A pertinent solution of helicopter rotor flapping stability

| ¢ & A PERT U RBATION SOLUTION OF HELI C OPTER ROTOR , : , , FLAPPI N G STABILITY _.

Wayne Johnson o F Ames Research Center and U.S. Army Air Mobility R & D Laboratory Moffett Field, Calif. 9403 5 (NA S A- T R - X - 6 Z 16 5 ) A PE R TI N E N T S OL U TI ON N 7 3 -1 _ 006 O F H ELIC OPT ER _ OTO B F L AP PI N G STABILIT _ W. Johnson (NA SA) dul. 197 2 1 2 3 p _ J _ t' CSCL 0 1A On c l as L G 3/ 01 5 2 2_ L j J _y 19 7 2

/ 'h

A PE RT U RBATION SOL U TION O F H E LI C OPT E R ROTO R F LAPPIN G STABILITY :. Wayne J ohnson _ R esearch Sc i entist _ " U.S. Army Air Mobility R&D Laboratory ' Moffett Field, California 94035 =._, A B STRA C T T he _t_ bil i ty o f the flapping m otion o f a si ng le blade o f a helicopter rotor i s exa mi ned usi ng the techniques of pe rturbation theo ry . The eq uation of motion s t u d i ed is l inear, with p eriod i c aer od ynam i c coeff i cients due to the f orward speed o f t h e rotor.

_' utions ar e f ound for four ca s es: sm al l _u-_d la rg e advance rat i o an d small and l a rge " L ock nu m ber. The pe rturbat i on techniq a es appropriate to each case are d i scussed and illustra te d i n the course o f the analy s i s . The app l ication o f pe rturbation techniques to o ther pro b lem s in rotor dynamics i s d i scussed. It i s c on cluded that pe rtu r bati on th e o ry i • is a powerful m athe m atic al technique wh i ch should prove very use f ul in anal y z i ng some o f the problems o f helicop te r d yn am i c s .

_ " T ABLE OF C ON T EN TS

P age

r.

NOMENC L ATURE 6 • • • • • • • • • • • • • • • • • o • • • • • • • • • • • • • • • • • • • • • • • S m all _ Case 9 • • • • • • • • • • • • • • • • • • • • • • • . • • • • • • • • • • • • • • • • • • • • • • 9 ThH o ve r .......................................

Expansi on . _r._ ....................................... I 0 _ Order _ results 1 3 Ord er / _ 2 results 25 Order 1 results 40 ' Order y results ....................................... 40 i ) Evaluation of the order y results ........................... 5 2 i Flap rate feedback ..................................... 6 5 Tabl e of Co n tents ( C ontinu e d) r AP P LICATI O N O F PERTURBATION TECHNIQUES T O REFERENCES 107 ° I oo l oeoeeo l o o eoooeoe l eooeoOeeeeoeoeoeeoeooo F I GURE S .............................................. 108 • f SUMStARY The stability o ft he f lapping motion of a singl e bl a deofa heli c opt e r ro t o r is e x a min ed us i ng t h e tech niqu e s o f p ert urb a ti o n t h e o r y . T h e eq u at ion of mo t ion stu di e d ' _ i s lin ear , wi t h pe ri od i c a e r od yna mi c c o e ffi c i e n ts d ue to the f or w ar d spe e d o f the rotor.

!_) Bl a d e pi tc h f ee db ac k prop o r t ion a l t o bo t h flappi ng displ aceme n t ( 53) a nd f l a pping c ate y_ :_ i si n c lud e d F our case s a r e c on s i d er e d : sm al l a n d l a rg ead v a n ce rat ioan d sma ll _._. ; an d la rg eLoc k n um b er. T he pertur b at i o n tec hn ique s appropr i ate toeac h ca s e are dis c us se d and illu strate d in t h e c ou r s e of the an aly s i s. An a l yt i c s olution s are ob tai n ed f o r each case , w ith p r i mar y em p ha sis on t he e i ge nv a lu e s (t h a t is , r oo t lo c i ) as indi - cat ors o f th e s y st em s ta bility and r esp ons e. T h e feat u re o f th e eq u at ion w hi c h m a k es - .

perturb a tion te c hniques u s e f ul is the per i odi c ity o f th e ae rodyn a mi c c o eff i c i e nt s .

T h e a ppli cab ili t y o f t h e four cases c on s id e r e d is dis c u sse d ; th e s m a ll a dv ance r at io r es ul tsi n pa rt ic u lar are v e ry usefu l ,bei n g v a li d o utto a n a d v an ce rat ioo f a bou t 0 .5. / • Th e a ppli ca tion o f perturb a tion t e chniqu es to probl e m s in rotor dyn a mi cs with mor e degrees o f freed o m o r be t ter aer odyn amic model s is di sc u ssed. It is c on c lu ded that f pert u r b at ion t h e o r y is a p ow erful, and yet no t ver y s oph i st i cated, m athe m at i c al te ch- '_ ni q u e w hi c h s houl d p r o ve v ery us eful in an aly zing s o me o f the p ro b l ems of h e li c o pter I d ynamic s .

INTRODUCTION This pape r c on si d ers t h e application of pertur b ati on te c hniquee to h el i copter rotor dyna m ics. Perturbation theory has been well d e v e loped i n rec e nt years, but has

i

not found much application to rotary wing problems. Classically helloopter eng inee ri ng - 2- r _ ha s made u s e o f th e sa me ne rt u rbation t heor ie s tha t f ixe d w ing e ngineer i ng has, for ex a mple lifting line theory and enginee r in_ be a m t h eo r y (bot h requ ir e a large blade aspect ratio). Another classical example i s actuator disk theory (a large number of blades is required). These theories were developed on an intuitive basis however, / and the more rigorous mathematical techniques of perturbation theory have not yet fo u nd widespread use for rotary wing s . The classi c al appli c ations are largely for aerodynami c problems; the mathemati c s of th es e problems can be very complicated however because the equatio n s involved are highly nonlinear part i al differe n tial equa- tions. The treatme n t of dy n amic problems can be more tractable s ince only ordinary differential equations are involv e d. Probl e ms with con s tant c oefficient linear differ- e n t ia l e qu at ion s can be s ol v ed exact l y with w e ll estab li s h e d me thod s , so fo r t h ese , pr o ble m s the extra effort of perturbation theory m ay not be ju s tified. O n the other h and for problems w i th t i me var yi ng or nonl in ear differential equations the o n ly solution procedure g enerally applicable i s the numerical integration of the equations o f mot i on.

Ho w ever, purely numerical so lut io ns are not entirely sat i sfact o ry f o r obtaini ng an under s tand ing o f t h e ph y si cal character o f the system, or for f o rmulat in g gener a l des i g n ru l es. Furt h er m ore , an a n al yt i c s ol ut i o n for the ge n eral case would b e d i fficult to obta i n ( i f p o s si ble a t al l ) and would be so c o mple x as t o be hardly better t h a n the n umer i cal s olut io n. The on ly s y s te ms t ha t c an be pr $ , ct i c ab ly h andled an al ytically a re th o se i nv ol v i ng li n e a r c o n s tant c o eff i c i ent di f fer en ti al equat io n s . Perturbati o n tec h n i que s are ava il ab l e whi ch are method s to s t udy time varyi ng or nonl i near syste m s suc h that at each ste p in t he an al y s i s only linear con s tant coeff i c i ent equat io n s must be han d l ed. T ime vary ing or nonl i n e ar d i ff e rent i al e quat i on s a re charac te ri s tic feature s of hel i co p ter dynamic s an d aerodynamic s , pr i m aril y du e to the rotat i o n of the wing.

\ Thus the possibilities for the use of perturbation theory in rotary wing problems are very extensive.

This paper considers the stability of the flapping motion of a single blade of a ' helicopter rotor. Thi : _ i_ a single degree of freedom, sccond order system, with analytic aerodynamic coefficients. The governing equation is linear with time varying coefficients; it is given below.

_'+ b' 28 = T _(M s -K R Mo) t }+ (M8-Kp Me) 8 ] (I) Regions 1 1 1 ) IVI_= -(_ + _ p _s i n $ + _-_ ( _sin_ ) 4 ( ii) 1 1 + _ Dsin $ (iii) t 1 M 8 - #co s@( 1 1 1 = _ + _ Dsin_ b - _ (Dsin @)3) (ii) I _ b ) (iii) _ t cos @ ( 1+ _' . s in 1 1 1 _+ :_Ds i n_ b (: _si n $) 2 (i ) . M e 1+ I 1 I ='8 , _!I s in$ + _ ( b Lsin$) 2 - _-_(Dsin@) 4 (ii) 1 1 1 ) (iii) _ _ + _ _ sin $ + "_(_ sin $)2 ' < -4 - where the coefficients have separate definitions in three regions of the disk defined by Region (i) 0"<_sin$< Region (ii) - l<_sin$<0 , Region ( iii) - _< b L_in $ < - 1 This is the homogeneous equation for small perturbations of the flapping motion of the blade about an equilibrium state ; the derivation of this e tuation may be found in the iiterature (Ref. 1). B is the degree of freedom representing the . blade flapping motion perturbation. The equation is nondimensionalized with the rotor speed, so the time variable is the azimuth angle _ b . v is the rotating natural frequency (non- dimensionalized w ith t h e roto r speed ) of the fl a pping mot i o n , whi c h may be grea t e r a than 1.0 for flapping hinge offset or cantilever root restraint of the blade. _ is the Lock number, defined by T = P acR4 / Ib (P is the air densit y , a the two-dimensional lift curve slope, c the blade chord, and R the blade radius); Ib is the equivalent mas s of Lhe flapping motion, given by the integral over the span of the s quare of the mode ' shape of the flapping motion weighted by the mass per unit length of the blade; for the rigid flapping motion of an articulated b_ade, the mode shape is proportio n al to the radial _listance from the hinge, and so Ib is j u s t the moment of inertia of the blade about t he flappi n g h ing e. K is th e flap p rop ortional f e edback gai n, b e t t e r known as P tan 53 ; KR i s the flap rate feedback gain. A fe e dba c k law A 0 = - Kpj 5 - KR_ } has b e e n used (AO i s the blade pitch change du e to flapping f ee dback control ) . _ i s th e rotar adv a nc e r a t i o (fo r war d ve lo city a ivi d ed by r ot or ti p s p eed). Th e coeff i cients M_ , M ff an d Me are the aer Gx tyn ami c forces on the b l ade, h e.ce the i r mu l t i p li cat i on by y and their dependence on / _ . Th e , three reg i on s f or the c oe f fi c i e r.$ re f lect the i nf l uence of t h e reverse flow region of t h e ro t or disk . I n re gion (i) t h e r e i s norm a l flo w over the entire blade span; in region (iii) rever s e flow over the entire span; and in region ( ii) normal flow outboard of r = - / _ sin _ and reverse flow inboard. Regio n ( iii ) is encountered only if / _ > 1. The aerodynamic coefficient s were obtained using a rigid I , . blade motion, and should properly be changed some to handle a blad e with c a n tilever root r e st r ain t. H o we v er t h e m a j o r e ffec t s of a ca nti l ev er r oo t o n th e d yna mic s of the / system ar e du e to th e change in v and 7 ( b o th ar e incr e a s ed, v to 1 . 1 5 s ay and Y t o ab ou t 5 / 3 t h e L o c k n um b er ba se d on th e r igid m ode in ert ial ) . Sin ce th ese are free " pa r a m e t ers i n t he a na l ys is t hi s f ormulat i on of the pro b le m s h ould g i ve re a sonab l e r es u l ts f o r all rotors.

• T h i s equation h as be en s t udi ed numer i ca ll y i n recent literature , priL _ ari l y in the :_ , ,

i

• context of Floquet theory (Re f . 2), wh i ch must be used because the aerodynamic coeff i c i ents are periodic i n $ i f / _ _ 0. The equat i on w i l l be studi ed in thi s paper u si ng t h e techn i ques o f pe rturbation theory. T h e mathemat i cal techn i qu es are not very s oph _ s - t i cated actu al ly; they are rather long, e sp ec iall y when the high e r order solut i ons are s ought; and there are some tr i cks to be l earned, but th e s t an dard ones work f or most sy s tems, i nclud i ng th i s one (Re fs . 3 and 4). It i s not m ai n tained t hat the diffe re nt i al equation stud i ed is a true m ode l of rotor dynamic s ; nonl i near aerodynam i cs and coupl ing w i th p i tch and lag m ot i ons ar e certainly very im portant. The purpo s e of this paper is not to present a study of true flapp ing d yn am i c s ; rath e r i t is i ntended to d em on I trate "" w h at information can be obta i n ed by the pert urbati o n technique s , and to exp l ore the ° method s w h ich are most u se ful for rotor dyna m i c s, so hel ic opter engineer s wi l l be able to decide whe th er to use th ese te chn i que s w i th m ore co m pl i cated or m ore real is t ic _" _ -6- systems. Th e dy n am i c prob l e m cons i dered here is t h e auest i on o f rotor f lapp in g sta- bili t y . T he st a bi l i t y o f th e m o ti on i s d e t erm ined by th e root s or e ig e nv al ue s of th e

! '

, cussed will be concerned with the roots. The equation considered is linear; perturba- i system (there are two for this second order equation) , and so most of the results di s - t i on theory is used because the aerodynam i c coefficients are time varying (specifically, periodic) for forward flight, I.e., when _ is greater than zero. A brief discuss i on of the characterist i c behavior of the etgenvalues of a pe riodic syste m i s given in Ap pe ndix I.

, NOME NCLATUR E Kp Flap proport i onal feedback gain I KR Fl ap rate f e edback gain " , M B Aer odynami c moment due to flapping displacement " • . ' b , Mb Aer od yna m ic mo m ent due to flapp i ng rate M e Aer odyn amic moment due to b l ade pitch B Flap motion degree o f freedo m t Y Bl ad e Lock nu m ber t ), E tgenv al ue or root of the syste m Rotor ad van ce ratio (forward s peed / ro to r tip speed) V Rotating D atural frequency of flap motion (centrifugal and structur al stiffening) , no ad imen s ion al ized wi th rotor rotational s peed P Rotor azi m u th a _ le, measured fro m down s tre a m

i o() " the order c_ ' *

( ) Conjugate of a comple x num b er

_ _ _ , "_ ' h LHP Left-hand plane RHP Right-hand plane LttS Left-hand side .

, RHS Right-hand side Re Real p a rt of a c omplex numbe c Im Imaginary part of a complex number ANALYSIS AND DISCUSSION Introduction to Perturbat:ton Techniques Fundamental to th e use o f pe r tur b ation te c hni q ue s i_ the existence of s ome parameter whi c h is either very small or very large ( how small or how large is deter- t mined during the analysi s ); for the moment repr e sent the s mall parameter (or the -., , inverse of the large parameter) by (. I n th e present p r oblem it is de s ired to find the roots of the motio n, whi c h means inv e stigating a s olutio n whi c h i s unif or ml y vali d ov e r ! long t im e p er i ods. The ap pr op r ia te pert u r bation techn i q u e i s the met hod o f multi ple time scales. This met hod as s um es t hat t h e b e havior o f t h e s y st em m ay be inv e s t i ga t e d i ov e r s evera l t ime s cale s , i .e. , Sn = ¢ n$ T h e ti me sc al es Sn a r e a ll a s s u med t o b e t h e s a me or de r ; t h e n fo r $ 1 = ¢$ the a ctu al t im e $ m ust be o f o rd er ( - 1 , i .e., very l arge co mp a red t o the b as i c t i me sca l e ! $0 = $ " N e x t t h e d e pen de nt v a ri a i4,_ s a r e expan d e d a s a s er i es i n c , ! = , , , ...)+¢ ,, , ... )_ ...

! B 8 0 ( $0 $1 $ 2 B 1 ( $0 $1 I where the temas 80, 81, etc. are all assumed to be the same o : der, and depehd on all the time scales now. The requir3ment that all the B be the same order for the long n time scale behavior of the motion is crutial "coobtaining the solutior; it leads, for certain valu ¢ .s of the free parameters, to critical regions characterized typically by a reduction J of the stability of the system. The details of this method will be given below in the context of the t_vatment of the rotor flapping equation.

Often an equatio n of motion is s u c h t h a t in t h e lim i t ( = 0 th e order of t he differ- ential equation is redu c ed. Su c h problems are c alled boundary layer problems, sin c e they are c hara c terized by narrow regions in wbieh the solution c hanges greatly. The outer solutio n m a y b e f ound h y u s e of a subst i tution o _ the fo=m = pd_ b e x p _$ f o ll owed b y an _ x pan s i on o f p a s a a er i e s ;_ ¢ : .,.

p = '-'_p n + .. .pO+¢Pl+. . .

( This m a in soluti,_n i s,, ot v a lidin ce rt a inna r ro w t r ar._,ition r eg ionso r bounda ry lay e rs .

A b a s i cpa rto fth i sue rtu r b_ t ion tec hniqu ei sm e thods _ o b ta insol ut ions t hrough t ho _' transition region , so th a t it is pos, , ible to m a t c h one m a h_ solution to another on the ,tg oth e rsid e o f th e tr a nsition re gion,or to bou nda r y c onditions atth e ba se o fth e boundary la y e r . Aga i n , d et a i l s o f t h e met hod w il l b e g i wm in th e co n tex t o f th e s olu ti on of t h e f lapping e qu a tion . _ For the f l a ppi n g e qu a tion th e r e a r e tw o pa r a m e t e r s wh i c h may be u se d for v er - • .

i tu rb at i o n q ua n titi es: the advance r a t i o _ a nd the L ( , _k num be r _ . The n the r e a r_ I -9- four cases t o be considered : small and l a rge bt , and small and large 7 . The flapping nat_tral frequ e n c y tJ is also a parameter in the p r oblem , bu t it v a ries tittle and further- mo , _e always has a value at or _lightly above unity ( i. e., is neither small nor large).

Each of these four ca s es will be e x amined in turn in the f o !lowin g sections.

J The Small # Case For the small bt ca s e (to 0(_2)) it is possible to ignore the reverse fl ' )w region, and t h e ae ro dynamic co e ffi c ie n ts in r egi on ( i ) can be used f or all _ b . Th e e quati o n o f motion is then (considering the case KR = 0 first): i • + ,_ sin $ + u 2 + bt cos_ b _ + i"sin + Kp _ + _.sin* + _'_sim b ) B = 0 (2 ) i c T h e sm al l parame te r is t he a d v an c e r a tio / a; the perturbation t ech ni q u e to b e us e d is '_ th e method of mu l tipl e t ime s c al es. The s olution w i ll b e e x am ined t o 0_2 ).

! Hov e r !

i i Fo r t h e h ove r c a se, i . e ., th e limit / _ = O , t h eeq u a tion re d ucest o i . wh i ch is a co nst a n tc o e f f i c i e nt e quationn o w . rhe r oots are obta in ed fro m

i

_ • )2 +8 _ -)'+( I_ 2+K p _ ) -0 ;]

_ -10-

aN

= - - (3)

i (and its con j u g ate).

i • _ Expansion in i U s ing the me th od of multiple time scales, the behavior of th e equation is exam i ned -1 -2 f o r ¢ of order 1, / _ / _ et _ ; th at is , let

"_ _0 = _

t N ext expand _ as a series i n / _ , w ith e a ch term depending on al l t h e time sc al e s C n: B = 8 0( ¢ O, ¢ 1' ¢ 2' "'" ) + / ' t _ 1( ¢ 0' ¢ 1' "'" ) + " " The tim e der iv at i ve no w be comes i : So t h at t h e o rdinary differential eq uat i on (Eq. 2) now becomes a p arti al differenti al i equation. Further m ore, the two remaining p ar ameters in the eq uat i on, v an d 7 , are also expanded as series in , } ,I V = V o + + + ...

7 = 7 0 + / 4 7 1 + / 42 7 2 + •..

- 11 - ( T hi s i s do ne beca u se i t i s t h e cha r acter is t ic of a sys te m wit h per iodic co effi c ients that for certai n values of u 0 and T0 ther e are s t a bility d e gradation regions d e scribed by boundaries in u 1 and T1, or v2 and T2, etc. ).

' _ No w / ], d / d_ , y , a nd T have all been ex pand ed a s s er i es i n / _ . T hese expans i on s a re s ubst i tu te d i n th e dif ferent ial eq u at i on . It is a ssumed t hat all th e coefficients in the e xpa n sion are of t h e s am e order; thu s _0' _ Jl ' _ b 2 ' etc. mus t all be of ord er 1 ; an d B 0 , B 1 , B 2, et c. must all be o f t h e s am e or d er fo r t h e beh a vio r o v e r all th e ti me sc al e s , , _ b n ( h ow l ar g e i s ar bi tr a ry s in c e t h e equa t ion is li n ear i n 8 , al t h o ugh i f 8 i s too l arge : the equation of motio n m a y n ot b e valid). The equ a t i o n of motion will then cont_ n terms 2 _ .r 2 ' • o f o rder 1 , / _, / _ etc. ; a ll t h e terms of l i ke o rder a re c o ll e cted and se pa r a tely e qu a ted ._ _ _: t o z ero , to g ive the equ a t i on th a t start s the anal ysis a t ea ch or d er. _ ' Order 1 Results : _..

