Perturbation solutions for the influence of forward flight on helicopter rotor flapping stability

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NASA TECHNICAL NASA TMX-62 , 361

M EMORANDUM

PERIi JRBATION SOLUTIONS FOR THE INFLUENCE OF FORWA R D FLIGHT ON HELICOPTER ROTOR FLAPPING STABILITY Wayne Johnson Ames Re sea r ch C e n ter an d U.S. A r m y Ai r Mobi l i t y R&D Labo r a t o ry Moffe tt F ield, Calif. 9 4 0 3 5 , , _ ' - _1_ . . 1 A u gust 1 9 74 _ - ,' _4 .......................................... II _ m . , CONTENTS Hov er ................................. 8 _ .

O r d er u 2 R e s u l ts .......................... 13 "- N e a r _ / rev F re quency ........................ IS N-BLADED ROTOR EQ U ATIO N S O F MOTIO N ................... 2 7 Equation o f MeLlon .......................... 2 8 Exp a n s ion in _2 ........................... 2 9 Ord e r I Results .......................... 2 9 Order u 2 R e sul t s .......................... 2 9 Near I / r e v F re quen c y ......................... 5 1 " Sum m ar y ................................ 3 3 Equations of Moti o n .......................... 3 5 I Hover ................................. 3 6 ! O rder 1 R e sul ts ............................ 5 8 , O r d er _ R e s ul t s ........................ 3 8 , Summary ................................ SO

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Order _t 2 Results .......................... 5 5 Summa r y ................................ 5 5 , G IMB ALL F D ROTOR , F IV E O R MORE B LA DE S .................. SS _ Summary ............................... 5 8 . __ E Q U ATIONS O F M OT I ON I N THE NON RO TATIN G FR A ME .............. 5 9 N _ 3 ................................. 6 1 , C o ning M o d es ............................ 6 6 High a nd Low Fre quen c y M od e s ................... 6 8 ' t J J' - _ , P age Summary ................................ 7 3 Fou r -bladed Rotor ........................... 7S T R ANSFERFUNCTION S ........................... 7 6 _ A Po in t About Exper im ental Technique ............... 80 , _ CONCLUSIONS .............................. 86 _ ' _ " " APPE N D IX C = SOLUTIO N OF T HE SECULAR EQ U A T IO N .............. 9 2 - RE FERE N CES ............................... 9 5 r t : ?

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P E RTURB A TION SOLOTIO N S FOR THE IN FLU EN C H O F F OR W ARD F LIGHT O N HELICOPTE R ROTO R F LAPPING ST A BILIT Y B y Wayne Johnson* J_ O. S. A r my Ai r M ob i l it y R _ D Labor at or y M o ff ett Field , Cal if . , 9403S S t_ RY The s ta bility o f the flapping m ot i on o f a hel i cop t e r r o t o r blade i n f or w ard f l i gh t i s i nvest i ga teds us i n g a pe rt u r ba ti on t_ chn f que wh i ch gives analy ti c exp r ess i ons f or the e ig env a lues s i nclud i ng the i n f luence o f t he pe ri - od i c ae r od yn am i c forces in f o rw ar d f l i gh t . The pe rt urba ti on solut i ons ar e based on s m al l ad vance rat i o _ (t he r a ti o o f the he li copte r f o rw ard s p ee d t o t he roto r ti p s peed ) . T he resul t s are val i d to a p p r ox imat ely _ = 0 . _ , wh i ch cove r s the f or w ard speed range o f m os t hel i cop t ers. Th e rotor con fi gura ti ons co ns i dered are a s i n g le s i ndependent blade; a t eete ri ng rotor; a gi a b a ll e d ro t or with three s four , an d fi ve o r m o r e blades; and a ro to r wit h N i nd e pende nt blad e s. The ei genvalu a s o f a constan t coe ff ic i en t app r ox im a ti on t o t he flapping equat i on are ob tai ned b y an e xpan s i on i n _s an d are compared _ w it h the pe r turbat i on solut i on including the pe ri od i c coe ffi c i en t s. The con- I s tant coe ffi c i e n t approx im a ti on w it h t he equat i ons and degrees o f fr eedo m i n t he nonro t at i n g fr ame rep r esen t s t he flap dyna m ic reasonably well f or t he . lower f requency m odes j althou g h it canno ts o f course , be co m p let ely cor r ec t .

The tr ans f e r f unct i on o f the rotor flap response to s i nuso i d a l p i t ch i nput i s exam i n e d s as an al te rnat i ve to t he e i genvalues as a represen t a ti on o f the ,- dyna mi c charac t er i stics o f t he flap motion.

_ ., IN TR O Dt _ IO N The f und a en ga l mo t ion o f helicopter ro t or blade i s flappin g motion: I first m ode ou t of pl _ e (ver t ical) d isplac em ent fro m the pl _ e o f rota t ion of t he ro t or. F or an ar ti cula t ed rot or t h i s m o t ion i s ri g id bod y rot a ti on o f t he , bl ad e about a hinge at o r near th e center of rota t ion. For a c an t i lever ro tor i t i s elastic bending motion , pr im arily about f lexib i lity at the blade root.

D u e to t h e high c e n t ri f u g al f orc es on th e bl ad e _ the m ode shape f or f irst m o d e i banding o f a blade wi th c an t i le v er root restraint is nearly th e sa m e as f or _ t h e r i g i d body flap _ti on of a h i n g ed b l ad e ; i t i s t h e d iffe rence i n t h e na t ural frequenci e s of t h e c ant ilev e r and ar t icula t ed bl a d e, s; t ha t i s j of " pri o ry si g ni f ic an ce f or the f l a p dyn am ics. Th is flapping m ot i on o f the rotor t ' °R e s earch Sc ie n t is t_ La rg e=Scal e Ae ro dyn am ics Bran ch, MA SA= Am as Re sear ch _! _ C e n t er.

I i , : J_ h a s a b a si c r ol e in h e li c op ter s ta bi l i t y a n d c on tr ol, bl a d e l o a ds, vib rat ion, a nd in mos t o t h er a r ea s of h e li c opt er b eha vio r . An impo r tant a sp ect of h e li- c o p t er r o t o r fl ap ping dyn a mi c s is th e dyn a mi c st a bility of th e mo t ion in f o r - w ar d f l igh t . B e sid e s the q u e stion of high s pee d s ta b i li t y, t his t o p i c a lso is of inter e s t for its impli c ations on the d e cay of t ransi e nt motio n s of th e blade , for example , in response to c ontrol inputs or a e rod y nam ic gusts. Any b significant degradation of the blade stability and response to control du e to forward speed will have important effects on all aspects of th e helicopter ! beh a vior.

In hov er (z er o fo r w ar d sp e ed ), t h e aer odyn a m ic fo rce s on th e r oto r b la d e pro ¢i d e t h e fl ap m otion wi t h high damp i ng , henc e , good s t ability a nd f a st re spons e. Th e mo tion of the b la d e in hov er i s d es cri bab l e by c ons ta n t co effi - ci en t li n ear d if f ere n tial e qu a tions , w h ic h m a y b e a n a lyz e d to obt ai n the dynam ic behav i o r by t h e standard m ethods of l i n e a r s y s t em theory . In fo r wa r d fl ight how e ve r , t he a ero d yn a m i c fo rc es on t h e bl ad e i ntr od uce tim e v ar y i ng - s pec ifi ca lly , per iodi c ar ound the r o t o r a z i mu t h - c oeff ic i e nts in t o th e l ine ar : diff e r e n t i a l eq u at i ons d e s c rib i ng the flap motion. Thes e periodi c c o e ff i - L_ c i ent s a r e due to th e on ce -p er - r evolut i onv ar ia ti on of t h e f ree s tr eam ve lo c - ity s e en by the rot a ting bl a de w hen the hel ic opter h a s for war d s p eed. Systems des c ribed by per i odi c c oef fici ent equ a t i ons requ ir e a c ons i d e r a bly more , involved a n a lysis in order to investig a t e the i r st a bility a nd response to c o n- trol. App e ndix A dis c uss e s the beh a vior c har ac teristi c of periodi c c o e ffi- cient syste m s, and p re sents some of t he r esul t s of the m a t_ _mat ic al theo r y of su c h systems. Of te n a dire c t num er i ca l in te gr a t i on of the e quat i ons of motion is used as either the most or th e only pr ac ti ca l solution method; the math e - , _ m atical th e o ry of p er i o di c c o e ffi c i e nt equat i ons has also bee n us e d fo r th e ' ,I . : { a nalysis of rot o r d ynami c s i n forward f lig h t .

The f lappin g s ta bility of a r o t or i n fo r w ar d fli g h t h a s b e en i nv e s t ig ate d .

in a numb e r of p ubli ca tions ( r e fs . 1 -2 0). T he prim ar y r ea so n for t h e fr e qu e n t i reex a mination of t h is on e probl e m is th e se a r c h for a s a tisf ac to r y t ec hnique f or t he trea tme n t of the i nfl u en c e of th e p e ri odi c co eff i c i en t s in t h e diffe r - en t ial eq uat i o ns, f or even t o de ter mi n e t he s t ability o f s u ch an eq u a t io n is a mu c h mor e diffi c ul t m at hem at i ca l p r obl e m t han u i t h c ons ta nt c o e f f i c ien t s.

Usu a lly t he fl a p s ta bili t y h a s been de t ermin e d by num e ri ca lly in t egra t ing t he _"A e q uat ions o f mo t i o n, and t he n u sing the res u lts of F l o q ue t the ory to f ind the I e ig e n va lues (r o ot s) f rom t he tr ansi e n t s o lution o ver o n e p e riod (re fs . 2 , 4 , 5 , 7, 1 0 , 12, 1 5 , a nd 1 8; th e re f erence s us u a l l y d e sc r ib e the m a them a ti cal 1 theo ry re q uire d fo r t h e meth o d) . Th ere ha v e a lso b ee n a n umb er of s oluti o n s u sing th e met h o d s o f p er t ur b a t i on th eor y ( r e f s. 5 , 6 , 9 , 14 - 1 7 ) , us% n g an , anal og c o m p uter t o sol ve t h e e q uat i on ( r e fs. 8 , 11 , and 20) , and a so lut ion u s i ng t he cl a ss ica l m et hods o f t h e a n al ys i s of Hi l l_s e q uatio n (re f. I). T h e reve r s e f l o w r e gi on of t h e r o t o r i s i m p o rtant i n ve r y hi gh sp e e d f orward , _ fl ig ht , a nd se v eral in ve s ti g a t i ons have co ns i d ere d the aero dyn amics of reverse _ f l o w and i t s imp a c t o n the f l app i n g sta bility (re f s. 5 , 7, 8 , I I , 1 2 j 1 4 , 1 5 , and 1 9 ). P i t c h / fl ap c ou p l i ng of t h e b la d e mo t ion i s ano t h e r im p o rt fact o r i w h i ch ha s b ee n treate d (re fs. 2 , 3 , 8 , 1 2 , 14 , 1 5 , an d 1 7 ) . A n u m b e r o f t h e " !. _ s tu di es ha v e e x te nd e d t h e s tab i l ity a n aly sis b y a dd in g o t h er d egrees of ; _ f ree d o m b e sid es ri gid fla p p in g to th e b la d e mo ti on d escr i pt i o n : flap wi se (_ _ elastic b e nd i ng of th e b la d e (re fs . 4 , 5) , b lade tor sion (re fs . 5 , 9 , 11) , _ r-'_ , _@_ ) bl ade l ag mo ti on ( r efs . I0 , 1 6 ) , o r the fl ap m oti o n of the o t h er b lades of t h e roto r ( r ef . 1 3) . Th ere is , h o we ve r , s ti ll wo r '_ t h at m a y be d o n e w i t h ju s t t h e p r ob le m of f l a pp ing s ta bil i t y , p art i c ul ar ly since m o s t o f t hes e stu d ie s ha v e o nl y p r e sented n tme r i cal results.

T his report investigates the e f f ect o f forward speed on the hel i cop t er rotor f lapping s tabilit y an d response to c ontrol. A solution f or t he s tab i lity i s o btained b y a perturbation techn iq ue which i s reas o nably straight fo rward , and wh i ch p r ovi des anal y t i c e x press i on s f o r the ei ge nvalue s o f the flap m oti o n. Th e so luti on m eth o d is direct e no u g h so t h at it is po ss i b le t o c on - s i d er h e r e a numb er of ro t o r co n fi gu rati ons , in a dd iti on t o t h e pr obl e m of a s i ng le i nd epe nd e nt bl a d e whic h i s t h e sub ject of mo st of th e literat u re . Th e p ert u r b ati on m et hod t o b e us e d h ere is known a s th e m et hod o f mul t i p l e ti m e s ca l e s. I t is d e s cri b e d b r i e fly i n Ap pe nd i x B, bu t in fact t h e m et hod i s b e s t d i s c uss e d b y e x a m p l e, of w h ic h t h ere w i ll b e s everal h ere . F o r mo re i nfo r ma - t i on on t h e m at h e m atic s of t h i s pe r t u r b ati on t ec hn i qu e a nd o t h er s, t h e rea d er i s d Lrecte d t o re f e r e n ce 2 1 (and t o re f. 1 4, whic h dis c- _ss e s t h e tec hn iq u e s s pec ifi cal ly in t he c ont e x t of t h e pr obl e m of r o t o r f lap p i ng s ta bi lit y). F o r the s pee d ra ng e of mos t helic o pter s , the a d va n ce rati o _ (t h e rati o of the h e l ic op ter f or war d s pee d to t h e r o t o r t ip s pee d ) i s a sm all para m e t er ; a maxi- m u m of _ = 0. 4 to 0. 5 m a y b e a ssum e d. Th ere fo re , a pert u r b ati on so l u ti on , b a s e d o n sm all _ m a y b e ex pec t ed t o b e a pp licable o ver t h e e n tire r ang e of i n tere s t fo r m os t helic o pter s . M o st o f the p re s en t w o r k will be c on cer n ed wi th t h e sm a l l _ re sul t s, a lthough th e s ta b ilit y at h i gh u (ar o un d 1 .0 to 2 . 5) wi l l b e b rie fly d i s c uss e d . Fo r th e r ang e of sp ee d i nvo l v in g sm a ll a dv a n ce ra t i o , u = 0.0 to 0 .5 o r so, a pert u r b a t i on so l u ti on t o o r d er _ 2 i s ' s a t i s fact o r y. It i s t h e re fo re p oss i b le to ne g le c t th e effe cts of th e re v er s e I I _ fl o w re g i on on th e r o t o r ( s i n ce t he y are of hi gh er o r d er th a n u 2 . s ee _ re f . 8 ) , which r es ult s in a c ons idera b le sim pli fi cati on o f t h e a e r ody namic i force s w h ic h m u s t b e c ons i d e r e d.

W it h the g reat d ifficultie s involve d in the m at h ematica l an al ysis of per io d i c c o ef fi c i e n t diff e r ential equa tions , it is n atural t o c onsi de r t h e u s e of a c onstan t c o e ff icie nt ap prox i m ati on t o th e e qu ati ons d e s cri b i ng t h e sys - t em . T hat is, the p e r i o d ic co ef f i c i ent s ar e r e plac ed by their avera ge value s , I so t h e e qu ati ons of mot i on are re du c ed t o co ns ta n t c o e ff icie nt e qu ati ons w h ic h b i l it y o f th e c o nst an t c o e ff icient appr o xi m a tion t o s everal o f t h e r otor c o n- _' I m a _ , b e ea s il y an al y ze d by t h e s ta nd ar d tec hn ique s . T h e v ali d it y and applica- figu rati ons c ons i d ere d h ere w i l l b e i nv e stig ate d . An e x p ans i on of th e ii ei g e n value s f o r s mall u wil l b e foun d , f or c o mp a ri son with t he pertur b atio n so lut ions i n clu d i ng the p e r iod ic c oef f i c ients .

• T he follo wi ng t o pics will b e c o nsidered i n t h is rep o rt. First, the } s ta b ilit y o f t h e flap mo t i on of an i n depe n de n t b la d e wi ll b e i nv e s t ig a t ed; I . that i s , t h e ca s e o f a rot o r wit h a fixed s ha f t , so t h at t h e m oti on o f an y o ne

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_ b la d e i s in d ependen t of tha t o f t he o t her bla d es o f the rot o r. A pertur b ation so lutio n (f o r s m al l _ ) w i ll b e fo und f or the i nfluence of forw ard f light o n

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the eig e nv alue s . T his a n al y si s wi ll be f ollowed by a s u mm a ry, wh ich w i ll : . co llect the r e sults fo r the e i g envalue s . The be havior o f t he s e r e sul t s w i ll } be di s cu ss e d , and exa m ple s give n of t h eir appl i catio n . N e x t t he flap st a b il- _ ity of a teetering r o tor will b e investi g ated, fo l lowe d b y a s u mm ary an d b , 3 j_ di s c ussion of th e r e sults. T he n a gi m ball ed rotor w ill be c o nsi der ed, for th e cases of three, four, a n d five or more blades. The desc r iption an d behavior in t h e nonrota t ing frame of a rotor with N independen t blades will be give n ; i t is more appropriate to make the constant coeffi c ien t approximation in t he nonro t a t ing fra m e s i nce some of the influence of t he rota t ing periodic c oeffi- cients is retained. Next, the constan t coefficient approxima t i o n in t he rotat i ng a n d nonrotating frames will be considered, a n d the results c ompared with the solution including the p e r io d ic coefficient influence . Then the transfe r fu n ct i o n , th at i s, the r esponse o f t h e f la p mot i o n to s in usoid a l co n- trol inputs, will be investigated, includi n g the influence of the perio d ic aer o d y nam i c f o r c es i n f o rwa rd flig h t . T h e t ransfer fun ction is an a lternative to the eige nv alu e s a s a r ep r esentatio n of t h e dynami c c h ar a cter i st i c s o f t h e s y s t em. F i n a ll y , some of the p revio u s wor k w i t h this pro ble m wi ll be d i scussed.

N O MEN C LATURE i A co nstan t in t he o r de r _ s o l u ti o n fo r 6 CI , C 2 co n s ta nts f o r the I / r ev c r iti c al re gion D se c u lar e qu at ion par a met e r , d e f i n ing t he root b e h av ior near th e cr i t i ca l re gi on s , H t r a n sf er func t i on . . , K p pi tc h / f l a p f eed b ac k gain, X p = t an 6 3 I. m bl a de i n de x , m = I , . .., N M R flap mom e nt d ue to flapp ing v el oc i t y ' . , M B fla p m o men t d u e t o fla p p ing disp lac em e n t 6 } (, Me f l ap mo ment d ue t o pitc h c on tro l e N numb er o f bl a d e s r bl a d e radia l s tat ion R bl a de r a d i us ;_ S b la d e flap d e g r e e o f f ree d o m 80 ,_ I , B 2 , ... c o e ffi c i e n t s in expan sio n of B a s s e ries in S lc rot o r t ip p ath p la ne pi tc h degree of freedo m @

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" 4 * B l s rotor tip path plane roll degree o f freedom B O r o t o r c o ning mode degree of freedom vector o f B l c,B l S degrees o f f reedom 7 loc k number o f th e blade _ O, T I , T2 ,... coefficients in expan si on o £ y as series i n 6 3 pi t ch / f lap coupling p ara m et p= A ( _ ) deno m ina t or o f the hover tran sf er f u nction A y Y " Y O _ " . , 0 air density n bla d e flap m ode s h a pe e blade pitc h con t rol ei g e nvalue _ 0 ord e r 1 ei genvalue ( t he hover l im i t ) a d v anc e rati o , heli c opter f orward speed d iv ided by rotor tip tt I s p eed i , _ c orn er _ at boundary o f c r it i cal region ° i _ 1 , _ 2 corner _ f or 1 / rev critical reg i on J blade f lap natural f requency (rotating , per rev) t . _ _ blade a z i m uth an gle; nondi m ens i onal time v ariable j _ 0, ¢ 1 , _ 2 , . .. tim e scal e s , _ n = u n _ I 1 _m azimut h an gle o f the m th blade ' _ fr e qu e ncy i fl rotor r ot at i onal sp ee d , . _ (' _ co m ple x conjugate; f or th e tr an sf e r func ti on sj m agn i tude of the input and response

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) i' _ ( ) time de z ivat i ve (when di me nsionl ess , the d e rivative with i r es pec t t o _ ) _' Th e e qu atio ns and p a ram ete rs us ed in t h i s an a lysis a re d i m e ns i o n l e ss , b as e d on t he a ir densi t y p, t he ro t or ro ta tion sp e ed fl, a nd t h e ro t or r a dius R.

P E RTU RB ATION SOL U TIO N F O R HELICO PTER ROTO R F LAPPING STA B ILI T Y j, E qu a ti o n of Mo t ion Consi d er th e firs t mo d e fl a p mo t ion of a singl e bl ade of a h e licop te r r oto r. The ro t or shaf t is fixed s o t here is no coupling of t he bla d es through . _ the shaf t or con t rol sys t em. Then each blade is independen t of t he m o t ion of the o t her blades. The equa t ion describing the flap mo t ion in t he ro t ating fr a m e is 8 + v2B = y[M_ + MSB + M o ( O - KpB)] (I) This is the equation for small perturba t ions of the flap mo t ion from a t rim s t ate. Only blade pi t ch con t rol (also a per t urba t ion from the trim sta t e, th a t is, cyc li c and collective con t rol as r e quired for the gi ve n thrust and forward speed) is included as a n inpu t . The degree of freedom r_:presenting t he per t urbed flap motion is 8. For an articulat e d rot o r 8 is t he a ngle of ro t ation of t he blade abou t t he flap hinge. In general, (i.e., for a c a n t i- lever r o t or, o r a n ar ticul ated bl a de wi t h flap hing e offse t ) t h e bl a de ou t of pl a n e d e fl ect ion is 8n, wh ere n(r) is th e m o d e sh a pe o f firs t m o d e fl a pping, ' normalized t o u nity a t t he tip (at r = I). Th e bl ad e pi t ch c o n tr ol per t urba- , tion is e; i t is inpu t by the c o ntrol system, as rigid pi t ch mo t ion of t he blade a bout a f eat h e ring a xis at the bl a de root. So e is t he pitch change of t h e bl a de a ll a long i t s sp a n. The dot d e no te s t he d e riv at iv e wi t h r e spec t t o t he blad e azi m u t h a ngl e _, whi c h is th e nondlm e nsion a l t ime v a ri a ble.

Th e firs t t e rm o n the l e ft-h a n d sid e o f e q uat i on (i), 8, is t h e fl a pping in e r t ia. Th e second t erm, _ 2 B, is the flap s pring , which ha s s tr u ct u r al and _ c e n tr ifugal con t r ibu t ions. Th e par a me t er _ is the r o t a t ing n a tu ra l fr e - quency of the flap mo t ion (per rev since frequencies ar e nondimensionalized by fl). For an articulated blade, tha t is, a flap hinge with no offset or spring , r e s t r a in, o -- i; t his is e n t irely the c e n t rifugal s t iffening. A bl a d e with s t ruc t ur a l r e str a in t (cantil e v e r roo t , or a hi n g e spri n g), or wi t h th e fl a p , h ing e o ffs e t f rom t he c e n te r o f rot a t i o n, ha s _ > I; _ - I.I to 1.15 is t ypic a l of a c a ntil e v e r ro t or, a nd u = 1 .03 to 1.05 for a n articu la t ed ro t or wi t h hin g e offs et . Th e ri g ht-h a nd sid e of e quation (I) is compo s ed of th e a ero d yn a m ic fl a p mome n ts, d u e to th e fl ap pin g veloci t y an d d is p l ace m ent , an d .

th e bl a d e pi t c h c on t rol. M ec hani ca l pi tc h / fl a p c ou p lin g is includ% d , th a+ is, ' , _ f ee db ac k co n t rol of th e form A8 = -KpS. This fe edb ac k is usually a c c om- : 1 plished ( fo r ar t icula t ed r o t o r s) by m e ch a nical a rra ng e m e nt o f th e f la p hing e , ;. a nd pitch for m equivalen t t o rota t ion of t he flap hinge by t he _ngle 6 3 , s_ I K p = t a n 6 3 . Po s i t iv e Kp . ( i .e., _ 3 > O) is n e g a ti v e f eed b a ck, which in t r o - d u ces a p o si t i ve a er od ynamlc sp r ing i n to t he flap equ at io n t hrough the ' c o e fficien t MO. '_ _ " " " i , ,_ , 6 '. : A Th e f act or v is the ro t or Lo c k numbe r , defined a s y = Q ac R_ / Ib (where is th e a ir d e nsity, a is t hc bl a de section lift-curv e slop e , c is t h e bl a de c hord, R the rotor r a dius, a nd Ib is th e mom e n t of inerti a of t he fl a p motion). Th e Lo c k numb e r represents t he r a tio of the aerod y n a mi c for c es t o t h e in e rti a for ce s on t h e bl a de; y typi ca lly h a s v a lu e s from 6 to 10 for cu , -renthel i copters. The le ft - h and side of equat i on (1) is th e bl a d e a ero - dyn a mi c fl a p momen t s due to fl a pping a nd pit c h c ontrol. N e gl ect ing r e v e rs e flow, a nd a ssuming the mod e sh a p e is n = r, th e n the ae rodyn a mi c mom e n t s a r e M_ = - + _ _ sin MB = g + g U si n , + (_ sin _)2

J

where u is the rotor a dv a n ce r a tio, the r a tio of the heli c opter forw a rd s p e e d to the roto r tip spee d ; fo r curr e n t hel ic op te rs, the a dvan ce r at io at m a ximum forward sp e ed is t yp ic ally be t ween 0.3 and 0.5. T h e n e gl ec t of re v er se fl o w i s v alid to about _ = 0.5, fo r r ev er s e flow a dds o r d er u_ te rms t o the h ar mo nic s of th e f la p mom e n t s. H e n ce , negl ecti ng r e v e rs e f l o w is c ons i s te nt w i t h th e pre s e n t sm a ll _ p e r t urb ati on a n a lysis, w h ic h w il l b e carrie d to ord er u 2 The equ at ion of motion then, substi t ut i ng for th e fl a p moments a nd dr o p p i ng the pit c h c ontrol for c ing terms to obt a in the homog e neous equ a tion, ' ' b ec om e s i

f

The der i v at ion of th i s e qu a tion may b e found i n t he l it er at ur e , f or ex a mpl e i n re fer e n ce 8 , wh i c h a l so con sider s th e rever s e f l ow aer o d yn a m ic s . T h e only pa r a m eter s are th e fla p na t u ral f re qu e n c y _, whi c h is al w a ys I or slight ly , a bo ve ; t h e p i tc h / flap co upli ng Zp, whi c h i s f re q ue n tly z er o for h e lico pter m a in r oto r s ; t h e Lo c k numb e r y , a m ea sur e of t h e relat iv e s tre ng * .hof t h e ae r o d yn am i c f o r c e s on t h e bl a d e ; an d t h e f o rward s peed U . Fo r t h e hov er i l i m it , u = 0 , e q u a t ion (5) r e du ce s to a const a n t co effi c i en t li n ea r di ffe r - ._ en tia l e qu a t i on. F o r forw ard f lig ht , i_ int r odu ce s per iod ic co e f f ic ie n t s i n t o the di ffe r e nt ial e qu a t i on . Th e s, a bili t y of a s y s t e m is d ef in e d by i ts e ig en- i v alu e s , o r roo t s; th ere a re two fo r thi_ s f, con d -order eq u ati on . T h e s ' c a bili ty o f th e Zlap m otion will b e e xamin e d f o r _ A ' eca s e of small a dvan ce r at io , by m eans of a pe r t ur batio n te c h niqu e t o ha nd : ,_ t h e in f lu e n ce o f t h e per io d i c [ ' / coe f f ici e n t s d u e to forw ard fl ig ht. _ , 7

t

Hov er In the hover l i m i t , _ ffi O . the equat i on reduces to T h e e i g e nval u es X o f th i s c c._t ant co eff ic ien t d iffe r e nt ial eq uat i o n ar e obt &in ed from th e c ha r a cter i st i c equ a t i on which has the s o luti o n T h e se r o ot s are usu a lly c om p lex c on j u gate pair, t h at i s , t h e radica nd is p o si t ive . F o r t ha t case, t he t w o roo ts lie on a c ircu la r a rc in the X pl a ne, w i t h ce n t er a t ReX = -Kp a nd r a dius d v 2 + K p' 2 ,so t h e c ir c le goes t hrough t h e im a ginary a x is at ±iv. The re a l p a r t of t he roots is given by y only, ReX - - y / 16, whi ch t hen de te rmin e s t he lo cat ion of t he t wo roo t s on t his a r c .

Th e root lo c us for varying y t hen c onsis t s of t h e por t ion of t his c ir c l e t o , ' t he left of t he im a gin a ry ax is (sin c e y is posi t ive), plus t he re a l ax is i I f rom - ® to - K p. A t y = 0 the root s are at X = ± i v on t h e i ma g i n ar y a xis .

