Analytical evaluation of tilting proprotor wind tunnel test requirements

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: FOR EW ORD , _ • o This report was prepared for the Na t ional Aeronau t ics and Space Administration, Ames Research Cen t er, Moffet t Field, Califor- nia. This work was performe a be t ween 1 April 1975 and 15 December 1975. The technical moni t or at NASA was Dr. Wayne Johnson. The principal SCI (Vt) investiga t or was W. Earl Hall, Jr. Project engineers were Richard Mohr and Rober t Walker. Repor t prepara t ion "" was done by Deborah Buenz.

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ii TABLE OF CONTENTS PAGE FOREWORD ........................... ii LIST OF FIGURES ....................... v LIST OF TABLES ....................... ix LIST OF SYMBOLS ..................... x I. INTRODUCTION AND SU_[ARY ................ I i.I Introduction ................... I 2.1 Review of Basic Rotor Wing Mathematical Model . . 9 3 .1 Frequency Response of the Rotor C antilever Wing 3 .1.1 Tr a nsfer Functions of Ro t or / Cantilever Wing 3 .1.2 Evalua t ion Me t hod of Transfer Functions . . 26 : 3 .2 Discussion of Tr a nsfer Fun c tion Charac t eristi c s . 3 4 3 . 3 Input Frequency Requiremen t ........... 3 9 a.

3 .4 The Effec t of Noise in Selec t ing Input Amplitudes 41 3 .5 Summary .....................

m_ IV. COUPLED WING DYNAMICS ................. 5 3 4.1 Frequency Response of the Coupled Wing Model . . . 5 3 - 4.1.1 Compu t a t ion of Coupled Wing R e sp o nses . . . SS 4. 1. 2 Asymme t ric M o de Charac t eris t ics ...... SS iii TABLE OF CONTENTS (CONCLUDED) PAGE 4.3 R e quir e d Input Amplitud e s to Achieve a Desired • ,•o e o•,,t••• ,ooeQooe 4 4 Sum m ar y 71 ' V. EXA_MPLES OF EV._UATION OF METHODS FOR MODAL IDENTIFICA- 5.1 Summary of Algorithms ............ 75 - APPENDICES 4•o • , e ,••• o • eeooaooQ,os, • REFERENCES 129 iv t I LIST OF FIGURES FIGURE NO. PAGE I.i The NASA / Army XV-IS ................ I 1.2 S t udy of ](V-IS Tunnel Test Procedures and Analysis . . 3 2.1 Comparison of Pole Locations for Symmetric Equations With and Wi t hou t Support Degrees-of-Freedom . . . iS 3.1 Modal Power Ra t io; Wing Flaperon to Wing Chordwise Accelera t ion (_w2 / _f), C a nt i lever Wing Transfer Func- 3. Z Noise- t o-Sign a l R a tio-- C olle c t ive Pi t c h to Wing Chord - wise Acceler a tion (_wl /8 o); Can t ilever Wing Tr a nsfer Fun c tion (For Frequen c y R a nge 4.20 to 6.3 5 Hz) ....

3 . 5 Noise- t o-Sign a l Ratio - -Collective Pitch to Wing Chord- wise Acceler a t ion (_w2 / 8o); C a n t ilever Wing Tr a nsfer Func t ion (For Frequency Range 4.20 to 6.35 Hz) ....

3.4 Noise- t o-Signal Ratio--Wing Flaperon to Wing Vertical Acceleration (_wl / _f); Cantilever Wing Transfer Func- 3.5 Noise-to-Signal Ratio--%_ing Flaperon to Wing Vertical Acceleration (_wl / 6f); Can t ilever Wing Transfer Func- 4 7 tion (For Frequency Range 3.04 to 4.60 Hz) ....

3 . 6 N o ise- t o-Signal R a tio--C o llective Pi t ch to Wing Ver t i- ca l Accelera t ion (_wl / 8o); Tr a nsfer Fun c t ion for Asym- metri c Motion (Fo r Frequen c y Range 3 .33 to 5.29 Hz) .

5 . 7 Noise- t o-Sign a l R a tio--Colle c tive Pi t c h to Wing Ver t i- ca l A cc elera t i o n (_wl / % o); Tr a ns£er Fun c tion for Sym- - me t ric Mo t ion (For F requency Range 2.77 to 3.49 Hz) . 49 4.1 Modal Power Ra t io (MPR)-- Collec t ive Pi t ch to Wing Chordwise Acceleration (_w2 / e o ); Coupled Wing Motion . 5 7 4.2 Mod al Power R a tio (MPR)--Wing F l a peron t o Wing Verti- i cal Bending Accelera t ion Coupled Wing Motion 57 4.3 Modal Power Ratio (MPR)--Wing Ver t ical Bending Accel- eration to Collective Pitch wl eo I ('_ / ); Coupled Wing I 4.4 Distortion Factor for Various Values of Control Cali-

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I V I LIS T O F FIGURES ( C ON TINUED) F IGURE NO. PA GE 4 .5 T h e Influ e n c e of Calibration Errors on th e Calculat e d Valu e s of Dampin g for _h e qwl Mode from th e _ 1 / 8 o Transf e r F uncti o n ...... w 66 .eeoeaoeloee 4.6 Noise - to-Signal R a t i o f o r Symmetri c V e r t ica l Wing B e nd- i n g (qlS) N o de fr o m _wl / 6f Transfer Func t ion (For Fre- quency Range 2.53 to 5.85 Hz Abou t the qlS Peak) . . . 70 4.7 Noise- t o-Signal Ra t io for Asy m me t ric Vertical Wing Bend- ing (qlA) Mod e from _wl / 6f T ransfer Fu n c t i o n (For Fre- quency Range 3.83 to 4.82 Hz Abou t the qlA Peak) . . . 7Z 4.8 Noise-to-Signal Ra t io for Asym m e t ric Ver t ical Wing Bend- ing (qlA) Mode from _wl / 8o Transfer Func t ion (For =re- quency Range 3.49 to 5.29 Hz Abou t the qlA Peak) . . . 73 5.I Time His t ory Fit of Reduced Order Model to Nine C.I _wl / 6f - Wing Ver t ical Accelera t ion to Flaperon (Sy m me t ric) ..................... Ii0 C . 2 _ w 2 / 6f - Wi n g C h or d wise A cc eler a t io n to Fl a peron (Symme t ri c ) ..................... Ii0 C . 3 _wl / 8o - Wing Verti ca l A cc eler a t ion to C olle c t ive (Sy m metri c ) ..................... Iii C.4 _w2 / 8o - Wing C h o r d wise A cc eler a ti o n t o C ollec t ive (Symmetri c ) . . _ .................. III C . 5 _w 2 / 81s - Wing Chordwise to Longitudin a l Cy c li c 11 2 (Sy m metri c ) .....................

C .6 _w 2 / 8 1c - Wing C hor d w i se to L a ter a l Cy c l ic (Symmetric) I 12 v i i LI S T OF FIGURES ( C ONTINUED) FIGURE NO. PAGE C. 7 _ w l / 81s - Wing V ert ical A cc el er ation t o Longi t udinal Cyclic (Symmetric) ................. 11 3 C.8 _wl / Slc - Wing Vertical Acceleration t o Lateral Cyclic J (Symmetric) ..................... I13 C.9 p / _f - Wing Torsion to Flaperon (Symmetric) ..... i14 C.ll p / S ic W i ng Torsion to La t eral Cy cl i c (Symmetric) . . 11 5 C.12 p / Sis _ing Torsion to Longi t udinal C y c lic (Symmetric) 11 5 C.1 5 8G C / 9 o - Fore / Aft Gimb a l Flapping t o Colle c tiv e <Sy m- metric) ...................... I16 C .14 S G S / S ° - Lateral G i m bal Flapping t o C ollective (Sym- -- C.1 5 8GC / 6 f F o re / Af t Gimbal Flapping t o Fl a peron (Sym- - metric) ....................... 11 7 ,, C.16 8GS / _ f - La t eral Flapping to Flaperon (Symmetric) . . 11 7 C.17 _s / So - Ro t or Perturba t ion to Collec t ive <Symmetric) . 118 C.18 _s / _f - Rotor Per t urba t ion to Flaperon <Symmetric) . 118 C .19 8(1) / _f - Blade Lead-Lag to Flaperon <Symmetric) . . . 119 Is C. 20 8(1) /% ois - Bl a d e Le a d - Lag to Colle c t i ve <Sym m etric) . . II_ C, 21 8_ 2 ) / _f Ou t -of - Plane Coning to Fl a peron (Symme t ric) i Z 0 C .22 8 2) / 8 o Ou t - o f-Plane Coning t o Colle c tive (Symm e t- _w C. Z3 _(1) / S o - Bl a de Lead-Lag to Collective (Symmetric) . 1 2 1 -- "I c - C. Z4 " Ic_(1) / _f - Blade Lead-Lag to Flaperon (Symmetric) . . . IZl qmb -- vii LIST OF FIGURES (CONCLUDED) FIGURE NO . PA G E C.ZS _wl / df Wing Vertical Acceleration to Flaperon C. 2 6 _w Z/ eo - Wing Chordwise Accelera t ion t o Collec t i v e (Asymmetric) .................... 12Z C.2 7 p / elc Wing T o rsion to Later a l Cyclic (Asymmetric) . 123 C.28 p / els - Wing Torsion to Longitudinal Cyclic [Asymmet- C.29 _wl / 8o - Wing Vertical Acceleration to Collective [Coupled Wing Response - Exci t e One Wing, M ea sure C.30 _w2 / 8o Wing Chordwise Acc e lera t ion to Collec t ive ( C oupled Wing Respons e - Exci t e One Wing, Measure C. 3 1 _wl / _f Wing V e rtical Accelera t ion to Flap e ron (Coupled Wing Response - Excite One Wing, M e asure C.32 _ w 2 / 6f - Wing Chordwise Acceleration to Flaperon (Coupled Wing Response Excite One Wing, Measur e Sa m e Wing) ...................... 12 7 viii LIST OF 'fABLES TABL E NO. PAGE i.i Frequ e ncy and Damping Information Availabl e in Various Test C o nf i gurat i ons f o r the N i ne D egree-of -F _eedom 2 .1 Ro t or, Pylon, Wing D e grees-of-Fr ee dom and Controls . . i0 2.2 Transfer F unc t ions of Coupled Wing Responses ..... 1 7 3 .1 Qualitative Comparison of Transfer F u nctions ..... 3 0 3 .2 XV-15 Ro t or / Can t ilever Wing Modal Decomposition . . . 3 1 3 . 3 Principal Symmetric Modes .............. 3 6 4.2 Calcula t ed Values of Dampin g for q2A Mode from _w2 / e Transfer Function ............ . . . . ..° 6 7 .. B. I C a lc u l a t e d D a mpin g as a Function of F requency Band- width Used for In t egr a tion .............. i00 i x i , LIST O F SYMBOLS A i _ p l i tude of fr eq u en c y r esp o nse A2,AI,A o Mass, damper, a nd spring m a tr i ces of dyn a mic equ at io n s a Accelera t ion, _ (Appendix B) a _, a l , a o M a s s , d ampe r, a nd sp r i ng matrices of ca nt i lever ....

w in g a n d s upport d yn a m ic e qu a tions (App e ndix A) C o e ffi c ient matrix for rot o r for c es and mom e nts in the c antilever wing and s upp o rt dynami c equa- t i ons (Appendix A) B C oeffici ent m a trix f o r co ntrol ve c t o r in dyn a m i c e q ua t i o n s BG Coefficient matrix for gus t vector in dynamic e q u a tio n s b " C oe ffic ient matr i x f or co ntrol vec t or in ca nt i lever w ing a nd support dyn a mi c equ a tions (Appendix A) b_ C o e f f i c ie n t m a tr i x for gust ve c tor i n ca nt il ever wing a n d support dyn a mi c equ a t i ons (Appen d ix A) c D a m pin g coeffi ci ent (A p pendix B) C xs,Cys,C_sup Da mp in g degrees of coe f f ic ien t s f ree d om for xs, Ys' and _su p (support) . .

D C ontro l di strib u t ion matr i x i n me a surem e nt equa - t i o n D _ Gu st di st ri b u t i o n matrix in m e a s ur e me nt e q u a t i o n • Base of n a tural exponent i als F S tat e d y na m ic s m a trix F i G e n e r a l i ze d f or c e f o r t h e i t h g e ner a l i z e d c oor d in - a te in La g ran g e 's equa t i o n ( App e ndix A ) f C o n t r o l varia b le ( App en dix B) i -- LIST OF SYNBOLS CCONTINUED)

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G C o n trol distribution matrix i n s tat e dynamics e qua- tion g G u s t ve cto r • . H S t ate d is tr ib u ti on mat r i x i n me as u r e m ent e q u a t i on H T ran sf er f unc ti on H I np l ane rotor f orce {Ap pen di x A) h D is ta n c e f ro m wi n g ti p e l a s t i c ax is to r otor hu b " " (A p pen d ix A) I I den ti ty m at r i x I(_) An in t egral us e d in damping cal c ulation I(_) - 2 2 2 + 2 d_ . . / 2 2_n_2 _1 (_n - _) (2_n_) j = / T[ , i mag i n a ry uni t Kx s ,Kys, K _sup Sprin g f r eedom c on sta nts f or x s , Y s ' and _sup degrees of " k S pr i ng coe ffic i ent k i R e si d u e for ith pol e of impuls e respons e k[ R e s id ue for ith pole of response to sinusoid a l i nput L Gus t co rr elation dis t a n ce " L La g r a n g ian, difference o f kine t ic and p o ten t ial e n e r g y (Appendix A) • . Z D ist a nc e fu s el a ge e. g. is ah ead of s uppor t y a w x f a xis , _yp D i s t a n ce t h e pylon e. g . is a h ead a nd a bov e th e £ X p w i n g ti p el a s tic axi s ' £ Y w D is ta n ce ca n t ilever r o o t res t ra i nt of wing a t t h e x w r oot el ast i c a _is is a he ad a nd to t h e r i g h t of su ppo r t y a w a xi_ xi LIST OF SYMBOLS ( C ONTINUED) Mx,My Rotor moments about the rotor's x and y axes MPR Moda l pow e r rati o m Ma ss m s Sum of fu se l ag e, w i ng, a nd pyl o n mas s e s N / S Noise-to- s ignal ra t io n Noise vector p W in g torsion de gree of freed o m Q Ro t or to rque ab o u t the z axis ( s haf t ) q Power s p e c t ral den s i t y of gu st qw I = q l Lowes t mode w i ng v e r t ic a l ben di ng d e g re e o f free 4 om m qw 2 q2 Low es t mod e wing chordwise b e nding degr e e of fr ee dom Ri R e sidu e of ith pol e r P o wer s p e ctral d e n s i t y of m e a s ur e m e nt noise S / N Si g n a l - to - no i se r a t i o s L a pl ac e i n de pen d en t v a ri a ble T Tr a n s fer fun c tion T Kine t i c e n e r g y (A p pen di x A) T Rotor thrus t (Appendix A) t Time UO Win g v e lo ci ty u C ont ro l v ec to r xii 1 !

I I LIST O F SYMBOLS [CONTINU E D)

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V V a riance V Potential e nergy (Appendix A) • I Xn'Xs Power under the noise and sign a l po_er spectra : x St a te ve ct or xs Suppor t longitudinal degree of freedom . x , y,z Ca rte si an p os ition coo rd i n a tes Y R o tor for c e a l o ng y ax is [App e ndix A) y M ea sur eme nt v ec tor Y s Supp o rt l at er a l deg re e o f f reedom YTw Wing sem is p a n, dis ta n c e from r u o t restr a in t to sh a f t a x i s al o ng wing el as tic a x i s a Angul a r p os ition vector [Appendix A) R ot or t i p-p a th - pl a ne p i t c h and ya_ deg r ees of 61 c 'Bls fre e d o m 8 o Rotor co ning d e gr ee of f r eedom _6 C ,8G Sa R ot or gimb a l p i tch a nd r oll an g les 8+I,S . I R oto r gi mb a l m ode s a bov e an d b e low on ce per rotor r ev olu t ion i n fr e q u en c y F G u s t dis tribution mat r ix i n s tat e dyn a mics equ a- tion y L oc k n umber [Ap pe n d ix A ) & Dis t o rt io n fa ct or of t r a nsfe r fun ctio ns due to calib ra t i on e r ro rs 6 f W i n g flaper o n defle c tion a ngle - - xiii

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LIST OF SYMBOLS (CONTINUED) 6wz,6w3 , _Wl , W i ng di h edral sweep, and incidence angles Con t rol calibration error D a mping f ac t o r _+i,_ I R o tor i nplane bending modes above and below once per rot o r rev o lut i on in frequency n Meas u remen t ca l i br a t i o n error q Blade bending mode shape (Appendix A) q_ Slope of bending mode (Appendix A) 8o,8coli Collec t iv e pi t ch 8c L a teral c yc l i c pit c h % s Lo n gi t u d in a l cycli c pit c h i. ith pole of t r a nsfer func t ion or, equivalen t ly, ith l e i genv a lue of F m a tr i x ith generalized coordina t e in La_range's equation _ i (App e ndix A) Frequ e ncy sp ect r um _u,¢v,Ow Fr e quen c y sp ec t ra of gus t c omponen t s Oi Phas e angl e of frequ e n c y r e sponse a Standard devia t ion Ro t or rotational speed per t urba t ion

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Suppor t yaw degree of freedom _sup Freque n cy _n Natural frequency xiv

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[ LIST OF SYMBOLS (CONCLUDED) I Subscripts A Asymmetric A Refe r ence frame centered at wing elastic a x i s roo t restraint (Appendix A) a Ac t ua l o r t r u e f Fusel a ge I Inertial reference frame (Appendix A) L Lef t m M e asured p Pylon p Peak R right rms roo t -m e an = square S Symme t ri'c w Wing Operators .. ('),(") F i rs t an d s ec ond d eriv a t ives w i t h resp e ct to t ime - ( )-i Matrix i n verse .. ( )T Matrix transpose ;- Im( ) I maginary part of c omplex quan t i t y [" _( ) Laplac e transform Re( ) Real part of c o mplex quant i ty i xv i.

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_D I. INTRODUCTION AND SUMMARY ) i.i INTRODUCTION The XV-IS is an advanced tilt rotor research aircraft (Figure i.i) currently under development by the National Aeronautics an_ Space Administration and the U.S. Army Air Mobility R&D Labora- tory. An important element of the aircraft development is an ex- tensive full-scale wind tunnel test to be conducted in the Ames Research Center 40- by 80-foot wind tunnel. Since previous wind tunnel scale model and full-scale tests of the tilt rotor concept have been effective in previous years, it is desired to conduct the forthcoming XV-I5 tests in a comprehensive manner to continue to minimize uncertainties in system characteristics.

° Figure i.i The NASA / Army XV-15 This study is a determination of specific test requirements which impact the conduct of these tests. This research is based on a mathematical model of the XV - 15. This model is used to analyze the dynamic characteristics of the vehicle at 190 knots in the wind tunnel.

1.2 METHOD OF APPROACH The method of approach used for this work is schematically shown in Figure 1.2. The mathemati c al model was converted into a simulation and a state _ector format. These two reformulations were then combined with existing data on the input c ontrol channels and instrumentation characteristics to provide a model for the test system.

The next steps were the evaluation of the test model to deter- mine basic requirements on the data analysis algorithms. In par- ticular, this step produced an evaluation of mode identifiability which set further requirements on instrumentation and inputs.

Originally, the effort w a s directed to a basic rotor / cantilever wing model. During the course of this effort, c oordin a tion with the test agency indicated the need for results which expanded this method of approa c h to coupled modes between both wings. The steps were repeated for this case.

1.3 PRINCIPAL CONTRIBUTIONS AND CONCLUSIONS The following analytical tools have been developed: (I) An XV-15 dig ital simulation incorpolating a simplified tunnel support model, c ontrol system loop, measurement l a gs, gust disturbances, and sensor noise. Time histor- ies a re gener a ted from this simulation which provide a correl a tion b a se for the tunnel tests, a s well as a me a ns of ev a lu a ting various dat a processing methods.

