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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 xv 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 vi 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 b B c Cj c M e e pc EI {£} F.G go>gs»Sc i lA [I] J 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) /T apparent inertia of air, slug-ft 2 identity matrix index referring to mode number J [K] [L] [L E ] ffi.niyy.myj M,[M'] dm dm m A n N [0] P r R S(0,i{i) H {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 R m *j dr Presented at the AHS/NASA-Ames Specialists' Meet- ing on Rotor craft Dynamics, February 13-15, 1974. generalized control vector <e o e s e c8o8sSc A o x s^c> 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 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 + K + v S R sin iji itoijj v o» v s» v c p a r T ,T S ♦J * u SI 8' + 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 ua (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. by 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 s -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] Mm (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 V s V . c Et L 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 V - [W] 1 - — m 2 yy 1 -2 m yy J 1 -Kj 1 2 yj. ft ?! 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 iu y - lu Hy iu (24) [K] 3*3J ai, -[L] [W] J 1 Hi 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 "0 m -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] m ~ K T -K I- 1 aa I 1-1 L, -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 I 1 .2 .3 Sst rod .4 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 86 8 =0 1 s _1_ 16 y (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 .2 .4 1 .05 V .10 I .12 i 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^^ 0/ -.02 i 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 .2 .8 1.0 1.2 132 -£. 0, oo .2 .4 .8 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 7 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 OR QUASI -STEADY INDUCED FLOW OR 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 NO INDUCED FLOW OR 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 _i 3.02 360 v ■a i 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. 10 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 04 .02 ^V-^ . . . . *r - Bv r= -?"7 . °. . , ,-180 1.2 360 180 1.2 .4 .8 u 1.2 .4 .8 \2 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. 11 .03 kj .02 o t _J s « .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 en XJ lj" 180 ■ CU3 =^et =: < ^^ == ^^ = ~ :=:: =Q~. I 0. 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. 9. 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. 12 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 13 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 14 ® 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 > © Tg » lift stall time constant t* = 6.0 . 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 © d - 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) © oKa. 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 m "mPOT — ORIGINAL LOOP o RECONSTRUCTED LOOP TIME DELAY PREDICTION «M- "' f = 75.25 cps 1.80 N O 1.40 A? »- J* / z JS J UJ 1.00 S'T u. yyy UJ 0.60 ^S 1- u. 0,20 £~~^ _l 12 16 20 ui 2 O 2 o {£ -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 15 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 ft =&a^ ^— < ^A STABLE ' tar' ^H UNSTABLE .0 ks 0.3375 < s& yy j k A <*> A r 1 1 ^< V\ f 1 !/ i -.2 UNSTABLE i i / <V Ji 7T -.6 A 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 16 2.4 2.0 Z UJ LlI o o UJ o 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 6 '// — ■ . — \ 1 ^ /> K x" \ ^ ^ Y" /' i 0.1 E o UJ o W -0.1 o 2 -0.2 — 5 -0.3 -0.4 ! i i i > "~~\ 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. 17 s s s 8 O 4 \ / 1- a f r S>s a v \ s 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 (9 W o 1 1 Li _l (9 Z < z S 3 UJ MAC A 0012 5? PA =0.25 S« = ! \"7 >* X tug ./ ft V 2 - < MACA 001 *PA=0.22 I . i^^ ■as /' f, -Og = 1<" / / f\ * .'" — •> '" ' 4 Ada =7 3 2h SC 1095 | Xp A = 0.25 F S« = 7 > // s' s .' L' "~ 6 SC (095 | 5! P a=0.22 ,«■ ^mm (SqsS //, s _J. A ^; . W 4 10 14 18 ~6 10 14 MEAN INCIDENCE AN~GLE,a. M , PEG. Figure 6. Model Airfoil Elastic Deflection. 18 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 18 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 V 9j l I \\ 4 \ J- X. \S, f— ^? J XZ ^ """ — "S ■4 / i ^ ___ i I i SC 1095 AIRFOIL ■ r . ■ i 'N CHAHGE II *■*» *»-«h i \ i \ \ — / ^ — ^ > N i > N -*C; -7 V / ■•> \V $ f- ^ V. J> L> V l i 4 \ J 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: (I 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: 19 TEST 40 20 L [ 60 40 20 80 - 60 - 40- 20 - TIME DELAY NACA 0012 AIRFOIL, ct M = I a,A,B N ACA C 012 AIRFO L, *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. 60 40 20 I=I REF ,XpA=0.25,Og = 5 TEST ,S H NACA 0012, i -SC 1 / / ' 1 1 I«l.5xI REF ,5? »«0.22. "8=5 - 1 TEST Ss'—~ -NACA / ' 'SC 1095 T~ s . z Ul TIME 3ELAY o S 40 -J < ..._. 3 §20 1- to / / -1 P n / - TIME DELAY ~/ / • 1 / 1 / ! a,/ *,B 40 ■- ---•• -- 20 / s ^ 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 20 '(D) Cc) 8000 Q-_ 4000 in -4000 -8000 4000 r/R = 08 • n as - TIME DELAt M if \ ^^ — 11/ f ^ ^ 7j \>^ z.^ J(V V / f\ / %tf\ \ \ y 60 120 180 240 BLADE AZIMUTH ANGLE , DEGREES 300 360 60 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 21 (a) 3200 m «2400 Q 3 ' Q £1600 O S 800 ... A 1 r ■ — TEST DATA — a,A,B 1 — Tl ME Dt .LAY ]// / /fi / Sv y s"- * <La s* 40 80 120 AIRSPEED, KNOTS 160 200 (b) 4000 .3200 2400 1600 < CD > 800 1 1 1 1 1 C T /cr TEST ANALYSIS 0.083 & PUSH ROD JOAD= ±220 LB \ _ / 1 n/i /m, / " 1 f A/ r ■( i i 1/ / / // ', / i 3 L^ f / / Y^" 40 80 120 160 ' 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 22 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. 