For y t he roo t s a ppro ac h X = -® and X = -Kp (whi c h no t i c e is the ce n ter ,, of th e ci rc le) . The y l ocu s in ter c ept s t he r e a l ax is a t X = - K . /_ 2 + Kpz , ( t h e ce n t e r o f the circ le plus it s r a d i u_), wh e n y l 16 = K p + ¢_ 2 P + K p 2 . Fo r , y s t ill l ar g er th e re ar e t w o r e a l root s, t h at is , th e r adi ca n d o f eq u a - ti on (S} is neg ati v e ; f or sm a ll e r y t h e roo t s a re a c om p lex c onjug ate p air .

For K p = 0 t his i n te r c ep t o cc urs at y / 16 - _, h e n c e at y a pproxim at ely 16, whi c h is qu it e la r g e f or c urren t hel ic o pt e r s. N e g ati ve pi tc h / f l a p _i t coup ling K p (p o sit i v e fee db a ck) is r eq u i re d for the in t e r cep t t o occ ur at -I i m or e usual v a l ue s o f y.

Wi t h tw o r e al roo t s, t he br a n c h o f t he lo c us going t o X = - K p will go in t o t he r igh t h a lf pl a ne - be c ome uns t abl e - if K p < O. Th i s r oo t is on th e real axis, so p a ss e s t hrough t he o r igin ( X = O) to go i n t o t he righ t h a lf , pl a n e . T hat is, a st atic i nst a b il i t y , o r d i v er g e n ce , o cc urr ing du e to t h e net s pri ng r ate b ei ng n e g ati v e . Th e s ta bili t y bound a ry i s c ross e d at y / 16 _ -v 2 1 K p , and th e f l a p mo t ion i s d i ve r g e n ce uns ta ble for l ar g er y. This m a y b ette r b e vi e w e d a s a l i m it of Kp : V 2 , Kp > - _ ( 6 ) g .

/ re q u i red f or s t a b ilit y . A t t hi s bo un d ar y , th e ot he r ro o t _ s at X - v 2 / K p _ _'_"_-' - ( he nce st a b l e , si n ce K p < 0 ) . _ . _ ' 8 _ :

i

1 " I ' i J H o ve r root l o c a tions w h ic h will be o f par tic ular in t eres t ar e tho se w her e th e f n r e qu e n c y is nea r a m ultiple of _ /c ev . The h o ve r ro o t has a fr eq u en c y of h / r e v for y / 16 - Kp _2 + Kp 2 - (1 / 4). For g p - 0 or s m all , t h i s mea n s y app= o xi m a t ely 14 , s o t he h ov e r root fre qu e n c y is ab o v e h / r e v usually the h o ve r r o ot has a fre qu enc y 1 / r ev for y / 1 6 • K p+__ -'-T-[. Fo r K p = 0 or small , this means y m u s t b e s mall . Sinc e _ _ I , t h ere is on ly o ne c ros s i n g of Im_ = 1 or h / r e v by t h e y l ocus ( ex cept f or t he c a se _ = 1 , whem t h e l o cus s t a rt s a t Im X = 1 for y • 0 , an d wi l l have a s e c o n d c ro ssin g i f g p > 0).

The natural fr eq u e nc y o f t h e fl ap mo ti on i s _n 2 = v2 + (y / 8)g p. Thls ts m ai nly given by t he s tr u ct ur a l a nd c en t ri f ug a l stiffen i ng, t ha t is, by t h e f requen c y v; it dep e nds on y only a s i t in f lu e n c es _ :h e effe ct iv ene ss o f pi tc h / f lap c oupling. Neg at ive pi tc h / f l a p f eedb ac k, K p • O , adds a posi t i v e k aerodyn a mi c spring and so in c re a ses t h e eff e cti ve f lap spring r ate . No t i v e t h at _ • 0 gives t he c ri te rion for d i v e r ge n c e of t he f l a p mo t ion. Th e d a m pin g r at io is _ = (y / 16) / _ n . Hen c e, th e fl a p mo t ion is ve r y h e avily ._ d a mp ed in hove r , with ¢ t ypic a lly S O p er c e n t c r i t i ca l damping, du e t o t he high a erodynami c d a mping.

Ex p an sion i n U Co nsider a p er t u r b a ti o n s o l ut i o n for t he stability in f o rwa rd f light at sm a ll u . Using the me t hod of mul t iple time s ca les ( a s described in Appen , di x B), the b e h av i or of t h e syst e m is ex m_ined f or _ o f th e or d er i, u-l , _- 2 , e tc .; t h at i s , l e t , ' @ , _I = _, _k • _ 2 _, . .. Th e n t h e tim e I d e ri vat iv e is i _ + _ _ _ 2

•

Exp a nd B as a s eri es in u , each t e r m def e n d ing o n a l l t h e t ime sca le s _n: I S • S o(_o,_l,_ 2, ..) + _Sl(_ o ,_1,.. . ) + .

, e md a lso ex p a nd t h e param eter v as a s e r ies in u: Y • Y O + I_Y l + 1j 2 y 2 + " " " T hi s exp an s ion of y i s a _ a y o f qu a n tif y i ng w h en y is n ear certai n ,_ " c ri t i cal val u e s; that is, i f Y0 i s som e cri t i cal valu e , an d Y1 • (Y'Y o) / _ - i s ord e r i , t h e n y i s or der _ ne ar Y o" T h e qu a n t i t y y is still t h e pa r am e t e r g iven , hence t h_ s d e compo s i ti on- i n t o Y 0 , YX , Y 2 , . c han ges with ta.

N o w 8, d / d _, and y are a l l ._ x pan d e d as se r ie s i n _ . _ h . es e e xpan si o ns ' are s u bs t i tute d in t o eq u a t ion ( 5 ), m : d a ll terms of t he sa m e o rd e r i n _ _"

| ' |

2 I

i J c nllected, ass_inR that all the co e ffici e nts i. the expansion are of the sam e order. A fundamental assumption of the me t hod of mul t iple tim e s cales is t ha t t he 8n mus t all be the same order for all tht t ime scale s _,.. All th e terms of like order in u are collected and separately set to zero, to obtain the equation that starts the analysis at e ach order.

Order I Results The order I tex,._ of _q a a t ion (3) give __ + + Kp 60 = O ( 7 ) a26o ¥ °a6° (v _ 9. ) " '" a,o 2 8 a , 0 The solution of this equa ti on is

_- 60= R e[ 60 1( _ 0 1 ' *2' " )eX°*° ] ( 8 )

wh e r e th e root X0 is X 0 = i-_ + i 2 -_- Kp - (9) i The convention will be foll o w e d th a t X 0 i s the ro o t w i th p o sitive frequ e ncy ; the other root is th e conjugate _0. Then to ord e r I the equation of motion, ! and so t he roots , a re just th e hover li m it. S i nce 80 is a fuuction of all the time s cale s , equation ( 7 ) i s a partial differential equati o n, which , determines 60 as a function of *0 only. T hus, 801 still depend s on _I' _2 ' etc.

_ . Ord=r u R e sults The ord e r u e qu a tion is ., 2 3 ,0 3' I 8 a, 1 _ - u sin *0 + c, _ S *0 + Kp s in , 8 0 (1 0)

" o )

: 1 Thxs i s a di f ferential equation f o r 61 (_ o ), forced by the ord e r I s o lution i 80( 0 0 ); s ub s tituti ng for B O , it b eco m es p * _"_ .

I0 . _ s , " ! • m = X° + _ + W r Xo + Kr ' _ao : , ,¢ + T_ 1 i(X0 - ' - 2KD 801 e(x 0 + i ) @ 0

,o [

+ _2 [i + i(_o + 2Kp)]801e(X0-i)*0 + conjugate (11) The right-ha n d side has a term of the form A1eX0$0 due to the B0 soluti o n, where A1 is independent of ¢0- But the left-hand side is the same as the order 1 equation (eq. (7)) , so it has the s ame homogeneous solution e X0 * 0 .

Then the solution for 81 is of the form (12) The forcing of equation (10) by its own homogeneous soIution pr o duces the sec o nd term in B1 , which is order _0 c ompared to the solution for 80. Su ' as @0 increases , 81 will become arbitraril T large compared to 80 , which , , v i olates the assumption that all the terms 8n in the expansion of g are the same order. This situation can only be avoided if the secular term A1, -- the coefficient of the homogeneou s solution forcing equation (i0), is ' identically zero.

Assu m e for now that X0 ± i # X0; then the periodic coefficien t s do not contribute to the secular term of equa t ion (!0). Se t ting the secular term to _. zero giv e s a differential equation for 801(_I): or B801 ., XlSO i : 0 (14) wher e I I J ! )'

! /

V . C Y1 -_ (t 0 + Kp) _1 = - 2iIm_ 0 The solution of this equation is B01 = 802(_2,. .)e I_I, so the solution for B so far is _

= Re[_o2c,2,. .)e_°_° OCt) C15)

The e igenvalue to order u is t hen X = X0 + U _I =" + i 2 + _ Kp - (16) L That is, _1 is just an order u perturbation of the h o ver root, due to the ' order u expansien of y . To order u th e n , the eigenvalue remains just the hover val u e, with no influence of forward flight at all.

The ass um ption tha t T O ± i # X 0 m e ans that Im X 0 _ ½ / rev . T his requirement then is t ha t the hover roo t X should no t have a frequency near ½ / rev; "near" means being able to write the hover roo t as _0 plus an incre- men t , such that when _0 has a frequency of ½ / rev, I - _0 is order , small. As U in c reases then, th e distanc e from ½ / r e v which c onsidered "near" in c reases. The analysis will return to the case Im X 0 = _ / rev la t er.

i

Wi t h t he secular ter m removed, equati o n (I0) becomes a_--_0 + 8 a_ 0 v 2 + Kp B1

_ o12 [i -iCXo

l _ YO12 [i + i(10 + 2Kp)]801 e(10"i)*0 i + conju g a t e (17) : The solution of this equation is + I A+e + A e (1 8 ) B, = R e S11(_i) e xo ¢ o 1 . 2 - 8o, ..

{ Y o ..r(x°+i)*° (x0-i),_} 12 _ ._ '_ t , where 1 ¥ i(t 0 + 2Kp) A ± = ±2Im10 + 1 The second term is the par t icular solution, which with the secular term _ dropped is now the same order as Bo. This completes the solution to order _; the solution so far is Bo($0,_l) and fll(_o).

Order v 2 Result s . . .

Th e orde r u 2 equati on is

]

a261 + --- + + sin _0

= 2 _¢o_¢_ 8 a_ -6-

+ COS _0 + Kp -_- + Kp -_- sin _ fll - F

(_ x_ Xo o)

a26 o Yo aB o a280- + + u sin _ _ \ T "6- sin _0 * _ + 2 'a#0 ;3¢'2 + 8 _}¢'2 + COS _0 + -_" sin 2_ 0 + Kp + + sin _0 - T cos 2_ flO

,o 0 )]

i

(19)

Th e s ol u ti on s t o ord e r u for 60 an d fll _r e sub st i t u te d in t o t his e quation, and the coeffici e n t of t he homogeneous solu t ion eI0¢0 - that is, th e s e cu- I ¢_ tar term - are collected. Assume for now t ha t 10 _+2i # 10, tha t is, [mA0 # i / rev; i t has already been assumed t ha t t he hover frequency is not near ½ / rev. Then the secular ter m is a611 _1811 = + k 2 + " 80 ekl_l _'_1 __--_--2 2ilml 0 (20) _. where _2 zs t he or der v 2 t erm in the expansion of t he h o ve r root : AO + BII + _212 = - _ + i 2 + Kp - + O( u 3 ) (21) - . 4 , 13 "< I : Regarding this as a differential equation for 811(_1), it is fo,ced by i ts own homog e neous soluticn eXl@l, qe.ting the secular term of eq u ation (20) (i.e., the entire right-hand side) _ o zero gives an equation for 602('P2): _802 Kp _ \_ l i(X 0 + 2Kp)(A+ - A _ ) + + 802 = 0 (22) %& 2 _2 2ilmX0

- "i

Thus, the eigenvalue to order u 2 is X = X0 + U X I _ _ 2 2 - 2iimXo + u 2i IK t, (,-, or to order u 2 this is }, = - -- Y - + i + (i+ _ Kp-i-'_6_ 2 (24) 16 - 9 _ 2 + _ Kp - - , Then to order U 2 - and for y and v such that the h ov er frequency is away from ½ / rev and 1 / rev - the i nfluence of forward fli g ht on the ei g envalues is simply a small (order u 2 ) change i n the frequency. The first effect in the ' frequenc l just co r rects the (y / 8)Kp spring term t o account f o r the increase of the mean of KpM o with _. The second effect of u is entirely due to the periodic coefficients. This order u 2 change in the frequency is in fact r quite small f o r u o ut t o 0.S or so, a s th e examples below will show. Th e re _&. is no influence of forward flight at all on the damping of the root.

( I This exp re ssion fo r the u igen v alue wa s d e r ived a ss u ming that the ho v er root . are a c omplex c onjug a t e p a ir; it m a y be shown, how e ver , t ha t th e same expr es sion is v alid f o r y lar ge en o u g h t h a t t he two roots are real, tha t is, when the radic a nd is nega t ive. A point of par t icular interest is where one branch of the locus on the real axis crosses into the right half pl a ne, t hat is, the divergence stabili t y bo u ndary. The cri t erion for t his bo u ndary is t ha t X = 0, which f r om e qua t ion (24) g i ves v2 2 , 14 I I I or sin c e -( y / 8)Kp is an or de r u 2 dis t an c e f r om _ 2 t h e c ri t erion on Kp for stability is

(1 + Kp, . 16 (2s)

The u eff e ct o n t he r i ght -h a nd s i de (the peri o dic c o e f fic i e nts ) do mi nates t hat on the le ft -hand side (t h e av e r age o f KPM O ) . So the c riti c al v al u e o f n e gat i ve K p , b e y o nd w h i c h t he l oc us li e s in t he r ight hal f pl a n e , i s a ctu ally ' - in c r ea sed by _; t h e hov e r c ri t erion on K p i s t h e n c ons er va t iv e i n f orw ar d fligh t . Th at is t h e oppos i t e c on c lusion as would h a ve b e en r eac h e d c onsid e r- ing jus t t h e a v e r a ged c o ef fi c i e n t s. In a ny ca s e , ho we v er , t h e i n f luen c e o f fo r w a rd f l i ght is only o r d e r _ 2 s mall .

Equation (24 ) also g ives the effects of p on the boundary bet w een w here _. th ere a r e two re al ro ots, and wh ere th e r oo ts a re corj u g ate pa i rs. T h i s boundary i s giv e n by ImX = 0, w h ic h t o o r d er _2 i s Y---= (Kp + / _'2 + gp 2 , p2 _ (26) 16 _¢_ + Kp 2 For Kp = 0 t his reduces t o simply J I Nea r ½ / r ev Fr equen c y l m X0 -- 1 / 2, t h e n X--0 + i -- h 0 a nd th e per i o d ic coef fi cie n t s c ont ribu t e to the t Now re turn to t h e c a s e when th e h ove r r oot fre q ue n c y i s n ear ½ / rev . If order u secu l ar te rm, T he c rit e rio n Im X 0 = 1 /2 means T Y0 _ r _ 1 I-'6-- K p + + K p 2 - _ .

- C inc e v = YO + U Y I + • ., t h at mea ns y m us t b e su ch t ha t _ - Y0 is _ ord e r u s m all. T he o rder i r oot is t h e n X0 = -(Y 0 / 16) + (i / 2), and th e o r der u s e c u lar t erm is now X0, _.A+ = , @*I -8- (xO + Xp)801 + I-' Z 1 - i(lo + 2 Kp _'01 0 2 Y 1 Y O --

IS

I }, • I Or _*I X 1 80 1 + 1-_ 6 Kp - _-_ i g0 1 = 0 (2 8 ) _801 /_O Y O Y O where here, since ImX0 = 1 / 2, the order _ e x pansion o f the hover root is J" xl = - I-- 6 -iT Y _- K .

Y1 YIIYo p) s- The _01 term is the perio d ic coefficient contribution to the secular term.

The solution of an e qua t ion of this for m is given in Appendix C. The solution for 801 depends on t he quantity - L\192 -6-Kp + =[7(_-_- KP)]2 _ (_7(_ 2 _ TK PYo + 4 Kp9 { 2 9 ) Now if D2 > 0, the solution for 801 has terms with time behavior like i -{Rell-+iD)_ 1 -u(y1 / 16)_+i u D_ , e = e an d then 8o has t e rms l ike (

_o_o -(y 1 16))_+i) [(i 1 2) +pD]

80 1 e - e

< I

due t o the period i c coe ffi c i e n t s w hen t he hover f requency is n ear _ / rev. If The d am p in g i s u nc hanged , an d there i s an order _ c h an ge i n t he f requen c y, D2 < O , t hen 80 1 h as t er ms l i ke I

e'(ReXI+D)*I = e['U(Yl / 16)-+nD] *

a nd th en 8 0 has term s li ke

[- (y / 16)"uD],l. , + (i / 2 ), 7

' T h ere is a n order _ c han g e in t h e da mping, b ot h m ore a nd l ess s t ab l e , whil e i t he f req u e n cy re ma ins fi_ ed a t ½ / rev.

i 16

# " 8Ol e _°i ° - e _i I The infl u en c e of _ o n t he root s nea r ½ / r e v frequen c y t hen is fi r s t an order _J c hange in the frequen c y t o wards ½ / re v , wi t h the damping at the hover value. Wh e n t he r oot s reach ½ / rev, t he freq ue n c y re m ains f ixed while there is an order _ c hange in t he damping. One r oo t is s t abilized , the lo c us mo ving to t he lef t on t he X plane, bu t th e o t her is des t abilized, i t m o ves t o t he right. This type of behavior of the roots is characteristic of periodic sys- tems (as dis c ussed in Appendix A). Indeed i t appears here due t o t he c ontri- bu t ions of the periodi c c oeffi c ien t s t o t he order u se c ular equation when the hover frequen c y is near ½ / rev. There is in t his c ase a c _ tical _ gion , inside of whi c h a c hange in the flap damping o cc urs. For many problems wi t h periodic c oeffi c ien t s, t he system is uns t able inside such a region. In this c a se, however, t he hover d amping, ReX = -¥ / 16, is quite larg e ; and the change in the damping is only ±uD, hence, ord e r u small compared to the hover damping. So the critical region is a region of stability degradation rather than instability (for small u, i.e.).

The boundary of the critical region is given by D2 = O, or

K W = - iz / TKP* 4 Kp 2 (30)

Sin ce Y = Y0 + _YI, write Y = Y0 + Ay w h e r e y is the v a lue su c h that the hover root is a t _ / rev. Then if th e hov e r fr e quen c y w i th y is n ea r ½ / rev , Ay = y - Y0 must b e order u sm a ll. The bound a ry of th e c riti ca l r e gion ' is t h e n ' _-_- K _ = _ " T Kp + 4K p 2 ( 3 1) ( Y o p) AY + i _ YO / x ) 2 Y O Considering the _ root loc u s , t hen the cr it ical region bound a ry i s the v a lu e of _ for w h ich the locus re a ches ½ / rev frequ c n c y, a nd is j ust abou t to encounter the st a bility ch a nge at _ / rev frequency. From th e b eh a v i or of the _N- _ r o ot l oc us, this b o un da ry va lu e wi ll b e d e not e d U co rner, w h e re the n 1 - 16 _ c o rn e r = (3 2 ) . . _ / v 2 YO - _ K p 2 Fo r K p = 0 t his redu c es t o U c orner = ±(A_ l 1 6 )(3 / 2 v). I n g e neral, thi s " eq uat i o n gi ve s the b o un dar y o f t h e cr i t i cal re gi o n o n t he Y - u pl a n e. Th e eigenv a lues a re i _ _ . ° / 1 7 '_ A = _0 + uReXl - i_D ( 3 3) whi c h m ay b e w r i tt en I = - 1-_+ _- i i-- 6 2 - K I - (3 4 ) A _ Y O _ co _ne r F o r v = O , this reduce s to X = .._--+ i _ - _- 2 Y(_-_ ) __ - 16 _- - i 16 - Kp ) r" which ts j ust an order u (i.e., order Ay) exp a nsion of the hover root fr o m _ / rev f r equ e n cy a t Y O ' N ear 1 / rev F re qu e n c y ' Co ns zd e r n o w the c as e with the ho ve r root fr e quency ne ar 1 / r ev . Wi t h ' i ImX o = 1 _ th e n T 0 + 2i = XO , s o t h e p e r i od i c c oe fficients c on t r ib u t e t o t h e or d er u * s e c u la r ter m . Th e cr it er i o n ImX 0 = 1 m ean s ' _0 1 16 = K p + _v 2 2 i, a n d th e or der I r oo t is then X o =- (Yo / 16) + i .

F or this c as e ImX0 # 1 / 2, s o t h e o r de r u r esults a re applic a b le ; t hen th e _4 root t o or der u is kn o wn (i.e., X = _ 0 + U Xl), a n d now t h e o rd_ _ 2 i n flu- e n ce is s o ugh'c. Th e s ecu l ar term o f e qu at i o n (19) b eco mes, f or Im Xo = I:

( )

¢ k ' ¢ a8 11 X l S z1 = | a-'_- + - _ '2 + ,. _ 2 . T Kp + 4Kp 2 ! - l p _-_ 8 02 oX I_ I

, o ] 1

, # " 3-_ ( I - il p ) C35) t " YO } _ -02e _ i * 1 , I f . | j 1 8 i ...................... I ........... rE j_ °_ The 802 term is the c on t ribution of th e p e rio d ic coeffi c ients to the secular term when ImX 9 = I. The secular t er m of th is equation is the coeffi- cient of e XI_I, hence, it is exactly ". sbefore (eq. (24)) - giving the same result for the eigenvalues to order u 2 - unless kI = kI. In tha t case, which requires that XI be real, the en t ire right-hand side is the secular ter m o f equation (35), including the _02 term. Since _" xl = T6 + i y6 -T 6 + K , YI " ¢ I(Yo p) requiring ImXl = 0 means that _I = O; and so Xl = 0 in fact. For the periodi c co effi c i e nts t o co n t rib ut e t o t h e o rd e r u 2 s ecu lar t erm r e quir e s then that ¥ = Y0 + u 272 + ., that is, y must be such that Y - Yo is order u 2 small; this quantifies what "near" l / rev frequency means. For a given u t hen the hover frequency m ust be closer to the critical value than was required for the 'i / rev region. The effect of forward flight near I / rev _. frequency is smaller than near h / rev, only order _2 compared to ord e r u for the latter.

With Y1 = 0, t he secular term of equation (35) becomes 2802 Y 2 i 2 + - K p 80 2 , _,---_ + i i-- 6 (xO + Kp) + _-_ - Kp 4 Kp _' 6 + _-_ + i K p g i _ . _- 6 + 2 1 T6" 2 K p + g02 = 0 ( 3 6) , [ - , ,0 - o ) ( - '0 0 < - -.

!

T h e s o lution f or 80 2 d e pends on t h e qu a ntity D (se e Ap pen dix C) giv e n b y "" ' " v2 Kp * 4 1 (p 2 . 2 -- T6 -K T6 + 1 2 i D2 " ) ' 0 Y 2 - _ - K p ' - \32 7 | " 9 2K + P " T - 2Kp (37) T he b eha vior of t h e roo t s is similar to that n ea r ½ / r e v, e x ce pt th a t a ll • _ cha nge s du e t o forw ar d f l igh t are h ere o r d er _2. Th ere i s a c riti ca l r e gion , insid e whi c h there is a n ord e r u2 c hange in the d a mp i ng of th e fl a p motion.

Fo r s m a ll u , t he d a mpi n g i s t he s a me a s t he hove r r o ot while th ere is a n } ord e r u 2 cha ng e in t he fre qu e n c y , towa rd s I / r ev . At t he boun dar y o f t he | cr it ica l r e gion th e r oot reac h e s I / rev fr e qu e n cy ; a nd fo r s till l a rg e r _, __,_.

J , 19 I_I .

i nside the cr i tic al r e g i on , the f re q u ency is fixe d at 1 / re v wh i l e ther e is an order _2 change in the damping , one root becomi n g mo r e stable and the oth e r l e ss.

The bound a ry of the critica l r e g i on i s g_ven by D 2 = 0. S i nce " ( = Y0 + uP Y2 w rit e Y = Y9 wh er e _0 is the value such that I the hove r root is at 1 / rev. Then , xf t he hover fr e quency w it h _ is near 1 / re v , _ : _ - Y0 must be o r der u 2 small. T he critica l r e gion bound a ry m a y t hen b e w r x tt en - Kp 16 -- { C 1 -+ C 2) '

Yo ) _ u 2 (s 8 )

where _ ) 2 Y O _" C 1 = _ (_)2 - _' 1 2 K P + 4K p 2 + K p 1-_ Y0

_ r 7 ( v 2 1) 2 Yo .... _ + Kp i_ (39)

16 1 6 (_) 2 , C2 " " _ - _ (_- 2 K p 2 K p

.0 / i >l.E .0. 0 >7

, 16 , _ 2 16 (40) ' = _ 1 + -if- , - I + gp - - t j- (v z " i) - 2 gp # T h e n th e u root lo c us r eac hes the c r i ti ca l region boundary at Ucor ner -- _ _ o r B 2 , w here

(

II U l , u 2 ( 4 1) I C l - + C 2 F o r K7 - 0 , t hi s b ec om es !

u , 2 u 22 = , ( 42 ) ; I -_ - I t I+ - - _.

20 "

' li

_ 2

m, T h e e igenvalu e s are n o w X = XO + _2ReX2 = i_2D Y + i - (_-_ Kp)- C - C2 2

-

_ ¢ + i - i A_, ' YO Kp 1 2 1 ( 43 ) ,-

:_ -

16 _ 16 - B1 _2 2 For u = O, this reduces to simply an order AO (order u 2) expansion of the hover root f-om 1 / rev at Y0" _" Summary, and Discussion of t h e R esults T hl_ sectton summarizes the results of the perturbation solution for the influenc e of f o rward flight on t he heli co p t er ro t o r blade flapping s t abili t y.

Fhe expressions for the eig e nvalues are collected from t he analysis above• The behavior of the roo t s is discussed, in terms of t he _ root loci and the crltical regions o n the _ - u plane. These results are for the shaft-fixed stabllitv of an individual rotor blade.

l ' Ho v e P.- The hover limit, _ = O, has t he eigenval u es , X = - ± i v 2 * _ Kp - _ (44 ) / (' ; T h es e r o o ts a r e us u ally a c ompl ex c onj u gate pa ir , lo c a te d a t ReX = -y / 16 o n th e c ir c ul a r a r c with r a dius v ¢ _ , Kp 2 and ce nter a t X = -Kp. F or wa rd _i flight , _ • 0, introdu c es p e riodi c aer o d y n a mi c for c es into the dyn am i c s , w hi c h r a di ca lly influen ce s the b e h a vior of th e eigenv a lues a nd the a n a lysis te c h- nique required t o obtain th e m. A perturbation method based on small u has b e en us e d to obt a in expli c i t e xpr e ssions for th e roots wh e n _ > 0 , in c luding I th e eff ec ts of t h e periodi c c o e ffi c ients. I t is a n order _2 an a lysis • ( c onszsten t w_th t he negl ect of t he r e v e rse flo w region effe c ts), w hi c h is va lid t o a pproxim ate ly _ = 0. 5 .

Fo r_ z , d _2 igh t, m oay from crit ical r egions .- Wh e n t h e h ov er ro o t fr e qu e ncy ts n ot t oo c lose to a multipl e of ½ / r e v, the ro ots t o o rder p 2 ir e 8 _ ( 4 S ) , ¢ t " 2 1 ,i i } .

J | Th ere i s o nly an o r d er _ 2 cha ng e i n the f reque n cy , wh ich i s qu ite sm all eve n up t o u = 0. 5. This e xp re ssion ap pl ie s i n p a r tic u lar wh e n th e r e are two r ea l roo t s , th a t is , w h e n y is l a rg e e no u gh so the rad i c and i s n e g ati v e , so i t giv e s the c rl teri on for d i v er g e n ce i ns ta bi l ity, in cl uding t h e influ e n ce o f f orw a rd fligh t . Th e d i v e rg e n ce bound a ry is g i v e n by wh e r e on e br a n c h o f t h e l o c us on th e r ea l a xis go e s th r ough th e o ri gin i nto t h e r i gh t h a l f p l a n e . Th e boundar_ c riterion is then _ = 0 , fo r whi c h e quat i on ( 4 5) giv e s th e c riter i on _" for divergen ce stabili t y a s ( 1 • _ 2 ) _ Kp > - 9 2 + b T - ( 46 ) ,_ Thi s i s a liml t on the pitch / f l ap f eedback a llow e d; the di v e rg e n ce in sta bili t y o cc urs wh e n K p is suffi cie n t ly l a rg e n e g ati ve (posi t iv e f ee db ac k). A c on - s t ant c o e ffi c i e n t a p p roxim ati on to t h e e qu ati on of mo t ion , us i ng th e a v e r a g e _. o f t h e co e ffi cie nts, in c lu de s only th e eff ec t o f the m e an of M0 on th e left- hand side of equa t ion ( 4 6); tha t is, a conservat i v e approxi m ation for th i s case.

