From the measurement equation
(8) we see that the matrix 3H is a

3a
measure of the sensitivity of acceleration

measurements to changes in the parameters.
For estimating parameters, a well designed
excitation is obviously one which causes
the elements of the P matrix to decrease
rapidly. If any elements of P are de-
creasing slowly or not at all , then a dif-
ferent type of excitation is needed. A
look at which elements of P are causing
the trouble will give a clue as to which
modes of the system are not being properly
excited.

Filtering the Response Data

In practice, we usually have some
knowledge of the character of the re-
sponse data. For example, because of the
damping present in physical systems, the
true response will not contain much
energy at high frequencies. We also know
that the acceleration is the derivative
of the velocity which is in turn the de:-
rivative of the displacement, etc. so that
these responses are not independent.

To remove high frequency noise with-
out effecting the signal a zero phase
shift low band pass digital filter was
used. This filter completely removes all
of the signal and noise above a certain
termination frequency ut without phase or
amplitude distortion below a cutoff fre-
quency 6) c . The digital filter used, due
to Graham, Reference 10, generates the
smoothed data as a numerical convolution
of the raw data and a set of numerical
smoothing weights, i.e.,

f(t„ + iAt)

= y~] w(jAt)f(t +

j=-N

(i + j)At)

(13)

where f(t Q + (i + j )AtJ are the sampled
values of the signal, f(t + iAt) are the
smoothed sampled values and where the
smoothing weights are given by

w(jAt) =

sin u-tJAt + sin <» c jAt

2jAt

(<fl t -w c ; 3

) n At

j = -N,...,+ N
J * o

w(o)

c((«3 t + w c )
2i

u c < (o t

(1<4)

where the constant c is chosen to satisfy
the constraint

w(jAt) = 1

(15)

j=-N

The continuous weighting function
w(t) , of which w(jAt) is a discretization,
has the Fourier transform, i.e., fre-
quency domain representation, shown in
Figure 1. Convolution of this function
with an arbitrary signal will obviously
result in a smoothed signal which has all
frequencies above m-\- completely sup-
pressed and all signal components below
w c undistorted. If w c and m-^ are pro-
perly selected then response data with
low frequency signal and high frequency
noise can be improved via digital fil-
tering, that is, signal to noise ratio
can be significantly increased.

In using the digital filter, it is
tempting to achieve a "sharp" filter by
taking o) c i w-f Graham, Reference 10,
has determined empirically that the
number of points N needed to achieve a
given accuracy is approximately inversely
proportional to |wt~ u cl at least over a
limited frequency range. Since N = 40
points were used to filter the data, we
selected | oj^-a)-. |>_ 1 which according to
Graham is sufficient to yield 2% accuracy.

In this study, the numerical convo-
lution was accomplished by using a moving
average, i.e., f(t '+ iAt) was computed
separately for each i using Eq. (13). For
long data records it is possible to
achieve considerable savings in computer
time by using the Fast Fourier transform
algorithm to do this convolution, see
Reference 11.

Improvements in the response data
can also be obtained by making use of
relationships among the various response
signals. For the coupled flapping-torsion
system considered in the next section the
displacements £, velocities n and accel-
erations c are related by

5 = n

n = c + v

(16)

We can use these equations as process
equations in a. Kalman filter along with
measurement equations

s +

W n v 2

(17)

where 5 and IT denote smoothed measured
values. In the process equation (16)
replace ? by its smoothed measured value
X and let R, the process noise covariance
matrix account for remaining errors .
Then the Kalman Filter is given by

+ PR

-1

5-5

ff - n

(18)

Note that n is available when solving the
above equations and can be used as an
improved estimate of X- Although this
technique has not been used in this
study, a similar procedure has been used
successfully in Reference 6 for heli-
copter derivative identification.

Computer Experiments

Coupled flapping-torsion vibrations
of a rotor blade at high advance ratio
are governed by the equations

8 + P 2 B = \ CM 9l (t)6 + M x (t)X +

M e (t)9<

C(t)B - K(t)g]

I + f 2 « = 4 e

3 Y F [C e (t)9 +

C 6 (t)6]
3YQU r 8(t)B + * r6 (t)B + fc rX (t)A +

*re (t)e o + K 6 (t)5]

(20)

where the periodic coefficients are
defined in Reference 7 . Responses to the
gust excitations shown in Figure 2 were
generated by solving Eq. (20) numerically
using a fourth order Adams Moulton method
with a time step of .05 and the following
parameter values:

p 2 = 1.69
f 2 = 64.
B = .97

4.0

U = 1.6

F = .24

Q = 15.

6 = 0.

(21)

Simulated noisy measurements were obtained
by adding samples from zero mean computer
generated Gaussian random sequences to
the computer generated responses. First
the noise was added only to accelerations
using the standard deviations

1.0

°S = 10

(22)

The following three parameters with the
values

a = y/2 = 2.0

b = -3yF = -2.88 (23)

c = _3 y q s -180 f

were assumed to be unknown.

They represent blade flapping and
torsional inertia numbers. Unsteady
aerodynamic inflow effects may possibly
be considered by modifications of these
inertia numbers from transient rotor
model wind tunnel tests. The linear
sequential estimator was started with
the initial values of the estimates and
errors of the estimates

40
P(o) 55
4000.

"a(o)"

b(o)

_c(o)_

(24)

The linear sequential estimator is, as
mentioned before, quite insensitive to
the initial standard deviations which
could have been selected still much
larger. The values for R used are the
following

R =

10
10

(25)

The method allows wide variations in the
assumptions of the noise covariance
matrix R. The integration scheme for
solving filter and covariance Eqs. (9)
and (10) was again a 4th order Adams
Moulton method with a time step^of ^. 05.
Fig. 3, shows the estimates a, b, c
normalized with the true values and the
3 diagonal terms of the error covariance
matrix P normalized with the initial
values vs. non-dimensional time t. The
excitation for this case was a unit step
gust at time t = o, as indicated in Fig.
2 by the dash line. In about one
revolution (t = 2tt) the diagonal compo-
nents of the covariance matrix P a P5 P c
are approximately zero and further
improvements of the parameter estimates
a 6 c are not obtained. There is a
small bias error (deviation from the
value 1) in two of the parameters , which
have been recovered within about 5% error.

The next case assumes that not only
the accelerations but also the state
variables are noisy. The following
standard deviations were used

a B = .2
■"l = - 6

<S-A = 1.0

.5
3.0

(22a)

qj = 10

The linear sequential estimator was first
applied to the raw data. In this case
the responses are far from smooth so' that
the use of a high order numerical inte-
gration scheme was unjustified. A first
order Euler's method was used for the
integration of the estimator equations.
The initial values were

|"a<o>
Mo)
c(o)

P(o)

35
1000

The values for the R used in the
estimator were

R =

22S

(24a)

(2 5a)

The excitation consisted of a upward unit
step gust at t = 2.0 followed by a down
step gust to X h -1 at t = 6.0, as
indicated in Fig. 2. The second gust
was added in order to provide to the
system another transient useful for the
estimator process. Fig. 4 shows that
though two of the diagonal covariance
terms go to zero after the second gust,
the associated parameter estimates
remain quite erroneous. The linear
sequential estimator cannot be used if
noise is present in accelerations as well
as in the state variables.

Next the same data were passed
through a digital filter with cut-off
frequencies w c = 12, u-t = 13, see Fig. 1.
These cut-off frequencies are about 50%
higher than the torsional frequency of
f = 8 . Applying now the linear
sequential estimator to the filtered data,
the initial values were the same as
before, Eq. (24a) , however R was reduced:

R =

1
9

(25b)

The results of the estimation are shown
in Fig. 5. All diagonal terms of the
covariance matrix go to zero soon after
the second gust, the estimates stabilize
in less than 2 rotor revolutions and have
only a small bias error of about 5% ; same
as for the case with zero noise in the
state variables. Digitally filtering the

data to remove high frequency noise has
thus appreciably extended the range of
applicability of the linear sequential
estimator. It might be argued that the
success of the digital filter is due to
the "white" character of the computer
generated noise whereas real data will
contain energy only at finite frequencies.
It should be noted that the digital
filter removes all of the signal above
the truncation frequency and hence
would be equally successful for any
other distribution of the energy above

In selecting the parameters for the
digital filter it is important to keep
w c large enough so that the responses
are not significantly distorted.
Initially, the noisy data was pro-
cessed using different digital filters
for the torsion and flapping responses.
A digital filter with high cut-off
frequency i.e., io„ = 12. and oa t = 13.
was used for torsion responses while a
lower bandpass filter with ui_ = 2. and
uk = 3. was used to filter flapping
responses. This resulted in poor identi-
fication of the parameter a in the
flapping equation. When the same
filter with high cut-off frequency was
used for all of the data, adequate
identification of all parameters was
obtained. Although w c = 2. is above the
natural frequency of flapping vibration,
the flapping response obviously contains
higher frequency components because of
the coupling with torsion. This can
easily be seen by inspection of the
flapping response in Figure 6. For a
good identification it is necessary that
these higher frequency components not
be removed from the signal. Fig. 6
compares the response without noise to
the response with noise but after
filtering. Also indicated are the
standard deviations for flapping and
torsion before filtering. It is seen
that the filter was very effective in
removing the noise corruption from the
data.

Conclusions

The linear sequential estimator, also
called equation of motion estimator,
has been successfully applied to
recover the system parameters of a
periodic system representing rotor
blade flapping-torsion dynamics at
high rotor advance ratio with noise
contaminated accelerations .
Filtering of the noisy acceleration
data was found to be not necessary.

If noise is present in the state
variables as well as in the acceler-
ations, the linear sequential

estimator performed very poorly.

3. Filtering both state variables and
accelerations with a Graham digital
filter with a low cut-off frequency
for flapping and a high cut-off for
torsion before estimation lead to a
poor estimate for the flapping
parameter.

t. Filtering both flapping and torsion
response with a high cut-off fre-
quency digital filter before esti-
mation resulted in an adequate para-
meter recovery both in flapping and
in torsion.

5. As compared to non-linear estimation
methods which are applicable also if
acceleration information is not
available, the linear sequential
estimator has the great advantage of
being insensitive to the assumption
of initial values for the estimate
and for the error of the estimate.

No matter what the actual measurement
noise is, the assumed noise covar-
iance matrix should be over-rather
than underestimated.

6. As compared to the usual form of the
least square estimation the linear
sequential estimator does not re-
quire the inversion of large matrices
but merely the numerical solution of
a system of ordinary differential
equations, thus allowing on-line
application. The digital filter
smoothes the data sufficiently so
that high order of accuracy predictor
corrector methods can be used for
the integration.

7. The computer studies were performed
assuming rather large measuring errors
with standard deviations for the
deflections of about 10% of the maxi-
mum measured values. The foregoing
conclusions assume the absence of
modeling errors , which would require
special investigations.

References

1. Curtis, H.C. Jr., COMPLEX COORDINATES
IN NEAR HOVERING ROTOR DYNAMICS,
Journal of Aircraft Vol. 10 No. 5 ,May
1973, pp. 289-296.

2. Berman, A. and Flannelly, W.G., THEORY
OF INCOMPLETE MODELS OF DYNAMIC
STRUCTURES, AIAA Journal, Vol. 9 No. H,
August 1971, pp. 1481-87.

3. Dales, O.B. and Cohen, R. , MULTI-
PARAMETER IDENTIFICATION IN LINEAR
CONTINUOUS VIBRATING SYSTEMS, Journal
of Dynamic Systems, Measurement and

10.

Control, Vol. 93, No. 1, Ser. G.
March 1971, pp. 45-52.

Sage, A. P. and Melsa, J.L., SYSTEM
IDENTIFICATION, Academic Press,
New York 1971.

Chen, R.T.N. , Eulrich, B.J. and
Lebacqz, J.V. , DEVELOPMENT OF
ADVANCED TECHNIQUES FOR THE IDENTIFI-
CATION OF V/STOL AIRCRAFT STABILITY
AND CONTROL PARAMETERS, Cornell Aero-
nautical Laboratory Report, No.
BM-2820-F-1, August 1971.

Molusis, J. A., HELICOPTER STABILITY
DERIVATIVE EXTRACTION FROM FLIGHT DATA
USING THE BAYESIAN APPROACH TO ESTI-
MATION, Journal of the American
Helicopter Society, Vol. 18, No. 2,
April 1973, pp. 12-2 3.

Sissingh, G.J. and Kuczynski, W.A. ,
INVESTIGATIONS ON THE EFFECT OF BLADE
TORSION ON THE DYNAMICS OF THE
FLAPPING MOTION, Journal of the
American Helicopter Society, Vol. 15,
No. 2, April 1970, pp. 2-9.

Hohenemser, K.H. and Yin, S.K. , ON THE
QUESTION OF ADEQUATE HINGELESS ROTOR
MODELING IN FLIGHT DYNAMICS, Pro-
ceedings 2 9th Annual National Forum
of the American Helicopter Society,
Washington D.C. , May 1973, Preprint
No. 7 32.

Bryson, A.E. and Ho, Y.C., APPLIED
OPTIMAL CONTROL, Ginn S Co., Waltham,
Mass. , 1969, p. 376.

Graham, R. J. , DETERMINATION AND
ANALYSIS OF NUMERICAL SMOOTHING
WEIGHTS, NASA TR R-179, December 196 3.

11. Gold, B. and Rader, C. , DIGITAL PRO-
CESSING OF SIGNALS, McGraw-Hill,
New York, 1969.

12. Ried, W.T., RICATTI DIFFERENTIAL
EQUATIONS, Academic Press, New York,
1971.

Appendix A

Solution of the Covariance Equation

The covariance equation of the linear
sequential estimator

is a matrix Ricatti differential equation.
It is well known that the general matrix
Ricatti Equation with all matrices being
time functions

P = -PA - DP - PBP + C (A-2)

of which (A-l) is a special case, has the
solution

p = VU x
where U and V satisfy
V = CU - DV
U = AU + BV

(A- 3)

(A-4)

This and other aspects of matrix Ricatti
equations are discussed in Reference 12.

By comparing Eqs. (A-l) and (A-2) we
see that Eq. (A-l) is of the form of Eqi
(A-2) with A=C=D=0 and B = /3H\ T R _1 3H

laa) 3a

Therefore, from Eq. (A-t) V = V Q , a con-
stant matrix and

U = BV Q
Integrating yields

U = U + /
Jo

Now since from (A- 3)

V Q Uo

B dtV,

-1

(A- 5)

(A- 6)

(A- 7)

we can satisfy the initial, condition by
taking V = I and U

Po" 1 -

Hence

U = ? r

B dt

(A- 8)

and

['.- 1 * /(-S) 1 *- 1

-1

(A- 9)

Minimizing the cost function Eq. (6)
with w = o, x = a and z = c, one obtains
the least square estimate

* ■ Cp o _1 + /({if*" 1 42 dtrl[p o _1 ao <

!5V

3a/

R _1 Cdt]

(A-10)

where the first factor is the covariance P
from Eq. (A-9). Eq. (A-10) is the equiv-
alent of solving Eqs. (9) and (10) and
has been used in Ref. 6 with P _1 = o
after replacing the integrals by sums.
In this case the result is independent of
R which cancels out.

Even in the general case of finite

P(o) the error covariance matrix R need
not be considered as a separate input.
If R is a diagonal constant matrix it is
evident that Eqs. (9) and (10) can be
written in the form

i ■ '-Off 1

C- H(5,a)l

P r =

3H p
3a **

(A-10)

(A-ll)

,-1

where P r = P R J ". This was pointed out
to the authors by John A. Molusis.

■« t - w c

Fig. 1. Fourier Transform of
Weighting Function

CdL

(U,

Fig. 3. Estimates 6 Covariance s vs.

Time, Acceleration Noise Only

A
10

-1.0

^ — *-

-£ ^ ^ t

Fig. 2. Gust Excitations

Fig. 5. Estimates S Covariances,
Filtered Data

Fig. i». Estimates S Covariances vs. Time,
Acceleration and State Variable
Noise

Fig. 6. Exact and Filtered Noisy Responses
(Solid S dash line respectively)

DYNAMIC ANALYSIS OF MJlTI-DfiGHEE-OF-FEEEaX)M SYSTEMS
USING HffiSIKG MATSICES

Kiehard L. Bielawa*

United Aircraft Research laboratories

East Hartford, Connecticut

Abstract

A mathematical technique is presented for
improved analysis of a wide class of dynamic and
aeroelastic systems characterized by several
degrees-of-freedcm. The technique enables greater
utilization of the usual eigensolution obtained
from the system dynamic equations by systematizing
the identification of destabilizing and/or
stiffening forces. Included, as illustrative
examples of the use of the technique, are analyses
of a helicopter rotor blade for bending- torsion
divergence and flutter and for pitch-lag/flap
instability.

Notation

[A], [B], Inertia, damping and. stiffness matrices, respec-
[0] tively, Eq.. (1)

A,v Elton-lag coupling for k'th edgewise mode
(=A9/A4 Vi )

A,, Pitch-flap coupling for m'th flatwise Bode

a Section lift curve slope, /rad

a i1' *11* Elemen ' , ' s of tllfi CA], [b] and [0] matrices

Viscous equivalent structural damping of k'th
edgewise mode

Blade chord, in.

EL., E^ Flatwise and edgewise bending stiffness, respec-
tively, lb-in. 2

Jp(t)} Dynamic excitation force vector, Eg.. (1)

f n Eesultant driving force for n'th degree of freedom,
E<1. (5)

[g(Xj)] ljynamic matrix for i'th eigenvalue, Eq. (3)

"no'

k "10

Torsional stiffness, lb-in.

Boot feathering spring, in.-lb/rad

Polar radius of gyration of spar about its center,
in.

Section thickness-wise and chordwise mass radii of
gyration, respectively, about spar center, in.

o 2
Section mass distribution, lb-sec /in.

Reference mass distribution, (= 0,000776 lb-sec 2 /in. )

*Senior Besearch Engineer, Botary Wing Technology
Group.

!>Ai3>
t?Ai]»

Iwi

4fi4

E
r
1
t

yio cg ,

y 10 c /4»
y l°3cA

'Vfc
'wi

\
*±a

V
9

»e
X

*1
P

{*<*>}

"Stability" Force Phasing Matrices for i'th eigen-
value, Eq.s. (6) through (8)

"Stiffness" fore* Phasing Mvtsrtces for i'th
eigenvalue, Eqs. (10) through (13)

k'th edgewise modal response variable

i'th flatwise modal response variable

j'th torsional modal response variable, (J = 1,
for rigid feathering)

Hotor radius, in.

Blade spanwise location, in.

Tension at r, lb

Time, sec

Vector of degrees of freedom

Chordwise positions forward of spar center of mass
center, quarter chord, and three-quarter chord,
in.

Spanwise variable section angle of attack about
which perturbations occur, rad

Blade pre-coning angle, rad

Angle defined in Fig. 1 (= arg X ± )

k'th assumed edgewise mode shape

i'th assumed flatwise mode shape

j'th assumed torsion mode shape

Coefficients describing quartie variation of profile
drag coefficient with angle of attack

Kronecker delta

Dumber defined in Eq. (9)

Geometric (collective) pitch angle at r, rad

Elastic torsion deflection at r, rad

(Uniform) rotor inflow

i'th eigenvalue, /sec

Air density, lb-sec e /in.

Blade solidity

Real part of i'th eigenvalue, /sec

i'th eigenvector of dynamic matrix equation

Hotor rotational speed, rad/sec

Imaginary part of i'th eigenvalue, /sec

<*)

( )•
(~)

E 3

Differentiation with respect to (fit)

Differentiation with respect to T

Indicates quantity is nondimensionalized using
combinations of R, iHq and ft, as appropriate

Diagonal matrix

I. Introduction

dynamic and aeroelastie analyses of aerospace
structures typically involve deriving and solving
sets of linear differential equations of motion
generally written in matrix form:

[A]{x} + Cb]{x} + [C]{xj- = |p(t)} (1)

In general, the A, B and C matrices are square
and real- valued. A recognized hallmark of rotary
wing and turbomachinery dynamics is an abundance
of nonconservative forces (usually involving rotor
rotation speed). Consequently, the resulting
analyses produce matrix equations of motion of
the above type which are highly nonsymmetrical,
and often of large orders.

Although a large part of the dynamic analyst's
job involves the calculation of dynamic loads and
stresses due to explicit excitations, the scope
of this paper will be limited to the equally
important eigenproblem (F(t) = 0):

This paper presents an easily implemented
technique for the improved analysis of dynamic
systems of the type described above. The technique
requires a reliable eigensolution and involves
manipulations of the given dynamic equations,
their eigenvalues and eigenvectors. Specifically,
the technique systematizes the identification of
destabilizing and/or stiffening forces by the
calculation of "force phasing matrices". Applica-
tions of the technique to analyses of bending-
torsion divergence and flutter and of pitch- lag/
flap instability of a helicopter rotor blade are
presented. Furthermore, this paper essentially
represents an expansion of a portion of an earlier
paper ^ . '

II. Mathematical Development

The principal function of the force-phasing
matrix technique is to identify those force terms
in the equations of motion which, for an unstable
mode, are so phased by the mode shape as to be
drivers of the motion. The technique is perhaps
nothing more than a formalization of the intuitive
use an experienced dynamicist would make of the
eigenvector information. The basis of the tech-
nique can be seen by writing any single equation
of the set represented by Eq. (3) as the sum of
the mass, damper and spring forces of the diagonal
degree-of-freedom and the remaining forces acting
as a combined exciting force.

W -sK'}-

a X?tpi 15 +t> ^•<i i) + c tPn 15
nn i T n nn iti nn T n

/£> nji nj i nj' Y j

(5)

X.t

(2)

[cA]xf + [B]X 1+ [c]l{cp (i >} = [GCXi)]^} - {0}

(3)

The eigenvalues X (= cr+io)), which give stability
and natural frequency information are obtained
from the familiar characteristic determinant:

|[A]X 2 + [B]X + [C]| .

(*0

(1)

by any of various well-established methods v ',
(2), w) # jije "flutter" mode shapes, <ffc\ are
obtained from Eq. (3) once the eigenvalues are
known.

For the usual case a^, bj^ and c nn are all
positive numbers; that is, each mass when un-
coupled from the others is a stable spring-mass-
damper system. Since the root, k±, is generally
complex, Eq. (5) can then be interpretted as the
sum of four complex quantities or vectors in the
complex plane which must, furthermore, be in
equilibrium. Assuming that the root with
positive imaginary part is used throughout, the
argument of the root, 7^, is the angle by which
the inertia force vector is rotated relative to
the damper force vector and the damper force
vector is rotated relative to the spring force
vector. For an unstable root this angle will
be less than 90 degrees. If a purely imaginary
value is assigned to the spring force vector,
unstable motion is assumed and it is recalled
that the four vectors are in equilibrium, then

the real parts of the damper and inertia force
vectors will be negative and the driving force
must always have a positive real part. Figure 1,
which demonstrates this argument, shows the four
force vectors in the complex plane for an
unstable oscillatory mode (Re(A^) = O" i >0) and
for unit imaginary displacement:

vwv

(SPRING FORCE)

7j '(mMXi) < 90deg.

(INERTIA FORCE)

Figure 1. Force- Vector Diagram for n'th Degree-
Of-Freedom, i'th Mode (Oscillatory
Instability)

A secondary function of the technique is to
identify those terms in the equations which, for
any coupled mode, act as stiffness so as to
increase the coupled frequency of the mode.
Reference to Figure 1 shows that driving forces
with positive imaginary parts will tend to rein-
force the diagonal spring term and, hence, raise
the frequency of the coupled mode. An interesting
observation that can be made from Figure 1 is that,
for unstable motion, the diagonal damper force
also has a positive imaginary part. Hence, it
tends to stiffen the (unstable) coupled mode in
contrast to the frequency lowering effect of
damping for stable motion.

Figure 2 shows the same forces as vectors
for an unstable aperiodic mode (divergence) for
negative unit real displacement:

SPRING, DAMPER AND
INERTIA FORCES

DRIVING FORCE

Figure 2. Force-Vector Diagram for n'th Degree-of-
Freedom, Divergence Instability.

Again, the driving force is always a positive
real number. Furthermore, for divergences, stif-
fening forces are by definition stabilizing; hence,
those components of the driving force which are
negative are also those that stiffen the coupled
mode.

These interpretations of unstable motion can
be quantitatively implemented first, by multiplying
each of the dynamic equations (i.e., each row of
the equation (l)) by a quantity which makes the
diagonal stiffness force (stiffness matrix element
x displacement) become pure imaginary and second,
by representing the modal vector as a diagonal
(square) matrix. This latter operation has the
effect of evaluating the magnitudes of the com-
ponent dynamic equation forces without numerically
adding them together. The resulting "stability"
force phasing matrices are then readily written as:

ffleCn/q^axftAtfq* 1 ^

Cp b J --fleCTi/qf^CBH 9 (1) 3

Cp_ ] = (Ret v/y a> Kelt <p c1)
c i

(6)

(7)

(8)

where

( i j for oscillatory instabilities
(-1. , for divergences

(9)

and where the eigenvalue in the upper half plane
is used.

la all cases, the real parts of the above
indicated matrix expressions give instability
driving force information. Forces defined by-
elements of the A, B and G dynamic equation
matrices which are phased by the mode shape so
as to be drivers of the motion then cause the
corresponding elements of the Pj^, Pg^ and P(Ji
"stability" force phasing matrices, respectively,
to be positive and proportional to their strength
as drivers.

Stiffening driving force information is
obtained differently for oscillatory motion and
for aperiodic motion. Those elements of the
dynamic equation matrices which are phased so as
to be stiffeners of the coupled mode will cause
the corresponding elements of the matrix expres-
sions to be either positive imaginary for oscil-
latory motion, or negative real for aperiodic
motion. The resulting "stiffness" force phasing
matrices are then expressed as:

[P Ai ] = JmF V<p (i) 3^WC <P (i) 3

[P B .] = JmC iy<P (i) ^i[B]£ cp( 1 )^

cp c .] = jmf uv^rae v (x) i

for oscillatory motion, and:

(10)
(11)
(12)

[*( )J - CP( )J

for aperiodic motion.

(13)

It should be stressed that these force
phasing matrices are no more than a more system-
atic and efficient interpretation of the all too
often voluminous eigensolution information. The
following sections illustrate the usage of the
force phasing matrix technique in substantiating
what is generally known of some rather fundamental,
classical helicopter rotor blade instabilities.

III.

Description of Illustrative
Rotor Blade Example

For illustrative purposes, relatively simple
linear equations of motion were formulated for a
generalized untwisted helicopter rotor blade and
then applied to a realistic nonarticulated rotor
configuration. The blade is assumed to be oper-
ating in an unstalled hover condition at some
collective angle and with a built-in coning angle.
Perturbative elastic flatwise, edgewise and torsion

motions are assumed to occur about the preconed
position. The resulting linear aeroelastic
equations are fairly standard *■''> ^ '; quasi-
static aerodynamics (uniform inflow) is assumed
and a normal mode description of the blade
elasticity is employed. Thus, for the chosen
configuration, two flatwise bending modes, one
edgewise bending mode, and the rigid feathering
degree-of- freedom are assumed. The resulting
response vector, |xj- , consists of the quantities
Iwij 4w2> 4vi> and- qg^ whose detailed dynamic
equations are given in the Appendix. The dynamic
equations then comprise a set of four differential
equations written as a k x k matrix equation of
the Eq. (l) type. The aeroelastic degrees-of-
f reedom together with the general parameters are
shown in Figure k:

TOTAL

PITCH

ANGLE,

Figure k.

PRE-CONING-

Schematic of Nonarticulated Eotor
Configuration and Aeroelastic Degrees-
of -Freedom.

The basic configuration incorporates a
counterweight over the outer 70 percent of the
blade, pitch- flap coupling (determined from the
geometry of the pushrod attachment and flatwise
modal deflection) and pitch- lag coupling of
arbitrary magnitude . The chordwise position of
the counterweight and the magnitude of pitch- lag
coupling are purposely varied in the following
analysis to establish known blade instabilities
in order to illustrate the phasing matrix
analysis technique. Table I below summarizes
the pertinent geometric and aeroelastic data
for the rotor blade configuration:

TABIE I - BLADE CHARACTERISTICS

17.

Hadlua
" Chord (0.1H to tip)
lip Speed

Siton-Flap coupling, A v ta |n = |
Hoof feathering spring rate
Blade coning

Airfoil: (HACA 0012; Mach Ho. - 0):
a
So
*U

(Uniform) maBS distribution
(Uncoupled) blade natural frequencies:

first flatwise mode

second flatwise mode

first edgewise mode

rigid feathering

210 in.
13.5 in.

650 fps

0.
0.188

3.55 x 10 6 in.-lb/rad
2 deg.

6.0/rad

0.01

3.30/^10^

O.OOO776 lb-sec 2 /in. 2

1.09e/rev
2.68l/rev
1.390/rev
3.820/rev

(edgewise mode structural damping) 0.01
(critical damping)

Flight condition (hovering)
collective angle, 8
inflow ratio, X

sea level, standard

10 deg.

-.0601

The normal flatwise and edgewise mode shapes
used are shown In Figure 5.

1.0 r

0.6-

-0.6*-

Figure 5.

3pamd.se Variation of Normal Mode
Shapes.

Application of Analysis to
Illustrative Example

Basic Configuration

For purposes of comparison, the data in
Table I was used together with a collective angle
of 10 deg, inflow ratio of -0.0601, and with zero
counterweight chordwise offset (from the quarter
chord), and zero pitch- lag coupling. This basic
case is stahle in all modes as is shown hy the
following list of resulting eigenvalues:

^1,2 =

J 5,6 =

X 7,8 =

-0.5CA- ± iO.960
-O.lHl ± 12.610
-0.027 ± il.398
-1.14-72 ± i3-506

While all the aeroelastic modes represented by
the various eigenvalues comprise responses in all
the four degrees of freedom, they could be
characterized as follows: mode 1 ( X^ 2) is first
flatwise bending, mode 2 ( X3 14.) is second
flatwise bending, mode 3 ( \<j } 6) is first edgewise
bending, and mode k ( X7 g) is rigid feathering.

Configuration With Rearward Chordwise
Counterweight

If the 70 percent outer span counterweight^ is
artificially shifted aft so as to place the chord-
wise section mass center at the 32 percent chord
point, reference to the dynamic equations (A-l,
A-2 and A-3) then yields the following A, B and
C matrices (given in E format):

A (Biertia) Matrix:

.2909-00 -.0000
-,0000 ,2006-00

*.oooo -.0050. „

-.1884-02 ,3162-03

B (Damping) Matrix:

•,0000

',0000
,25*2-08
•tOOOO

«, 1884-02
,3482*09

•.0000
,1133-03

.2B87-00
.7992-01
.3099*01
,1536-07

,7757-01
,1665-00
,1943-01
,6326-01*

-, 1*500-02 -, 1256-01

•,6166-02 -.$309-03

,7901-02 -.2020-02

•,1068-03 ,3370-03

C (Stiffness) Matrix:

.3382-00 -.5763-01 ,4633-01 -.3067-00
-.8232-03 , 1^22+01 ,7464-02 -.7080-01

.4633-01 -,2974-02 ,4968-00 -.1396-01
-, 2300-02 -,5423-02 -,2919-03 ,1646-02

The first two rows of the matrix equation
are the equations for the first and second
flatwise bending modes, respectively. The third

row is the equation for the first, edgewise mode,
while the fourth is for rigid feathering.
Correspondingly, elements in the first two columns
of the matrices are terms multiplying the flatwise
■bending responses and their derivatives. Simi-
larly, third and fourth column elements are terms
multiplying edgewise bending and rigid feathering,
respectively, and their- derivatives .

The eigensolution for these matrices (see
Eq. (k)) reveals the configuration to he unstable
in "both divergence and flutter:

*1,2

*3,4

X 7,8

0.1(08, -4.466

0.300 +1I.789

-0.0088 ± il.402

-0.578 ± i3.099

Since the equations are in nondimensional form,
the units of these eigenvalues are per rotor
revolution frequency or "p". Using Eqs. (6)
through (9) the following "stability" force
phasing matrices are written for the unstable
divergence mode:

A Phasing Matrix, Paj_

-.1*840-01 -.0000 -.0000 ,5061-03
-,0000 -.3338-01 -.0000 -,1726-02
-.0000 -.0000 -.4229-01 -.0000
.1941-03 -.1944-05 -.0000 -.1885-04

B Phasing Matrix, Pg^

-.1178+00 -.1715-02 .4408-04 ,3272-02
-.6016-00 -.6791-01 .1476-02 .4022-02
-.5265-00 -.1789-01 -.3223-02 .5540-01
-.3681-08 -.8660-06 .6418-06 -.1375-03

C Phasing Matrix, P^,

-.3382-00 ,3123-02 -.1113-02 ,4953-00

,1519-01 -.1422+01 -.3316-02 .2110+01

-.1929+01 .6710-02 -.4988-00 ,2956+01

.1424-02 .1620-03 ,4341-05 -.1646-02

The larger of the destabilizing driving
forces, which show up as positive terms, have been
underlined for clarity. A reasonable yard-stick
for measuring the size of the destabilizing forces
is to compare them to the size of the stabilizing
element in each matrix equation row. For
oscillatory instabilities that element would be
the diagonal damping force; for divergences, it
would be the diagonal stiffness force. As would
be expected of a divergence instability, the
major destabilizing forces are displacement de-
pendent (i.e., appear in the C matrix). By making
additional reference to Eqs. (A-l) and (A-2), and

(A-3) and to their evaluation given above, the
following interpretation can be drawn from these
results :

1. The unstable mechanism involves a
coupling mainly between first flatwise bendirig
and rigid feathering. The mode shape, ^w =
(O.619, O.O336, 0.0149, and 1.0), confirms this
result .

2. The position of the chordwise mass
center behind the elastic axis, as indicated by
negative dynamic equation elements a^_ jj. and cij 1
and reference to the explicit statements of the
equations in the Appendix, is a major link in
the unstable coupling chain of events . This
result confirms well-known results concerning the
divergence of rotor blades ^J. specifically,
that the torsion modes drive the flatwise modes
aerodynamically (elements Cj ^ and c 2 ^) while the
flatwise modes drive the torsion modes with
centrifugal inertial forces through the rearward
mass center position (elements c^ -j. and c k ?)•

3. The first edgewise bending mode is being
driven by the rigid feathering through aerodynamic
and inertia terms (element bo ij.) but is not
actively participating in the unstable mechanism.

In a similar manner the following force
phasing matrices are written for the unstable
oscillatory (flutter) mode, \~

x 3"

A Phasing Matrix, P^_

•, 3125-00 -.0000 -.0000 -.1904-01

■,0000 -.2155-00 -.0000 -.1101-01

■.0000 -.0000 -.2731-00 -.0000

.1421-02 -.1114-04 -.0000 -.1217-03

B Phasing Matrix, R

••5163-00
,6947-00

.7587-02

•, 2973-00

.1659-02

,2749-02
-.1561-01
-.1413-01

.1968-01
,1282-01
,4186-05

-.1493-00
,5201-09 -.5155-05 -.6870-05 -.6027-03

C Phasing Matrix, Pq_

-.1260-08 .6779-02 -.4336-02 .8548-00

-.4997-02 -.1325-08 .1751-01 -.1802-00

.3814-01 .1173-02 -.0000 .3956-00

-.6696-03 .2794-04 -.3155-04 -.0000

Again the major destabilizing terms have
been underlined for clarity. With few exceptions
the same interpretations can be made of the
flutter force phasing matrices as were made for
the divergence ones. While the feathering

degree-of-freedom again drives the first flatwise
mode aerodynamically (element c-^ j,.), the flatwise
mode now drives the feathering degree-of-freedom
with vibratory inertia forces (element a^ jj •
Again these results confirm well-known findings.

Configuration With Pitch-Lag Coupling

Using the reconfirmed knowledge that an aft
chordwise center of mass is destabilizing, the
configuration is altered back to the original
quarter chord balanced configuration. In
addition, unit pitch-lag coupling (Ay... = 1.) is
introduced into the configuration. Hie resulting
dynamic equations are as follows:

A (inertia) Matrix

.2909-00 -.0000 -.0000 -.0000

-.0000 ,2006-00 -.0000 -.0000

-.0000 -.0000 .25ft2-00 -.0000

-.0000 -.0000 -.0000 .1133-03

B (Damping) Matrix

.28B7-00 .7757-01 -.1622-01 -.1256-01

.7992-01 ,1665-00 -.3596-02 -.3309-03

.3015-01 .1953-01 .1003-01 -.2169-02

,1536-07 ,6326-04 ,3801-03 ,3370-03

B Phasing Matrix, Pg

-.3822-00 ,5003-02 -.2424-01 .8736-02
.2114+01 -.2204-00 .2079-00 -.4936-02
.1546-01 -.3974-03 -.1327-01 -.2946-03

-.3235-07 .6967-05 .4427-03 -.4461-03

C Phasing Matrix, Pq

.1260-08
,2728-02
,2015-01
■,0000

•.4643-03
.5299-08
.7210-04

-.0000

.3315-00 .7077-01

-,1886>0i -.2069-00

-.0000 -.1373-01

-.0000 -.0000

C (Stiffness) Matrix

By referring to the explicit dynamic
equations given in the Appendix, the following
observations can be made:

1. The instability appears very similar to
classic pitch-lag instability ' '' and is mainly a
three-way coupling between first flatwise and
edgewise bending modes and the rigid feathering
degree-of-freedom. The resulting coupled mode,

0(3) = (-O.383 - 10.435, 0.015 + i0.024, 1.,

-0.100 - i.0317).

2. The edgewise bending mode is being driven
by inertia forces generated by flatwise bending
motion: coriolis forces proportional to precone
and flatwise bending rate and forces proportional
to pitch angle and flatwise bending deflection.

,3382-00 -.5763-01 -.2563-00 -.30*9-00
-.8232-03 ,11*22+01 -.5358-01 -.7113-01

.4633-01 -.2974-02 .4431-00 -.4337-01
-.0000 -.0000 -.0000 ,1646-0?

The eigenvalues for this configuration reveal the
configuration to be unstable in the edgewise
bending mode:

-0.573
-0.408
Xjjg = 0.0119
X 7j8 = -1.449

L l,2
L 3,4

iO.977
i2.609
il.324
13-505

Again, the stability force-phasing matrices
are formed for the unstable mode ( ka) and the
larger positive terms are underlined;

A Phasing Matrix, P&

-.9137-02 -.0000 -.0000 -.0000

-.0000 -.6302-02 -.0000 -.0000

-.0000 -.0000 -.7985-02 .-,0000

-.0000 -.0000 -.0000 -.3558-05

3. The flatwise bending mode is being
driven by aerodynamic forces generated chiefly
by pitch- lag coupled edgewise bending and to a
lesser extent rigid feathering deflection.

4. The rigid feathering degree-of-freedom is
being driven principally by a centrifugal force
moment involving chordwise mass radii of inertia,
pitch angle, and edgewise bending rate.

The stiffness force-phasing matrices for this
mode are formed and the significant terms for the
edgewise bending equation are underlined:

A Phasing Matrix, P A

-.5097-00 ,0000 .0000 .0000

,0000 -.3516-00 ,0000 .0000

,0000 ,0000 -.4455-00 .0000

,0000 ,0000 ,0000 -.1985-03

B Phasing Matrix, Pg_

.3*25-02

-,3696-00

.1722-01

-.1*69-07

.7*27-03 .2799-01

.1975-02 -.339*-00

-.6235-03 .1190-03

-.1870-05 -.1*50-02

C Phasing Matrix, Pq_

,3362-00

,16*7-01

-.1777-01

.0060

.280^-02
,1*22+01
■,*505-0*
,0000

.2923-00
■. 1179*01
.**M-00
,0000

,3780-02
-.1230-02
-,'906*-n3

.3996-05

■, 1596-00
.7995-00
,*327-02
.16*6-02

It can be seen that the principal stiffening
terms are, not unexpectedly, the diagonal mass and
stiffness terms. Mae only other significant
stiffening terms are those involving flatwise
tending rate and deflection which are also the
drivers of the unstable edgewise motion.

That the flatwise tending deflection term is
negative and numerically greater than the rate
dependent term can be appreciated by noting that
the unstable coupled edgewise mode frequency,
1.32*, is lower than the original corresponding
stable mode frequency, 1,398-

V. Concluding Remarks

The "force-phasing" matrices technique
provides yet another tool for understanding
dynamic/aerodynamic phenomena. While it does
not, by itself, indicate stability levels such
as are provided by the eigensolution, it does
complement the eigensolution by giving insight
into the details of the dynamic configuration
which are not director available from the
eigenvalues and eigenvectors alone-. Moreover,
the technique requires, in particular, eigenvector
information as a starting point. Hence, it is
inherently incapable of answering the more
fundamental question of why, for any one mode,
the eigenvector elements are indeed phased as
they are. It should also be stressed that the
technique is a tool to be used with, and in
support of, engineer/analyst judgement; the
results have to be Interpreted properly, generally
in the context of the specific application.
Finally, the relative simplicity of the formula-
tion makes the incorporation of the technique in
any aeroelastic eigensolution program a straight-
forward and easily implemented task.

Appendix - Details of Dynamic Equations

The linear dynamic equations used to
represent the aeroelastics of the rotor blade in
hover are formulated using an assumed modal
approach; the derivation is standard and uses
the nomenclature of Reference 6. The lineariza-
tion and subsequent simplifications are based
upon the following assumptions:

1. quasi-static, incompressible, nonstalled
airloads , /

2. coincident spar center, shear center
and tension center.

3. zero twist.

*. two flatwise bending modes, one edgewise
bending mode and the rigid feathering
degree-of- freedom .

5. normal uncoupled bending mode shapes
(zero twist and pitch angle).

The flatwise bending equations are then written
as:

J {(^wiYwm) *4 m + (» yi0 0g YwiYe 3 ) *<& 3
+[2mY Wi (PY Vlc + y 10og sin9Y^.)] l^

-(a I i^cose yq^) qe d (a-i)

•K^VV* 116 oose ) ^ + csy Wl (yio og ooB2e

-r9 S ine)Y 9;J ] *8j + *gj V^fre^j

The edgewise equations are written as:

where:

-[2 MYwmCPYvfc + yiOcgSine Yvfc)3 fw^

-[2 s(Pyio cg Yv k + % s ^ e y^) y 9; .3 3^

+(3y Y w sine cos6) q w (A-2)

v k m m w m

+ CEi z Y^ n + ry^ n - 5 cos 2 e Yvk Y Tn ] q^

+[mY V:k .(?P cose + yio cg sin2e)Y 9; .] q^

+ ^ Y^[-r2(2 « - U Jarg)(Y ej qej+ *^%

+ 2 *<? - ^M

to + v^

The torsion equations are written as:

(» yio cg Y 9;j Yw m ) X + M^ 10 + k z 2 10 )Y ejYe k ] *4 k

+ (2 m k^cose y^J q^

+[2 m(py 10cg Y Tn -+ kf^sine Y^) Y e ^ *v n

+» yiO^C^Yw " sin2 9 Y w )y 9 .] fc

J cg ™m

«m" B j J

+[«w + i&r) Ye* vi + m (^ 10 -J§ 10 ) cos2 9 y 9 ,y 9 ^

3 j' a k N z 10 Tio
+ ( 5 yiO Qg sln e cos 9 Y 9;J Y Vq ) q Vn
pacR r _o_

J'«k J ^k
(A-3)

+ ~2^ Ye/-^yiQ cA (Ye k qe k + ****** + VW

+ * yiO c /u(Y W Jw m - *oY v Jv n + yio3 C /^(Y ek ^ k
+ %%+ %%)) - iS * yio 3cA (Ye k qe k
+ A Wa | Wm +A Vn | Vn )]J<a? + ^ 3 ^qe 1 = o

* = + \/r

(A-^)

Beferences

Wilkinson, J. H. : Ihe Algebraic Eigenvalue
Problem. Clarendon Press, Oxford, 1965.

Programmer's Manual: Subroutines ATEIG and
HSBG. IBM System/360 Scientific Subroutine
Package, Version III, GH20-0205- 1 *, August 1970.

Leppert, E. 1., Jr.: A Fraction Series
Solution for Characteristic Values Useful in
Some Problems of Airplane Dynamics . Journal
of the Aeronautical Sciences, Vol. 22, No. 5,
May 1955.

Bielawa, S. L. : Techniques for Stability
Analysis and Design Optimization with Dynamic
Constraints of Nonconservative Linear Systems.
AIAA/ASME 12th Structures, Sturctural Dynamics
and Materials Conference Paper No. 71-388,
Anaheim, California, April 1971.

Miller, R. H. and C. W. Ellis: Blade
Vibration and Flutter. Journal of the
American Helicopter Society, Vol. 1, No. 3,
July 1956.

Arcidiacono, P. J,: Prediction of Rotor
Instability at High Forward Speeds; Vol. I,
Differential Equations of Motion for a
Flexible Helicopter Rotor Blade in Steady
Flight Including Chordwise Mass Unbalance
Effects. USAAVIABS Technical Report 68-18A,
U. S. Army, February 1969.

Chou, P. C: Pitch- lag Instability of
Helicopter Rotors. Journal of the American
Helicopter Society, Vol. 3, No. 3, July 1958.

SOME APPROXIMATIONS TO THE FLAPPING STABILITY OF HELICOPTER ROTORS

James C. Bigger s

Research Scientist

Ames Research Center, NASA

Moffett Field, California 94035

Abstract
The flapping equation for a helicopter in for-
ward flight has coefficients which are periodic in
time, and this effect complicates the calculation
of stability. This paper presents a constant
coefficient approximation which will allow the use
of all the well known methods for analyzing constant
coefficient equations. The flapping equation is
first transformed into the nonrotating coordinate
frame, where some of the periodic coefficients are
transformed into constant terms. The constant
coefficient approximation is then made by using
time averaged coefficients in the nonrotating frame.
Stability calculations based on the approximation
are compared to results from a theory which cor-
rectly includes all of the periodicity. The com-
parison indicates that the approximation is reason-
ably accurate at advance ratios up to 0.5.

Notation

a blade lift curve slope

B tip loss factor

c blade chord

I blade flapping inertia

i /T

kg flapping spring stiffness

N number of blades

R rotor radius

t time, sec

V forward velocity

a angle of attack of hub plane

J3^ flapping of ith blade relative to hub plane

B vector of rotor degrees of freedom in non-
rotating coordinates

B rotor coning angle

6 lc rotor tilt forward (longitudinal flapping)

B ls rotor tilt to left (lateral flapping)

B 2 rotor differential flapping

V blade lock number, pacRVl ■

X eigenvalue or root, nondimensionalized by fi,

a 3 a

V rotor advance ratio, V (cos a) /OR

v flapping natural frequency of rotating blade

a real part of eigenvalue

p air density

iji azimuth angle, Bt

S2 rotor rotational speed

to imaginary part of eigenvalue

Ug flapping n atural frequency of stationary

blade, ]/Icg7l"

( ) derivative, d( )/di/i

(") derivative, d 2 ( )/di|/ 2

For helicopter stability and control studies,
it is desirable to use as simple a math model as
possible while retaining reasonable accuracy, both

Presented at the AHS/NASA-Ames Specialists 1 Meeting
on Rotorcraft Dynamics, February 13-15, 1974.

to reduce computation effort and to gain insight
into system behavior. However, for a helicopter in
forward flight, the rotor flapping motion is
described by a differential equation having coeffi-
cients which are periodic in time (azimuth) . This
fact complicates the solution of the equation,
requiring methods which use considerable numerical
computation and which give little insight. Thus it
is desirable to find a differential equation with
constant coefficients (hence an approximation)
which adequately represents the forward flight
flapping dynamics of a helicopter rotor. If such
an equation is found, all of the well known tech-
niques for analyzing constant coefficient equations
may be used.

The flapping equation may be transformed into
the nonrotating coordinate frame, as done in
References 1 and 2, where some of the periodicity
is transformed into constant terms. This result
suggests that the use of constant coefficients in
the nonrotating frame will retain some of the
periodic system behavior. The constant coefficient
approximation examined herein is made by using time
averaged coefficients in the nonrotating frame. A
comparison is made between the eigenvalues (sta-
bility) obtained from the approximation and the
results from a theory which correctly includes all
of the periodicity. The comparison indicates that
the approximation is a useful representation of
helicopter flapping dynamics for both hingeless and
articulated rotors. This approximation was briefly
discussed in Reference 1 for one set of rotor
parameters. The present paper discusses the
approximation in a more general manner and gives
more insight into its features, limits, and
applicability.

The rotor math model used here is for fixed
shaft operation and includes only first mode
(rigid blade) flapping, with spring-restrained
flapping hinges at the hub center. Flapping
natural frequency may be matched by selecting the
spring rate. Thus the only approximations are in
the use of the aerodynamic terms for rigid blade
motion. Uniform inflow is used, and for the
advance ratios considered here (u < 0.5), reverse
flow effects are not included.

Equations of Motion

In this section, the single blade homogeneous
flapping equation is presented for a rigid, spring-
restrained, centrally hinged blade. This equation
is then transformed to a nonrotating coordinate
frame, using a coordinate transformation which is
briefly discussed. Insight into the fundamental
behavior of the rotor is gained by examining the
hovering (u = 0) eigenvalues of the equation in
nonrotating coordinates.

For the single blade, the homogeneous equation
of motion is

where

iL + Mgij + (v 2 + M g )B i =

M. = I B" + y I B 3 sin i^

= u X B 3

2l R 2

HtB 3 cos 4. + p* 4- B z sin 2$

D 1 O

1 +

"a 2 " in 2

Note that reverse flow has not been included here.
Although it could be included, it would not, signifi-
cantly affect the results for u < 0.5, since the
additional terms are fourth order in u.

By a coordinate transformation of the Fourier
type, the single blade equation may be written in
terms of nonrotating coordinates. The transforma-
tion accounts for the motion of all blades, and the
number of degrees of freedom is equal to the number
of blades. For example, with a three-bladed rotor,
the degrees of freedom are coning (all blades flap-
ping together), rotor pitching (cosine i)» flap-
ping) , and rotor rolling (sine i|j flapping) .
Adding a fourth blade adds a differential flapping
degree of freedom, where blades 1 and 3 flap in one
direction while blades 2 and 4 flap in the other
direction. This type of differential motion is a
degree of freedom with rotors having any even num-
ber of blades. Adding more blades adds degrees of
freedom which, in the nonrotating frame, warp the
plane described by the sine iji and cosine ty
flapping motion.

The coordinate for the single blade is p..
For a three-bladed rotor, the corresponding
nonrotating coordinates are

where B , g, , and g ls are rotor coning, pitch-
ing, and rolling motions. For a four-bladed rotor,

where 2 is the differential flapping motion
discussed above.

In general, the blade degrees of freedom in
the transformation are

i=l

B nc =

i=l

8 i

cos

nipj

B ns =

i=l

sin

rof^

B N =

(-D

, N eve

i=l

Then the motion of the ith blade is

'■

B i ■ e o + £ (e nc cos »*i + e ns sin n *i) + s n ( - 1)i;

n=i

K =<

j (N - 1), N odd
j (N - 2), N even

The equations of motion (that is, eq. (1)) must
also be converted from a rotating to a nonrotating
frame by a similar procedure. This process is
accomplished by operating on the equations with the
summation operators

IJ(...) > |S(...)cosn+. )
i i

This is virtually the same procedure used in
Reference 1.

It may be seen that the transformation
involves multiplication by sin ty, cos ty, sin 2$,
cos 2ip, etc. This changes some of the periodic
terms of the equations in the rotating reference
frame into constants (plus higher harmonics) due to
products of periodic terms, and vice versa.

Performing the indicated operations for N = 3
yields the following equations for a three-bladed
rotor.

J B» . u ^ B 3 sin 3* 2 - u j

» J B 3 -I.iiiB'eoslt t B" - i

r B 3 sin 3*

v 2 u 2 ^ b ! sin 3* -u 2 ^ B 2 cos »

■J «>♦»'$ B 2 sinW v 2 _ 1 . „ X B 3 cos 3* | (B'.iu 2 B 2 ).|jjB 3 sin3*

-v 2 jB ! cos3» - J(B»-ju 2 B 2 ).ujB 3 sin3» » 2 - 1 - M J B ! cos 3*

(2)

Similarly, operating as above with N = 4, the
equations of a four-bladed rotor are obtained.

a -^ B 3 sin 2iJ>

W z £B z sin2j.

v z - 1 * y 2 ^r B z sin 4$
U Xb 3 cos 2*

i(B" + iy z B2-|BVcos4*3

v 2 - 1 - y 2 ^ B z sin 4i^
u 2- B 3 sin 2*

u 2 £ B 2 sin 2*
u I- B 3 cos 2*

u J B 3 sin 2*

(3)

The thrtee- and four-bladed rotors have similar
behavior except for the terms which are periodic in
<fi. The periodic terms are 3/rev for the three-
bladed rotor, but are 2 and 4/rev for the four-
bladed rotor.

The main advantage of the transformed equations
is that it is easier to express the combined rotor
and airframe motions because the rotor equations
are now in a nonrotating reference frame and
include the motions of all blades. Furthermore,
rotor motions are more intuitively understood,
since the degrees of freedom are those seen by an
observer in or beside the helicopter.

In the nonrotating coordinates of equa-
tions (2) and (3), the equations are coupled by
off-diagonal terms. Note however, these are actu-
ally independent blades (unless some sort of feed-
back is added) and the coupling is due to the
coordinate transformation.

To gain understanding of these degrees of
freedom, the hovering (v = 0) behavior is examined
next. The hover equations for four blades are
given below.

| B 4 2

IB*

v 2 - 1 $ B*
- J B* v 2 - 1

8 ■

(4)

For three blades, the hovering equations are iden-
tical, except that the B 2 equation is then absent.
The Bg and 6 2 equations at hover are completely
uncoupled and are both identical to that of the
single blade in rotating coordinates.

(1), for u =

B. + X B 4 B, + v 2 B. =

3. o 1 X

1-0

K + 1- ^K * v2 b„ = o

6 2 + % b^B,, + v 2 B 2 = 0,

from equation (4)

Eigenvalues of these equations are easily calcu-
lated, and are shown on figure 1. These will be

J3+-^B 4 j3 + I/ 2 /3 =

Figure 1. Hover eigenvalues of coning and reaction-
less modes (v ■ 1.2, y - 8).

called coning and reactionless modes. The reaction-
less mode is so named because at hover it produces
no net reaction at the hub. The equations for
rotor pitching and rolling are,

x>*

i B *

and the characteristic equation is then,

(x 2 + £ B"X + v 2 - l) 2 ♦ (2X + } B 1 *) 2 -

The eigenvalues for this equation are shown in
figure 2. By analogy to a gyro, these modes will
be called precession (the lower frequency mode) and
nutation (the higher frequency mode) . The damping,
-•y/16, is the same as for the single blade of equa-
tion (1) and for the &„ and B 2 modes discussed
above. However, the coordinate transformation has
resulted in the precession mode frequency being ft
lower than the single blade mode frequency in the

Figure 2. Hover eigenvalues of precession and nuta-
tion modes (v = 1.2, y = 8) .

rotating frame. Similarly, the nutation mode fre-
quency is G higher than that of the single blade in
the rotating frame.

The coning mode (fig. 1) will excite vertical
motions of the vehicle, while the precession mode
will excite pitch and roll motions. Thus vehicle
responses are more intuitively understood by use
of the nonrotating coordinates. Also, these equa-
tions may be used to study feedback control systems
such as rotor tilting or rotor coning feedback,
which were discussed in Reference 1. Note (from
eqs. (1) and (2)) that the performance of such sys-
tems will depend on the number of blades used,
since the blade motions become coupled by the feed-
back terms and the coupling will vary with the

number of blades.

To compare the various modes with each other
and with other theories, it is necessary to trans-
form all eigenvalues into the same reference frame.
The obvious choice is the rotating coordinates of
equation' (1), since most other theories are appli-
cable to this frame. As may be seen by comparing
figures 1 and 2, the precession and nutation modes
may be transformed back into the rotating frame by
adding and subtracting Q respectively. This
process results in four identical eigenvalues, as
expected, since the rotor is composed of four iden-
tical blades, each described in the rotating frame
by equation (1). As noted above, the frequencies
of the 0. and 3, modes do not change

The equations for u = have been easily
solved and the nonrotating coordinate system has
been presented and discussed. In nonhovering
flight, however, the equations have periodic
coefficients, which makes the equations more diffi-
cult to solve, as well as giving the solutions some
special characteristics. These will be discussed
in the next section.

Periodic Coefficient Solutions

Floquet Theory

Eigenvalues of equations such as (1) may be
found with Floquet theory, as for example in
References 3 and 4. The equation is integrated for
one period Op = 0,..,, 2ir) for each independent
initial condition to obtain the state transition
matrix. The frequency and damping of the system
modes are then obtained by taking the logarithms of
the state transition matrix eigenvalues.

This technique has been applied here to three
cases, and the results are shown on figure 3 for

-1.5

a. v-\.\ , y = 6.0

b. !/=l.O, y = 6.0

c. v-\.0, y=l2.0

-- -.5

-1.0

.2 .3 .4 .5

x -1.5

Figure 3. Floquet theory root loci for varying y;
single blade in rotating coordinates.

varying u. Note that as u is increased, the
frequency (w/fi) decreases, while, the damping (a /Si)
remains constant at -y/16 until the frequency
reaches an integer multiple of 1/2 /rev. As. u is
increased further, the frequency remains constant
while the damping both decreases (the upper roots)
and increases (the lower roots) as shown for cases
a and c. This behavior may be surprising to those
accustomed to constant coefficient equations, but
is typical of periodic systems. The nonsymmetry
about the real axis is analogous to the behavior of
a constant coefficient equation root locus when the
locus meets the real axis. At that point, the
roots separate (no longer complex conjugates) , one
becoming less stable and the other becoming more
stable. With periodic coefficient equations, the
separation can occur at any multiple of 1/2 the
frequency of the periodicity. Actually, the con-
stant coefficient equation is a special case of the
periodic One, where the frequency in the coeffi-
cients is zero. This behavior may be seen in more
detail by plotting the eigenvalues versus y, as in
figure 4, which again shows results from Floquet
theory.

\:<l

1.0

a. i/ = i.i,y = 6

b. !/ = l.0,y = 6

^^^Nc

c. i/ = l.o,y-l2

I
(a) FREQUENCY
1.0

(b) DAMPING

Figure 4. Floquet theory variation of frequency and

damping with
dinates.

u; single blade in rotating coor-

Figures 3 and 4 have shown the eigenvalues in
the rotating coordinate system. These may be
transposed into the nonrotating system to examine
the behavior of the nonrotating modes. Choosing
case c (v = 1.0, y = 12) as an example, the root
locus is plotted on figure 5. The coning mode has
the same eigenvalues shown in the two previous
figures. The nutation and precession modes have
the same damping, but as mentioned before, their
frequencies are Q higher and lower, respectively,
than the coning frequency.

The regions where the frequency remains constant
while the damping changes, called critical regions,
may be illustrated by constructing the y - \i
plane as in figure 6 (and discussed in References 3
and 4). In the 0/rev region, the behavior is like
that of a constant coefficient equation when the
root locus meets the real axis; there are two real
roots, with order u 2 changes in damping. In. the
|/rev region, the frequency is exactly half of the
rotational frequency (fig. 3, case b), and the
damping changes somewhat more rapidly. In the
1/rev region (fig. 3, case a) the frequency is the
same as the rotational frequency (fi), and again
the damping changes are order y 2 . As previously
noted, damping is constant at -y/16 outside of the
critical regions. Note that varying v has little
effect on the boundaries of the 0/rev and ■j/rev
regions, but as v is increased the 1/rev region
moves upward.

In this section, the characteristics of the
periodic coefficient solutions have been discussed

NUTATION

-■--2.0

Figure 5. Floquet theory root loci for varying u;
three-bladed rotor in nonrotating coordinates
(v = 1.0, y = 12 j case c).

'CASE a

-J_

.1
(a)v = l.l

.2 .3

Figure 6. y - v plane for single blade in rotat-
ing coordinates based on Floquet theory.

Note that the 6 2 equation is not coupled to the
others and is the same as the 2 equation for
hover; hence it yields only the y = roots.
Therefore the 6 2 equation will not be discussed
further or included in subsequent figures. The
B equation has only one u-dependent term,
coupling it to the 3 1S motion. The pitch and
roll equations are coupled by both damping and
aerodynamic spring terms.

Comparison

As noted earlier, eigenvalues may be compared
by adding Q to the precession frequency and sub-
tracting f2 from the nutation frequency. In
examining the constant coefficient approximation,
any differences in eigenvalues will be due to the
dropped periodicity. That is, all of the roots
should approximate those obtained by using Floquet
theory to solve equation (1) . Using the comparison
method mentioned above, the constant coefficient
approximation is compared to Floquet theory results
in figures 7, 8, and 9. The frequency scales have
been expanded to exaggerate the effects of forward
speed. Each of the three cases is discussed below.

Case a.

1.1, y = 6. (fig. 7)

Figure 6. Concluded.

for nonhovering flight. The next sections will
discuss an approximation which has constant coeffi-
cients, yet gives some of the behavior of the peri-
odic coefficient system.

Constant Coefficient Approximation

In equation (1) the periodic coefficients are
all of the speed (y) dependent terms, and a con-
stant coefficient approximation yields only the
hover solution. However, in the nonrotating frame
of equations (2) and (3) , these periodic terms
have been transformed into constants plus higher
harmonic periodic terms. This result suggests that
the primary effects of u may be determined by
using the average values of the coefficients. The
constant coefficient approximation thus obtained
for a four-bladed rotor is given in equation (5) .
The corresponding equation for three blades is
identical, except that the B 2 motion is absent.

^B* 2

X B 3

*!» 3

x(b^,^)

I (B" - I V W) v* - 1

(5)

This case corresponds to a hingeless rotor
similar to the Lockheed XH-51. For this rotor the
variations with y of frequency and damping are
small but significant since the 1/rev critical
region is encountered (see fig. 6). All three
modes of the approximation agree well with Floquet
theory at low advance ratios, where the influence
of the periodic coefficients is small. As the
advance ratio is further increased, the precession
mode displays the same type of behavior as the
Floquet theory results, but the other two modes do
not. For the precession mode (and the Floquet
theory), the frequency becomes constant at 1/rev,
and the damping then has two values as previously
discussed. It is useful to examine why the con-
stant coefficient approximation displays periodic

1.075

1.050

1.000

O PRECESSION
D NUTATION
A CONING
FLOQUET THEORY

.1
(Q) FREQUENCY

Figure 7. Comparison of constant coefficient
approximation to Floquet theory (v = 1.1, y = 6).

-.5

-.4

O PRECESSION
D NUTATION
A CONING
FLOQUET THEORY

,-Q

«=*#=

CASED
V'U

y--e

(b) DAMPING

NUTATION

CASE a

y-e

•FLOQUET THEORY
■ APPROXIMATION

CONING

/j. = 0.5

4 .w

2.05
-- 2.00

1.05
1.00

-- .05

PRECESSION /

1 — — «--K

-.5 -.4\

■V

CONING

NUTATION

.05

---1.00
-1.05

-J- -2.05

Case b. v = 1.0, y = 6. (fig. 8)

This case corresponds to an articulated rotor
having relatively heavy blades, such as might be
used for a high speed helicopter. This case is
well removed from critical regions, and there are
no significant changes in the eigenvalues for the
u range shown. The constant coefficient approxi-
mation agrees well with results from Floquet theory.

Figure 7. Concluded.

system (critical region) behavior. In this case,
the precession roots at hover (u = 0) are very near
the real axis due to the coordinate transformation.
As \i is increased, the precession roots move
toward the real axis and then split when they reach
the axis, as usual with constant coefficient sys-
tems. Thus the damping both increases (the left
branch) and decreases (the right branch) .

1.0

O PRECESSION
D NUTATION
A CONING
FLOQUET THEORY

CASE b
I/=I.O
y = 6

(Q) FREQUENCY

-.2

-.1

r °

PRECESSION

□

NUTATION

CONING

FLOQUET THEORY

Pi -.PI rS 9 -

CASE b

v=\.0

y-6

1 1 1 1 1

(b) DAMPING

Figure 8. Comparison of constant coefficient
approximation to Floquet theory (v = 1.0, y = 6).

CASE b

!/=I.O

y-6

-FLOQUET THEORY

-« APPROXIMATION

/x= 0.5

-- 1.95
1.90

.95
.90

.10
-- .05

-*«

-- -.05
.10

-.90
-.95

-- -1.90
1.95

(C ) ROOT LOCI

Figure 8. Concluded.

Case c. v = 1.0, y = 12. (fig. 9)

This case corresponds to a typical articulated
rotor with blades similar to many aircraft flying
today. The Floquet theory indicates that the 4/rev
region is encountered at u = 0.215. It is seen
that the nutation mode is a poor approximation.
Apparently, the constant coefficient approximation
is not adequate for higher frequency modes if a
critical region is encountered. The precession and
coning modes (combined), however, do display the
correct type of behavior: the frequency approaches
■j/rev and the damping both increases (the precession
mode) and decreases (the coning mode) . In this
case, the correct behavior is obtained because two
modes are involved. As may be seen in figure 9(c),
the two sets of Floquet roots approach each other,
meet at ^-/rev, and split (no longer complex con-
jugates) . This behavior is approximated by the
coning and precession modes, but in the approxima-
tion, the roots remain complex conjugates as shown
in figure 9(c). The frequency of the precession
mode does not agree well with Floquet results, but
its damping is increasing; hence it is of less
interest. The coning mode agrees well with the
Floquet results, predicting the reduced damping
very accurately.

Perturbation Theory

Equation (1) has also been studied in
Reference 5, using a perturbation technique known
as the method of multiple time scales. Analytic
expressions are derived for the eigenvalues, with
expressions valid near and within each of the
critical regions and ones which are valid away from
the critical regions. These results are very

.75

.70

r °

PRECESSION

NUTATION

CONING

FLOQUET THEORY

CASEC

f = 1.0

y = l2

.1
(a) FREQUENCY

3. .3

-1.0

-.8

-.7

-.6

-.5

O PRECESSION
□ NUTATION
A CONING
FLOQUET THEORY

(b) DAMPING

Figure 9. Comparison of constant coefficient
approximation to Floquet theory (v = 1.0, y = 12),

-2.0

CASE
l/=I.O
y = l2

NUTATION

- 1.5

FLOQUET THEORY

-•—

—

APPROXIMATION

= 0.5

/coning

- i.o

■ ».

- .5

/ I *

" 1 PRECESSION

1 a .

-1.5

-1.0 -.5

- -.5

- — ■*.

- -i.o''

- -1.5

--2.0

Figure 9. Concluded.

useful; they give additional insight into the behav-
ior of periodic systems in general and equation (1)
in particular. A comparison between the Floquet
results of the present work and the analytic results
from Reference 5 indicates that the latter are also
useful quantitatively. An exception is near the
i/rev region, where the perturbation solution was

carried only to order
extended to order u 2
perturbation solution.

y. It should evidently be
as was the rest of the

Discussion

Based on the cases described above, it is
apparent that the constant coefficient approximation
may be used to calculate rotor eigenvalues at
advance ratios up to 0.5. A range of rotor param-
eters (y and v) have been studied which are repre-
sentative of most conventional helicopters. The
lower frequency modes agree well with Floquet
results and display behavior approximating that of
the Floquet theory critical regions. Therefore,
there are many cases where the approximation may be
used instead of more complicated methods.

The higher frequency modes of the approxima-
tion, however, do not display the correct behavior.
Where these modes are important, for example, in
using high gain feedback, the approximation should
be used with caution.

The perturbation theory of Reference S is very
easy to use for rotor stability calculations.

However, the solutions are for uncoupled blades in
the rotating coordinate frame. To account for
inter-blade coupling (as with certain feedback
schemes) one must either use another technique
such as that described herein or rederive the solu-
tions with the coupling included.

Conclusion

Transforming the flapping equation of a heli-
copter rotor in forward flight into the nonrotating
coordinate frame results in a set of differential
equations where some of the periodicity due to
forward flight is transformed into constant terms.
Using the time-averaged values of these, i.e.,
dropping the remaining periodicity, gives a con-
stant coefficient approximation which retains some
of the periodic effects. Comparison between results
of the approximation and those of Floquet theory
indicates that the approximation should be accept-
ably accurate for calculating flapping stability of
most helicopters for the advance ratios shown
herein. Use of the nonrotating coordinates has
given insight into rotor behavior and indicates how
the vehicle motion would be affected by the rotor
modes .

The higher frequency modes of the approximation
do not agree well with Floquet theory. Where these
modes are important for example, in using high gain
feedback control systems, the approximation should
be used with caution.

References

1. Hohenemser, K. H. and Yin, S-K., SOME APPLICA-
TIONS OF THE METHOD OF MULTIBLADE COORDINATES,
Journal of the American Helicopter Society ,
Vol. 17, No. 3, July 1972, pp 3-12.

2. NASA CR- 114290, RESEARCH PROGRAM TO DETERMINE
ROTOR RESPONSE CHARACTERISTICS AT HIGH ADVANCE
RATIOS, Kuczynski, W. A. and Sissingh, G. J.,
February 1971.

3. Peters, D. A. and Hohenemser, K. H., APPLICATION
OF THE FLOQUET TRANSITION MATRIX TO PROBLEMS OF
LIFTING ROTOR STABILITY, Journal of the American
Helicopter Society , Vol. 16, No. 2, April 1971,
pp 25-33.

4. Hall, W. Earl Jr., APPLICATION OF FLOQUET THEORY
TO THE ANALYSIS OF ROTARY-WING VTOL STABILITY,
SUDAAR No. 400, Stanford University, February
1970.

5. NASA TM X-62,165, A PERTURBATION SOLUTION OF
ROTOR FLAPPING STABILITY, Johnson, W., July 1972.

FLAP-LAG DYNAMICS OF HINGELESS HELICOPTER BLADES AT MODERATE AND HIGH ADVANCE RATIOS

P. Frledmann
Assistant Professor

and

L.J. Silverthorn

Research Assistant

Mechanics and Structures Department

School of Engineering and Applied Science

University of California, Los Angeles

Abstract

Equations for large amplitude coupled flap-
lag motion of a hingeless elastic helicopter blade
in forward flight are derived. Only a torsionally
rigid blade exicted by quasi-steady aerodynamic
loads is considered. The effects of reversed flow-
together with some new terms due to forward flight
are included. Using Galerkin's method the spatial
dependence is eliminated and the equations are
linearized about a suitable equilibrium position.

The resulting system of equations is solved
using multivariable Floquet-Liapunov theory, and
the transition matrix at the end of the period is
evaluated by two separate methods. Results
illustrating the effects of forward flight and
various important blade parameters on the stability
boundaries are presented.

Notation

A
k

^i'^i

^Fi'*Li
b

Two dimensional lift curve slope
Tip loss coefficient

Periodic matrix with elements A.
defined in Appendix B

*J'

C T
C

Generalized aerodynamic force for
i tn flap and lag mode respectively

Same as above, in reverse and
mixed flow regions.

Semi-chord nondimensionalized with
respect to R

Tip loss coefficient

Generalized masses defined in
Appendix A

Thrust coefficient

Constant matrix

Profile drag coefficient

Presented at the AHS/NASA-Ames Specialists'
Meeting on Rotorcraft Dynamics, February 13-15,
1974.

_
Presently, Dynamics Engineer, Hughes Helicopter
Company, Culver City, California.

C(k)

FEE 6

^Gi'^cr B ik'

Is _cs „cs

im'

(EI)
3

A *k
8 SF ,S SL

A \
i- yCT

h
i

L ,L

y z

im' ik

Theodorsen's lift deficiency
function

Defined in Fig. 1

Terms associated with elastic
coupling defined in Appendix A

Stiffness for flapwise bending

Stiffness for inplane of rotation
bending

Flap coefficients defined in
Appendix A

Generalized coordinate, k
normal flapping mode

Static value of

Perturbation in

in hovet

about

Viscous structural damping in
flap and lag respectively

Generalized coordinate, m
normal inplane mode

Static value of h, in hover

, o

Perturbation in

about

Unit vectors in x,y and z direc-
tions (Fig. 1)

Mass moment of inertia in flap,
defined in Appendix A

Unit matrix

Length of blade capable of
elastic deflection

Aerodynamic load per unit length
in the y and z directions
respectively

Lag coefficients, defined in
Appendix A

Mass of blade per unit length

M,N
^i>\i

( Vikr ( Vimr

P »P »P
r x r y r z

ikm
P(t)

Q
T
u,v,w

V ,v
e eo

w ,w
e eo

x,y,z

V Y G

Number of modes in lag and flap
respectively

Generalized mass for the i*-"
flap and lag mode respectively,
defined in Appendix A

Defined in Appendix A

Resultant total loading per unit
length in the x,y and z direc-
tion respectively

Defined in Appendix A

Periodic matrix

Blade radius

Constant matrix used in Floquet-
Liapunov theorem

Constant matrix

Common nondimensional period

x,y and z displacement of a point
on the elastic axis of the blade

Component of air velocity w.r.t.
the blade at station x perpendic-
ular to x-y plane (hub plane) ,
positive down

Same as above, in the x-y plane,
tangent to a circle having a
radius x

Elastic part of the displacement
of a point on the elastic axis of
the blade parallel to hub plane,
(see Fig. 1), subscript o de-
notes the static equilibrium
value

Velocity of forward flight of the
whole rotor

Elastic part of the displacement
of a point on the elastic axis of
the blade, in the k direction,
approximately, (Fig. 1)

Rotating orthogonal coordinate
system

Running spanwise coordinate for
part of the blade free to
deflect elastically

Defined in Appendix B

Angle of reversed flow region
(Fig. 2)

Angle of attack of the whole
rotor

T1 SF 1 ' n SL 1

m,%)

Droop, built in angle of the
undeformed position of the blade
measured from the feathering axis
(Fig. 1)

Preconing, inclination of the
feathering axis w.r.t. the hub
plane measured in a vertical
plane

Lock number (y=2p bR a/I. ) for
normal flow

m tn inplane bending mode

Symbolic quantity having the same
order of magnitude like the dis-
placements v and w

Real part of the k character-
istic exponent

k flapwise bending mode

Viscous structural damping coef-
ficients defined in Appendix A

Collective pitch angle measured
from x-y plane

Critical value of collective
pitch at which the linearized
coupled flap-lag system becomes
unstable in hover

Inflow ratio, induced velocity
over disk, positive down, non-
dimensionalized w.r.t. Rfl

Diagonal matrix, containing
eigenvalues Aj^ of R

Diagonal matrix containing eigen-
values Aj^ of J,(T,0)

Advance ratio

Critical value of advance ratio
at which flap-lag system becomes
unstable

Density of air

Blade solidity ratio

State transition matrix at ty ,
for initial conditions given at

Azimuth angle of blade #=fit)
measured from straight aft
position

■"C

Flutter frequency

ith

Imaginary part of k character-
istic exponent

%l'\l

Natural frequency of l" 1 flap or lag
mode , rotating

Speed of rotation

Special Symbols

( )

( )'
(*)

(.)

()

Nondimensionalized quantity, length
for elastic properties nondimensional-
ized w.r.t. A; all other w.r.t. R
frequencies w.r.t. £2; mass properties
w.r.t. ^

Differentiation w.r.t. x

Differentiation w.r.t. \|)

Subscripts, denoting real and imagin-
ary parts of the appropriate quantity

The symbol beneath a quantity
denotes a vector or a matrix

Denotes the inverse of a matrix

The dynamics of a helicopter blade in forward
flight are usually described by a system of linear
differential equations with periodic coefficients.
A growing acceptance of hingeless helicopter
blades for conventional helicopters flying at
relatively high forward flight speeds has intensi-
fied the need for fundamental research on the
aeroelastic stability of such systems.

Studies dealing with the effect of forward
flight (or periodic coefficients) have been
primarily devoted to the study of flapping insta-
bility at high advance ratios. 1 " 8 A limited
number of studies dealing with the effect of
periodic coefficients on coupled flap-lag ' or
coupled flap-lag-pitch 11 motion were also con-
ducted. The case of coupled flap-lag motion has
been, somewhat inconclusively, investigated by
Hall using multivariable Floquet theory, the
same problem was also considered by Friedmann and
Tong 9 but the treatment was limited to low advance
ratios (y<0.3). The coupled, linearized, flap-
lag-torsion motion has been investigated by
Crimi 11 using a modified Hill method. In both
cases 10 ' 11 only a limited number of numerical
results were obtained and the physical mechanism
of the aeroelastic instabilities has not been
clearly identified, in particular the degree of
freedom which triggers the instability was not
identified and the results for forward flight were
not compared with those for hover.

Recent investigation of the aeroelastic sta-
bility of hingeless blades in, hover 12 indicated
that the aeroelastic stability boundaries are
quite sensitive to the number of degrees of free-
dom employed in the analysis. Therefore it is
important to determine how the flapping behavior
of a blade at high advance ratios is modified by
the lag degree of freedom. This important problem,
which has not received adequate treatment before,
is one of the main topics of the present study.

The mathematical methods used in previous
studies dealing with the effects of forward flight
were: (a) The rectangular ripple method 1 , (b) Ana-
log computer simulation, (c) Various forms of
Hill's method, 2 ' 11 (d) Multivariable Floquet-
Liapunov theory, 6 ' 7 ' 1 (e) Perturbation method in
multiple time scales. 8 ' 9 The mathematical method
employed in the present study is the Floquet-
Liapunov theorem, and the transition matrix is
evaluated by two separate methods. It is also
shown that careful use of this method enables one
to circumvent problems associated with identifying
the results encountered in previous studies.

In addition, a new and convenient approxima-
tion for the reversed flow region is developed,
this approximation is believed to be adequate for
most blade stability analyses. Finally, the effects
of various important parameters such as collective
pitch setting, structural damping, droop and pre-
coning on the instability associated with forward
flight is investigated.

1. The Equations of Motion

1.1 Basic Assumptions

The present study is based upon a consis-
tently derived system of equations of motion for
the linearized coupled flap-lag motion of a
cantilevered rotor blade at arbitrary advance
ratios .

The derivation itself is algebraically
tedious, thus only a brief outline will be given
in this paper, the complete details of the deriva-
tion can be found elsewhere.

The geometry of the problem is shown in Fig.
1. The following basic assumptions were used in
deriving the equations of motion: (a) The blade is
cantilevered at the hub. It can have an angle of
droop Bj) at the root. In addition, the feather-
ing axis can be preconed by an angle 3p. The
angles 3n and 0p are small, (b) The blade can
bend in two directions normal to the elastic axis
and is torsionally rigid, (c) The deflections of
the blade are moderately small so that terms of
0(ej}) can be neglected compared to one. (d)
Moderately large deflections have only a small
effect on the tension due to elastic effects on
the blade since one of its ends is free, thus a
linear treatment of the elastic restoring forces
is adequate, (e) Two dimensional quasi-steady
aerodynamic strip theory is used C(k)=l and
apparent mass effects are neglected, (f) Reversed
flow is included using an approximate model for
reversed flow described in Appendix C. (g) Stall
and compressibility effects are neglected.

Using the assumptions given above a system
of nonlinear partial differential equations for
the coupled flap-lag motion of the blade is
derived, with respect to an x,y, and z coordinate
system rotating with the blade. The derivation
follows essentially along the lines of Reference
14, all the details can be found in Reference 13.

1.2 Brief Derivation of the Equations of Motion

The differential equation for the dynamic
stability of a cantilevered rotor blade can be
written as '

3x'
o

3x 2

( 3 2 w

a 2
3 v

+ E

C2 3x 2
o

3x u 3x J z
o o

[(HI) z - E cl ]

3x'

3x 2 j
o /

^L-L-r

3 v 3 w^ , „ „

-/ +E C2 7^-3x-tix-^y
o o '

(1)

where the quantities E(^, E(j2 are given in
Appendix A.

The distributed loading terms in the x,y and
z directions with terms up to 0(ei) in displace-
ments can be written as

p = _?_ = nin 2 [(x + e n ) + 2*]

x x o 1 '

9 isis it

P = L - mft [v - (e + v) + 2u]

y y o

9 ieit it

P z = L z ~ m Q W " ^F 12 W

8 SL n *

(2)

The boundary conditions for this kind of blade are
well known. The displacement field of the blade
with sin6s9 and cos8 = 1 can be written as 13 ' 11 *

\Z]

-wv-W-iCI®'*®]

v=v - x 8_9
e o D

w-w e + x (B p + 3 D )

Hx x
(3)

and

» B - E* vx^ct) - * w

k=l
M '

v — yi& Y (x )h (t) = -X, y h
e *—! 'm v o' m ' 'm n

(4)

m=l

where it is understood that repeated indices imply
summation unless otherwise stated.

The aerodynamic loads L» and L^ are given

L z " a P A tR V U T 9 ~ V c
L y - ap A bR|u p (U T e - U p )+ -f-

where the velocities D_ and U_ are given by

Up - fiw + ffitfx + u cos^) l^-j

o '

* /- 3v \

U„ - S2v + fiRlx+p sinifi + u cos^) -^-j

(5)
(6)

(7)
(8)

The last term in Equation (8) is due to the
radial flow along the blade. This term has been
neglected in some previous analyses. For arbi-
trary advance ratios this is an important and non-
negligible term.

Combination of Equations (1) through (8) and
application of Galerkin's method to eliminate the
spatial variable reduces the problem to a system
of ordinary differential equations.

** — _ * _ s —2 cs
*V±H+ 2 W Fi M Fi T1 SFi S i +E ik 8 k + "FAi 8 ! = E im h m

+2 WkV ( VV 5 ^ 2 <VV 5 L h m + h± (9)

— rfesfe _ _ 4> — —2 =5 =cs

\l\ +2 \±\±\L± h ± + \i\± h ±- E im h m " E iA

-2 B? k ( VVlX 1 ^D 9 * 2 [ § imr-< fi Y >imr ] \ h r

" 2 ^mM V* 2 B LA> e \ + ^i (10 >

where the various quantities Mp^, Pi^, Mli, S± m: ,

0*r\)lkSL are generalized mass integrals given in

References 13 and 14, and also in Reference 12

-1 -3
(for i=k=£=m=r=l) . While the quantities B^, B im ,

E ik> 5 im. Km' E ik> Si 1 , I7 k , Bf amd If m etc.
are given in Appendix A. The quantities Ap^, A L i
are generalized aerodynamic forces defined by

(ID

(12)

Equations (9) and (10) are coupled nonlinear
ordinary differential equations. In the present
study these equations will be linearized about a
suitable equilibrium position, which is taken to
be the steady state equilibrium position of the
blade in hover . Through this process of lineari-
zation many nonlinear terms are transformed into
coupling terms. At this stage one encounters a
considerable number of terms which are small and
therefore negligible. In order to neglect the
appropriate terms a rational ordering scheme is
used which enables one to neglect terms in a
systematic manner. In this scheme all the impor-
tant parameters of the problem are assigned orders
of magnitudes in terms of a typical displacement
quantity Ej) thus:

v
R

0(e D );|

0(e D ); x=0(l); y-O(l);
X=0(e D ); 9=0 (e D )

fr=0( eD );|^=0( eD );3 D -e p -0( eD );
o o

do
a

»0(e 2 )

(13)

An order of magnitude analysis of the equa-
tions indicates that in general terms up to and
including O(ep) must be included in the linearized
flap equations, while for lag equations some 0(e-h
terms have to be retained.

The process of the linearization consists of
expressing the elastic part of the displacement
field as

v = v + A v =-Y(h° + Ah) (W)
e eo e 'mm m

where the static equilibrium condition in hover
is given by

s iA +52 Fi M FA - ^X=-(^v 5 i-i<|) 2 ( 9F i- XF i>

where i = 1,2,.

11 IRk

+ J (|) 2 [X(8lJ- XL 2 ) + -|°- I*] 1-1,2,... H (15)

The various quantities F , L are defined in
Appendix A. Next, for the sake of simplicity,
the equations are specialized to the case of one

elastic mode for each degree of freedom, i.e. one
flapping and one lead-lag mode.

Furthermore for mathematical convenience the
equations of motion have to be transformed into a
system of first order equations. This is achieved
by using the following notation

Ag l = *1

Ah 1 -y A

(16)

For the stability analysis, only the homo-
geneous part of the equations of motion is
required, thus the equations of motion in their
final form can be written as

£ = A i*)y < 17 >

where A is a 4x4 matrix defined in Appendix B.

The equations of motion (17) will have a dif-
ferent form for the normal flow region and for the
reversed flow region. The representation of the
reversed flow together with its effect on the form
of Equations (17) is described in Appendix C.

2. Method of Solution

The stability investigation of the blade
motions is based upon the Floquet-Liapunov
theorem 15 which states the knowledge of the state
transition matrix over one period is sufficient in
order to determine the stability of a periodic
system having a common period T. Based upon the
Floquet-Liapunov theorem, the transition matrix
for the periodic system can be written as 15

SOMO

*(*.*„) = £

OJOe

£<*„>

where

PflrtT) = P(i|>)

(18)

(19)

where R is a constant matrix and £(t) is a periodic
matrix. Clearly the stability of the system is
determined by the matrix R, where R is given by
following relation

*(T,0)

*&-£.

(20)

A direct result of the Floquet-Liapunov theorem is
that the knowledge of the transition matrix over
one period determines the solution to the homo-
geneous system everywhere through the relation

RT V

&0P+ST.0) = £0|»,-0)(e~ )
where < i|i <_ T, s any integer.

(21)

In general R is a fully populated (nxn) square
matrix. If it has n independent eigenvalues, it
is well known from elementary linear algebra 5
that a similarity transformation can be found such
that

OR Q = A

(22)

where the columns of Q are the n-linearly inde-
pendent eigenvectors of R and A, is a diagonal
matrix whose elements are the eigenvalues of g,.
Combining Equations (20) and (22) and using the
definition of the matrix exponential 15 one has

Q ^ Q -l

J? . A = g~ l C £

jf^cr.ooij

(23)

where A, is a diagonal matrix containing the eigen-
values of the transition matrix at the end of one
period. The eigenvalues of £(T,0) or the char-
acteristic multipliers are related to the eigen-
values of R, denoted characteristic exponents,
through the relation

e = A^ k=l,2,...r (24)

Clearly Aj. and A^ are both complex quanti-
ties in general, thus

\ = \ + iW k
\ = A kR +lA kI

from which

and

<* - 2? fa[i 4R + 4*

\ = T^ A^

(25)

(26)

(27)

the quantity to^ can be determined according to the
Floquet-Liapunov theory only within an integer
multiple of the nondimensional period.

The stability criteria for the system is
related to the eigenvalues of g. or the real part
of the characteristic exponents f^. The solu-
tions of the Equation (17) approach zero as iJj •*■ °°
if

+ AJ^I < 1 or Zy.< k=l,2,..

. ,n

Finally a brief description of the numerical
implementation of the scheme described above will
be given. The transition matrix at the end of
one period $£T,0) is evaluated using direct numer-
ical integration. Equations (17) are integrated
for the set of initial conditions corresponding
to $£0,0) = £. The numerical integration is per-
formed using a fourth order Runge Kutta method.
The eigenvalues of the transition matrix are
evaluated by a Jacobi type eigenvalue routine.
For some of the cases the value of 4>XT,0) has
been evaluated using Hsu's method. 13 ' 17 This was
done in order to obtain results by two different
numerical schemes and also because Hsu's method
was found to be more efficient numerically. Both
methods yield identical results, therefore it is
not specified on the plots which scheme was used
to evaluate $0,0).

3. Results and Discussion

3.1 Humerical Quantities Used in the Calculations

In computing the numerical results the fol-
lowing assumptions were made,

Mass and stiffness distribution was assumed
to be constant along the span of the blade. Two
different kinds of mode, shapes were used:

(a) For most of the cases for which essen-
tially trend type studies were conducted an
assumed mode shape in flap and lap was used as
given by the appropriate expression in Reference
12. When an assumed mode shape is used the elastic
coupling effect 16 is neglected.

(b) For a few cases an exact rotating mode
shape in flap and lag was employed. These mode
shapes were generated by using Galerkin's method
based upon five nonrotating cantilever mode shapes
for each flap or lag degree of freedom. For these
cases the elastic coupling effect was included.

The coefficients F x , L and B defined in
Appendix A and in References 12 through 14 were
evaluated using seven point Gaussian integration.
For the region of reversed flow these coefficients
were treated in a special manner as explained in
Appendix C.

For the cases computed the inflow was evalu-
ated using an expression for constant inflow ratio
in hover , given by

, aO
A = 16

1 +

248
acr

- 1

(28)

This inflow relation is equivalent to taking
the induced velocity of 3/4 blade radius as repre-
sentative of a constant induced velocity over the
whole disk. It is clear that for forward flight
one should use the expression

X = u tano^ + C T /2 Yu 2 +X 2

(29)

Use of this relation would have required the
use of a trim procedure in the calculations. It

can be seen from Reference 14 that the require-
ment of trimmed flight at a fixed C T results in
an increase of 8 at advance ratios of p > .3 and
it also requires continuous changes in B-± c and
e^g. The experience gained when using this
approach in Reference 14 indicates that when the
trim requirement is included in the calculation,
the value of y c at which instability will occur
will be usually lower. Furthermore, when using
this approach it was found that it is difficult to
determine which part of the degradation in stabil-
ity is related to the increase in 8, 8 ls and 8i c
and which part is due to the periodic coefficients.
This added complication is not warranted in a
trend study such as the present one, and it is
not consistent with the stated purpose of this
paper, which is; a clear illustration of the
effects of the periodic coefficients when the lag
degree of freedom is included in the formulation
of the problem.

Finally, in all the computations the follow-
ing values were used:

C do = .01; a=2ir; 0= .05; e^O; A=0. ; I"=l

Various other pertinent quantities are specified
on the plots.

3.2 Results

The results obtained in the present study
usually are given in form of plots representing
the variation of the real part of the character-
istic exponent £fc with the advance ratio y. Most
of the cases presented in this study were evalu-
ated using an assumed mode shape, as described in
the previous section, and neglecting the elastic
coupling effect.

For some cases an exact rotating mode shape
in flap and lead-lag was used and the elastic
coupling was included, when this approach was used
a statement to this effect appears on the appro-
priate plots. When no such statement appears it
is to be understood that the assumed mode shape
is used and the elastic coupling is neglected.

A typical case is shown in Figure 3 for a
collective pitch setting of 8= .15. Starting the
computation at y=0, enables one to easily identify
the instabilities encountered, by using results
previously obtained for hover. As shown the lag
degree becomes unstable and the frequency of the
oscillation is <% = 1.28119. This result clearly
indicates that by neglecting the lag degree of
freedom one could obtain completely incorrect
stability boundaries.

The importance of the reversed flow region is
illustrated by Figure 4. As shown with the
reversed flow region the instability occurs at
higher values of y than without the reversed flow
region. Similar trends were observed in previous
studies when only the flapping motion was con-
sidered, 5 indicating that by neglecting the
reversed flow region one could expect conservative
results from a stability point of view. It also

*8 7 .Sj cyclic pitch changes.

indicates that in this particular case the
reversed flow region starts being important above
advance ratios of p = 0.8.

It is important to note that the frequency at
which the lag degree of freedom becomes unstable
is not 1/2 or 1 as is usual for the case of para-
metric excitation. Thus it seemed important to
identify the source of this destabilizing effect.
The results of this study are presented in Figures
5 and 6. The effect of neglecting the radial flow
terms on the real part of the characteristic
exponent, associated with the flap degree of free-
dom, is shown in Figure 5. As shown, the radial
flow terms have a stabilizing effect on the flap-
ping motion with the radial flow forms suppressed
the flap degree of freedom becomes unstable at
p=1.33. The effect of the radial flow terms on
the lead-lag degree of freedom is illustrated by
Figure 6, as shown without the radial flow terms
the instability in the lag degree of freedom is
completely eliminated and the system becomes
unstable in flap. When the radial flow terms are
included, the lag degree of freedom is the crit-
ical one and the system becomes unstable at u= . 754 .
This matter was pursued further by identifying the
actual destabilizing term in the equations of
motion, which was found to be an aerodynamic
coupling term. This term couples the flap motion
with the lag motion in the flap equation, its form
is

2 2, 9w 8u

v cos * Si" ST

o o

This term is due to the U T Up term in Equation (5) .
The term shown above is the complete nonlinear
one, clearly the one retained in the equations of
motion is the coupling term obtained from linear-
izing this expression.