To o rder 1 t h e eq ua ti o n is ,_ -- ,_ : .

, _0 2 8 0 +' _ - _ B 0 + P 0 2 + K p 80 ffi 0 ( 4 ) _':_ 2 The solution o f thi s equat i on i s

B o - a e [ .So _ ( _ r _ 2' "" ) ex ° _ ° ] ( 5)

whe re the root )'0 is g i ven by

, o " X o 2+ _ ^ o + (_ o +K p - o

i o r

, 0 2 0 , , o ( , _ o / '

)'0 ffi " 1 " 6+ + Kp _ -" \ 16 / (6)

• l

t \ -12- "2 T _ and i ts conjugate . Si nce B depends o n a ll the t i me scales , _0 = 80 ( _0 ' _1 .... )' ; E q. 4 i s a par t i a l d iffe r e n tial e q ua tio n , a nd only d e t erm ine s th e b e ha v io r of a s a fun c tion of _0 " Th us th e quan t i t y B 01 st ill d e pe n d s on _ b 1, _2' etc.

The o rder 1 equ a tion i s identical with that o btained for _ = O , i.e., the h o ver - _ limi t, an d ind eed ) '0 is e x act ly th e h o ver ro ot ( t o o r d er 1). T he v a r iation of t hese # } r oo ts w i th _ fo r v = 1 an d sever al K p is sh own in Fi g. 1 ( t h e / _ = 0 lo c i ) . Th e c orn- } pl e x p o rt io n o f the r o ot l o c u s i s a c i rc ul ar a rc , wit h ce nte r o n the re al axi s at ) , = - K p , _ and r a d ius of J p 2 + T he c o rre s p on di ng T fo r a p oin t on the c o mp l ex portion of , K p 2 .

t he _ lo cus may be o bt ai ned f r o m t he re al p ar t of X since Re } , = - _ / 16 (n o de pe n d ence ! o n u o r Kp ) . For T = 0 t h e lo cus i s at ) , = i v ( there is no ef f ect o f K p s i nce t here ar e no ae ro dynami c te rm s i f y = 0). F o r K p > 0, Ira ) , incre a ses as _ incre a ses f r om " ! z ero; a pe ak in Imk is reached a t T/ 16 = K p w h ere ) , = - Kp + Jy 2 + K p2 (the pea k _ o ccurs j u st over the cen t er o f the c i rc l e s o t he f req u e nc y i s gi ven b y th e circ l e rad i us ). , F o r K p < 0 , Im } , decre a ses immed ia te l y as _ ' incre as e s f r o m zero. The l oc u s inter- cepts the re al ax i s at 7/ 16 = K p + _ v 2 + Kp 2, where ) , = - K p + j W 2 + Kp 2 . Then a s - " _ . t h e roots rema i n on the real ax i s , o ne g o ing t o k = - _ a nd the other t o ) , = - Kp i (th e cen t er of the circ l e). Thu s one br an ch of t he _ l ocus cr o sses int o the RHP if _ i s a t ) , = v 2 / K p ( w hi ch is l ess th a n zero sinc e Kp < 0). When the so l uti o n i s e xa m i ned ! t o h ig her o rder in / _ ( as be l ow) , spe cial prob l em s occur when the f re q uency o f th e hover Kp < 0 ; the crossover po i nt occurs f or _ / 16 = - y 2 / 2Kp; at th i s y the o ther br a nch _i r o ot i s a t o r ne a r a mu l tip l e o f _ / rev . The order 1 root cr oss e s I m k = _ fo r { ' T/ 16=Kp+ J _ 2 Theloc u sw i ll c rosslmk=lfor _ / 1 6=Kp+ JV 2+Kp2 - 1 . " Since V a I there can be on l y one cross i n g of Im k = 1 or _ by the locus (except when p = i, i n wh i ch cas e t h e l ocus star _ _t Im k - I for _ = 0) . _ _ ' _ _ ........ - - _" e "_ " _ ' _ _.......... ,_ - - : •--- r -----,- - - _ . - . ................... T , _ , , _ , _ , ,_ a e ,, • -13- The root loci for fixed y and varying v or Kp would be somewhat simpler than the y loci. What is being varied is the natural frequency of the system, a_ = , // _2 + Kp _- _ n 8 for (k c o mplex, 1_,1 = Wn - ' ) For the complex portions of the loci, Re _ = - 7 / 16 is fixed, _ so the locus would be a vertical line in the LHP. For y 2 + K p ( 7/ 8) = = the locus would !_ be at Im X = ¢o; for y 2 + Kp (_ / 8) = (7 / 16) 2 the locus would intercept the real axis, i.e., : would I)e at Im X = 0. For smal l er y 2 + Kp ( 7/ 8 ) the locus would have two branches on the real axis. For v 2+KP(7 / 8) =-_ the locus ,: ould be at X=:_. The locus " would go through the origin , crossing into the RH P, at v 2 + Kp (7 / 8) = 0 (the other i : branch woul d be at ), = - (7 / 8)); since V - ' - 1 this can occur only for Kp = -(8 / T) V 2 -< -8 / Y, _ i.e., for sufficiently negative Kp.

} The preceding paragraphs have diacussed the behavior o f the hover root loci, i . e . the / _ = 0 roots . For / _ #0, X has t he s a me form as t he hover root s , but in _ " ' O ,_ terms o f _0 a nd v 0 ; so it is not th e e n tire r es t bu t r at her on ly the order 1 part o f it . i { Thu s fo r ex a m p le, i f Im " %0i s ex a ct l y at a m ul ti pl e of _ y' r e v fo r some # thi s on ly implies t ha t the hover root is ne a .__.Lr that point; the ho v er roo t , ba sed o n y = Y0 + / a Y1 + "'" and y=v 0+ / a v 1+ ... mus t be a small d is t an c e ( 0 (_) if _1 a nd Y l are n ot z e ro ) away f r o m k 0, w hi c h is b ased on _0 a nd Y 0 onl y. T h e o rder _ , _ 2 et c. p ar ts of the roots for N # 0 will be obtained in the analysis below.

Or der /_ Results ' The ord e r / a t erms in t he differential eq u a tion a r e (d r opp ing the common factor i ¢ : of / _) : - -1 4- -1 5 - whe r e A 1 i s a c om p l e x con s ta n t ( whi c h r eally dep e nd s o n _ 1 ' _2 ' et c.). T h e s o lu ti o n to this equation is i _ 1 = R e 11 3 / 0 % e k 0' $ 0 + i !

; 2 x 0

w h ere 8 11 e k 0 _ b 0 i s the ho mo g ene o us s o lu tion . The part i cu la r so lu tion fo r 8 1 has a term proport i onal to A II_ 0 e k O _ O; compare this with the sol uti on for B O: B 1 (constant)Al_ b 0 eX 0_ b 0 = = - - , (constant)Al _ 0 • _ 00 (c o nstant) e k0 _0 T h e n B 1 w i ll b e come ar bitr a r i ly l a r g e comp a red to B 0 i f _ 0 is la rg e eno ug h, wh i ch vio la te s the assumption t h at B 0 and 8 1 are o f t h e sa m e order for a ll % . Th e o nly _ ", _ w a y such a term i n B 1 can be avoided as if it i s required that A 1 be zer o . Recal l _ however th a t the equation f or B 1 i s really a partia l differential equat io n , and A 1 h a s terms l ike _B 01 / _ b 1 an d _ 01" Thus setting A 1 = 0 gives a diff e rent i al equation fo r ' i B 01 in terms of _ 1' the solution of which carries the solution for B 0 out to time I scales of the order of _ -1. In general, the forc i ng terms o n the RHS of the equation i come from the homogeneous so l utions for the lower orders of the B expansion. It is I the nature of th e pert ur bation e xpansion (not of the parti c ular equation b e i ng studied) , that to each order the equat i on for B n always has the same homogeneous solution (in this case e)'O ¢ O). Thus th e equation for B n is be i ng forced by its own homogene ou s solution, which gives rise to s olu ti ons of the form _ b 0 times the homogeneous solution " ( _ 0 e)'O _ O here), unles s the coefficient of the homogeneou s solution i s s et to zero. It i s a fund am ental feat ur e of the method of multip l e time s cale s th at s etti ng this

V

I g .

l _!j / _ . _ , -16- c o eff i c ien t to ze r o gi v e s anot her diffe r en ti al eq uati on, whi c h m a y be u se d t o fi n d th e The method of multiple time scales, a s outl in ed abo c e and descr i bed i n more i b e havio r of _n -1 t o th e next t im e sc al e.

"i det ail i n the l iter a ture ( R e f . 3) , i nv ol ves t h e n th e follo w in g s t e p s.

"I _ a ) Exp and t ] , d / d a!_, and a ll pa r a meters a s ser i e s i n / _ .

i b) O bt ai n t h e p a rti a l d ifferent i a l equat ion fo r o rder / _ n; wr i te it as a n o r d inary d i ffere n t i a l eq u at io n fo r 8 n i n terms o f _ b 0; s ub s t itu te t h e s olution s ob ta i ne d f or 8 1, B2, . . ., 8n _ 1 into th e RH S.

c) F i nd in the forc i ng terms the coeffic i ent of the homogeneous solut i on; this _ c o eff i c i e n t is c all ed t h e secular term. Set the secular term to zero, thereby obta i ning a di ff erent i al e q u a t i on f or B n_ 1 i n terms of _ 1" This i s do n e in order t h a t the so l uti o n be uniformly valid for all time. Usu al ly it i s the behav i or of the solution to longer .... " time s cale s (e . g . , _ 0( _ 0, _ 1 )) th a t i s of interest, r a ther t h an the h i g her order cor - rect i ons to the solution (e. g., B I( $ 0) , wh i ch i s an order D corre c tion to 8 0 ) . So i t i s re al ly the different ial equat i on re s ult i ng from the secular term that i s s oug h t.

F o r hi gher orders, 0 ( D2 ) and above, t h e so l u t ion p rocedure is a bi t more invo l ved ( t here are t h en s ecular terms i n t h e s ecular terms), but t h is is best e x pl ai ned by e x ample.

# Re tu rn ing no w to the f la p equat i on to o rd er / _ (Eq. 8), the secular term (the coeff i cient of e )'0 _ 0) i s, if ) _ 0 _ l _ )'0: _ 0 + + _ 0 + 2 u O U l + 8 B O 1 I p .

-1 7 - o r to o rder _ . T h us t o o rder . the root rema in s the h o ver root; there i s no effec t of advance rat i o or of the per i odic coeff i c i ent s . This i s the case for most y and v , the exception be i ng when y and p are near Y O an d v 0 s uch th at v _^ _ i = )'0" If _ 0 + i = X0 then the pe r i od i c coeff i c i ent s contr i bute to the s ecular ter m ; _ 0 + 1 - _ '0 mean s that YO and _ 0 are s uch that Im X0 = _ t or ; i u 0 _ I _ • _ 8 4 , t I t / -1 8 - - S o t h i s c a se o cc ur s when th e ho ver r o ot has a f requenc y ne ar _ / r e v . ( No t e that O_ - i = ) ' 0 i s no t po ss i b l e be ca use ) '0 ha s been d e fi ne d to h a ve p o s iti ve imagina r y i t part . ) T he 0(1) roo t in thi s c a se i s !

i 7 0 i i :- 1 6 2 - X0 = - --+- i T he secu la r ter m is now

+ T A o+ + 8 O l

x0 _ ( 2_o_1

+ x o +_+_ z 1-o i

'(1-_' - "Y o 'o P )B o

!

,J or i ',\ . ___C Z B o I + 0 70 YO - _ 1 - _ - Zp - _ " / 301 = 0 112) where =_ + i_ i 1 2 Y O U 1

[ , c, 0 , )] r = -' _ + - _ - \_ ' _ - K

The 80 1 term ar i ses due t o the p s riod l c coefficients. The so l uti on t o t he equati on (see App e n d ix II) de p ends on the qua n t i ty t

i " 2V o _ ' 1 s \._e" K -( _/, , .4 , tl + - 113 )

[ I I 4 1

l

• ..2)_

# - i -19- . _ N ow ff > 0 , h a s terms w i th t ime behav io r like '_ -( r d r ± i D )_ I - _ _ ± i D# _ I e -- e ' __ and t h en 8 0 ha s term s li k e

Y--_ ±i, ( _+ _ D . )

le_ .0¢0 - 16 _ " _ 0 =e ; F :, the damping is unch anged by the additional 3 ecular terms, and there i s an 0 ( / _ ) change ', [ in the frequency. If D2 < 0, _ 01 has term s like e-( a r *#D) $ 1 _ -±# ¢

!

so B 0 has terms like _ i there ia an 0(_ 1 cha n ge in the d am pi ng ( both more an d less stablel, while the frequency re m a ins f ix ed at _ / rev. D 2 - 0 m ust g iv e the bo un da ry l_ ,tw een the t w o t ype s of I behavior. Con s ider next the interpretat i on of the quantity D . Con s tant D2 implies, for a given _ 0' V 0 ' and Kp, that _ 1 is a constant , i . e . ,

, " ° " . K : ]

i " " = * _g /

i , i = _ l con atan t) 1 141 | i T his equation represent s a straight line of V versus _ wi th a slope of _ U 1 _ ' 0 / 16 - Kp f ,

>

!

-20- Compare this with the slope of th e line represented by ' I mX 0= 2+Kp _ _ _1 6 1 2: _ v 0 7 0 / 16 - Kp

y o / 1 6 - V o

Thus the lines given by D2 = co nstant are parallel to the line given by I m )'0 = 2" (these can be considered lines of u as a function of 7 ). Furthermore, in this case 71 and u 1 give an 0( # ) perturbat i on from 7 0 and u 0' whic h are such that I m )'0 = 2 ; thus it follows, since a given v al ue of D2 gives two lines of U l versus 7 1, that D2 = c o nstant represents two lines an 0 _ ) distance either side of an d parallel to the 1 are of particular interest, f l_ ' s t the bounda l T D2 = 0, _ , line Im ) _ 0 = _ " Tw o v al ues of D2 and second the maximum p o ssible negative v al ue o f D2. The latter is given by c _ . =0, i .e., 7 1 (7 0 _ Thi s llne runs thro u gh v I = Y l = 0 and ha s the s ame s lope as th e Im )` 0 = _ llne there; thus this llne si mply represents, to 0( / _ ), the Im )` 0 = 1 _ line, and it i s s ufficient to u s e = -. 2 i the po i nt U 1 7 1 0 for the minimum D . Return now to the solution for _ 0 in terms • 2> o f D; it has the character i stics ex pe cted of a pe r i odic system (see Ap pe ndix I). D 0 and very large implies v I and 7 1 very la rg e, which means a root far from Im )` = 1.

. As D2 decreases, Re), remains at the basic value (- 7/ 16) but there is an 0( _ ) change " _ in the frequency, until at D2 = 0 the frequency has reach ed exactly Im k = 1. Th _ ' boundary D = 0 occurs for nonzero v al ues o f U 1 an d 7 1, and so the root h as reached 3 . , -21 - = the I m k --1 2 li n e w h ile t h e hov e r r oot i s s ti ll a n 0( _ ) di s tanc e a w ay . Fo r D2 < 0, _i t h e f re quen c y rem a in s f i x e d at _ / r ev wh il e there i s a n 0( D ) ch a n ge in the d a mp i ng, _. b ot h po s i t iv e a n d ne ga tive . T his t ype . o f c han ge in the s ta bili ty of the sy s te m i s ch a r - _' a c t er i st ic if per i od i c syste m s; i nd e ed i t appe a rs h ere du e to the ter m s in th e se cul a r

)

_. e qua t io n t h at come f rom t he p e riod ic c oe f fi c i e n ts i n t h e eq ua t i on of mot io n; i t i s not ) ! se e n in th e b e ha v i or o f t h e b asic root to 0 O _) . Th ere i s a critical region, bo un ded b y D2 = 0 , i n s ide which the change in the damp i ng o cc u rs. I n m an y p r o blems wi th period i c coe ffi c i ents, t he syste m is unstab l e in s i de suc h a reg i on; in t h is case h o wever t h ere is _.

i the b asi c (h ov er ) damp in g , represe n ted by Re k = - T / 16, w h ic h i s large and sta bl e. __ The change in the da m p i ng is _ / _ D, which is 0( D ) compared t o the bas i c d amp i ng, s o _: _, th e cr i t i ca l reg i on i s a reg i on of stab ili ty degradat i on rat h er th an o f in stab ili ty. W ith th i s discuss i on as a guide, the solution of the flapp i ng eq u ation near I mk 0 = _ wi ll be cons i dered i n m ore detail.

The bo un dary of the cr i t i cal re g ion i s given by D2 = 0, or

K,)" ' 0 / . z ,)

T h e f i rs t t erm o n th e RHS mal_ e stheli n ep ar allel to Im )tO = 2 ' _ the sec ond t erm gi ves the w i dth o f th e region; t h e cri ti c al region Is a n ar row band, of width 0 0 _ ), about 1 !I,: Im _ 0 ffi 2' O ut s ide the crit i ca l / _i on ther e is an 0 0 _ ) change in t _ requency while the real part of the root is un changed fro m th e hover v al ue. In si de , _ e frequency is fixed at _ / rev wh i le there is an 0 0 _) ch ange in the damp i ng. The m ax im u m stab il i ty change occurs at th e center of th e crit i c al r, _ gion , where D2 ha s i t s m axi m u m nega ti ve v a lue , i.e. , at v 1 ffiF1 ffi 0. The root there i s . _ . _ 2 16 m ax -22 - So the 2 _ a x i mu n i _ t ab i l it y d e g rad a t i on ( and enh a n cer , lc,'_i_ i s

a_ = _ , ' , 6 : -__ - " _ - : ' _e +_K p (17)

ma x I " w hi c h is an O t u ) small reduction; the s y s t em r ema i ns stab l e becau s e of t h e l a rge h over d a m pin g. In gener al the root i s given by k--k 0+_r-iD D ( 18) where f o r f ixed V ( i . e., V 1 = 0) h a ve \ 16 / 4 "_ g - _ 24 / + (_ - 4 K Let A T/ 16 = D 0 ' l / 1 6 ) , so T = V 0 + A T ; recall T0 i s g i ven by the requ i rement Im ) _ 0= 2; T must be such that _T i s 01 _ 1 s m ' _ l l , I .e., T must be such that the hover locus i s an _ ( _ ) distance from I m ), = 1 Then 2" Th e critical region L _ undary i s crossed when th e quantity under the _ luare root s ign : is zero, th at i s w h en I i ' / _ " / Zcorner " _ 1 1 91

i

J _

) for a fixed 7 (i.e., for the _ locu s ), or when . ' - 16 = (2 o ) :: corner _ K • I t I

. i

! for a fi x ed _ (i. e., for : he y lo _as ). Then the r o ot l o cus is g i ven by •161 2 _t t _ - - (# / P c o rne _ 2 (2 1 ) or , - 16 + _ - 1II + - 4 - 1 ( 22 ) ) , = . Z . l ( 2_ 4 . 4) d / 1 (Z Kp) 2J ( _ly / A _ corner )2 t i

i

For Kp = 0 the Pe expre ss ion s simplify to i The (- I ) in the last term o f ). become s ( i l) for # > P c o rner o r 4 7 < A Y c o rner.

I P corner = 16 2 _ _ 1 6 o i :er and o

i

Furth e rmor e , for _ << _c orn e r or A y > > A T cor n er, i.e., far outsid e t he critical re g io n , th e expre ss ion for th e root be c o me s -2 4 - whi ch is ju st a n 0( _) e xp an s io n of t h e ho v e r r oo t wh en Im k i s near _ J re v . F i nal l y , the s olution for _ 01' with D 2 > 0, is /

e

[- ( _ ( '0 T O )YOi ) ] e - i D'_) ' _ with _ . an d D gi ve n ab o ve; a s i mi la r expr es sion m a y be o bt ain ed f o r the case '_ D 2 < 0 (see Ap p end i x I I).

These r e sults h a ve been u sed to pl o t the root l o c i f o r varyi n g / _ an d y ; the :, r e s ul ts o bta i ned s o fa r ar e valid f o r s mal l / _ . F i gure 1 sh o ws typic al r oo t lo ci f o r i ..