Forward f light, near _ /rev freq uency . - It is c ha r a cte risti c of a syst e m with periodic coeffi c ients tha t for ce rtain valu e s of t he parame t ers t here occurs a degradation of the stabili t y. Typically this occurs where the basi c eigenvalue - here the hover root - has a fr e quen c y c orr e sponding t o a multipl e of one half the f un damen t al frequency of the e qua t ion coefficien t s. Specifi- I c al l y, co n si d er wh en the f r e quenc y of th e ho ver r o ot o f t h e f l a p mo t ion is n e ar ½ / rev. Th e r e o cc urs t h e n a n ord er u i nflu e n ce of forw ar d f ligh t. When the f requ e n c y Is n ea r _ ] r e v , e qu ati on ( 45 ) i s no long e r valid Cor _, a nd t h e f o ll o wing r e s u l t m u s t be u s e d ins te a d. L et YO b e t h e value o f y fo r whic h t h e hov e r root would (wi t h th e given v and Kp) h a ve a frequen c y exa c tly ½ 1 r e v; # t ha t is, Yo l l6 = Kp + / _ 2 . Kp 2 . (I / 4 ). Wr ite Ay = y - Y0" Th e n, if Ay / 16 is ord e x u sm a ll , t h e e i genvalue is g i ven by " " I-_ i 16 - Kp) / I _ ) - Uc o -rne r (4 7 ) wh e r e i _ c _m er - . _ ( 48 ) • ,_ . _ - / _ 2 YO - - _-K p + 4Kp 2 Th e s ub scr i pt "c or n er " re f ers t o the b eh a vior o f t h e u root l ocus on th e p l ane , as d i sc u ssed below ; it _ s the bound a r y of th e cr i t i cal re gi on. For K p • 0 th x s res ult red u ces to Y O / 1 6 = ¢ _ 2--TF W , an d U co rn e r • (Ay / 1 6)(5 / 2 v ) , _ -- _ , for t h e b ou nd a r y . _ - ; , 22 ,_ t .... ..................... _ ......................... -# - _ . = .

-- = .

lhls soluti o n e xhibits the .,". ': ,_il:g behavior. If u < u u m e r, th e re is .:_ _r d cr _ chonge in the freo:.. _ - , to , _ard _ / rev, while the real part of the root ret,_ai n s fixed at the _;_,- , i;_e of -y / 16. At u = Ucorner the roots r e ach !-m_ : ½ / rev. ! : r,r . : n er t h e f r e e ' e ncy r emains fix e d at ½ / r ev while the . _ . ..'s m l or der L. _.: . .. , ge in the d a m ping; ReX is increas ed f o r o n e root and J oe'-eased for the , , ,e_.. This behavi o r o f the l o cus near _" lm), :- _ / rev is character' :.._t _ c of p e rio d ic systems (Appendix A). _q_ile th e re is a stabillty degradatiou tf u is large enough (greater than V corner , which decreases with 8-,, : . e . , as th e hover ro o t approaches '_ / rev) , the reduction tn damping is order u small. The hover damping , ReX = -y / 16 , is quit e large for u sual values of y , and so stability is maintained for small u , even with the influence of the periodic coefficients.

Fo r_ apd f _£ gh t , neap i/ r, ev fpequenoy .- Similar behavior is exhibited when the hover root frequency is near I / rev. Let A y = y - Y O , wher e now Y0 is the value of v f o r which the hover root (with the given v and Kp) is exactly at I / rev , that is , ¥0 / 16 = Kp * I v 2 + Kp 2 - I. Then if Ay / 16 is order small the roots are given by X = -_._ + i - - K 1 - ( 49 ) p2 2 "_ wh er e the c o rner u a re a 16 (s o ) ! U 1 2_ 22 = CI + C2 E_p r essi o ns fo r Cl an d C 2 a r_ g iv en ab ov e ( e qs. ( 3 9) and (4 0 )) , a lo n g w it h th e r e sults fo r the l i mit Kp = O. The b e havio r o f the v lo ci is like that n ear ½ / r ev , except t ha t h ere all chan ges a re onl y ord er u 2 F o r s mall u _'( th e re ts _ o rd er U 2 chan ge in t he f r eq ue ncy , t o war d l / re v , w hi l e t h ere i s n o ctange tn t h e rea l pa rt fro m th e hov e r v a lue. For u = U l or u 2 (onl y one will be real) the roots rearh ImX = l / rev , the critical re g ion boundary. For still larger _ , the frequency is fixed at 1 / r ev while there is an increase o f the real part of one r o o _ and a decrease of the o ther. The stability degrada- txon t s o nly order v 2 small , s o again the flap motion will remain s table f or small u and reas on a b le y .

, Co mp a ris o n s wit h nu mer ica l so l ut i o ns f o r the f l ap r o o ts (f r oN c a lcula- ,_ t i on s by t he aut h or , a nd f rom th e li te rat ur e, e . g . , ref. 1 2) i n dica te t h a t t h e i " pe rtur bati o n sol ut i o n t o or d er _2 is a cc urate to ab out u = 0.5. T h is ?

Roo t _ _ ov _ ovo ap dfi ' l, £ gh _. - T y pic a l _ r oot l oc i are s h own i n s olu t i o n th e n c o v ers th e r an ge o f in tere s t f or mo st h e lic opter s. _ t_.

fi gure 1 , for se ve ra l cases of _ an d y . The c ases c onsi d ere d are : ( a ) v • 1 _ , ' .i.!_ I _ d y - 1 0 ; (b) v = 1 .1 an d y = 6 ; an d (c ) v = 1 an d y = 5 . T he p it c h / f la p f eedbac k Kp = 0 f or all t hree c a ses. T hen t he h o ver ro o t s, u O , a re l o c at ed o n a circle wi th r a d x u s _ and cen t er a t the o ri _k n. C a _ e ( a ) is a typzca l art i culat e d blade; wi t h t h e large y and _ - 1, the h o ver frequ e nc y is w ell belo w 1 / rev. H e n ce, wi t h u > 0 t he l o cus e nc o unt e r s the _ / rev crit i ca l reg i on. C a se (b) i s a t y pi ca l c an tJleve r b l ad e ( i. e. , a hing e l ess ro t or) ; _ w ith v • 1 and a lo we r y, the ho v er frequency is abo v e 1 / r eJ . H en c e, t he l ocu s e nc oun t er s t he 1 / re v c ri t i c al r e gion. C a s e ( c ) i s an exa m ple -_ th e beh avio r w h e n t he h ov er frequen c y is a w ay fro m an y m ul t ip l e o f ½ / rev. S o the on l y tnf l u e nce of _ is a ve r y sma ll - o r d er u 2 - c h ang e xn th e f re quenc y.

T he loci shown i n f i gure 1 thus co ,, er all t he c a ses o f aw a l f rom t he crit i cal reg t o ns, n e a r ½ / re v , and n e a r 1 / re v. Th e y il lus trate th e fo rm a n d ma g n i tud e . _ of the tnfluenc e of for w ard f l ight. Spec i ficall y , thr b e hav ior in the crzttcal regions at _ ' r e, - and 1 / r e v freq ue n cie s is sh c _n ; and th e l arge - order u - eff e c t cf th e ½ / r e v r e g ion , ;L o .,Ith e qu i t e small ef f e ct awa y fro m all critical regions may be se e n.

- u / 9_ a r u _.- T he e i g e nvalu e s d e pend pri'.ar il / o n y and u , s o the abov e results m ay be pres e nt e d a s contours o f cons t ant ReX and t h e co n stant Im_ o n t h e Y - u p la n e. Th i s i s t h e p re s e ntat io n fo und i n m u ch o f t h e li t er a- tur e . Suet, pl o ts are sh ow n in fi g ur e s 2 t o 5 f o r K p = 0 and _ = 1 , 1.05 , 1 . 1 , a n d 1.1 5 , re sp e cti v el y; a nd f o r u - 1 a nd Kp ffi 0 .1 i n f igure 6 . Th e y are b a s e d o n th e per t ur b at ion solution giv e n h ere. T h e exp re ssions fo r t h e ._ cr i tical reg i on boundar i es in t e rms of th e c or ner u for thu u root toc i m a y b e r e ar r ang e d , t o g i v e th e boundary i n t erms of y for a giv e n valu e of ,. Writing v = Y0 * Ay , th e t w o r ea l root r e g i on boundary i s

)

Ay l a 2 ; ' 0-'dJ- Kp + + Kp2 + K p = -- (Sl) w here Y0 / 16 = K p + /v 2 - K p2 . T h e ka / rev c r t i cal region bo un da r y is Yf ,] _ 2v2 Y O 2 _ . A , ¢ _ , " " T K p + 4KI..

- ' _ - _ - + u -- -- -- ('_2) -_- K! ) w he r e _ 0 / 16 = Kp + / _ '2 , K p 2 . I / 4 . An , i t he l / r e v cri,'. _ ca l r e g i on bo un dary i s ay u 2 Cl ± C2 (s 3 ) , . * Tg " Y o

Kp

I w here V o / 16 - Kp , / £2 , Kp2 . 1 , an d th e constan t s Cl _m C 2 ar e g i ven ' .

above (eq s . (39 ) an d (40)) . This expressi _ m sho w s t hat C I gives the offse t ' : , .

• I ;4 ii t

: N5

i

- ' " - .......... - ..... : , ; =- ...." - _ __i_ " -

of the 1 / rev critical region boun d ary frGm YO' and C 2 gives the width of the region; the ½ / rev region has no offset to o r der v, so is s y ngnetrical in A_ about Y0" For Kp = 0 these boundaries reduce to

Vo A__16 = _2 ._ v 3 with 1-- 6 = v (5 4 )

Ay _ _-u 2 YO / _2 1 ( 5 5) I-- 6 - _ _ with i-' 6 = -

l--g = - 1 I) +-IFi- A "

Ay U2f 16 2_ V _ 1+ 1 / _ = 6 1

with Y__O = _ / _. i (56) _ 16 r esp e ctively. The bounda r ies in figu re s 2 to 6 we r e c o nst ru cted using these expresslor.s Only the critical region an d real root boundari e s are shown here , but expressions for constant ReX and ImX may also be obtained from the perturbation solutions. The prese n tations of the Y - U plane results given in the l_terature usually include the ReX contours at least (since only , numerical results are usually avail a ble , the y - _ plane is the m ost t efficient way to indicate the stability trends with y and V variations).

No i nstabilities a re e ncount ere d on th e Y - u pl_l e , f or the r ange o f parameters shown. A divergence (static) instability is encountered at hig, y if Kp • 0 , and a n instability in the critical region (usually the 1 / rev re g ion first) is encountered for much higher u (arou n d 2). The critical regions , combinations of y and u where the frequency is fixed at ½ or 1 / rev, are the principle effect of the periodic coefficients. Notice they encompass tN I( m ore and more o f the y ra n ge as v incr e ases, that is, as the periodi c coefftctents incr e ase. The two real root region (ImX = 0 , i.e. , two roots on I the ReX axis) is due to y b e ing lar g e enough s o the flap m oti o n has super- critical damping; w hile it is influenced by the period i c c o efficients, i t is not th e same type of ph e nomenon as th e critical regi o ns (the roots for this cas e a r e g _ven b y the same expre s si o n as f or two co m pl ex r oots aw a y fr o m th e _.

uritical re gi o ns).

A h or iz o n ta l line o n t h e Y - u pl a n e i s a l i n e o f c onstant y, hence , ?:_ the v a rxa t io n of ReX and ImX a s s u ch a line is t r a v e rsed g iv e s the foot _ locus for v ar y ing V . F or ex am pl e , cons ider t he V loc u s f o r c a s e (a) abov e , ' l _ _ = 1 and y = 10; the y = 10 li ne i s i n di cate d on fig ure 2. A_ u inc rea ses _ from zero, th e line remains para l l e l to ReX = consta n t line s , so R e X re mains fixed at the hov e r valu e . Th e ImX = q / r e v cr i tical r e gion comes close r to th e ho r izo nta l lin e , indi cati ng that t h e f re qu e n c y of the r oot _' a p pr o a ch e s ½ / re v . A t th e corner U , th e l o cus c r oss e s i nt o th e ½ / rev regi on. _'_ 2 5 . _; • Fo r highe r u the lo c us is in the c r itic a l r e g ion, so t he f r e q uen cy r_ a ins fixed at ½ / rev while for each point in the region there will be t wo values of Re_, one more and one less stable t h a n t he hover roo t . The con s t a n t y line s for cases (b) an d (c) are also shown in figures 2 and 4; ti le b e havior of the Y - u plane along these lines ma> be co m pared lo the _ loci shown in figur e 1.

Figure 7 s how s t h e y - _ pl,ne for v = 1 _md K p = O, from a perturbation solution equivalent to t h e presen t one but including th e order ' u2 c o rrection t o the ½ / rev critical region bo u ndar y . That solution, fro m reference 6, gives the ½ / rev bo u ndary as f Ay ±u 2 _2 i0 1-'_= _ (ssa) to order u2. Figure 7 may be compared t o figure 2 , which is based on the present solution , hence, only to order _ in the ½ / r ev region. While the L order _2 influence is not negligible for _ = 0 . 5 or s o , th e m ajor effects of forward flight are contained in the order _ solution. The res u lts of reference 6 are discussed further below.

High u behavior.- Fo r _ lar ge r t h a n 0.5 or so , a n um erical me t ho d -_ must be used to calculate the eigenvalues of the flap motion. It is also necessary then to include the reverse flow effect in the aerodynamic flap u oments. At very high _ (above 3 to 4, say) a perturbation solution based on ' an expansion in _- 1 is possibl e (r e f. 14). Such a s o lution is of less use e ver, give some ins i gh t in t o t he high v behavior ( refs. 14 and 15). For t han t he small _ s o lution, a t l east f o r curr e nt he l i co p ter s; i t d o es , h o w- aro u nd 1 , only n u meri c al so lu ti o ns are possib l e, unl e ss so m e o t her para m e t er is used for the p ert urba t ion vari a ble ( s uc h as y, as in ref. 14).

A t u = 2,2 or so - t he ex act v alue dep e n ds on v, y , a nd K p , b ut t here i s not much varia t ion for t he range of p a ramete r s o f c ur r ent h e li c op t ers - a f l apping i ns t ability is en c oun t ered. I t o ccu rs in t h e 1 / rev critical region, I t h e roo t being d es ta bi li z ed b y t h e p eri o d i c c o effi c ien t inf l uen ce c r o s si n g the _" i mag xn ary ax_s in t o th e r ight hal f pl a ne. T h e y - u p l ane fo r v = 1 a nd Kp = O, with u out t o 2.5 , is sho w n in figu re 8; t h i s p lo t is a c om p osi t e of th e r e s u l ts av a i lable in th e l i t e ra t ure . Figu re s 2 and 7 giv e t he y - pl a ne for th e s a m e v a n d Kp, b ut fo r B to 0.5 only. The b e h a vio r of th e cr i tical region bo u ndari e s at high u , an d t h e high _ inst a bili t y in th e 1 / r e v r e gion a re shown. Th e high u b e h a vior of t he u roo t l oci in fig u re 1 may also be i nf e rr ed from fig u re 8 .

For c a s e ( c) , v - 1. 1 an d y = 6 , th e _ lo c us e nt ers t h e 1 / re v c r i t i cal i " re gion a t sm al l _. Th e lo c us re m a ins in that reg ion, on e r o ot movin g t o tile i '_ l e f t an d th e oth er t o th e r igh t, until th e latter b ra n c h b e com e s un sta%l e _!

(cr oss e s th e im a gina r y a xis in t o the r igh t h alf pl a n e) a t high p. Fo. i c a s e _b), _ = I and y = 6 , the l oc u s shows a de crea s e i n f reque n c y up t o L , u = O.S . Bu t at so me wh at hig her _ t he l oci t u rn ar o u nd a nd t h e f re qu e n c y L ,_ b eg xn s t o in crea se , fo r at a b o u t u - i. I th e roots e nc o unt er the I / rev ' ;._ 26 c ritical reg io n - see the y = 6 lin e in figure 8 . I n th e i / rev c ri t ical r e gion for higher u, one branch of the locus is destabilized, until an instabilit7 is encountered at about _ = 2.35.

For c ase (a), _ = I and 7 z 10, the u l oc i also encoun t er t he high p ins t ability in the i / rev critical region, so firs t the roo t s mus t get from the ½ / rev region t o the I / rev region (see the y = i0 line in figure 8). As increases above 0.5, t he loci in t he ½ / rev region eventually turn around, the real parts of the t wo branches t hen approaching each o t her instead of diverg- ing. At u abou t 1.55 t he roo t s ge t back to the hover value of damping, and t hen break away from t he ½ / rev region. The frequency of the roo t s increases toward i / rev the n , while t he damping is fixed; t hat is, the roots are complex conjuga t es again. When t he roo t s reach / rev they en t er the ! / rev cri t ical region. The t ransi t ion from ½ / rev to I / rev frequen c y occurs very quickly, during a very small p increase, because t he corridor be t ween the two criti- _ cal regions is very narrow at this point (fig. 8). In th e i / rev cri t ical region for higher _, one branch of t he locus is destabilized then, and even t ually a fl a pping ins t abJT[ty is encountered a t ab o u t u = 2.25.

N-BLADED ROTOR EQUATIONS OF MOTION Consider a ro t or with N independent blades, wi t h no coupling by chart mo t ion and only the excitation due t o the blade pi t ch con t rol. The flap mo t ion of the ro t or is described by a set of N equa t ions, each of t he for m , o f e q ua t ion (31:

I

, _m) U sin _m)8(m) ccs ,m(_*_U sin *m) ' I wh er e 8 ( m) is t h e f l ap degree of f ree dom f or th e ru th b l a de , m = 1, . ,N. _ Th e a z i m ut h l o cat ion o f th e m t h bl ade is _m = _ + mA_, A_ = 2_ / N. :' Fo r a te e ter ing o r gimb a ll e d r o t o r , th e bl ad es d o no t ac t in depe n de n t ly , 1 so the r o t o r m o t ion i s no t d escr i be d by thes e e q uat i on s. T h e y m a y b e us e d , " -' _ - how e v e r, t o d er i v e t h e ap p r opria t e e qu at ions o f mo t ion.

% 2 7 • + _ r t TEETERING ROTOR Eq u a tion o f M o t i o n Consider a t ee t e r i ng ro t o r: a t wo - bl a ded r o t or with a sin g le fl ap hin g e at t he cen t er of rot at ion. The t wo bl a des are not i n dependent t hen , r a t he r t he ro t o r flaps as a who l e , o n e blade up a n d one down. The equation of motion mus t be ob ta ined from equilibrium of m o me nt s on t he en t ire ro t or rather t h a n on individual blades. The coning m otio n of t he r o tor - both bl a des up or both do w n a t t he same time - is reacted by the s tru ct ura l r es t r a in t a t t he blade ro o t. Tha t is, it is a can t ilever mode t y pe of mo t ion , fo r a ver y st iff blade, so with a very hi g h frequency. In t h e te e t e r i ng motion, however, t he bl a de ac t s like a hin g ed r o t or, usually,in fact, with no hub sprin g at a ll so th e f lap natu r al fr e que n c y _ = 1 / re v ( a lt houg h the gen e r al ca s e of _ _ 1 will be considered ) , Consequentl y , th e co n ing m otion of the rotor will be neglected as a higher f r equency motion, and only the t e ete r ing d e g r ee of freedom conside re d.

_ " Let 8 be t he de gr ee of freedom fo r t he r o t o r t e e te r i n g m o ti on , s o 8 (2) = 8 and 8 ( I ) = - 8 . Eq uilibrium of fl a p m om ent s on t he e nti re ro t o r i s gi ven by half t h e diff er ence betw e e n t h e 8 (2) and 8(1) eq u at ion s of m o t ion, i Then, t he e qu a tion of motion for the tee t e rin g ro t o r flap mo t ion is : t 8 + _ _ + 2 + _ _2 sin 2_ + K p (_+ _ _2. _ _2 co s 2 8 = 0 (58) I Notice that all 1 / rev harmonics drop from the coefficients of 8 , because ' with the t ee t e rin g mo t ion the rot o r has a perio d of only _ with re s pe c t to the aerodynamic environment. As a consequence , all the order u terms have d ro pped, le avin g ol_!y t he o r d er v 2 inf l uen c e o f fo r w a r d f lig h t on t he f c o eff i cients of t he f l ap e qu at i on .

' I Hover I 1 Fo r t he hove r limit _ = O , t h e r oo t s a re a s b ef o re (f or ind e pen de n t bl a des) : L - X = .. -_-+ i _ 2 + K p - (59) i 'F 16 - t f

i '

, ,, 28 I E x p ansi on in u 2 Only ord er _2 te r ms app e ar in t he flap e qu a tion, so a n e xp a nsion in _2 is used now: _ _ _ + _2 _ _ _ _ _ - _ 2 + . .

8 = 8 0 + _ 2 8 2 + t e_ J _ Y = YO + I' 1 2y2 + " " O rder 1 Resul t s Th i s is t he hove r limi t a g a in: + =

0,,0 0

_ 0 0 2 8 _0 wi t h solution , . 130 = Re [8 02 0# 2 ,. .) e ) ' 0 50] (61) wh ere t h e order 1 e i ge n va lu e i s 7 0 YO (7 o _ 2 X° = _' 6 i 2 + _ (62) 'r_ , / v - + T Kp \16 ] ,,i O rder g 2 Res u lt s The order _ 2 equation i s : 2 , I

-L_ __, o

_2B O YO _ } B O Y2 _B O

= + + # _

2 _b O ;) kb 2 8 ;) _b 2 8 _b 0 + sin 2 _0 + K p + K p T - lp T c os 2 _ 8 0

o)

_B0 2

=I(_o + _)_. [_<_ o . <,_. _,_ ] _ o _ } . _ o _ o

Y_ 2e ( XO+2£ )_ o + _-_ ( - £ - K p)B 0 ( ,- YO 0 - 2 i)_0 + _-_ (i - K p)80 2 e (k * c o n jug a te (6 3) A ss um £ ng t ha t T 0 + 2 1 # k 0, th a t i s, Im X0 _ I / r e v , t h en the s ec ular t e r m i s

' (_ o . _ ) _o_

o r t a80 2 _ 0 2 X 2 B 02 = 0 (6 4) wher e YO Y O T ()'o + Kp) + T l(p _2 = " _. 2ii m Xo Th e solutio n i s ; e ),2 _ 2 " i SO 2 = 6) O t_(_Jm+ ,. • •) ( 6 5 ) " ' - .' _ ..; f 3 0 m - .....h_ ............................. lJ , .... , m q • T So X O _ O+ X 2 _ 2 B = R e ( 8og e ) + O(p 2 ) (66) T h e n, f o r Im X0 # 1, the eige nva l ue to o r d er _ 2 is X = X 0 + g 2 X 2 Th e o n ly i nf lue n ce o f f o r w ard f l igh t on the eig en va lu es of t h e t ee t e ring r otor is then the i , crease i n the mean of KpM 0 (to order u 2 , and f o r ImX 0 # 1).

_ This result is substantially s i mp l e r than f o r the independent blade , wi th mu c h less inf l ue nc e of the p e r iodic c oef f i c i ents. T he dif fer e nc e is t he resu l t o f th e i nte rn a l c an c e ll ing o f t he 1 / rev , order u fla p m om en ts f o r th e t ee t er ing r o t o r. [ n fact , if Kp = 0 there is n o o rder u 2 influence o n the r oo ts at ' a l l , t hey remai n a t the h ov e r v a lu e s ; i f K p > 0 t here is an ord e r U 2 i ncre ase in t h e f re qu enc y , due t o t he i n crea s ed e ffe ct iv e ne ss of Kp ac t ing t h roug h _e .

. The dive rg e nce crite r i o n ( X = 0 ), bec om es no w, f or s ta b ility ( I 2 ) _ K p > -v 2 (6 8 ) . This is a str zc t e r requirement on Kp th a n for the i nde p end e nt bl a de ( i . e . , the b o und ar _ ts re a c hed at a n e g ati ve Kp o f small er magni t ude), b ut t he [ differen ce ts only order u 2 . The t w o rea l root bound a ry (ImX = 0) i s _b 16 K p 2 / v 2 , fp 2 (6 9 ) or f o r Kp = O , _ / 16 = _ . _ : Near l / roy F re q ue n c y R et urn now t o t he ca s e X 0 + 2 i • X0, t h at is , ImX0 • I , t h e h o ver ! , , ,, I freque n c y n ear l / re v . Th i s m ea n s _ 0 / 16 • l p + / _ 2 l p 2 _ I , a n d the o r d er I i r oot _s _0 = - ( v 0 / 1 6 ) + i . N o w th e peri o d i c c oef f i c i e n t s contri b ute t o t h e _ i se cular t e r m , to g_ v e - -,__ : _ _'_ , 3 1 _ It , . f _ ; '_ _._ 8B 02 X 21 B 02 i Y0 2 ¢ 2 + _-_ (i + Kp)_ 02 = 0 ( 7 0 ) wh e re since Im_ 0 = I , _ ' _ > ' 2 = Y2 i Y2 YO K p K p - 16 + _ 16 + + _ "

[(_ 1, o ]

rf_ " The solut i o n de p e n ds on

(_, o , o l + Kp) + Zp "r'6"J " k 3 -- 'Z / (1 , Kp 2 ) (71 1

The c r itica l reg i o n b o u n da r ) is g iven b y D 2 = O , o r w r i t in g Y = YO + Ay, w ith Ay ord e r _ 2 small : . _ In t e r ms of t he u lo c u s, this give s the co rner _ : l i U12 U 2 2 = T_ Y O YO _ + ; Kp i-- 6 ± _" K p 2 A d T his expr es si o n is s im ila r t o th at f or th e 1 / r e v r e gi o n b o u ndary o f the '_ i ndepend e n t bl ade ( e q. ( 4 1)), b ut w i th ' ( 74 ) ; C I i Kp_ Y O / I+ I Y o } i C2 _ 2 " l p 2 s that i s , wit h c onsider a b l y l ess in f luen c e of the per i odic co e f f ic ients. The ,¢ t eigenvalue i s . .. _.

; I I = ZO + g 2 R e X 2 " i_ 2 D = " 1-- 6 + i - i _-_ _' 6 " K I - i _2 ( 7 5)

2 5

' w h ich is x de ntical to the in dep end e nt b l ade resul t ( eq. (4 3 )), e xc ep t t hat th e corner u have different v a lues. For Kp = 0, the criterion Iml0 = 1 red u c e s t o Yo / 16 = _ - I, t he I / r ev boun da ry is p 12,_2 2 = ±2(A y / 16), a n d t he el g envalue r educes to

:- _ + i- i _ v C_- 1 T ( 7 6)

For , = 1 ( a nd Kp = O) this is the same b o u ndary, as f o r the independent blad e , Fo r _ > I, th e c r itical reg ion has a smaller width, and is n ot offset (CI = 0 still, since Kp = 0).

S u mm a ry The influence of forw ar d flight on the teete r ing r ot o r diffe rs f ro m t hat ' on the xndivldual bla d e principally in that: there is no ½ / rev critical re g ion a t all; and the periodic coefficient influence is much si m pler. Awa y from th e I / rev reg i o n, th e r o ots a re g i ve n b y = 1 /6 6 ± i 2 + (I + U 2 ) _Kp - ( 77 ) / , inclu d in g the f o rw ard fli ght infl uence t o order u 2. The c orre s po n d i ng _._ c ri t er i o n f o r dive rg e n c e stability is | I (I • U 2 ) _ Kp > _v 2 ( 78) I T here i s n o _ / rev cr i t i ca l reg i on , so t h i s ex p ress i o n ho l d s a l so w hen t h e hover frequency i s near _ / rev. A cr iti cal reg i on i s en c oun t ered onl y if t he hover f re q ue n c y i s near I / rev. T hat i s , i f AY = Y " Y O is order _ 2 s m all , # : , _ wh er e v 0 / | 6 = K p + / _2 + K p 2 _ 1 , th e n th e roo t s are I =- / _ + i- i _(_- K I- _) -U 2 ( 79 )

P

L i, I / - wher e t h e com er _ a r e

°

_ 12, U 2 2 = '_ - "_ ' - K Ci ± C2 ( 80) and the constants CI and C 2 a re g iv e n ab o v e L eq . (74)). F or Kp = O, this r e duc es to u 2 = ±2(A y / 1 6) co _ er t The 7 - u plan e bo _ daries a re , writing Y = Y O + Ay, f o r th e t wo rea l root region A y u 2 Yo Kp

= ( 8 1)

_2 + Kp 2 ..

wher e Y o / 16 = Kp + _2 + Kp2; _d f o r th e 1 / rev critical regi o n Kp + _ Kp2 Y O _ + Ay _ 2 _ "- (8 2 ) 1- _ = Yo

1 - - Kp _

w he r e Y0 / 16 = Kp + _2 + K p 2 _ 1 . Fo r K p = 0 t hese be_ d ari e s r e du ce t o ' Ay VO I ] _ = 0 with ] -_ = v ( 83 ) _y i _ 2 w ith Y O _ ( 8 4)

i - G = ± T =

r e sp e ctiv e l : '. T h ese r e sults m a y b e c o m p ar_ i with th e Y - U p l ane b o u n dar i es ( o f the independen t bla d e, a s illu s trat e d in f igures 2 t o 6 . Fo r X p = 0 , the -I two real r o ot b o undary is n o w a c o n s tan t y li ne , th a t i s , a h or izo n t al l i n e ; I t her e is n o ½ / rev reg ion a t a ll; a nd t he I / r e v r egion is the sa m e wh e n v • I, but n a rrow e r and no t offse t (CI = O) when v > I.