23 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. *1 If J P(t) or P Q= (I 1 /itI f )c/R R R a,b,c a c f t v w(jAt) w X z 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. 25 Subscripts o 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 26 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) 27 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& W 3H 3a dt (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 28 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., N 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 +N 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) 29 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 30 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) 30 35 1000 The values for the R used in the estimator were R = 16 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 2. 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 31 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) 32 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 15 3a dt (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. H L ■« t - w c Fig. 1. Fourier Transform of Weighting Function CdL (U, Fig. 3. Estimates 6 Covariance s vs. Time, Acceleration Noise Only 33 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) 34 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) 2 GJ "no' k "10 "o 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 8, 'Vfc 'wi \ *±a V 9 »e X *1 P {*<*>} 35 "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 *y (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 36 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. 37 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: 38 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: 39 .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 B3 .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 40 \ 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 41 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 42 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: ,1 * (» 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. 43 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. 45 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^ CD = 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 N B nc = 2 N E i=l N 8 i cos nipj B ns = 2 N i=l N h sin rof^ B N = 1 N E h (-D , N eve 2 i=l Then the motion of the ith blade is K '■ 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. 46 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. i* | B 4 2 J B* 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, J* 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 47 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. i 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 a .5 -- -.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. 48 \:<l a 1.0 b .8 a. i/ = i.i,y = 6 b. !/ = l.0,y = 6 .6 ^^^Nc c. i/ = l.o,y-l2 t 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). 24 20 '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. 49 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 t .1 (Q) FREQUENCY .3 Figure 7. Comparison of constant coefficient approximation to Floquet theory (v = 1.1, y = 6). 50 -.5 -.4 O PRECESSION D NUTATION A CONING FLOQUET THEORY ,-Q «=*#= CASED V'U y--e .1 (b) DAMPING NUTATION CASE a y-e •FLOQUET THEORY ■ APPROXIMATION CONING L /j. = 0.5 4 .w 'a 2.05 -- 2.00 1.05 1.00 -- .05 PRECESSION / 1 — — «--K -.5 -.4\ ■V a ~1 CONING NUTATION .05 ---1.00 -1.05 -J- -2.05 (C) ROOT LOCI 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 .7 O PRECESSION D NUTATION A CONING FLOQUET THEORY CASE b I/=I.O y = 6 1 (Q) FREQUENCY -.2 -.1 r ° PRECESSION □ NUTATION A CONING FLOQUET THEORY Pi -.PI rS 9 - ¥ CASE b - v=\.0 y-6 1 1 1 1 1 .1 (b) DAMPING Figure 8. Comparison of constant coefficient approximation to Floquet theory (v = 1.0, y = 6). 51 CASE b !/=I.O y-6 -FLOQUET THEORY -« APPROXIMATION /x= 0.5 'a -- 1.95 1.90 .95 .90 .10 -- .05 4- -*« -- -.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 D NUTATION A CONING FLOQUET THEORY - CASEC f = 1.0 y = l2 t .1 (a) FREQUENCY 3. .3 .5 -1.0 -.8 a -.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), 52 i 'a -2.0 CASE l/=I.O y = l2 C NUTATION - 1.5 FLOQUET THEORY -•— — APPROXIMATION = 0.5 /coning - i.o / ■ ». - .5 / I * " 1 PRECESSION cr 1 a . -1.5 -1.0 -.5 .5 ^ - -.5 - — ■*. - -i.o'' ~L - -1.5 --2.0 (C) ROOT LOCI 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. 53 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 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 c do 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 th 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 55 M,N ^i>\i ( Vikr ( Vimr P »P »P r x r y r z ikm P(t) R R Q T u,v,w V ,v e eo w ,w e eo x,y,z V Y G a 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 o Azimuth angle of blade #=fit) measured from straight aft position ■"C Flutter frequency ith Imaginary part of k character- istic exponent 56 %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 o 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. 10 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. 57 a i 3h 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. by 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- 4 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 58 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 Q + 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 RT 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 V 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 59 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. 60 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 61 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 2. 3. 5. 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. 8. 9. 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. 