As mentioned in the previous section the
results presented in Figures 3 through 6 were
obtained by neglecting the elastic coupling effect.
In order to asses the effect of this assumption
the typical case has been also recomputed with the
exact mode shape and the elastic coupling effect,
the results are shown in Figure 11. From Figure
11 it is clear that use of the exact rotating mode
in flap and lag reduces the value of p c to
p c = 0.653, when the elastic coupling is also
included p c is further reduced to p c = .583. Thus,
for this case, p c seems to be more sensitive to
the type of mode shape used than to the inclusion
of the elastic coupling effect. It is also
interesting to note, that for this case the elastic
coupling effect is destabilizing, while for hover
its effect on 9 C is quite stabilizing.

Previous studies 12 dealing with the effect of
viscous type of structural damping on the stabil-
ity boundaries for hover indicated that this para-
meter has an important stabilizing. The effect
of this parameter for forward flight is shown by
Figures 7 and 8. The stabilizing effect of
structural damping in the lag degree of freedom
is evident from Figure 7, where the real part of
the characteristic exponent associated with the

lead-lag degree of freedom is plotted as a func-
tion of the advance ratio p , again only the struc-
tural damping in lag is important. A summary of
these results is presented in Figure 8 showing the
variation of p c as a function of the structural
damping. It is interesting to note that this plot
indicates that the greatest stabilizing effect
due to structural damping is obtained in the
range < risLl < -^ 2 &% o£ critical damping),
after which, the gain in stability tends to level
off. Similar trends were obtained from stability
studies in hover. 12

Again in order to illustrate the sensitivity
of the results to the mode shape and elastic
coupling, the results have been recomputed with
these effects included; these results are also

shown in Figure 8. As seen the use of the correct
mode shape and the elastic coupling effect reduce
the value of p c , at which instability occurs.

The sensitivity of the results, to different
collective pitch settings is illustrated by
Figure 9. Comparison of Figures 3 and 9 indicates
that by decreasing the collective pitch setting
from 9 = .15 to 8 »' >05 eliminated, the instability
associated with thelead-lag motion. The instabil-
ity in this case occurs at p c • 1.88 with a
frequency of or 1. This is a typical flapping
instability due to the periodic coefficients.
Comparison of Figures 3 and 9 seems to indicate
that the assumption of nonlifting rotors used in
some forward flight studies 7 can be
nonconservative .

Finally, Figure 10 shows the dependence of
Pc on the angle of droop 6n- As shown p c is
relatively insensitive to B D . On the other hand
3j) has a very important effect on the value of 8 C
at which the linearized system in hover becomes
unstable.

It should be also noted that a considerable
number of additional numerical results, including
the effects of elastic coupling can be found in
Reference 13.

4 . Conclusions

The major conclusions obtained from the pre-
sent study are summarized below. They should be
considered indicative of trends and their appli-
cation to the design of a helicopter blade should
be limited by the assumptions used.

(1) Flapping instability and response
studies at high advance ratios can be inaccurate
and misleading due to the neglection of the lag
degree of freedom. The effect of the periodic
coefficients on the coupled flap-lag system shows
that in general instability can occur at lower
values of advance ratios than when the flap
degree of freedom is considered by itself.

(2) In addition to the instabilities associ-
ated with the periodic coefficients (i.e. with
frequencies of 0, 1 or 1/2) the coupled flap-lag

system has the tendency to become unstable due to
an aerodynamic coupling effect associated with the
radial flow terms. This instability which has a
frequency close to the rotating lag frequency of
the system, occurs usually at values of p c much
lower than those for which the flapping degree of
freedom becomes unstable.

(3) Viscous type of structural damping in the
lead- lag degree of freedom has a stabilizing
effect on the instability discussed in previous
conclusion.

(4) The value of the collective pitch setting
has a considerable effect on the value of the
advance ratio at which instabilities due to the
periodic coefficients or the radial flow
aerodynamic coupling terms occur. Increase in
collective pitch is destabilizing, therefore high
advance ratio studies which do not include this
effect (nonlifting rotors) may yield nonconserv-
ative results.

(5) The numerical results obtained in the
present study agree with the analytical results
obtained previously 9 indicating that hingeless
blades with a rotating lag stiffness of 1/2 or 1
can easily become unstable due to the effect of
periodic coefficients.

(6) While droop has a very strong effect on
the stability boundaries of hingeless blades in
hover, it has a very minor effect on the stability
boundary in forward flight.

References

Horvay, G. and Yuan, S.W., STATILITY OF ROTOR
BLADE FLAPPING MOTION WHEN THE HINGES ARE
TILTED. GENERALIZATION OF THE 'RECTANGULAR
RIPPLE' METHOD OF SOLUTION, Journal of the
Aeronautical Sciences , October 1947, pp. 583-
593.

Shulman, Y. , STABILITY OF A FLEXIBLE HELI-
COPTER ROTOR BLADE IN FORWARD FLIGHT, Journal
of the Aeronautical Sciences , July 1956,
pp. 663-670, 693.

Sissingh, G.J., DYNAMICS OF ROTOR OPERATING
AT HIGH ADVANCE RATIOS, Journal of American
Helicopter Society , July 1968, pp. 56-63.

Sissingh, G.J., and Kuczynski, W.A., INVESTI-
GATIONS ON THE EFFECT OF BLADE TORSION ON THE
DYNAMICS OF THE FLAPPING MOTION, Journal of
the American Helicopter Society , April 1970,
pp. 2-9.

R & M No. 3544, THE STABILITY OF ROTOR BLADE
FLAPPING MOTION AT HIGH TIP SPEED RATIOS ,
Lowis, O.J., 1968.

Peters, D.A. , and Hohenemser, K.H. , APPLICA-
TION OF THE FLOQUET TRANSITION MATRIX TO
PROBLEMS OF LIFTING ROTOR STABILITY, Journal
of the American Helicopter Society, April
1971, pp. 25-33.

10.

11.

12.

13.

Hohenemser, K.H., and Yin, S.K., SOME APPLI-
CATIONS OF THE METHOD OF MULTIBLADE COORDI-
NATES, Journal of American Helicopter Society ,
July 1972, pp. 3-12.

Johnson, W. , A PERTURBATION SOLUTION OF ROTOR
FLAPPING STABILITY, AIAA Paper 72-955.

Friedmann, P. and Tong, P., NONLINEAR FLAP-
LAG DYNAMICS OF HINGELESS HELICOPTER BLADES
IN HOVER AND IN FORWARD FLIGHT, Journal of
Sound and Vibration , September 1973.

SUDAAR No. 400, APPLICATION OF FLOQUET THEORY
TO THE ANALYSIS OF ROTARY WING VTOL STABIL-
ITY, HALL, W.E., Stanford University,
February 1970.

NASA CR-1332, A METHOD FOR ANALYZING THE
AEROELASTIC STABILITY OF A HELICOPTER ROTOR
IN FORWARD FLIGHT, Crimi, P., August 1969.

Friedmann, P., AEROELASTIC INSTABILITIES OF
HINGELESS HELICOPTER BLADES, AIAA Paper 73-
193 January 1973, (also Journal of Aircraft,
October 1973).

UCLA School of Engineering and Applied
Science Report, AEROELASTIC STABILITY OF
COUPLED FLAP-LAG MOTION OF HELICOPTER BLADES
AT ARBITRARY ADVANCE RATIOS, Friedmann, P.,
and Silverthorn, J.L., to be published
January 1974.

NASA-CR-114 485, DYNAMIC NONLINEAR ELASTIC
STABILITY OF HELICOPTER ROTOR BLADES IN
HOVER AND FORWARD FLIGHT, Friedmann, P. ,
and Tong, P., May 1972.

Brockett, R.W. , FINITE DIMENSIONAL LINEAR
SYSTEMS, John Wiley and Sons, 1970.

Ormiston, R.A. , and Hodges, D.H. , LINEAR
FLAP-LAG DYNAMICS OF HINGELESS HELICOPTER
ROTOR BLADES IN HOVER, Journal of the
American Helicopter Society , Vol. 17, No. 2,
April 1972, pp. 2-14.

Hsu, C.S., and Cheng, W.H. , APPLICATION OF
THE THEORY OF IMPULSIVE PARAMETRIC EXCITA-
TION AND NEW TREATMENTS OF GENERAL PARAMETRIC
EXCITATION PROBLEMS, Journal of Applied
Mechanics , March 1973, pp. 78-86.

Appendix A. Definitions of the Generalized
Masses, Aerodynamic Integtals and other Quantities

The quantities, ^11,Mf1,M^ 1 ,'sX 11 , (M^) lu ,

(My) in are generalized masses given, in Appendix
A of Reference 12, with the general i,m,k indices
these quantities can be found in References 13
and 14.

14.

15.

16.

17.

5 L =)i3 X Ti i[^ mY » d5 i] d V i b
i x °

o x

11 x °

3£r ** AU »<*i +i i> di ] d Vv

o 1

x dx

o o

Structural damping is represented by

g SF = 2 n 5 Fi n SFi ; ^'28^^

The elastic coupling effect is represented by
E C1 = [(EI) z -(EI) y ]sin 2 e; E C2 = [ (EI) a -(EI) 1

sin6cos9

j(\i« d *o

"i^ 1

/ E C2« d *o
*^o

" ^ V^*

/ Yi' Y" E_-dx
y 'i 'n CI o

' 5m V 1 **

1
/« E C2 d *o

- ^ V&

When using these expressions in a one mode analysis
for each degree of freedom the lower indices are
deleted for these expressions and the expressions
for the generalized aerodynamic integrals. The
generalized aerodynamic integrals F*, iA can be
found in References 12,13 and 14. For this study
some additional expressions had to be defined, only
these are given below.

■J 1

F 23.
ikm

C 20
im

/ x ruY'dx ; F 22 - / r].y' d:
I o I'm o im I I'm

A A

A
B

Y'dx ; L'
'm o

21 ,

'ikm

/vM^o

t 22 -
ikm

L 24 =

,Y'dx ; L 23
k'm o ' im

Y'x dx
'm o

£\w

Appendix B. Elements of the A ~ Matrix

The elements of the A - matrix, which defines
the equations of motion when written as first order
differential equations, are given below:

A 21 - 1; A 22 = A 23 = A 24 =

A 43 - lj A 41 - A 42 - A 44 -

A ll ""®D1 + 2 ( l } H<- F9sin, (' + F 24 h°cos\J) )

hi' ~(4 + f) + \ ¥\^«* *♦ + ^ 6cos *)

2 "1

+ | F 23 h° (1+cos 2M

A 13 = y | (|) 3 I -2euF 11 siniiH-p(e p +e D )F 11 cosi(-
+ F 1A g°u cosJ

A 14 - ^ + | (|) ? Je(-2F 21 uco8i( ) -u 2 sin2tpF 22 )

+ XuF 22 cosi|; +ij (l+cos2i)»)[F 23 g°+F 22 (ep+e D )]

A 31 = Y G + \ (f) 3 [9L 8 u sinip-2uL 17 g°cos*-6yL 22 h°cosij)
- 2vKe p +e D )L 8 cosij/]

_£8 . 2

+ i(i

* 32 ~ *u ' 2

L U sin 2ifH-L 10 y cosij;)

-2AL 1:L y costfi-e^ (1+cos 2^)L 21 h°-

-y 2 (e p +3 D ) L 11 (1+cos 2i|>) -L 24 ]i 2 g° (l+cos2i|0j

A 33 = -ijj^ X (£) 3 [-9uL 16 g°cos*-2 -22- uL 13 siniJ )

0p(e p +6 D )L 13 coaiil

^34

(4" ^) + i <|) 2 '{-e^ 20 cos^ + f 2 (i

"li'

+cos2i|>)[-L 21 gi- (g p +B D )L 20 ]e+ -£■ (-)A 20 sin2.J.
-2u L 23 cosi|))|

where
21

2P e°
z lll g l

f- (|) 3 [26F 10 -F 11 X]+ S- 2(e_+fi )
«F1 2M F1 V

ZklH!l + _X_ (|) 3 [ L 7 9_2AL 8 ]- ll 2(g 1J +e_)
2M L1 ^1

"ta

Appendix C. Approximate Reverse Flow Model and
the Associated Aerodynamic Loads

the circular region of reversed flow, which
exists over the retreating blade, is quite well
known. In past treatments of reversed flow it has
been customary 3 to define three separate regions:
(a) normal flow, (b) reversed flow, (c) mixed flow,
and evaluate the appropriate aerodynamic expres-
sions for each region. When this model is used
together with a modal representation of the blade
the evaluation of the generalized aerodynamic
expressions &■,!?■ becomes quite cumbersome, and a
more convenient procedure had to be devised.

The approximate reverse flow model developed .
for the present study consists of replacing the
circular region, by an approximate region which is
a circular sector as shown in Figure 1. The
approximation is based on the assumption that the
area contained in the circular sector must be equal
to the area contained in the approximate region.
Two separate cases must be considered: (1) y < 1,
(2) u > 1.

Case (1). For u < 1, the radius of the circular

part is taken as y. Simple geometric consid-
erations show that the angle a is always a
constant .

given by a = ir/2

Case (2) . For y >^ 1 simple geometric considerations
show that

j. 2 -11 r 2
a = it - 2 sin (— ) + y sin (— ) - vp -1

Thus, for y < 1 the generalized aerodynamic
loads are calculated from

Api

" / Vk dx o + / Vi dx o
- J k *y J

- H. 5 n

- / L Y, dx + / L Y. dx
J y 1 o J y'i o

La I, J

while for y >_ 1.0

hi " _A Fi and *Li = -^Li

These expressions are based on the assumption that
the lift curve slope in the reversed flow region
is equal to the negative value of the lift curve
slope in normal flow.

« r — ' . »

A v .j X, i

TOP VIEW

Figure 1. Displacement Field of a Torsionally Rigid Cantilevered
Blade with Droop and Preconing.

<a)(*<1

(b)n>1

Figure 2. Geometry of Approximate and Exact Reverse Flow
Regions.

1. ALL (cjj) TERMS IN LAG EQ. NEGLECTED

2. ALL TERMS INCLUDED

3. ONLY (eg) TERMS ASSOCIATED WITH
DAMPING INCLUDED

Effect of Third Order Terms in the Lag Equation on
Characteristic Exponent for Lag.

WITH REVERSE FLOW

<3pj« 1.175

u u - 1.28303

y - 10.0

o - 0.05
Op - 0.0

WITHOUT

REVERSE FLOW

e - .«

"SF, " •<"

"SL,- 01

Figure 4. Effect of Reversed Flow on Characteristic Exponent
for Lag.

-1.6
-1.2
-0.8

"F
U L
7
o

h
Po

- 1.176

- 1.28303

- 10.0

- .06

^7- RADIAL FLOW TERf
y / INCLUDED

-0.4

— — u k CONTINUOUS — W

' V RADIAL FLOW TER!
/ NOT INCLUDED

1 1 1 1

1 T v ^ 1.

0.2 0.4 0.6 0.8

1.0 1.2 L4»

Figure 5. Effect of Radial Flow Terms on Characteristic
Exponent for Flap.

Figure 6. Effect of Radial Flow Terms on Characteristic
Exponent for Lag.

5 F1

- 1.175

"L1

- 1.28303

- 10.0

- .05

- 0.0

0„

- 0.0

- .16'

Figure 7. Effect of Viscous Structural Damping on Characteristic
Exponent for Lag.

.005

.010

.015

ELASTIC COUPLING,
EXACT MODE SHAPE
NO ELASTIC COUPLING.
EXACT MODE SHAPE
NO ELASTIC COUPLING.
ASSUMED MODE SHAPE
_1
.025

.020

'SF^-'SL,

Figure 8. Critical Advance Ratio /u c vs Structural Damping
Coefficients jj sf , 1J SL ■

,f c

-1.0

cc
o

-0.8

-0.6

£
X

-0.4

-0.2

<
z
u

0.2

X
X

a F1"

1.175

X
X
X
X

S L1"

1.28303

10.0

.05

h -

0.0

X
>»

%> -

0.0

w k CONTINUOUS

.05
.20

cJi, - 0.0 OR 1.0
IT 1 1

N^ 1

I B

0,2

0.4 0.6 0.8 1.0

1.2 1.4 1.6

1.8 2.6'

-2.2

1.0

0.8

0.6

0.4

Opi» 1.09
GJ L1 = 1.00
y = 10.0
a - .05
(Jp - 0.0

e = .is

-L

-2.0

-1.5

0.0

Figure 9. Effect of Collective Pitch on Typical Case.

-1.0 -0.5

D (DEGREES)

Figure 10. Effect of Droop on u

0.5

ASSUMED MODE SHAPE
NO ELASTIC COUPLING

WITHOUT ELASTIC
COUPLING

Figure 1 1 . Effect of Exact Mode Shape and Elastic Coupling
on Characteristic Exponent for Lag.

CORRELATION OF FINITE-ELEMENT STRUCTURAL DYNAMIC
ANALYSIS WITH MEASURED FREE VIBRATION CHARACTERISTICS
FOR A FULL-SCALE HELICOPTER FUSELAGE

Irwin J. Kenigsberg

Supervisor - Airframe Dynamics

Sikorsky Aircraft

Stratford, Connecticut

Michael W. Dean

Dynamics Engineer

Sikorsky Aircraft

Stratford, Connecticut

Ray Malatino
Helicopter Loads and Dynamics Engineer
Naval Air Systems Command
Washington, D. C.

Abstract

Both the Sikorsky Finite -Element Airframe
Vibration Analysis Program (FRAN /Vibration Analy-
sis) and the NASA Structural Analysis Program
(NASTRAN) have been correlated with data taken
in full-scale vibration tests of a modified CH-53A
helicopter. With these programs the frequencies
of fundamental fuselage bending and transmission
modes can be predicted to an average accuracy of
three percent with corresponding accuracy in
system mode shapes .

The correlation achieved with each program
provides the material for a discussion of modeling
techniques developed for general application to
finite-element dynamic analyses of helicopter
airframes. Included are the selection of static
and dynamic degrees of freedom, cockpit structural
modeling, and the extent of flexible- frame model-
ing in the transmission support region and in the
vicinity of large cut-outs . The sensitivity of
predicted results to these modeling assumptions
is discussed.

cut-outs and concentrated masses such as the trans-
mission, main rotor, and tail rotor, which are
unique to helicopters, play a major role in con-
trolling vibrations .

Although advanced analytical methods based on
finite-element techniques have been developed for
studying the vibration characteristics of complex
structures , a detailed correlation of such methods
with test data is not available in the general
literature. Further, little information is avail-
able on the accuracy of various modeling assump-
tions that might be made to reduce the cost and
time of applying these vibration analyses .

As a result a research project was establish-
by Naval Air Systems Command with Sikorsky Aircraft
to:

a) Determine the accuracy of the Sikorsky
Finite-Element Airframe Vibration
Analysis in predicting the vibration
characteristics of complex helicopter
"airframe structures.

Introduction

Helicopter vibration and resulting aircraft
vibratory stress can lead to costly schedule
slippages as well as to problems in field service
maintenance and aircraft availability. At the
core of vibration control technology is the require-
ment to design the helicopter structure to minimize
structural response to rotor excitations . Both the
complexity of the structure and the increasingly
stringent mission and vibration control specifica-
tions demand development of airframe structural
vibration analyses that can be used rapidly and
economically to evaluate and eliminate vibration
problems during the preliminary design phase of
heli copters .

The complex helicopter structure consists of
sections that differ considerably in structural
arrangement and load carrying requirements . These
sections include the cockpit, cabin, tail cone,
and tail rotor pylon. In addition, large fuselage

Presented at the AHS/NASA-Ames Specialists '
Meeting on Rotorcraft Dynamics, February 13-15,197^

and

b) Develop and evaluate general helicopter
dynamic modeling techniques that could
be used to provide accurate estimates
of vehicle dynamic characteristics while
at the same time minimizing the com-
plexity and cost of the analysis .

Due to the increased usage of NASTRAN
throughout the industry as well as the efficiency
resulting from employing a single analytical sys-
tem for both static and dynamic analyses , a par-
allel correlation study using NASTRAN has been
performed. The results of these correlation
studies are the subject of this paper.

Phase I - Stripped Vehicle

Test Vehicle

At the initiation of this effort, the phi-
losophy guiding the development of modeling tech-
niques was based upon the concept of gradually in-
creasing the complexity of the analytical repre-
sentation. It was decided that the first

correlation study would be conducted on an air-
craft stripped of all appendages. It was believed
that the modeling techniques for representing air-
frame response characteristics could be identified
and developed most easily in this manner. Then,
as various appendages were added to the basic
vehicle, only the modeling techniques required for
the structure or masses added need be developed.

The vehicle used in this test and correla-
tion study was the CH-53A Tie Down Aircraft, Ve-
hicle designation number 613. A general arrange-
ment of the structure is illustrated in Figure 1.
For initial correlation, all appendages were re-
moved. These included the nose gear, main landing
gear, main landing gear sponsons, fuel sponsons,
tail pylon aft of the fold hinge, tail rotor and
associated gear boxes, engines, cargo ramp door,
horizontal stabilizer, and all remaining electri-
cal and hydraulic systems. The main rotor shaft
and all gears were removed from the main transmis-
sion housing and only the housing itself was re-
tained for the test configuration.

Testing

The ground test facility employed to estab-
lish the dynamic characteristics of the test vehi-
cle was a bungee suspension system that simulates
a free-free condition, a rotorhead-mounted uni-
directional shaker, and the Sikorsky shake test
instrumentation console. Instrumentation con-
sisted of lit fixed and 10 roving accelerometers .
A complete description of the test apparatus and
the instrumentation is provided in Reference 1.

All accelerometer signals and the reference
shaker contactor signal were transmitted to the
console. The signals were processed automatically
by the console resulting in a calculation of the
in-phase and quadrature components of the acceler-
ations. The accelerations were then normalized to
the magnitude of the shaker force at the particu-
lar frequency. As frequency was varied, the re-
sulting response of each accelerometer was record-
ed on a XYY' plotter, Figure 2, as g's/1000 lbs.
versus frequency.

Ideally, a fuselage mode can be identified
by a peak in the quadrature response and a simul-
taneous zero crossing of the in-phase response.
Once a mode is located, all quadrature responses
at this frequency can be recorded to define the
mode shape. The modes defined in this manner from
the shake tests are listed in the left-hand column
of Table I. It should be noted that this tech-
nique is applied more easily at lower frequencies,
where sufficient modal separation exists so that
the forced response in the vicinity of a resonance
is dominated by a single mode. As shown in Figure
2, the mode shapes at higher frequencies must be
extracted from the coupled response of many modes.

Analysis and Correlation

The shake test data indicated that the
natural modes of vibration of a helicopter can be
categorized as beam-like modes controlled by

overall fuselage characteristics (e.g., length,
depth, overall bending stiffness, mass distribu-
tion, etc.) and those controlled by the transmis-
sion support structure. Therefore, the overall
helicopter structure was modeled utilizing three
modules :

and

center section including the transmission
support region

forward fuselage and cockpit

aft fuselage and tail.

The center section was modeled in greatest detail
by applying finite-element techniques . The struc-
tural characteristics of the forward and aft fuse-
lage were derived from beam theory. These equiva-
lent beams were located at the neutral axis of the
airframe section and were assigned the bending and
torsional properties of the total section. The
beam models of the forward and aft fuselage were
cantilevered from rigid frames at the respective
forward and aft ends of the center section, Figure
3. The influence coefficients of these beams with
respect to their cantilevered ends were then com-
bined with the influence coefficient matrix of the
remaining structure.

The Phase I correlation was performed using
the Sikorsky Finite-Element Airframe Vibration
Analysis (FRAN /Vibration Analysis) . This analysis
consists of two programs: PPFRAH and a 200 dynamic-
degree-of-freedom eigenvalue/eigenvector extraction
procedure. PPFRAH is derived from the IBM/MIT
Frame Structural Analysis Program, FRAN (Reference
2), a stiffness method, finite-element analysis
limited to two types of elements , namely bending
elements (bars) and axial elements (rods). This
limitation necessitated further development of FRAN
for application to stressed skin structures. This
development consists of the addition of pre- and
post-operative procedures linked to FRAN. In the
pre-operative procedure (Pre-FRAN), the fuselage
skin is transformed into equivalent rod elements.
This transformation is developed by satisfying the
criterion that the internal energy of the skin
structure under an arbitrary set of loads be the
same as that of the transformed structure under the
same set of loads. The post-operative procedure
(Post-.PRAN) extracts the influence coefficient
matrix corresponding to the selected dynamic
degrees of freedom. A detailed description of the
FRAN /Vibration Analysis is provided in Reference 1.

The elements used to represent the airframe
structure are:

1) bending (bar) elements for fuselage frames
and for the nose and tail beams

2) axial (rod) elements for. the stringers
and

3) equivalent, diagonal rod elements for skin
panels .

For dynamic analysis, the structure is as-
sumed to be unbuckled, so that all skin panels are
considered fully effective in resisting axial
loads. Thus, the total axial area of each skin
panel is lumped with the areas of adjacent string-
ers.

During Phase I correlation, three modeling
parameters were varied: the number of bays over
which the finite-element (flexible -frame) model
extends (Figure k) , the number of nodes per frame
(number of stringers), and the number of dynamic
degrees of freedom assigned to each frame (Figure
5). The results of the correlation are presented
in Table I, which shows the sensitivity of the
analysis to each of the above parameters and the
accuracy of the predicted frequencies and mode
shapes. The criteria for establishing the level
of mode shape correlation are:

E (Excellent) - Correct number of nodes, nodes

less than 2.5 percent of fuselage
length from measured location,
local modal amplitudes within 20
percent of test values.

G (Good) - Correct number of nodes, nodes

less than 2.5 percent of fuselage
length from measured location,
difference between actual and pre-
dicted local modal amplitudes ex-
ceeds ±20 percent of test values .

F (Fair) - Correct number of nodes, nodes

more than 2.5 percent of fuselage
length from measured location,
difference between actual and pre-
dicted local modal amplitudes ex-
ceeds ±20 percent of test values.

P (Poor) - Incorrect number of nodes, nodes

located improperly, difference be-
tween actual and predicted local
modal amplitudes exceeds ±20 per-
cent of test values.

A comparison of the 30- and 60-stringer anal-
yses indicates that there is no change in the re-
sults when modeling the structure with half the
number of actual stringers. In addition a compari-
son of results obtained with the basic and reduced
dynamic degree of freedom allocation indicates that
no more than 16 dynamic degrees of freedom per
frame are required for dynamic modeling.

Although mode shape correlation resulting
from the analysis in the frequency range of inter-
est (below 1500 cpm) is encouraging, see Table I,
the absence of a representative mass distribution
made the analysis overly sensitive to certain mod-
eling assumptions. This sensitivity appears to
account for the less than satisfactory frequency
correlation. For example, the frequency of the
transmission pitch mode is normally controlled by
the mass of the fully assembled transmission and
the properties of the structure in the transmission
support region. In the absence of a mass distribu-
tion representative of a fully assembled vehicle,
however, any element of the structure and any

lumped mass can contribute significantly to the
control of the dynamic characteristics. In this
case, the analytical representation appears to be
too stiff because of the beam model used for the
fuselage forward of F.S. 262, which constrains the
upper and lower decks to deform equally. This
constraint is not imposed by the actual structure.
A comparison of the results of the 9- and l8-bay
analyses indicates that due to the local nature of
the transmission pitch mode, extension of the
flexible-frame model aft beyond the limit of the
9-bay model has no significant effect on the pre-
diction of this mode.

The poor frequency correlation for the first
lateral bending mode persisted throughout this
phase of correlation. This mode was characterized
by differential shearing of the upper and lower
decks of the aft cabin, Figure 7. The 6-bay and
9-bay flexible-frame model represented most of this
structure experienceing the differential shearing
as a beam capable of only bending and torsion.
This overly constrained model resulted in predicted
frequencies substantially higher than test values.
Extending the flexible-frame representation to 18-
bays appears to be the solution. However, size
limitations in PPFRAN required that the l8-bay
flexible-frame model be generated in two 9-bay sub-
structures , married at a rigid intermediate frame
at F.S. kk2, Figure 3. Although the extended model
improved the correlation of the first lateral bend-
ing mode, absence of a representative mass distribu-
tion again appears to make the model overly* sensi-
tive to the presence of the rigid frame at F.S. hk2.
This accounted for the remaining difference between
test and analysis.

Many of the higher frequency modes are con-
trolled by the structure in the area of the rear
cargo ramp. This accounts for the failure to pre-
dict the Transmission Vertical mode until the flex-
ible-frame model was extended into the ramp area,
see Table I. Although this extension of the model
improved correlation, the high frequency modes
above 1500 cpm are difficult to identify analytical-
ly due to the coupling of overall fuselage modes
with local frame modes . This difficulty is com-
pounded in this investigation, because the fre-
quencies of the basic fuselage modes are raised due
to the stripped condition of the vehicle, while
frequencies of the local frame modes are lowered
due to the lumped-mass modeling used to represent
each frame. Tests of a more representatively load-
ed fuselage can be expected to minimize the problem
of mode identification.

From the results of this phase of the corre-
lation, it is concluded:

1) The selection of static degrees of freedom in
the flexible frame model can be based on a
structural model that contains stringers num-
bering one half the number of actual
stringers .

2) No more than sixteen dynamic degrees of free-
dom on each flexible frame are required for
dynamic analysis. The typical location of

these degrees of freedom is Illustrated in
Figure 5-

Transmission modes can be predicted by a
flexible-frame representation of the trans-
mission support region extending about 1.5
transmission lengths forward and aft of the
corresponding transmission supports, about
9 bays. If the vehicle contains large cut-
outs, such as the cargo ramp of the test
vehicle, the flexible-frame model should ex-
tend through this region as well.

PHASE II - BALLASTED VEHICLE

Testing

Shake tests were performed after adding bal-
last to provide a more realistic representation of
a helicopter mass distribution, Figure 6. At the
transmission mounting plate, two lead blocks hav-
ing a total weight of 1*570 pounds were mounted so
that the mass and pitching moment of inertia of
the simulated transmission and rotor head approx-
imated that of the actual CH-53A. At the tail, a
1500-pound block was mounted to simulate the re-
moved tail pylon, stabilizer, and tail rotor. At
the nose, a 3000-pound block was mounted on the
nose gear trunnion fitting to balance the vehicle.

The natural modes of vibration identified by
shake tests are listed in Table II along with the
frequencies measured during Phase I. Not only did
the ballast succeed in lowering the fuselage modes
into a frequency range more representative of that
encountered on a fully assembled aircraft, but ad-
ditional modes were also identified that are
strongly controlled by the ballast. In fact,
these modes were identified as local modes of the
ballast blocks themselves. Due to the complex
structural nature of the ballast, Figure 6, these
appendages did not lend themselves to simple ana-
lytical representations. Therefore, the flexi-
bility of each ballast block was measured by in-
strumenting both the block and the adjacent air-
frame structure and then measuring the accelera-
tions occurring at both locations near the modal
frequencies of interest. The mass of each ballast
block and its absolute acceleration resulted in a
force which produced the relative motion between
the two instrumented parts. The empirically de-
fined flexibilities of the ballast were then used
in the dynamic model.

Analysis and Correlation

The modeling techniques developed in Phase I
of this study were applied to both the FRAN/Vibra-
tion Analysis and NASTRAH.

The finite-element model analyzed in Phase
II was identical to the 18-bay model analyzed in
Phase I, except for adding the mass and structural
characteristics of the ballast blocks. The FRAW
model was formed with rod and bar elements, as
discussed previously, while the HASTRAM model used
CROD, CBAR, and CSHEAR elements (Reference 3). As
before, all skin panels were assumed fully

effective in reacting axial load and this effective
area was lumped into the adjacent stringers.

Including ballast, to replace removed appenda-
ges resulted in a' substantial improvement in the
correlation, particularly in frequency prediction
as shown in Table III. Significantly, ballast
eliminated the difficulties identified as sensitivi-
ty to modeling assumptions and local frame modes in
the absence of representative mass distributions.
The average error in predicting the frequencies of
fundamental fuselage bending modes and the trans-
mission pitch mode was 3-W for both the FRAN/Vibra-
tion Analysis and NASTRAH. In addition the shape
correlation for these modes varied from good to ex-
cellent. The analyses also were able to predict
accurately the significant changes in the charac-
teristics of the fuselage and transmission modes
resulting from the addition of the ballast, Figures
7, 8 and 9- To achieve this degree of correlation,
modeling of the ballast flexibilities was required.
This modeling was successfully accomplished in the
vertical/pitch direction, Figure 10, but did not
prove successful in the lateral/torsion direction,
Figure 11. The contrast between these two results
establishes the ability of finite-element analyses
to predict accurately the characteristics of fuse-
lage and transmission modes when the structural
data base is defined with sufficient accuracy. Fur-
ther improvement in the correlation could have been
achieved if a more detailed definition of the bal-
last flexibilities had been acquired from measure-
ments of static deflections .

Reasonable success has been achieved in pre-
dicting higher frequency, ramp-controlled modes,
Figures 12 and 13. However, some margin. does exist
for further improvements in shape and frequency
prediction. From the standpoint of modeling, it
appears that the 200 dynamic degree of freedom
limit established in this study is inadequate for
predicting the shell-type modes of the cargo ramp
structure. In addition, the test procedure em-
ployed, namely the use of a single rotorhead
shaker, does not provide a means of uncoupling the
forced response characteristics of modes at the
higher frequencies, Figure 2.

Conclusions

Finite element analyses can predict accurate-
ly the frequencies and mode shapes of complex
helicopter structures, provided the structur-
al data base is defined accurately.

Complete stripping of a vehicle for correla-
tion purposes may make the analysis overly
sensitive to normally minor modeling assump-
tions .

Significant changes can be predicted accurate-
ly in the character and frequency of fuselage
and transmission modes due to changes in mass
distributions and structural characteristics.

The modeling techniques established by this
study can be used during aircraft design re-
gardless of the finite-element analytical
system' being used.

Recommendations

References

1) A full-scale shake test correlation should
he performed on a fully assembled flight
vehicle to establish and validate modeling
techniques for those appendages removed
during this study.

2) Appendages not amenable to accurate or eco-
nomical structural analysis should he
tested statically to determine flexibility
data required for dynamic analysis.

3) Integrated structural design systems should
he developed to couple static and dynamic
analyses and thus provide the accurate
structural data required for defining vibra-
tory response characteristics as early as
possible during aircraft design.

k) Use of additional shaker locations should be
incorporated in the test procedure to pro-
vide a means of uncoupling higher frequency
modes. Alternatively, more sophisticated
means of processing shake test data (e.g.,
system identification techniques described
in Reference k) should be employed.

1) Kenigsberg, I. J., CH-53A FLEXIBLE FRAME
VIBRATION ANALYSIS/TEST CORRELATION,
Sikorsky Engineering Report SER 651195,
March 28, 1973.

2) IBM 7090/709!+ FRAN FRAME STRUCTURE ANALYSIS
PROGRAM (7090-EC-OlX).

3) McCormick, C. W. , ed. , THE NASTRAN USER'S
MANUAL, (level 15), NASA SP-222(0l), June
1972.

1+) Flannelly, W. G., Berman, A., and Barnsby,
R. M., THEORY OF STRUCTURAL DYNAMIC TESTING
USING IMPEDANCE TECHNIQUES, USAAVLABS
TR 70-6A,B, June 1970.

5) Willis, T., FRAN CORRELATION STUDY, Sikorsky
Report SYTR-M-36, July 1969.

Illustrations

Horizontal
Stabilizer

Tail Sotor
Attachment

Cargo
Ramp Door

-Nose Gear

Sponsons

Figure 1 CH-53A General Arrangement

o
o

CO
_i
o
o
o

o
-H

0-.

% •■

FULL SCALE

3G/I000 LB.
FULL SCALE"

SECOND TRANSMISSION
VERTICAL VERTICAL

FULL SCALE
I I

■4-

100

200 300 400 500

600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700
FREQUENCY, CPM

Figure 2 Typical Vertical Response to Vertical Excitation

TABLE I
PHASE I SHAKE TEST CORRELATION SUMMARY

TEST

ANALYSIS

18 Bay
30 Stringer
Reduced DOF

9 Bay
30 Stringer
Reduced DOF

6 Bay
30 Stringer
Reduced DOF

6 Bay
30 Stringer and
60 Stringer
Basic DOF

Mode

Freq.
(CPM)

Freq. Error

Shape

Freq.

Error

Shape

Freq.

Error

Shape

Freq. Error

Shape

1st Lateral

910

1207 33$

1U66

60%

1U35

58%

11*1*0 58%

1st Vertical

1155

1175 2%

1282

11%

121*2

121*1 8%

XSSN Pitch

1U90

1709 13%

1710

13%

17^8

17%

1758 17%

2nd Vertical

1950

2150 10%

2390

22%

2505

28%

2577 32%

XSSN Roll

2000
2150

21405 20%
2250 k%

2870

1*3%

2900

1*5%

2891* 1*5%

XSSN Vertical

Torsion

2300

2763 20%

2l*28

P/F

2U22

P/F

21*1*5 6%

P/F

632 /V- RIGID FRAME

Figure 3 Modular Representation
of Helicopter Structure

-RIGID FRAME

F.S. tlO

Figure *t Finite-Element Model, Transmission
Support and Bamp Areas

WL 163
BL -5L91

WLJ39.
8L-52.92

WLI09
BL-52

WL 189.7
BL 31.81

WL I9t
BL ^20

BASIC

WL 163
BL^!98

WL 139
BL 52.92

WLI57

BL-52.38

iiy]

WL JEt
BL- 52.97

71 T4

WL 87.55 WL87 WL 87.55

BL-35.7 BL BL 35.7

V — L

"7^

REDUCED

1 A^ I

mj57

BL 52.!

WL 139
BL 52.9

WLJ2I
BL 52.97

Figure 5 Degree of Freedom Locations for Basic and
Reduced Dynamic Degree of Freedom Models

WL 87.0J
BL 22.86

Figure 6b Tail Ballast
Installation

Figure 6c Nose Ballast
Installation

Figure 6 Phase II Ballast Installations

TABLE II - SHAKE TEST FREQUENCIES

MODE

1st Vertical Bending

1st Lateral Bending

Transmission Pitch

Hose Block Lateral/Roll

Hose Block Vertical/Coupled

Forward Cabin/Nose Block Lateral

Hose Block Vertical

Second Vertical Bending

Torsion

Transmission/Ramp Vertical Bending

Ramp Vertical Bending

PHASE II
TEST FREQUENCY CPM

PHASE I
TEST FREQUENCY

kko

1155

615

910

7^0

1U90

930

970

990

1050

1290

1950

1310

2300

lte5

2150

16^0

TABLE III

PHASE II CORRELATION SUMMARY

VERTICAL/PITCH MODES

MODE

1st Vertical Bending

Transmission Pitch

Hose Block Vertical/
Transmission Pitch

Hose Block Vertical

Second Vertical

Transmission Vertical/
Ramp Vertical

Ramp Vertical

Frequency

Test

FRAN

kko

1*38

TkO

751

970

933

1050

101*3

1290

1523

1U25

1563

161*0

139 s *

Error

1.555

10?
15?

Shape

HASTRAN

1*53

785

956

1063

1608

F/G

181*3

P/F

1355

Error

1.5%

155
25?

29?

17?

Shape

E
G

G
F
F

F/G
P/F

1st Lateral Bending 615

Hose Block Lateral/Roll 930

Forward Cabin Lateral/

Nose Block Lateral 990

Torsion 1310

LATERAL/T0RSI0H MODES
659 7? &
735 21? P

858
1601

13?
22?

P
P

595

812

970
1325

3?
13?

G
P

FRAN-1207 opm

Figure 7a Correlation of First Lateral Bending
Mode, Phase I - Stripped

NASTBAU-595 cpm

Figure 7b Correlation of First Lateral Bending
Mode, Phase II - Ballasted

FHM-11T5 opm

Figure 8a Correlation of First Vertical Bending
Mode, Phase I - Stripped

NASTBAN-453 opm

•Figure 8b Correlation of First Vertical Bending
Mode, Phase II - Ballasted

PRM-1709 cpm

Figure 9a Correlation of Transmission Pitch Mode,
Phase I - Stripped

NASTRAN-785 cpm

Figure 9t>

Correlation of Transmission Pitch Mode,
Phase II - Ballasted

HASTRM-956 cpm

Figure 10 Correlation Hose Block Vertical/Transmission
Pitch Mode, Phase II - Ballasted

MASTRAU-812 cpm

•Figure 11 Correlation of Hose Block Lateral/Roll
Mode, Phase II - Ballasted

Phase II test-1290 cpm

HASTMM-1608 cpm

NASTRAH-1355 cpm

Figure 12 Correlation of Second Vertical Bending
Mode, Phase II - Ballasted

Figure 13 Correlation of Ramp Vertical Bending
Mode, Phase II - Ballasted

COUPLED ROTOR/AIRFRAME VIBRATION PREDICTION METHODS

J. A. Staley
Senior Dynamics Engineer

J. J. Sciarra
Senior Structures Engineer

Boeing Vertol Company
Philadelphia, Pa.

Abstract

The problems of airframe structural
dynamic representation and effects of
coupled rotor/airframe vibration are dis-
cussed. Several finite element computer
programs (including NASTRAN) and methods
for idealization and computation of air-
frame natural modes and frequencies and ,
forced response are reviewed. Methods for
obtaining a simultaneous rotor and fuse-
lage vibratory response; determining
effectiveness of vibration control devices,
and energy methods for structural optimi-
zation are also discussed. Application of
these methods is shown for the vibration
prediction of the Model 347 helicopter.

Notation

A - airframe mobility matrix

B - rotor impedance matrix

EI - blade bending rigidity

F - force

GJ - blade torsional rigidity

I - identity matrix

k - rotor frequency multiple, 1, 2, etc.

K - stiffness matrix

M - mass matrix

q - airframe mode generalized coordinate

X - airframe displacements

rn - airframe mode generalized mass

w - airframe mode natural frequency

$ - airframe mode shape (eigen vector)

fl - rotor frequency

[] - matrix

{ } - column vector

Subscripts

A - absorber , airframe

c - cosine component amplitude

H - hub

k - rotor frequency multiple, 1, 2, etc.

n - airframe mode number

o - zero hub motion

R - rotor

s - sine component

Presented at the AHS/NASA-Ames Specialists'
Meeting on Rotorcraft Dynamics, February
13-15, 1971.

Part of the work presented in this paper
was funded by the U.S. Army Research
Office - Durham, North Carolina under
Contract DAHC04-71-C-0048.

Superscripts

. - velocity
. . - acceleration
T - transpose

Prediction of helicopter airframe
vibration involves two major problem
areas s

• Prediction of rotor vibratory
hub loads

• Prediction of airframe dynamic
characteristics .

The effects of vibratory hub motion on
vibratory hub loads and effects of vibra-
tion control devices and resulting air-
frame fatigue stresses must also be con-
sidered.

Methods for independent prediction of
vibratory hub loads and airframe dynamic
characteristics have been developed pre-
viously and are discussed briefly below.
Independent determination of rotor vibra-
tory loads and airframe vibratory response
to these loads does not account for any
interaction between airframe vibratory
motion on rotor vibratory loads. One
approximate method for accounting for
these interactions is to assume that an
effective rotor mass is attached to the
airframe at the rotor hub. A more direct
method is to compute (or measure) the
rotor hub impedance and determine compat-
ible vibratory hub loads and hub motions.
This method is discussed below. A simple
example of compatible rotor load-hub
motion is given for a single rotor heli-
copter with vertical hub motion. In
addition, flight test results for the
Model 347 helicopter are compared with
vibration predictions obtained using a
coupled rotor/airframe vibration computer
program.

Rotor Vibratory Hub Loads

Methods and digital computer programs
have been developed for prediction of
rotor vibratory hub loads for constant
speed level flight conditions 1,2,3.
Rotor blades are represented by lumped
parameter analytical models as indicated

8.1

in Figure 1. Iteration techniques are
used to compute individual blade deflec-
tions and aerodynamic and inertia load
distributions at integer multiples of the
rotor rotating frequency. The total
rotating and fixed system rotor vibratory
hub loads are obtained by summing indivi-
dual blade root shears and moments. The
vibratory hub loads may be computed assum-
ing no hub motion. If the vibratory hub
motions are known, effects of these
motions may be included when computing
blade aerodynamic and inertia loads.

Airframe Dynamics

Structural Model, Natural Modes and Fre-
quencies , ' ana Forced Response

Finite element methods have been used
in the helicopter industry for some time
for prediction of airframe dynamic charac-

teristics^

As indicated in Figure 2,

developing a finite element airframe model
consists of:

• Defining nodal data

• Defining elastic properties of
members connecting nodes

• Defining mass properties asso-
ciated with each node.

Nodal data and properties of struc-
tural members are used to develop stiff-
ness matrices for individual members.
These matrices relate forces at each node
to nodal displacements. The stiffness
matrices for individual members are super-
imposed to obtain the stiffness matrix
for the entire airframe.

Most of the degrees of freedom are
reduced from the airframe gross stiffness
matrix. Mass properties are concentrated
at the remaining (retained) degrees of
freedom. Equations (1) are the airframe
equations of motion with the gross stiff-
ness matrix. Equations (3) are the air-
frame equations of motion, in terms of
the reduced stiffness matrix.

(1)

fcj = [ K n] - L K 12] fri "^21] < 2 >

[M] {xj + [Kj {X X } ={f r } _ (3)

M 0"

Xl"

hi K i2

• —

1*2

?21 K 22.

l X 2j

The solution for natural modes and
frequencies is made using the reduced
stiffness matrix and the mass matrix
associated with the retained degrees of
freedom.

The airframe motions are expressed
in terms of natural modes s

M = [♦] M

(4)

and, after assuming sinusoidal motion
with no external forces, Equation (3)
becomes :

-1

(5)

The modal generalized mass is then
computed. A value of modal damping is
assumed for each mode, and these modal
properties are used to compute airframe
response to vibratory hub loads:

*» = N T I M ] W

(6)

q n + 25 n w n q n + w n q n = [4^ {Fr}/^ (7)

Substructures Method

A large saving in computer time can
be realized by performing the matrix
reduction process on several smaller sub-
structure stiffness matrices instead of
on the large stiffness matrix for the
entire airframe. In one application, use
of the substructures method reduced com-
puter running time from about ten to two
hours on an IBM 360/65 computer.

The airframe is divided into several
substructures, and all but mass and
boundary degrees of freedom are reduced
from the stiffness matrix of each sub-
structure. The stiffness matrices of the
substructures are then merged (super-
imposed or added just as they are for
individual members) to form a stiffness
matrix for the entire airframe. Any
degree of freedom on the boundaries may
be reduced after merging the substructure
matrices (Figure 3) .

NASTRAN

New developments in finite element analy-
sis have been occurring on a continuous
basis. New programs and new structural
elements, both dynamic and stress analy-
sis capability, FORTRAN programming cap-
ability by the engineer within the finite
element program, and greater problem size

capability have been developed . NASTRAN
(NASA Structural Analysis) 6 is a govern-
ment developed, maintained, and continu-
ally updated finite element program which
has apparently provided a solution to the
difficulties of developing and maintain-
ing finite element programs by private
contractors. NASTRAN is similar to other
finite element computer programs except
that it generally provides additional
capability:

• More types of structural elements

• Common deck for stress and
dynamic analysis

• User programming capability

• Transient vibration analysis,
buckling, non-linear, and static
capability

• Unlimited size capability for
mass and stiffness matrices.

For a nominal fee, this program and
manuals describing the program and its
use are available. NASTRAN provides a
standard for airframe dynamic analysis
and relieves contractors of some of the
problems of maintaining the most up-to-
date methods for airframe structural
analysis.

Energy Methods for Structural Optimization

One further development related to
airframe dynamics is the Damped Forced
Response Method 7 / 8 . The airframe forced
response is computed, and structural
members with significant strain energy
are identified. These members are
changed to reduce vibration response for
modes with frequencies above and below
the rotor exciting frequency. This
method is outlined in Figure . 4 .

Vibration Control Devices

Vibration control devices such as
absorbers are often used to reduce vibra-
tion in local areas of the airframe.
The force output for an absorber may be
computed by expressing the vibration as
the sum of vibration due to rotor forces
and the vibration due to the force output
by the absorber.

{ f a} - -[ a aa] _1 |^ar| { f r}

(9)

X r

A AA a ar

a ra a rr

(8)

The absorber force output required to null
vibration at the absorber attachment
point is

The corresponding motions at the rotor
hub are

{%} " [ A RR- A RA a aa A AR ] { F f} ("J

The mobility matrices in the above equa-
tions may be obtained analytically using
computed modal properties (Equations (1)
through (7)) or by applying unit vibra-
tory loads to the airframe in a series of
shake tests.

This method was applied to predic-
tion of cockpit vibration with a vertical
cockpit absorber for the Model 347 heli-
copter°. Analytical and flight test re-
sults are compared in Figure 5*.

Coupled Rotor/Airframe Analysis

Theory

Any vibratory motion of the rotor
hub will change the rotor blade vibratory
aerodynamic and inertia forces which are
summed to obtain vibratory hub loads.
Changes in hub loads will in turn cause
changes in vibratory hub motions^'-*-".

Airframe Motion is assumed to be
related to vibratory hub loads by a
mobility matrix for a particular exciting
frequency:

^ks

A kll A kl2
A k21 A k22

r ks
F kc

= [Ak]

■ks

-kc

(11)

where
{ x k}={ x ks} sin knt + ( X kc( cos knt

{ F k} = { F ks} sin knt + { F kc} cos knt

The airframe mobility data are air-
frame responses to unit vibratory hub
loads; these data may be obtained analy-
tically by using theoretical modal proper-
ties (Equation (4) through (7)), or by
conducting an airframe shake test. It is
emphasized that these are airframe
response characteristics for no blade
mass attached to the airframe at the
rotor hub. All blade inertia effects
will be included in the rotor vibratory
hub loads as modified by vibratory hub
motion.

In general, six sine and six cosine
components of shaking forces and moments
exist at each rotor hub; a tandem rotor
helicopter would have a total of 24

components of vibratory forces. If only
the rotor hub motions are considered, the
relationship between hub motion and hub
forces is:

24x1

Tiks

Hike

24x24 24x1

= [%k]

; ks

? kc

(12)

The vibratory hub loads are assumed
to be loads with no hub motion plus an
increment of hub loads proportional to hub
motion :

24x1
F kso
F kco

F kso
F kco

24x24
B kll B kl2
Bk21 B k22

24x1

x Hks
x Hkc

(13)

*-Hks

x Hkc

The coefficients of the B matrix are
obtained by making several computations of
vibratory hub loads :

• Components of vibratory hub loads
are computed assuming no hub
motion

• Components of vibratory hub loads
are computed assuming a small
vibratory hub motion at the fre-
quency for each degree of freedom
of hub motion at each rotor

• Changes in sine and cosine com-
ponents ' of vibratory hub forces
per unit vibratory' hub motion in
each rotor hub degree of freedom
are then computed.

The coupled rotor/airframe solution
for compatible rotor hub motions and rotor
hub loads is obtained by substituting '
Equation (13) in Equation (12) and solving
for vibratory hub motions:

*Hks

x Hkc

-r-1

[I]-[A][B]

[A]

b kso

e kco

(14)

Once a solution for Equation (14) is
obtained, the total vibratory hub loads
may be computed using Equation (13) and
the vibration for the entire airframe may
be computed using Equation (11) .

Single Rotor Example

Figure 6 shows a simple example of
the coupled rotor airframe method applied
to a single rotor helicopter vertical
vibration analysis. Hub vertical vibra-
tion response and the vertical vibratory
hub loads are computed at a frequency of
four times rotor speed (4/rev) . The air-
frame is represented by its rigid body
vertical mode and one flexible mode.
Figure 6b shows airframe mobilities vs
flexible mode natural frequency for 4/rev
vertical hub forces. Hub vertical shak-
ing forces vs hub vertical motion are
shown in Figure 6c. The vibratory hub
loads are seen to vary approximately
linearly at least up to .005 inches of
motion at the 4/rev frequency. Figures
6d and 6e show compatible rotor hub
vertical vibration amplitudes and rotor
hub shaking forces vs flexible mode
natural frequency.

For this example, the rotor vibra-
tory hub motions and forces both peak
when the flexible mode natural frequency
is just above the rotor hub force excit-
ing frequency. This is not a general
result, but depends upon the relation-
ships between hub shaking forces and hub
motions .

Coupled Rotor /Air frame Analysis Computer
Program (D-65)

Figure 7 shows the flow-diagram for
the Boeing Vertol D-65 Coupled Rotor/
Airframe Analysis computer program. This
program links three major computer
programs-*^.

Trim analysis program A-97

Rotor vibratory hub loads analy-
sis program D-88

- Airframe forced response analy-
sis program D-96.

Compatible fuselage motions and
vibratory hub loads are obtained using
this program with the method discussed
above. In its current state, the D-65
program computes three vibratory rotor
forces and three vibratory rotor moments
at each rotor for either single or tandem
rotor helicopters. Response to trans-
lational and rotational vibratory hub
forces is computed for the airframe, but
compatibility of hub forces and motions
is satisfied for hub translational
degrees of freedom only in the current
version of the program. The program will
be modified in the near future to provide
compatibility for hub rotational degrees
of freedom.

Analysis vs Test Results for the Model 347
Helicopter

The D-65 coupled rotor/airframe pro-
gram was used to predict Model 347 flight
vibration levels. Figure 8 shows the
model used to predict airframe dynamic
characteristics. Figure 9 compares pre-
dicted vertical and lateral cockpit
vibration levels vs vibration levels
measured in flight. Vertical vibration
levels are in reasonably good agreement
at high airspeeds where vibration levels
may become significant. Lateral vibra-
tion levels are higher than predicted.

Conclusio ns

Methods have been developed indepen-
dently for prediction of rotor vibratory
hub loads and airframe dynamic character-
istics. Methods are available for in-
cluding effects of vibration control
devices on airframe vibration and for
optimizing the airframe structure. The
substructure method is available for
minimizing computer running time in
analysis of airframe structures, and
NASTRAN now provides a common finite
element structural analysis program avail-
able to all aerospace contractors. Rotor
hub vibratory motions can modify rotor
hub vibratory forces acting on the air-
frame. A linear coupled rotor/airframe
analysis method provides an approach for
determing compatible hub motions and hub
shaking forces . This method should be
studied further to determine its
validity. A method of this type should
be considered in applications of NASTRAN
for prediction of helicopter vibration;
the user programming feature in NASTRAN
should permit a coupled rotor/airframe
solution of this type within NASTRAN.

Figure 10 shows a scheme for solving
for rotor trim, rotor forces with no hub
motion, and the rotor impedance matrix
using a rotor analysis program. NASTRAN
would be programmed to use these mobili-
ties and the rotor analysis results to
solve for compatible rotor/airframe loads
and motions. The NASTRAN airframe analy-
sis could include airframe installed
vibration control devices either in the
initial airframe analysis or in the
coupled rotor/airframe solution. Finally,
results of these analyses could be used
to determine optimum changes to the air-
frame structural elements for minimizing
airframe vibration.

10.

D.C., 1954.

Boeing Vertol Company, D8-0614,
AEROELAST1C ROTOR ANALYSIS, D-95,
Thomas, E., and Tarzanin, F., 1967.

Boeing Vertol Company, D210-10378-1,
& -2, AEROELASTIC ROTOR ANALYSIS,
C-60, Tarzanin F. J., Ranieri, J.
(to be published) .

Sciarra, J. J., DYNAMIC UNIFIED
STRUCTURAL ANALYSIS METHOD USING
STIFFNESS MATRICES, AIAA/ASME 7th
Structures and Materials Conference,
April 1966.

The Boeing Company, D2-125179-5, THE
ASTRA SYSTEM — ADVANCED STRUCTURAL
ANALYSIS, Vol. 5, User's Manual.

NASA SP-222 (01) , NASTRAN USER'S
MANUAL, McCormick, Caleb W. ,
National Aeronautics and Space
Administration, Washington, D.C.,
1972.

Sciarra, J. J., and Ricks, R. G. , USE
OF THE FINITE ELEMENT DAMPED FORCED
RESPONSE STRAIN ENERGY DISTRIBUTION
FOR VIBRATION REDUCTION, ARO-D
Military Theme Review, Moffett Field,
California, U.S. Army Research
Office, September 1972.

Sciarra, J. J., APPLICATION OF IMPE-
DANCE METHODS TO HELICOPTER VIBRA-
TION REDUCTION, Imperial College of
Science and Technology, London,
England, July 1973.

Gerstenberger, W. , and Wood, E. R. ,
ANALYSIS OF HELICOPTER CHARACTERIS-
TICS IN HIGH SPEED FLIGHT, American
Institute of Aeronautics and Astro-
nautics Journal, Vol. 1, No. 10,
October 1963, pp 2366-2381.

Novak, M. E ., ROTATING ELEMENTS IN
THE DIRECT STIFFNESS METHOD OF DYNA-
MIC ANALYSIS WITH EXTENSIONS TO
COMPUTER GRAPHICS, 40th Symposium on
Shock and Vibration, Hampton,
Virginia, October 1969.

References

Leone , P . F . , THEORY OF ROTOR BLADE
UNCOUPLED FLAP BENDING OF AERO-
ELASTIC VIBRATIONS, 10th American
Helicopter Society Forum, Washington,

ACTUAL BLADE

APPROXIMATION

BLADE SECTION
BOUNDARIES

EQUIVALENT SYSTEM

CONSTANT EI 6 GJ
ELASTIC BAY

APPLIED
AIRLOADS

\rrttYfrmtmroj j mw((rmtt^- n( tmli( mf ^--^-Y

/ MASS BAY

^EQUIVALENT MASS

-BLADE SECTION
BOUNDARIES

-PITCH AXIS

Figure 1. Rotor Blade Analytical Model

• AIRFRAME INPUT DATA

-NODAL COORDINATES AND CONSTRAINTS
-STRUCTURAL ALEMENT PROPERTIES
-MASS AND INERTIA PROPERTIES

• STRUCTURAL ANALYSIS

-FORM MEMBER STIFFNESS MATRICES

AND ADD TO OBTAIN THE AIRFRAME GROSS

STIFFNESS MATRIX
-REDUCE NON-MASS DEGREES OF FREEDOM

FROM GROSS STIFFNESS MATRIX

• COMPUTE AIRFRAME NATURAL MODES AND
FREQUENCIES AND GENERALIZED MASSES

• ROTOR VIBRATORY
FORCES

■ . . . ....

• COMPUTE AIRFRAME FORCED RESPONSE

Figure 2. Uncoupled Airframe Dynamic
Analysis

INPUT

• NODE NUMBERS, CONSTRAINTS
RETAINED, REDUCED DEGREES
OF FREEDOM

• STRUCTURAL PROPERTIES OF
MEMBERS CONNECTING NODES

• MASSES AND INERTIAS TO BE
CONCENTRATED AT RETAINED
DEGREES OF FREEDOM FOR
MASS MATRIX

ELEMENTS

AXIAL

1« «2

BEAM

SKIN

GENERATE STIFFNESS MATRIX, K

• GENERATE MEMBER STIFFNESS
MATRICES AND ADD TO OBTAIN
AIRFRAME GROSS STIFFNESS MATRIX

• REDUCE GROSS STIFFNESS MATRIX
TO RETAINED DEGREES OF FREEDOM

N =

K 11 K 12

_ K 21 K 22

~ 2000 X 2000
1

[Kll] = [Kii-K 12 K22~ 1 K2i]~200 X 200

• LARGEST COMPUTER TIME ASSOCIATED
WITH REDUCTION PROCESS

COMPUTE NATURAL MODES , ^
AND FREQUENCIES , u n r

INPUT

ROTOR FORCES, F R

COMPUTE FORCED
RESPONSE , {x}

A n n n n

n n

{ >nR} (Fr) -fn

K} T [>] {♦„}" W n

• COMPUTE RESPONSE
OF EACH MODE;
ADD TO OBTAIN
TOTAL RESPONSE

q n = MODE GENERALIZED
COORDINATE

Figure 2. Continued

AIRFRAME INPUT
DATA, SUBSTRUCTURE i

OPTION 1

• GENERATE SUBSTRUCT.
i GROSS STIFFNESS
MATRIX

• REDUCE ALL BUT MASS
AND BOUNDARY DEGREES
OF FREEDOM

• MBRGE SUBSTRUCTURE REDUCED
STIFFNESS MATRICES

• REDUCE NON -MASS BOUNDARY
DEGREES OF FREEDOM

• COMPUTE AIRFRAME NATURAL
MODES AND FREQUENCIES

• ROTOR FORCES

• COMPUTE AIRFRAME
FORCED RESPONSE

. - i

K AA |

_o-_J

1
1

I . _

1 K BB-

1
1

r~+--~

-4- J

| K cc
I

Figure 3.

BOUNDARY DEGREES
OF FREEDOM

Substructure Method for Gener-
ating Airframe Reduced Stiff-
ness Matrix

S-74

DYNAMIC ANALYSIS
NORMAL MODE
METHOD

INTERNAL LOADS
TAPE

F s = Kx s
F c = KX C

STRUCTURAL
DATA

S-74

STRESS

ANALYSIS

OPTION 2

ALL DEFLECTIONS
OBTAINED

» GENERATE ELEMENT
STIFFNESS MATRICES

• PICK UP ELEMENT
END DEFLECTIONS

NASTRAN
NORMAL MODE
ANALYSIS-
CALCULATE
ALL DEFLECTIONS

COMPUTE MAX

X T KX FOR

EACH ELEMENT

ACCEPTABLE
OPTIMIZATION

STRAIN ENERGY

SORT - CALCULATE

WEIGHTS AND

STRAIN DENSITY

- SORT

X£it>

CRITERIA

1. VIBRATION LEVEL

- MIN.

2. WEIGHT PENALTY

- MIN.

3. WITHIN ALLOWABLE.
STRESSES

MODIFY STRUCTURE,
RUBINS METHOD, RE-
RUN

RESPONSE OF
.ORIGINAL FUSELAGE -
WITH RIGID BODY MOTION

RESPONSE USING RUBIN
METHOD FOR STRUCTURAL
MODIFICATION-RIGID BODY
MOTION INCLUDED

Figure 4. Damped Forced Response Method
for Airframe Optimization

MODEL 347 COCKPIT VERTICAL
(NO ABSORBERS)

.5 r

+1.4 - OBJECTIVE

H
Eh

H
U

FLIGHT DATA

ANALYSIS

40 80 120 160
AIRSPEED - KNOTS

MODEL 347 COCKPIT VERTICAL
WITH ABSORBER

O
+ \

H
U

OBJECTIVE

ANALYSIS

FLIGHT DATA.

— . * * 5=gF

40 80
AIRSPEED

120
KNOTS

160

z H4s

A ll A 12

A 21 A 22

! Z4s

Z4c

A 22 = A ll

A 21 A 12

A ll' A 12
10"" 4 IN/LB

.4
.3

*ll/

^Exciting

Frequency
i = 4fi
<
t

.2
2

\
\
\
\

/
/

/ v \^
1 ^

i 4 £

A 12 j

>v / K

(b) Airframe -Hub Mobilities

Figure 5. Predicted Vs. Measured

Cockpit vibration Reduction
with a Vertical Cockpit
Absorber

iy a = 44.5 RAD/SEC

T F , 2
1 Zk Eh

(a) Single Rotor Helicopter Vertical
Vibration

F Z4 " F Z4c cos 4S2t + F Z4s sin

4f2t

Z H4 = Z H4c COS 4flt + Z H4s sin 4fit

-200.

200.

H4S, .001 IN.

Figure 6. Coupled Rotor /Airframe Analysis
for a Single Rotor Helicopter
Vertical Vibration

Figure 6. Continued

Z = (Z 2 + Z 2 )H 2

.0010

EXCITING -i

.08

I frequency!

.06

.0005

.04

IN.

.02

G's

F
Z4

LB.

(d) Vibratory Hub Motion

F Z4 = ( F l 4 c + Fg.s)

II 2

(e) Vibratory Hub Force

A- 9 7 TRIM
ANALYSIS

D-88 INPUT
FORWARD AND AFT
ROTORS

PARAMETRIC STUDY
READ D-88 AND D-96
PARAMETER CHANGES

i~_

BASIC D-88
AEROELASTIC ROTOR ANALYSIS

Vibratory Hub Loads
x' My'

F x' ?y *"z'

MATRIC
CAPABILITY

VIBRATION
DEVICES

D-96 INPUT
(Fuselage)

1. Masses

2. Natural Frequencies

3. Degrees of Freedom

BASIC D-96
DAMPED, FORCED RESPONSE OF
A COMPLEX STRUCTURE

OUTPUT FOR EACH
DEGREE OF FREEDOM

1. Displacement

2. Phase Angle

3 . G-Levels

Figure 6. Continued

Figure 7. D-65 Coupled Rotor/Airframe
Program Flow Diagram

. STRUCTURAL IDEALIZATION

HAS:" 1061 STRINGERS
' 1089 SKIN ELEMENTS

38' BEAMS

SZ1 NODES (STRUCTURAL! ,
' 1849 DEGREES OF FREEDOM
51 MASS NODES
139 RETAINED O.O.F.

Figure 8. Model 347 Airframe Dynamic Model

STA. 95 C/L VERTICAL
4/REV VIBRATION
-NO ABSORBERS

.5
.4

G's

"40 60 80 100 120 140 160 180

AIRSPEED, KNOTS

STA. 95 C/L LATERAL
4/REV VIBRATION
-NO ABSORBERS

FLIGHT TEST

4o d rt ioo ria i4o iso i

AIRSPEED, KNOTS

Figure 9. Model 347 Flight Data Vs. D-65
Coupled Rotor/Airf rame Analysis
Results

• TRIM ANALYSIS

• VIBRATORY ROTOR LOADS ,
NO HUB MOTION {f r0 },

• VIBRATORY ROTOR LOADS
WITH UNIT VIBRATORY
HUB MOTIONS

• ROTOR IMPEDANCE
MATRIX, B

, AIRFRAME SUBSTRUCTURE ANALYSIS
(INCLUDE MODELS OF VIBRATION
CONTROL DEVICES)

• MERGE; SUBSTRUCTURE STIFFNESS
MATRICES

• COMPUTE" -AIRFRAME MODES, FREQUENCIES,
AND GENERALIZED MASSES WITH NO BLADE
MASS AT ROTOR HUBS

•COMPUTE AIRFRAME RESPONSE TO UNIT
VIBRATORY HUB LOADS:

A RR
. A AR.

F R

• COMPUTE COMPATIBLE VIBRATORY HUB
MOTIONS AND FORCES

{X R } -[[l]-[A RR ] [B]]~ [A RR ] {f ro ]

compute total hub forces
{fr} = {fro} + [b]{x r ]

•compute motions at other airframe
degrees of freedom

{x A } - [a ar ] (f r )

•IDENTIFY STRUCTURAL CHANGES
TO MINIMIZE AIRFRAME
VIBRATION USING STRAIN
ENERGY METHODS

Figure 10. Coupled Rotor/Airf rame/NASTRAN
Analysis

HELICOPTER GUST RESPONSE CHARACTERISTICS
INCLUDING UNSTEADY AERODYNAMIC STALL EFFECTS

Peter J. Arcidiacono

Chief Dynamics

Sikorsky Aircraft Division of United Aircraft Corporation

Stratford, Connecticut

Russell R. Bergquist
Senior Dynamics Engineer
Sikorsky Aircraft Division of United Aircraft Corporation
Stratford, Connecticut

W. T. Alexander, Jr.

Aerospace Engineer
U. S. Army Air Mobility Research and Development Laboratory

Eustis Directorate
Fort Eustis, Virginia

Abstract

The results of an analytical study to eval-
uate the general response characteristics of a
helicopter subjected to various types of discrete
gust encounters are presented. The analysis em-
ployed was a nonlinear coupled, multi-blade rotor-
fuselage analysis including the effects of blade
flexibility and unsteady aerodynamic stall. Only
the controls-fixed response of the basic aircraft
without any aircraft stability augmentation was
considered. A discussion of the basic differences
between gust sensitivity of fixed and rotary wing
aircraft is presented. The effects of several
rotor configuration and aircraft operating param-
eters on initial gust-induced load factor and
blade vibratory stress and pushrod loads are dis-
cussed. The results are used to assess the accu-
racy of the gust alleviation factor given by MIL-
S-8698. Finally, a brief assessment of the rela-
tive importance of possible assumptions in gust
response analyses is made and a brief comparison
of gust and maneuver load experiences in Southeast
Asia is presented.

The results confirm that current gust alle-
viation factors are too conservative and that the
inclusion of unsteady stall effects result in
higher initial load factors than predicted using a
steady stall aerodynamic analysis.

Notation

An gust alleviation factor; see Equation (l)

l *

and (3)

ft. two-dimensional lift curve slope of rotor
blade section

b number of blades

B tip loss factor

XL blade chord, ft

C' T vertical force coefficient, Thrust/jjpn. t R i *'

GW gross weight, lbs

Presented at the AHS/NASA-Ames Specialists' Meet-
ing on Rot or craft Dynamics, February 13-15, 197^.

Xg blade mass moment of inertia about flapping
hinge, slug - ft 2

R blade radius , ft

S fixed wing area, ft 2

"t a , partial derivative ^ ^ e '

V forward velocity, knots or ft /sec

V,Lta average characteristic velocity for helicop-
ter rotor

Vbl maximum vertical velocity of gust, positive
up, ft/sec

(X angle between shaft and relative wind, posi-
tive tilted aft, radians

ft .
)f blade lock number, f <*-<cft/X

. ,- , MA* THMST ■

^ n incremental rotor load factor ; — — ■ B — •

j&fij incremental rotor load factor predicted by
linear steady theory for sharp edge gust
instantaneously applied to entire lifting
device .

A inflow ratio, ( VSin* - ^)/xiK

ytx advance ratio, V c-* 4 -'* /siH.

1/ rotor induced velocity, positive up, ft /sec

f air density, slugs /cubic foot

<T rotor solidity, ^"^/n R

XL rotor angular rotational velocity, radians/
second

three dimensional lift curve slope for fixed

wing
Subscripts
P" denotes fixed wing
" denotes helicopter

<
o

M .0.5 -

2 .4-6 8 10.

.'.'- DISC LOADING, LB/FT 2

Figure 1. Current Gust Alleviation Factor.

Current procedures for predicting helicopter
gust-induced loads involve computing rotor loads
by means of a simplified linear theory and modify-
ing these loads by a gust alleviation factor de-
fined in Specification MIL-S-8698 (AEG). The al-
leviation factor is shown in Figure 1 and is a
function of rotor disc loading alone. Further, no
alleviation is allowed for disc loadings greater
than 6.0 - a value exceeded by many modern heli-
copters. Attempts to verify the accuracy of this
approach through flight test have been complicated
by uncertainties regarding the gust profiles.
This has led to side.-by-slde flight tests of fixed
and rotary-wing aircraft (Reference l) in order to
build a semi-empirical bridge between the rela-
tively straight forward fixed wing situation and
the more complex situation associated with rotary
wings. Limited qualitative results on aircraft of
comparable gross weight indicated that the heli-
copter was less gust sensitive than the fixed wing
aircraft, but extensive quantitative data from
this type of test are, obviously expensive and
difficult to obtain. Analytical confirmation of
the MIL-S-8698 (ARG) gust alleviation factor has
been hampered by the lack of an analysis which can
handle both the transient response of the helicop-
ter and the aeroelastic response of the rotor
blades, while, simultaneously, providing a reason-
ably complete modeling of the rotor aerodynamic
environment. An improved gust response analysis
(described in Reference 2) has indicated that cur-
rent procedures are too conservative. The primary
objectives of this investigation were (l) to
develop a similar computerized analysis based on
the rotor aeroelastic and unsteady stall aerody-
namic techniques developed at Sikorsky Aircraft
and the United Aircraft Research Laboratories and
(2) to apply the analysis to predict rotor gust
alleviation factors for comparison with those
given in Specification MIL-S-8698 (ARG) and in
Reference 2. The principal contribution of this
analysis relative to that of Reference 2 is the
inclusion unsteady stall aerodynamics. The re-
sulting computer program was designed to function
on the CDC 6600 computer and is catalogued at both
the Langley Research Center and the Eustis
Directorate.

Comparison of Helicopter and Fixed
Wing Gust Response

Before proceeding with a detailed analysis
of the helicopter gust response characteristics,
it is instructive to compare fixed wing and heli-
copter characteristics in relatively simple terms.
Such a comparison follows.

In analyzing the response of fixed wing air-
craft to discrete sharp edge gusts, (eg. Reference
3) the concept of a gust alleviation factor is em-
ployed. The gust alleviation factor is simply the
ratio of the "actual" incremental load factor pro-
duced by the gust to the incremental load factor
computed from simple steady-state theory. The
"actual" load factor may represent a measurement
or may be computed from some more rigorous theory
applicable to the unsteady gust encounter situa- .
tion. Thus, if Ag is defined as the. gust allevia-
tion factor, we have

for fixed wing aircraft we have

(l)

(2)

Following the same approach for a helicopter
having a rotor as its sole lifting element, we can
write:

(a*) h = (*» s ) H (A<j) H

(3)

Using steady, linear rotor theory results from
Reference k, and assuming a sharp edge gust in-
stantaneously applied to the entire rotor,
is given by:

(a» s ) h . - kf C^iMM*

&W/bcR

k« ^V ft -^

6^/bX!R

(k)

(5)

(6)

Hence, the actual load factor is given by

^/ bA R

(7)

Equation (7) is of similar form to the correspond-
ing fixed wing equation (Equation 2). Further, by
comparing the two equations, it is clear that the
characteristic or average velocity for the rotor
is given by tj , XJ-R and that the characteristic
area for the rotor is the total blade area. The
characteristic velocity V<wj, of the rotor is pre-
sented in Figure 2. A typical value of Va.^ is
about 0.5ilR and the effect of advance ratio (or
forward speed) is seen to be small. This con-
trasts with the fixed wing case where the charac-
teristic velocity is equal to the aircraft's for-
ward velocity.

0.8

7 = 15,B=0.97 ;

o a

lE;f

O o

fcF

0.6

■ __

m 5

— - — ""

tr. or

0.4

o t-

£s

X UJ

o >

0.2 0.4
ADVANCE RATIO, /J-

0.6

Figure 2. Rotor Characteristic Velocity Ratio.

Now, attempts have been made to measure the
gust alleviation factors of helicopters through
side-by-side flights with fixed wing aircraft.
However, the relative alleviation factors so de-
termined are only meaningful if the Gust Response
Parameter for the two aircraft are equal. This
equality of Gust Response Parameters is shown in
Equation 8:

(8)

6-W/bxiR

If the relation above is satisfied, Equations (2)
and (T) indicate that the following equality also
holds :

(9)

Assuming the two aircraft encounter the same gust
velocity profile, Equation (9) reduces to

do)

Thus, the ratio of the gust alleviation factors
will be in proportion to the measured load factors

for the two aircraft . If the fixed wing gust al-
leviation factor is known, (Ag)g can then be de-
termined. • '•

If Equation (8) is not satisfied, then the '
gust alleviation factor for the helicopter can be
determined from the following relation:

M^^ri v §

(A*),

%'w

IS Aw \\

*"L*>

.(ID-

Typical values of the Gust Response Parameters of
Equation (8) are presented in Figure 3. The re-
sults of Figure 3 indicate that a helicopter having
a blade loading of 100 lb /ft 2 and operating at a
forward speed of 250 fps will exhibit approximately
the same sensitivity to a gust as a fixed wing air-
craft having a wing loading of about 60 lb /ft 2 ,
provided, of course, that the gust alleviation fac-
tors for both aircraft are equal. In practice,
for this example, the gust alleviation factor for
the fixed wing will be significantly higher (mean-
ing higher acceleration) than that for the helicop-
ter.

SEA LEVEL

FIXED WING 6C L /3a =4.5)

HELICOPTER (a = 5.73, JIR = 700 FPS)

100 200 300

FORWARD VELOCITY, V- FT/SEC

Figure 3.

Fixed Wing and Helicopter Gust
Response Parameters .

Factors Influencing Helicopter Gust Response

The computation of gust induced loads for
helicopters is a difficult analytical task because
the rotary wing lifting system is a complex aero-
elastic mechanism operating in complicated aerody-
namic environment. Principal factors which can be
expected to influence the gust response of a heli-
copter are described briefly below.

a. Rotor blade response - Helicopter rotors

differ from fixed wings in that the blades
(wings) of the rotor are relatively flexible
and, in many cases, are articulated relative
to the fuselage. The blades, therefore, are
much more responsive to gust loads than is
the aircraft as a whole and react in such a"
way as to reduce or isolate (at least tem-
porarily) the fuselage from the impact of
the gust. Thus, while the blades axe re-
sponding to the gust, the fuselage has time
to build up vertical velocity which, in
turn, redUces the effective velocity seen by
the rotor. A simple example illustrating

the magnitude of the various forces con-
tributing to the fuselage acceleration is
shown in Figure k for a sharp edge gust
applied instantaneously to a rotor having
nonelastic flapping blades and operating in-
hover. In this extreme case, because of the
overshoot of the blade flapping response,
the peak acceleration experienced by the
body is about the same as it would have been
had the blades been completely rigid (i.e.
equal to the acceleration given by the gust
term alone). As seen in Figure h, the
forces associated with the blade dynamic re-
sponse are large; hence any factor influenc-
ing the blade is potentially important.

r.io HOVER

SHARP-EDGE GUST, ZERO TIME PENETRATION
NONELASTIC BLADES

GUST TERM

100 200 300

BLADE AZIMUTH, DEG

Figure h.

Comparison of Terms Contributing to
Fuselage Acceleration.

Fixed wing response - If the helicopter is
fitted with fixed wings (compound configura-
tion), additional gust loads are, of course,
generated. These can be treated using
available fixedr-wing techniques and are not
of primary concern in this study.

be phased so that the peak loads for each
blade occur at different times (see Figure 5)>
As a result, finite-time penetration of the
gust reduces the peak fuselage accelerations
produced by a given gust profile.

SHARP-EDGE GUST
FINITE TIME PENETRATION
NONELASTIC BLADES

BLADE AZIMUTH, DEG

Figure 5. Finite-Time Penetration Causes Peak

Blade Forces to be out of Phase.
Control system inputs - The ultimate effect

of gust on the helicopter must be influenced
by any reaction of the pilot or stability
augmentation system to the initial loads pro-
duced by the gust. It is possible (but un-
likely with a properly designed system) that
the largest loads produced by the gust will
not be the initial loads but, rather, those
associated with the longer term response of
the coupled system represented by the air-
craft, pilot, and stability augmentation sys-
tem (See Schematic in Figure 6). These longer
term effects depend on the specific design
characteristics of the aircraft system and no
attempt was made to model them in the present
study. Hence, the gust-induced loads con-
sidered are the initial loads caused by the
gust for a controls-fixed rotor operating con-
dition.

Rotor Aerodynamic Modeling - The ability of
a rotor to generate load factor during a
gust encounter will depend on the proximity
of the blade trim angle of attack distribu-
tion to stall. A rotor operating on the
verge of stall prior to a gust encounter can
be expected to generate less additional lift
due to the gust than can a rotor initially
operating further away from stall. The
modeling of stall aerodynamics is important;
therefore, the impact of unsteady aerody-
namics on rotor stall was investigated in
this study.

Gust Characteristics - Gust profile and am-
plitude are, of course, potentially impor-
tant factors. In addition, the speed of the
helicopter as it penetrates a given gust
front can be expected to be significant.
Figure k indicated the fuselage acceleration
for a gust applied instantaneously to the
entire disc. With a finite-time penetration
of the gust front, the contribution of each
blade to the fuselage loading will not be
identical (as in Figure It) but rather will

SECONDARY LOAD PEAK
RESULTING FROM INTERACTION
OF GUST AND AIRCRAFT-
PILOT- SAS SYSTEM^

INITIAL MAXIMUM

LOAD PRODUCED

BY GUST (CONTROLS -FIXED)

Figure 6.

Schematic of Possible
Load Factor Time Histories.

Brief. Description of the Analysis'

.Simple,' Linear Gust theory

Complete documentation of the equations used
in the analysis is given in Reference 5, while
procedures for running the associated computer
program may be found in Reference 6. Both of
these references can he obtained from the Eustis
Directorate of USAAMRDL.

Briefly, the analysis is essentially a digi-
tal flight simulator that can be used to determine
the fully coupled rotor - airframe response of a
helicopter in free flight. This is accomplished
by the numerical integration of the blade - air-
frame equations of motion on a digital computer.
The principal technical assumptions and features
of the analysis are listed below.

1. The blade elastic response is determined
using a modal approach based on the equa-
tions defined in References 7 and 8. The
number of modes used consisted of three
flatwise,, two ehordwise and one torsion for
each .blade.

2. The .aerodynamic modeling of the blade in-
cludes unsteady aerodynamic effects based on
the. equations ■ and tabulations defined in

'.Reference 8 which assume that the lift and
. moment coefficients can be expressed as
as functions of instantaneous angle of at-
tack and its first two time derivatives.
Steady-state drag was used, however, because

■ of a lack of data on unsteady drag in stall .

3. Rotor inflow is assumed constant for this
study although provision for time-varying
induces velocities is available. The con-
stant value is determined from classical

' momentum theory and was invariant with
either position on the disc or with time.

■ In view of the short times required for
peak loads to be achieved, this assumption
is considered reasonable.

h. The response of each individual blade is
considered.

5. The fuselage is a rigid (nonelastic) body
having six degrees of freedom. Provisions
for fixed wings are included. The aerody-
namic .forces on the wings are computed using
simple, -.finite-span wing theory, neglecting
stall and unsteady effects.

6. Fuselage aerodynamic forces and moments are
determined using steady-state nonlinear,
empirical data.

7. The gust is assumed to be both two dimen-
sional (i.e. does not vary along the lateral
axis of the rotor) and deterministic in
nature. • Although three dimensional and ran-
dom gust effects may prove important, their
inclusion was beyond the scope of this
study.

, As stated earlier, it was des'ired to cast the
results obtained in this investigation in terms' of ,
correction factors (gust alleviation ' factors ) that
could be applied to results dbtained . f rom a simple,
specification eventually evolved.

The simple "theory used is that defined in '- .
Reference k, in which blai# stall and'compressi-'
bility effects are; neglected. In -addition, it is •
assumed here that the' gust is" sharp'^edged and is
instantaneously applied' 'to -the, entire rotor. The,
increment in rotor load f actpr prtiduced by the gust'
is then given by Equation (5) . ' Using the relation,

ew «f^r i$y

(1?)

the incremental rotor, load factor given by simple
theory is

The ratio of the ( A*\ ) H . computed by the more
complete analysis described herein to the value
given by Equation (13) represents a gust allevia-
tion factor which can be used to correct the load
factors results given by Equation (13). Thus:

(i>0

Values of Ag presented in this paper are based on a
rotor blade lift curve slope, a, of 5.73. Hope-
fully, if Ag shows reasonably consistent trends, it
can be used with some confidence to rapidly predict
rotor load factors for combinations of parameters
other than those considered in this study.

Scope of Study

Gust load factors , blade bending moments , vi-
bratory hub loads, and rotor control loads were
calculated for a range of values for rotor thrust
coefficient- solidity ratio, blade Lock number, ad-
vance ratio, and blade flatwise and torsional
stiffness. The effect of adding a wing was also
investigated. The responses associated with three
types of vertical gusts were investigated: sine-
squared, ramp, and sharp-edged. The sine^squared
gust and the ramp gust reached a maximum value of
fifty feet per second at a penetration distance of
ninety feet. The gust profiles are displaved in
Figure 7- Three types of rotor systems were eval-
uated: articulated, nonarticulated (hinse3.ess) ,
and gimbaled. Emphasis in this paper is placed on
the results for the reference articulated rotor.
The reader is referred to Reference 10 for details
of the other configurations studied. The articu-
lated rotor properties can be found in Table I,
together with the natural frequencies of the
blades. As indicated, the number of modes used

consisted of three flatwise, two ehordwise, and
one torsional modes.

SIN

1 ■

! -SQUARED OUST

RA1

P 61

JST

#»=

0.5,REFER£NC£ ARTICULATED ROTOR,

:,/.

..06

__ — =— — Sf HE -SQUARED CU
SHARP-EDGED GUS1

SUIT

| 24 000

•"'

,■!■

„'

■'

•&

V'-.N

\ ,'

":»\ /;

1 6 ODO

~v 17

==.

I 8 000

160

320
ROTOR BLADE

480 ft
*IAZiMUTf

640
, DEC

800

HARI

1 1

-ED!

ED I

UST

,»

40 80 120 160 200

PENETRATION DISTANCE , FT

Figure T. Gust Profiles.

Discussion of Results

Effect of Oust Profile

The effect of gust profile on incremental
rotor thrust force for the reference articulated
rotor was evaluated by the penetration of three
gusts with profiles as shown in Figure 7. The
helicopter was assumed to penetrate a stationary
gust with a velocity of 350 feet per second. This
corresponds to an advance ratio of 0.5.

The time history of the rotor thrust asso-
ciated with each of the gust profiles is shown in
Figure 8. It may he seen from this figure that
while the actual gust wave form has little impact
on maximum rotor force and consequently on rotor
load factor, the particular time histories behave
differently, although in an expected manner. Ini-
tially, the sine-squared and the ramp gust shapes
result in similar peak rotor loads at approximate-
ly the same time. The sharp-edge gust induces a
greater peak load with a faster build up. As the
penetration distance increases, the loads produced
by the sharp-edge gust and the ramp gust tend to
merge since their respective velocities are hoth
50 fps while the value of the sine-squared gust
velocity has dropped back towards zero.

Analysis of the computed results forming the
basis for Figure 8 indicates that at the time the
maximum rotor vertical force and load factor is
reached, the helicopter fuselage has had time to
develop only a modest amount of vertical velocity.
The vertical velocities at the peak load points of
the helicopter associated with the sine-squared
and sharp-edged gusts are 6fps, and 3 fps,3 fps,
respectively. These compare to the '50 fps gust
velocity, indicating that little gust alleviation

GUST FRONT IS GUST FRONT REACHES MAXIMUM GUST VELOCITY

TANGENT TO HELICOPTER CENTER REACHES HELICOPTER

ROTOR DISC OF GRAVITY CENTER OF GRAVITY

Figure 8. Rotor Force Time Histories.

is being produced by fuselage motion for the con-
dition analyzed.

The three types of gust profiles evaluated did
not produce greatly different peak rotor loads.
While the sharp-edge gust does produce the largest
loads.it is probably the least realistic of the
three profiles. Since other studies, such as
Reference 2, have used a sine-squared gust, the
remainder of the results presented are based on
this profile.

Effect of Rotor and Flight Condition Variables

The variation of gust alleviation factor, (as
computed from Equation .1*0 , with initial rotor
loading is shown in Figure 9 for the three types

/x= 0.002378 ftR=700FPS o-=0.085 SINE 2 GUST

1.2

NOH

ART

CUL

ITEt

' ROTOR

itlL-

S-8(

CUR

REN

' ST

UDY

AR'

ICUl

ATE

D R

3 TOR

«IL-

>-8i

CUR

SEN

JDY

.04 .06 .08 .10

ROTOR THRUST COEFFICIENT/ SOLIDITY, Ct/o-

Figure 9. Gust. Alleviation Factors for
Different Rotors.

types of rotors analyzed. Rotor iloading in this
figure has been expressed t>oth in terms of rotor
thrust coefficient solidity ratio and rotor disc
loading. It should he remembered that disc load-
. ing is not a unique function of Op/8" hut rather
depends on the value of density, tip speed and
solidity of the rotor. Values for these quanti-
ties are noted on the figure.

The results of Figure 9 indicate that in-
creasing Oj^r leads to a large reduction of gust
alleviation factor, This is similar to the trend
noted in Reference 2 and is believed to be re-
lated to the loss in average additional lift
capability at the higher Op/a* due to the occur-
rence of stall. The influence of rotor configura-
tion is seen to be of rather secondary importance.
Rotor configuration would be expected to influ-
ence fuselage motion through the transmittal of
differing rotor pitching moments to the airframe,
depending on the degree of rotor articulation.
The relative insensitivity of the results to con-
figuration is believed to be due to the short
time in which the initial, controls-fixed load
factor is generated. As a result, the fuselage
response to the differing moments is not large
and the load factor tends to be dominated by the
rotor blade dynamic response, which is roughly
the same . for all rotors . This result is also
similar to that observed in Reference 2.

It is also evident from Figure 9 that the
gust alleviation factors defined in MIL-S-8698
(ARG) are too high (i.e. result in loads which
are too high) . The conservatism of the current
specification is particularly evident at the high
thrust coefficient-solidity ratios where rotor
stall becomes a factor limiting gust-induced
thrust generating capability. On this basis, one
would expect the gust alleviation factor for up-
ward gusts to be different from those for down-
ward gusts (i.e. gusts which unload the rotor).
While downgusts are not critical from a structural
loads viewpoint, they could prove more important
from a passenger - comfort point of view.

The results of Figure 9 are for typical
reference rotor configurations (see Reference 10
and Table I herein) . As part of this study, cal-
culations were made to examine the sensitivity of
the computed gust alleviation factors to separate
variations in blade Lock Mumber, bending stiff-
ness and torsional stiffness. Ranges of the
parameters considered are noted below:

ratios and higher Lock numbers being associated
with the lower gust alleviation factors. The rela-
tively small effect of blade stiffness variations
is perhaps not surprising inasmuch as the total
blade stiffness tends to be dominated by centrifu-
gal stiffening effects. The variations shown for
Oj/t of 0.06 are believed to be representative of
those at other Of/tr values; however, this should be
verified.

Articulated rotor load factors predicted using
the results of Figure 9 are presented in Figure 10
where they are also compared to the results of
Reference 2. Load factors predicted by the current
study are seen to be higher than those of Reference
2. This increase appears to be due to the use of
unsteady aerodynamics in the current study.

articulated' rotor
sine^gust

■-.

-^^

***

3-1.

CURRENT STUDY (UNSTEADY STALL AERO)

CURRENT STUDY{STEADY STALL AERO

L L_^_J ! J J

ROTOR THRUST COEFFICIENT/SOLIDITY, C T /«•

DISC LOADING, LB/FT 1

Figure 10. Comparison of Results with Reference 2

Gust-Induced Blade Stresses, Control Loads and
Vibration

The effects of a gust encounter on other
quantities of interest to the designer such as
blade stresses, control loads, and aircraft vibra-
tion were briefly examined. In examining these
effects, an attempt was made to generalize the re-
sults to a limited degree by relating the maximum
values produced by the gust to the initial trim
values. Results are based on the trim condition of
Cy/<r = 0.06 and an advance ratio of 0.3 are pre-
sented in Figure 11. Detailed analysis of the
trends shown were beyond the scope of this paper.
The reader is referred to Reference 2 for a more
detailed discussion.

Lock Humber: reference, 0.7 ref. 1.3 ref.

Bending stiffness: reference, 0.5 ref.

Torsional stiffness: reference, 0.5 ref.