Consider t he u roo t lo c i of t he t ee t ering r o t o r . Wh e n Kp = O , as i n figure i for t he ind e penden t bl a de, t hen for ca se C a ) w ith v • I and y • I0, and ca se (b) wi t h v = 1 a nd y • 6 - t hat is, a way from t he i / r e v c ri t i ca l region - there is no e ffe ct o£ u a t a ll to order _ 2 . The lo c us r e m a ins a t th e h o v e r valu e . Fo r cas e ( c) wi t h v = i. I and y • 6 , that is, n e a r I / rev , ' t h e b e h a vi or is b a si c ally th e sam e a s f or the i n d e p e nd e n t bl a d e. H o we v er , t h e ' _- cr i t i cal region is n a rrow e r now , a nd no t of f se t , re su lti ng i n a h i gh e r v a lu e fo r P co rn e r (0. 4 07 c om p ar e d t o 0. 28 6 f or th e ind epe nd e n t bl a d e ). No t i ce t h at if t h e hov e r roo t fre qu e n c y we r e b el ow i / r e v , t h e n _ c o r n er would prob a bly d ecrea s e b e cau se o f th e abs e nc e of th e cr iti c al r egi on off s et w ith th e t ee t e rin g r oto r ; t h is i s b es t s e en on th e y - p p lan e, as i n fig ure 4. _ " ; 3 4 k I There is t h e n con si d erably less i n fl u en c e o f t he peri odi c a e r od y namic forces on the teetering r o t o r flap stability, as compa_ed to the independent blade. T his is the result of the internal cancelling of all the 1 / rev, or d er flap moments. Specifically, there is no ½ / rev critical region, because of the absence of 1 / rev harmonics in the coe f ficients; the 1 / rev critical region is narr o wer, in general, an d no t o ffset if K p = O ; and t here is less infl ue n c e _h of u on t he two real r oo t boundary. If Kp = O , t he t wo real r oo t bou n dary does not c hang e a t all wi t h u; t his c orresponds t o the absen c e of a ½ / rev cri t i c al region, whi c h for t he independen t blade pushes t he O / rev boundary up (a s in f igure 2) .

In fact, unless the fr e quency o f t he ho v er root is n ear I / rev - a n d the " _ ' 1 / rev critical region is quite narro w , being only orde r u2 wide - then using a constant coefficient approximation gives exactly the correct eigenvalues to order uz . That is , if the period i c coefficients i n equation (58) are simply dropped - and they are the reason for all this perturbation analysis - then the only effect of u retained is the increase of the mean of M0 by the i fac t o r (1 + u2 ) ; bu t t ha t is t he o nly eff ect t ha t app e a r s in the c orr e ct ro o ts for the teetering xotor an y way. T his suggests that such a constant coeff i - cient approximation may be an adequate representation for the teetering rotor flapping dynamics (if the fr e quency of the root is kept away from 1 / rev).

GIMBALLED ROTOR , THREE B LA D ES Eq ua tions of Motion Consider a gimballed rotor: thre e or more bl a des a tta c hed to a hub w i th c antilev e r root restraint, a nd th e hub to th e rotor sh a ft by a univers a l joint. Th e possibility of a hub spring is in c lud e d , so u L 1 is allow e d i s t ill . The bl ad es do no t mov e ind e pendently now, ra t h e r t he entire rotor mov e s as a w hole a bout the g i mbal bearin g s. T her e are t w o degrees o f fre e do m des c ribing the rotor fl a p motion: longitudin a l tip p a th plan e tilt 81 c a nd l a t e r a l tip p a th pl a n e tilt Sls. Th e v a riables des c ribe the rotor motion in b , .4 t he non r ot a t in g f r ame; _tc is defin e d pos i tive for ti lt for w ard, an d Sls is positive for tilt t o w ard t he re t rea ti ng sid e . All other modes of motion of are as c oning by i t h e rotor blad e s su c h the mot i on re ac ted the ca nt i lever roo t s tl ffness , so will be n e gle ct ed as h i gh er f requen c y mo t ions.

The flap mo ti o n o f t he m t h bl a d e is given no w by B(m) = Sl C CO S _m + Si s s in _m ( 85 ) , alp T he eq ua t i ons of mo t ion for t h e gimb a l l ed ro t o r tip pat h pl ane t i lt a r e ob ta in e d f r om equ i l ib rium of t h e p i tc h and r o ll m o m en ts o n t h e h u b . The equa- ti ons m a y b e ob tai n e d f r om t h e fl ap moment eq u il i b r ium f or t he m t h bla d e (e q. (57)) b y t h e op er at or s I _ ° , b I _1 % , e - N N 2 2 _ (. .) c o s _m , _ E (" • .)sin _m (86) _= 1 m= l The resu lt of the su mmatio n o perati o n depends o n N wh en , as h ere , the e qu a- _ tion s have periodic coe ff ic i ent s . For a three bladed g i m balled rotor, N = 3 , the re s ult i s lS L-2 - IJ I" _ CO S 3_ Y--- 8 U _ sin 3%_J_8 1s I: 8 . sin3 / B, t '_' 1 - + U sin 3 _ - 1 - U _"c os Iv2- 1+ _ _ c os 3_ Y--( 1+ _ , 2 2 - ) _1% c l = ( 87 ) U _ c o s 3_ _ (I + _IJ 2 ) - ] j _sin 3 0 ls l( pBl s / Th e cases o f N = 4 and N _ _ 5 wi l l be cons i dered i n lat e r sect i ons . The ; no tation I r_ ,. \ e 1 s / '

I

1 w ill be us ed f or t h e d e gr ees o f fr eed o m .

Hover .- : , In the hover li mi t , u • O , the d ifferen t ial equation is :. + 0 CSS ) I _

L-

i 2 _ 2 1 + Kp i _ . 36 Th e characteristic eq u at ion £ s _ 2 + 8 _ _ + _j 2 - 1 + K p + 2 _ + = 0 ( 89)

(

wh i ch ha s the sol u t i o n s _ = _ 0 ¢ £ a nd the i r co njugat es, w h er e II is the ro t a t i ng ho v er root. So th e h o ver r oo ts fo r t he g i mb a l de g rees o£ f r e e d o m, w h ich are in the nonrot a t i ng f ra m e, ar e j us t t h e ro t a ti ng root s +-l / rev freque n c y d ue t o the tr a nsfer fro m rotatin g to no nr o tati ng coor d inates.

Th is hover re s ult _NR ffi _R + i is tr u e f or an y n u m b e r of b la d e s , in f act. In f or w ar d flight the re s ult is not th is s i m p l e becau s e the blades of _. g im balled i ro t o r a re no t i nd epe nd e n t.

Th e r e are two d e g ree s o f free dom , h e n ce, a t o t a l of f our eigen v a l u e s.

Th er e is a h i gh fre qu e n c y mod e wit h roo t s _0 and it s c on j ug ate , t h at is, at fre qu e n c y Im_0 + i / r e v; a nd a l ow f r e qu e n c y mod e wi t h r oo t s k0 i a n d l ts c onjug a t e , t h at is , at fre qu e n cy I mX0 - i / re v . Th e c o rre s p ond e n ce J b et w ee n th e ro tati ng and nonro ta t i ng r e g i ons is th e n a s follows : .Rotatingfrequency No.nrotatlngfrequency , 0 / rev 1 / rev l _ / rev 1_ and 3 / 2 / rev l / tee 0 an d 2 / rev i The 0 / rev rotati n g f requency m e an s tw o real roots; it include s the case o f diver g e n ce ins ta b ili ty . Th e region s , boun d arie s , and b ehavior o f t he roots • wi ll b e di sc us s e d i n te rm s o f t h e ro # a ti_ fr e q uen c ie s , s o that th_ r es ults m a y b e c om pa r e d wi t h t hos e of t h e sing le bl a d e an al ysis.

' l b. Th e ei g envector s for t h e roots _ 0 + i are Sl c (91)

( :)

that i s, B lc / els • -+i; th i s c o rresponds t o a wobbling or whirlin g m _ tion of the tip path plane.

: / i Exp an sio n i n t i : L e t " ; 37

I

tip - _ - " ' _ - - ' I IN IIIB - -- ..................... iliililli J_ ............................ - "" _ + _ _ g 2

_- T = _ , _ - "; _ ' " _ + " '

j_

¥ = Y O + _ 2¥2 + " " " ( O rder 1 Re s ul t s The order 1 equa t ion i s just t he hover l i,oi t a s usu a l: , + 2 _2 . I + K p --_- , + a ¢ O_ _ + 8 0 = 0 (92) _ 2 _ 1 + _ p Th_ order 1 e i genvalues are t hen _ " X 0 _ i an d t h e c onjuga tes, w here Y O / ( Y o ) 2 (93) t _ 0 " " 1- 6 + i ° 2 + "_'K P - tl " _ ' i s t he order 1 rotating hover root. + Th e corre s ponding eig e nv e ctor s are ' 8 t c / 8 1 s = - +i, s o t he solu ti on for 8 0 i s ) Th e f i rst t erm (sub s cri pt +) i s t}. e hig h grequ e ncy mode , th e s econd (su b s crip t - ) th e lo w f requ _, c y mode; g O l± are complex £ _ c t ion s o i th e h igh e r t ime scales _lJ_ 2, et c _ , Order' u Results Th e order U equati on is L

P

o 2 _;_ - F Y 7 -_

_- _% • 2 _;o.[T'_ sin 3,0 - n cos 3_o Jo

YO

.o 1 28'08'i - _J a-_l L-_ c o= 3,o T- Y I Y012 s i n 3, 0 j]_-_O

8 + T sin 3 @0- K p T c ° s 3 @ 0 l I _0 " " I | Y I +Y0 sln " . . Y0 . Y0 YI Y 0 L- _ " _ - 3 _0 - _ p- _ " c° s 3 _0 - -6 -c °s 3 % 0 + Kp' _- -K P-6 - si n3 ¢ ° _ *'. _ k = 2X0+_) 3-_--i+-_- e (x O +i)_O - i _- (X0 - i + 2 l p)_Ol+ e (XO- 2 i)O0 I

. ( , )

i [(2 a'Ol - + _] (-:)e (XO - i)'O J Y 0 e(XO+ 2 i)@o + +i _ (_ ' 0 + i + 2 Kp ) 801_

I:)

._i_" i + conjugate (95) T he s e cul ar t e rm s a r e t he co ef fi c ien ts of t he h o m og e neous so lu t i o n i (-+:) e (_ O+ i)@o j - , ; on t h e r ight-hand si d e of e q u at i o n ( 9 S). N ot i c e t he r e are t w o h o mogeneous .J _ . solut ions no w ( r eally fou r , i nc luding t h e co m p lex co nj u g a tes) , as in eq u a- 1 t i on (94), a nd t h e c oe f ficien t s of eac h m u s t b e s et t o ze ro . Th e r es u l t in g ' fr equenc y m ode s.

eq uat i on s give t h e or d er u c o rrect i o n s t o the ro o t s fo r t h e high a nd lo w ] A ss u ming f or n ow that ImX0 # _ / rev , the n the secular ter m s are (2 80) @s o 1 + Y1 X 0 + _ + T (x0 + K p ) S 01 _+= 0 t or { @8 °I+ X I B0 1+ = 0 ( 9 6) a ¢ l wh ere Y S -_ ( k 0 + Xp ) . , 2 iIm_0 Th e sol u tio n is 801+ = 802+( ¢ 2 ,. .) e 1 1 ¢ 1 , so th e so lut ion f or 60 is + S O2- e S O = Re 0 2 + _ - _

[_, OoCx o 'i)_ o 'l_ _ (i) (_ o -i)_ o +_i_ (9,)

Hence , t h e e i genvalue s t o order p are $ I. I = _ 0 + i + V l I

/

, =-_ + i / U2 +8 _ KP - ] -_ - + i ( 98 ) w h ic h is ju st t h e r o t a t in g h ove r r oo t +i s t i11. Wit h t h e sec u l ar terms s e t ' I to z e ro , t h e sol u tio n of e qua tio n (95) is ' , _ I = R e 11 + e( L 0 +i ) _0 + 811- e( 1 0 " z) kb 0 :

C ) C) " '

I +A + 8 0 1+ e( x0 - 2 i) o 0 +A-B 0 1- * ( 99 ) I : _. w h e r e _ , , / b i , ( X0 _, Y 0 + 2 Kp ,_i) (_i) .

I " A_ = - I ./ +_ 2 1 m l 0 - I '_ , i , _ , i ! w h ic h co mpl e t es the so lution to o rd e r _.

I

O r d er v 2 Re s ult s T he o r de r u 2 e quat i o n i s . 8 _2

[ " ° _" _'L-_ _' - ,._

L- i T. Y ° cos 3_o Yx ¥0 si n s . ' .l

_ _ o _

+ / Y1 YO Y n Y O ] _ 1 -- -- si n 3_0 .

Y0

- _ -- _ sin S _ oJ

Y. i Y o ... o 7

] T +y_sx n 3_ o - Y T c °s 3_o _ o Kp+ y GK p -_ + Y 1 / I _ / --+ ; 0

i b . I L - i -2 CO S 5 _ 0 Y 2 Y l . " 8_0 I Y 2 Y o Y o c o s 3_0 + K p _ si n 5 _0 s i n 3 _0 - K p _ cos 3 @ 0 + 13 0 si n 3_ o - lp co s 3_0 - - _ -co s 3_ O -K P s i n 3_

' !

, 41

' 1

j_ _ 2 BO l++ Y1 @8 o1+ _ . (_0 +xP ) +T Xp+] - _ ( X0+2xP ) iA 6o 1 + e (_ 0+ i) O 0 ¢5 + _ 8 @¢ 1

,o , o .]}C)

+ -_ (i + K p)B 0 1 + • (_0 + i ) @ 0

[, o ]C)

+ _ (-i)(I 0 + 2K p - i)Bil++ ( - i)+ + i)A+ @*I

Y1 YI e( XO- 2i)* o

[ . ] } -

( ** )

: i _ + _'_ (-i)(X0 +Z K p -i) + T (X0 +Kp-i ) A 801 + + -_-_ + () , 0 +lp ) +T lp + - _ () , 0+2lp )(-i) A 8 0 1 -

,,, , o, . .o .] IC)e. O -., o

[ .o ] o. o -. . o .

, _ + _ 'K (' i+ KP)_ o l-

( ** )

+ n i ( 10 + 2K p+ i)811- + i + A O+T+ 2 A a_l

{ Yo Y[ n (2 Y ° i)_] _B °I -

+ _- i(X 0 + 2 X p+ i ) + ..- _ - (X 0 + K p + £ ) A B 0 1 4,. [Y1 Y! .] .} (i) e ( X0+2 i) ¢ 0 t + conjugat e ( 100) A ssm. i n g th a t I m_0 _ I , th e secular t erm i s 1 ' 42 : _!

811+ ll_ll+) _80 2 + + A2 + "_- K p + -_ (X 0 + 2Kp)(+i)A = [ _--_2- - 2i_m_o 80 2+ e_ l _ l (I01) )

, - "I }

a n d t h e secu la r term o f t hi s is v ,8 02+ -_- Kp + _ (X O + 2K p )( + i) --_-+ 12 + 2i_ m _o 802+ = 0 (102) I t follows t ha t t he ei ge nva l ues t o o r der p2 are = t i+ _ 0 + u _ I + u 2 2 +i -_- K p +_ ( X 0 + 2 K p )(+i)A+ 21m10

[ ,o , o l

YO

= +-i-_+i 2+(1 ._2i - (i o _)

/ _ (-Y-) 2 16 i-'_(XO + 2Kp] (+-i)A+ 2 I m_ 0 , I W hic h m a y b e re d uced t o 2 _ 2 ._ . K p+4Kp 2 " ' v 2 -_ - K p +4Kp 2 2 4 ' ( - i-_6 +i 2+(I+u2) _K p - +U 2 ....

, : + __ . _ - ( -

, } I 1 2 , and t h e c on ju g ate s.

,a - _ Ithou gh d eri v ed h ere o n the b a s i s of co m plex root s , i t m a y be i de mo nstra t e d t h at th is expres si on is also vali d whe n y i s lar g e en o u gh so t hat t h e rotatin g hover rc , ) t s l i e on t h e real ax i s . T his si n g le co m pos i te , ex p ressi on t h e n is vali d _ r all ca s es except wh e n t h e ho ver root f requency is _ i , near ½ or I / rev. Th i s result h as so m e o f t h e ef f ec t s t h at have been seen for _, .

' _ 43 I / - t he in depe n dent bl a de: t he fac t o r (1 + _ 2) c orr ec tin g f or t he in cre a se i n th e m e an of KpM 8 a p p ears as usual, a n d the second order u2 te r m in the r a dican d is like the corresponding term for t he independe nt blade (eq. ( 24 )), al t hough h a lf t he m a gnitude . There are also no w some order _2 effec t s t h a t give qui t e new behavior, ho w ever. There is an order V 2 ch an ge i n t he ±i shif t of th e n o n rot ati n g root s. Secondly, the las t o rde r u 2 te r m in t h e radi c and _h pr oduces, fo r the c ase of complex r oots (i.e., as written), an orde r u 2 ch an ge in the damping: an increase in t he damping o f t he high f requen c y mode, a n d a decrease for t he low f requency mode. This is quite diffe r ent f r om t he behavior seen so far, where t he damping has always r emained fixed a t the ho v e r value -v / 16 when the r oots a r e ou t side t he c r itical r egions. For t he case of the ro t ating roots being real (i.e., bringing the i inside th e radica n d) _ there is an order uz cha n ge in the frequency of the roots: the more stable root going to highe_ frequency , and the less stable root to lower frequency.

T hese new effects will be seen again when the constant coefficient approxima- t i o n is co nsidered; a dis cu ssion o l t heir origin will be put off to tha t time, since t hat is a problem of more general interest. T he present discussion will concen t ra t e on ou t line t he behavior of t he eigenvalues produced by these new ter m s.

Consider t he order u2 influence of f orward fl i ght on t he di v erg e nce boundary. When t he ro t a t ing hover roots are real - t he divergence boundary being a special case of t ha t - t he last t er m in t he radicand of equa t ion (104) produces an order _2 change in the frequency. Hence R e _ = - 1_. _ . ±_ + U 2 9 4 ........ _ (.. _ _. _ 2 _ - _ 2 _ ( l+p2 ) _ K p S et ting Reh -- 0 f o r t h e dive r gen ce b o u nd a r y , t here follows t he c r it er ion f o r d iv e rgence stab i l i ty: T he o rd er u 2 in f luence o n t he r i g h t -h a nd s i d e i s j u s t h a l f t h e resu l t fo r t he independent blade, so t he c ri t erion on ne g ative Kp is mor e s trict in th is ca se . The difference i s on l y or der u2, h ow e v er . I n t he non r otat ing !

fra m e, this divergence instability occ urs a t a frequency o rder u 2 ab o ve 1 / rev.

C on s ider now t he beh avi or of t he ro ot s o f t h e t h ree - bl a de d gimball e d r o tor ne ar t he b o und a ry fo r two r ea l r oo t s ( r o tati ng). R ecall t h e resul t fo r t h e h o ver roo ts o f individu a l b l ades, i n th e r otat i n g f r am e ; figur e 9(a) _ . i ll u s t r ates th e beh a vio r , f o r v ar i at ions in y . For y / 16 = K p + ¢ u 2 + Kp2 t he t t wo root s m e e t at t he r e a l a xi s , and t h en f o r lar ge r y t hey p roc ee d i n ' _ o pp o site di r e ct ions alo ng th e real axis. The tr an s form ati on to the no n rot a ting fr a m e si m ply s h if t s this b e h a vio r by ±i , so that t he a bo ve b e havio r o cc u r s a t ±I / r e v ins te ad o f on th e rea l axi s . lqa e hov er r oo t s of indiv i du a l bl a des , will i n the non r o ta ting f ra m e t hus h a v e t h e b e h a vior sho w n i n f igure 9 (b ). This is a lso th e b e h a vior o f th e hov er roo t s of t h e t h ree - bl a d e d g i mball e d r o t o r , fo r y n ear t h e ro tat ing rea l r oo t bound ar y. This should not be c onfused with the behavior of roots ne a r a c riti ca l region; it j_ o cc urs here a t ±i / rev be ca use of the tr a nsform a tion to th e non r ot a ting frame.

The c riti c al region b e h a vior involves two roots, one a t posi t ive frequen c y and one a t neg at ive fr e qu e n c y , whi c h a fter c rossing t h e c riti ca l region bound ar y pro c eed in opposite dire c tions on the X pl a ne. The b e h a vior n ea r _be re a l root bound a ry involves four roots, two of whi c h me e t a t 1 / r e v and two a t -i / re v ; hen c e, the roots o c cur a s c omplex c onjug a te pairs a lw ays. " In for wa rd flight (v> 0 ) the beh a vior of the roo t s for individu a l bl a d e s in th e rot a ting frame is the sam e a s that of th e hov e r roots (i.e., as s hown in fig. 9(a)), just with a n order _2 shift. Sin c e t he bl a des are ind ep end- - ent, the transfo r mation t o the nonro ta ting f rame ca n only shi f t t he lo c us by ±i in for wa rd flight as it did in hover; so figure 9 (b) pr e sents the beh a vior _L of th e ro o t s of i nd i v idual b la d e s i n t h e n on r o tati ng f r am e for forw ar d f li gh t " a s w e ll a s for hov er . The t h ree -bl a d e d gimb a lled r o t o r , how e v er , e xhibi t s diff e r e n t b e h a vior n ear th e r ea l roo t bo u nd ar y in fo rwar d flight. Ins tea d of t h e tw o roo t s me et ing and t h e n pro cee ding in opposi te di rect ions a t I / re v f re - qu e n c y , the t wo b ra n c hes of t h e l o c i on l y p a ss c los e to e ac h o t he r a s il l u s- tr a t e d in figu re 9(c ). Th a t is , a s _ i n crea s e s, th e lo c us inte r s ec tion pu l ls a pa rt . In te rms of t h e y lo c us at a giv e n u > 0, a s t h e bran c h e s a ppro ac h I / r e v t h ere is a ± o r d er _2 c hang e in t h e damping ( w i t h a c o r r e - , spending c hange in the c onjug a te roo t s, so t his is no t critica l r eg i on behavio r) ; ev e ntu a lly the r oots tr ansition t o more like th e t wo re a l r oot behavior ( a t I / rev frequen c y nonrot a t i ng), but with a ± or d er _2 differ- en c e in the frequ e n c y. Su c h beh a vior is, in f ac t, typi ca l of the dynami c s of c oupled degrees of freedom; t he root lo c i do not c ross , but r a ther only p a ss c lose to one a nother, pulling f a rther a p a r t a s t h e c oupling in c r ea ses (in t his c a se , a s u i nc r e a s e s). T he sou rce of t his beh avi or i s the l a st t erm in the ; r a d ic and of equ a t i ofl (104) for X . There is , in add iti on, the order _2 ch a nge in the ±i t e rm of equat i on (104), so all this oc c urs r e ally at not r qui te ±l / rev.

i I t foll o ws t h e n, t h at w h en u > 0 t h e t h ree -bl a d e d gimb a lled r o t o r d o es no t h a v e a d ef ini te r e a l root bound ar y. R at h er t here is a g ra du a l t ra ns it ion f r om p r im ar ily c om p l e x r oot b e h a v i or to p ri ma r ily rea l root beh a v i o r . Examine 1 now the locu s in the neighborhood o f the r eal root bound a ry. Writ e Y = Y0 + _ 2 y^ w here Y 0 = Kp + iv 2 + Kp 2 , so y i s o r d er u 2 fr om th e hov er r ea l root bo ( Ind a ry. Look at th e r oot n ear th e i / re v f requ e n c y , an d n ear t h e -y / 16 re a l pa rt; that is, let A_ be def i n e d by i + v " I ' _+ _h ( I 0 6 ) ; Th e n , from equ ati o n (i 04 ),

I

A L = P 1-_" Z - Kp - T _ - y / _(Y o p) _ 1 6 _ i Yo v 2 ( 1 0 7 ) Th i s m a y be sep ar a te d i n t o r e a l a nd im ag i nar y p art s: - = 1-- 6" Kp - T Kp - T _ (10 8 ) (ReA_) 2 {imAm) 2 U 2 [_(Y0 ) Y0 1 6 41 _ Y.

U 2 -_ v 2 ( 109 ) (ReAL)(ImA_) = - T he l a st e qu a tion show s th a t t h e l o cu s on the AL p l a ne ( i .e., o n the L plane , bu t wi th a s hif t e d o ri g in ) i s a hype r b o l a . The po i n t o £ c lo se s t app r oach of the two b r anches occu r s at " ' R e A L = -ImA X = ±_i_8_ ( 1 1 0 ) a n d the mi ni mum s e p ar a t ion o f t h e b ran c hes i s th ere J 21ALlmi n = U g _ (iii) { Th e s e pa r ation of th e b r anche s th e n is o r d er _. Th i s p o int o f cl ose s t i a ppr o a c h m a y b e tak e n a s t h e d ef ini t i on of th e re a l root bo un dary for t hi s c a se. W i th ReA L = - ImAL , equa t i o n ( I 0 8 ) th en gives t h e boun dar y as ' " Y_ g i-_ _" 2K _2 + 1--6"KP

,o

S 16 YO i-_" Kp _ : ! o r

t

• , J .-4 I Y Y0 u 2 9 ( I 12) !

16 16 + _ 2 + gp 2 i : s a me a s wo u ld b e ob t a i ned i f t he las t ter m _ : ,' , ._t e r a d z cand o f L ( e q. (I 04 )) - :_" w h i ch i s ca u sing all t h i s b e h a v i o r - w ere s, . .._ _ , i gno re d . S et t i ng t h e ', radi c a n d to z e r o w i thou t this t erm g i v es exa ' ' the a bo v e bo_d ar y T he :. bound ar y obt ai ned i s s imilar in beh avi o r t o t h a _ o f t h e in d i vidu a l bl a d e s; but F or gp = 0 t his re duc e s t o y / 1 6 - _ _ + _ 2(8 / 9_, o . _ ' . Not i c e th a t t hi s i s t h e " i th e ma gn it u d e. He nc e , th e Ay o f th e boun da ry f o r Kp = O , fo r e xam p l e , i s _ - only half as l a rge as fo r the i nd e pendent bla d e . " " , a g ai n t he secon d o r d e r u 2 i n t he ra d i cand o f e q uati on ( i04) h a s o nly h a l f !

• J

I

Nea r ½ / r ev Frequency Consider now t he case Im_ 0 = 1 / 2, when t he non r ota t ing roots a r e a t 3 / 2 / rev and -½ / rev frequency. The order v secular equation fo r BO1- is u nchanged for t his case, hence, t he low frequency mode (k0 - i) does no t have a c ritical regi o n when the rotating f r equency is near ½ / re v . The o rde r _ _ r oo t is the same as equati o n (98) then , that is, t here is n o f or wa rd flight _ n fluence o n the l o wer frequency r oot t o o r d er u when Imk 0 = 1 / 2. The , secula r equation fo r B 0 1+ i s, h o weve r , change d fo r t his case, an d s o t h e high f r equency m od e d oe s enc o unt er the _ / rev c rit i ca l re gi o n. The co m p l et e r otor b e havi or is giv e n , o f c o urse, by b ot h t h e l ow an d h i gh f r equen c y m ode s t a ke n t oge t he r ; when t he high f r equency r oo t s enc o un t e r t he ½ / r e v regi o n , the en tir e roto r do es , e ve n t ho u gh o nly two of t h e f our root s p art i c ip ate in t he crit ic al r e gi o n behav io r.