- va rying y , with / _= 0 a nd / _ = 0 .1. T h e beh a vi o r o f the h o v e r lo ci { p = 0 ) h a s been i described , ab o ve. The h o ver l oc i cr os s ImP . = _ t for y = 37.7, 13.9, a nd 5. 2 f o r !

i

_ Kp = 1, 0 , an d - 1 re s pectively; f o r u s u al r o tors th e n the I mk = _ t cr i t i cal region is _ .

! li ke ly t o b e en c o untered onl y if Kp _ 0 . The poi n t D on the h o ver lo c us (K p = 1, i n ] Fi g . 1 ) i s where the l o cus crosses Imk= _ . A s y increase s , since this po int is at th e center of ; h e cr i t i c al region (A T / 16 = 0 ) it receives the maximum st ab i li ty chang e , a : d so i s pulled o ut t o the po in t B. In term s o f t h e y locu s , a s y i ncreases and the hove r locus near s Imk= _ , the ro o t ha s an 0(#) change in the frequency, p ul ling the lo c u s t ow ard I m k = _ . When th e lo cus cr o sses i nto th e cr i t i cal region the f reque n cy ha s j u st reached _ / rev, and t h e r oo t l o cus is a t th e p oi nt A. F o r sti ll l arger _ the _-'. -25- 5:- f re quen cy re m a i n s fix ed w hil e the rea l p art o f on e root decreases and that of th e ot h er inc r ea ses. Whe n 7 re aches t he val u e for wh,_. c h th e ho v e r r o o t h a s a frequenc y of i . _ / re v , t h e l ocus i s at t h e c e n t er of t he cr i t i c al re gion ; t h ere th e r oo t s h a v e t h e ir maxi - , mu m st a bi l i ty c hang e (w h i c h i s 0 ( _ ) so t h e lo c u s is at t he poi n t B. A s 7 i n cr e ase s _ more , t he l o c u s m ov es to war d t he o th er bo und ary of the cr it ic al re g i on . Th e lo cus _ reac h es tha t boun da r y a t the po in t C , a nd f or st il l l ar g er 7 the f reque n cy is no l on g er # _ fix ed a t __ / rev; rather the re al part of t he ro ot i s t he s am e a s the hov er v al ue, wh il e i there i s an 0( g ) ch ange i n t h e f r equ ency w hi ch decreases i n s iz e as 7 in c re a s e s. Wh en _ 7 / 1 6 i s n o l o n g er 0( D ) , t h e l ocus is again ide n t i c al (to 0( D )) t o the hov e r lo c u s.

_ g F igu re 2 s h ows t ypi c al r oo t l oc i fo r s e ver al 9 / an d v ary i n g _ . T h e ci rc le the ._ locus star t s from i s the 7 locu s for hover _ = O) and the appropriate Kp: The 7 for _ each l ocus ma y be f ound f ro m ReX at / _ = 0 , s i nce fo r th e h o ver r oo t R e X -- - y/ 16.

A s # incre a ses f rom zero, for the roots n e a r Im k = _ t h ere i s an 0( _ ) ch ang e in the f r e q ue nc y p u ll i ng th e r oo t t o w ard I m k = _ , w h i l e the d a mpin g re mai n s fixed at the h ov er _ va lu e. T h e locus reaches I m k = _, i.e., cros s es the boun da ry of the cr i t i c al re g i o n , moves t o the l e ft (i n cre a sed s tabi l ity) and the other t o the r ig h t (decre a ses st a bil i ty } .

! when # = #e°rner" F°r l ar g er #' the f re qu ency re ma i n_ fix ed a t _ / rev w hil e ° ne l °cus i A gai n , this is t he c h aracterist i c beh a v i or o f roots o f a system w ith periodic coeff i cient s .

Order p2 Result s • In o rd er to find the roots to 0( _ 2), it i s first necessary to complete the 0(#) s olution. After the s ecular ter ms have be en removed, the equation for B 1 becomes, for Imk 0 _ _ : -26- 2 7 0 Y O _ + __ b _ _ . B1 + ( v 0 2 +_ 'KP ) ¢1 b_ b 02 1 8 b_ O 7 o = - _ - ¢ 01 ((k0 + 2Kp) s in _ )0 + c os $ 0 ) e )` 0 5 0 + conj ug ate ( 24 ) Th e s olutio n o f thi s i s ¢ 1 = Re [ ¢ 11 ( $ 1 ) e) ' 0 @ 0 + _¢ 0 1 (A1 s i n _ )0 + A 2 cos _ 0)e )`0_ 0] J (25 ) w h ere i k0 + 2Kp - 2i Im )` 0 1 " r ()'0 + 2K p ) 2i Im )` 0 - _ .

1- (2Ira 1 (2Ira A 1 - ),0) 2 ' A2 ffi ),0) 2 or recall in g t ha t ¢ 0 1 = ¢ 02 e ), l _ b l' the soluti on so f a r is i " " , _ ¢1 = Re ¢1 1( _ )1) e k0 $ 0 + - _ - ¢ 02 (_ 2) (A 1 s in _ b 0 + A 2 cos _0 ) e )`1 _1 e)`0 _ 0 (2 6)

[ , o ]

t Th e ord e r N 2 terms in the equatio n of motion are (droppin g the commo n factor ' _ o f _2 ): -27- 52 70 5 + Kp f _2

: - 2 _ b 0_)_ b I ¢1 + --_ B 1 + + -- sin _ b 0 _ f _l { 8 _i 6 _o

J i + Y O Y l + -_-cos _ b 0 + Kp _- + K p ,_- sin$0 _I

+ _o + 2 _o 'r--_ o + V + v_n_ o ° o

_ b l2 5_ b oa_ b 2 b$'2 + + "_-sin_0 _0 ?

+ + 2 Y O U 2 + c os _ b 0 + _ sin 2 _ b + K p T + T sin_ b 0 + T (sin _0 _ " [(2 T0 _ 5Bll !

_ , = Xo+8 / _ 1 i

I ,o o)l ]o,0 o l

+ ' - ' o +_- : o o _+ (' : + (r r +

_0

2%v 2 , - _ o o _ . i _ 2 * 0 + Y l + _0 '+ _ -

_ I _ 0 + K p (_-_ + -_- s i n $0 + -_.- (sin$0) 2 ) ) , 0 2 e kl$1 e )' 0 $ 0 • + "_- B 0 2 _ ' 1 + '_- + "_-sin e 0 ) (0- 0 A1 - A2 ) sin$ 0+ ( k 0 A2 +A1 ) c o s $ 0)

, o [( , ,,,o

i

A 1 . , "_ =s in_ b ( Al S ln_ 0+ A 2c o_ b 0 e X l¢lek 0$ 0 , a te _ .......

-28- Considering n o w Im )'0 _ 1 (already have assumed Im )'0 _ ½)' t h e secular te r m is _ _B ll_ )' 1 _ I = _ 2 i Im )` 0 E 0 e )` l _ b l ( 28 ) \ _l_1 [ ' _l_2 + )`2 + # w here )'2 i s the 0 1 _ t 2) term i n a n ex p an s i on o f the hover ro ot , i . e . .

)CO +ti)`l +"2 )` 2 - "_16+I q /y 2+KP8 _- 16 +0( I t3) t Rega r ding t h is i s a n o r d in a ry d iffe ren tial e qu ation fo r El l i n terms o f $1 ' t h e R H S i s a + consta n t times the homoge n eou s s olution. I n order that th e s oluti o n be uniformly valid, i.e._ , that Ell be no more s i n gular than E01, the s e c ular term of _this equation mu s t also be set t o ze ro. _ he re s ul t is a diff erenti al equ a t i on f or E 0 2 in terms o f _ 2 ' wh i ch ; ,, wi ll gi ve t he r oo t to 0( bt 2 ): i

_E02 (_ _0 (7)2)`0A1 (7 ; 1 ) > ! - _+ _+(_r + 7 +x_ _ / _.,,_o_ _o_ :° <_> i

" 1 ' i T h u s th e r oot i s i )` = X0 + bt )` l 2

k

= X ho ver + h i 21 ,V02" _" KP + : _ K _'_ 2 + _ / 0 1 4 Im _ t 0 ( 1 ' ( 2 I m X 0 )2 ) Kp I' 6 i'_0 / To o r d er _ t 2 th is resu lt is

# .z.;2 (1 . it _ 8 _ " icr,,+

• x:-_+i + < _+_ xr" _s _ _ 2_ _ _3 o _

i " " K p •

() t) 8

_a '' " t ; - 29- T h u s t o o r d er # 2 the r oo ts fo r m o s t _ / 0 and Y 0 ( a w a y f r om Im k = _ o r 1 t ha t is) ar e just the hover root s with a n 0(# 2 ) change i n the frequency. There are two effects of # ; the fir s t corrects the term _ / 8 Kp in the hover fr e quency to properly ac c ount for , t he av er ag e o f KpM 8, i .e. , m ultipli es thi s t er m b y th e fact o r ( 1 + # 2). The second " ; effe ct, t h a t in th e la s t t er m of t h e f reque n c y, is en t ire ly d u e to t h e p eriod i c a er o dyn a mic i c o eff i cients ; th i s is t h e f irst effect of t he p eri odi c coeffic i ent s seen in the mlal ys i s , : e x ce pt f or the critica l regi o ns near I mk = _ . Typ ic al ro ot lo c i fo r v ar y ing # , con- s tru c ted from Eq . 30, ar e s hown in Fig . 2. The s e ar e the lo c i that are not n e ar Im k = _ o r 1; th e frequ e ._cy c ha_ ge i s small even at # = 0.5. Equatio n 3 0 may al s o b e us e d for the bra n che s of th e root lo c i o n th e real a xis wh e n the qu au tity u n der th e , . squar e ro ot s ig n i s ne gativ e ( i. e ., for _ , l arge enough ). Th e r e ar e two real r o ot s then,

i

I the (+ i ) in the f requenc y b eco m ing ( _ 1). A p o int on the l ocus of s pec ia l in terest is where c, ne br a nch o f th e l ocus o n th e r eal a x i s cr o sses in to t he RHP , i.e. , be c om es un st a b l e ,, - - ( s ee Fig. 1). The criterion for thi s diverge n ce boundary is that k ffi 0, or 2 2 - 8 _ Kp + 4Kp 2 , T o o rder #2 t hi s be c om es (s i nce _ / 8 K p must be an 0 ( _ 2) d i stance f rom 2 ) 16 + 4 ' . _ e ,__. N _ , , " -3O- _ c r = _ u 2 + 2 1 6 16 + y - - (I + / 2) 8 _K p 9 _ 2 ( 31 ) 1 6 + _ I The ef f ect of / _ on the Rt l S (due t o the perio di c coeff i cients) dominates that on the LHS w : ( d ue t o the average of Kp M_ f o r all va l ues of 7 and y . Th u s the cr i tic al v al ue of ne g ative Kp, b eyond wh i ch the lo cus lies in the LHP , is actu all y i ncre a sed by increas- , ing _ . The criter io n f r o m the hover case i s c o nserv ati ve then; this is the oppo si te of , th e c o nc l usion th a t would h a ve been re a c h ed f rom a c o nsider a ti o n of t h e av erage d [ ": coe ffi cien t s only .

If _ 0 + 2 i = k 0 then the periodic coefficients c o n t r i bu t e to t h e sec ul ar te rm; ' : _ 0 + 2 i = 7, 0 me an s that v 0 and 7 0 a re such that Im kO =1 I i or !

Thus there will be an 0 _ 2 ) cr i tic al region where the hover root is near Im ). = 1 The 0(1) r oo t in this case is 7 0

x 0 = i

J

J

% , _, - 3 1- a = The s ecular term f or the equatio n f or / 3 0 2 is then

11 , o , o 1

o - o - 1 / 1 , o

, Kp /T 0 _ 2 ] - i The first term on the RHS is the same as the RHS of Eq. 28 (with Im X 0 = 1). Unless k 1 i s real, so th at _ 1 ffi kl' the sec ul ar equation for _ 02 will there f ore be exactly the __!

I sam e as above (Eq. 29). Recalling that _

. _

' 1 ' 1 1 . ' 0 ) : _ " _ '1 = - G (_'0i+ImKp) _ .0 - p01_l = "" _ T 1 + i _'_ _I m + KP _ 0+ lj0 v l : { " Im k 1 = 0 r eq uires 2 U 0 U l- T ' _ Kp)=0

"i ,_ ( , 0 _

{

I This i s recogn i zed as a line of con s tant Im k; since it goes through Pl = _ 1 "O, it i s th erefore s uffici e nt com mi d e r i Ju s t th e I m _ . - 1 line to 0(_ ), and It is to only the cas e I _ I = _ I = 0 ( S o)'I = 0 also). Th e n the s ecular term in Eq. 32 i s

-

i _ , _ _ , o _ _ , + - +iKp s i _ +i - 3 i 6+ _ jp+ 2 = 0 1831 =--- 0 . . - _ ......... _ "_ ....... _...... _ . - ........ _ _ - , - T _ , . ._ ..

!

-3 2 - The solut i on d e p e nd s on D2 _ ' 0 _2 _02 - "_-Kp + 4Kp 2 _0

= _-_ _-_ o _ + i_ -_P

- + _-_: - > K + - - 2 K _34_

Th e be ha v i or o f t he so lutio n ne ar t h e crit ical re gio n i s simi la r to t hat n e a r I m _'0 = ½ " The boundary i s giv e n by D 2 = 0, which g i ve s a narrow ba n d, of width 0 ( D 2 ) her e ( as oppo s ed to 0 0 _ ) for th e Im _' 0 = _ c a s e ) , about Im k = 1. Out s id e th e c riti c al r e gion ( D2 > 0) , th e dampi n g is th e s am e a s th e hover root mid th e r e i s a n 0 ( D 2) c hang e in the f req u e n cy ; at the bo un dary o f the critic al reg io n th e f re q ue n cy re a ches 1 / r e v. I nsi d e (D 2 O q _ 2 ) , , , the cri tic al region < 0) the f re q uency i s f ixed at 1 / rev w hil e there i s an J change i n the d a mping , wi t h one roo t becoming l ess st abl e a nd t h e ot h er m ore .

- The bound a ry o f the _ rltical regi o n (D2 = 0) i s _' 0 i , T hi s i s a l ine p ar al l e l to Imk = 1. The ter m s on t h e LHS give a l ine par allel to I m k 0 = 1; the f irst two terms on the RHS giv e the correction required due to the 00 _ 2) change in t h e frequency found above (Eq. 31) , so the l ine is parallel to Im k - 1 for the basic root to 00 _ 2). The last term on th e RHS gives the width of the critical region; so the critic al region i s an 00 _ 2) band ar o un d I m k - 1.

, ' , 4 i+ ': -33- f r'- _ D2' The maximum stability change occurs f or the maximum negative value o f i.e., at

i

- K " _ V 0 U 2 U 02 - 8_KP+4KP 2 7 0 } - =- _ + Kp _ (36) J ¢ 0 _ 2) which is just where Im _ , = 1 for the bas i c root (the hover root plus the correction _ to the frequency). At th is p oint the root i s - 34- L et ( A) ,/ 1 6) = _ t 2 _ ' 2, / 1 6 ) , s o ) , = Y 0 + A T ; reca l l / 0 i s giv e n by the req ui rement I m ) '0 = 1 ; _ must b e su c h th a t A7 '/ 1 6 is 0_ 2) small, i. e .,_ ' m ust b e s u c h th a tthe hov e r lo c usis a n 0_ 2) dist a n ce f rom Im k = 1. Th e n . . _ 2 D )2 = \1 6 \ 16- - C2)2 wh ere - -_- Kp + T 0 !

( T 0 _ 2 2 Y0 4Kp2 > C 1 = - \ 6 / 1 2 + K p - _ ,o c,o , )' ] ' C 2 = _ '2 + 9 \16 - "9 "X ' _- 2K .

"l T h e cr i t i ca l reg ion boundary is cr o s _i e d w h e n ( _ 2 D)2 = 0, that i s w h e n 2 2 16 x 16 - K 2 2 = / _ c o r n er = C1 i C 2 = / _ 1 ' _ 2 (4 0 ) f o r t h e _ l o c u s, o r w h en 16 16 y ,., 16 ' 16 ; . _'_ - Kp

I •

for the y l ocu s . The n the root loc u s is given b y ). = - + i - i kle l _,le " 1 - (42) |

I Kp) _ ( _ 2 // _ 12)(1 _ 2 // _ 22) ' • i

i -35-

I

i o r i X = - _' + f -J 16 2 J C 21 - C2 2 v f (A _ /A _ I - 1)(A _ /A _ 2 - 1 ) (43) t , i The (-1) i n t h e l ast term i n X beco m es 1 _ : 1) inside the critical region. These results have been used to pl ot i n F igs . 1 an d 2 typi c al roo t l oci near Im X = I f or var y i ng _ ' and / _.

Flap Rate Feedback The use of flap rate feedback, KR = 0 , does not change the behavior of the 4 solution qu al itat i ve l y. The hover root becomes ' _ '0 ffi "_0 (1 R) + 1 2 + T Kp "L 16 (1 + KR ( 44) and there ar e c, : l tic al reg i on s about Im ) ' 0 " _ and 1 again. T he c ritica l region boundar i e s i - and s_ ablllty degradation depend c m KR now. It is nece ss ary that KR > -I f or the hover root to be s table, but KR > 0 will be the us u al c ase an yway.

!

- p Plane The re s ults of th e sm all u analy sis may be u s od to plot l i n es o f con s tant R e _, and t I m _. on the T - / _ plan e . Typi cal re s ult s are s hown i n Fig s . 3, 4, and 5 for I _ = 1.0 and Kp = 0, 0. 1, and -0. I re ape_ dv e ly. Th e cri ti cal reg in n m appear in the T - / _ pl ane as reg i ons in whi c h Im k I s c o uts nt ( _ / r e v or 1 / rev); th e y ar e indi c ated in the fig ure m by the c i rcled value s of lm _ . (th e region wh e re Im k - 0 is where th e r e are two real roots, not a cr i ti cal reg i on). The n flgur u a, " e in terpreted a s f ollow s . A horizontal l in e 15 a l ine of con s tmnt _ , and eo a m _ varie s i t giv es the corre s po nd ing v al u e o g Re _ .

i .

-36 - an d I m k as a _ root lo cus d oes. S i milarly a vert i cal l i ne i s a c o nstant / _ lin e , and so g i ves k as a function of y just as a y root l ocus does. For example, co n sider a h o r i zo n tal l i ne i n Fig. 3 (K p = 0 ) _ A th y = 8, i.e., 7 / 16 = 0.5. As _ incre a ses, the li n e re in _ a ins p arallel to the Re k = constant l i n os so Re k re m a i n s fi x ed at the hover ?

value. The Im k = _ region comes closer to the horizontal l i ne a s / _ increases, which m e ans that I m k moves toward _ / rev. Eventually th e con s tant y l i ne crosses into t h e I m ) , = _ reg i on; then I m k is fi x e d at _ / rev wh i le for each po i nt in the reg i on t h ere ar e two value s of Re k, giving the damp i ng for the t w o branch e s (one more and one less s table than the hover root ) . Th is behav i or is just th at s een al ready i n the / _ loc i (F i g. 2 ) . F i gure s 3, 4, an d 5 may be co m pared with simi lar ones in Ref. 2 , wh i ch w e r e co l, s tructed from n umer i cal c al culat i on s ; o n t h e bas is of this co m p aris o n , t h e ,

2)

anal ytic re sul t s are quite accurate up to _ ffi 0. 5 or s o. There i s some di s crep- ancy be tween th e res ul ts for the Im k = I region however, part i cularly w i th K _ p = -0 . i, _ .

_ although the change in scale ( Re f. 2, shows re s ult s out to _ = 2.5) exaggerate the differ- ence. For p exactly 1 the an alytic result s ind i cate no I mk = 1 cr i tic al region if < 0 (for _ 0 = 1, Eq. 35 shows that th e cr i t i c al region at T0 / 16 = 0 ha s z ero w i dth unless Kp = 0); but only a s lightly larger I ,, (for e xam ple v = 1.01) i s nece ss ary to get a sizable cr i t i c al reg i on with Kp = -0.1 ( s ee Fig. 5). The analyt i c r esul t s show the !

v = 1 case is a very sensitive one for s mal l y, and it is unl i kely that a nu m eric al e al cula- _ t i on w o uld treat th e case accurately, 0[ course an actual rotor will al way s have p at | least sl i ghtly grea te r th an 1, s o the nu m er i cal c alcul ations w ould be reliable then; furthermore, the di s crepancy m ay al m o be an indication that tor very s mall y the anal ytic res ul t s ar e act v al id out to as large a p as they are for mo re retmoaabls y. [ t

! !