1 I I : I i ! (2) Sp ecialization of existing data analysis programs t o t h e high order XV-15 dynamical model. The programs so speci- alized at various stages of t his work are: (a) the trans- fer func t ion progra m , (b) a t ime series analysis program (FFT , au t o-correlation, transfer functions), and (c) SCIDNT, P an advanced maximum likelihood parameter id e ntifica t ion program.

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( 3 ) Several auxiliary programs have been developed to provide es t ima t es of damping from t ransfer functions, as well as calculation of modal decomposi t ion of system response (to identify modes), par t icularl y useful for anal y sis of coupl e d wing responses.

Applica t ion of t h e s e programs to th e basic sym m e t ric, nin e degree-of-freedom m athematical model have produced the following conclusions (see Table I.i): (I) If the test c o nfiguration at 190 knots is such that con- t rol inpu t s are limi t ed to collec t ive pitch and flaperon at frequencies below approximately 5 Hz and measuremen t s a r e m ade orly of th e wing bending accelerations, qwl a nd qw z' t hen suff ici ent infor m a t io n f o r ca lcul a ting m o d a l fr e q u e nc y and da m ping accura t e l y i s availabl e onl y f or t h e q wl a n d q w2 mod es. (N e c e ss ar y i npu t ampli t u d e s t o achi e v e a d es ir e d n o i se - t o- s i g nal ra t io fo r di f f e r e n t l e v e l s o f w ind gu st and s p ee d o f ac t ua t or r es pon se can b e o b t ain e d f rom Fi g ur es 5 . 5-5 .7 f or t h e can t il e v e r wing and Figur e s 4.6 -4 . 8 fo r t h e c o upl e d w ing .)

( 2 ) If th e a b o v e te s t c onf i gur at ion co uld b_ impr oved t o a llow f l a per o n in p ut freq u en ci e s i n th e vi cin i t y of I0 Hz, t h e n in f or ma ti on would b e suffi ci ent to co mp u te m o d a l frequ e n c y a n d damping of the wing t orsion mo d e from mea s u rements of the w i ng ben d ing acce lerations.

I _' (3) Similarly, collec t ive pitch input frequencies near 20 Hz would provide information for the frequency and damping of the ro t or coning mode, B.

(4) If instrumentation is provided t o measure the wing t o r- sion degree-of-freedom, p, and if czclic pi t ch con t rol at 18.6 Hz is possible, then the frequency and damping of the upper rotor inplane bending, _ be determined.

( 5 ) Ev e n u nd er th e b es t o f circums t ances (i .e. , an y of the four c on t rols c ould be ex c i t ed a t a ny frequency and me a surements c ould be t aken of all nine degrees-of-freedom) suffi c ien t infor m a t ion is not presen t in the tr a nsfer functions to ca lcul a t e the modal f r equen c i e s a nd d ampings of the upper a nd lower gimb a l modes, B 8 1 , a n d the lower ro t or inpl a ne bending mode, _-I" It is be c ause t hese modes a re well da mped t h a t t here a re no reson a n t pe a ks asso c i a t ed wi t h t hem in the t r a nsfer func t ions.

C o n s id e ration o f t he c o upl e d wing m o t i o ns in t he pr ese nc e of control and m e asurem e nt calibra t ion e rrors produc e d th e following conclusions: (i) The t es t configura t ion of e xciting one wing and m e asur- ing the r e spons e of tha t wing mak e s the d e t e rmination of fr e qu e ncy and damping of individual mod e s e xtrem e ly difficult since th e symm e tric an_ asymm e tric mod e s usually r e sul t in a singl e r e sonant p e ak in th e fr e qu e ncy r e sponse.

{ 2 ) Symm e tric a nd a sy mmetr ic m o t i o ns ca n b e s epa rated i f ei t her both wings are excited and / or the responses of both wings are measured.

- (3) Errors in calculating a mode's frequency and damping ,- from its resonant peak ma y s t ill occur if the motion of o t her m ode s i s sign i ficant.

, ° m (4) Measuremen t and con t rol calibra t ion errors less t han 20% do not produce significant errors in calcula t ing damping values compared t o the problem s t a t ed in (3) abo v e.

i . 4 SUMMARY OF REPORT T h e ob j ecti v e o f t his repor t is t o presen t pr i ncipal modeling and analysis approaches and reference ma t erial for t unnel t es t planning. Chap t er II discusses the modeling of the aircraft, instrumentation, and con t rols. Chap t er III reviews resul t s of the rotor / cantile v er wing model, and Chap t er IV presen t s the coupled wing resul t s. Chap t er V gives some examples of da t a prediction wi t h system iden t ifica t ion t echniques. Chap t er Vl presents de- tailed conclusions and recommenda t ions.

T he app e ndices pr o vide th e d e t a iled analysis an d figures up o n which the text draws. Appendix A is the derivation of the support equations. Appendix B presen t s the principal derivations for t ransfer func t ion analysis. Appendix C is the plots of transfer functions.

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II . TILTING PROPROTOR MATHEMATICAL MODEL An a n a lyti ca l e v a lu a tion o f tiltin g proprotor test require- ments is b a sed on a c omprehensive d e s c rip t ion of the dynamics an d aero d y nami c s of the sub j e c t v ehi c le. This s e c t ion discusses th e b a sis of t hat description (the simula t ion, d et a iled in [2 ] ), a nd modifica t ions to this simulation whi c h approximate typi c al test con- figur a tion c onstr a in t s and dis t urban c es. S e c tion 2.1 revi e ws ess e n- t i al elements of the basi c rotor / wing mo d el. Se c t ion 2.2 discusses th e i nfluen c e of the air c r a ft t unnel suppo rt res t r a in t . Coupled w i n g i nt er ac t ions a re a ppr o xim a te d i n S e c t ion 2 . 3 , followed b y Sec t ion 2.4, ou t lining the gus t simul a tion method, Se c tion 2. 5 , modeling of i ns t rum e n t ation, an d S e c tion 2.6, control sys te m repr e senta t ions.

Z.1 REVIEW OF BASIC ROTOR WING MATHEMATICAL MODEL T he basic m a t h e mati ca l mo de l of th e XV-I 5 aircr a f t dyn a mics h a s b ee n establish e d by Johnson [ 2-5]. A c onclusi o n of R e fer e nc e 5 was t ha t , in gen e r a l, the basic ro t or dyna m ics w e re sa t isf a ctorily d e scrib e d b y a nine d e gree-of-fr ee dom mod e l (th e firs t b e nding mod e and rigid pi t ch mode of e ach blad e , gimbal pitch and roll angl e s, and ro t or speed p e rturbation), but that in some cases it may be reduced to six d e gree-of-freedom by using the quasi-static blade- t orsion approxi m a t ion, which is discussed in Reference 2.

For the purposes of this s t udy, it was de t ermined t ha t the six degree-of-freedom model was sa t isfactory• Hence, the complete ro t or / pylon / win g model consis t s of th e six ro t or and pylon degrees-of- freedom d e scribed above, plus t hre e degre e s-of-fr ee dom for th e win g , _hich are lowest mode ver t ical bendin g , chordwise bendin g , and t orsion. These nine degrees-of-freedom are lis t ed in Table 2.1 for _ ase of re f e r e nc e.

i t • {' Table 2. i Rotor, Pylon, Wing Degrees-of-Freedom and Con t rols i Degrees-of-Freedom I SYMBO L DESCRIP T ION

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B1c Longitudinalcomponentin nonrotatingframe of lowes t blade bendingmode Bls L ateral c omp o nentin nonrotatingframe o f lowest blade bending mode B o C o nin g o f the rotor BGC Gimbal pitch angle BGS G imbal ro l l a n gle _S Rotor rotationalazimutl, perturbatio n ql Wing vertical bending q2 Wing chordwisebendi n g p Wing to r s i o n Controls eo Blad e c o llecti v e p itch es B la d e longit ud inal cyclic p itch e c Blade lateralcyclic pitch 6f Wing flaperon deflection i i , S e v e ral aspec t s of this mod e l d e serve furth e r con_n e nt. The low e st bl a de b e n di ng mo de s a r e e s se n tia lly inpl a ne bend i ng {lea d- lag) during high speed, axial flo w . The effec t of collec t ive lag of t he blades a ppears in t he ro t or speed per t urba t ion degrees-of-freedom.

The e f fec t s of engine and transmission dynamics a re inclu d ed in the equations of mo t ion, bu t t hey do no t introduce ad di t ional degrees- i0 : i g . . of- f reedom . Th e equ a t i ons of mot i on i nclude t h e e ffects o f g u sts a n d f o ur control var ia bles (s ee T a ble 2 . 1): collect i ve, l o ng i tud i n - al cyclic, and lateral cyclic p i tch of the blades and wing flaperon de f lection.

T h e computer pr o gram developed by Johnson [3] calcul a tes the equ a ti o ns o £ mot i on i n the general form A2R + AI_ + AoX - Bu + BGg (Z.l) wher e x is the state vector composed o( either the nine sym- metr ic o r asym m e tr i c degrees - o£ - £reedom, u is the v e ctor o £ control v a r ia bles , g is the g u s t vector, Az,AI,A ° are the "ma ss ", "d a mper" , and " s pring" matri c es, r e spe c tively, and B,B G distribute the c ontrol s and gu s t s , respe c tivel y , a mon g the s tates.

T hi s equ a t i o n i s re a dily convert e d to t h e s t a nd a rd st a t e- s pa ce form

= F x + Gu + rg (z. z )

w h e r e

"Ix] (z.3) x- ._j

F- ° 1

[ 0 I 1 (2.¢)

! G - (z . s)

B

[°I

an d

r ,, (z.6)

I°]

BG Measurements , y, of the degrees - of - freedom are modeled as lin - ear combinations of the s t ate and control: y " Hx + Du (2. 7 9

J

E x c e pt w here s ta t e d o t h e rwise, it i s a ssum e d th at the a ccel erat ions o f th e wing b e nding d e gr ee s-of-fr ee d o m, ql and q2' ar e m e asur e d by accelerometers and that direct m e asurem e nts of other degrees - of- freedom are available as need e d for the sake of discussion.

The frequency response of the measurem e nts to the controls or g u s t s a re a v ai lable from t he L a pla c e transfer func t ions u_ - HCsI - F)'IG (2.8) g_ - HCsI - F) ' IF. D" (2.9) wh i c h follow di r ectly f r om Eq. (2.2). T h e results ob tai ned f rom t h e se fun c t i ons a r e dis cu ss e d in C h a p t ers II I a n d IV.

2 . 2 EFFEC T O F SUP P ORT F L E XIBILITY A p r elim i n a ry r equ ir emen t f or th i s study w a s t o de ter m i ne i f t h e f lex ib il it y o f the s u pport s t ru ct u r e i n t h e 4 0x 80 wind t unn e l wo ul d i n tr odu ce mo da l fr e qu e nc ie s n ear f req ue n c i es o f i n te r est. I f IZ | f

J

_' so, the isolat io n of t h e modal frequenc i es of <nterest and the I de t e r m i nation of the modal da mp i ng f a c t or would b e m a de more diffi- i c ult.

D i s p lacement of t h e support attachment p o ints in a horizonzal pl a ne w a s c o nsidered. Vertical displacement was not con s idered be ca use the support is f a r more ri g id in t h a t direction. Thus, the basi c equations of motion were expanded to in c lude two more degrees-- of-free d om: longitudin a l a nd la t er a l tr a nsl_tion of the a ircr a ft, x s a n d v s' respec t ively. Air c r a ft y a w i s a lso possible, but data were not a vail a ble to a dequately specif y the frequency and d a mping par a me t ers for the y a w degree-of-freedom, so it w a s omitte d from the stu d y.

The support equations will be only briefly described here; a detailed derivation of the equ a tions appears in Appendix A. The direct aerodynamic effects on the x s and Ys degree-of-freedom are negligible compared to the structural kinetic effects; Zherefore, only the latter are retained in the equations. The result may be written simpl y as ms _s + C xsXs - FXs (Z. I O) m s Ys Cy s Y s * K ys Y S " Fy s ( 2 .ii) wh e r e t h e asterisk superscript s d e n o t e n o ndimensional quantiti e s.

Th e m a ss c oe ff i ci e nt, ms, i s e s t im a t e d from a w e ight analysis of _he XV-15. The terms FXs and Fy s are composed of a ll the c ross-coupling terms between the support a nd the remaining de g rees- o f-fr eedom , p l us th e e ff ec t s o f c o ntr o ls and g usts, all of which were availabl e from sh e e xisting coraputer pro g ram. Therefore, only t h e da mp i n g c o e ff i c ie nts , Cxs an d Cys, a nd th e spr i n g c oeffi c ient s , KXs a n d Kys, r e m ai n to b e s p e ci fi ed. Th i s w a s done by a ssum i n g 2 %

| !

• I !

s tr u ctural damping and s pec i fy i ng t h e frequency of the s upp o rt oscillati o ns with the rot o r off. Th e se frequencies were t aken as 3.5 Hz in the longitudinal direc t ion and 4.0 Hz in the lat e ral, based on earlier experience in the 40 x 80 tunnel [6].

The result is shown i n Figure Z . l, w hi ch com p ares t he pole ' locations of the sy m metric nine degree-of-freedom model to the same m o del with t he support degrees - of-freedom added. As can be see_, the eff ec t is a n i n c r e ase in the mo da l frequencies of the rotor- coni ng (S O ) m od e ( f rom 2 .6 to z .g per re v ); o f the u pp e r b la de la g (_ ¤ m ode ( f rom 2 . 4 to 2 .6 pe r r e v); a nd of th e w i ng c ho rdw i se ben d ing (qw 2 ) m od e (from 0.67 to 0.84 per rev). The frequencies of the remaining modes are unchanged. The two new modes associated wi t h x s and Ys are at low frequencies (0.1Z and Q.IS per rev, r e s p e c t i v el y ) a nd a re not e xpe c ted t o ca use a n y diff i c ul ties i n identifying the freq u encies a nd damping of other modes. Also, even tho u gh t h e 80 , 4+ 1 , and qw2 mo dal freque n c i es are a ltere d , there -_ still remains a d equ a te sep a r a tion of frequen c ies to permit d is c rim- in a t ion of individu a l m o d es. Therefore, the c onclus i on is th a t the support flexibility do_s'hot_h_nge the basic ch a r a c t eris t ics o f the . _ / -1 5 s t ru c tur a l d a mping or frequen c ies a nd nee d not be c onsider e d in the rem a inder of this stu d y.

2.3 MODELING OF COUPLED WING RESPONSES A ma jor pa rt of t hi s stud y was to as cert a i n the effects of ex ci ting on_ or both rot o rs (or wines), th e effe c ts of measur i n g the _response of one o r both wines, a nd the effe c ts of c a libr a tio n e rrors in the mea sur e m e n t s o f th e c ontro l inpu t s a nd th e wing responses on the ability to correctly compute the fr e quency and da m ping of important modes. The nine degrees-of-freedom of the right rotor / pylon / wing sys t e m are coupled to the nine 4egre_--of- fr e edom of the le ft syste m , pr i n c ip al ly by me a ns of the transmission I

] I !

_m 3.0 0 POLE FOR CASE WITHOUT SUPPORT

(9DOF)

"" _ B O /k POLE FOR CASE WITH SUPPORT (11DOF) 2.5 NOTE: ONLY ONE POLE OF A CONJUGATE COMPLEX PAIR IS SHOWN. _+1 2.0

(25B+I

e 1.0 0.5

qh

B. i_ 9 Ys_

I ' I I ' I i I ' " I I _ I ' ,.,5. / ..-P -I.0 -0. 8 -0. 6 -0.4 -0.2 0 REAL PART, "{mn _ REV'I Figu r e 2 . 1 Compa_'ison o E Pole Lo catio n s E or S y mmetric E quation s _ith and With o ut Support De._roes -o f-Freedom !

i I i i

!

i nt e rcon n ect sh a ft. This left and rig h t system of eighteen degrees- of-freedom may be decoupled into symmetric and asymmetric systems of nine degrees-of-freedom each. This procedure is advantageous because it aids the und e rstanding of the complete system, simplifies the mathematics, and reduces the computational effort.

There are nine cases to be investigated (Table 2 . 2). It is assumed that the sym m eZric and asymmetric responses are separated whene v er possible to better i sol a te t h e frequencies of individual modes.

S o me defini t ions are h elpful before proceeding with t his analy - sis. Six subscripts which will be used are: S _ symm e tric A _ asymmetric R _ right L _ l e f t a _ a ct ual or t r u e m _ measured The basi c symme t ric and asymmetri c transfer fun ct ions are: YS (s) Y A (S) Ts( S ) - _ , T A(S) - U A _- (2. 1 2 ) where I (s)), measurement of s y mmetric Ys(S) -__ (YR(S)+Y L respons e ( 2 .13) YA( S ) _ ½ ( Y R(S)-YL(S)), m e asuremen t of asymmetric r es ponse (2.14)

!

0 . - J , , '-Jl _ . i P_" L P . _I: l a . l _ _ - I - cM Z v -I - c _l E " .. I _ j ,- ",- w a {'M .. . l j I.- 4- - l - - I- -I - -I - _ 4 - _w ¢ M ' _ _ II II II II II II - I - _- , "I - I I I I P " l _ _ II II II II

I ° _ _1_ _1= = _:_

E" , I N I = "

% ' _ - ; _ +_- i _

-_,1 == _ v + I =" - i - -- { _1 = I.l.I I I II ¢ _ _ C _ I -,- I -- / " _ . A _ . A _ - \_ _ + If II II II II II

_I = _ _I = _ _I# _I_ _I# _I= _

l ,|, _" 1 . 1 . 1 Z 1, -- Z _ Z I -- .. I _ usCs) -_½ (URCS)eULCS)), symmetric con t rol input C2.15) UA(S) _ _ (UR ( S)-UL(S)), asymm e tric c ontrol input (2.16) Also, ri g h t and lef t c on t rol c alibr a t ion error s ( _R and _L' respe c - tively) and the right and lef t m easuremen t calibration e rrors C _R and qL' respe c tively) a re def i ned such that: (UR) ac tu a l = (l+¢R)(UR) c ommanded or measured ( 2 . 17) (YR)measured = (l+qR) (yR)actual ( Z .18) a nd the left variables a re related similarly.

An exa m ple of the firs t case, t hat of exciting one wing and measuring one wing where no calibra t ion errors are pres e n t , is the _J following: YR(S) = Ys(S) 1 (s) (s)] (s) - Ue(S) ] (Z.19) = _ TSCs) [uR +uL [uR

YR(S)

•" _ " { [Ts(s)+TA(S) ] (2.20)

y ( s )

UL _ -{ [Ts(s) - TAt s )] (2.21) Next, c o nsider the last case, that o f exciting b o th wings either symmetrically or asymmetrically and measuring the responses of both wings in the presence of both control and measurement cal- ibration errors. For symmetric excitation: Y R = (I _ " m a = + TAU A YRa TSUSa a YRa = TS{½ [URm(l } (l+ZR)- (l 22) - " [URm ULm L a nd similarly f o r YLm and yLa . From the above, it is easily shown th a t and, s i milarly f o r asym m etric ex c ita t i o n, uA m T he results f o r th e r e m ainin g c as e s are found in a similar manner and presented in Table 2.2 Note tha t many cases follow directly from the above two equations when the calibration errors are set to zero as appropria t e. For example, the case of measuring both wings and exciting bo t h wings wi t h con t rol (bu t not measurement) calibra t ion errors follows simply by se tt ing oR = nL = 0 in the above equation.

T h e foll o w i ng observa t ions a r e m ade co n c erning these r esults: (1) Ex c iting o ne w i ng and m easuring o ne wing do e s not allow t h e s ep ara t ion of symm et ric and asymm e tri c responses.

| i L I

r I 1

I , I 1 i i (2) Exci t ing bo t h wings with calibration error and measuring I one wing means t ha t the measured response will consist of bo t h sy m metric and asy m metric modes when only one or the other was wan t ed. A similar case exis t s when one wing is exci t ed and both are measured in the presence of m e asuremen t e rr or.