When ImX0 = 1 / 2, t he o rder u sec u lar t e r m f o r BO I + i s _)B O I + X1BO 1 + + --+ 2K p + _ ' 0 1+ -- 0 (113) , This i s t he sa m e se c u l a r eq u a ti on an d D 2 as f or t h e _ / rev cr it i ca l r e gio n o£ I t r oot s am e as i ndependen t the b la de , foll o ws t h en t ha t t he beh av i or i s th e th a t s olu ti o n, s pe c i f i ca ll y t he e x pre ss io ns f or t he _r it ic a l re g io n b o u n d ar y, and for _ near the r eg ion (e q . (52) and ( 34 )) are ap p li c a b le here as well - a dding i t o X t o t r ans f orm it t o t he high fr e q uen cy m o d e in th e n onro t a t in g f rs m e.

I N ear 1 / re v F r eq uenc y .

C o n s ide r t h e cas e Im_0 - i , so t h e n o n rotat ing roots are a 0 and 2 / rev.

The or d er _ anal y si s i s ap p lica b le , since ImX0 # 1 / 2 , and t h e order _2 analys is i s correct f or th e h i g h f requency m ode e v en w he n Im_ 0 = 1. He n ce , t h e high f re q uency ro o ts ( n ear 2 / rev f requenc y i n the n onrotat i n g f rame) are _ g i ven b y th e s am e ex p re s s i on as f or th e roots awa y f rom th e cr i tical re g ions _ ( e q. ( 1 04)). The l ow fr e q ue n c y mode , h o weve r, ( n ea r O / rev nonrotating) h as an _ a d diti on al term i n t h e or d er _ 2 se cular t e rm w h e n Im_ 0 = 1; i n s t ead o f : _ _*_ , eq u ation ( I 0 1) , _ I I- i s n o w g ive n b y

l

, 4 7

I

3el l - Y1 _l l _ 1 -[ ( 2 X O * _/ T_ 1 * T (X o + KP) = {( 2xO + _ ) @B 0 2- + [ x1 2 + _ L8 X1 + "_ 8 (XO + Kp)+ Y 8 -E K p , _ ( X0 + 2 lp)(- i )A B0 2 _ e Unless X 1 i s real , s o _ 1 = Xl , t h e se c u l a r equ a tion f o r B 02 -(_ 2 ) i s t h e _ s a m e a s be f o r e, ho weve r. T he n , e q uation (10 4 ) ho l d s f or t he lowe r fr e quency m o de a l so , e xc e pt w h e n th e f req u en cy is o rder v 2 f r o m 1 / re v ( r o ta ti n g ).

Requiring k l b e real mean s Y1 = 0; the secular term of equation (115) i s then + X2 " i T'6KP + kl-' Z] _ + _"\'iT ) (- "IT +

_Bo__ I ' o ( Y o _i 1 ( _o_ _ ' o 2Kp)

-i - - 8-- Kp + 4K p Y0 ' , (_) 2 u2 ¥ 0 2 2 6 -1602- + _ (i - i Kp ) _' 02 - = 0 (11 6) | ]

O

• SO ' 2 = T6- Kp T6 *2 " \ i l l * 2 4 K p - (1 * lp 2 )

t_ (117)

which ha s s imilaritie s t o t he results f or both t he i nd i vidual blade and the .

t ee t erin g ro t o r . The b ound a ry o f t he c rit ic al r egion is de fi ne d b y D2 = O; _' l etting Y = Y0 w it h Ay order u2 s m all, then the boun d ary is

. ' 1"_"N " K - u a(cl * ca) (it s ) !

l or in t er m s of t he corner u i I i, ! , i s i.

l-

Ay

c i - + ¢ 2 (119)

This is the u sual form , w i th n o w ' C l = - _ - - - + Kp 1 -- 6 ( 120 ) _ r , bJ

Yo / i

Cz = _ + Kp 2 ( 1 2 1 ) C 2 , which g ives t he width o f t he c ri tical re g i o n on the y - _ p lane, is th e s a m e as fo r the teeterin g roto r ; hen c e, the width is, in g e neral, s malle l th a n ,_ for t he individual blade , e xc e p t fo r _ = I, in which ca s e the wid t hs are e qual. C I , which g ives th e offse t of the r e gion, is s imilar to the indi v idual blad e r es ult. The first t e rm is new, how e v e r, c orre s po n ding to th e n ew order u 2 shift in the ±i frequ e nc y of t he roots away from the criti c al re g ion ( e q. (104)J; a nd t he second t el m_ has half th e ma g nitud e , c o r r e sponding to the se c ond order _ 2 tena in the radi ca nd of equation (104). Th e eig e nvalue of the low frequen c y mode is now YO _ _ 2 Y1 _ 2 + + i_2 D ' X = - 1 -- 6 T6" " T6 2K p _ i ; 8 - 1 1] + -i - - I (1 22 ) I S o the e i genv a lue al s o h a s a ne w t e rm , an order u 2 d a m ping c ha nge , c o r r espond in g t o th e thir d o rder u 2 t e rm in t he ra di c an d o f _ on ( 104 ).

) Fo r K p = O, th e cr it e rion fo r Im_0 = I r ed uc es t o Y O / 16 = ¢ v2 - 1, 7 equat i on s (1 2 0) a n d (1 2 1) b e c ome 1 /Y ° _2 (_ ) 2 _ 2 8 (_ 2 1 )( V2 + 5) (1 2 3) ' _

c_. - gk_ ) - _ r 'T r " !

• "v _ Y O 1 _ (1 24 ) i

I ' c2 - _ - g

i ,

" L

so the boundary redu c es t o " 4 9 _ Ay _1 2 ,P2 2 = I-'E ( 12S ) - _ a n d th e eigenv alu es to 1 _

r

L l- _ _

Comp ar e d t o t h e ind i vid u a l blade re s u l t , th e bo u ndar i e s are ident i c al i f - I, bu t i f _ > I thi s b o undar y i s o f f s e t mo r e, wi t h a sm a ll e r widt h , C o m- p ared to t he t e e t ering r oto r , the w idth i s L . _ same but the teetering r o t o r is not o ffs e t even when v > 1.

Summ a ry Fo r t he hove r limit , _ = O , t h e fo ur e igenv a l ue s of t he t h r ee - b l ade d gim balled rot o r are J a n d their conju g at e s. Fo r the h o ver frequency away fr om ll / rev o r I / rev (ro tatin g ) , t h e roo t s t o o rder _ are t h e s a m e as t h e h over r oo ts, a nd t o ord e r _ 2 t h e y a r e g i v e n by e qu ati on ( 10 4) a bo ve . The i nfluen c e o f for wa rd , fligh t cons i sts of : an o r de r _ z f re q u en c y ch a ng e , sim i l a r t o t h e i ndiv idual i b la d e bu t w ith reduce d m a g nitu d e; a n o r d er ' _2 c h a ng e illth e +_i f re quen c y ; sh i f t du e t o t h e tr ansfo r m ati on t o t he n on r o tati ng f r am e; a nd a + - o rder _ 2 c hang e i n t h e d a mpi n g of c om ple x r oo t s of th e fr e qu e n c y fo r r ea l r oots. T h e latter c ha n g e h a s the e ff ect of pul l ing a p art t h e real r oo t in ter s ecti on of . . t h e lo c us ( at - + I / r e v f r e qu e n c y nonro tat ing), so t h at t h e roo t l o c i only pa ss , i c l os e wh e n u > 0 rather than a c t u al ly i nt e r s ecti ng (t h e s eparati on i s o r d er .

' I u ) . T h e cr i teri on fo r diver g ence s ta bi li ty i s , in t h i s c a s e i , , _ : Th e ll / rev cr i t ic a l r eg io n i s id ent ic al to that o f the i n d iv i d ual b la d es, j _ Only t h e h ig h f requency m o d e , at 3 / 2 / rev non r o tatin g freq u e n cy , participates in th is r eg i on , h o we v e r. The low fr e qu e nc y mode , at _ / rev , has the sa m e ' - b ehavi o r a s it does awa y fr om t h e c ritical re g i o n - t h at is, n o f or w ard f l ig ht i ,_ i n f l uence at a l l t o o r d er u .

i so

j_ If Ay = 7 - YO i s order u 2 sn m !l , wh ere Y O is t h e va lu e re qu ired f or I m X0 = 1 , t h at i s, 70 / 16 = K p + / _ 2 + Kp2 - 1; th en t h e lo w f requenc y mo d e, at O / re v nonrota t ing f requenc y , encounter s the 1 / rev critical reg i on.

Th e hig h f reque n cy m ode d oe s not part i c ip ate i n t h e cr i t i cal re gi o n be h av i or; i t s root s ar e g i ve n b y t h e s a m e e xp r e ss i o n a s aw a y from the cr i t i ca l re gi o n , th a t is, e quatio n ( ]04). Th e low f re qu e ncy r o _ 5 a r e giv e n by i

_

, , y<

(17 9 )

wh ere th e cor ne r _ 2 _ re

1 - 6 \16 - KP )

gZ2, _ 2 2 = CZ ± C2 ( 13 0) wi t h C1 a n d C2 g i ven by e q u at ions (1 20) an d 4121) above. The beh a v i o r o £ t he loci in t he cr it ic a l r e g i on is s imi lar t o wh at h as been s e en be f ore wi t h t he in d ivi dua l bl ad es and t he t ee t erin g r o t or. Th ere ar e now , however , o r der _ 2 chang e s in th e f req u ency and d a mp i ng t in C1 , an d ex p l i c i tly in R e X a bove) cor r es p on d ing t o t he c han g e s seen a w a y f rom th e c rit ic a l reg i on , a s w ell as some ch an ges in t he ma gni t ude o f t he f o _a rd £ 1igh t i nfluence.

I I , The 7 - _ plane bo und ary f o r the t w o rea _ root regio n (± I / rev I in tersect ion n o n rotat in g), w r i t in g 7 = Y 0 + A Y , i s T

' : (- Kp . / _2 K : ) . K p

;. 16 / _ 2 _ K p2 t ' % .. where Y0 / 16 = Kp + / _ 2 + Kp2. Thi s is only a so£ t boundary , ho w ev e r , the I point of closest a pp roach of the branc h es w hen U > O. The _ / rev critical region bo u n dar i es (5 / 2 / rev nonrotat i ng) are J Y 0 /_ 2 YO &y _ -_ - -_ - Kp * 4 K p2 1 -- ' 6 " ± _ (132) YO

z-'_ " K p

_ w here 70 / !6 - Kp + / v 2 + 'K p 2' _ '(1 / 4 ) ; and t h e 1 / rev critical region boundari es (O / rev nonro t atin g ) are C ± C 2 | _ , _ Y _ 2 1 • _.- Kp 4133) , $!

. . .... . . .

v _ P I where Y 0 / 16 = K p . / v '' _ + y .p2 _ i. For Kp ffi 0 these bo,mdaries reduce to 1,1 2 8 3 YO 16 • . _ . v wi t h 1 -' 6 " v (13 4 ) ,_ . J

, , o F _

16 " + _ _ _ w it h _ - _ - - (155) 16 = p2 _ O • _ ( lZ 6 ) ' respect ively.

GIM BA L LED ROTOR , FOURBLADES The equa t ion of mot ion f o r a four-bladed gisb a lled ro to r are

, 8 - '2 - zc

= (13 7 )

The o nl y f o rw a rd flight effects re m aining are order u 2 , the order u m o m en ' , 1 cancell i ng in t ernall y a t t he hub as the y did fo r the teetering rotor. The onl y periodic coefficien t s remaini n g are order _ 2 hence , only a I / re v critical re g ion i s expected.

The hover resul t i _ identical t o that of t he N • 3 case; that result , : p in f act , h o lds for all N >_ 3. Only order p2 ter m s appear in the e quat i on s . _ , . of mo t ion , so a . _ e xpans i on in u 2 i s u s ed: _ ? .

,. : 52

. !

I

B 8 + 1J 2 @ _ - _ = _ -- _ _ .

Y = YO + _ 2 X 2 + " " The order 1 so l u t io n i s th e h o ver li m i t again, s o is th e sam e as t h e N = 3 r esu l t . T h e sol u ti o n fo r _0 is m ¢ 80 : R e 02+(¢ 2, ..-) e( l O+i ) ¢o + 802- ( ¢2,. • . ) e( X O-i)_ ?.

(13 8 3 '_ Or de r _2 Resul t s Th e o r d er U 2 e qua t i o n i s ' " [ 8 ¢ 02+ @TO . 2 _ I + Kp 8 280 T 2 _ o + 72 b S0 P T+ Kp _-_ -_- + _ -_ 8 0 = 2 _¢0 _¢2 Y 0 _ -8" - _ ' _ 0 +/ Y 2 Yo

2 L T + T6" Kp -_-+ Kp 3 Y0

I_ J , Y 0 s i n 4 ¢ 0 -K p i-6 co s 4 ¢ 0 - I- - 6 cos 4¢, 0 - Kp i- 6 sin 4_ I._. Y0 Y0 Yo 07 t + _l 0 ; Kp " 1 - 6 sin 4 _ 0 " "['6 s in 4 ¢ 0 + Kp i-- 6 c o s 4 ¢ " _

oJ "

,-,.

, !

, , 53 jr + ( i + K p) B0 2 e + ( -i + K p )8 02 e [ X 0" 3 i ) 90 + (-i + K p ) B 02 e ( l O 'i)$ O + 1 - T ( i + K p ) B0 2 •

-](;/ [,o

+ conju gat e ( 1 39 ) A s s um i ng th a t ImX 0 _ 1 , t he secu l ar t erm i s _8 02 ± _$ 2 X2B O 2t = 0 (140) whez e 8 (_0 + Kp) + _Kp 1 2 = . 2iiml 0 l T he s olution i s 8 02 ± = C 0_ ± ($_,. . .) e X 2 _ 2 . Hen ce , th e e ig e nv a lu es to or d e r ! _ 2 ar e X • X 0 + _2X 2 ± i = ± J " i _ + i 2 + (I + _ 2 ) Kp - (141 ) , and the con j ugates. Thi s i s the same res u lt as f or the teeter i n g rotor root s awa y f r o m t he 1 / rev crit i cal regi o n. I t fo l lows t hen th at the real root boundary ( at ± l / r e v nonrotat i ng) and th e d i vergence stab ili ty cr i terion are the sa m e as f or the teetering rotor, i Near 1 / rev Frequency :, If t he hover roo _ s are order _ 2 from 1 / rev frequ e nc y ( ro t ating) - tha t 1 [ i s , Im X0 = 1 - t h e n , the p e r i od i c coe ffi c i e n ts co n tr i b u te t o the order _2 .

sec ular t e rm s: - - : _ ' 54 _"_ - I @80 2 ± YO -- @_2 _ 2 802± - i , _ (i + Kp)B0 2 ± = 0 (14 2 ) C o m par ing this wit h the te e te r i ng r oto r 1 / r ev secul ar e q u a tion , i t f o ll o ws t h a t the fou r -bladed gimballed roto r h a s id e ntic a l e xpr e s s ions fo r the r oo t s a nd bound a ries of t he I / rev cri t ic a l r egion.

S u mma r y # The b e havio r of t he fou r -bladed gimball e d roto r is th e s a m e as that of t he t ee t e r in g roto r flap sta b ility - exc e pt , of cou r se, tha t t he g imball e d roto r has four roots , in t he nonro t ating fr a m e , hence, shif t ed by ±i from t he t e e t ering rotor r oo t s (rotatin g ). As for the t e e t er ing r oto r , it al s o follows .

he r e t ha t the constan t coeffici e nt approximation t o t he e qua t ions of mo t ion, e ven in t he ro t ating frame so the only forwa r d fligh t influence re t ained is the fac t or of (I + _ 2 ) in t h e mean of M8, g ives exac t ly t he correc t eigenvalues excep t n e ar the i / r ev critical r eg ion.

Away f r om t he cri t ical r e gion , th e eig e nval u es of the four-blad e d gimballed ro t or are

,I

and their conjugates. There is no _ / rev critical region. The I / r e v critical region is the same as for th e teet e ring rotor; both the high frequ e ncy and low fr eq uency modes (at 2 / rev and O / rev nonrotating) participate in the critical region beh a vior. T he Y - U pl a n e boun dar ies a re t he s a m e as given for the tee t ering r o t o r.

G IM B ALLE D R O T OR , FI VE O R M ORE BL A DES I F or a g i mb al l e d r oto r w it h f iv e o r m o re bl ade s, the e q uat io n o f m o ti on i s _ '°.

= ( 144 ) 8 + _ - U \ O l s Kp Bl s / S o to ord e r u 2 the re a re n o peri od ic c o e f f i c i ent s w h e n N __ 5. A l l t h e p er io d i c f la p m o me nt s can c el i n t ernall y at the ro tor hub , an d do no t co nt rib - u t e to t he n e t p itc h an d roll moment s on th e rot o r d isk. T he re s u l t is a con - s t ant c o ef fici ent di f fere n t i al e q uat io n fo r t he gim balle d r o t o r wit h f i ve or more blade s . T he e i g envalue s o f th is rotor are g i v en by the ro o t s o f the ch a ract e r is tic e quat i on of j , _

L

or - 2 + _ + v 2 - I + K p (i + p 2) + 2X + - U 4 (I + K pz ) = 0 ( 1 4 5) As a c onst a n t c o e ff ic i e nt e qu ati on, t h e ei g e nv a lues are sim p ly the r oo t s o f a _ t polynom ial , a lthough i n t h i s ca s e it is a f our t h-o r d er p oly n om ia l, w h ic h m a y, i n g e n era l, on l y b e solv e d num erica lly. S i nc e th e r oo t s fo r th e limit U = 0 ar e kno w n , how e v er , t h at i s, th e hov er r oots - a per tu r b a t i on tec hn i qu e m a y b e us e d to ob tai n e x p li ci tly e x pre ss i ons fo r X i n c lud i ng th e in f lu e n ce of i _ f o rw ar d fli gh t.

• ' On l y or d er U 2 a nd ;,4 ter ms a pp ear in t h e charac t eri st ic e q uati on , s o a n expa nsion i n U 2 is us ed . W r i te X • _ 0 + i + B 2 1 2 + . . an d t h e conju - i g ate s, wh ere _ 0 is th e rotati ng h o ver ro o t a s u s ual : _ 0 = " 1 - - 6 + i 2 + Kp- ( 146) j Th e Lo c k nu mb e r is a lso e xp a nded a s a se rie s i n _ : Y = YO + P 2 Y 2 " + _ i _" The , or d e r I character is t i ce q uat ion t h e n h a s ju s t the s olut i o ns X = _ 0 Y O + i. Th at i s, t h e or d er 1 te rm in t he ex pa ns io n o f _ a bo ve is ind e ed th e i cor r ect h over limi t .

t ! Or d er _ 2 R e s u lt s Th e o r d er u 2 te r m s of th e c h arac t eri s tic e qu ati on are + ( X0 ±i) + K p 0 + i) 2 + - 0 :.-4

I s6

... tN

o r (link 0 ± 1) ( 2 ilm k O) k 2 + K p + -_-(k 0 , K p) = 0 (1 4 7)

[ ]

Assuming th a t I mk 0 _ 1 wh e n the low er frequen c y root k 0 - i i s b ei ng c ons i d ere d, so Im k 0 - 1 _ O, t h e n the solut i on is Y---_- 2 (X0 + Kp) + Kp _ 2 = i ( 148) _" 2 1m k 0 It follows th e n t hat t he eigenv a lues t o o r de r _ 2 a r e

/ (, : X = -+i-_6 + i 2 + (I + U 2 ) _ Kp - _-_ (1 4 9)

an d th e c o n ju g a t e s. The r e a l ro o t bou ndary a n d t he di ve rg ence criteri o n ar e ° the s am e a s g ive n f o r t he t ee t e r in g r o t o r .

N ot i ce that t h i s is t he same r esul t a s w o ul d h a ve been o bta i ned if a , c o n stant coeff i cient a pprox im atiou were made in the rotating fr a me before finding the net pi t ch a nd roll moments. Ac t u a lly, the expansion of _ as a necessary, s e ries in U 2 is not really f o r to order _2 the last term in the ' c h ara ct er is t i c equat io n ( t he u4 te r m in e q. (145)) drops and l may be foun d dir e ctly, e x ac t ly as fo r t he hov er c a s e . However, such a p r oc e du re do e s not show t h a t t h e solu t ion is not v a lid for t h e low fr e quency mod e n ear I / r e v i (ro ta ting fr e q uen cy ) .

• f N ea r i / re v F re quen cy ; J This is w h ere t he cri t i ca l r e gion is e n co u n t ered when the eq u a t ions h a ve pe r io d i c c o e f f i c ie n ts, th at is, w h e n ImP 0 = I. T h e high frequ e n c y mo d e is at 2 / rev nonrotating freq u en c y, and the roo t s mus t remain c omplex c onjug a t es 1 sin c e t his is a con st a n t co effi c i e n t eq u a t i o n. He nce, th e i / rev c r i t i ca l r e gion beh a vi o r c an n o t b e e n c ount ere d f or t his mod e . I n d ee d, the o rder _ 2 ,, re s u l t ob t ain ed a bov e hold s fo r t h e _0 + i r oot, e ven wh e n Im_ 0 = 1 ( t h e uppe r s ign is u s ed i n e q. (147) , s o t he fi rst f a c to r h a s t h e va lu e 2). Th e low f re que nc y m od e, how eve r, is a t 0 / r ev non r o t a t ing f r equency, so t h e s e roo t s a r e abl e t o mee t and p r o c ee d in opposi t e d i r ec t ions alon g t h e re al a xis. This ','_ is t h e i / r ev c r i t i ca l r e gion beh av ior which i s a llo w e d fo r t h i s c on st an t , c oe ffi c i e nt e q uat ion b e c au se t he tr a nsforma t ion to the non rotat ing f ra m e pu ts t t he s e r oots ne ar th e rea l ax is i ns tead of 1 / r ev.

i C onsid er th e l ow fre q u en c y mo de w he n Iml 0 - I; th at is , = -(YO / 16 ) + _ 2 _ 2 + .... Th e o r d er _ 2 c h a r acter is t i c e qu at ion i s ) 5 7 , :C I , ; ' t'

I

/ - i d e n tially ze ro then (eq . (1 4 7), us in g the lo w e r sign so t he fi r s t fa ct or has t he value 0 whe n Iml 0 = 11, a n d i t is necessa r y t o g o t o o r der u4 to find k 2 . The order u4 t e r ms in t he c hara ct e r is t ic equa t ion are fo r t his c ase Or ( X2 + = - D 2 ( 1 5 0 1 wh e r e

. : -_ + z 16j " k32 1 (1 Kp2 )

S e tt i ng D 2 = 0 gi v e s k2 = - 7 2 / 1 6, so _ = -y / 1 6 t o ord e r _ 2 . H e nc e , D 2 = 0 -, i s th e b ou ndary f or tw o r e a l r oo t s i n th e n o nr o tat i n g f r a m e , o r th e 1 / r ev bou ndary in th e r o t a t i n g fr a m e . In gene r al , th e r o o t is ' ), = - T _ + v 2 y_±i = - y_ Y O <- Y 2 D> Y _+ u2 i D (1511 Th e e xpr e ssions for _ a nd D a r e i d e nt ica l to th e c orr e spond i ng on e s for th e t ee t e r in g rotor, or for the four-b la d e d g i mb a ll e d rotor, n ea r th e 1 / r e v c riti- i cal reg ion. T h us, th e gimb a lled rot o r with five or m or e blades en c ounters a 1 / r e v c rit ica l r e g i on , a lthough only the l o we r f r e quen c y mode (ne a r th e r ua l a xis nonrot a ting) p a rt ici pates i n th e c r i ti ca l r e g i on beh a v i or. _ . , bound a ry of the reg i on, and th e so l ution fo r _, a re th e s a me a s we r e given Jr th e _ b ./ teeterzn g r o t o r ( - i to get t o t h e low f reque n c y no nrot at ing mode l. , S u mmary i Th e gimba ll e d rot o r wi t h f i v e or mo r e b l ades i s desc rib e d b y c ons tan t _, coeff ici e n t diff ere n tia l equati ons . T he b e h av ior of t he ro ot s i s , ho w e v er , !_ t n e a rl y i d e nt ica l to that o f a t ee t er ing roto r or a fo ur -b l a de d gimba lled roto r ; t h e latter i s p er haps a b etter c omp ari son sin ce it is a l s o a c as e w it h no n- '_ ro tat ing d e g r ees o f free dom . Awa y f r om I / r ev ro t ating freq uen c y, t h e roots ; _ ar e given by

i

= + i -1 _ + i 2 + + t1 2 1 _ Kp = (1 52) i

D

! s , and the conjuga t es. The real root bounda r y (±l / roy nonrota t ing) and the divergence criterion are t he same as for the t eetering rotor.

A critical region is encountered by t his rotor, when the rotating frequency is near 1 / rev. Th e expressions for the boundary and the roots near t he critical region are the same as for the teetering rotor. Only the lower _ frequency mode participates in the 1 / roy critical region, however, for it is the fact that this mode is near the real axis (nonrotating) that allows the critical region behavior to occur with a system described by constant coeffi- cient differential equations. The high frequency modes are near 2 / rev for - t his c ase , and s o ar e still given by equa t ion (152) , tha t is , ha v e t he same _, behavior as away from the critical region.

The eq u a t i on s o f m ot i o n f or t he g i mballed r ot o r wi t h N L 5 ar e id e ntic al to t h e cons t ant c oeffi c i e n t approximation o f t he eq u a t io n s f o r t he t hr e e o r - - • four-bladed gimballed r ot ors (i.e., e q. (87) or (1 5 7) with the periodic t er m s dropped). Henc e , the solu t ion for N _ 5 may be consid e red as a c onstant coefficient approximation to the d y namics of the N = 3 or 4 cases.

As a constant coefficient approximation to the N = 4 case, this present solution gives exactly the correct roots with the exception of the high fre- quency mode near the 1 / rev critical regi o n. Th e se roots for the N = 4 case encounter the critical rugion along with the l o wer frequency roots; but for the constant coefficient approximation s_eh behavior is n o t all o wed f o r the high frequency roots if they are to remain complex conjugates. The behavior , of t he low frequency roots is given correctly everywhere , including the 1 / rev i , critical region, because in the non r otating frame that behavior oc curs o n the f real axis. Away from the 1 / rev critical regi o n, it is , in fact , possible to make the constant coefficient approximation in the rotating fra m e , before ' finding the net pitch and roll moments, and still obtain the corr e c t i expressions for the roots of the four-bladed gimballed rotor.

As a con s t an t co effi c ien t appr o x i ma t i o n to t he N = 5 case, this presen t solution is not really good anywh e re. The errors are not even order u2 only, because the t hree-bladed gimballed rotor encounters the hJrev critical region i_. w here t here a r e or de r _ effects o f forward f l i g h t . T h e or d er _2 effe ct s [ on the roots , bo t h a w ay f r om a n d nea r the 1 / r ev c riti c a l reg i on, a r e , in ad d i - I t ion, quite different f o r t h e N = 3 case.

i E Q UA TI ON S OF M O TI O N IN T H E N ON RO T A TIN 6 FRAME i C o n s i der a r ot o r w it h N i n d epen de n t blad es , e a c h w i t h rot a t i n g n at ur al : , • f r equ en c y v . The flap mo tio n o f t h e e ntir e r o to r i s d e scrib e d by t h e N :_ d egree s o f fr eed om _(m) , and the N r ot at i ng equa t ions o f motion giv e n abov e i (e q . (5 7)), i n c ont r ast to th e g i mballed ro to r , w h ic h i s de s c r i b e d b y onlt t w o degr e es of freedom and e qu a t i ons for all N > 5. As for th e gimballed r o tor, : ' ho we ver , t he m o ti on of t h e r oto r wi th i nd e p e n - dently flapp i ng blad e s ma y a l so d t , i _ b e d e s c rib e d i n : Le n o nr otat i ng f r ame . Th e fol low in g n ew de g ree s of f ree do m _ a r e i n t rodu c ed : ii, ' ' 59 1 _ B(m) Bo = _" m= I N 8 no = _" _E_ B ( m) cos n _ m m=l ( 1 5 3 )

} , ' 2 8 (m) s i n n_b m

ISns = _ " m= 1 Bim)(_l)m B N / 2 = _- m= 1 where 0 ; m = _ + mA C i s t he azi mu th l oc ati o n of the ru th bl ade a n d A_ = 2 _ / N i s t h e i n ter v al b etwee n t h e bl a d e s. T h e fl ap mot i on of th e ruth b la d e i s then gi ven by Bim) = B0 +_= _(B nc cos n¢m + 6ns sin ngm) + 8N / 2 ( - 1) m (154) wh e r e t he sum ov e r n g oes f ro m 1 to iN - 1) / 2 for N odd , and fr o m 1 to ' i N - 2 ) / 2 f or N e v e n. The 6 N / 2 d egree of free dom o n l y a pp ears if N is ev e n .