62 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 1 sin6cos9 j(\i« d *o "i^ 1 i / E C2« d *o *^o " ^ V^* 1 / 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 im 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 A B Y'dx ; L' 'm o 21 , 'ikm /vM^o t 22 - ikm L 24 = fa ,Y'dx ; L 23 k'm o ' im k Y'x dx 'm o £\w dx 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 ) 2 + 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;) 2 -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 c 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|))| 63 where 21 2P e° z lll g l V" 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 v Thus, for y < 1 the generalized aerodynamic loads are calculated from Api hi a\ u\ " / 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 k (b)n>1 Figure 2. Geometry of Approximate and Exact Reverse Flow Regions. 64 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 T - 10.0 a - .05 "P - 0.0 0„ - 0.0 B - .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 ■ 65 ,f c < -1.0 cc o u. -0.8 z UJ z -0.6 £ X Ul -0.4 UJ 3 -0.2 < z u 0.2 r X X a F1" 1.175 - X X X X S L1" 1.28303 10.0 .05 - X h - 0.0 X >» %> - 0.0 w k CONTINUOUS ' .05 .20 i 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 I _L -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. 66 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 67 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 : 1) and 3) 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 . 68 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 69 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 3. 70 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 71 o o 3 x (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$ G 1U66 60% P 1U35 58% P 11*1*0 58% P 1st Vertical 1155 1175 2% E 1282 11% E 121*2 8% E 121*1 8% E XSSN Pitch 1U90 1709 13% E 1710 13% E 17^8 17% E 1758 17% E 2nd Vertical 1950 2150 10% G 2390 22% F 2505 28% F 2577 32% F XSSN Roll 2000 2150 21405 20% 2250 k% P F 2870 1*3% P 2900 1*5% P 2891* 1*5% P XSSN Vertical Torsion 2300 2763 20% F 2l*28 6% P/F 2U22 6% P/F 21*1*5 6% P/F 72 632 /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 >' -© : — : — S "§- 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 74 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 E 1*53 G 785 G 956 F 1063 F 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 75 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 76 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 77 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 78 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 79 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 80 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 M • — Fr 0_ 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 82 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* 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 %c 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 83 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 .1 x Hks x Hkc (13) H *-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. 84 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, 85 ACTUAL BLADE APPROXIMATION BLADE SECTION BOUNDARIES EQUIVALENT SYSTEM CONSTANT EI 6 GJ ELASTIC BAY APPLIED AIRLOADS A \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 1 i ■ . . . .... • 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 T COMPUTE NATURAL MODES , ^ AND FREQUENCIES , u n r INPUT ROTOR FORCES, F R COMPUTE FORCED RESPONSE , {x} A n n n n T 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 86 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 1 . - i K AA | rn 1 _o-_J 1 1 I . _ 1 K BB- 1 1 1 r~+--~ 1 -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 * 1 STRAIN ENERGY SORT - CALCULATE WEIGHTS AND STRAIN DENSITY - SORT X£it> CRITERIA 1. VIBRATION LEVEL - MIN. 2. WEIGHT PENALTY - MIN. 3. WITHIN ALLOWABLE. STRESSES I MODIFY STRUCTURE, RUBINS METHOD, RE- RUN ' NO 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 87 MODEL 347 COCKPIT VERTICAL (NO ABSORBERS) to .5 r +1.4 - OBJECTIVE 8 H Eh H H U 3 FLIGHT DATA ANALYSIS 40 80 120 160 AIRSPEED - KNOTS MODEL 347 COCKPIT VERTICAL WITH ABSORBER O + \ § H w H U 3 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 / / \ \ \ \ .1 n / / / v \^ 1 ^ i 4 £ .1 A 12 j 2 .2 >v / K .3 (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. H4S, .001 IN. (c) Hub Forces Vs. Hub Motion Figure 6. Coupled Rotor /Airframe Analysis for a Single Rotor Helicopter Vertical Vibration Figure 6. Continued 88 Z = (Z 2 + Z 2 )H 2 .0010 j EXCITING -i .08 I frequency! j/ .06 z .0005 .04 Z 4 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' M. ML 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 89 . 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 .5 /4 STA. 95 C/L VERTICAL 4/REV VIBRATION -NO ABSORBERS .5 .4 .3 G's .2 .1 "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 80 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 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 ) r •IDENTIFY STRUCTURAL CHANGES TO MINIMIZE AIRFRAME VIBRATION USING STRAIN ENERGY METHODS Figure 10. Coupled Rotor/Airf rame/NASTRAN Analysis 90 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 B . ,- , 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 id three dimensional lift curve slope for fixed wing Subscripts P" denotes fixed wing " denotes helicopter 91 < 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) 92 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- <u £s 0? 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 93 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. 94 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 95 consisted of three flatwise, two ehordwise, and one torsional modes. SIN 1 ■ ! -SQUARED OUST 40 *s / 1 r 1 RA1 1 P 61 1 JST 40 f #»= 0.5,REFER£NC£ ARTICULATED ROTOR, :,/. ..06 __ — =— — Sf HE -SQUARED CU SHARP-EDGED GUS1 SUIT - A r :- | 24 000 !