The parameter variations listed above were made
at an advance ratio of 0.3 and 0.5 for a Oj/a- of
0.06. The range of results is also shown in
Figure 9 and as can be seen, the effect of blade
Lock number and stiffness is relatively small.
Lock number and advance ratio account for most of
the small variation shown, with the lower advance

.' 40

-w 20

■' q:

w 10

§ 6

in 4 H
o

i 2

I °

° 3.0

2 2.0

■ 4P

VERTICAL

VIBRATORY' FORCE

- '.

•^ y= 7

Reference 2 and previously presented in Figure 10 .

vibratory control load
St

y=i3^-

i.o -

VIBRATORY FLATWISE MOMENT

7= 7, 10,13

0.2

0.3 0.4

ADVANCE RATIO

0.5

Figure 11. Effect of Gust on Vibration,

Control Load and Flatwise Moment,
C T = 0.06, 50 fps SINE 2 Gust

Sensitivity of Results to Assumptions

As discussed in an earlier section of this
report, many factors could potentially influence .'
rotor gust response characteristics. To account .
for all of these factors leads to a time consum-
ing, complex, digital analysis. In the following '•
paragraphs, the results of a brief examination of
the importance of some of these factors are dis-
cussed. Only the reference articulated rotor at
one operating condition is considered. Any con-
clusions drawn from these results must, therefore,
be considered preliminary and should be substanti-
ated by further investigation. A summary of the
results obtained is presented in F'igure 12. Shown
is the percentage change in the predicted gust al-
leviation factor resulting, from the separate elim- '
ination of fuselage motion, blade elastic torsion^
finite time penetration j and unsteady stall ef-
fects in the analysis. The baseline value cor- ' ■'.
responds to value for' the complete' analysis. A -.
positive change in Ag means that the effect
eliminated causes an increase iri predicted load- ..
ing. It is evident that the- unsteady stall aero-'
dynamic and finite-time gust penetration effects
are most important* Excluding- unsteady aerody-
namics reduces the predicted value of Ag by about
29%.: This; is because the maximum lift capability
of the rotor base"d on' steady aerodynamic "stall, .

'characteristic's is lower than that "based on un- .
steady- characteristic's (see Reference 9). The re-!

■ duction is consistent with the observed lower' '
values 'of -predicted -load factors obtained in

70 1—
60 -
50 -
40 -
30 -
20 -

10 ■

-10
-20

-30

TRIM C T Aj- =0.06

50 FPS GUST AMPLITUDE

M-0.5

ARTICULATED ROTOR

. NO FINITE
TIME
PENETRATION
. (SHARP-EDGE)

FUSELAGE t&HJIu

MOTION , TORSION

(SINESgUST) (SINEgQUST)

Figure 12.

Sensitivity of Gust Alleviation to
Analytical Assumptions.

The largest change in predicted gust allevi-
ation factor was produced by the elimination of
the finite time penetration of the gust front. As
might be expected, when the gust is assumed to af-
fect all blades simultaneously, the blade forces
are all in phase and large values of Ag (and hence
loading) result.

It should be emphasized that the results pre-
sented in Figure 12 were determined for only one
reference trim condition. Further worK is required
to substantiate the generality of the- results.

Gust Load Factor Experience .in SEA

The earlier portions of this .paper have been
• devoted to analytical techniques, appropriate for

determining- the effects of gust encounters on heli-
1 copter response variables . One point' which has
' been made is that rotor blades , because of their
•f.lexib."ility,. tend to reduce the impact of the gust
on the fuselage. Resulting gust alleviation fac-
tors have been found to be low and, hence, one
.would expect that gust-induced loads on the fuse- -
lage could be' reduced in importance. Experimental
evidence supporting this contention has been ac-
•qulred .by the U. S. Army in SEA. A brief discus-
sion' of .that 'data is presented below.

' • ' The U. S. Army has been acquiring usage data;
on ' its combat operational helicopters' in Vietnam .

," since early 1966. Beginning with both, the cargo
arid armored versions of the CH-1*7A, the CH-5l*A,
Aft-IG; ■ nd.UH-6A helicopters were instrumented to'
rec6r'd .the history of their actual combat Usage.

-• Since .control positions and c;g. accelera-
tions were among the parameters 'measured 'and the '■'
data were, recorded in analog format, .occurrences •

, of gus'trinduced loads' were- identified- and isolated .

.'from pilot-induced (maneuver) accelerations by

•analyzing those particular trace recordings. Gust-
induced acceleration, peaks, therefore, were-identi-

. f ied as" those, accelerations occurring when both

the cyclic and collective stick traces were steady
or, if stick activity was present, the sense of
the peaking acceleration had to be in opposition
to that expected from the stick control motion.

A total of lVf7 hours of flight data were
acquired during the measurement programs for the
cited aircraft (References 11-13). The conclu-
sive finding in each of these programs was that
normal loads attributed to gust encounters were
of much lesser magnitude and frequency than
maneuver loads. Further, when the total load
factor experience was statistically examined for
each aircraft, the loads directly attributed to
gust encounters were found to be only a small
percentage of the total experience. These points
are graphically illustrated in Figure 13. The
maneuver load scatter band was obtained from
References Ik and 15.

It should be pointed out that while gust-
induced load factors are smaller than typical
maneuver load factors for military aircraft, gust
loadings can be an important consideration from a
ride comfort standpoint in commercial appliea-
tions .

10,000

SUST INDUCED
O CH-54A
* AH-IG
« 0H-6A

* '

A / £

\ V fk

A h/ /

\ \ \

A \\\W i

^ m\\\\\\\\k

m\\W t>

\* I Bi

A\\\\W /«

\ \

• 411/ '*

°\ \

MANUEVER '

Mo- SCATTER

BAND

1 '

•5 .5

INCREMENTAL LOAD FACTOR

Figure 13.

Gust-Induced toads are Significantly
Less than Maneuver Loads.
Conclusions

The following conclusions were reached as
a result of this study. It should be noted that
Conclusions 1-3 are based on the computation of
initial gust-induced load factors for various
rotor systems mounted on a single fuselage and
operating with the controls fixed throughout the
gust encounters.

1. The results of this study generally con-
firm those of Reference 2, indicating that
the current method for computing gust-in-
duced load factors for helicopter rotors
(Specification MIL-S-8698 (ARG)) results
in realistically high values and should be
revised.

cause retreating blade angles of attack
greater than the two-dimensional, steady-
state stall angle, the inclusion of unsteady
aerodynamic effects based on the model of
Reference 8 results in gust-induced load fac-
tors which are higher than those based on a
steady aerodynamic model such as that used in
Reference 2,

Principal parameters influencing gust-in-
duced load factor appear to be nondimen-
sional blade loading, proximity of the rotor
trim point to blade stall, and rate of pene-
tration of the rotor into the gust.

Gust loadings on military helicopters appear
to be significantly lower than those due to
maneuvers .

References

Crim, Aimer D., GUST EXPERIENCE OF HELICOP-
TER AND AN AIRPLANE IN FORMATION FLIGHT,
NACA Technical Note 335**, NACA, 195U.

Harvey, K. W., Blankenship, B, L. Drees,
J. M. , ANALYTICAL STUDY OF HELICOPTER GUST
RESPONSE AT HIGH FORWARD SPEEDS. USAAVLABS
Technical Report 69-I, September I969.

Bisplingoff, R. L. , H. Ashley and R. L.
Halfman, AEROELASTICITY, Addison-Wesley Pub-
lishing Company, Inc., Cambridge, Mass. 1955.

Bailey, F. J., Jr., A SIMPLIFIED THEORETICAL
METHOD OF DETERMINING THE CHARACTERISTICS OF
A LIFTING ROTOR IN FORWARD FLIGHT. NACA
Report No. 716.

Bergquist, R. R., Thomas G. C. TECHNICAL
MANUAL FOR NORMAL MODES AEROELASTIC COMPUTER
PROGRAM, July 1972.

Bergquist, R. R., Thomas, G. C. USER'S
MANUAL FOR NORMAL MODE BLADE AEROELASTIC
COMPUTER PROGRAM, July 1972.

Arcidiacono, P. J. , PREDICTION OF ROTOR IN-
STABILITY AT HIGH FORWARD SPEEDS, VOLUME 1.
STEADY FLIGHT DIFFERENTIAL EQUATIONS OF
MOTION FOR A FLEXIBLE HELICOPTER BLADE WITH
CHORDWISE MASS UNBALANCE. USAAVLABS Techni-
cal Report 68-18A, February I969.

Arcidiacono, P. J., Carta, F. 0., Cassellini,
L. M. , and Elman, H. L., INVESTIGATION OF
HELICOPTER CONTROL LOADS INDUCED BY STALL
FLUTTER. USAAVLABS Technical Report 70-2,
March 1970.

Bellinger, E. D., ANALYTICAL INVESTIGATION
OF THE EFFECTS OF UNSTEADY AERODYNAMICS VARI-
ABLE INFLOW AND BLADE FLEXIBILITY ON HELI-
COPTER ROTOR STALL CHARACTERISTICS. NASA
CR-1769.

If the gust amplitude is sufficient to

10. Bergquist, R. R. , HELICOPTER GUST RESPONSE
INCLUDING UNSTEADY STALL AERODYNAMIC EF-
FECTS. USAAVLABS Technical Report 72-68, -
May 1973.

.11. Giessler, F. Joseph; Nash, John F.; and

Rockafellow, Ronald I., FLIGHT LOADS INVES-
TIGATION OF AH-1H HELICOPTERS OPERATING IN
SOUTHEAST ASIA, Technology, Inc., Dayton,
Ohio; USAAVLABS Technical Report 70-51,
U. S. Army Aviation Materiel Laboratories,
Fort Eustis, Virginia, September 1970, AD
, 878039. ■..•;'

12. Giessler, F. Joseph; Nash John F.; and
Rockafellow, Ronald I., FLIGHT LOADS INVES-
TIGATION OF CH-5U HELICOPTER OPERATING IN .'
SOUTHEAST ASIA, Technology, Inc., Dayton,
Ohio; USAAVLABS Technical Report 70-73,
Eustis Directorate, U. S. Army Air Mobility
Research and Development Laboratory, Fort
Eustis, Virginia, January 1971, AD 881238. ,

13. Giessler, F, Joseph; Clay, Larry E.; and
Nash, John F., FLIGHT LOADS INVESTIGATION \
OF 0H-6A HELICOPTERS OPERATING IN SOUTHEAST.
ASIA, Technology, Inc., Dayton, Ohip;
USAAMRDL Technical Report 71-60, Eustis • '.
Directorate, U. S. Army Air Mobility Re- .

. Search and Development Laboratory, Fort

, Eustis, Virginia, October 1971, AD 7308202;'

Ik. Porter fields John D., and Maloney, Paul F., '
■ .. EVALUATION OF HELICOPTER FLIGHT SPECTRUM
DATA, Kaman Aircraft Division, Raman Corpo-
' ration, Bloomfield, Connecticut; USAAVLABS
Technical Report 68-68, U. S. Army Aviation
Materiel Laboratories , Fort Eustis , Vir-
ginia, October 1968, AD 680280. '

15. Porterfield, JotmD., Smyth, William A. and
Maloney, Paul F. , THE CORRELATION AND EVAL-
UATION OF AH-1G, CH-5ltA, and 0H-6A FLIGHT
. SPECTRA DATA FROM SOUTHEAST ASIA OPERA-
TIONS, Kaman Aircraft Division, Kaman Corpo-
ration, Bloomfield, Connecticut; USAAVLABS
Technical Report 72-56, Eustis Directorate,
U. S. Army Air Mobility Research and De-
velopment Laboratory, Fort Eustis, Virginia,
October 1972, AD 75555^-

' : TABLE I, Reference Articulated .
■ ' . Rotor Characteristics

Density Slugs/ft 2 .002378

Tip speed, ft/sec 700.

Radius, ft 25

; No. of blades h.

Blade Chord, ft : 1.67

Flap hinge off set ratio • 0.0k

Twist, deg •'. . -8.0

Young's Modulus, psi 10'

Mass per unit length at. 0.75R slugs/ft 0.18

Lock Number 10.0

.Rigid body flatwise frequency 1.03P

First bending flatwise frequency 2.66P

Second bending flatwise frequency 5.06P

Third bending flatwise frequency * 8.50P

Rigid body chordwise frequency 0.25P

First bending chordwise frequency 3.68P

Second bending chordwise 10.20P

First bending torsional frequency 5.72P

100

APPLICATION OF ANTIEESONANCE THEORY TO HELICOPTERS

Felton D. Bartlett, Jr.
Research Engineer

William G. Flannelly
Senior Staff Engineer

Kaman Aerospace Corporation
Bloomfield, Connecticut

Abstract

Antiresonance theory is the principle
underlying nonresonant nodes in a struc-
ture and covers both nonresonant nodes
occurring naturally and those introduced
by devices such as dynamic absorbers and
antiresonant isolators . The Dynamic
Antiresonant Vibration Isolator (DAVI)
developed by Kaman Aerospace Corporation
and the Nodal Module developed by the Bell
Helicopter Company are specific examples
of the applications of transfer anti-
resonances. A new and convenient tech-
nique is presented to numerically calcu-
late antiresonant frequencies. It is
shown that antiresonances are eigenvalues
and that they can be determined by matrix
iteration."

Novel applications of antiresonance
theory to' helicopter engineering problems,,
using the antiresonant eigenvalue equation
introduced in this paper, are suggested.

Notation

f force vector

K stiffness matrix

M mass, matrix

y response vector

Z impedance matrix

6 antiresonant eigenvector

on forcing frequency

to antiresonant frequency

In forced vibrations an antiresonance
or "off- resonance node", is that frequency
for which a system has zero motion at one
or more points. A nodal point in a normal
mode is a special case of an antiresonance.
Driving point antiresonances have a
readily grasped physical interpretation
since they are the resonances of the
system when it is restrained at the
driving point. However, transfer anti- .
resonances are not all real and, in
general, have not been susceptible to
analysis except in special cases. The
eigenvalue equation for antiresonances
used in this paper renders them as amen-
able to analysis as are resonances. The
mathematics for analyzing resonances are . •
conventional and well-known^.

Although general analytical methods
for transfer antiresonances were not here-
tofore commonly used, the existence of
both driving point and transfer antires-
onances in the forced vibration of a
string were described by Lord Rayleigh 2 .
The invention of the dynamic vibration
absorber in 1909 gave antiresonances some !
practical engineering importance^. The
absorber is an appendant dynamic system
which has a driving point antiresonance
at its fixed base natural frequency and it
therefore reacts the forces at its base in
the direction in which it acts. Isolating
devices based on transfer antiresonances
were not invented until this decade^.
Sometimes natural fuselage transfer anti-
resonances for major hub excitations
occurred near a critical point and at the-
proper frequency (e.g., the pilot's seat
at blade passage frequency) by fortuity
of helicopter design. Occasionally,
engineers have manipulated transfer
antiresonance frequencies and positions
in design through lengthy trial-and-error
response analyses." • However, the industry .
has not used a direct analytical method
for calculating the positions and fre-
quencies of natural. antiresonances.

Presented, at the AHS/NASA-Ames Special-
ists' Meeting on Rotorcraft Dynamics,
February 13-15, 1974.

101

Structures have antiresonances as an
intrinsic "natural" property much as they
have "natural" resonant frequencies.
Natural transfer, or "of f -diagonal", anti-
resonances are as important to structural
dynamics engineering as are resonances.
Unfortunately, many of the theorems which
underly conventional analyses do not apply
to transfer antiresonances. The anti-
resonant dynamical matrix is in general
nonsymmetrical and therefore not positive
definite. This results in both left-
handed and right-handed eigenvectors which
are unequal and require a new orthogonal-
ity condition for the calculation of
successive eigenvectors. The antiresonant
frequencies of the transfer antiresonance
determinants are not necessarily real and
the imaginary roots do not have a simple
physical interpretation. These matters,
along with the lack of an engineering
eigenvalue formulation for antiresonances,
may, in part, account for the relatively
little attention given to natural anti-
resonances over the years.

Z m f k i Z .
mn J irrj

2_?f_i I _SJ_^_IS

Z kn n * * ! Z kj

.fA__j_ Z C_
Z R ] Z kj

(4)

If the impedance matrix is similarly
partitioned so that the upper left-hand
matrix does not contain the j-th row or
the k-th column, then

Z =

(5)

It follows from Equations (4) and (5) that

Z A = Z A

From Equation (3) we obtain

z A y- o

(6)

(7)

Theory

The steady-state equations of motion
for an undamped spring-mass system vi-
brating in the vicinity of equilibrium
are:

A kj antiresonance is defined such
that for a force at k alone, the response
at j is zero. Normalizing y and sub-
stituting for Z A in Equation (7) results
in the antiresonance eigenvalue equation.

(K - to M)y = f

(1)

where the impedance matrix is defined as

(2)

Z = [8f i /3y j ] = (K

a) 2 M)

Let all the forces be zero except the
force acting at the k-th generalized
coordinate and further impose the re-
straint of zero motion for the j-th
generalized coordinate. The resulting
eigenvalues are jk antiresonances of
Equation (1) . Since Z is real and
symmetric the antiresonance eigenvectors
are real and the jk and kj antiresonance
eigenvalues are real (positive or nega-
tive) and equal.

Partition Equation (1) so that the
kj-th element of the impedance matrix
appears in the lower right-hand corner.

Z__ m f k I Z .

mn ' i mj

iLL'sL^fe

-i-

. kn

where

it, .

(3)

102

M A 9 r = -T- K A 9 r
w r

(8)

A jk antiresonance eigenvalue equation is
similarly defined by considering Equation
(5) and making use of Equation (6).

l sW " ~Tf *,\

(9)

Equations (8) and (9) constitute a
set of right-handed and left-handed eigen-
vectors. Since Z^ is not symmetrical,
the jk eigenvectors are not orthogonal
but instead are biorthogonal with the kj
eigenvectors-'-. Premultiply Equation (8)
by ~ T, postmultiply Equation (9) by 9 ,

s
and subtract to obtain

T* VVr

r s
when s ^ r we have

9 s K A 9 r = °

(10)

(11)

Thus , the kj antiresonance eigenvector is
biorthogonal to the jk antiresonance
eigenvector.

When s = r the corresponding gen-
eralized' mass "and stiffness are defined as

3 M-9-
r A r

9 K,9 = K
r A r r

(12)

(13)

Successive antiresonance eigenvectors are
found by applying the biorthogonality
condition and using classical matrix
iteration techniques. The (n + l)st jk.
antiresonant eigenvector is obtained from
Equation (14) ,

(K,

- E

. , K.
1=1 l

-)M„e j_,

A n+1

u n+l

n+1

(14)

which establishes the method of sweeping.5

Discussion of Theory

Each antiresonant eigenvector con-
sists of a pair which is biorthogonal with
respect to both mass and stiffness. For
driving point antiresonances (j = .k), the
two eigenvectors are, obviously i the same.
An N-degree-of-freedom system has N 2
possible antiresonant eigenvectors cor-
responding to all possible forcing and
response coordinates.

Since the mass and stiffness matrices
are nonsymmetric in the antiresonance
eigenvalue problem and consequently, not
positive definite when j ^ k, the anti-
resonant generalized masses and stiff-
nesses may be either positive or negative.
In other words, the antiresonance fre-
quencies are not necessarily real. When
j = k the antiresonant mass and stiffness
matrices are symmetrical and positive
definite, resulting in at least U-l,
positive real antiresonances. As shown
in Reference 6 the driving point anti-
resonances lie between the natural
resonant frequencies.

Applications of Antirfesonance.' Theory

To illustrate- the practical potential
of antiresonance theory,- consider a ten-
degree-of- freedom beam specimen with .
springs to ground at stations- 3. and 9 and
mass and stiffness parameters simulating.'
a 9000 pound helicopter. Antiresonances
are continuous functions of frequency and
position and Figure 1 presents- a typical
position spectrum plot of the specimen
forcing at station 3 alone. ' The dashed
vertical lines are the natural resonant
frequencies determined conventionally.-

When an antiresonance line crosses a
natural frequency line there is a nodal
point in the "natural mode".

Figure 1. Antiresonance Lines Forcing at Station 3

With the same techniques of altering
masses and stiffnesses to avoid undesir-
able natural resonances , the engineer can
manipulate natural antiresonances.- The
stiffness between stations 2 and 3 was
increased by 11.8% in the K 2 3 term of the
stiffness matrix and Figure 2 illustrates
this effect in the natural frequencies
and antiresonance lines. Similar changes,
in the mass of the structure have a sim-
ilar effect. This possibility for re- .
sponse control indicates a profitable' ■'..
area for further exploration.

-i 1 — t — i — i i 1 1 i

10
■ . FREQUENCY -HZ

Figure 2. Antiresonance Lines with Stiffness Change
Forcing at Station 3

Conventional Use of the Dynamic Absorber

A dynamic absorber .is an appendant
dynamic system attached to a helicopter,
usually at a point, as shown in Figure 3.
When we eliminate the i-th. row and column.,
corresponding to the 'attachment point
(see Figure 3) we obtain two uncoupled
•systems.

103

o j z

I aa

is not at the tuned frequency of the
(15) "resonator" and does not necessarily
produce an antiresohance- at j for
excitations along generalized
coordinates other than k.

N — ** "

— - —

Zff

! ^

- 1 ■ — ■

1
1

--t- -

z if

! z h

1
1

-4--
1
1

Z ia

►

! Z ai

z aa

L J

The aforementioned system and equa-
tions of motion are shown in Figure 4.
To obtain an antiresonance at j for a
force at k we eliminate the k-th row and
j-th column from the equations of motions.
This results in the antiresonant eigen-
value equation,

z ff

fVj i z i

1 1
f jfk j f £ k I

fVj i z. . i z.
! z . j z

1 ai I aa

(17)

which is of the form of Equation (7)

Figure 3. Conventional Absorber

The antiresonant eigenvalue equation is
obtained from Equation (15) as

[K M ]6
aa aa J r

2 r

(16)

which is of the form of Equation (7).

If the absorber system were attached
at I points, instead of one, we would
eliminate the I rows and columns corres-
ponding to the attachments and find the
simultaneous antiresonant frequencies of
all I points.

Unconventional Use of the Dynamic Absorber

In some instances there may be only
one significant unreacted force on the
helicopter as, for example, when an in-
plane isolation system or in-plane hub or
flapping absorbers leave small hub moments
but a relatively large vertical oscilla-
tory force. We can use a dynamic absorber
in the fuselage at some point i as a
"resonator" to shift antiresonance lines
so that there exists an antiresonance at
another point j (e.g., the pilot's seat)
for the one remaining large force or
moment along the k-th generalized coor-
dinate. This is creating a jk antires-
onance by manipulation of a "resonator"
at point i. The jk antiresonant frequency

( i

V. .i . —

r iiii

— —

i i i
z «i i z fj i 2 * i °

f*k | f ^ k| f5>k|
**\ | I I

Y f1

i __• __!__

Z kf 1 | Z kj | Z ki | °

f 1 ! L + .._4.__ + __

Z if1 , 1 Z ij I Z « | Z ia

o ; o ! z ai j 2 aa

l Yi I r
._.

L Y aJ L

Figure 4. Antiresonance at Station j from a Resonator at Station i

This technique of using a remote
dynamic absorber as a "resonator" allows
the engineer to obtain an antiresonance,
to a given excitation, at points where
structural limitations prevent installa-
tion of an absorber. When the new res-
onant frequency introduced by the
"resonator" cuts across a natural anti-
resonance line, the shifts are dramatic
as shown in Figure 5. Figure 5 illustrates
the antiresonance lines in the specimen,
forcing at station 3, when an absorber of
77.2 pounds tuned to 7.7 Hz is added to
station 2. The natural frequency intro-
duced by the absorber intersects the
antiresonance line of Figure 1 and pro-
duces new antiresonances at all stations,

104

forcing at station 3

FREQUENCY -HZ
Figures. Antiresonance Lines with Dynamic Absorber at
Station 2, Forcing at Station 3

The effect at station 5 of the 77.2
pound absorber located at station 2 and
tuned to 7.7 Hz, in terms of both anti-
resonant frequency and bandwidth, is the
same as the effect produced by a 193 pound
absorber located at station 5 itself and
tuned to 8.0 Hz. Bandwidth is here
defined as the difference between the
antiresonance frequency and the nearest
natural frequency. This comparison is
presented in Figure 6. The approximately
two to one reduction in absorber weight
does not imply that such savings are
always obtainable.

iu 5

— 77.2 LB ABSORBER AT

STATION 2 TUNED TO 7.7 HZ

1\-193 LB ABSORBER AT

I \ STATION 6 TUNED TO 8.0 HZ

i — i — i — i 1 1 1 — | r

7.0 7.S 8.0 8.S

FREQUENCY -HZ

Figure 6. Comparison of Antiresonance Lines
for Two Absorbers

Antiresonant Isolators

Passive antiresonant isolation
devices have received considerable
attention from the industry in recent
years. Notable among these are Bell
Helicopter's Nodal Module, Kaman's BAVI
series, and the Kaman COZID.

Figure 7 illustrates the antiresonant
isolation system and corresponding equa-
tions of motion. The excited structure

105

r- —

*- —

Y .

Y
Y

,=<

L J

L .)

Figure 7. Antiresonance isolation

is coupled to another structure through,
and only through, the antiresonant
isolation system which has inertial and
elastic elements. Any isolator with a
single input and single output, or a
symmetrical arrangement having the same
effect, has antiresonant frequencies given
by the eigenvalues of

fDl_j^DD
K OI J K OD

-1

M I M
T>I | OD

(18)

where I, .0, and D represent the input,
output and internal isolator degrees-of-
freedom, respectively. The two-dimen-
sional and three-dimensional DAVIs have,
respectively, each two and three un-
coupled equations of the form of Equation
(18) . Two outputs displaced with dynamic
symmetry from a given input, or the con-
verse, are also described by Equation (18)
because the roots are not changed by
transposing a matrix.

It is possible to solve for simul-
taneous antiresonances on arbitrarily
placed multiple outputs for an equal
number of arbitrarily placed multiple
inputs by letting I and O be greater than
one in Equation (18) . However, such
simultaneous antiresonances will, in the
general case, occur only for those dis-
tributions of input forces given by the
product of the rectangular impedance
matrix of rows corresponding to the forced
degrees of freedom and the vector of dis-
placements. This is the reason why
multiple input-output antiresonant isola-
tors are not used in engineering. It is
observed that the impedance matrix of

Figure 7, is, in general, nonsymmetric

while the impedance matrix of Figure 3

is necessarily symmetric. That is the 1.

mathematically distinguishing feature

between absorbers and antiresonant

isolators .

2.
It is obvious from Equation (18) that
an infinite number of mechanical systems
exist which will produce antiresonant
transmissibilities at more than one fre-
quency. Such systems can be analytically 3.
synthesized using desired antiresonant
frequencies , the biorthogonality conr
dition, and the methods of Reference 7.
However, not all such synthesized systems 4.
will be physically realizable and not all
of the physically realizable synthesized
systems will be practical from an
engineering standpoint.

An immediately practical application
of Equation (18) would be the investiga- 5.
tion of physical multi-input antiresonant
isolators with internal coupling using
simpler engineering arrangements for
multi -harmonic antiresonances than has yet
been achieved. 6.

Conclusion

This paper has presented a solution 7.
to the antiresonant eigenvalue problem.
It has been shown that antiresonances can
be determined by < matrix iteration tech-
niques. Antiresonant nodes introduced by
dynamic absorbers and antiresonant iso-
lators have been discussed to illustrate
the novel application of the theory to
helicopter engineering problems.

' References

Meirovitch, L. , ANALYTICAL METHODS IN
VIBRATIONS, McGraw-Hill Book Co., New
York, 1967.

Strutt, J.W. , Baron, Rayleigh, THE
THEORY OF SOUND, 2nd Edition, Volume
1, Sec. 142a, Dover Publications,
New York, 1945.

Den Hartog, J.P. , MECHANICAL VIBRA-
TIONS, 4th Edition, McGraw-Hill
Publishing Co., New York, 1956.

Kaman Aircraft Report RN 63-1,
DYNAMIC ANTIRESONANT VIBRATION
ISOLATOR (DAVI) , Flannelly, W.G. ,
Kaman Aircraft Corporation,
Bloomfield, Connecticut, November
1963.

Rehfield, L.W. , HIGHER VIBRATION
MODES BY MATRIX ITERATION, Journal
of Aircraft , Vol. 9, No. 7, July 1972,
p. 505.

Biot, M.A. , COUPLED OSCILLATIONS OF
AIRCRAFT ENGINE-PROPELLER SYSTEMS,
Journal of Aeronautical Society ,
Vol. 7, No. 9, July 1940, p. 376.

USAAMRDL Technical Report 72-63B,
RESEARCH ON STRUCTURAL DYNAMIC
TESTING BY IMPEDANCE METHODS,
Giansante, N. , Flannelly, W.G. ,
Berman, A., U. S. Army Air Mobility
Research and Development Laboratory,
Fort Eustis, Virginia, November 19 72.

106

THE EFFECT OF CYCLIC FEATHERING MOTIONS

DYNAMIC ROTOR LOADS

Keith W. Harvey

Research Engineer

Bell Helicopter Company

Fort Worth, Texas

Abstract

The dynamic loads of a helicopter
rotor in forward flight are influenced
significantly by the geometric pitch
angles between the structural axes of the
hub and blade sections and the plane of,
rotation.

The analytical study presented in-
cludes elastic coupling between inplane
and out-of-plane deflections as a function
of geometric pitch between the plane of
rotation and the principal axes of inertia
of each blade. In addition to a mean col-
lective-pitch angle, the pitch of each
blade is increased and decreased at a one-
per-rev frequency to evaluate the dynamic
coupling effects of cyclic feathering mo-
tions. The difference in pitch between
opposed blades gives periodical coupling
terms that vary at frequencies of one- and
two-per-rev. Thus, an external aerody-
namic force at n-per-rev gives forced res-
ponses at n, n±l, and n+2 per rev.

The numerical evaluation is based on
a transient analysis using lumped masses
and elastic substructure techniques. A
comparison of cases with and without cyclic
feathering motion shows the effect on com-
puted dynamic rotor loads. The magnitude
of the effect depends on the radial loca-
tion of the pitch change bearings.

Introduction

For a stiff -in-plane rotor system,
the blade chordwise stiffness may be 20
to 50 times greater than the blade beam-
wise stiffness. The elastic structure
tends to bend in the direction of least
stiffness, resulting in dynamic coupling
between out-of-plane and inplane motions
as a function of the geometric pitch
angles due to collective pitch, built-in
twist, forced cyclic feathering motions of
the torsionally-rigid structure, and elas-
tic deformation of the blade and control
system in the torsional mode.

Typical cruise conditions for a mod-
ern helicopter require collective pitch
angles of 14 to 16 degrees at the root,
depending on the amount of built-in twist.
Cyclic feathering motions of 6 to 7 de-
grees are required to balance the one-per-
rev aerodynamic flapping moments. In cur-
rent design practice, elastic torsional

deflections of the blade and control sys-
tem of a stiff -in-plane rotor are generally
less than one degree. The largest part of
the angular motion in the blade-torsion de-
gree of freedom, therefore, is the forced
feathering motion due to cyclic pitch.

Periodic variations of the inplane/
out-of-plane elastic coupling terms are
caused when the geometric pitch angle of
each blade is increased and decreased at a
frequency of one cycle per rotor revolution.
When one blade is at high pitch and the op-
posed blade is at low pitch, an asymmetri-
cal physical condition exists with respect
to a reference system oriented either to
the mast axis or to the plane of rotation.
One-half revolution later, the reference
blade is at low pitch and the opposed blade
is at high pitch. Thus, periodic dynamic
coupling occurs at the principal frequency
of one-per-rev with respect to a rotating
coordinate system. The coupling terms are
nonlinear functions of blade pitch; hence,
these terms also have 2-per-rev content.

Both the steady and periodic coupling
terms have been treated in an analytical
study of the effects of one-per-rev cyclic
feathering motions on dynamic rotor loads.
Equation's have been derived and programmed
for a digital computer solution of the
transient response of an elastic two-bladed
rotor.

The rotor is modeled by elastic sub-
structure elements and lumped masses, for
which the accelerations and velocities are
integrated over small time increments to
determine time histories of deflections,
inertia loads, bending moments, etc. The
time-variant analysis includes the capa-
bility to calculate rotor instabilities.
The present computer program has been
tested for this capability, but further
discussion of instabilities is beyond the
intent of the paper.

Dynamic rotor loads have been calcu-
lated for a parametric series of rotors,
where the coupled natural frequencies were
tuned over the range of contemporary de-
sign practice for teetering rotors. A de-
scription of the analysis and a summary of
computed results is presented.

107

Objective

A primary consideration in the design
of a helicopter rotor is to minimize os-
cillatory bending loads, or at least to
reduce the loads to a level that will en-
sure satisfactory fatigue life. During
early stages of design, the principal
method of evaluating the dynamics of a
proposed rotor is to calculate its coupled
rotating natural frequencies. If required,
design changes are made to achieve suffi-
cient separation between the natural fre-
quencies and harmonics of the rotor .oper-
ating speed.

Current practice at Bell Helicopter
Company is to require a separation of 0.3
per rev for all flight combinations of
rotor speed and collective pitch. One
purpose of the present analytical develop-
ment is to determine whether the separa-
tion rule may be relaxed due to beneficial
effects of cyclic feathering motions on
rotor dynamic response.

Collective and Cyclic Modes

The calculation of natural frequen-
cies for semi-rigid rotors uses a coor-
dinate system that is based on the plane
of rotation. The orientation of the cen-
trifugal force field, the angular motion
allowed by the flapping hinge(s), and the
constraints of opposing blades lead to the
segregation of natural frequencies into
collective modes, cyclic modes, and (for
four-bladed rotors) scissor or reaction-
less modes. This procedure allows the use
of continuous-beam theory for a single
blade, where the centerline boundary con-
straints are imposed from conditions of
symmetry or asymmetry to match deflections,
slopes, shears, and moments for the other
blades .

The centerline boundary conditions for
the collective mode (Figure 1) are:

- zero vertical (out-of-plane) slope
change ,

- vertical deflection constrained by
mast tension/compression,

- inplane slope constrained by mast
torsion, and

- zero inplane translation.

The centerline boundary conditions for
the cyclic mode (Figure 2) are:

- vertical slope change unrestrained
(except with flapping springs),

- zero vertical deflection,

- zero inplane slope change, and

inplane deflection constrained by
mast shear.

ROTOR SPEED 300

FIGURE 1. TYPICAL COLLECTIVE MODE

FREQUENCIES AND MODE SHAPES.

ROTOR SPEED
FIGURE 2

300

TYPICAL CYCLIC MODE FRE-
QUENCIES AND MODE SHAPES.

108

For the reactionless modes, the cen-
terline boundary conditions are:

- zero slope change and zero trans-
lation in both the inplane and
vertical directions.

Uncoupled frequencies are determined
by setting the geometric pitch angle of
each elastic element to zero. The un-
coupled frequencies are shown in Figures
1 and 2 by the labeled curves. Note that
the frequencies of the vertical (out-of-
plane) modes are highly dependent on rotor
speed, and that the frequencies of the in-
plane modes are only slightly dependent
on rotor speed.

Coupled natural frequencies are shown
as small circles in the figures. Typical
collective modes have very small frequency
shifts as a function of collective pitch.
However, the cyclic modes (Figure 2)
couple significantly with collective pitch.
Note that the inplane frequency decreases
and the vertical frequencies increase with
collective pitch. The method of deter-
mining these coupling effects is given in
Reference 1.

By using only one blade plus appro-
priate boundary conditions, this method of
calculating rotor natural frequencies is
based on one explicit assumption, i.e.,
all other blades are at the same geometric
pitch angle as the reference blade. If
the blades are at different pitch angles,
then the conditions of symmetry or asym-
metry are not present. The inclusion of
cyclic feathering motion, therefore, re-
quires that the analysis treat separately
each blade of the rotor and provide a
means of matching the centerline slopes
and deflections.

Elastic Substructured Rotor Analysis

A digital computer program has been
developed to study the effects of cyclic
feathering motions on dynamic rotor loads.
The Bell Helicopter computer program is
identified by the mnemonic ESRA for Elas-
tic Substructured Rotor Analysis.

The analysis is a transi
of elastic rotor blade motion
coupling terms for each blade
separately and the necessary
are imposed on each blade to
and deflection continuity at
centerline. Each blade is co
being divided into a discrete
segments, with uniform weight,
ness properties over the leng
ment. The geometric pitch an,
segment is a function of roto
position and input values of
lateral cyclic pitch. To rep
hub structure that is inboard

ent solution
s , where the

are treated
constraints
insure slope
the rotor
nsidered as

number of

and stiff-
th of a seg-
gle of each
r azimuth
fore/aft and
resent the

of the pitch-

change bearings, the inboard elastic ele-
ment may be specified as an uncoupled
element (geometric pitch equals zero).

All forces are applied at the ends of
the elastic elements. Slope and deflection
changes over the length of a segment are
based on linear moment distributions versus
span. For compatibility, shear over the
segment length must be constant, which re-
quires concentrated forces for both inter-
nal and external forces. In its simplest
form, the analysis follows a lumped-mass
approach. All of the important rotor dy-
namic characteristics may be retained with
this method, however, by using the con-
cepts of equivalent structural segments.

The dynamic response equations are
solved by a step-by-step iterative method in
order to include transient conditions. If
the initial deflections and velocities are
specified (spanwise distributions for each
blade), then the internal bending moment
distributions are found with respect to the
rotating reference system. Internal shear
distributions are obtained from the moment
distributions, and summed with applied air-
load forces and inertial components of the
centrifugal force field to determine span-
wise distributions of accelerations.

The first estimates of deflection and
velocity changes are calculated for con-
stant acceleration during the integration
time step. Then bending moments, shears,
and accelerations are calculated for the
end of the time step. Subsequent deflec-
tion and velocity estimates are based on
accelerations changing linearly with time,
and the iterations continue until a pres-
cribed error limit is satisfied for the
entire set of accelerations, or until a
limit is reached on the number of itera-
tions .

In recognition of the problems inher-
ent with this type of numerical integration,
the initial development of the ESRA com-
puter program has been limited to a quali-
tative study of cyclic feathering effects.
The current program represents each blade
with four elements, each with beamwise and
chordwise bending elasticity. Only the
forced rigid-body motion is allowed in the
blade-pitch degree of freedom, i.e., elas-
tic blade torsion is not considered.

The current computer
ited to two-bladed rotors,
al impedance of the drive
to be zero. In practice,
a two-per-rev torque from
proportional to the drive-
times the Hooke ' s- joint an,
which is a function of rot
rotor is the predominant i
of the drive system, and a
mation for two-bladed roto

program is lim-

and the torsion-
system is assumed
the rotor senses
the mast that is
system impedance
gular oscillation,
or flapping. The
nertia component
good approxi-
rs is to assume

109

that the true axis of rotation remains per-
pendicuLar to the tip-path plane even when
the tip path plane is not perpendicular to
the mast. Thus, Coriolis accelerations
equal to the product of coning times flap-
ping are not appropriate in a two-bladed
rotor analysis.

Bending deflections of the elastic
elements are linearized; therefore, Cori-
olis accelerations from radial foreshort-
ening are excluded also. Vertical and in-
plane translational motions of the rotor
center are not included in the current
version of the program.

Referring to the description above,
the formulation of the analysis allows the
removal of these limiting assumptions.
For instance, nonlinear bending deflections
and Coriolis accelerations may be included
by a direct addition to the inertial forces
acting on each mass. Translation of the
rotor centerline, additional blades, con-
trol system flexibility, elastic blade
torsion, and nonlinear hub and control
kinematics also may be added within the
existing computational method.

With the limitation of four elastic
elements for each blade, plus provisions
for slope and deflection continuity at the
rotor centerline, the current ESRA pro-
gram allows 15 distinct vibration modes
for the rotor:

- 3 rigid -body modes (flapping, mast
torsion, blade pitch)

- 6 coupled elastic collective modes
(3 vertical, 3 inplane)

- 6 coupled elastic cyclic modes (3
vertical, 3 inplane)

In attempts to predict rotor loads for
two-bladed rotors, emphasis is placed on
response components at least up to the
third harmonic of rotor speed. Three-per-
rev airloads excite the cyclic mode that
derives from the first elastic asymmetric
mode in the out-of-plane direction. Four
elastic elements for each blade should
provide a very satisfactory dynamic repre-
sentation for this frequency range. At a
frequency of f ive-per-rev , the second elas-
tic mode would be excited and computed
loads may be marginally valid. Current
design practice is to minimize higher fre-
quency loads by proper tuning of the rotor
natural frequencies, as discussed earlier.

Numerical Evaluation

A parametric computer study was ac-
complished to resolve a basic question:

With respect to the natural fre-
quency of the first coupled vertical

elastic cyclic mode at or near
3 per rev, how much does cyclic
feathering motion affect 3-per-
rev dynamic rotor loads?

Selection of Rotor Dynamic Characteristics

Corresponding to a Huey main rotor,
the computer study was based on a 48-foot
diameter 2-bladed semi-rigid rotor, oper-
ating at 300 RPM. Two basic design ap-
proaches were selected as end points for
the evaluation. ;

1. A constant blade weight distri-
bution of 1.20 lb/in. with no dynamic -
tuning weights was picked to simulate the
early production Huey rotors. Uniform
beamwise and chordwise stiffness values
were determined to locate the two lowest
coupled cyclic mode frequencies at 1.40/rev
(inplane) and 2.60/rev (vertical) for a
collective pitch of 14.75 degrees. The fan
plot of cyclic mode natural frequencies for
this rotor is shown in Figure 3.

BASIC SECTION WT
=1.20 LB/IN

NO TIP WEIGHT

_ 14.75° COLL. PITCH
o

25
U

-1/REV

ROTOR SPEED, RPM

300

FIGURE 3. CYCLIC MODE, TUNED
BELOW 3/REV.

2. A very recent rotor development at
Bell (the Model 645 rotor) was simulated by
a configuration with a constant blade weight
distribution of 1.00 lb/in plus a dynamic
tuning weight of 100 pounds located at the
blade tip. Uniform beamwise and chordwise
stiffness values were determined to locate
the coupled cyclic mode frequencies at
1.40/rev (inplane) and 3.40/rev (vertical),
as shown in Figure 4, again for 14.75 de-
grees of collective pitch.

110

ROTOR SPEED, RPM

300

FIGURE k. CYCLIC MODE,, TUNED
ABOVE 3/REV.

Between the two basic configurations,
a series of intermediate rotor parameters
was established by stepping the uniform
blade weight from 1.20 down to 1.00 by
increments of 0.025 lb/in., while increas-
ing the tip weight from 0. to 100. by in-
crements of 12.5 pounds. Beamwise and
chordwise stiffnesses were varied to hold
the coupled inplane frequency at 1.^0/rev
while tuning the coupled vertical fre-
quencies from 2.60/rev to 3.40/rev in in-
crements of 0.10/rev. Thus, to compensate
for the program restriction of no hub mo-
tion, proper placement of the coupled in-
plane frequency was maintained by the
selection of rotor stiffness. This ap-
proach affects the spanwise. distribution
of inplane bending moments, but is entire-
ly adequate for a qualitative evaluation.

All of the above frequencies were
tuned with the first segment uncoupled
(hub structure to .25 radius), which maxi-
mized the coupling of the vertical mode
near 3/rev and minimized the coupling of
the inplane mode near 1/rev.

Additional input data was taken
directly from the Bell Helicopter Rotor-
craft Flight Simulation, program C81-68
(References 2,3), for a Model 309 King-
Cobra flying at 150 knots. Data used in
the present computer evaluation included
a collective pitch setting at the root of
lif.75 degrees, a total cyclic pitch of
6.30 -degrees, and the spanwise distributed
airloads up to and including the third*
harmonic components.

The study results presented below are
based, therefore, on full-scale parameters
that are realistic with regard to current
helicopter design practice. Although di-
rect correlation with measured loads is not
possible because of the simplifying assump-,
tions , it may be noted that the magnitude
of calculated bending moments is well with-
in the expected range.

Computed Results

The forced response was computed for
the series of nine parametric rotor con-'
figurations, where the inplane coupled fre-
quency was held at 1.40/rev and the verti-
cal coupled frequency was varied from 2.60/
rev to 3.40/rev. The dynamic rotor loads
for each configuration were calculated
twice, once with cyclic feathering and
once without cyclic feathering.

Figure 5 shows the 2/rev vertical
bending moment at the rotor centerline as
a function of natural frequency of the ver-
tical elastic cyclic mode. The 1/rev vari-
ation in structural coupling due to cyclic
pitch, and the 3/rev applied airloads pro-
duce a 2/rev component of bending moment.
This additional component peaks and changes
sign as the vertical mode is tuned through
3/rev. For the two-bladed rotor, 1/rev and
3/rev vertical bending moments at the cen-
terline are negligible.

3
i

o
o

45-

4tt

S 25|

z
w

2/REV

WITHOUT CYCLIC FEATHERING

WITH CYCLIC FEATHERING

2.6 2,8 3,0 3.2 3.4
COUPLED VERTICAL NAT. FREQ. , PER REV
FIGURE 5. VERTICAL MOMENT AT CENTERLINE

111

InpLane bending moments at the rotor
centerline are shown in Figure 6. The
large peak in the overall oscillatory mo-
ment occurs as the coupled vertical mode
is tuned through resonance at 3/rev. Note
that the coupling associated with cyclic
feathering increases the 1/rev response by
about 5 percent for the vertical frequency
tuned to 2.6/rev. In other respects, the
effect of cyclic feathering appears to be
minimal.

Beamwise moments and chordwise mo-
ments at midspan are shown in Figures 7
and 8, respectively. Two-per-rev moment

WITHOUT CYCLIC FEATHERING
WITH CYCLIC FEATHERING

340

320-1

300

280-

260-

240-

220

3 200-1
i

'/ OVERALL N
'' OSCILLATORY

--,_ 1/REV

o
o
o

180
160

g 140-1

s ••

5- 120

a
z
w

« 100

80
60

3/REV

V—

2.6 2.8 3.0 3.2 3.4
COUPLED VERTICAL NAT. FREQ. , PER REV
FIGURE 6. INPLANE MOMENT AT CENTERLINE

components are not shown in the figures
because of their small magnitudes. The
significance of the cyclic feathering ef-
fects at midspan is consistent with that
indicated in earlier figures for the rotor
centerline.

►J

o
o
o

1—1

WITHOUT CYCLIC FEATHERING
WITH CYCLIC FEATHERING

OVERALL
OSCILLATORY

°VS

n 3/REV

2.6 2.8 3.0 3.2 3.4
COUPLED VERTICAL NAT. FREQ. , PER REV
FIGURE 7. BEAMWISE MOMENT AT MID SPAN

140

•J 120

glOO

I— I
Q
Z
W
«

60
40
20

OVERALL
OSCILLATORY

WITHOUT CYCLIC FEATHERING
WITH CYCLIC FEATHERING

2.6 2.8 3.0 3.2 3.4
COUPLED VERTICAL NAT. FREQ. , PER REV
FIGURE 8 . CHORDWISE MOMENT AT MID SPAN

112

As discussed in a previous section,
the aeroeLastic effect of blade bending
velocity was excluded from this study by
basing the response- calculations on a pre-
scribed set of airloads. No inference is
intended regarding the magnitude of aero-
dynamic damping that may be associated with
elastic bending velocities. Conversely,
the procedure was selected so that the
time-variant structural couplings could be
studied in an analytical environment that
does not include other sources of damping.

The computed responses appear as un-
damped resonances centered at 3/rev, from
which it follows that cyclic feathering
motions do not provide any significant
amount of equivalent damping to suppress
3/rev dynamic loads. Regarding the verti-
cal cyclic mode near 3/rev, in particular,
the effect of cyclic feathering motion
does not provide relief for the design rule
that requires 0. 3/rev separation of coupled
frequencies from excitation harmonics of
rotor speed.

The results presented above are all
based on rotor structural simulations with
the inboard 25 percent radius treated as
non-feathering hub structure. This option
of the program was selected to maximize
the coupling (as a function of collective
pitch) of the vertical cyclic mode near
3 per rev. As noted, the largest change
in rotor loads due to the inclusion of
cyclic feathering motions was a 5 percent
increase in inplane bending moments at the
rotor centerline.

The pitch-change or feathering bear-
ings of production two-bladed main rotors
are located typically at about 10 percent
radius. In this respect at least, the
above results are based on a dynamic model
that is not representative of actual de-
sign practice.

To evaluate the importance of the
radial location of the bearings, another
set of rotor loads was computed for a
case where the entire radius is in the
feathering system.

A constant blade weight distribution
of 1.20 lb/in with no dynamic tuning
weights was selected, as before, to simu-
late the early production Huey rotors.
The structural properties of the rotor
were modified to maintain a 1.40/rev
natural frequency for the coupled inplane
cyclic mode. For the modified parameters,
the natural frequency of the coupled verti-
cal cyclic mode is 2.87/rev.

The computed results are shown in
Figures 9 through 12 for the case in which
the feathering bearings are located at
zero percent radius. The bar< graphs show
first, second, and third harmonics plus

overall levels of oscillatory bending mo-
ments. The open bars are for the condition
of no cyclic pitch, i.e., the geometric
pitch of the elastic elements held fixed
at the specified value of collective pitch.
The closed bars are for the condition that
the geometric, pitch of the elastic struc-
ture is a function of both collective pitch
and cyclic pitch.

Vertical and inplane oscillatory bend-
ing moments at the rotor centerline are
shown in Figures 9 and 10, respectively.
The vertical moments are not changed

o
o
o

z
w
PQ

100-
80-
60
40
20

WITHOUT CYCLIC
WITH CYCLIC

1 2 3 OVERALL
HARMONIC (PER REV)

FIGURE 9. VERTICAL MOMENT AT CENTERLINE

300

2 200

100-

CZ1 WITHOUT CYCLIC
5 WITH CYCLIC

FIGURE 10.

1 2 3 OVERALL
HARMONIC (PER REV)

INPLANE MOMENT AT CENTERLINE

113

signif icantLy by the inclusion of cyclic
pitch. However, the inplane centerline
moments increase by 57 percent, with both
the first and third harmonics contributing
to the increase.

Beamwise and chordwise oscillatory
bending moments at 50 percent radius are
shown in Figures 11 and 12. Most of the
increase in the overall oscillatory mo-
ments at mid-span is due to an increase
in 3/rev response.

Due to cyclic feathering motions, a
significant increase (57 percent) in rotor
loads is indicated with the feathering
bearings at zero percent radius; a mini-
mal increase (5 percent) is indicated with
the feathering bearings at 25 percent
radius. This suggests that the radial lo-
cation of the feathering bearings may have
a controlling influence on the magnitude
of the cyclic-feathering effect. Further
study of this relationship is in progress.

Conclusions

« 100

o
o
o

H
55

80-

60-

20-

WITHOUT CYCLIC
WITH CYCLIC

12 3 OVERALL
HARMONIC (PER REV)

FIGURE 11. BEAMWISE MOMENT AT MID-SPAN

1. The cyclic feathering motions of
a helicopter rotor cause time-dependent
elastic coupling due to asymmetrical pitch
on opposed blades. The effect of these
motions on dynamic loads may be calculated
by modeling the rotor with elastic sub-
structure elements, by providing individual
treatment of each blade , and by matching
slopes and moments at the rotor centerline.

2. Cyclic feathering "motion of the
elastic blade structure does not cause any
significant damping effect on the 3-per-rev
dynamic loads of a two-bladed semi-rigid
rotor. The design rule requiring 0.3? rev
separation between coupled natural fre-
quencies and aerodynamic excitation fre-
quencies should not be relaxed on the
expectation of beneficial effects from
cyclic feathering.

3. The inplane one-per-rev rotor loads
of a stiff -in-plane rotor are affected sig-
nificantly by cyclic feathering of the
elastic structure. The magnitude of the
effect is decreased as the feathering bear-
ings are moved radially away from the rotor
centerline.

a 100

M
O

E-t

55
W

80-

20-

WITHOUT CYCLIC
WITH CYCLIC

m n^ rm i m

1 2 3 OVERALL
HARMONIC (PER REV)

FIGURE 12. CHORDWISE MOMENT AT MID-SPAN

References

Blankenship, B. L. and Harvey, K. W. ,
A DIGITAL ANALYSIS FOR HELICOPTER PER-
FORMANCE AND ROTOR BLADE BENDING MO-
MENTS , Journal of the American Heli-
copter Society , Vol,! 7 , No . 4 ,
October 1962, pp 55-69.

Duhon, J. M. , Harvey, K. W. , and
Blankenship, B. L. , COMPUTER FLIGHT
TESTING OF ROTORCRAFT , Journal of the
American Helicopter Society , Vol. 10,
No. 4, October 1965, pp 36-48.

USAAVLABS Technical Report 69-1,
ANALYTICAL STUDY OF HELICOPTER GUST
RESPONSE AT HIGH FORWARD SPEEDS, Harvey,
K. W. , Blankenship, B. L. , and Drees,
J. M. , U.S. Army Aviation Materiel
Laboratories, Ft. Eustis, Virginia,
September 1969.

114

CONTROL LOAD ENVELOPE SHAPING BY LIVE TWIST

F. J. Tarzanin, Jr.
Senior Engineer, Boeing Vertol Company, Philadelphia, PA

P. H. Mirick
Aerospace Engineer, Eustis Directorate, USAAMRDL, Ft. Eustis, VA

Abstract

For flight conditions at high blade
loadings or airspeeds, the rotor control
system experiences a rapid load growth,
resulting from retreating blade stall.
These loads frequently grow so large that
the aircraft flight envelope is restricted
long before the aircraft power limit is-
reached. A theoretical study of one flight
condition and a limited model test have
shown that blade torsional flexibility
plays a major role in determining the mag-
nitude of these large, stall-induced con-
trol loads. Recently, an extensive
analytical investigation* was undertaken
to determine the effect of changing blade
torsional properties over the rotor flight
envelope. The results of this study showed
that reducing the blade stiffness to intro-
duce more blade live twist** could signi-
ficantly reduce the large retreating blade
control loads. Too much live twist, how-
ever, may increase the control loads by
introducing a large nose -down advancing
blade torsional moment. Still, signifi-
cant control load reductions can be
achieved and the flight envelope can be
expanded by increasing live twist to reduce
retreating blade stall loads, but not
enough to greatly increase advancing blade
loads.

Introduction

For any practical' helicopter design,
the level-flight, steady-state loads should
be below the endurance limit (infinite life
load) so that sufficient life will be avail-
able to absorb the larger maneuver loads .
A major design objective is to produce an
aircraft with a flight envelope limited by
aircraft power and not by structural limits
Frequently, however, the operational flight
envelope is limited by a rapid growth of
stall- induced control loads that exceed the
endurance limit. Therefore, the flight
envelope is limited by control loads, and
the available power cannot be fully util-
ized.

* Work performed under Contract DAAJ02-
72-C-0093, Investigation of Torsional

Natural Frequency on Stall-Induced
Dynamic L o ading , by The Boeing Vertol
Company, U. S. Army Air Mobility Researc
and Development Laboratory (USAAMRDL) .

** Live twist is the steady and vibratory
elastic pitch deflection that results
from blade torsional loads.

The rapid control load growth is at-
tributed to stall flutter which is a
consequence of high angles of attack and
resulting blade stall. Visual confirmation
of the large stall loads can be found in
pitchlink or blade torsional gage waveforms
on which characteristic stall spikes appear
in the fourth quadrant of the blade azimuth.
These high loads result from an aeroelastic,
self-excited pitch motion in conjunction
with repeated submersion of a large portion
of the rotor blade in and out of stall.

An aeroelastic rotor analysis program^
was developed, using unsteady aerodynamic
theory that could preduct the large stall -
induced control loads. Limited analytical
studies of a single flight condition, using
this program 2 and another study by Sikorsky
Aircraft?, indicated that modifications to
the blade torsional properties could
significantly reduce the stall-induced
control loads. These encouraging theoreti-
cal results led to a model test3 to verify
the control load reduction. The test re-
sults showed that, by reducing the blade
torsional natural frequency from 5.65 to
3 per rev, the model stall flutter torsion
spike was reduced 73 percent, giving a
first verification of the analytical trend.

Next, an extensive study was under-
taken to explore the impact of modified
blade torsional properties on blade tor-
sional loads over the flight envelope. The
study had two major parts- -the first part
compared model test results of blades with
different torsional properties with analy-
tical results to evaluate the analysis;
while the second part analytically explored
the variation of control loads for flight
conditions of hover and 125, 150, and 175
knots with blade loadings (C T /a) from 0.05
to 0.18. This paper summarizes the results
of this study.

Theory and Test Comparison

For useful analytical results, confi-
dence in the theory must be established to
show that the fundamental phenomena are
properly accounted for. The aeroelastic
rotor analysis has been successfully cor-
related with control loads obtained from
full-scale CH-47C flight data for both
stalled and unstalled conditions. Addi-
tional correlation with the model rotors

115

test was performed to further evaluate the
analysis.

The model test used three six-foot
diameter rotor sets. Each rotor set had
three articulated blades with identical
airfoil and planform, but each set had a
different torsional natural frequency. The
first set of blades had a torsional natural
frequency of 4.25 per rev and was con-
structed of fiberglass, using conventional
crossply torsion wrap. The second set of
blades had mass properties similar to the
first blade set, but had a torsional' nat-
ural frequency of 3.0 per rev. These
blades were constructed with fiberglass,
using a uniply torsion wrap which substan-
tially reduced the blade torsional
stiffness. The third set of blades had a
torsional natural frequency of 5.65 per
rev and was constructed of carbon composite.
Although the carbon blades were not signi-
ficantly stiffer than the first set of
blades, they had significantly lower tor-
sional inertia which accounted for the
higher torsional natural frequency.

A number of runs were made for each
rotor set at full-scale tip speeds and an
advance ratio of 0.3. Due to the differ-
ence in torsional properties, the blade
live twist of each rotor set was different,
resulting in propulsive force and thrust
differences for identical collective,
cyclic, and shaft angle. One run for each
rotor set was selected such that the rotors
would have similar propulsive force varia-
tions with thrust. From each of these runs,
five test points were selected which covered
the range of available blade loading (Cx/a)
and which provided at least one flight con-
dition below stall, one condition in
transition, and two stalled conditions. A
detailed description of the test conditions
and all the model blade physical properties
are given in Reference 4.

The variation of test and analytical
blade torsion amplitude with blade loading
(Cf/a) is shown in Figures 1, 2, and 3 for
the low stiffness, standard reference, and
carbon blades, respectively. Each blade
was instrumented to record blade torsion
data. Due to gauge failures, only two
gauges on the standard reference blades and
only one gauge on the carbon blades were
operational. In general, the analysis
correctly preducts the trend of blade tor-
sional load amplitude with blade loading
for both stalled and unstalled flight
conditions. The preducted stall inception
agrees well with test for the low stiffness
blade. For the standard reference blade
and the carbon blade, stall inception is
predicted about 0.01 Ct/ct too early (see
Figures 2 and 3) . The analysis predicts
approximately the correct torsional load
growth rate in stall; but, because the
stall inception is predicted early, the

« 16

•

J * *

• ANALYSIS

, RANGE OF TEST

I DATA FOR 3
1 INSTRUMENTED
1 BLADES

O TYPICAL TEST
VALUE

0.04

C T /o - BLADE LOADING

Figure 1 . Comparison of Measured and Calculated Blade Torsion Amplitude for
the Low Stiffness Blade (3 per rev).

_ ______

ft 1

•

• I

[

z
o
w 16

h-
a

r-
<

1 12

LLI

Q
<

co 8

•

,I lX

C ANALYSIS

T RANGE OF TEST

[ INSTf
BLAD

1UMENTED
ES

0.O4 0.06 0.08 0.10 0.12 0.14

C T /o - BLADE LOADING

Figure 2. Comparison of Measured and Calculated Blade Torsion Amplitude for
the Standard Blade (4.25 per rev).

116

w —

•

°;

n ,

» /

•

ANALYSIS

TEST VALUE F
NSTRUMENTS
INSTRUMENT
ON OTHER BL
HASFAILEDI

OR ONE
D BLADE
ATION
ADES

■

C T /a- BLADE LOADING

Figure 3. Comparison of Measured and Calculated Blade Torsion Amplitude
for the Carbon Blade (5.65 per rev).

stall loads are overpredicted for the two
stiff er blades.

A number of possible explanations for
the analytical overprediction are discussed
in Reference 4. The two most likely expla-
nations are first, questionable model blade
physical properties (possible 0.02 chord
error in the carbon blade center of gravity,
0.06 chord error in the standard blade
shear center, and -0.07 chord error in the
carbon blade shear center) secondly, the
unsteady aerodynamic plunging representa-
tion is inadequate (a lack of cyclic pitch
led to model blade flapping of 12 , the
theory may be overly sensitive to these
large plunging velocities) .

Even though correlation is not as good
as desired, it is clear that the analysis
predicts the large stall-induced control
load increase (stall flutter) , approximates
the control load increase with increasing
rotor thrust, and defines the proper load
trend for changes in blade torsional prop-
erties. These results are sufficient to
provide a degree of confidence in the
theoretical trends, indicating that the .
qualitative results of the flight envelope
investigation are meaningful.

Analytical Investigation of
Stall -Induced Dynamic Loading

The analysis and test of Reference 2
showed that changes in blade torsional prop-

erty can change the stall -induced control
loads. For this discovery to have a prac-
tical application, it must be shown that
realistic changes in the blade torsional
properties will lead to a significant
reduction in the large stall-induced con-
trol loads throughout the flight envelope.
To determine the variation of control loads
over the flight envelope, an extensive
analytical study was performed. The air-
craft used for this study was a single-
rotor helicopter with CH-47C blades. A
20.1 square-foot frontal area was assumed
for the fuselage and a tail rotor suffi-
cient for trim purposes was added. The
main rotor consisted of three articulated
30- foot radius blades with a constant chord
of 25.25 inches. The blade cross-section
is a cambered 23010 airfoil with 9.137
degrees of linear twist along the blade
span.

To perform this study, blade torsional
properties were modified by changing blade
torsional stiffness (GJ) , changing control
system stiffness, and changing blade pitch
inertia. Using the CH-47C blade's tor-
sional natural frequency of 5.2 per rev as
a baseline, the frequencies were adjusted
from 3 per rev to 7 per rev.

Natural frequencies of 3 rev, 4 rev,
and 7 rev were obtained by multiplying the
torsional stiffness distributions by 0.25,
0.5 and 3.3, respectively. A control
system stiffness of 1650 pounds per inch
generated a frequency of 3 per rev; 11,850
pounds per inch was used for the basic
blade, and an infinite stiffness produced
a 6 per rev frequency. Pitch inertia
changes resulted in blades with frequencies
of 3, 4 and 7 per rev due to scaling the
pitch inertia by factors of 3.08, 1.7 and
0.55, respectively. All changes were made
by only varying the blade properties in-
dicated, while holding all other parameters
at the nominal CH-4 7C values.

Figure 4 shows the relationship be-
tween pitch-link load amplitude and
torsional frequency (3 per rev to 7 per
rev) at an airspeed of 125 knots and a
blade loading of 0.115 for the three meth-
ods of varying frequency. Each method
produced approximately the same trend of
increasing pitch-link loads with increasing
torsional natural frequency. Figure 5
illustrates the variation of pitchlink load
amplitude with natural frequency at 150
knots airspeed and blade loadings of 0.115.
For both airspeeds, the variation in tor-
sional stiffness leads to larger changes
in pitch- link loads, than do changes in
control stiffness or pitch inertia.

Since blade torsional stiffness changes
resulted in the largest change in control
load and the lowest loads, the effect of
blade torsional stiffness changes will be

117

* 2000
1

BLADE LOADING = 0.115

TORSIONAL STIFFNESS VARIATION

PITCH INERTIA VARIATION

CONTROL SYSTEM STIFFNESS
VARIATION

2 3 4 5 6 7

TORSIONAL NATURAL FREQUENCY - PER REV

Figure 4. Variation of Pitch-Link Load Amplitude with Natural Frequency
at 125 Knots.

6000

5000

I-

3000

1000

3 4 5 6 7

TORSIONAL NATURAL FREQUENCY - PER REV

Figure 5. Variation of Pitch-Link Load Amplitude with Natural Frequency
at 150 Knots.

explored in greater depth. It is apparent
from these results that it is not the
reduced torsional frequency alone that
reduces pitch-link loads, but also the in-
crease in blade live twist, resulting from
reduced torsional stiffness.

Four blades with different torsional
natural frequencies (i.e., 3 per rev, 4
per rev, 5.17 per rev and 7 per rev) were
analyzed for 24 flight conditions to in-
vestigate the interactive effects of

torsional stiffness, blade loading (Ct/ct) ,
and airspeed. The airspeeds ranged from
hover to 175 knots and blade loading from
0.05 to 0.018.

Hover

Figure 6 shows the variation of pitch -
link load amplitude with blade loading
for four sets of rotor blades. One degree
of cyclic pitch was used to provide some
means of introducing a cyclic load varia-
tion. If this were not done, the analysis
would predict only steady loads. At blade
loadings of 0.115 and 0.12 the pitch-link
load has a 1-per-rev waveform with an amp-
litude of about 100 pounds for all four
blades. These loads represent an unstalled
condition, and there is virtually no load
variation with blade loading or torsional
natural frequency. At a blade loading of
0.15, the loads increase to between 200
pounds and 300 pounds, with the 3 per rev
and 4 per rev blades having the lowest
load. At this condition, the rotor power
is around 4000 horsepower which is well
beyond the available rotor power.

At a blade loading of 0.165, the pitch-
link load for the 3 per rev blade increases
sharply to 1000 pounds. The major portion
of this load is a 950 pound, 8-per-rev
component. Since the blade torsional
natural frequency is 3 per rev, it was
surprising to observe that there was little
3-per-rev load and a very large 8-per-rev
load. Further examination revealed that
the blade second torsional natural frequency
is almost exactly 8 per rev, explaining the
source of the large load. It is not known
why the torsionally soft blade prefers to
oscillate in its second mode. Further
investigation is necessary.

Cjh - BLADE LOADING

Figure 6. Variation of Pitch-Link Load Amplitude with Blade Loading in Hover
for One Degree of Cyclic Pitch.

118

The 4-per-rev blade, at the same
flight condition, has a pitch -link load of
340 pounds which is the lowest of the four
blades. The 5.2-per-rev and 7-per-rev
blades had approximately the same load at
about 400 pounds. The required rotor power
for all blades is approximately 5000 horse-
power which is 66 -percent more than the
available rotor power of a CH-47C rotor.
Since the required power is so high, res-
ults at this (0.165) and higher blade
loadings probably have no practical appli-
cation.

At a blade loading of 0.18, the 3 per
rev blade pitch-link load increases to
4500 pounds, with the 8-per-rev component
again providing the largest load. The 4-
per-rev blade also shows a large load in-
crease, reaching a load of about 4000
pounds. However, this blade's large tor-
sional loads occurred at the first torsional
natural frequency (3600 pounds at 4 per
rev). The 5.2-per-rev blade has a load of
650 pounds and the 7-per-rev blade load is
540 pounds. At this condition, the loads
reduce with increasing torsional frequency.
The required rotor power for this flight
condition is over twice the available power,
indicating that rotor stall has reached a
larger portion of the blade.

These results indicate that in hover,
increased torsional frequency (i.e., tor-
sional stiffness) delays the inception of
stall flutter. This conclusion generally
agrees with propeller experience. However,
the large power required at a blade loading
of 0.16 (i.e., 50 percent above available
power) implies that this flight condition
and higher blade loadings do not apply to
current aircraft. If current power to
rotor solidity ratios are therefore used,
there is very little difference between
the torsional loads of the four blades up
to reasonable blade loadings (for this
discussion, approximately 0.15 blade
loading) .

125 Knots

a large, high-frequency torsional load
component which generally appears between
an azimuth position of 270 degrees to 60
degrees and usually determines the load
amplitude as shown in Figure 9. The
stalled pitch-link load continues to
rise to 2650 pounds at a blade loading
of 0.11. Increasing the blade loading
beyond this point results in a load
reduction. This reversal of the load
trend may at first appear surprising, but
it has been observed in model data (see
Figure 1) and full-scale results (see
Figure 10) .

The 7-per-rev blade has generally the
same pitch-link load trend, with blade
loading as the basic blade. There is an
unstalled load region up to a blade loading
of 0.09 (with a typical waveform given in
Figure 8) , a stalled load region typified
by a large load increase with blade loading
(with a typical stalled waveform at a blade
loading of 0.10 as shown in Figure 8), and
a load reversal at a blade loading of 0.11.
However, as far as control loads are con-
cerned, the 7-per-rev blade is significantly
worse than the basic blade. In the un-
stalled region, the loads are about the
same; in stall the 7-per-rev blade loads
are 65-percent larger. Stall inception
occurs at a blade loading of about 0.095
which is 0.008 before the basic blade.

The 4-per-rev blade has a significantly
different pitch-link load trend with in-
creasing blade loading than the two blades

The variations of pitch-link loads
with blade loading for each of the four
different torsional frequency blades are
shown in Figure 7 for an airspeed of 1.25
knots. The basic blade (with a torsional
natural frequency of 5.2 per rev) pitch-
link load increases slowly with increasing
blade loading up to a value of 0.10. In
this region, the pitch-link load waveform
is predominantly 1 per rev (see Figure 8)
and the loads are classified as unstalled
(even though some stall is present) . Stall
inception occurs at a blade loading of
about 0.103. The stall inception repre-
sents the flight condition in which the
control loads begin to exhibit the rapid
increase, due to blade stall. In this
region, the pitch- link load waveform has

0.06 0.07 0.08 0.09 0.10 0.11 0.12

C T /o - BLADE LOADING

Figure 7. Variation of Pitch-Link Load Amplitude with Blade Loading at 125 Knots.

119

AIRSPEED 125 KNOTS
BLADE LOADING

TORSIONAL
FREQUENCY

Cf/o = 0.09

4/REV 1,

5.2/REV "

270 360

BLADE AZIMUTH - DEG

270 360

BLADE AZIMUTH - DEG

Figure 8. Pitch-Link Load Waveforms for 125 Knots, at Blade Loadings of
0.09 and 0.10.

AIRSPEED 125 KNOTS

TORSIONAL 3
FREQUENCY

C T /o =

.12

A /

3/REV

■W

r v

-2

-3

■

-4

-5

-6

■

-7

4/REV <

Cy/o».12

90 ISO 270 360

BLADE AZIMUTH - DEG

90 180 270

BLADE AZIMUTH - DEG

BORON
BLADE

6.B3/REV

GLASS BLADE
5.45/REV

Figure 9. Pitch-Link Load Waveforms for 125-Knots at High Blade Loadings.

0.07 0.08 0.09 0.10 0.11 0.12

NONDIMENSIONAL BLADE LOADING - C T /o
(IN WIND AXIS SYSTEM)

Figure 10. CH-47C Advanced-Geometry Blade Flight Test Data at an Advance
Ratio of 0.2.

previously discussed. While there is the
typical unstalled region with little load
growth up to a blade loading of 0.09 (a
typical unstalled waveform is given in
Figure 8), there is an irregular, but mod-
erate, load growth between 0.09 and 0.12.
At a blade loading of 0.12, the torsion load
is only 1641 pounds and the waveform just
attains a fully stalled characteristic (see
Figure 9) . There is a large load increase
(1360 in-lbs) as the blade loading increases
from 0.12 to 0.13. Examining the 0.13
pitch-link load waveform (see Figure 9)
shows that the large load is not caused by
stall flutter; instead, it is due to a
large nose -down moment generated by the
advancing blade combined with moderate
stall spikes.

The 3-per-rev blade has a load trend
similar to the 4-per-rev blade. The un-
stalled load region extends to a blade
loading of 0.115, and a typical unstalled
waveform is given in Figure 8. Even at a
blade loading of 0.12, the pitch-link load
waveform does not show a fully stalled
waveform (see Figure 9). However, at
0.125, the pitch-link load increases by
150 percent to 5273 pounds, by far the
largest load of any blade. The waveform
at 0.125 (see Figure 9) shows large stall
spikes with an amplitude of 2500 pounds;
however, the large load increase is due to

120

a 5000 pound load at 90° azimuth which re-
sults from a large nose -down pitching
moment. The load growth is so large at
this condition that it may represent the
lower boundary of an instability.

It is clear from these results that
stall inception occurs earlier as the
blade torsional frequency is increased.
Further, the maximum retreating blade
stall-induced pitch-link loads are larger
for blades with higher torsional frequen-
cies (i.e., the 7-per-rev blade has the
largest stall-induced loads) .

The CH-47C flight test data substan-
tiating the conclusion that stall inception
occurs at higher blade loadings as blade
frequency is decreased. Figure 10 shows
the results of a CH-47C advanced- geometry
blade flight test for aft rotor blades with
a boron filament spar and a fiberglass spar
at an advance ratio of 0.2. The glass
blade has a stall inception delay of 0.0085,
due to reducing the torsional frequency
from 6.53 per rev to 5.45 per rev. The
single-rotor study results show a stall
delay of 0.008 for reducing the torsional
frequency from 7 per rev to 5 . 2 per rev at
an advance ratio of 0.3.

The large advancing blade loads ex-
perienced by the 3-per-rev and 4-per-rev
blades, beyond a blade loading of 0.12,
are not due to the stall -flutter phenomenon
which results from retreating blade stall
and unstall cycles. This load is associated
with the negative lift on the advancing
blade tip and appears to be a divergence-
like phenomenon. Large, negative tip lift
causes the blade to bend tip down; the
high tip drag coupled with the flap de-
flection causes a nose -down moment. The
moment causes elastic nose -down pitch which
leads to more negative lift, resulting
eventually in even larger loads.

150 Knots

At 150 knots ( see Figure 11) , the
basic CH-47C blade has an unstalled load
region up to about 0.08 which is typified
by a slow increase of pitch-link load with
blade loading and a predominantly 1-per-
rev waveform. In the region between 0.08
and 0.1, a different load trend is observed.
The load increases gradually from 1500 to
2000 pounds, but at a faster rate than the
sub-stall load growth, and the waveforms
show significant evidence of stall spikes
for the retreating blade (see Figure 12) .
Stall inception (i.e., rapid load growth)
appears to occur around 0.10 3, reaching
maximum load near 0.11 (see Figure 12).
The load drops at a blade loading of 0.115,
showing a load reversal as observed at 125
knots .

POWER
LIMIT

0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

C T /o- BLADE LOADING

Figure 1 1 . Variation of Pitch-Link Load Amplitude with Blade Loading at
150 Knots.

TORSIONAL
FREQUENCY

AIRSPEED 150 KNOTS

90 180 270 360

BLADE AZIMUTH - DEG

90 180 270 360

BLADE AZIMUTH - DEG

Figure 12. Pitch-Link Load Waveforms at 150 Knots for Blade Loadings of
0.09 to 0.11.

121

The 7-per-rev blade shows a similar
load trend with blade loading as the basic
blade, but with significantly larger stall
loads. The unstall loads occur up to 0.07.
Stall inception occurs at approximately
0.075, reaching a fully stalled waveform
at 0.09 (see Figure 12). The loads level
out at 0.10, reach a second stall inception
near 0.105, and the load begins to grow
again. Examining the pitch- link load wave-
form at 0.11 (see Figure 12) shows that the
load increase is due to a large stall spike
occurring in the to 50-degree azimuth re-
gion, not to retreating blade stall spikes.

With an expected unstalled waveform
the 4-per-rev blade has a typical unstalled
control load growth up to a blade loading
of about 0.09. Between 0.09 and 0.115,
there is an irregular load growth. In this
region, the waveforms show evidence of
retreating blade stall (see Figure 12) , but
no large load increase. At a blade load-
ing of 0.11, the torsion load is 2100
pounds and the waveform just attains a
fully stalled characteristic (see Figure
12). There is a 1230-pound load increase
as the blade loading increases from 0.115
to 0.12. Examining the 0.12 pitch-link
load waveform (see Figure 13) shows that
the large load increase is caused by a
large, advancing-blade nose -down spike
combined with retreating blade stall spikes.

The 3-per-rev blade shows a reasonable
pitch-link load through a blade loading of
0.11. However, at 0.115, the blade is
apparently unstable since the loads have
grown so large that the blade would proba-
bly fail. The pitch-link load waveform
at 0.11 (see Figure 13) contains relatively

TORSIONAL
FREQUENCY

AIRSPEED 130 KNOTS
BLADE LOADING
■0.11 Ct/o-0.12

small retreating-blade, stall-induced
spikes. There is, however, a large com-
pression load for the advancing blade at
90 degrees blade azimuth. By examining
the pitch link load waveform for the un-
stable flight condition, it appears that
the biade divergence involves a large ad-
vancing blade compression load that con-
tinually increases with each rotor
revolution."

The 3-per-rev blade is experiencing
an additional problem which is not apparent
by simply observing the load trend. For
all the load conditions calculated at 150
knots, the required power exceeds the
available power. Apparently, the blade is
experiencing so much live twist that there
is a significant increase in rotor drag.
The other blades, by contrast, exceed the
available power only at a blade loading of
0.115. It is, therefore, obvious that the
3-per-rev blade is not an acceptable con-
figuration for the 150-knot flight condi-
tion.

175 Knots

At 175 knots (see Figure 14), the
basic blade pitch-link load trend shows
unstalled loads continuing to a blade
loading of 0.07 and stall inception occur-
ring about 0.075. The stalled load
increases with a moderate growth rate up
to 0.09. Figure 15 illustrates the pitch-
link load waveform at 0.09, showing the
retreating blade stall spikes and a large
nose-down load at 90 degrees azimuth. Be-
yond a blade loading of 0.09, the load
does not reverse as it does for previous
airspeeds (even though the retreating
blade stall spike is significantly reduced
at a blade loading of 0.11 as shown in
Figure 15). Instead, the load continues
to increase at about one half the previous
growth rate, due to an increasingly large
nose-down load at 90 degrees azimuth.

The 7 -per
trend is almost
blade trend up
shows , the wave
frequency stall
shows a slight
but then resume
typical stalled
growth at 0.11
large stall spi
azimuth and an
at 90 degrees a

rev blade pitch-link load
identical with the basic
to 0.09. As Figure 15
form exhibits typical high-
spikes. The torsion load
load reversal beyond 0.09,
s the load increase at the

load growth rate. The load
is due to a combination of a
ke at around 30 degrees
increasing nose -down load
zimuth (see Figure 15) .

90 180 270 360
BLADE AZIMUTH - DEG

90 180 270 360

BLADE AZIMUTH - DEG

Figure 13. Pitch-Link Load Waveforms at 180 Knots for the 3-per-rev and
' 4-per-rev Blades at High Blade Loadings.

The 4-per-rev blade has a typical
substall load growth up to 0.07 and gener-
ally follows the load trend of the 7-per-
rev blade and the basic blade up to 0.08.
Beyond this point, the load growth rate
drops significantly. At a blade loading of
0.09, the pitch link load is 500 pounds

122

TORSIONAL
FREQUENCY

AIRSPEED 175 KNOTS

BLADE LOADING

0,7s -0.09 Of/a -0.11

POWER
LIMIT

0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Cjh - BLADE LOADING

Figure 14. Variation of Pitch-Link Load Amplitude with Blade Loading at
1 75 Knots.

below the other two blades. Also 0.09,
the pitch-link load waveform shows little
evidence of retreating blade stall (see
Figure 15) , but does show that the major
component of the load results from a nose-
down moment at 90 degrees blade azimuth.
Beyond this point, the load growth rate in-
creases sharply from 2500 pounds at 0.09 to

4200 pounds at 0.11.

The 3-per-rev blade is not seriously
considered at this airspeed. The loads
are 1500 pounds beyond any of the other
blades, and an advancing blade instability
is apparent at a blade loading of 0.09.
Further, the required rotor power exceeds
the available CH-47C power for all 175-
knot flight conditions examined.

Examining the 175-knot pitch-link load
waveforms at a blade loading of 0.11
clearly shows that all four blades experi-
ence increased advancing-blade compression
loads when compared to the 15 -knot wave-
forms (compare Figures 12 and 14) . The
7-per-rev blade shows a 1'300-pound advan-
cing-blade load increase for the 25-knot
airspeed increase. The basic 5.2-per-rev
blade load increase is 2200 pounds, the
4-per-rev blade load increase is 3500
pounds, and the 3-per-rev blade has di-
verged. Therefore, the blades experience
advancing blade load problems which are
intensified as airspeed is increased and
blade torsional stiffness is reduced.

270
BLADE AZIMUTH - DEG

90 180 270

BLADE AZIMUTH - DEG

Figure 1 5. Pitch-Link Load Waveforms for 1 75 Knots at Blade Loadings of
0.09 and 0.11.

When comparing the 175-knot results
for the 4-per-rev, 5.2-per-rev and 7-per-
rev blades, it should be noted that the
rotor power limit for a single CH-47C rotor
is reached just beyond a blade loading of
0.09. For conditions below lthe power
limit, the 4-per-rev blade is slightly
better than the others, since the maximum
pitch-link load is 500 pounds lower. The
three blades appear to have adequate man-
euver margin, although the 4-per-rev blade
may experience larger maneuver loads.

These results show that a significant
reduction of the basic blade control loads
can be realized over a considerable range
of advance ratios and blade loading and
that these reductions lead to a significant
extension of the control load-limiting
aircraft flight envelope. These results
can be summarized by obtaining the flight
condition (as a function of Cj/a and p)
that first experiences a 2500-pound pitch-
link load. The 2500-pound load approximates
the original pitch- link endurance limit load
for the CH-47C control system. These flight
conditions lead to a blade loading versus
advance ratio envelope for the 2500-pound

123

pitch-link load or an endurance -limit
flight envelope. Figure 16 compares the
endurance limit flight envelopes for each
of the four different frequency blades
investigated.

As Figure 16_ shows , the blade with a
torsional natural frequency of 4 per rev
(dashed line) has the best flight envelope
and represents a significant improvement
: over the basic blade configuration. The
4-per-rev flight envelope has essentially
the same shape as the basic blade, but it
occurs at a higher blade loading. The
basic blade envelope occurs at a blade
loading of 0.016 below the 4-per-rev blade

at an advance ratio of 0.29 . At an.

advance ratio of 0.38, the basic and 4-per-
rev blades are approximately equal; but at
an advance ratio of 0.4, the 4-per-rev
blade envelope is expanded beyond the basic
blade by a blade loading of 0.005.

The 3-per-rev blade (short dashes)
shows a different flight envelope. At an
advance ratio below 0.29, the 3-per-rev
blade reaches the endurance limit at a

blade loading of' 0.123. However, the 3-
per-rev envelope drops sharply with
increasing advance ratio and eventually
falls below the three other blades at a
0.375 advance ratio. The sharp boundary
reduction of this blade at the higher ad-
vance ratios is due to the large advancing -
blade load growth which eventually becomes •
an instability.* These instabilities show
that the 3-per-rev blade is clearly un-
acceptable, at least for the current pitch-
link-controlled configuration.

The 7-per-rev blade clearly has the
poorest flight envelope up to an advance
ratio of about 0.37. At the higher advance
ratios above 0.35, the 7-per-rev blade has
the smallest reduction of blade -loading
capability with increasing advance ratio.
In this region, the 7-per-rev blade sur-
passes the 3-per-rev blade at an advance
ratio of 0.37, surpasses the basic blade
at 0.40, and will probably surpass the
4-per-rev blade around 0.44. Therefore,
a torsionally stiff blade may be required
to attain a reasonable flight envelope
beyond advance ratios of 0.44.

o.i3r

0.12 -

0.11 -

9 o.io
q

0.09 -

0.08 -

0.07

0.06 -

0.2

4.04/REV

3/REV-'*\

\ PITCH LINK LOAD

> ABOVE 2500 LB

5.165/REV

\ \

ON V

■^\«v \

•\l

~A^

\ X

9 "ft

6.97/REV

\ l\

\ *

PITCH LINK LOAD

BELOW 2500 LB

NATURAL FREQUENCY

GJ SCALING FACTOR

3.0

.25

4.0

.50

5.2

1.0

7.0

., „ ,_!_„, L_

3.3

i L_ i

0.3
ADVANCE RATIO - (i

0.4

Figure 16. Control Load Endurance Limit Boundaries for Blades with Torsional
Natural Frequencies of 3, 4, 5.2 and 7 Per Rev.

Conclusions and Recommendations

The results of the theory-test com-
parison performed for the 6- foot-diameter
model blades and the study of varying
torsional properties for the full-scale
CH47C size blades have lead to:

1. The theory-test comparison with the
6 -foot-diameter model data indicates that
the aerolastic rotor analysis reasonably
represents the large stall induced control
loads, the, control load change with blade
loading, and the load variation with
changes in blade torsional properties.
Therefore, the analytical study of the
CH47C size blades should provide at least
a qualitative evaluation of the control
load variation.

2. Changes in control system stiffness,
pitch inertia, and blade torsional stiff-
ness vary the large, stall-induced control
loads. However, the control load change
is not a simple function of torsional 3
natural frequency as previously suspected,
since torsional frequency changes, due to
varying the blade torsional stiffness,
produce control load changes larger than

* It may not be possible for an actual
rotor to experience blade divergence.
Before large divergence loads result,
there" is a significant increase in re-
quire^ rotor power. Therefore, a real
rotor may simply run out of power and be
unable to attain a flight condition for
which divergence would occur.

124

other methods of changing torsional fre-
quency.

3. A blade with a torsional natural
frequency of 4 per rev represents a com-
promise between significantly reducing
stall flutter' loads, while allowing
moderate increases in the advancing blade
loads at high speeds. This compromise 1.
provides the best endurance limit flight
envelope up to an advance ratio of 0.45.
Beyond this advance ratio it appears that

a torsionally stiff blade will provide a
better endurance limit flight envelope. 2.

4. Additional work is required in the
following areas.

• A model test program is needed to
validate the analytical results

over a wide range of flight con- 3.
ditions with remote collective and
cyclic pitch to insure trimmed
flight.

• Theory improvements are needed to
eliminate deficiencies discovered 4.
in the theory-test comparison.

• Continue analytical studies to in-
vestigate mechanisms of the load
generation, .maneuver and high-

speed load trends and other means
for expanding the endurance limit
flight envelope.

References

F. J. Tarzanin, Jr., PREDICTION OF
CONTROL LOADS DUE TO BLADE STALL,
27th Annual National V/STOL Forum of
the AHS, Preprint No. 513, May 1971.

F. J. Tarzanin, Jr. and R. Gabel,
BLADE TORSIONAL TUNING TO MANAGE
ROTOR STALL FLUTTER, Presented at the
A1AA 2nd Atmospheric Flight Mechanics
Conference, AIAA Paper No. 72-958,
September 1972.

F. 0. Carta, L. M. Casellini, P. J.
Arcidiacono, H. L. Elman, ANALYTICAL
STUDY OF HELICOPTER ROTOR STALL FLUTTER,
26th Annual Forum of the AHS, June 1970.
the AHS, June 1970.

F. J. Tarzanin, Jr. and J. Ranieri,
INVESTIGATION OF TORSIONAL NATURAL
FREQUENCY ON STALL- INDUCED DYNAMIC
LOADING. Performed under contract
DAAJ02-72-C-0092, USAAVLABS TR-
(Not yet released) .

125

APPLICATION TO ROTARY WINGS OF A SIMPLIFIED AERODYNAMIC LIFTING
SURFACE THEORY FOR UNSTEADY COMPRESSIBLE FLOW

B. M. Rao* and W. P. Jones**

Department of Aerospace Engineering

Texas ASM University, College Station, Texas

Abstract

In a recent paper, Jones and Moore have deve-
loped a simple numerical lifting surface technique
for calculating the aerodynamic coefficients on
oscillating wings in subsonic flight. The method
is based on the use of the full lifting surface
theory and is not restricted in any way as to fre-
quency, mode of oscillation or aspect ratio when
M < 1. In this study, this simple but general met-
hod of predicting airloads is applied to helicopter
rotor blades on a full three-dimensional basis.
/The general theory is developed for a rotor blade
at the <|j = tt/2 position where flutter is most lik-
ely to occur. Calculations of aerodynamic coeffi-
cients for use in flutter analysis are made for
forward and hovering flight with low inflow for
Mach numbers and 0.8 and frequency ratios p/H=l •
and 4. The results are compared with values given
by two-dimensional strip theory for a rigid rotor
hinged at its root. The comparisons indicate the
inadequacies of strip theory for airload predicti-
on. One important conclusion drawn from this stu-
dy is that the curved wake has a substantial effect
on the chordwise load distribution. The pitching
moment aerodynamic coefficients differ appreciably,
from the results given by strip theory.

Introduction

In a survey paper, Ref. 1, Jones' et al. give
a detailed account of significant developments in
the field of unsteady aerodynamics of helicopter
rotor blades. One of the problem areas surveyed
was that of blade flutter as it has been found that
under certain operating conditions, rotor blades
can flutter in both hovering and forward flight.
This phenomenon has been investigated by several
researchers in Refs. 2, 3, 4, and 5 and the results
of their studies have improved our understanding
of the problem. For the case of hovering flight,
J. P. Jones in Ref. 2 applied a method developed
by W. P. Jones in Ref. 6 to derive the approximate
aerodynamic coefficients for an oscillating single
rotor blade for use in his flutter analysis. He
approximated the actual flow conditions by neglect-
ing curvature effects and assuming a simple two-
dimensional mathematical model cosisting of a ref-
erence blade and an infinite number of wakes lying
beneath the reference blade extending from -°° to °°.
He considered flapping and pitching motions and com-
pared his results with those obtained experimenta-
lly by Daughaday and Kline in Ref. 3. On the basis

Presented at the AHS/NASA-Ames Specialists' Meet-
ing on Rotorcraft Dynamics, February 13-15, 1974.
The funds for computation were provided by the
U. S. Army Research Office, Durham.

* Associate Professor

** Distinguished Professor

127

of this work it was concluded that the wake is pri-
marily responsible for some of the vibratory pheno-
mena found on helicopters in practice. For low
inflow conditions, Loewy in Ref. 4 used a similar
mathematical model to that of J. P. Jones and inve-
stigated the variation in the pitching moment damp-
ing coefficient of a particular blade section as
p/fl varied for specified positions of axis of osci-
llation and a range of values of wake spacing. He
found that the damping coefficient became negative
whenever p/S was slightly greater than an integer
for axis of oscillation forward of quarter-chord.
Similarly he found that the damping coefficient
for a flapping oscillation dropped sharply at inte-
gral values of p/S2 but did not actually become neg-
ative. Tinman and Van de Vooren in Ref. 5, on the
other hand, assumed that there was no inflow thro-
ugh the rotor disk and developed a theory for cal-
culating the aerodynamic forces on a blade rotating
through its own wake. Their results agree with
those obtained in Refs. 2 and 4 in the limit when
zero spacing between the wakes is assumed. All
this theoretical work confirms the conclusion that
the proximity of the wake is a contributing factor
to rotor blade flutter.

All the theoretical work described above is
based on the assumption that the flow is incompre-
ssible. However, with the advent of helicopters
capable of flying with blade tip speeds ranging up
to and in excess of the speed of sound, compressi-
bility effects need to be taken into account when
determining coefficients for use in flutter analy-
sis. Jones and Rao in Ref . 7 were able to do this
on the basis of two-dimensional theory and have
derived coefficients for a range of Mach numbers,
reduced frequencies, and wake spacing. Their ana-
lysis is based on the use of Loewy's model, Ref. 4,
of the helical wake and the application of a theory
developed earlier by Jones in Ref. 8 for an oscilla-
ting airfoil in compressible flow. The values of
the coefficients given in Ref. 7 agree with those
obtained in Refs. 2 and 4 for zero Mach number but
differ appreciably when the Mach number is varied.
Hammond in Ref. 9 also developed a theory for det-
ermining compressibilty effects by using a differe-
nt model of flow from that used in Ref. 7. In his
model, the wake of the qth blade of a Q bladed
rotor after n revolutions extends from -2ir(n+q/Q)
to »; in Jones and Rao's model it extends from -°°
to o°. His aerodynamic coefficients for several
Mach numbers and inflow ratios are in general agr-
eement with "the results of Jones and Rao in Ref. 7.