-37- In any case this discussion illustrates th e kinds of problems that r. _ ay be hidden in a purely nu m er i cal solution; ' _ .heycan only be fo u nd and s tu d ied by _ n al yt i -_ procedures (which at least tell where to look for problems).

I Reductz " s to MathieuWs E q uation The equation for rotor flapping stability may be stud i e.] by con _ ,erting it to Mathieuts equation; Matbieu's equation is an equation o f the form d2 +(a_2 q cos2z)y =0 dz 2 It i s the classic example of a different i al equation with per i od i c coeffic i ents, m s d the functions s _ tlsfying z t (the purely pe r i odic ones are called Mathieu functions) have be e n w ell studied and documented. To tr a n sf orm the flapping eq uation to Ma th ieu l s equat i o n it is first nece ss ary to remove the _ ter m ; fl u s is done by the su be_ tution .r _ =y e i t This is equiv al ent to separating out the hover damping from the solution. Then th e { classic i nstability regions of Mathleuls e _ uat i on (for cert ai n values of a ver R us q) are just th e crlt / c a l regions wi th out the large (stabill _ .,Ing) hover d am ping. To get th e flap- ' p i ng equaUon i n tt - e r eq u i , 'ed form it is al so necessary however to noglect al l O _ 2) ter m s; w hen this i s done o n e obtain s (w ith Kp ffi KR = O) i .

IW i - 38- and 2z -- _ +tan -1Z Thus using Mathieu's equation implies only an O(D) analysis. There are of co u rse classical techniques for handling the more general equation with periodic coefficient s , of the form d2 y + f(2z) y =0 dz 2 wh ic h is c all e d H i ll' s eq uation; f( t ) is a g e n er al pe r iodi c fun ct ion wi t h p eri od T = 2_ .

T h es e tec hniqu e s c ould be us e d t o s t u d y the f lapp ing e q u at ion , t o al l o r de r s in / _ .

Howev er t h ese ge n era l tech nique s are al l unsat i s f acto r y in t h at t h e y t end t o o b sc ur e the physics o f the syste m be in g s tu died, both because so m e tr an s f ormation is necessary to arrive at the required f orm o f the equation, an d be ca u se us in g st an dard solut i ons or f ormal c alculation tec hn iques m eans the great a m ot m t of information gained in the process o f deriving the solution is lost. Fur the r m ore, the classic treatments have only con- s id ered a s ingl e de g ree off ree d om sys t em , s o t h att h e y are no timmed i ately app l icab l e to m o re ge n eral pr ob lem s .

The Smal l 7" Case _! , ' j i Consid e r fo r t h e small ) _ c ase t h e fla p p in g e qua t ion wit h bot h p r opo rt ional an d :_ _ ra t e f ee dba c k . No w / a i s ar bi trar y , s o the g e n er al e qu at ion is c on s id ere d , of t h e fo rm ¢ _ + Y 2 _ = T F.(M_ - KR M 0 ) _ + ( M _ - K p M_ _] (4 5 ) - 39 - w here the a erodynami c coefficien t s a re func t i on s o f D and _ b ( peri o dic in $ ) , a n d include the effect of the reverse flow region ( see Eq. 1 ) . The small parameter is the Lock number 7. The p e rturbation technique to be used is the method of multiple t i me , sc ale s. Th e s o l u tio n w ill b e e xamined pr im a r i ly t o 0( 7 ).

: Z ero Lock Number In the a bsence of a er od yn a mic forces, i.e., the limit 7 = O, the solution is

=Re eiV)

s o the r oo ts a re _ ' X = i V (46 ) ,_ a n d it s co nju gat e ; i. e . , th e s olution i s an un damped osc illa ti on at t he rot a t ing na tur a l , f req u e ncy of t h e flap motion. W i th no ae r od yn a_d cs, there i s of c o urs e n o effect of / _.

L Thi s res ult fo r th e r oot ag rees with th e low l z res ult fo r 7 = 0 ( se e F ig. 1 ) .

x , E xpansion in _ ' :_ i U s ing t h e met h od o f mul t ipl e t i mes sc al es , w rite :, :

: '

• • __.,_ o ,. _ : _ _ , , o t . -. ...... T .----_- - _.: . : . - '_': -_.:_ ' _ ,, . . ""_. ", -40- and th e n expa n d f 3 a n d _ o/ cp_ b a s series in y : = _ 0 ($0' $1' "'" ) + ' 7 B1 (_0' _1 .... ) +"" A l s o expand t h e f r e e para me t e r v as a ser i es: U = U O + ' Y U l + . . .

Order 1 Results To o rder 1 t h e e q uatio n is _2 ' -- J 30 + Y 02 _0 = 0 (47)

a ¢ o2

The s o l utio n o f t his e quatio n is _ _0 =R e 1 ...) e th e r o ots are ) ' 0 = i u 0 (48) _ ' _ and i ts c o n j ugate. Th i s s ol ut i on i s of c o urse Just th e _ = 0 li m i t. _ t : O rder _ Results i T h e order _ , terms i n the d i fferent i al equat i on give (dropp i n g t h e common factor t i : t i

of y ) : I

- 41 - o 2 o 2 2 0 l +vO _1 = -2 _O-2VOVl_O

°

I _01 iPO_ b O = - 2iV 0 _-_-1e + [-2 U O U l+(M E-KRM_ i U o + ( M B - K p M_] f 301eivO_ b O + co n j u ga te (49) " Now exp a n d t he a er o dynam i c coeff i c i e nts as c o m pl e x F o ur i er series : f ' E n i n_ _:_' ME = ME e L C whe r e " 2_

l--J" e-in*

$ / M_ = 21 t 0 ME ¢p_ :_ _ a n d sim ila r l y fo r M_ a nd M o . , } If - u 0 + n _ u 0 , t. e ., u 0 ¢ n / '2 f or a n y i nte g er n (t. e ., Im _' 0 n ot eq ua l to a \ s m ul t ipl e of {_ / rev ), t h en the s ecular term ha s n o co n tr i b u t i o ns fr om a n y ha r m onic s of _;_ _._ th e a erody ha m i c c o e ffi c i ent s e x ce p t t h e zero th h ar m o ni c ( t h e a v er age o v er t h e a z imu t h); i'_: set t i ng the sec ula r term t o zer o g ive s .a _, B@ 1 _ - K RM + M - Kp M _0 1 = 0 (50) _ + i U l _ _ 0 _ 0 -42- The solution of this equation is

f i / 0 0 0 ) ]

_01 = EO2(_ b 2... ) e (51) I - a nd t he solutio n fo r _ is

=Re 802 e %_ _ +o(v) ( 5 2)

{ }

T h e r oots ar e t and i ts con j ug a te. Be f ore p ro c eeding f ur t her, it is no t ed t hat for the par t icular a ero- dynami c c oe ffic i en ts c o ns id ere d here man y of th e h a rm oni cs a re z er o. A c tuM c al c ula- tion of tbe harmonics or symmetry arguments can d e m o nstr a te th a t

_ , l c 2s 3.c

! : M_ = M_ = M E = .. . = 0 .{ }i 0 l s 2 c _ , _ : M E = M E = M E =... = 0 1 53) t !_ l c 2s M 3C = 0 ' _ M e =M e = e = ' '" ' _ 0 _i (the s uperscr i pt s re l a t e t o a cos i ne / sine Four i er ser i es) . U s in g t h e f act that M B = O , _,_, ,_ : a nd a ls ot hat t o Off) y = y 0 + 7y I, t h e ro otb ec o me s i ¢ _ 7 _ 0 _ i k ffi M - +, 1 + 154) ) _"_ "_ .... ........ t ¢ % ...... t ......... =" :_"" * " - 43- -- F lap r ate f ee d ba ck on ly e ffe cts ( to 0 0 ' )) the d am pi ng ; th e re al pa r t of ) , may b e writ ten L

[

0 0 The r atio -M_ / M 8 de t er m i n e s th e re lative effect of K R ; this ratio is shown in Fi g. 6 _ as a function of g. This parameter is a positive number, which varies little with _ ( f ro m 1 at _ = 0 to _ a t _ = _ , w ith m ost o f the ch ange b e l ow _ -- 1). The n e g at ive 0 0 of t hi sr a t io gi v es a cr iti c al K R , s in ce i ti s n ece ssary t h a tK R > K R = M_ / M 0 • c ri t in o r de r t ha t the system be s ta bl e fo r s ma ll y . K R > -_ ins u resst a b ili ty fo r all ( to 0 ( _ ) a nd wit h V 0 # n / 2 ); fo r very sm all _ the cr i ter ion i s KR > - 1 , w h i ch ag ree s . - _ i withthe re sultf r o m th eh ove r r oot ( Eq . 44) . Fo r K R = 0 the rootis "

,:. x=- SM +i u + 8M (a 5)

-e- j : _ : The ae rodyn ami c c o effici ent s (-8M a n d ( 8 M are a l way s po s iti ve ; the y h a ve t he ;,, v M ue 1 f or N -- 0 an d a re a s ympto t i c to 8 / 3 _ N a nd 16 / 3 _' _ res p ec tiv el y for l ar ge _ ; _ ii" t hese co e ffi c i ents a re sh own in Fi g . 7. To o rder the r oo t is _ ) ,=-1_66 + i u 1+ 16 - _" (I + 0 0 21, ; w h i c h agr e es wit h the O ff ) e xpan s i on of th e sm al l g res ul t s f o r ) . (t o b ut w i th _: ,: _: Im A0 / _ or 1 ). !)

T he roo t l oci for va r y i n g _ an d v a ryi ng _ a re sh o w n in F i_I . 8 f o r Kp = I , _i KR = 0, an d u = I; the l o c u s f o r Kp = -I is o b t ained b y reflec ti ng th i s l oc u s ab ou t the

i" !

"'_ ' _ f ._ , ................ _ F _ .-_-- ......... . _ . _ , . _- "_r- _ , ....

- 44 - Im ) , = v l ine . The locus for Kp = O i s difficul t t o plo t since Im )_ = v for a ll T and / _ ( t o orde r 7 ); it may be v isu a lized by pro j ecti ng t he Kp = I loci on t o the l i ne Im k = v.

These loci should be compared w it h t h e small _ port i o n s of the cur v e s i n Fig . l , which are for small / g. In F ig . 8 , the _ locus for a gi v e n / g star t s ou t a t )k = i v always, and l is a s t raigh t l ine w it h slope Im _ K p I 0 0 R e ) , Y _ M_ / M 0 wh ich v a r i es fr o m -Kp / V t o -2Kp / Y fo r D f r om 0 t o _ . Th e s t e p s iz e on th e _' lo c u s, for a unit change i n _' / 16, i s M 8 M , w hi c h va ri e s f rom + 25 8 _1 4 ( Kp / Y) 2 for D from 0 to I to ® respe c tively , Th e D locus for a give n _ starts out v ertically f r o m t he _ = 0 li ne, an d is as ym ptot ic t o t h e _ = ® l i n e, wi th th e s t e p s iz e o n th e l ocus for a un i t ch a n g e in D i ncreasi ng as D increases. For reason a ble D the locus does n o t va ry mu c h f r o m t he s m a ll _ resu l ts. An 0 (' t ) an alys i s can only o bta in t h e s lop e o f ?

t he _ / l ocus at _ / = 0, s o t h e lo cus i s a str ai ght ll ne, as fo und ab o ve; t o fi n d t h e curva - _ t ure e f fect it is necessary to go to order _ , 2 . T h e si g ni f ic an ce of the curvat u re (0 0_2 )) m ay be jud g e d f ro m a c o m p arison o f the _ ' l oc I of Fig. 8 (al l _ , s m al l _ , ) an d Fi g . 1 i (smal l _ , all _ / ); on t h is bas i s the sm all _ results shou l d be li mited to _ / 16 = 0. 2 !

• t

-45- o r less. O n the basis of neglect of the curvature e ffe cts _ lone the results might be a ccept ed t o hi g her _ , , but t h e 0 _) results will a l so be li m it e d by t h e effects of t h e \ critical region, which will be examined next.

,. The di sc u ss ion of t he 0( _) analy s is h a s so f ar o nly be en c on cer n ed with the basic roots, me an i ng the r oo ts awa y from the i nfl uence of a critic al re g io n . In this problem, the criterion for being away from a critic al region is that v 0 _ n / 2 for any integer n; this ma y be wri tt e n v _ ( n / 2) + 0 _) , i.e., the rotatin g na tu r al f reque n cy m ay n ot be a distance of order 0( y ) from (n / 2) / rev. Since V is aLnost always just slightly above 1 / rev ( V = 1. 2 ", vould be ve ry lar g e for a rotor; it would r eq uire very st if f blades an d thus al so m e an high blade loads) thi s criterion is seldom fulfilled, and the cr i tical regions may be expec te d to dominate the root loci behavior for small _ . Furthermore, if Kp } is large enough positive or negative, the basic locus will also cross Im X = 3 / 2 or _v!\ , _ for _ / 16 still sm al l ( s ee Fig. 8), so these critic al regions may affect the loci even if i the I m X = 1 region does not.

L If - v 0 = v 0 for some in teg er n, then the h ig her harmon i c s of the aerodynamic 'i coefficients contribute to the secular equation, and there arise cr i tic al region s with _ behav i or of the root loc i s imi lar to that enco un tered already i n the s mal l N case . The _, criterion - V 0 + n= v 0 m ean s u 0=n / 2, i. e. , V = _ 0 + T v l +''" is 0( T ) from a multiple of _ / rev; the only cases likely to be encounteref for rotors are v 0 = _ , 1, an d 3 / 2 (n = 1 , 2 , an d 3). O nly the case KR = 0 will be _ : _ ider ed for now , and u s e will _ al so be m ad e of the fact th at M _ = 0. Setting the seculm term in Eq. 49 to zero gives _ then -46 -

, KpM )

+ _ ' M]}+ M -KpM BO1 = 0 (56) I where recall that Mn i s the nth har m onic in the complex Fourier series expan s ion of the aerodynam i c coef f icient, and n i s here given by n = 2 _ 0. For this equation = _ 1 n _ Kpl _ e _ 0 D2 + -- M 2 1 M _ -i (57)

1 2 U o ( 2 U o )

Rec al l that for the aerodyn am_ .c coeffic i ents considered here, either the cosine or si ne term f o r each harm o n i c is z e r o (Eq. 53 ) , s o this par am eter may be wr i t te n Kp M 0_ 2 _ 1 n 2 Fo r n odd, M _-i Y o l_ = _ . _ _ - v 0 M O=- _M 0 ; fo r n e v e n , n M_ i / 'ns n c) _ n c MB - iV 0 = - _ L M_ + v0M _ and = t M 0 • T h e boundar y of the cr i tic al reg ion is give n by D 2 ffi0 ; ou ts i de the reg ion ( I )2 > 0) th e re al part of ), re m ai ns at th e bas i c • r o ot value 0 '/ 2 M ) while th ere i s an 0 0" ) c h ange in t h e freq u e n c y ; in s i de ( D2 < 0) i ! th ere i s an 00_ ) change ( both pos i t i ve and n egat i ve) in t h e re al part of ), w hil e the f re- !

; qu e n c y r e m ains fixed at (n / 2) / r ev . Co n stant m ean s + M = c on st an t, o r i D2 (2 Y Ot l I Kp _ ' ; _ ... _V = -Kp 0 I i _ " _ U l = 2 U 0 M0 I ,

i '

i

-47- " Compare this with the slope of the basic root (Eq. 55) when Im ) , = constant: K Im k = v -t Z--p 0 2 V M O = constant I_ S O b Y -Kp 0 _ ly = 0 - 2 v 0 M S So t he cc it i c al re gion is a na rr o w ban d , of wid t h 0(y), ar o und I m ) k = (n / 2) / re v . T he boundary ( D2 = 0 ) is

: -Kp_ 0 2_0 _ / _I_ 12 Kp21 e l 2 " : : _" e _ - i_ o M _ ( ,_)

Th e max imum st abil it y chan g e oc c u rs at the ce n ter of t he cri t ic a l region: • - K p 0 _:_ Y l = - 2V 0 M O .

whi c h is w h ere the ba si cr oo tw ould cr o s s I m ) , = v 0 = n / 2 ; t here th e r oo t i s ',_ , ) k = iVo + M _ Y D max i_ ; _ ' : 7 : 6 : ' In s i d e the c ri tical reg ion the f re q ue n cy is f ixed at v 0 = n / 2 whi m there is an 0(' / ) change _ i n the dam pi n g . T he d am p in g of th e b as i c r oo t is itself Off) h o wever, so In c ont ras t t o :_i_ the smal l / _ ease, t h ecrit i ca l reg io nca n here l eadt o act u a l i ns ta bili ty , n ot Justst abil ity i -4 8 - degradation. In general the roct i s given by i[ ).=i y 0+ M_ +iyD (62) , L e t A V = y P I , s o Y = y 0 +A v = n / 2 +A y and v mu st be s u ch that A v i s 0 ( 7 ) s mal l, i. e . , an 0(' / ) di s tan ce from ( n / 2) / r e v. Then Z 0 ) , = i Y O + 2 ME t 2 +i Y 0 _ Kp2,Mo, / (63) - \z u 0 - i + F a r ou t side thecr i tica l reg iont h i sres ul ta pp r oa chest h e ba si cr oo t( E q. 55);how ev e r forthesmall y c as e itisdi ff i c ult togetf a r away fro m all c riti c al r e gions,sin ce 1_ whi c h isnotv e ry s mall. T he c riti c al r e gionis c ross e d IAPI = I v - n / 21 isat most 4' wh e n y D = 0 , i. e .,wh e n 2 Y 0 Av 7 = 7 c ° r n er ffi 0 _M_ 2 n 2 = 7 1' 7 2 (64) So t h e _ lo c us i s g i ve n b y ) , = i V 0 + M + iA y4 _l -T / T1) (1 -y / V 2 ) ( 65) _ .

i.

In sid e t h e c ri t i c al reg ion th e ( + i) in the l ast ter m in ), bec om e s 1_ 1) . T h e / a l o c u s i s i t best found fr o m Eq. 63 d i rect l y, s in ce the h arm o n i cs o f th e aerodynam i c coeff i c i ent s i are rat h e r c o mp l e x funct ions o f _ .

_ , , . _, ......... t - --" _ - _,,_ - ..... . " . . .... 7 __ ' L_. "' The _ loci show the beh av ior characteristic of periodic systems, a pd familiar from the discussion of the loci for s m all 1_ . For small 1_ , Re k is fixed at the bas i c root value while the frequency moves toward (n / 2) / rev. For v near 1 an d s m all 7 , , _ the locus moves toward I m k=l for sm al l Kp, toward Imk=3 / 2 for Kp= 1 or s o, ! and toward Im k = _ for Kp = 1 or so. For s o m e i_ the locus crosses the b o undary of the critical region; at this point the frequency has reached (n / 2) / rev. For larger 1_ I m k remains at n / 2 while the effect of the critical region is to decrease the stability of one root an d increase that of the other. The m axi m um stability change occurs at th e center of the critic al region. The center is reached when (see Eq. 60) n Kp Y -- A l a - l a - _+ - - n- _ 16 18M (66) ! Th e aerodynamic coefficie n_ ( SM is a positive number g r e at e r than 1 and m onotonically incr e asi ng with N ; it is s ho wn in Fig. 7. If K n ¢ 0, this cr i terion will alway s be satisf i ed _ for some 7/ 16, so the 7 locus always reaches the center (just a s the 7 l ocus for s m all , / _ always goe s through the center -- see Fi g . I; however in this case the v al ue of 7/ 16 required may be outside the range of v al idity of the solution if Kp is too sm al l). For _ i the _ locus, again if Kp _ 0, the center will be reached for some _ provided A U has i the same sign as -Kp, and I A vl > [Kpi / (n / 2 ) 7/ 16. H o wever, if Kp = 0 the center is i n ever reac h ed for eit h er t h e 7 o r / _ loc i unles s A V --0, i .e., v i s exactl y n / 2; i n !

that case the locus ls alway s at the cen te r, s ince the frequency of the basic Ictus is fixed at V = n / 2 then. Thi s di s cus si on ha s Just been a more quantitative examination of the criterion that the center of the critic al regio n Is reached when the basic root would have cro s sed lm k = n / 2; i t i s i llustrated grap h ic al ly in Fig. 8 for Kp - 1 and n / 2 = 3 / 2.