\ A r •"' K H [/ ,■!■ „' r ^ ■' vy ft 1 // •& \ / \ V'-.N \ ,' ":»\ /; 1 6 ODO £ vw ~v 17 r \ ==. '/ \ \ £ ^K ? V i I 8 000 l. 1 1 160 320 ROTOR BLADE 480 ft *IAZiMUTf 1 640 , DEC 800 9C s HARI 1 1 -ED! ED I UST ft ,» 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( 98 CUR REN ' ST UDY AR' " ICUl ATE D R 3 TOR «IL- >-8i 98 CUR SEN ST JDY .04 .06 .08 .10 ROTOR THRUST COEFFICIENT/ SOLIDITY, Ct/o- Figure 9. Gust. Alleviation Factors for Different Rotors. 96 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 97 .' 40 30 -w 20 ■' q: "- o w 10 § 6 H in 4 H o i 2 X I ° u_ ° 3.0 o 2 2.0 ■ 4P VERTICAL VIBRATORY' FORCE - '. i •^ y= 7 Reference 2 and previously presented in Figure 10 . vibratory control load St 10 y=i3^- 4 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 n 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 i M u 4 / i * ' 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 99 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 T 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, . o (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 (- 0) 0) 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 M_ 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, n - 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 — ** " r- 1 - — - — Zff ! ^ - 1 ■ — ■ 1 1 --t- - Yf f z if ! z h 1 1 -4-- 1 1 Z ia < Yj Ki ► ! Z ai z aa Ya L 1 L J 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 10 10 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, M I M T>I | OD e V e (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 ON 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 g. -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 z o o o z 45- 4tt 35 30 S 25| o z a z w 20 2/REV WITHOUT CYCLIC FEATHERING WITH CYCLIC FEATHERING 15 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 25 '/ OVERALL N '' OSCILLATORY --,_ 1/REV o o o H z 180 160 g 140-1 s •• o 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. 80 70 m ►J z 60 o o o 1—1 50 z 40 1 30 o z M Q 20 W 10 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 pq •J 120 z glOO H Z o o z I— I Q Z W « 8a 60 40 20 OVERALL OSCILLATORY oU 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 cn z o o o Z s o z M 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 9 300 o 2 200 z 3 o z 100- w 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 z o o o H 55 I w 80- 60- 40 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 55 M O O o E-t 55 55 M Q 55 W pq 80- 60 40 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 • • 6 • 1 J * * xU • 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). 28 _ ______ ft 1 I 24 • • I EQ 1 z [ z o w 16 EC o h- a z r- < 1 12 LLI 1- < 1 I I Q < co 8 • • ,I lX i i C ANALYSIS 4 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 " O 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 o 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 4 TORSIONAL 3 FREQUENCY C T /o = .12 2 \ A / 3/REV % J\ ■W r v -2 \y -3 ■ -4 -5 -6 ■ -7 L 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 5 0.09 - 0.08 - 0.07 0.06 - 0.2 K \ 4.04/REV 3/REV-'*\ \ PITCH LINK LOAD \ > ABOVE 2500 LB \ \ \ 5.165/REV \ \ ON V ■^\«v \ •\l ~A^ \ X f\ 9 "ft 6.97/REV \ l\ \ * PITCH LINK LOAD \ BELOW 2500 LB \ \ V ' * 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 3Z 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 ' 3z ,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 P P K(x'.y') = k'(x',y') e" U ' X ' „2 -iic'r' „ ■ ' e r-> = -e 2 £ G = 3z ,2 (- 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 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) w 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) tj ./[e + 2i- K tj - V Eqs. (20) and (21), then yield u: 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 ij 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 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;) r' w (26) 2 2 (1- 3g *' ■ ) + k ,2 B 2 z' 2 ] ds'dn* r ,Z w 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' •• w (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 p 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 1. 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 1 K&- "A tai.y'i) — | \ \ i \J» ii , / V 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 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 S s \ I 1 ^^ 1 1 t 1 t 1 0.1 0.25 0.40 0.55 0.70 0.85 1.00 x R 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 1 HELICAL WAKE / CIRCULAR WAKE • STRIP THEORY / 0.20 / / M. 0.15 4 — / S" — \ / ,' \ \ / / 0.10 _ S /V~ ■' >' / 0.05 S — -'"L-/ o 0.1 0.25 0.40 0.55 0.70 0.85 1.0 y "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 y "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 ro 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 "°* V 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 O d STABLE MARGINAL UNSTABLE Eh W 200 o o o o o o o o o 160 - o o o o o o o o o o o o o o o o o o o ooo o o \0 o o o ooo 120 §! »o lo o o o o o o o o o odd od ooo o o o o o o» ooo o o o o o ood 80 ~o o o o o. o^ ooo o o o o o o o "ooe ig o o o o o o o ooo 40 o o o o o o o o o o o o o ooo o o o o o o o o o - o o o o o o o o o o o o o o o o -40 o ~ o o o o o o o o o o o o o 1 o o _1 o o ,„l o 1 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 _, *G — \/PI ' — <fM0 TCH-PREDOMINANT DE 1 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 |U Q H a « S Ai -1 A A i i A 3< a < s \ Ul -1 m l{ 3 7 A A TEST — ANALS POINTS SIS 50 60 70 80 ROTOR SPEED NORMAL ROTOR SPEED 90 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 k- 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 A BTS 6-FT ROTOR V23010 -7° D RTS 6-FT ROTOR VR7 -9° tf RTS 6-FT ROTOR VR7/8 -9° 14-FT ROTOR V23010/13006 -10.5" a UHM COMPOSITE V23O10 -10.5° o 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 ft ill I ff TEST DATA AT () t = -9° >. cjj 7j» 7^~~TEST DATA AT t = 0° V j/% F THEORY AT t =0° TEST DA AT 0t = rA 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 r\ 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 Hi h pa a w PQ PRECONE * DEGREES SWEEP =2.5 DEGREES AFT CONTROL STIFFNESS =90 IN. -LB/RAD 360 AZIMUTH ANGLE DEGREES !