( 5 ) Whe n b o t h wine s ar e exc i t e d and bot h m e a s ur e d and b ot h m e a su r em e n t and con t rol e rror s a r e pr es e n t, th e n again t h e unwan te d mod e s c o ntribu t e t o t he r e s pons e . H ow e v e r , t he f rac t ional a m oun t o f t hi s unwan te d c on t ribu t ion i s r e duc e d t o t h e ord e r o f t he sq uar e o f t he c alibra t io n e rro rs .

Ho w muc h the presen c e o f t h ese unw a nted m o des affect t h e es t i - m a t e of the damping f a ctors of individual modes a s compu t ed from t es t da t a will be the dis c ussed in C h a pter III.

2.4 MO D EL I NG OF GUST EFFE C TS I n general, t h ere a re two princip a l s o u r ces of rand o m effe c ts whi c h ca n degr a de the inform a t ion con t en t of da t a. These a re pro ce ss noise (e.g., gus t s) a n d m e asurement noise. For a well i nstr um ented a ir c r a f t , wi t h prefi l t e ring o n d a t a , t he pro c ess n oi se effe c ts a re of mos t c oncern. P a r t i c ul a rly for the w i nd t unnel t es t s at high sp e eds, t hese gust effe c t s ca n obs c ure essenti a l s t a bili ty information.

T h is study r e qu i r e d a method f o r emul a ting gust effects in bo t h th e fr e qu e ncy and th e tim e domain. Gus t spec t ra charact is- t ics of t h e 40- by 80-foot wind t unn e l ar e larg e ly unquantifi e d, and it is not cl e ar as to t h e r e lation b e t w ee n tunn e l rando m n e ss and corr e sponding flight gusts.

2O

I

.. The approach used is based on use of an atmospheric gust spectrum. The yon Karman spectrum is known to be one of the most accurate of iso t ropic atmospheric gust models, although such a spectrum is not convenien t for matching with linear filters (due to noninteger factors in the spec t rum). An approximating spec t ru m is the Dryden spec t rum, which is "close" to the yon Karman spectrum for low frequencies. The Dryden spectra used for longitudinal, lateral, and vertical gus t s are:

O uL

_u{W ) . __ Z

\ % !

% (w) -

\ % / J

O.w2 L I + 3 [wL I 2\uO /

%(w) = -- Uo

" F1 + { wL_21-_

L" k / J

wh ere L i s t h e co r re l at io n d i stan c e, u o th e w ind vel o ci t y, a nd L = _ 1 u-_ Tcorr _n {_n is t he ban dw idth of t h e n ois e ).

For t hi s w o rk, t h e b a ndw i dth of th e noise w a s est i mated at 2 Hz (wh ich corresponds t o a correl a t io n di st a n c e of 2 5 . 5 ft) f o r a ll t hree d irections. The v a ri a n c e of the la t er a l a nd ver t ical gust w a s c hosen to be a fixe d fr a c t ion of the longitu d in a l gust v a ri a n c e, but t his ra t io w a s c hosen c onserv a tivel y . This a llowe d par a meteri- z at ion of the noise-to-sign a l r a t io (see Appen d ix B.2. 3 ) as a fun c - t io n of inpu t ampl i t u d e on the longitu d in a l gust rms velo c i t y.

The resul t ing v a ri a n c e is

1 F I r '

i

:I

Clong a_ a t = 0 .172 2 0 3 4 3 avert where cT is the t ot a l rms g us t velo c i ty .

As dis c ussed in Se c tion B.2. 3 , these spec t r a and v a rian c es were u s ed w i t h the a ir c r a f t d y n a mi c response to r a n do m g usts to determine the power d ue to r a n d om gusts in a par t icul a r measuremen t .

Viewing the t hree gust di re c t ions a s _ c orrel a te d wi t h e ac h o t her, t heir power con t ribu t ions to a p a r t i c ul a r measuremen t c an be adde d i n an ms sense. Represent a t ions of t hese gust effec t s in t ime d o ma in s i mul a t ion were a pproxim a t e d using whi t e noise p a sse d through a f irs t order fil t er w i th a bre a k frequenc y a t 2 Hz.

2. 5 I NSTRUMENTATION MODELING Me a su r emen t s were t aken from the p o s i tion s t a t es excep t t h ose me a surements with which the XV-1 5 is now instrumente d --w i ng ver t ic a l ben d in g ac celer a t i on and wing chor d wise bending a cceler a t i on. The measuremen t s were cre a t e d from the appropri a t e line a r combin a t ions of the st a t es a n d c on t rols. The simul a t ion d i d not inclu d e a lag in me a suremen t (the str a in g a uge ac celerometers on the p y lon h a ve b a n d wi d th of 1 0 0 Hz (13.1 per rev), which is far above any mo d es of interest). Ini t i a lly , r a ndom measurement noise was a dde d to simulations, but i n gener a l the process noise ( t unnel gust) w a s the only noise cons id ere d , d ue to the expected severity of su c h t unnel d is t urb a nce.

The t ra nsfer functions a ll included a first o rd er l a g a t I00 Hz ( 13 . 1 per rev), but its effe c t is difficult to isolate since t h e prim a ry mo d es of i nterest l i e i n the re g ime 0 .i to i0 per rev.

i i !

t _w . , , . 2 . 6 CONT RO L MOD E LING T w o cases f o r t he o rder o f t he c o ntr o l sys t em actuat o r were ch o sen, a first o rder and a f o ur t h o rder lag, t o represent an o pti- mi s tic case and w o rst case f o r the steepness with which t he actuat o r gain falls o ff wi t h frequency. The break frequency wa s S Hz.

(This corr es ponds to a good quality con t rol se rvo.)

Th e fr e qu enc y re s p o ns e s di sc ussed in C hap te rs II an d I II a ll in c lud e a firs t ord e r lag a t 5 Hz. The f o ur t h o rd e r co n t r o l lag was a c hie v ed by applying th e f o ll o wing third ord e r fil te r to t h e transf e r fun ct i o n data: w 3 n . wi t h s - j_ .

(S + _n)( S Z + 2 ( 0. S )_n S+ _ _) (5 Hz = 0.6 5 4 per r e v). As will b e see n in Chap t e r s Ill and IV, t h e r e is li tt l e diff e r e nce in t h ese two repre se n t a t ions of th e act uator below 5 Hz, for ex a mple in s t udying th e qwl mode (sy mm e tric) mo t ion at 0.398 per rev). Of course, the four t h order lagged con- t rol would make the s t udy of high fr e quency modes (greater t han 1.5 per rev) prohibi t ive. Cyclic inputs were s t udied wi t h a first order con t rol lag ac 2 Hz and I0 Hz.

T i me d om ai n simul ati ons o f th e me a sured r e sponses w e r e made using different inputs (mul t iple sinusoids, random inputs, and swep t sine), all using a first order lag, primarily at 5 Hz band- width but a l so at i0 flz.

Scal e e rrors b et w ee n t h e command e d and a c t ual con t rol wer e mod e l e d as d e scrib e d in S e ction 2.3.

._d r III. ANALYSIS OF ROTOR / CANTILEVER WING DYNAMICS This se c tion presents the analy t ical evalua t ion of test con = siderations for ro t or c an t ilever wing modes. This evaluation is based on the frequen c y resp o nse c hara c t eris t i c s of the nine degree- of-freedom model, dis c ussed in Se c t ion 3.1. Se c t ion 3.2 reviews the wing m odes whi c h are of prin c ipal signifi c ance in defining s t abili t y t es t s at maximum tunnel speed of 190 kn o t s. Se c t ion 3. 3 m shows the inpu t c onsidera t ions whi c h are required to isola t e these prin c ipal modes, and Se c t ion 5 .4 dis c usses the effe c t of tunnel in- duced random dis t urban c es in the measured responses. A su m mary is presen t ed in Se c tion 3 .5.

3.1 FREQUENCY RESPONSE OF THE ROTOR / CANTILEVER WING MODEL As dis c ussed in Chap t er II, t he t ransfer fun c t ion u_ = H(sI-F)'IG+D can be used t o e valuate th e fr e quency response charac t eris t ics of th e sys t e m with dynamics ma t rix, F; m e asurem e n t dis t ribu t ion matrix, H; con t rol distribu t ion ma t rix, G; and direc t m e asur e d inpu t s, D.

Th e s t ability of t his transf e r function is compl e t ely describ e d b y the roo t s of th e charac t e ris t ic e quation of F. He asur e m e nt of t hes e s t abili t y charac t e ris t ics, how e v e r, dep e nds not only on t h e s e roots, but a lso on the roo t s o£ t he numera t or of t he t ransfer func t ion.

These numerator roots are governed no t only by the sys t em dynamics (F), but also the mesurements and the con t rols which d efine the t e st c onf igu ra t i on .

. . P A C ' EI INTE " ' -,. " : , "" 2 5 6.

I I '

I 5.1.1 Transfer Funct ions of Ro t or / Cantilever Wing Model The primary controls available on th e XV-15 are collec t ive p i t c h [80 ) and w i ng fl a peron (_f), at l e as t for the fully c onv e r t ed configuration. For r e ference, the frequency r e sponse to these inpu t s for all degrees of freedom of the basic ro t or / cantilever wing, are given in Appendix C.

3 .1.2 Evaluation Method of Tr ansfer Functions The t ransfer fun ct ions of Appendix C serve as a useful refer- e nc e d es crip t ion of the fr e qu e ncy r e spons e of th e nine degre e s-of- freedom to collective and flaperon inpu t s. In this sec t ion, we exa m ine the t o t al informa t ion con t en t of the t ransfer func t ions, assuming t ha t all states are measured and inpu t bandwidth is above the highes t frequency mode of t he sys t em. This "ideal" case demon- s t ra t es t he relative ranking of modal information independent of test input or instrumen tation limits . In Sections 3 .Z- 3 .4, we dis- cuss the effec t of t hese limi t s in m ore de t ail. The purpose of presen t ing these "ideal" t ransfer function characteristics is to formula t e a basis for discussing the desirability of more s t ringen t requirements on ins t rumen t ation and exci t a t ion hardware.

The b a sic ob je c t iv e of s ta bil it y test i ng i s the de t e rmin a tion of frequen c y a nd da mping of system responses. For a multiv a ri a ble ro t orcr a ft s ys te m, h oweve r , it is n e c ess ary to i sol a t e the fr e quency a n d da m pi ng o f th e e l e m e n ta l mo des wh ich pr od u c e t h at s y ste m r es pon s e . T h e pro b l e m is , how e v er , t h a t t h e system r e spons e (as ind ic a t e d by t he transf e r fun ct ions o f Appendix C ) represen t th e sum of c on t r i b u t i ons of all these el e ment a l mo d es a t a p a r t icul ar freq u en c y. In m a ny ca ses, t he elemen t al modes a re essentially un- c oup l ed, a nd a p a r tic ul ar pe a k c orrespon d s to a un i que degree of fr ee dom ( e .g . , wing v e rti ca l b e nding). For t h e rotor c r a f t, how e v e r, signifi c an t mod es ar e highly c oupl e d and m ea sur e m e n% of syst e m 2 6

..... I I I

frequency response may not provide the desired stability character- istics for a particular degree-of-freedom.. A corollary to this is that determination of syste m damping for a particular peak, itself a nontrivial calculation, may not define the stability of a particu- lar degree of freedom.

T he re a re s e v e ral an a lyt i c a l tec h n i qu e s wh i ch may be used to decompose syst e m response i nt o contr i but i ons from v a r i ous mod e s (such decompos i tion i s equ i valent to quant i f y ing the ident i f i ab i l i ty of a p a rt i cular d e gree of freedom). These techn i ques i nclude the follo wi ng : ( a ) M ode Shape Anal ys is : Co rre s ponding to a p a rt i cul a r n a tur- a l fr e qu e n c y of re s pon se i s a mode s h a p e a ss o c iat e d with th a t fr e qu e ncy. This mode s hap e , o r e igenv e ctor, may b e c a l c ulated. The eigenvector con s i s t s of a ve c tor of compon e n t s of th e e l e m e n t al d e gr ee s-of-fre e dom. Th e rela- t iv e size of thes e compon e nts quantifi e s th e participation of e ach d e gr ee of fre e dom at a natural fr e qu e ncy. U nfor- tunately, highly coupled modes yield eigenvectors which show several componen t s of nearly equal contribution, and i t i s not cl ea r h ow to isolate the mo s t s ignificant d e g r e e s-of-freedom.

(b) Residu e Analysis (Appendix B): Th e natural response of a system mode may be written as a sum of elemental modes. The terms of this expansion are of the form I n i e A i cos (wni / [T_i t +¢i ) w he r e A i is th e combin e d r e sidue of the mod e e igenv a lue, I • ;i _J_i (Ai = 21Ril' where R i is residue of li ). Modal con te nt c a n b e e stim a t e d by r a nkin g th e r e sidues of all mod e s at each frequency.

(c) N odal Power (Appendix B): The m o dal p o wer o f a par- t icular transfer function peak is the power con t ribu t ed by th e c omplex pole pair a ss o c ia t ed wi t h th e vibra t i o n (in a selec t ed frequenc y range about the peak).

J (d) Mo dal Power Ratio (_lPR) (Ap p endix B): T he ratio of a par t icul a r d e gre e s of fr e edom p o w e r to total power of a r e spons e . Th e MPR can b e positiv e or n e gativ e d e p e nding o n wh e t h e r a par t icular d e gr ee of fr ee dom is adding or sub t rac t ing pow e r fr o m a r e sonant p e ak. Th e sum of t h e MPR's f o r e ach mod e is unity.

rhe mo da l p o wer and mod a l pow e r r a tios a r e us e d t o quantify t h e fo l lowing: (I) Wh i c h tr a nsfer function is more des i r a ble to identify a particular mode?

(2) Which _ode is most significan t in a resonant peak when more than one mode is c o ntribu t ing?

The mo d al power r a tio does not include inf o rmation on the m a gnitude of the peak. (For example, t ransfer function B may show a resonant peak of mu ch less m a g n i t ude than transfer func t ion A, yet their MPR's are nearly equal.) Thus, f o r selecting which transfer func- tion is more desirable t o identify a par t icular mode, t he modal power should be us e d. This can b e done since: (I) the system ^f equa ti ons is norma li zed so t h e t r a nsfer func t i o n ev a lua t ions ar e in- de p e n de n t of t h e u nits, an d ( 2 ) th e sa m e in teg r at ion int e rval w as u sed on ea ch t r an sf e r fun c tion. W he n i nv es tig a ting on e p ea k o f a par ticula r tr a n s f e r f u n c tion to d e t e rmin e wh ic h mod e i s b e ing i d e n - tif i e d , e i ther the MPR or mo dal p o wer m a y b e used . W_en comparing th e uniq uene ss (m ea n i ng l ac k of o t h e r mod e s t ha t c o nt a minate) of Z8

li

differen t peaks of one t ransfer fun c t ion, the MPR mus t be used ' , : because the modal power _nay be compu t ed a t different frequency bandwidths for each peak.

3.1.3 Transfer Func tion Comparisons Table 3 . 1 is a qualitative comparison of the t ransfer func t ions to collezti v e and flaperon. Such a t able is useful for rapid refer- - ence to determine the impor t ance of measuring a par t icular degree- of-freedom or using a par t icular inpu t . For example, the t able indicates t hat th e qwl' qw2' p' and 8 mod e s can b e isola t ed (if , , sufficient control bandwid t h is a v ailable).

Table 3.2 is the quan titative evaluation of the t ransfer func- t ions, upon which Table 3 .1 is based, and summarizes t he modal power compu t a t ions for t he cantilever wing [symme t ric mo t ion) transfer func t ions. Transfer func t ions to all nine degrees-of-freedom exci t ed by collec t ive pi t ch and wing flaperon , as well as selec t ed c y clic t ransfer func t ions, are included. In the horizon t al direc t ion of the t able are the different t ransfer func t ions; in the ver t ical direc t ion of the t able are the differen t modes. As developed in Appendix B.2.2 and B.2. 3 , every mode con t ribu t es ei t her posi t ivel y or nega t ively by some grea t or small amoun t t o the to t al power of each peak in a par t icular t ransfer func t ion. Only con t ributions (in t erms of MPR) grea t er t han 5% are included in Table B.I.

Each box ssseciated wi t h a par t icular t ransfer function con t ains t hree values. The firs t is the frequency of the peak for which the modal power is compu t ed, and the following two values are the modal power and modal power ra t io (MPR). Each transfer function does not exhibi t peaks at each of the modal frequencies.

W_en a mode is not exci t ed in a par t icular t ransfer func t ion, the block for t ha t mode is used to indica t e t ha t mode's contribution i to the neares t adjacent peak, which is at the indica t ed frequency.

3 0 i q N ,C :* "_ _N _" '_J _ C ) ' , :: , C : , ,C =" ( :: . O C : i " "" i W :) _ r " i . , B l

-_-I_ " _ ........ _ _ = _° = " _ - _- " --

_ o " " = _, = = = , • • " . .... _ .....

•-_ _ ? .... = = 6 - = _" = =, " , = ' ,h 'J _ .,.4 ... ; _ ' ....

i "_ , , ,; , . . , _ • . • . . _ ....

r w _ • . m ,- _ ; w _- _ _ , . . _ , _ , ,_ _ . . .

¢. I " ,-4• 6 ,4 . _ ,.4. & - " : ..4 = . .. o ":. _ _ " : _ O R I GI NA L P _ G _ IS 3 . . OF POO R Q U A LITY . .

, I

I I

Thus, the table can be used in the following wa y s: • C o mparis o ns can be m a de h o rizont a lly of the modal power v a lues f o r a desired mode to choose the transfer function giving the bes t information for that mode. This compari- son can be m a de because the equations were normalized and the same frequency bandwidths were used.

• C o mparisons o f MPR can be m a de ve r t i c a lly t o d eter mine which peaks of a particular transfer func t ion arise mostly from the r e s o n a n c e o f a s ingle m o d e and are l eas t o b sc ured by o t her modes.

Figures 3 . 1 a a nd 3 . 1b are i llustrative of the manner in w hi ch : modal pcwer ratio is used t o evalua t e sys t em response. The wing chordwise acceleration to wing flaperon (_w2 / _f) transfer func t ion is shown in Figure C.2 (Appendix C). Four dis t inc t peaks are evi- dent. Figure 3.1a shows the modal power ratios for each of t . tese peaks for con t ribu t ors above 5% of the total power in the response.

(N ote t ha t o t h e r d e gre e s of fre edo m not shown a r e below 5% of th e t o t al power and sum wi t h the plot te d NPR's to uni t y.) The following Figur e 5 .1b shows th e effe c t of additional c ontrol sys t em lag b e yond 5 Hz and the alt e rations in MPR associat e d with t his lag.

T h e m o dal p o wer ratio h as even more ut i l i ty when considering t he coupled wing responses. This will be discussed in Chapter IV.

3.2 D I SCUSSION OF TR A NSFER FUN C T I ON CHARACTERISTICS The pr i mary m o des o f interest for XV-IS structural st a bility te s t ing a r e th o se a sso c i a ted wi t h th e wing. For the pu r p o ses of this study , these modes include the wing vertical bending {qwl), w i ng c h o rdw i se bending (qw2), a nd wing torsion (p) m o d es . Th e s e I

1 I

] mo d es are lightl y d a mped and, therefore, o f significan t i nterest I i for fligh t test predic t ion.

: l

I _I In addi t ion to the wing m o des, there is another lightly i damped mode whose response is impor t ant. This is t he upper inplane I I mode, _+I' although t his mode is at a high frequency. These four modes are summarized wi t h respect to frequenc y and damping in Table 3.3. The da t a in Table 3.3 is the "ideal" frequency and damping, based on an analysis of the characteristic equa t ion. Sta- bili t y parameters de t ermined from various t ransfer func t ions will, in ge n e ral, b e diff e r e nt , as dis c ussed b e low. I Of these four modes, further analysis was focused primarily on the qwl and qw2 modes. The p mode has the greatest damping of the four a t 0.0 5 8, an d the a n al ysis pe r fo r me d on th e qwl a nd qw2 modes T ab le 3. 3 P r incipal Symmetric Mo d es MOD E w D •L , , I 2.4 3 Per Re v _+I (18.6 Hz) 0.02990 1. 3 4 Per R ev p 0 .05 7 8 3 (10.3 Hz) 0.666 Per Rev q w2 0. 04258 (5 . 09 Hz) 0.$98 Per Rev qwl 0. 0479 8 (3.04 Hz)

j I

• can be easily ex t ended to the higher two modes when consideration of con t rol excita t ion at their frequencies is desired (p at 10.3 Hz _ and _+i at 18.6 Hz).