- 50 - As such, the criterion is limited by the fact only an 0 (7 ) an aly s is ha s been used; the curvature of th e root loci due to 0( 7 2) effects can be quite important, particularly for small Kp. For example, for Kp = 0 the basic locus ha s a sl ope of zero to 0 (7 ), i.e., , I m X is e q u al to g for al l _ ' and #, and in general for s m _ l l Kp the 0( 7 2) cha n ge in th e frequency will be more i m port an t th an the 0 (7 ) cha n ge. An 0 (7 2 ) an al ysis w ould change si g nificantly the conclur. i on s about whether the center of the critic al region is reached under certain conditions; for example, F i g . 1 indi ca te s that with v s! _ htly greater th an 1 the _ locus (for sm al l #) would nev e r cross I m _ = 3 / 2 for Kp = 1 while it wo ul d always cross I m _ = 1 for Kp = 0, j u s t th e op p o s i t e of the conclusions indica te d by the 00') res ul ts (Fig. 8). In an y case, since the m axi m um stability ch ang e occurs at the center of the critic al region, it i s u s eful to ex am ine it as a worst possible case, which m ay perhaps be approached but never reached f e.r certain v al ue s of l_ .

Return in g no w to th e behavior of th e # locus, the m axi m um change in the d am p - i n _ gfrom the v al ue of th e b a s ic root Is (Eq. 61) 7) ; al w ays rA_ a t iv e The contribut i on from the basic r oo t damping, _ / 2 ) H _ , i s i / J as for the sm al l # case it i s 0 _ ), but here th at means th e basic d am pi ng i s s mall. In fact it is th e same order as th e critical region contribution, s o the de s tabilized r oo t i t

: 1

0 may be actu al ly unst ab le, rather than just a small perturbation from th e b as ic da m pi ng i as for the sm all _ case. The behavior of the locus de pe nd s on the relaUve effect s of

i

'!

1 l.

-5 1 - ME a nd t h e nth harmoni cs of the ae r od y namm c oeffi c ient s unde r the sq uare ro ot i n Eq. 67. For mr , st eases the critical region effect dominates, so that as _ increases it eve n t u a!iy r ea c hes a cr i t i c al value, at " hi c h point ( for t h e ca se of the maximum sta- bili ty c ha n g e) o ne roo t cr o sses int o .... RH P, i .e., b ecomes u nst abl e . F r om E q. 6 7, I it follows that increasing (K p ) 2 (K p either positive or negative) alway s increases the effe c t of t h e cr iti c al re gi o ns, w h i ch me ans d ecre a s ing t he cri ti c al- _ fo r w h ich t he r o _ , t becom e s un s table° Thus the critical N i s _ function of Kp, for e ach of the critical r e gio ns; t h is f un c ti on ma y be f ou n d from E q. 67 by se tt i ng R e k = 0 1th e r equire me n t for cro s s i n g th e Im k axi s) . S i : m e the ae r ody nam i c co effi c i e nt s are r a th e r c o m pl e x f unc t ions of / _, i t i s more conven i ent t o f in d t h e cri ti c al [Kp I a s a f un ct io n o f u : : Thi s ma y be regarde d as a maximum I K p [ fo r a g iv e n _ ; fo r large I Kp _ J _ e lo c u s i s ' . , i n t h e RHP at that / _ . Th ese bo und arie s o f K p ! ver s u s _ ar e sho wn i n F i g. 9 f or I m ), near _ , 1, and 3 / 2. With t h e except i on of roots n e ar Im ), = _ (w hi c h requ i res Kp < 0 since U i s :'ear 1) wi th _ . above C. 5 c r so , Fi g . 9 s h o w s t he cr i ter io n on I Kp I _ i s n ot v er y str i nge n t; a value fo r I Kp[ o f 2. 0 for e x amp l e i s quite l ar ge, c or respond- : _ ing to t 53 ] = 63.a degrees. Figure 9 shows also that _m ar Im), = 1 the roots are al way s _ / stable, r eg ardless of p , ff [Kp l < _ / 2:, the Iocl may _ expected to be ne a r Im ), = I : _ for zero or sm al l lI _ p l . In ter m s of th e _ l ocus, this mean s t h at as / _ i ncrease s the t_, locus doe s not cro ss in to th e RHP. Just a f ter the locus cro s se s the cr i tical r egi on __ bo undary, the effect of the critical region i s seen and one br an ch moves t o the right and __ L - _ . o .

-52- £ the o ther to the left (as d o the l o ci in Fig. 2). As D increases further however, the damping of the basic root (which is always stable, and increases with / _) eventually domi a _- , ' . es the effect of the critical region, and the root which was becoming less stable turns around before reaching the RH P . So for larger _ both branches of the locus will I b e mo vin g to _h_ i _ / _ , i . e . , be c omi n g m o r e s tabl e as _ in cre a ses. For the ro ot loci near I m = _ or 3 / 2, the effe c t of th e c ritical r e gion remains dominant, and so one root e_e n tually crosse s into th e RH P a s D i s increa s ed. The criti c al D is cons i derably lower for Im = ½ than for Im _, = 3 / 2. This point s out an und es irable feature of negative pi*.ch fl ap coupling, K p < 0 : not so much that it r e duces th e critical D, bu¢ rather that it moves the bas i c root nearer to Imk = _ .

Flap Rate Feedback Th e u s e of flap r at e fe edba c k , KR _ 0, r esu lt s in no qu alitat ive c hang es i n th e behavior of the loci. K i s how e ver a u s e f ul de s ign paramet e r; it may be used for R e xample to rals_th e c ritical D o r ]K p l° E valuation of the Order _ Results Numerical c alculation s w e re made of the D root loci for moderat e and small value s of _. On th e ba s i s of a comparis c n of the numeri c al a n d a n alyti c re s ult s , it i s conclud e d that th e small V analy s i s to order 0 ( 7 ) i s u s eful only for truly s mall 7 , i e. g . , 7 = 2 or 3 ( _ / 16 = 0 .2 o r s o ). Pr o b l ems are enco un ter e d w ith bot h the b a sic roots a n d the e ff ects : _ f ,*h e cr i tical reg ion . T h e ba s ic r oot t o o r d er ) ' n eg l ect s t h e _ curv atu r e of t he _ lo cus , w hi ch i s es p ec ia l ly im po rtan t fo r zer o o r small Kp, s in ce i 0( 7 2 ) i then t h e c h a ng e of Im _, fo r sm all _ is du e more t o terms than t o the 0 0 _} term.

: I

....... ,[-.- v - - - - 53 - F o r example, a ' _ l oc us based o n the O ff ) analysis wo ul d start out ( _ = 0 ) a t the wrong point, the error being tke difference between the circle giving the exact 7 locus at _ = 0 a n d a line tangent to +i_e circle at 7 = 0 (see Figs. 2 a n d 8). The damping o f the basi c root is 0( 7 ) always, no matter what order the a n alysis is carried to; e.g., I the 0 O _2) re s ults give Re k = - 7 / 16 for all y . But while the ba s ic damping is 0( 7 ), Lhe co n tribution te the damping due to the critical region will have te r ms that are 0( 7 2).

Thus for large enough 7 the conclusio n s i n t h e discussion above of the effects of the critical regi o n on the D root lo c i will not be valid, since they depend on the basic a n d critical region damping being of the same order in 7 . I n particalar, the behavior of t he lo c u s in w hi c h th e r oot b ein g d est ab i l i z e d by the c r i t i c al re g io n turn s a r ou nd an d _ b ec o mes more st ab le due to t h e event ual d o min a nce of t h e b asi c da m ping is no t po ssi bl e _ ", 0 ( 7 2_ _: ex c e pt f or very smal l 7, f o r wh i ch . effects a re i n fa c t n eg li gi b l e . In de ed, it wa s _ found i n the numer i c al c al c ul ati o ns th a t w i t h 7 = 6 ( 7/ 1 6 = 0 .375 , i.e. , not v er y s mall ), :_ v ne a r 1, and Kp zero or sm all s o t h e r oo t is near 1 / rev, tha t the / _ I ocus doe s n o t : t u r n a r o und b ut r a ther eve n tu all y cr o sses i n t o the R HP. The st abil i t y boun da r i es gi ve n P, in F ig . 9 a re o nly v alid t h e n fo r tr uly s mall val u es of 7/ 16.

' Order 7 Results _o To c a rry t h e so l u t i on to o rder 7", it i s f irst necessary to f i n i s h t h e order 7 ,: i s olut i o n. C o nsider i ng o nly the case KR = 0 an d Y 0 _ n / 2 f or any n (t. e. , t h e b a sic _' ro o t), re m ov al o f the s e cu l ar term from the equation f or _ 1 leaves _'_"_'i _ _ 2 + n) i!" -- _I + B1 ffi P O M_ e + o onJuga S e (69) .... _ ,, . r -" _ " ' ' -54- The solution of this equation is

_o*o Mi _0 _ %% + "]

_l = Rel_ll(_ b 1...)e - f ]01 n_0 n2 + 2 v 0 n e (70) I -55- The secular term is, since v 0 i_ n / 2: )`l_l 5 fl 02 + 2 _---- _ k i]9 1 1 = _ 2 i Y 0 5_ 2 its secular term is '- R e ga rd ing t h i s as a di fferent ial e q u a t i o n fo r E ll i n t e r m s of _ ) 1 ' 5 fl 02 ( 75 ) - ---- - ) '2 f1 02 = 0

_$ 2

wi t h s ol u t i on k 2@2 ( 7 6)

fl o2- fl o 3 % ' " " ) e

w here y i k 2 = - 2 i Y 0 r ' 2 Y o _ - " _ 00 K p M0 - 2 y 0 T hus the basic root, to 0if 2 ) , i s )` = k0 + V )` 1+ T 2 k2 t = 2 2 Y 0 + - " _ 2

2 i _ j _. r O Y 2 v o _ P M 0 \ 2 y 0 _

2 X .o. i_ M? )

2 Y 0

(Kp / v) z

= + i Y + K p M0" 8 v 2 \ _ ..... T = ..... _ - ----_ • ___ . _ - --'_ -- __ . . •

"'" ,- - "" 1973005279-(

- 56 - As repor t e d ab o ve, t h ere is no 0( 72) cha n ge in Re k , a n d the 0 (72) change in t h e fre- quency is dominant for small Kp, indeed for Kp = 0 the only change in the frequency is 00 ' 2). To order D thi s root is - ) , = _.Z_+i v I+: L --P _ 1 1 ( 2 1 _ +_ y / J 16 16 y 2 u 2

/

whic h ch e c ks with th e e xpansion to 0( V 2) of th e r oo t f rom th e s m all / _ analy si s (t he hover root for 0 0 _)).

Th e o r de r 7 r e sul ts would sign if i can tly al ter th e plo ts o f the ba s i c r oot lo ci shown i n Fi g . 8. Exten d in g the res ul ts f or the crit i c al re g io n to 0 0 , 2 ` wou l d b e much more i n volved bec a use o f the gre a ter complexity of the solution for fl 01( _ b l) when U = n / 2 .

Th e Large _ /Ca se Fo r t h e l arg _ 7 case, co n s id er the g e n er al equati o n of m o t ion, of t h e fo rm: _ j' + 2 fl= y [(M_ - K R M{ _ _ + (M fl - KpM O fl ] (78) : T h e sm all p arameter in t hi s c as e is t h e inv erse of t h e Lock n umber. F o r 7 v er y la r g e , t h e aerodyn ami cs do mi nate t he sys tem . F o r ) , = ® the in erti a an d ce nt rifu gal s p r ing _ t erms (t h e LHS of E q . 78) a re n e gli gib l e , l e aving a f i r s t o r d er s ystem w h i c h do es no t +_ _, d e p e n d on y, nam e l y - KRM s ) ( M fl - Kp M e) i!: ( M _ _ + fl ffi 0 ( 79)

'

r-_ .... _-_ " t .'_ -__ _' _ '" 2 ............ 2 7 ..... 2 _...... _ __ --- __ . . ....

Reduction of the order of the equation of motion when the sm a ll parameter (1 /7 ) is set equal to zero is a characteristic of a bou n dary layer type of problem. The solution of the reduced equation is v a lid over most of the range in $. As a first order equation however, its solution can involve only one free constant; thus it is not possible to start I the solution from the two general initial conditions allowed for the original second order s y stem. Furthermore, at certain points the solution of the reduced equation will exhibit singular behavior, indicating that the assumptions used to derive it must be reexamined.

In general, there must be narrow regions in which the higher time derivatives are very l ar ge , so t hat inside th e regio n the inertia terms a r e of t he same o r de r as t he aero- dynami c te r ms a n d m a y n o t ; _e n egle c ted (o th e r s i mp lif i c a t ions o f the e q ua t io n of mo t i o n ar e often possible t ho ugh) . L ° su c h a n a rr o w r egi o n i s used to c on n ec t a solut i on o f the re du c ed equ ation t o t wo in it i al c o n di t ion s i t i s called a boundary layer; if it i s us e d to co_mect a s olution of the reduced equation to another such solution on the other side of \ the layer, it i s c al le d a transition region. T h e s o lu ti o n of t he r edu c ed e q uat i o n is called a , the mai n solution. Mo r e ge n e r al te r mi n ol o gy i s in ner and outer r egions, and inner and o ut e r s olut io ns. Bec a u s e the pr o cedure for connecting t he s olution s i n the inner and N , o uter region s i s cen t r a l t o the ana l y s i s of b o und a r y l a y er pr obl ems, th e e n tire an al ysis _ ' tec hniqu e has bee n n a me d t h e me t hod of matched asymptotic expansions; th e nam e _ ; pro p er ly re f ers to the p r o c e ss of c o nnectin g th e inner and ou ter so lut i o n s but is us u al ly !_ } u s e d t o i n c lud e t he en t ire ana lysis. Many techn i ques m ay be used t o f i n d th e main s o lu - i_ t ion s. Us u ally t he tec h nique u s e d is a d irec t ex p ans io n of the de p e n de nt v a r i ab l e (_ ) "_!_ i : a s a series i n the smal l param e ter (1 / _) . Th is tec hn ique i s n ot satisf a c to ry f or the i prese nt p rob l em because i t does not yie ld a so l ution which i s un ifo rmly v ali d for all -58- (which i s required in order to find the roots); details of the appl i cation of this technique to problems are given in Ref. 3. It is also possible to use the method of multiple time scales to find the main solution. This technique is not entirely satisfactory either how- ever, since the equation obtained to lowest order is just the reduced equation given by setting 1 / 7 = 0 (Eq. 79). Since this is a first order equation, the solution gives only one root, which to lowest order (i. e., for _ = =) is independent of _. The reduction of t he o rder of the eq u atio n m eans that on e o f t h e r oot s g o es t o - = as 7 goes t o = ; i.e . , the solution c orresponding to this root is exponentially small compared with the solution of the reduced equation. There is no way that the method of multiple time sc a les ( as described here anyway ) c an find this root. The perturbation te c hnique useful for finding both solution s in the outer regions is the use of a sub s titution of the form B = e xp y$ pd '_, follow e d by an expa n sion of p ( _ ) as a series in the small parameter.

The la r ge _ c a s e i s a bound ar y laye r type o f p r oble m, w h i ch me ans that i n gene r al ther e will be several outer regions around the azimuth with a separate expression obtained for the s olution in ea c h region. Thu s it will not in general be possible to find a single solution, u n i f ormly valid for all _ b , fr o m which the eigenvalues of the system may be foun d b y ins pe ct io n , a s w a s po ssib l e fo r t he c a se of sm all _ o r s mall : y . In ste ad i t i will b e necess a ry t o u se the g e n er a l t ec hniqu es o f t h e analy s is of a s y ste m wi th p er iodi c I c oe fficie nt s ( a s o ut l ined in A p pendi x I ) . Th i s e n tai l s obtai ni ng the s ol uti on fo r on e • re volu t ion of th e r o t o r (on e p eriod ), i n p ieces ff necess a r y, e a ch pi ec e v al i d i n a p a rticular i nn er o r out er reg ion . Firs t th e m ain s olutio ns must b e o b tain ed in th e oute r re gion s , b ut here i t i s n eces sa ry t o f in d bot h m ain s olu t io ns r a ther t h a n j u st the ii so l u t ion of the re du ce d e quation . N ext t h e meth od o f matc h e d as ym p tot i c e xpa n s i ons is ! I \ used t o c onne c t the main s o lutions a cro ss the trans i tion r egions , o r a c ross boundary layers to initial or final conditions• Finally with the solution constructed over a corn - plete pe r iod, th e results of Floquet th eo r y (see Appendix I) may be used to find the eigenvalues of the system from the initial and final values of the solution.

I Expansion in T C o nsi d er f irst the equ atio n wit h K R = 0; write = exp _ b P d_ b (80) and P = ? P-1 +P0 + T Pl + "'" SO t = p exp p d_ ), Su b s ti tut i ng these ex p ressions for _ , _ , and B into the e q u a ti o n of mo t ion, and c oll ect - i ng al l terms of l i ke order i n 7 _ ives ( exp J ' _ pd l_ is a common factor in the e n t ire e qua t i o n , s o dr op s o u t; th e c o mm o n fa c to r of 7 n has a l so been dr op pe d f r o m t he follow- _ ) i ng e q uations): 0( 7 2):p2_ l =M _ P-I "_ O ff ): 2 P-1 P 0 + P -1 = M _ Po + M_ - KpM 8 (811 : :_ ' i , 0(I): PO + 2P-I Pl + I _ 0 + = 2 V 2 M _ Pl etc. !

•

\ -60- The or der T2 equation gi v es P-1 = 0 or P-1 = M_, whi c h give the tw o m a in solutions.

First Solution ' _ T l l e o r d er y 2 e quatio n g ives P -1 -- 0 so t o orde r "/ fi n d M_ - KpM e PO M_ a n d to o r d er 1

_0 _ 0 _ "'_ M_ )

Pl = M_ - M_ The n t h e s ol ut io n fo r _ is d_ b = _ 1 e x p M _ _ bMi8 - X pM 0 1 M_ d_ b + O_ ( 82 - KpM F - K p M e - _2 ) I V2_( M _ 'i +(M_ M_ ,,, ' -2

+_ M _

wh e re BI i s a c ons tant.

I Second Solution T he o r d e r y_ equation gi ves P-I = M _ so t o order _ f i nd \ M_- KpM e - (M_)" Po = '_rr_ --._-.e--, . ,j_- + + --- ' 2 ' .... _" ..,. 'r"" - 61 - Then noting that

- (- Mi)--

'- th e so l ut io n f or _ is I -- d_ b + 0 ( } ' -1) ( 83) I T * $ M_ - K p M e ] wh_ re 82 i s a c on st an t.

Eigenvalues - - Fr o m the co n t i nu al app e ara nce of M_ in th e den omi n at or , i t is e vid e n t t ha t a ,, transiti o n r e gion o c cur s where M_ = 0. This criterio n means a tr a nsition region o cc u r s wh ere t h e da m pin g go e _ thr o u g h z er o . Alter na t iv e l y , ff M_ i s n e a r zer o , t h e _ _ t erm in th e reduced e q u a t ion ( E q. 79 ) i s muc h sm al l er th a n t h e _ term , whi c h i mp li es t hat the in ert ia (_ ter ms must be i nc l ude d i n or der t o ob tain a di ffere n t ial equ a t ion w ith all terms of the s am e order; that i s, t h ere must be a tr ansi t i on reg i o n about the po i nt A wh ere M_ = 0 . A s i t happ e n s how ever , M_ ($ , # ) i s a n e ga tive q ua ntity w h i c h never

2-1 / 3) _

re a ches z er o ; i n fact, 8M_ < - ( 1 - = -0 .2 06 a nd e v en th a t v al ue i s ne v er re a c h ed _ unle ss # > 0 . 795; for # = 0, 8M_ = -1 f o r a ll $. Thu s f o r the case considered _ : % (KR = 0) , the main solu ti ons are uniformly val i d over the w hole az i mu th , and i t i s not _i_ ne c essary to deal w ith trans i tion r egi ons an d boundary layers to f i nd the comp l ete _ so lu t i on. }_ -6 2 - With th e solution ove r all $ , F loque t theo ry may now be used to find the eigen- values. Consider the first solution, to order 1.

- d_

M fl - KpM 8 ' 0 M _ - _ = _ 1 e Thi s may be w ri tte n 2 _

o Mh

a = a 1 e (8 4)

wher e M 8 - Kp M 0 d_ 1 2 , r M_ - KpM 0 d , _ " 0 0 U si ng the fa c t tha t the aerodyn a m i c coeff i c i e n t s M ff M _ , and M e a re po ri od ic , it ma y be establi s hed th a t f is a l s o per i odic in _ . Now Floquet the o ry states th a t th e solution , to a different ial equ a tion w i th peri od ic c oe ff i cie n t s m ay be writ t en in th e form = B 1 eM b u(¢) (85) w h ere _ 1 is a co n st a nt, ) , is t h e etgenv al ue, an d u( _ ) is a pe ri od ic f unct io n ( s e e Ap pe nd i x I). Co m pari s on of Eq u . 84 and 85 shows th a t the elgenvalues must be

f _ d _ i

w \ ! I , -63- T hi s r e s u lt may be easily e xtend e d f or bot h sol u tio n s to any or d er in _ . Th e n th e two main s o l utions g ive two root s : ' 1 [ 2f l MB - K p M0

, x z -- - _j M -_ ---- d_

- 0 2 _ 2 _ 11 11 M_ -KPMO -.