While the aerodynamic derivatives predicted
by two-dimensional strip theory are widely used in
predicting the flutter speeds of helicopter rotor
blades, the method does not allow for curvature
and finite aspect ratio effects. For incompressi-
ble flow, Ashley, Moser, and Dugundji in Ref. 10
developed a three-dimensional model in which they

modified Reissner's theory, Ref. 11, for oscillat-
ing wings in rectilinear flow by including the free
stream- velocity variations along the span. Their
results indicate a negligible difference between
two and three-dimensional solutions up to 95% of
the span. Jones and Rao in Ref. 12 similarly stu-
died tip vortex effects in compressible flow and
they also concluded that such effects are negligi-
ble except in regions close to the tip. In some
of his earlier work, Miller in Refs. 13, 14, and
15, developed a helical wake model in which the
rotor wake was divided into a "near" wake and a
"far" wake. The near wake included the portion
attached to the blade that extend approximately
one-quarter of a revolution from the blade trailing
edge. The effects of the near wake include an in-
duced chordwise variation in downwash and were for-
mulated using an adoptation of Loewy's strip theory.
The chordwise variation in the velocity over the
airfoil induced by the far wake was neglected.
Miller extended his model to study the forward fli-
ght case and found that the nonuniform downwash
induced at the rotor disk by the wake vortex system
could account for the higher harmonic airloads en-
countered on rotor blades in forward flight. He
also showed that under certain conditions of low
inflow and low speed transition flight the return-
ing wake could be sucked up into the leading edge
of the rotor which would account for some of the
vibration and noise. Piziali in Ref. 16 has deve-
loped a"n alternative numerical method in which the
wake of a rotor blade is represented by discrete
straight line shed and trailing vortex elements.
He satisfied the chordwise boundary conditions, but
the rotor blade was limited to one degree of free-
dom in flapping. Sadler in Ref. 17, using a model
similar to Piziali 's, developed a method for predi-
cting the helicopter wake geometry at a "start up"
configuration. He represented the wake by a fine
mesh of transverse and trailing vortices starting
with the first movement of the rotor blade genera-
ting a bound vortex, and, to preserve zero total
vorticity, a corresponding shed vortex in the wake.
Integrating the mutual interference of the trailing
and shed vortices upon each other over small inter-
vals of time, Sadler was able to predict a wake
geometry. Although his model showed fair agreement
with the available experimental data for advance
ratios above one-tenth, Sadler's method is limited
due to the large computational time required tp
represent the wake by a finite mesh.

soning led them to represent the blade motion by a
series of oscillatory pulses, where each disturba-

TT TT

nee occurs over the range, -r- - Aij). <$<■=■+ A$„.

Corresponding to each burst of oscillation, packets
of vorticity are assumed to be shed into the wake.
With increasing forward speed, the spacing between
the packets of vorticity also increases and it was
found that the flutter speed became constant when
y, the advance ratio, was above 0.2. The approach
used in the present study differs from that adopted
by Shipman and Wood in that continuous high freque-
ncy small oscillations are assumed to be superimpo-
sed on the normal periodic motion of the blade.
The airloads and aerodynamic derivatives associated
with the perturbed oscillation of the rotor blade
can then be calculated by the method described in
this paper. Since the rotor blade will first att-
ain its critical speed for classical flutter at
i|i = it/ 2, the aerodynamic derivatives corresponding
to this value of ifi only have been calculated. The
method takes finite aspect ratio and subsonic com-
pressibility effects fully into account. Typical
results for a rotor blade hinged at its root desc-
ribing flapping and twisting oscillations are given
for a range of Mach numbers and frequency values.

Basic Equations

In the development of the analysis of the
Jones-Moore theory, Ref. 18, for oscillating wings
in rectilinear flight, the space variables x, y, z,
and t are replaced by X, Y, Z, and T, respectively,

s0 that „„ ** *z ^ n ,,.

x = iX, y = j-, z = — , t = — (1)

where % is a convenient reference length, U is

the uniform velocity, M is the Mach number and

2 1/2
8 = (1-M ) . The velocity potential of the flow

around a surface oscillating at a frequency p

can then be expressed as . . .

<j>(x,y,z,t) = U»(X,Y,Z)e li " + Wi; (2)

2 2
where co = p«7U, X = M co/B . The function * may be

regarded as a modified velocity potential. Fur-
thermore, it can be shown that it satisfies the
wave equation

2 2 2

a $ O. , O.

2 2 2

9X 8Y Z 3Z

+ K $ =

(3)

where k = Mu/g

Though many forms of flutter can occur on ro-
tor blades, attention in this report is concentra-
ted on the determination of appropriate aerodynamic
coefficients for use in the analysis of blade flu-
tter of the classical bending- torsion type. Ship-
man and Wood in Ref. 20 have considered this prob-
lem but they did not take compressibilty and fini-
te span effects into account. The two-dimensional
mathematical model used is similar to that employ-
ed by other authors except that they assumed that
flutter would first occur when the relative velo-
city over the rotor blade reaches its critical va-
lue when 4> = ir/2. For greater or lower values of
ij), the relative speed would be reduced below the
critical speed for flutter and any incipient grow-
ing flutter oscillation would be damped. This rea-

Since in this problem the motion of the sur-
face is assumed to be prescribed, the downwash
velocity at any point on it must be the same as
the downwash induced by the velocity (or doublet)
distribution over the surface and its wake. This
condition must be satisfied in order to ensure
tangential flow over the surface at all points.
It is also assumed that the rotor blade is a thin
surface oscillating about its equilibrium position

in the plane z = 0. If i; = £'e pt defines the
downwash displacement at any point (x,y) at time
t, this boundary condition requires that the down-
ward velocity and 8<f>/3z must be equal. In the
transformed coordinates, this implies that

128

e -i(XX + uT)

(4)

where w = -r~ + Dr* Is known.

at dX

A further condition that must be imposed on
any solution is that it leads to zero pressure dif-
ference across the wake created by the oscillating
surface. From the general equations of flow it can
be established that the local lift JE(x,y,t) at any
point is given by

j.(x,y,t) = p(||+uf|) (5)

where k » $^ - $ „ , the discontinuity in the veloci-

(5) , it immediately follows

•u 'tf

ty potential. From Eq.
that on the surface

S,(x,y,t) = pU (iyK + — ) e

(6)

where v = co/@ and K =
ivK + f =0

This yields

(7)

everywhere in the wake since the lift must then be
zero. From Eq. (7), it can be deduced that at any
point in the wake

K(X,Y) = K(X t ,Y) e" iv(X-X t )

(8)

where X

edge of the section at Y

X denotes the position of the trailing

As shown in Ref . 19, the solution of Eq. (3)
may then be derived from the integral equation

-iKC

4irW(X ,Y ,0) = // KCX.Yy^F-r-

p p z^o az K

-)dXdY

(9)

where W is the modified downwash at the point X ,Y

P P
given by Eq. (4), K has to take the form specified
by Eq. (8) at points in the wake and

5 = [(X-X ) 2 + (Y-Y ) 2 + Z 2 ] 1/2 .

The double integral in Eq. (9) must be taken
over the area of the oscillating surface and its
wake. It should be remembered, however, that K =
along the pleading edge and the sides of the area of
integration.

In the numerical technique developed in Ref.
18 for calculating the airloads on oscillating
wings in rectilinear flight, the wing is divided
into a number of conveniently shaped boxes and K
is assumed to be constant over each box. The wake,
on the other hand, is divided into a number of cho-
rdwise strips and K over each strip is defined by
Eq. (8). The contribution of the wake to the down-
wash W(X ,Y ) is then derived by direct numerical

P P
integration.

The application of the method outlined above
to determine the airloads on rotor blades presents
certain difficulties, the principal one being that
the flow velocity over the rotor blades is not con-
stant as assumed in the derivation of Eqs. (3) and
(8) for wings in straight flight. To overcome this
difficulty, it is assumed that the rotor blade can
be represented by a number of spanwise segments

over every one of which the flow is taken to have
its average value and appropriate Mach number. On
this basis the above analysis can be modified for
application to rotor blades as outlined in the
next section.

Rotor Blade Theory

In the present analysis, the rotor blade is
taken to be fixed at the ifi «■ ir/2 position and its
helical wake is assumed to extend rearwards as
indicated in Fig. 1. Normally, one would expect
the vorticity shed by the perturbed blade to be
carried downstream by the distorted wake of the
loaded rotor blade. However, in the present preli-
minary study, uniform inflow is assumed and any
distortion of the wake due to blade-tip vortex
interference is ignored. The aerodynamic coeffic-
ients corresponding to any prescribed motion can .
then be calculated for forward and hovering flight
by the method described below.

a) Forward Flight (Rotor Blade at $ = tt/2)

Let R denote the tip radius of the blade and
assume x = Rx' , y = Ry' , and z = Rz 1 . For forward
flight with velocity V, the relative local velocity
at section y will be denoted by U(=V+«Ry') and U'
(=p+y'), where fi is the angular rotation and u(=W
S2R) is the advance ratio. It then follows that at
the section y' , the downwash w(x',y') is given by

(10)

wCx'.y*) = flR(i|c' + II'IC) e ipt

P"P

pax 1

where 5 = R?'e ™ is the displacement of the blade
at the point, (x',y'). When the blade is describ-
ing flapping and twisting motions, 5' may be expr-
essed as

C' = Y'f(y') + x'a'F(y') (11)

where y' and <*' ar e the amplitudes at the reference
section and f (y') and F(y') are the modes of flap-
ping and twisting oscillations, respectively. If
the blade is assumed to be rigid and hinged at the
root, f(y') = y', and F(y') = 1 in the above equa-
tion. For convenience, the reference section is
taken to be at the tip but, in actual flutter cal-
culations, the section at 0.8R would be a better
choice.

To obtain the distribution of K corresponding
to the motion prescribed by Eq. (10), Eq. (9) is
first expressed in terms of the original variables

and K is replaced by Rk'e p . It may then be
written as

/ t 1 p X p ..,1 f ~2 -iic'r'
4irw'e F v _ ,,, F ,, , ... -iX'x'3 ,e

= //k'Gc'.y^e
^ P dx'dy '

,2^-r^

(12)

where k' = -&— , X' = Me' , w = w'e lpt , g 2 = 1-M 2 ,
6 fiH'

and r' = [(x'-x^) 2 + B^y'-y^) 2 + e 2 z' 2 ] 1/2 .

The above equation can be used to obtain the
solution to the problem of determining the flow
over a rotor blade with a rectilinear wake. Since

129

the wake can withstand no lift, the condition,

8k &k

t— + Ur~ = 0, must be satisfied. For a rectilinear

dt oX

wake, this yields

k'(x',y')

-i-

p(x'-x')

k[(y')e

S2U'

(13)

However, if the wake originating from a blade strip
is assumed to be curved

p(s'-s')

-X ■■ - ,

k'(s',y') -.- k^(y')e **' , (14)

where q' = (p 2 + y' 2 + 2viy'sin *) 1/2 , s = Rs' is
the distance along the vortex path and y' specifies
the spanwise location of the blade strip.

For computational purposes, Eq. (12) may be
conveniently expressed as

4ttW(x' y*) = //K(x',y') f dx'dy',
P P

(15)

where W(x' ,y') =

w(x',y') -iA'x'

P P

K(x'.y') = k'(x',y') e" U ' X '
„2 -iic'r' „ ■

' e r-> = -e 2 £

G =

and
iic'r*
— t— )[ (1+iic'r*)

2 2

(i_2L«l.) + k*Vz' 2 ].

r' Z
It should be noted that in the wake

v'(s'-s')

-* r-^-

K(x',y') - K t (y«) e q

where v' = -*-=■•

fig

(16)

b) Hovering Flight:

For the simplest case of hovering flight,
u = and s' = y'8. Hence Eqs. (10), (11), (12),
(13) , and (15) can be simply modified by replacing
•u with zero. Eqs. (14) and (16) then become

-i|(6-6.)
k''(6,y') - k^(y') e H •*

««<i - - -±v'(e-ej

K(6,y') - K^(y') e fc ,

(17)

(18)

where y* defines the location of the blade strip
from which the wake originates.

Method Of Solution

The schematic diagram of the oscillating rotor
blade is shown in Fig. 1. Eqs. (15) and (16) are
combined and expressed as

4irW(x^,yp

dx'dy'

-iic'r'

/ /K(x',y')(3( r
blade r

surface

, -Xl-HLie'r')

|J s

-i-

v'(s'-sp

/ / K t (y')e
wake

3< a

-IK'r;

..3

2 2
[ (l+i K 'r*)(l- ^_5l_) + K 'Vz' 2 ] ds'dn', (19)

where r' = [ (x'-x') 2 + B 2 (y'-y') 2 ] 1/2
s p p

r; - I (x'-x^) 2 + B 2 (y*-y p ) 2 + B 2 z' 2 ] 1/2

ds* - (dx ,2 +dy ,2 ) 1/2
and dn' is perpendicular to ds' and approximately
equal to dy' on blade. The rotor blade is divided
into a number of rectangular boxes (M x N) on which
the doublet strengths are assumed to be constant as
in Ref . 18. Based on this assumption, Eq. (19) can
be expressed as

4«W - I 2 S K . + S T K (20)
** i=l j=l id ij j=l j tJ
In Eq. (20) S.. and T. may be interpreted as the

aerodynamic influence coefficients and the actual
expressions are given later in this section. S. .

T. are the downwash velocities induced at the box

mn due to the unit strength doublets located^ at the
box ij and the wake strip j respectively. K. . and

K . are the doublet strengths at the box ij and the

trailing edge of the wake strip j , respectively.
With the use of the wake boundary condition, K^
can be expressed as

u! v'(x' -x' )
j A „ 1 A Hj. 3 (n)

./[e

+ 2i-

K tj - V

Eqs. (20) and (21), then yield

AirW mr , -2 S A K
i=l 1=1

where A

ij °ld

for i ^ M

v*(x;

t. rv

and Aj

S ij + T j /[e

D J

(22)

v'(x' -<,)

+ 21-

t rv

u j

for i - M. _

For a given mode shape (W 's are known), Eq. (22)

represents a system of M x N linear algebraic equa-
tions, the solution of which yields the values for

K 's. M and N denote the total number of chord-
mn

wise and spanwise stations respectively.

Once the appropriate K distribution has been
found, it is then relatively easy to determine the
aerodynamic forces per unit span acting on the rotor

blade. If, in Eq. (5), k - Rk'e ipt = RKe ipt e iX ' X ' ,

it then follows that the local lift L(y) - L'(y)

e p and the nose-up pitching moment, M(y) - M 1 (y)

e , referred to the mid-point of the chordwise

section at y are given by

2z'

¥£& - (L.+iL.H— £) + (L,+iL,)a'

p(U R ) 2 (c/2) 12c 34

130

■ x t

<rt>(ir><i£ / k'dx'+U'k')

(23)

M'(y)

2z!

p (0,^/2)

= CMj+lM 2 ) (-^-) + (M 3 +±M 4 )a*

X t X t

= ( a )( |R ) 2 [u , ( f k , dx , _ k . , _ ± £ / k ' x ' dx ']

R X I X 2

Ji * A (24)

where c is the local chord, D_ is a reference velo-
city, z' and a' are the local amplitudes of the
flapping and twisting motions respectively, and L 1 ,
L_, M_, M„, and L„, L,, M-, M,, are the in phase
and out of phase airload coefficients, respectively.

Expressions for Aerodynamic Influence Coefficients

Forward Flight (Rotor Blade at i[> = it/2)

The influence coefficients are calculated by
the method outlined in Ref , 18. For a box not con-
taining the collocation point

y'+d 2 x'+d -i K 'r

1^.1

S,, = - / / (- 5— )e(l+i K 'r')dx'dy' (25)

d y'f A 2 x i- d i n

where ic* = -f— , r' - I(x*-x') 2 + g 2 (y'-y') 2 ] 1/2 ,
(S 2 JHJ! s m n

d = Ax'/2, d„ = Ay '/2, and Ax' and Ay' are the

chordwise and spanwise spacings of the rectangular
grid on the surface of the rotor blade. When the
collocation is inside the box considered, the value
of-S. must be calculated by the method of Ref. 18.

For the curved wake

n^ 2
T, = - / / e
' ^ d 2 S t.

v'(s'-s') . . , ,

. t -iK'r'

-i 1 _ w

q B(- r - )[(l+iK'r;)

(26)

2 2
(1- 3g *' ■ ) + k ,2 B 2 z' 2 ] ds'dn*

r ,Z

where r' = [(x*-x') 2 + f3 2 (y'-y') 2 + 6 2 z' 2 ] 1/2 ,
w m n

x' = u9 + y' sin 9, y' = y! cos 9, z' = d'9/2-ir,

ds' = (dx' 2 +dy' 2 ) 1/2 = (u 2 +y' 2 +2uy' cos 9) 1/2 d9,

and d(=Rd'), the downward displacement of the wake
per revolution, is assumed to be small. T. 's are

evaluated numerically at the j ' th spanwise strip by
taking small increments of 9 and n.

Hovering Flight (Low Inflows)

For hovering flight, u = 0, s' = y'9, and

ds' - y!d9. The expression for S .. , Eq. (25), can

be simply modified by replacing u with zero and the
wake integration for the j ' th strip

n j«2- -iV(e-v - lK ' r w

/ / e C 6 (- r - )[(l+i K 'r')

n'-d, .9,. r' 3 w

j 2 t w

2 2
(1- 3g *' ) + k'Vz' 2 ] y! d9dn'

r' ••

(27)

where x' = y' sin 9, y' = y' cos 9, and z 1 - d'9/2ir.

The effect of the helical wake in hovering
flight is estimated by two different methods. In
the first method, a Helical Wake Model is used and
the actual helical path is taken in evaluating the
T. coefficients. In the second method, a Circular

Wake Model is employed and the helical wake is re-
placed by its near wake, which is assumed to extend
over 9 £ 8 £ ir/2, and a number of regularly spaced

circular disks of vorticity below the reference
plane. The formula for k' for the n'th disk at
z' = nd' is taken to be simply

-i£[(6-9.)+2nir]
k'(6,y',nd') = k£<y')e B ,. (28>

the actual spacing between consecutive disks being
Rd\

Results and Discussion

A rectangular rotor blade of R/c = 10 was cho-
sen and the. blade Was assumed to extend from 0.1R
to R. For the computation of the airload coefficie-
nts, a grid of thirty six rectangular boxes consist-
ing of six chordwise and six spanwise stations were
used. The convergence of the results was tested by
taking grid sizes of 6x8 (chordwise x spanwise) and
8x6. Rigid mode shapes for flapping and twisting
oscillations are assumed so that

? = yy' + ax'
w'(x'.y') = £5R[a*(u+y,I+i£0 + Y'if(v+y^)]

P"P

p S p

S v

2z!

^p^x-ir^

(29)

It should be noted that the above equation is valid
only for the blade position at $ = it/2. For hover-
ing flight, u = in Eq. (29).

The airload coefficients for Mach numbers
0.8 for several values of *- and wake spacing of two

chords were obtained with reference to the blade's
quater-chord axis. In Figs. (2) thru (5), selected
airload coefficients for slow forward flight (u =
0.1) are compared with the results obtained for
hovering flight using both a helical wake model and
two-dimensional strip theory. For this particular
comparison, the reference velocity in Eqs. (23) and
(24) was taken as the relative local velocity, U,
and the tip Mach number was 0.8. From these plots,
one can conclude that strip theory predicts substa-
ntially larger values for the airload coefficients.
One of the most important observations one can make
is that the curved wake changes the chordwise load
distribution in such a way that the center of pre-
ssure shifts forward of the quarter-chord axis
position (see Fig. 4).

131

Figs. (6) and (7) compare the results by seve-
ral mathematical models used for the hovering fli-
ght case. The airload coefficients are referred
to the tip velocity (fiR) and this choice was made
to indicate the trends of spanwise load distribut-
ion. The Circular Wake model representation resu-
lts in a substantial saving in computational time.
For example, to obtain the airload coefficients for
one set of geometric and flight conditions using
6x6 grid on the blade, the Circular Wake model took
only 1.5 minutes of computing time on IBM 360/65
while the Helical Wake model took 2.5 minutes.
Although the Circular Wake model seems to indicate
the general trends of the airload coefficients, one
should use the full helical wake to compute the
airload coefficients accurately.

Some typical results for hovering flight using
the Circular Wake representation, compared with the
results of two-dimensional strip theory, Ref. 7,
are shown in Figs. (8) thru (11). The results for
the curved wake are in good agreement with the re-
sults for strip theory for the inner blade sections;
however, the agreement is poor towards the tip.
Figs. (12) and (13) show the variation with axis
position of M, , conveniently referred to as pitching
moment damping airload coefficient, at spanwise
stations of 0.475R and 0.925R, respectively. From
these results, one can conclude that the agreement
between the curved wake results and strip theory
is good near the quarter-chord position for M =
and M = 0.8 but it becomes very poor as the axis
is moved towards the trailing edge.

References

6.
7.

8.
9.

Jones, W. P., McCrosky, W. J., and Costes, J.
J. , "Unsteady Aerodynamics of Helicopter Rotor
Blades," NATO AGARD Report No. 595, April 1972.
Jones , J . P . , "The Influence of the Wake on
the Flutter and Vibration of Rotor Blades,"
Aeronautical Quarterly , Vol. IX, August 1958.
Daughaday, H. and Kline, J., "An Investigation
of the Effect of Virtual Delta-Three Angle and
Blade Flexibility on Rotor Blade Flutter,"
Cornell Aeronautical Laboratory Report, SB-86
2-5-2, August 1954.

Loewy, R. G., "A Two-Dimensional Approximation
to the Unsteady Aerodynamics of Rotary Wings,"
Journal of the Aeronautical Sciences , Vol. 24,
No. 2, February 1957, pp. 81-92.
Timman, R. and Van de Vooren, A. I. , "Flutter
of a Helicopter Rotor Ratating in its Own
Wake , " Journal of the Aeronautical Sciences ,
Vol. 24, No. 9, September 1957, pp. 694-702.
Jones, W. P., "Aerodynamic Forces on Wings in
Non-Uniform Motion," R&M No. 2117, 1945, Bri-
tish Aeronautical Research Council.
Jones, W. P. and Rao, B. M. , "Compressibility
Effects on Oscillating Rotor BladeB in Hover-
ing Flight," AIAA Journal , Vol. 8, No. 2,
February 1970, pp. 321-329.
Jones, W. P., "The Oscillating Airfoil in
Subsonic Flow," R&M No. 2921, 1956, British
Aeronautical Research Council.
Hammond, C. E., "Compressibility Effects in
Helicopter Rotor Blade Flutter," GITAER Report
69-4, December 1969, Georgia Institute of

Technology, School of Aerospace Engineering.

10. Ashley, H. , Moser, H. H. , and Dugundji, J.,
"Investigation of Rotor Response to Vibratory
Aerodynamic Inputs, Part III, Three-Dimensional
Effects on Unsteady Flow Through a Helicopter
Rotor," WADC TR 58-87, October 1958, AD203392,
U. S. Air Force Air Research and Development
Command.

11. Reissner, E. , "Effects of Finite Span on the
Airload Distributions for Oscillating Wings,
Part I - Aerodynamic Theory of Oscillating
Wings of Finite Span," NACA Technical Note
No. 1194, 1947.

12. Jones, W. P. and Rao, B. M. , "Tip Vortex Effe-
cts on Oscillating Rotor Blades in Hovering
Flight," AIAA Journal , Vol. 9, No. 1, January
1971, pp. 106-113.

13. Miller, R. H. , "On the Computation of Airloads
Acting on Rotor Blades in Forward Flight,"
Journal of the American Helicopter Society ,
Vol. 7, No. 2, April 1962, pp. 55-66.

14. Miller, R. H. , "Unsteady Airloads on Helicop-
ter Rotor Blades," Journal of the Royal Aero-
nautical Society , Vol. 86, No. 640, April 1964,
pp. 217-229.

15. Miller, R. H. , "Rotor Blade Harmonic Air
Loading," AIAA Journal , Vol. 2, No. 7, July
1964, pp. 1254-1269.

16. Piziali, R. A., "A Method for Predicting the
Aerodynamic Loads and Dynamic Response of
Rotor Blades," USAAV-LABS Technical Report
65-74, January 1966, AD 628583.

17. Sadler, S. G. , "A Method for Predicting Heli-
copter Wake Geometry, Wake Induced Flow and
Wake Effects on Blade Airloads," presented at
the 27th Annual National V/ST0L Forum of the
American Helicopter Society, Washington,

D. C, May 1972.

18. Jones, W. P. and Moore, J. A., "Simplified
Aerodynamic Theory of Oscillating Thin Surfa-
ces in Subsonic Flow," AIAA Journal , Vol. 11,
No. 9, September 1973, pp. 1305-1309.

19. Jones, W. P., "Oscillating Wings in Compressi-
ble Subsonic Flow," R&M No. 2885, October 19
55, British Aeronautical Research Council.

20. Shipman, K. W. and Wood, E. R. , "A Two-Dimen-
sional Theory for Rotor Blade in Forward Fli-
ght," Journal of Aircraft , Vol. 8, No. 12,
December 1971, pp. 1008-1015.

132

K&-

tai.y'i) — |

\J»

, /

III

/ jth wake/
\/ strip /

S 1

*■ y ,n,l

♦

x',m,i

Pig. 1 Schematic Diagram of Rotor Blade and
its Wake.

4.5

4.0

3.5

3.0

2.5

2.0

FORWARD FLIGHT (p=0.l)

HOVERING FLIGHT

STRIP THEORY _

0.10

M'OB.jp I, Ur'U

FORWARD FLIGHT (^-0.1)

HOVERING FLIGHT

STRIP THEORY

Fig. 3 Spanwise Variation of L »

2.0
1.5

-^ y^-' \

/ M = 0.8, -jj =1 , U R =U ' (

1.0

0.5

/ FORWARD FLIGHT (^ = 0.1) |

/ -— — HOVERING FLIGHT 1
STRIP THEORY j

—

-~UaJ_. 1 1 1

0.10

1.0

0.8

0.25 0.40 0.55 0.70 0.85 1.00

R
Fig. 4 Spanwise Variation of M,.

0.6

-M 4

Fig. 2 Spanwise Variation of Lj.

0.4

0.2

\\ M-0.8, "f.UFcU

-\ FORWARD FLIGHT (^

\\ HOVERING FLIGHT

■0.1)

\\ STRIP THEORY

\ ^

\j*«O.I J

1 1

133

0.1 0.25 0.40 0.55 0.70 0.85 I.C

1
R

Fig. 5 Spanwise Variation of M. .

0.25

0.20

M = 0.8, ■£ = !, U„=flR

HELICAL WAKE

CIRCULAR WAKE y

STRIP THEORY y

\
I
1

1
t
1
t

0.1 0.25 0.40 0.55 0.70 0.85 1.00

Fig. 6 Spanwise Variation of L. for
Hovering Flight.

0.7

U H =flR

-CIRCULAR WAKE MODEL /
-STRIP THEORY / ,

/ /

Fig. 8 Spanwise Variation of L 2 for
Hovering Flight.

0.25

M=0£, jy»l, U R = flR

HELICAL WAKE /

CIRCULAR WAKE •

STRIP THEORY /

0.20

M. 0.15
4

— / S"

—

/ ,'

\
\

/ /

0.10

_ S /V~

■' >' /

0.05

S — -'"L-/

0.1 0.25 0.40 0.55 0.70 0.85 1.0

"R~

Fig. 7 Spanwise Variation of M, for
Hovering Flight.

3.5

3.0

CIRCULAR WAKE MODEL /

STRIP THEORY /

01 025 0.40 0.55 0.70 0.85 1.0

"R"

Fig. 9 Spanwise Variation of L, for
Hovering Flight.

134

I 20

0.8

Ur'OR

— CIRCULAR WAKE MODEL
STRIP THEORY

Fig. 10 Spanwise Variation of M 3 for
Hovering Flight.

|-=0.475, U„=GR

CIRCULAR WAKE MODEL

-— STRIP THEORY

-M„

Fig. 12 Variation of M^ With Reference Axis
Position for Hovering Flight.

-M,

Fig. 11 Spanwise Variation of M, for
Hovering Flight.

2.5

2.0

-M„

-i =0.925, U R =»R

CIRCULAR WAKE MODEL

STRIP THEORY /

/
/
/
/
/
/

Fig. 13 Variation of M. With Reference Axis
Position for Hovering Flight.

135

ROTOR AEROELASTIC STABILITY
COUPLED WITH HELICOPTER BODY MOTION

Wen-Liu Miao

Boeing Vertol Company

Philadelphia, Pennsylvania

Helmut B. Huber

Messersehmitt-Boelkow-Blohm Gmbh

Ottobrun-Munieh

Federal Republic of Germany

Abstract

A 5. 5-foot-diameter, soft- in-plane, hingeless-

rdt-or system was tested on a gimbal which allowed the
helicopter rigid-body pitch and roll motions. With this
model, coupled rotor /airframe aeroelastic stability
boundaries were explored and the modal damping ratios
were measured. The time histories were correlated
with analysis with excellent agreement.

The effects of forward speed and some rotor de-
sign parameters on the coupled rotor/airframe stability
were explored both by model and analysis. Some phys-
ical insights into the coupled stability phenomenon were
suggested.

Introduction

The coupled rotor-airframe aeroelastic stability
phenomenon of air resonance has received considerable
attention in recent years. A scaled model of the BO-105
helicopter was built and tested to explore this phenom-
enon and its sensitivity to design parameters. 1 An ex-
tensive analytical study was performed and correlated
with BO-105 flight test data. 2

To further explore this coupled stability phenom-
enon, a large scale model having different resonance
characteristics than the BO-105 was built and tested.
Parameters that were influential to the stability 1 ' 2
were incorporated into the model and their effects were
examined. An improved test technique enabled the de-
termination of modal damping ratio at every test point,
providing better data for correlation and better assess-
ment of stability.

Description of Model

The model, shown in Figure 1, consisted of a
Froude-scaled model rotor mounted on a rigid fuselage,
■which in turn was mounted on a two-axis gimbal having
* 10 degrees travel in pitch and roll. The model had a
5. 5-foot-diameter, soft-in-plane, hingeless rotor with
pertinent hub parameters such as precone, sweep, and
control system stiffness being variables to enable in-
vestigation of their effects on coupled rotor-airframe
stability.

A proportional (closed-loop) control system
equipped with a cyclic stick provided lateral and longi-
tudinal control to fly the model in the pitch and roll de-
grees of freedom. In addition, a shaker system was
installed in the longitudinal and lateral cyclic system

Presented at the AHS/NASA-Ames Specialists Meeting
on Rotorcraft Dynamics, February 13-15, 1974.

Figure 1. Dynamic Model Helicopter With 5.5-Foot-
Diameter Single Rotor

(see Figure 2) to allow excitation of the model at the
desired frequency. This enabled the measurement of
the modal damping ratios at each test point. The meas-
ured modal damping permitted the precise determina-
tion of the stability boundaries and also showed the
sxtent of stability when the model was stable.

Figure 2. Details of Model Rotor Hub and
Swashplate

137

The stability and control augmentation system was
based on position feedback. Position potentiometers on :
the helicopter gimbal axes provided position feedback
signals which were amplified, filtered, and fed into the
cyclic actuators for automatic stabilization of the model.
The filter was designed to block any feedback at a fre-
quency of ft-u)£ and thus eliminated any control inputs
that would tend to interact with the air-resonance mode.

Collective pitch was set by means of an open-loop
control and a pitch-angle indicator. Other controls pro-
vided for the operator included mounting-pylon pitch
attitude, stick trim, and quick-acting and slow-acting,
self-centering snubbers to lock out the pitch and roll
degrees of freedom. The horizontal stabilizer was
manually trimmable and rotor speed was controlled by
the wind tunnel operator.

Signals from the blade flap, torsion, and chord
strain gages, along with body pitch and roll motion,
cyclic stick position, and l/rev, were recorded on os-
cillograph as well as on multiplex tape recorder. One
of the chord-bending traces was filtered to display the
chord bending at the critical lag natural frequency to
allow quick determination of modal damping on line.
Most of the testing was performed in the wind tunnel at
the University of Maryland.

Test Technique

As discussed in References 1 and 2, the air-
resonance mode stability is determined by the blade
collective pitch as well as the rotor speed. Therefore,
for every airspeed, a comprehensive variation of rpm
and collective pitch was conducted.

SET UP
TEST CONDITION

RECORD
REASONABLE
DIVERGENCE
OF MODEL
MOTION

SET UP NEW
CONDITION

RECORD

CONVERGENCE
OF MODEL
MOTIONS

STABILIZE

MODEL BV: DROP RPM,

DROP COLLECTIVE, OR SNUBBING

ANALYZE DATA
OBTAIN MODAL
DAMPING RATIO

"°*

COLLECTIVE »
PITCH I

Figure 3 shows the test flow of events for each
data point taken. After the test conditions had been set
up (rpm, tunnel speed, and collective pitch), the model
was trimmed and was held at the trim attitude with the
stability and control augmentation system (SCAS). The
shaker and the tracking filter frequ< ■ were set to
0-w e and oj ? respectively, with the absolute magni
tudes dependent on the rotor speed. Both the multiplex
tape recorder and the CEC recorders were turned on to/
record the steady-state response of the model. The
swashplate was then oscillated in the lateral control
direction. After the termination of the excitation, re-
cording was continued until steady- state conditions weje
again reached, when practical. The decay of the filte/ed,
in-plane, bending-moment trace was reduced to obtai$
the modal damping ratio.

SYMBOL CONDITION

STABLE

MARGINAL

UNSTABLE

Eh
W

200

160

o
o

ooo

120

§!

»o

o
o

odd
od

ooo

o»

ooo

ood

ooo

"ooe ig

ooo

o
o

ooo

- o

-40

o
o

o
_1

,„l

60 80
ROTOR SPEED

100

120

140

NORMAL ROTOR SPEED

PERCENT

Figure 3. Plow Diagram of Test Technique

Figure 4. Typical Map of Test Points in Hover

Test Results

Figure 4 shows a typical map of test points taken
at a constant tunnel speed, in this case in hover. Two

138

stability boundaries were present: one at about 70 per-
cent of normal rotor speed and 120 percent of normal
collective pitch and another at about 135 percent rpm
and 100 percent collective. Examination of the coupled
frequency variation with rotor speed while holding con-
stant thrust, Figure 5, reveals that the low-rpm bound-
ary corresponds to the resonance with the body-pitch-
predominant mode and the high-rpm boundary with the
body-roll-predominant mode.

1.0

jgg

gs^

0M1NANT ,

R
M

M>E _,

— \/PI
' — <fM0

TCH-PREDOMINANT
DE

80 90
ROTOR SPEED

NORMAL ROTOR SPEED

100 110
- PERCENT

Figure 5. Coupled Resonance Characteristics

Figures 6, 7, and 8 show the time histories of
three hover air-resonance points which are at constant
collective pitch of 133 percent 6NOR 0-& hover collec-
tive at Njjor) w i*h rotor speeds of 100 percent, 72
percent, and 67 percent NNOR respectively. At NNOR
the chord bending decayed after the excitation termi-
nated, at a rate of approximately 1 percent of critical
damping, and the body participation was barely detect-
able. Approaching the stability boundary at 72 percent
rpm, the chord bending took longer to decay compared
to the 100 percent rpm case. Body participation was
quite pronounced in both pitch and roll. While the fil-
tered chord-bending gage in the rotating system was
indicating at the blade lag natural frequency, ca^ , the
body pitch and roll motions responding in the same air-
resonance mode were at the fixed-system frequency of
8-id. . It is of interest that these Q-us^ body motions
are superimposed on some very low-frequency, flying-
quality-type motions.

At 67 percent rpm, Figure 8, the air-resonance
mode started to diverge after being excited; when the
body was snubbed, the blade motion decayed and re-
turned to the l/rev forced response.

The response characteristics described here held
true for all airspeeds tested up to a scaled test speed
limit of 225 knots.

HOVER, 100% N Nnn , 133% 6 Mr)p , RON NO. 8

"NOR

NOR'

BLADE CHORD

/V\/1%\/\a/W

\W\AWVWvW\aa

BLADE CHORD FILTERED

-MA/VW\/W\/WWWAV.v u ^,^^

"\y\y\jr^^

-H u c|«-

BODY PITCH

BODY ROLL

LATERAL EXCITATION

1/REV

I I I I I I I I I 1 I I I I I I I I > I M I I I I I 1 I I I I I I II 1 I I i I 1 I I I I 1 I I I I I I 1 I I 1 I 1

Figure 6. Response Time Histories in Hover at %qr

139

BLADE CHORD

HOVER, 72% N N0 R, 133% e NOR, - RUN N0 - 13

1/REV
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIMIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIMIIIIIIIIIIIIIIIMIII

Figure 7. Response Time Histories in Hover at 72
Percent N N0R

Analytical Model

To treat the dynamically and aerodynamically
coupled rotor-airframe air-resonance problem, the
analytical model shown in Figure 9 is used. In this
model, the elastic cantilevered blade is represented by
a spring-restrained, hinged rigid blade. Three hinges
are used to simulate the first flap, first lag, and first
torsion modes, in that order from inboard to outboard.
In addition, a pitch degree of freedom is provided in-
board of the flap hinge to facilitate the simulation of any
torsional stiffness distribution relative to the flap and
lag hinges. The blade model includes built-in pitch axis
precone, blade sweep, kinematic pitch-flap and pitch-
lag coupling, and a variable chordwise center-of-gravity
distribution over the blade span.

The airframe has five rigid-body freedoms:
longitudinal, lateral, vertical, pitch, and roll; and two
flexible freedoms: pylon pitch and pylon roll. The
equations of motion are nonlinear and are solved by a
numerical time-history solution technique. The blade
: degrees of freedom are calculated for each individual
blade.

To evaluate the aeroelastic stability, the aircraft
can be perturbed from the trimmed state. For air-
resonance investigations this is usually done by oscil-
latory stick excitations, which Can be simulated in any
frequency. The time history of each degree of freedom
is then subjected to an oscillation analysis program to
obtain the frequencies, amplitudes, phases, and
damping coefficients. A more detailed discussion of
this analytical procedure can be found in Reference 2.

The aerodynamic model is based on current blade-
element theory and can handle all hover, fo I .light,
and maneuver flight conditions. It uses two-dimensional
airfoil data with stall, reverse-flow, and compressibil-
ity effects.

Using a linear lift-curve slope, this coupled
analysis in hover can be reduced to a set of second-
order differential equations with constant coefficients
by applying the quasi-normal coordinate transformation

140

HOVER, 67% N N0R# 133% 8j, 0Rf RUN NO. 12

BLADE CHORD

1/REV

iiiiuiiiiiiiii iii iii i i ii i iiii i i iii i i i iiiiii ii i ii ii iiii i i iii iiii i i i ii ii i ii ii i iii i ii ii i i i iiiii i ii i iii i i i i i iiii i iiiii i i i iii i iiiniiiii ii iiiuiiiii

Figure 8. Response Time Histories in Hover at 67
Percent Njjqr

for the rotating eoordinates3. This enables the closed-
form solution. The eigenvalues and eigenvectors thus
obtained yield the information on frequencies, damping,
and mode shapes.

PITCH FLAP LAG TORSION

«. \ \ c, ^

'LONGITUDINAL VERTICAL

PITCH

LATERAL

Figure 9. Coupled Rotor-Fuselage Analytical Model

Correlation

Rotor Thrust

Figure 10 shows the air-resonance mode modal
damping ratio variations with thrust at NnoR in hover.
The agreement between test and analysis is quite good.
The propitious trend with increasing collective pitch is
due partly to the increase of aerodynamic damping, but
is mainly a result of the favorable piteh-flap-lag
coupling. A typical blade elastic coupling is shown in
Figure 18 where the blade flap, lag, and pitch torsion
responses to a cyclic-pitch input are indicated. The
type of elastic coupling of this rotor system is discussed
in more detail in a subsequent section.

Rotor Speed

Shown in Figure 11 are the test correlations of
the air-resonance mode damping variation with rotor
speed at constant collective pitch (133 percent SjJOR)
in hover. The analytical results are in good agreement
with test points over the whole rotor speed range. The
stability boundary corresponding to the resonance with
the body-pitch-predominant mode at low rpm is pre-
dicted well by theory. The somewhat higher level of
damping of the test points might indicate that the struc-
tural damping of the real model blade is higher than the
0. 5 percent damping assumed in the analysis.

141

6.0

HOVER

COLLECTIVE PITCH = 133% 9

ROTOR SPEED

= 72% N,

EST]

NOR
[ NOR

LATERAL EXCITATION

BLADE CHORDWISE MOMENT

50 100

COLLECTIVE PITCH
HOVER lg COLLECTIVE PITCH

150 200
~ PERCENT

Figure 10. Effect of Thrust on Air-Resonance
Stability

HOVER

COLLECTIVE PITCH = 133% 6 N0R
IN-PLANE DAMPING . = 0.5% CRITICAL
CONTROL STIFFNESS = 642 IN. -LB/RAD

O H
SSfrl
H H

Q H

a «

S Ai

-1

a
<

-1
m

A TEST
— ANALS

POINTS
SIS

50 60 70 80

ROTOR SPEED
NORMAL ROTOR SPEED

PERCENT

100

110

Figure 11. Effect of Rotor Speed on Air-Resonance
Stability :

| A N A L Y S I S|

LATERAL EXCITATION

[ -W\A

BLADE IN-PLANE MOTION

Hhr.

BODY ROLL

AAA/W

n-a) r

•TIME

ONE-PER-REV MARK

Figure 12. Correlation of Test and Analysis of
Time Histories in Hover

The good agreement of Figure 11 is merely a
reflection of the excellent correlation between test and
analysis in the time-history waveform of blade and body
motions. One example is shown in Figure 12. For this
case the oscillation analysis program yields a damping
coefficient of 0. 39 percent at blade lag natural frequency
for the rotating blade.

Forward Flight

The test trend of air-resonance mode damping
with airspeed is also verified by analysis in Figure 13.
Test points shown in this diagram were obtained with
constant collective pitch, so that they do not correspond
to a lg-thrustAevel-flight condition. The analysis was

142

performed under the same colleetive/shaft-angle set-
tings to get an exact simulation of the test conditions.
At 150 knots, the collective pitch is slightly reduced,
from 133 percent to 111 percent, which produces a
sharp decrease in rotor thrust. Therefore the air-
resonance mode is less stable than for a normal lg-
thrust condition.

COLLECTIVE PITCH = 133% 6 N0R (111% 9 N0R )
ROTOR SPEED = 100% % 0R
IN-PLANE DAMPING =0.5 PERCENT CRITICAL
CONTROL STIFFNESS = 642 IN. -LB/RAD
31 1 1 1 <—r

T E S T

80 KNOTS FORWARD FLIGHT

COLLECTIVE PITCH = 133% 6 N0 R
ROTOR SPEED =100% NjjOR
SHAFT TILT ANGLE = -4 DEGREES

[

LATERAL EXCITATION

^AAAAAA^ —

BLADE CHORDWISE MOMENT

NOR

BODY ROLL

80 120 160

VELOCITY - KNOTS

Figure 13. Effect of Forward Speed on Air-
Resonance Stability at Constant
Collective Pitch

Theory' shows some influence of cyclic control on
air-resonance stability at high speed. As longitudinal
cyclic also controls rotor thrust in forward flight, this
variation of air-resonance stability comes solely from
the change in rotor thrust. With thrust held constant the
stability is insensitive to steady 1/rev cyclic-pitch vari-
ation. This is shown in a later section.

In Figure 14 one example of a typical time history
at a scaled airspeed of 80 knots is compared between
test and analysis. When one considers the complex fre-
quency modulations during this excited air-resonance
case, the correlation can be said to be excellent. This
should indicate that theory allows a definitive and reli-
able view of a helicopter's stability characteristics.

Additional test results of air-resonance stability
in forward flight are illustrated in Figure 15. This
trend, which was obtained for a lg/level-flight condition,
follows the rotor power curve quite well. As shown in
Figure 10, for a moderate range of thrust variation, say
around lg, the air-resonance mode becomes more
stable with increasing thrust and less stable with de-
creasing thrust. The forward-speed trend here simply
reflects this thrust (and aerodynamic coning angle) de-
pendency. This trend, which shows that the air-
resonance mode stability improves significantly at high
forward speeds, is also apparent in the BO-105 flight
test data2.

ANALYSIS

LATERAL EXCITATION

FiAAAA/W^-

BLADE IN-PLANE MOTION

ONE -PER-REV MARK

'iiifiiiiiiiiiiiiiiiiiiiiiiiii

TIME

llllllllllllllll

Figure 14. Correlation of Test and Analysis of
Time Histories in Forward Flight

143

80 120 160

VELOCITY - KNOTS

200

Figure 15. Effect of Forward Speed on Air-
Resonance Stability in lg Level
Flight

Physics of Air Resonance

General

The mechanism and the stability characteristics
of air resonance have been well described in numerous
papers. !> 2 > 4 > 5, 6 n suffices to say here that the soft-
in-plane hingeless -rotor system derives its inherent
stability mainly from the powerful flap damping. While
rotors with untwisted blades may have substantial reduc-
tion in the flap damping near zero thrust, the damping
available remains essentially unchanged for blades with
nominal twist. Figure 16 shows the test data for various
blades of different twist. Above a thrust coefficient of
0. 005, the twisted blade and the untwisted blade both
have the same thrust-per-collective slope. While the
untwisted blade has a drastic reduction in slope with re-
duction in thrust in both theory and test, that of the
twisted blade remains the same.

SYMBOL

TEST

AIRFOIL

°t

BTS 6-FT ROTOR

V23010

-7°

RTS 6-FT ROTOR

VR7

-9°

RTS 6-FT ROTOR

VR7/8

-9°

14-FT ROTOR

V23010/13006

-10.5"

UHM COMPOSITE

V23O10

-10.5°

UHM 6-FT ROTOR

VR5

-14°

•

MBB TIEDOWN

0012

0°

▲

AMRDL MODEL

23012

0°

g 0.03

° 0.02

« 0.01

<J =0

061

E/1

ill
I ff

TEST DATA
AT () t = -9°

>. cjj

7j»

7^~~TEST DATA
AT t = 0°

j/%

F THEORY AT

t =0°

TEST DA
AT 0t =

2Ba?

//
//
»/
/

-4-2 2 4 6 8 10
COLLECTIVE PITCH, 6.75 - DEGREES

Figure 16. Effect of Blade Twist on Thrust
Coefficient

Let us examine the coupling terms that are inher-
ent in the hingeless rotor system with an equivalent
hinge sequence of pitch-flap-lag from inboard to out-
board. One term that stands out is the perturbation
pitch moment produced by the induced drag (steady
force) acting through a moment arm of vertical-flapping
displacement (perturbation deflection) . This flap-pitch
coupling term due to the induced drag has the Sense of
flap up/pitch noseup. Figure 17 compares the air-
resonance mode damping of the same rotor system with
this particular coupling term suppressed. With the
induced-drag term suppressed, the air-resonance mode
does not become unstable at high collective where the
induced drag dominates.

By the same consideration, the air-resonance
mode should become more stable in descent since during
descent, the induced drag acts toward the leading edge
producing a flap-pitch coupling of flap-up/pitch-nose-
down sense, which is stabilizing.

HCTIOR SPEED - lOOt 1% 0E

HftNEUVER q LEVEL

Figure 17. Stability Characteristics With Sup-
pression of Flap-Pitch Coupling Term
Due to Induced Drag

144

Pitch- Flap- Lag Coupling Characteristics

For a complete understanding of the elastic-
coupling characteristics of a hingeless rotor with a
pitch-flap-lag sequence of hinges , all blade motions
must be considered together. For this purpose it is in-
structive to analyze a simple cyclic-pitch case in hover.
In Figure 18 the elastic flap, lead-lag, and pitch mo-
tions are shown over one rotor revolution. It can be
seen clearly that the flap and lag motions are accom-
panied by an elastic pitch torsion, the resultant coupling
being in the sense of flap up/lead forward/pitch nose-
down. For a clear understanding this complex coupling
can be divided into two distinct coupling phenomena: the
one equivalent to a negative pitch-flap coupling (flap up/
pitch nosedown) , the other equivalent to a positive pitch-
lag coupling (lead forward/pitch nosedown). The cou-
pling factors are 0. 4 degree pitch per degree flap and
0. 6 degree pitch per degree lag.

flexibility. Some of these design rules have already been
applied to this model rotor design (low precone, aft
sweep, soft control systems).

Parametric Sensitivities

The following paragraphs describe the air-
resonance mode stability sensitivities obtained from the
model test.

Climb and Descent

Figure 19 shows the sensitivity with lg climb and
descent at a scaled airspeed of 80 knots. With normal
control system stiffness (90 in. -lb/rad) , descent sta-
bilizes the mode as discussed in the previous section;
conversely, climb has a destabilizing effect.

O
H
Q

3
O
H

n
o
w

pa
a

PRECONE * DEGREES

SWEEP =2.5 DEGREES AFT

CONTROL STIFFNESS =90 IN. -LB/RAD

360

AZIMUTH ANGLE

DEGREES

!** 2

J 1

1 —

= 80 KNOTS

. ,

SYMBOL

CONTROL STIFFNESS

= %OR

90 IN. -LB/RAD

642 IN. -LB/RAD

_VV_

i-l

-20 -10 * 10

t CLIMB | DESCENT |

ROTOR ANGLE OF ATTACK - DEGREES

Figure 19.

Effects of Climb, Descent, and Control
System Stiffness on Air-Resonance
Stability

Figure 18. Blade Elastic Coupling

Control System Stiffness

Also shown in Figure 19 are the test data obtained
with the control system stiffness seven times stiffer
than normal. The effect of climb and descent almost
disappeared. Since the stability is affected by the pitch-
flap-lag coupling, a stiff control system minimizes the
coupling effect, be it favorable or unfavorable.

Precone

Precone of the pitch axis directly alters the pitch-
flap-lag coupling. The beneficial effect of lower precone
has been evaluated many times. 1.2,7 Figure 20 shows
the test confirmation of the favorable effect of the low
precone.

Besides the well-known stabilizing effect of pitch-
flap coupling, the pitch-lag part of the total coupling is
of utmost importance for the in-plane motions of the
blade. Positive pitch-lag coupling (decrease of pitch as
the blade leads forward, increase of pitch as the blade
lags back) has a highly stabilizing effect on the lead-lag
oscillations. Recent investigations!. 2 have shown that
these coupling characteristics can be influenced by sev-
eral hub and blade parameters, for example, by feather-
ing axis precone, blade sweep, and control system

Cyclic Trim

An evaluation of the cyclic trim on the air-
resonance stability was accomplished by varying the
angle of incidence of the tail. The tail incidence angle
was varied from 2 degrees through 45 degrees. As
shown in Figure 21, the stability is insensitive to the
range of cyclic-trim variation at constant thrust. This
suggests that the steady 1/rev cyclic-pitch variation in
forward flight can be ignored with respect to the air
resonance.

145

K
J W
<C 0.

3.0

2.0

1.0

V =
N N0R =
SYMBOL
A

150 KNOTS
100 PERCENT

PRECONE
DEGREES —
1.5 DEGREES
3.25 DEGREES

200

COLLECTIVE PITCH
HOVER lg COLLECTIVE PITCH

SXMBOt

THRUST

V = 150 KNOTS

lg THRUST AT HOVER

)R = 100 PERCENT

92 PERCENT

116 PERCENT

139 PERCENT

162 PERCENT

2.0

- PERCENT

os<

ZH
OjU
§H

ass

H
< «

a a

Oft

10 20 30

STABILIZER ANGLE

Figure 20. Effect of Blade Precone on Air-
Resonance Stability

Figure 21. Effect of the Incidence Angle of the
Horizontal Tail on Air-Resonance
Stability

Conclusions

1. The air-resonance mode stability is sensitive 1.
to collective pitch (thrust).

2. Air-resonance mode stability is also sensitive
to climb and descent; that is, descent is stabilizing while .
climb is destabilizing.

3. The prime coupling term in the rotor system 2 -
which causes the degradation of stability at high thrust

is the induced drag. This coupling also provides the
trend versus climb and descent.

4. Air-resonance mode stability in lg level flight „
shows the rotor-power-curve trend with highly stable
characteristics at high speed.

5. The elastic-coupling behavior of the model

rotor with normal control system stiffness is charac- 4,

terized by a pitch-flap coupling (0. 4 degree pitch per
degree flap) and a pitch-lag coupling (0.6 degree pitch
per degree lag).

6. High control system stiffness minimizes the
flap-pitch coupling effectiveness and reduces the sensi- 5.
tivity of the air-resonance stability to design parameters
which are otherwise influential.

7 . Less precone is stabilizing for a soft-in-plane
hingeless-rotor system with an equivalent hinge sequence

of pitch-flap-lag from inboard to outboard. 6.

8. Variation in cyclic trim does not affect air-
resonance stability.

9. The testing technique to define air-resonance 7 *
modal damping discretely at many operational conditions
proved highly successful. Use of these methods to define
modal damping, rather than defining only the boundaries,
allows for a more definitive view of an aircraft's stability
characteristics.

References

Burkam, J.E; , and Miao, W. , EXPLORATION OF
AEROELASTIC STABILITY BOUNDARIES WITH A
SOFT-IN-PLANE HINGELESS-ROTOR MODEL,
Preprint No. 610, 28th Annual National Forum of
the American Helicopter Society, Washington, D. C. ,
May 1972.

Huber, H.B. , EFFECT OF TORSION-FLAP-LAG
COUPLING ON' HINGELESS ROTOR STABILITY,
Preprint No. 731, 29th Annual National Forum of
the American Helicopter Society, Washington, D. C. ,
May 1973.

Gabel, R. , and Capurso, V. , EXACT MECHANI-
CAL INSTABILITY BOUNDARIES AS DETERMINED
FROM THE COLEMAN EQUATION, Journal of the
American Helicopter Society, January 1962.

Lytwyn, R.T., Miao, W. , and Woitsch, W. , AIR-
BORNE AND GROUND RESONANCE OF HINGE-
LESS ROTORS, Preprint No. 414, 26th Annual
National Forum of the American Helicopter Society,
Washington, D. C, June 1970.

Donham, R.E., Cardinale, S.V. , and Sachs, I.B.,
GROUND AND AIR RESONANCE CHARACTERIS-
TICS OF A SOFT INPLANE RIGID ROTOR
SYSTEM, Journal of the American Helicopter
Society, October 1969.

Woitsch, W. , and Weiss, H. , DYNAMIC BEHAVIOR
OF A HINGELESS FIBERGLASS ROTOR, AIAA/
AHS VTOL Research, Design, and Operations
Meeting, Atlanta, Georgia, February 1969.

Hodges, D.H. , and Ormiston, R.A., STABILITY
OF ELASTIC BENDING AND TORSION OF UNI-
FORM CANTILEVERED ROTOR BLADES IN
HOVER, AIAA/ASME/SAE 14th Structures, Struc-
tural Dynamics , and Materials Conference,
Williamsburg, Virginia, March 1973.

146

AH APPLICATION OF FLOGPET THEORY TO PREDICTION OF MECHANICAL INSTABILITY

C. E. Hammond

Langley Directorate

U.S. Army Air Mobility R&D Laboratory

NASA Langley Research Center

Hampton, Virginia

Abstract

The problem of helicopter mechanical insta-
bility is considered for the case where one blade
damper is inoperative. It is shown that if the hub
is considered to be nonisotropic the equations of
motion have periodic coefficients which cannot be
eliminated. However, if the hub is isotropic the
equations can be transformed to a rotating frame
of reference and the periodic coefficients elimi-
nated. The Floquet Transition Matrix method is
shown to be an effective way of dealing with the
nonisotropic hub and nonisotropic rotor situation.
Time history calculations are examined and shown
to be inferior to the Floquet technique for deter-
mining system stability. A smearing technique
used in the past for treating the one damper inop-
erative case is examined and shown to yield uncon-
servative results. It is shown that instabilities
which occur when one blade damper is inoperative
may consist of nearly pure blade motion or they
may be similar to the classical mechanical
instability.

Notation

lag damping rate

effective hub damping in x-direction
effective hub damping in y-direction
lag hinge offset

X b

^b
N

second mass moment of blade about lag
hinge

lag spring rate

effective hub stiffness in x-direction

effective hub stiffness in y-direetion

blade mass

effective hub mass in x-direction

effective hub mass in y-direction

number of blades in rotor

characteristic exponent corresponding to
j th eigenvalue of the Floquet Transition
Matrix

x, y

x c> v c

*Q»

*1>

e±

[A(t)]

CD(t)]
[Q]

[0(t)]
fs(t)|

force acting on hub in x-direction

force acting on hub in y-direction

first mass moment of blade about lag
hinge

period of the periodic coefficients,
T = &c/n

coordinates of hub in rotating reference
frame

coordinates of rotor center of mass in
fixed reference frame

coordinates of hub in fixed reference
frame

coordinates of elemental blade mass dm
in fixed reference frame

lag deflection of i th blade

defined by Equations (l8)

defined by Equations (7)

j th eigenvalue of the Floquet Transition
Matrix

defined by Equations (18)

defined by Equations (7)

distance from lag hinge to elemental
blade mass dm

azimuthal location of ith blade

rotor speed

defined by Equations (l8)

defined by Equations (7)

characteristic matrix, periodic with
period T

state matrix, periodic with period T

Floquet Transition Matrix

state transition matrix

state vector

Presented at the AHS/NASA Ames Specialists' Meet-
ing on Rotorcraft Dynamics, February 15-15, 197 1 t-.

L-9l£lv

147

The problem of mechanical instability of
helicopters on the ground has been recognized and
understood for many years. The analysis by Coleman
and Feingold- 1 - has become the standard reference on
this phenomenon although it was not published until
many years after the first incidents of mechanical
instability, or ground resonance as it is commonly
known, were encountered on the early autogyros.
The mechanical instability phenomenon is most com-
monly associated with helicopters having articu-
lated rotors; however, helicopters using the soft-
inplane hingeless rotors which have became popular
in recent years are also susceptible to this
problem. Machines employing these soft-inplane
hingeless rotors are also known to experience a
similar problem, commonly known as air resonance,
which occurs in flight rather than on the ground.
The air resonance problem has received much atten-
tion in recent years (see, e.g., Refs. 2 and 3).

From the analysis of Reference 1 and others
it is known that the ground resonance problem is
due primarily to a coupling of the blade inplane
motion with the rigid body degrees of freedom of
the helicopter on its landing gear. These analyses
have shown that with the proper selection of blade
lag dampers and landing gear characteristics the
problem of mechanical instability can be eliminated
within the operating rotor speed range. All of the
mechanical instability analyses conducted to date
have one assumption in common - all blades are
assumed to have identical properties. This is a
reasonable assumption under ordinary circumstances j
however, the U.S. Army has a requirement on new
helicopters which invalidates this assumption. The
requirement is that the helicopter be free from
ground resonance with one blade damper inoperative.
As will be shown later, this one blade damper inop-
erative requirement has a serious impact on the
classical method of analyzing a helicopter for
mechanical instability. Further, there is at pres-
ent no published method available for treating the
case where each of the blades is permitted to have
different properties. Thus the designer is faced
with the dilemma of trying to satisfy the require-
ment with an analysis method in which one of the
basic assumptions is severely violated.

Two methods have been used to circumvent this
difficulty. The first of these involves a physical
approximation so that the classical analysis
becomes applicable. In this approach all blades
are still assumed to have identical lag dampers
even when one blade damper is removed, but the
value of each of the dampers is reduced by the
amount ci/W where N is the number of blades
and c^ is the original lag damper rate. As can
be seen, with this approach a system is analyzed
which is quite different from the actual situation
of a rotor with no damping on one blade. The sec-
ond method which has been used is to reformulate
the equations of motion allowing for differing
blade characteristics and to obtain the stability
characteristics of the system using a time history
integration of the equations. This second approach
has the drawback that interpretation of stability
characteristics from time history calculations is
often difficult and open to question. The method

will yield correct results, however, provided the
equations are integrated over a sufficiently long
time period.

The purpose of this paper is to present a
method of obtaining the mechanical stability char-
acteristics directly for a helicopter operating on
the ground with one blade damper inoperative. As
will be shown later, the equations governing the
motion of this system have periodic coefficients.
This fact suggests the use of Floquet theory as the
means for determining the stability characteristics
of the system. In the following, the one-damper-
inoperative problem is formulated and the resulting
equations are solved using the Floquet Transition,
Matrix method described by Peters and Hohenemser.
Results obtained using this method are compared
with results obtained from the two previously used
methods and recommendations are made concerning
the future use of the three methods described.

Equations of Motion

The equations of motion for the mechanical
instability problem will be formulated using an
Eulerian approach. It will be assumed, as is done
in Reference 1, that the helicopter on its landing
gear can be represented by effective parameters
applied at the rotor hub. It will be further
assumed that only inplane motions of the hub and
blades are important in determining the ground
resonance characteristics of the helicopter. Thus
the degrees of freedom to be considered consist of
two inplane hub degrees of freedom and a lead- lag
degree of freedom for each blade in the rotor. The
mathematical model to be used in the analysis is
shown in Figure 1. Note that in the figure only a
typical blade is shown. The analysis will be
formulated for a rotor having IT blades, and each
blade is assumed to have a rotational spring and
damper which act about the lag hinge.

The blade equations are developed by summing
moments about the lag hinge. The coordinates of
the elemental mass dm in the fixed system are

x. = x, + e cos +. + p cos(+. + £.)
y i = y h + e sin 1/ ± + p sinC^ + t, ± )

(1)

where

t ± = At + 2fl(i - l)/N i = 1,2,..., H

These expressions can be differentiated twice with
respect to time to yield the accelerations exper-
ienced by the differential mass

x 1 = \ - en 2 cos t ± - P (n + C i ) 2 cos(t i + q)j

- pl ± sinC^ + C ± )
y x = y h - efl 2 sin t ± - p(fl + ^^sin^ + t, ± )\

+ (j[ t cosC^ + S ± )

Using D'Alembert's principle the Bummation of
moments about the lag hinge can be written as

(2)

148

Jp sin (tj + 5 i )x jL 3m - fp cos (+ i + ^y^cbn

k.£, - Q.X. =
iM i b x

i = 1,2,. ..,U (3)

where the integrals are evaluated over the length
of the blade. Introducing the expressions for
5c. and y. and defining the following

%- So

(4)

the blade equations become

1^ + ea\ sin q - S^ sin(+ i + ^

- y h cosC^ + 5i)] + \^ + \i % = o (5)

i - 1,2,..., N

If small displacements are now assumed the blade
equations may. be linearized to obtain

k + \k + (a> °i + ^o^i = (%/ e > [2 h sin +i

- y^ cos t.^

i=l,2,...,N (6)

where the following parameters have been introduced

1i = CA

Under the assumptions stated earlier the hub
equations of motion can be written directly as

m x, + c x, + k x, =P
x n x n x n x

Vh + C A + Vh = P y

(8)

where the coefficients on the left side of these
equations are the effective hub properties in the
x- and y-directions, respectively. The determina-
tion of these properties depends on an extensive
knowledge of the helicopter inertial character-
istics and the stiffness, damping, and geometrical
characteristics of the landing gear system. These
properties may be determined either by ground shake
tests of the helicopter, as suggested in Refer-
ence 1, or by direct calculations. The right-hand
side of the above equations are the forces acting
on the hub due to the fact that the rotor is
experiencing accelerations in the x- and y-
directions. If the accelerations of the rotor
center of mass are x and f c , respectively, the
P x and Py are given by

P.,

p y = "^c

(9)

The equations as written also indicate that in the
absence of the rotor the hub degrees of freedom are

uncoupled. This is an approximation, but it is

an assumption made in Reference 1 and one generally

used in helicopter mechanical stability analyses.

If all blades in the rotor are assumed to have
the same mass distribution, the coordinates for the
total rotor center of mass may be written as

= *h + I E x i

1=1 c

(10)

y c = ^h + | E y i

1=1

where x^ and y^ are the coordinates of the
individual blade center of mass, measured with
respect to the hub. If the center of mass of the
ith blade is a radial distance p c from the lag
hinge

x i c = e cos \ + P c eos^ + t. % )

y ± = e sin ^ + P c sin^ + ^)

Making the observation that, for H > 1

N N

£ cos t k = E sin t k

(11)

k=l

the rotor center of mass coordinates become

x c = *n - ( p c / N) £ k sin *i

y c = y h + (p c /n) £ t, ± cos ^

(12)

These expressions may now be differentiated twice
with respect to time and the forces P x and P y
obtained as

P x = - Bn b S h +8 b ^k- a \ )Bin V 2 < cos *i

J. r- 2

P y = -H^h-^ SPi-^i^ 8 +i- 2Q 5i Sin *i

(15)

The hub equations of motion thus become

(m x + Hrn^ + c^ + k^ =

h r.. p

s^ £ M x - n^)sin t ± + 20^ cos t.
1=1 L

(a y + ^)r h + y h + y h =

(i»0

-% E (?i - ^iJcos ^ - 2fl£ ± sin ^

The equations of motion for the system thus con-
sist of (N + 2) coupled second-order differential

149

equations with the coupling terms having periodic
coefficients. The periodic coefficients arise
because the hlade equations are written in a rotat-
ing reference system whereas the hub equations are
in a fixed system. As is shown in the Appendix,
if all the blades have identical lag springs and
lag dampers, the periodic coefficients may he
eliminated through the use of multiblade coordi-
nates. The effect of these coordinates is to
transform the hlade equations from the rotating to
the fixed system of reference. The resulting con-
stant coefficient system of equations is the set
normally solved in the classical ground resonance
analysis. As is shown, however, if the blades are
allowed to have different lag springs and dampers,
the periodic coefficients cannot be eliminated in
the usual manner.

An alternative does exist, however, for
eliminating the periodic coefficients even when
the blades are allowed to have differing character-
istics. The alternative consists of transforming
the hub equations into the rotating system of
reference. In order to eliminate the periodic
coefficients using this approach, the additional
assumption must be made that the hub is isotropic.
That is

y + r^y + (a£ - fi 2 )y + 2f2x + Oi^x

N r
- - v l E KCj-flPCj)** f^Q-l)- 2Qi 3 sin §5(3-1)

(17)
where the following parameters have been introduced

v h = V (m x + ^V

\ - c x/ (m x + fc b )

(18)

Introducing the rotating coordinates into the blade
equations, Equations (6), results in

2 2v

l i + "A + << + ° < K i

= (#•)

(x - fl x - 2f2y)sin 21 (j - l)

- (y - fi 2 y + 2&) cos |L(j - l)

A = 1,2,

(19)

.,N

c => c
x y

This is the approach used in Reference 1 for treat-
ing the two-bladed rotor which is another case
where the periodic coefficients in the equations
of motion cannot be eliminated by transforming the
blade equations to the fixed system.

The transformation from fixed to rotating
coordinates is given by

x = x, cos fit + y, sin Sit .
y = -x, sin At + y. cos fit '

(15)

Differentiating these expressions allows the fol-
lowing identities to be established

x. cos fit + y. sin fit =* x - fiy

-x. sin fit + y. cos fit = y + &

- 2- *

x. cos fit + y, sin fit = x - fi x - 2f!y

-x. sin fit + y. cos fit = y - fi y +

2fix

The hub equations in the rotating system are then
obtained by appropriate combinations of the xjj
and yjj equations, Equations (l4). The resulting
equations are given below

- - 2 2

x + TjjX + (a£ - fi )x - 2fiy - fiiyr

= v l E p j - fl 2 ^)sin %{ j - 1) + 2fl£j cos §£( j - 1)1

Since modern helicopters do not in general
have isotropic hubs, the above equations can only
be used to approximate the effects of a noniso-
tropic rotor. They are, however, easily solved for
the stability characteristics of the system and
thus they might be used to obtain a first approxi-
mation to the mechanical stability boundary for a
helicopter with one blade damper inoperative.

Prom the foregoing discussion it can be seen
that if either the rotor or the hub is isotropic,
the mechanical stability characteristics of the
system may be obtained using conventional tech-
niques. If both the rotor and hub sire nonisotropic
the equations of motion of the system contain
periodic coefficients and thus the standard eigen-
value techniques cannot be used to determine
whether the system is stable or unstable. It is
the purpose of this paper to demonstrate that
Floquet theory can be used to analyze this general
situation of a nonisotropic rotor coupled with a
nonisotropic hub.

Solution of the Equations

If the periodic coefficients in the equations
of motion are eliminated by assuming either an
isotropic rotor or an isotropic hub, the stability
of the system can be determined using standard
eigenvalue techniques. The general case of a
nonisotropic rotor coupled with a nonisotropic hub
will be treated using Floquet techniques as
described by Peters and Hohenemser,* and Hohenemser
and Yin. 5 A brief description of the technique
will be presented here for the sake of completeness.

In state vector rotation the free motions of
the system may be written as

B = [D(t)] 8

(20)

150

where the state variables for the problem being
considered consist of

» • »
?!» i z > •••> 5 H * V V ^1' ^2' ■••' 5 H' V K

and the equations which describe the motions of the
system are Equations (6) and {lk) . The matrix
[D(t)] is periodic with period T and for the
mechanical stability problem T = 2rt/fl.

Floquet's theorem states that the solution to
the above system of equations has the form

ill -CA(t)]U>«»)*|

(21)

where [A(t)] is the characteristic matrix and is
also periodic with period T. The column of
initial conditions jZ(o) is used in determining
\a\ as

a}- [A(0)]" 1 JZ(0)J

(22)

The matrix [A(o)], the modal damping A, and the
modal frequency cu are determined from the Floquet
Transition Matrix [Q] which is defined by the
equation

Z(T) - [Q] 2(0)

(23)

for all sets of initial conditions Z(o)J . It is
shown in References k and 5 that the eigenvalues
A* of the matrix [Q] can be used to determine Aj

and cui since

Aj - e ( V la3 J )T

(210

and the modal matrix of [Q] is just [A(o)]. The
characteristic matrix [A(t)3 is then Bhown to be
given by

CA(t)] - [0(t)][A(O)][e- ( ^ +ia)) *l (25)

where the state transition matrix [0(t)] is defined
ty

B(t)J = C0(t)]jZ(o)

(26)

The characteristic multipliers A., of the
system are uniquely defined since the matrix [Q]
is realj however, only the real parts of the

characteristic exponents
defined uniquely since

Aj + to,

^■?. (ta hl +lar sV

(27)

The imaginary part can only be determined within an
integer multiple of 2jr/T. This indeterminacy of the
cuj causes no particular difficulty if one is only
interested in the stability of the system. However,
if one is interested in understanding the mechanism
involved in any instability which might be found,
this indeterminacy can be quite troublesome.

The Floquet Transition Matrix which is the
basic element needed in the stability analysis is
easily determined by a numerical integration of
the equations of motion over one period T. If
one desires to compute the characteristic functions

[A(t)] the matrix [0(t)] is saved at each time
point in the numerical integration to obtain [Q]
For the calculations of this paper, the fourth
order Runge-Kutta method with Gill coefficients'
was used for the numerical integration.

A comment is in order concerning the charac-
teristic functions [A(t)]. The matrix [A(t)] is
a complex valued matrix and is determined at as
many time points as desired. The computation of
these functions can be relatively expensive and
intepretation can be difficult. The interpreta-
tion is made easier by the procedure outlined in
Reference 5 for converting the complex functions
into real functions which may be plotted as func-
tions of time. The scheme used is essentially the
same as that used when it is desired to plot as a
function of time the modes of a system having con-
stant coefficients. That is, for a conjugate pair
of characteristic exponents

P j = h + iCD j

P j = *j

io>.

the characteristic functions are also conjugate
pairs. Thus the real modal function column for
this conjugate pair of characteristic exponents
will be given by

{« (t)} - JA (t^ty*^* + |I d (t)|e^J- 1CU J 3t

(28)

A,(t)j is the jth column of [A(t)] and
Is the complex conjugate of this column,
he purpose in performing these manipulations is
to be able to plot the modal functions to deter-
mine the relative magnitudes and phases of the
various degrees of freedom in each mode. A discus-
sion of this technique as it applies to constant
coefficient systems is given by Meirovitch.T In
this paper the exp(Ajt) is omitted from the above
equation since it is simply a constant which multi-
plies each component of the mode and causes each
component to damp at the same rate. Thus the
plots of the characteristic functions which are
presented later in the paper will appear to be
neutrally damped.

In making the calculations for this paper it
was found that the output from the calculation of
the modal functions became so voluminous and these
calculations became so expensive that the modal
functions were only computed for selected points.
Generally a sweep of rotor speed was made and the
results examined. If an unstable region was indi-
cated the rotor speed corresponding to the maximum
positive Aj was rerun and the modal functions
calculated.

Discussion of Results

In order to demonstrate the application of
the above-mentioned techniques and to obtain a
general understanding of the effect of one blade
damper inoperative on mechanical stability, a set
of parameters were chosen. The parameters in the

151

mechanical stability analysis were chosen so as to
be in the general range of interest for a single
rotor helicopter and were such that the system was
stable with all dampers functioning up to a rotor
speed of ^00 rpm. The parameter values chosen for
the calculations are shown in Table 1.

The parameters presented in Table 1 correspond
to an isotropic rotor and a nonisotropic hub. In
the following discussion results are presented for
the case of an isotropic hub coupled with a non-
isotropic rotor and a nonisotropic hub coupled
with an isotropic rotor as well as the case of
interest which involves a nonisotropic hub coupled
with a nonisotrpic rotor. When an isotropic hub
is mentioned, this means that the hub parameters
in both the x- and y-directions were assigned the
values shown in Table 1 for the x-direction. An
isotropic rotor implies that all dampers are
operational and a nonisotropic rotor is meant to
indicate that the lag damper has been removed from
blade number 1. The analysis has been formulated
in such a way that any number of blade lag dampers
or lag springs may be removed to make the rotor
nonisotropic. The results presented here, how-
ever, only involve the removal of the lag damper
from one blade.

The ease of an isotropic hub was first run in
an effort to become familiar with the nonisotropic
rotor results before proceeding with the more
complicated Floquet analysis. The isotropic hub
permits the equations to be transformed into the
rotating reference frame and results in a system
of equations with constant coefficients, Equa-
tions (l6), (l7)j and (l9)> even with a noniso-
tropic rotor.

Figure 2 shows the results of the calculations
for the isotropic hub with all blade dampers work-
ing. Note that since the equations were solved in
the rotating system, the frequencies in the lower
portion of Figure 2 are plotted in the rotating
system. The numbers attached to the different
modes in Figure 2 and in subsequent similar figures
have no significance other than' to provide a label
for the various modes. In Figure 2 the dashed
lines represent the uncoupled hub modes. The
uncoupled rotor modes follow along the curves
labeled 1,2 which also represent, in the terminol-
ogy of Beference 5, the rotor collective modes.
Note that the uncoupled blade frequencies are zero
for rotor speeds less than about 65 rpm. This is
due to the fact that the blades are critically
damped for these low rotor speeds. At the higher
rotor speeds modes 3 and k are essentially rotor
modes and modes 5 and 6 are essentially hub modes.
At the lower speeds, however, due to the coupling
between rotor and hub, mode h changes to a hub
mode and mode 5 changes to a blade mode. Note
from the damping plot that all the modes indicate
stability over the entire rotor speed range.

The results for one blade damper inoperative
and an isotropic hub are plotted in Figure 3.
Note that the removal of a blade damper has caused
the appearance of a mode which was not present in
Figure 2, namely the mode labeled 3 in Figure 3,

and that this mode exhibits a mild instability .
between 160 and 200 rpm. At rotor speeds below
about 100 rpm this mode has a frequency which cor-
responds to the uncoupled frequency of the blade
which has no damper. At rotor speeds above 100 rpm
this mode begins to deviate in frequency from the
uncoupled frequency. Another interesting point is
that mode 1 in Figure 3 is precisely the same as
the collective modes of Figure 2, and in Figure 3
there is only one such mode. Thus it appears that
the unstable mode in Figure 3 has evolved from one
of the two collective modes shown in Figure 2
because of the removal of one of the blade dampers.

A time history calculation was made for the
point of maximum instability in Figure 3 which
occurs at approximately 175 rpm- The results of
the time history calculation are shown in Figure h.
These results were obtained using the same inte-
gration scheme as that used for generating the
Floquet Transition Matrix. The top portion of the
figure represents the individual blade lag motions
whereas the lower portion represents the hub
response in the x- and y-directions. Note from
the figure that each of the degrees of freedom was
given an initial displacement but the initial
velocities were zero. The equations were inte-
grated for 17 rotor revolutions. The figure indi-
cates the blades which have lag dampers are .well
damped, but the blade on which the damper is
inoperative experiences large lag excursions.
Also, the hub motions, although not large, do not
appear to have a high degree of damping. From the
time history one would conclude that the system is
stable since the motions of the various degrees of
freedom do not appear to be increasing in ampli-
tude with increasing time. The eigenvalue analysis
has shown, however, that an instability exists.
The problem with the time history calculations is,
of course, that the equations of motion have not
been integrated over a sufficiently long time
period for the initial conditions chosen. Herein
lies the difficulty with using the time history
approach for calculating the stability character-
istics of systems. One can never be sure if a
sufficiently long integration period has been
used, and the choice of initial conditions which
will minimize the integration time required is a
trial and error process. It has been observed on
an analog computer that for the ground resonance
problem the choice of initial conditions has a
strong bearing on the conclusion inferred from the
time history traces. The time history integration
is also much more time consuming on the digital
computer than the eigenvalue analysis. The time
to generate Figure k- which is for only one rotor
speed was much greater than the time required to
generate the eigenvalue results for all of Fig-
ure 3- It is thus concluded that whenever it is
at all possible the eigenvalue approach to sta-
bility calculation is to be desired over the time
history approach.

Having examined the case of one blade damper
inoperative an an isotropic hub, the next logical
step is to examine the more realistic situation of
a nonisotropic hub. Before examining the one
damper inoperative situation it was first desired

152

to confirm that the system was stable with all
dampers working. The modal damping and frequency
of the -various modes with all dampers working and
a nonisotropic hub are shown in Figure 5- As can
be seen from the damping plot, all the modes are
stable. In this ease the equations of motion are
solved in the fixed frame of reference and hence
the frequencies are plotted in this frame. The
dashed lines on the frequency plot represent the
uncoupled system: the horizontal dashed lines
being the hub modes and the slanted dashed lines
being the rotor modes. Note that because the rotor
modes become critically damped at low rotor speeds
the two uncoupled rotor frequencies come together
before reaching the origin. The uncoupled rotor
lines also represent the collective modes for the
rotor. These modes are completely uncoupled from
the other modes and hence are not included in the
eigenvalue analysis of the nonisotropic hub
coupled with an isotropic rotor. The damping for
the collective modes is exactly the same as that
shown for modes 1,2 in Figure 2.

The validity of the Floquet analysis was
verified by comparing results from this analysis
with results from both the rotating system analysis
(isotropic hub) and from the fixed system analysis
(isotropic rotor). In each case the results from
the Floquet analysis were identical to results
from the other analyses.

Having thus established the validity of the
Floquet analysis, results were obtained for the
nonisotropic hub and one blade damper inoperative.
These results are shown in Figure 6. Note that
these results are very much similar to those shown
in Figure 5 except that, as was the case with the
isotropic hub and one blade damper inoperative,
there are additional modes introduced. Also indi-
cated is a relatively strong instability between •
210 and 305 rpm. The frequencies of the addi-
tional modes which are introduced correspond, at
low rotor speeds, to the frequencies of the
uncoupled blade which has no damper. In the rotor
speed range where the instability occurs, however,
the frequency deviates from the uncoupled value as
indicated by the mode labeled 3- In this range
and at higher rotor speeds the mode labeled 5 is
nearer the uncoupled blade frequency. It thus
appears that for this case the instability is more
a coupled rotor hub mode than a pure blade mode as
was indicated for the isotropic hub.

This conjecture is further strengthened by an
examination of the modal functions. The modal
functions for a rotor speed of 255 ^V^t which' is
the point of maximum instability, are shown in
Figure 7. The functions are plotted over a time
period corresponding to one rotor revolution. Note
from this figure that blade 1, the blade without a
damper, has a significantly higher contribution to
the mode than the other blades. Also from the
plot of hub response it can be seen that the par-
ticipation of the lateral hub degree of freedom,
which has the higher of the uncoupled hub fre-
quencies shown on Figure 6, is considerable. It
is thus concluded from Figures 5 and 6 that the

one damper inoperative situation can lead to a
classical mechanical instability.

Time history traces for this same condition
are shown in Figure 8. These traces show the same
general trends as observed in the case of the
isotropic hub, that is, a large response of the
blade having no damper and moderate responses from
the other blades and the hub degrees of freedom.
Again the time history traces are inconclusive
regarding the stability of the system.

One of the methods used in the past for
treating the one blade damper inoperative case
involves a smearing of the total blade damping.
The reasoning for this approach is as follows.
If the rotor has N blades then the total damping
available in the rotor is Nc^ where ci is the
damping on one blade. If one damper is removed,
the total damping becomes (N - l)c^. Thus, using
this approach, each blade in the rotor would be
treated as if it had a lag damper equal to
Ci(N - 1)/N.

After an examination of the preceding one
damper inoperative results it would be expected
that this approach would lead to unconservative
results. This is due to the fact that the insta-
bilities encountered in the previous results
involved large motions of the blade which had no
damper. The smearing technique results in damp-
ing, which is not greatly different from the
original value, being applied to each blade and
thus the true situation is not adequately modeled.

To illustrate this method, the nonisotropic
hub case was analyzed using the smearing approach.
The results from these calculations are shown in
Figure 9- Note that although mode 3 becomes
lightly damped the system remains stable through-
out the rotor speed range considered. The fact
that mode 3 approaches instability is attributable
to the fact that this mode was not heavily damped
in the original calculations. A run of the iso-
tropic hub ease, where all the modes were origi-
nally well damped, indicated that the smearing
technique resulted in well damped modes for one
blade damper removed. The smearing technique is
thus not recommended for treating the one blade
damper inoperative situation since it leads to
unconservative results.

Since one way for eliminating the classical
mechanical instability is to increase the blade
damping, it was decided to attempt this approach
on the instability indicated in Figure 6. The
approach was to leave the damping identically zero
on one blade and increase the damping on the
remaining three blades. The results of this series
of calculations are shown in Figure 10 where the
region of instability is presented as a function
of blade lag damping and rotor speed. As can be
seen from the figure, increasing the blade damping
on three of the blades has very little effect on
the stability boundaries when one blade has zero
damping. This result was somewhat expected since
from the previous calculations it was observed

153

that the blade with zero damping responds more or
less independently of the other blades in the
rotor .

During the increased damping calculations no
attempt was made to determine whether or not the
nature of the instability had changed. That is,
whether the instability had changed from one
involving both blade and hub motion to one con-
sisting of primarily blade motion with only small
amounts of hub motion, Further delving into pos-
sible corrective actions for the instability which
occurs with one blade damper inoperative was
beyond the scope of this paper and thus more
research is needed to determine how the instability
may be eliminated.

Conclusions

There are several conclusions which may be
inferred from the preceding results. First of all,
the fact that a helicopter is free from mechanical
instability with all blade d amp ers working does
not guarantee that it will be free of instabilities
with one blade damper inoperative. The instability
encountered with one blade damper inoperative may
be a blade mode instability or it may be the
classical mechanical instability.

The Floquet Transition Matrix method can be
used effectively in examining the mechanical sta-
bility characteristics of helicopters with one
blade damper inoperative. When both the hub and
rotor are considered to be nonisotropic, the equa-
tions of motion contain periodic coefficients and
the Floquet approach provides an efficient means
for dealing with this situation. Since the
Floquet approach yields the stability character-
istics directly, it furnishes a more desirable
approach to stability problems than time history
calculations .

Time history calculations can lead to erron-
eous conclusions relative to the determination of
system stability. The erroneous conclusions stem
primarily from the fact that the time history
calculations require considerable computer time
and the tendency is to integrate over as short a
time period. as possible. Thus, if the initial
conditions are not chosen properly, the time
history traces may still contain transients when
the integration is terminated. The time history
approach to stability problems is thus recommended
only when no other recourse is available, and then
several different combinations of initial condi-
tions and integration periods should be examined
before making a conclusion regarding stability.

The smearing approach which has been used in
the past for treating the one blade damper inop-
erative situation leads to unconservative results.
Therefore, this method is considered to be an
unacceptable means for determining stability under
these conditions.

References

OF SELF-EXCITED MECHANICAL 0SCILIAT3DUS OF
HELICOPTER BQTOBS WITH HINGED BIADES, NACA
Report 1351, 1958.

2. Donham, R. E., Cardinale, S. V., and Sachs,

I. B., GROUND AND AIR RESONANCE CHARACTERISTICS
OF A SOFT IN-PLANE RIGID-ROTOR SYSTEM, Journal
of the American Helicopter Society , Vol. 14,
No. 4, October 1969, pp. 33- J H-

3. Lytwyn, R. T., Miao, W., and Woitsch, W. ,
AIRBORNE AID GROUND RESONANCE OF HTNGELESS
ROTORS, Journal of the American Helicopter
Society, Vol. 16, No. 2, April 1971, PP- 2-9-

k. Peters, D. A., and Hohenemser, K. H., APPLICA-
TION OF THE FLOQUET TRANSITION MATRIX TO
PROBLEMS OF LIFTING ROTOR STABILITY, Journal
of the American Helicopter Society , Vol. 16,
No. 2, April 1971, PP- 25-35-

5. Hohenemser, K. H., and Yin/s. K., SOME
APPLICATIONS OF THE METHOD OF MCJLTIBLADE
COORDINATES, Journal of the American Helicopter
Society , Vol. 17, No. 3, July 1972, pp. 3-12.

6. Carnahan, B., Luther, H. A., and Wilkes, J. 0.,
Applied Numerical Methods , John Wiley & Sons,
Inc., New York, 1969.

7. Meirovitch, L., Analytical Methods in Vibra-
tions, The Macmillan Company, New York, 1967,
p. 411.

Appendix

If the rotor is considered to be isotropic
the periodic coefficients appearing in the equa-
tions of motion can be eliminated through the use
of multiblade coordinates similar to those
described in Reference 1. These coordinates
essentially transform the blade degrees of freedom
into a fixed reference frame. The transformations
are given by

Si - E ?i sin *i
i=l

l II = ^ ^i COS *i

i=l

(Al)

Differentiating these expressions leads to
the establishment of the following identities

i=i

(A2)

154

£ L sin t = 6 - air - 2fl|

i=l

£ '( ± cos t ± = in - 21 n + 2J2IJ

i=l

2t can be seen from these identities that the
transformation is made by multiplying the blade
equations, Equations (6), by either sin % or
cos ilr^ and adding the equations. Crucial to this
operation is the ability to remove the i\± and
ccg. from the summations. This can only be done
if all the blades have identical lag springs and
lag dampers. If one or more of the blades have
differing characteristics, the tjj and/ or og^
cannot be factored from the summation and hence
the identities above cannot be applied. Thus, if
one or more of the blades are permitted to have
different lag springs or lag dampers, the periodic
coefficients cannot be eliminated using the pro-
cedure described in this Appendix.

If Equations (6) are first multiplied by
cos i|f^ and summed and then multiplied by sin f^
and summed, the following equations are obtained
after introduction of the identities (A2)

6 n + \ ! n

2 2 (1 - v 2 ) - a^Ji^ + Sfilj.

+ an^ = (v^/e)
- jr h E cos 2 ti

i=l

% 2 sin +i cos +1

. 1=1

(A3)

*t + I**

i*!

n 2 (i-v2)-<|| T - 2 «i IT

Xfc E sin +J..

1=1

"Mil = ( v o/ e)

B - 1

^ £ sin +i cos *i
i=i j

Making the following observations that for N > 2

N
^ sin f . cos \|r =

i=l

J, P i P

X) cos^ i|r = X) sin + = N/2

1=1 1 i=l X

the equations become

'II

*i*

fl 2 (l

- v 2 ) - c 2 ],

Ijj + 20IJ.

+ Sit) i l I = -(Hv^/2e)3r h

*I + \h - [ fl2(l " V o } " 4J'

(A*)

Oil"!

2J2|

^i 6 II » K/ 2e >*h

These two equations describe the rotor motions in
the fixed frame of reference. In terms of the
variables described by Equations (Al) the hub
equations, Equations (1*0, became

(m x + Bn^ + c^ + kx^ = Sjj
(m y + ""tX + Vh + Vk = " S ^H

(A5)

The stability of the rotor-hub system can now
be determined using Equations {Ak) and (A5) which
have constant coefficients. This set of equations
or a set similar to it is the one normally used in
helicopter mechanical stability analyses.

As a final observation, note that if. the
blade equations, Equations (6), are simply summed,
the following equation

«o + Vo + (£D °i + ^X - ° (A6 >

is obtained, where

6 o = Z 5

(A7)

1=1

This equation represents the rotor collective mode
and it may be observed that this equation is com-
pletely decoupled from the hub degrees of freedom.
Hence, the collective mode cannot influence the
stability of the system and it is therefore not
normally included in the mechanical stability
analysis .

155

TABLE 1. PARAMETERS USED IN THE SAMPLE CALCULATIONS

Number of blades

Blade mass, m.

Blade mass moment, S,

Blade mass moment of inertia, X.

Lag hinge offset, e

Lag spring, k^

Lag damper, c.

Hub mass, m

Hub spring, k

Hub damper, c

6.5 slugs (9^.9 kg)

65. slug-ft (289. 1 kg-m)

800.0 slug-ft 2 (1084.7 kg-m 2 )

1.0 ft (O.30W m)

0.0 ft-Ib/rad (0.0 m-N/rad)

3000.0 ft-lb-sec/rad (^67. 5 m-N-s/rad)

550.0 slugs (8026.6 kg)

225.0 slugs (3283.6 kg)

85000.0 lb/ft (121KA81.8 N/m)

85OOO.O Vo/ ft (I2i«)li8l.8 N/m)

3500.0 Ib-sec/ft (5IO78.7 N-s/m)

1750.0 lb- sec/ft (25539.3 N-s/m)

Figure 1; Mathematical representation of the rotor
and hub.

200
ROTOR SPEED, RPM

3 4

ROTOR SPEED, Hz

Figure 2. Modal damping and frequencies for iso-
tropic hub, all blade dampers working. Fre-
quencies plotted in the rotating system.

156

MODAL
FREQUENCY,
radlsec 12

200

ROTOR SPEED, RPM
J_

3 4

ROTOR SPEED, Hz

Figure 3- Modal damping and frequencies for iso-
tropic hub, one blade damper inoperative. Fre-
quencies plotted in the rotating system.

1
2
■3
-4
28

24
20

MODAL 16
FREQUENCY,
rad/sec 12

/ /

s>y

s S
s x
/ /

V-2

"V >*^

Z" 1

-V- — :*S^ —

- /

1 1

100

200
ROTOR SPEED, RPM

_L_

300

400

3 4

ROTOR SPEED. Hz

Figure 5- Modal damping and frequencies for non-
isotropic hub, all blade dampers working. Fre-
quencies plotted in the fixed system.

BLADE
RESPONSE

HUB
RESPONSE

-.5

-1
1

-.5
-1

BLADE 1

J-±

-^^t£Pr<

J L

J I I _J I L

TIME, sec

Figure k. Time history calculations for isotropic
hub, one blade damper inoperative, fi = 175 rpm.

24
20

MODAL 16
FREQUENCY,
rad/sec 12

6 A //

4 y\r

4?/

gr-

£r

V-2

" <&

r»

- /Z^

1 1

100

200
ROTOR SPEED, RPM

300

400

3 4

ROTOR SPEED, Hz

Figure 6. Modal damping and frequencies for non-
isotropic hub, one blade damper inoperative.
Frequencies plotted in the fixed system.