Observa t ions abou t the transfer functions are made here pri- marily to point out modes not available in the wing bending transfer functions from instrum e n t ation presently planned for the XV-I5 (_wl / _f and _wZ / %o are presently available). For example, it is difficult to obtain information on the upper inplane mode. Direc t measuremen t s of th e inplane mo t ion show a resonant peak in the fre- quency vicini t y of the upper inplane mode (_+i). However, the ro t or coning mode (8) is very close to this frequency. Exci t ation of the inplane degrees of freedom by collective gives a response wi t h magnitude I0 "I, but the collec t ive also excites t he 8 mode consid- erably (see Table 5 .2). Excitation of the inplane degrees of freedom by the flaperon does not excite the 8 mode significan t ly; however, the resonant peak has a mag n itude of only 10"2. Cyclic inputs would exci t e the _+I mode sufficien t ly without significant par t icipation of the 8 mode. Resonan t peaks wi t h good informa t ion on the _+I mode occur in the t ransfer func t ions of wing torsion (p) and rotor inplane mo t ion (8(I) and 8 (1)) to cyclic inputs (81C and 81S) IC IS Thus, by measuring wing t orsion, it would not be necessary to add s t rain gauges on the ro t or blades t o measure the _+i _.ode.

C o llec t ive pitch t o the rotor coning d e gree of freedom do e s not show sharply distinguished resonances for modal decomposition.

Excitation of this degree of freedom with wing flaperon does show resonance; however, the 8+1 mode corrupts the _ (see Table 3.2). Also, t his peak is rather small in magni t ude (10" 5 ), again indicating t ha t a ro t or blade measurement would not add t o the information available from present ins t rumen t ation.

t ! i I !

Gimbal measurements excited by flaperon (_f) gi v e no additional modes. Collec t ive pi t ch (90) does exci t e t he T ho_=v er, it is corrup t ed somewha t b y the B mode (see Table 3 . 2). The rotor azimuth perturba t ion t ransfer func t ions gi v e no additional modes.

Of the wing bending measuremen t s presen t ly instrumented on the XV-15, investigation of the cross transfer functions shows that _w2 / _f is qui t e attenuated, with peaks at a magnitude of 10 .3 or below. However, _wl / 8o is ac t ually more desirable than _iwl / _ f for iden t ifying the qwl mode, showing a greater magni t ude, greater modal power and also a greater modal power ra t io (Table 3.2). Both measuremen t s, qwl and qw2' will be recorded when either 80 or _f is exci t ed. Ob t aining t he cross transfer func t ions (_wl / 8o or _w2 / 6f) is merely a ma t ter of addition_l data processing. For iden- tifying t he damping of t he ql mode, t he estima t es from the _wl / _f and iw! / So tr a nsfer func t ions can be combined to give an improved estimate. The _wl / _f t ransfer func t ion can be used to identify the p mode. (There are o t her transfer func t ions where the p mode is excited more such as the gimbal angle, inplane bending, and torsion transfer func t ions_ but these require additional instrumentation.)

The _w2 / 8o t r ansfer func t ion can be used t o identify the 6 mode (given t h at the control sys t em could be excited at this high fre- quency: 19.5 Hz). As wi t h the p mode, the 8 mode could be identi- fied from other trar-fer func t ions by adding measuremen t s. The wing bending measurements exci t ed with cyclic yield no additional modes (_+I and 8 mode obscure each other).

Torsion measurements excited by either collective pitch or wing flaperon give no additional modes. The significant observation about a torsion measuremen t , however, is t hat the upper inplane mod e _+i ca n b e i de n ti f i e d w it h e x c i ta t ion by c y c l ic pit c h (if the control sys t em could be excited at this frequ e ncy: 18.6 Hz). A t orsion m easurem e n t could be ob t ain e d from lin e ar combina t ions of v ertically aligned a c celerometers on t he fore and aft ends of the

r ! I I I '

I p

p y lon. This ins t rumen t ation would be much easier than adding a s t rain gauge to the ro t or blades and passing the signal through a commu t a t or at the ro t or hub.

3 . 3 INPUT F R E QU E N C Y R E QUIREN E NT The overall s ys t em transfer fun c tion analysis of Se c tions 3 .1 and 3 .2 is now s pe c ialized to analy s i s of the wing t ransfer fun c - tion s . F rom t he results of Tables 3 .1 and 3 .2, wing response trans, - fer fun c t ions are isolated whi c h would give the be s t informa t ion about parti c ular degrees of freedom.

The da mping of m o des from p a r t icular transfer func t ions i s no w presented from t he frequency response data plo t ted in Figures C.I t o C.12. The method used is that suggested by Johnson in Ref. 6, Appendix E. A de t ailed derivation and explanation of the method is included in Appendix B of this repor t .

Table 3.4 shows the damping fac t ors calculated for each mode using this me t hod. The transfer function was selec t ed using the modal power as a criterion. The t able includes the effec t s of t wo differen t control lags--a first order and a four t h order lag-- both with a bandwid t h of 5 Hz. It is seen tha t , wi t h existing wing input channels, wide band inpu t frequencies can be used to isola t e coning (Be) , upper blade out-of-plane (8+I) and inplane (_+I) modes, and wing t orsion. Only wing vertical bending and chordwise bending can be excited with a one / rev limit on 8o and _f.

Note th a t the _ c ould be excited by a c y c l ic input ( a lthough at a h i gh input fr e quency).

The calculation of the dampings of Table 3.4 was achieved by integra t ing over t he frequency band 0.9 _n t o I.I _n' where _n is the peak frequency. Damping obtained by this method was found very sensitive to this frequency band. A wider band, 0.8 _n " 1.2 _n was found too wide, introducing other modes and degrading the damp- a.

J_ I 4 O i ng to S O _ o f t he t he or e t i ca _ ,_ue fo r t h e qwl an d qw2 mo des.

R e du c in g the fr e quen c y bandw1_'_ t o t h e half-p o w er valu e of 0 . 95 _n" 1.05 _n fur t h e r sh o w e d a d e gr ' ,_=d a cc ura c y.

The r e sp o ns e c on c lusions based on the tr a nsfer fun ct ions apply p r inc ipally t o i npu t a nd me a s u remen t f r equency r equirem e nts. . _pli- r ud e spe c ifi c ations for inpu t s and instrum e nta t ion a r e bas e d on ac t uat o r and s e nsor limi t s not fully availabl e at t his t im e , bu t e asily d ete rmin e d fr o m t he t ransf e r fun ct ions of Appendix C. A n imp o r t an t additional sour c e of in p u t r e quir e m e n t aris e s from t h e co nsid e ra t i o n o f t unnel induc e d t urbul e n ce , discussed in th e f o llow- in g Sec ti o n 5 .4.

3 . 4 THE EFF EC T OF NOISE IN S ELECTING INPUT _IPLITU D ES Havi n g sele ct ed th e tr ansfer func ti ons t o iden t ify the p r imary m o des o n a de t erminis t ic basis, we now address t he effec t o f ran do m disturbances in t he wind t unnel in fur t her specifying input ampli- t udes. There are two questions o f in t eres t here: (i) _a t i s t h e r e s pon s i ve n e ss (o r sus cept ibili t y to dis- t urban ce ) of a par t i c ular measurem e n t to rand o m gusts?

(2) Giv e n a p a rt icula r measur e m e n t e x ci t ed by a p a rt icular input, wha t inpu t magni t ude is required t o achie v e a desired signal- t o-noise ra t io (SNR), or, alternatel y , minimi:e a noise- t o-signal r a tio (NSR)?

T h ese q ue s t i o ns c an b e a n s w e r e d by c onsidering t h e mo d el of th e X V-1 5 dynami cs i !

= F x rv dy state response t o co ntrol e xci- D] u(j_) s e d t o pr e dic t th e magnitude and al exci t a t ion (a discrete fr e - xcitation in a frequency range of re Bod e plo t would b e the response in the range of th e Bode plot or e. Thus, w e can _ ;o nsid e r r andom ; and, sp e cifically, w e can consid e r of a peak of in t er e st.

.2. 3 ), tha t the n o i s e - t o-s ignal i can be written

(3.1)

in t h e fr e qu e ncy band pow e r from a sinus o idal signal in b a nd p ect ral densi t y nois e power spe c tr a l d e nsit y quar e valu e of the inp u t : ] [ _ i 1 I !

i I

[ I I

I i ; i i I _" The second term of this equa t ion is negligible compared to high speed wind tunnel turbulence, and may be neglected. (Note " that the effect of measurement noise can be simply added.)

Plots for the selected transfer functions of NSR, as a fanc- - tion of the rms inpu t ampli t ude, are plo tt ed as families of curves parameterized on the rms veloci t y of the gust in the wind tunnel and on the control actuator bandwid t h (Figures 3.2 through 3. 7 ).

The results are presen t ed f o r two cases. -" (i) First order lagged control with the Dryden wind gust model (see Section 2.4).

(2) Four t h order lagged control with the Dryden wind gust model.

Two general observations are eviden t in the NSR plo t s [Figures 3. 2 -3. 7 ). Firs t , t he Z Hz con t rol bandwidth case shows a greater de v ia t ion from the base case of S Hz than the I0 Hz control band- wid t h case. This is to be expec t ed since the S Hz break frequency is close to t he upper limi t of t he wing modes under consideration, while t he 2 Hz break frequency is comple t ely below t hem. Second, t he four t h order lagged co nt rol s y stem makes t he bre a k frequency more significan t , and indica t es a significan t increase in rms amplitude to achieve a desired NSR for a m o de abo v e the break fre- quency.

An example use o f t h ese charts is as f o llows. To achieve a I NSR between 0.I a nd 0.2 for the _wZ / eo transfer func t ion when t he expec t ed t otal rms wind t unnel gust is i0 ft / sec, the following ] c ol lect i ve p i tc h i npu t w o uld be required:

i

]

]

I )

I

' i L t+ T i l t R oto rat 1 9 0 k no t s F ir s tOrder Contr ol L ag at S H Z FirstOrder Measuremen t Lag at 100 Hz Dr y d en G us t Mo d e l J C o rr el ation Di s tan c e 2 5 .5ft Figure 5 . Z N oi se-t o -Sig na l R a tio-- Co llect i ve Pit c h to Wing Ch o r d- w i se A cc eler a t i o n (_w2 / eo); Can t ilever Wing Tr a nsfer Fun c t ion ( F or Frequen c y Range 4. Z 0 to 6.3 5 Hz) ,) I , T ilt Rotorat 190knot s __ Fou rthOr d e r Co n tr o lLagat 5 Hz FirstOrder _ea s urt_nent Lag at _00Hz _ . DrydenGu s tM o de l C or r elat io n D i stance 25 . 5ft I( 50 .\ \ I ......

\

_ --

\

1 \

'

N S R \ k

L '\

2 Hertz | 5 Her tz I \ • I il i_O IO0 e O rm s ,DEGRE E S Figure 3.3 Noi s e - to-Signal R at io -- C o llective Pi t ch to Wing Chord- wise Accelera t ion (_w2 / Bo); Cantilever Wing Transfer Function (For Frequency Range 4.20 to 6. 3 S Hz) 4S I I

\

\ •

I. \

\

\

. R \

b ,Z --

\

\ CO NTR O _ B_O= = ".- - --

J "

Fourth Order C ontrol La g at 5 Hz Ttlt Rotor a t 190 Kts I 2 H e r t z .....

Fi rst O rde rMea s ureme nt L a g a t 1 00 H z I 5 He r tz , , C or r e l at i on Ot stance 25.S f t • "yde. G ust M odel _ [10 Hert z - - , .0 1 ....... l .............

I I . 10 . 1 00.

6f , DEGREE S Fi g u r e 3. 4 No i se - t o- Sign a l Ra t i o --W i ng F l a pe r on t o W i ng Ver t i ca l A cc eler a t ion [_wl / _f); C a ntilever Wing Tr a nsf e r Func- tion (F o r Frequen c y R a nge 3. 0 4 t o 4.60 Hz) J I 0 I I -.,---_ Gus tV eloc ityin f t / sec BA N DWIDTH: T i l t R oto r a t 1 9 0k no t s _ I ICON T ROL F ir s t Ord e rC o ntr ol Lag at 5 Hz \ I 2 Hertz F ir s t Or de rMea s urement Lag at loe H z k I 5 Her t z Dryde n Gu s tM o de l 11 10 Hertz Co rre l ation D i s tan ce 25. 5 ft | . 0 1 , \ , .! l _ if, D EGREES l O 100 Figure 3. 5 N o ise-to-Sign a l R a ti o- -Wing Flaperon t o Wing Vertical A c celeration (_wl / _f); Cantilever Wing Transfer Fun c - tion ( F or Frequency Range 3.04 to 4.60 Hz) I

rl l ! '

I i T ilt Rotorat ! 90 knot s F irstOrder Co n tr ol L ag at 5 Hz F ir s t OrderMea s ur e me n t Lag i00HZ DrydenGust Model Correlation Di s tance 25.$ ft \ \ ,

\ , \ _

MS Gu s t Ve lo cityin , _ ft / se ¢

\ \

o \ ,

' \ \ \

\

_, \ , \

, \ \

. X- _ \

\ \

, )z H ertz -- .

5 H e rt z I I 0H e r tz -- I i i' " $ • 1 1 eO, OEGREES 10 100 Figure 3 . 6 Nois e - to -S i gnal R a ti o -- C oll e c t ive Pi t ch to Wing V e rti- cal Accelera t ion (_wl / 8o); Transfer Function for Asym- me t ric Mo t ion (For Frequency Range 3.33 t o 5.Z9 Hz)

1 r 1 ! F

, i

\ \

I- k 0----- RMSft / secG U S t Ve loc it y in

\

\

N S R • I CON T ROL BANDWIO T H: Ttl t R o t o r at 1 9 0 k no t s I Z H e rtz Fi r s tOrd e r C o n tr ol L a g at 5 H z F ir s t O rder Mea s u r ement L ag 10 0 Hz I S H e rtz Co rr ela tion O l s t 4nc e Z $ .5 ?t O_en _ s t _de I _I0H e rtz .01 , , !

•1 1 I 10 10 0 COrm s, D E G R EE S Figure 3.7 N o ise - to - Sign a l Ratio--C o llec t ive Pitch to Wing Ver t i - c a l Acceleration (_wl / eo); Tr a nsfer Function for Sym- metric Motion (For Frequency Range 2 .?7 to 3. 4 9 Hz)

1 ' I

CONTROL BANDWIDTH INPUT AMPLITUDE IN DEGREES - RMS (HERTZ) (FIRSTORDER LAG IN CONTROL) 2 5.80 - 8.10 5 2.97 - 4.20 10 2.31 - 3.30 , (From Figure 3.2) Th e NSR plo t s can b e used i n t he foll o w in g m an n e r (n e gl e ct i ng measuremen t noise): (i) A range of accep t able SNR's and, hence, NSR's, and a range of e xpec t ed rms longitudinal gus t should b e selec t ed.

(Z) T h e c o rresponding l i m i t in g upper and l o wer rms i nput amplitudes from the NSR plo t s give the required ampli- tude to achieve the desir e d SNR.

( 3 ) Th es e values s h oul d b e d etermined f or the f i rs t o rde r lagged con t rol mo d el and the four t h or d er lagged model (at the d esired break frequenc y ) to give a best and worst case (as d iscussed in Sec t ion 2.6).

3 . S SUMMARY This sec t ion h a s presented the results of a de t a iled study of a model of XV-I5 ro t or / can t ilever wing dynamics at 190 kno t s.

The objec t ives of t his stud y have been the following: (I) Calcula t ion of t ransfer functions from the two principal con t rols (collective and flaperon) to the nine de,tees- of-f re edom of t h e m o del.

SO /

! I

p d {

(

( 2 ) Ranking of the transfer func t ions wi t h respec t to the i r u t ili t y in isola t ing principal modes, and input frequency bandwidth requirements.

(3) Quantifica t ion of the effects of tunnel noise on required input amplitudes.

It is concluded that the principal limitation to mode isolation_ for the nine degree-of-freedom model is input bandwid t h. Instru- mentation of the wing is sufficien t to identify all but the [+i and lower (-I) rotor modes if collec t ive and flaperon input channels are used. To iden t ify t he _+i m ode, measurements of wing torsion excited by longitudinal cyclic would be effective. In all cases, however, input frequencies required to iden t ify the upper (+I) ro t or modes and wing torsion mode are above 1 per rev. The lower (-i) ro t or modes are not identifiable wi t h wing instrumentation because t hey are heavily damped and obscured by o t her modes.

Evaluations of the quality of transfer func t ions from this ni n e degree-of-freedom model are based on a control system band- wid t h of 5 Hz. A firs t order and fourth order lag were used with this bandwidth. The principal effect of increased control system rolloff beyond 5 Hz is increase in inpu t a,,_plitude required to achie v e an adequate noise-to-signal ra t io of output data. Inpu t amplitudes required to achieve a desired noise-to-signal ra t io for differen t levels of wind gust were determined for a control system bandwid t h of 2, 5, and i0 Hz.

I V 1 ! ,

I i I

, i I

I t . IV. COUPLED WING DYNAMICS This sec t ion presen t s the analy t ical evaluation of the coupled wing dynamics. Sec t ion 4.1 discusses the frequency response for the measurements presently available [both wing bending accelera- t ions) exci t ed by collec t ive pi t ch and wing flaperon. Sec t ion 4.2 suntmarizes the modeling of calibration errors in measurement and - exci t a t ion and discusses the results. Sec t ion 4.3 presen t s the effec t of t unnel-induced random disturbances and the inpu t ampli- t udes and frequency ranges which are required to iden t ify principal modes of in t eres t with par t icular noise- t o-signal ra t ios.

4.1 FREQUENCY RESPONSE OF THE COUPLED WING MODEL 4.1.1 Computation of Coupled Wing Responses The d i scussion of the c o upled wing m o del will be limi t ed t o the _wl / _f and _w2 / So transfer func t ions for identifica t ion of th e following modes: qlS: symmetric ver t i c a l wing bending m o d e qlA: asymme t ric ver t ical wing bending mode q2s: symme t ric chordwise bending mode q2A: asymmetric chordwise bending mode As discuss e d in Section 2.3, th e r e spons e of th e right wing by e xcita t ion of th e right wing has a symmetric and an asymm e tric por- ti o n. Exci t a t ion of a wing, and m e asur e m e nt of that wing, is no t suffici e nt t o s e parat e t he symm e tric and asymm e tric coupl e d wing mod e s if th e y ar e "cl o s e in fre._ , " As discussed in S e c t ion Z.3 and again in S e ction 4,2, addi t ional instrum e nta t ion mak e s it "" possible t o separate t he symme t ric and asymmetric frequency F : ' .."_I ' ,t TEN T IO N A' . LY _, LANK !

r e s ponse s , e xc e pt for s ome fraction of th e o ppo s it e m otion (asym- m etr ic or s ym m e tric) du e t o ca libr a ti o n errors.