) ' 2= ) _ _ M _ d_ + ' 2- _ M / } d_ + 00 ' ' 1 ) 0 0 !i T h e s ymmetry of M_ an d M_ mean s that ......

21 1 ' ,_ • 'd_=O . - _ 0 _ _ a i Wi th this relat i o n , th e roots simplify to 2 _ _ kl " - KP 2 _ -M--' _ d ' * _.

I O M O 2Zr / M B " KpMo ¥ / M'0" KpMo _ 2 i | _i +7 _ 1 1 M _ d _ +0 0" - _ ) (86)

i I _ - I_l + l.-L- , - r - _ ' l ,_

) -6 4 - and 27 2T t 1 I M_d$+K p 1 I M O k 2 = _'_ 2 " _ - M '-'_- d$ + O (v -l ) 1 87) 0 0 I Thu s t here a re t w o real roots, one 0 .2 ) a pproach i ng - _ as _ inc r eases t o ® ( M_ is negat i v e ) , the other 0 ,1) approachi ng a con s tant; )` 1 i s the root f rom the reduced e quat i on. This behavior o f the T root loci i s expected from the s mall / _ results; Fig. 1 shows that for large enoug h T th e locus i s on the re al ax is , i.e., there are two re al roots, one approaching - co and the other -Kp for _ / -, = . To l owest order )'1 doe _ _o t d epend on y , because it repre s ent s the b al ance e f the aerodynam i c dampi ng and the aerod ynami c s pr i n _ , only. The value of ).I / ( -Kp) for varyi ng #, an d T = _, i s s hown i n Fig. 10; th e move m ent s ho wn take s place ent i rely on the re al ax is i n the )' r _an e.

As for the smal l _ case ( Fig. 1 ) th e root is on the re al ax i s, in the LH P if Kp > 0 an d in the RHP -- unstable -- ff Kp < 0. The value o f )'l / ( -Kp ) vari 2 s f ro m 1 to 7 / 8 for _ = 0 to =, w i th most of the change between _ = 0.5 and _ = 1;, thu s th ere i s l ittle var i ation of the r oo t with / _( to 0 (1)) . The s i ze of the 00 _ -1 ) ter , . a in )' 1 i s indicated by the re s ult for # = 0, w hi ch i s ea si ly obtained (sin ce the aerodyn am i c c v eff i c i ent s are con s tant th en) as ._L. 1 1 _ 2+Kp 2

"" Kp - / 1 6 ; + ° ( v -2)

Th i s res ul t agree s with an O0 _ 'I) expans i on of the hover root fr , m the s m al l anal y als.

| _ To lowest order )'2 I s : 2 f fJ %. r ) t - 65 - The aerodynami c c oeffi ci e n t - 8 M_ i s g i ve n i n F ig . 7 . F o r _ t < 1 it has t he value -S M _ = l + _4 / 8; for large _ t it is asymptoti c to 8 / 3? t D. This root becomes increas- i n gly nega_2;'e a s y i nc rease s , m_d also as A t i:_creases. The order 1 term in k 2 is the negati - -8 of the lowe s t orde r ' (also 0 ( 1 )) term in k 2; thu s the behavior of this t term is _2so given by Fig . 10.

Flap Ra te Feedback Wh en K R = 0, there a re no tr a n si t io n reg i ons bec au s e M_ < 0 al_ ' ays . W i th fl ap rate fe e dba c k, KR _ 0 , the s am e e xp r ess io ns for th e main s olutio ns are obtai ne d e xcept that M _ is rep la ced b y M_ - KRM0. Th e aer od y nami c coeffic i en t - ( M_ - K R Me ) "- ca n b e com e n e gativ e ov e r r e gin ns of the di s k for c e rtain c ombira t i on s of $ t a n d KR; : _ i . e ., there m ay be negative damp i ng o ve r part of t h e az im ut h ran ge . W hen suc h reg i ons _ of n egat i ve damp in g exi s t i t m eans t h ere m u st be tran s it i on regions a bo u t th e po int s where the d am p i ng goe s th rough zero. The m ain s olut i ons obta i ned above are val i d st i l l in th e outer reg i on s , but i n each reg i on t h ere are two constants, w hi ch must be matched th r o ugh the inner reg i on to the two co ns tant s of the next m ain solution.

T h e criterion for t he existence o f trans i tion r eg ions i s that th _ re be negative damp- _ , in g on some portion of the disk, i.e., -(M _ - KRM _ < O. M _ is a l way s n eg at i ve; M { ) ; % i s u s u al ly po s i ti ve, but m ay be n eg at i v e on the retr e at i ng s ide f or large e nough / z ._.

i ( _ > 0. 64 1 ). If KR i s too larg e pos iti ve, t he n e gativ e va l ue s of M 0 on the r e treat ing !_ sid e e v e ntual l y dom in at e M _ a a / _ Is incr e as e d, so th e r e wi l l be n e gat i v e d am ping on . " th e r e tr ea t i ng a i d e ; if KR i s too large n e ga ti v e , KRM 8 e v e ntua l ly dom l nat em M _ on _ the advancing aid e and th e r e will be n e gativ e d am ping the r e if _ i s l a rg e e no u gh. _ := ;_ , n -66- Quantitati:re values of max i nmm and minimum KR as aft, n c tion of # are given in Fig. 11. For the cases with negative damping ther e will be transition regions (of width 0(7-2 / 3)) near where M_ - KRM 8 = 0, which greatly complicates the analysis.

For these cases it is also expected that there will be other pr o blems, including mate- ] rial computation problems, physical control problems, and large flapping amplitud u s.

The situation may be compared with stall flutter of a r o tor in hover or forward flight, where a limit cycle oscillation is reached with the negative pitch damping in stall balanced by the positive damping below stall, resulting in high amplitude pitch motions and large control loads. Thus while a region of negative damping does not necessarily mean there is a flapping instability, it does mean that there are many problems -- analytical, computational, and physical -- so requiring -(M_ - KRM _ > 0 is a reason- able design criteri o n. This criterion pr o vides a maximum and minimum KR for a given D. The limits of KR from this rule are much easier to obtain than actual stability boundaries; and Fig. 11 s hows that although conservative, it is not a serious restriction for b _ less than 1 or 2. For large # it is a serious limitation (for large D, KR _ 2 / 3D (1 + 7 / 1 2 #) and KR -_ - 2 / 3D (1 - 7 / 12D)), indicating that M e max sin (blade pitch) is not very good for flapping rate feedback then. Time varying KR might work better but it would have to be programmed with D probably. Although the deriva- tion of this rule ha_ been based on the . large y case, the criterion of no negative damp- ,! ing ha s not hin g t o do w ith y , a n d so sho ul d be a re a son a b l e criter ion for all 7, Indeed , _ .

the criterion KR > - 1 for D = 0 is the s a me as from the sma ll # case , wh e re it is a true st a bility criterion , a nd v ali d for all y.

\ -67 - The L arge # C a s_ _ ..._e F or the l a rge # c ase the general flapp i ng e q ua t i o n o f m o tion, as given in Eq. 1, ' i s u s ed ; t he sm all pa r amete r in th i s c a s e i s t h e in v erse of the advan c e r atio, For # I _ : ve ry larg e , t h e a e r odynami c s agai n d o min a t e th e sy s t e m. Wh en # g oes t o inf inity, the B and _ terms are arbitrarily large compared with the _° term because of the influen c e of g in th e a er odynami c c oe f f ici ent s. T hu s th is p r ob lem i s al so o f b ounda r y lay e r type , and th e s o lu t io n i s sought as fo r the l ar ge y c ase i n t e r m s of o u ter solu t i on s and tr an si ti on re g "c n s. Wh en KR _ 0 , the a er od yn am ic d amping , KRM o, is the s a me : or de r i n # as t h e a er odynam ic s p ri ng term, M B - KpM 0, (n a me l y 0 (#2) , see E q. 1) , s o se tting 1 / # to z er o re d uces t h e o r d er of th e syste m . T he re d uce d eq u a t io n g ives ,: o n e m ai n so l u t i o n, an d t he o ther w ill b e e xp o nent i al l y s malle r (or la r ge r). Wh e n ; , : - _: KR = 0 ho w ever, th e a er odynami c d amp i ng, M_ , is of 0 ( _ ) w h il e the a er odynami c &" _i : sp r ing , M_ - Kp b l 0 is 0{D 2) (E q. I ), so i no r de r to ob t ainan e qu at ionfo rt heou t e r _ solu t ion w i th the pr o per o r d er ing of terms it _ s n ece s s ary to in c lud e the in e rtia te r m ' _ ; ( ¢ ) ev en in th e equ at ion f or the o ut er r e g i o n . T h at is , f or # - - ® the ae r odynami c ) sp r ing m u st b e balan ce d b y the in ert ial fo rce _ % wh i ch lea d s t o an equati on of t h e fo rm ::_ *o :

B+ # c + # 2 Z ¢=0 1 8 8)

} ?,; T he s ol uti on o f th i s equat ion is e i ther a rap id s tnu s oi d al o scil l ati on w i t h f r equ e n cy o f ....

_ _ : 0{D ), o r a s um of e xpon e n t ials wit h t i me c on st an ts of 0_ -I), depe nding on whe th er _¢' : :_ L " t h e a erod yna m i c s p r i ng i s n eg a t i ve o r po s i t iv e (the cr i ter ion is a b i t m o re c o m pli c at ed _i ! - i¢_ _- rea lly , bu t t h a t st atem e n t w i l l d o for t h e prese n t di sc u ss ion ). N ow the a er od yn ami c s p ring changes sig n in t h e m i ddle o f th e adv an c i ng s i de an d again i n th e m i dd l e o f th e , J i , , ,r . _ = .... .... ,r*.. _ ; _- :. t....... _ .... " ' -' : "_ : .....

B _ - 68- retre a ting side, a nd at each p o in t t h ere i s a t rans it i o n regi o n (of width 0(_-2 / 3)) across which the solutions must be matched. There are also transition regions (of width 0(_-2 / 3)) between the advancing and retreating sides of the disk (i. e., around _ b = 0 and 180°), through which the main solutions must be matched. Thus in contrast with the I l ar ge 7 ca se, f or la r ge / _ it is n o t possi b le to f in d an ou t er s olu t ion uniformly valid for all $. Rather it is n ecessary to go througl : the entire procedure o f matching the main solutions through the tran s ition regions and by that proc e ss construct a solution over one rotor revolution. Then the re s ult s of Floquet theory (Appendix I) may be used to obtai n the eigenvalues from the initial and final values of the solution. The procedure for finding the main solutions will again be ba s ed on a s ubstitution of the form f ] = exp _ b pd _ b with p now expanded as a s eries in / _-1 The matching techniques of the method of matched a s ymptotic e xpansions will be illustrated in the treatment of the transition regions. The procedures requir e d h e re are reason s bly straight forward, but in general the matching t e chniques c an b e quite c omplicated, particularly when solutions ar e s oug h t to high er or d er. The r e ader i s di rec t ed to R e f . 3 f or m o re d e tail s of t he m ethod of matched a symptot i c exp ansi on s .

In re g io n s ( 1 ) and ( i ii) of t he ro to r di s k the diff erent ial e quation h as the form _ '+ y 2 _ =_r 7 + _ s l n +K R + _D s i n $+_ ( _ s l n _ b )

+ cos ¢ +7. s i n + Kp 0 9 ) "i

e : wh ere r i s a c on st an t with the val u e +1 i n re gion (t) an d - 1 in re gion ( i i i ). R ec all _.

. ; fro m t he di scussi o n o f E q . 1 that re gion (t) i s t he ad van c i n g s i de o f the di s k , wh ere t he i -6 9- blade has n o r mal flo w ove r it s e n ti r e s pa n, and r eg ion ( i i i ) is th e r a n ge of $ on t he retreath_g side where the blade has re v erse flow over its entire span. As N - ' _, r egion ( i i i ) o cc upi es near ly all t h e retre ating side , wi t h t h e e xception of 0_ -1) bands n ea r _ = 180 ° and _ = 3 6 0 ° . I n Eq . 89 , _ t appea r s in t h e a e r ody n ami c c o e fficients I nearly always in the comb i nat i on g sin_; any assumptions made about the order of terms based o n th e o r d er of tt wi ll b e v io l a ted t h en if sin _ b is small eno ug h; t his i s t he o ri gi n of the transition regions near _ = 0 and 180 ° . When KR = 0, there ar e also transition r e g i on s i n the m i d dl e o f t he ad v an c ing an d retre at in g s i des a r ound t h e p o int w h e r e th e aerodynamic spring goes through zero. These regions arise in the analysis because the a er odyn am i c s p ring bein g z ero o r very s mall w i ll a ga in v iola te t he a ss umpt i on s m ad e a b o ut the or d er of the terms ; p hysical l y the y a rise bec a us e th ere must b e a tr a nsitio n , be tw ee n t he s ol u t i o ns on ei t her si d e s in ce they h a ve very d ifferent be ha v i or , n ame ly s in us oi dal o sci ll at io n an d e xp one n t i al d ec ay o r g r o wt h . If K R _ O, such tr ans iti on re gio ns a re n o t r e qu i red; on l y the t ransiti o n regi o ns be t wee n t b _v ancin g an d re t re at - i ing si de s are nee d e d . T h e case with KR = 0 wi ll be examine d f trb _ . • E xpansion in _ _ Consider th e outer re g ions, where / _ s in _ b is of order / _ and all t i me deriv a t i ves _ : o f _ a re o f t h e sa me order; thi s m e an s t h e re gi on s (i) and (iil), a w ay from the boundar i es ne a r _ = 0 an d 180 ° . The equation o f motion i n t h e ou t er region I s then Eq. 89 w i th _ K R = 0 . F o r KR = 0, the an a l ysis is s i mplifi e d i f f irst t h e 0_ ) aerGdynam i c damping _ , is r e moved from the equ a t i on o f mot i on (wh i ch is o f the f orm of Eq. 8 8 to the lowe _ t ' _ _ order); t his is accomp l is h ed by the f o l low i ng sub s titution i

- e y (901

!, With this substitution, Eq. 89 for KR = 0 becomes y = -r y -r_y _ 12Dsin ' +_C O SS +_N s in +Kp +_Nsin$ 4 (Dsin_ b ) (91) Then the main solutions a r e found by use of th e subs t itution - p d S ( 92} y exp f S -1 w i th p e xpan ded in a se r ies i n / _ : P = /z P -1 +P o +_ Pl + "'" Main Solu tions Subs t i t u t ing for y a nd c oll ect ing te r m s o f lik e order, t h ee qu at ionof mo t ion gives !

= - 8 P- 1 _ - { cosS + _Kps i n 0(D ): l__l r_ - r7( - 9_66 ]sinS[+ 1 1 S) F r o m t h e o rder p 2 equat ion, o n e o b t ai ns : The s olution fo r _ has a fa c to r of t h e fo r m e xp _ P 1 d S ; t h e do ub l e sign in P- 1 g iv es , _

- i

t he two m ain solutions . Whe n the q ua nt i ty under the sq u are r oo t i s p osi t i ve , P - 1 is real _ ; and t h erere s u l ts m a in solu ti ons with e xpon e n t ial deca y o r g r owt h ( wit h t i me c on s t a nts , - - - - " • -- ....... . , ,' W'_ ., _ , _,_ _ _, . ......

_ = .

-71 - imaginary of 0(_-1)); and when the quantity under the square root is negative, P-1 is and there results main solution s with sinusoidal oscillatory behavior (with frequency of O(D)) . When the qu a n t ity under t h e squ ar e ro o t is zero. th ere is a tra n s i t i on region .

, On the advan c ing si d e (regio n (i)) . P-1 is z er o at -1 1 _ b tl _ ( 9 4) = ta n _36 Kp and on t he re t re ating s id e ( re g i on ( iii )). P-1 is ze r o at -1 1 So ther e i s a hi g h f requen c y o scil lat i o n on the rear of th e disk . exp o n en t ial s oluti on s o n • the f r o nt, an d t r an s i t io n re gio ns bet w een t h e two types o f b ehavi o r. P - 1 is als o z ero t w h eu sin _ b = 0, i .e., at _ b= 0 or 18 0" , s o transiti o n re g i o ns wi ll al s o be req ui red near _" the ed g e s of re g ions ( i) an d (ii I) . In _l dit i o n there is re g ion (ii ) , which for l arg e N i s , " an order / a- 1 sm all b an d on t h e retreat i n g si d e near _ b = 0 and 180 " ; i n c o nnec ti n g the main so l ution f ro m the adv an cing si d e to t h e retre a tin g s id e it is necessary to g o thr o ug h t h is re gio n as we ll as t h ro ug h the tr a ns it ion regions. In g eneral, w h enever P - 1 is : smal l the a as uml _tio ns m ad e about t h e order of t h e terms in o btaining E q . 92 are v i olate d , ,: . 2 .

: so the m ai n solut i on c an n o l onger be v al id there• Th i s cr i terion gives the f our transi- t -_L I t i on re g ions. _ : T h e o r d er _ equat io n g i ves _:_., ir P0 = _'_ cos _ b + _ K p sin _i_ i , 2P - 1 ; _ b _l. r 8_P_l _ r ,( _9_6 ., s ln,,+ 1 1 ,) i - 7 2- fr o m whi c h, using t h e relati on P-I , o n e o b ta ins

+ - _ Isin_l +:: cos+ +:_psin+

POd_ =-:_P-I! - rfi_ b - r d_ b P-1 With t he above expressions for P-1 and P0' t he solut i o n fo r B (to 0(1) in p) is obtained by using the substitutions for B and y, i.e., Eqs. 90 and 92. There are main solutions in four re gions , whic h will be call e d q uadrants ( al th ough t hey a r e not r e all y so, sin c e Stl and $t3 are not equal to 90° and 270° ) . Ea c h quadrant Js bounded by transition r e gions; th e ranges of the four quadrants and the main solution valid in each are a s fo l lows .

-73 - 3 rd q uadrant: y < $ < i_t3

-_o_+_ __ i _ _ _++

, _t = e 5 °

° ° °1

+ c6 e 1 3 _s ( 98 )

4 th quad r a n t : I_t 3 < _ < 211 _t f e - / i 1_ 22 c°s _ b + 1"_6 l_ ( - f) - 1 / 4 C 7 +i C 8) ei 4 v / -:'f dl_ + _ b 4_d -.

i + con ' jugate ] (99) ' w h ere

f =Zlsin ¢l(_lsin_1-cos _- K p sin _ )

i ; ¢ ,

7 g = 2 co s _ + 7 Kp s in I - Isi n I

, and C1, C2, ..., C8 a re constants. The match i ng procedures w il l result in ; onnection .i f ormulas throug h th e trans i tion regions, which w ill give the two const an ts of one main _, " so l ution i n terms o f the two c o nstants o f the ma i n so l ut i on on t he ot h er side of the 4 ' ; transit i on reg i on. In add i tion, the c o nstants w il l be matched to arb i trary i nit i al cond l - :_ t io n s at a certain point on the d i sk. The quant i ties I_ 1, !_2 , I_ 3, an d I_ 4 i n Eqs. 96 to 9 9

[

are al so constant s , which must be in the appropriate qu ad r an t; they are n o t free !, consta n ts since a chan g e in them must be accompanied by a change in the value of the C's. These a n gles may be given any v a lue convenient to the analysis; it is most con- venient here to leave them arbitrary since they will drop out of the final result anyway.

I Transition Regions Consider the trar.sition region near _ b t (mea n i-g _ b tl in region (i) or St3 in region (iii)). This tran s ition region has a width of 0(D-2 / 3). The reader is directed to Refs. 3 and 4 for illustrations of methods for finding the prop e r width of a transition region or boundary layer. The technique involves assuming d$ = 0(D-n); then B is of -2n order D , and similarly the order of all terms in the equation of motion may be found in terms of n. The exponent n is det e rmined from the criteria that the resulting equation , to lowest order in D , must a) include the highest time derivative (_) and b) must produce solutions capable of being matched to the outer solutions. It also helps to know what to expect of certain typ e s of problems; for example , a width of 0(_ "2 / 3) is typ i c al of tr a nsit io n re gi o n s for equat i o n s of t h e form of Eq. 88.