157

BLADE
RESPONSE

HUB
RESPONSE

1.000
.500

-.500

-1.000

1.000

.500

-.500
-1.000

EIGENVALUE - 0.32508 +i 8.04396

J I L

J I I L

_1 I

I I I I I I L

J I I 1

.050 .100 .150 .200 .250 .300

TIME, sec

MODAL 16

Figure 7- Modal functions for nonisotropic hub, frequency.
one blade damper inoperative, SI = 255 rpm. rad/sec 12

BLADE
RESPONSE

HUB
RESPONSE

TIME, sec

Figure 8. Time history calculations for noniso-
tropic hub, one blade damper inoperative,
J) = 255 rpm.

200

ROTOR SPEED, RPM

3 4

ROTOR SPEED, Hz

Figure 9. Modal damping and frequencies obtained
for nonisotropic hub, one blade damper inopera-
tive, using the smearing technique.

m-N-s/rad
22000

19000

16000

BLADE
LAG 13000
DAMPING

10000
7000

4000

fHb-sec/rad
15000 r

_ 12000

9000

6000

b 3000

STABLE/^ UNSTABLE ^STABLE

200 250 300 350
ROTOR SPEED, RPM

CT3

4 5
ROTOR SPEED. Hz

Figure 10. Instability region as a function of
blade lag damping for the nonisotropic hub and
one blade damper inoperative.

158

THEORY AND COMPARISON WITH TESTS OF TWO FULL-SCALE PROPROTORS

Wayne Johnson

Research Scientist

Laige-Scale Aerodynamics Branch

Ames Research Center, NASA

and

U.S. Army Air Mobility R&D Laboratory

Moffett Field, California

Abstract

A nine degrees-of-freedom theoretical model has been devel-
oped for investigations of the dynamics of a proprotor operating
in high inflow axial flight on a cantilever wing. The theory is
described, and the results of the analysis are presented for two
proprotor configurations: a gimballed, stiff-inplane rotor, and a
hingeless, soft-inplane rotor. The influence of various elements of
the theory is discussed, including the modeling used for the blade
and wing aerodynamics and the influence of the rotor lag degree
of freedom. The results from full-scale tests of these two prop-
rotors are presented and compared with the theoretical results.

Notation
en blade lift-curve slope

p wing torsion degree of freedom

q t wing vertical bending degree of freedom

q 2 wing chordwise bending degree of freedom

R rotor radius

V forward velocity

blade flap degree of freedom

(3 rotor coning degree of freedom

0! c rotor tip path plane pitch degree of freedom

(3, s rotor tip path plane yaw degree of freedom

7 blade Lock number

8 3 pitch/flap coupling

f damping ratio of eigenvalue, -ReX/ 1 X |

f blade lag degree of freedom

f rotor collective lag degree of freedom

f time derivative of f ; rotor speed perturbation degree of
freedom for autorotation case

?! c rotor vertical cyclic lag degree of freedom

?! s rotor lateral cyclic lag degree of freedom

X eigenvalue or root

Vo rotating natural frequency of blade flap motion

eg- rotating natural frequency of blade lag motion

to frequency of eigenvalue, ImX

J2 rotor rotational speed

The tilting proprotor aircraft is a promising concept for
short haul, V/STOL missions. This aircraft uses low disk loading
rotors located on the wing tips to provide lift and control in

Presented at the AHS/NASA-Ames Specialists' Meeting on Rotor-
craft Dynamics, February 13-1!>, 1974.

hover and low speed flight; it also uses the same rotors to provide
propulsive force in high speed cruise, the lift then being supplied
by a conventional wing. Such operation requires a ninety degree
change in the rotor thrust direction, which is accomplished by
mechanically tilting the rotor shaft axis. Thus the aircraft com-
bines the efficient VTOL capability of the helicopter with the
efficient, high speed cruise capability of a turboprop aircraft.
With the flexible blades of low disk loading rotors, the blade
motion is as important an aspect of tilt rotor dynamics as it is for
helicopters. When operated in cruise mode (axial flight at high
forward speed), the rotor is operating at a high inflow ratio (the
ratio of axial velocity to the rotor tip speed); such operation
introduces aerodynamic phenomena not encountered with the
helicopter rotor, which is characterized by low inflow. The
combination of flapping rotors operating at a high inflow ratio
on the tips of flexible wings leads to dynamic and aerodynamic
characteristics that are in many ways unique to this configura-
tion. The combination of efficient VTOL and high speed cruise
capabilities is very attractive; it is therefore important to estab-
lish a clear understanding of the behavior of this aircraft and
adequate methods to predict it, to enable a confident design of
the aircraft. Experimental and theoretical investigations have
been conducted over several years to provide this capability.
However, much remains to be studied, both in the fundamental
behavior and in the more sophisticated areas such as the design
and development of automatic control systems for the vehicle.
This paper presents the results of a theoretical model for a
proprotor on a cantilever wing, including application to two
proprotor designs." a gimballed, stiff-inplane rotor and a hingeless,
soft-inplane rotor. Using these two cases, the influence on the
system dynamics of several elements of the analysis was exam-
ined, including the effects of the rotor blade lag motion and the
rotor and wing aerodynamic models. The predicted stability
characteristics are then compared with the results of full-scale
tests of these two proprotor designs. The development of this
theory is presented in detail in Reference 1, together with some
additional applications to the analysis of proprotor aeroelastic
behavior.

Analytical Model

Figure 1 shows the proprotor configuration considered for
the theory and for the full-scale tests. The rotor is operating in
high inflow axial flight on a cantilever wing. For the tests, the
rotor was unpowered, hence operating in autorotation (really in
the windmill brake state). This configuration incorporates the
features of greatest importance to the aircraft: the high inflow
aerodynamics of a flapping rotor in axial flow and the coupled
dynamics of the rotor/pylon/wing aeroelastic system. Many
features of the aircraft-coupled wing and rotor motion may be
studied with such a model, theoretically and experimentally,
with the understanding that the model must eventually incorpo-
rate the entire aircraft.

159

The theoretical model of the proprotor developed in Refer-
ence 1 consists of nine degrees of freedom: the first mode flap
(out of disk plane) and lag (inplane) motion for each of three
blades; and vertical bending, chordwise bending, and torsion for
the cantilever wing. The degrees of freedom of the individual
rotor blades are combined into degrees of freedom representing
the motion of the rotor as a whole in the nonrotating frame.
Thus the rotor flap motion is represented by tip path plane pitch
and yaw (fii c and /3 X s ) and coning (/3 ) degrees of freedom. The
rotor lag motion is represented by cyclic lag ^ c and fi s (lateral
and vertical shift of the rotor net center of gravity) and collective
lag f Q . Wing vertical and chordwise bending of the elastic axis (qj
and q 2 ) and torsion about the elastic axis (p) complete the set of
nine degrees of freedom.

The rotor blade motion is represented by first mode flap
and lag motion, which are assumed to be respectively pure
out-of-plane and pure inplane deflection of the blade spar. For
the gimballed and hingeless rotor blades considered here (except
for the flap mode of the gimballed rotor), there is, in fact, some
elastic coupling of the flap and lag modes, so that there is
actually participation of both out-of-plane and inplane motion in
each mode. In the coefficients giving the aerodynamic forces on
the rotor, it is further assumed that the mode shapes are propor-
tional to the radial distance from the hub, i.e., equivalent to rigid
body rotation about a central hinge. The model based on these
two assumptions, which considerably simplify the aerodynamic
and structural terms of the rotor equations, proves to be an
adequate representation of the proprotor dynamics.

The theoretical results presented here will usually be for the
rotor operating unpowered, i.e., windmilling or autorotation
operation. An important element of autorotation dynamic
behavior is the rotor speed perturbation. With no restraint of the
rotor shaft rotation, this degree of freedom has considerable
influence on the aeroelastic behavior of the proprotor and wing.
The rotor speed perturbation is modeled by using the collective
lag mode f Q . By setting the rotating natural frequency of this
mode to zero, i.e., zero spring restraint, f becomes equivalent to
the rotor speed perturbation. (The natural frequency for the
cyclic lag modes, f a c and f i s , is not set to zero in the representa-
tion.) The other extreme case is that of the hub operating at
constant angular velocity (fi) with no speed perturbation, with
f then the elastic inplane deflection of the blade with respect to
the hub. This case will be considered to represent powered
operation of the rotor, although it is really the limit of operation
with a perfect governor on rotor speed.

The proprotor operating in high inflow has simpler aero-
dynamics than the low inflow rotor in forward flight. As for the
case of low inflow (i.e., the hovering helicopter rotor), the
symmetry of axial flow results in a corresponding symmetry in
the equations of motion; it also means that the differential
equations of motion have constant coefficients. In high inflow
there is the additional fact that both out-of-plane and inplane
motions of the blade produce significant angle-of-attack changes
at the sections, and the resulting lift increment has significant
components both normal to and in the disk plane. Hence the
rotor aerodynamic forces are primarily due to the lift changes
produced by angle-of-attack changes, i.e., the co^ terms in the
aerodynamic coefficients. This is in contrast to low inflow,
where, for example, the inplane blade motion produces signi-
ficant contributions to the forces by the lift and drag increments

due to the dynamic pressure changes, i.e., the eg and c^ terms in
the aerodynamic coefficients. As a result, high inflow rotor
aerodynamics are well represented by considering only the cg„
forces. If, in addition, the lift curve slope is assumed constant,
then the aerodynamic coefficients depend on only two param-
eters, the Lock number y and the inflow ratio V/JZR. Refer-
ence 1 presents the aerodynamic coefficients also for a more
complete theoretical model of the rotor aerodynamics, namely a
perturbation about the local trim state, including the cg^, eg, c,j,

and Cj terms (and also derivatives with respect to the Mach

"a
number). Such a model is in fact little more difficult to derive

than with the cg a terms alone. The influence of its use in the

theory is examined below.

This nine degrees-of-freedom model will have nine roots or
eigenvalues (really nine pairs of complex roots) and correspond-
ingly nine eigenvectors or modes. Of course, each mode involves
motion of all nine degrees of freedom. The modes are identifiable
by their frequencies (which will be near the uncoupled natural
frequencies, nonrotating for the rotor modes), and also by the
participation of the degrees of freedom in the eigenvector. The
nine modes will be denoted as follows (the approximate
uncoupled, nonrotating natural frequency of the mode is given in
parentheses):

p wing torsion (w p )

q x wing vertical bending (w q )

q 2 wing chordwise bending C« q )

coning (i^)

P + 1 high frequency flap (va + fi)

P - 1 low-frequency flap (vo - fi)

f collective lag (fg-)

i +. 1 high-frequency lag(i>>. + S2)

f - 1 low-frequency lag (v> - Q.)

The basic theoretical model will consist of all nine degrees
of freedom, autorotation operation, and just the cg a rotor aero-
dynamic forces. The wing aerodynamic forces are also included
(based on a strip theory calculation). The dynamic stability of
the system, specifically the frequency and damping ratio of the
modes, will be examined for variations of the forward velocity V
and the rotor rotational speed S2. Both V and fi sweeps change
the inflow ratio V/J2R, and hence the rotor and wing aero-
dynamic forces. A variation of the rotor speed £1 also changes the
values of the wing and rotor blade nondimensional (per rev)
natural frequencies. The rotor frequencies may also vary with
Y/CIR due to the change in the rotor collective pitch angle.
Several elements of the theoretical model will be examined to
determine their influence on the predicted proprotor dynamics:
the influence of the blade lag motion (by dropping the f , c and
Ji s degrees of freedom), the wing aerodynamics (by dropping the
wing aerodynamic coefficients), the rotor speed perturbation
(i.e., the autorotation and powered cases), and the more com-
plete model of the rotor aerodynamics (compared to just the eg
terms).

Two Full-Scale Proprotors

The theory described above will be applied to two full-scale
proprotors. The first is a 25-ft diameter gimballed, stiff-inplane
proprotor, designed and constructed by the Bell Helicopter
Company, and tested in the Ames 40- by 80-ft wind tunnel in
July 1970. The second is a 26-ft diameter hingeless, soft-inplane

160

proprotor, designed and constructed by the Boeing Vertol Com-
pany, and tested in the Ames 40- by 80-ft wind tunnel in August
1972. The configuration for the dynamics tests consisted of the
vrindmilling rotor operating in high inflow axial flow on the tip
of a cantilever wing, as shown in figure 1 . As far as their dynamic
characteristics are concerned, the two rotors differ primarily in
the placement of the rotating natural frequencies of the blade
flap and lag motions. The Bell rotor has a gimballed hub and
stiff-inplane cantilever blade attachment to the hub, hence
Vo= 1/rev (nearly, for it does have a weak hub spring) and
vy> 1/rev; it also incorporates positive pitch/flap coupling
(5 3 < 0) to increase the blade flap/lag stability. The Boeing rotor
has a cantilever or hingeless hub with soft-inplane blade attach-
ment, hence va > 1/rev and i>*. < 1/rev. The different placement
of the blade frequencies, at the opposing extremes of the possible
choices, results in quite different dynamic characteristics for the
two aircraft.

The rotors are described in References 2 to 5. Table I gives
the major parameters of the rotor, and of the cantilever wing
used in the full-scale tests (a more complete description of the
parameters required by the theory is given in Reference 1). The
wing frequencies in the theory were match to the experimentally
measured values by adjusting the spring constants. The typical
wing frequencies given in table I are for the coupled motion of
the system (including the rotor) at 100 knots and design £2. The
blade rotating natural frequencies are shown in figures 2 and 3
for the Bell and Boeing rotors, respectively. The variation of the
Bell lag frequency (fig. 2b) with V/fiR is due to the collective
pitch change. The Boeing rotor blade frequencies vary little with
collective pitch (V/J2R) since the blade has nearly isotropic
stiffness at the root.

The damping ratio of the wing modes was measured in the
full-scale tests by the following technique. The wing motion was
excited by oscillating an aerodynamic shaker vane on the wing
tip (visible in fig. 1) at the wing natural frequency. After a
sufficient amplitude was achieved, the vane was stopped. Then
the frequency and damping ratio were determined from the
decay of the subsequent transient motion of the wing.

Results and Discussion

Gimballed, Stiff-inplane Rotor

The effects of several elements of the theoretical model will
be examined for a gimballed, stiff-inplane rotor (the Bell rotor).
The theoretical results will then be compared with the results of
full-scale tests. The test results and results from the Bell theories
are from Reference 2. The predicted variation of the system
stability with forward speed V at the normal airplane mode rotor
speed (£2 = 458 rpm) is shown in figure 4, in terms of the fre-
quency and damping ratio of the eigenvalues. The wing vertical
bending mode (q x ) becomes unstable at 495 knots. The damping
of that mode first increases with speed; the peak is due to
coupling between the wing vertical bending (qi) and low-
frequency rotor lag (f - 1) modes (it occurs at the resonance of
the frequencies of these two modes). Figure 5 shows the influ-
ence of the rotor lag motion, comparing the damping of the wing
modes with and without the fj c and Si s degrees of freedom. The
rotor low-frequency lag mode has an important influence on the
motion, particularly on the wing vertical bending mode; the q[
damping is increased when its frequency is below that of the

?- 1 mode (low V), and decreased when its frequency is above
that value. The high speed instability is relatively unaffected,
however, indicating that the mechanism of that instability
involves primarily the rotor flap motion. Therefore, the net
effect of the reduced damping at high speed due to the lag
motion is a reduction of the rate at which the damping decreases,
which is beneficial since the instability is then less severe.

Figure 6 shows the influence of powered operation (stabiliz-
ing) and of omitting the wing aerodynamics (destabilizing). The
powered state effect is the influence of dropping the rotor speed
perturbation degree of freedom. The wing aerodynamics effect is
mainly the loss of the aerodynamic damping of the wing modes
due to the angle-of-attack changes during the motion. Figure 7
shows the influence of the more complete theoretical model for
the blade aerodynamics, compared with the results using only the
eg terms. The basic behavior remains the same, but the better
aerodynamic model reduces slightly the level of the predicted
damping ratio. The predicted speed at the stability boundary is
significantly reduced, however, because of the gradual variation
of the damping with speed. It is therefore concluded that for the
prediction of the characteristics of an actual aircraft, the best
model available for the rotor aerodynamics should be used.

Figure 8 shows the variation of the dynamic stability with
velocity at the normal rotor speed (J2 = 458 rpm), in terms of
the frequency and damping ratio of the wing modes; the full-
scale test results for the Bell rotor are compared with the pre-
dicted stability. Also shown are predictions from the Bell linear
and nonlinear theories, from Reference 2. Figure 8 shows reason-
able correlation between the predicted and full-scale test stability
results. Additional comparisons with the full-scale test data are
given in Reference 1 .

Figure 9 shows the influence of the rotor lag motion. The
predicted and measured stability is shown for the Bell rotor on
the full-stiffness wing, and on a quarter-stiffness wing. (By oper-
ating on a quarter-stiffness wing at one-half design rotor speed,
n = 229 rpm; the wing frequencies and inflow ratio are modeled
for an equivalent speed twice the actual tunnel speed.) Also
shown is the predicted stability for the rotor on the full-stiffness
wing, but without the f j c and f i s degrees of freedom. Figure 9a
shows the variation of the wing vertical bending mode damping.
The full-scale experimental data show a definite trend to higher
damping levels with the full-stiffness wing, and this trend corre-
lates well with the results of the present theory. Figures 9b and
9c show the predicted stability of all the wing modes. The
difference in damping at the same inflow ratio is due to the rotor
lag motion. Figure 9d shows the frequencies of the f - 1 , q, , and
p modes for the full-stiffness and quarter-stiffness wings. The
full-stiffness wing has a resonance of the f - 1 and q, modes
which produces the peak in the damping. Slowing the rotor on
the quarter-stiffness wing greatly increases the lag frequency (per
rev), removing it from resonance with the qj mode. Another way
to remove the influence of the rotor lag motion, in the theory, is
simply to drop the f , c and fj s degrees of freedom from the full-
stiffness wing case. When these degrees of freedom are dropped,
the predicted wing vertical bending damping is almost identical
to that for the quarter-stiffness wing (figs. 9a and 9b). Figure 10
examines further the influence of the rotor lag motion on the
wing vertical bending mode damping. Predicted stability with and
without the ?i c and J k s degrees of freedom is compared with
experimental results from tests of a 0.1333-scale model of a

161

gimballed, stiff-inplane proprotor. The test results are from Ref-
erence 6; this rotor is a model of the Bell M266, similar in design
to the full-scale rotor considered here. The experimental data
correlates well with the predictions, including the influence of
the rotor lag motion.

Hingeless, Soft-Inplane Rotor

The effects of several elements of the theoretical model will
be examined for a hingeless, soft-inplane rotor (the Boeing
rotor). Then the theoretical results will be compared with the
results of full-scale tests, and with results from the Boeing theory
(the latter are from Reference 3). The predicted variation of the
system stability with forward velocity at normal rotor speed
(Q, = 386 rpm) is shown in figure 11. The low-frequency flap
(/3-1) mode becomes unstable at 480 knots. By the time this
instability occurs, the mode has assumed the character of a wing
vertical bending mode (i.e., the q x motion, and the associated p,
fie. fis. and j motions); hence this instability has the same
mechanism as does the Bell rotor. With the soft-inplane rotor,
v* < 1/rev, the proximity of the f - 1 and qi mode frequencies
significantly reduces the wing mode damping at low speeds; this
effect is the air resonance phenomenon. A similar influence
occurs with the resonance of the J - 1 and q 2 modes, leading to
an instability of the, wing chord mode (this instability can occur
because the wing chord mode aerodynamic damping remains low
even at high speed). At higher £2, this q 2 mode instability is, in
fact, the critical instability. The influence of the rotor lag motion
is shown in figure 12. The substantial decrease in the damping of
the wing vertical and chordwise bending modes due to the rotor
lag motion is the air resonance effect. Figure 13 shows the
influence of powered operation and of omitting the wing aero-
dynamics, and figure 14 shows the influence of the better theo-
retical model for the rotor aerodynamics on the predicted
stability. The effects, and hence the conclusions from figures 13
and 14 are similar to those for the Bell rotor.

Figure 15 shows the variation of the predicted stability of
the Boeing rotor with rotor speed at 50 knots. At this low speed,
the resonance of the £ - 1 and qj mode frequencies actually
results in an instability of the wing vertical bending mode.
Figure 16 shows the stability variation with rotor speed at
192 knots. The reduction in wing vertical bending mode damping
due to air resonance is still present, but the increase in the rotor
lag aerodynamic damping and wing vertical bending aerodynamic
damping with flight speed has been sufficient to stabilize the
motion even at resonance. Figure 17 summarizes the air
resonance behavior of the Boeing rojor.

Figure 18 compares the predicted and full-scale results for
the stability of the wing modes for a velocity sweep of the
Boeing rotor at £2 = 386 rpm. Figure 19 shows the variation of
the wing vertical bending mode damping with rotor speed at
V = 50 to 192 knots. These runs were conducted to investigate
the air resonance behavior of this configuration, i.e., the influ-
ence of the rotor lag motion. Reasonable correlation is shown
between the predicted and measured stability, except at the
higher speeds where tunnel turbulence made extraction of the
damping ratio from the experimental transient wing motion
difficult. Also shown are predictions from the Boeing theory,
from Reference 3. Additional comparisons with the full-scale test
data are given in Reference 1.

Concluding Remarks

This paper has presented theoretical results for the stability
of a proprotor operating in high inflow on a cantilever wing :
Some experimental results from full-scale tests have been pre-
sented, showing reasonable correlation with the predicted stabil-
ity. The nine degrees-of-freedom theoretical model has been
established as a useful and accurate representation of the basic
dynamic characteristics of the proprotor and cantilever wing
system. The significant influence of the rotor speed perturbation
degree of freedom (i.e., windmilling or powered operation), the
wing aerodynamics, and the rotor aerodynamic model on the
predicted stability have been shown, indicating the importance of
including these elements accurately in the theoretical model.
From a comparison of the behavior of the gimballed, stiff-inplane
rotor and the hingeless, soft-inplane rotor, it is concluded that
the placement of the natural frequencies of the rotor blade first
mode bending — i.e., the flap frequency va and the lag frequency
vy — has a great influence on the dynamics of the proprotor and
wing. Moreover, the theoretical and experimental results have
demonstrated that the rotor lag degree of freedom has a very
important role in the proprotor dynamics, for both the soft-
inplane (y* < 1/rev) and stiff-inplane (v* > 1/rev) configurations.

References

1. NASA TN-D (in preparation), THE DYNAMICS OF TILT-
ING PROPROTOR AIRCRAFT IN CRUISE FLIGHT,
Johnson, Wayne, 1974.

2. NASA CR 114363, ADVANCEMENT OF PROPROTOR
TECHNOLOGY TASK II — WIND TUNNEL TEST
RESULTS, Bell Helicopter Company, September 1971.

3. Boeing Vertol Company Report No. D222-1 0059-1, WIND
TUNNEL TESTS OF A FULL SCALE HINGELESS
PROP-ROTOR DESIGNED FOR THE BOEING
MODEL 222 TILT ROTOR AIRCRAFT, Magee, John P.,
and Alexander, H. R., July 1973.

4. NASA CR 114442, V/STOL TILT-ROTOR STUDY
TASK II - RESEARCH AIRCRAFT DESIGN, Bell Heli-
copter Company, March 1972.

5. NASA CR 114438, V/STOL TILT-ROTOR AIRCRAFT
STUDY VOLUME II - PRELIMINARY DESIGN OF
RESEARCH AIRCRAFT, Boeing Vertol Company, March
1972.

6. Kvaternik, Raymond G., STUDIES IN TILT-ROTOR
VTOL AIRCRAFT AEROELASTICITY, Ph.D. Thesis, Case
Western Reserve University, June 1973.

162

TABLE I - DESCRIPTION OF THE FULL-SCALE
PROPROTORS, AS TESTED IN THE AMES 40- BY 80-FT
WIND TUNNEL.

Bell

Boeing

Rotor

Type

gimballed, stiff-

hingeless, soft-

inplane

Number of blades

'Radius, R

3.81m (12.5 ft)

3.96 m (13 ft)

Lock number, y

3.83

4.04

Solidity ratio

0.089

0.115

Pitch/flap coupling,
5,

-15deg

Rotor rotation direc-

clockwise

counterclockwise

tion, on right wing

Tip speed, SIR

183 m/sec

160 m/sec

(cruise mode)

(600 ft/sec)

(525 ft/sec)

Rotation speed,

458 rpm

386 rpm

(cruise mode)

Wing

Semispan, y w /R

1.333

1.281

Mast height, h/R

0.342

0.354

Typical frequencies

Vertical bending

3.2 Hz 0.42/rev

2.3 Hz 0.36/rev

Chordwise bending

5.35 0.70

4.0 0.62

Torsion

J 9.95 1.30

9.2 1.48

<D g -

\
\

\
\
^

a, rpm

^"^^,

_^-229

^' — .^

(b)

I 1 I I

.-350
J^458
">550,

v/a rpm

Figure 2. Blade rotating natural frequencies for the Bell rotor,
(a) flap frequency vq (normal SI = 458 rpm), (b) lag
frequency v> (V/S2R = 1 at 355 knots and normal SI).

fl, rpm
,l 93

•300 ,386

(a)

^550

V/flR

Figure 1. Configuration of analytical model, and for full-scale
tests: proprotor operating in high inflow axial flight on a
cantilever wing.

Figure 3. Blade rotating natural frequencies for the Boeing rotor
(V/S2R =1 at 311 knots and normal St, 386 rpm). (a) flap
frequency Pa, (b) lag frequency v* .

163

200 400

V, knots

600

V, knots

Figure 5. Effect of deleting the rotor lag degrees of freedom (f i c
and f is ), Bell rotor velocity sweep at fi = 458rpm.

(a) damping of wing vertical bending mode (q t ),

(b) damping of chordwise bending (q 2 ) and torsion (p)
modes.

VELOCITY SWEEP

»H 1 — I 1 — I — h—

25 200 400 600

V, knots

2J0

1.0

Rex

AUTOROTATION, WITH WING AERODYNAMICS

POWERED

MO WING AERODYNAMICS

-.05 FOR q,, q 2 , p

600

Figure 4. Predicted stability of Bell rotor, velocity sweep at Figure 6. Influence of powered operation, and wing aerodynamic
12 = 458 rpm. (a) frequency of the modes, (b) damping forces, Bell rotor velocity sweep at Q. = 458 rpm.

ratio of the modes, (c) root locus. (a) damping of wing vertical bending mode (qj),

(b) damping of chordwise bending (q 2 ) and torsion (p)

modes.

164

.05

MORE CQHPLETE MODEL FOR ROTOR AEROOTIIAitlCS

OKLY C La TERMS Id ROTOR AERODtHAHIC C0EFF1GIEHTS

H H- HELICAL TIP MACH NUMBER LIMITS

CRITICAL SONIC

POKERED

.15

s~/ \

^^^AIITOROTATION \

POWERED // / KT~ V \\

.10

yv / \*s\\-

/ / / ' \\

A // \

/' / / T\

/' /"' w

.05

~j^^^=r^-_l'0*ERED \\

AUT0R0TATI0in\^ N^-\A "^

(b, ,

i i i \ \i \ r^-^ i

200 400

V, knots

600

Figure 7. Influence of a more complete model for the rotor
aerodynamics, Bell rotor velocity sweep at SI = 458 rpm.

(a) damping of wing vertical bending mode (q,),

(b) damping of chordwise bending (qj) and torsion (p)
modes.

(a!

.05

(b)

j05

(c)

.05 -

td)

PRESENT THEORY

NONLINEAR

LINEAR

o EXPERIMENT

BELL THEORY

" ^ HJ Ltj ' t! ■wWaaBua ^Basr- sf i " "rr i n*

_ g_

100

200

V, knots

Figure 8. Comparison with full-scale experimental data, Bell
rotor velocity sweep at S2 = 458 rpm. (a) frequency of the
modes, (b) damping of the wing vertical bending mode (qj ),

(d) damping of wing torsion mode (p).

.05

(a)

i i i i ii i i i i

.5
V/flR

1.0

Figure 9. Influence of the rotor lag motion. Bell rotor velocity sweeps on the full-stiffness wing, on a quarter-stiffness wing, and on
the full-stiffness wing without the $Vc and ?is degrees of freedom (theory only), (a) damping of the wing vertical bending
mode (q, ), comparison with full-scale experimental data, (b) damping of wing vertical bending mode (q, ), (c) damping of wing
chordwise bending (q 2 ) and torsion (p) modes, (d) frequency of the modes.

165

.15

.10

.05

EXPERIMENT THEORY

o FULL STIFFNESS WING

• QUARTER STIFFNESS WINS

FULL STIFFNESS WING, WITHOUT £ |C , £, s

-^-_.

(d)

A-

1.0

V/flR

Figure 9. Concluded.

o EXPERIMENT (0.1333 SCALE MODEL)

.08

WITHOUT £ |C , £ |S

/ \

°^^-i

> \ \

o\ \

.04

o\ 1

\ ° 1

^^^

(a) 238 rpm

1 1

1 A

200 x N

V, knots

400

+ 1

£ + 1

T~~

Qi\

C-K-—

-— ^-1

(ah-

1 1 1

,A ,

-Rex

-.05 FOR q |r q 2

Figure 10. Comparison with experimental data from tests of a
0.1333-scale rotor and cantilever wing model of Bell M266

aircraft (experimental points from Reference 6), velocity Figure 11. Predicted stability of Boeing rotor, velocity sweep at
sweeps at (a) 12 = 238 rpm, (b) £2 = 298 rpm, $2 = 386 rpm. (a) frequency of the modes, (b) damping

166

WITHOUT J |C , C (S

WITH £.., {

.10

.06

AUTOROTAT10H ,

WITH WINS AERODYNAKICS

POWERED

HO WING AERODYNAMICS

.15

.10

.05 -

600

200 400

V, knots

600

Figure 1 2. Effect of deleting the rotor lag degree of freedom (f t c
and ?! s ), Boeing rotor velocity sweep at to = 386 rpm
(a) damping of wing vertical bending (q t ) and flap ((3-1)
modes (the (3 - 1 mode is shifted by 250-300 knots to
higher speed by the removal of the lag influence, beyond
the scale shown), (b) damping of wing chordwise bending
(q 2 ) and torsion (p) modes.

Figure 13. Influence of powered operation, and wing
aerodynamic forces, Boeing rotor velocity sweep at
to = 386 rpm. (a) damping of wing vertical bending (q a )
and rotor flap 03-1) modes, (b) damping of wing
chordwise bending (q 2 ) and torsion (p) modes.

600

v, knots

Figure 14. Influence of a more complete model for the rotor aerodynamics. Boeing rotor velocity sweep at to = 386 rpm (a) damp-
ing of wing vertical bending (q, ) and rotor flap 03—1) modes, (b) damping of wing chordwise bending (q t ) and torsion (p)
modes. -

167

.15 r

600

Figure 15. Predicted stability of Boeing rotor, rpm sweep at Figure 16. Predicted stability of Boeing rotor, rpm sweep at
50 knots, (a) frequency of the modes, (b) damping ratio of 192 knots, (a) frequency of the modes, (b) damping ratio of

the modes. the modes.

400

a, rpm

600

Figure 1 7. Air resonance behavior of soft-inplane hingeless rotor. Boeing rotor at 50 to 192 knots, variation of damping of

wing vertical bending mode (q, ) with rotor speed.

168

•PRESENT THEORY

2 r

JUL
a

BOEING THEORY

O EXPERIMENT

-g— o-

(0) 4 I

400

Figure 18. Comparison with full-scale experimental data, Boeing rotor velocity sweep at SI = 386 rpm. (a) frequency of the modes,
(b) damping of the wing vertical bending (qi ), chord bending (q 2 ), and torsion (p) modes; the experimental data is for q! only.

• 60 knots

EXPERftiEHT

o 50 knots

.03, 80 J* fto t S ) PRESEKT THEORY p

50 knots I

BOEING THEORV, 60 knots

On o

o \»P

(b) IOO knots ^X
1 1 ^ii-

/ /
//

f i

PRESENT THEORY

BOEING THEORY

o EXPERIMENT

J03

.02 - \>

" s -^. ,o ©

\>oo
0H50

°° //

\P[

f 1

► 1

(c! 140 knots o

" \ 1

300

400 500

a, rpm

\ G oo o

-\v °^° /

(d) 192 knots

600 300

400 500

a, rpm

600

Figure 1 9. Comparison with full-scale experimental data, Boeing rotor rpm sweeps, damping of wing vertical bending mode at

(a) 50-60 knots, (b) 1 00 knots, (c) 140 knots, (d) 192 knots.

169

EXPERIMENTAL AND ANALYTICAL STUDIES IN TILT- ROTOR AEROELASTICITY '

Raymond G. Kvaternik

Aerospace Technologist

NASA Langley Research Center

Hampton, Virginia

Abstract

An overview of an experimental and analytical
research program underway within the Aeroelasticity
Branch of the NASA Langley Research Center for
Studying the aeroelastic and dynamic characteris-
tics of tilt-rotor VTOL aircraft is presented.
Selected results from several joint NASA/contractor
investigations of scaled models in the Langley
transonic dynamics tunnel as well as some results
from a test of a flight-worthy proprotor in the
NASA Ames full-scale wind tunnel are shown and dis-
cussed with a view toward delineating various
aspects of dynamic behavior peculiar to proprotor
aircraft. Included are such items as proprotor/
pylon stability, whirl flutter, gust response, and
blade flapping. Theoretical predictions, based on
analyses developed at Langley, are shown to be in
agreement with the measured stability and response
behavior.

Notation

e Blade flapping hinge offset

H Rotor normal shear force

cfflt/oNx Rotor normal shear force component in
phase with pitch angle

5H/5q, Rotor normal shear force component in
phase with pitch rate

Blade radius

0.75 blade radius

Rotor perturbation thrust

Airspeed

v F /nR

Flutter advance ratio

Vertical component of gust velocity

Mast angle of attack

a Oscillation amplitude of airstream
oscillator

B Blade flapping angle 1

3(3 /da Blade flapping derivative

8, Pitch-flap coupling angle

Presented at the AHS/NASA Ames Specialists' Meeting
on Rotorcraft Dynamics, February 13-15, 197^-

e Gust- induced angle of attack
g

£_ Hub damping ratio

+ Aircraft yaw rate

SI Rotor rotational speed

Q Frequency

oo Blade flapping natural frequency

u> Pylon pitch frequency

co. Pylon yaw frequency

The feasibility of the tilt-proprotor com-
posite aircraft concept was established in the mid
1950' s on the basis of the successful flight
demonstrations of the Bell XV-3 and Transcendental
Model 1-G and Model 2 convert iplanes. Flight
research conducted with the XV-3 identified
several dynamic deficiencies in the airplane mode
as technical problems requiring further atten-
tion.-'- A more serious proprotor dynamic problem
was identified in a 1962 wind-tunnel test of the
XV-3- In that test, conducted in the Ames full-
scale tunnel, a proprotor /pylon instability simi-
lar in nature to propeller whirl flutter was
encountered. Clearly, to maintain the viability
of the tilt-proprotor concept it remained to
demonstrate that neither the whirl flutter anomaly
nor the major flight deficiencies were endemic to
the design principle. An analytical and experi-
mental research program having this objective was
undertaken by Bell in 1962. Results of this
research, which defined the instability mechanism
and established several basic design solutions,
were reported by Hall. 2 Edenborough? presented
results of subsequent full-scale tests at Ames in
1966 which verified the analytical prediction tech-
niques, the proposed design solutions, and demon-
strated stability of the XV-3 through the maximum
wind-tunnel speed of 100 m/s (195 kts).

In 1965 the U.S. Army inaugurated the Com-
posite Aircraft Program which had the goal of
producing a rotary- wing research aircraft combin-
ing the hovering capability of the helicopter with
the high-speed cruise efficiency and range of a
fixed-wing aircraft. Bell Helicopter Company,
with a tilt-proprotor design proposal, was awarded
one of two exploratory definition contracts in
1967 . The Model 266 was the design resulting from
their work (Fig. 1). The research aircraft pro-
gram which was to have been initiated subsequent
to the exploratory definition phase was never
begun, however, primarily due to lack of funding.

171

Figure 1. Artist's conception of Bell Model 266
tilt-proprotor design evolved during the Army
Composite Aircraft Program.

Concurrent with the developments described
above, various VTOL concepts based on the use of
propellers having independently hinged blades were
proposed with several reaching flight-test status.
These included the Grumman proposal in the Tri-
Service VTOL Transport competition, the Vertol VZ-2
built for the Army, and the Kaman K-l6 amphibian
built for the Navy. A vigorous investigation of the
whirl flutter phenomenon peculiar to conventional
propellers had been initiated in i960 as a result
of the loss of two Lockheed Electra aircraft in
fatal accidents. The possibility that hinged
blades could adversely affect the whirl flutter
behavior of a propeller undoubtedly contributed
considerable impetus to examine the whirl flutter
characteristics of these flapping propellers. Wort
related to these efforts was reviewed by BeedA

The foregoing constitutes a resume of
proprotor-ralated experience through 1967. This
paper will present an overview of a research pro-
gram initiated within the Aeroelasticity Branch of
the NASA Langley Research Center.- Included in this
program are joint NASA/contractor wind-tunnel
investigations of scaled models in the transonic
dynamics tunnel and the in-house development of
supporting analyses. For completeness, motivating
factors leading to the work and the scope of the
investigation are outlined below.

A 0.133- scale semispan dynamic and aeroelastic
model of the Model 266 tilt rotor built by Bell in
support of their work pertaining to the Composite
Aircraft Program was given to Langley by the Army
in 1968. The availability of this model and the
interest of both government and industry in the
tilt- rotor VTOL aircraft concept suggested the use-
fulness of continuing the experimental work ini-
tiated by Bell with the model to further define the
aeroelastic characteristics of proprotor-type air-
craft. Because both the XV-3 experience and studies
conducted during the Composite Aircraft Program
identified certain high-risk areas associated with
operation in the airplane mode of flight,

specifically proprotor /pylon stability (whirl
flutter), blade flapping, and flight mode stability,
it was judged that the research effort would be
primarily directed to these areas.

The experimental portion of the research pro-
gram was initiated in September 1968 in a joint
NASA/Bell study of proprotor stability, dynamics,
and loads employing the 0. 133-scale semispan model
of the Model 266, Several other cooperative
experimental studies followed this investigation.
The models employed in these studies are positioned
in chronological order in the composite photo giverj
in Figure 2. Briefly, these other studies include^:
(l) A study of a folding proprotor version of the/
tilt-rotor model used in the first study, (2) a
parametric investigation of proprotor whirl flutter,
(3) a stability and control investigation employing
an aerodynamic model, and (k) a "free- flight"
investigation of a complete tilt-rotor model.

TILT-ROTOR AEROELASTIC RESEARCH

LANGLEY TRANSONIC DYNAMICS TUNNEL

Figure 2. Tilt-rotor models tested in the Langley
transonic dynamics tunnel.

The results pertaining to the above-mentioned
studies are quite extensive. The particular results
to be presented herein have been selected with a
view toward highlighting some of the dynamic
aspects of proprotor behavior, delineating the
effects of various design parameters on proprotor/
pylon stability and response, and providing valida-
tion of analyses developed at Langley. The results
pertaining to investigations conducted in the
Langley transonic dynamics tunnel are presented
first. These are arranged in chronological order
according to Figure 2. To provide additional data
for correlation, some experimental results obtained
by Bell in tests of a semispan model and a full-
scale flight-worthy proprotor are also included.
In each case both experimental and analytical
results are for the pylon fully converted forward
into the airplane mode of operation and the rotors
in a windmilling condition. Equivalent full-scale
values are given unless noted otherwise.

172

Model Tests in Langley Transonic Dynamics Tunnel

Bell Model 266

(a) September I968

Although the 0.133- scale semispan model of the
Bell Model 266 was not designed to permit extensive
parametric variations, in that it represented a
'specific design, it did permit a fairly diversified
|test program. The principal findings of this inves-
tigation have been published and are available in
the literature. 5, 6 seme results adapted from
Reference 6 pertaining to stability and gust
response are discussed below.

Proprotor /Pylon Stability. To provide an -
indication of the relative degree to which stabil-
ity could be affected, and to provide a wide range
of configurations for correlation with analysis,
several system parameters were varied either indi-
vidually or in combination with other parameters
and the level of stability established.

A baseline stability boundary, based on a
reference configuration, was first established.
The degree to which stability could be affected was
then ascertained by varying selected system param-
eters (or flight conditions). Stability data were
obtained by holding rpm constant as tunnel speed
was incrementally increased, transiently exciting
the model by means of lightweight cables attached
to the model, and analyzing the resulting time
histories to determine the damping. The reference
configuration consisted of the basic Model 266
parameters with the pylon yaw degree of freedom
locked out and the wing aerodynamic fairings
removed. A 100$ fuel weight distribution was
maintained by appropriately distributing lead
weights along the wing spar. The hub flapping
restraint was set to zero and the S3 angle to
-0.393 radian (-22.5°). The reference stability
boundary as well as changes in this boundary due to
several parameter variations are shown in Figure 3-

For the reference configuration instability
occurred in the coupled pylon/wing mode in which
the pylon pitching angular displacement is in phase
with the wing vertical bending displacement. A
characteristic feature of this coupled mode is the
predominance of wing bending (relative to pylon
pitch) and the frequency of oscillation, which is
near the fundamental wing vertical bending natural
frequency. For descriptive purposes this flutter
mode is termed the "wing beam" mode herein. Negli-
gible wing chordwise bending or rotor flapping
(relative to space) was observed. The pylon/rotor
combination also exhibited a forward whirl preces-
sional motion, the hub tracing out an elliptical
path in space. However, because of the large ratio
of pylon yaw to pylon pitch stiffness the pylon
angular displacement was primarily in the pitch
direction. The flutter mode of the model in each
of its perturbations from the reference configura-
tion was essentially the same as for the reference
configuration.

The proprotor /pylon instability described
above is similar in nature to classical propeller

RPM

4001-

1 Flight condition
for gust response

Measured Calculated

q _^_ — i — ,_ Reference boundary

A *-' Altitude, 3668 m (12000 ft)

O __-.__-— Pylon yaw unlocked (fy aw » 6 Hz)

O -- — :.., Hub restraint

□ — ^ — Wing aerodynamics

300 400

Airspeed, knots

Figure 3-

Airspeed, meters/sec

Effect of several system parameters on
proprotor/pylon stability.

whirl flutter. However, because of the additional
flapping degrees of freedom of the proprotor the
manner in which the precession generated aerody-
namic forces act on the pylon is significantly
different." Specifically, while aerodynamic cross-
stiffness moments are the cause of propeller whirl
flutter, the basic destabilizing factors on
proprotor/pylon motion are aerodynamic in plane
shear forces which are phased with the pylon motion
such that they tend to increase its pitching or
yawing velocity and, hence, constitute negative
damping on the pylon motions.

(1) Altitude - Altitude has a highly benefi-
cial effect on proprotor/pylon stability. This
increased stability is a consequence of the fact
that the destabilizing rotor normal shear forces
decrease with altitude for pylon pitch frequencies
near the fundamental wing elastic mode frequencies.
This means that a given level of these destabiliz-
ing shear forces is attained at progressively
higher airspeeds as altitude increases.

(2) Hub Flapping Restraint - A stabilizing
effect due to moderate flapping restraint is also
indicated in Figure 3- Increasing the flapping
restraint increased the flapping natural frequency
from its nominal value of about 0.8o/rev bringing
it closer to the "optimum" flapping frequency in
the sense of Young and Iytwyn.f They showed that
this increased stability because the pylon support

173

stiffness requirements were reduced as the optimum
flapping frequency was approached.

(3) Wing Aerodynamics — Figure 3 indicates
that wing aerodynamic forces have a slight stabi-
lizing effect. Now the stiffness of a strength-
designed wing for tilt-rotor application is
generally sufficiently high to relegate the flutter
speed of the pylon/wing combination (with blades
replaced by lumped concentrated weights) to speeds
well beyond the proprotor mode flight envelope.
This suggests that wing aerodynamics will contrib-
ute primarily to the damping of any coupled rotor/
pylon motions. This is substantiated in Figure It-,
which shows the variation of the wing beam mode
damping with airspeed through the flutter point for
the reference configuration and the corresponding
configuration with the wing airfoil segments
installed. The damping of the mode is increased;
however, the magnitude of the increase is small
indicating that proprotor aerodynamic forces are
predominant in the ultimate balance of forces at
flutter. This provides some justification for
neglecting, in this flutter mode at least, wing
aerodynamics as a first approximation.

a - 5 Hz (298 EPM)
Measured Calculated

Without wing aerodynamics
With wing aerodynamics

■Blade inplane flexibility included

200 300

Airspeed, knots

_1 l_

100 .150

Airspeed, meters/sec

Figure h. Comparison of measured and calculated
wing beam mode damping for reference
configuration.

The initial increase in the stability of the
wing beam mode before instability occurs is asso-
ciated with the fact that dH/dq, the component of
the normal shear force associated with pylon pitch
rate, initially becomes more stabilizing with
increasing airspeed until 5H/ck%, the component of
the normal shear force in phase with pylon pitch
angle, becomes sufficiently large to lower the
coupled pylon pitch frequency to a level where
cffl/dq becomes increasingly destabilizing with
increasing airspeed. " The increased damping
response at about 103 m/s (200 kts) is due to
coupling of the blade first inplane cyclic mode
with the wing beam mode. Uote, however, that the

predicted flutter speed is not sensitive to blade
inplane flexibility for the Model 266.

{k) Pylon Restraint — When the pylon yaw
stiffness was reduced by unlocking the pylon yaw
degree of freedom and soft-mounting the pylon in
yaw relative to the wing tip the stability
decreased slightly (Fig. 3). The particular yaw :'
flexibility employed in this variation effectively f
produced a more nearly isotropic arrangement of the f
pylon support spring rates. Since the region of !f
instability in a plot of critical pylon yaw stiff-
ness against critical pitch stiffness is extended
along the line representing a stiffness ratio of
unity, the configuration approaching isotropy in
the pylon supports is more prone to experience an
instability than one in which one of the stiff-
nesses is significantly less than the other.

The general trend of decreasing stability with
increasing rotor speed shown in Figure 3 was found f
for all values of the adjustable parameters of the*
model. In each case the, predicted flutter mode
and frequency were in agreement with the correspond-
ing measured mode and frequency.

Gust Response . Analytical methods for deter-
mining aircraft response to turbulence are usually
based on power spectral analysis techniques which
require the definition of the aircraft frequency
response function, that is, the response to sinu-
soidal gust excitation. A study to assess the
feasibility of determining these frequency response
functions for fixed-wing aircraft utilizing models
in a semi-free-flight condition using a unique air-
stream oscillator system in the transonic dynamics
tunnel has been underway within the Aeroelasticity
Branch for several years. ° This system (Fig. 5)
consists of two sets of biplane vanes located on
the sidewalls of the tunnel entrance section. The

Figure 5. Langley transonic dynamics tunnel air-
stream oscillator showing cutaway of driving
mechanism.

vanes can be oscillated in phase or l80 c out of
phase to produce nominally sinusoidal vertical or
rolling gusts, respectively, over the central por-
tion of the tunnel. The gusts are generated by the
cross- stream flow components induced by the trail-
ing vortices from the tips of the vanes. With a

174

view toward the possible application of this tech-
nique to rotary-wing aircraft the airstream oscil-
lator was employed to excite the model for several
"flight" c, Llity

boundary. Alt i '■ee" the

data so obtained did give an indication of the
frequency response characteristics of the c&ntl-
levered mc m of the

effects of airspeed, rotor speed, and rotor and
wing aerodynamics on the overall dynamic response.

A measure of the gust- induced angle of attack
(or stream angle) was provided by means of a small
balsa vane flow direction transmitter (see Fig. 6)
which gave readings proportional to the stream
angle. The variation of the vertical component of

been normalized by the maximum amplitude of the
stream angle using the curve of Figure 7«

Figure 6. 0.133- scale semispan tilt-rotor model in
simulated conversion mode showing boom-mounted
flow direction transmitter.

the stream angle for in phase (symmetrical) oscil-
lation of the biplane vanes is shown in Figure f.
The curve shown is actually an average of data
obtained from runs at several tunnel speeds and
air densities. The amplitude of the stream angle
has been normalized on the maximum amplitude of
oscillation of the biplane vanes arid plotted
against the frequency parameter os/v, where m is
the frequency of oscillation of the biplane vanes
in rad/sec and V is the tunnel speed in m/s
(ft /sec). This parameter is proportional to the
reciprocal of the wavelength (spacing) between
vortices shed from the tips of the oscillating
vanes.

The frequency response of wing vertical bend-
ing moment was taken as one measure of system
response to vertical gust excitation. To ascertain
the relative influence of rotor and wing aerody-
namics, three model configurations were employed:
wing only, with the rotor blade weight replaced by
an equivalent lumped weight; rotor only, with the
wing aerodynamic fairings removed; wing and rotor
combined. For the "flight" condition indicated in
Figure 3 the r fects of rotor and wing
aerodynamics are & ■ -, . I ' rjures 8 and 9. In
each of these figures the wing bending moment has

/ \

Proprotor

Flow direction '
transmitter-

rad/ft

_i L

.3 .6 .9 1.2 1.5 1.8

rad/m

Wavelength parameter, w/V

Figure 7- Measured variation of vertical component
of gust angle with frequency parameter for vanes
oscillating in phase.

Comparison of the rotor-on and rotor-off
response curves for the wing panels on configura-
tion is shown in Figure 8. Two proprotor-related
effects are indicated: first, the significant
contribution of the rotor inplane normal force
(H-force) to wing bending response, as indicated by
the relative magnitudes of the bending moments; and
second, the rotor contribution to wing beam mode
.damping,* as indicated by the relative sharpness of
the resonance peaks. The peak amplitudes occur
when the gust frequency is in resonance with the
wing beam mode frequency. The peak for the blades-
off condition is shifted to the higher frequency
side of the rotor-on peak because the rotor H-force
decreases the frequency of the wing beam mode. For
the rotor-on case the bending moment is consider-
ably larger than for the. rotor-off case throughout
the range of gust frequencies investigated. The
wing chord mode frequency (about 2.8 Hz) is within
the gust frequency range but is absent from the
.response curves because the gust excitation is

At this particular airspeed, the rotor was
still contributing positive damping to the wing
beam mode.

175

Wing airfoil segments installed
V . 200 knots (102.8 m/s)

Measured Calculated

O a - 238 RPM (4 Hz)

- Rotor off (rotor weight replaced by
equivalent lumped weight)

~u\5 O O 275 CS O -

Simulated full-scale gust frequency, Hz

Figure 8. Effect of proprotor aerodynamics on wing
root tending moment amplitude response function.

in-lb/deg

N-m/rad

a « 238 HPM (4 Hz)

V - 200 knots (102.8 m/s)

Measured Calculated

O Wing airfoil segments removed

A Wing airfoil segments installed

_1_

_L_

_i_

0.5 1.0 1.5 2.0 2.5 3A~

Simulated full-scale gust frequency, Hz

Figure 9> Effect of wing aerodynamics on wing root
bending moment amplitude response function.

primarily vertical and there is very little
coupling between the wing beam and chord modes.

Figures 8 and 9 quite clearly illustrate that
proprotor s operating at inflow ratios typical of
tilt-rotor operation in the airplane mode of flight
are quite sensitive to vertical gusts. This sensi-
tivity is due to the fact that the proprotors,
being lightly loaded in the airplane mode of flight,
operate at low blade mean angles of attack (a) and
any gust- induced angle of attack is a significant
fraction of a.

Hote that good correlation is achieved for
frequencies up to about 2 Hz beyond which the cal-
culated responses are much lower than the measured
values. This discrepancy is thought to be a con-
sequence of the deviation of the induced gust from
its nominally one- dimensional nature to one which
is highly two-dimensional (i.e., varies laterally
across the tunnel) at the higher frequencies. The
analytical results shown are based on the assump-
tion of a one- dimensional gust. Unsteady aerody-
namic effects may also be a contributing factor to
the discrepancy. v

A comparison of the wing panels-on and wing
panels-off response curves for the rotor-on con-
figuration is given in Figure 9- As might be
expected, the wing response for the case in which
the wing airfoil segments are installed is higher
than for the rotor alone. The reduced magnitude of
the response at resonance for the rotor-plus-wing
combination relative to the rotor alone is due to
the positive damping contributed by the wing aero-
dynamics. This increased damping is evident by
comparing the widths of the resonance peaks.

Close examination of Figures 8 and 9 reveals
a very heavily damped, low amplitude resonance
"peak" at a gust frequency of about 0.8 Hz. This
resonance is a manifestation of the low-frequency
(i.e., £1 - fita) flapping mode. Analyses have indi-
cated that the flapping modes are generally well
damped for moderate or zero values of flapping
restraint." These results constitute an experi-
mental verification.

These results indicate that "free- flight"
tilt-rotor models could be used to measure the
frequency response functions needed in gust
response analyses. This would be a fruitful area
for future analytical and experimental research.

(b) January 1970

A joint HASA/Bell/Air Force test program was
conducted in the transonic dynamics tunnel in
January 1970 for the purpose of investigating any
potential problem areas associated with the folding
proprotor variant of the tilt-rotor concept. The
model used in this study was the same model
employed in the first investigation but modified to
permit rapid feathering and unf eathering of the
proprotor and to include a blade fold-hinge. The
main objectives were to investigate stability at
low (including zero) rotor rotational speeds,

176

during rotor stopping and starting, and during
blade folding. All objectives of the test program
were met. No aeroelastic instabilities were
encountered during the blade folding sequence of
transition, the blade loads and/or the feathering
axis loads inboard of the fold hinge being identi-
fied as the critical considerations from a design
point of view. The stop- start portion of the test
indicated that additional flapping restraint would
be required to minimize flapping during rotor
stopping.* Stability investigations conducted over
a wide range of rotor speed identified an apparently
new form of proprotor instability involving the
rotor at low and zero rotational speeds. The
influence of several system parameters on this
instability was established both experimentally and
analytically. 6

Proprotor /Pylon Stability . For the stability
investigation a reference configuration was again
established. This consisted of the basic Model 266
configuration with the pylon locked to the wing tip
in both pitch and yaw, a hub restraint of
117,685 N-m/rad (86,800 ft-lb/rad), 03 = -0.595 rad
(-22.5°), a simulated wing fuel weight distribution
of 15$, and the wing aerodynamic fairings installed.
The flutter boundary obtained for this configuration
and that for 03 = -0.558 rad (-52°), are shown in
Figure 10 as a function of rotor speed. Open sym-
bols denote flutter points. Excessive vibration
resulting from operation near resonances with the
pylon/wing or blade modal frequencies often limited
the maximum attainable airspeed. These points are
indicated by the solid symbols. The annotation to
the right of the flutter boundaries indicates that
the model experienced several modes of flutter.
The predicted flutter modes and frequencies were
in agreement with the experimental results. The
nature of these flutter modes is discussed below.

For Q greater than about k Hz (2^0 rpm)
instability occurred in the wing beam mode and had
the characteristics described earlier for the
September 1968 test. For fi between about 2 Hz
(120 rpm) and k Hz (2^0 rpm) the motion at flutter
was predominantly wing vertical bending and rotor
flapping with the hub precessing in the forward
whirl direction. Examination of the root loci
indicated that this instability was associated with
the low- frequency (i.e., 0. - cup) flapping mode root
becoming unstable. The subcritical response through
flutter for 83 = -0.558 rad (-52°) and a = 2.86 Hz
(172 rpm) is shown in Figure 11 where, in addition
to the measured wing beam mode damping and frequency,
the calculated variation of both the wing beam and
low-frequency flapping modes is shown. These
results illustrate an interesting modal response
behavior similar to that described by Hall. 2 The
wing beam mode, being least stable at low airspeeds,
is at first dominant. As airspeed increases, how-
ever, its damping continually increases. The damp-
ing of the fl - <»q flapping mode meanwhile is
continually decreasing. Crossover occurs analyti-
cally at Ikh m/s (280 lets) at a damping of 17$ of

These aspects of this investigation are given
detailed treatment in Reference 9-

300 400

Airspeed, knots

100 200

Airspeed, meters/sec

Figure 10. Model 266 flutter boundaries showing
variation in character of flutter mode as rpm is
reduced to zero.

critical. Beyond 280 knots, the fi - fflp flapping
mode is the dominant mode and very abruptly becomes
unstable as airspeed is increased. Hence, a tran-
sition from a dominant wing beam mode to a dominant
flapping mode with an accompanying change in fre-
quency. Since the flapping mode frequency is only
slightly less than the wing beam mode in the
vicinity of flutter there is only a gradual, albeit
distinct, transition in the frequency of the wing
beam mode as the flapping mode begins to predominate
over the wing mode. Examination of the Q - cup
flapping mode eigenvector indicated that a larger
amount of wing vertical motion was evident in this
mode than in the wing beam mode eigenvector. This
implies that the predominant motion in the flutter
mode is not necessarily determined by the root
which analytically goes unstable as airspeed is
increased but the frequency at which a root goes
unstable.

Below about 2 Hz (120 rpm) instability is in
the high-frequency (i.e., JJ + coo) flapping mode and
is characterized by large amplitude flapping, the
rotor tip-path-plane exhibiting a precessional
motion in the forward whirl direction. The modes
of instability at zero rotational speed were similar
in character to those at low rotor speeds but with
larger amplitudes of flapping. Although the rotor
was not turning, the flapping behavior of the blades

177

200 300

Airspeed, knots

Wife"

100 150

Airspeed, meters/sec

Figure 11. System response characteristics for
flutter at Jl = 172 rpm and S3 = -3?.".

was patterned such that the tip-path-plane appeared
to be wobbling or whirling in the forward direction.
Negligible wing motions accompanied the flapping
motion. Figure 12 shows the variation of flap
damping with airspeed. A hub damping of £g = 0.015
was originally used in calculating the stability

6 3 » -.393 rad (-22.5°)

? R - .020

O Measured
Calculated

300

„sr

Airspeed, knots

_l_

100
Airspeed, meters/sec

150

boundaries, leading to very conservative values for
the flutter speed at the low rotor speeds. Based on
the results of Figure 12, which indicate that the
rotor hub structural damping is closer to £ R = 0.025,'^
the stability boundaries were recalculated using
£r = 0.025. The predicted boundaries in Figure 10
reflect this change.

The small region of increased stability in the
region of 0.8 Hz (kQ rpm) is due to a favorable
coupling of the flapping mode with wing vertical
bending.

The instabilities encountered at low and zero
values of rotational speed were quite mild and had
a relatively long time to double amplitude. The
necessity of limiting the flapping amplitude during
the feathering sequence of transition dictates that
significantly increased values of hub restraint are
needed as rotor rotational speed is reduced to zero.
Since increased flapping restraint was found to
stabilize this mode° this instability is probably
only of academic interest, at least for the config-
uration tested. However, since it was a new
phenomenon and was not understood a,t the time of
the test, attention was directed to assessing the
effect of the variation of several system parameters
on the flutter speed. Both experimental and analyt-
ical trend studies were conducted for this purpose. °
Based on these studies it was concluded that rotor
precone was the primary cause of the instability.

Blade Flapping . In the feathering sequence of
transition flapping sensitivity to a given mast
angle of attack varies with rotor rotational speed.
A typical variation of steady- state one-per-rev
flapping response is given in Figure 1>. These
data were taken to establish a steady-state flapping
response baseline for evaluating the transient

rad

.ior

«; .04

Measured

- .02618 rad (1.5°)
-a_ - .01745 rad (1.0°)

100 200

Proprotor speed, RPM

1 1

Proprotor speed, Hz

Figure 12. Variation of fl + (Da flapping mode Figure 13.
damping with airspeed for zero rpm.

Variation of blade flapping with rotor
rpm.

178

flapping response during the feathering portion of
the test. Since the proprotor mast was not affixed
to a rigid backup structure the wind- on mast angle
of attack was not known (it was nominally 1° ) . The
important conclusion following from Figure 15 is
that the measured trend is predicted correctly.
The peak in the flapping response occurs when the
rotor rotational speed is in resonance with the
flapping natural frequency in the rotating system.

Grumman Helicat (March 1971 )

A wide variety of technical considerations
confront the structural dynamicist in the design
of a proprotor VTOL aircraft. Perhaps the most
celebrated consideration has been that of prop-
rotor/pylon whirl flutter, having been the concern
of many investigators in both government and
industry. Several years ago Baird 1 - raised the
question of whether proprotor whirl flutter, in
particular forward whirl flutter, could be pre-
dicted with confidence. His skepticism was
prompted by the lack of agreement between the
experimental results obtained with several small
models of flapping-blade propellers and the corre-
sponding theoretical predictions.^ To provide a
large data base from which to assess the predict-
ability of proprotor whirl flutter, a joint NASA/
Grumman investigation was conducted in the tran-
sonic dynamics tunnel employing an off-design
research configuration of a 1/4.5- scale semispan
model of a Grumman tilt-rotor design designated
"Helicat" (Fig. 14). This design is characterized

Figure 14. Grumman "Helicat" tilt-rotor model in
whirl flutter research configuration.

by a rotor which incorporates offset flapping
hinges in contrast to the Bell rotor in which the
blades are rigidly attached to, the hub which is in
turn mounted on the drive shaft by a gimbal or
universal joint housed in the hub assembly. The
Helicat model was specifically designed to permit
rather extensive parametric changes in order to
provide a wide range of configurations. These
variations included pylon pitch and yaw stiffness
and damping, hinge offset, and pitch- flap coupling.
To obtain flutter at low tunnel speeds, a reduced-
stiffness pylon-to-wingrtip restraint mechanism

which permitted independent variations in pitch
and yaw stiffness was employed. The resulting
pylon-to-wing attachment was sufficiently soft to
insure that the wing was effectively- a rigid backup
structure. Details concerning this model as well
as a summary of results are contained in
Reference 11.

Some whirl flutter results are given in Fig-
ures 15 to 17, where flutter advance ratio Vp/QR
is plotted versus pylon frequency nondimensionalized
by the rotor speed. The effect of 85 on stability

1.4

Symmetrical pylon frequencies
e/R » .05

1.2

6 3 > .349 rad (20°)

1.0

6- « .118 rad (6.75°) /

i / 6 3 " - 524 rad ( 30 °)

P / !

I of 4

.6
.4

■■/ /*/

/ A^ 7

1 f£{ Measured

'/ / °

' / A

/ □

1 1 1

Calculated

1 1

1.0

Pylon frequency, cycles/rev

Figure 15.

Effect of pitch-flap coupling on whirl
flutter.

is shown in Figure 15 for the case in which the
pylon pitch and yaw frequencies are identical and
e/R set to 0.05- Many of the configurations were
not exactly symmetrical in the frequencies. These
data were adjusted to reflect a symmetric frequency
support condition using Figure 18 of Reference 11.
The results show a strong increase in flutter
advance ratio (and hence flutter speed for a fixed
rpm) with increasing pylon support stiffness and
decreasing 63. All flutter was in the forward
whirl mode except for the two points denoted by the
solid symbols, which were in the backward mode.
The analytical results shown assumed a symmetric
frequency configuration and, since the structural
damping varied somewhat, an average value of damp-
ing of i = 0.01 in pitch and I = 0.02 in yaw.
The analytical results shown were obtained using the
theory of Reference 6 which is based on the assump-
tion of a gimbaled rotor. For analysis purposes the
restoring centrifugal force moment from the offset
flapping hinge was represented by introducing an

179

equivalent hub spring which preserved the blade
in-vacuum flapping natural frequency in the manner
indicated in Appendix B of Eef erence 6.

The beneficial effect of increased hinge off-
set is demonstrated in Figure 16. The results for
the 13$ hinge offset are particularly noteworthy

1.4

Symmetrical pylon frequencies
6 3 • .349 rad (20°)

1.2

e/R

. .05

1.0
.8

e/R m .13 /
/ /

^ /
/ /

/ /
/ °/

A / /
/ (J

/ y

/ Measured

' O
A

Calculated

1 1 '

.2 .4 .6

i.o

Figure 16.

Pylon frequency, cycles/rev

Effect of hinge offset on whirl
flutter.

in that both forward and backward whirl motions
were found to occur simultaneously; in effect, the
flutter was bimodal. Theory also predicted this
bimodal behavior, the forward and backward whirl
modes being within a few knots of each other
analytically.

The effect of asymmetry in the pylon support
stiffness is shown in Figure 17 . Again the sym-
metric frequency data reflect adjustments to true
symmetry for configurations which were nearly, but
not exactly, symmetric. The nonsymmetric results
reflect actual measured values, the lower of
either the pitch or yaw frequencies being plotted.
It was analytically shown^ that for sufficient
asymmetry in the pylon support stiffness increas-
ing the asymmetry more does not increase the flut-
ter speed. The data for the nonsymmetric fre-
quencies are an experimental demonstration of this
fact. Flutter in all the asymmetric conditions
was in the backward whirl mode.

1.4

1.2

l.o-

6 3 - .349 rad (20°)

e/R « .05

/
/

1 .

1 1

p I

1 /

£<V

/ y

Measured Calculated

/<§7

*-* W B' ty

1
1

A W

1 1 1

1.00

1.50

.75

.4 . .6

Pylon frequency, cycles/rev

1.0

Figure 17.

Effect of pylon support stiffness on
whirl flutter.

Bell Model 300
(a) August 1971

A joint NASA/Bell investigation employing a
1/5- scale aerodynamic model of a Bell tilt-rotor
design designated the Model 300 was conducted in
the transonic dynamics tunnel in August 1971 for
the purpose of providing the longitudinal and
lateral static stability and control characteris-
tics and establishing the effect of proprotors on
the basic airframe characteristics in both air and
freon. Use of freon permitted testing at full-
scale Mach numbers and near full-scale Reynolds
numbers.. Flapping was measured in both air and
freon for several values of tunnel speed over a
range of sting pitch angles. The resultant flap-
ping derivatives, obtained by evaluating the slopes
of the flapping amplitude versus pitch angle curves,
are shown in Figure 18. Since the range of inflow
ratios over which the derivatives were measured was
the same in air and freon and the test medium
densities at the simulated conditions were about
the same, an indication of the effects of Mach
number on the flapping derivatives can be obtained
by comparing the air and freon results. The speed
of sound in freon is approximately half that in
air so that for a given tunnel speed (or inflow
ratio) the Mach number in freon is about twice that
in air. The calculated results reflect the

180

Measured Calculated

Air
Freon

shafting is also employed in wind-tunnel models.
The availability of thrust damping to provide a
stabilizing force for yawing motion is dependent
on the structural integrity of this cross- shafting
and has implications which are pertinent to both
full-scale flight and model testing. Consider the
case of a windmilling "free-flight" model. A fully
effective interconnect maintains synchronization of
the rotor speeds during any motions. A yawing
motion of the model to the left, say, as might
occur during a disturbance, generates blade angle-
of -attack changes which decrease the lift of blade
elements on the right rotor and increase the lift
of blade elements on the left rotor. This produces
resultant perturbation thrust changes which tend to
damp the yawing motion, as depicted in the sketch
in the right-hand portion of Figure 19. If the

Rotor interconnect shaft
Engaged (AT > 0)
Disengaged (AT * 0)

Measured Calculated (Bell)

Inflow ratio at 0.75 Wade

radius,

v/m

.10 .20

Mach number in air

.1 i

.30
1 1

■a so
|

.40 .60

Mach number in freon

.80

Figure 18. Effect of Mach number on proprotor
flapping.

variation of Sx with blade pitch. Drag was
neglected in the calculated results shown for air
but was accounted for, in an approximate manner,
in the results shown for freon." The drag rise
associated with operation at high Mach numbers is
seen to reduce flapping as Mach number is increased
and suggests that calculations based on the neglect
of blade drag will predict conservative values of
flapping at Mach numbers where drag is important.

These data are believed to be the first which
provide an indication of the effects of Mach number
on blade flapping.

(b) March 1972

The most recent investigation conducted in
the transonic dynamics tunnel utilized a l/5- scale
dynamic and aeroelastic "free-flight" model of the
Bell Model 300 tilt rotor for the purpose of
demonstrating the required flutter margin of safety
and to confirm that the aircraft rigid-body flight
modes are adequately damped.- 1 - 2 During this test
the importance of rotor thrust damping on stability
of the Dutch roll mode was investigated. This
damping is associated with rotor perturbation
thrust changes which can be generated during axial
oscillations of the rotor shaft and constitutes a
positive damping force on aircraft yawing motions.

The rotors of tilt-rotor aircraft are gener-
ally designed to have an interconnecting shaft
between the two rotor/engine systems to provide
synchronization of the rotor speeds and to insure
that in the event of an engine failure either
engine may drive both rotors. Interconnect

180 220 260 300 340

Tunnel speed, ft/sec

50 ii loo

Tunnel speed, meters/sec

Figure 19. Thrust damping effects on tilt-rotor
Dutch roll mode stability.

interconnect is absent, the rotors are able to main-
tain their inflow angle and, hence, angle of attack
by increasing or decreasing rotor speed. The per-
turbation thrust changes thus go to zero and the
stabilizing contribution of this damping to the air-
craft yawing motion is lost. The effects of thrust
damping on the stability of the Dutch roll mode was
investigated by measuring the Dutch roll mode damp-
ing as a function of tunnel speed for the cases in
which the model interconnect was engaged and dis-
engaged. Some typical results are shown at the
left of Figure 19 along with the damping levels
predicted by Bell. The substantial contribution of
thrust damping to total damping is quite apparent.
It is of interest to point out that for the rotors
contrarotating in the direction indicated in the
sketch at the right of Figure 19 (inboard up) the
perturbation thrust changes accompanying an aircraft
rolling angular velocity are destabilizing on Dutch
roll motion. For contrarotating rotors turning in
the opposite direction (inboard down) the &T due
to both yawing and rolling motion are stabilizing
on Dutch roll motion.

Rotor rpm governors of the type which maintain
rpm by blade collective pitch changes while main-
taining constant torque are being considered for
use on full-scale tilt-rotor aircraft. With the
interconnect engaged, full thrust damping is avail-
able (assuming a perfect governor). However, in
the event of an interconnect failure, the governors

181

would respond to any rpm changes by varying blade
collective pitch in a manner which tends to main-
tain the original blade angle- of- attack distribu-
tion and hence torque. This is aerodynamically
equivalent to the windmilling case with no inter-
connect. It is axiomatic that tilt-rotor aircraft
must be designed to have stable Dutch roll charac-
teristics should an interconnect failure occur
anywhere within the flight envelope.

Some Additional Results Applicable to the
Bell Model 300 Tilt Rotor

- nnrHS-nOFQ o <~>

Measured

D Design stiffness test stand

O Ifi design stiffness test stand

- A 1/5, scale aeroelastic model

Calculated (design stiffness test stand)

Blades rigid inplane

A dynamic test of a flight-worthy proprotor
for the Bell Model 300 tilt- rotor aircraft was
conducted in the NASA Ames full-scale wind tunnel
in July 1970 (Fig. 20). Two different test stands

Figure 20. Bell 25-foot flight-worthy proprotor
in NASA Ames full-scale tunnel for dynamic
testing.

were used. One duplicated the actual stiffness
characteristics of the Model 300 wing; the other
was one- fourth as stiff. By using the reduced
stiffness spar and operating the proprotor at one-
half its design rotational speed it was possible
to preserve the per-rev natural frequencies of the
wing and simulate, at any given tunnel speed, the
inflow of flight at twice that speed. This expe-
dient did not, however, maintain the blade per-rev
elastic mode frequencies or simulate compressibil-
ity effects on rotor aerodynamics.

Some results from the full-scale test are
compared with data obtained from a test of a 1/5-
scale model and theory in Figure 21. Note that

Airspeed, knots

100 200

Airspeed, meters/sec

Figure 21. Model/full-scale comparisons of wing
beam mode damping and frequency variation with
airspeed for Bell Model 300.

the calculated results are based on the use of the
design stiffness test stand characteristics. To
provide for an indication of the effect of blade
inplane flexibility on stability, the predicted
results for the case in which the blades are
assumed to be rigid inplane are also shown. The
predicted increase in damping at about 103 m/sec
(200 kts) for the case in which blade inplane flex-
ibility is included is associated with coupling of
the blade first inplane cyclic mode with wing verti-
cal bending. For the range of tunnel speed over
which full stiffness test stand data are available,
the results are in good agreement with theory
assuming flexible blades. Note that a significant
stabilizing effect is predicted for the Model 300
as a consequence of blade inplane flexibility.
This trend is in contrast to that predicted for the
Model 266. The data for the quarter- stiffness test
stand are in agreement with theory assuming rigid
blades because operation at half the design rpm has
effectively stiffened the blades by a factor of h.
The 1/5- scale model data are also seen to be in
better agreement with analysis based on the assump-
tion of rigid blades. This is because the model
hub employed at the time the data were obtained was
too stiff. If this increased stiffness is taken
into account the predicted damping is in agreement
with theory (Fig. 22). The model/full-scale com-
parisons shown in Figure 21 indicate that assessment
of full-scale stability can be made on the basis of
results of small-scale model tests.

182

,„- 4

Q-

MEASURED CALCULATED

O
A

WING BEAM MODE
• WING CHORD MODE

O O

(P'zoo

'100

300
AIRSPEED, knots

400

500

150 200

AIRSPEED, meters/sec

250

Figure 22. Variation of wing beam and chord mode
damping with airspeed for 1/5- scale aeroelastic
model of Bell Model 300.

Conclusions

An overview of an experimental and analytical
proprotor research program being conducted within
the Aeroelasticity Branch of the NASA Langley
Research Center has been presented. On the basis
of the particular results shown herein the follow-
ing basic conclusions can be drawn:

(1) A proprotor /pylon/wing system can exhibit
a wide variety of flutter modes depending on the
degree of fixity of the pylon to the wing, rotor
characteristics, and rotor rotational speed. In
particular, for pylons which are rigidly affixed
to the wing tip, the instability can occur in
coupled pylon/wing, pylon/wing/rotor, or rotor
modes j for pylons which are soft-mounted to the
wing, a true whirl instability akin to classical
propeller whirl flutter can occur.

(2) Lightly loaded proprotors operating at
inflow ratios typical of tilt-rotor operation in
the airplane mode of flight exhibit a marked
sensitivity to gust excitation.

(3) Blade inplane flexibility can have a
significant effect on stability.

(k) A significant contribution to aircraft
lateral-directional (Dutch roll) stability arises
from rotor thrust damping. Since the availability
of this thrust damping is dependent on the integ-
rity of the rotor interconnect shaft, tilt-rotor
aircraft must be designed to have acceptable
lateral-directional response characteristics should
an interconnect failure occur anywhere within the
operating envelope.

(5) Proprotor whirl flutter, both backward
and forward, can be predicted with simple linear-
ized perturbation analyses using quasi- steady rotor
aerodynamics.

(6) For strength designed wings, wing aerody-
namics have only a slight stabilizing effect on
proprotor flutter speeds.

(7) The drag rise associated with proprotor
operation at high Mach numbers reduces blade flap-
ping and suggests that calculations based on the
neglect of blade drag will predict conservative
values of flapping at Mach numbers where drag is
important.

The analytical portion of this research pro-
gram is continuing. Attention is presently being
directed toward refining the existing stability and
response analyses and extending them by including
additional degrees of freedom.

Acknowledgment s

The author acknowledges the assistance pro-
vided by Bell and Grumman in preparing and testing
the models employed in the investigations conducted
in the transonic dynamics tunnel. Particular thanks
are extended to Troy Gaffey of Bell for his general
advice and assistance since the initiation of this
research program and to Jerry Kohn of Grumman for
performing the correlations with the data obtained
during the whirl flutter investigation using the
Helicat model.

References

1. Deckert, W. H. , and Ferry, R. G. , LIMITED
FLIGHT EVALUATION OF THE XV-3 AIRCRAFT, Air
Force Flight Test Center, Report TR-60-lt-,
May i960.

2. Hall, W. E. , PROP-ROTOR STABILITY AT HIGH
ADVANCE RATIOS, Journal of the American Heli-
copter Society, June 1966.

3. Edenborough, H. K., INVESTIGATION OF TILT-
ROTOR VTOL AIRCRAFT ROTOR-PYLON STABILITY,
Journal of Aircraft, Vol. 5, March-April 1968.

k. Reed, W. H., III., REVIEW OF PROPELLER- ROTOR
WHIRL FLUTTER, NASA TR R-261+, July I967.

5. Gaffey, T. M. , Yen, J. G. , and Kvaternik, R. G. ,
ANALYSIS AND MODEL TESTS OF THE PROPROTOR
DYNAMICS OF A TILT-PROPROTOR VTOL AIRCRAFT,
presented at the Air Force V/STOL Technology
and Planning Conference, Las Vegas, Nevada,
September 1969.

6. Kvaternik, R. G. , STUDIES IN TILT-ROTOR VTOL
AIRCRAFT AEROELASTICITY, Ph. D. Dissertation,
Case Western Reserve University, June 1973.

7. Young, M. I., and Lytwyn, R. T., THE INFLUENCE
OF BLADE FLAPPING RESTRAINT ON THE DYNAMIC STA-
BILITY OF LOW DISK LOADING PROPELLER- ROTORS,
Journal of the American Helicopter Society ,
October 1967.

183

8. Gilman, J. , Jr. , and Bennett, R. M. , A WIND-
TUNNEL TECHNIQUE FOR MEASURING FREQUENCY-
RESPONSE FUNCTIONS FOR GUST LOAD ANALYSIS,
Journal of Aircraft, Vol. 3, November-
December 1966.

9. Yen, J. G. , Weber, G. E. , and Gaf f ey, T. M. ,
A STUDY OF FOLDING PROPROTOR VTOL AIRCRAFT
DYNAMICS, AFFDL-TR-71-7 (Vol. I), September
1971.

10. Baird, E. F. , CAN PROP-ROTOR STABILITY BE

PREDICTED?, presented at the Aerospace Flutter
and Dynamics Council Meeting, San Francisco,
California, November 12- l^, 1969.

11. Baird, E. F. , Bauer, E. M., and Kohn, J. S. , ,
MODEL TESTS AND ANALYSIS OF PROP-ROTOR DYNAMICS
FOR TILT-ROTOR AIRCRAFT, presented at the Mid-
east Region Symposium of the American Helicop-
ter Society, Philadelphia, Pennsylvania,
October 1972.

12. Marr, R. L. , and Neal, G. T. , ASSESSMENT OF
MODEL TESTING OF A TILT-PROPROTOR VTOL
AIRCRAFT, presented at the Mideast Region
Symposium of the American Helicopter Society,
Philadelphia, Pennsylvania, October 1972.

184

COMPARISON OF FLIGHT DATA AND

ANALYSIS FOR HINGELESS ROTOR

REGRESSIVE INPLANE MODE STABILITY

W. D. Anderson

and

J. F. Johnston

Lockheed California Co.

Burbank, California

Abstract

During the development of the AH-56A, a considerable
amount of analytical and experimental data was obtained on the
stability of the regressive inplane mode, including coupling with
other modes such as body roll and rotor plunge. The data were
obtained on two distinctly different control systems; both gyro
controlled, but one with feathering moment feedback and the
other with direct flapping feedback. The paper presents a review
of the analytical procedures employed in investigating the
stability of this mode, a comparison of analytical and experi-
mental data, a review of the effect of certain parameters,
including blade droop, sweep, 63, aj , vehicle roll inertia, inplane
frequency, rpm and forward speed. It is shown that the stability
of this mode is treatable by analysis and that adequate stability is
achievable without recourse to auxiliary inplane damping devices.

Notation

B subscript referring to blade feathering

Cj j2 measure of damping, cycles to half amplitude

F subscript referring to fuselage

g structural damping ratio

I imaginary part of root, rad/sec

Kg collective feathering stiffness, ft-lb/rad/blade

Ko root flapping moment per unit of blade flapping, ft-lb/rad

L rotor lift, pounds

moment, ft-lb

Presented at the AHS/NASA-Ames Specialists' Meeting on
Rotorcraft Dynamics, February 13-15, 1974.

N R

€ x

blade product of inertia about feathering axis, slug-ft 2

normal rotor speed

real part of root, per second, subscript referring to rotor,
or rotor radius, ft

airspeed, knots

airframe longitudinal motion, ft

airframe lateral motion, ft

airframe vertical motion, ft

pitch lag coupling - positive nose up feather due to lag aft
of blade

rot6r blade collective flapping or coning, radians

pitch flap coupling angle - tan"' (-8/p)

to indicate partial differentiation

rotor blade cyclic inplane motion sine component, positive
forward, radians

rotor blade cyclic inplane motion cosine component,
positive to the right, radians

fraction of critical damping

pitch motion, radians

blade collective feathering, radians

blade effective sweep angle, radians

servo time constant, sec

♦

roll motion, radians

frequency, rad/sec

nip

inplane natural frequency, rad/sec

rotor rotational speed, rad/sec

185

In hingeless rotors two fundamental types of coupled rotor
body inplane mode stability problems exist. One is associated
with a soft inplane system having the inplane frequency less than
rotational speed, and the other with a stiff inplane system where
the inplane frequency is above rotational speed. The soft inplane
system when coupled With a basic body mode is unstable in the
absence of aerodynamics, and therefore its stability must be
provided by aerodynamic or auxiliary damping. This type of
system is discussed in References 1, 2, and 3. In contrast, the
stiff inplane system does not exhibit this inherent mechanical
instability, and so its stability is less dependent upon aero-
dynamic or auxiliary damping.

Both types of modes, however, are subject to aeroelastic
phenomena which can be stabilizing or destabilizing. Also, both
types can exhibit response characteristics caused by pilot and/ or
gust inputs which are undesirable. The critical inplane mode in
the soft inplane system is advancing in the stationary system,
whereas for the stiff inplane system, the mode is regressive. The
frequency of the mode in each case is the magnitude of (w .

n)or(n-w mp ).

nip

This paper deals specifically with the stiff inplane system.
The various types of coupled rotor body regressive inplane
stability /response problems associated with this type of system
are discussed. The paper deals with both a feathering moment
feedback gyro-controlled system and a direct flapping moment
feedback gyro-controlled system.

These two types are described in more detail in Reference 4.
The inplane mode characteristics of the direct flapping moment
feedback type system would be more characteristic of any direct
control hingeless rotor system employing a stiff inplane rotor.

The stiff inplane hingeless rotor system is worthy of serious
consideration because of its inherent characteristics of being free
from ground/ air resonance mechanical instability type
phenomena and its ability to provide a stable, highly maneuver-
able, rotary wing vehicle.

The absolute level of the stability of the inplane mode is not
the only consideration in establishing design criteria. An equal or
even more important criterion is that of response of the mode as
a result of pilot or gust disturbances. The stability of the mode
can appear adequate, but if it is easily excited by either pilot or
gust inputs, the mode can be unsatisfactory. Conversely, the
mode may exhibit very low damping characteristics, but not be
easily excited by either pilot and or gust excitations, and be quite
satisfactory because no high loads or undesirable body motions
occur.

Besides the basic stability considerations of the regressive
inplane mode, certain other basic types of coupled rotor body
regressive inplane mode stability /response problems may be
encountered. Some of these may be either low or high airspeed
phenomena or virtually independent of airspeed.

It is not intended here to go into a complete theoretical
treatise describing each of these types of phenomena, but the
effects of some parameters and flight conditions on stability/
response characteristics for particular rotor vehicle configurations
are presented. Because stability and response characteristics
depend on considerations of the detail design, generalized
conclusions cannot always be drawn as to the effect of each
parameter discussed.

The fundamental types of regressive inplane mode stability/
response problems discussed include those associated with:

• The basic regressive inplane mode.

• The coupled regressive inplane body roll mode,

• A coupled regressive inplane-roll-rotor plunge mode.

The first type can exhibit itself to the pilot as an apparent
rotor weaving or rotor disc fuzziness with very little body
response. The second type appears to the pilot as rotor tip path
plane response and body roll or just body roll oscillation.
Depending on the frequency of this mode, and the gain of the
feed forward loop of the control system, this mode may be
subject to pilot-induced or pilot-coupled oscillations. The last
type basically exhibits itself to the pilot as a rotor umbrella
mode and a vertical plunge of the vehicle. This mode can exist
in the absence of the inplane mode but can be seriously
affected by its presence. The mode has been characterized as
a "Hop" mode because of the plunge response of the airframe.

Analytical Method

The analytical method employed in the study consists of a
fundamental 13-degree-of-freedom representation of the coupled
rotor body control system. The body is characterized by 5
degrees of freedom — yaw being ignored. Likewise, the rotor is
represented by eight multiblade coordinates including rotor disc
plunge, pitch and roll; lateral and longitudinal inplane; and pitch,
roll, and collective elastic feathering/ torsion. The model is shown
schematically in Figure 1. The equations are solved as linear
constant coefficient equations, modified as required to represent
the control system being considered.

For the solutions shown, certain simplifying assumptions
were made. These include neglecting the effects of retreating
blade stall, reverse flow, and advancing tip Mach number.

COLLECTIVE FLAPPING
AND FEATHERING

FUSELAGE MOTIONS

Figure 1 . Description of Analytical Model.

186

The model includes the effects of elastic coupling
phenomena of inplane moments times flapping deflections
causing feathering moments. A simplified inflow model is used
which characterizes the induced velocity from low transition
speed to high speed as a trapezoidal distribution withupwash at
the front of the rotor and downwash at the back of the rotor, for
positive rotor lift

To gain a fairly comprehensive understanding of such modes
as the regressive inplane mode as well as the other coupled rotor
body fundamental aeroelastic modes, it is felt this type of model
is a necessity.

Effect of Parameters

Following is a discussion of each type of mode and the
relevant parameters which affect the stability /response of the
mode together with some parametric effects.

Basic Regressive Inplane Mode

The basic regressive inplane bending mode can be lightly
damped without any problem, provided it is not easily excited
by the pilot or by gusts. The mode if its frequency is well
separated from any other rotor body control mode frequencies,
behaves very much as a single blade would behave.

Figure 2 shows a complex plane plot of a typical mode of
response of the regressive inplane mode. It is noted that motions
of all other degrees of freedom are small compared with the
response of the blade inplane € x and 6y. This would be for a
case where the inplane mode frequency is well separated from
other rotor-body control mode frequencies, and the inter-
coupling with these modes is not large. In this case the inplane
mode is reasonably above the body roll mode, with a frequency
ratio of 1.32 in nonrotating coordinates.

| UP

FWD

VIEW LOOKING IN AT
TIP OF BLADE

For this case, the effect of several parameters is examined.
The stability/response of the mode is largely controlled by such
parameters as discussed in Reference 4 and 5. That is, parameters
such as blade kinematic and elastic pitch-flap-lag couplings are
extremely important. Also, such items as precone, feather
bearing location, hub/blade stiffness distributions and control
system flexibility play important roles in the stability of the
mode. A discussion of each of several parameters affecting the
stability of the mode follows.

Inplane Damping. Figure 3 shows a locus of roots as a
function of equivalent structural damping in the inplane mode.
Identified on this figure are lines of constant damping in terms of
one over cycles to half amplitude, 1/C| m. This figure shows the
expected results. That is, computing the approximate change in
fraction of critical damping from the root locus plot by taking an
increment of change in the real part of the root due to the
change in modal structural damping, g§j, and dividing it by the
sum of the imaginary part of the root and rotational speed
results in a value of damping roughly half the change in
equivalent structural damping. This is consistent with the well
known relationship of g =«2C where g « 1.

Since some centrifugal stiffening effect is existant in the
inplane mode, the actual change in damping, 24 , is less than the
change in equivalent structural damping in the mode.

-3 -2 -1

REAL PART OF ROOT

Figure 2. Complex Plane Plot of Typical Regressive
Inplane Mode, V = 20 KN, w € /u Roll = 1.32.

Figure 3. Locus of Roots
Damping, V =

-Effect of Structural
= 20 KN.

187

Kinematic Pitch Lag Coupling. Figure 4 shows the change in
damping due to a variation in pitch lag coupling. The indicated
sense of this coupling for improved stability is nose down
feathering due to lag aft of the blade for the stiff inplane system
whereas Reference 6 showed that the opposite coupling is stabi-
lizing for articulated or soft inplane system. The effect of this
parameter on the stability of the regressive cyclic inplane mode is
similar to the effect on the stability of the reactionless inplane
mode as indicated in Reference 5. The fundamental mechanism
of the a j coupling is to cause blade flapping to couple through
coriolis forces to damp the inplane mode. This can be deduced
by examining Figure 2.

of up flapping velocity at the time the blade is moving aft,
causing a Coriolis force forward to reduce the inplane motion.
The effect of blade droop on the regressive inplane mode is
shown in Figure 5.

-3 -2 -1

REAL PART OF ROOT

Figure 4. Locus of Roots - Effect of Pitch Lag
( a j) coupling, V = 20 KN.

A schematic of the response of a single blade for this mode
looking in at the blade tip is shown in the lower left corner of
Figure 2. The response shown is a stable response. With the
inplane frequency above the basic flapping frequency, nose
up blade feathering when the blade is forward, positive ct|, will
cause the blade to flap up as the blade is going aft. The up
flapping velocity of the blade generates a Coriolis force which
reduces the inplane motion.

Blade Droop. Blade droop is the built in vertical angular
offset of the blade below the feathering axis (see Reference 5)
and causes an elastic pitch lag coupling which is similar in effect
to stabilizing aj coupling. The droop effect though is somewhat
more effective in stabilizing this particular mode since some
additional phase lag results in the response of the elastic blade
feathering which improves the amount of flap induced Coriolis
damping in the mode. This is accomplished through an increment

-3 -2

REAL PART OF ROOT

Figure 5. Locus of Roots
Angle, V =

- Effect of Blade Droop
20 KN.

An additional insight into the effect of droop on the charac-
teristics of the system is shown in Figure 6. Shown is a predicted
frequency response of inplane response and of vehicle roll rate
response due to lateral stick excitation as a function of excitation
frequency. It is noted that the inplane becomes quite responsive
at low values of blade droop. It is also noted that even with the
fairly large separation of the roll mode and inplane mode
frequency, an influence of blade droop is seen on the roll mode.
This influence is seen to make an increase in the peak response of
roll rate at its peak response frequency with increasing blade
droop.

Other Parameters. Other parameters such as built in blade
sweep forward or aft of the feathering axis, 6 3 coupling, control
system flexibility, stiffness distribution of the blade and hub and
location of the feather bearings influence the stability of this
mode. Again, it is pointed out that the influence of each
parameter depends to a large part on the detail design. However,
in general, for a stiff inplane hingeless rotor, couplings which
result in nose down feathering due to lag aft of -the blade add
damping to the regressive inplane mode. Also, with the inplane
mode frequency above the flapping mode frequency, couplings
which act as a negative spring increment to the flapping mode or
a positive spring increment to the inplane mode are stabilizing to
the inplane mode. These couplings may, however, influence the
stability /response characteristics of other modes, in particular the
roll mode.

188

RESPONSE PER INCH OF
LATERAL STICK

tr uj

-jSS

O ui
CC D

s 1° \
\ J

"^3° -^

o z

UJ j

2 ■

< 5
-j

Q. in
2 2

\ DROOP
J ANGLE

1 10

J&^?

2 3

FREQUENCY - CPS

Figure 6. Effect of Blade Droop on Inplane and Vehicle
Roll Frequency Response Characteristics, V = 20 KN.