This is a Fo u r i er c oo r d i na te t ran sf or m a t ion f ro m th e N degree s of fr ee d o m B(m) de sc r i bing th e rotor m o t i on i n t h e r ot ati ng f ra m e to th e N , d egr e e s of free dom B0, 8nc , Bns, B N / 2 descri b i ng t h e r o t o r m otion in t he non rotat ing fra me (for f u rthe r discuss i on, se e , i. e. , r ef . 13). Th e 8 0 va r iable is t he roto r c oning mo t ion; 81c and 81s a r e the tip p at h plane longi- tudinal and l a ter a l tilt deg r ees of freedom a s fo r t he gimballed ro t or. The ; I BN / 2 m o tion i s s imila r t o th e co ni n g ex cept t h a t th e bl a de s al ternate i n u p " -- an_ d ow n motion. This coor d i na te transf or mation m ust b e acc om pa n ied b y a c on - l v ers i o n o f t h e equat io ns of m o t io n fr o m t he r ot a ti ng fr am e ( eq. ( 5 7)) t o th e nonrotat ing f ra me . T h is i s a ccomp lis he d b y o p er a t ing o n th e equ a t i on s o f m ot i on wi th the fo ll o w in g s u m matio n o p er at or s : ' 1 2 2 1

El...), i...)c os n m , m

m m m m (INS) The s u m m ati ons co mb ine the e q uati ons f or f lap mo men t e q ui lib r ium into th e ; r o t o r m o m e n t equi l i b ri um a p pr o priate to th e n o nr ot ati ng d egree s of f re edom . _ I. a n d fo r th e 81 c an d B l s m otio n , t he operat o r s f in . _ th e ne t p i tc h a nd r o ll : . For e xa mp le , fo r t he co nin g mo t i on , th e f irs t o _e rator f inds th e conin g m om en t; ! ,, ! , V . _ _ - - , i m o ments o n the disk _ was done for t he gimballed r otor. The r esul t in g N e quations in the nonrotating frame are coupled, even for the case of a shaft fixed rotor as here in contrast to the rotating equations, which are not coupled at all (eq. (57) ) .

The r esult o f t he su m m at ion oper ato rs depe n ds o n th e n umbe r of bl a des (N) where , as here, the equations have periodic coefficients. Hence, the set of N nonrotating equa t ions depend on N. The complete sets of differential equa t i o ns describing the r ot or flap m ot ion o f N independent blades excited by blade pitch inpu t s only are given below for the cases N = 3 and N = 4.

d - N=3 "Z 0 _--Y-, 1 2 " I + 0 _ + _ sin 3¢ 2 - u 1-_2 cos 3_ Blc

_ y Y

U _PlS /

-2 - _ ___ cos 5, _- _ _Tsin _, \_ls /

i, I !

rod. _ (

i I

J N=4 jr The derivation and discussions o f these equations may be found in the literature, for example in references 13 and 18.

The t r a nsfor m ation t o nonrotating degrees of freedom and equa t ions of mo t ion has t he effec t of sweeping the periodic coefficien t s from the lower degrees of freedom, especially as N increases. There are, however, always periodic coefficien t s in the nonro t ating equations where _ > 0, jus t as there ar e in the ro%ating equa t ions, for the same sys t em is being described and a , cons t an t coefficien t set of equations could never give t he behavior that has been found with the periodic coefficients.

The flap d y n a mics described by these sets of differential equations is nothing new. This is still a set of N independent blades and simply observ- ing the motion in t he nonrotating frame does no t change the nature cf the s y s- tem. Hence, the eigenvalues m ust still be as calculated above for the case of a single independent blade; the only c han g e is a -+ni for the 8nc / Bns modes, that is, a _+n / rev shift in the freq u ency to account for the transfor m ation ! f r om rotating to non r ota t ing f r ames. Th e p er turba t ion solu t i o n f or t hese e quations, in c luding t he influen c e o f t h e peri od ic c o effici e nts in f o rwa r d flight, has already been obtain e d th e n. The value of t h e desc r ip t ion of t he r otor by t h e se nonrotating degrees of f r eedom and equa t ions lies in i t s use for d y namics involving the helicopter b o dy or shaf t motion , ae ro dynamic gust, o r any o t he r exci ta t ion from t he n o n ro ta t ing f r am e . Th e rot o r res pon ds t o s uch e x c i t a t ion a s a whole, in non r o t ating modes of mo t ion, s o t his d e s c rip t i o n o f th e ro t or m o t ion and momen t e quilibri u m is a pproFria t e for s t udyin g su c h prob- lem s . Fo r t h e cu r rent inves t i gat ion, the us e of t h e se non r o t a t ing equ at ions ' is th a t t hey provide a b a sis f o r a c onstant c o e fficien t a pproxim at ion for the rotor flap dynamics.

CO N ST AN T CO EFF ICI ENT AP PR OXIM A TION As should be q ui te a pp a re nt b y now, d iffe re n t i a l e q u at i o ns w i t h pe r io d i c co e f f icients requir e co n si d er a bl y mo r e an a l y si s th an con s ta n t coeffici e nt I eq u a t ions, even to s imply find t h e eigenv a lues. Co ns t a n t c oe fficie nt differ- en t ial equations a r e much preferable f o r t he s t udies of dynami c sys t ems. C an { the fl_ d y nami c s of a h eli c opt er rotor i n forward flight be adequ a tely d e s c ribed b y s om e con st a nt co e fficient ap p r o xi ma t i o n to the equations of i moti o n? The con s ta n t coef f icie n t appr o ximations f o r the t eete r in g and gim- _.

balled rotors were discussed along with their perturbation solutions including the periodic coefficients; this section will be concerned with the case of individually flapping blades.

Co ns i d er the r ota ti n g e q u ation o f motion f o r 8(m) (e q . ( 5 7)). Th e t • ,_ c onst a nt c oeffi c ient a pproxim a tion to th a t equ a tion , th a t is , using only the ' " mean v a lues of t he c oeffi c ients , is i { _ , _ + = , (l , _2 ) S = o (I SS)

' q

, t % j_ The eigenvalu e s o f whi c h ar e The o n ly inf l u e nc e o f f o rw a rd fligh t is th e o rd e r _ 2 incr ea se in t he m e an o f M0 , so in the ef f ec t iveness of K p . Consider this equat i on a n d its roots _-_ a 1 representa t ion of the flap d ynamics o f an in d epen d ent bla d e in f orward fli g h t.

A correc t es t ima t e o f t he ei g envalues to order u is ob t ained wi t h s uch a represen t a t ion, ex c ept near t he ½ / rev c ritical region, simply becaus e nei t her t hi s cons t an t coefficien t appr o xima t ion nor t he correct s o lu t ion away f rom t he _ / rev re g ion sh o w any order u eff e c t s. T o order u 2 , t hi s approxima t ion i s not c orrec t even away From t he cri t ical re g i o ns.

With t he con sta n t coefficien t appr o xima t i o n m a de in t he r ot a t in g equa t ion, t he result i s vir t ually jus t the h o ver r oot ; so t he que st ion of t he appli c a- bili t y of t ha t appr o xima t ion is really whe t her t he hover roo t s may be u se d a s a mea s u r e o f t he flap dynami c s in for wa rd flig ht a s we ll a s in hov e r . Th i s a pproxima t ion miss e s t he ½ / r ev and I / rev c riti ca l regi o ns en t irely, a nd s o pro v id e s no infor m a t i o n at all a bou t th o se e ffe ct s. Aw a y f rom th e c ri t i ca l regi o ns, how e v e r, t he corre ct ro ot s show only quit e small influ e n c e of f o rward f l i g h t , as for ex a m p l e ca s e C c ) in f igu r e 1. This con s t an t coe f f i cient app rox ima t i o n = th a t is, the hover root - m a y b e con si dered a re as o n able r e p r e s ent a t i o n f or s uch c as e s . I t s hould be n ot i ce d, ho w eve r, that the influen c e o f t h e cri t i ca l r egi o n s b e c o mes g reat er as _ i n cre a s e s, s o t his ' approx im at ion _ ' . st ev en tu ally break d o wn for eve ry case.

!

I The nonr otatin g eq uat ions of mo t i o n pro v i d e a n ot h e r s ou rc e o f a c ons t a nt c oe ff i c i e n t a pprox i m at ion t o t he fl a p dyn a m ic s of a ro t or w i t h N indep e nden t , blades . T he transforma t ion to the n o n rot a t ing fr a me is a cc omplished first, i and t hen t he av e ra ged c oefficien t s ar e fo und. The ha rm o ni c s o f th e co e ffic i- . en t s of t he ro tat ing e qu a tion c on t ribu t e t o t h e c ons ta n t c oeffi c ien t s in the i" nonro ta ting frame, h e n c e, th is pro c edure r eta ins mor e of t he influen c e of t he p e riodi c c o e ffi c ients. For N = 3, t he re sulting equ at ions of mo t ion a re l + 0 -yM- - M. 2 1

0 2c

B 0 M.2c_B,s / _i V -yM_s -2 -YMA +_ a D Ic v2 I y 2s 0 2 c + -yM8 - - _M a -yM. -_X_ I B 0 _ 2 c 0 2 c v2 i V1V ' It# ' _ - _ M a + yS_-_Mg - - - , ' y H 0 0 _ " HO 0 - KpS o 0 2c , ' = 0 YM0 + _M 0 0 V 1 c - KPBlc l (16 0 ) ls 0 2c k

o o Y"o - "o - KpB, , /

w h ere u se h a s be:n made of t h e f o llowing F ourier series re p re se nt a tio n s )f : h e flap moments i n forw a rd fli gh t: I i o I S 2c !

1 MB : M0 + M o sin _ + M O co s 2_ + .

0 MlS 2 c M. - M. + . sin _ + M. c o s 2_ + ! S _ a B I C 2s MB = MB c os _ + Ms sin 2 ¢ + f _ i T he mi s sin g h a rmo n ics i s a ge n e r a l r es ult val l d for al l _ even incl udi ng th e ' , I rev er se f lo w reg ion ef fec t s. Su b stitu t_n_ fo r _Y e harmo n i c s o f t h e f l ap mom en t s fro m eq u atio n ( 2), th a t i s, n egle c ti ng r ev e , : e fl ow a s us ual f or th e I ! p r ese nt or d e r u 2 a n al y si s , th e eq u at i on s are p t.

I ' ', 65 j_ /' _o \'" v_ 0 _' I-_ Bo " g lc \BlS / _- -2 _ + Y v2-1+ 1+ Y=-Kp ; =0 (1 6 1 ) Kp_ _ -_{ I -_ ) _ 2 - 1 +( 1 _ 2) _Kp ,BlsJ _ 3 8 m # , This is t h e c onst a n t c o eff i c i e n t a pproxim at ion , in t h e nonr ot ating f ram e , i or t h e f l a p dyn a m i _.s o£ a ro t or wi t h thr ee ind e p en d e n t bl a d e s. It m a y a lso b e ob ta in e d from e qu a tion (1 5 6), by dropping t h e 3 / r e v t e rms in t he c o ef fi cie a t s .

Th e a p p roxim ate e ig e nv al u e s a r e th e n solu t io n s of t h . c har acte ris t i c e qu a tion . J

[ 1'

t + (2X +_ _ )2 . tjL , (l _ _6 )2 ( 1 + Kp 2 )} _p 2 (1_ )2 2 ( ,k + 2 Kp) [2 ;k + _ ( 1- -_-_-- ) ] -t_ 2 (1-_-) 2 2( ) . + 2Kp) 2 D ,2 _- ). _-(1 + - _ "0 (1 , _2 ) Only u 2 a nd _h ter ms a p pear, so co n sider a n expan s ion i n _ 2: = _ 0 •. , or _ = _ 0 + - i + u2 X 2 + . .. , a n d the conjugates , for t h e con in g mode and B lc / B l s m odes , re s pec ti vel y . Th e Lock nu m ber i s al s o . , exp a ad e d as _ serie s : _ _ ¥ 0 + _ 2Y2 .; a nd _ O i s t he ro t at ing hover I roG t as u s ual: / ,, ;_ o " " 1- ' 6 Kp - _ 16 J To order I , t he s ol u¢ion f or t he r _ ot s i s ju s t X • t 0 , _ 0 ± J a n d t heir con j u gat es , verif y in g t ha t t he above exp a n si on is co rr ec t f or t he hover limi t .

,_.

Co ni n g M odes , _ , . " C on si d er the co n i ng mo de roots in forward flight , that i s , the expansion (: _ ( X • _ 0 + u 2 _ 2 .. The order _ 2 l ena s in th e charac t eri st ic eq ,_ a t ion _ . +_ gi ve t hen ' 66

•

r of #) r , - _- / 2010 + 2Kp) + + + = _) 2(A 0 2 K p) 2 0

or + - -_- K p + 4 K p 2 = 0 ( 1 6 3 ) Assuming that In a k 0 # 1 / 2 , t h e root to or d er u 2 is then -- -Y-- + i + (1 + _2) _ Kp-(_ + _2_ .

(164) T his is ex a ctl y t he correct res u lt for t h e eigenv a l u es a w ay fr o m t h e critical regio n s, th a t is, eq ua tio n ( 2 4). Th e re a l root bo unda r y an d div e rgence crit e - rion for this root are also correct then. This expression also applies for the coning mode approximate root when Im k 0 = I / rev, however, so it misses t he ' i / rev criti cal region .

' Now co nsi d e r th e co ning mod e root n ear Im k 0 = 1 / 2, that i s , to order 1 k 0 = -{Y 0 / 16) + i / 2. The o r de r u 2 expansion did no t w o r k t he r e, s o c o nside r an o r der _ exp a nsion: h = k 0 + H A 1 + a nd y = Y0 + _Yl + .. The ord er u term of the cha r a c te ri stic equagion is iden t i c all y ze r o t h e n, a n d , ( _. the orde r u 2 ter m is I + _--" 0 ) _0+ o / kl+ , 1 - ' 2 "1 2{ k O 2 Kp) k 0 o - or

,)] ( , , k l : - i- 6 - K + k1 2/ - TKP

/ = - D 2 (1 05 ) ,. )j Th e root to order p is then A = X0 * p A1 = - + _ ± i _ D (1 66) T h e s e a re e xa ctl y t h e sa me e x pre ss io ns f or D and _ a s were fo u n d for t h e _ _ J re v cr i t i cal reg i on by the pe rtur b at i on analys is i n the rot a t i ng f r am e , i nclud i ng the per i od i c c oef fi cient s . So the coning mode o f th i s con s t a nt c oe ffici e n t a pprox i m a t io n e n coun t er s the _ / rev c r i t ical reg i on , wi th t h e correct boundar y and root s . N o w the ro o ts i n a cr i tical reg i on h a ve a p iu s and m i nu s i ncr e ment i n t h e damping due to t h e p er i od i c coe ffi c i ents, t h at i s, _" t he root s a re not complex conju ga te s . W hat i s happenin g to a llow the roots o f a con s t an t c o ef fi cient a pproxi ma t i on to exhibit thi s cr i tic a l re gi on be havi or is the follo w in g . Th e low frequenc y m ode root s k 0 " i an d the co n ju g ate are ~ . , a lso a t ±_ J rev w hen Im_ 0 = 1 / 2 . H ence , w i th the con i n g a nd lo w frequency mode roots there a re a total o f f our roots, two a t _ J rev a nd two a t -_ / rev , w hich mu s t pa rt i c i pate in th i s _ J rev cr i tic a l re g ion beh a vior. H ence, the _ ,. cr i t i c a l reg i on be h av i or ca n occur whi le t h e root s re m a i n a s complex conjugate pai r s as is required f or consta n t co e f f ic i ent equ a t i on s .

High and Low Fre qu e n cy M ode s Consid e r the lo w frequ e n c y a nd h i gh fr e qu e n c y rotor modes, th a t is, the ex p a nsion _ = _ 0 ± i + u _2 + .... Th e n t h e or d er u2 e qu atio n is Y 2 (ImX 0 ± i ) (_2 Im X 0 - I ) 2 iImX o )X 2 + "_- ( _ 0 + K p) + l I • - k_- ] ( X 0 ± i +2K P ) (x 0+2K P ) = 0 ( 167) Ass u me for now th a t ImX0 # 1/ 2 or 1 fo r th e low fr e qu e n c y rotor mo de; th e n : Y2 i [._ . _ Y O p _ Y9 , ___0,2 (X0+ i + 2 K p)(_ 0+2 K p '.] Jk 2 =- I- 5 21 m ) _ 0 _-_+K +T K P '_12 / T +2_m_0 - 1 (16 8 1 t Th e root s to or d er "o 2 ar e the n i 68 _

{ E2 , KpP2]}

Y 2 4 - _ K p + 4 Kp 2 2 (169) an d the con j ug a tes. _is result is similar to what was found for the th r ee- bla d ed gimballed rotor away from critical regions , so the beh a vior is famili a r , fr om that discussi o n. The o nly change, in f a ct, is a reversal o f the signs o f the ± order B2 terms, hence, all the analysis of th a t case may be used h er e .

The m a in fe a tu r e of th e beh a vio r is t h a t wh e n _ > 0 th e re a l r oo t in t e r s e ction of th e y loci -.at ±i / r ev non r ot a tin g - pulls ap ar t. The o r d e r tJ 2 ch a nges in the f re quency and d a mping a re in th e oppo s ite dir e ction from what is sh o _ in figure 9(c). The closest approach o f the two b r _ches has an order _ width: 8 Y O , 2{ AXl m in = _ _- v (170)

I I

' w her e Yo / 16 = Kp + / v 2 + Kp 2 is the y for the b o un d ary wh e n _ = O . • Takin g th e p o in t o f c l o s e st ap p ro ach as th e re al roo t b ounda r y , it is th e n ' I y Y O V 2 9 ' 1--6 = i-6 + ....... (171) --, J r2 + Kp 2 - ' The criterion for divergence stability for these four roots is b ,. T here i s th i s im por tan t differe nc e from the c as e o f th e t hree - b lad ed ' g im ba l l e d rot o r : this sol u t i on d o es n o t repre s e n t t he b eh av i o r o f a ny real , s yste m , but rat he r is b e ing c o n s id er ed a s an ap p rox i m a t ion to the r o tor flap < : d yn a mi c s in f o rw ar d f li g ht. A s s uc h, it ha s th e n o r d er v 2 e rr o rs in both _

i 69

• ° _ % . _s-' ,. ' I t the fre q uenc y an d da m pi n g comp a red t o the correct s olution ( e q. ( 24)). Aw ay f r o m c ri tic al regi ons , t h e c o rrect so luti o n s h o w s only a s m a ll in fl uence o f f or w ar d fligh t anywa y , s o t hese e r r or s a r e p r obab l y no t lar g e ei t her.

This beha v ior of t he r oots n ar the real root bounda r y has been seen t wice n ow: w i t h the t hr ee-bl a de d gimballe d r otor , a n d he r e f or th e c on sta nt _ coe ff ic i e nt appr o ximati on to the i n depe n den t blade d y namics . The orig in o f th i s behav i or is the order v, 1 / re v t erm in the flap dampi ng M.. T h at term B g i v es the 3 / r e v pe r i o dic c o effi c ients for t he g im ba ll e d ro t o r which produ c e !

the t e r m in t he radicand o f k t hat gi v es t his behavio r . F or t he c ons t an t co effi c ie nt appr ox i mat i o n, the flap d amping harmoni c M_s gi v e s te rm s cou - "_ B pling the _lC and _lS mo t ion , pr o ducing t he observed be h a v i o r. W ith inde- pendent blades observed in the rota t ing f rame, t he lo c i in t erse ct at the real axis as i s charac t eris t i c of r o ot l oc i; observed i n the non rot ating frame , t he behavi or is shif t ed by ±i so it oc c urs a t ±l / rev now , but the blades are still independen t so the r oo t l oc i still intersec t . The r oo t l oc i of coupled _ . . de g rees of f r eed om d o n ot show t ha t behavi o r , h ow ever; rather t he y si m pl y p a ss c l o se to o ne another, t he minimum separa t ion being a measure of t he ma g ni t ude of the couplin g . The gimballed ro t or couples t he blades by requi r ing t h at the rotor move as a whole, in 81c and 81s motion only. The cons t an t c o e f ficient approxima t ion couples t he blades by d r opping t erms from the equa t ions of moti o n describing t he independent blades , cre a ting a desc r iptio n of some new sys t em wi t h coupled degrees of freedom. (How closely t he d_mamics o f this new sys t em migh t rep r esen t the r oto r wi t h independen t blade s is what is being ' examined here.)

t When I m _ 0 = 1 / 2 , t ha t i s , wh en th e hove r r ota ti ng f r eque n c y is n ea r ½ / re v , the low f re q ue nc y ro t or m ode ( at -½ / r e v no n . _ .i n g f r eq ue n cy) encoun t ers the _ / rev cri t i c al regi o n . It i s n ot _ _ary t o g o l oo king fo r i this s o lu t i o n, be c a u se t he o nly w ay t he co n i ng m ode ald exhibi t t he c ri t i c al region beha v ior with a cons t an t c oeffi c ient equa t ion is if t he lo w fr e qu e ncy m ode joins it , s o t he ro o ts may remain complex conjugates even inside t he c ri tic al regi o n . The c ri tic al region beh avior fo u nd i s the same as tha t of t h e perturbation solution inc l uding the period i c c o e ff ic ients. S o the c on- --, s t ant co t;ff ici en t approx i m a tion g i ves the c orre c t beha v ior of these four r o o t s l n e a r the _z / rev c riti c al reg i on . T he h i gh frequen c y rotor mode (at 5 / 2 / rev non ro tating f r e q ue. cy l does not, h o w e ve r , pa r t i c i pate i n th i s b e ha vi or.

Indeed, the above result (eq. (16911 is st i ll v alid for the X0 + i root e v en I when Im h 0 = 1 / 2 ; the e rr o r in that r oo t is then o r de r _ .

N ow co nsider the l o w fr e quen c y m o d e (X0 - i) wh e n ImX0 = 1; th es e r oo ts are at 0 / rev nonrotating frequenc y, that is , to order 1 X = -Yo / 16. Th e high I freq u en c y m o de g i ves n o pr o blems i n th e o r der u 2 ch a r a ct e r is t i c equa tio n ' even w hen Im_0 = 1 , s o equation ( 16 9) gi _ e s the ro o ts th e re. F o r the l ow '_'_ f r eq ue nc y m od e , howev e r , th e or de r u 2 ch ara ct eri st i c equa t i on (eq . ( 1 67 11 is i identically z ero when ImX0 = 1, so it is ne c essar y to go to o r de r _ to f i nd _ 2. T he or de r u_ cha r acte ri st i c e quat io n i s , w h e n ImX 0 = 1 : t J , 70 ?

A . i _-_+ Kp) + Kp = 0 _'5 - 112 ] _16 - 2K _- _ I 11-6- 2Kp (_ YO YO or i

Y2 1 )

whe r e

" f 2 ( )! ' }

_ ] + -_- _- 2 K p P _ YO ( 174 ) - + - ]-6" 2Kp .._ and the r oot to o r d er u 2 is = - y la 2 Y O YO + :-_ _- 2Kp + i u 2 D (175) This is the critic a l r egion behavior a g ain, possible with t he cons ta nt c oeffi- i cient equat i ons beca u se f or t his m ode the nonr o ta ti ng roots a re at the real ax i s. The criti c al region boundary is g i ven b y D 2 = O, or the corner _ : ! b (_-_ -Kp )_ (176) :" • Ia12 ,1J 2 2 = el 4" C 2 '

i Y o Ay

' where here _ I (._) 2 , _2 _ Kp + 4Kp2 1 /yo _2 Y O < C l = - g + 2 _12 / + Kp :-_ 8 YO = - g (, o 2 - 1) 2 + Kp _ (177) I , 71 = 5- 2 - + -- ( v2 - 1 + p _ _ . (v 2 _ 1 ) _ - _ - 2Kp (1 78 ) and Yo / 16 = Kp + / v 2 + Kp2 - 1 fo r t he 1 / r ev _e g i o n as usual. The r oo t m a y _ - be w r i tt e n t h e n ( v 2 1 - + i (_ Kp ( 17 9 ) " _ a n d t h e bou ndary o n t h e y - _ pla ne is CI +C a _ _ = _2 - 2 (18 0 ) 1 6 TO

- Kp

F o r K p = O , these r educ e t o Y 0 / 16 = _ - 1 , • 8 C 1 = - _ (v 2 1) 2 (1 8 1) _0 / i - - i C2 = _ -- + _ 9 _(v 2 1 ) + - _ 1 6 _ 2 (v 2 1) 2 (1 82 ) i ; a n d t h e I / rev cr i tical r e gio n b ou nd ar y o n t h e _ - u p l _e be co me s 1 1-"6 " - 9 (v2 ± _ 1 + . ( V 2 - i) +- v2 (V 2 - 1) 2 (1 8 3 ) The g e n eral beh a vi or o£ t h e r oo ts in this l / re v cr i t ic a l re gion i s c or rec t bu t th ere are o r d e r _ 2 e r r ors in both t h e fre qu e n c y a nd d a m pi ng c om - p are d t o th e e x act solution. C 2, t h e wid t h of t h e cr i t i ca l re gion , i s e x ac tly t_ co rre c t. _ ere is a n o rd er B 2 c h a nge in t h e d_p i ng ( e q. ( 1 7 9 )) ; and th e -: of f se t o £ t he c ri t i ca l r egi o n , C l, i s n ot corre c t , l ea d i ng t o an or d er 11 2 ch a ng e in t h e fre qu e n c y . T h e s e or de r _ 2 err o r s c orre s po nd to t h ose in t h e , ex p re ss io n for t h e ro o t s a w a y fro m t h e critic al re gi o n (e q . (16 9)) . For v = I t h e d_ pi ng ch a ng e goe s to zero , an d for v = I a nd K p = 0 t h e o f fset ._ .

) , : , 7 2 ) C I r e du ces to z e ro fo r both the c orr ec t solut i on an d for this ap p r o x im a tion; in g e n c ral, ho we v e r, th e r e ar e sm a ll er rors in the 1 / r e v c riti ca l r e gion b e h a vior a s th e r e w e r e for the roots a w a y from th e c riti_:al r e gion.

Th e c oning mod e ( a t I / r ev nonrotating fr e qu e n c y) a nd th e high fr e qu e n c y m o d e ( at 2 / r e v ) d o n ot parti c ipate at all in th e 1 / r e v cr iti ca l r eg i o n . Th e s e _,_ roots are given still by their respe c tive expressions for away from the c riti ca l regions (eqs. (16 4 ) and (169)).

Summ ar y Th e fl a pping d y n a mi c s of a rotor with three ind e p e nd e nt bl a d e s a re d e s c ribed by a tot a l of six eigenv a lu e s. In the rot a ting fr a me, th e re a r e thr ee independ e nt e qu a tions; then the e igenv a lues o cc ur a s two tripl e pol e s. _ In the nonrotating frame, the degrees of freedom a nd e qu a tions of motion a r e : c oupled. There a re two eigenv a lues a t th e rot a ting value for the c oning mode, a nd four eigenv a lues a t the rot a ting v a lues ±i / rev for th e h i gh a nd low fr e - qu e n c y rot o r mod e s . T h u s, if kR i s t he ro ta ting e l g en va lu e , the n v nrot a tin g roo t s are kNR = _R , hR ± i (for N = 5) . The t otal number o f e ig e nvalu e s i n t h e nonro t a t ing frame is still six for the thre e- blad e d rotor. The cons ta nt c o ef fi c ien t a pproximation in the nonrotating fram e gi v es resul t s for all six eig e nv a lues at on c e (in general , f o r all 2N roots of th e rotor flap mot io n).

Th i s approximation gives the following behavior for the roots.

, Away from cri t i ca l r e gions (i.e. , t h e hov e r r o t a ting frequ e n c y no t n ea r ' h / rev or 1 / rev) t he two c oning m o d e root s are giv e n by eq u ation (16 4 ) , whi c h ( I is e x ac tly t he r e s u l t o bt a in e d in c l u ding the p er i o di c c oeffi c i e n t s . Th e four . 1 roots of the low frequ e n c y a nd high frequen c y modes a re giv e n by e qu a - ; tion (169) , whi c h h a s o r der _ 2 errors in both the frequ e n c y and d am p ing.

The effe c ts of forw a rd flight a w a y from th e c riti ca l region a re, in g e ner a l,

i

sm a ll , however , so these errors a re not too signifi c ant.

The four c oning a nd low fr e qu e n c y mode roots en c ount e r th e _ / r e v c riti ca l ; r egion (at ±½ / rev non r ot a t i ng f re quen c y ) , with e xa c t l y th e same beh a v i o r a s T . , the correct so lut ion (to or d er _ at lea s t) . T h e two ro o t s o f t h e high f re - quen c y m o d e ( at 3 / 2 / re v) do n ot p a rt i c ip a t e in t his b e h av i or , whi c h m e ans a n o r d er _ err o r . !