** 2 J 1 V 1 — = 80 KNOTS . , SYMBOL CONTROL STIFFNESS a = %OR D 90 IN. -LB/RAD O 642 IN. -LB/RAD _VV_ H i-l a f Ul -20 -10 * 10 t CLIMB | DESCENT | ROTOR ANGLE OF ATTACK - DEGREES 30 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 H zu M as w nu K J W <C 0. a 3.0 2.0 1.0 V = N N0R = SYMBOL A O 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 o 92 PERCENT A 116 PERCENT a 139 PERCENT ^ 162 PERCENT 2.0 - PERCENT M H os< ZH OjU §H ass H < « a a Oft E 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 e X b *i *x ^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» ^h *1> n e± % ii *i [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 dm dm (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 k=l the rotor center of mass coordinates become H x c = *n - ( p c / N) £ k sin *i N 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) a - (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 "3 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. .6 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 N Si - E ?i sin *i i=l N l II = ^ ^i COS *i i=l (Al) Differentiating these expressions leads to the establishment of the following identities N i=i (A2) 154 £ L sin t = 6 - air - 2fl| i=l II £ '( ± 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* II 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| II ^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 mass, m Hub spring, k Hub spring, k Hub damper, c 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 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 1 100 1 200 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 1, .5 -.5 -1 1 .5 -.5 -1 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 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 fr 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 inplane Number of blades 3 3 '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 I 1 I I .-350 J^458 ">550, 1 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 (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 ), (c) damping of wing chordwise bending mode (q 2 ), (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 a -^ /V -^-_. - A ^- (d) i A- i i 1.0 V/flR Figure 9. Concluded. o EXPERIMENT (0.1333 SCALE MODEL) .08 WITHOUT £ |C , £ |S /\ / \ / \ °^^-i / y/ > \ \ 77 a/ o\ \ .04 // o\ 1 \ ° 1 yS \ ° 1 ^^^ V (a) 238 rpm 1 1 1 A 200 x N V, knots 400 3 + 1 ? £ + 1 P 1 _ A T~~ Qi\ C-K-— -— ^-1 (ah- 1 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 (c) £2 = 358 rpm (equivalent full-scale V and J2). ratio of the modes, (c) root locus. 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 600 600 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- "2 (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 /T o \»P (b) IOO knots ^X 1 1 ^ii- / / // f i PRESENT THEORY BOEING THEORY o EXPERIMENT J03 .02 - \> " s -^. ,o © 1 \>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 R Blade radius R 0.75 blade radius AT Rotor perturbation thrust V Airspeed v F /nR Flutter advance ratio w g Vertical component of gust velocity a m 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- * I ft 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 J_ 300 400 Airspeed, knots I 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 o A Without wing aerodynamics With wing aerodynamics ■Blade inplane flexibility included V 50 200 300 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 J_ .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) A - 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_ J_ J_ _1_ _L_ _i_ 0.5 1.0 1.5 2.0 2.5 3A~ Simulated full-scale gust frequency, Hz J 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 50 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 a «; .04 Measured o - .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 K > 1 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 °) .8 P / ! I of 4 .6 .4 ■■/ /*/ / A^ 7 1 f£{ Measured '/ / ° ' / A / □ 1 1 1 Calculated .2 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 fa > 1.4 Symmetrical pylon frequencies 6 3 • .349 rad (20°) 1.2 e/R . .05 1.0 .8 e/R m .13 / / / ^ / / / / / / °/ .6 .4 - A / / / (J / y / Measured ' O A Calculated .2 1 1 ' 1 .1 .2 .4 .6 .8 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- > I 6 3 - .349 rad (20°) e/R « .05 / / 1 . 1 1 p I _ 1 1 1 / o< I Bl o 1 1 1 OT $1 mi £<V / y uy Measured Calculated /<§7 *-* W B' ty 1/ 1 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 o to 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) O 1 Inflow ratio at 0.75 Wade radius, v/m 1 1 .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 10 ,„- 4 Q- s MEASURED CALCULATED O A WING BEAM MODE • WING CHORD MODE O O (P'zoo n^ '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 by 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. M. 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 M moment, ft-lb Presented at the AHS/NASA-Ames Specialists' Meeting on Rotorcraft Dynamics, February 13-15, 1974. xy N R Xc Po 63 a € 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 t, fraction of critical damping e pitch motion, radians % blade collective feathering, radians \ blade effective sweep angle, radians is servo time constant, sec ♦ roll motion, radians 10 frequency, rad/sec nip inplane natural frequency, rad/sec Q 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. *F | UP ^T 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. U) -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. CO -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 CO -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 16 <o tr uj -jSS O ui CC D 12 ,0 ^ r " s 1° \ \ J t "^3° -^ 1- 2 Ui o z UJ j 2 ■ < 5 -j Q. in 2 2 1 \ DROOP J ANGLE 1 10 1 r 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 OJ -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 i 13,000 & 16,000 16 14 ■ 12 10 REAL PART OF ROOT Figure 1 0. Locus of Roots - Roll Inertia, V = - Effect of Airframe 20 KN. OJ (j0 33 > REAL PART OF ROOT, 1/SEC Figure 9. Locus of Roots - Effect of Blade Droop Angle, V = 20 KN. CJ -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: M, gyro M, 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. 