Coupled Regressive Inplane Body Roll Mode

Next is considered the coupled regressive inplane bending-
body roll mode where the frequency of the inplane mode and of
the roll mode are nearly coalescent. Figure 7 is a complex plane
plot of a typical coupled regressive inplane-roll mode for a direct
flapping moment gyro control type system where the inplane to
roll mode frequency ratio is 1.1. Comparing this figure with
Figure 2, it is noted that the roll response of the airframe relative
to the inplane is significantly larger in this mode. In this case, the
phase relationships between inplane, the rotor pitch and roll, and
the cyclic blade angle are still in a damping phase for the inplane
but the rotor roll-airframe roll phasing is such as to provide a
slight driving to the airframe roll motion. For this particular case,
the net damping of the regressive inplane mode would be some-
what reduced. Again, a discussion on the effect of significant
parameters which influence the characteristics of this mode
follows.

Inplane Frequency . Figure 8 shows the influence of inplane
frequency on coupled regressive inplane bending-roll mode
damping. Data are shown for a low-speed, 20-knot case and a
high-speed, 235-knot condition (compound helicopter flight
mode). It is interesting to note that at low speed, the roll mode
loses damping due to frequency coalescence whereas at the high-
speed condition, it is the inplane mode that tends to lose
stability.

Figure 7. Complex Plane Plot of Typical Coupled Regressive
Inplane Roll Mode, V = 1 60 KEAS, w e / w RoU = 1.1.

235 KNOTS
20 KNOTS

-3 -2 -1

REAL PART OF ROOT

Figure 8. Locus of Roots — Effect of Inplane Frequency.

189

Blade Droop. The influence of blade droop, where the
inplane mode and roll mode frequencies are close, is shown in
Figure 9. These data show a significant effect of droop on the
tradeoff of damping between the two modes. It is noted that
increasing droop has a significant effect in increasing the damping
of the inplane mode, but an equally significant effect in reducing
the damping of the roll mode.

Vehicle Roll Inertia . Figure 1 shows the influence of
vehicle roll inertia. In the case shown, a small reduction in damp-
ing of the regressive inplane mode and a significant improvement
in damping of the roll mode result from increasing the roll
inertia. A reduction in roll frequency is also seen. Ordinarily, a
roll mode in the 0.6 to 1 .3 Hz frequency region, with the damp-
ing sufficiently low, can be subject to pilot-coupled oscillations.
As can be seen from Figure 10, this problem is avoided by the
corresponding large increase in damping of the roll mode as the
frequency decreases into this range with the increasing roll
inertia.

Pitch Flap (63) Coupling. The influence of pitch-flap
coupling on coupled regressive inplane-roll mode stability is
shown in Figure 1 1 . This figure shows the inplane mode to be
little affected by decoupling with flap-up, pitch-down coupling
being slightly stabilizing. The effect on the roll mode is to
increase its frequency with positive coupling and also to improve
its damping. The inplane mode frequency is decreased as was
expected, but the damping increase was not expected. For a case
(not shown) Where the inplane frequency was considerably above
the roll mode frequency, the influence of the more positive pitch
flap coupling was to destabilize the inplane mode slightly with a
more significant effect of improving the stability of the roll
mode.

REGRESSIVE
INPLANE MODE

7,000

13,000

& 16,000

■ 12

REAL PART OF ROOT

Figure 1 0. Locus of Roots -
Roll Inertia, V =

- Effect of Airframe
20 KN.

(j0
33
>

REAL PART OF ROOT, 1/SEC

Figure 9. Locus of Roots - Effect of Blade
Droop Angle, V = 20 KN.

-3 -2

REAL PART OF ROOT

Figure 1 1 . Locus of Roots — Effect of Pitch Flap
(63) Coupling, V = 20 KN.

190

Feedback Ratio . Feedback ratio, \ , is the ratio of the
moment applied to the control gyro by rotor cyclic flapping
moment or shaft moment to the corresponding rotor shaft
moment. This parameter is described in detail in Reference 4.
It is used both to prevent excessive rotor shaft moments while
the vehicle is in contact with the ground, and to aid in tailoring
the vehicle handling qualities. This ratio is defined by the
following equation:

gyro

shaft

The influence of this parameter on coupled roll regressive
inplane mode stability is fundamentally on the roll mode. As
shown in Figure 1 2, increasing the magnitude of this parameter
increases the frequency of the roll mode and reduces its damping
while increasing the damping of the inplane mode.

Servo Time Constant. For the configuration being discussed,
the blade cyclic feathering is obtained through irreversible servo
actuators which are slaved to the control gyro. The lag in the
servos then causes a lag in the response of the cyclic blade feath-
ering as commanded by the control gyro. The influence of the
cyclic servo time constant is shown in Figure 13. It is noted that
the effect of increasing servo time constant is to reduce the fre-
quency and damping of the roll mode and the damping of the
inplane mode. The effect of the cyclic servo time constant
becomes increasingly important with increasing speed in deter-
mining the damping of the third type of mode, discussed below.

-4 -3 -2 -1

REAL PART OF ROOT

Figure 1 3. Locus of Roots - Effect of Cyclic Servo
Time Constant, V = 20 KN.

Coupled Regressive Inplane-Roll-Rotor Plunge Mode

This mode is most critical in high-speed flight. It has been
characterized as a Hop mode because plunging of the rotor disc
results in a vertical bounce of the airframe. The parameters
strongly influencing the stability of this mode in a feathering
moment feedback system are inplane frequency, collective
control stiffness, pitch-flap coupling, blade product of inertia
relative to the feathering axis and blade sweep. A typical mode
shape for this type of mode is shown in Figure 14. It is noted
that a considerable amount of rotor inplane pitching, rolling and
plunging, and airframe vertical and rolling motion occurs. The
mode may become critical with increasing speed if the rotor
plunge mode and coupled roll inplane mode are allowed to
approach coalescence.

\4-r fx

) ^s

Z F

REAL PART OF ROOT

Figure 12. Locus of Roots — Effect of Feedback
Ratio, V= 20 KN.

Figure 1 4. Typical Mode Shape of Coupled Regressive

Inplane Bending - Roll — Rotor Plunge

Mode, V= 180 KEAS.

191

This coalescence can be caused by the influence of several
factors. First, any couplings that cause the rotor plunged mode
to decrease in frequency with increasing forward speed may
cause coalescence. This can be due principally to collective con-
trol system flexibility, blade sweep, adverse pitch-flap coupling
and blade product of inertia effects; all affecting the collective
pitch response to vehicle normal acceleration or rotor coning. On
a first-order basis, a negative or positive aerodynamic spring on
the collective plunge or coning mode of the rotor can be
expressed by the following equation:

9L
80,,

*4( K *o

5, - JTZMxy

- K /?)'

where
knots.

3 L approximately doubles between hover and 1 20

Another source of coalescence or near coalescence can be
due to the coupled roll-inplane mode increasing in frequency
with increasing speed. As the lift due to collective blade angle
increases with speed, so do the aerodynamic derivatives associ-
ated with the cyclic motions of the rotor disc, and the aero-
dynamic coupling terms between cyclic and collective rotor disc
motions. Any kinematic or aeroelastic couplings that phase these
aerodynamics to act to stiffen the coupled roll inplane mode
with increasing speed will cause an increase in the frequency of
this mode with speed.

Absolute coalescence of these two modes is not necessary
for instability to occur. Both coupling between the modes and
frequency proximity are key to stability. Couplings which cause
cyclic aerodynamic forces or moments due to lift or plunge res-
ponse of the rotor, which in turn result in cyclic response of the
rotor disc which cause rotor disc lift or plunge driving forces, can
be destabilizing. When these couplings are sufficiently strong and
properly phased, the system will be unstable.

The significant aerodynamic coupling terms between these
two modes, which are strongly affected by forward speed, are a
rolling moment on the rotor due to change in collective blade
angle, a pitching moment on the rotor due to change in coning of
the rotor, and lift or plunge aerodynamic loadings due to roll
velocity of the rotor or to change in longitudinal cyclic blade
angle. These are direct aerodynamic couplings between these two
modes.

Indirect aerodynamic couplings exist through the inplane
response of the rotor system. This is particularly true in a feath-
ering moment feedback system, because relatively high inplane
exciting forces are generated as a result of changes in rotor lift.
The resulting inplane responses can couple through blade static
and elastic coning relative to the feather axis and cause perturba-
tional cyclic feathering responses. These cyclic featherings result
in aerodynamic forces which can be either stabilizing or
destabilizing.

Again Figure 14 shows a typical mode shape or eigenvector
for this type of coupled roll-regressive inplane bending rotor
plunge mode. In this case, which happens to be stable but lightly
damped, the collective feathering is at an amplitude and phase
with respect to O , collective coning of the rotor, to act as a

negative aerodynamic spring on the coning mode. Likewise, Q ,
collective blade angle, acts in conjunction with longitudinal
cyclic blade angle in causing the rotor to pitch up. As can be
seen from this figure, the rotor pitch response is lagging the
collective blade angle response by approximately 45 degrees,
whereas the coning response is actually leading the collective
blade feathering response by a small phase angle.

Further examination of Figure 14 shows that the coning
response, in addition to the rotor pitch response, is also being
driven by longitudinal cyclic blade angle. It is interesting to note
that the lateral inplane response is leading the rotor coning res-
ponse by approximately 90°. Positive lift on the rotor combined
with lateral cyclic blade angle causes a lateral inplane excitation.
Positive lift results in an increase in lateral inplane bending to the
left, which is aft bending on the aft blade and forward bending
on the forward blade. The fact that the lateral inplane response is
lagging its excitation by approximately 90° and the response is
virtually pure regressive indicates that the inplane mode is very
close to being in resonance. The inplane response, in coupling
through the feathering axis, is a prime source of the longitudinal
cyclic blade angle.

Even though the mode shown in Figure 1 4 is stable, one can
see the potential for the mode to lose damping, which it does for
the case shown, with increasing air speed. Lift and rotor disc
rolling moment due to 6 and longitudinal cyclic both increase
with air speed, as well as rotor disc pitching moment, due to con-
ing of the rotor. These aerodynamic terms in conjunction with
the inplane aerodynamics due to rotor coning are the principal
coupling terms between rotor disc plunge and coupled roll regres-
sive inplane response. It is through these terms and the choice of
rotor/control system parameters that the coupled rotor vehicle
system can be rriade to have adequate damping at high speed.

Figure 1 5 shows the effect of pitch flap coupling, blade
product of inertia, control system collective stiffness, and blade
sweep which, as indicated earlier, are key parameters in influen-
cing the stability of this mode.

Studies are also presented for the stability characteristics of
this type mode, for the direct flapping moment feedback type
control system. In this system, one other parameter was intro-

+1r

O
O

E£
Li.
O
H
CC
<

12 3

SWEEP ANGLE-DEGREES

10 20 30

K„ 10 3 FT-LB/RAD

+1f

-1
-20

-10

v MXY - SLUG-FT*

-.6 -.4 -.2

S3 COUPLING, -9/p

Figure 15. Effect of Parameters on Stability of Coupled

Regressive Inplane Bending — Roll - Rotor Plunge

Mode, V= 180 KEAS.

192

duced which has a significant effect on this mode. The parameter
is the time constant or frequency response characteristic of the
main power cyclic actuators. It is through these actuators that
the control gyro commands cyclic blade feathering. As indicated
earlier, a lag in the servo response results in a lag in the cyclic
blade, feathering, which can have an adverse effect on the stabil-
ity of the hop mode. Inasmuch as the hingeless rotor depends on
corrective control such as by stabilizing gyro to prevent pitchup
at high speed, it is recognized that lag in the corrective control
may lead to dynamic instability. This influence or effect is shown
in Figure 16.

Figure 1 6 shows also the effect of 63 coupling as well as
collective control system stiffness and inplane frequency.

REGRESSIVE INPLANE MODE

ROLL MODE

1
2

S 3

SERVO TIME CONSTANT
.01 .02 .03 .04 .OS

S3 COUPLING, -0/(3
-0.1 0.1 0.2 0.3

- —

—

2
3

<
a

K(J -% NOMINAL
20 40 60 80 100

INPLANE FREQUENCY-P
1.2 1.3 1.4 1.5 1.6

■» mm •■"

Figure 1 6. Effect of Parameters on Rotor Vehicle High Speed

Dynamic Stability — Direct Flapping Moment Feedback

Control System, V = 280 KEAS.

Experimental and Analytical Comparison

The experimental and analytical comparison is based upon
aata obtained during the development of the AH-56A. Early in
the development of the AH-56A, a vehicle equipped with an
experimental rotor system in which the blades had been modified
by adding torsional doublers encountered a dynamic Hop
phenomenon. The principal effect of the torsional doublers on
this mode was to lower the inplane frequency and cause it to
become more critically coupled with the rotor plunge/body roll
mode. An analytical study was undertaken to define this
phenomenon which extended the coupled rotor body linear
analysis method available at that time and led to the
development of the linear math model discussed earlier.

Figure 17 shows a comparison of the experimental and
analytical data obtained for the vehicle configuration which
initially encountered the Hop or coupled roll-regressive inplane
bending-rotor plunge mode phenomenon. Additionally, the
following table summarizes the normalized roll rate, chord
moment and collective control load comparison obtained for this
condition. In both the experimental and analytical data, the
responses are due to a roll doublet excitation and are normalized
on vehicle e.g. vertical acceleration.

95% N R
Test Analysis

100% N R
Test Analysis

Roll Rate,
deg/sec/g

20 19.1

15.5 ■ 9.2

Collective Control
Load, Ib/g

2700 2820

2370 2610

Inplane Moment,
in. Ib/g

424K 420K

770K 670K

Frequency Ratio, co/n

0.52 0.51

0.54 0.54

Speed, KEAS

190 180

178 180

- ANALYSIS

97% N R "I

95% N R V TEST DATA

94% N R J

i100% N

^95% N R

120 160 200 240

FORWARD SPEED - KNOTS

280

Figure 1 7. Comparison of Theory and Test Damping vs Speed

For Initial Encounter With Hop on Early AH-56A

Development Configuration.

The loss in damping was caused by a coalescing of the rotor
body roll mode, the inplane mode (with both modes exciting
blade cyclic feathering), and the rotor plunge mode. The terms
discussed in the equation for 9L/gg previously given were such as
to cause the rotor coning or plunge mode to decrease in fre-
quency with increasing speed. In hover, the frequency of this
mode was close to IP. With increasing speed, the frequency drop-
ped into the 0.5 to 0.6P frequency range in the 200-knot speed
regime and coalesced with the lower-frequency body roll, regres-
sive inplane modes. This resulted in the observed reduction in
damping of the Hop mode with increasing forward speed.

A modification was made to the system which included
approximately doubling the collective control system stiffness,
reducing the pitch-flap coupling from a value of 0.27 to a value
of 0.05 at a collective blade angle of 5 degrees, increasing the
blade sweep from 2.5 to 4 degrees sweep forward, and reducing
the inplane frequency from approximately 1.55P to 1.4P. The

193

reduction in pitch-flap coupling and the increase in collective
control system stiffness were done specifcally to eliminate the
Hop phenomenon within the flight envelope. The increase in
sweep and reduction in inplane frequency were done to improve
certain handling quality characteristics. These changes resulted in
the frequency of the collective coning mode remaining virtually
constant with increasing forward speed. The changes also resulted
in the coupled roll regressive inplane mode remaining at nearly a
constant frequency with speed. The resultant effect was to
increase significantly the speed at which the predicted coupling
between these modes became critical.

Figure 1 8 shows a comparison of the predicted damping of
the coupled roll-regressive inplane bending mode with test results
for this modified configuration as a function of speed. This figure
indicates fairly good agreement between the measured and pre-
dicted values. Figure 1 9 shows a comparison between the
predicted and measured chord-bending response due to a lateral
stick doublet at 170 knots. As can be seen, good agreement
between the two responses was obtained.

The rotor system was then modified to increase the blade
droop from 2°20' to 3°10' (Reference 5). this configuration
change had little effect on the high-speed coupled roll-regressive
inplane mode stability characteristics, and the vehicle was sub-
sequently flown to 240 knots' true airspeed with no indication of
a high-speed dynamic stability problem.

This latter configuration change however, did, lower the'
damping of the coupled roll regressive inplane mode in hover and
low-speed flight because of the increase in blade droop. The
mode was characterized by roll oscillation and inplane response
due to pilot lateral stick inputs. The frequency of the mode was
approximately 1 Hz. This, coupled with the roll oscillation of the
airframe, made the mode susceptible to pilot coupled oscillation.

Figure 20 shows a comparison of the experimentally
determined and predicted roots of the coupled roll-regressive
inplane mode for the two different blade-droop configurations.
Again, fairly good agreement is seen between experimental and
analytical results.

ANALYSIS

• EXPERIMENT

P 1

o
>

— m (

120 160 200 240

FORWARD SPEED-KNOTS

280

Figure 18. Comparison of Theory and Test Damping vs Speed
For Modified Rotor — Control Configuration.

PO
top
-JO •

< w 5

< Q ANALYSIS

---EXPERIMENT

A major revision was then made to the control system
which replaced the feathering-moment feedback system with a
direct flapping-moment feedback system. This change necessi-
tated placing the main cyclic power actuators between the
control gyro and blade feathering instead of between the pilot
and the gyro.

2° 20' DROOP

3° 10' DROOP

OPEN SYMBOLS - ANALYSIS

CLOSED SYMBOLS - TEST

10 u>

ELAPSED TIME - SEC

Figure 1 9. Comparison of Experimental and Analytical

Transient Chord Bending Response Due to Lateral

Stick Doublet, V = 175 KN.

REAL PART OF ROOT

Figure 20. Effect of Droop Angle on Low Speed Roll Mode

Stability - Comparison of Experiment and Analysis,

V = 20 KN.

194

The rotor and control system parameters were selected to
provide a system that was completely free of either the undesir-
able Hop or roll mode characteristics discussed earlier. The
various parameters, which were established to be critical by
extensive parametric studies, using the linear analysis method
adapted to computer graphics, were established and controlled
very carefully. These parameters included both cyclic and collec-
tive pitch flap coupling, inplane frequency, main cyclic power
actuator time constant, gyro to blade-feathering gear ratio and
phasing, blade sweep and droop, and shaft-moment to gyro-
moment feedback ratio.

The initial configuration, when tested on the whirl tower,
was determined to have met all criteria except that the inplane
frequency was below the criteria value by about 0.05P, or 0.21
Hz. Some limited flight testing was performed with this configur-
ation to validate the criterion, after which the final configura-
tion, conforming to the original criteria, was reached by
removing 6.8 pounds tip weight from each blade. The results
with both configurations, are discussed in the following.

i m

o -I

-100K l \^y ^

100K
100K

rf 1-
x 5 -i° 0K

a! 100K

95% NR

100% NR

105% NR

Figure 21 shows the effect of predicted rotor vehicle
responses as a function of rotor speed for a lg, 1 60-knot flight
condition with the degraded inplane frequency. The excitation in
each case is a lateral stick doublet at 1.5 Hz which is the tech-
nique used in flight test for exciting coupled rotor-body dynamic
modes to determine their stability characteristics.

It is noted that with increasing rotor speed, the damping of
all responses decreases, and the magnitude of the pitch response
of the rotor disc in the mode increases. This was noted by the
pilot as a characteristic of the mode in that, with similar excita-
tions, the rotor disc tip path oscillations would be imperceptible
at lower rotor speeds but would become increasingly responsive
at higher rotor speeds.

Figure 22 shows a comparison between the calculated roots
and the experimentally determined damping and frequency for
this configuration at 1 60 knot airspeed. The actual vehicle res-
ponses contain a varying mix of the two roots, increasing in
inplane content with increasing rpm.

-H ANALYSIS

• EXPERIMENT - FROM RESPONSE OF ROLL RATE

2 3

ELAPSED TIME - SEC

3 -2 -1

REAL PART OF ROOT

Figure 21 . Rotor Speed Effect on Transient Response Due Figure 22.

to Lateral Stick Doublet, V= 160KEAS. Analysis -

Locus of Roots — Comparison of Test and
Reduced Inplane Frequency, V = 160 KN.

195

Figure 23 shows the predicted effect of increasing the
inplane frequency by removal of tip weight on the frequency and
damping of the coupled roll-regressive inplane modes. Figure 24
shows the corresponding predicted transient response at 105
percent of normal rotor speed with the tip weight removed. The
comparison in roots shown on Figure 23 and the comparison of
the 1 05 percent rpm transient response in Figure 24 with the 95
and 105 percent rpm cases in Figure 22 show a significant
improvement in the damping and transient response character-
istics at rotor overspeed for the configuration with the tip weight
removed to give the desired inplane frequency placement.

For the final configuration, Figure 25 shows a comparison
of the measured and predicted damping as a function of forward
speed. The data show good agreement in measured and predicted
damping levels from hover through transition and in higher-speed
flight. Experimental data on damping of the regressive inplane
mode consist of only one point because even though the mode
was not excessively damped, it was extremely difficult to excite
by the pilot with lateral stick doublet type excitations to amp-
litudes sufficiently large to obtain a reliable determination of its
stability. This final configuration was tested over a very large
flight envelope covering speeds to 220 knots true airspeed and
maneuvering load factors from -0.2g to 2.6g in the 1 80 to
200-knot true airspeed flight regime. The pilot reported "excel-
lent" to "deadbeat" damping and minimal responses to air
turbulence in high-speed flight.

X TIP WEIGHT REMOVED
O TIP WEIGHT IN

280

REAL PART OF ROOT

Figure 23. Locus of Roots - Effect of Tip Weight,
V=160KEAS.

12 3

ELAPSED TIME - SEC

Figure 24. Transient Response Due to Lateral Stick Doublet
Tip Weight Removed, V = 160 KEAS.

ANALYSIS

O EXPERIMENT
FORWARD SPEED - KNOTS
80 120 160 200 240

n 1

z
a.
S
<
Q

Figure 25. Damping vs Forward Speed — Comparison of Test
and Analysis For Final AH-56A (AMCS) Configuration.

Conclusions

Several types of modes can exist in a stiff inplane hingeless
rotor which involve coupling with the regressive inplane mode.
These include phenomena where the inplane mode is not well
coupled with the rest of the system, phenomena where the
inplane mode and body roll mode are the primary participants,
and even phenomena where the rotor plunge mode is heavily
involved in the total system dynamic behavior. These phenom-
ena, particularly the first two types, are not limited to any
particular flight regime but can be critical in either stability or
response in either low- or high-speed flight or can exhibit char-
acteristics which are virtually independent of speed. Each of the
modes is treatable by analysis, and certain parameters such as
blade droop, control system stiffness, pitch-flap and pitch-lag
couplings, blade sweep, blade product of inertia, inplane fre-
quency and control system parameters are influential in control-
ling the stability and response characteristics of these modes.
Additionally, a totally satisfactory system can be achieved
without recourse to auxiliary damping devices.

196

References

1. Burkam, J.E.; Miao, Wen-Liu; EXPLORATION OF
AEROELASTIC STABILITY BOUNDARIES WITH A
SOFT-IN-PLANE HINGELESS-ROTOR MODEL, Journal
of the American Helicopter Society , Volume 1 7, Number 4,
October 1972

2. Huber, H.; EFFECT OF TORSION-FLAP-LAG COUPLING
ON HINGELESS ROTOR STABILITY, Preprint No. 731,
Presented at the 29th Annual National Forum, Washington,
D.C., May 1973

3. Donham, R.E.; Cardinal, S.V.; Sachs, LB.; GROUND AND
AIR RESONANCE CHARACTERISTICS OF A SOFT
IN-INPLANE RIGID-ROTOR SYSTEM, Journal of the
American Helicopter Society , Volume 14, Number 4,
October 1969

4. Potthast, A.J.; Blaha, J.T.; HANDLING QUALITIES
COMPARISON OF TWO HINGELESS ROTOR CONTROL
SYSTEM DESIGNS, Preprint No. 741, Presented at the
AHS 29th Annual National Forum, Washington, D.C., May
1973

5. Anderson, W.D.; INVESTIGATION OF REACTIONLESS
MODE STABILITY CHARACTERISTICS OF A STIFF
INPLANE HINGELESS ROTOR SYSTEM, Preprint No.
734, Presented at the AHS 29th Annual National Forum,
Washington, D.C., May 1973

6. Chou, P.C.; PITCH-LAG INSTABILITY OF HELICOPTER
ROTORS, Journal of the American Helicopter Society ,
Volume 3, Number 1, July 1958.

197

HUB MOMENT SPRINGS ON TWO-BLADED TEETERING ROTORS

Walter Sonneborn
Grp. Eng. Mechanical Systems Analysis

Jing Yen

Grp. Eng. VTOL Technology

Bell Helicopter Company

Fort Worth, Texas

MAGNITUDE OF HUB MOMENT

Two-bladed teetering rotors with
elastic flapping hinge restraint are
shown to be suitable for zero-g flight.
The alternating moment component intro-
duced into the fuselage by the hinge
spring can be balanced about the aircraft
center of gravity by alternating hub
shears. Such shears can be produced in
proper magnitude, frequency, and phase
by additional underslinging of the hub
and by judicious choice of the location
of the first inplane cantilevered natural
frequency. Trends of theoretical results
agree with test results from a small scale
model and a modified OH-58A helicopter.

Centrally hinged rotors have traditionally
relied upon thrust vector tilt for gener-
ating control moments about the helicopter
eg. All present production two-bladed
rotors have central teetering (flapping)
hinges. Such rotors, without hub restraint,
have no control power in zero-g flight.
Recent military specifications for
transport and attack helicopters call for
the ability to sustain zero-g flight for
several seconds. Helicopter control under
this condition of no rotor thrust requires
hub moments which in two-bladed rotors can
be generated by springs restraining the
flapping hinge. The resulting flapping-
dependent hub moment, when observed in the
fixed system, has a mean value in the
direction of and proportional to the
maximum flapping relative to the shaft.
A 2/rev oscillatory moment with an ampli-
tude equal to this mean value results in
both the fore and aft and lateral direc-
tions. This paper discusses methods for
producing 2/rev hub shears for balancing
the oscillatory component of the spring
moment about the helicopter center of
gravity. Practical magnitudes of hub
moments are defined by minimum control
power requirements for zero-g flight, and
maximum values are limited by a variety of
factors. Test results from a 1/12- scale
Froude model and flight test results from
an OH-58A helicopter with variable hub
restraint are presented.

The basic benefits of hub moment are
better aircraft rate damping and positive
control power in zero-g flight. Minimum
hub moment requirements have been investi-
gated by analysis and testing of an OH-58A
helicopter. Zero-g flight was demonstra-
ted with this helicopter using only stiff
elastomeric bearings in the see- saw hinge
for hub restraint increasing the 1-g
control power by 10%. Figure 1 shows a
record of the maneuver. Only small

oo
§

1-1 H

Eh ci

t-H

00 B^

Ph -

►J §

q fa
ei

d
o

H
O

100

i) Q Q

1-3

(

i
o

20'

LONGITUDINAL
9 <t>

[| □ tl

LATERAL

a A a

~(> J 2.9 SEC < .25 g 1 ©
°t> « t\ <a /» °

-20

ii_©ja.

ROLL.

-B-f-

PITCH

12 3 4 5

•TIME - SECONDS

FIGURE 1. Model OH- 58 A pushover with
elastomeric flapping bearings.

199

lateral inputs were made during the maneu-
ver, and the roll angle did not exceed +5
degrees. The lateral SCAS was engaged and
contributed significantly to roll stabili-
zation. This test and analysis indicated
that some 25% of the 1-g control power is
adequate for zero-g flight. The OH- 58 A
helicopter was subsequently fitted with
ground- adjustable hub torsion springs
which added 23% or 37% to the 1-g control
power (see Figure 2). The pilot's reac-
tions were favorable with regard to the
lower spring value, but the stiff er spring
made roll control power excessive and also
increased the gust sensitivity noticeably.

LENGTH FOR 1
ll32 FT-LB/DEGI

LENGTH FOR
1210 FT-LB/DEG

SLIP RING

EFFECT OF UNDERSLINGING
AND CHORDWISE FREQUENCY

The effects of underslinging on 2/rev
hub shears and of hub restraint on 2/rev
hub moments are shown by an analysis of
the simple rigid body model shown in
Figure 3. The kinematics of an underslung,

<t MAST

VIEW
FROM
TOP .

2/REV CIRCULAR PATH
OF ROTOR eg

FIGURE 3. Simple rigid body math model
of two-bladed rotor.

flapping, two-bladed rotor with a teeter-
ing hinge cause the center of mass of this
rotor to travel in a circular path at
2/rev if the mast does not oscillate about
point A (this assumption will be partly
justified below). The resulting centrif-
ugal forces introduce hub shears S for
fore and aft (F/A) flapping:

2
Sp/ A =2ai<u u m cos2<ut . (la)

S, . 1 ,=2a, <u u m sin2&)t (lb)

FIGURE 2. Experimental Model 640 rotor
with ground-adjustable torsion tube
flapping restraint.

Other considerations limiting the hub
restraint of two-bladed rotors are:

- Fuselage vibration caused by
oscillatory hub moment at
2/rev.

- Increase in beamwise bending
of flexure.

- A weight penalty of about 90
pounds per 1000 ft-lb/deg of
effective control moment.

- Instability of the coupled
pylon/rotor system at extreme
spring stiffnesses. 1

These tend to discourage the designer from
introducing significantly more hub moment
than that equivalent to 25% of the 1-g
control power from thrust vector tilt.

where

a-, = F/A flapping angle

m = rotor mass
cot = rotor azimuth
The moments M from the hub spring are:
M F / A =- O.SajKjjQ + cos2«t) (lc)

M LAT =-0.5a,Kr, sin2eut

(Id)

Now taking moments about point A below
the rotor (M. = M + Sh) , it is evident
that all oscillatory components can be
cancelled in both the fore and aft and
lateral direction if

— a l K H = (2a,&> u m) h

(2a)

200

and du _^

dK,

4w hm

(2b)

When this condition is met a rigid
mast will not oscillate, but merely
experience a steady tilt, the amount of
which is determined by the mean value of
: the hub spring moment and the stiffness
Kp of the pylon spring.

The dynamic analysis was extended to
include the effects of the first inplane
mode and the rotor coning mode. Also
included were aerodynamic calculations at
the 3/4 blade radius and a modal repre-
sentation of the pylon support system.
The set of five differential equations
\was solved on a hybrid computer (Bell
J program ARHB2). The solution" showed that
,the location of the first inplane canti-

\levered blade natural frequency ok. has
k prounounced effect on hub shears,
figure 4 shows how the requirement for
uhderslinging u of the eg changes with

Since the loads induced by spring
restraint are not in phase with the loads
of the unrestrained rotor, small amounts
of hub restraint can reduce chordloads
(see Figure 5). In general, the spring
induced loads are small when compared
with the + 7000- inch- pounds loads occur-
ring in tEe unrestrained rotor at V v in
level flight.

K H = 132 FT-LB/DEG

VECTORS INDICATE
MAGNITUDE AND
PHASE OF MAX.
FWD. BLADE BENDING

w = 0.94/REV
K H =

= 210 FT-LB

FWD FLAPPING

Rigid Blade Theory

»-<y;

FIGURE 4. Underslinging requirement
versus blade cantilevered first inplane
natural frequency.

The blades act like a dynamic absorber
whenever alternating hub spring moments
begin to induce pylon motion. They retain
this absorber function over a large range
of blade frequencies because of the
relatively large absorber (blade) mass.
The chordwise bending moments induced at
the blade root in this manner have been
computed for the experimental Model 640
rotor, (see Figure 5. The pylon param-
eters used are representative of an OH- 58 A
helicopter). This rotor has a cantilever-
ed blade natural frequency of 0.94/rev.
(This frequency is raised to 1.4/rev in
the coupled rotor/pylon system. )

FIGURE 5. Blade root chord loads as a
function of hub moment spring rate.

The ideal moment balance about point
A, as suggested by equation (2a) and the
above discussion, is actually not fully
achieved. When the underslinging on a
flapping hub- restrained rotor is varied,
the complete calculations show a residual
pylon oscillation remaining and the phase
of the pylon response changing in a contin-
uous manner (see Figure 6). 'The reason

FWD FLAPPING = 6 DEG

=132 FT-LB/DEG

20.-

1
m

o w
< to

g +9<£

6-"

1-1 W

04 w o

Bs .go 01

132 FT-LB/DEG\
0.94

132 FT-LB/DEG

.10

.15

.20

UNDERSLINGING, u - feet

FIGURE 6. Pylon response versus under-
slinging.

201

for this is that the airloads of the free-
flapping rotor are slightly modified when
the airload moment due to hub spring is
considered. Figure 7 shows the lift and
drag increments on a lifting rotor
resulting from this, and it is evident
that an inplane shear 90 degrees out of
phase from the desired shear results.
(If the rotor were not producing any

%0K R / .75R

/^~ ^^^\

4 AL

/ \

1 + 1

L Ca

Wad adW

VIEW
SIDE,

FROM

VIEW FROM
TOP

FIGURE 7. Change AL in blade lift tp
balance hub moment, and resulting drag
components.

net lift then both blades would experi-
ence a drag increase, leaving no net hub
shear) . Figure 8 shows records of
computed pylon responses with and without
consideration of the inplane shears due
to airloads.

AIRLOADS INCLUDED AIRLOADS OMITTED

IN HUB SHEARS

FROM HUB SHEARS

^ =

u = BASELINE

210 FT- LB
DEG

u = OPTIMUM

:::;

.'.".

;•■:

■■:

— i

[

I!::

:<£

-r.

::■:

<£

s ;

•':'.':

'.i.:

! >L

::'.':

j::i

'■•':.

: . -/

.: .

■■■■■if

■ \

U- '

:-:^

i ■' ■

— -u

W^. = 1.4/REV

FIGURE 8. Comparison of hub acceleration
responses.

EXPERIMENTAL RESULTS

Model Tests

A test was conducted on a 1/12- scale/
Froude model. Figure 9 shows the appara-/
tus also shown schematically in Figure 3./
The model was operated at a fixed cyclic

and collective pitch and at 900 rpm. Th€
mahogany blades were heavier than typical
for helicopter design practice, hence
little coning took place, and even at the
smallest possible amount of under slingin$
(determined by the bearing diameter of
the see- saw hinge) some hub spring was
required for smooth running of the mode]
Accelerometers above and below the plane's
of the pylon gimbal support detected thijs
smooth condition. The amplitude of the/
2/rev acceleration was a function of thg
hub spring rate and the amount of under?-
slinging, as shown in Figure 10. The data
scatter near the equilibrium position is
indicative of the residual ( oscillation.
The phase change of the pylon response
occurred in the. gradual manner found in
the analysis. However, it was noted that
additional underslinging was only about
half as effective as anticipated in
balancing hub moments. (The cantileVered
blade frequency is 1.5/rev). A partial
explanation is in the different mast bend-
ing due to a shear and a moment (see
Figure 11a).

■■■■■■■HMHHPHP

■■aDJUSTA! i . j\l

feT""*"

eromete:

FIGURE 9. (Model (1/12 Froude Scale) with
variable underslinging and hub restraint.

202

w .

P-i

UNDERSLINGING -IN.
(FULL SCALE EQUIVALENT)

KjjXlO J FT-LB/DEG
(FULL SCALE VALUES)

. FIGURE 10. Model pylon accelerations
•versus hub spring.

The mast deflection of the tip of the
mast under an applied moment is greater
than that resulting from the balancing
shear force by an amount equal to 20% of
the radius of the circle described by the
eg of the underslung rotor. In addition,
the excursions of the rotor's center of
mass are reduced when part of the total
disk flapping occurs in blade flexures.
The magnitude of this effect is dependent
on rotor coning (see Figure lib). Both
of the above effects were omitted from
the analysis.

SPRING
MOMENT

HUB
SHEAR

(a) Moment Diagrams

M J '

"blade »

(b) Shift of Rotor C.G. Due
to Flexure Bending

FIGURE 11. Factors reducing the effect of
unde r s 1 ing ing .

Flight Test Results

The OH-58A helicopter shown in Figures
2 and 12 was flown at 3250 pounds gross
weight with hub restraints of 0, 132, and

FIGURE 12. Modified 0H-58A helicopter
with restrained flapping hinge.

210 foot pounds per degree. The eg was
varied from station 106.1 to 111.8
Figure 13 shows the 2/rev vibration
measurements at the pilot's seat.

60
+1

,1-

G.W.

= 3250
e.g.

LB
K H -FT-LB./DEG

106.1
111.8
106.1
111.8

132

a
ft

60 80 100
AIRSPEED - KIAS

120

FIGURE 13. Vertical vibration of pilot's
seat.

The influence of hub restraint (132 ft- lb/
deg) is negligible compared with the
increase in vibrations with forward speed.
There are several reasons for this. The
helicopter has a focused pylon isolation
system^ which is effective for isolating
inplane hub shears and hub moments.
(There is no vertical isolation) . In
addition to the moment and horizontal
shear isolation, the predicted dynamic
absorber effect appears to take place.
The first cantilevered inplane natural
frequency of the test blade is located
at 0.94/rev and it was shown in Figure 4
that for this frequency placement nearly
no additional underslinging is required.
The dynamic absorber effect of the blade
is reflected in the oscillatory chordwise

203

loads. The measured change in these loads
in hover as a function of hub restraint
and flapping is similar to the computed
values :

K H

FT- LB

DEG

AFT

FLAPPING

OSG. CHORD MOMENT @

STA 7. 8- IN. LB
MEASURED COMPUTED

4.1°

2900

2175

132

3.4°

900

2000

210

2.6°

3000

No computations were made for the forward
flight case. It is helpful, however, to
compare the oscillatory moments introduced
by the hub spring with the moments about
the aircraft eg due to the 1/rev hub
shears from airloads. These shears are
estimated from the modal shear coeffic-
ient of the first inplane cyclic mode.
The + 7000- inch-pound chordwise moment
at V„ corresponds to a hub shear of + 80
pounas per blade, which is equivalent to
a 2/rev moment about the helicopter eg of
+ 450 ft- lb. The maximum oscillatory
spring moment was only 50% of this value
when flapping reached 3.3 degrees in
hover at the forward eg and with the 132
ft-lb/deg hub spring. (This amount of
flapping is usually not exceeded in normal
maneuvers). This comparison shows that
the vibratory excitation introduced by
the hub spring is relatively small to
begin with.

The pylon isolation system and the
placement of the blade first cantilevered
inplant frequency near 1/rev made
additional underslinging unnecessary in
this aircraft for vibration isolation.
(The baseline underslinging for the
experimental Model 640 rotor was 2.375
inches).

CONCLUSION

(1) Two bladed rotors with hub restraint
are suitable for zero-g flight.

(2) Hub restraint which added some 27%
to the one-g control power of an
OH-58A helicopter with a Bell Model
640 rotor caused a negligible
increase in 2/rev vibrations during
hover and level flight.

(3) The 2/rev oscillatory moment compon-
ent due to hub restraint in a two-
bladed rotor can be balanced about

a point below the rotor hub by
additional rotor underslinging. The
amount of this underslinging depends
on the location of the natural
frequency of the first cantilevered
inplane blade mode.

REFERENCES

Gladwell. G.M.L. and Stammers, C. W./,
On the Stability of an Unsvmmetricaj
Rigid Rotor Supported in Unsymmetrical
Bearings , Journal of Sound and
Vibrations, 3,(3), (1966), pp. 221-'
232.

Balke, R. W. , Development of the
Kinematic Focal Pylon Isolation
System for Helicopter Rotors . The ',
Shock and Vibration Bulletin, 38,(3),
November (1968), p. 263.

204

OPEN AND CLOSED LOOP STABILITY OF HINGELESS ROTOR
HELICOPTER AIR AND GROUND RESONANCE

Maurice I. Young*

David J. Bailey**, f and Murray s-
The University of Delaware
Newark, Delaware

Hirschbein**

Abstract

The air and ground resonance insta-
bilities of hingeless rotor helicopters
are examined on a relatively broad para-
metric basis including the effects of
blade tuning, virtual hinge locations, and
blade hysteresis damping, as well as size
and scale effects in the gross weight
range from 5,000 to 48,000 pounds. A spe-
cial case of a 72,000 pound helicopter air
resonance instability is also included.
An evolutionary approach to closed loop
stabilization of both the air and ground
resonance instabilities is considered by
utilizing a conventional helicopter swash-
plate-blade cyclic pitch control system in
conjunction with roll, roll rate, pitch
and pitch rate sensing and control action.
The study shows that nominal to moderate
and readily achieved levels of blade in-
ternal hysteresis damping in conjunction
with a variety of tuning and/or feedback
conditions are highly effective in dealing
with these instabilities. Tip weights and
reductions in pre-coning angles are also
shown to be effective means for improving
the air resonance instability.

Notation

C = landing gear equivalent viscous

damping coefficient, lb/ft/sec
C = pneumatic shock strut viscous

damping coefficient, lb/ft/sec
C. = tire viscous damping coeffi-
cient, lb/ft/sec
CG = helicopter center of gravity
I = moment of inertia about x axis,

x slug-ft2
K = landing gear equivalent spring

e rate, lb/ft
K = non- linear, pneumatic shock

s strut spring rate, lb/ft
K = tire spring rate, lb/ft

M = mass of helicopter

M.. = control moment acting in later-
1 al swashplate equation of mo-

tion, ft- lb

Presented at the AHS/NASA-Ames
Specialists' Meeting on Rotorcraft Dyna-
mics, February 13-15, 1.974.

Acknowledgement is made of the support of

the U.S. Army Research Office,

Durham, N. C.

under Grant DA-ARO-D-1247G112.

•Professor of Mechanical and Aerospace
Engineering. **Graduate student and re-
search assistant. tCurrently U.S. Army
Transportation Engineering Agency, Fort
Eustis, Virginia.

M„

T/W

XYZ

t
x,y,z

V a 2
«1' «2

8'k

n j

control moment acting in longi-
tudinal swashplate equation of
motion, ft- lb
number of blades
thrust to weight ratio
inertial coordinate system
decibels

offset of virtual flapping
hinge , ft

offset of virtual lead-lag
hinge , ft

distance between center of mass
of helicopter and coordinate
system axis, ft

lateral and roll coupling para-
meter

helicopter longitudinal, lateral
and vertical displacements, ft
helicopter pitch and roll angu-
lar displacements, rad
helicopter pitch and roll rate,
rad/sec

flapping angular displacement
of kth blade, rad
lead-lag angular displacement
of k th blade, rad
logarithmic decrement
non-dimensionalized (by rotor
radius) displacement of virtual
flapping hinge from rotor cen-
ter of rotation, ft/ft
generalized fuselage and rotor
system degrees of freedom
constrained swashplate-blade
pitch degrees of freedom
percent of uncoupled critical
roll damping

percent of uncoupled blade lead-
lag damping

azimuthal coordinate of the kth
blade , rad

out-of-plane or flapping fre-
quency ratio, cycles/revolution
in-plane or lead-lag frequency
ratio, cycles/revolution

Introduction

In recent years an intensive re-
search and development effort within gov-
ernment and industry has focused on hinge-
less rotor helicopters with a view towards
mechanical simplification, improved flying
qualities and greater aerodynamic clean-
ness. The approach being employed capi-
talizes on modern structural materials and
technology which, in principle, permit the
hingeless rotor blades to flap and lead-
lag by flexing elastically, rather than by
the use of mechanical hinges. In order to
keep cyclic bending fatigue stress and

205

blade weight within bounds, the in-plane
or lead-lag hingeless blade fundamental
natural frequency ratio, as a practical
matter, inevitably falls within the range
.6-. 9 cycles per revolution, although fre-
quency ratios as small as .5 or as great
as 1.2 are possible. As a consequence of
this .6-. 9 range of frequency ratios, both
ground and air resonance instabilities can
still occur which stem from this frequency
ratio being less than unity.

There arises the added concern that
slight amounts of internal blade structural
damping to be expected in hingeless rotor
blades can cause such instabilities to be
much more severe and difficult to control
than in an articulated rotor case, where
mechanical lead-lag dampers would be a
standard design feature. On the other
hand, the elastic flapping of hingeless ro-
tor blades and the presence of large blade
structural moments which are aeroelasti-
cally coupled to the fuselage oscillations
both in hovering and on its landing gear,
and the aforementioned relatively high fre-
quency ratio of hingeless blade lead-lag
oscillations compared to those of conven-
tional articulated rotors (.2-. 4 cycles
per revolution) , present the favorable
possibility of significant alterations in
the ground and air resonance stability
characteristics. This is in contrast to
centrally hinged, articulated rotors,
where flapping motion would be expected to
have negligible effect on such instabili-
ties. Several recent investigations! #2,3
have contributed to increased understand-
ing of hingeless rotor helicopter ground
and air resonance characteristics, but in
each case were directed principally at de-
sign and development of a particular ma-
chine with its unique size, structural and
operational characteristics, rather than
at broad development of parametric trends
and general principles, as well as the
possibilities for enhancing system stabil-
ity by application of modern control engi-
neering/techniques in conjunction with ex-
isting/ conventional blade pitch control
systems .

In this study, the effects of the
various design and operating parameters
which traditionally influence the ground
and air resonance instabilities of articu-
lated rotor helicopters have been con-
sidered, but with the addition of the uni-
que hingeless rotor helicopter parameters
such as blade internal damping and virtual
hinge locations. The effect of scale on
stability is investigated by considering
aerodynamical ly similar designs which
range in gross weight from 5,120. pounds to
48,000 pounds by keeping tip speed and
mean rotor lift coefficient constant. Sev-
eral other cases of general interest are
also considered, such as off-loading, rpm

reduction, increasing blade number, etc. /
In view of the enormous control power availf
able with a hingeless rotor due to its /
structural characteristics and the possible
need for or desirability of full artificial
stabilization or stability augmentation of
certain design configurations or operating
conditions, a closed loop stabilization
approach is also investigated. It is
viewed as an evolutionary approach which
would employ a conventional helicopter
swashplate type of control system of blade
collective and cyclic pitch. A variety of
output variables and their derivatives are
examined as possible sources of closed
loop feedback information for control ac-
tuation. The roll and the roll rate vari-
ables are seen to be highly effective.
The dynamics of cyclic and collective
pitch change are also examined4 as part of
such a closed loop stabilization system
for ground and air resonance where the
control process is seen to be that of a
multiple input-multiple output, interact-
ing control system5.

Detailed parametric studies of the
ground and air resonance stability bound-
aries are carried out using a standard
eigenvalue routine. The parameter combin-
ations which can result in the instabili-
ties are examined with a view towards com-
paring designs with inherent stability
with those that are a result of artificial
stabilization. Finally those combinations
of design, operating and stability augmen-
tation parameters yielding hingeless rotor
type aircraft free of the ground and air
resonance instabilities are obtained.

Analysis

The analysis is carried out with the
objective of developing a broad understand-
ing of the influence of the principal de-
sign and operating parameters on both the
system air and ground resonance instabili-
ties. Consequently the degrees of freedom
chosen for the analytical model are those
which can be expected to be common to all
hingeless rotor helicopter designs in hov-
ering and on the ground, irrespective of
size and gross weight, operational require-
ments or specific structural design ap-
proaches .

The fuselage body degrees of free-
dom are taken as those which would repre-
sent both the hovering and ground oscilla-
tions of a single rotor helicopter either
in the air or on a three point, conven-
tional oleo-shock strut type of landing
gear. These then follow as the lateral,
longitudinal and vertical translational
degrees of freedom and the angular roll
and pitch degrees of freedom. A yawing
degree of freedom is not included, since
it is deemed an unnecessary and unproduc-

206

tive complication of marginal significance.
This follows from the large yawing inertia
of the body, the close proximity of the
aircraft center of gravity to the two main
landing gear and the rotor thrust line,
the net effect of which is to virtually
decouple the yawing freedom from the
others, and thereby effectively eliminates
its influence on the air and ground reson-
ance instabilities.

The landing gear type and arrange-
ment used in the analysis of ground reson-
ance are viewed as typical, but by no
means universal. However, the effective
spring and viscous damping restraints
which are arrived at in the landing gear
analysis are sufficiently broad in charac-
ter to be representative of the many diff-
erent landing gear systems currently in
use. The two most prevalent systems are
the skid type, and pneumatic shock strut
and tire type configurations. Since the
skid-type landing gear represents a spec-
cial case of the more general shock strut
and tire formulation, an analytic model of
the latter has been employed. This formu-
lation has the added advantage of permit-
ting various effects, such as the shock
strut damping, the non-linear pneumatic
spring rate and the combined spring rate
of the tire and landing surface to be more
easily studied and is developed in detail
in Reference 6.

The hingeless rotor blades are flex-
ible, cantilever structures which flap
elastically in oscillations normal to the
plane of rotation and lead-lag elastically
in the plane of rotation. A generalized
coordinate, normal mode type of analysis
could be employed effectively for the
structural dynamic aspects. However this
does not lend itself well to a simple de-
termination of the aerodynamic forces and
moments which play a central role in the
stabilization process because of the blade
bending curvature during the oscillations.
Consequently the concept of virtual springs
and hinges 7 ' 8 for the flapping and lead-
lag oscillations of the blade is used,
where quasi-rigid body blade motions are
introduced to replace the continuous, elas-
tic bending deformations of the real
blades. These degrees of freedom are il-
lustrated in Figure 1. An isometric view
of the body degrees of freedom is also
shown .

The blade pitch changes are treated
as constrained degrees of freedom in the
stability analysis. That is the blade
pitch can be changed collectively or cy-
clically by displacement or tilting of a
swashplate mechanism. In the open loop
case this is done by the pilot displacing
the collective or cyclic pitch control
sticks. This results in a transient re-
sponse of the aircraft either about its
initial hovering state or on its landing
gear by altering the aerodynamic forces
and moments produced by the hingeless rotor.

Since it takes the form of a reference in-
put or external disturbance, it has no ef-
fect on the system stability as long as

these disturbances are reasonably small.
In the general closed loop case the air-
craft roll position, roll rate, pitch posi-
tion and pitch rate are sensed and used to
drive a system of swashplate actuators with
a view towards employing the enormous con-
trol power inherent in the cantilever blade
design of hingeless rotor systems. This
can yield full stabilization, if required,
or it can augment the inherent stability of
the system when design and operating condi-
tions permit. The swashplate-blade pitch
change arrangement and the system block
■ diagram are shown schematically in Figure 2.
More sophisticated closed loop control
system arrangements offer the possibility
of enhanced performance and optimization
,of the system at the expense of complexity
or possible reduction in reliability. For
example an inner control loop on rotor
blade bending deflections by strain gage
techniques, as well as sensing of body
translational displacements and velocities
offer interesting possibilities which are
considered in Reference 9. Needless to
say, departure of blade pitch from the set-
tings called for by the control system com-
plicates and may degrade the stability and
controllability of the system. For example
blade torsion which is not included in this
study is an important factor considered in
Reference 16.

The combination of the fuselage,
landing gear (when applicable) and the ro-
tor blade systems yields 5+3N freedoms in
the closed loop case and 5+2N freedoms in
the open loop case where the blade number
N is at least four. The minimum number of
four blades follows from the possibilities
of a dynamic instability unique to two-
blade systems^ and resonant amplification
of three blade aerodynamic loadings in the
case of three bladesll which must be avoid-
ed by using a minimum of four blades in a
hingeless rotor system.

The number of blade freedoms is re-
duced by introduction of quasi-normal ccr,
ordinates to describe the rotor motions iz ' J - J .
This approach reduces the complexity of the
analysis by eliminating all blade motions
which do not couple with the body in a co-
herent manner during open and closed loop
oscillations. These coordinates describe
the various significant patterns of blade
motion by five degrees of freedom in the
open loop case. These are the rotor cone
vertex angle, the lateral and longitudinal
tilt of the rotor cone, and the lateral
and longitudinal displacements of the blade
system center of gravity with respect to
the geometric center of the rotor (due to
lead-lag motion in the rotating frame of
reference) . In the closed loop case three
freedoms are added through the displace-
ments of the swashplate for blade collec-
tive pitch changes and by the angular tilt-
ing of the swashplate for blade lateral
and longitudinal cyclic pitch changes.

207

The analysis proceeds assuming
that the rotor system has four blades.
The single exception to this is the con-
sideration of a very heavy helicopter
(72,000 lbs.) air resonance behaviour.
In this case a six blade design obtained
by adding two blades to a four blade,
48,000 lb. design is examined. This
leads to a final quasi-normal coordinate
model which has ten degrees of freedom
for the open loop case and thirteen for
the closed loop case. These equations of
motion are then reduced to a canonical
form suitable for application of a stan-
dard digital computer routine for deter-
mining the complex eigenvalues and eigen-
vectors of the system. In effect twice
the number of first order, linear differ-
ential equations with constant coeffi-
cients result. This is a twenty-sixth
order system in the closed loop case, if
ideal actuators are assumed. As more
realistic models of the control hardware
are employed (due to leakage across hy-
draulic seals, imperfect relays, ampli-
fier frequency response characteristics,
etc.) the order of the system would in-
crease further. This is deemed to be a
specialized design problem which needs
attention on an ad hoc basis.

Discussion of Numerical Results

The discussion of the numerical re-
sults begins with the open loop stability
or stability boundary characteristics of
the hingeless rotor helicopter ground re-
sonance problem and is then followed by an
examination of the potential influence of
closed loop, feedback control in system
stability. This approach is then repeated
for the air resonance problem. The dis-
cussion closes with an overview of the po-
tential of closed loop control for both of
these hingeless' rotor helicopter instabil-
ities.

Ground Resonance

In order to develop insight into
the nature of the ground resonance insta-
bility as it might occur for a typical
helicopter employing a hingeless rotor, a
12,000 pound reference case based on the
S-58 helicopter^ i s considered first.
The rotor is modelled as one with four
hingeless blades with a flapping frequen-
cy ratio of 1.15 cycles per revolution,
and a lead-lag frequency ratio of .70 cy-
cles per revolution at a rotor tip speed
of 650 ft/sec. The wheels are first as-
sumed to be locked, preventing the air-
craft from rolling freely in a longitu-
dinal direction. The uncoupled lateral
and longitudinal translation modes of the
aircraft are assumed to have five percent
and three percent of critical damping, re-
spectively, as a result of tire hysteresis

losses. As the thrust to weight ratio is
varied from zero to unity the vertical
loading on the landing gear decreases.
The stability of the small, coupled oscill-
ations about a series of initial steady
states determined by the thrust to weight
ratio (T/W) is then studied as a function
of oleo-shock strut damping for several
small, but typical values of blade hys-
teresis lead-lag damping. Both damping
parameters are expressed in terms of per-
cent of equivalent viscous critical damp-
ing.

The unstable mode of oscillation is
found in all cases to be dominantly a
fuselage rolling mode with a small amount
of lateral translation coupling, and still
lesser amounts of pitching and longitu-
dinal motion. Release of the brakes, per-
mitting the aircraft to move freely longi-
tudinally, has a slightly stabilizing ef-
fect, but of minor importance compared to
the influence of oleo-shock strut damping
and blade internal damping. The numerical
results of the study with brakes on are
presented in Figure 3. Equivalent viscous
damping of the uncoupled rolling mode ex-
pressed in percent of critical damping is
taken as the abscissa, while thrust to
weight ratio is the ordinate. The hori-
zontal dash line at T/W = .9 is a visual
reminder that this is an unrealistic condi-
tion and that the stability data beyond
this value is probably unreliable, since
the analytical modelling of the landing
gear depends on the questionable assumption
of an initial steady state for thrust to
weight ratios greater than nine tenths.
The aircraft is, of course, in the trans-
ient condition of landing or take-off.

It is seen that if blade hysteresis
damping should be equivalent to one percent
of critical lead-lag damping, then slight
amounts of oleo-damping of the rolling
mode produce stable oscillations . If the
blade internal damping is as little as one
quarter of a percent of critical, stability
can still be achieved for all thrust to
weight ratios, if roll damping is equiva-
lent to fourteen percent of critical damp-
ing. Internal blade damping of one percent
or greater is found to eliminate the in-
stability entirely, if only slight amounts
of landing gear damping are available, for
example from tire hysteresis . Thus the
ground resonance instability for the refer-
ence case is found to be quite mild and
easily eliminated with the moderate amounts
of blade and landing gear damping normally
present.

In order to understand the influence
of the tuning of a hingeless rotor on this
desirable result, the lead-lag frequency
ratio is varied about reference frequency
ratio of .7 cycles per revolution as the

208

flapping frequency ratio is held constant
at 1.15 cycles per revolution. Blade damp-
ing is taken at one-half percent of criti-
cal while roll damping is held fixed at
eight percent of critical. Figure 4 shows
the effect of this tuning on the unstable
mode by plotting the log decrement of this
mode versus thrust to weight ratio. It is
seen that increasing the lead-lag frequen-
cy ratio above .7 makes the system stable,
while decreasing it below this reference
value makes it progressively more unstable.
Figure 5 considers the effect of the off-
set of the virtual flapping hinge and tun-
ing of the flapping frequency ratio on the
instability with respect to the reference
case. It is seen that a flapping frequen-
cy ratio of 1.0 corresponding to a conven-
tional, articulated rotor is considerably
more unstable than the reference case. It
is seen that increasing the offset and
frequency ratio to progressively higher
values is beneficial and stabilizing al-
though tending to reach a point of dimin-
ishing returns at a flapping frequency
ratio of 1.20 cycles per revolution.

Size and scale effects are investi-
gated by considering the coupling of the
lateral and rolling motion as the distance
between the rotor hub and the center of
gravity of the aircraft is varied with re-
spect to the reference case, where it was
assumed to be at a distance of seven feet.
As this distance is decreased to five feet,
the instability is observed to change in
relationship to the thrust to weight ratio,
but not in general character. On the
other hand as the coupling increases by in-
creasing the distance to nine feet, there
is a stabilizing effect. This is illus-
trated in Figure 6- This result can be
understood in terms of the coupled rolling
natural frequency, which tends to decrease
as this distance increases. Thus if the
lead-lag natural frequency ratio is held
fixed at .7, stability can be improved by
detuning the fuselage coupled rolling mode
to a lower frequency. This result is typ-
ical of all helicopter ground resonance
instabilities .

The influence of large size and
scale changes is considered by studying
the stability of two additional hingeless
rotor helicopters of 5,120 and 48,000
pounds, respectively, which are obtained
from the reference case by aerodynamic
scaling. That is the rotor diameter and
overall proportions of the aircraft were
altered to accomodate the gross weight
changes at the same mean rotor lift co-
efficient and tip speed. It is seen in
Figure 7 that aircraft smaller than the
reference case of 12,000 pounds tend to-
ward inherent stability with the blade
tuning and nominal amounts of damping as-
sumed. On the other hand the relatively

heavy machines tend to a more severe insta-
bility at slightly higher thrust to weight
ratios than the reference case, but still
well within the range of achieving inher-
ent stability with moderate amounts of
blade hysteresis damping and oleo-shock
strut damping of the unstable, coupled
rolling mode.

Ground Resonance with Feedback

As an alternative or as a supple-
ment to parameter selection which results
in stable oscillations, closed loop feed-
back control is considered. Since pro-
portional control action (at least quali-
tatively) alters the frequency of oscilla-
tion of simple systems by adding or sub-
tracting a virtual spring effect, depend-
ing on whether feedback is negative or pos-
itive, the reference case was used as a
basis for investigating this possibility.
Figure 8 shows the effect of proportional
roll feedback and control action (in this
case positive feedback is actually employ-
ed) in detuning an unstable coupling by
depressing the critical fuselage roll mode
frequency. It is seen that this is very
effective in stabilizing the system. It
should be noted that in the case of other
design reference parameters, proportional
feedback and control action of opposite
sign might be beneficial, if the detuning
of the critical fuselage roll frequency
required increasing, rather than decreas-
ing. The application of this control ac-
tion is deemed beneficial, but is best de-
cided on an ad hoc. basis.

A more conventional use of feedback
control is considered in Figure 9 which
shows the effect of negative feedback with
derivative or rate control action. This
tends to augment the damping of the criti-
cal fuselage rolling mode. This is seen
to be highly effective also, and, at least
to a first approximation, is interchange-
able with oleo-shock strut damping of the
unstable roll mode.

A logical extension of the fore~
going application of feedback control to
the stability of ground resonance is the
blending of both proportional and deriva-
tive control action. In this case the
critical roll mode can be both detuned and
damped to approach an optimum. This is
shown to be the case in Figure 10. Here
the system is made progressively more sta-
ble over the entire range of thrust to
weight ratios. It is not the intention
here to optimize the stability boundary,
but to show that this is possible even
with small values of blade internal hys-
teresis damping and the normal amounts of
landing gear damping of the reference case,
In view of the relatively unimportant in-
fluence of the pitching, and longitudinal

209

degrees of freedom for the reference case,
pitch rate feedback and control action was
not deemed effective. However, this re-
mains a potentially useful and important
tool in the event that special design or
operational requirements modify the open
loop system.

Air Resonance

The basic reference helicopter of
12,000 pounds gross weight is examined for
its air resonance stability as a function
of lead-lag frequency ratio for several
values of flapping frequency ratio. It is
seen in Figure 11 that lead-lag frequency
ratios of .70 or less result in instabil-
ity over the structurally feasible range
of flapping frequency ratios between 1.10
and 1.20. It is also to be noted that in
the neighborhood of neutral stability (for
the assumed blade equivalent viscous in-
ternal damping of 1/2%) , increasing flap-
ping frequency ratio is stabilizing. This
interaction effect between these two key
blade natural frequency ratios is further
illuminated in Figure 15. It can also be
seen that the lighter blades (i.e. an Over-
all mass fraction of 4%% rather than 6%%)
require higher frequencies for neutral sta-
bility. It is shown in Reference 12 that
in the stable range of lead-lag frequency
ratios, an increasing helicopter blade
mass fraction improves stability further.
On the other hand, it is also shown that
for an unstable configuration, increasing
blade mass fractions can further degrade
stability.

The critical effect of internal
damping of the blade lead- lag motion is
presented in Figure 13 for the reference
case with a flapping frequency ratio of
1.10 (comparable results were obtained at
frequency ratios of 1.15 and 1.20). It is
seen that increased internal damping en-
hances air resonance stability and inter-
nal damping levels of 1% of critical vir-
tually eliminate the air resonance insta-
bility for a lead-lag frequency ratio of
.75 or greater (since hingeless rotor
flapping frequency rates greater than 1.10
improve stability further) .

Although the frequency ratios for
lead-lag and flapping motions of hingeless
rotor blades have a fundamental effect on
the air resonance stability boundaries, de-
sign differences in structure, materials,
and proportioning of such blades can re-
sult in differences in the virtual hinge
locations and stiffness with important
modifications in the stability boundaries.
These effects are presented in Fiaure 14,
which show that more outboard location
of the virtual hinges for lead-lag mo-
tion tends to stabilize, although not
by a substantial degree. This effect

is believed to stem from a decrease in the
relative energy level of the blade in-plane
motion, just as in classical ground reson-
ance.

Size effects as distinguished from
gross weight are presented in Figure 15.
It is seen that the reference helicopter
air resonance stability is virtually un-
affected by large changes in the body
pitch and roll moments of inertia, pro-
vided that the lead- lag frequency ratio is
sufficiently large for stability {<»)££=. 75) .
However relatively large machines are seen
to be less unstable, if an air resonance
instability exists. The influence of gross
weight changes through aerodynamic scaling
is presented in Figure 16 for 5120, 12,000
and 48,000 pound machines which operate at
the same mean rotor lift coefficient and
tip speed. It is seen that very large in-
creases in gross weight tend to be stabil-
izing with respect to the minimum lead- lag
frequency ratio required for neutral sta-
bility, although gross weight effects for
machines in the 5,000 to 12,000 pound class
are not clear-cut because of the greater
sensitivity to all the other system para-
meters. In fact, it may be difficult to
obtain a rational trend when blade mass
fraction is held constant, when in reality
the very small machines will tend toward
larger blade mass fractions. In contrast
to this, if the gross weight of the refer-
ence machine is decreased by off-loading
(cargo, for example) , there is a clear-cut
improvement in the air resonance stability.
This is shown in Figure 17 and stems from
the reduction in blade initial coning.
The effect of coning is discussed further
below.

Built-in pre-coning angles are nor-
mal in the design of hingeless rotors", to
minimize steady bending stress is a rou-
tine consideration. Figure 18 shows the
stability boundary for the reference case
and the effect of deviating from the nom-
inally ideal case of built-in pre-coning
matching the coning that would result from
a 1-g load of a centrally hinged, articu-
lated rotor. It is seen immediately that
"over-coning" destabilizes and "under-con-
ing" stabilizes for the entire range of
lead-lag frequency ratios. This suggests
that a direct, profitable trade-off between
steady bending stress and air resonance
stability exists. That is reduce coning
by structural action and enhance stability.
Figure 19 continues this theme by showing
the influence of a concentrated tip weight
on the air resonance instability. In this
case it is seen that tip weight is benefi-
cial and stabilizing, providing the lead-
lag frequency ratio is of the order of
three-fourths or greater (^£^.75) . Fig-
ure 20 shows the design effect of an RPM
reduction at fixed gross weight. This

210

would increase coning and the data shows a
consistent loss of stability for the var-
ious lead-lag frequency ratios .

Aerodynamic scaling for very large
helicopters appears to be barred by the
adverse trend of coning at constant mean
rotor lift coefficient and tip speed, un-
less blade number is increased beyond four
blades. For example increasing gross
weight from 48,000 pounds to 72,000 pounds
was considered by increasing disk loading
and solidity by fifty percent - that is
adding two blades to the original four-
blade design. This yields the beneficial
effect of no increase in coning angle and
only a minor modification of the stability
boundary. This is illustrated in Figure
21. The dash or ghost line on this figure
represents a second mode of marginal sta-
bility at a high frequency. This is dis-
cussed at length in reference 15. The im-
plication is that a high frequency air re-
sonance instability might become a factor
in very large hingeless rotor helicopters.
However, the effect of including the addi-
tional rotor degrees of freedom suppressed
by the "quasi-normal" or "multi-blade" co-
ordinate transformations requires addi-
tional, careful study since the current
analysis limits rotor flapping type mo-
tions to those which result in either col-
lective or cyclic flapping of the indivi-
dual blades .

Air Resonance with Feedback

Proportional feedback and control
action proves to be a very effective means
of stabilizing air resonance. Figure 22
shows the influence of proportional roll
control action for the reference helicop-
ter; roll corrections alone are found to
be highly effective over the entire range
of lead-lag frequency ratios, whereas air-
craft pitching motion is found to be a
relatively small component of the air re-
sonance instability mode and not a produc-
tive avenue for closed loop stabiliza-
tion. 15 Figure 23 examines the efficacy
of proportional roll control for a case of
maximum air resonance instability when
u)££=.60. It is seen to be very effective
and virtually a linear influence on sta-
bility over the range of practical inter-
est.

Sensing aircraft roll rate is also
found to be highly effective in closed
loop control, but less so for pitch rate
because of the relatively small participa-
tion of pitch in the air resonance insta-
bility. However the complex phase rela-
tionships which exist in the mode of air
resonance instabilityl 5 make it very de-
sirable that aircraft roll and pitch con-
trol actions be mixed (i.e. the interact-
ing control actions referred to above 5 ) .

This is illustrated in Figure 24 which
shows the influence of pitch control ac-
tion for several levels of roll control ac-
tion (where both are based on roll rate
feedback information) . The linearity of
this stabilization method is made evident
by cross-plotting the influence of pitch
control action for a magnitude of roll
control action which results in (almost)
neutral stability.

Closed Loop Stabilization

The foregoing data illustrates that
an appropriate mix of aircraft roll and
roll rate information, in conjunction with
aircraft roll and pitch control action,
permits straightforward artificial stabili-
zation of both the air and ground reson-
ance instabilities of hingeless rotor heli-
copters under very adverse design condi-
tions. More importantly, perhaps, the da-
ta indicates that the marginally stable
configurations resulting from the lead-lag
frequency ratio being tuned to .70-. 80
and/or internal damping levels for this os-
cillation being of the order of %% of cri-
tical or less can be easily stabilized by
utilizing existing, conventional control
systems .

A significant difference between
the ground resonance and air resonance
modes of instability is the phase rela-
tionship between rotor cone tilting and
fuselage rolling motion. Also the fact
that a slight positive or regenerative roll
feedback and control action can be benefi-
cial in stabilizing ground resonance. The
reverse is true for air resonance. The
common, beneficial element for both insta-
bilities is in sensing aircraft roll rate
and utilizing this information for nega-
tive feedback to implement roll control
action. This in effect is stability aug-
mentation of the aircraft roll damping both
on the ground and in the air. The addi-
tional control action for aircraft pitch
has been found to be beneficial for sta-
bilizing air resonance 3 - 5 and not detrimen-
tal for stabilizing ground resonance. 6
Thus the interacting, closed loop control
system driven by roll rate information
emerges as a simple, evolutionary approach
to complete artificial stabilization, or
stability augmentation of the hingeless
rotor helicopter air and ground resonance
instabilities .

Conclusions

1. The ground and air resonance in-
stabilities of hingeless rotor helicopters
are marginal ones, but they will persist
as design considerations because of the
natural tendency of the lead-lag frequency
ratios to be less than unity (and conceiv-
ably as small as .60), while internal damp-

211

ing levels will be slight, unless special
materials and design measures which in-
crease internal damping can be found and
which are acceptable with respect to other
design and operating constraints.

2. The air resonance instability
is very sensitive to blade coning, while
ground resonance is not. Reductions in
coning by a variety of means are benefi-
cial, but the possibility of accepting a
modest level of steady bending stress in
lieu of other approaches (such as tip
weights) is worthy of more consideration

(since this would also reduce Coriolis-
type fatigue loads in steady forward
flight) .

3 . Completely artificial stabiliz-
ation of both the air and ground resonance
instabilities is possible, by utilizing
the concept of interacting controls. This
is not suggested as a serious approach to
design, but as an indication that a modest
stability augmentation approach, in con-
junction with adherence to simple design
criteria and objectives, can eliminate
both the air and ground resonance instabil-
ities.

4. The ground resonance instabili-
ty which was studied exhaustively in Ref-
erence 6 is seen to be inherently the same,
whether conventional oleo shock strut or
skid type landing gear are used, providing
the effective stiffness and damping are
properly represented in the overall system
design. On the other hand, failure or mal-
function of a single element of the system
which destroys the assumed symmetries (e.g.
a single blade damper on an articulated ro-
tor system) must be evaluated on an ad hoc.
basis since the system might then become
unstable despite the stability of the nor-
mal system.

References

1. Donham, R. E., Cardinale, S. V., and
Sachs, I. B., "Ground and Air Reson-
ance Characteristics of a Soft In-
Plane Rigid-Rotor System," AIAA/AHS
VTOL Research, Design and Operations
Meeting, Georgia Institute of Technol-
ogy, Atlanta, Georgia, February 1969.

2. Woitsch, W. and Weiss, H., "Dynamic Be-
havior of a Fiberglass Rotor," AIAA/
AHS VTOL Research, Design and Opera-
tions Meeting, Georgia Institute of
Technology, Atlanta, Georgia, February
1969.

3. Lytwyn, R. T., Miao, W. , and Woitsch,
W., "Airborne and Ground Resonance of
Hingeless Rotors," 26th Annual Forum
of The American Helicopter Society,
Washington, D.C., June 1970, Preprint
No. 414.

4. Young, M. I., "The Dynamics of Blade
Pitch Control," Journal of Aircraft,
Vol. 10, No. 7, July 1973.

5. Ogata, K. , Modern Control Engineering
Prentice-Hall, Englewood Cliffs, N.J.,
1970, pp. 377-396.

6. Bailey, D. J., "Automatic Stabiliza-
tion of Helicopter Ground Instabili-
ties," University of Delaware, Master
of Mechanical and Aerospace Engineer-
ing Thesis, May 1973.

7. Young, M. I., "A Simplified Theory of
Hingeless Rotor Helicopters," Proceed-
ings of the Eighteenth Annual National
Forum of The American Helicopter Soci-
ety, Washington, D.C. , May 1962,

pp. 38-45.

8. Ward, J. F. , "A Summary of Hingeless
Rotor Structural Loads and Dynamics Re-
search, " Journal of Sound and Vibra-
tions, 1966, Vol. 4, No. 3, pp. 358-377.

9. Young, M. I. , Hirschbein, M. S., and
Bailey, D. J. , "Servo-Aeroelastic Pro-
blems of Hingeless Rotor Helicopters,"
University of Delaware, Department of
Mechanical and Aerospace Engineering,
Technical Report No. 155, August 1972,
(Revised October 1973) .

10. Kelley, B., "Rigid Rotors vs. Hinged
Rotors for Helicopters," Annals of The
New York Academy of Sciences, Vol. 107,
Article 1, 1964, pp. 40-48.

11. Marda, R. S., "Bending Vibrations of
Hingeless Rotor Blades," University of
Delaware, Master of Mechanical and
Aerospace Engineering Thesis, April
1972.

12. Young, M. I. and Lytwyn, R. T. , "The
Influence of Blade Flapping Restraint
on the Dynamic Stability of Low Disk
Loading Propeller-Rotors," Journal of
The American Helicopter Society, Vol.
12, No. 4, October 1967, pp. 38-54.

13. Hohenemser, K. H. and Sheng-Kuang, Y.,
"Some Applications of Multiblade Co-
ordinates," Journal of The American
Helicopter Society, Vol. 17, No. 3,
July 1972, pp. 3-12.

14. Seckel, E., Stability and Control of
Airplanes and Helicopters, Academic
Press, 1964, pp. 456-457.

15. Hirschbein, M. S., "Flight Dynamic
Stability and Control of Hovering Heli-
copters," University Of Delaware, Mas-
ter of Mechanical and Aerospace Engi-
neering Thesis, October 1973.

16. Huber, H. B. , "Effect of Torsion-Flap-
Lag Coupling on Hingeless Rotor Sta-
bility," Preprint No. 731, 29th Annual
National Forum of the American Heli-
copter Society, Washington, D. C. ,
May 1973.

212

V flapping axis of

rotation

STABLE REGION

2 4 6 8 10 12 14 16 18 20

Fig. 1 XYZ - coordinate system, x, y, z, u\ t &2
displacements.

Fig, 3 Stability boundary as a function of roll
damping .

.BLADE PITCH
CONTROL AXES

DISTURBANCES

ira>

3LAPE PITCH CHANGE

AND SWASH-PLATE
DYNAMICS

FUSELAGE AND ROTOR
SYSTEM DYNAMICS

FEEDBACK DEVICE
CHARACTERISTICS

1
1
1
1
1
1

UNSTABLE

s 1

\. r~~

•~~ f

STA

3LE

-SO

1 »«

2 m tt

3 -u

AU H

5 u u

T/W

.B 1.0

Fig. 2

Fig. 4 The effect of lead-lag natural frequency
ratio .

213

.8 1.0

Fig. 5 The effect of flapping natural frequency
ratio and virtual flapping hinge offset.

Fig. 7 Size effects.

1 t - .61

2 t = .75

3 t - .83

/ »» \

I
1
1
1

I r x + ffi2 l

UNSTABLE^

1 /" ""

1 \]

i > f>

STOBLE

-20

-40

2 / *■

-——_______

3 ^/

i
i

-60

.2 .4 .6 .8 1.0
T/W

Fig. 6 The effect of coupling between lateral
and rolling motion.

Fig. 8 The effect of roll position feedback
control.

214

Wt. - 12000 lb.
6 1/2%

UNSTABLE

/ ' % y //

3
1

-1

5 -20

1
STABLE

I/ y/

-40

f 1 NO FEEDBACK
da-

2 M 2-"-3»-

da,

3 »2 " 20 -dF

-60

da,
4 M 2 .. BO.jJ

.6 .3

Fig. 9 The effect of roll rate feedback control.

■a

UNST

HBLE

STA

<L£

\ /

2 / ^^

-20

^X \

-40

' 1

"^NOFEEDBACK I
M 2 - -2. « 2 +20. j^-

'da
M 2- -*■ "2 +2I> - W

. 1

.8 1.0

Fig. 10 The effect of roll position and roll rate
feedback control.

c/c c

- 1/2%

e l

» .10

e 2

- .10

■ .

u £ - 1.15

u f « 1.20

.50

.60

.78 \

v°\

.90 1.00

Fig. 11 The effects of lead-lag natural frequency
ratio .

Wt. " 12000 lb
C/C - 1/2%

1.25

e l * * X0

e" - .10

ll)j = 1.15

1.20

1.15

^ Stable

<0^

1.10

Unstable

1.05

^ » 6 1/2% '

^ - 4 1/2% ,

1.00

.40 .50 .60 .70 .80 .90 1.00

Fig. 12 Flapping-in-plane natural frequency
ratio stability boundary.

Wt. - 12000 lb.
^ - 6 1/2%

" - .10

Fig. 13 The effect of in-plane blade structural v
damping.

215

wt.

- 12000 lb.

■h

- 4 1/2*

c/c c

- 1/2%

u f - 1.10

e x - .05

*- <t 2 ' -IS

5j - .10

'-•- - -10

n^ - 6 1/2%
0/C o » 1/2%
= .10

- .10

- 1.15

Wt.- 48000 lb.

Wt.= 12000 lb.
Wt.- 5120 lb.

Fig. 14 The effect of virtual hinge offset.

Fig. 16 The effect of scaling.

Wt. - 12000 lb.
ij, - 6 l/2»

c/c c - 1/2%

- .10

Fig. 15 The effect of changes in the body
moments of inertia.

n^ - 6 1/2%
C/C c = 1/2 S
= .10
= .10

12000 lb.
Wt. = 8000 lb.

t$ f.00

Fig. 17 The effect of off loading without
changing Ixq and Iyg.

216

Wt. - 12000 lb.
«fe - 10%
C/C - 1/2%

Fig. 18 The effects of pre-cone angle.

Fig. 20 The effects of a 10% RPM reduction.

Hjj - 6 1/2%
C/C c - 1/2%

.» .10

- .10

- 1.15

■Lower Frequency Instability
«* - <!-»„)

Higher Frequency Instability
o* - (l+o„)

wt. » 72000 lb.
Wt. = 48000 lb.

Fig. 19 The effect of tip weights.

Fig. 21 Stability of heavy helicopters.

217

wt.

' 12000 lb.
■ 4 1/2*

c/c c

- 1/2*

- .10

e 2 - .10
u f " 1.15

M l /o 2 " 1 '°

B 2 /a 2 =0.0

*-^— .^6 .60

.70

.80.

.90

-1

•

-2

-3

Open Loop — —

-4

■

-5

-6

Closed Loop —~

-7

■

-8

Fig. 22 Comparison of open loop stability to
closed loop stability with roll
position feedback.

wt.

12000 lb.

•

4 1/2*

VC r

1/2*

•l

.10

e 2

.10

1.15

"..

.60

Wt.
"b

c/c

12000 lb
4 1/2*
1/2*

•l

.10
.10

1.15

.60

Fig. 24 The effect of roll rate feedback with
pitch and roll interacting control
actions.

Fig. 23 The effect of roll position feedback
control at maximum instability.

218

VERTICAL-PLANE PENDULUM ABSORBERS FOR
MINIMIZING HELICOPTER VIBRATORY LOADS

Kenneth B. Amer
Manager, Technical Department

James R. Neff
Chief, Dynamics Analysis

Hughes Helicopters
Culver City, California

Abstract

This paper discusses the use of pendulum dy-
namic absorbers mounted on the blade root and
operating in the vertical plane to minimize heli-
copter vibratory loads.

The paper describes qualitatively the concept
of the dynamic absorbers and presents results of
analytical studies showing the degree of reduction
in vibratory loads attainable. Operational experi-
ence of vertical plane dynamic absorbers on the
0H-6A helicopter is also discussed.

Introduction

In a helicopter it is important to maintain a
low level of vibration for two reasons; first for
the comfort of the crew and passengers, and
secondly to minimize maintenance problems. During
early flight tests of the 0H-6A helicopter (see
Figure 1) in 1963, a high level of 4/rev fuselage
vibration was encountered primarily during ap-
proach to hover and during high speed flight.

WKKm

Figure 1. 0H-6A Helicopter

Various analytical studies and experimental pro-
grams were conducted in an effort to alleviate
this problem. The configuration finally adopted
was vertical-plane pendulum absorbers mounted at
the roots of the main rotor blades (see Figure 2) .
It is the purpose of this paper to describe the
concept of the vertical-plane pendulum dynamic
absorber and to present the results of analytical

studies and flight tests showing the degree of re-
duction in vibratory loads attained.

Figure 2. Pendulum Absorbers on 0H-6A

Over 3 million flight hours of satisfactory
experience have been obtained with the use of
vertical-plane pendulum absorbers on the 0H-6A
helicopter and on its commercial counterpart, the
Model 500 helicopter. This operational experience
is also discussed in this paper.

Sources of Fuselage Vibration

The 0H-6A helicopter has a 4-bladed main rotor.
Table I summarizes the sources of 4/rev fuselage
vibration from the main rotor. It can be seen from
Table I that vertical shears at the blade root with
frequencies of 3/rev, 4/rev, and 5/rev can induce
4/rev vibrations in the fuselage. The 3/rev and
5/rev blade root shears induce 4/rev fuselage vi-
brations by producing 4/rev hub moments. The 4/rev
blade root shear produces a 4/rev hub vertical
force. With regard to in-plane blade root shears,
both the 3/rev and the 5/rev component of in-plane
root shear produce a 4/rev hub horizontal force. A
further discussion of the mechanism by which rotor
blades induce vibration in the fuselage can be
found in Chapter 12 of Reference 1, particularly
the tables on pages 318 and 319.

219

Table I. Sources of 4/Rev Fuselage
Vibration - 4-Bladed Rotor

Vertical

In-Plane

Shear

Load Path

Shear Load Path

3/ rev

Hub moment

3-rev Hub horizon-
tal force

4 /rev

Hub vertical
force

5 /rev

Hub moment

5/rev Hub horizon-
tal force

Table I indicates that there are 5 possible
sources of excessive fuselage 4/rev vibration in
the 0H-6A helicopter. The next step was to estab-
lish which of the 5 possible sources of vibration
were the most important. Tables II and III pro-
vide an answer to this question.

Table II. 0H-6A Main Rotor Blade Natural
Frequencies (per rev) - 100%
RPM - Pendulums Off

Flapwise

Chordwise (Cyclic Mode)

2.72
4.87

5.14

In Table II are listed the main rotor blade
flapwise and chordwise natural frequencies near
the 3/rev through 5/rev frequency. It can be seen
from Table II that the two frequencies most likely
to cause a 4/rev vibration in the fuselage are the
first and second mode flapwise bending frequencies
which are very close to 3/rev and 5/rev. The
blade chordwise natural frequency is also close to
5/rev (see Table II). However, Table III con-
firms that the blade flapwise first mode and second
mode frequencies are the primary source of the
vibration problem, in that the fuselage vibration
is much more responsive to hub moments than it is
to hub vertical or horizontal forces.

Thus blade vertical bending at a frequency of 4/rev
and blade chordwise bending at frequencies of 3/rev
and 5/rev can be ignored and the primary sources of
vibration can be concluded to be blade flapwise
bending at 3/rev and at 5/rev.

Concept of Vertical-Plane Dynamic Absorbers

Based on the above evaluation, it was con^-
eluded that it was necessary to reduce the level of
blade 3/rev and 5/rev flapwise bending. After in-
vestigating a number of possible approaches,* it
was decided to pursue the concept of a dynamic vi-
bration absorber which is discussed in Reference 2
in the section starting on page 87.

The concept of a dynamic vibration absorber
consists of adding a small mass to a large mass.
The uncoupled natural frequency of the small mass
(vibration absorber) is chosen to be equal to the
frequency of the disturbing force. Thus, for the
OH-6 vibration problem, it was concluded that it
would be necessary to incorporate two dynamic
vibration absorbers; one tuned at 3/rev and the
other tuned at 5/rev. Furthermore, inasmuch as
rotor speed can vary somewhat, it was necessary
that the vibration absorbers maintain the proper
frequency relative to rotor speed. In order to ac-
complish this, it was decided to use the concept of
a tuned centrifugal pendulum discussed on page 219
of Reference 2. This concept has been used for
many years to minimize the torsional vibrations of
piston engines. Thus, the final configuration that
evolved consisted of two pendulums mounted at the
roots of the main rotor blades; one tuned to a
natural frequency of 3/rev, the other tuned to a
natural frequency of 5/rev. Inasmuch as the shear
force and blade motion which were to be minimized
were in the vertical plane, the dynamic pendulums
were oriented to oscillate in the vertical plane.

Figure 3 shows schematically the pendulum
motion relative to the blade deflection for the
case of response to 3/rev excitation. It is evi-
dent that the centrifugal force from the pendulums
is directed such as to cancel most of the trans-
verse shear due to blade modal response. The net
result is a significant reduction in the 3/rev
vertical shear force transmitted to the hub.

Table III. 0H-6A Cockpit Response to Rotor
Excitation, V = 100 Knots
(No Pendulums Installed)

4/Rev
Vertical

4/Rev

Pitching

Moment

4/Rev

Rolling

Moment

4/Rev

Longitudinal

Shear

4/Rev

Lateral

Shear

Excitation

130

*86

**112

Force , lb

Unit Response
at Cockpit,
in/sec/ lb

.0012

.00265

.0106

.00193

.0077

Response at

Cockpit,

in/sec

.16

.23

1.19

.019

.27

* Blade vertical shear force causing pitching moment.
** Blade vertical shear force causing rolling moment.

* Other approaches evaluated included providing
control of blade first and second mode natural
frequencies by means of anti-node weights and by
use of preloaded internal cables. Flight tests did
not show these methods to be sufficiently effective.
Hub-mounted vertical plane pendulums were flown and
proved to be effective, but considerations of drag
and weight were unfavorable for this configuration.
Fuselage-mounted non-rotating dampers were elimi-
nated because of the difficulty of tuning to a
sufficiently wide range of frequency. Fuselage-
mounted centrifugal pendulum dampers were con-
sidered impractical from the standpoint of space
requirements and mechanical complexity.

220

I 1 1

U« —

i FLAPPING HINGE

v /

'^V

tv.

1.0
.8
.6
.4
.2

MODAL

DEFLECTION _ 2

-.4

-.6

-.8

-1.0

20 40 60 80 100 120 140 160
BLADE STATION - INCHES

Figure 3. Pendulum Motion Schematic

Basic Physical Parameters

The pendulum configuration that was estab-
lished, flight tested, and put into service has the
following characteristics :

3/ rev pendulum

weight: 1.8 lb

actual mass ratio: .048

modal mass ratio: .64

Analytical Studies

Analytical studies were conducted to investi-
gate the effectiveness of vertical plane pendulum
absorbers in minimizing the blade vertical root
shears and the fuselage vibration levels. The re-
sults of these analytical studies are presented in
Table IV for the 0H-6A at a forward speed of 100
knots. It can be seen from Table IV that the addi-
tion of the 3/rev pendulum dynamic absorber reduces
the 3/rev vertical root shear by 75%. The addition
of the 5/rev vertical dynamic absorber reduces the
5/rev vertical root shear by 85%. The net result
is a 72% reduction in the vibration level in the
crew compartment.

Table IV. Effect of Vertical-Plane Pendulum
Absorbers on Root Shear and
Cockpit Vibration - 0H-6A

(Analytical Studies, 100 Knots)

Configuration

Root

Shear

3/Rev

5/Rev

Cockpit

Vibration, amp.

in/ sec

Undamped Blade
Damped Blade

91
23

42
6

1.8
.5

5/rev pendulum

weight: .7 lb

actual mass ratio: .019

modal mass ratio: .52

The pivot axis of both pendulums is located at
15% of the blade span from the center line of the
rotor, and 29% of the chord from the leading edge.
This location was chosen so that existing bolts in
the blade root fitting could be used, thus pre-
venting the introduction of stress concentration
points into critical sections of the blade.
Analysis indicates that a location further out-
board would be more favorable, but this has not
been confirmed by test, because of the structural
considerations cited above.

Damping of the pendulums due to friction in
the pivot bearings is estimated to be equivalent to
1% of the critical viscous damping ratio for the
3/rev pendulums at an amplitude of -16 . For the
5/rev pendulums at the same amplitude the damping
ratio is 3% of critical.

The dampers are "bench" tuned, by means of
shims, to the correct pendular frequency within
0.5% of the length of the 3/rev pendulums and to
within 1% of the length of the 5/rev pendulums.
The effect of mis-tuning has been investigated only
to the extent of showing that - one shim does not
have a consistently observable effect on either
qualitative or measured cockpit vibration.

The analytical procedure used to achieve the
results of Table IV is designated SADSAM. This
analytical procedure is described in Reference 3
and was conducted in two steps. In the first step,
SADSAM was used to calculate the blade root shears
for a forward speed of 100 knots both without and
with the pendulum absorbers. The analytical model
of the blade used in this step was a ten station,
fully coupled representation with aerodynamic ex-
citation forces obtained from flight measured
pressure distributions (Reference 4) . In the second
phase of the analysis, a 41 degree-of-freedom fuse-
lage mathematical model, adjusted to agree with
shake test results, was analyzed using SADSAM to
obtain the effect of the resulting hub moments on
the response in the crew compartment.

Flight Test Results

The favorable analytical results referred to
above led to a decision to fabricate an experi-
mental set of pendulum dynamic absorbers. These
absorbers, similar to those shown in Figure 2, were
installed on the flight test 0H-6A helicopter.
Tests were conducted measuring the vibration level
in the crew compartment, both without and with the
vertical-plane dynamic absorbers installed. The
measured vibration levels at the pilot's seat are
presented in Figure 4. It can be seen that the ad-
dition of the vertical-plane vibration absorbers
reduces the vibration level at the pilot's seat
approximately in half. The qualitative assessment
by the pilot was also very favorable. Based on
these results the decision was made to incorporate
vertical-plane dynamic absorbers in the production
0H-6A helicopter.

221

VIBRATION

VELOCITY,

IN./SEC

2.0

1.8

1.6

1.4

1.2

1.0

I 1 II 1

WITHOUT VIBRATION

ABSORBERS

HTH VIBRATION
ABSORBERS

1 1 1

90 100 110
Vi, KNOTS

120 130

Figure 4. Measured Vibration Level of 0H-6A

Without and With Pendulum Absorbers

Operational Experience on 0H-6A

The vertical plane pendulum absorbers were
incorporated on all production 0H-6A helicopters
and on its commercial counterpart, the Model 500.
Over 3,000,000 flight hours have been accumulated.
Up to a service life of between 300 and 600 hours,
the absorbers did a good job of controlling the
vibration level of the helicopter. However, after
approximately 300 to 600 hours of service, the
bearings and shafts on which the absorbers are
mounted exhibited excessive wear, resulting in in-
creased vibration level in the helicopter. Re-
placement of the bearings and shafts generally
returned the helicopter to an acceptable level of
vibration. The premature wearing of the bearings
and shafts was attributed to the high PV value.

Laboratory tests were conducted on various
combinations of bearings and shaft types with the
objective of selecting a combination that would
have the desired service life of 1200 hours. It
was also required that any new shaft and/or bearing
materials be interchangeable with the initial pro-
duction bearings and shafts. Thus no change in
geometry was permitted.

The results of these laboratory tests showed
that all combinations of shafts and bearings tests,
with the exception of one, were inferior to the
original configuration (which consisted of a
bearing consisting of a stainless steel outer race
with a bonded self-lubricating teflon liner, and a
stainless steel shaft with an 8 RMS finish) . The
only improved configuration consisted of an Astro
AM1282 bearing, which was specially made for the
laboratory test operating on the original shaft.
This Astro bearing is currently under consideration
for retrofit.

Conclusions

This paper has demonstrated both analytically
and by operational experience that the use of pen-
dulum dynamic absorbers, mounted on the blade root
and operating in the vertical plane, can success-
fully reduce helicopter vibratory loads. The
specific application on an 0H-6A helicopter was a

4-bladed rotor with the pendulums tuned to 3/rev
and 5/rev. The pendulums reduced the vibration
level in the cockpit to approximately one half of
the level that existed prior to the installation of
the pendulums.