1 Th e fou r c onin g a nd high fre qu enc y mo d e root s do not e n c o u nt er th e l l rev : cr it ical re gion , whi c h mea ns o r d er _ 2 err o r s for t h e s e r oots. Th e two r oo t s i ' of t h e l o w f reque n c y m o d e , a t t h e rea l a xis in the non r ot a t i ng fra m e do show !

th e c r i tica l re gi o n b e h a vi or. T h ere are , how e v er , in g e n era l or d er _ 2 4 e r r o r s in both th e fr e qu e n c y a nd dam p ing c omp a r e d t o th e c orr ect solu t ion , : c o rre s p on di ng to the b e h a vio r of t h e root s of th is mod e a w a y from the c riti ca l _ r egions . T h e m ag ni t ud e of th e s e err o r s is dis c uss e d bel ow in t er ms of th e ( , - _ p la n e.

: ' , C onsid er the _ - _ pla n e fo r _ = I a nd Kp = 0 (f i g . 2). Th e c ons ta n t ' ; c o e ffi c i e n t a p pr oxim ati on h as a n id e n t i cal i / rev re gion bou nd ary but o nly fo r _\ , _ i : th e lo w f re qu e n c y r o tor m o d e, th at is , t w o ou t o f t he si x r o ot s. It h a s th e = . . ........

lli i iiii i ,, ...........

|

sam e ½ / rev regi o n, bu t on l y for the c oning an d low frequency mod e s {four out o f six). The re al r oo t b o un d ary f or the c o ning m ode is the same but the b ou n d ary f or the high and l ow frequency m od es sh o ws o nly half the influence o f _. The latter b o un d ary is, _n fact , o nly the po int o f cl o sest ap p r o ach o f these tw o m od es at ±l / r ev n o nr o tating frequ e ncy, n o t an intersecti o n at all.

F o r _ = 1.1 an d Kp = 0 (fig. 4), the y - _ p lane o f the c o nstant c o efficient ap p r o ximati o n has a I / rev re gi o n b o undary , but o nly f or the l o w fr e quency m ode again. The regi o n has the same wi d th f o r a giv e n _ as the c orr ect s o luti o n , but is o ffs e t highe r . The er ror in th e b ou n d a ry is rou gh l y Ay = 4_ 2, or ab o ut Ay = 1 at U = 0 .5 (co mpa r ed t o Y 0 ab o u t s e ven f o r t he 1 / re v r egi o n with v = 1.1 , as shown in fig. 4) . As a result, r otors with y _'" su c h that th e h o v er f re qu e nc y is ab ove 1 / re v will ha v e a highe r U co rn er by t his appr oxi ma ti on. F or e x ample, wi t h ca se (b) a bove, _ = 1.1 an d y = 6 , th e I co ns tant c oeffi c ie nt a p prox im at i o n give s _ corn e r = 0.32 6 , compar e d to t he _ corre c t s o lution of Vc o rner = 0 .28 6. T hi s err o r in the 1 / rev region boun da r y - w hi c h co rr es p o n ds to the f requ e n c y error a w ay f rom the cr itic a l r e gion , an d so is a mea sure of th e ma g n i tu d e of th a t a lso - is not n e g ligible, of but is small in terms o f the Ay shift of the b oun da ry , or even in terms the _c o rne r , w hich is m o re sensitive t o the bounda r y shifts. The c o m p a r isons o f the ½ / rev regi o n an d real roo t b o un d a r ies f o r this case f o ll o w exactly as f o r v = 1 , disc u ssed ab ov e.

: T h e co n stant coe ffi c i e n t approx ima t ion to t h e rot o r f l ap d y nam ic s in f or w ard f li g ht produc es d iff ere nti a l equa ti ons th a t do no t ac t ua ll y descr ib e th e rea l ro t or any more , a n d mus t a 3w ays g i ve some erroneou s resu lt s. As a I [ i r ep re s e nt a tion o f t h e actua l ro to r dynami c s , h o w ever, t he co nst a nt coeffi c i e nt a ppro ximati o n (in th e n o nr o tating f r ame] is act ua ll y r ema r kably good . The : I' i n fluence of forw a r d flight on the roo t s is g i v en r a the r well by this ap pr o xi- i m at i o n, the p r i mary e rro r bein g t h a t t he r oots of the hi g h f r equ enc y rot o r m od e s do not en c ount e r the c riti c al r eg ion s . For this cas e o f N = 3, th e ½ / rev regi on is s e en by t h e four roots at ±h / rev , b ut n ot b / th e t wo r oot s a t , - ± 3 / 2 / re v; the 1 / re v r e gi o n is se e n b y th e two roots at the r e a l axis, but not b y the four roots at ±l / rev and ±2 / rev. T hi s b e ha v ior is f und a mental to the c onstant c o effi c ient a pproxim a t i on; the root s from that a p proximation m us t i _ , alw ay s b- complex conju g ates , so th e critical re g ion beh av ior c a n enly be ; exhi b i t ed b y t wo r oo ts o n t h e rea l ax i s ( as t h e I / rev r e g ion h e r_) ; o r at a of when there f our the mul t ipl_ ½ / r e v a re ro ot s to p a rt i c ipate, t w o a t p os i- tire frequency an d tw o at the negative frequency (as the ½ / r e v r egi o n he r e ).

J With these r e s t ric t ions, t h e c on st a n t coe ffici e n t a p prox i ma tio n pic k s up t he cri t ical region behavior of t he periodic sys t em whenever possible. The con- ' i stant c oefficient app r oxi m a t i on i n t he n onro tat in g frame is be tte r th an i n the [ r ot atin g fr a m e bec a use t he t r a nsf o rm at i o n t o the no n r ot a t in g f ram e shif t s t he i fr equ en cy of th e r o ot s t o a ll o w s uch o ccurr en c es. A t t h e least, the se re sul ts ; I s u gg e s t t h at t h e con s ta n t coeff i c ien t a p p ro x i mat ion is a d e q uate fo r t h e l o w _ . ! fre que ncy dynami c s o f a r o t o r a s f o r h eli c o p t er s ta bi l ity an d con tr ol i nve sti g ations.

; ( 7 4

i

Four-bladed Rot o r Comparing the constant coefficient approximation of equations (156) and (1S7) for the N = 3 and N = 4 cases , it follows that the four-bladed rotor only adds the 82 degree of freedom and equation which are completely decoupled for the constant coefficient approximation. So , to equation (161) t must be added the equation for the four-bladed rotor. The roots for this mode are X = - i -- _ _ + This is t he s ame as t he co ns t ant co effi c ient app_. ox ima t ion in t h e r ot a t ing frame which is not very good. All the critical regions are missed and there are even order _2 e rrors away from the critical region. The solution for the other six roots is the same as for the N = 3 case.

Fiv e or Mo re Bla d e s As the, number of blades increases , more and more modes are added to the t non r otating rep r esentation of the ro t or . For example, N = S has degrees of f ree dom 80, 81c, 81s , and , 2

i B2 c = cos 2,m

m

_- 2 (m)

'rl p I ., 82s ,_ E 8 sin 2_m , m wi t h c orr es ponding r o o t s X N R = XR + 2i . C ons e qu ent ly , more of th e low f re - q u enc y mo d e s w ill b e abl e to pi c k up t h e c riti ca l r e gion b e havior i n t h e c o n - . _ s t a nt coe ffi c i e n t app r o x ima t ion. T he high e s t f reque n c y m o d e s a l w ays d o no t ' en c ount er t he c riti ca l r e gions, of c ours e , s o t he c o n s t an t c o e ffi c i e nt appro x i - ._ m at ion c a n n ever give c ompl e t el y c o rrec t be ha v ior . This e ff ect of i ncrea sing N pa r a llels t h e e ff ect on th e diff e r e n t i a l e qu at ions. In c r ea sing N t e nds : _ to sw ee p th e p er i o di c coe ffi c ients f ro m t he lo w er f reque n c y mod e s; t h ere a re *_ . a l ways p er iodi c coe ffi c ien t s present in t he d e g ree s of f ree d o m an d e q uat i o ns of t h e high fr e qu e n c y mod e s , however.

t

+ ' 75 !

° _ ..... _ TRANSF E R F UNCTIONS Th e e xp e r i me n t al d e t e rmination of the e i ge nvalu e s, e s p e cial l y t he small inc re men t s (orde r u o r ev e n u2) f ound here, in th e har s h a e r o d y namic envi - r o nmen t of th e heli co pter in f o rward flight, is a diffi cu l t task t o a cco m p lish _ wi t h a cc ura c y. A mor e dire c t measur e m e n t - hen c e, fundamentally m ore a cc u- ra t e - is t he transfer func t ion, t ha t is, t h e response of t he blade flap motion due t o sinusoidal exci t ation.

Co n s id er t h e transf e r f u n ct i o n o f t h e fl a p r esp o n se to bl a d e pi tc h con tr ol, parti c ularly the influence o f th e p er i o dic c oe ffici e n ts in forwa rd w _ fligh t , whi c h h a s led t o th e special b e havior of the e igenvalu es . Th e e qua- t ion of m o t ion fo r an indep e nde n t bl a de exci t e d by pitch control inpu t s only, in t he ro t a tin g fr a m e, is _.

+ [v2+_ _ cos @+ _ _2 sin 2@+Kp[_ (i+_ 2 )+ _ _ sin @ -_ _2 cos 2@]]8 I I ] Th e t r ansf er func t ion is d e fin e d as t h e r e spons e to sinusoi da l inpu t I 8 = _ei_@, where _ is a compl e x con st ant. T a kin g only the r e al p a r t s of b o t h the input and th e ou t put is implied. With a c onstant co e fficien t differ- ential equa t ion, the output will also be a sinusoid at frequency m, that is, i 8 = _e i_@. The differential equa t i o n relates the output E to the input ; by a singl e complex func t ion H(_), which is th e transfer function: m ' r 8 = H(_) (1 8 7) With a periodic coeffi c ient differential e quati o n, the respons e to e at j fre quenc y _ i s n o t j u s t 8 a t f r e qu e n c y _ . Th e sinus o i d a l in p u t a t _ is mult ip l i ed b y the coeff i c i e n ts w h i ch have te rm s that are a ls o s i nu s o i dal n o w wi th freque n c i e s 1 / rev , 2 / rev , etc. (fl, 2 fl, . . ) . T he product o f t w o sin us- o i d a l fun ct i o ns is a s u m o f s in us oid s a t the s u m and d i ff ere n ce o f the f re - quencies. It = ollo w s then th a t f or a period i c coe ffi cien t s y ste m an i nput a t : , fr equ e n c y _ lea d s to an ou t pu t wi th te rm s at f r eq ue nci es _, _ ± l / r ev , _ . _ _ ±2 / re v , etc. T he o u t pu t i s t h e n a s um o f t h e for m i , 7 6 8 _ Ho(m)eim_ ei(m+l)_ ei(m-1)_ -- - - + H+I(_) -;H l(m) e + H+2(_ ) e i ( _ +2 )_ + H _ 2 ( _ ) e i ( _-2)_ + . ( 188) w h e r e Ho, Hil, H ± 2 , a re a ll th e tra n sf e r func t ions o _ the syste m • No t ice t his m a y be wri t t en as 8 / _ = H(m , _) e im_, whe r e He , H± I , H± 2 , e tc .

a re t h e har m o nics of a F o u r ier se r ies repre s e n t a t ion of a _ m cti o n H(_, _), which is p e riodic in _ ( H (_, 9) is complex, how e v e r, so H +n a nd H _ n are no t , - c onj u ga te s)• Equ at ion (188) is a general ,esult for periodic coe fficien t differ e n t i a l eq u a t ions: the dynami c behavior is d e scribed by no t a single t r a nsfer fun c t ion, bu t rather a seri e s of t ransfer fun ct ions. Ther e is t h e direc t res pons e H e , t he ou t pu t a t t h e same f r eq u en c y a s th e i n pu t ; a nd t here ar e al so now s id e band r esponses Hzn , ou t pu t a t t he inpu t fr e qu e ncy _ + n / r e v .

.: Si n c e fo r the hove r limi t u = 0 t he p r es e n t d iffe re n t ial equa t i o n red uc es t o c o ns t ant coe ffi c ients, all th e t ransf er fun ct ions bu t H o m u s t be ze ro t h e n.

I t will b e fo u nd, in f act , t h a t Hzn a r e o r de r un.

Analysis Assume the input is 0 = _e i_. I t is possible t o simply su b stitut e in to e qua t ion (186) the expansiov for 8 as in equa t ion (188), coll e ct like hat- , monies, and t hus solve for th e t ransfer functions. Following t he r e s t of the r presen t investigation, however, an expansion in u will be used which mak e s ou t put I t h e analysis m o re orde rl y• Expand th e as a series in = 80 + u 81 + _282 + .... Since it is known t ha t t he ou t pu t has only time ,. be h a v io r like e i(_Zn)_, th e re a re no o ther t im e sc al es bu t # in this pr ob- i le m . T hi s m ea ns t h ere i s n o cr i t ic a l reg ion be h av i or in t his problem t ha t is a f e a t u re of t he e i ge n va lu es only. The cri t ic a l r eg ion be h a vio r i s r _pl aced j - h ere b y t he sid eba nd t r an sf er func t ion s which will b e shown t o ca rry eq ui va - le nt inform at ion a bou t t he sys t £m dyn a mic s . It is t h e r e fo re a lso no t I n ece s sar y to e x pan d y as a s e _c_ i n v .

'_ t To o r d er 1 the diff ere n t i a l e q uat i on i s ' _ 0 + _fiO + (v2 _Kp)80 = _0 (189)

' 1

S o 0 = Oe i_ ¢ giv es 80 = gO e1_* w i t h _ I

.= . = (1 9 o)

O t ' ' 77 I ,wi q = ,,L.- . , .. . r -- .............. J W .... / J_ wh er _

A_) = - : +_ i_ + _2+ Kp_ (191)

Th e or d er 1 so l ut io n i s th e hov er l i m i t as usual ; he r e it i s the hove r t r ans- j _ fe r function fo r the r esponse of blade flapp i ng to pitch inpu t . The transfe r function has the usu a l fo r m fo r a second orde r system re s ponse , i n th i s case a heav i l y damped s y stem with natural frequency _n 2 = v2 _ (y / 8)Kp. Not i ce that A(i_) is t he h ov e r c ha r a cter is t i c e qua t i o n .

The ord er _ diffe re ntial e qua t i o n is _ " = Y--- [-2i 0 - (_+ 1- 2ikp)80]ei(_+l)_ + 1- _ [2i 0 + (_- 1 - 2iKp)$o]e i (_-l)_ (1 9 2) He n ce , the sol u tion is 81 = 81 + 81_ei(_-I)_ with the t r ansfe r f u nctions Y [-_ + 2 i(u 2 v2)] B I + - = 1 -2 - - ( 195 ) !

_- The order u response is then just the _ -+I / r ev sideband functions. The fo r cing terms, the right-hand side of equation (192) , a re the same terms which gave the order u, _ ff revcritical region fo r the eigenvalues. That is, nea r _ ff revthese t e rms contribut ed to th e or d er u sec ul a r equa tion . H e nce, th e order u sid e b a nd tr a nsf e r fun c t i ons a t _ -+I / r e v c onstitute b e h a vior of th e s y stem e quivalent to the _ / r e v criti ca l r e gi o n of th e e ig e nv a l ues .

T he o r der _ 2 diffe r enti a l equ ati on is :_ -[_ sin ,_l (_ COS , + Kp _ sin *)81 _._ Subs t itu t ing f o r 8, 6 0 , and 8l, th e s o l u ti o n is 8 2 = B2e i _ + _ 2+e i( = , B2 - e i ( =-2 ) $ ( 1 95) wi t h A(= ) 8. = _ _ Kp A((_) A(uOAI ( ¢ _) i A 1 (_) ( 1 96) wh e r e i AI(_) = A(_ + 1 )A(. , - I) = [ A(_ ) - 1 1 2 + (2i = + _ ) (1 9 7 ) I ! h ove r r o ots. T he dir e ct re s pon s e t o o rd e r _ 2 i s then !

N ot ic e t hat AI(i_} i s jus t the cha r ac te r ist ic equ a t i o n f o r th e 61c / 8 1s ;- _ AI ! c19, . , So th e order u 2 response c ontr i butes to the d i re c t transfer function m uch as i t do e s t o t h e e i ge n val u es a w ay f rom the c r i t i cal reg i ons . The order _ 2 : = corre c tion f or the mean of M B i s recogniz ab l e as usual , and there are other t_ order _ 2 effects due t o t h e period ic coeffi c ients . T h e _ _ 2 / re v s_ deband ! tr _n. fer fu nc t i ons are t I , 7 _ r -- lib - - I I I ill j_ (1 99) | '_ , S u mmary r The :ransfer functions describing the response of the b l ade f l ap moti o n to pitch excita t i on a r e _-_

S o+ 2B2

Y = U e _ '" ' E1 ± H±] = U _ ( 200 ) e

g

H ± 2 = U 2 -- 2 +- , B y meas u r i ng t h e r e s po ns e to a sinusoidal input, i t i s t hen possible to verif y th e equation re p r ese nting t he fla p d y na m i c s. Sp ec i f i cally, it is po ssi b l e to r' t ve ri f y t he pe ri o di c coeff i c i e n t s whi c h pro du c e t h e cr i t i ca l regio n b e h av i or o f ' the eig e nvalu es. T h e r e is not, ho w ever , an y cr i t i cal region be havior for the t ra nsfer funct i ons , t hat i s, spe c i fic r a ng e s of th e p ar ame t e rs w here t here is " mo re crit i cal b e ha vi o r of the res p o ns e . T h at is re pla c e d by t h e . _d e band tr a ns fe r fun c tions. He n_c, th e tr a ns fe r fun c tion m ea sur e m e nt do e s not prov i d e e xp e rimental d e monstr a tion o f the c riti cal r e gion be h a vior, al though it do e s equ i v a l e nt in f or_at ; _u , v e ri f ying th e e qu a tion whi c h produ ce s th e c r i t ic al j reg i o n f or the ei genv ai u es.

A P o in t Abou t Exper !m en t al Techniq ue i Sin ce the resp ons e of a p er i o di c s ys te m to i n p ut at fre q ue n c y _ i s outp ut a t ¢ -+ n / rev _ for all i nte g ers n (a ltho ugh the m agnitu d e of the _, respo n se d ecreases wit h n), i t f o llo w s that the ra nd o m exc i tat io n technique I " for m eas ur i ng the trans fer fu nct i on of a consta n t coef fi c i ent sy ste m i s not ' i d i rectl y app li cable to p er iodi c coefficie n * , sy ste m s . A ra n do m in p ut ha s all | I fre qu enc i es at once , hence , th e ou t p ut i s also co mpo sed of al l fre qu enc ie s.

, .- t4 F o r cons t an t coefficien t s y ste ms t h e k no w ledge tha t the respo nse at u cam e ,_ o n ly from the inpu t at _ c an b e use d to d et e rm in e the tr a nsfer func tio n when the i n p ut is ra n do m - the entire f un ction a t once , i n fact. W ith a p eriodic _ .

/ coeff l c i ent s ys te m , how e v e r , the o u tput at _ has co n tr ib u tion s fro m in p uts I ,- at u * - n / r ev f or a l l n , so not e n oug h i Pfo rma t io n i s a vai labl e to d e ter m ine 2 _ _ the tra n sfer f un ct ion .

J An input at a single fr e quency _ must be used then and the ou t pu t a t all harmonics _ ± n / rev measured; t his ta_k is performed for the rang e of required. This technique is slower than with the random exci t ation; in addi- tion, the noise filtering feature of the random input is lost, that is, a c curacy of the m ea surements is los t .

The problem is that there is not a single transfer function to measure but rather se v eral• For exa m ple, the ro t or flap response up t o _ = 0.5 or so would require t he m easurement of Ho, H±I, and H±2 a t least. It is plobabl y pos s ible t o e x t end th e random exci t ation t echniques to Feriodic sys t ems. It would be necessary, however, t o use a n u mber of inp u ts wi t h independen t sp e ct ra s o t h at enu u gh independen t m e a s ur em e n t s are made at eac h f r equency to , _ determine the required nu m ber of t ransfer func t ions. In any case, the peri- odlc sys t em will requ,re more experimen t al effor t t han the cons t an t c oe ffi c ien t s y s t e m.

Nonro tat ing Resp o nse L F or t he dynami c s of the h e li c opter as a wh o l e , it i s t h e r e s pons e of th e roto r in t he nonro t ating f r ame t hat is of inter e s t , t ha t is, the response of the 80 , 81 c , 81s, e t c . degre es of fr ee dom. Similarly, t he inputs usually ava i labh for th e ro t or are c olle c tive and c yc li c pitch c on t rol : O 0, 0 1 c _ s. He: ;, th e tran s f e r fun ct ions in t h e nonrot a ting f ra me m ay b e e xamined, L_ r . , .J i t _on t o t h e r ot a t ing r espons e gi ¢ e n above. The an a l y sis p r o c e e ds mu c h _ove. It i s s t raightforward s in c e th e re a re no c ri t i c al re gi o n s t o b e ' con c erned with o r any othe r singular behavior; it is co nvenient to u se ve ct ors ' of th e input a nd output v a riables and matrices of the transfer functions (_ ( due to 8). The analysis and results are not very illuminating, however, l bec a u s e o f th e i r co m plexi t y.

¢ DISCUSSION OF PREVIOUS WORK Hor v a y (r e±. I) co nsid e red t he flap dynamic s o f a r o t or wi t h v = I, b'l Kp = 0, and n o reverse flow eff,--ts. By the subs t i tu tion ' - 7F _ + u c o s _ . _ 8=y e , , he tr a n s f o rmed t h e f l a p equ a t i o n to th e s= a nd a rd f o rm o f Hill's e qu a ti o n: Y = 0 I,, wh e re f is a p e riodic function. T his equ a ti o n h e solve d by the infinite '_

' L

, d e te r minant me t h o d of cl a ssical theory. Th e so lution pr o du c ed n u m e rical , _<

8 1

results presented as constant Rel lines on the y - _ plane. T he boundaries found were essentially those given in figure 7 although he did not investigate small enough y to find the 1 / rev re g ion. Horvay used the nota- tion n = y / 8, since this is the coefficient of tke flap damping in hover.

Then , in hover and outside the critical re g ions , ReX = -¥ / 16 = -n / 2. Hence , he intr o duced the "apparent damping coefficient" nap, = -2REX. Then _ F napp / n = -ReX / (V / 16) takes the values 1 outside the critical regi o ns , 1 to 0 in the stable p o rtions of the critical regions, and less than 0 in the unstable portiom_ of the critical regi o ns. T he use of the notati o n napp / n and the parameter nap p for Re ½ is found in much of the literature on rotor flap stability. Horva_ found approximate solutions at high u (to _ = 8) for several equations which are not true descriptions of the flap d ynamics but d o '" illustrate some o f the high _ behavi o r o f the real r o t o r.

Horvay and Yuan (ref. 2) considered blade flap stability for the cas_ o f ..

= 1 and n o r everse fl o w but including Kp = O . The roo ts were f o und %r o m the transient s o luti o n f o r B, using the results o f Fl o quet the o r y (a_ out- lined in Appendix A here). Much o f the literature uses this appr o ach , the _- real differences being the method used t c find the transient solution for B.

Hor v ay and Yuan used the ripple method to integrate the differential equation.

They pres e nted the numerical results on the _ ' - v plane for , = 1 and Kp = 0 , - f 3 / 12 , and ¢ r 3 / 4 (these v alues o f Kp were chosen because they place exactly at ¥ / 8 = ¢_ the ½ / rev region, the real root , and the 1 / rev _gion boundaries, respectively, when _ = 0). A large azimuth interval was used in the ripple method but the results compare fairly well with the solution of Horvay (ref. 1), at least at small _.

Parkus (ref. 3) considered the flap stability f o r _ = 1, no reverse flow but Kp _ 0 (with the r,otation k = -Kp). He found the roots from the Floquet result using an expansion of _ as a series in _ to find the transient solution. The expansion is the same as is used here away from the critical , regions since the m_ltiple time scales and expansion of ¥ are not needed there. Parkus left his result in the f o rm o f a quadratic equati o n %r o = e 2_[_+(Y / 16)]', since it is only an order _2 solution , howev_ - it is ) ; possible to solve explicitly for h. The result is exactly the sm a ¢ as _. o btained here for the roots away from the critical regions (eq. (24), with = 1). Parkus did not recognize, how e ver, that this solution is not valid for ImX near 1 / rev or ½ / rev. The assumption that the Bn in the expansion of 8 are all the same order is violated near the critical regions. Parkus 1 calculated the roots for varying _ , for y = 12 avd 14; the behavior of the roots looks like that in critical regions but really the expressSon for the roots away from the critical regions is breaking down as the roots approach _; the ½ / rev region. When ImX 0 is near _ , the order _2 term in the radicand :_ of X (eq. (24) here) is large so for large enough _ the radicand is ' !

negative. The result is two real _oots and an order _2 effect which is far _-_ from he correct ½ / rev region behavior.

Gessow and Crim (ref. 19) considered the flap stability at high numerically integrating the equations of motion to find the transient behavior _ ,

!

_ of _. Reverse flow was included , as well as the effects of stall, compressi- . : 4,_.,.

bility, and large angles. They found the flap motion to be stable at _ = 1 _ _ "__ . • q5 - , ] for y = 2.4 to 14.9 , at u = 2.2 for 7 = 2.4 to 7.1 , and at U = 3 for y = 0.6 t o 2.4. The m otion was unstable at _ = 3 for ¥ = 14.9.

Shulman (ref. 4) considered the flap stability for 9 = i , Kp = 0, and no reverse flow. He added, however, a second degree of freedom, the first elastic flapwise bending mode of the articulated blade. His principal solu- _ tion method was simply to numerically integrate the equation of motion. He f o und a significant effect of the second degree of freedom for high advance ratio, _ = 1.0 say, but small influence for U up to 0.5. The equations were solved for u = 1 and 1.2, but with the neglect o f reverse flow the true flap dynamics are no longer represented at that _. His conclusion that the flap ,_ instability occurs at _ = 1.5 to 1.8 is not correct then; but the importance of the second degree of freedom at high u is probably qualitatively correct.

Perisho (ref. 5) considered the stability of the flap and pitch dynamics -.

of the blade at high v, for an articulated rotor , Kp # 0, and including reverse flow aerodynamics. He solved the equations by numerical integration for the case _ = 1.05 , y = 5 , and a pitch natural frequency of 10 / rev. For , just the rigid flap degree of freedom , he found the instability boundary at = 2.2 with Kp = 0, and at _ = 2.45 with Kp = 2, in the I / rev region for both cases. Adding the torsion degree of freedom reduced the _ for instability to _ = 1.8 for Kp = 0; and to _ = 1.65 for Kp = 2, in the ½ / rev region now. Adding flapwise bending , that is, three degrees of freedom, reduced the speed for instability still farther to _ = 1.43 for Kp = 2. The reduction in the stability at high v due to the torsion degree of freed o m appeared to be largely a torsion divergence Jn the re v erse flow region.

Shutler and Jones (ref. 6) considered the flap stability for _ = 1 and Kp = 0, with no reverse flow. They solved for the eigenvalues and for _ by : a perturbation solution based on the Floquet th e o ry result that the solution of a periodic coefficient differential eq u ation may be written in the form = C leklcul(¢) + C2e_2_u2(¢) ' I where the ei_envectors u I and u2 are periodic (Appendix A). The substitu- -'I tion 8 = e_u is made into the flap equation and then X and u expanded as 1 series in _ The requirement that all the functions in the expansion of u(_) must be periodic is equivalent to the requirement in the method of multi- I _ ple time scales that the successive functions grow no faster than t he earlier ones (the order 1 term of u(_) is the hover limit so it is constant; then ' requiring successive terms in the expansion grow no faster means they can at _ ! most be periodic); _he result has the same secular terms as o btained here.

This is basically the method used by Shutler and Jones, although the details , differed considerably. T o treat the critical regions it is necessary to _ '_ - quantify the requirement that the frequency be near a multiple of ½ / rev. The present investigati o n used an expansion of y to d o this, while Shucler and Jones essentially expanded Imk itself (their eq • (25); wi%b the present n o tati o n for k). So contrary to their statement, this expansi o n (eq. (25)) 0_ d o es have physical significance, namely, what "close" t o a c ritical regi o n -,_' means• Similarly , their parameter o also has physical significance: it ;' _'_ ' 8 3 _ _ _5 gives contours of c o nstant Rel in the c r i t ical regions; for example, the cri t ical region boundary is such a con t our. They found the eigenvalues and 6 for the c_ es of _he ½ / rev region, t he I / rev region, and awa y from the criti- cal regior to order _2 everywhere, including in the ½ / rev region. Their resul t s m" , be shown t o be equivalen t to t hose ob t ained here, for u = 1 and Kp = O; excep t that here tile ½ / rev region is only carried to order _. Their result for the ½ / rev region boundary to order v2 has been given abo v e (eq. (55a)) and illustra t ed in figur e 7. They did no t , how e ver , take advantage of th,_ fact they had a perturbation solution to put together explicit analy t i_ expressions for t he eigenval u es.