T h e nine de gr e e - of - fr e e do m rot o r / c an t i l eve r w i ng mo de l c a n be us ed to a n a lyz e th e couple d w i ng ( e .g., r i ght a n d left w i ng couple d m o de s} i n th e f ollo wi ng mann e r: (I) compute t h e sym m e t ric t r a nsfer func t i on, Ts(J_], from - the l ine a r equ a t ions repres e nting the symmetri c mot i on; ( 2) c o mput e th e a symm e t r ic t ra nsf er f u nct i on, TA(J_), s i mi- la rly , an d (3) co mb i n e t he s e t rans f er fun c t i o n s a t d i s c r e t e v a l ues o f t he fre q uenc y, _, by th e f o r mu l a T C j_) " ½ [Ts C j c _) +'TA C it)] • Be c a us e th is n e w tra nsf er f un ct i on T now ha s twic e a s m a ny mo de s i n th e s a m e fr e qu e n c y r a n ge , mor e pronoun c e d mo d al couplin g o cc ur s i n a sma l l f re qu e n c y ra ng e and t he ph a s e m a y a pp e ar to b e d is c on t inuous (App e nd i x B. 2 .1}. However, if a suffi c iently small fre qu e n c y i nc re m en t w ere u sed, it woul d s h ow t he p ha se t o b e c on- tinu o us , sin ce both tr a nsf e r functions a r e a n a ly t i c .

C l e arly, t he coupl e dw i ng m o t i on w i ll e xh i b i t cha r act eri st i cs of b o th th e symm e tr ic a nd a symm e tric motions. Th e charact e ristics o f th e symm e tric t ransf e r functions w er e discuss e d i n som e d e t a il i n the pr e vious ch a pt e r. B e for e continuing th e discussi o n of t he m c o upled wing motion, th e c ha r a ct e r i st ic s of a symm e t r ic moti o n w ill be p re s e n te d b rie fly.

5 4 4.1.2 As ymmetric M o de Characteristics The principal asy mmetric mo d es a re li s ted in T a b le 4.1. The a symme t ric frequency responses for the w i ng bend i ng acc elera t i o ns ar e shown in Appendix C (Figur e s C.2 5 to C .28).

T h e l i ne a r m od el f o r t he a symmetric mo t i o n d iff e rs from the sy m metric m o t ions in tha t the coupling due to the driv e shaft be t ween t h e two r o t ors i n t ro d u c es an o s c ill a t ory mode a ss oc i a t ed with the rotor speed perturb a tion. The wing , tiffnesses, associated w i t h th e two a sy m me t ric w i ng b e n d ing de gre e s of freedom, were i n - cr e a se d to sh i f t t h e qlA a n d q2A m o d es up i n frequen c y from the qlS a n d q2s m o d es i n or d er to d istinguish the m . The q2A mo d e w a s s hi fte d up such t h a t i t w a s a lmost coi n c i d en t with the _oA m od e (t h e r o t or speed perturb a tion mo d e). Th i s w a s c ons id ered o ne t ype o£ wors t ca se f o r ana l y z i ng the effec t of t h e a symm e tric m o d e on me a sure d frequen c y a nd d ampin g .

T a ble 4.1 Pri n cipal Asymme t ric Mod e s DA M PED DAMP ING F RE QUENCY FAC T OR MO D E _D . . , . ,, 2.44 P e r R e v 0.0297 8 ; * i 1 8.6 H z 1.42 P e r R e v p 0 .0 5 3 8 2 I 0 .8 Hz , , . m • 0.7 2 7 P er R e v q w 2 0. 0 2124 5.55 H z ..... , ,, , -- 0. 56 2 P e r R e v 0. 0554 3 "_ qwl 4. 2 9 Hz ' ' ' 1 [ 1 ' 1 r 4.1. 5 Coupled W ing M ode Characteristics !

T h e t ra n sfe r f unct i ons f or the coup le d wi n g re s po ns e s are c al cu- ' late d f ro m the plo t s p re s en ted in Appendix C . Fo r t h e t w o p ri ncipal m o des of mos t inte r est, qwl an d qwz' t h e qlS a nd qlA mo d es ar e prac- t i c a lly unobsc ur e d and th e q2s a nd q Z A mo d es are obsc u re d to a _o d e r - a te a n d gr ea t er ex t en t , r es p ectivel y . T h is ill u st ra tes bot h ex- tre m es w hich can occ ur in the a c t u a l d y nami c s o f t he XV-IS.

° T he _w 2 / eo t r a nsfe r fu n ct i on sh o ws a pro mi n ent r es o n a n t p e ak f or q2 s ( at 0.66 per re v), bu t t h e q2 A m o de (at 0 . 72 p e r r e v ) sh o ws n o resona n t p e ak at all. I t is obs c u r e d by th e q 2 s a n d { o A modes . Th e q2 s p e ak i t self is obs cur e d somew ha t b y the _ oA m od e (a t 0 . 7 3 pe r re v ) , as i s s h ow n b y t he mo dal powe r ra t i o ( N P R ) in Fi g ur e 4.1. Th e symm e t r i c a n d a symm e t r i c fl a p p ing mo de s B S a n d B A are ver y c lo s e in f req u e nc y (S S = 2 . 5 5 a n d 8A = 2 .61 per r e v) , a nd a l t hou g h th e Z a re not p rinci pa l mo de s, it i s int e restin g t o no te t h e ir eq u a l c on t ribu t ions t o th e s ec ond r e son a n t R ea k on t | _w2 / 8o tr a nsf e r fun ct ion from t h e MPR shown in Fi g ur e 4.1. B eca us e th e s y m m e tri c a nd a sym m et ri c 8 mo de s re s u l t in a sin g l e r e sonant pea k, t h e s e mod e s ca nno t b e dis t in g uish e d in this t r a nsf er fun ct ion, wh ich illus t r at es t he v a lue o f b e ing a ble t o se p ar at e the symm e t r i c a n d a s ym metric mo t ions.

The _wl / 6f tr an sf e r fu n c ti o n sh ows di s ti nc t r es o n a n t p e aks f o r th e qlS a nd q lA mo d e s a nd o n e r e sonan t p ea k wh e r e t h e tw o t o r s i on mod e s o c cur (P s a % 1 .5 4 a n d P A at 1.41 7 per r ev). Th e MPR's f or th is tra nsf e r fun ct ion, wh ic h a re shown in Fi g ure 4. 2 , i nd i cat e t h at ql S is un obscu r ed an d t h a t q lA is sligh t ly obscu red b y qlS" The to rsiou mo d es o bs c u r e ea c h ot he r in t he t hir d peak as ex p e c te d . The m o dal po we r ra t io fo r co lle ct ive p i tc h t o wing ve rt i c al bending a cc ele r ation (_wl / e o ) wi t h c ou p le d wing mo t ion is sh o wn in Fi g ure 4.3.

At th is po in t , it mus t b e co nclu de d that th e di £ ficulties with t h e cou p led wing mo de l a r e as follows: 5 6

i I

i I ,' PEAK I PEAK 2 .755 mpeak " 0.661 mpeak = 2.51 t ' I I I I I I I I I I qw2 go qw2 go go (S) (A) (A) (S) (A) F i gure 4.1 M oda l Power R a tio (MPR)--Collective Pi t ch to Wing Chordwise Acceleration (_w2 / 8o); Coupled _ng Motion PEAK i PEAK 2 1.00 u mpeak = 0.398 .858 _peak = 0.575 I Hpeak I = 0.016 . 0 9 _ IHpeakJ = 0.0098 r .--'_ I I !

I I I I I I I i I q wl q wl q wl (S) (S) ( A) Tilt rotor at 190 knots I .550 PEA__._.KK3 5 Hz WPeak = . L .44 First Order Lag in Measurement = 0.0121 at 100 Hz First Order Lag in Control at I __'394_ IHpeak I ' I I I I I I P P (S) (A) F igure 4 _ ..odal Power Ratio (NPR)--Win_ Flaperon to W : n_ Vertical B e nding Acceleration (Ciwl / Sf); Cou_le / '.%'ing _,Iotion S T

P II ! I

ij !

' PEAK I 1.04 PEAK2 : .84 _peak = 0. 3 98 _ _ = 0 5 7 5 ' _ -- - p e a K " . 5 3 IHpeak I= 0.0232 IHpeakl= 0.0636 -_ , I ii I i " "" I - . 19 1 - .0 8 l, _ I t I I I -- a. I I I _I t o q wl _ o q w l - ( S ) (A)(A) (A) (A) R PEAK3 . 7 9 PEAK 4 _ , • 1" 37 !' 3 7 _ IH p e a k ! 0 .00 68 [ H pe ak I = 0. 0 0 49 .. '" I | I I "" I.. I I - - i l l : I J I I , B o S O -- I -. 63 (S ) (A) . ; I i I I I . : P P Z ;o qwl (S ) (A) (A) (A ) T ilt R o t o rat 19 0 knot s - ; F ir st OrderLag in C o ntr o lat 5 Hz F ir s tOrderLag in Measurement at 100 Hz Figure 4.3 Mo d al Po w er R at io (M P R )-- _ ? _ nE Ver t i c a l Be n din g A c c e l er - a t i on to Collect i ve P i tc h (qwl / eo); Couple d W in g M ot i on

r 1 I ; L L

,. 4 • - (I) T her e ar e so many modes in th e same frequency region that it is diffi c ult to deter m ine whether a particular resonant peak is d ue to sy_etric or asymmetri c motion (i . e., d ue to the excitation of the wing which is being measure d , or the reaction for c e of the res t of the ai r - c raft on the wing being measure d ) .

(2) Eve n more im portantly, g i ven a part i cular res o nant peak on the frequen c y response, one does not know how man y , whi c h one, or how mu c h o t her modes a re contributing to th a t peak. The model e d d ynamics used in t his a nalysis show ex a mples where some prin c ipal mo d es a re obs c ured a nd o t hers a re not. The measure d d y namics of the XV-1 5 c ould be be tt er or worse, bu t the signifi ca nt point is t h a t one will not be a ble to d e t ermine t his unless the symme t r ic and asy m metr i c mo t ions a re sep a r a t e d . It is impor t an t to know w h ether a principal mode is obscured or not a nd to know how much. The d a mping c al c ula t ed from a reson a n t pe a k coul d show the XV-IS t o be qu i t e s t a ble, when in f ac t t here a re two or m ore modes c on t rib- u t ing to this pe a k, one of whi c h is safel y s t a ble, but a no t her of whi c h h a s ver y ligh t d amp i ng tha t c ould be c ome uns t a ble in some flight c ondi t ions.

4.2 HVALUATION OF CALIBRAT I ON ERRORS " " The ap p r o ac h f o r m od eling ca libr a t ion err or s f o r v a r i ous wi l l tunnel t es t c onfi g ur a t i ons (e. g ., ex c i t e one wine an d me a sure one wi nE , or ex ci te one wine a nd me a sure both wines , etc.) is d evelope d in Se c t i on 2 .3. I n t his se c t ion, three tes t c onf ig ur a tions a re c onsid e r ed in de t a il to a sses s the e ffe c t o f unwant e d m o d es o n the dampin g estimates ' of d e sired modes. The three cases are: (I) the p r esent test c onfigu ra t i o n (exc i t e o ne wing a n d me a sure that wing); S9 (2) A n i mpr ov e d test c o nf i g u rat io n wh i c h w i ll separate sy m - metr i c a n d as y m met ri c m o t i on (exc i te o ne w i ng and measure b o th w in gs); and ( 3 ) th e mos t a ccu ra t e t es t c on f i g u r a t io n f o r sep a rat in g sym - I me t r i c a n d asummetric mo t i on (ex c i t e both wings and measure bo t h w i ngs).

_e Fo r each c on fi g ura t i o n , t he ac t ual en v i ronm en t i s c on s i dered; that -r is, ca lib ra t ion erro r s i n bo t h m ea surem e n t s a nd con t rols are modele d .

F i rs t c o nsi d e r the effec t of c a lib r a tio n e r rors i n bo th "" m ea s u re m en t and c on t rol of th e pr e se nt t es t co nf i gurat i on. Th e n o t a t ion of Se c t ion Z. 3 i s u s ed h e re except t ha t fun c t ional _- dep e ndence on th e L a pl a c e c omplex v a ri a ble s is n o t shown exp lici t l y . The m e a sure d respo n s e o f the ri g h t w i ng i s . .

- (i R) . .

YRm YR a where n R is the me a suremen t ca lib ra t ion error a n d the a c t ual r esponse of the right w i ng is / = T s • YR a Us a + TAUAa Thus,

- ]

Y R m (i + n R ) [ T SUS a YRa ' UL m • Ts _ Rm ¢ 6O

! r I

whe r e CR and eL are the con t rol calibra t ion e r rors.

• i YR _ = ½ (1 R)(1 R)[T s , TAI # - [I + cR + nR + nR ¢R ] ½ [Ts + TA] T hu s, calibrat i on errors i n bot h exc i tation an d measure m ent for the present configura t ion have the effec t of scaling the measured transfer function by a gain factor, which does not affec t the determina t ion of t he resonant peak frequencies. "This gain fac t or also do e s no t affect the damping calculation becaus e the square of the gain enters in bo t h t he numera t or and denomina t or and, there- fore, cancels out.

T h u s , for t h e p resent wind t unnel te s t c o nf i gur a ti o n, c a l i br a - t i o n errors have a negligible effec t i n t h e c al c ula t i o n of t h e f requen c y an d damping associ a t e d with a p ar t icular resonan t peak on t he frequency response. However, the d i f fi c ulty wi t h t he presen t "" tes t configura t ion, as discussed in the previous sec t ion, is t ha t "- th e s y mm e tric and a symm etr i c m o d e s obscure each o t h er, m a k i ng t h e -- d e t ermination of th e damping of principal modes for assessing the -- d y na mi c s t a bili t y of the XV- 15 i n effe c t ive.

Nm .. Th er e fore, i t i s ve r y d esir a ble to sep a rate t he sy m m e t ric a nd .. asymmetri c motion to determine wi t h confidence which modes are .. pr e sen t in t he r e sonan t peaks of the frequency r es pons e . This can : : be acco m p l i she d i n t hree w a ys, as dis c ussed in Section 2.3: • ° ( 1 ) M e as u re b oth w i ngs e xcit ing on e w i ng.

( 2 ) M ea s u r e on e w in g exci t i ng both w i ngs.

( 5 ) M eas ure bo t h w i ngs e x ci _ing b o th w i ngs.

!

t It will be shown later that the t hird alternative, al t hough it in- volves more tes t equipmen t and, hence, more cos t , ac t ually reduces the effec t of calibration errors t o a negligible level. The firs t alternative will be considered here since it is the easiest to im- , L plement.

Ag ai n, us i ng t h e not a t i on of S e ction 2 . 3, _ 4 B YSm " ½(YRm +yLm) = ½ [(l+nR)YRa + (I+nL)YLa] " C o n s id er in g t he c a s e w he r e t he co n t r o l in put i s t o t he r i gh t wing °.

on l y ( i . e. , u L - 0 ) ..

U n der t h e ass u mp t i on t h at p r o du c t s o f error s may b e n e gl e ct ed , "_' URm • - Li k e w ise, Y Am = ll / n R'nL_ nR I

( , .,. '

Sin ce a g ai n consta n t t i mes the m e as ur e d t ransfer funct i o n d oe s not aff ec t t he ab i l i ty t o i d en t if y th e fre q uenc y a nd d am pin g o f p r in- cipa l mod e s, Eqs. (4.1) and ( 4.2 ) can b e w r itten a s : 6Z

i 1 1

m_ • = - _ = {[I+ {CnR+nL+2_R)] TS+ 2+nR+nL+ZCR i URm I YAm- {[i+ { C nR+nL + 2_R)]I_ 2 nR nL + 2_R ] Ts + TA 1 URm Le t nR'n L 2 +nR+nL+2¢ R (4.5) b e ca lled th e " d ist o rtion f a ct o r" b ec aus e it is this fa c tor tim e s the asymm e tri c transf e r fun c ti o n in Eq. ( 4 .3) which "distorts" th e desir e d symm e tric transf e r function. S imilarly, this same factor tim e s TS distorts TA in Eq. (4.4). That is, YSm -- _ T S + A T A ( 4. 6) .. U R m . o Y A m - -- cc TA + A TS (4. 7 ) . . U R m . .

.. For v a lu e s o f t he mea su rement a n d c o n tro l sc a ling f a ct or s be twee n -0.20 a nd + 0 . 20 , th e disto r tio n f a cto r v a r ies in the ran ge - 0.25 < _ < + 0.25 Clearly, the v al u e o f t h e dist o rtion fa c tor is m o st str o ngly d e p e n den t upon th e diff eren c e o f t he c a libra t ion er r or s in th e l _' ri g h t a nd lef t w i ng r es p ons e m easuremen t s , q R- n L, and less up o n t he righ t con t rol calibration e r ror, _R" Figur e 4.4 shows how A v a ries a s a fun c t i on of th e r i gh t and l e f t me a sure m en t c al i br a t io n err o rs for dis c re t e v a lu e s of the c on t rol ca libr a tion error.

To s t udy t h e effe c t o f c a l i br atio n errors o n the d a mping esti- "- m a t es, t h e qw l mo d e wa s c hos e n s i n c e bo t h the q wl symm e t ric and qwl a symm e tri c mod e s a re not obscured s i gn i f ican t l y by o t h e r m odes. i T h e symmetric or a symm e t r i c t r an sfer fu nctio n w a s d egr a de d b y ad d i ng _ or s u btra ctin g th e o t h er tr an s f e r f u n c t ion w i t h varying lev el s of T dis t or t io n fa c tor. The d a mpings c al c ula t ed for t hese var i ous levels of dis t or t ion fa c t or are plo tt e d in Figure 4. 5 . As c an be seen, ca libr a tio n errors do not have a si g nificant effec t on the cal c u- l a ted values of the q l S an d qlA modal dampin g . The error in the c alc ul a t e d va lue is pr ima r i l y d ue t o t he i n fl uen c e of o t he r mo de s whi c h a r e pr e s e n t e v e n wh e n th e c a libr a t io n errors a r e zero.

Th e r e a r e two import a nt consider a t ions to b e k e p t in mind.con- "" c erni n g t h e effe c t of c ali br a t i on e rr or o n t h e cal c ula t e d da mp in g - - values. First, the c a lc u la t e d d am p ing valu e s pr e sen t e d here a re . - e m piri ca l i n the sense t ha t th e c ompl e t e a lgori t hm for accura t ely d eter m ining _ is a sub je c t for fur t her s t u d y. N a t urally, t h e r e ar e . .

s e veral d ata p rocessing consid e ra t ions affe c t ing the a c cur ac y a t (see Appen di x B for a d is cu ssio n of the me t ho d us ed to cal c ula t e damping). Th e s e c o nsi d era t ions inclu d e th e d ist a nc e between t h e fr e q u en c y r e spons e data poin t s ( a fun ct ion of the da t a r e cor d l e ngth), the siz e of th e b a n d wid t h abou t th e pe ak use d in th e ca l- c ulations, a nd w i t h a c t u a l d ata, th e sign a l- t o-nois e r a t io of e ach dat a rec o rd and th e number of d a t a r e co rd s a veraged. I t w a s found t ha t t h e d a m p i ng value (_) w a s som e wh at sensi t iv e to the siz e of th e b and w id t h c h o se n a bou t t h e p e a k. Too w id e a b and w id t h incl u ded too much influenc e fr o m o t her m o d es and te nd e d to und e r e s ti m a te _; too narr o w a ba ndwi t h di d n ot con t ain s u ff ici e n t i n f orma t ion a n d al so und e r es t ima t e d _. B et te r damping valu e s could be produc e d b y i i 1 [ I I , i i : ,"L

/

. _ _ • Z t_ i i ( c) cR " *O .Z Figu re 4 . 4 Dis t ortion F actor for Various Values of C ontrol C a libra- t ion Error I t -- 4_ m,., • ,. IU C A L CU LATE D VA L UESFOR qls MODE 0 AC T UA L VA L UEFOR qIS MOD E ---- CALCU L ATED VALUESFOR q I A MODE ACTUALVALU E FOR q i A MODE i| i im • o

.o 6 o

Z , . 0 55 .,... ...... _ - - TA+ AT S ,=, =,= , = w _ ,mm , _ ,=, , ===, , =m=== _ , _= _ _ . 050 - .