1 >R \4-r fx R ) ^s Z F k 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 /?)' 3L 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 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 o o z SERVO TIME CONSTANT .01 .02 .03 .04 .OS S3 COUPLING, -0/(3 -0.1 0.1 0.2 0.3 - — — 1 2 3 k, < 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 O 97% N R "I Q 95% N R V TEST DATA A 94% N R J c 3 C o 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 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. RT 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. a z i m o -I -100K l \^y ^ 100K 100K rf 1- x 5 -i° 0K a! 100K 9S 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 a 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^ O Ph - ►J § q fa ei o o d o H O 3 100 80 60 i) Q Q Q 1-3 o 00 < 2 ( i o 20' LONGITUDINAL 9 <t> [| □ tl LATERAL a A a (I o ~(> J 2.9 SEC < .25 g 1 © °t> « t\ <a /» ° I* -20 ii_©ja. ROLL. -B-f- O 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, H 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. "H 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; *c 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 >* 04 132 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 AL %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 "H 210 FT- LB DEG u = OPTIMUM f :::; .'.". ;•■: ■■: — i - [ I!:: :<£ -r. < ::■: <£ s ; « •':'.': '.i.: ! >L K ::'.': j::i '■•':. cr : . -/ r .: . ■■■■■if i ■ \ 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 if feT""*" eromete: FIGURE 9. (Model (1/12 Froude Scale) with variable underslinging and hub restraint. 202 w . g 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. .3 60 +1 « > > w PS 2- ,1- G.W. O a A = 3250 e.g. LB K H -FT-LB./DEG 106.1 111.8 106.1 111.8 132 132 a ft a 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 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„ N T/W XYZ db e, t x,y,z V a 2 «1' «2 8'k n j a2 JM JU 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 80 X 1 1 1 1 1 1 1 40 UNSTABLE 1 1 s 1 s \. r~~ •~~ f 1 STA 3LE i_ 40 .5 4/ 1 -SO 1 »« 2 m tt " .6 | 3 -u - ,7 1 AU H - .8 1 5 u u s= .9 . i 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. ** 40 1 t - .61 2 t = .75 3 t - .83 / »» \ I 1 1 1 I r x + ffi2 l 20 UNSTABLE^ 1 /" "" 1 1 \] i > f> STOBLE 1 -20 1 -40 2 / *■ i -——_______ 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% 40 20 UNSTABLE ' / ' % y // c 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 C .2 .4 .6 .3 1. Fig. 9 The effect of roll rate feedback control. ■a 40 20 UNST HBLE 1 STA <L£ \ / 2 / ^^ -20 ^X \ -40 ' 1 2 3 "^NOFEEDBACK I M 2 - -2. « 2 +20. j^- 4 'da M 2- -*■ "2 +2I> - W 60 . 1 .8 1.0 Fig. 10 The effect of roll position and roll rate feedback control. ft 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 S ^ 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 ft 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. "b ' 12000 lb. ■ 4 1/2* c/c c - 1/2* - .10 4 e 2 - .10 u f " 1.15 3 M l /o 2 " 1 '° 2 B 2 /a 2 =0.0 1 *-^— .^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. m 12000 lb. "h • 4 1/2* VC r « 1/2* •l » .10 e 2 " .10 "f . 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 10 35 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 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 .8 .6 .4 .2 I 1 II 1 WITHOUT VIBRATION ABSORBERS V HTH VIBRATION ABSORBERS 1 1 1 70 80 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. GW aircraft gross weight Presented at the AHS /MSA- Ames Specialists' Meeting on Rotorcraft Dynamics, February 13-15 > 197 1 *. I K 75 (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 / s / V / \ / / A / \ ^ ,/ / ±30001* \ / \ t 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 a! a o J! kO 80 120 160 200 2l+0 280 320 360 Azimuth, Degrees Figure 5a. Measured Flight Test Result. -1000 -2000 -3000 -1+000 -5000 /I ^ \ ! f n \ \ /-* v / ^ / f , \ / \ j \ ' J \ \ 1 \ / \ 1 J ) J \ / I ,/ Ar - 3xlh 3 ] .b->-- 1+0 80 120 160 200 2l+0 280 320 3.60 t3 o 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-*— *—*■- o Azimuth, Degrees Figure 7a. Conventional Pushrod. t- 1000 -1000 -2000 -3000 A \ 1 1 1 r \ -^ * / \ } \ / \ Si J \ rs _^^g: V/ N \A \J ' 1+0 80 120 160 200 2l+0 280 320 360 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' u & 1 CD 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- 90 180 90 270 360 Blade Azimuth, Degrees 180 Figure 12. Rotating Pushrod Load Comparison 110 KT 96% N„ Level Plight, 1*700- lb. a a) 3 o 60 « -P 0) •P O K S-i O IS U •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 it 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 13 o o 3500 3000 2500 2000 ' H 1500 3 ■p m 1000 o ■p u 500 °l 1° , » ! ** U of 1 ft / 1 i |A| V o —A- — -J^ C 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 i 70 60 50 fof' T7T02 4& r fc J^t J& 60 70 80 90 100 110 120 Calibrated Airspeed, KT Figure 17a. 100? Rotor Speed. 100 •n 90 e J§ 80 70 60 50 • 2, 'j© -A—" 60 70 80 90 100 110 120 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. 1. 