Lowis (ref. 7 ) ct,,'_ideredthe flap s t ability for 9 = I and Kp = 0, a t high _ particularly, first wi t hout and then with reverse flow effects. He solved for t he roots by the results of Floquet theory , after numerically inte- gra t ing the equation to find 6. Wi t h o u t reverse flow he found an ins t abili t y .-.

at about B = 1.42 ill t he i / rev region. With reverse flow, t he instability occu r red a t abou t B = 2.3, stil l in the I / rev region. The represen t ation of the reverse flow aerodyna m ics used w a s only approximately correct, however: he ,_ did n o t acc o unt for the azimuth 7 ange where the blade is partly in normal flow and partly in reverse flow. This model has the correct li m it for very large but it is no t good at all below _ = i. The behavior of the solu t ion around _ = 2 , t hat is, including the flap instability, is probably correc t but neither of the solutions (withou t and wi t h reverse flow) he obtained is good in the range u = 0.5 to _ = I.S or so.

Wilde, Bramwell, and Smn_erscales (ref. 20) considered flap s t ability a t J --*b-- U , including the effects of reverse flow, and presenting results mainly I I for v = 1 and K, = 0. They considered a teetering rotor also. The s o lution • 1 was obtained by use of an analog computer. They found a flapping instability at about u = 2.25 f o r _ = 6. The teetering rot o r was stable t o _ = 5 at -_ least.

i

Sissingh (ref. 8) c o nsidered the flap stability at high u, including u k 1 , Kp _ O , and re v erse fl o w (with the notation P = v , C 1 = -K p ; and o nl y K p = 0 was used f o r the results). The aer o d y namic c o efficients o f r everse I f low were deri v ed and discussed . The s o lution o f the equ a ti o n o f moti o n w a s i o btaine d b y use of an analog c gmputer . The results were presented as stabil- .

I ity boundaries fur several _ , on the V - \, plane , with the emph a sis on quite large v, typical of a slowed or stopping rotor. For v = 1, he found i _ = 2.2 for instability at 7 _ 8 (hi g her for other v alues of y) .

, Stammers (ref . 9) considered the influence of forward flight on the flutter of the helicopter rotor blade. He found a pertuibation soluti o n by me t h od s similar t o t h o se o f Shu t ler and Jones (ref. 6) nther t han t he m e t h ods i used in the present wor k .

H a ll (ref . 10) in v estigated the dyn a mics of the flap / la g / torsion motion of the bl a de . He used Floquet theory to obtain the roots from a numerical ". integrati o n o f the equati o ns o ver o ne pe r i o d. He discussed Fl o quet the or y an d , it_ r esults fo r a multi-degree of freed o m system . He als o presented results _ : i tot a single d egree o f freedom, th a t is, just rigid flap. _ ' 8 4 - ?

Sissingh and Kuczynski (ref. ii) extended the earlier work of SissJngh , (ref. 8) to include the torsion degree of freedt , m; the emphasis again was on the results for very high u. They showed a qutte significant effect of torsion ollthe blade stability reducing the _ for instability considerably.

Fer exa m ple, with y = 6 the _ for instability was reduced to about 1.7 for a torsion frequency of 8 / rev and to about _ = 1.35 for a frequency of 5 / rev.

These boundaries were just a little below those for torsion only; the effect was largely due to torsional divergence in the reverse flow region where there is a negative aerodynamic spring and reduced aerodynamic damping on the pitch motion. Their results compared well with those of Perisho (ref. 5).

Peters and Hohenemser (ref. 12) considered the flap stability including L i, Kp _ 0, and reverse flow aerodynamics. They used a continuous repre- sentation of the reverse flow aerodynamic coefficients as obtained by Sissingh (ref. 8). They solved for the roots by integrating over one period and then _.

using Floquet theory. Their solution compared well with the analog computer solution of reference 8. They presented the numerical results on the y - • plane for _ = 1 and Kp = 0, 0.I, -0.i, out to u = 2.5. They found approxi- L .

,, mately _ = 2.3 for the flap instability in the i / rev region (at y = 9, higher for other values of y ); Kp > 0 increased the _ for instabilxty.

There were possibly some numerical problems with the i / rev boundary at low u (below 0.5), since the boundaries appear more like order _ than the correct order u 2 behavior (their fig. 3). In addition, the _ = 0 point of the I / rev region really must shift for Kp _ 0 and not remain always at y = 0 as shown (their figs. 4 and 5).

, _ Hohenemser and Y_t (ref. 13) investigated the flap dynamics using the I nonrotating degrees oY freedom and equations including the periodic coeffi- cients in forward flight. They gave the equations for the case of N = 4.

They solved for the r_ot_ by the methods of reference 12. They discussed the constant coefficient approximation to the nonrotating equations in forward i flight but only in the context cF tip path plane tilt feedback control, how- : - ever (at v = 0.4 with N = 3 , an o v = 0.8 with N = 4) They found the con- stant c_efficient approximation was not bad at low gain, especially for the low frequency modes but for high gain it could be unconservative.

t _"j J o hns o n (ref. 14) considered r o t o r flap stability including v _ I, I Kp _ 0, and reverse flow. He obtained perturbation soluti o ns for the cases o f small and large u and small and large _ The small _ solution formed the " 1 basis for the analysis of the individual blade case in the present work. This rep o rt was summarized ir, reference 15, a n d reference 1 7 presented a s y n o ptic o f , the small _ results.

Tong ( r ef . 16) c onsidered the blade flap / lag dynami c s. He o btained a solution by perturbation techniques to handle the nonlinear features of this ,_ p ro blem (i.e., limit cy c le instabilities), as well as the influence of fo r ward " fl" ght.

I t '

J

,. Biggers (ref. 18) c o nsidered the flap dynamics f o r Kp = 0 and n o , rFverse fl o w but including u _ 1. tie constructed U r oo t loci and y - i , _.anes for several cases using a numerical calculation of the exact i _ 8s j_ eigenvalues by means of Floquet theory . He presented th e nonrotating equations of motion including the periodic coefficients i n forward flight for N = 3 and N = 4. The n , he considered the constant coefficient approximation in the nonrotating frame, solving the characteristic equation for the roots of the system numerically and comparing this constant coefficient approximation with the exact results (Floquet theory). He examined several cases out to _ p = O.S. The results of this comparison of the exact and constant coefficient e;genva]ues (both obtained by numerical techniques) are in agreement with the results of the comparison given here (of corresponding perturbation solutions).

r CONCLUSIONS _ This report has considered the influence of forward flight on the flapping stability of several helicopter rotor configurations. The eigen- values of the motion have been obtained by a perturbation technique which gives analytic expressions for tt, e roots. Comparison between numerical solu- . tions f o r the exact roots and the present perturbation solutions indicates that the latter are quantitatively accurate to about v = 0.5. (An exception is near the ½ / rev region, where the perturbation solution was carried to only order u. It should evidently be extended to order v2 as was the rest of the solution.) In general, for this range of v, the flap motion retain:: the high aerodynamic damping of hover and so remains very stable in forward ilight.

There exist, however, critical regions due to the periodic coefficients which are encountered if the hover root frequency is too near a multiple of ½ / rev.

' The influence of the critical regions increases with u as the periodic . regions of interest are the ½ / rev and the i / rev regions. In a critical region t coefficients increase. For the usual values of v, Kp, and y, the critical there is a plus and minus increment in the real part of the root from the hover value while the frequency remains fixed at a multiple of ½ / rev. Hence, there is a decrement in the stability of the system when a critical region is encountered. However, the damping change is only order _ in the ½ / rev region, and order _2 in the I / rev region, so the stabilit y decrease is small and the flap motion remains highly damped in forward flight.

, I -- , For _ order 2 or so, that is , beyond the range ot validity of the I perturbation solutions obtained here, there can occur a sufficient stability degradation in a critical region (usually in the I / rev region) so that a flap instability is encountered.

, These conclusions about the flap stability may generally be found in the i existing literature on this problem. What the present work adds is explicit analytic expressi o ns for the eigenvalues of the flap moti o n, including the periodic coefficient influence in forward flight Also the behavior of J • , : ' cantilever (_ > 1), teetering, and gimball,_d rotor configurations is examined • _1 in addition to that of an independent, articulated blade.

The transfer function of the flap response to blade pitch has been i c o nsidered as an alternative to the e i genvalues fo_ describing the dynamic _..

'_ characteristics of the system. The transfer funct i on indeed is found to r ep r e s e nt equiv a l e n t infor m at ion a bout t he d yn a mi c s , b u t wi t h qui te d iffe r en t b e h av io r . Th e cr i t ic a l r e g ion beh a v io r of t he e igen ', a lues i s re pla ced by sid e band tra nsf er func t ions (respon s e a t f r equen c y _ ± n_ t o inpu t a t f re qu e ncy _).

T h e co ns ta n t coe ff i cien t a pp r o x i ma t ion t o t he f la p equat ions o f m o t i o n in _ t h e no n r o tat in g fr a m e w a s inves t i g a t ed and the e i g en v alue s comp are d w% 1: h t he solu t ion s in c ludin g t h e pe r i o di c coefficien ts . Su c h an a pproxi m ation ( :anno t be e n t ir e ly c orr ect , o f c ours e , but it is re m ar kably g ood, e spe c i a lly for th e lower f r e q u e n c y nonrotatin g mod e s of th e ro t or. This implie s t ha t for cer t ain p r o b l e ms - s u c h as t he low frequ e ncy d y nami c s of t he r o t o r involvin g t he heli- copter bod y m o ti o ns - th e co ns t an t coe fficien t app rox im a tion is an a de qua t e " _ r epre s e n tat i o n o f th e sys t em . T h e p o ssi b ility o f usin g th e c o nstan t c oe ffi- ci e n t approxima t ion invol v e s a c onside r a b le redu ct ion in the e ffor t r e quir e d t o ana l yz e and un der stand th e r ot or d ynamics. I t is sugg e ste d , the r efore, - that t he first s t ep i n an inves t igation involving helicop t er ro t or forward fligh t dyna m ics shoul d be t o c h ec k t h e v a lidi t y o f t he co ns t an t co efficien t __ approximation b y c o mpa ri ng wi t h an e xact so l u t ion (ob t ained by num er ical t ech- r, niqu es pro b ab ly) f or the particular p r o b l e m . The pe r iodic coeffi c ients (in the nonrotati n g f r ame) may not even be n e eded. If they are required, then the m e thods of p er turbation th e ory are very useful in ex a mining the fun d amental be h a vior.

L i ra O !

J

|

APP E N D IX A PERIODIC SYSTEMS The forward flight of the helicopter introduces pexiodic coefficients into the differential equations describing the flap motion due to the periodic variation of the free-stream velocity seen by the rotating blade. F or large enough u the periodic aerodynamic forces radically influence the behavior of the root loci, and the analysis techniques required to find the e igenvalues.

The root loci of a constant coefficient system characteristically exhibit behavior in which two roots start as complex conjugates, meet at the real axis, and then proceed in opposite directions along the real axis. The existence of q periodic coefficients in the differential equations describin g the motion generalizes this behavior so that it can occur at any frequency that is a multiple of ½ / rev, that is, at ImX = n / rev or n + ½ / rev where n is an '_- integer, not just at O / rev (the real axis ) . The property ef the solution that _" allows this behavior is the fact the eigenvectors are themselves periodic (instead of constant as for a constant coefficient syst e m; see the mathematics below). Th e analysis which demonstrates that periodic systems show this behavior is called Floquet theory. -_ Thus,the following behavior of root loci is characteristic of periodic systems (refer to fig. i0). If the parameter being varied, for example, the advanc e ratio _ in the present problem, is such that at _ = 0 the system is not periodic, then the roots start out as complex conjugates (point A on the loci in fig. i0). As _ increases, the periodic forc e s increas e , and the roots move toward n / rev (or n + ½ / rev) frequency, remaining complex conju- gates. At some critical _ the loci reach Im_ = n / rev (point B on fig. I0), i and then for still larger _ the frequency remains fixed at n / rev while the ..

real palt of one root is decreased and that of the other is increased. The root being destabilized may cross into the rig v * half plane for some (point C on fig. 10), indicating that the system has become unstable due to the influence of the periodic coefficients.

"., ' A general system of differ e ntial equations with periodic coefficients may b e reduced to a set of first-order equations , and may therefore be written 1 (in matrix notation) as _x A_ (AI) _ where x is th e st a te ve c tor of the syst e m and A(t) is a periodi c m a trix of c _effi c ients: A(t + T) = A(t). It may be sho_m that the solution to this _ : differential equation can be obtained in the form _.t _(t) = _. qi(0)eXit_i(t ) (A2) ' I _ 88 _ b T he _ i a r e t he ei g enva lu es; th e ei g en vector s u i a r e p erio d ic: ui(t + T) = ui( t ); and the nu m bers qi(O) a r e constants obtained from the ini t ial conditions. The t heo r y that shows t his is c a lled Floquet t h eo ry . Th e solu t ion in this foln m is a direct extension of the normal solutiop for a con- stan t coefficient differential equation which is characterized b / cons t ant e igenvectors (_i independent of time).

The eig e nvalu es Xi may b e obtain e d b y th e following p r oc ed u re . Th e equa t ion ¢ = A _ ( A3) ' wher e _( t ) is a ma t rix, is int eg rat e d ov e r one period, fro m t = 0 to t = T, with t he ini t i a l c ondi t i o ns _(0) = I ( t he unit matrix). Then, if X c i ar e e igenvalu os o f t he ma t rix C = _(T), t he ro ots Xi are given b y Xc = elT , or I = (Zn l c ) / T. While the ro o ts Xci (a s ei ge nvalue s of a r e al ma t rix C) mus t appear as real numb e r s or c o m plex c onju gate pairs, the eigen- £- va l ue s Ii a r e un d er n o su c h res tr i ct ion, leading t o the behavior of the r oo t r" l oc i as describ e d a b ov e .

, Fo r a sin g l e degr e e of f ree do m , s e cond-or de r syst e m, let x R b e t h e s o l ut ion o b ta in ed f ro m in tegrat in g t h e equation with t h e initial co n d i t i o ns _ x(0) = I, x{0) = 0; and le t Xp be t h e solu t ion wi t h ini t i a l condi t ions x(0) = 0, x(0) = I. Th e n, t he roots _c are g iven by t he qu a dra t ic equa t ion: i [ t _c2 - [XR(T) + xp(T) ] _c + [XR(T)xp(T) - XR(T)xp(T)] = 0 (A4) / , e.

) I 1 - !

) 89 APPENDIX B M E T H O D OF MULTIPLE TIME SCAL E S This a ppendix describes briefl y the perturbation technique known as the method of multiple time scales . I t should be used in parallel with the anal y- sis of the flap equation, for many of t h e steps are more clearl y expre s sed in the context of a specific example. More details of the method, and example s of its application may be found in reference 21 (and also ref. 14).

Fundamental to perturbation methods is the existence of a small parameter , here the advance ratio u; and for the solution of periodic coefficient differ - ential equations, that the periodic terms be functions of u such that the equations reduce to constant coefficients when ,, = O. For a study of the : stability of a system, a solution is required that is uniformly valid over long time periods, so the long time behavior may be assessed. This leads to r, the use of the m e thod of multiple time scales. Define a series of time scales Cn = un4" The time scales 4n are all assumed to be of the same order. Then for 41 = P$ the actual time 4 must be order u -1 , that is, very large com- pared to the basic scale 40 = 4. The behavior of the solution over several time scales 4n will be investigated , each implying successively longer time behavior. The derivat z ve with respect to time becomes then I J 2 B B B + P + +

: "

So the a s sumption of the time scales is equivalent t o an expansion of d / d4 as a series in u.

, N e xt the dependent v ariable is also expand e d as a serie s ip p: ' B = Bo(4o, _I, _2, .) + pB z (_o, _I, .) + • ' The 8n now depend o n all t h e t ime s c ales _n" Th e Bn are all a s sumed t o b e of t h e same order, for all the lon g ti m e s cale b e havior . Thi s requirement / is critical to obtaining the s olution; it le a d s , for certain value s o f the ' fr ee pa r ameters, to criti c al r e g i o ns in whi c h ther e i s a r edu ct i o n in th e stabili ty of the s y stem. In order t o i nves t i g ate t he influenc e of the f r e e parameters, the y aiso ar e expanded a s s eries in p. In thi s ca s e, f o r the ! L o ck number y h av e _4 Y = Y 0 + _Yl + P2Y 2 + , ; , ¢ I , 9 0

f

i , i w her e t h e Yn are all o f the same order. F or c e rt ain cr i tic al v a lues of Yn - that i s , w hen y i s order p or order p2 from certa in cr i tic a l v a lue s - there wi ll occur a s tab i lit ) de g radatio n due to the in flue n ce o f the periodic coe f f i cients.

So n o w 8, d / d _, an d the free parameters are all expanded a s series i n p .

T hese a re substituted in t o t h e d i ffere n t ia l equation, and t h e n a ll ter ms of li k e order in u a re collected and se p aratel y s et t o zero. T hu s, d iff er - e n t ia l e q u ati o ns a re o btain e d f or o r d er 1 , _ , p2 , ., _n , .... With th e expansion of d / de , that is, the use of multiple tim e sca le s , th e s e equation s a re n o w pa r tia l diff er ent i al e q u ations fo r 8n as func t ions of _n. Th e order un equation has t he following form : it may be writt e n as a diff e r- ential equation for _n ( ¢0) , forced by the lower order solutions. Tha t is, on the right-hand side there are 8n- l , 8n-2, . , 80 and their deri v ati v es with respec t to cn , cn-1 , ., _0. The set of p ar t ial differential equa- tions is solved progressively , beginning with the lowest order equation . ' (o r d er 1), t h e n o r de r _ , et c.

_" I t is t he c ha r acte r istic of this expansion in p that the u n order equati o ns, w r itten as or dina r y differential equati o ns f o r 8n(_ 0 ) , all ha v e the same hom o gene o us s o luti o n. Since the equation f or 8n(_ 0 ) is f o rce d by the l owe r or de r so lu tions 8n- 1 , ., B0, it fo ll ows t h e n that when th e known solu t ions for 8n_1, . , 80 are substituted, the equation for 8n will be forced by its own homog e neous solution. Thi s wo u ld give ri s e to solutions for Bn of the form @0 times its homogeneous solution, that is, , s ol u t i o ns f or Bn o f or d e r _ 0 8n_l, . , _ 0 8 0. Th e n 8n / Bn_ l , . , Bn / 8 0 w oul d be order _ 0, that is , wou l d b e com e a r b i t r arily l a r ge if _0 is l a r g e I enough . This violates the assumption that Bn is of the same order as 8n - 1, • . , 8 0. The o n l y w ay such a s o lution may b e avoid e d is if th e co e f- 1 fi c i e nt of the homogeneous so lu tion on th e r ight-h a nd sid e of th e _n order i e qua t i o n f or 8n ( _ 0 ) is itse l f s et t o ze r o. This coeffici e nt of th e homog e n e - ous solu t ion for c ing the equation for Bn ( ¢0 ) is c all e d t he s ec ular t er m . Th e . secular t erm is set t o zero so that the solution for fln wil l b e u nifo r m ly va l id for all time, tha t is, for large _0.

J '-._ Now t he r igh t -hand sid e of th e 8n e quation i t s e lf in v olves d er iva t ives i of 8n- 1 , . , B0 with r esp e ct to the time scal e s _n , ., _0- At this stag e in t h e analys i s , conside r ing t he _n o r de r equ ati on , some of th e so lu - ti o n fo r B is kn o wn al r eady. What is n o t kn o wn is t he behavi or o f Bn-1 I w i t h re sp ec t to _1, _ 2 , . , t h e b e ha v i or o f 8n- 2 w i t h respect to r _ 2 , ., an d so o n d own t o 80 with re s o e ct to _n" H e nc e , th es e c o mbin a - ' t ions o f the d epen d e nt an d in d ep e nd e n t va r iabl e s re m ai n o n th e r i g ht-h a n d s i de of t h e 8n ( ¢0 ) e qua t ion when th e known _ l uti o n is substitut e d. Sp e ci f ic a lly, : : t h es e c o m b in a ti o ns wi ll b e in th e c o e ff ici e n t s o f t he h o m og ene o u s so l ut i o n , : " that is, in t h e s e cula r t e rm . T h e n, s e t t ing t h e s ec u la r term to z e ro resu l t s _ i n a dif fe r e ntia l equation , _hich m ay be w r itten as an equ a ti o n for Bn_l(_l) , _ " i for c ed on th e ri ght-hand s ide by t h e l owe r ord e r s o l uti o ns. This e qu at i on } , will a lso b e fo rce d b y its own ho m og e n e ous s olution , an d s o its sec u l ar ter m , : m us t be s e t t o zero in order t o m ai nta in t h e u ni for m va l i d it y o f 8n-1 ov e r ' th e _1 ti m e sc a le. Th e re s u lt is a n equ a t io n f o r 8n_ 2( _ 2 ); t his pro c ess L is c o ntinu ed do wn t o a n e quat i on f o r 8 0( ¢n).

b

, 91 So the o rder _n equ a ti on pr o d uc e s a set of differe n tial e q uatio n s fo r 8n ( _ 0) , 8n_ l( _ l) , 8 n - 2( _ 2), - , 8 0( _n ). T h i s set o f eq u a t io n s is t h e n s ol v e d , f or th e beh av ior o f 8 n w i t h respect to _ 0, Bn- 1 w ith respe c t to $ I , a n d s o on do w n to 8 0 w ith respect t o _ n, thu s co m pletin g t h e so l ution t o order vn . T he a nal ys i s then pro c eed s to the o rder g n+l equ at ion. For th e s t abi l i t y o f t h e sys t em , it i s th e be hav ior o f 8 0 wi t h re s pect to _n _ t ha t is of interes t since t hat gi_s t he eigenvalu e t o order un. The nes t ing behav i or of the differen t ial equa tio ns makes each success i ve ord e r solution more involved bu t t he basic procedure rema i ns t he sam e .

t

• w" i

i

I, APPENDIXC SOLUTION O F TH E SEC U LAR E Q U AT I O N T he ap pli cat ion of t he m e t h o d of mul tip le time s ca l e s to p er i o di c - , coe f f i cie n t d if fe re n tial e qu at io n s r e sul t s in se c u la r e qu at i o ns o f t he f o r m _BB+ (a + id)B + (b + i c )B = 0 ( e l) wh ere B is a c omp l ex quan t i t y , a nd th e c ons ta nts a , b, c , a n d d a re r e a l .

Writing I D2 = d2 - (b2 + c2) = d2 - I b + ic l 2 (CZ) D = _ ( C3) it ma y b e verif i e d that th e sol u t i o n of t h is e q uat io n i s , f or D 2 > O: ' , 8 = e- a *{A[d - D + i ( b + i c) ]eiD* + A[d+D.i ( b + i c )] e - i u * } (C4)

t '

w h e re A is a c o m p lex co n s t a nt; for D 2 = 0: B = e' a* ( A{[d + i ( b _ ic) ]¢ + i} + B[ d + i(b (C5) l wh ere A and B are r e a l c o nsta n t s; a nd for D 2 < 0 : S = e -a * A[d D* + B[ d - iD + i( b+ i c) ] e- D* ( C6) w her e A a nd B a re re al c ons ta nts. T he l imi t ing ca s e b = c = 0 gives D = d, so th e so l u t ion is " B ': A e"(a _,, here A is a : omp le x c on sta n t.

' The reg i o,, of d ecrea s e d st abilit y - the cr it i cal r e g i on - i s gi ven by i D 2 < 0 . Th e bou n d ary o f . ' he c ri tica l reg i o v i s D 2 = 0. The b ehavi o r of the , 93 I j_ solut i on of th _ s equation when b or c # 0 i s that described f o _ per i od i c systems; indeed it wi l l be found that the _ t e r m co m es from the periodic coefficients.

|_ m J - J z

i

, 94 _ - I 1S . Johnscn, Wayne: A Perturbation Solution of Rot o r F lapping Stability.

AIAA Paper no. 72-955, AIAA 2nd Atmospheric Flight Mechanics Conferenc_ Sept. 1972.

lv Tong, Pin: Nonlinear Instability of a Helicopter Blade. AIAA Paper n o. 7 _ 9 5 6, A IAA 2nd A tmo spheric Flight Mechani c s Co nf er en ce , Sep t.

1972.

1_ . J ohnson , Wayn e : A Perturbation Solu t ion of He licopter Ro to r F l a pping Stability. J. Aircraft Synoptic , vol. 10, no. 5, May 1973.

18 . Bigge r s, Jam e s C . : So me Appr o ximations t o th e Fl a pping St ab ility of Helicopter Rotors. AHS / NASA-Ames Specialists' Me e ting on Rotorc r aft Dynamics, Moffett Field, Calif., Peb. 15-13, 1974.

19. Gessow, Alfred; and Crim, Almer D.: A Method for Studying the Transient Blade-Flapping Behavior o f Lifting R oto rs a t Ex t r e me Opera t ing Conditio n s. NACA TN-3366, Jan. 1955.

20. Wilde , E.; Bramwe11 , A. R. S.; and Sunnnerscales , R.: The Flapping Behavior of a Helicopter Rotor at High Tip-Speed Ratios. Aeronautical Research Council C. P. no. 877, April 1965.

21. Cole, J. D.: Perturbation Methods in Applied Mathematics. Blaisdell Pub. Co., Waltham, Mass. , 1968.

d !

.

J _ J t i 96 _ ' ImX I

:o .l .2.3 ._ .5 / (c)

# fo r ca se ( c) / / _ ,

/

(a) I I = I' ! l_- R eX "( I l 4 ,

i '

-', (a ) I (e) . _ : _= .5 _ .

' 1

i ' - I t _ , _ Figure 1.- Root loci for varying u, based on the perturbation _,olution. The '.; i cases shownare (a) v = 1 and y = I0, (b) v = 1.1 a n d y = 6, and (c) _ = I '_ and y = 6 ( Kp = 0 for all t h reecases). _ , .-

| i

28- r 24- 20 2 real roots

;i

i Y " I 12 - i Y = I0 locus 8- :, 4 _ )- : 6 locus l } 4- i ' I ,, i I I I r '. }_ _ 0 . I . 2 / _ .3 .4 . 5 I'_ " Figure 2 . - Cr i tical reg i on and real root bound a ries on the y - _ plane for _" '_ f v = 1 and Kp = 0 (b os ed on the pertur b ation s o l u t i o n ). I-'" ; ' , .

f I , t " i 4 I / rev

J

' / t _ e 4

i . _ . 5

Fig ur e 3.- C ri tica l re g i on an d re _l roo t " b ou nd aries o n th e Y - _ plan e . ; _ v = l . OS a nd Rp = 0 (b a s ed on for t h e pert u rba t io n s o lutio n ). J |! : 2g- 2 reol roots ............. ] .... ] ..... [ z .... J _ " 0 .I ,2 .3 ,4 . 5 . I F i g u re 4 ,- Cr iti cal re gi o n a n d real root b o u n d ar i es o n the ¥ - u p lane _ or ! ' _i . _ :, : _i '. ( _ . , v = ! .1 a n d Kp = 0 (based on the p e_ . turba t io n solution). I

1974023388

28- 24- 20- , i I / rev 16 2 + y +1 14 - 8 -

II +- !

' 2 _p L I I , l I l , 0 .I .2 .3 .4 .5 + ',e -I p .

-[ Figure 5.- Criti cal r e g i on and real root boundari e s on the y - I_ plane for _ , v = 1.15 and Kp = 0 (based on the perturbat i on solut i on) .

i ' ?. r e a l roots 2 O

t / r , v

Y 1 4 t 8 { - _=_ I / r e v I i .4 . 5 0 j .2 . 3 i " . _ plane f o r _ bo _ da ri e s o n the soluti o n ] f_ 6.- Fi g u r e i Cr i t i ca l r e gi o n a n d rea l r o o t v = 1 a n d Kp = 0 .1 (b ased on the pert u rbat i o n

1974023388- '

I / rev ., , , ,I ,,, t , , _1., I 0 .I . 2 . 3 . 4 .o

i -

- I F igur e 7 . - C riti c alr e gion an d re a l roo t bound aries on t he Y - u pl an e fo r _ = I a n d K p = O , i nc luding the or d er p2 infl ue n ce o n the ½ / r e v r e gion boun d ary(fr o m Te l . 6) , I I m X J,

(a)

t _ v y ) ,

T

Im x - -- --_ . _- -- I / rev (b) " _ t '

-Ii

' _ R e X I _ Im x d • t Re X _._- t i _ Figure 9 B ehav i or of t he _ roo t locus o f t he t hree - bladed gimbal l ed ro t or, ! , _ near the t w o real roo t boundar y . The three cases are discussed in the text J - .

• i _ .

i j_ [ m X J n / rev

A l

B

i c

_ -_ R eX a ( i - - - n / rev

,., [

I • k Figure I0.- Sketch of the characteristic behavi o r of root loci of periodic i coe ff icie nt d if f eren t ial e qu ations i t- i , I i