• o -.0 45 T S + A T A -. 040 i l i l _ • • i .m | I I i I -- I I I I I -. 25 - . 2 0 -.15 -.10 -.0 5 0 . 05 .10 .15 . 2 0 . 25 DI ST OR T ION F A CT OR, F igur e 4. S Th e Influ e nc e of C al ibr a ti o n E rror s on th e C a lcul a t ed V alues of D a m pin g f o r th e Mode f r om t he _wl / O o Tr a nsfer Function qwl

I

T , calculating the transfer function wi t h smaller frequency increments, ._ and varying the bandwidth used for each peak (within what is known to be a reasonable range from the plot of the frequency response) v e to achieve an accurate value for _. The purpose of resul t s pre- sented here is to provide sufficien t information to evalua t e vari- • ) ous tes t instrumentation configurations.

A s e cond probl e m in analyzing th e eff e c t of calibration e rrors on calculated damping values is knowing what mode is actually being measured from a particular resonant peak. The qw2 asymmetric (q2A) peak gives an excellent example of what happens to the damping calculation when a mode is being obscured by other modes. The q2A mode (at 0.72 per rev) is obscured somewhat by the q2s mode (at 0.66 per rev) and significantly by the rotor speed perturbation, which is an asymmetric mode (at 0.73 per rev). It is not valuable to plot the damping of the q2A mode as a function of calibration errors since it is obscured by other modes. However, the calculated values for the damping for different levels of distortion are given in Table 4.2 for reference.

Table 4 . Z Calculated Values of Damping for q2A Node from _w2 / 8o Transfer Function D ISTORT ION CALCULATED FACTOR '" 0.0 0.0429 "" 0.25 0.0439 •- 1.0 0.0426 ; Actual Value of _-0.0212 • m, _o e_ . e 6 7 L i

1 [ T 1

Th e e ff ec t of c alib r a t i on err o r s o n t h e m o s t complet e t es t i n - s trument a t i on con fig u rati on will no w b e dis cu ss ed. M e as ur i n g b oth wi n gs w h il e exc i t i n g b oth wi n gs wi t h c alib r ati on error s on b o t h m e as ure m ent a nd contro l will b e c all ed the f u ll te s t i n s tru m ent a t i on "" co nfigu r a t i o n .

J I t foll o ws fro m th e e xp re s s i o n fo r the theo r e t i c al transf er _.

func t ion in the abo v e c as e in Tabl e 2.3.1 that "T, s 4 " Y Sm k T S ( 2+ ¢R + CL) ( 2 +rlR A { (eR'CL) (nR "qL) 1 USm i Y A m i (¢R ' _L)(nR ' nL)( 2 +ER I I U-_m" k T S A

!

wh ere k = 1 + ½ (nR+nL+2¢ R) - The distor t ion fa c tor h e re be c om e s -- eu (¢R'¢L) (nR - nL) - - A - Consid er ing t ha t ea c h c alib r a t ion is 20% and t hat t hey a re combined in the wo rs t c ase, t he magni t ude of A is bounded a t 4%. For t he pre v ious te s t configura t ion {measure bo t h wings, e xci t e one wing) the magni t ude of A was bound e d at 25 %. Th u s, b y e x citing bo t h wings, e ven t hough a n a dditional sour c e of c alib rat ion e r ro r is introduced, the maximum dis t or t ion fa c tor is redu c ed b y a fac t or of six. B a sed on t he variation observe d in the damping as a func - t ion of t he dis t or t ion fac t or in t h e previous discussion, it is c on c lud e d th a t th e e ff e ct of c alibr a tion e rror on damping d e t e rmin- 6 8 I ": _ . a t i o ns is negligible using t his f u ll t e st ins trume n t a t i o n c o nfig- u r a ti on.

4.3 REQUIRED INPUT AMPLITUDES TO ACHIEVE A DESIRED SIGNAL-TO-NOISE RATIO FOR THE COUPLED WIN G .

The inpu t ampli t ud e necessa r y t o achieve a desi r ed noise- t o- signal ra t io will be discussed using the ver t ical wing bending sym- me t ric (qls) and asymme t ric (qlA) m o des as examples. I t is assumed t ha t t he symmetric and asymmetric m ot i o ns are no t dec o uple d ; t ha t is, t he transfer fun ct ions f r om which t hese resul t s we r e ob ta in e d are t he sum of the symme t ric and a symme t ri c t ransfer func t ions.

This simula t es t he case when only one wing is exci t ed and only one wing measured.

Th e n o ise- t o-signal (NSR) rat io c al c ul at i o n f or a p a r t icul a r "" m o de is based upon the frequenc y r e sponse t o t he desired control "- and th e f re quen c y r espons e to gusts in a f r equency ban d ab out t h e -- resonant peak for that mode. This calculation is described in - Appendix B. For example, assume information on the qlS mode is , s o ugh t f r om t he frequen c y r e s p ons e o f t he v er t ical ben d ing accele r a- t ion t o flaperon input (i.e., t he _wl / _f t ransfer func t ion). The bandwidth ab o u t t he qlS resonant peak fo r which t he NSR calcula- t ions apply is 2.5 3 t o 3.8 3 Hz (0.335 t o 0.501 per r ev).

i

The r e s ul t i s s h o wn i n F i g u re 4 . 6 . Fro m t h i s figu r e, i t c a n be seen, f or e xample, t ha t , if t h e root- mean - squa r e g us t velocity is 1 f t / s ec, a n o is e -to-s ig n al ra ti o o f 0.i w o uld r equi r e 3 . 6 d e- "" g r e e s r ms fl a peron inp u t. As a n ot her exa mple, f or a g u st v e l o ci t y -- of I 0 ft / sec rm s , t he s ame fl a p e ron in p u t re su l ts in a no i s e- to - .. sign a l rati o o f i 0 . ( T h e se e xamples are for a first or d er co nt ro l . . lag wi t h a break fr e q u e nc y a t S Hz).

: : I t I Fi g ur e 4. 6 No i se- t o- Sig n al R a ti o for Symmetric V ertical %Vi ng B en d- ing (qlS) M o d e from _wl / _f T ra ns f e r F u nct i on (F or F r e - qu e ncy R an ge Z .5 3 to 3. 8 3 H_ Ab o ut the q l s Peak) 7O

f T

l ' I p ?

[

]. N e x t , a ssum e information on th e qlA mod e is sought from th e sam e _wl / 6 f transfer function. Th e refore, the calculations ar e ._ based on a bandwidth about the qlA resonant peak (3.83 to 4.82 Hz).

The r e sults are shown in Figure 4.7.

A c o mparison of Figures 4.6 and 4.7 shows that the same rms flaperon input in the presence of the same rms gust velocity would mean a higher (i e., worse) noise-to-signal ratio (NSR) for th e qlA mode than for the qls mode. This was to be expect e d b e cause the qlA resonant peak has a lower amplitude than the qlS resonant peak.

Large input._ are required to lift the signal above the noise.

. ) A more favorable situation e: _'sts when information on the ql_ mode "s sougL- from the response of _.., to collective pitc]; inputs b e cau_ , e the resonant peak of the qwl mode has a higher amplitu,_e in the _wl / 8o transfer function. Th e NSR re_u!ts for this case, shown in Figure 4.8, are ccnsiderably improved over the result_ in •" I Figure 4.7, which were based on the qwl / Sf transfer function.

4.4 SU_IARY The test configuratlon of exciting one wing and measuring •. that wing does not allow separation of the symmetric and asymmetric modal response s . It follow_ t ' _ determination of frequer_v and .. damping of particular modes is difficult, requiring very special- ized input designs or more complicated instrumentation.

If b ot h wi n gs are excited and / or the res) onses of both wi n gs a r e measured, s ymmet r ic and asymmetric motions can be separ_,ted.

'" In this co n figuration, con trol a n d measurement calibration errors will result in some fract.onal part of unwaPted : n odes tc be p r esent with desi r ed modes.

1 0- \ __i\ '

\ \

\ \

: 4 NSR

\ \

\ ; I C ONTROL BANDWIDTH: F ir s tGr d er Con tr ol La,] at 5 H z T i lt R o t o ra t IgOk no ts I 2 Hert z , Flr s tOrderMea su remen t L ag at I 0 0 HI \ ) 5 H ert z ; Co rr el ati o n D i s t ance 25. 5 ft Dryde n Gu s tM o del i l OH e rt z .0 1 I 0 iOO 6f, DEGREES Figure 4. 7 No i se-t o -Signal Ratio for Asymmetric Vertical 1Ving Bend- ing (qlA) Mode from _wl / _f Transfer Function (For Fre- quent) , Range 3.83 to 4.82 Hz Ab o ut the qlA P eak) Tilt Rotor at 190 knots FourthOrderControlL_g at 5 Hz FirstOrder Measurement Lag at 100Hz OrydenGustModel Correlation Distance 25.5 ft IO

, \

C \

'\ \ \ \

NSR \

\ , \

-- \

• - 5 H e rtz .. 10 Hertz " " ] _ _ 2 Hertz . O 1, m "' \ i . I i I0 _ O 0 0O, DEGREES •- Figure 4.8 Noise-to-Signal Ratio for Asymmetric Vertical Wing Bend- ing (qlA) Mode from _wl / @o T ransfer Functien {For Fre- quency Range 3.49 to 5.29 Hz About the qlA Peak) " 73

I i 1 '

o Measurement and control calibration errors less than 20% do not pr o duce significant err o r in calculating damping values compared to the attenuation effects o f w i ng cross-coupling.

_e ,. V. EXAMPLES OF EV P LUATION OF METHODS FOR MODAL IDENTIFICATION FROM DATA T h e preceding tw o sec t ions have summarized analyt i c a l resul t s on inpu t and measurement sys t em requirements for XV-I5 wind tunnel t es t s. For any specified t es t configu{ation, however, the final accuracy of t he modal frequen c y and damping from t es t data depends on the processing methods used. This sec t ion discusses a prelim- inary evaluation of two such methods--spec t ral analysis and maximum likelihood parameter iden t ification.

Th e o bje c t ive o f t h is p ha se o f t h e studywas t wofold. First, it was desired to use the digital simula t ion as a data generator for evaluating algorithms to ob t ain more accurate estima t es of frequency and damping. Second, use of an advanced parameter iden- t ifica t ion algorithm was t o be p_rformed on such da t a, and compared wi t h the more conven t ional spec t ral analysis approaches.

?

5.I SUMMARY OF ALGORITHMS The spec tral analysis of t his da t a was performed by a Time "" Series Analysis Program, which compu t es an es t ima t e of the t ransfer -_ function, H(f), by dividing t he estima t ed cross-spec t rum of the -- in p u t and ou t put channels by t he e s t ima t ed a u t o-spectrum of th e .. input c hannel. T ha t i s, . .

. , uz (f)

Suu ( f)

The spec t ra were compu t ed by the Fas t Fcurier Transform algorithm.

Details o f t his computer program are f ound in Reference 7.

The m aximum likelihood parame te r identificat :o__nn was p er fo r med " wi t h a modific a t ion of an exis t ing pr o gram, SCTDNT [ 8]. The modi- fic a tion w a s t o use a simplified model of a second order frequency- damping model wi t h me a sur e men t s only of accelera t ion. No t e t ha t , in general, it is necessary to perform a de t ermina t ion of transfer T f i function o rder pr i or t o ac t u a l iden t ifica t ion. Developmen t and .I pr o gr a n_u in g of a n ana lys i s t o d o t his f o r the c urren t problem is T beyond t he scope of t his progr a m, a nd so the assump t ion of the L : second order model was used. For the purposes of this s t udy, this model was considered satisfac t ory. I The model was as follows: !

= + u, xC0) - "_ LXzj -_ -2_;_ x 2 x2(O) . .

= _" = [-_ - 2 _1 + gu 2 m eas x 2 where the parame t ers to be identified are _, m, g, and possibly "" the initial conditions, xl(0) and x2(0).

Note t ha t this model differs from the s t andard state variable 2) , model fo r a second order system (where FI2 = I and F21 = -_ . This form is more amenable to identification than the standard canonical f o r m.

5.2 SIM U LATED TES T DATA The data were generated by a digital computer simulation based on the XV-!5 mathematical mode_ discussed in Chapter II. It was assumed that a means existed which allowed separation of the sym- 7 6 .. metric and asymmetric responses of the coupled wing motion. Tha t is, ei t her bo t h wings were excited or the responses of both wings were me a sured. Hence, t h e s i mul a t ion used t he equa t ions of motion .

for the nine degree-of-freedom symme t ric model. Two tes t cases w e r e g en e r a t e d. I n t h e f i rs t s i m ula t ion case, a Gaussian random ' • inpu t in collec t ive pi t ch was passed through a firs t order lag wi t h a break frequency of 5 Hz (0.654 per rev). This lag w a s in t roduced to a ppr o xima t e t h e d y n amic s of t h e c on t r o l system. The d a t a l e ng t h_ w a s 20 revolutions of the ro t or, and d a t a samples were t aken every 0. 0 4 of a rev o lution ( 0.25 tad); I n the se c ond cas e , th e c ollec t ive pi t c h inpu t w a s the sum of five sine waves whose frequencies were in the neighborhood of . the resonant pe a k (specificall y , t hey were 0.615, 0.628 , 0.698, 0. 7 3 9, and 0.?85 per rev). The steady s t a t e response to t his inpu t was simula t ed for 12 revolu t ions of the rotor and the da t a sampled every 0.02 3 9 of a revolution [0.1 S tad).

_u 5. 3 RESULTS "" The spect1"al anal>,sis method was only used on the firs t set of t es t da t a ( e.g., random inpu t ). When applied to the m easurement 3 of wing chordwise accelera t ion, an es t ima t e of the _w2 / 8o t ransfer func t ion w a s ob t ained. It displayed a resonan t peak for the qw2 mode at 0.66 3 per rev wi t h an associa t ed damping factor 0.091.

T h e ac t u a l v a lues i n the s im ulation f o r t h e qw2m°dal frequency and damping were 0.666 per rev and 0.043, respec t ively.

The maximum likelihood me thod was applied to the same da t a.

Th i s id ent i f i c a tion m e t h od res u l ted i n an e s t im a t ed qw2 m od al fre- que nc y of 0.671 per r ev a nd da mp in g o f 0 .03 6. T hese est i m a tes c omp a re f a vorably to the true values use d in the simul a tion.

7 7 & , I ! [ I f I : J I Figure 5. 1 shows t h e t ime histories f o r t he collec t ive pi t ch ' r a nd o m inp u t and the wing ch o rdwi s e accelera t i o n measuremen t from the nine degree-of-freedom simula t e d data. Superimposed on the qw2 ,. me a suremen t i s t he estimated measurement t ime his t ory from the simple second order model.

Sin c e n o d ata w i nd ow s w e re us e d f o r the t ime series analysis me t hod, it should be c on c lud e d th a t the app a ren t dis c rep a ncy of the result (rela t ive to the simul a ted a nd m a ximum likelihood esti- m a t es) c ould be redu c ed signifi ca n t ly.

In order t o evalua t e the maximum likelihood method further, t he second set of da t a (using sum of sines inpu t s) were processed.

The resul t s were an es t ima t ed qw2 modal frequency of 0.663 per rev and damping of 0.033. These results and t hose of the spec t ral analysis are summarized in Table S.I.

Table 5.1 Frequen c y and Da mping E s t ima t es of qw 2 Mode , Val u es u sed in sim u lation 0.666 0.043 Values from spectralanalysis of _.

random input case 0.663 0.091 ML values identifiedfrom random i np ut ca se 0.67 1 0.0 36 " ML v al ues i den tifi ed f r om s u m o f 5 sines i npu t c a se 0.663 0 . 033 m 5. 4 CONCLUSIONS This b r ief example ha s demons t ra t ed: (I) the p o s s ibility and usefulne s s o f generating simula t ed w i n d t unnel da t a a s a da t a b a s e foz wh ic h t h e tr u e values of modal frequency and damping are known, 7 8 J L

,I

t _ ( 2) a procedure for assessing the accurac y of methods (e.g., spectral analysis or parameter identification) for esti- mating modal frequencies and damping factors, and (3) the ability to assess which types of control inputs will _m resul t in be t ter estimates (random and sinusoidal inputs were used above). -" t Furthermore, t he introduc t ion of aerodynamic gust effec t s to the -- si m ulation, which was demons t rated in Chapt er II, adds th e possibil- ..

ity of s t udying their e ffects on t he res u lts of v ar ious da t a redu c - ..

t ion techniques.

8O I

F

/ I I :

(

i I r :

I"

•. Vl. RECOMMENDATIONS This study has achieved the following objectives. First, several advanced analytical tools have been successfully demonstrat- ed on the XV-IS mathema t ical model, including high order transfer function evaluation and maximum likelihood parameter identification techniques. Second, the application of this analysis has produced a preliminary guideline for selec t ing tes t inpu t s and instrumenta- t ion. Third, specific problems have been isola t ed which may limit t he informa t ion which can be ex t rac t ed from the t unnel t ests.

I t is re co mmended th at several aspe ct s of th is p r el im inary ' s t udy be exp a nded. T h e s e include the following: (I) Incorporation of exis t ing compu t er programs t o de t ermine t h e s e ns itivi t y of t ran s fer func t ions (e . g . , frequency res p onse) to par t icular par a m ete rs such as t unnel v e loci t y, wing stiffness, collec t ive pitch, or measuremen t error.

This would produce a valuable guide to estimating the most significan t error sources in de t er m ination of fre- quency or damping.

( 2 ) Investigation of the effec t s of nonlineari t ies in wing - - s t ruc t ural p a r a m ete rs, t h e con t rol sys t em, o r a erodynam- ics. Th e s e nonlin ea ri t i e s would includ e backlash, hys- .. t eresis, or d ea db a nd.

( 3 ) D etermin a tion of e ffects of fu se l a g e mode s on w i ng r e spons e . The pr e s e nt suppor t a dmi tte dly lim i ts t he ef fe ct of fuselag e c oupling, but t h e s e c ouplings m a y b e impor ta nt in a n a lys e s of fligh t d a ta.

A I' i One fur t her rec o mm en da ti on i s f o r t he devel o pmen t o f a n ad - v a n ced t ech n i que f o r p r oce ssing da t a t o d e t e rm i ne, on-line, fre- quen c y a nd d a mping. This re c omm e n da t ion is b a se d on the resul t s of Chap t e r s III to IV of t hi s repor t , an d ev a luat io n of existing m e t h o d s of spe c t r al a n a lysis. The ba s ic req ui rem e n t is to develop J th e ca p a bili t y t o de t er m in e a n es t im a t e of th e r e quire d an a ly t i ca l mo d e l s t ruc t ure a nd p a r a me t ers whi c h best a ppr o x i m a t e a m u ltiv a r i - a ble mo da l respo n se. A fund a ment a l resul t of this repor t , for ex a m p le, i s t ha t m ul t i v a ri a ble response p oses a d i ffi c ult pr o b l e m i n es ta bl i s h i n g mo da l co n t ri b u t i ons f rom a p a r t i c u la r t r a ns fe r fun c t i on. This problem will be fur t her magnifie d b y the require- L m e n t to de t erm in e freq u e n c y and dam p ing from m ul t i v a r ia ble r es p o n se da t a . - T h e followi n g proced u re i s, t h erefore, s u ggested to i mplement -- t his re commend a tion: (i) D e v e l o p a lg o ri t hm s which pr o vid e o n-line e s ti m a t e s of mod e l s t r uctur e (e . g . , t ra n s f e r f u nc tio n) and parame t ers . .

o f t ha t mo del .

( 2 ) Evalua t e t hi s alg o ri t hm o n t he s imula t i o n di s cu sse d in Chap te r II , a nd t h e t ran sf e r f unc t i o n evalua t i o n of App e ndix B .

( 3 ) I mpl eme nt t h e a l g o r ith m on - l i n e at Am es R es e ar ch C e nt er an d e v a lu ate it on he licopt er mo de l da t a.

(4) Perf o r m a n y m o difica t i o ns o r f u r t her e x te n si on s of t he al go rith m .

It i s a nt icipated that su ccess fu l dev e lopment of s uch a t ech- n i q ue, co m b i n ed wi t h t he res u l ts of th i s re po rt, w i ll p r o d uce a sta t e - of - the - art te c h no l o g y for i mp r o v i ng the res ult s of ad v a n c ed r oto rc r a ft t un ne l te s ting .