A ssu ming d$ = 0 (D-2 / 3 ) , E q. 9 1 bec o x m s t o lo west or d er ( 0 (D4 / 3 )): --"y ( $ - _ b t) $'_ b t (100) Y w here _ t rt =r4

t

g '7_ _. .... "¢i t Alter na tively, write z = (r (y / t) 2)1 / 3 ($ _ Ot) , substitute fo r $ in th e differential equation for y, and then obtain to lowest order the equation d-_2 " z y = 0 ( 101 ) dz / Thi s e qua t i o n i s a stan dard f o rm , the s ol u t ions of w hi ch ar e c all ed A i ry fu nct i on. T h ere are two indep e ndent s olution s , denoted Ai ( z ) and Bi ( z ) . Thes e fu n ction s may be wr i tt e n in terms o f Bess e l fun c tio ns ; how e ver, the g e neral b e havio r o f th e s o lution in the tra n si- tion r e gion is not of int e r es t h e r e . Rather th e s olution in th e tran s itio n region is only to be u s ed to find a c onnection formula bet w ee n the n e ighboring ma i n solutions. For this purpo se all that is requir e d i s th e behavior of the solution for v e ry large z.

W r iting th e solution of Eq. 1 0 1 as y = 2v / _- a Ai (z) + J _'b B i (z ) , ,. ' " w he r e a an d b are con s tants, th en the be hav io r fo r l ar g e z i s: _ - 1 / 4 _ e _ ) _. z '_ : y "_ z (ae +b _ z-' - : y - , , I z l - t e +c o njuga _ , w here ,,L = 9. i,. 18 / 2 /9. .....

Y , r , T h e match i ng procedure cons i sts of f i nd i ng the l i m i t of t h e outer solut i on as _ : : . _ b - , _ t' an d the lim i t of the I nner solution as z -* J : ® , a n d requL r ing that the t w o l i m i t s i,

!

- 76- have identical behavi o r. This criterion gives the constants in the inner region in terms of the constants in the outer region. The matching procedure is considerably more complex if higher order solutions are involved. Consider first matching from the first I quadrant to the second quadrant, through the transition region at @ = @tl. The outer solution in the first quadrant is given in Eq. 96. As _ - * Stl,

=? t / _ r -_3 ] $- _t l 3 / 2

so t he main so l u ti o n fo r _ b -_St 1 is • ( / _ ? t f _ b t g _ Y o uter I' Z . _ _ t ) { - 1 / 4 _1 _ b l -.. - (C l +iC 2) e e + conjug a te To th e i nner s oluti on, t h e o u ter region in the fi rst qu adrant appears as t h e li mit z -_ - = , and i n th i s l i m i t I (4_D 2 ) ,- 1/4 / b + a) -i 4 Yinner ' * - 1 / 1 2 1¢ - $_ , 1 k _" i e ff e- I_ 3 _ - 1 ¢ - ¢_ t13 / 2 _ + co n juga t e { Yc ut er ) _ = ( Y i_ ner ) z g i ves ,_ Then requ i r i ng = _ b t = - ® _ t / : ' { d e - a¢ 1 ¢, _ d - i- (C1 + i C 2 ) e " e (_+ i a) ( 1 02) i _ ¢ 1 _)1 / 6 4b " _": " , ',_._.. , W_ t -=..-. - - _ _. I _ ' :._.,.. ....

\ I I , -77 - Simi la r l y, th c m a i n solu t ion in t he secon d q u a d rant ( E q . 97 ) b e comes, f or _ _ _ t

v ,J ' " a d , - ]" + V

Y ou te r 3 e o* t g .El, - / _ y S t v q'd * + J¢ _ d * - * tl 3 / ' - 3 ) + C 4 e '2 2 and the inner so l ut i on bec o mes, for z - * Yinner - " (_ 17)- 1 / 1 2 I* - * t 1 -1 / 4 e- " 3 +b e ll_ 3 d _ - [* - * t l 3 /

a _1' -* t l 3 / 2 2 /

and the m atching criter i on gives l , g : II d trd ¢ - _ d e _:' < C3 e ¢ 2 ¢ 2 = ( 4 _ - _ / e b , _. (103) !. : , -li 4' _'d ¢ + "_d ¢ : : c4 e * 2 ¢ 2 =\ 4 _ /[ Y - _1 / 0 a ;; Next combining t h e re sul ts of matching between the first q u adrant an d t h e inner _ olutio n : _' • ( Eq. 1 02) , and be twee n th e inner sol u tion and the se cond qu ad r an t (Eq. 103 1 , gives th e _ 4 i c onnection formula be tween the f lret an d _e c cn d quadr an t s : n ,... i - " ........ | t '- ...... = -- - 2 " # " ........ _ _ "- _ - " -7 8 - (C 1 + i C2) = e - e ' + i C4 e _2 (104) Simi l arly, the m atching procedure on the retreating side around _ t3 give s the connection formula between the third and fourth quadrant s .

? '± d -. - --

(C7 + iC8)e \ $4 J$ 4 v / _ = e i4 C6 e $ 3 $ 3 / ' / + _ ) __ - f_ C T d_ _ tg d_ j - / _ J" _ b t ¢ c _ d _ b + J 3" _ d _ - i C 5e _ 3 ( 1 05 ) Now cons id er t h e trans i t i on regions near $ = 0 o r 1 80"• The main so lu t i ons indi - " cate that there must be transition regions at t h e edge s of regions (i ) and (iii ) , but there is als , , ,- , _ gion (ii) in between regions (i) m i d (i i i). The extent of the th ree regions is defined by (see Eq. 1): r eg . _ on (i) , " s in ¢ > 0 _ r eg ion ( U ) - I < _ s in _ < 0 i l r egl o _ ( Ill ) _ sin _ b < - 1

: i

i The tran s ition regio n h as a width of order / z-2 / 3; thu s region (ill , which has a width t

i " 1

| _ of only order _ -I, lies entirely within the tran s i ti on region. Region (ill appear s th en i ; i t .

'

_ _ _ ,, - .............. _ _-- , , =. , - / _ _ - --_ _.- -79- - a s an interior region w h ich h as no effect on t h e solution to lowest order. This me a ns it i s not necessary to use the more complicated aerodyna mi c coefficients of region (ii).

It also mea n s that the appearance of #4 terms in the aerodyna m ic coefficients i s " d eceptiv e , since these d o n ot appear except i n reg i o n (ii ) , in which reg i o n (# si n _ ) n t 4 : . is of 0(1) or s mal ler for all n. The true order in # i s given by the aer od ynamic

" 0 2 )

coefficients in regions (i) a I_ l (iii), and these are of at most, a s expected of / - aerod yn amic f o rces.

- The proper matching pro c edure is to find the s o l u ti ons in the transit i on regions ; at t h e edges of regions (i) and (itl); in addition, the solution is f o und In the Interior region t :_ includ i ng region (il) and the neighbori ng part s of region s (i) an d (ill) where # s in i s of 0 (1). The n the match ing pr o ce _ pr o ceed s fr o m the ma in s o l u t ion i n r e gion ( 1 ), ?

to the tran sition r eg ion at the edge of r eg ion (i), tc the Interior region, to the second _ " tr ans ition region In the edge of r eg ion (ii), and finally to th e main solution in region (ill); by this process the connection formula between region s ( 1 ) and (fill i s established. Wi th 1)) the substitution x = # sin _ (x i s as s um ed to be of 0(I) s o O _ b = 0(#- the equation of motion for the int p rior region i s found to be, to lowest order (0 )): t s _ _ 0 (106) _' ' dx 2 . _ the solution of whl e h is : where B 1 and _ 2 are e onstan te . Thu s m atching th roug h th e in te r i or re gi on is Just a , ,: m atter of matching th e displaceme n t and 8 lope of the n e i ghboring tran s i ti o n ]:egions.

......... , . _ . -_ I - - - .........

= _ ,,, -80- q To the tra n sition region ( o f width 0 (tz-2 / 3)) the interior regi o n appears 0(_ -1 / 3) small; then in terms of the tr a nsition region variabl¢ z - 1 / 3 = x, tl,_ matching process is to be c _r ri e d o ut a t t h e l im it z -, 0.

2 / 3 00Z 2 / 3 ) With th e s u bstitutio n z = sin _ (wh i ch impli e s an re gi on aro u nd = 0 or 180 ° ) , th e equation of motion f or re g ions (i) and (iii) (Eq. 89) b e comes, to lowest order in _ (0( _ 4 / 3)): d2_ 4 " 4 _ I zl B = 0 (10 7 ) d z 2 T he pl us s ign a p pli es on t h e bac k o f the d i sk and th e minus s ign o n t he f r o nt . In terms o f th e v a riable z , the ad _,_ c in g sid e i s gi ven by z > 0; an d the retre a ti ng side by z < 0; o n f i4 sr _ re g ion ( ii) ap p e a rs a s a n e g li gi bly sm a ll (0 (_- 1 / 3 )) a rea at z = 0.

I C o ns id er the b a c k of t he d i s k , i .e. , f rom the f o urth t o the f irst q ua d r a nt . T he d i ff e _- enti al eq uatio n for th e t r an s i t ion re gion i s ( Eq . 1 0 7) !

+ Izl B = 0 dz The sol utio n ag a i n invol v e s Ai ry f unct i ons , _d m ay be w r i tten 7,

':[., . , E( . ] 0

z = 0 gives a* = _ b and b* = :i / _ / 3, so _ where a , b, a*, and b * are con s t an ts. M atching the d i spl a cement and s l ope at - 81 - = _ / 3 b Ai 1 / 3 + _ B i z < 0

\

T h e a sym pt o tic be h av io r of th e solu ti on i n t h e tr a ns i t i o n re gio n is _ he n , _ z o: Z - _ - _ : / _" 1 / 3 ' _ 2 _ / 3 / el 4 ei _ * co nj ng 3 / 2 where _ = _ /7/ 3 z , n o t e t ha t the s ol uti o n ha s o sci llatory b e ha v ior, whi c h i s t h e proper b e hav i or for ma tc hin g th e ma i n s ol ut io n s o n t he r ea r of t h e di s k. Now th e mai n solution in the f ourth q uadr an t (Eq. 9 9) becomes, as 1_-, 2_ _ outer _ , e 12 16 (4 _ - 2 / 3 [z , )-l / 4 7 + i.. , 8 ) e + conjug a ' a nd the main s ol ution in th e f irst qu ad r a nt (Eq. 96) becomes, as @=, 0 ..

5 :' , _ " 1"_ 2 + _' 6 2_' -2 / 3 )- 1 / 4 : ' : iii':i : , 1 + 1C 2 ) _ 1 _1 + c o n j u g at _:-_,_" i -82- Then the matching process, requiring (Bouter)$=2_ r = ( f linner)z = __ in the fourth quadrant and ( f _inner)z=¢_ = (_outer)$ = 0 in the first quadrant, gives the connection formula:

e i J" _d_- J" d

' _ (C l + i C2) _ b l _1 eD 6_ - _ 4 .

i 2 _d_+ 2 4_ d

=2 (C7+iC8) e _b 4

- i d '-Td$+ _ _d$ I

- i ( C7 - i C a) e ¢ 4 _b 4 / (108) m Fin a l l y conside r t he f r o nt of th e di s k, i.e ., f rom t h e sec on d t o t he thir d q u adr ant.

The dif f erenti a l equ al , )n is ( E q. 10 7) :

d_2- Z Izl ,8 = 0 (109)

dz , ' Th e s ol ution is

,:) z> o

2; + b * Bi z < 0

=,Ai (_(z)1 / 3) (_(z)_ / 3)_)

Ma tc hi n g the dis pla cement and sl ope a t z = 0 g ives a* = ¢r 3 b an d b* = a / / 3 , s o

_l:_b,_t,-t_- ) +_ B _ .< o

i

"-x_

-83- T h e a s ymptot ic b e h avio r of th is s o lu tio n is

, 7 . -. -_ : B .._ az b +

/ w h e r e _ = / 7 / 3 IZl 3 / 2 ; t his s olution has t he e xpo nen t i al beh a v io r re q ui r ed f or matching on the front of the di s k. Th e mai n s olutio n i n th e s eco n d quadrant ( Eq. 97 ) b e com es, a s _ - D_ _out er ._ e - / _ 1_22- 1_66 3_ (4__-2 / 3 i z _ - 1 / 4 - 3 e _2 _ 2 3 ' [ C _ _ g -' _ lzl3 / 2 _, - / _ , / ' i' d _ b + + C4 e 2 _ outer ) ffi (_i nner ) z=o , g i ves : Ma tchi ng th i s to t h e i nner solut io n , i. e., requiring ( ___ c 3 e 2 e _ 4 . ] _ ,._: It V ( 110 ) _a

_ - _ - 3 s 1 1 / e _

c 4_ 2 . - _ b _ , i_

i i;}_

The main solution in t h e third q uadrant (Eq . 98) be c omes, as $ - - Bouter - , e it (4 Z # - 2 / 3 i zO - 1 / 4 *4 g 3 1 2 5 e _3

+ c 6 e _ 3 _ 3

t1' g ¢ _ 3 / l M a t c hing thi s t o t he i nne r solutio n , i . e . , r e q ui r i n g ( B inn er) z = .= = ( _ o u t er)_ b ___q f g i ves C 5 e _ 3 3_ d_ e # 1 _ 2 + 1. _ 6 _ . 1 / -Z . . _l / 6 a ? T T ? " Then co mb ining E q. 11 0 and l _ q. 11 1 g ives t h e co nn ect i on f or mula e: e = - _ -C5 e - / 4 J _d$ + y d_ b - / 4 - 4 f r = 2 $ 3 ¢ 3 C4e _ P 2 e _ C6e

' I

-85- The de ri vatio n of the c onnec t i on fo rm u lae a cr o ss t he fou r tr an s i t io n regi ons co m pl e te s the constructior, of the solution aro u nd the disk . It i s also n e ce s s ary however to s t art and finish the so l ution with i nitial and final values of B and _ at some point. It is mo s t conveni en t to start and finish the s olution at _ ) = y; thi s is th e middle of the i n terior regi o n, where it i s e a s i er t o m a tch th e tw o c o nst a nts in the so lu t io n t o a r b itr a ry v al ues of _ a nd _ th a n a t an y other po i nt on the disk (to lowest order, the solution to Eq. 1 06 in the interior region i s line a r in x, so the two const a nts a re e a sily rel a ted to _ a nd _ ). In terms of the tr ansi t i on region, the slope an d m ag nitude of the solution are to be m a tched to B a nd _ at z = 0. Now the solution of Eq. 109 becomes, for sm al l z : B_d 1 ( a + ¢ _b)+(4_] / 3 d 2 ( -a+ ¢ _ b ) z : _ fr o m whic h _ B (_ ' ) = dI (a+ 4' 3 b) !i B ( _ ) = d2 (a - / 3 b ) _ o r i_I _._ _ _(. _/_ ) - 1 / 3 " 2 a - dl + d2 (113) _ " _ 2_ b _ _. _ _ . _(#) 1 / 3 _:: _ d 1 d2 , w here d 1 and d2 are c on sta n ts as so c tat e a w i th the A i ry functio n s ( d 1 - _ 0. 355, _ ( d _ . -_ 0 .259). The n f o r l ni _ . ial c on d t t l_L , Eq. 111 re la t i ng C5 a nd C6 to a a nd b l _ g i ves t h e ma in s o l u t i o n in the th i rd q u a d ran t in ter m s o f i n i t i a l va lu es of ¢ ( ? D an d _ ).

-8(;- Fo r ending th e s olution, Eq . 11 0 re lat i ng C3 a nd C4 t o a a n d b gi ve s th e final v alue s of B(y) a n d _ (? T ) in terms of the main s olution in the second quadrant.

Eig envalues I With the s olution in each of the four o uter regions, and the connect i on formulas between them, the complete s olution may b e c on s tructed over on e rotor revolution.

Floquet theory for a single degre e of fre e dom, s ec ond order equation (s ee A ppe n dix I ) show s that th e eigenvalue s are given by the quadratic equation where _ p= _ ( _ +r) due to _ ( _ )=1, B ( tr )= 0 _ p = _ ( _ Πto _ ( _ )=1, _ ( l r)=0 _ R = _ ( y + T) a ue to B ( lt ) = 0 , _ ( _ ) = 1 _ R = _ ( t r + T) due to _ ( _ ) = 0 , _ ( y ) = 1 an d T is the per i o d o f t he system (T = 2 _ here).

Co m b i ni ng t he connect i on f or m u la e (Eq. 104, 105, 108, an d 112) and the i n i ti al i an d final val ue f ormu l ae (Eqs. ii 0 , iii. an d 113) results in the fo ll ow in g express i on _ : f o r _ f and at _ b = _ + T i n te rm s of an d at _ = _ : 5 - 87 -

/

=e e d2 \ 4 1 / ' 2_ e F3 + e 4 1] where _ tl i_ tl g- _ - _ d , 0 0 @

tl R I 1116)

; "_t 3 ! d,

: F3- _ 3 / _ d ,+3 / _

_: # # 2# 2# g d, _

y a --

" _ t * t

t Ob ta i n i ng _ P " ' . , fro m E q . 115 and su b s t i tut i ng i nto E q . 114 , there _ ,.,. _ , h f o llowing equation for the etgenvalues: ,_i_' .esults ( after some man ipu L_i on, - --e : _ k 2_ + - 2b _ + + 1 / ; e C!

' . a'_.

where :: :,._ • b= _| 4e coslF 1 +F 4)'c°slF 1 F4 i!_ +_e co s ( F 1 + c ° sl F I F4 ._ - _ t

- ._ .... - " 3005279 091

r - -88- S o b is a bm ct ion of # , T , a n d Kp . T h e so lut io n f or t he r oo t s is th en -#6 _ _ cosh b +ni forb> 1 . _ L.4. 1 -1 6 _ ffff 1 - 1 , k = / _ • i _ - _ cos b + ni for -1 < b < 1 (119)

" c °s h -1 Ib [ + 7 + ni f or b < -1

where n i s some integer. This result shows the typical behavior o f the roots of periodic systems. For b < 1 the damping i s fixed at - / _ ( _ '/ 6v) with a change due to b in the frequency; for b > 1 the frequency is fixed at n / rev with a positive an d nega- tive ch an ge due to b in the da m ping; for b < - 1 the frequency is f ix ed at n _ / rev with ] a positive and negative change in the damping. The critical region boundaries are given b y b = 1 and b = -1.

The gener al character of the cr i tical regions and in s tab i lity boundaries in the -# plane, a s obtained from the solution of Eq. 11 "/ , i s sketched in Fig. 12. Because t # is large, it happens that I b l is much greater than 1 al most al way s , so th e critical regions dominate the behavior of the roots. Becau s e of the cos i ne term s in b , th e sign of b c h anges regularly; b m ust o _ course go through zero then, but it does s o very quickly, s o there i s only a very narrow band be tween the Im k = n / rev an d t h e Im n + _ / rev reg i o n s in which lb[ < I. When lbl < i , th e real p art o f k i s - p ( '// 6 # ), +_.

i.e., t h e root is stable for al l / _ an d V ; thus th ere must alw ays be a band of stability _ surro un di ng the transition from n / rev to n _ / rev. These characteri s tics are illus- _ ; _ : trated in F ig . 12. The locus be tween tL _ , critical r eg ions ha s a rat h er fine structure , which wo ul d be d iffic ul t to obtain numerically. A root locus for varying / _ or - 89 - ( a verti c al o r h orizontal sec tion in Fi g . 1 2) in the vic inity of a cr itical r eg io n bounda r y would in quick succession mov8 from the RHP (u n stable) to the LHP (s table) with fre- quency fixed at n / rev, rapidly move from Im k = n / roy to Im 1 ---n + _ / rev in the , RH P with damping given by - / _ (_ / 6y) (which would be nearly con s ta n t becau s e the c r itical region boundari es are s o clo s e) and then move s from the LH P into the RHP with frequency fixed at n + _ / r e v.

F igure 12 show s that for a given / _ th e s y s tem i s s table for a larg e e no u gh 7 .