References

1. Gessow, Alfred and Myers, Garry C, "Aerodyna-
mics of the Helicopter," The MacMillan
Company, New York, 1952.

2. Den Hartog, J. P., "Mechanical Vibrations,"
Fourth Edition, McGraw-Hill Book Co., New York,
1956.

3. Peterson, L., "SADSAM User's Manual," The
MacNeal-Schwendler Corp., 7442 N. Figueroa St.,
Los Angeles, CA, Report MSR-10, December 1970.

4. Scheiman, James, "A Tabulation of Helicopter
Rotor Blade Differential Pressures, Stresses,
and Motions as Measured in Flight," NASA
TMX-952, March 1964.

Acknowledgment

The contribution of R. A. Wagner and other
Hughes personnel to the development of the vertical-
plane pendulum absorbers is hereby acknowledged.

222

EVALUATION OF A STALL-FLUTTER SPRIIG-DAMPER
PUSHHOD IN THE ROTATING CONTROL SYSTEM OF A
■CH-5^B HELICOPTER

William E. Nettles
U.S. Army Air Mobility Research & Development Lab.,
Eustis Directorate, Ft. Eustis, Va.

William F. Paul and David 0. Adams

Sikorsky Aircraft, Division of United Aircraft Corp.

Stratford, Conn.

Abstract

This paper presents results of a design
and flight test program conducted to define the
effect of rotating pushrod damping on stall-
flutter induced control loads . The CH-5to hell-
copter, was chosen as the test aircraft because
it, exhibited stall-induced control loads . Damp-
ing was introduced into the CH-5^B control system
by replacing the standard pushrod with spring-
damper assemblies .

Design features of the spring-damper
are described and the results of a dynamic
analysis is shown which defined the pushrod stiff-
ness and damping requirements. Flight test
measurements taken at ^7,000 lb gross weight with
and without the damper are presented.

The results indicate that the spring-
damper pushrods reduced high-frequency, stall-
induced rotating control loads by almost 50?.
Fixed system control loads were reduced by k0% .
Handling qualities in stall were unchanged, as
expected.

The program proved that stall-induced
high-frequency control loads can be reduced
significantly by providing a rotating system
spring-damper. However, further studies and
tests are needed to define the independent
contribution of damping and stiffness to the
overall reduction in control loads. Furthermore,
the effects of the spring-damper should be
evaluated over a range of higher speeds and with
lower-twist blades .

A0B
CAS
C

C M

c/c„

Notation

angle of bank

calibrated airspeed, kt

damping rate, lb-sec/in.

blade section pitching moment
coefficient

damping ratio

ERITS equivalent retreating indicated
tip speed, kt.

aircraft gross weight

Presented at the AHS /MSA- Ames Specialists'
Meeting on Rotorcraft Dynamics, February 13-15 >
197 1 *.

I
K

(ll/iJ

torsional moment of inertia

spring constant

damper spring rate, lb/in.

rotor speed

blade section angle of attack

blade angle at 75$ rotor radius

torsional natural frequency, cycles/sec

ratio of natural frequency to rotor
frequency

Introduction

Control system loads can limit the
forward speed and maneuvering capability of high
performance helicopters. The slope of the con-
trol load buildup is often so steep {Figure l)
that it represents a fundamental aeroelastic
limit of the rotor system. This limit cannot be
removed by strengthening the entire control
system without incurring unacceptable weight
penalties .

Control
System
Vibratory
Load

Control System

Endurance Limit

Control System
Airspeed Limit

Stall Region

Figure 1.

Airspeed
"ontrol Load Characteristic

Studies of the problem reported in
Reference 1-7 indicate that the abrupt increase
in control loads is induced by high-frequency
stall-induced dynamic loading. This loading
is attributable to a stall-flutter phenomenon
which occurs primarily on the retreating side
of the rotor disc in high advance ratio and/or
high load factor flight regimes. At the
relatively high retreating blade angles of
attack which occur under these conditions, the
blade section experiences unsteady aerodynamic

223

stall and the moment coefficient varies with the
time-varying angle of attack as shown in Figure
2. Inspection of the moment hysteresis loops
exhibited in this figure indicate that positive
work can be done on the system as the blade
section oscillates in torsion. This aeroelastie
mechanism, by which energy is added to' the system,
can be termed "negative damping" and produces
pitch oscillations of increasing amplitude at the
blade/control system natural frequency. The
rotor system is therefore more responsive to
rotor loading harmonics which are close to the
blade torsional frequency, and the end result is
a rapid buildup of higher harmonic control loads
during maneuvers and high-speed flight.

was available to the program. Rotating pushrod
dampers were used instead of fixed system dampers
because they provided the required damping
directly at the blade attachment. The program
was limited in scope to an analytical and
experimental feasibility study of the concept,
and was conducted in four phases .

(1) Dynamic Analysis

(2) Functional Design

(3) Ground Tests

(k) Flight Test Evaluation

Blade
Pitching
Moment, 0-

C M

Damping Area

Pitch Down

Constant oc

Postive Work or
"negative Damping" Area-

Figure 2. Pitching Moment Hysteresis Loops.

The response of the rotor system is
usually stable, because the blades are moving
into and out of the negative damping region once
per revolution. However, during maneuvers in
which a significant portion of the rotor disc is
deeply stalled, very large oscillations can exist
(Reference 7) and the negative damping region can
increase to a point where blade oscillations can
continue into the advancing portion of the rotor
disc .

Efforts to understand the problem have
centered on defining unsteady aerodynamic
characteristics of the blades in stall (References
h and 6) and on incorporating this data into
blade aeroelastie computer analyses (References 6
and 9). Results of the studies are encouraging.
The buildup of control loads and high-frequency
stall-induced loads is predicted with reasonable
accuracy .

Recognizing that the basic cause of the
problem was insufficient pitch damping, the
Eustis Directorate contracted with Sikorsky
Aircraft to evaluate the effects of pushrod
spring-dampers on control loads of the CH-5UB
helicopter. This helicopter was selected for the
study since it exhibited high-frequency stall-
induced control loads during maneuvers at
maximum speeds and U8,000 pounds gross weight and

Dynamic Analysis

An aeroelastie analysis of the CH-5te
rotor was performed to evaluate the effectiveness
of spring-dampers in reducing the control loads
associated with retreating blade stall-flutter
and to evolve design criteria. The primary
mathematical analysis used was the Normal Modes
Rotor Aeroelastie Analysis Y200 Computer Program.
This analysis, which is described in Reference 8,
represents blade flatwise, edgewise, and torsion-
al elastic deformation by a summation of normal
mode responses and performs a time-wise integra-
tion of the modal equations of motion. This
analysis can also be used to study blade transient
response following a control input or disturbance.
Aerodynamic blade loading is determined from air-
foil data tabulated as a function of blade
section angle of attack, Mach number, and first
and second time derivatives of angle of attack.
Unsteady aerodynamics and a nondistorted helical
wake inflow were used throughout this investiga-
tion.

The version of the 1200 Program used for
this study is a single-blade, fixed-hub analysis.
The assumptions were made that all blades are
identical and encounter the same loads at given
azimuthal and radial positions and that blade
forces and moments do not cause hub motion. Any
phenomena which are related to nonuniformity
between blades or to the effect of hub motion on
blade response are not described by this analysis .

Free Vibration Characteristics

For a blade restrained at the root by a
pushrod, the first step in the aeroelastie
analysis is the calculation of the undamped
natural frequencies and modes for a blade
rotating in a vacuum. In order to analyze the
spring-damper/blade system using the normal modes
procedure, the damped free vibration modes and
frequencies were calculated based on the model
shown in Figure 3. The torsional system was
represented by fifteen elastically-connected
lumped inertias restrained in torsion by a spring-
damper at the blade root. The eigenvalues and
eigenvectors of the system response were calcu-
lated using a Lagrangian formulation of the
damped free vibration equations . A radial mode

224

shape, natural frequency and modal damping were
calculated and used in the Y200 Program.

Rotor

Spr ing-Damper

Control System

Figure 3. Schematic of the Spring-Damper Free
Vibration Problem.

Spring-Damper Behavior

The behavior of the CH-5^B spring-
damper was determined by employing the free
vibration analysis to determine the general
relationship between the properties of the damper
itself and those of the blade first torsional
mode. Figure k shows the variation of blade first
torsional natural frequency and percent critical
damping with changes in the spring and damping
constants of the spring-damper.

20 1*0 60 80 100 120 lUO
Spring-Damper Damping Constant, C , lb-sec/in.

Figure k. Effect of Spring-Damper Properties on First
Torsional Mode Frequency and Damping.

Three trends are evident from this figure:

1. For a given damper spring constant, K D ,
high levels of damping can increase
the root dynamic stiffness enough to
result in torsional natural frequencies
which are close to those obtained with
a rigid pushrod. It is clear from

Figure k that as the damping constant,
Cjj, is increased, the damper spring is
effectively bridged so that the
torsional natural frequency approaches
the standard pushrod value (7.^ per rev.)

2. For each spring constant, Kj), a
specified value of the damping constant,
Cp, maximizes the modal damping.
Increasing or decreasing the damping
constant decreased the percent critical
damping ratio of the torsional vibra-
tion.

3. The variation in the percent critical
damping parameter with damping constant
is relatively gradual, so small manu-
facturing differences between the six
production dampers will not cause great
differences in first torsional mode
damping .

Rotor System Analysis

For the initial analytical comparison of
the control system loading with and without damp-
ing, prior to design of actual hardware, a repre-
sentative flight condition was selected for which
experimental data existed for the conventional
system. This data was extracted from the
structural substantiation flight tests of the
CH-5UB and represents a condition in which stall-
induced dynamic loading was experienced. The
specific flight condition used - gross weight
1+7,000 lb, 100$ Rotor Speed (l85 RPM), sea level
standard, 30° angle of bank right turn- was
selected because it was the condition which
consistently produced stall-induced high-frequency
loading. The plot of rotating pushrod load
against azimuth for this condition is shown in
Figure 5a.

The pushrod load resulting from the Y200
Wormal Modes Program for the same flight condition
is compared with flight test results in Figure 5b.
To account for the increase in rotor lift ex-
perienced in the turn, a lift of about 60,000 lb
and a propulsive force of 3,300 lb was calculated.
Although the calculated pushrod load shows a
significantly greater steady nose-down load, the
vibratory amplitude and frequency content of the
analytical result match the test reasonably well.

To study the effectiveness of the
spring-damper in reducing vibratory control loads,
the flight condition described above was simu-
lated using several spring-damper configurations .
Each of these cases was run with the same control
settings as the standard case. The results are
shown in Figure 6. As shown, the combination of
5000 lb/in. and damping between 50 and 90 lb-sec/
in. was about optimum. Referring back to Figure
k, it is seen that a damping value of 90 lb-sec/
in. would provide a frequency of 7P which was the
same as the standard aircraft. This configura-
tion was therefore selected because the test
results could then be used to evaluate the spring-
damper at the same torsional frequency as the

225

standard aircraft. Also it would provide an
option to reduce the damping in follow-on
programs to allow an evaluation at 5-5/rev and
20$- critical damping.
+ 3000

+ 2000

+ 1000

-1000

-2000

-3000

±30001*

result, the overall peak-to-peak control load is
reduced by only 25%, while the high-frequency
retreating blade control loads are reduced by
more than 50%. It is these high-frequency loads
that cause the 6 per rev control system loads in
the fixed system.
1000

kO 80 120 160 200 2l+0 280 320 360

Azimuth, Degrees
Figure 5a. Measured Flight Test Result.

-1000
-2000
-3000
-1+000
-5000

f n

/-*

)

3xlh

3 ]

.b->--

1+0 80 120 160 200 2l+0 280 320 3.60

to
&

<• 1000

-1000

-2000

-3000

-l+ooo

-5000

z^ v ^ w .

t ^vA «- rtl

_j V i . A if

a 1ii_/\7j _.tt

3 v \z H :1 t

u -i

± 3100 lb-*— *—*■-

Azimuth, Degrees
Figure 7a. Conventional Pushrod.

t- 1000

-1000

-2000

-3000

1 1 1

}

rs _^^g:

V/ N

1+0 80 120 160 200 2l+0 280 320 360

Azimuth, Degrees
Figure 5b. Derived Result.

Figure 7b.

Azimuth, Degrees
Stall-Flutter Spring- Damper,

K„

5000 lb/in., C D = 90 lb-sec/in.

Figure 5- Comparison of Measured and Derived
Conventional Pushrod Load - CHS'tB,
1+7000 lb G.W., Sea Level, 100 KT, 30 c
A0B Eight Turn.

1+000

$ 3000'

2000

1000

Spring-Damper Damping Constant, CD, lb-sec/in.

Figure 6. Effect of Spring-Damper Parameters

on the Amplitude of Vibratory Control
Loads

The plots of pushrod load against
azimuth shown in Figure 7 compare a standard
pushrod with a spring-damper having a spring rate
of 5,000 lb/in. and a damping rate of 90 lb-sec/
in. For this configuration the free vibration
analysis gives a torsional frequency of 7 per rev
and 0.20 critical damping ratio. The Figure
shows approximately equal amounts of one-per-rev
variation occurring in the control load time-
histories since the pushrod spring-dampers do not
affect the low-frequency torsional motion. As a

Figure 7 .Comparison of Derived Conventional Pushrod

Load and Spring-Damper Load - CH-5I+B, 1+7000 lb
G.W., Sea Level, 100 KT, 30° A0B Right Turn.

It is clear from this analysis that
(l) damping at the blade root is effective in
reducing control loads for a given root stiff-
ness and (2) reducing root stiffness tends to
decrease the loads for a given damping constant
(at least for the ranges investigated).

Functional Design

Design Requirements

The aeroelastic analysis indicated
that spring and damping introduced at the blade
root could significantly reduce stall-induced
loads. The most favorable location for the test
of a blade root spring-damper is at, the pushrod
connecting the rotating swashplate to the blade
horn, since the existing pushrod may be replaced
easily with the spring-damper. It was determined
that a spring-damper device could be fabricated
to replace the conventional pushrod, provided
that the restrictive size limitations could be
met. The use of an elastomer as the primary
structural member met the size and spring rate
requirements .

The design requirements, based on the
aeroelastic analysis and the planned test
programs, are summarized as follows:

Replace Conventional Pushrod

Life - 50 hr

226

Load - ±5,000 lb

Spring Rate - 5.000 lb/in.

Damping Rate - 90 lb-sec/in.

Maximum Elastic Deflection - ±1/2 in.

Adjustable for Blade Tracking

Fail-Safe Design

Principles of Operation

The final configuration of the stall-
flutter spring-damper pushrod designed to meet
the above requirements is shown in Figures 8 and
9.

Orifice Slot

5.1

Figure 8. Stall-Flutter
Spring-Damper Pushrod Assembly.

Figure 9. Stall-Flutter Spring-Damper Pushrod.

The concept consists basically of a piston
restrained in a cylinder by two natural rubber
elastomeric bushings which provide the required
spring rate. Damping is obtained by displacement
of fluid through orifices . The bushings are
mounted in parallel, thereby providing a fail-safe
design. In addition, physical stops are incorpor-
ated to limit spring-damper deflection to ± 1/2
inch in the event of overload or complete rubber
failure. Ho sliding action takes place as the
spring-damper is deflected. Elastomeric elements
were chosen because of their high allowable

227

strains, integral hydraulic sealing, and compact-
ness. An integral air-oil accumulator was found
to "be inadequate and an external accumulator
system was used in the ground and flight tests .

Ground Tests

A comprehensive ground test program was
conducted to develop the required performance of
the spring-damper, to demonstrate structural
adequacy and safety for the flight tests, and to
evaluate the performance of an installed spring-
damper system. This was accomplished by the means
of single unit dynamic performance and fatigue
tests, flight unit proof and operation tests, and
an installed system whirl tests utilizing the
flight test spring-dampers and rotor blades.

Flight Test Evaluation

The performance of the stall-flutter
spring-damper pushrod system installed on a CH-5^B
helicopter was evaluated in a series of flight
tests consisting of: (l) base-line flights of
the CH-5te helicopter in standard configuration,
and (2) comparison flights with the spring-damper
system installed.

The investigation was limited to the
feasibility of the damper and did not extend to
an extensive evaluation of the overall effect on
the CH-5to operating envelope.

Baseline Flights

A short series of baseline flights was
conducted on the instrumented test aircraft in
standard configuration in order to obtain up-to-
date performance and control load data.

Of the several conditions flown, the
115 kt, 96% rotor speed, level flight point was
the best stall condition from the standpoint of
uniformity and repeatability. The maximum pushrod
vibratory load observed was about ± 2,100 lb.
This is lower than some stall results observed in
the past on this aircraft, but the typical stall-
flutter characteristic was observed in the push-
rod time histories and was therefore adequate for
baseline purposes.

Spring-Damper Pushrod Tests

The spring-damper pushrods were in-
stalled on the CH-55b rotor head as shown in
Figure 10 and 11. Flight test time histories of
rotating pushrod load for rigid pushrods and for
the spring-damper pushrods at 1*7,000 lb gross
weight are shown in Figures 12 and 13. These
segments of data which depict the time history for
approximately 1-1/2 revolutions were selected as
representative samples from oscillograph traces
in which the waveform was continuously repeated
for more than 15 revolutions.

Figure 10. Spring-Damper System Flight
Aircraft Installation.

Figure 11. First Flight of the Spring-Damper
System, February 6, 1973.

228

Tension

Pushrod
Load, lb

-Spr ing-Damper

Rigid Pushrod-

180

270 360
Blade Azimuth, Degrees

180

Figure 12. Rotating Pushrod Load Comparison

110 KT 96% N„ Level Plight, 1*700- lb.
a

-P
0)
•P
O
K

S-i
O

•H
>

I Spring-Damper Pushrod
I Rigid Pushrod

1 2 3 It 5 6 7 8 9 10

Tension

Pushrod
Load, lb

Harmonic Frequency, Per Rev

Figure ih. Comparison of Spectral Analyses -

CH-5to, 1*7000 lb G.W., 115 KT 96$ H
Level Flight, 2000' Altitude.

Comparison of Stationary Control Loads

Flight test time-histories of right
lateral stationary star load for rigid pushrods
and for spring-damper pushrods are shown in Figure
15. These records show the expected dominance of
the 6 per rev response in a 6-bladed rotor. As
shown, stationary control loads were reduced by
k0% for the spring-damper case.

Blade Azimuth, Degrees

Figure 13. Rotating Pushroa Load Comparison

115 KT 96% H E Level Flight, 1*7000 lb GW.

As shown, the rigid pushrod record ex-
hibits the high-frequency oscillation beginning on
the retreating side which is characteristic of the
stall-flutter phenomenon. This frequency was
between 7 and 8 per rev and compares well with the
calculated system torsional natural frequency of
7.1* per rev. As seen, the high-frequency loads
were significantly reduced with the spring-damper
pushrods. The overall reduction was smaller
because the low-frequency response was not reduced.
This was expected because the high twist blades
produce large lp loads and the spring-damper was
not designed to reduce these loads. As shown, the
results demonstrate a reduction of almost 50$ in
high-frequency loads. A spectral analysis of the
data burst which contains this cycle is shown in
Figure lit.

Test Condition;
1*7,000 lb GW, 115 KT,
± 3320 lb

N_, 2000* Alt

Right
Lateral
Stationary
Star Load
With Rigid
Pushrods

Right
Lateral
Stationary-
Star Load
With Pushrod
Spring-Dampers

Figure 15. Comparison of Stationary Control Loads.

229

A plot of stationary control load
against ERITS (Equivalent Retreating Indicated
Tip Speed) is shown and defined in Figure 16.
The sharp increase in load as stall is entered is
seen to be unchanged by the damper installation,
but as the aircraft goes deeper into the stall
region, the loads are reduced.

CH-S^B Structural Substantiation

_^^ Flight Test Results

o Base-line Flight Test Data
fl - — ~—a Spring-Damper Flight Test Data

■d 1+000
o

o
u

o
o

3500

3000

2500
2000

' H 1500

■p

m 1000

o
■p

500

°l

1°

! **

ft
/ 1

|A|

V o

—A-

— -J^

320

300

280

260

2U0

Erits-Knots

Figure 16. Stationary Control Load Against Erits

Note: Erits - Equivilent Retreating
Indicated Tip Speed

Rotating Tip Speed x /Air Density Ratio

-CAS

/ Load Factor x Gross Weight
37,500

Comparison of Aircraft Handling Qualities

The handling qualities of the aircraft
were unchanged with the spring-dampers installed.
Pilot's reports state that the aircraft exhibited
the characteristic increase in vibration,
difficulty in maintaining airspeed, and forward
control motion required when approaching a stall
condition in both the baseline and spring-damper
flights. The stalled condition of the rotor
appears unaffected by the installation of the
spring-damper. Blade stresses and blade motions
(except for the stall-flutter torsional oscilla-
tion) are virtually the same in each case. Cock-
pit vibration levels are unchanged. This was
expected because the stall was not changed, just
the local torsional response of the blade was
changed.

The effect of the damper on the control
system can be seen in plots of control positions
against airspeed (Figure 17). The lateral control
is unaffected, but as much as 10$ more forward

longitudinal control is required when flying at
the 115 kt, 9&% N s reference. stall condition.

LEGEND
— «- "^^-^^D Base-Line Flight Data •

.^& Spring-Damper Flight Data

100

t 9 °
I

■2 80

fof'

T7T02

J^t

60 70 80 90 100 110 120

Calibrated Airspeed, KT
Figure 17a. 100? Rotor Speed.

100

•n 90

J§ 80

•

'j©

-A—"

100

110 120

Calibrated Airspeed, KT
Figure 17b. 96% Rotor Speed

Figure 17 . Longitudinal Control Positions.

Aeroelastic Analysis of Flight . Test Data

Following completion of flight testing,
three additional computer analysis conditions were
run, using test conditions actually observed in
the flight tests . The methods used were the same
as described earlier with the exception that a
calculated lift higher than* the gross weight
actually flown was used. The amplitudes of
pushrod load predicted were much lower than
observed using the correct lift, and since the
comparison with and without the spring dampers
was of primary interest, the calculated lift was
increased, This shows that improvement in the

230

analysis is needed.

Figure 18 shows pushrod load vs azimuth
for the 115 Jet, 96% % reference condition for
conventional pushrods as generated by the aero-
elastic analysis and as observed in the baseline
flight. The analysis again shows a good correla-
tion in wave shape with test result. Based on
analysis of force-displacement phase shifts seen
in the flight test results, a damping rate of 70
lb-sec/in. was determined to be a likely Talue
actually achieved. Figure 18 also compares the
analytical result with the flight test result.

A good correlation in wave shape is
obtained. However, the sharp reduction in peak-
to-peak amplitude over the rigid pushrod case as
predicted by the aeroelastic analysis is again
not achieved in practice. It should be noted
that the aeroelastic analysis assumes that all
blades and spring-dampers are identical, which is
known not to be case. Difference among spring-
dampers would at least contribute to the dominant
one-per-rev component and perhaps the harmonics
as well.

Conclusions

Tension

It is concluded that:

Derived Conventional
Pushrod Load at
115 KT, 96% N R

1 (Lif t = 51,925 lb)

Actual Conventional
Pushrod Load at
115 KT, 96% N R
»(GW = 1*7,000 lb)

90 180 270 360 90 180

Tension

Derived
Spring-Damper
Pushrod Load,
C=70 Ib-sec/in.
(Lift=l+9,969 Vo)

Actual Spring-Damper
Pushrod Load
(GW=Vf,000 lb)

90 180 270 360 90 180
Blade Azimuth, Degrees

Figure 18. Comparison of Measured and Derived
Pushrod and Spring-Damper Loads.

Stall-flutter spring-damper push-
rods located in the rotating
control system effectively reduced
stall-induced high-frequency
rotating control loads on the
CH-5te by almost 50$ and overall
stationary control loads by more
than k0%.

The spring-damper pushrod system
does not significantly alter the
performance or handling qualities
of the CH-5te helicopter.

Rec ommendations

The test results were very encouraging,
but as usual raised more questions than it
answered. Some of these are stated below:

1. The combination of a spring and
damping worked well, but
quantatively what was the
contribution of each?

2. Would lower twist, higher mach
number and lower frequency provide
different results?

3. Would a high-speed aircraft show
some improvement in performance in
stall with the spring-damper?

To help answer these questions, the
CH-54B rotor system could be installed on an
H53 helicopter and flown to high speed. Damping,
torsional frequency, and twist could easily be
varied to qualify their effects . Plans to
accomplish this are underway.

References

1. Harris, F. D., and Pruyn, R. R., BLADE STALL
- HALF FACT, HALF FICTI0H, American Helicopter
Society, 23rd Annual National Forum Proceed-
ings, AHS Preprint No. 101, May, 1967.

231

2. Ham, N. D., ana Garelick, M. S., DYNAMIC
STALL CONSIDERATIONS IN HELICOPTER ROTORS,
Journal of the American Helicopter Society ,
Vol. 13, No. 2, April 1968, pp. U°-55.

3. Ham, N. D., AERODYNAMIC LOADING ON A TWO-
DIMENSIONAL AIRFOIL DURING DYNAMIC STALL,
AIAA Journal, Vol. 6, No. 10, October 1968,
pp 1927-193**.

k. Liiva, J., et al., TWO-DIMENSIONAL TESTS OP
AIRFOILS OSCILLATING NEAR STALL, Vol. I,
Summary and Evaluation of Results, The Boeing
Company, Vertol Division; USAAVIABS TR 68-13A,
0. S. Army Aviation Materiel Laboratories,
Fort Eustis, Virginia, April 1968, AD 670957.

5. Carta, F. 0., et al. , ANALYTICAL STUDY OF
HELICOPTER ROTOR STALL FLUTTER, American
Helicopter Society, 26th Annual National
Forum, AHS Preprint No. 1+13, June, 1970.

6. Arcidiacono, P. J., et al., INVESTIGATION OF
HELICOPTER CONTROL LOADS INDUCED BY STALL
FLUTTER, United Aircraft Corporation,
Sikorsky Aircraft Division; USAAVIABS

Technical Report 70-2, U. S. Army Aviation
Materiel Laboratories, Fort Eustis, Virginia,
March 1970, AD 869823.

Carta, F. 0., and Niebanck, C. F., PREDICTION
OF ROTOR INSTABILITY AT ffl - TORW © i r "3BS,
Vol. Ill, Stall Flutter, United Aircraft
Corporation, Sikorsky Aircraft Division;
USAAVIABS Technical Report 68-18C, U. S.
Army Aviation Materiel Laboratories, Fort
Eustis, Virginia, February 1969, AD 687322.,

Arcidiacono, P. J., STEADY FLIGHT
DIFFERENTIAL EQUATIONS OF MOTION FOR A
FLEXIBLE HELICOPTER BLADE WITH CHORDWISE
MASS UNBALANCE, USAAVIABS TR-68-18A, February
1969, AD 685860.

Carta, F. 0., et al., INVESTIGATION OF
AIRFOIL DYNAMIC STALL AND ITS INFLUENCE ON
HELICOPTER CONTROL LOADS, USAAVIABS TR72-51,
Eustis Directorate, U. S. Army Air
Mobility Research and Development Laboratory,
Fort Eustis, Virginia, September 1972,
AD 752917.

232

MULTICYCLIC JET-FLAP CONTROL FOR ALLEVIATION OF HELICOPTER
BLADE STRESSES AND FUSELAGE VIBRATION

John L. McCloud, III* and Marcel Kretz**
Ames Research Center, Moffett Field, California 94035

Abstract

I Results of wind tunnel tests of a 12-meter-
diameter rotor utilizing multicyclic jet-flap control
aef lection are presented. Analyses of these results
are shown, and experimental transfer functions are
determined by which optimal control vectors are
developed. These vectors are calculated to eliminate
specific harmonic bending stresses, minimize rms
levels (a measure of the peak-to-peak stresses) , or
minimize vertical vibratory loads that would be
transmitted to the fuselage.

Although the specific results and the ideal
control vectors presented are for a specific jet-flap
driven rotor, the method employed for the analyses
is applicable to similar investigations. A discus-
sion of possible alternative methods of multicyclic
control by mechanical flaps or nonpropulsive jet-
flaps is presented.

a, b, c,

Aci

CLR/c

CXR/a

Cyr/o

F l> F 2» F 3

V *c

V *s

<* s

6 3p = i tan'

6 4P = i tan

Notation

matrix elements

number of blades

chord of blades

blade section lift coefficient

increment of blade section lift coeffi-
cient due to multicyclic jet-flap
deflection

rotor average lift coefficient (6Cm/a)

rotor lift coefficient (L/p(fiR) 2 bcR)

rotor propulsive force coefficient
(X/p(SR) 2 bcR)

rotor side-force coefficient
CY/p(SR) 2 bcR)

forces measured below the rotor hub

rotor lift

rotor radius

transfer matrix

forward flight velocity

cosine component of the summation of
forces F for the nth harmonic

sine component of the summation of
forces F for the nth harmonic

rotor propulsive force

rotor side force

rotor shaft axis inclination

jet- flap deflection angle

-1

(<5 /<$
3S' 3C

4s' i»c

:!}

azimuth angles for max-
imum deflection

blade bending stress (or rotor solidity

for rotor coefficient definitions)
air density
azimuth position
rotor rotational velocity

Subscripts

P
s

0, 1, 2, 3 .
Superscript

cosine

variable parts
primary control
sine
.n harmonic number

transpose of matrix or vector

(Units are as noted, or such as to produce unitless
coefficients.)

Introduction

To achieve its full potential as the most
effective VTOL aircraft, the helicopter must dras-
tically reduce its characteristic vibrations and
attendant high maintenance costs. As shown in
Reference 1, helicopter maintenance costs are twice
those of fixed-wing aircraft of the same empty
weight. With the same basic elements — engines,
gear boxes, pumps, propellers, and avionics equip-
ment — in both aircraft, this difference is
assuredly traceable to the high vibration environ-
ment of helicopter components. Coping with this
environment, helicopter designers are forced to
provide heavier systems, which result in higher
ratios of empty weight to payload. These ratios
combine to yield maintenance costs per unit payload
that are greater than twice those of fixed-wing
aircraft. The relationship between oscillating
loads — hence vibration — and maintenance costs
has been dramatically demonstrated and reported in
Reference 2. As shown in that report, the Sikorsky
bifilar system reduced rotor-induced vibratory loads
by 54.3%, which in turn reduced failure rates so
that 48% fewer replacement parts were required, and
overall maintenance costs were reduced by 38.5%.

Many vibration suppression systems are being
investigated by various groups. These systems are
characterized as either absorption, isolation, or
active control. The multicyclic jet-flap control
is an active control system, which controls or
modulates the oscillating loads at their source,
that is, on the blades themselves. That we can
effectively change the loading distribution of a
helicopter rotor in forward flight so as to reduce
cyclic blade stress variations, or to reduce vibra-
tory loads transmitted to the fuselage, has been
demonstrated by large-scale wind tunnel tests of
the Giravions Dorand jet-flap rotor at Ames Research
Center. The rotor, its design, and performance
characteristics have been reported on in Refer-
ences 3 and 4. Its supporting wind tunnel test
equipment and some of the results of the multicyclic
load alleviation tests were presented in Reference 5.
Some of that multicyclic test data will be shown
herein also.

*Research Scientist
Ames Research Center, Moffett Field, Calif. 94035

**Chief Engineer

Giravions Dorand, 92150 Suresnes, France

233

The main purpose of this paper is to show the
method used to analyze the multivariable data, and
how it is possible to develop several "ideal" con-
trol schedules "or vectors to achieve specific blade
stress and vibratory load reductions. A simplified
analysis of the results is presented, indicating
that multicyclic systems that do not employ propul-
sive jet-flaps may be feasible.

Rotor and Test Apparatus

The Dorand Rotor is two-bladed, with a teetering
hub and offset blade coning hinges, but no feather-
ing hinges. The rotor is driven in rotation by a
jet-flap, of the blown mechanical flap type, on the
outer 30% of the blade radius. The mechanical flaps
are deflected by a swash-plate and cam system, which
provided both collective and harmonic control.
Swash-plate tilt provided the longitudinal and lat-
eral control, whereas the cams introduced second,
third and fourth harmonic variations. The rotor is
shown, mounted in the NASA-Ames 40- by 80-ft wind
tunnel, in Figure 1. Further details of the rotor
and test apparatus are given in References 3, 4, 5,
and 6.

Results and Analysis

The wind tunnel tests, their range and the
modi operandi, are described in Reference 6. The
tests simulated forward flight conditions at blade-
loading coefficients Clr/cf somewhat greater than
conventional rotors employ.

Figures 2 and 3 (taken from Reference 5) show
some typical results from the multicyclic tests.
Figure 2 shows three sets of jet-flap deflection
angle and blade-bending stresses with and without
multicyclic control. Some control distortion is
affecting the "without multicyclic control" in that
the deflection is not purely sinusoidal. The basic
bending stresses are predominantly three per revolu-
tion (3P), typical for a relatively stiff, heavy
blade. The peak-to-peak stress reductions are 29,
21, and 36%. Figure 3 shows the effect of the
multicyclic control on the forces below the hub in
the nonrotating system: on the left, traces for
three vertical force transducers for the condition
of zero multicyclic control; on the right, traces
for the same transducers for multicyclic control
applied.

These tests produced data for a large number of
flight conditions and multicyclic deflection com-
binations. More of these data are presented in
Reference 6, which includes both time histories and
harmonic coefficients of blade-bending stress, ver-
tical forces, and jet-flap deflection.

Blade-Bending Stresses

As discussed in Reference 5, the relationships
between the time histories of jet-flap deflections
and the resulting blade-bending stresses can be
expressed by a transfer matrix.* The time histories

"This method of analysis was first suggested and
developed by Dr. Jean-Noel Aubrun of Giravions
Dorand.

of jetrflap deflection and blade-bending stress are
both expressed as harmonic series. If the harmonic
coefficients of the stress variation (Eq. 1) are
related to the jet-flap deflection harmonic coeffi- /
cients (Eq. 2), as shown in Eq. 3, they can be
expressed in the matrix form as in Eq. 4.

a = o Q + oi cos * + 0i sin * ♦ 02 cos 2^ + 02 sin 2<|» + ... lj

f
5 = S + 61 cos ij) + 61 sin i|i + 62. cos 2$ + «2 S sin 2$ + ... (2Q

"ns - («n,)(«o) * (bn s )(«l c ) * («ns) («I„) •♦ KX^) * - ("%„) (3)

then

°lc
"Is

ao b c d • • oo
ai c oi c c, c d, c • • o l( . o

a 's b ls c "s dl s " ' O1 s
an s >>n s % s d„ s • • o nso

«o

«n
1

(4)

The last term of Eq. 3 and the last column of the
transfer matrix represent the harmonics of stress,
which are due to the flight condition. With the
column matrices or vectors of the harmonic contents
of jet -flap deflection and blade stresses known for
several conditions, computer routines can solve for
the transfer matrix elements,

A sample result of this method was shown in
Reference 5, together with correlation plots showing
very good agreement between stresses calculated
using the transfer matrix and measured stresses.
The matrix, based on 15 flight conditions, showed
large amounts of interharmonic coupling, particularly
for the third and fourth harmonics of stress.

It is apparent from Eq. 4 that it is possible
to determine multicyclic jet-flap deflection ampli-
tudes that will eliminate the corresponding higher
harmonic stress coefficients. These higher harmonic
stress terms are set to zero and the equation is
then solved for the required jet- flap deflection
coefficients. These coefficients will be hereinafter
called the "ideal harmonic control vector." Refer-
ence 6 presents some of these control vectors.

Although the objective of zero higher harmonic
stresses was achieved, the requisite multicyclic
jet-flap deflections produced different amounts of
IP stresses and, in some instances, the peak-to-
peak stresses were increased. The changes in IP
stresses imply a change in the rotor's thrust and
inplane forces. (Note that the ideal harmonic
control vector as determined in Eq. 4 may be consi-
dered to be for "fixed stick" conditions as existed
in the wind tunnel tests.) Therefore, a second
transfer matrix (Eq. 5) was defined as shown below.

234

°>c

°»s

°2 C

»2 S

■

f»e

*>3 S

?»c

<=0 *«,

lis. "1.

"1,

80 ho *<)

ClrA>

Cyr/o

(5)

«irms = -CT^TJ-lCT/TpJ S p

(7)

Notice that the columns of the transfer matrix
and the elements of the control vector have been
rearranged. The first column represents stress
levels for the condition of zero rotor shaft inclin-
ation, zero rotor force coefficients, and no jet-flap
deflections. The second through fourth columns
represent the changes in stress level due to rotor
angle of attack and the rotor's force coefficients.
The remaining columns correspond to stress deriva-
tives with respect to the multicyclic jet-flap
deflections. The control vector has been realigned
to reflect the column changes. Note that the matrix
elements are no longer defined by Eq. 3, but by
Eq. 5 itself, and the basic "collective" and "IP
cyclic" terms have now been replaced by the rotor's
force coefficients, C^r/o, Cxr/u and Cyr/0 (multi-
plied by 10 3 for numerical convenience) . This can
be considered the transfer matrix for "fixed flight"
conditions. Correlations for this matrix are not
as good as those for the "fixed stick" conditions,
probably because of the greater scatter in the force
data. However, for 30 test conditions, the corre-
lation is very good, comparable to the 15-test con-
dition correlation shown in Reference 5.

The matrix, based on 30 flight conditions, is
shown in Figure 4. Again, it is possible to deter-
mine multicyclic jet-flap deflections to produce
zero higher harmonic stresses. These deflections
also define an ideal harmonic control vector, this
time for fixed flight conditions. Although the IP
stresses may still change, and the peak-to-peak
stress increase, the rotor's force output is
unchanged, at least to the accuracy of the basic
methodology.

While elimination of a particular harmonic, or
all higher harmonics of stress, may be beneficial,
it may be more desirable to reduce other stress
parameters, such as the root -mean-square, or the
peak-to-peak values. It is difficult to relate
peak-to-peak values to the harmonic coefficients,
and the iterative algorithm necessary to affect
peak-to-peak minimization would be considerably
more complex, for example, than one to minimize the
root-mean- square values. The rms value of the
variable portion of the stresses will be minimized
when the sum of the squares of the harmonic coeffi-
cients is also minimized. This sum is given by

?(\

(6)

where irms indicates an ideal root-mean- square, and
the matrices and vectors are defined by partitioning
Eq. 5, as shown below:

b c do e f go ho io

'=0

c °s

a 'c

; d 'c •

J l

T P

Tin

id» s .

ii.

C Y r/<»

This product will be minimized when the multicyclic
deflections are given by

These ideal vectors have also been calculated
for the 30 cases with resultant rms reductions
between 40 and 66%. Figure 5 shows a few of these
cases, with stress calculated for "zero" multicyclic.
These stresses have been, in effect, extrapolated,
whereas the data in Figure 2 were measured. As
indicated on the figure, the ideal rms control also
reduced peak-to-peak stresses. For the 30 cases
investigated, the ideal rms control vectors reduced
peak-to-peak stresses from 39 to 65%.

The ideal multicyclic vectors given by Eq. 7
are a function of the flight condition as defined
by shaft axis inclination, advance ratio, and the
rotor's lift, propulsive, and side-force coeffi-
cients. The elements of these ideal rms control
vectors have been plotted against propulsive force
coefficient in Figure 6. Different symbols denote
the corresponding lift coefficient levels. The
effects of Clr/o and Cxr/o and shaft axis inclin-
ation are quite apparent. (The range of side-force
coefficients was insufficient to deduce its effect.)
The third and fourth harmonics were quite constant
in phase; hence, only their amplitudes have been
plotted. Note that these harmonics do not appear
sensitive to rotor lift coefficient.

Transmitted Vibration Forces

The rotor suspension system for the wind tunnel
tests incorporated a six-component balance and a
parallelogram support discussed in References 4
and 5. The parallelogram support absorbed inplane
vibratory loads very effectively, so that the verti-
cal vibratory loads were the only ones of interest.
These loads are due to thrustwise hub shears in
combination with the motions of the hub due to the
parallelogram support. For this two-bladed rotor,
the transmitted loads contained only even-order
harmonics as shown in Figure 3. These loads may
also be related to the harmonics of the jet-flap
deflection by a transfer matrix, as shown by Eq. 8.

With this transfer matrix it is possible to
eliminate the second and fourth harmonics of the
vertical vibratory loads by the same procedures
used to eliminate the higher harmonic blade-bending
stresses if two of the harmonic components of the
control vector are specified. The resulting

235

°0
V 2„„

Po 1o r o
P2. 12 C r 2

Cxr/°
Cvr/o

«2 S
5 3 s

(8)

where

V A (Fi

+ F 2 + F3)n r

Vn s A (Fi + F 2 + F 3 )n s

deflection harmonic components would define ideal
vibration control vectors whose elements would depend
also on the flight condition. Such vectors have been
calculated for the third harmonic jet-flap deflec-
tions set to zero and are shown in Reference 6.
These vectors (calculated for 12 cases) show the
second and fourth control components to be constant
in phase, but they are significantly different in
phase and magnitude from the ideal stress control
vectors. As might be expected, the lack of third
harmonic jet-flap deflection, and a large fourth
harmonic requirement, resulted in very large third
harmonic blade stresses, when these ideal vibration
control vectors were input into Eq. 5.

When ideal rms (stress) control vectors are
input into Eq. 8, the vibratory loads sometime
increase. A sample case is shown in Figure 7.
Shown are the stress and vibratory loads for "zero"
multicyclic, the actual multicyclic used in the wind
tunnel test, and the ideal rms control vector. The
actual peak-to-peak stress reduction is 39% and the
ideal stress reduction is 47%. The ideal rms con-
trol vector increased the vibratory loads 78%, while
the actual control increased them by only 48%. The
upper portion of the figure shows the actual and
ideal multicyclic component amplitudes and phases.
The actual phases are quite close to the ideal
phases, but the actual third and fourth harmonics
are too low. It is also apparent, however, that
these third and fourth harmonics caused the increase
in vibratory loads.

It is apparent from the foregoing that some
sort of combined matrix is needed to effect reduc-
tions in both stress and vibratory loads. It would
not be possible to eliminate all of the harmonic
components since for this test rotor, we only have
six elements in the control vector, 62 » $z s
through 6u . It is possible, however? to eliminate
six of the response elements. For example, one may
select both harmonics of vibratory loads and the
third and fourth sine components of stress — the
largest of the stress components — and construct a
transfer matrix such as shown below. The multicyclic
deflections required are determined by the solution
of this equation for the condition that V2 C , V2 S ,
Vu c , Vu s , 03 s and ai, s are all equal to zero. The
remainder of the stress coefficients and Vo can be
determined from Eqs. 5 and 8 after the multicyclic
control vector has been evaluated.

V2 c„

V2 c

P2 C

'«..

V2 s

P2,

V 'c„

P"<

V *S0

p»

° 3 s„

°3,

»3

■*•,„

°»s-

q Zc r2 c
b3 s c 3s

1
Clr/°

CxR/o
C YR /o

(9)

Of course, other ideal control vectors are also
possible, and these would depend quite obviously on
the particular rotor and flap control system and the
number of blades, etc. The blades' natural frequen-
cies, the position and extent of the flaps will all
affect the blade stress transfer matrix. The num-
ber of blades will have a definite effect on the
harmonics of blade loads transmitted to the nonrota-
ting system; hence, the compromise between loads
and stress control would differ in each case. How-
ever, the basic method for analysis used herein
can be applied to any such investigation, experi-
mental or theoretical.

Multicyclic Lift Requirements

The results presented here correspond to a
specific jet-flap driven rotor. The question arises
to what extent other circulation control means would
permit a similar reduction of stress levels in the
blades and of vibratory loads. Such systems as
mechanical flaps, servo flaps controlling the twist
of the blades, low-powered jet-flaps, conventional
rotor blades having multicyclic control in addition
to swash-plate control may introduce multicyclic
lift effects and are, at least conceptually, capable
of producing some amount of stress and vibration
alleviation. This capability being assumed, the
problem then becomes one of degree rather than one
of nature. The systems differ only by their
unsteady flow characteristics but have to offer the
similar capability of producing high frequency lift
inputs up to at least the fourth harmonic of rotor
frequency. The remaining question is "How much
incremental lift is needed?"

There was no instrumentation on the blades to
determine the local lift variations, and had there
been, it would not be possible to determine the
amount due to the multicyclic jet-flap deflection
directly. However, knowing the jet-flap deflection
and the average jet momentum coefficient, it is
possible to calculate an incremental lift coeffi-
cient, assuming a nonvariant alpha. This has been
done for several of the wind tunnel test cases and
the Aci ranged from ±0.12 to +0.68 for the higher
harmonic components. Figure 8 shows the variation
of the local blade element coefficient Aci for an
ideal rms control vector. The corresponding stress
reduction projected for this case would be 50%.
(Note that Acj is approximately ±0.68.) The figure
shows that the highest lift variation occurs on the
retreating blade, a fact that proves favorable for
the jet-flap, whose capability increases in low
Mach-number flows.

236

It is believed that these magnitudes of Aci
are obtainable with low powered jet-flaps. Assuming
that somewhat lesser incremental lift variations
would be necessary for softer conventional rotor
blades, multicyclic mechanical and/or servo-flap
control appears feasible. Two study contracts
underway also support this contention.

The sensitivity of the blade stresses and
vibration to multicyclic control and our present
inability to predict harmonic loading, stresses, and
and vibration, leads to the desirability of com-
pletely automating multicyclic control such as would
be attained by feedback control systems. The Gira-
vions Dorand firm is engaged in a basic research
program to develop such a feedback system and early
results are quite encouraging.

CONCLUDING REMARKS

Wind tunnel tests of a jet-flap rotor simulat-
ing forward flight have shown that it is possible
to modulate the rotor's loading by means of a multi-
cyclic control system so that rotor blade stresses
and vibratory loads transmitted to the fuselage
can be reduced. A method of analyzing the multi-
variable problem has been presented and several
"ideal" control schedules are presented. The sched-
ules themselves are applicable only to the specific
jet-flap rotor tested, but the method of determining
the schedules is applicable to similar systems. It
was shown that it is not possible to eliminate all
oscillatory blade-bending and vibratory loads with
a system such as the test rotor, which had only
three higher harmonics of azimuthal control. Such
limited systems can, however, be used to eliminate
specific selected harmonic component stress and
vibration responses.

A simplified estimate of the incremental lift
coefficient being generated multicyclically by the
test rotor indicates that similar multicyclic
mechanical or low-powered jet-flaps could also be
sucessful in reducing blade stresses or vibratory
loads .

References

1. Aronson, R. B. and Jines, R. H., "Helicopter
Development Reliability Test Requirements,
Vol. I - Study Results," USAAMRDL TR 71-18A,
February 1972.

2. Veca, A. C, "Vibration Effects on Helicopter
Reliability and Maintainability," USAAMRDL

TR 73-11, April 1973.

3. Evans, William T. and McCloud, John L., Ill,
"An Analytical Investigation of a Rotor Driven
and Controlled by a Jet-Flap," NASA TN D-3028.

4. McCloud, John L., Ill, Evans, William T., and
Biggers, James C, "Performance Characteristics
of a Jet-Flap Rotor," in Conference on V/STOL
and STOL Aircraft , Ames Research Center,

NASA SP-116, 1966, pp. 29-40.

5. McCloud, John L., Ill, "Studies of a Large-Scale
Jet-Flap Rotor in the 40- by 80-Foot Wind Tunnel,'
presented at Mideast Region Symposium A.H.S.
Status of Testing and Modeling Techniques for
V/STOL Aircraft, Philadelphia, PA, October 1972.

6. Kretz, M., Aubrun, J.-N., Larche, M., "March 1971
Wind-Tunnel Tests of the Dorand DH 2011 Jet-Flap
Rotor" NASA CRs 114693 and 114694.

MULTICYCLIC MULTICYCLIC

Figure 1. Jet-flap rotor in the Ames 40- by 80-Foot Figure 2. Effect of multicyclic jet-flap deflection
Wind Tunnel. on blade stresses.

237

|— I REV— -|

IVArW

|—l REV— )

/WV\A-

WITHOUT MULTICYCLIC
CONTROL

yWA/^Af

WITH MULTICYCLIC
CONTROL

Figure 3. Effect of multicyclic jet-flap deflection
on vertical forces below hub.

-441

-36

287

-t9

-230

-12

-409

-16

660

-5

-2

-3

-2

-13

1 12

-4

1 °

-2

-3

1 " 5

-13

-20

-18

1 10

-15

-52

1 "

-5

-21

-20

1 °

-7

-78

a s

d.,/0-

C XR /er
C VR rtr

h c

8 «s

(T 45 RADIAL STATION
30 CASES AT V/flR*.4

Figure 4. Transfer matrix for fixed flight
conditions .

Clr/ctxIO 3
o 110— 121

a loo— 109

O 90—99

' 4 _ A 80 — 89
2 L k 70 — 79

-I L-

NO FLAG
ONE FLAG
TWO FLAGS

a 5

-10°
-12°
-15°

Figure 6. Ideal rms vector relations.

MULTICYCLIC
DEFLECTION PHASES

8 2 ,
S3'

BLADE
STRESS

A\A\

I/O MULTICYCLIC

comw.

ACTUAL MULTICYCLIC IDEAL r m s CONTROL
MIND TNKEL TEST) FOR STRESSES

r\ a /^

-\A A/

6
2

: vyv

P-P REDUCTION
65%

f\ a A r*

r"\

t 8

■\A A/

J V\^

X 6

l-VW V

'1 4
z 2

- V

51%

rx /\ A _.

A /

^\\\r

. \

^/u J

- \ v

\y \y

4
2

47%

WITHOUT MULTICYCLIC WITH "IDEAL rmS CONTROL

Figure 5. Calculated blade bending stresses using
equations 5, 6, and 7.

Figure 7. Calculated blade stresses and vibratory-
loads using equations 5, 6, 7 and 8.

Figure 8.

90 180 270 360

AZIMUTH ANGLE, * deg

Variation of the estimated increment of
blade section lift coefficient due to
multicyclic jet-flap deflection.

238

IDENTIFICATION OF STRUCTURAL PARAMETERS FROM HELICOPTER DYNAMIC TEST DATA

Nicholas Giansante
Research Specialist

William G. Flannelly
Senior Staff Engineer

Kaman Aerospace Corporation
Bloomfield, Connecticut

Abstract

A method is presented for obtaining
the mass, stiffness, and damping param-'
eters of a linear mathematical model,
having fewer degrees of freedom than the
structure it represents, directly from
dynamic response measurements on the
actual helicopter without a priori knowl-
edge of the physical characteristics of
the fuselage. The only input information
required in the formulation is the approx-
imate natural frequency of each mode and
mobility data measured proximate to these
frequencies with sinusoidal force excita-
tion applied at only one point on the
vehicle. This dynamic response informa-
tion acquired from impedance testing of
the actual structure over the frequency
range of interest yields the second order
structurally damped linear equations of
motion .

The practicality and numerical sound-
ness of the theoretical development was
demonstrated through a computer simulation
of an experimental program. It was shown,
through approximately 400 computer ex-
periments, that accurate system identifi-
cation can be achieved with presently
available measurement techniques and
equipment .

Notation

C
d
f
f

i
J
K
m

Presented at the AHS/NASA-Ames Special-
ists' Meeting on Rotorcraft Dynamics,
February 13-15, 1974.

N
P
Q
R
S
Y

n
[*]

Subscripts

J, k

( )

number of degrees of freedom

number of forcing frequencies

number of modes

residual

modal mobility ratio

displacement mobility, 3y/3f

natural frequency

matrix of modal vectors

modal index

degree of freedom index,
generalized coordinate index

a subscripted index in
parentheses means the index
is held constant

Superscripts
(q) q-th iteration
* modal parameter
R real

influence coefficient

damping

force

-1

force phasor

-T

structural damping coefficient

imaginary operator (i = /^T)

Brackets

number of generalized coordinates

[ ], ( )

stiffness

r J

mass

{ }

imaginary

transpose

inverse

transpose of the inverse

pseudoinverse , generalized
inverse

matrix

diagonal matrix
column or row vector

239

capital letters under matrices indicate
the number of rows and columns,
respectively

a dot over a quantity indicates differen-
tiation with respect to time

The success of a helicopter struc-
tural design is highly dependent on the
ability to predict and control the
dynamic response of the fuselage and
mechanical components. Conventionally/
this involves the formulation of intu-
itively based equations of motion.
Ideally, this process would reduce the
physical structure to an analytical
mathematical model which would predict
accurately the dynamic response character-
istics of the actual structure.
Obviously, the creation of such an
intuitive abstraction of a complicated
real structure requires considerable
expertise and inherently includes a high
degree of uncertainty. Structural
dynamic testing is required to substan-
tiate the analytical results and the
analysis is modified until successful
correlation is obtained between the
analytical predictions and the test
results .

Until a prototype vehicle is avail-
able, intuitive methods are the only
choice for describing an analytical model.
However, once the helicopter is built,
the method of structural dynamic testing
using impedance techniques can be used to
define directly a dynamic model which
correlates with the test data. Such a
model, synthesized from test data,
succeeds in unifying theory and test,
minimizing the intuitive foundation of
conventional analyses.

System Identification has been de-
fined as the process of obtaining the
linear equations of motion of a structure
directly from test data. In System
Identification the objective is the ex-
traction of the mass, stiffness and
damping parameters of a simple mathemati-
cal model directly from dynamic response
measurements on the actual helicopter
without a priori knowledge of the physical
characteristics of the fuselage. Figure 1
presents a pictorial representation of
the System Identification process.

This paper describes the theory of
System Identification using impedance
techniques as applied to a mathematical
model having fewer degrees of freedom
than the structure it represents. The
method yields the mass, stiffness and
damping characteristics of the structure,
the influence coefficient matrix, the
orthogonal modes, the exact natural
frequencies, the generalized parameters

associated with each mode and dynamic
response fidelity over the frequency range
of interest. The only information nec-
essary to implement the method is the
approximate natural frequency of each mode
and mobility data measured proximate to
these frequencies with the excitation
applied at a single point on the vehicle.
This data can be readily obtained from
impedance type testing of the helicopter
over the frequency spectrum of interest.

TBE0B&TZOU.

DEVELOPMENT

-MBH*M.H

Figure 1. System Identification
Process

The usefulness and numerical sound-
ness of the theoretical development was
demonstrated through a computer simulation
of an experimental program, including a
typical and reasonable degree of measure-
ment error. To test the sensitivity of
the method to measurement error, a series
of computer experiments were conducted
incorporating typical and reasonable
degree of measurement error. The results
indicate that accurate identification of
structural parameters from dynamic test
data can be achieved with presently
available measurement techniques and
equipment .

240

Description of the Theory

Derivation of the Single Point Iteration
Process

As presented in References 1 and 2,
the mobility of a structure at forcing
frequency, io, is given by

[y 3 = r*] ty* j r$] T

(1)

With excitation at station k, the respon-
ses at station j, including k, are
obtained. These provide the k-th column
of the mobility at a particular forcing'
frequency o> 1 :

t{Y j(k)u 1 HY j(k)a) 2 }1

= Wr^jUY^HY*^}]

JxN NxN Nx2

Generally, for p forcing frequencies
where 1 < p < P,

tY j( k)p^ - t*ir*kiJ [Y i P ]

JxP

JxN NxN NxP

(4)

(5)

j (k)u.

If J > P, Equation (5) is set of more
equations than unknowns for which there is
no solution. In this situation, Equation
(5) can then be written as

3y J /3f 1

[Y. ,.. ] = [*] M>, -J [Y. ] + [R. ]
3 (k)p J L J lT ki^ L lp u jp

(6)

J x Y L/ki { * } i = [ ^ {Y L/ki } (2)

where R. is the residual associated with

DP
the j-th station and the p-th forcing
frequency .

where 1 £ j £ J and 1 _< i £ N.

This represents a column of mobility
response each element of which is the
response at a generalized coordinate on
the structure with excitation at station
k and at forcing frequency u,. Similarly,
with the exciter remaining at station k,
the k-th column of the mobility at
another frequency, w 2 , can be obtained.

3 Yl /3f 2 '
{Y j( k)u, 2 >= {3 y 2 / 3f 2>

8yj /3f 2

N
= 2

i=1 Y L 2 w*>i = [$]{Y L 2 *ki } < 3)

The columns of mobility response
represented by (2) and (3) may be com-
bined into one matrix

As described in References 1 and 2,
the imaginary displacement mobility is
usually significantly affected by modes
associated with natural frequencies in
proximity to the forcing frequency.
Reference 3 indicates that accurate
estimates of the modal vectors may be
obtained by considering only the effects
of modes . proximate to the forcing fre-
quency. Therefore, the analysis will
employ only Q modes, where Q is less than
N. The imaginary displacement mobility
may be expressed as:

[y;

**■>

j(k)p

] = [•] f» ki J [Y ± ;] + [R. p ] (7)

Since each column of [Y. ] is

associated with a particular frequency,
the dominant element of each row of the
matrix will be the modal mobility measured
at the forcing frequency in proximity to
a particular natural frequency. Nor-
malizing the rows of the aforementioned
matrix on the largest element yields

I V ■ riA in J [Y Ip 3

(8)

where Y. is the maximum value of the i-th
m

row. Equation (7) may be rewritten,

incorporating Equation (8)

241

[Y j(k) P i - [♦]W kl *2ns lp i+.[H 3p ]

(9)

The matrix Equation (9) has no
solution, however, an approximation to a
solution may be defined as that which
makes the Euclidian norm of the matrix of
residuals a minimum. The modal vector
matrix with respect to which the Euclidian
norm of the residuals is a minimum is ob-
tained through use of the pseudoinverse ,
and is given by

w = [Y D T (k)p ] [s . p ] + r -i-

f ki in

(10)

where [S. ] is defined as the generalized

inverse or pseudoinverse of [S. ] and is
defined by lp

[S ip ]+= [S ip ]T([S ip ][S ip ]T)_1 (11)

In Equation (10) each diagonal element of

[= jf=J simply multiplies the correspon-
ds .Y.
Y ki in

ding column of the modal matrix. Since
each modal vector is normalized on the
largest element in the vector, the effect
of the aforementioned multiplication is
negated and Equation (10) can be reduced
to

M = [Y j(k)pHs ip ] + (12)

The [S] matrix can be accurately
estimated from knowledge of only the
forcing frequencies and the natural fre-
quencies. Equation (12) will be solved
utilizing matrix iteration techniques.
At each successive iteration a solution
is found that minimizes the Euclidian
norm of the residual matrix with respect
to the newly found matrix of either [S]
or [<{>]. The basic algorithm used in the
matrix iteration procedure for the q-th
iteration becomes

H> (g) ] = tY I ][s (q " 1, ] +

and

[S (< J } ] =

[<f> (g) ] [Y X ]

(13)

Determining the Modal Parameters

The real modal impedance at forcing
frequency u can be written as

5 iw

Y *R
id)

5-E ■ - ■ ■■

*-d 2 *_ 2

(Y. ) + (Y.; )

10) 10) '

p ' p

(14)

Substituting the real and imaginary dis-
placement mobility as given in Reference 1
yields

\l =K.(l-a>X 2 )
p F

(15)

From Equation (15) it is observed >
that the modal impedance is a linear
function of the square of the forcing
frequency. The forcing frequency at which
the modal impedance becomes zero is,
therefore, the natural frequency. From a
least squares analysis of modal impedance
as a function of forcing frequency
squared, proximate to the natural fre-
quency, the generalized stiffness of the
i-th mode and the natural frequency of the
i-th mode can be calculated.

The generalized mass associated with
the i-th mode is given by

it it O

m i - vv

(16)

The structural damping coefficient may be
determined from

*i =

(-£-

1 2

10)

(17)

Models

There are two basic types of dynamic
mathematical models describing structures.
The first type described as "Complete
Models" considers as many modes as degrees
of freedom. The second type labelled
"Truncated Models" considers fewer modes
than points of interest on the structure.
Using the methods described herein, it is
possible to identify either a complete
model or a truncated model.

For the completed model the modal
matrix [4>] is square. However, in the
case of the truncated model the modal
matrix [<H is rectangular having J rows
corresponding to the points of interest
and Q columns associated with the mode
shapes, where J > Q.

242

Truncated Models

Consider a rectangular identified
modal matrix which has J rows indicating
the points of interest on the structure
and Q columns representing the modes being
considered where J > Q. The influence
coefficient matrix for the truncated model
is given by

[C TR ] = [4,]p4-j[0] T
K i

(18)

The above matrix is singular being of rank
Q and order J. The mass, stiffness and
damping matrices for the truncated model
are

[m^] = [<f>] rm.jj [<j>]

[K™! = [<j>] +T tK*J[(|)] +

"TR

[d TR ] = [(!>] +T rg i K*4f<J>] + (19)

The classical modal eigenvalue equation
has the analogous truncated form

i c ra"»W { *i>-:rr<*i>

(20)

Complete Models

For the complete model the identified
modal vector matrix is square, having the
same number of degrees of freedom as mode
shapes, thus J = Q. The influence matrix
is given by

N T

[C] = [<!>]n/K*J[(i.] T = t 1/K*{(f>. H(j>.}
1 i=l 1 1 1

(21)

The mass, stiffness and damping
matrices for the complete model are simi-
lar to those of Equation (19) , except
that the matrices are square.

[m] = Wl^muJEt))]" 1

[k] = m T tt/K*jr<i>r ;L

[d] = M -1 ^*.]^]"" 1

(22)

Full Mobility Matrix

The full mobility matrix of either
complete or truncated models is given by

[Y] = I*] tY*J [<j>] T
Computer Test Simulation

(23)

The usefulness and numerical sound-
ness of the theoretical development was
demonstrated through a computer simulation
of an experimental program. Approximately
400 computer experiments were performed in
the study. A twenty-degree-of-freedom
lumped mass beam type representation of a
helicopter supported on its main landing
gear and tail gear was used to generate
simulated mobility test data. Each of the
coordinates was allowed a transverse de-
gree of freedom. The concentrated mass
and stiffness parameters of the beam are
shown in Table I, with EI varying
linearly between stations and with 5
percent structural damping.

Simulated Errors

System Identification theories of any
practical engineering significance must be
functional with a reasonable degree of
experimental error. Therefore, a typical
and reasonable degree of measurement error
ranging to +15% random error uniformly
distributed and 15% bias error, was incor-
porated in the simulated test data. Both
random and bias error were applied to the
real and imaginary components of the dis-
placement mobility data. The levels of
error applied are consistent with those
inherent in the present state-of-the-
measurement art.

Models

The number of degrees of freedom of
a physical structure is infinite. There-
fore, the usefulness of model identifica-
tion, necessarily with a finite number of
degrees of freedom, using impedance
testing techniques , depends on the ability
to simulate the real structure with a
small mathematical model.

Several size models , containing from
5 to 15 degrees of freedom, were synthesi-
zed from the simulated test data incor-
porating the specified experimental error.
Table II describes the various models
used in the analysis. The model stations
used in the models refer to the corres-
ponding stations in the twenty point
specimen .

Identified Models

Typical generalized mass identifica-
tions are shown in Tables III, IV and V.
Table III presents results for several
different five point models. The model
designations refer to the descriptions
presented in Table II. Data are also

243

presented for the twenty point specimen
with zero experimental error. Thus, a
basis of comparison is established with
the theoretically exact control model of
the beam representation of the helicopter.
It is apparent that no outstanding dif-
ferences exist among the identified
generalized masses for the models con-
sidered for comparison. Table IV presents
similar data for the nine-point models
studied. The generalized mass distribu-
tion associated with each of the models
is in excellent agreement with the twenty
point model results.

Table V describes the results of the
computer experiments conducted employing
the twelve point models. The results are
satisfactory except for the identification
of the generalized masses of the tenth and
eleventh modes. However, the generalized
masses associated with these modes are

extremely small in comparison with the
remaining modal generalized masses. An
examination of the tenth "mode shape re-
vealed a lack of response at all points
of interest on the structure other than
the first station. Therefore, the effect
of the tenth mode is difficult to evaluate
in the calculation of the generalized
parameters. Computer experiment 309
yielded a negative generalized mass for
the tenth mode. All computer experiments
that failed in this respect gave dras-
tically unrealistic values of generalized
mass. Ordinarily, in a situation where
the generalized mass was unrealistic, use
of different stations for the model
improved the identification.

Sta No.

TABLE I.

20-POINT SPECIMEN DESCRIPTION

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Sta (In.)

60 120 160 200 240 280 320 370 430

30 100 140 180 220 260 300 340 400 460

Mass
(Lb-Sec 2 /In. )

.029 3.67 2.18 2.385 2.08 .910 .170

1.05 3.71 2.18 2.59 1.56 .260 .085

.070 .095 .210

.060 .120 .150

|EI , n .35 .35 1.95 4.37 5.80 4.425 3.07 2.05 .975 .55
(Lb-In7 x 10 10 )

.35 1.20 3.00 5.70 5.60 3.6 2.60 1.60 .65 .50

Springs to
Ground (Lb/In.)

10000

TABLE

II.

MODEL

DESCRIPTION

Stations

Used

Model

12A

12B

12F

244

TABLE III.

IDENTIFICATION OF
5X5 MODEL OF 20

GENERALIZED MASSES,
X 20 SPECIMEN

Model

j **

Computer

Experiment

Number

296

297

292

295

Random Disp.

Error

+5%

+ 5%

Bias Disp. Error

+5%

+ 5%

Random Error

Seed

Mode

Generalized Masses
(Lb-Sec 2 /In.)

8.544

8.538

8.543

8.568

534

4.506

4.619

4.610

449

.494

.49 3

495

1.048

1.047

1.050

.994

.087

.653

.651

.629

.630

** From 20 x

Sp<

scimen

TABLE

IV.

IDENTIFICATION
9X9 MODEL OF

OF GENERALIZED MASSES
20 X 20 SPECIMEN

Model

20 Pt

Computer

Experiment

Number

300

303

304

^**

Random

Disp.

Error

+ 5%

Bias Disp. Error

+ 5%

Random

Error

Seed

Mode

Generalized Masses
(Lb-Sec 2 /In.)

9.000

9.015

9.043

8.534

4.350

4.335

4.513

4.449

.472

.495

1.042

1.138

1.087

.551

.549

.584

.6 30

.786

.783

.723

.743

1.154

1.243

1.120

1. 177

1.401

1.411

1.396

J.412

.787

.708

.791

.78(>

** From 20 x

20 Specimen

245

TABLE V.

IDENTIFICATION OF GENERALIZED MASSES,
12 X 12 MODEL OF 20 X 20 SPECIMEN

Model

12B

12F

12A 20 Pt

Computer

Experiment

Number

312

311

309 1**

Random Disp.

Error

+5%

+ 5%

Bias Disp. Error

+5%

+ 5%

Random Error

Seed

Mode

Generalize
(Lb/Sec

d Masses
2 /In.)

8.474

8.464

8.518 8.534

4.556

4.510

4.492 4.449

.488

.487

.487 .495

1.150

1.151

1.103 1.087

.596

.597

.595 .630

.722

.724

.777 .744

1.182

1.113

1.159 1.177

1.232

1.242

1.215 1.412

.797

.743

.789 .786

1.203

1.043

-.564 .043

,09 3

.104

.0103 .172

1.177

1.119

1.147 1.050

** From 20 x

20 Specimen

Response From Identified Model

One of the most essential requisites
of relating a discrete parameter system
to a continuous system is model response
fidelity over a given frequency range of
interest. The finite degree of freedom
model must accurately reproduce the dy-
namic response of the infinite degree of
freedom structure over a specific number
of modes. Figures 2a and 2b show typical
real and imaginary driving point accel-
eration response respectively for the
five point model. The "exact" curve

represents the simulated experimental
data for the twenty point structure ,
obtained with zero error. The frequency
range encompasses the first five elastic
natural frequencies. Figures 3 and 4
present similar results for typical nine
and twelve point models, respectively.
The computer experiments for which results
are presented incorporated a +5 percent
random and a +5 percent bias on the real
and imaginary displacement mobility data.
As evidenced by the figures, the various
models yielded satisfactory reidentifica-
tion of the twenty point specimen simu-
lated dynamic response data.

246

CASS SEED ERROR

290 . HO

292 13 YES

293 421 YES

CASS SEED ERROR

290 SO C

292 12 YES a
292 421 ITS '

Figure 2a. Effect of Error on Five-Point
Model Identification of Real
Acceleration Response; Driving
Point at Hub

Figure 2b. Effect of Error on Five-Point
Model Identification of
Imaginary Acceleration
Response; Driving Point at Hub

CAIE 1SSD ERROR

299 NO o

300 13 YES a

301 421 YES A

CASES WIOT ERR'.R
±99 RANDOM, St IZAS
100

Figure 3a.

Effect of Error on Nine-Point
Model Identification of Real
Acceleration Response; Driving
Point at Hub

300 400 soo

Figure 3a - Continued

nm ZOSHTZFUD

«» RANDOM, St EIAS

CASE SEED ERROR
305 HO O

in 13 yes a

307 421 YES »

Figure 3b .

Effect of Error on Nine-Point
Model Identification of
Imaginary Acceleration
Response; Driving Point at Hub 9

Figure 4a.

Effect of Error on Twelve-
Point Model Identification of
Real Acceleration Response;
Driving Point at Hub

247

200 300 400 500 600 700

CMC SEED ERROP

.305 NO o
ill D YES a
307 421 YES A

Figure 4a - Continued

Figure 4b. Effect of Error on Twelve-
Point Model Identification
of Imaginary Acceleration
Response; Driving Point at
Hub

Conclusions

Single point excitation of a structure
yields the necessary mobility data to
satisfactorily determine the mass,
stiffness and damping characteristics
for a mathematical, model having less
degrees of freedom than the linear
elastic structure it represents .

The method does not require an in-
tuitive mathematical model and uses
only a minimum amount of impedance
type test data.

The eigenvector or mode shape
associated with each natural
frequency is also determined.

Computer experiments using simulated
test data indicate the method is in-
sensitive to the level of measurement
error inherent in the state-of-the-
measurement art.

References

1. USAAMRDL Technical Report 70-6A,
THEORY OF STRUCTURAL DYNAMIC TESTING
USING IMPEDANCE TECHNIQUES,
Flannelly, W.G. , Berman, A. and
Barnsby, R. M. , U. S. Army Air
Mobility Research and Development
Laboratory, Fort Eustis, Virginia,
June 1970.

2. USAAMRDL Technical Report 72-63A,
RESEARCH ON STRUCTURAL DYNAMIC
TESTING BY IMPEDANCE METHODS - PHASE I
REPORT, Flannelly, W.G. , Berman, A.
and Giansante, N. , U. S. Army Air
Mobility Research and Development
Laboratory, Fort Eustis, Virginia,
November 1972.

3. Stahle, C.V. , Jr., PHASE SEPARATION
TECHNIQUE FOR GROUND VIBRATION
TESTING, Aerospace Engineering, July
1962.

248

ENGINE/AIRFRAME INTERFACE DYNAMICS EXPERIENCE

C. Fredrickson
Senior Engineer

Boeing Vertol Company
Philadelphia, Pa.

Abstract

Recent experience has highlighted the
necessity for improved understanding of
potential engine/airframe interface
dynamics problems to avoid costly and time-
consuming development programs. This
paper gives some examples of such problems,
and the manner in which they have been
resolved. It also discusses a recent pro-
gram in which contractual engine/airframe
interface agreements have already proven
helpful in the timely prediction and
resolution of potential problems.

In particular, problems of engine/
drive system torsional stability, engine
and output shaft critical speeds, and
engine vibration at helicopter rotor order
frequencies are discussed, and test data
and analyses presented. Also presented is
a rotor/drive system dynamics problem not
directly related to the engine.

General

This paper is an attempt to highlight
some recently encountered problems in the
area of helicopter engine and drive system
dynamics. In comparison to the number of
technical papers published in the area of
rotor and blade aeroelasticity and
stability, and fuselage vibration reduc-
tion schemes, there are relatively few
indeed dealing with engine/airframe
dynamics .

The paper does not present highly
sophisticated methods of solution for
these problems . It instead shows that
solutions were attained by the application
of basic engineering principles to state-
of-the-art analytical and test techniques.
Also, having encountered these problems,
we are more cognizant of these potential
"show-stoppers," the manner in which they
manifest themselves, and the available
courses of corrective action. It is
essential that the knowledge gained
through these programs be judiciously
applied to new helicopters, and growth
versions of existing models.

Engine/Drive System Torsional Stability

The usual stability requirements that
dictate fuel control gain limits are com-
plicated by the flexibility of the heli-
copter drive system and by the dynamics of

a gas turbine engine. The interaction of
the helicopter rotor and drive system,
engine, and fuel control requires careful
attention if a good or even workable fuel
control is to be achieved. In the case of
the T55-L-11 engine and the CH-47C air-
craft, these items were growth versions of
existing components. There was no require-
ment for new control concepts since opera-
tion had been successful on previous
models. However, the fuel control gains
had to be carefully re-evaluated for the
new power levels.

Computer simulation of the CH-47C
rotor system with the T55-L-11 turbine
engine was accomplished before initial
flight tests began. The simulation
indicated favorable engine/control stabil-
ity. However, as pointed out in Reference
(1) , unacceptable oscillations in engine
shaft torque and rotor RPM were observed
during initial flight tests (Figure (1) ) .
These torque oscillations were audible,
disconcerting to the flight crew, and were
observed only in hover and on the ground
(not in forward flight) . The frequency of
the oscillation was also higher than the
predicted drive system torsional natural
frequency .

Since the torsional instability was
not predicted by the computer simulation,
a study of pertinent system parameters was
undertaken. It was discovered that the
only parametric change having a significant
effect on torsional stability was the
slope of the blade lag damper force-
velocity curve below the preload force
level. When this curve was artificially
"stiffened" beyond its actual limits, as
shown in Figure (2) , the oscillation was
reproduced. This fact suggested that by
"softening" the actual damper preload
slope, the oscillation might be suppressed.
Once analytically reproduced, the oscilla-
tion could be eliminated by simulating a
fuel control with a reduced steady state
gain and a slowed time constant. The
computer analysis, therefore, revealed two
potential solutions to the torsional
oscillation problem: a lag damper modifi-
cation and a fuel control modification.

Flight tests with a set of lag
dampers with significantly reduced preload
slope, together with the original fuel
controls, were conducted. These tests
revealed that the torque oscillation was

249

apparently suppressed. However, since the
lag dampers were on the aircraft for
ground resonance reasons, this significant
load change reduced damping capacity and
produced some degradation in the ground
resonance characteristics of the helicop-
ter. Therefore, damper modification to
remedy torque oscillation was rejected.

Fuel controls with a 30% reduction in
steady-state gain were flight tested, and
yielded acceptable torsional stability.
However, this degraded to marginal insta-
bility in colder ambient temperatures.
Controls incorporating a gain reduction
plus an increase in time constant provided
acceptable engine torque stability in the
cold and over the entire engine operating
envelope. Fuel control frequency response
curves are shown in Figure (3) . Pilots
also noted that engine response to input
power demands was not perceptibly degraded
with these slowed-down controls. There-
fore, this fuel control modification was
considered an acceptable production fix.

Representation of the lag damper with
just the force-velocity curve in the
engine/drive system/fuel control simulation
had been shown to be insufficient to
accurately reproduce the torque oscilla-
tion phenomenon. Therefore, a more
accurate math model of the damper was
deemed necessary for further analysis,
and for a more complete understanding of
the problem. The derivation of the up-
graded lag damper math model is shown in
Reference (1) . Inclusion of this lag
damper math model into the torsional
stability computer simulation accurately
reproduced the torque oscillation with
the original fuel controls. Frequency of
oscillation, phasing and magnitude of
damper force, shaft torque oscillation,
and fuel flow fluctuation were now sim-
ulated accurately. Final simulation may
be seen in Figure (4) . The primary
difference between this damper simulation
and the earlier version is that the new
model included the hydraulic spring
effect of the damper.

The reduced gain-increased time con-
stant fuel control fix has provided
satisfactory torsional stability for the
CH-47C production fleet. However,
several early production aircraft reported
instances of a "pseudo- torque oscillation".
This phenomenon is a torque split,
followed by a low amplitude torque oscil-
lation of the high torque engine. The
problem was traced to high levels of
vibration affecting the internal workings
of the fuel control. Vibration at cross
shaft frequency caused an instantaneous
increase in the effective gain of the
control, increasing its torque output
with respect to the other engine, and

making it susceptible to torsional
instability. The problem was resolved by
closely monitoring cross shaft vibration,
and with minor fuel control component
modifications .

During the latter part of the torque
oscillation program, it became apparent
that the engine and airframe manufacturers
can easily coordinate their efforts to
prevent this type of incompatibility.
Lycoming has now provided Vertol with a
mathematical model of the engine and fuel
control system, so that rotor/drive system
design changes may be evaluated for their
effect on torsional stability. It is
equally important that as accurate a
representation as possible of the rotor
and drive system be given to the engine
manufacturer.

There has been some mention in recent
years about the possibility of using a
zero torsional stiffness coupling
(Reference (2)) to effectively isolate the
engine from the rotor drive system, there-
by precluding torque oscillation. At this
time, potentially high developmental costs,
uncertainty of transient behavior, and
added weight to the drive system seem to
rule out the z.t.s. coupling. However,
continued research may yield an acceptable
concept that may be the design solution
for torsional instability for the next
generation of increasingly larger, faster
and more complex VTOL rotorcraft.

Engine Vibration at
Helicopter Rotor Frequencies

The CH-47/T55 engine installation is
"hard-mounted", as shown in Figure (5).
It employs two front mounts on a yoke at
the engine inlet housing, and an aft
vertical support link at the engine
diffuser flange. The outboard yoke air-
frame point is connected to take out high
f ore-af t maneuver loads . Engine vibration
had rarely been a problem on the CH-47A
and B models with this type installation.

However, field service reports
indicated an increase in engine, engine
component and engine mount vibration-
related problems with the installation of
the T55-L-11 and -11A engines in the
CH-47C helicopter. These problems led to
a full scale engine and strain survey, the
purpose of which was to determine the
dynamic characteristics of the engine
installation, especially the vibration/
strain relationships. The engine survey
(Reference (3)) provided a wealth of
information concerning the CH-47C engine/
airframe interface dynamic characteristics.
In particular, the survey identified
rotor 3/rev as the predominant excitation
frequency in the engine mounting system.

250

Also, inlet housing stresses and drag strut
load increased significantly with frequency
(rotor speed) , as if approaching a res-
onance, as shown in Figure (6) • As a
result of this discovery, a ground shake
test was recommended to define the char-
acteristics of the apparent engine/air-
frame mode being excited by rotor 3/rev.
The shake test setup is shown in Figure
(7).

The CH-47C/T55-L-11 engine shake test
revealed a 14.2 H z rigid body yaw mode.
Installation of -11A engines (an addition-
al 40 lbs.) caused a .4 H z downward shift
in modal frequency, and a twofold increase
in 3/rev inlet housing strains. Addi-
tional testing showed that reducing drag
strut bolt torque could lower the engine
yaw mode frequency into the CH-47 operat-
ing range (11.5 to 12.5 H z ) . Complete
elimination of the drag strut lowered the
mode to 7.5 H z , well below the CH-47C
operating range. Shake test frequency
sweeps are shown in Figure (8) . Removal
of the drag strut, however, is not a
practical solution. It is needed to
assure acceptable cross shaft alignment
under high maneuver G and jet thrust
loads. The solution, therefore, was to
retain the drag strut, but slot one end to
eliminate dynamic stiffness for small
amplitude motions, resulting in a struc-
turally detuned installation.

Flight evaluation of the slotted drag
strut was desired, and the Model 347
research helicopter was available as a
testbed. The 14 H yaw mode fell within
the operating n/rev frequency range (14-16
H z ) of the four-bladed Model 347 and,
therefore, it would be possible to verify
the inflight placement of the mode. How-
ever, rotor speed sweeps of from 210 to
240 RPM with the standard strut failed to
show a peak inlet housing stress response
in the expected frequency range. Reducing
rotor RPM still further finally located
the engine yaw mode at 13.2 H .

Installing the slotted drag strut on
one engine completely eliminated the 13 H z
peaks, and resulting 4/rev inlet housing
stresses were reduced by as much as 75%.
Lateral 4/rev vibration at the engine
diffuser showed as much as an 85% reduc-
tion. These load and vibration reductions
are illustrated in Figure (9) .

It is noteworthy that analytical
efforts to predict the installation
dynamic characteristics met with limited
success. This analysis first made use of
assumed values of fuselage backup struc-
ture stiffness, and later used values
calculated from a finite element struc-
tural model of the entire fuselage.
However, the accuracy of these stiffness

values is a function of idealization
accuracy and validity, and end condition
assumptions. The analytical predictions
began to resemble the actual test results
only when static load-deflection test data
at the engine support points was used in
the analysis. It is important here to
point out two other factors that con-
tributed to the CH-47C engine vibration
stress problem; the increase in normal
rotor RPM from the A to C model to improve
the flight envelope resulted in a higher
forcing frequency, and the increasing
engine weight and inertia of the more
powerful engine moved the resonant
frequency downward.

Engine bending was not a contributing
factor in this installation. In engine
installations where it is a factor, the
analysis becomes much more complex. Close
coordination between engine and airframe
manufacturers, through engine/airframe
interface agreements, will be necessary to
accurately describe the installed engine
dynamics in this case.

In the overall design of an engine
installation, it is imperative to choose
the engine dynamic characteristics
(isolated, detuned or hard mounted) such
that output shaft alignment is not
jeopardized. Or, conversely, output shaft
couplings must be tailored to the vibra-
tory environment of the engine. In an
isolated engine installation (where most
engine modes are placed well below pre-
dominant forcing frequency) , output shaft
couplings with high misalignment capa-
bility must be employed. In a hard-mounted
or detuned installation, low misalignment
couplings, such as the Thomas coupling,
may be utilized.

Rotor/Drive System B/Rev
Torsional Resonance

The Boeing Vertol Model 347 research
helicopter is a derivative of the CH-47C
Chinook helicopter, the primary differences
being a 30 inch higher aft pylon, a 100
inch longer fuselage, and an increase in
rotor blades from 3 to 4 per rotor
(Reference (4) ) . A Holzer torsional
analysis of the CH-47C revealed natural
modes at roughly .3/rev, .9/rev, 4.1 and
4.2/rev; therefore, the Chinook was con-
sidered to be free from b/rev torsional
resonance (3/rev in this case) . A similar
analysis on the Model 347 revealed almost
identical non-dimensional torsional
frequencies, despite a lengthened aft
rotor shaft and forward synchronizing
shaft, and a reduction in rotor RPM.
There was some concern about the proximity
of the third and fourth torsional modes to
b/rev (4/rev in this case) . However, it
was believed that forcing levels and

251

phasing would not be sufficient to excite
these modes. The Model 347 drive system
torsional modes are shown in Figure (10) .

The Model 347 program was flown
successfully, until the aircraft was flown
at high gross weights. Here, high 4/rev
blade chordwise bending moments in transi-
tion and high speed forward flight became
a structurally limiting factor. Examina-
tion of flight test data revealed that the
chordwise bending moments of all four
blades on each hub were exactly in phase.
Data also revealed substantial rotor. shaft
4/rev torque fluctuations, with the for-
ward and aft rotor systems opposing each
other as shown in Figure (11) , and 4/rev
chordwise bending moments increasing
sharply with RPM, as if approaching a
resonance (Figure 12) .

Analytical parametric studies were
conducted to evaluate the effect of various
system modifications on the apparent 4/rev
resonance. Modifications such as forward
and aft rotor shaft stiffness changes,
synchronizing shaft stiffness changes and
effective lag spring stiffening were all
found to be effective to some extent.
However, these changes were rejected due to
the magnitude of change required to move
the resonance and sensitivity to RPM
changes. A much more acceptable modifica-
tion was found to be raising the blade
uncoupled chordwise bending natural
frequency. On the CH-47C, this blade
frequency was just above 5/rev; conse-
quently, the largest blade bending loads
are at 5/rev. However, with these same
blades on the Model 347, the largest blade
bending loads were at 4/rev, indicating
the blade/drive system coupling effect.

Both blade softening and stiffening
were investigated. It was found that
decreasing the blade chordwise bending
frequency was more effective in moving the
drive system resonance than the same per-
centage increase, as shown by the Figure
(13) analysis. But it was felt that this
blade softening would present too great a
structural degradation problem in the
blade. Hence, raising the blade chordwise
frequency, and with it the coupled blade/
drive system torsional resonance, was the
design goal. Analysis revealed that a 4 H 2
increase in blade natural frequency would
result in satisfactory detuning of the
blade/drive system resonance.

The most effective location to attempt
a chordwise frequency increase is at the
trailing edge. It was necessary in this
case to add on a material of high stiffness
and minimum weight, such that chordwise
balance and CF loads are not grossly
affected. The design selected consisted of
top and bottom boron fiber doublers bonded

to the stainless steel trailing edge from
30% to 70% span, and boron skins applied
to several blade boxes. The benefit of
the boron stiffening is twofold, for in
addition to increasing the chordwise
frequency to avoid resonance, strength is
increased.

The addition of boron stiffening moved
the blade uncoupled flexible chordwise
frequency from 5.26/rev to over 6/rev.
This resulted in a shift in the blade/
drive system natural frequency to over
4.2/rev (at 235 RPM) or to 4.3/rev (at 220
RPM) . This was sufficient to preclude
high 4/rev amplification, since blade
chordwise trailing edge loads are now
highest at 6/rev (the uncoupled blade
frequency) .

This problem does not fall strictly
into the category of engine/airframe inter-
face dynamics. However, the influence of
the engine in the drive system dynamics,
and the potential impact of such a problem
on the engine cannot be ignored. For
example, to accurately predict drive
system modes, the power turbine inertia
must be accurately known.

Engine Output Shaft
Critical Speed Analysis

The Boeing Vertol Heavy Lift Helicop-
ter prototype will incorporate three
Detroit Diesel Allison XT701-AD-700 turbo-
shaft engines. These engines have been
developed from the Allison 501-M62B as
part of a program to procure representative
engines for the HLH Advanced Technology
Component (ATC) dynamic systems test rig.
Many helicopters built in the past were
designed around existing engines. However,
in the case of the HLH, initial development
of the engine is to be fully coordinated
by the prime contractor; hence, development
of both engine and airframe will- be in
parallel. The HLH engine program is dis-
cussed in Reference (5) .

A development problem was encountered
during the program which involved the
engine/airframe output drive shaft inter-
face. The original design of the engine
output shaft was a short splined shaft
with the torquesensor mounted within the
main frame of the engine. Based on more
detailed engine nacelle design, it was
requested that the splined shaft interface
be moved forward to reduce inlet blockage
and to facilitate inspection of the shaft
coupling. This change was agreed upon,
and the drive shaft connection was moved
to a point 17 inches forward of the front
face of the engine. The torquesensor
was also housed in the resulting engine
"nose". A cutaway view of the torque-
sensor and housing is shown in Figure (14) .

252

The original shafting concept on the
HLH was to drive into the main transmis-
sion directly, without right angle gear-
boxes, resulting in a substantial weight
savings. A layout of the original HLH
engine/ntixbox shaft configuration is
shown in Figure (15) . The original
engine-to-mixbox shafting consisted of two
7.25 inch diameter sections of equal
length with a single bearing support
point. However, in an attempt to further
reduce inlet blockage and reduce weight,
the shaft diameter was reduced to 6 inches.
This decision was based on preliminary
analytical trade studies which used an
initial estimate of engine flexibility.
Critical speed placement was analyzed to
be more than 25% above normal operating
speed (11,500 RPM) .

As the detailed design of the engine
progressed and was included in the critical
speed analysis, it became apparent that the
anticipated critical speed margin would not
be realized. The analysis was expanded to
include the torque sensor, its housing,
bearings, and effective engine radial and
moment flexibility. This more detailed
analysis, performed at Detroit Diesel
Allison and confirmed by Boeing Vertol,
revealed the shaft/ torque sensor whirl
mode in the area of 12,500 - 13,000 RPM,
or only about 10% above normal operating
speed. The analytical mode shapes and
frequencies are shown in Figure (16) .

Working together, both companies
conducted parametric analyses to evaluate
various potential fixes. Prime candidates
were inlet housing and torquesensor
housing stiffness increases, a shorter
engine nose, auxiliary support struts,
stiffened torque sensors, plus combina-
tions; however, when they were analyzed in
combination with a complete engine dynamic
model, none proved satisfactory. In fact,
with the complete engine model, the
critical speed of the original configura-
tion was around 10,200 RPM, below normal
operating speed. The mode involved sub-
stantial whirl of the torquemeter housing,
some shaft bending and some case bending,
and was very sensitive to output shaft
coupling weight and unbalance.

This analysis revealed that the only
practical solution was a drastic shorten-
ing of the torquesensor and housing, such
that the shaft adapter is an integral part
of the engine output shaft, and the
flexible coupling is now only 5.3 inches
from the front face of the engine. Due to
the increased distance between the engine
and combining transmission, the output
shaft was changed to a 3-section configura-
tion. This also reduced the amount of
weight hung off the engine. Analysis of
this configuration placed the natural mode
at about 14,200 RPM, which was basically

power turbine conical whirl interacting to
some extent with the torquesensor shafting.
Another mode at about 17,200 RPM showed
compressor conical whirl with rotor, power
turbine and case participation. Forced
response analysis showed both these modes
were only mildly responsive to mass unbal-
ance at the output shaft coupling, as
shown in Figure (17) . This indicates that
the desired shaft/engine dynamic decoupling
has been accomplished.

It is interesting to note how design
decisions not directly related to engine
shaft dynamics provided constraints to the
solution of the interface problem. For
example, the decision to move the shaft
interface well forward of the engine front
face led to the long torquesensor housing
design, which brought about the shaft/
torquesensor whirl problem in the first
place. Also, the engine/shaft interface
could not be moved very much closer to the
engine front face without shortening the
torquesensor. Since torquesensor accuracy
is a function of length, the decision to
drastically shorten the torquesensor and
housing was made with reluctance, since
torquesensor accuracy had to be compromised
to some extent.

Another interesting aspect of this
problem is the fact that the critical speed
of the engine-to-mixbox shafting could not
be accurately analyzed until the complete
engine dynamics were included. This is
where the engine/airframe interface agree-
ment in effect between Boeing Vertol and
Detroit Diesel Allison has been instru-
mental. It has led to excellent working
agreements between the companies that have
helped to reveal, analyze and solve this
potential problem before it reached the
hardware stage. Preliminary shaft critical
speed work was done at Boeing Vertol. How-
ever, when it became apparent that engine
dynamics must be included to accurately
predict the critical speeds, all work was
done jointly with Allison.

Conclusions

(1) Helicopter engine/drive system
torsional instability may be pre-
vented if care is taken to accurately
represent both engine and rotor
systems in the analysis, including
such effects as hydraulic compress-
ibility of the blade lag damper.

(2) Accurate analysis and/or shake testing
of all engine installations, whether
hard mounted, detuned, or isolated, is
required to determine potential engine
vibration and stress problem areas.

(3) Helicopter rotor blades and drive
systems must be designed such that
blade lag flexibility does not couple

253

with drive system torsional flexibil-
ity to produce a resonance at the
number of rotor blade's frequency
(b/rev) .

(4) Formal engine/airframe interface
agreements have already proven
beneficial in the timely resolution
of potential interface dynamics
problems .

References

1. Fredrickson, C, Rumford, K. and
Stephenson, C, FACTORS AFFECTING FUEL
CONTROL STABILITY OF A TURBINE ENGINE/
HELICOPTER ROTOR DRIVE SYSTEM, 27th
National American Helicopter Society
Forum, Washington, D.C., May 1971.

2. Vance, J. M. and Gomez, J., VIBRATORY
COMPATIBILITY OF ROTARY-WING AIRCRAFT
PROPULSION COMPONENTS, 29th National
American Helicopter Society Forum,
Washington, D.C., May 1973.