8 2

r I I I

i I l _ ; .. APP E ND I X A SUP P O R T E QUA TI O N S OF M O TIO N The addition o f the s up por t degre e s o f f reedo m to th e e qu ation s of mot i o n is acc o m pl i s he d i n a m a n n er v er y si m i lar to th e in c lusion of th e win g d e g r ee s of freedom a s d es cribed in R e f. 3. The equa- t i on s of mo ti on of the su pport a nd w in g w il l b e fou nd a s fu nc t ions of t h e forc e s a nd mom e n t s ac t in g on th e hub du e to t h e ro t or.

T he s e ca n t h e n b e c omb in e d wi t h exis ti ng equ a tions (in Ref. 3) of mot i o n for t h e rotor, w hich req ui re the motion of the hub, to com- pl et e t he equ at ions.

T h e e quati o ns for t h e wing and suppor t d e gr ee s of fr eed om may b e f ound by u s e o f Lagra n g e 's equ a t i on wh ere L • T - V T = Kin et ic e n e rg y o f t h e s y ste m V = P o te nti al e n erg y of t h e s ys te m "" F i - Gener ali ze d fo rc e for i t__h ge n er a lize d v a -lable _i " i t ch gen e r a liz ed coo r din a t e Co m p uti ng the ki netic e n erg y of the s y ste m re q u ir e s t h e in e r- tia l l i n ear a nd rotati on a l v e lo cities of t h e pyl o n c e n t e r o f m ass and t he fu s elag e c e nter o f ma ss, which ar e c om pu t e d b e low .

F i g ur e A. 1 s ho w s t h e geom e tr y a nd c oo r d i na te s o£ t h e w i ng a nd f use l age s y s t e m . T h e : o a xis is t h e r o tati on a l a xis of t h e '_su p P YLON . L C .G.

• T ,-Q C Z p HU i Z w [ NGTIP h \ _w 3 I_wl I Y Tw X Zo Fi g u re A.1 Sup port and Ca ntite v e r W i n g G e o m e t r y _4 A !

i Y l degree of freedom. T h e f u sealage center of grav i ty i s a d i stance _xf forwar d of th i s ax i s. The cant i lever root restra i nt o f th_ wing elastic a xis is at point A, which is ZXw forwar d and _ Y w q l a te r _l l y ( pos i t i ve ri ght) from t he "o a xis. Th e wi n g has sw ee p ang l e _w3 (posit i ve a ft), dihe d r a l _Wl (p o sitive up) , and in- cidence a ngle _Wz (positive le a ding edge up). The wing semi- = sp an i s YTw. A t t h e wing tip t h e _Wl , _w2 , _w3 angles are reverse d so t h a t the rotor hub a xis, :w' is par a llel to xo.

The ro t or f or c es a nd moments a c t ing a t the hub a re T, H, Y, Q, : Mx, My, a s s h o wn.

The system degrees-of-free d om are.

x - t r an s la t i on i n x d irection s o A __: t_ on Ys = t ransl a tion in 7 o .......

Ws u p I r _ation ab o ut zO axis qw I - l o wes t mode w i n g v e rtical bending (positive up] qw 2 - lowes t mo d e win g chbrdwise ben d ing (posi t i ve up) p - l o_t mo d e win E t orsion (posi t ive l e a ding edge u p) The iner t ia l referen c e fr a me is t a ken to be c olinear with the frame c e n t er e d a t po i nt O a s shown in P igure A.I a n d _ixed in inerti a l sp ac e.

• Gi v e n t h e po si t i on of a po i nt expre s sed in " h e A fr a m e , de- not e d x, i ts pos i t ion in t he ine r t i a l f ram e i s : A x I " * _s C '# s } _Yw , 0 1 0 0 0 j 1 0 0

[: oo 011 I ll ° °

(A. Z ) _S l S i m i l a rly, gi ven a n a ng ul a r p o si t io n vect o r expressed i n the A f ra m e , deno t ed _, it is exp r essed in t h e ine r tial f r ame as .: _ - + 0 1 a (A .3) I _s -I 0 A . ! i

E:jE 00 I

} U sing th e se two e quations and equations in Refs. 1 and 2 , which "_ i ex p ress the p o si t i o n of t h e hu b , p ylo n c. g . , a n d an ar bi t r a ry ele- i m en t o f ma ss in t he win g w i t h re s pec t to t he A fra m e , it is _ st ra i g htforw ar d to e _r e ss t h e s e posi t ions in th e in er tial fram e .

. o Th e t a sk of computin g th e kin e t ic e n e rgy will b e much simpl e r if _ T _ i s f i rst ca l cula te d for an a rb i t r a ry x. La t er, t h e p y lon . .

II A A and w ing pos i t ions, A X p and _ _, w i l l be s u bs t i t u t ed f o r x. - ; ge L e t xT = ix, y, z], then from E q. (A.I) _- A ..

x - + ( = + Z xw) S , s + ( y+Zyw)c_ ( A. 4 ) _.

I

[ii cz w ca yyws i

- _,s (Cz+_ , xw) c ,s- (y+_ . y ,, ,) s,s) (A .s ) _-

: Th e p os i t i o n of th e p y lon ce n ter of ma ss in th e A fr a m e is : " " "T v: 8 6 } r :

i L

;F

@ 0 • .

: i! "YTw6wl + qWl (y 'h pn_3)+q w 2 ('Y_2+hpn S l) -- :,i + P w [ hp h p) + Z E A6 2 ] Y T w + qwl [+ hp n6 2 "Y61 ] + qw 2 [ h pn -y6 3]+Pw [-hp6l + hp_ 3_ 2] .i "YTw _ 3"qwly6 2"qw 2 Y + Pw ["((hEA " hp) _ 2+ ZEA ) ]+ hp (A . 6) - w here the s h ort h an d notation of Ref . 3 has bee n ad o pted . S imi lar - l y , -nql + - 63p a = - n d3q I + n61q 2 + p (A . 7) i p nd2q 2 1 nd2q I _sup + + nq2 61P To compute the k ine tic energy o f t h e wing, the velocity o f a differential mass element will be found, and the result integrated over the wing. Let this differential element be a distance z ahead of the elastic axis and a distance r from the root measured along the elastic axis. Then the position of this element in the A fra m e is: I r6 w l+Z_w2+qwlCn ( r ) - zn" (r)63)+qw 2( - n (r)_Z + zn " ( r }_l) +P w (_ C r ) - _ w (Y Bw ) r S 3 ) X = ii Aw r*qwl (zn " (r)6 2- n (r)61) + qw 2 (zn " (r) " n (r)63) "Pw_ (r)_ l+Z_ 3 _- - r6w s *Z - qwln (r)_ 2 -qw2n (r) -pw _ (YBw)r_I t

, . CA .8 )

t i i 8 7 F ina lly, th e pos i t ion o f t h e fusel a ge i n the ine r t ia l fr a m e i s _f " s ( A. 9) . ..

-F _f " 0 CA.I0) T 0 su p

!

All t h e pa rts a re n ow a vail a ble to comp u te th e k inetic energy.

Th e p o si tions i n t h e A fr am e must be expr e ssed i n t h e inertial f r am e i t h rough Eqs. ( A. Z ) a n d CA.3). T h e n t h e pos i tions must be differen- tiated w i t h re s p ec t t o t i me, a n d t h e re su lting vel oc ities s ub s ti- i tut ed in to t h e k i ne ti c e n e r gy of t h e sy st em, which i s

i

-Y T w L.E. ..

+ {_ _T.E. f AmwxTwxw dz dr (A.II) ..

T h e p otent ial energy is s i mp l y

v.{

Kpp KxsX s Kysy s 2 "" + K_sup_su p (A.12) The gen e ra l i z ed f o r ces i nc l ud e a e r o d y nam i c f o rc e s o n t h e w i ng and fusel ag e an d str u ct u r al d am pi ng te r ms. T he a erodyn a mic fo r ces f o r t he wing w e re d erived in Ref. 5. T h e ne ro for c e s on ",he f u se la ge w ill be neg l ected bec a use t h e y are sm a ll by comparison to t h e wi n g an d rotor f orces .

_7 8 8 i I i By subs t itu t ing Eqs. (A. II) and (A. 1 2 ) into Eq. (A.I) and _ performing t h e i n di c a t ed dif f erenti a t i o n a n d by n egle c t ing sm a ll ± ter m s , t h e follow i ng vec t o r e qu a t io n res ul t s: .

a2_w + alxw + aoX w = b ' _f + b_g + _F (A.13) a where Xw " [qwl qw2 Pw _sup Xs Ys IT _f = wine f lap e r o n d e fl e c t ion = [u G lo n gitud in al gus t g = I vG lateral gus t w G _ v e rtic a l g u s t i: F = -Y-a a [ C T 2C H 2 C y CQ 2CMy 2C Mx ]T [ b - 3 x 4 matr i x I ' . o b_ - 3 x 5 matrix • _ a nd t h e matr ice s a 2 , a I, ao, an d i a r e g i ven o n th e foll o w i ng • . p a g es .

i i " '

i I 1 '

I I ++ Y ++ t , I Is - _ I is t otal yaw mom e nt of t he aircraf t _Ib exclu s ive of the ro t ors.

, 2 mw - mw YT w ..

, m i s ma ss o£ t h e w i ng N w - Ib . .

, 2 m f - mfYT w , mf i s m a s s of th e fus ea l a g e • ii _w N 9 ,y w - _y w / YTw , 2,y w is l at er a l com p onen t of dis tanc e ..

fro m wing roo t el a s tic a x i s to c .g., £X w " £x w / YTw, _Xw i s long i tud i n a l c ompon e n t of d i s t a n c e " fro m w i ng root el a stic a xis to ai rcr a f t , - C.g . , a nd a ll o t her pa r a me te rs a r e a s defined in Ref. 5. ..

The t ot a l equ ati ons of mo t ion are found by c ombin i ng th e ..

rotor e qu at ions a nd E q. (A.13) a s d e s c rib e d in Ref. 5 .

Th e num e ri ca l results o£ Ch a p t er II w e r e gen e r a ted with the follo wi ng simpl i fi c ations : Q (a) _ s up = 0 (no t ors i on of suppor t s) (b ) £Yw = 0 (w i ng c antil e v e r att ac hem e nt a t c .g.)

;. APPEND I X B TRANS FER FUNCTION ANALYSIS B .1 REVIEW OF M ATHE M ATICAL FOR M ULAT I ON As expl a i n ed in Cha pter II , t h e m a t h em a ti ca l model for t h e X V-I5 , as deve l o p ed by J o h ns on [ Z-5 ] , e xpr esse d in vec t o r no tat i o n,.

is - F x - Hx + D u ( m eas ur e m en ts e xc i t e d by th e contr ol ) i y* - Hx + D *v (m ea sur e me n ts exc i te d by ra nd o m g ust) Th e r e i s an F , G, r , D , and D " a s so c iat ed w ith th e symm e tr ic mot i on of th e rotor / ca ntil e ver wi ng, and anoth e r F, G, r, D, a nd D" asso cia t e d with asymmetr ic mot i on of th e rotor / ca nt i l e ver wi ng.

Th e symmetri c a nd asymm e tr ic mod e ls are c ombin e d ( a s e xpl ai n e d i n S ec t i on 2 .3) for a c omplete d e s c r i pt i on of th e dyn a m ic s at th e left and r i ght side of the a i r c r a ft.

. . Th e fre qu enc y r e spons e s

Z(s ) -iG

• . _ = H( s I-F) * D (ref e rr e d to as sign a l tr a nsfer funct i o n s ) (B.I) and l ' (s ) -I • _ = H(sI-F) r ( r e f erre d to a s no i s e tr a nsf e r fun c t i ons ) (B. Z) we r e ca l c ul a t e d us i ng th e L e v e rri e r m e thod [ 9 ,10 ] tO e v a lu a t e th e a djoint. Th e tr a Esfdr fun c tion w as th e primary a n a lyt ica l tool t used t o ev a lu a te t he dyn a mic s o f th_ X V - I 5 and , hen c e, evaluate the instrume n tat i on r equ i reme n ts an d test g ui del i nes fo r the w in d t u nnel t e s ting of the XV-15 tilt r o to r aircraft.

B .2 DERIVAT I ON OF FO R MUL A E T ran sf er f u nc t i o ns f or m easure m e nt of t he st a t es o r lin ear c oor di na t es o f t he s t a t e s w ere u s e d i n se v era l w a y s: ..

(I) T he t rans f er f u n ct ion indicates i f a par t icu l ar m easure- m en t exci t ed by a p ar ticular c o ntr o l sh o ws a res o nan t v_ peak to i den ti f y a des i red m ode.

(2) To s t ud y t he st a b i l i ty o f the X V- 1 5, est i m a t es of t he dam p i n g ( _ ) of t he leas t da m ped mo de s are des i re d . F o r - " " m ulae u s ed f or ca lc u l a t in g _ fr o m t ransfer f u nct i o n da t a are d is cussed i n Se ct i o n B.2. 1 . " (5) T o quan ti f y t he v alue o f di f ferent t ransfer f unc t i ons _ .

f or ide nti f y i n g t he f r e q ue n c y a n d damp i ng of a de si re d mode, and to clarif y wh ich m o des co n t r i bu te s ubstan t ia l l y _ to a r e sonant peak (when m o re t han o n e mode ar e clo s e to the peak frequency), t he m o dal p o wer and m o dal p o wer ra t i o were derived . Thes e f o rmula e ar e pr e sen t ed in Se ct i o n B . 2.2.

( 4) T o analyze t h e e ffe ct o f vari o u s g u st l e vels in t he wind t unnel o n t h e a b ili ty to i denti fy t h e frequency and damp- " i ng o f a de s ired m o de, t he n o i s e- to-s ignal ra t i o was derived a s e x pla i ne d in Se ct i o n B.2.3. The s ignal p o wer an d n ois e p o wer were calcula t ed fr o m t he signal a nd n o is e t ran s fer f un c ti o ns o ver t he same frequency range.

9 4 i I •- B.2.1 Damping Calculation [ 5 ] . T o evaluate t he damping o f a par t ic u lar m o de, t he transfer func t i o n i s appr o ximat e d b y a g en e ral s e c o nd- o rder s y st em in a r eg i o n ab o ut t he r eso nan t peak fo r tha t m o de. Th e se c o nd o rder s y s t e m i s

m_ * c _ + k x - f C B . 3 )-

where _ n = _ is th e n a t ura l f re q ue n c y - c / 2 _ n m i s th e dampin g factor f = cont rol i n pu t .

This e qua t i o n m a y b e rewritten as mR + m _ n2X - f (B .4 ) Th e tr an s f er functi o n fr o m f t o t he syst e m accelerati o n, a = R , is

- T(s ) - ....Z z ( B .S )

s2 m s + c s n T he Bode m ag n i tud e plot re qu ires t he m agni t ude of this f u n c- tion evaluated on t he j _ axis o f t he s- p lane.

T C J_ ) - "_ - C B . 6 )

m C _ n Z-_ z) * jcw

!

ITCj_) I z - _ C B . 7)

m 2 2 2 2 2

C _ n- _ ) * c Zw

Eqs. C B.6 ) and C B.7 ) m a y be c o mbi n ed to y ield

2 m C _ . 2) . jc CB. S )

Thi s f or m i s pre f erred over E q . (B.6) because the co m plex deno mi na- to r i s e limi na t ed. ..

Taking the i m a g inar y parts o f bo t h sides o f Eq . (B.8) , inte- grat i n g both s id e s , an d sol vi n g f or the averag e value o f c g i v e s _w _2 U

_ = 1 C S .9 )

2 _ d u w h ere t he int er v al (U l, U 2 ) i s c hose _l such t ha t u n i s at its mid poin t and f or wh i ch t he approxima t ion of t he frequency response curve by t he seco n d-order s y s t e m is v a l id ( e.g., u I - 0 .8 _n a n d u 2 = 1.2 _ n ). The frequenc y at the peak, U p, is used as the est im a t e o f u n , which i s reasonable f or s m all v al u es o f q since U p - _ n _ (B . I O ) f or a second order s y s t e m .

T [ [ I '

) i I i ) _"

' T

It is n o w n e c essar y to c o mp u te a n a v erage v al ue f o r m i n o r d er to compu t e th e da mp i n g f ac to r f r om _o "-

- -- ( S. ll)

• - 2_ n _ To t his en d , Eq . (B. 5 ) is written a s . s2 /m (B 12 )- T( s ) l 2 Z s n Th is l e a ds to a n e qu at ion anal ogo us to E q . (B. 7 ) ; that i s ,

(8.13 )

l t 3 l'T"w ) ' 2 1 S u bst i tuting 2 _nm f o r c i n Eq . ( B.8) giv e s Sub s tituing E q. (B. 1 3) i nt o (B. 1 4) fo r [T(j_ )[ 2 and t he n t a king t he im a gin a r y par ts o f both sid e s o f Eq. ( B .1 4), i nt e gr a ting both sid e s , an d s olving £ or th e a v e r a g e v a lue of m giv e s - I m • ( B. l S ) w 2 W

Im(T)d_

w1 whe r e w 2 I • I( _ ) = ( w. . w Z)Z n d w ( 8 .1 6 ) L 1 2_W n W 2 9 7 i This in t e g ra l m a y be c om put e d ana lyt i cally as f ollo w s:

I(_ , _) = t an" l + ---3 --_n

\_ n -_ / z-_ T _ / -: ._ j j

" 1 F inall y, s u bst i t utin g Eq s . ( B. 9) and ( B.15 ) in to E q . C B.11 ) r e s ult s i n a n i mp li cit e q u a t ion fo r th e d amp in g ,

_ dw

1 1 J g " "_. - _r_ " . , (8.177 o r P

- _ C B. 1 8 3

w h e re P i s d ef ined appr o pr i a t el y fro m Eq. ( B . 17) . P i s a functio_ o f th e tr a n sfe r f un c tio n b e tw ee n th e integrati o n li m it s of _I an d w 2, an d i s ao. _ t a f unction o f _ . T h e r e for e , _ m a y b e found by i t e r at i v e u s e o f E q . ( B.18 ) . It w as f ound tha t , g iv e n a good i nitia l guess f or _ , conv e rgenc e wa s achie v ed in thr e e o f f o ur it e rations .

A g o od ini tial v a l ue for _ is

C o " L r o " IT - 2 o. p ( S. l_)

- " *

w h ich w a s f ound fro m Eq. ( B.I8) by expandi n g th e ana lytic e xpr e s s i on for I( _) in a power s e r i e s an d n e g lec ting s eco n d a nd hig h er order term s i n _ .

I }

i I I k t t T h e t ra n sfe r fun ctio n T(j_) was appr o x i m at e d by a se c o n d ._ or d er s y s t em i n t he re g i on o f the reson a nc e . A s e co n d or d er sys t em ha s a phase angle of - 90 ° a t the p e ak. Thus, th e e ffec t o f o t her : : mo d es o n the r es o n an c e c a n b e l es s e ned by s hif t ing the phase o f th e mea sur e d t r a nsf e r fun c t ion b e fo r e a pply i ng E q. (B.1 7 ). This ph a s e shif t is p e rfor m ed by the following eq u a t ion: Tne w = T • -j ]_ (B .2 0) where Tp i s the m e a s ured response at the re so nan t peak.

Data Processing _ T h e calcul a ted value o f _ i s a fun c t io n o f bandwidth since other modes interfere a s the bandwidth in c reases a nd the a c cura c y o f t he i nt e grals (averages) us e d in t h e c al c ul a t io ns may det e riorate as bandwid t h decreases, par t icularly if the number of da t a points in the bandwid t h is small. To inves t iga t e t his dependency, the following procedure was used.

U s ing the _wl / _f tr a nsfer function as an example, T(j_) wa s co mp u t e d f o r I00 val u es o f _ b e tw ee n 0.3 5 an d 0 . 4 5 p e r r e v for th e " " s ym m etric ca s e (q l s is 0.398 per rev), and i00 values of _ be t ween "_ 0.4 9 a nd 0.62 for the a symmetric cas e (qlA i s 0. 5 62 p e r r e v). U s ing - - th ese data , _ was c al c ulated for various bandwidths containing e a c h .- p ea k. T h e r e sults a re summ a rized in Table B.I.