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, b c ci Aci CL CLR/c CXR/a Cyr/o F l> F 2» F 3 L R T V V *c V *s X Y <* s 6 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 -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 c m 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 if "ns - («n,)(«o) * (bn s )(«l c ) * («ns) («I„) •♦ KX^) * - ("%„) (3) then "0 °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 "o °>c °»s °2 C »2 S ■ f»e *>3 S ?»c K <=0 *«, lis. "1. to "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 b c ; d 'c • J l 's T P Tin 1. 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- a* 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 12 2 -230 -12 -409 -16 660 42 1 1 2 -5 3 5 -2 6 iO 1 -3 -2 -13 7 1 12 4 -4 2 6 1 ° 14 -2 1 10 -3 1 " 5 -13 32 -20 -18 18 1 10 6 -15 50 -52 32 1 " -5 18 27 -21 -20 1 ° 7 -7 5 59 -78 I 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 10 r\ a /^ 8 -\A A/ 6 2 : vyv P-P REDUCTION 65% 10 f\ a A r* r"\ A t 8 ■\A A/ -\ J V\^ X 6 l-VW V '1 4 z 2 - V 51% 10 rx /\ A _. r^ A / 8 ^\\\r . \ ^/u J 6 - \ v \y \y 4 2 V - 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 g 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 i 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 I damping T 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) {Y 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) DP 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 = 2 (-£- 1 2 a* i) Y. 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 10000 TABLE II. MODEL DESCRIPTION Stations Used Model 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 5A x X X X X 5B X X X X X 5C X X X X X 5D X X X X X 9A X X X X X X X X X 9B X X X X X X X X X 9C X X X X X X X X X 12A X X X X X X X X X X X X 12B X X X X X X X X X X X X 12F X X X X X X X X X X X 244 TABLE III. IDENTIFICATION OF 5X5 MODEL OF 20 GENERALIZED MASSES, X 20 SPECIMEN Model 5A 5B 5C 5D j ** Computer Experiment Number 296 297 292 295 _ Random Disp. Error +5% +5% +5% + 5% Bias Disp. Error +5% +5% + 5% + 5% Random Error Seed 13 13 13 13 - Mode Generalized Masses (Lb-Sec 2 /In.) 1 8.544 8.538 8.543 8.568 8 534 2 4.506 4.506 4.619 4.610 4 449 3 .494 .494 .494 .49 3 495 4 1.048 1.047 1.050 .994 1 .087 5 .653 .653 .651 .629 .630 ** From 20 x 20 Sp< scimen TABLE IV. IDENTIFICATION 9X9 MODEL OF OF GENERALIZED MASSES 20 X 20 SPECIMEN / Model 9A 9B 9C 20 Pt Computer Experiment Number 300 303 304 ^** Random Disp. Error + 5% + 5% + 5% Bias Disp. Error + 5% + 5% + 5% Random Error Seed 13 13 13 - Mode Generalized Masses (Lb-Sec 2 /In.) 1 9.000 9.015 9.043 8.534 2 4.350 4.335 4.513 4.449 3 .472 .472 .472 .495 4 1.042 1.042 1.138 1.087 5 .551 .549 .584 .6 30 6 .786 .783 .723 .743 7 1.154 1.243 1.120 1. 177 8 1.401 1.411 1.396 J.412 9 .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% + 5% Bias Disp. Error +5% +5% + 5% Random Error Seed 13 13 13 Mode Generalize (Lb/Sec d Masses 2 /In.) 1 8.474 8.464 8.518 8.534 2 4.556 4.510 4.492 4.449 3 .488 .487 .487 .495 4 1.150 1.151 1.103 1.087 5 .596 .597 .595 .630 6 .722 .724 .777 .744 7 1.182 1.113 1.159 1.177 8 1.232 1.242 1.215 1.412 9 .797 .743 .789 .786 10 1.203 1.043 -.564 .043 11 ,09 3 .104 .0103 .172 12 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 IV 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 En u <u & 2000 n) 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. & s B i.O .8 .7 .6 .5 .4 .a ■2 /ORIGINAL PROD. (.0 3 SEC. T/C) -30% REDUCED GAIN '" ■ » ■ -,' —-^ jo % ---Ox *•.. \ ^-sINCRE LAG DAMPER LOAD + 400 LB. J 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 W 3 EH TO O a H CO D O En H 2000 1500 1000 500 3/REV / ^STANDARD STRUT NO STRUT / 1 \ %. -— -^S^K 10 15 FREQUENCY - Hz 20 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 12 13 14 Hz 15 16 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 a En H @ 20,000 g Hi H ^ 4/REV GW 32,100 LBS TAS 150 KNOTS 10,000 o lis A 200 210 220 ROTOR RPM 230 240 Figure 12. Model 347 Blade Chordwise Bending Moment vs. RPM 5.0 m u H ■a S d) -P to >i > (!) 0) > M ■H u u a a) ■d H 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 0) o a <d a D N o I a < a .20 .15 .10 • 5.5" Overhang • Torquesensor in Inlet Housing • 1 In-Oz Unbalance @ Thomas Coupling h HP Rotor LP Rotor V Torquesensor © B/V Shafting 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 SI rotor angular velocity, sec' ■1 azimuth position of master blade, rad C„„ Blade Root Moment, STA (o) RM 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 M K m % T 2 K 3 T 3 K 4 r 4 K 5 '5 K 6 '6 L K 7 r 7 K 8 T 8 K 9 r 9 K 10 r 10 K ll r ll K 12 T 12 T 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 = Y 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 CS " T s _0 C c. 3. (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 M ^nominal a Cj/ff 0.191 12° -5° 0.102 0.239 4 -5 0.028 0.443 4 -5 0.011 0.849 10 -5 -0.005 0.851 4 -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. V 0.191 0.239 0.443 0.849 0.851 Ms 0.3805 - 1.7207 2.6149 20.0483 3.5349 M c - 0.5301 -0.4113 -0.5208 -4.5724 -8.4341 h 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) p 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 1 5.617 42.3 1.099 125.6 2.236 120.5 4.798 72.0 4.094 116.5 2 6.126 44.0 1.141 149.1 2.791 129.3 4.787 72.6 3.487 135.6 3 17.571 -9.6 52.416 -30.1 42.237 -28.7 18.537 -19.8 43.319 -5.1 4 26.019 -45.4 47.991 -37.3 40:073 -30.1 20.329 -41.5 37.081 12.7 5 30.696 155.7 59.416 182.9 45.186 188.4 33.002 183.4 26.170 214.2 6 32.505 181.7 77.408 193.2 61.144 180.8 21.085 180.0 38.661 184.5 7 2.856 136.0 4.246 81.9 8.166 86.5 2.472 102.1 10.097 93.1 8 1.507 98.4 5.083 67.1 8.077 66.9 3.412 144.7 7.979 62.9 9 35.384 213.4 59.420 198.8 43.846 181.4 44.506 200.5 48.081 176.2 10 41.674 185.8 51.280 198.6 39.383 195.7 48.473 201.0 40.850 187.7 11 45.953 116.6 76.875 108.3 78.512 101.8 67.268 134.4 88.540 94,17 12 61.589 131.5 86.361 99.3 80.995 95.7 61.288 141.5 90.934 95t3 13 6.879 45.6 5.420 51.4 8.928 39.2 8.188 35.8 9.340 38.5 14 7.211 43.7 6.195 46.4 8.999 35.9 8.906 36.1 9.651 35.6 15 