3. Boeing Vertol Company, D210-10348-1,
CH-47C/T55-L-11 ENGINE VIBRATION AND
STRAIN SURVEY, Rumpel, M. , October
1971.

4. Hooper, W. E. and Duke, E., THE MODEL
347 ADVANCED TECHNOLOGY HELICOPTER,
27th National American Helicopter
Society Forum, Washington, D.C., May
1971.

5. Woodley, D. and Castle, W. , HEAVY LIFT
HELICOPTER MAIN ENGINES, SAE Technical
Paper 730920, October 1973.

254

4.1 HZ

FWD, ROTOR
SHAFT TORQUE

AFT ROTOR
SHAFT TORQUE

FUEL FLOW
ENG. TORQUE

Alt. ± 116K in- lb.
(12.4% of Max.)

A A It A

Alt. ± 10 3K in- lb.
(11.0% of Max.)

WVK

AA/WW\M

DAMPER FORCE

± .455 GPM
(8.95%)

Steady 80.1%
Alt. t 7.6%

± 2800 lb.

Original Fuel controls
Standard Lag Dampers

Figure 1. Torque Oscillation Plight Test
Data

5000

. 4000

<u
o

£ 3000

u
<u
&

2000

1000

Damper "Stiffened" to
Reproduce Oscillation
on Computer

t ' Damper "Softened" to
Eliminate Oscillation
in Test

1.0 2.0 3.0

Peak Damper Velocity, In/Sec

Figure 2. Lag Damper Force-Veiocity Curves

+74K IN-LB.

ROTOR SHAFT
TORQUE

+10K IN-LB.

i.O
.8

.7
.6

■2

/ORIGINAL PROD.
(.0 3 SEC. T/C)
-30% REDUCED GAIN

'" ■ » ■ -,' —-^ jo %

---Ox

*•.. \ ^-sINCRE

LAG DAMPER
LOAD

+ 400 LB.

INCREASED TIME
CONST. (.10 SEC.)

FINAL FIX-^ \ \ X s
(.10 SEC. T/C, \ \ \
30% RED. GAIN) \ \ \

• Si ■ - v ■ v ■ 2"

2 3 A 5 6 7S5IO to 80

FREQUENCY, CPS
Figure 3. Fuel Control Frequency Response

+2500 LB.

y\/\/w

+.33 GPM

ORIGINAL
FUEL CONTROLS

FUEL FLOW

REDUCED GAIN AND
"SLOWED DOWN"
FUEL CONTROLS

Figure 4. Final Torque Oscillation Simulation

255

2500

230 240 250
ROTOR SPEED - RPM

Figure 6. Inlet Housing Stress &

Drag Link Load vs . RPM

256

^ W3 BP* : ¥'^ s *^ 1 Si

■Figure 7. CH-4

EH
TO

H
CO
D
O

En
H

2000

1500

1000

500

3/REV

^STANDARD
STRUT

NO STRUT
/

1 \

-— -^S^K

10 15

FREQUENCY - Hz

Figure 8. CH-47C/T55-L-11 Engine Shake Test Results

257

800

INLET HOUSING 4/REV STRESS

LATERAL DIFFUSER 4/REV VIBRATION

180

200 apjyj 220

240

14
Hz

ROTOR SPEED - RPM

180

200

220

240;

Figure 9. Vibration and Stress Reductions with Slotted Drag Strut

Fwd.

Aft

Modal Deflections Normalized to
1 Degree @ Fwd. Rotor Hub

Figure 10, Model 347 Drive System
Torsional Modes

Fwd. Blade
Chordwise
Bending
-In-Lb. -

Fwd. Lag
Damper Load
-Lb.-

;ip FWD

ULUl 100,000 t

Torque J yVVV> ftftfl^ /WV

'*" -ioo^oo Idsivino I I

Dtor "•••«• r

Torque *f-
b.- -ioo.oooIc

Fwd. Rotor m,ooo
Shaft
-In-Lb

-IDO^OO «D8IVIN0

Aft Rotor «*-•»><>%

Shaft Torque °r — i^^V^/V j s^/lvlvH/VV^

"In-Lb.- -100,400 1 DRIVISS
SOOOf TENSION

Aft Lag
Damper Load
-Lb.~

Af t Blade
Chordwise

Bending
-In-Lb • — -401000

vvv^

1 ROTOR REVOLUTION

0,3. 0.4 0,6 ad

SIMS -MM--

Figure 11. Model 347 Rotor/Drive System
Flight Test Data

258

i 40,000

H
+1

§ 30,000

H
W

@ 20,000

Hi
H

4/REV

GW 32,100 LBS
TAS 150 KNOTS

10,000

lis

200

210

220
ROTOR RPM

230

240

Figure 12.

Model 347 Blade Chordwise
Bending Moment vs. RPM

5.0

m
u

■a

S
d)
-P
to
>i

(!) 0)
> M
■H

u u
a a)

■d

m
■d

<D
H

§•

o
o

Uncoupled Blade Ghordwise Bending Freq.
{per rev)

Figure 13. Effect of Chordwise Blade Bending
Frequency on Model 347 Drive
System Modes

Output Drive Shaft
Torquesensor Shaft -
Extended Housing —

B/V Shaft

Bearings
Figure 14. Original 501-M62B Torquesensor Configuration

259

Figure 15. Original HLH Engine to Combiner
Box Shafting

Engxne
11500 EPM t 1 ?

215. Hz (12900 RPM)
i

Figure 16. Preliminary Engine/Shaft Dynamic

Analysis showing Torquesensor/Shaf
Conical Whirl Mode

o
a
<d

D
N

o
I

<
a

.20

.15

.10

• 5.5" Overhang

• Torquesensor in
Inlet Housing

• 1 In-Oz Unbalance
@ Thomas Coupling

8000

12000 16000 20000

Engine Speed - RPM

Figure 17. Final HLH Engine/Shaft Analysis
Response to Unbalance

260

fflNGELESS ROTOR THEORY AND EXPERIMENT

ON VIBRATION REDUCTION BY PERIODIC VARIATION

OF CONVENTIONAL CONTROLS

G. J. Sissingh and R. E. Donham

Lockheed-California Company

Burbank, California

Abstract

The reduction of the n per rev. pitch-, roll- and vertical
vibrations of an n-bladed rotor by n per rev. sinusoidal variations
of the collective and cyclic controls is investigated. The
numerical results presented refer to a four-bladed, 7.5-foot model
and are based on frequency response tests conducted under an
Army-sponsored research program. The following subjects are
treated:

• Extraction of the rotor transfer functions (.073R hub
flapping and model thrust versus servo valve command,
amplitude and phase)

• Calculation of servo commands (volts) required to
compensate .073R hub flapping (3P and 5P) and
model thrust (4P)

• Evaluation of the effect of the vibratory control inputs
on blade loads

• Theoretical prediction of the root flapbending
moments generated by o to 5P perturbations of the
feathering angle and rotor angle of attack.

Five operating conditions are investigated covering advance
ratios from approximately 0.2 to 0.85. The feasibility of vibra-
tion reduction by periodic variation on conventional controls is
evaluated.

Summary

For several operating conditions covering advance ratios
from approximately 0.2 to 0.85, the control inputs required to
counteract the existing 4P pitch, roll and vertical vibrations are
calculated. The investigations are based on experimental vibra-
tion and response data. As the tests were part of and added on to
a larger hingeless rotor research program, only a few operating
conditions with essentially zero tip path plane tilt were investi-
gated because of limited tunnel time. At the test rotor speed (500
rpm) the rotor blade mode frequencies were 1 .34P, first flapping,
6.3P, second flapping, and 3.6P, first inplane.

This work was conducted under the sponsorship of the
Ames Directorate of the U. S. Army Air Mobility R&D Lab-
oratory under Contract NAS2-7245.'The authors gratefully
acknowledge the assistance of Mr. David Sharpe, the AMRDL
Project Engineer, and Messrs. R. London and G. Watts of
Lockheed in conducting the experimental portion of this work.

It should be noted that there was no instrumentation to
measure the vibratory pitching and rolling moments. These
moments were obtained by properly adding up the flap-bending
moments of the four blades at 3.3 in. (0.073R) which were
measured separately. This means, the effects of the inplane
forces, vertical shear forces and blade torsion have been ignored.
These are important influences in current hingeless rotor designs.
The inplane 3P and 5P shear forces are of particular
interest. However, the experimental data obtained for a model
hingeless rotor system provides the beginning of at least a partial
data base for the investigation of vibration attenuation of such
systems through periodic variation of conventional controls.

Generally speaking, the control inputs required for flapping
(hub moment) sourced vibration elimination are smaller or about
of the same magnitude as those used for the frequency response
tests. Their amplitudes lie, depending on flight condition and
advance ratio, between 0.2 and 3 degrees. With the exception of
the m = 0.85 1 case, for which the results are somewhat in doubt
(the response tests to lateral cyclic pitch and the corresponding
baseline data were inadvertently run with 0.3-degree collective
pitch differential), the control inputs required for vibration re-
duction drastically reduce the 3 and 5P, and have only a minor
effect on the 2P flexure flap-bending moments. Chord-bending
moments and blade torsion generally increase.

The theoretical predictions mentioned refer to forced-
response influence coefficients. They are based on the first two
flapping modes. The blade root flap-bending moments (OP
through 5P) which result from unit perturbations of blade
feathering angle and rotor angle of attack have been calculated.
The solution provides for intermode coupling through the 17th
harmonic by analytic solution of the two-degree-of-freedom
system, utilizing constant coefficient and loading descriptions
over ten-degree azimuth sectors. In each solution case, the rotor
reached steady-state motion in eight revolutions. In that time the
least converging second mode flapping motion converged to a
minimum of four significant figures.

Evaluation of the test data reveals two types of short-
comings, which should be avoided in future tests. First, the data
given are based on a single test and have not been verified.
Second, in some cases, the baseline and frequency response tests
were not run successively.

From the data available, the approach is promising,
especially for the low and medium advance ratio range. At higher
advance ratios (n~ 0.8), the control inputs required for vibration
reduction may become prohibitive.

Notation

Presented at the AHS/NASA-Ames Specialists' Meeting on A, B

Rotorcraft Dynamics, February 13-15, 1974.

quantities describing cos 4^ and sin 4^ components
of actuator input for frequency response tests, volt,
see Table II and Equation (1)

261

C, D quantities describing responses to A and B, in.-lb

and lb, respectively, see Equation (1)

E, F, G, H blade loads due to unit actuator input, in.-lb/ volt,
see Equation (13)

Kj . . . Kjg gains of rotor response, see Table I

m calculated flapbending moment at 3.3 in., in.-lb,

m = m + 2m ns sin n# + 2m nc cos n^

M, L, T 4P vibratory pitching moments, rolling moments
and thrust variations, in.-lb and lb, respectively;
subscript e denotes existing vibrations to be com-
pensated, subscript control describes effects of
oscillatory control inputs.

M e = Mg sin Aii + M c cos 4^

L e = L s sin 4^ + L C cos Aii

T e = T s sin Aii + T c cos Ai>

^nominal nominal collective pitch, degrees

O , 6 S , C oscillator inputs for collective, longitudinal
and lateral cyclic pitch,*volt

6 o = e os sin 4 * + e oc cos 4 *

S = SS sin Aii + 8 SC cos Aii

C = 0™ sin Aii + 6 rr cos Aii

T l • • • T l 8 * a 8 aa ^ es °f response, degrees, see Table I

rotor angular velocity, sec'

■1

azimuth position of master blade, rad

C„„ Blade Root Moment, STA (o)

a<T ttR 3 p(«R) 2 aa

where

a = 5.73

p = 0.002378 slugs/ft 3

a = 0.127

"Compensating Control Inputs" define those which reduce
the existing 4P pitching moments, rolling moments and vertical
forces of a given flight condition to zero.

The analysis deals with the concept of vibration reduction
by oscillatory collective and cyclic control applications. Several
related aspects of this problem are treated. The foremost are the
determination of the proper control inputs and their effect on

the vibratory blade loads. These studies are based on frequency
response tests conducted on a 7.5 foot-diameter, four-bladed,
hingeless rotor model, the results of which are published in
Appendixes C and D of Reference 1. The subject matter covered,
apart from the items listed below, is an abridged version of these
appendixes.

Other subjects treated are (a) the calculation of blade loads,
based on test data, due to vibratory control command applica-
tions; (b) the theoretically determined eigenvalues, at 10-degree
azimuth intervals, of the first and second flapping modes, at
M = 0. 1 9 1 , 0.45 and 0.85 1 ; (c) the computed single-blade root
flap-bending moment, Sta 0, harmonic influence coefficients
at m = 0. 1 9 1 , 0.45 and 0.851; and (d) a limited comparison of the
theoretical loads with experiments.

The general case of vibration control will include the effects
of lateral and fore-and-aft shear forces at blade passage frequency.
These forces can be as influential as the pitch and roll moment
and thrust oscillations in causing fuselage vibrations. Thus, in
general, five rotor vibratory inputs are to be controlled by mani-
pulation of three controls. Although the five vibratory inputs
cannot be nulled individually with three controls, their combined
contribution to the fuselage vibration can be controlled. Thus,
the general application will involve control of fuselage vibration
at three points; say two vertical vibrations and one roll angular
vibration. This general application implies the use of adaptive
feedback controls. Although the present paper is limited to the
more simple case outlined herein, the general application to the
control of any three suitable quantities will be apparent.

Although prior investigations of the use of higher harmonic
pitch control on teetering and offset hinge rotors have been con-
ducted to investigate improved system performance and also for
vibration attentuation (References 2 , 3 and 4 ), this is believed
to be the first experimental and theoretical hingeless rotor study
of the use of periodic variation of conventional controls for
vibration attentuation. The use of 2P feathering to improve rotor
performance is not included as part of this work.

Transfer Functions Involved

As a distinction must be made between control applications
in phase with sin Aii and cos 4^, there are six control quantities
available, i.e., QS , QC , &s , d sc , 8 CS and CC , to monitor the
pitching moments, rolling moments and vertical forces. This
means the dynamic system investigated, which consists of rotor,
control mechanism and oscillators used, is characterized by 18
gains Kp and lag angles r p . The subscripts p (p = 1 through 1 8)
are defined by Table I.

TABLE I

GAINS AND LAG ANGLES OF RESPONSE

TO OSCILLATORY CONTROL APPLICATIONS

e os

Ooc

fl ss

9 sc

9 cs

fl cc

K m

% T 2

K 3 T 3

K 4 r 4

K 5 '5

K 6 '6

K 7 r 7

K 8 T 8

K 9 r 9

K 10 r 10

K ll r ll

K 12 T 12

K 13 T 13

K 14 r 14

K 15 T 15

K 16 T 16

K 17 T 17

K 18 T 18

262

As indicated, Kg is defined as the amplitude ratio M/0 SS and
t 3 is the lag angle of M with respect to 8 SS . For convenience, the
dimensions used are identical with those of the computer output,
i.e., oscillator voltage for input, in.-lb for M and L, lb for the
thrust variation T. This means the dimensions of Kp are

Kj through Kj 2
Kj3 through Kjg

in.-Ib/volt
lb/volt

See also Figure 1 which shows the oscillatory pitching moments
due to combined 8 SS and SC control applications. The moments
generated are presented by rotating vectors where cos 40 is posi-
tive to the right and sin 40 positive down. This means, the vector
positions shown refer to = 0. By definition, the quantities Ry
characterize the responses in phase with the excitation and Iy
those out of phase. The latter are positive if the response leads.
As indicated, there are altogether four responses involved which
are combined to the resultant M.

The phase angles r p are given in degrees, r p is positive if the re-
sponse lags.

Although the investigations deal exclusively with 4P control
variations, some general remarks may be in order. The general
case involves sinusoidal collective and cyclic control variations
with the frequency n£2 where n can be any positive number.

If n is an integer, the rotor excitations repeat themselves
after each rotor revolution which means that the responses of
each revolution are identical. This is true for any number of rotor
blades but does not necessarily mean that all blades execute
identical flapping motions. The latter is true only if n equals the
number of rotor blades or is a multiple of the blade number.
Only for these cases does a truly time independent response with
invariable amplitude ratios K and lag angles r exist.

Extraction of Gains and Lag
Angles from Experiments

As for all response tests conducted, the oscillator input con-
tained both sin 40 and cos 40-components; always two amplitude
ratios K and two lag angles t sis involved. Therefore, each time a
set of two tests must be evaluated. According to Table II, the
input is characterized by the quantities Aj B j A2 B2 and the
response by Cj Dj C2 D2.

If the rotor responds to cos 40 excitations with the gain K;
and the lag angle t-. (j = even number) and to sin 40 excitations
with Kj and r- x (i = odd number), input and output are related by
the equations

Aj Kj cos (40 - rp + Bj Kj sin (4* - t { ) = C j cos 40

+ Disin40

A 2 Kj cos (4* - Tj) + B 2 K } sin (40 - Tj) = C 2 cos 40

+ D 2 sin 4^

TABLE II
INPUT AND OUTPUT NOTATIONS

(1)

Test

Input

Response

#1
#2

Aj cos40 + Bj sin 4*
A 2 cos 40 + B 2 sin 40

q eos40 + Dj sin 40
C 2 cos 40 + D 2 sin 40

To calculate the unknowns Kj Kj Tj and v., a component
analysis is used. The gains Kj Kj are expressed as

Kj = (R? + 1?) 1/2
Kj - (R?+ I. 2 ) 1/2

(2)

SS'3

SIN 40

Figure 1. Vector Diagram Showing Pitching Moment
Due to SS and 9 SC Control Applications

Inserting Equation (2) into Equation (1) leads to

A,D 2 -A 2 D 1

AiBj-AsBj

A 1 C 2 -A 2 C 1
A 1 B 2" A 2 B 1

tan?;= llj/Rj I 0<Ti<ff/2

(3)

and

Rj =

CiB 2 -Bj[C 2
AjB^B!

BiD 2 -B 2 Pl
AiB 2 -A 2 B,

tan?j= llj/Rjl 0<Tj<jr/2

(4)

263

In both cases

r=+T for R>0 I<0

= -t R>0 I>0

= 7T+? R<0 I>0

= rr-r R<0 I<0

Check of Calculated Kj K: Tj and r.- Values

If so desired, Equation (1) can be used to check the calcu-
lated values of Kj Kj Tj and' t-.. Splitting up these equations into
sin 4# and cos 4v components leads to the following four
expressions which must be satisfied

Aj Kj cost: -Bj Kj sin Tj = Cj
Aj Kj sin Tj + B j Kj cos-tj = Dj
A2 K: cos Tj - B2 Kj sin tj = C2

A2 & Sin T: + B2 Kj COS Tj = D2

Oscillatory Control Inputs Required

(5)

The six oscillator inputs available have to be selected so that
their responses satisfy the requirements, whatever they may be.
By definition, the vibratory control inputs result in the following
pitching moments, rolling moments and vertical forces (n = 4):

M control = + e os K l sin ( n *- T l)
+ e oc K 2 cos(n^-T 2 )
+ 9 ss K 3 sin(n*-r 3 )
+ sc K 4 cos(n<J'- r 4 )
+ cs K 5 sin(n*-r 5 )

+.fl cc K 6 cos(n*-r 6 ) (6)

L control =+e s K 7 sin ( n *- T 7)
+ 9 oc K 8 cos(ni//-Tg)

+ ss K 9 sin(n^- T g )

+ e sc K 10 cos(n*-T 10 )

+ cs K 11 sin(n*-T 11 )

+ cc K 12 cos(n>/'-r 12 ) (7)

T control = + e os K 13 sin ( n *" r 13>

+ e oc K 14 cos ( n *- T 14)
+ ss K 15 sin(n*-r 15 )

+ e sc K 16 cos(n*-r 16 )

+ 6» cs K 17 sin(n'/'- t 17 )

+ cc K 18 cos(n</'- t 18 )

M control = ^ sin 4 * " M c cos 4 *
L control = " L s sin 4 * " L c cos 4 *
T control =-T s sin44'-T c cos4*

(8)

(9)

To reduce the existing vibrations, the moments and forces
generated must counteract M e , L e and T e , i.e.,

Equations 6 through 9 lead to six linear equations, (10),
for the unknowns OS , OC , 8 SS , SC , CS and 6 CC .

Effect on Blade Loads

An objective of the investigations is to determine the effect
of the compensating control input on the blade loads, i.e., on the
following measured quantities:

• flapbending at 3.3 in.

• flapbending at 1 3. 1 5 in.

• chordbending at 2.4 in.

• torsion at 9.28 in.

In all cases the 2 to 5P content of the loads is of interest.
The first task is to determine from the response tests the contri-
bution of each of the six possible 4P control inputs to these
loads. Again, two sets of data are required. The vibratory control
applications used and the resulting n" 1 harmonic of the load con-
sidered are written as follows :

Test

Input

Resulting Load (in.-lb)

#1 Aj cos 4^ + B j sin 4^ C n j cos n^ + D n j sin nf
#2 A 2 cos 44* + B 2 sin Ai> C n2 cos n* + D n2 sin n*

(11)

+Kj cos Tj +K 2 sin t 2 +K3 cos T3 +K4 sin t 4 +K 5 cos r 5 +Kg sin Tg

-Kjsin Tj +K 2 cos t 2 -K3 sin T3 +K4 cos t 4 -Kg sin tj +Kg cos Tg

+K7 cos Tj +Kg sin Tg +Kg cos 19 +Kjq sin Tjq +Kj j cos tj j +Kj2 sin t j2

-K7 sin Tj +Kg cos Tg -K9 sin Tg +K jq cos Tjq -Kj j sin tj j +Kj2 cos r ^

+Kj3Cosfj3 +Kj4sinrj4 +Kj5COsrjj +KjgsinTjg +KJ7C0STJ7 +Kjg sin Tjg

-Kj3sinTj3 +KJ4COSTJ4 -KjgsinTjj +Kjg cos Tjg -Kj^sinrj^ +Kjg cos Tjg

264

#OS

-M s

9 oc

-Mc

e ss

-h

9 SC

"L c

" T s

_0 C c.

(10)

If nonlinear effects are excluded, the n per rev load vari-
ation due to unit control application in phase with

(a) cos # amounts to (E n cos ni^ + F n sin n# )

(b) sin 4* (G n cosn<HH n sinniJ>)

In these expressions

(12)

These moments were obtained by properly adding up the
flap-bending moments of the four blades at 3.3 in. which were
measured separately. This means, the effects of the in-plane
forces, vertical shear forces and blade torsion have been ignored.

TABLE IV

VIBRATORY MOMENTS AND FORCES

TO BE COMPENSATED

E n =

B 2 C nl- B l C n2

% =

H„

A 1 B 2"

A 2 B 1

B 2 D nl

- B l D n2

A 1 B 2"

A 2 B 1

A l c n2

" A 2C„1

A 1 B 2"

A 2 B 1

A lDn2

- A 2^1

A 1 B 2 -A 2 B 1

(13)

If 0£ S , 6t c (t, = o, s, c) denote the vibratory control inputs
used, the increments of the n tn harmonic of the load considered

(14)

(Aload) n = (0g c E n + 0t s G n ) cos n^

+ (0| C F n + £s H n )sinn*
Evaluation of Experiments
Flight Conditions Investigated

The methods outlined in the previous sections are applied to
the following five operating conditions for which test data are
available:

TABLE HI
OPERATING CONDITIONS INVESTIGATED

^nominal

Cj/ff

0.191

12°

-5°

0.102

0.239

-5

0.028

0.443

-5

0.011

0.849

-5

-0.005

0.851

-5

-0.013

In all cases the shaft angle of attack is a = -5° and the rotor is
trimmed so that essentially a| =b| =0. As can be seen, the tests
cover the advance ratio range from approximately fi = 0.2 to
H= 0.85. The case u= 0.191 is characterized by nomuia i = 12°
and C T /ff = 0.102, the latter figure indicates a relatively high
specific loading. In contrast, at the advance ratios ji = 0.849
and 0.851 the rotor is practically unloaded, i.e., no steady lift-
ing force is generated. The 4P vibrations associated with the
various test conditions are listed in Table IV. The moments are
given in inch-pounds and the vibratory forces in pounds.

0.191

0.239

0.443

0.849

0.851

0.3805

- 1.7207

2.6149

20.0483

3.5349

M c

- 0.5301

-0.4113

-0.5208

-4.5724

-8.4341

12.2080

1.3725

-6.7626

9.4647

-10.5154

L c

2.2180

-1.9145

- 3.7399

-31.1214

-17.2626

T s

0.1979

-0.1089

0.0304

1.9247

0.8838

T c

-0.2013

- 0.0865

0.0556

- 0.0048

- 0.8626

Gains and Lag Angles

The rotor response characteristics are calculated by applying
equations (2, 3,4) to the test data available. The results
available are listed in Table V. As pointed out previously, the
values given include the effect of the actuator used. Some
general statements can be made. It is obvious that for n = 0.
the gain and lag angle of the responses to sin 4V- and cos
4^-type control applications must be the same. For p. j= this is
no longer true, and one would expect that the spread between
KjKj and tjTj (see equations (3), (4)) widens with increasing
advance ratio. Further, according to classical rotor theory which
neglects blade stall, the nominal collective pitch setting has no
effect on the frequency response characteristics.

Generally speaking, the KjK: and Tj t: values given in
Table V differ very little. It appears however, that at higher
advance ratios (compare columns for n - 0.849 and 0.851) the
collective pitch has a larger effect than anticipated. It is also
possible that the error of the baseline data described in the sum-
mary may play a role.

Oscillator Inputs Required

Equation (10) is used to calculate the inputs required to

(a) generate unit amplitudes of pure pitching moments,
rolling moments and vertical forces and

(b) compensate the existing vibrations

The results are given in Tables VI and VII. They show that, as
to be expected, the oscillatory.inputs required for vibration
reductions generally increase with increasing advance ratio.
Surprisingly, the rotor collective pitch setting seems to play a
larger role than the steady lift generated. See also Table VIII
which summarizes the results obtained and lists the operating
conditions investigated in the order of decreasing vibrations.
The first column shows the relative magnitude of the vibratory
moments generated and the last column the approximate
amplitude of the blade pitch variation required to compensate
the vibrations. The amplitude of the pitch variation produced
per volt oscillator input changes with the control loads and

265

TABLE V
GAINS AND LAG ANGLES DERIVED FROM EXPERIMENTS

CKp-

in.-lb/volt, T p - degrees)

M = 0.191

H = 0.239

H = 0.443

/u= 0.849

M= 0.851

K p

T P

K p

T P

K p

T P

K p

T P

K p

T P

5.617

42.3

1.099

125.6

2.236

120.5

4.798

72.0

4.094

116.5

6.126

44.0

1.141

149.1

2.791

129.3

4.787

72.6

3.487

135.6

17.571

-9.6

52.416

-30.1

42.237

-28.7

18.537

-19.8

43.319

-5.1

26.019

-45.4

47.991

-37.3

40:073

-30.1

20.329

-41.5

37.081

12.7

30.696

155.7

59.416

182.9

45.186

188.4

33.002

183.4

26.170

214.2

32.505

181.7

77.408

193.2

61.144

180.8

21.085

180.0

38.661

184.5

2.856

136.0

4.246

81.9

8.166

86.5

2.472

102.1

10.097

93.1

1.507

98.4

5.083

67.1

8.077

66.9

3.412

144.7

7.979

62.9

35.384

213.4

59.420

198.8

43.846

181.4

44.506

200.5

48.081

176.2

41.674

185.8

51.280

198.6

39.383

195.7

48.473

201.0

40.850

187.7

45.953

116.6

76.875

108.3

78.512

101.8

67.268

134.4

88.540

94,17

61.589

131.5

86.361

99.3

80.995

95.7

61.288

141.5

90.934

95t3

6.879

45.6

5.420

51.4

8.928

39.2

8.188

35.8

9.340

38.5

7.211

43.7

6.195

46.4

8.999

35.9

8.906

36.1

9.651

35.6

6.635

245.2

4.275

205.9

2.571

195.2

5.976

215.0

3.623

184.0

6.033

218.3

3.962

208.1

3.123

188.7

4.775

229.5

1.977

185.4

13.000

127.3

7.596

94.3

7.632

76.7

13.261

133.1

11.188

86.9

10.057

128.6

8.176

97.4

8.381

92.2

7.953

126.3

11.101

90.7

the type of control (0 , 6 s , C ) used. Therefore, the con-
version factor varies and the last column of Table VIII is
given only to indicate the approximate amplitudes involved.

With one exception, the vibratory control applications re-
quired were smaller than those used for the frequency response
tests. The exception is the case with the highest vibration level
encountered for which the compensating controls required were
approximately 1 5 to 20% higher than the inputs used for the 4P
frequency response tests.

Blade Loads

The calculation of the effect of the compensating control
inputs on the blade loads is based on Equations ( 1 3 ) and ( 1 4). The
first step is to calculate, for each specific case, the quantities E n
through H n (n = 2, 3, 4, 5). See Table IX which refers to n = 0.849
and lists the sin n 4* and cos n^ components of the various
loads due to unit control (volt) application. The table shows, for
instance, that at the advance ratio n = 0.849, a ±1 volt variation
of „, produces 3P chordwise bending moments of the magnitude

(-91.77 sin 3^+7.15 cos 3*) in.-lb

As the control inputs required for vibration reduction have been
previously calculated, their effects on the blade loads can be de-
termined by adding up the various contributions. The reader is
referred to Table X which applies to the flapbending moment
at 3.3 in.for the case m= 0.849. Given are the original loads
without vibratory control application, the individual contribu-
tions and the sum. The last column shows the amplitudes with-
out and with compensating control input. A summary of the

loads is represented in Table XI. Generally speaking, chord-
bending, blade torsion and the 4P flap-bending moments of the
root flexure increase with increasing advance ratio. The 3 and 5P
flap-bending moments of the flexure are, by nature, reduced and
the 2P flap-bending moments are least affected. From the limited
data available, it appears that the 4P chordwise- and 5P torsion
moments may be the critical load for this configuration,
inasmuch as the natural frequencies are close to these
values.

As mentioned previously, it is assumed here that
the pitching and rolling moments are solely caused by the flap-
bending moments of the root flexure which were individually
measured and properly combined by a sin-cos potentiometer.
This means, the only source for the troublesome 4P moments in
the nonrotating system are the 3 and 5P flap-bending moments at
3.3 in. For four identical blades, it follows that elimination of
the 4P pitching and rolling moments requires that the sin 3^, cos
3^, sin Si> and cos 5^ components of the flap-bending moments
at 3.3 in. are reduced to zero. As the four blades behave dif-
ferently, this ideal condition will practically never be fulfilled.

In the preceding paragraphs the flapbending moment of a
specific blade, with consideration of the compensating control
input, was calculated. To a certain extent, these predicted loads
can be used as an independent check. As an example, the case
H - 0.849 is treated. According to Table IV the amplitudes of
the 4P pitching and rolling moments to be compensated are

M= 20.56 in.-lb
L = 32.52 in.-lb

(15)

266

The calculated 3 and SP flap-bending moments with considera- The amplitudes of the resulting 4P pitching and rolling moments

tion of the compensating control input amount to (see Table VII), are

m 3s = 0.6233 in.-lb
m 3c = -1.1833
m 5s = -1.9266
m 5c = 0.3099

(16)

M= 3.14in.-lb
L = 5.91 in.-lb

(17)

TABLE VI

OSCILLATOR INPUTS REQUIRED (VOLT) TO GENERATE PURE sin #- AND cos #- COMPONENTS

OF PITCHING MOMENTS, ROLLING MOMENTS AND VERTICAL FORCES

"■control

e os

fl oc

9 ss

e sc

9 cs

9 cc

0.191

"s, control = '

+0.0143

- 0.0485

+0.0508

+0.0290

- 0.0296

+0.0241

**c, control = '

+0.0117

-0.0123

- 0.0055

+0.0283

-0.0219

-0.0098

*% control = *

-0.0177

- 0.0236

-0.0113

+0.0052

-0.0169

+0.0073

^c, control = *

+0.0042

-0.0071

-0.0209

-0.0200

+0.0003

-0.0147

* s, control = *

+0.0922

+0.1380

- 0.0490

-0.0302

+0.0252

-0.0232

^c, control = '

-0.1044

+0.1164

+0.0123

-0.0210

+0.0235

+0.0081

0.239

™s, control = '

+0.0028

-0.0069

+0.0299

+0.0219

-0.0111

+0.0211

"*c, control ~ '

+0.0109

+0.0028

-0.0096

+0.0206

-0.0154

- 0.0070

^s, control = '

- 0.0023

-0.0108

- 0.0056

+0.0203

-0.0167

+0.0078

^c, control = '

+0.0128

-0.0029

- 0.0245

-0.0243

- 0.0008

-0.0210

l s, control *

+0.1356

+0.1337

-0.0053

-0.0155

+0.0072

-0.0128

* c, control = *

-0.1436

+0.1085

+0.0168

+0.0070

+0.0091

+0.0100

0.443

™s, control = '

-0.0019

- 0.0053

+0.0255

+0.0116

- 0.0069

+0.0145

"*c, control = *

+0.0053

+0.0011

-0.0023

+0.0331

-0.0168

-0.0004

^s, control = '

- 0.0057

- 0.0067

-0.0021

+0.0253

-0.0135

+0.0126

*% control ~ '

+0.0120

-0.0028

-0.0155

-0.0084

- 0.0093

-0.0112

*s, control - *

+0.1020

+0.0732

-0.0088

-0.0094

-0.0018

-0.0138

* c, control = *

-0.0714

+0.0941

+0.0071

-0.0108

+0.0171

-0.0024

0.849

^s, control - '

+0.0049

-0.0240

+0.0338

+0.0179

- 0.0229

+0.0182

"*c, control = *

+0.0149

-0.0149

-0.0109

+0.0487

-0.0271

- 0.0222

*% control ~ *

-0.0124

-0.0137

-0.0120

+0.0074

-0.0118

+0.0024

**c, control = *

+0.0052

-0.0056

-0.0072

-0.0121

+0.0006

-0.0123

l s, control *

+0.1050

+0.0698

-0.0211

+0.0037

+0.0017

-0.0214

*c, control = *

-0.0772

+0.1079

-0.0034

-0.0305

+0.0221

+0.0031

0.851

™s, control = *

+0.0001

-0.0081

+0.0191

+0.0122

- 0.0077

+0.0109

™c, control = '

+0.0082

-0.0055

-0.0126

+0.0290

-0.0135

- 0.0055

*% control = *

- 0.0080

-0.0107

-0.0043

+0.0117

- 0.0057

+0.0098

""c, control = '

+0.0113

-0.0102

- 0.0069

+0.0028

-0.0137

-0.0037

* s, control = *

+0.1016

+0.0599

-0.0087

+0.0109

-0.0130

-0.0091

T — 1

1 c, control ~ '

-0.0682

+0.0998

+0.0034

-0.0143

+0.0189

- 0.0058

! in.-lb

267

TABLE VII

OSCILLATOR INPUTS REQUIRED (VOLT)

TO COMPENSATE EXISTING 4P- VIBRATIONS

TABLE VIII
VIBRATION SUMMARY

V-

0.191

0.239

0.443

0.849

0.851

6 OS

0.1683

0.0394

0.0146

0.0457

0.0300

e oc

0.3121

0.0224

-0.0490

0.2354

-0.2726

9 SS

0.1746

0.0090

-0.1400

-0.7980

-0.3275

e sc

-0.0133

-0.0293

0.1273

-0.5881

0.3498

9 CS

0.2052

-0.0026

-0.1176

0.4610

-0.3549

9 CC

-0.0651

-0.0180

0.0056

-0.8308

-0.0428

Rel. Vibration
Level

flnomi-
nal

C T /ff

Ampl. of Pitch
Variation

0.849

10°

-0.005

-3.0°

0.58

0.851

-0.013

2.0

0.32

0.191

0.102

0.8

0.21

0.443

0.011

0.5

0.08

0.239

0.028

0.2

Decreasing
Vibration
Level !

TABLE IX

EFFECTS OF UNIT 4P OSCILLATOR INPUT ON BLADE BENDING

AND TORSION MOMENTS (in-lb). ju = 0.849

M= 0.849

Input

sin 2ii

cos 2^

sin 3*

cos 3^

sin 4^

cos Ai>

sin 5*

cos 5*

"os

0.3815

- 2.6028

- 1.1212

+ 1 .9467

+ 0.0022

1.6252

- 0.4640

+ 0.2286

#oc

- 0.7265

- 0.7428

- 2.1170

- 0.9082

- 1.7646

0.1744

+ 0.4336

- 0.2014

e ss

-20.1796

- 7.1252

0.4843

10.9746

9.2290

- 1.4705

-12.1221

-16.4408

Flapbending

9 SC

1.4455

- 18.6069

-11.8793

0.8771

1.9670

9.2946

+18.4710

-13.1116

3.3 in.

#cs

-15.0717

19.2091

- 1.7568

+ 13.3006

4.4390

13.0827

24.2022

-18.4700

e cc

-11.0041

- 12.5052

-12.2451

- 3.9250

-11.5818

6.8481

17.1269

+18.2863

#os

- 3.1446

0.01156

+ 0.0644

- 6.4289

0.5673

- 5.5966

- 2.6912

- 5.2806

#oc

0.4488

.- 3.3139

+ 5.7587

- 0.6033

7.2213

1.7289

4.4109

- 1.9638

Flapbending

*ss

-13.1131

- 1.6401

- 9.4439

11.4718

2.7493

1.6368

20.3552

30.4485

13.15 in.

e sc

- 3.1093

- 10.4663

-13.7168

- 7.3647

- 0.7250

4.6008

-31.6355

23.4534

#CS

-15.3541

3.9011

- 20.8842

- 14.1583

- 4.0272

- 4.6816

-53.1766

36.9531

e cc

- 3.7738

- 10.2279

7.2742

- 11.8491

2.4534

1.0036

-30.9619

-33.3918

e os

- 5.2318

5.1653

18.4997

- 66.4765

8.5046

- 2.0555

6.0027

8.6689

#oc

- 0.3311

2.6008

55.9170

15.3823

8.5503

12.5308

-10.1401

4.6381

0ss

- 23.2604

3.6649

-91.7693

7.1537

-12.9172

- 5.1116

-13.8450

7.4174

Chordbending

9 SC

4.7043

- 8.0015

-37.9514

- 71.7419

6.5301

-16.8130

- 4.2184

-12.8505

2.4 in.

0cs

-25.0714

15.3009

- 59.7492

- 177.5673

41.5059

-80.7110

- 5.8153

- 27.4052

e cc

- 2.0059

- 7.7253

77.1483

- 7.0902

68.5358

26.5134

7.7451

-28.5566

60s

0.1891

0.0544

- 0.2460

0.5652

- 1.0733

0.2665

0.1925

0.0465

0OC

0.0788

- 0.1531

- 0.1960

- 0.2328

- 0.6076

- 1.0110

0.0102

0.01822

" d ss

0.4975

0.2685

- 0.9271

- 1.5838

- 0.0498

1.4606

15.6374

13.1496

Torsion

e sc

- 0.6976

- 0.7498

3.0700

- 1.4345

- 1.0039

0.9952

-11.8807

15.1709

9.28 in.

#cs

0.8756

- 0.0250

- 1.5421

- 0.9968

- 1.9762

1.0423

- 14.6088

21.3914

#CC

- 0.8745

- 0.9375

2.1226

- 2.5792

- 1.3255

- 1.2713

- 17.9657

-13.8937

268

Comparison of Equations (15) and (17) shows that the vibratory
pitching moment is reduced to approximately 15 percent and the
rolling moment to approximately 18 percent of its original value.
This indicates that the various blades behave differently and that
the goal of zero 4P pitch-roll and vertical vibrations is achieved
by cancellation of the effects of the four blades.

Analytical Formulation and
Calculated Results

The aeromechanical characteristics of the High Advance
Ratio Model (HARM) has been analytically described in 2
degrees of freedom. These are based on the first and second flap-
ping modes which have been approximated by polynomial fits of

finite element determined mode shapes. The first and second
mode shape approximations used are given by

and

where

#2 = 2.292x 2 - 1.292x 3

2 = -10.21x 2 + 20.78x 3 -9.57x 4

x = r^ the non-dimensional radial station.
R

The aerodynamics are based on classical quasi-steady incom-
pressible strip theory. The reverse flow region is fully accounted
for, but stall effects have been neglected, as described in Refer-
ence 5.

TABLE X

EFFECT OF VIBRATION COMPENSATION ON FLAPBENDING

MOMENT (in-lb) AT 3.3 in. y. = 0.849

cos n

sin n

Amplitude

W/O Vibration Control

-92.7652

17.2338

94.35

Contribution of O

- 0.0559

- 0.1536

16.6165

15.2507

0c
TOTAL

19.2393

2.2002

- 56.9653

34.5311

66.61

W/O Vibration Control

- 1.1732

- 14.7883

14.83

Contribution of Q

- 0.1248

- 0.5496

- 9.2715

6.5928

6c
TOTAL

9.3862

9.3684

- 1.1833

0.6233

1.34

W/O Vibration Control

- 0.1403

- 3.5448

3.55

Contribution of g Q

0.1153

- 0.4152

- 4.2868

- 8.5191

TOTAL

0.3317

11.6713

- 3.9801

- 0.8078

4.06

W/O Vibration Control

3.2312

2.2658

3.95

Contribution of d s

- 0.0370

0.0809

<?s

20.8199

- 1.1807

»c
TOTAL

- 23.7042

- 3.0926

0.3099

- 1.9266

1.95

269

TABLE XI

SUMMARY OF OSCILLATORY BLADE LOADS (IN.-LB)

WITHOUT AND WITH VIBRATION COMPENSATION

Operating
Condition

Flapbending at 3.3 in.

Flapbending at 13.15 in.

Chordbending at 2.4 in.

Torsion at 9.28 in.

0.191
0.239
0.443
0.849
0.851

n=2

n = 3

n = 4

n = 5

n = 2

n = 3

n = 4

n = 5

n = 2

n = 3

n = 4

n = 5

n = 2

n = 3

n = 4

n = 5

Without
Oscillatory
Control
Input

r
i

30.1
10.5
16.4
94.4
18.9

4.4
0.6
2.7
14.8
8.6

1.6
0.2
0.1
3.6
1.5

3.5
0.9
1.6
4.0
3.1

16.0

5.3

9.2

55.9

17.7

1.9
1.7
3.2
3.6
4.6

3.0
0.9
0.4
9.5

3.4

4.3
1.2
3.5
5.9
5.8

21.0

4.6

9.4

31.5

17.4

2.2

2.0

1.7

31.4

10.9

8.3
11.0
10.5
13.1
18.9

19.4

2.6

7.7

14.6

10.7

1.2
0.5
0.9
6.8
3.3

0.7
0.2
0.6
4.1
2.4

0.4
0.3
0.3
0.9
0.7

0.6
0.2
0.2
0.3
0.4

With

Oscillatory
Control
Input

•-

0.191
0.239
0.443
0.849
0.851

29.6
10.3
12.3
66.6
20.2

1.1
0.4
1.3
1.3

2.4

2.9
0.3
1.3
4.1
6.5

0.4
0.7
1.1
2.0
4.1

16.1

5.3

7.5

41.7

16.5

4.4
1.9
2.7
1.3
3.8

5.0
0.8
0.5
2.4
7.0

3.0
1.5
1.3
2.1
2.5

19.2

4.7

7.7

15.7

17.5

22.7

3.0

3.5

68.8

13.6

10.9
11.5
13.6
38.9
75.6

3.9
2.1

8.7

22.3

7.0

1.0
0.5
0.8
6.5

2.3

1.2
0.3
1.4
4.4
4.0

0.4
0.3
1.4
0.8
0.7

4.5
1.2
1.7
3.1
3.5

The method of solution provides for intermode harmonic
coupling through the 1 7th harmonic. This is accomplished by
obtaining transient solutions of the 2-degree-of-freedom descrip-
tion of the rotor system described as constant coefficient linear
differential equations over 1 0-degree sectors of the rotor
azimuth.

The values of the coefficients for the system of differential
equations evaluated in this work have been determined at the
center of the sectors i. e., at 5°, 15°, 25°, etc.

The basis for the analytical formulation is founded on
Shannon's sampling theorem which says that the discrete signal is
equivalent to the continuous signal, provided that all frequency
components of the latter are less than 1/2T cycles per second, T
being the time between instants at which the signal is defined,
(References 6 and 7 ). Since the solution also provides for a com-
pletely general transient solution, it can be used to calculate a
Floquet solution by specializing the initial conditions. This has
been done for the square spring oscillator case studied by M. A.
Gockel and reported in the AHS Journal in January 1972. The
problem statement which is exactly describable by this theo-
retical method was shown to yield the identical Floquet solutions
as those reported. It is important to note that should the system
be unstable, the harmonic balance method of solution would not
directly reveal this instability.

Briefly, the initial conditions at the beginning of a sector are
determined by calculating the terminal conditions for the pre-
vious sector which are then used to initialize the new sector. It
has been found that essentially arbitrary conditions can be used
to start the solution and that excellent steady-state conditions
have been obtained for the conditions examined in six rotor revo-
lutions. For each solution case presented, the rotor has been
solved for eight revolutions to ensure that the second flapping
mode contribution to the response has converged to a steady-

state value accurate to at least four significant figures. The pro-
gram is used to calculate closed-form analytic solutions over each
10-degree sector and therefore is not dependent on a particular
method of numerical integration. (See Appendix A.) The
method, however, when applied to the analysis of steady-state
conditions, does require that sufficient solution time be calcu-
lated so that initial transients are dissippated to ensure that
steady-state equilibrium is achieved (Reference 8).

The test configuration experimentally examined with re-
spect to IP flapbending distributions sX\i = 0, including center-
line measurements, has been compared with this analysis
procedure on Figure 2, utilizing the two-mode description. This
is a limited use of the analysis technique to establish test/analysis
correlation. It is believed that the absence of time-dependent
aerodynamics quasi-steady, largely accounts for the phase error
in response. The centerline shaft moment measured was 0.75 of
the calculated (a = 5.73). This may be due to the relatively low
inflow of the test condition.

In general this correlation, including the spanwise distribu-
tion, appears reasonable.

The eigenvalues of each 10-degree sector are evaluated as
part of the method. These are summarized in Tables XII, XIII, and
XIV versus azimuth the ju = 0.191, 0.45, and 0.85 where the real
and imaginary parts of the eigenvalues have been normalized by
the noted natural-mode frequencies. The negative aerodynamic
spring effects over azimuth 90 < *< 270 as well as the positive
stiffening from 270 < %<90 are as expected more pronounced
on the first mode frequency. The effects of reduced aerodynamic
spring and damping are also seen on the retreating side. These
results show that both damping as well as frequency variations
occur around the azimuth which influence the rotor response
with harmonic excitations.

270

3.0

§2.0
cc

x
°- 1.0

) o-

— o|

t •-

NOTE: PHASE MEASURED IN

DIRECTION OF ROTATION
FROM 4/ = 0°

1 1 1

TABLE XII

NORMALIZED EIGENVALUES* AT EACH 10-DEGREE

AZIMUTHAL SECTOR FOR *i= 0.191

0.2 0.3

x = r/R

Figure 2. One-Per-Rev Blade Radial Flap-Bending Moment
Distribution at h-=0.

The rotating frequencies and properties of the flapping
modes noted in Tables XII, XIII, and XIV analytically describe
the 7.5-ft-diameter rotor, configuration (5), 500-rotor-rpm con-
dition for which all harmonic feathering tests were conducted.
In an effort to further improve analytic correspondence with
test data the slight change of the second flapping mode fre-
quency resulted from matching collective blade angle selection
at the test conditions. Details of the test model are given in
References 9, 10 and 11.

The harmonic components of the blade root flap-bending
moment (OP through 5P) were calculated for these advance ratios
for unit perturbation of blade feathering angle at 6j c , 0j s , $2v

e 2s> e 3c< 9 3s» 9 4c> "4s> e 5c> 9 5s> as we . U as for unit chan 8 e in
6 and a

The single non-dimensional blade root, centerline flap-bending
moment harmonic influence coefficients resulting from harmonic
feathering are summarized in matrix form in Tables XV, XVI, and
XVII for ft = 0.191, 0.45, and 0.85. These are based on har-
monic analysis of the moment at each condition for 36 equally
spaced (10-degrees apart) azimuth intervals. Single-blade

P =

1.34

P =

6.38

SECTOR

xJrO

R l

R.o

-.204

1.024

-.155

1.002

-.212

1.022

-.163

1.002

-.220

1.019

-.170

1.002

-.227

1.014

-.177

1.002

-.233

1.007

-.183

1.001

-.238

.999

-.187

1.001

-.242

.990

-.190

1.000

-.244

.980

-.192

1.000

-.245

.970

-.193

.999

-.244

.960

-.192

.998

:105

-.242

.951

-.190

.998

115

-.239

.943

-.186

.997

125

-.234

.937

-.182

.997

135

-.228

.933

-.176

.997

145

-.221

.930

-.169

.996

155

-.213

.930

-.162

.996

165

-.205

.932

-.154

.996

175

-.197

.935

-.146

.997

185

-.188

.940

-.138

.997

195

-.180

.945

-.130

.997

205

-.172

.952

-.123

.997

215

-.165

.958

-.116

.998

225

-.159

.965

-.111

.998

235

-.154

.971

-.106

.998

245

-.150

.978

-.103

.999

255

-.148

.984

-.101

.999

265

-.147

.989

-.100

1.000

■ 275

-.148

.995

-.101

1.000

285

-.150

.999

-.103

1.000

295

-.154

1.005

-.107

1.001

305

-.158

1.010

-.111

1.001

315

-.164

1.014

-.117

1.001

325

-.171

1.018

-.124

1.002

335

-.179

1.021

-.131

1.002

345

-.187

1.023

-.139

1.002

355

-.195

1.025

-.147

1.002

♦SECTOR EIGENVALUES ARE GIVEN BY:
(Rj+Ij i) (1.34fl)
AND (R 2 + I 2 i) (6.38S2 )

computed root flap-bending moment influence coefficients
at n = 0.45 are compared with experimental 0.O73R
single-blade data, in parentheses, from Reference 1 and 1 2.
in Table XVIII. '

These appear reasonable when shear effects are considered.

It is important that the general character of these influence
coefficients be established in future tests. These tests should be
. structured to permit measurement to confirm these distributions.

271

TABLE XIII

NORMALIZED EIGENVALUES* AT EACH 10-DEGREE

AZIMUTHAL SECTOR FOR n= .45

TABLE XIV

NORMALIZED EIGENVALUES* AT EACH 10-DEGREE

AZIMUTHAL SECTOR FOR M = .85

P =

1.34

P =

6.2

SECTOR

#°

R 2

-.215

1.087

-.167

1.007

-.234

1.088

-.186

1.007

-.252

1.084

-.203

1.007

-.269

1.075

-.218

1.006

-.283

1.059

-.232

1.005

-.295

1.037

-.242

1.004

-.303

1.011

-.250

1.002

-.309

.982

-.255

1.000

-.311

.951

-.256

.998

-.310

.920

-.254

.996

105

-.305

.891

-.249

.995

115

-.297

.867

-.240

.993

125

-.285

.850

-.229

.992

135

-.271

.839

-.215

.991

145

-.255

.837

-.200

.991

155

-.237

.842

-.182

.991

165

-.218

.854

-.164

.992

175

-.197

.870

-.145

.992

185

-.177

.889

-.126

.993

195

-.158

.909

-.108

.994

205

-.139

.928

-.092

.995

215

-.123

.945

-.078

.996

225

-.109

.960

-.068

.997

235

-.098

.972

-.061

.998

245

-.089

.982

-.057

.998

255

-.085

.990

-.056

.999

265

-.083

.997

-.055

1.000

275

-.084

1.003

-.056

1.001

285

-.089

1.011

-.058

1.001

295

-.078

1.018

-.062

1.002

305

-.108

1.027

-.069

1.003

315

-.122

1.038

-.079

1.003

325

-.138

1.049

-.093

1.004

335

-.156

1.061

-.110

1.005

345

-.175

1.072

-.129

1.006

355

-.195

1.081

-.148

1.006

♦SECTOR EIGENVALUES ARE GIVEN BY:
(Rj+Ij i) (1.340)
AND (R 2 +I 2 i) (6.20 U)

P 1 =

1.34

P 2 =

6.20

SECTOR

\]/0

R 2

-.231

1.192

-.040

1.014

-.267

1.209

-.048

1.015

-.301

1.212

-.055

1.016

-.332

1.200

r.061

1.015

-.360

1.171

-.067

1.013

-.382

1.126

-.071

1.010

-.399

1.065

-.074

1.006

-.409

.992

-.076

1.002

-.413

.911

-.076

.997

-.411

.826

-.076

.992

105

-.402

.745

-.073

.988

115

-.387

.675

-.070

.984

125

-.366

. .625

-.065

.982

135

-.339

.603

-.060

.980

145

-.308

.611

-.053

.980

155

-.274

.645

-.040

.981

165

-.237

.698

-.039

.983

175

-.199

.759

-,031

.986

185

-.160

.822

-.023

.989

195

-.123

.879

-.016

.992

205

-.090

.925

-.012

.993

215

-.062

.954

-.011

.994

225

-.043

.970

-.012

.996

235

-.034

.977

-.014

.997

245

-.032

.983

-.015

.998

255

-.032

.990

-.015

.999

265

-.033

1.000

-.016

1.000

275

-.032

1.009

-.015

1.000

285

-.032

1.016

-.015

1.001

295

-.034

1.021

-.014

1.002

305

-.043

1.028

-.012

1.004

315

-.061

1.040

-.011

1.006

325

-.088

1.063

-.013

1.007

335

-.120

1.094

-.017

1.008

345

-.156

1.130

-.024

1.010

355

-.193

1.165

-.032

1.012

*SECTOR EIGENVALUES ARE GIVEN BY:
(Rj± Ij i) (1.34S2)
AND (R 2 ±I 2 fi) (6.20 £2)

272

TABLE XV

:EM. _ BLADE ROOT (STA 0) BENDING MOMENT INFLUENCE COEFFICIENT MATRIX FOR n = 0.191
a " • (Pj = 1.34, P 2 = 6.38)

cpo

cp lc

c Pis

CP 2 C

C P2S

C P3C

C P3S

C P4C

cp 4 s

C P5C

C P5S

.0034

.0009

-.0016

-.0001

.0132

.0049

-.0111

-.0007

-.0005

-.0001

A61S

.0036

.0057

-.0225

-.0018

-.0003

.0002

AG1C

-.0005

-.0213

-.0045

-.0001

.0018

.0002

A62S

-.0056

-.0004

.0018

.0031

.0014

-.0009

.0002

.0003

A62C

-.0003

-.0005

.0057

.0031

-.0018

-.0009

-.0014

.0003

-.0002

A63S

-.0001

.0004

.0001

.0002

-.0046

-.0012

.0015

-.0014

.0001

.0003

A63C

.0004

.0002

-.0001

-.0012

.0046

-.0014

-.0015

.0003

-.0001

A94S

.0001

-.0010

.0017

-.0076

-.0026

.0017

-.0020

A64C

.0001

-.0001

.0017

.0009

-.0026

.0076

-.0020

-.0017

A65S

.0119

.0002

-.0015

.0024

-.0101

-.0033

A65C

.0002

-.0002

.0024

.0015

-.0033

.0101

TABLE XVI
C RM _ BLADE ROOT (STA 0) BENDING MOMENT INFLUENCE COEFFICIENT MATRIX FOR n =.45
aff (Pj = 1.34, P 2 = 6.20)

CPo

cp lc

CP 1S

cp 2C

C P2S

C P3C

C P3S

cp 4C

CP4S

CP5C

■ c Pss

.0085

.0053

-.0089

-.0010

-.0012

-.0004

-.0001

.0160

.0135

-.0276

-.0038

-.0029

-.0009

-.0004

-.0001

-.0002

A61S

.0087

.0102

-.0292

-.0048

-.0016

-.0004

.0007

-.0006

A61C

-.0011

-.0226

-.0034

-.0004

.0046

.0011

.0002

.0001

.0003

^G2S

-.0130

-.0006

.0006

.0046

.0036

-.0020

.0009

.0015

-.0002

.0002

A62C

-.0015

-.0024

.0142

.0043

.0001

-.0020

-.0035

.0015

-.0009

.0001

.0002

A63S

.0023

.0004

.0010

-.0057

-.0032

.0038

-.0036

.0008

.0016

A63C

-.0002

.0022

-.0002

.0007

-.0001

-.0033

.0057

-.0036

-.0038

.0016

-.0008

A64S

-.0001

.0003

.0007

-.0026

.0042

-.0090

-.0046

.0043

-.0048

A64C

.0004

.0042

.0026

-.0046

.0090

-.0048

-.0043

A65S

.0003

.0008

-.0038

.0058

-.0118

-.0055

A65C

.0003

.0008

-.0009

.0058

.0038

-.0054

.0118

TABLE XVII
d|M. - BLADE ROOT (STA 0) BENDING MOMENT INFLUENCE COEFFICIENT MATRIX FOR ju = .85

(Pi = 1.34, P 2 = 6.20)

C PlC

C PlS

C P2C

C P2S

cp 3C

C P 3 S

C(3 4C

CP4S

C P5C

C P5S

Ace

.0201

.0227

-.0296

-.0056

-.0102

-.0037

-.0039

-.0021

-.0015

.0253

.0378

-.0598

-.0141

-.0155

-.0061

-.0039

-.0032

-.0003

-.0018

A61S

.0192

.0278

-.0490

-.0117

-.0114

-.0036

-.0004

-.0019

-.0014

-.0005

-.0001

A81C

-.0024

-.0258

-.0015

-.0012

.0085

.0035

.0008

.0003

.0022

.0009

Afi2S

-.0006

-.0229

-.0007

-.0026

.0079

.0081

-.0034

.0024

.0054

-.0019

.0016

A62C

-.0056

-.0110

.0308

.0084

.0067

-.0026

-.0064

.0052

-.0019

.0013

.0023

A63S

-.0003

-.0014

.0076

.0010

.0035

-.0088

-.0082

.0100

-.0084

.0033

.0049

A83C

-.0009

.0060

.0029

-.0009

-.0084

.0089

-.0084

-.0100

.0048

-.0033

A64S

-.0005

-.0004

.0003

.0013

.0008

-.0067

.0087

-.0135

-.0100

.0112

-.0100

A94C

-.0004

.0002

-.0002

.0007

-.0014

.0087

.0067

-.0100

.0134

-.0101

-.0112

A65S

.0001

.0006

.0002

-.0003

.0029

.0023

-.0088

.0124

-.0167

-.0115

A85C

.0006

-.0002

-.0003

-.0002

.0023

-.0029

.0124

.0088

-.0116

.0167

273

TABLE XVIII
BLADE ROOT (STA 0) BENDING MOMENT (IN-LB)/DEG INFLUENCE MATRIX FOR n •■

( £2= 52.36, Pj = 1 .34, P 2 = 6.20)

.45

Pic

Pis

P2C

P2S

P3C

Pas

P4C

P4S

P5C

P5S

LIFT

-20

-2

-3

-1(1)

-1 (-2)

0(1)

0(-D

-62

-9

-7

-2(1)

-KD

0(1)

0(0)

A61S

-66

-11

-4

-KD

2 (-2)

-1

0(1)

0(0)

A61C

-2

-51

-8

-1

3(0)

0(1)

0(0)

A92S

-29

-1

-5

-1

A02C

-3

-5

-8

-2

-1

A63S

-13

-7

-8

A63C

-1

-7

-8

-9

-2

A64S

-6(0)

9 {6)

-20 (-8)

-10 (-5)

10 (-5)

-11 (-1)

A64C

9(6)

6 (-2)

10 (-4)

20(7)

-11 (-6)

-10(3)

A65S

-9

-27

-12

A65C

-2

-12

Full-Scale Control Loads

The feasibility of active vibration attenuation depends on
the capability of the rotor to generate cancelling shaft moments
and shears while control forces and displacements remain within
acceptable limits.

Since full-scale data are the most relevant from the stand-
point of hardware test background, the CL 840/AMCS
(Advanced Mechanical Control System) Cheyenne rotor
configuration, at a gross weight of 20,000 and with a rotor shaft

moment of 100,000 in.-lb, was analyzed for hovering flight to
gain a numerical measure of how loads compare with limits. In
this analysis three higher harmonic blade-feathering excitations,
3P, 4P and 5P, were examined to determine the relationships
among control loads, shaft moments and shear forces. The
Lockheed Rotor Blade Loads Prediction Model was used for
this analysis; 68 finite elements were used to describe the system.
The calculated results, based on 1 -degree excitation levels, are
summarized in Table XIX.

TABLE XIX

CL 840 ANALYSIS -

SHAFT AND BLADE LOADS DUE TO ONE-DEGREE

OF HIGHER HARMONIC BLADE-FEATHERING MOTIONS

FEATHERING FREQUENCY

Endurance
Limit, in; -lb

Amplitude

Phase

Amplitude

Phase

Amplitude

Phase

Shaft Forces

4P H-force

4P Y-force

4P Pitching Moment

4P Rolling Moment

4P Thrust

3801b
3801b

22,000 in.-lb

61°
84°
83°
16°

401b
401b

3000 lb

59°
83°

40°

3101b
3101b

108,000 in.-lb

34°

12°

8°

76°

\ 325,000

Blade Root Torsion *
Harmonic

Steady
IP
2P
3P
4P
5P

-3800 in.-lb
210 in.-lb
80 in.-lb
1500 in.-lb
130 in.-lb
20 in.-lb

11°
49°
15°
47°
27°

-4000 in.-lb
210 in.-lb
50 in.-lb
70 in.-lb
13,300 in.-lb
80 in.-lb

11°
42°
82°
88°
35°

-3900 in.-lb
220 in.-lb
50 in.-lb
40 in.-lb
400 in.-lb
7800 in.-lb

11°
39°
84°
57°
10°

\ 15,500

Pitch link forces are internal loads between the blade and
swashplate and therefore self-cancelling.

274

The calculated root torsion moments shown in the table
reflect both the feathering moments at the primary exciting
frequency and the interharmonic coupling terms; as expected,
the latter are considerably less. Pitch link loads can be
determined by multiplying the root torsion moment by 0.1 (to
account for all applicable geometry); endurance limit of the
pitch link load is 1550 pounds.

The 7.5-foot hingeless rotor model data showed that 0.2 to
0.6-degree cyclic angle excitation levels were required. Study of
CL 840 test data indicates that similar blade excitation would
be expected with a full-scale, four-bladed rotor. The CL 840
data are not yet published in documents that can be referenced,
however, this material is expected to be published during 1974.

In summary, full-scale data founded on endurance limit
considerations indicate that internal blade loads and control
loads will not limit the trim flight use of periodic variation of
conventional controls for vibration attenuation.

Conclusions

The present report is a preliminary evaluation of the con-
cept of vibration reduction by properly selected oscillatory col-
lective and cyclic control applications. The investigations are
based on experimental frequency response data covering advance
ratios from approximately 0.2 to 0.85.

Because there was no instrumentation for the measurement
of the pitch and roll vibrations, these values were obtained by
properly adding up the flap-bending moments at 3.3 inches. Any
other quantity representing pitch/roll vibrations can be
compensated for in the same fashion.

The calculated control inputs required for vibration reduc-
tion stay within acceptable limits. For four of the five conditions
tested they are smaller than the values used for the frequency
response tests. The blade pitch variations required for vibration
alleviation vary, depending on the advance ratio, less than 1 ° for
.2 < m < -45 and ~ 3° for /u = .85.

As to be expected, the compensating controls greatly affect
the blade loads, i.e., torsion, flap- and chordwise bending. With
regard to flap-bending at 3.3 inches (root flexure), the following
statements can be made :

• 3 and 5P flap moments were, by command,
drastically reduced

• 2P flap moments were least affected. These
were the largest oscillatory loads.

• 4P flap moment increments generally increased
with increasing advance ratio, but were small
relative to the 2P flap moments.

As a general rule, chordwise bending and blade torsion
increments also increase with the advance ratio. At lower \x
values the loads are not critical. It is concluded that the
concept investigated is primarily suited for low and medium
advance ratios, i.e., for the speed-range of present day
rotary wing aircraft. The latter application appears promising
and further studies and tests are suggested. Instrumentation

to determine rotor vertical and inplane shear forces should
be incorporated in such future tests. Also a system with a
first inplane frequency in the vicinity of 1 .5P in combination
with a flapping frequency of 1.1 P should be tested at con-
ventional advance ratios to provide experimental data
representative of current designs.

References

1 . London, R. J., Watts, G. A., and Sissingh, G. J., EXPERI-
MENTAL HINGELESS ROTOR CHARACTERISTICS AT
LOW ADVANCE RATIO, NASA CR-1 148A, December
1973.

2. USAAVLABS Technical Report 69-39, SUPPRESSION OF
TRANSMITTED HARMONIC VERTICAL AND INPLANE
ROTOR LOADS BY BLADE PITCH CONTROL, Balcerak,
J. C, and Erickson, J. C, Jr., Ft Eustis, Virginia, July 1969.

3. ASRL - Technical Reference 150-1, HIGHER HARMONIC
BLADE PITCH CONTROL FOR HELICOPTER, Shaw,
John Jr., Massachusetts, December 1968.

4. USAAVLABS Technical Reference 70-58, WIND TUNNEL
INVESTIGATION OF A QUARTER-SCALE TWO-
BLADED HIGH-PERFORMANCE ROTOR IN A FREON
ATMOSPHERE, Lee, Charles; Charles, Bruce, and Kidd,
David, Ft Eustis, Virginia, February 1971.

5. Sissingh, G. J., DYNAMICS OF ROTORS OPERATING AT
HIGH ADVANCE RATIOS J. American Helicopter Society,
13(3) July 1968.

6. C. E. Shannon, COMMUNICATION IN THE PRESENCE
OF NOISE, Proc. IRE, Vol. 37, January 1949, p.ll.

7. DeRusso, P. M., Roy, R. J., and Close, C. M., STATE VARI-
ABLES FOR ENGINEERS, New York, John Wiley and
Sons, Inc. 1967, p. 6-9.

8. Donham, R. E., Subsection titled "RESPONSE OF HELI-
COPTER ROTOR BLADES TO GUST ENVIRONMENTS"
in NUCLEAR HARDENING SURVIVABILITY DESIGN
GUIDE FOR ARMY AIRCRAFT. This report is being pre-
pared by the B-l Division of Rockwell International under
Army Contract DAAJ02-73-C-0032.

9. Kuczynski, W. A., Sissingh, G. J., RESEARCH PROGRAM
TO DETERMINE ROTOR RESPONSE CHARACTER-
ISTICS AT HIGH ADVANCE RATIOS, LR 24122,
February 1971, prepared under Contract NAS 2-5419 for
U. S. Army Air Mobility Research and Development Lab-
oratory,Ames Directorate, Moffet Field, California.

10. Kuczynski, W. A., Sissingh, G. J., CHARACTERISTICS OF
HINGELESS ROTORS WITH HUB MOMENT FEEDBACK
CONTROLS INCLUDING EXPERIMENTAL ROTOR
FREQUENCY RESPONSE, LR 25048, January 1972. pre-
pared under Contract NAS 2-5419 for U. S. Army Air
Mobility Research and Development Laboratory,Ames
Directorate, Moffet Field, California. (Volumes I and 11).

275

1 1 . Kuczynski, W. A., EXPERIMENTAL HINGELESS ROTOR where cr { , o- 2 , a„ are all distinct, this yields

CHARACTERISTICS AT FULL SCALE FIRST FLAP

MODE FREQUENCIES LR 25491, October 1972, prepared

under Contract NAS 2-541 9 for U. S. Army Air Mobility p(t)

Research and Development Laboratory,Ames Directorate,

Moffet Field, California.

n P(o- k )

w W 6

<*kt

(4)

1 2. Watts, G. A. and London, R. J., VIBRATION AND LOADS
IN HINGELESS ROTORS, Vol. I and II, NASA
CR-1 14562, September 1972.

Appendix A

The transient response solution of a system described by
constant coefficient linear differential equations is developed
in this appendix. The single-degree-of-freedorn case with arbi-
trary initial conditions and solution of the general case for an
nth order system with both zero and nonzero initial conditions
is reported.

Given the single degree of freedom:

In the case cited

Q(s) = A(s)(s-or)(s- Y )

where

«j =

o/2 ~ ot
<>3 = V

and

P(s)

Therefore

L + p(0) A s l + p(0)B + p(0)A s

A 14+ B ^r + C P = F W

(i)

where A, B, and C are constants, then

AX^f) +BX(§) + CX(p) = £(F(t))

where X. is the Laplace transform operator. This yields
(As 2 + Bs + C) p(s) = F(s) + p(0)(As + B) + p(0)A

(2)

B(s) = F(s) + P(0) (As + B) + p(0)A
As 2 + Bs + C

If a positive constant step load of magnitude + L is the form of
F(t), then

£(F(t)) = F(s) = ±i

and

p(s)

p(0)As

A(s) (s - a) (s - -y) A(s - a) (s - y)

, P(0)B + p(0)A
A(s- a)(s-y)

Where p(0) and p(0) are the values of the variable p at time
t = and c, y are the roots of s 2 + Bs/A + C/A, p(s) trans-
formed back into the time plane is accomplished through use
of the inverse Laplace transform of the form P(s)

Q(s)

where

P(s) = polynomial of degree less than n

and

Q(s) = (s-a 1 )(s-a 2 ) (s-of n )

p(t) = ^ W e

p(t) =

A(-a)(- Y )

t [L + [p(0)A] a 2 + [p(0)B + p(0)Al a\ at (5)
[ A(+ o)(+ o, - y ) J 6

| [L + [p(0) A] V 2 + [p(0)B + p(0)Al y

(A)(+y)(y-«)

Extension to the general case is accomplished as follows.
Given the general determinantal equation:

Js 2 [a] + s[b] + [c]J jp(s)j = (f(s)

(6)

Where the elements of matrix A, B, and C are constants,
using Cramer's Rule:

PjGO

Denominator with
Column i replaced by F(s)

(7)

s 2 A + sB

[c]

' ' Expanding
yields
where

| S 2[A] + s[B] + [C]|
AqCs-ojXs-^) • • • (s-a n )

(8)

Coefficient of highest power term

cxj (i = 1 . . . n) are the eigen values (roots) of
the determinantal equation

276

Case 1 - Zero Initial Conditions

Case 2 - Nonzero Initial Conditions

Assume p^O) and (3^0) for all i are both zero and that a
positive unit load acts on p e and that the response of Pf is to
be determined. Then

+ 1/s in row e with all other
rows equal 0_

Defining

F(s)

s 2 [a] + s[b] + [c]

(e,fj

(9)

as the original determinantal equation with Row e and Column
f removed and all the remaining rows and columns moved up
and to the left, respectively, this forms a determinantal equation
of one less order.

Based on the earlier development in the s-plane

X ( Pf(t) ) -J2. + _!L. + 4_ + ... + j2_

*"- v 1 / s s-aj s-<*2 s ' a n

and in the time plane

«it

0">t

o„t

where

3f(t) = a + aje ' + a 2 e l + . . . + a n e n

(.I)(e + f)D(o) ej f

and

l o -

A o n«i

i=l

(-l)(e + f)D( 0j ) e;f

l 3

ot: A n («; -a,)

i=l

ir J

A is determined by the relationship

D(o) = A Q n^i
i=l

(10)

The general form of F(s) now becomes:

F(s)j =j^[ + j 4 A ) + [B][ jpjCO)
+j[A]jp>)

(11)

where Lj are the forces applied at each coordinate pj and pj(0)
and pj(0) are the positions and rates of the coordinates at time
zero (initiations of the solution). In this case place the column
s |F(s)j into the column location of the coordinate for which the
response is desired without reduction of the order. Then

P(s)

Column i

s F( S )

(12)

where all other terms are

and

s 2 [a] + s[b] + [c]

Q(S) = A (s-«o)(s-ai).,.(s-« n )

(13)

where the a's are the eigenvalues of the determinantal
equation

s|s 2 [a] + s[b] + [c] |j --=

Then

(14)

D(o) e f and D(<a:) e j- are formed from the original determinantal

equation with Row e and Column f removed and all the remaining where s = and the remaining eigenvalues of the general deter?

minantal equation form the set of aj-'s, and Aq is determined

by the relationship

rows and columns moved up and to the left, respectively,
evaluated at o and a:. The a: are the roots of the original deter-
minantal equation before Row e and Column f were removed.
These roots are assumed distinct, an unimportant limitation for
most physical systems. Note that this solution does not preclude
instability either aperiodic or oscillatory.

In practice the eigenvalues are obtained prior to the forma-
tion of the coefficients and are examined to verify the distinct
character of the eigenvalues.

Scalar multiplication of this solution provides the result
for the nonunit loading case. Summation of solutions obtained
for loadings at each coordinate can be used to provide the
general solution for this case where Pj(0) and Pj(0) for all i
are both zero, i.e., that the initial conditions at time zero
are all zero.

In most applications the restriction that the initial condi-
tions are zero is an unacceptable constraint and this condition
has been relaxed; the solution follows.

D(0) = A n c4
i=l

(15)

as given in Case 1 .

277

FOREWORD TO THE SUPPLEMENT

This supplement includes questions and answers following the papers of Sessions I
through IV and all of the material of Session V. Questions and answers, as well as panel members'
prepared comments were transcribed from tape recordings. This material has been carefully
checked and minor changes have been made for clarification. Where the meaning may have been
ambiguous, editorial comments are bracketed. Panel members have checked their comments for
accuracy and made minor corrections in the transcript where required.

R. A. Ormiston
Technical Chairman

279

WELCOME

Clarence A. Syvertson

Deputy Director

Ames Research Center, NASA

Dr. Mark is out of town, so I have been asked to substitute. I guess most of you have been here at Ames before but in spite of that I
would like to welcome you to the Ames Research Center. I'm very pleased, the whole center is very pleased, that the specialists in this
field of rotor dynamics picked Ames for the site of this meeting. I'm certainly not an expert in your field but from all I can see the
field of rotor dynamics represents one of the most technically difficult and most challenging fields in modern aeronautics. I looked
over the papers that will be presented and it seems, in spite of all the difficulties, that some real progress is being made in the field. I
think that is very encouraging. I also noticed that you have papers by representatives of virtually every major rotorcraft manufacturer
in the country, by representatives from the Ames and Langley Research Center, and by representatives from the Ames, Langley, and
Eustis Directorates of the U. S. Army Air Mobility R&D Laboratory. I think that it's this broad representation from all the
organizations throughout the country concerned with these problems that really makes meetings like this especially fruitful. I hope
you find the papers interesting and the meeting productive. And again, in spite of the fact that you've probably been here before, I'd
like to welcome you to Ames Research Center. Thank you.

OPENING REMARKS

E. S. Carter, Meeting General Chairman

Chief of Aeromechanics

Sikorsky Aircraft

First, I would like to acknowledge the indispensable contributions from NASA that have made this meeting possible. The AHS is
not an affluent organization and NASA has provided not only the facilities, but the printed brochure, the manpower to staff the
registration tables, the bus service for the tours, and our hardworking administrative chairman, Jim Biggers. When the meeting was first
conceived by Bob Wood's AHS Dynamics Committee, it was hoped that the Army could also co-sponsor this meeting. A significant
factor in selecting Ames as a location was the presence here of the rotary wing dynamics research team in the Army Ames Directorate
which has probably the most sharply focused rotorcraft dynamics program to be found anywhere within the government research
agencies. It developed that the Army cannot officially co-sponsor a meeting such as this, but they have provided the technical
chairman, Dr. Bob Ormiston. To Bob must go the credit, not only for having a good bit to do with initiating the meeting in the first
place, but for following through with the excellent technical program which you are about to hear.

The rotorcraft dynamics problem, which this meeting addresses, is perhaps the most challenging, most complex and technically
sophisticated challenge that can be found short of the biological sciences. The problem is well illustrated by Slide 1 which I have
borrowed from a paper by Bob Tapscott at the Civil Transport meeting at the Langley Center in November 1971. It also illustrates all
of the ingredients of the problem that we will be addressing in the next two days: the air mass dynamics problem , which because of its
four dimensional (time variant) characteristics, virtually defies visualization; the lifting surface problem with its skewed flow, unsteady
effects, and centrifugally pumped boundary layer which can't possibly be reproduced in two dimensional wind tunnel tests; the blade
dynamics, complicated not only by the centrifugal field, but by the difficult coupling effects introduced by built-in or elastic twist;
the fixed to rotating coordinate transformation problem that immensely complicates the airframe and rotor interactions; and finally,
the aeroelastic characteristics of the body itself with its large concentrated masses, unsymmetrical offsets and very large cutouts.

DYNAMICS COMPLEXITY

IMPACT LOADS

FLEXIBLE BLADES
COUPLED MODES

STRUCTURAL MODES

For the next two days we'll be assessing the state of the art
and our ability to handle each of these problems and on Friday
we will have a chance to back off and overview the whole situa-
tion. Bob Ormiston's paper in this final session is, as far as I
know, unique in the comparison it makes between all of our
competitive methods addressed to a single problem. Finally, in
the panel sessions, our ultimates customers, the designers and the
service users, will be given an opportunity to tell us what we're
doing wrong.

^^ BLACK BOXES

Slide 1.

280

DINNER ADDRESS

WHAT CAN THE DYNAMICIST DO FOR FUTURE ARMY AIRCRAFT?

PaulF. Yaggy

Director, U. S. Army Air Mobility R&D Laboratory

Moffett Field, California

The terms "rotary wing aircraft" and "dynamics" are synonomous. All dynamics are not rotary wing, but all rotary wings are
dynamic. Every consideration of the rotorcraft structure includes dynamic phenomena in some form. One cannot talk of the utility
and economy of rotorcraft without considering the impact of dynamics on structural vibrations, passenger comfort and ride quality,
pilot handling qualities, safety, and wearout and life cycle of critical components. Some have been so derisive in their comments as to
say that the rotorcraft in many respects is an effective inherent fatigue testing machine.

Now you may object to an analogy as harsh as that, but you would be hard pressed not to admit that there is much semblance in
fact to support it. For this beautifully sophisticated and integrated machine with its vertical flight capability has a nasty attribute of
creating its own hostile environment as it attempts translation flight. We speak of the rotor operating through the vortices shed from
its rotating blades, producing large amplitude nonsteady loads; not unlike a wheeled vehicle bouncing over a corduroy road. This
would be problem enough, but these first order loads produce highly interactive, coupled phenomena throughout the structure from
the rotor through the shaft to the drive system, the fuselage, the control system, the instruments, and even the pilot's posterior resting
on his seat. All of these motions are important, but their relative importance is not easily determined. Ability to adequately account
for dynamic phenomena has had a pronounced influence on the development of rotorcraft, particularly on its most prevalent
derivative, the helicopter.

It is both interesting and revealing to consider the role of dynamics in the history and development of rotary wing aircraft. For
convenience, let us consider them in five decades from that before 1 940 to the current decade of the '70's.

Prior to 1940, development efforts were more of a novelty than an orderly plan. The Berliner, Focke, Flettner, Sikorsky, and
others made brilliant achievements in vertical flight, while Cierva identified some of the basic problems and restrictions of rotary wings
with his autogyro. The immediate goal of this period was simply the demonstration of some type of successful, controlled flight, an
elusive goal which many failed to attain. It is interesting that many of the dynamic problems we wrestle with today were readily
identified in that period. Rotor vibration, high blade vibratory stresses, "ground resonance," air resonance, blade flutter, stick
vibration, and short life of critical components were all readily apparent. Lacking sophisticated analytical tools and methods, the
stalwart pioneers of the day turned to empirical approaches which further proved their genius as they incorporated dampers, blade
balancing techniques, and other modifications to improve system dynamics of their marvelous machines.

Based on these early efforts, the decade from 1940 to 1950 witnessed a great acceleration of more orderly development for the
helicopter. So many advances were made in this period that it is almost startling to consider them in retrospect. The full range of rotor
configurations was investigated; single, coaxial, tandem, and jet driven rotors were experimented with by industrial groups such as
Bendix, Platt-Lepage, McDonnell, Aeronautical Products, Piasecki, Hiller, Kaman, Bristol, Cierva, and others. Some that reached
production included Flettner, Focke, Sikorsky, and Bell.

Novelty gave way to utility in this decade, even though much was exploratory in nature. Included were wire laying, shipboard
operations, courier duty, observation and air ambulance operations. Utility was limited by the reciprocating engine with its awkward
volume envelope and high specific weight. This was but one of the limitations with which the designer had to cope and which
restricted operational vehicles to maximum capabilities of 80 knots, 6 passengers and significantly reduced altitude and hot day
performance. This decade, too, was plagued by the now all too familiar dynamic problems of high vibration levels, rotor instabilities,
blade weaving, blade flutter, and a very limited life of dynamic components. Typical lifetimes ran about 75 hours before removal for
discard or overhaul; a high price to pay for the unique vertical flight capability.

Significant research and development efforts were undertaken in this decade to cope with blade structural and dynamic analyses
and rotor airflow and wakeflow analyses. However, the complexity of the mathematical models overtaxed the computational capabil-
ities of the day and necessary linearization of the problem masked many of the important characteristics.

In the decade from 1950 to 1960, the number of variants in helicopter design began to diminish as the most optimal designs
began to be apparent. Utility was increased by applying new found technology and methodology for design. A new source of power,
the turbine engine was appearing which would give a more optimum volume envelope and eliminate the vibration input of the
reciprocating engine. This transition to turbine power, with its improved specific weight and fuel consumption, was destined to
revolutionize helicopter utility and capability.

The helicopters of the '50's were produced in large quantities under the impetus of the Korean War and their new found military
utility. Payload capability increased to 20 passengers. Performance reached the ability to hover at 6000 feet and 75-degrees F.
Allowable vibration criteria were quantified and defined by military specification. Bold new life goals for dynamic components were

281

set at 1000 hours before removal. Although some reached these levels, many still barely attained 250 hours. Utility of the vehicle was
the primary gain of this period, but the plaguing restrictions resulting from dynamic phenomena still produced high costs and
restricted performance.

In the decade just past, 1960-1970, the gas turbine completely supplanted the reciprocating engine in helicopters with a resulting
increase in payload and range. Payload exceeded 5000 pounds, carried at speeds in excess of 200 miles per hour in test flights. Hover
was attained at 6000 feet and 95-degrees F. Life of dynamic components was somewhat improved, even to on-condition removal for
some parts, but the spread of lifetimes for dynamic components still ranged from 250 to 1000 hours. Although the established
vibration criteria was met in part by some helicopters, vibration problems still plagued the designer. Unpredicted rotor instabilities and
coupled phenomena still occurred with surprising regularity, quite often with disastrous impact to vehicle and development program.
Flight speeds were increased by improved power ratios. However, this only served to further increase the dynamics problem, since at
these higher speeds stall induced loads, stresses and vibrations became the key limiting factors in determining critical speeds and
maneuvers. Use of the helicopter for combat, with requirements for greater agility, further sharpened the awareness of these limita-
tions and focused greater attention on the dynamic constraints.

Now what of this decade in which we reside. Our research and development efforts, based on our newly acquired computational
capability and advanced technology, have given us promise that we can achieve greatly increased performance, utility, and agility from
our new helicopter system developments. Driven by military requirements, we have accepted the challenge to also survive in the hostile
environment of the battlefield. To do this, reliability, maintainability, availability, and detectability all must reach new levels. Among
these criteria, nonsteady phenomena become most critical factors. Improvements in capability and cost now become the challenge
primarily of the dynamicist and he will determine success or failure.

Then where are we now in making these advances? Early helicopters had accepted low component life, high vibration, and
marginal performance. Predictive techniques at that time were based on relatively simple analytical and experimental models. Techno-
logical advance was slow, based on the empiricism of rudimentary experiment. We have exploited to the maximum that past
technology with its simple representations of rotor dynamics and flow fields. The old barrel of empiricism is bare. The greater demands
for agility, longer life, and lower vibration demand new advances in dynamics from basic and applied research and development. A
rededication to innovative methodology for aerodynamics, rotor flows, dynamics, and their coupling in interdisciplinary systems is
necessary to meet the demands of the forthcoming generations of helicopters. Only significant advances in the comprehension of
dynamic phenomena will meet the need for a technological base for desired future growth of capability.

Three specific areas of emphasis suggest themselves in considering the scope and direction for future technological development.

First, intensive effort is required in both the evolution of global analytical models which describe the dynamics of the helicopter
and in the more specific interpretive models which describe the physics of the phenomena in more detail. Results of both of these
modelling efforts must be verified continually by experiments with both model and full-scale tests. The improved comprehension of
physical phenomena from interpretive models must be integrated into the global models in a timely manner as they are verified.

Second, the adequacy of the forcing functions, which are the inputs for the foregoing models, must be improved. These forces
result from aerodynamic loads generated on the rotor blades, which are often highly nonlinear in nature. Unlike fixed wing aircraft, in
helicopter flight these regions of nonlinear loading are penetrated deeply and periodically by the rotor blade. The generation of these
loads results from the complex flow field in which the rotor operates. Here again, adequate mathematical models must be generated to
predict these dynamic loadings and the models must be verified by experiment. When correlation is attained, we must assure timely
efforts to include these descriptions into the inputs of the global and interpretive dynamic analyses.

Last, but by no means least, we must demonstrate true professional acumen to sensitize our efforts to the gain to be expected.
Expressed in other terms, we must assess the gain to be made by more accurate models against the cost to obtain that gain. The last
few percent of accuracy may well not be worth the cost. Our goal should be to produce predictive techniques for the designer to
assure sufficient accomplishment of performance goals without the surprises of instabilities and shortened component life which have
plagued us for now five decades, but without the cost of even one degree of sophistication more than required for that goal.

The discipline of dynamics shares the preeminence with that of structures as offering the greatest potential for advanced
helicopter capability in this decade and the next. The assemblage at this symposium is the hope of that achievement. May you rise to
the challenge and show its worth!

282

SESSION V

APPLICATION OF DYNAMICS TECHNOLOGY TO HELICOPTER DESIGN

Panel 1 : Prediction of Rotor and Control System Loads

Peter J. Arcidiacono, Moderator
William D. Anderson
Richard L. Bennett
Wayne Johnson
Andrew Z. Lemnios
Richard H. MacNeal
Robert A. Ormiston
Frank J. Tarzanin, Jr.
Richard P. White. Jr.

Panel Members

Head of Rotor Systems Design and Development, Sikorsky Aircraft

Research and Development Engineer, Lockheed California Company

Assistant Group Engineer, Aeromechanics, Bell Helicopter Company

Research Scientist, USAAMRDL, Ames Directorate

Chief Research Engineer, Kaman Aerospace Corporation

President, The MaeNeal-Schwendler Corporation

Research Scientist, USAAMRDL, Ames Directorate

Chief, Rotor Loads Unit, Boeing Vertol Company

Executive Vice President and Director of Engineering, Rochester Applied Science
Associates, Inc.

283

COMPARISON OF SEVERAL METHODS FOR PREDICTING LOADS ON A HYPOTHETICAL HELICOPTER ROTOR

Robert A. Ormiston
Research Scientist

Ames Directorate

U. S. Army Air Mobility R&D Laboratory

Moffett Field, California 94035

ABSTRACT

Several state-of-the-art methods for predicting helicopter rotor loads were used to calculate rotor blade loads including airloads,
bending moments, vibratory hub shears, and other parameters for a hypothetical helicopter rotor. Three different advance ratios were
treated: n = 0.1 , 0.2, and 0.33. Comparisons of results from the various methods indicate significant differences for certain parameters
and flight conditions. Trim parameters and flapwise bending moments show the smallest variations, while chordwise bending moments,
torsional moments, and vibratory shears show moderate to large differences. Torsional moment variations were most sensitive to
advance ratio. Analysis of the results indicates that the differences can be attributed to all three fundamental parts of the problem:
numerical solution methods, structural dynamics, and aerodynamics.

INTRODUCTION

The prediction of rotor loads is one of the most difficult analytical problems in rotary wing technology since it involves a highly
nonlinear aeroelastic response problem. Rotor loads, however, are basic to helicopter design because vibratory forces and moments
from the rotor largely determine fatigue life, reliability, flight envelope limits, and ride comfort. Ultimately, rotor loads have a large
impact on the cost of the vehicle. Much effort has been devoted to the development of sophisticated methods for calculating rotor
loads, but these methods necessarily depend on empiricisms and approximations because the aerodynamic and structural phenomena
involved are not completely understood.

Needed progress in the development and refinement of these methods is often hindered for several reasons - the inherent
difficulties of the problem, the specialization of different methods to treat different rotor types, the scarcity of reliable experimental
data, and the difficulty in transferring experience gained by different investigators using different methods. As a result, it is difficult to
accurately assess the state-of-the-art or to reach a consensus on the areas that require special attention. One proposal to partly
overcome these difficulties is to specify a standard problem for calculating and comparing results of several loads prediction methods.
This approach would focus attention on common, as well as individual, problem areas, permit sensitizing or "calibrating" different
methods with respect to one another, and provide a new and broader basis for transferring experience.

Several rotor loads specialists jointly agreed to undertake a project of this type for presentation at the AHS/NASA-Ames
Specialist's Meeting on Rotorcraft Dynamics. This paper summarizes the main results. The project was made possible only by the
enthusiastic cooperation of these specialists and the support of helicopter manufacturers and other organizations. The principal
individual contributors were Wayne Johnson, USAAMRDL, Ames Directorate; Richard L. Bennett, Bell Helicopter Co.; Frank J.
Tarzanin, Jr., Boeing Vertol Co.; James R. Neff, Hughes Helicopters; A. Z. Lemnios, Kaman Aerospace Corp.; John Gaidelis, Lockheed
California Co.; Michael P. Scully, M.I.T.; J. J. Costes, O.N.E.R.A.; Peter J. Arcidiacono, Sikorsky Aircraft; and A. J. Landgrebe, United
Aircraft Research Laboratories.

Experimental data are not available for establishing the accuracy of any one loads prediction method; therefore, all
interpretations and conclusions are based solely on relative comparisons of the results.

STANDARD PROBLEM SPECIFICATION

The standard problem was defined on the basis of inputs from all contributors. The basic philosophy was to emphasize
aerodynamic phenomena by choosing a simple structural configuration. Most of the analytical difficulties are associated with
aerodynamics, and interpretations of the results can be made simpler by removing unnecessary structural details.

Presented at the AHS/NASA-Ames Specialist's Meeting on Rotorcraft Dynamics, Moffett Field, California, February 13-15, 1974.

284

A conventional articulated rotor was chosen, with three rectangular blades having 10° twist, an NACA 0012 airfoil section, and
coincident flapping and lead-lag hinges with 4% offset. The blade is uniform in stiffness with coincident mass center, aerodynamic
center, shear center, and feathering axis to minimize aeroelastic coupling effects. Complete details are given in the Appendix. A tip
speed of 750 ft/sec (M = 0.672) and rotor lift coefficient C L /a = 0.0897 were chosen for three basic flight speeds: Case A at
250 ft/sec 0* = 0.333) to emphasize retreating blade stall flutter, Case B at 150 ft/sec Qi = 0.20) for a typical unstalled flight
condition, and Case C at 75 ft/sec (p. = 0.10) in the transition flight regime to emphasize vortex wake-induced blade loads.

In addition to the three basic cases, several additional specialized cases were defined. These include selective combinations of rigid
blade motion, linear aerodynamics, and uniform downwash in contrast to the most general case that includes elastic blade response,
nonlinear aerodynamics, and nonuniform downwash. The particular combinations treated are listed in Table 1. Nonlinear
aerodynamics is defined here to include the effects of unsteady stall, compr

Internet Archive Audio

Featured

Top

Images

Featured

Top

Software

Featured

Top

Books

Featured

Top

Video

Featured

Top

Mobile Apps

Browser Extensions

Archive-It Subscription

Save Page Now

Full text of "Rotorcraft dynamics"

See other formats