T h e ca l cu l a te d v a lues o f damp i ng pr ese nted i n th e body o f thi s r e por t w e r e bas e d t ypi c ally on i0 to 15 da t a point s in th e fr e qu e n c y rang e of int e gration (i. e ., in th e bandwid t h _2-_i). Th e purpos e of ob t aining i00 points in t his small rang e was to r e du ce the numer- i c al e rr o rs in t h e c al c ulation to a n e gligibl e l e v e l. Th e r e for e , th e error in t h e c al c ula t e d valu e s of _ ar e du e t o th e int e rf e r e n ce of o t h e r m od e s and to th e approximations in th e t h e or e ti c al method : 99 • I !

I j j 1

Table B. I C alcul a te d Damping as a F unc t io n of F requency Bandwidth Use d for I ntegr a t ion _ - Je FOR qwl MODE FROM -- i BANDWIDTH i_wl /_ f TRANS:ER FUNCTION _.

SYMMETRICMOTION ASYMMETRICMOTION - " uI u2 _ o = 0.4002 _ = 0.5616 P P -- O .99_ p - I . 01 U p O.0 409 8 O . 0 4200 "" mw O. 9 8_ p - I .0 2 Up O.04 33 1 O.0 4 619 " O .0 4486 O .049 77 O. 9 7 _ p ,. 0 3 U p . .

O . 96 m p - 1 . 04 _ p O . 04567 O.0 51 94 - : O.9 5 _ p - 1 . 05 (_ p O.04 5 84* O.0 53 02 " " O . 94 _ p - 1 .0 6 _ p O.0 4 5 78 O .05 3 4 8 : O. 93 _ p - 1 .0 7 U p O .045 03 O. 0 536 1 * - T O . 92 U p - 1 . 08 _ p O . 04537 O . 05359 . , _ O . 9 1U p - 1 . 09 _ p O .0 4511 O . 053 49 " O .90_ p - 1. lO O p 0.0 4 48 3 O.0 533 6 " II T r ue V a lue o f _ 0.04 8 00 0.0 5 540 i . .,L i i i i i , • % E rr o r (B a sed on B es t Value o f _) 5 .0% 3 . 2 % Maxim u m v alu e o f of t h e c alcul a t i o n. F u rt h er st u dy o£ d a ta pro c ess in g te ch n i q u es sho ul d in cl ude i nv es t ig at i o n o f ot her met ho d s , s uch as th a t s u gge s t e d by K e nn e y an d P an cu [ii ].

I t wa s foun d t ha t w i t h i0 to 1 5 d a t a poin ts in t h e re g i o n 0. 8 U p to 1 . 2 Up t ha t 0 . 9 _ p to i . i Up ga v e t he b e s t v alu e f or d a mp- ' / 1 0 0 4"_ I I 1 r

, i

L

mo

I

o- in g an d wi t h i00 data p oi nts in t h e 0. 8 Up t o 1 . 2 Up ba n d, t h at 0.9 5 U p t o 1 . 0 5 U p g a ve the bes t v a lue f o r d am pi n g. Given a sm a ll f re quency in c remen t b e t ween d a t a po i n t s, the calcula t ed damping d oes not v ar y signif ica n t l y in the r an ge 0 .9 Up - i.I Up t o 0 .9 5 Up - 1. 05 Up.

T h i s ana lys i s was p e rf o rm ed o n t he q ls (0 .598 p e r rev) an d qlA (0. 5 6 2 p e r rev) m o des. Th e t r ends show n i n T a ble B.I m a y c h a n g e - somew ha t f o r high freq u e nc y mo d es (such as the _+ i mode a t 2.4 3 per rev), where the same frequen c y r a t io abou t Up gives a bandwid t h several t i mes grea t er.

B .2.2 Mo dal Power and Modal Power Ra t io T o best id en t i f y p a r t icul a r mo de s from t h e frequenc y responses o f the s y stem, it was desired to: • Q uan t i f y t h e q ua l i t y o f d i f feren t t r a nsfer fu n ct i o n s to i d ent i fy a p ar t i c u l a r m o de • Qu an t i f y t h e ex t ent to wh ic h a mo d e i s o bsc u re d when mo r e .. t ha n o ne are c ontr i buting to a res o nant peak on the fre- .. quen c y response.

The quantif i c a t ion of a pa r t icular mode's con t ribu t ion to the t r a nsfer func t ion c an be a ccomplishe d in the follow i ng manner.

Co n si d er the response of a t r a nsfer func t ion to sinusoid a l excit a - t i on a nd express t his r esponse in t erms of t he response to an im- p ul se func t io n. In o t her wor d s, ex p ress t h e st ea dy st a te response t o a s i nus o i dal f o r cin g f un c t i o n in terms o f t r a nsien t response qu a n t i t ies, n a m ely the resi d ues a t the poles of the p a r t ic ul a r t r a nsfer fun c tion.

I01 i

° ! I

!" i I T h e response to these two i nputs w i ll be expanded i n a p a r t ia l fr a c t ion exp a nsion. Let th e subscrip t s t d e n o t e transien t , a ssoci- "" ; a ted w i th a n i mp u lse i np u t ; s d en o t e s i nuso i d a l, a ss o ci a t ed with a _ sinusoidal inpu t ; and ss denote steady state, associa t ed with a °" s in uso i d a l i n pu t af t er t h e t r an s i en t s have died out.

' For a n n th or d e r s y s t em, C B . Zl) Yt Cs ) " TC s ) _{ SC t ) } _ TC s ) _ _s n k.

; yt Cs) - z z C BZZ ) ° "

i - 1 i s'x i ) "

w here N (s) and D (s) ar e t he nu m e rat o r an d d e n . o m in ator polyno m i a l s , -- r es p ec tively, a nd t he li's a r e t he p oles of T(s) and the ki's ar e -[ t he r e s idue s a t t h o se po les . . .

i -m F orc in g t h e s y s t em w i t h a c o m p lex s i nuso id of u ni t m ag nitu d e . _ give s YS( S) - T(s) _{e J _ o t} . " " [ I T ( s ) N ( s

" ( s - j %) " " ( _-j%)b( s) ( _ . z3)

- _ 24)

i -1 Fr om E q s. ( B . 2 1) and (B.23 ) , i t i s seen t ha t

yt (s) - ( s- j% ) Y s(S) - o ( B . Z S)

1 0 2 [

L

v - *" The n e x t s tep i s t o s u bs tit u t e Eq. (B. 2 2) f o r Y t and Eq. (B.24) - - f or Ys in E q. ( B. 2 5 ) a nd to e xp ress th e r es u l t as the r a tio o f t w o _, po ly n om ia ls, w hos e numer ato r p o lyno mia l m us t be - 0 id e nt i c ally 7 : f or all v a lues o f s. Hen c e, th e c oeff icie n t of e ac h p o wer o£ s J m u s t = 0. T h e f o llow i ng rela t ionships r e sul t .

i .

' J n , kss : i=l k , z ( B.2 7 ) k _ = ix.j U o Theref o re, n k i

- z _i ( B. 28) k ss i = l " J Uo

T h e stea d y state r es p onse is th e n = = Z (B . 29 ) Ys s (S ) ksse JW° t e J_ O t n - ki i=l li'J_o A c omp l ex s i nuso id was use d b e caus e i t simplifies th e alg e bra .

T he stea d y s t a t e response t o a p ur ely si nu soi d al ex c i t ation, u - si n _ o t , can be ob t aine d by merely ta k ing Im[Yss(S) ].

T he ma gnit u d e o f th e s te a d y stat e r es po nse is

- IT Cj _, o 9 I - ks s ( B.3 0)

w h ere T(s) i s th e syst e m tran s fer fun ctio n. Th e po w e r in a n , , int e r v al (_ 2 ,U l ) abo ut a r es o nan t pe ak is d e f ine d as i!

I i 0 3 | I

I I 1

t , P _ [T(j_)[ 2 d_ : i i e _ 2 . .

(B. 31)

=[ T (j_)• T (j_)d_

J W 1 " " where _(jm) i s a ve c tor in the c omplex plane and (.9 indica t es the "" v ec tor inn e r [ do t) p rod uc t . '" From E qs. ( B. 2 9) a nd ( B .3 0 9, -- n __

T (j_)- E T i(j_) (B. 3 Z) *

i=1 " whe r e "" k i - -

Ti(J_) " i'_ (B . 33)

o_ T hi s is eq ui v a le n t to sa y i ng th a t t h e t o t al s y s tem r esponse at . .

fr e qu e nc y _ i s t h e s u m of t h e r e sp onse s o £ t he in d i v idua l m o d e s . .

!

(i.e ., Ii s). ..

F or ea ch o sc illatory mode, t h ere exi s ts a conjugate complex p air o f p oles , w h ic h w i ll be de n oted li an d lZ = [ i ' w h o se a ._so c i- ated r e sidues a re k i and k& = _i" L e t

T _- T i + T _

ki _ i

- - --_ (B . 3 4 )

_i"J_ _[i-j_ 1 0 4

!

/ ,'7 tm -- T h e n th e con t rib ut ion o f thi s o s c ill ator y mode to t h e tot al power •- is 1 • e

• -

an d r a t io o f th i s fr ac t i o na l pow e r t o t h e t o t a l p o w e r ( E q. (B.31)) - is ca ll ed the mo da l pow e r r a t io for t h e i t h oscill a t ory m od e : Pi MP R - -- @ - (B. 5 6) B. 2.5 Noise-to-Signal Ra tio T he tran sfe r fun c t i o n s g i v en in S ecti o n B . I u_ _ Hs(J_ ) (sign a l tr a nsf e r fun c t ion) v_ _ Hn(J_ ) (no i s e transf e r fun c tion ) g i ve th e g ain a nd p has e o f t h e system when excit ed a t a d iscrete fr e qu e n c y, a nd s i n c e the model i s lin ear , a ny numb e r of fr e qu e n c y r e s p ons e s c an b e a dd e d to g i v e t he r e spons e to th e sum of th e i n p uts. Theo re t ica lly, wh i te no i se w ould g i ve t h e ent i r e tr a nsf e r fun c t i on sin c e i t i n c lud e s a ll fr eq u e n cie s.

T hu s, t o e s t a bl i sh t h e r a t io of how a pa rti cu l a r m ea su rem en t r e sponds to r a ndom gu s ts i n the tunnel c omp are d to how i t r e sponds .. to co m m a nd e d co n t r ols• Th e sig na l squared, o r po we r o f th e two . t ra nsfe r functions, c a n b e com p ar e d in th e fr e qu e n c y r a ng e of int ere st .

i" lOS & J.

L _02 ]Ul 2 IHs 12 d_ S / N = , . .

f W2 q *n d_ C_2 - _ I ) i ' _o I ..

wh e r e q i s the gus t co va r ianc e and r i s th e me asur e m e n t c ovariance " (co n si d e r ed as wh i t e noise i n t he frequency r ange o f in t er e st ) " _ - q = 2 Tc o rr X(t) -_ where . .

Tc o rr i s t h e c orrel ati o n t ime of t h e noi s e = L / u o _.

(correla t io n di st a n ce divided by t he w i n d velocit y ) x ( t ) is the au t oco r rela t io n o £ the n o i se . _

x(O) - v

r m s , , Cn is a composi t e power sp e c tr um obtain e d in the £ollow- i n g way: i

#Xo u t _l in I n l Iz

¢ - ¢ I H 21 2

2o u t 2 in

, . , IH Iz

3o u t 3in wh e re ¢ 1 ' $ 2' _5 correspond to th e long it udinal , lat e ral, an d ver tic al spo ct rum, respe cti vel y , of t he Dryd e n mod e l, and H I, H 2 ,

l t i

C / ; H 3 a re t h e g u s t tra n s f e r fun ct ions from th ese th ree sou r c e s to the de s ired mea su remen t . Assu m i n g t he gust f rom t he t hree d i re ctions " is uncorre lat e d m e a n s that th e po w er f ro m t he s e t hre e sig n als ca n " " b e a dded i n a n RM S se n s e ; t here f ore , t he s e s u ms ca n b e i n t e g r at ed Sn • Z i = 1% l o ut

UZ r ms I H I z d_

I S I N = ( B.3 7 ) / = 2 % d_ * r [_z-_ 1] w Th e rec ip r o ca l of t his re l a t i o nship, th e nois e -to-s i sn a l ra tio ( NSR], i s m o r e c onv e ni e n t to wo r k wi t h sin ce th e gus t e ff ec t s ca n b e s t u d ied al on e w i t h o ut t he mea s ureme n t no i s e .

q Xn r ( _ 2-_I ) N / S • Z + Z ( B. 38) U r msXs U r msXs wh ere X n a nd Xs are t h e po w er un der t h e n o is e a nd signal pow er s pectra. T h e no is e-to- sig na l ra t i o for the m ea su re m en t can b e a dd ed i n if it's c h ara c ter is t i c s a r e k no wn . It sh ould b e p oi nte d out th at t his is t h e no is e - to- sig na l r a ti o f or o n e d ata record.

N or m a lly, k record s are recorded and ave r a g ed in t h e f re q uenc y " d o m a i n , wh ere t h e sig na l a d ds an d the ran d o m noi s e ten ds t o cance l, "" g reatl y im prov i n g t h e sig na l- t o -no is e ra ti o. B en d at an d P i e rs o l [ 1 2 ] -- sugg e s t u s i ng 10 rec o rd s or mo re. Th e pr in ci p a l r e s tric ti on is t he iL tot al samp l e time.

:. 107 i I ' i T h e poi n t to be emp ha s i zed i s t ha t the s i E na l-t o -no i se r at io in Eq. (I0) i s for o ne re c ord, a nd t hus a l l the n oise-to-si E nal [ (N SR ) pl o t s in C hapt ers I I I and IV are f or o ne data r e c o rd .

J .4 .£ I ' I i . . A PP EN DIX C .. T RA N SFER FUN CTIO N FREQUE N CY RES PO NSES T_i s ap p end ix pr ese nts th e B o d e p lo t s fo r th e ro tor / ca n ti l e ve r wi ng a nd c o up l ed wi ng m odel s . T h e s e plot s are outputed f r o m t h e tr ansf er f un c t i on ana lysis program. O ther data o utputed are the poles, zer o e s , an d re si due s o f t h e s e tran sf e r fu nct i on s . The s e l a t t e r o u tp u ts ar e no t p r ese nt ed h e r e.

Each B o d e plo t is given with a re f erence scale which in d icates, according to t he poles o £ t he trans f er f unct ion , which d e gre e of freed o m is pr e d om inan t a t a par t icular f requency.

e _ . o tQ : 109 1 1 0 I I

i r, T I

' l

i 12 I 11 5 / i i

J

ii l

! • I ' 11 7 !, I , I I _ L 1 18 1 2 0 1 22 Figu re C.29 ._w l / eo - W i n g V er¢ i cal A ccelera ti on t o Collectiv e (Coupl ed Wi ng Response - Exc i te One Wi n g , _le as u re Same Wi ng ) I Z4

, r I 1

i 1 , i I I P , i 8 0 0( P L_T -o ,A GNI T UO( OF r _ l_ _ SI* _ )NSI[ / • .

t 10° '"I ' ' ' ' ' '"" + I0 < I0 " : I0-4 I0 -s 10 " _ I0 ° tO: L L I _ ¢ L.I i-,,L ., ..,. , , , ''' _

A s "_m e t : "i ¢: '+ , , e _] _" , ,_-q

_ - 'L ¢ - _, q+ - - P B+zi i

I

q : _o *x So .- S ym metr ic' . ,. . , , , , ,, ,. ,, t 6 - i g - z q _ q+, _+i _o .. _ + 1 * + Q

+ +

. 9 0 1 [ Q g e " 9 0

-

-18 0 _ : ' ' ' I0"l I0° I 0 ' IrlI ( O U I'M C ' r -- P ( R _ [v Pigu re C. 5 0 q w2 / 9o - W in g Ch ordw i se A cc e l e r a ti on t o C o ll ect iv e (Coup l e _ Wing Respo n s _ - E x c it e On e Wing , Heas u re _ ame Win g) 8eOl[ P L e T -- rIR ON I T UOE O F T _ *[ I _ I[S POe *S( 1 0 " L - ; i ; ! _ e i f I r t _ * I * i _ - 1 O - t

i

m E . | I0 10 .4 I i i ,,i,' I il l , , _ , J , ,, I0 "s 10 ° I0 i As,. _m e t l* i c: , ., ,. , ,., ,.,,I -, -., , , , L _,1 - - 8 .z _ - x ql F S + l

' '

q : ¢ o : S y mme t r i c: , .. t , , ,.,,,l . ., , , , : ,., ,l J ,"_ i liP-- , 8-i _ ' l q l q : P _z J ' S o _ ; J , l 1 8 0

! 90

i °

Fi sure C.3 1 qwl / 6 £ - Wi ng V er ti c al A cce l er ati on t o Fl aperon (C o up l ed Win g Response - Ex ci_ e One Wing , M e a su r e Sane Wi ng) 1 2 ' 6

i I I T I

_ 800 ( PL . _ I' -- rl F I G NI TU0( _f THI[ , R _S P O N S ( ,, 10 - 2 _ = • _ ,- 10 I 0 -6 10 -'7 , _ 10" 10 ° I0 I Asymmetric - L ., ,. , ,.,,.,,I . ., .., , , , , ,, t 8._ _ ' * q* P B + I q2 _0 + 1 S o , t _ 1. _ _ I I I I t [ t [

• Symmet r i c : ' I' _ t .,' i'''t i I I'm

8o

IZ -180 ,I K L , t ,,,, !0 "L i0 ° I0 z , Figure C .12 _wi / _f Wing Chordwise Acceleration to Flaper o n _ ( C ou p le d Wing Respon s e - Exc i te One W i ng, M ea s ure Sa me Wing) , 12 7

I I I

i 1 I i , I REF E RENCE S i. T i l t R o tor Pr o ject Off i ce Staff, "Tilt Rotor Research Aircraf t Familiariza t ion Documen t ," NASA TMX-62,40 7 , Jan. 1975.

2. Johnson, W., "Dyna m ics of Tilting Proprotor Aircraf t in Cruise Fligh t ," NASA TND- 7 6 7 , May 19 7 4.

3. Johnson, W., "Analytical Model for Tilting Proprotor Aircraft Dynamics, Including Black Torsion and Coupled Blade Bending Mod e s, and Conversion Mode Operation," NASA TMX-6236, Aug.

1974.

4. Johnson, W., "Analytical Modeling Requirements for Tilting Propro t or Aircraf t Dynamics," NASA TND-8013, July 1975.

5. Johnson, W., "The Influence of Engine / Transmission / Governor on Til t ing Propro t or Aircraf t Dyna m ics," U.S. Army Air M o bili t y R_ D L a bor a t ory, Moffe t t Field, Calif. (In t ernal Documen t ).

6. Johnson, W. and Biggers, J., "Shake Test of Ro t or Tes t Ap- para t us in the 40- B y 80-Foot Wind Tunnel," NASA TMX-62418 , Feb. 1975.

7. Brown, T.J., "Program for the Anal y sis of Time Series," NASA TMX-2988, Sept. 1974.

8. Hall, W.E. and Gup t a, N.K., "Me t hods for the Real Time Iden t i- fication of Vehicle Parameters," TR No. 4 to Office of Naval Research, Feb. 1975.

9. Gantmacher, F.R., The Theory of Matrices, Vol. I, Chelsea Publishing Co.. New York, 1960.

I0. Bosle y , M.J., et al., "The De t er m ina t ion of Transfer Functions fro m S t a t e Variable Models," Automatica, 1972, Vol. 8, pp. 2 15- 2 1 8 .

i i. K e nn e d y , C.C . a nd P a ncu, C.D.P., " U se of V e ctor s in Vib ra tion Me a surement and Analysis," Journal of the Aeronautical Sciences , Vol. 14, No. II, Nov. 194 7 .

1 2 . B enda t, J . S . a n d P ie r s ol, A.G., R anaom Da t a: _l a l y sis a nd Measure ment P rocedures, J ohn Wile y 6 Sons, Inc., New York, " " 19 7 1.