6.635 245.2 4.275 205.9 2.571 195.2 5.976 215.0 3.623 184.0 16 6.033 218.3 3.962 208.1 3.123 188.7 4.775 229.5 1.977 185.4 17 13.000 127.3 7.596 94.3 7.632 76.7 13.261 133.1 11.188 86.9 18 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 f "■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 M flnomi- nal C T /ff Ampl. of Pitch Variation 1 0.849 10° -0.005 -3.0° 0.58 0.851 4 -0.013 2.0 0.32 0.191 12 0.102 0.8 0.21 0.443 4 0.011 0.5 0.08 0.239 4 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 n cos n sin n Amplitude 2 W/O Vibration Control -92.7652 17.2338 94.35 Contribution of O - 0.0559 - 0.1536 h 16.6165 15.2507 0c TOTAL 19.2393 2.2002 - 56.9653 34.5311 66.61 3 W/O Vibration Control - 1.1732 - 14.7883 14.83 Contribution of Q - 0.1248 - 0.5496 h - 9.2715 6.5928 6c TOTAL 9.3862 9.3684 - 1.1833 0.6233 1.34 4 W/O Vibration Control - 0.1403 - 3.5448 3.55 Contribution of g Q 0.1153 - 0.4152 9s - 4.2868 - 8.5191 TOTAL 0.3317 11.6713 - 3.9801 - 0.8078 4.06 5 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 M 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 ui < x °- 1.0 ) o- — o| "C 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 h R.o h 1 5 -.204 1.024 -.155 1.002 2 15 -.212 1.022 -.163 1.002 3 25 -.220 1.019 -.170 1.002 4 35 -.227 1.014 -.177 1.002 5 45 -.233 1.007 -.183 1.001 6 55 -.238 .999 -.187 1.001 7 65 -.242 .990 -.190 1.000 8 75 -.244 .980 -.192 1.000 9 85 -.245 .970 -.193 .999 10 95 -.244 .960 -.192 .998 11 :105 -.242 .951 -.190 .998 12 115 -.239 .943 -.186 .997 13 125 -.234 .937 -.182 .997 14 135 -.228 .933 -.176 .997 15 145 -.221 .930 -.169 .996 16 155 -.213 .930 -.162 .996 17 165 -.205 .932 -.154 .996 18 175 -.197 .935 -.146 .997 19 185 -.188 .940 -.138 .997 20 195 -.180 .945 -.130 .997 21 205 -.172 .952 -.123 .997 22 215 -.165 .958 -.116 .998 23 225 -.159 .965 -.111 .998 24 235 -.154 .971 -.106 .998 25 245 -.150 .978 -.103 .999 26 255 -.148 .984 -.101 .999 27 265 -.147 .989 -.100 1.000 28 ■ 275 -.148 .995 -.101 1.000 29 285 -.150 .999 -.103 1.000 30 295 -.154 1.005 -.107 1.001 31 305 -.158 1.010 -.111 1.001 32 315 -.164 1.014 -.117 1.001 33 325 -.171 1.018 -.124 1.002 34 335 -.179 1.021 -.131 1.002 35 345 -.187 1.023 -.139 1.002 36 355 -.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 #° Ri h R 2 h 1 5 -.215 1.087 -.167 1.007 2 15 -.234 1.088 -.186 1.007 3 25 -.252 1.084 -.203 1.007 4 35 -.269 1.075 -.218 1.006 5 45 -.283 1.059 -.232 1.005 6 55 -.295 1.037 -.242 1.004 7 65 -.303 1.011 -.250 1.002 8 75 -.309 .982 -.255 1.000 9 85 -.311 .951 -.256 .998 10 95 -.310 .920 -.254 .996 11 105 -.305 .891 -.249 .995 12 115 -.297 .867 -.240 .993 13 125 -.285 .850 -.229 .992 14 135 -.271 .839 -.215 .991 15 145 -.255 .837 -.200 .991 16 155 -.237 .842 -.182 .991 17 165 -.218 .854 -.164 .992 18 175 -.197 .870 -.145 .992 19 185 -.177 .889 -.126 .993 20 195 -.158 .909 -.108 .994 21 205 -.139 .928 -.092 .995 22 215 -.123 .945 -.078 .996 23 225 -.109 .960 -.068 .997 24 235 -.098 .972 -.061 .998 25 245 -.089 .982 -.057 .998 26 255 -.085 .990 -.056 .999 27 265 -.083 .997 -.055 1.000 28 275 -.084 1.003 -.056 1.001 29 285 -.089 1.011 -.058 1.001 30 295 -.078 1.018 -.062 1.002 31 305 -.108 1.027 -.069 1.003 32 315 -.122 1.038 -.079 1.003 33 325 -.138 1.049 -.093 1.004 34 335 -.156 1.061 -.110 1.005 35 345 -.175 1.072 -.129 1.006 36 355 -.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 *i II R 2 h 1 5 -.231 1.192 -.040 1.014 2 15 -.267 1.209 -.048 1.015 3 25 -.301 1.212 -.055 1.016 4 35 -.332 1.200 r.061 1.015 5 45 -.360 1.171 -.067 1.013 6 55 -.382 1.126 -.071 1.010 7 65 -.399 1.065 -.074 1.006 8 75 -.409 .992 -.076 1.002 9 85 -.413 .911 -.076 .997 10 95 -.411 .826 -.076 .992 11 105 -.402 .745 -.073 .988 12 115 -.387 .675 -.070 .984 13 125 -.366 . .625 -.065 .982 14 135 -.339 .603 -.060 .980 15 145 -.308 .611 -.053 .980 16 155 -.274 .645 -.040 .981 17 165 -.237 .698 -.039 .983 18 175 -.199 .759 -,031 .986 19 185 -.160 .822 -.023 .989 20 195 -.123 .879 -.016 .992 21 205 -.090 .925 -.012 .993 22 215 -.062 .954 -.011 .994 23 225 -.043 .970 -.012 .996 24 235 -.034 .977 -.014 .997 25 245 -.032 .983 -.015 .998 26 255 -.032 .990 -.015 .999 27 265 -.033 1.000 -.016 1.000 28 275 -.032 1.009 -.015 1.000 29 285 -.032 1.016 -.015 1.001 30 295 -.034 1.021 -.014 1.002 31 305 -.043 1.028 -.012 1.004 32 315 -.061 1.040 -.011 1.006 33 325 -.088 1.063 -.013 1.007 34 335 -.120 1.094 -.017 1.008 35 345 -.156 1.130 -.024 1.010 36 355 -.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 Aa .0034 .0009 -.0016 -.0001 -.0001 A6 .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 .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 Aa .0085 .0053 -.0089 -.0010 -.0012 -.0004 -.0004 -.0001 -.0001 A6 .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 .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) CP 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 A6 .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 Po Pic Pis P2C P2S P3C Pas P4C P4S P5C P5S LIFT Aa 19 12 -20 -2 -3 -1(1) -1 (-2) 0(1) 0(-D 6 A6 36 31 -62 -9 -7 -2(1) -KD 0(1) 0(0) 10 A61S 20 23 -66 -11 -4 -KD 2 (-2) -1 0(1) 0(0) 6 A61C -2 -51 -8 -1 10 3(0) 0(1) 1 0(0) 0(0) A92S -29 -1 1 10 8 -5 2 3 -1 A02C -3 -5 32 10 -5 -8 3 -2 -1 A63S 5 1 2 -13 -7 9 -8 2 4 A63C 5 -1 2 -7 13 -8 -9 4 -2 A64S 1 2 -6(0) 9 {6) -20 (-8) -10 (-5) 10 (-5) -11 (-1) A64C 1 9(6) 6 (-2) 10 (-4) 20(7) -11 (-6) -10(3) A65S 1 2 2 -9 13 -27 -12 A65C 1 2 -2 13 9 -12 27 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 3V 4V 5* 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 22,000 in.-lb 61° 84° 83° 16° 401b 401b 3000 lb 59° 83° 40° 3101b 3101b 108,000 in.-lb 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) df dt 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 f (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 - n A o n«i i=l (-l)(e + f)D( 0j ) e;f l 3 n 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