Po s itive Kp i s s tabilizing, tend i ng to decr e a se the size of the in s tability r e gion s ; neg a tive K p is dest a b il izi ng in th J s sense• The rotati ng natur a l f requenc y o f th e fl a p mot i on, y , does not enter the hi g h D case to order P0 ( the aerodyn a m i c spri ng dom i na tes the cen t ri f ug al s p r ing unt il o rder p l ); t h is is c o ns i ste n t w i th t he fa ct t ha t th e cri t ic al re g io n s d o m inate th e h igh be ha vior , s o th e f re q uency o f the mo tion is fix e d at a multi p le 5f _ y t rev.

f A compar i son o f these anal ytic al res ul t s with the results of numeric al c a lculat i o n s ; indic a te s t h at the hi gh _ s ol u t io n is go od down t o _ = 2 .5 or s o . Thu s nu m er i c al c al cu - lations are re q uired to J o in t h e loc ! f rom D = 0.5 t o 2 .5 sa y ( for 7 ne i t h er smal l nor l arg e ) . The be havior the o re ti cally pred i cted fo r th e locus a t l a r g e D (in pa rticular the r a pid m o vements between I mk = n / r e v a nd n+_ rev , a nd p er h ap s -- fo r V n o t too lar g e -- be tween t h e RH P and th e LHP ) actually d o e s show u p in th e numeric al c al c ula- +i ons of th e s t a bility (above D = 3.0 sa y ) ; s uc h be havi or o f a n um eric al s olution mi g ht be que stion ed w i t h out t h e p erturbation sol ut ion t o p rovide a g uid e t o w ha t t o e xp ect. I t

i

is unfortunate that t h e boundary o f th e i nstab il ity reg i on for _' / 16 of order 1 is f i r s t en c ounte r ed at mode r ate _ ( a ro und D = 2 .25 for s mal l Kp; se e Fig . 12) a nd so ca nno t be obtained by perturbation techniques ( to the order explored anyway ). Because of the small time constant in ;he main solutions ( 0 ( _-1 )) an d the four tran s ition regions ( of , width 0 ( _-2 / 3)) , a n umerical c alculation of the roots for truly large would be difficult; the p ertur bat io n t h eor y h a ndle s the s e s i n g u la r p r obl ems anal yt i c ally , and the calc tt la - tions th a t rem ain are no n s ingula r, short, and s i mple.

Flap Rate Feedback T he use _. llap r ate fee d b a ck ( KR _ 0 ) c ha n g es the sc l u t i o n f o r l a r g e D f u nd a - m e ntally, be c au se the a e rody n amic d a mping ( -K R M _ i s th en th e s am e ord e r a s tb * aerodynami c s prin g (M s-K p M_ . 2h e d e r i vat i o n of th e mai n so lution s i s s impl e r th e n, and only th e tran s itio n regio ns n e ar $ = 0 a n d 1 8 0 " ar e r eq uired. The m a i n s olut i on s ar e obtai ne d u sing th e s ub s t i tut i o n

e xp S p

w it h p e xp anded as a s er i e s in D : 2 1 P -- / _ P 2+ / _ p_ l +P0+ _ Pl+...

Mak ing t his substitut i on in E q . 89, th e t erms of l ike order in g i ve 0 _ 4): P 2 =-r T R i{s in _ )2p-2 : 0 { / _ 3 ) : 2P_2P_l = - r T sin _, + P-2 + KR i (S in $ )2 P-1 L g _, .- 0 ( D 2): p_l+2P 2p _ + p 2--r T (I+KR) P_2+ _ s i n , +KR ) P. 1 _ + K R _" (s i n , )2P 0 sin * ( 0 0. * sin *)

, , ]

-91- 0_ ): 2 P_lP0 +2 P_ 2 Pl + _ ) _l--'- r Y (l + KR) P_l + _s i n_ + Z P0 + K R _ (sin,) 2P l + c os _ + _ Kp sin , The first solution is given by P-2 = 0. Then the order # 3 equation gives also = 0, and the order #2 equation gives P-1 P 0 = " K- - R(c o t _ + K p ) or Th e o rder # e qu ati on g i ves _ 3_ _( 1 1_ c os# 2Kp 1 _ 3KR2 s in _ b _ . or L ' he n the s ol uti o n f o r _ i s

k _ ae- - ---- 1+_ + or. - s ) " _ ) lb; 2 _ _ .l n _b 3KR KR ( 1 2 "

. _ " B 1 (sin , 1 (tan _ e " i A,, \ I -92- where B 1 ;s a con s tant. This i s the _ olution of the redu c ed equation, and s o i s indepvnd- ent of _ to this order, i.e., i s the result of a balance of aerodynamic d am ping an d ae r od yna m ic s pring terms only. The solution i s not valid when s in _ b i s too s mall; an d it i s exponentially growing (unstable, at l e,. s t in the outer region) if Kp / K R < 0.

I The second solution i s given by P-2 = - r (' // 4) KR ( s in _ )2 or The order _ 3 equation g i ves P-1 = - r 6 Z (1 + 2KR) sin _ b o r Y_ P-1 d _ = r 6 _ (1 + 2KR) co s an d the order / _ equ dti on give s 1 - 2KR Kp PO =- r 8 _ (I+KR )+ K; c°t * + _ R or P0d _ . KR _ nsint+ -r 11+K R t Then the s olution fcr _ is _ _ ' I _ - _ (sin , ) exp - r _ KK ( ? , - si n 2 _ ) - _ -8 (1 + 2KR) _ o s , - .

!_ + 2 (I .]" + KR * + 0(P "11 1121) - 93- w he re _ 2 i s a c on s tan t. Th is s ol utio n does i n vo l ve _' , i.e., it in v ol v e s t he iner ti a terms in t he e q u at i on o f motion; th is s olut io n i s al s o not v al i d whe n s in ¢ i s too s mvll.

T h ere are tr a nsition regio n s bet w een the adv a ncing an d retreatin g sides , ne a r I _ / _ = 0 a n d 180 " . As for the l r = 0 case, region (i l ) is an iuterior regio n lying entirely -o withi n t h e 0 0 _ - 2 / 3) tr a nsitio n regio n ; the solution i s rnat e hed thr oug h th iz interior region ag ain by matching me - l isplace m ent a nd slope o f the tr an sition re gi on so l ut i ons at z = 0. Wi th the s ub s t i tution z = 2 / 3 s i n _ , th e equation of m o ti on i n the tra n sit i on reg i on beco m es, to lowe q t order in # (0(#4 / 3 ) ) dz 2 where t he plus sign app l ies on the bac k of the di s k a n d the m in u s si g n on _h e front.

It can be s ho wn th at the sol u t i o n s of th i s ,. :l u atlon wil l h ave t _ . _ 9 ro per asymp t o t i c , be havior for match ing to the two ma i n so l ut i on s . U n fort u nate ly the , _ o l u ti on of this eq uat i o n is n ot av ail able in ter ms of c l as si cal function s . The behavior o l the solutio n s ._ for l ar g e and s mall z would have to be fou n d, probabl . v lar g ely by nume z lc al metho _g , be fore the matchin g procedure co ul d be c a. • i ed out.

?

g

|

L i

._ -94- % _ Ap Plic a bility of the Four Cases . i: This s ectio n wi ll c on sider the r a nges o f / _ an d Y over which the f o u r ._ cases investig a ted above are use f ul. Perturbation th eory is based on the , ex #a nsion of q, an tities in terms of a very small or ve ry large parameter.

In r:a ny prob l e m s h o wever, th e results a re usef ul, even quite a ccurate , f a r bey o nd the l imits for w hi ch th ey a re th e o retically v al id. It is th e s e perturbati on s ol ut io ns, extend abl e up o r d o wn fr om tr ul y s m al l o r l ar ge values o f the pert urbati o n par a me te r, which are of most va l ue. They ma y be f o und o nly by c om parison wi th e xa ct s ol uti on s, usu all y obtained by n umeric al me tho ds, f o r mo derate values o f the perturbation p a rameter. _ An oth er question ab o ut th e range o f validity o f th e s olut i on s arises in th is 7_ proble m because th ere a re two par am e te r s , _ an d _ ' , which are avail- I - ab l e as perturbation par am e te rs. In th e perturbation an al ysi s based on I , one par ame ter, say _ , 9ma ll or l arge, th e so lut ion i s derived under th e assumption that the o th er par am eter is of order 1, e.g., _ = 0(1). This raises the question of th e v al idity of the solution when the other pa r am eter is itself very small or very large. It may ha p pe n that the results ar e still valid when th e other par am e te r is outside its a s sumed ran ge , but this must be checked in each case. The ordering process of pe rturbation te chniques provides a quantitative fr am ework for making th is check. It is u s u al ly qui te simple to de te rmine what ran ge the other parame te r must ha ve so that the assumption s made aboutthe ord e r of terms are still valid, ,.m , u , _ . . ,,.. _ ,. _ , - -.,. _ = . w ,. _ - _ %,, q _ .. ........ T --- _ - .......... -,---- - - ......... ,,- r -95- Extendin g the analysis into ranges of the o th er parameter where the ordering / assumptions were violated is an other mat te r; it of course means an entirely i new case to be considered an d an alyzed by perturbation techniques. The _ , ! results of th e an alytic solutions obtained above were compared wi th numerical calculations ( pe rformed by the author) of the roots of the flapping equation, primarily for m oderate and sm al l Y over a wide r an ge of _ ; these c al cula- 1 , . ._ n s were similar to those re.uorted in Ref. 2.

I •

The small / _ results a re good out to _ = 0.5, which is a very useful r an ge. These re sults are v al id for al l T , since th e order of _ does not ch an ge the te rm s retained in the s mal l _ analysis. The lar ge _ results a re valid above / _ = 2.5 or so, which Is als o a good ran ge . Here however _ either very small or very large will violate th e ass um ptio n s made about _ th e order of the terms in deriving th e large N solutions an d so th e resu l ts may not be v al id in th ese corners of th e 9 / - _ plane.

The small and large V results, to the order in vestigated, a re really usef ul only for truly sm al l or tr ul y large T , alth ough the results are qui te informative. The small T results are accurate up to T = 2 or 3. The | [ limitations of th e 0( T ) solution , as discussed in th e sm all _ an alysis, p re- vent the accura te use of the s olution for mod era te or even reasonably small P • _ V ( _ = 6 s ay). Th e large _ r es ult s giv e two re al root s , so a .'r , obviously l imi ted in u s efulne ss . It i s unlike l y that _ wou l d be l arge e nough to requi re i

! L!

-96- r_ this solution. The large _ solution is good down to y / 16 = 3 or so, which _! is actually a very good range in the lerturbation parameter; it just happens that for practical rotors T falls far b e low th is limit. Letting / _ go to zero , does not ch an ge the order of th e aerodyn am ic coefficients (because of th e constant te rms ) so th e resu l t s f or lar ge and sm al l _' sho ul d be g o o d for all _ of order 1 or smaller. Lett in g / _ go to inf in ity does change th e order of terms in the anal ysis, so both th e s mal l T and large T results m a y be in vali d f o r very l arge / _ (ab o ve / _ = 10 say).

It would be ve ry desirable to be ab le to use th e small / _ and sm al l ) ' results to construct composite root loci which are reasonably accurate for all values of / _ an d ) ' likely to be encountered in helicopter ro to rs. The s mal l / _ res ul ts would be used up to ab out / _ = 0.5; then th e s m all ) ' res ul ts would be used up to _ = 5 or so. The major obstacle to th is is th e lack of an 0( T 2 ) an alysi s for the critical regions; th e s m all T an alysis presented here, which was carried only toO(' / ) in the critic al re gio n s, is not adequate for the accurate constru ct ion of loci for any except very sm al l Y .

Pos s ible ext e nsions of the solutions described he re inc lu de the fo l lowin g : a) Extend th e sm al l _ re sults to 0( _ 2 ) for Imk = _ , and to 0 (_ 3 ) or 01 _ 4) to h an dle th e I m = 3 / 2 critic al re gi on.

,m b) Extend the s m al l _ results to O ( T2 ) in th e critical region s .

l " -97- r _ + : c) Exte n d the large _ results to 0 0 ' - 1 ) .

d) Extend th e large _ t results to 0(_ t- _ . /' The most useful would be the 0 1_ 2 ) and 0 ( _ -I) results. A s m all _] solution

' i

- reliable to _ / 16 = _ or so ( which an 0(_ 2 solution should accomplish) could + be combined wi th the small _ t s o lution to con st ruct accurate comp o si te + 1 ,J ci. Together these results would th en cover most of the r ang e o f _ t and _ ._ +_ o f conventi o nal rotors. The large _l case extended to 0(_I "I ) should be able ;i to p redict accurately the first in stability boundary of th e _ t lo o t, which o ccurs at _ t = 2 to 2.5. These two cases are however also th e o nes involvin g _ r the most work. ii APPLICATION O F PERTURBATI O N TECHNI Q UES T O HE LI C O PTER DYNAMICS i ....

This sectio n returns to the question of whe th er perturbation te chniques

+ l

m ight be profit ab ly applied to m ore complicated or mo re re al i st ic dynamic systems th an the one cor _ sidered here. A s pa rt of th e an swer, consider what these techniques will not do: obviously th ey c an not give results for cases whe re th ere is no parameter th at is ei th er s m all or large, for ex am ple , when . _ = 16 and _ = 1. However, the four cases consi dere d toge th er cover a g ood de al of the ran ge s of _ an d Y , and wi th primarily analytical re sults.

For ma _ helicopter s the s m al l _ ca se will be quite satisfactory al one.

_ ' What the te chu / ques can do als o includes: L , / ;_: a) Sin c e they give an a lyt ic solu t .%ns the y provid e more i ns i ght into , 2 _ . _ th e pr o blem, a s w ell a s specif ic des i g n c ri te ri a fo r th e system; this feature ; ?

L , _L is part i c u larly impo r ta n t f or nonline a r or t i m e-v a r yi n g sys t ems , w h ich !_ hav e p r op e rt ies m u c h d ifferen t f r o m tho s e of const a n t coef fi cien t , l ine a r sys te m s .

& i h;, Pe rtu rbati o n me tho ds c a n find, a nd h a nd l e, cases th at are ve ry sens iti ve to th e pa rameters , o r tha t a re di f ficult to s ol ve a cc u r a te ly b y ! n um eric al m e th o d s.

i c ) T he m e thod s p rovide m ore insight in t o the r ath er unusual be - f h a vior of th e s ol ution of per i odic syste m s, by sh o win g e x p l icitly how th e l ', i peri od ic _ oeffic i en t s m o di f y th e tr a nsi e nt so l utions a nd why th ey g i v e til e ! root lo ci th eir ch a r a c te ristic be ha v i or in th e critic al re gi on s .

i i d) Finally, even if the technique s are not used to find the comp l e te

t

' , so l ution, it only takes a l ittle work to f ind out w here th e problems a re ! (e. g ., critic al re g ions and tr a nsitio n re g io n s) and w h a t t h e order of th in g s is, which information would be of invalua bl e he l p in the numerical an alysis t of a sys te m.

_ i_ _ .

The extension to more degrees of freedom or m ore real istic aero- t dynamic coefficients would certainly make the analysis m ore comp l icated.

,o I n gener a l however _ st u d y - ana l ytic, cem pu t_ t t_ l al, or experimenta l -- / - 99- of a system becomes more complicated as the accuracy of the modelling of the true system increa s es, and perturbation techniques are not expected t o be an exception to this rule. Regardless of the s ys te m being studied, the position perturbation techniques occupy between simple linear an alyses I an d complex nonlinear numerical calculations makes th em a very po werf _ fl tool for providing both exact solutions and increased underst an ding of problems in rotor dynamic s .

The problems in rotor dyn am ics to which perturbation te chniques might profitably be applied am ount to all those involving nonlinear or pe riodic coefficie n ts, and there are m an y o f tho se. There is s o me additi on al t work that m ight be d on e wi th the flap p ing dyn am ic s proble m ( on e de g ree of f re edom), in clud in g for ex am ple _ " a) te e te ring rotor; " b) cantilever blade, wi th correct aerodynamic _ oefficients; c) incl u sion o f stall an d c o mpres s ibility in the a er odynam ics.

The solutions found in th i s paper m i gh t be extended to 0( T 2) for th e s m all 7 case and to 0( _' 1) for th e large _ case. These extension s m ight prove very useful, or only slightly more s o than the s olutions to th e order pre- sen te d here; but th e y should b e examined for th e simple s ing l e de gree of • freedo m probl em before being consi de r e d for m o re co m plicated s yste ms .

i

! L

-100- Even the small / a solution could be t a ken a little farther -- for example, to 0( / _ 2) in the I mk = ½ critical re g ion. Starting with two degree s o f freedom, pos s ible probl e m s in c oupl e d dy n amics i nc lud e , a ) flap d_aam ics o f a gim b all ed rotor; b ) pitch / flap dy n am i cs; c) fl ap / l a g dy n am ic s .

T he pi t ch / fla p system is the prob l em o f rotor flutter. T he fl a p / lag s y stem is pa rticu la r ly rich in possib l e vari a tions; th e p roblem h a s periodi c co - e ffi c t ents i f / a > 0 of course , but it _ s also nonline a r (even in h o ver) d u e t o the in e rt ial cou pl in g of the degrees o f freed om . I t is moreover very sensitive to b l ade root g eo m et ry , so th at a n art i cul ate d an d a cantilever b la de h a ve q uite different dynamic c h aracterlstlc g . Problems in co upl ed dyn am ics wi th more de grees o f f reedom incl ude " a ) pit c h / flap / l ag dynamic s ( th r ee de grees of fre e dom); b) fl ap dynamics of an N - bla de d rotor (N _ 3) with flapping f ee dback contr ol in th e f ixed system (at l ea st four degrees of fr ee do m ).

Whi l e for these pr ob le ms the advance ratio / _ would probably be of m o st v al ue as a pe rt urbation parameter , there wil l likely arise prob l e m s whe re oth e r param e ters are al so usef ul . A s long a s a reas on ab l e m o de l i s chosen for the syste m , and as m uch e f fort i s given to the in terpretati on I of the solution as to its derivation, pe rtu rb ation techniques sho ul d prove - 101 - q u it e use f u l in p ro v idin g inf o rm a tion abo u t these problems, and many oth e rsin rotordyn amic s a nd a e r odyn am i cs.

i T h i s p a p er h a s d_ , monst ra t e d th e m e thods o f pe rt u r b a tion th e o r y a n d " ; ha s p r ovi de d e x am pl es o f t h e i n f o rmati on ab o ut dynamic s y stems w hich i m ay b e ob ta in ed us i n g th e m . The te ch ni q u e s have p r ov e d very us ef u l fo r t h e p r ob lem st udi ed. I t sh ou ld no t be c on cl ud ed ho w e v er th at th e techniques p rese n t e d are al l there i s t o pe rt u rba t i on the o ry; th ere are r , _ an y mo re m e thod s tha t have no t b ee n touc h ed on h ere. P ert u rbati on the o ry is I a p owe r f u l,a nd yet n o tve ry s o p h isticated , m a th emat i caltec hn i qu e which sh oul d p ro v e ve ry u sef ulin a n a l yz in g so me o f th e p roblem s of hel i c opt er dynam ic s.

)

/ -102- Appendix I. Eigenvalues of a Periodic System Con si d er a ge n eral sys tem of differe nt ia l e qu a t i on s wi th p e r i od ic c o e ffi c i e nts ; thism a y b e r e du ce d to a syst e m offir s t ord e r e qu a tions, a nd l - m ay th eref o re b e w ri tte n ( in m a t rix no ta t i on ) a s X = where A (t ) is peri odi c: A (t + T) = A(t ) . It may b e s h ow n that t h e s olu ti o n t o thi s dif fere nt ia l e qu a t i o nca n be o bta in edi nth e f o rm = Z l qi(O ) e)'l t u _t (t ) _A . 1 _k Th e _'i are the e i genvalue s ; th e e i ge n vector s ui are periodic, ut(t + T) = ui(t); an d t he n umbers q i (0 ) are co ns tant s obtained from th e i ni tia l cond i t i o ns .

T he theory th at sh o ws this i s cal l ed F l o q uet the ory. The s o l utio n in thi s for m i s a di rect exte ns io n o f the nor mal sol u tio n f or a co n stant coe ffi c i ent di ff e re nt ial equat i on, wh i ch is cha r ac.eri z ed by constant e i ge nv ectors.

Th e e igenval u es ki may be ob tained by th e fo l l o wing p roc e du re .

T he eq u at io n _ = A_ where #(t) i s a m atrix, i s integrated over one period, f rom t = 0 to t = T, with in itial conditions _ (0) = I (the u n it m at rix). Then if Xci a t e the

i

eig _ envalues of th e m atrix C = t (T) -103 - _2 r o ots X i are give n by or / 1 inXc

x---¥

Whi l e t h e ro ot s _ ci (a s e ig envalues of a r e al matrix C) must appear as r eal numbers or complex conju g ate pairs, th e ei g envalues Xi are under no such restriction. The root loci of periodic systems are th us characteri z ed by th e type of behavior sketched bel o w, Im X ?

4 .

/ -1 0 4- T he hor iz on ta l port i on s o fthe lo cica n onl ya pp ear atIm _ = n / re v or n + _ / r e v. I f th e p a r ame t e r b e l vg v a ri e d, f or e x am pl e t he a d va n ce r a t i o / _, i s su c h th a t a t_ --- 0 th e syst e m is notp e riodi c ,th e nth e roots at pointA , _r e c o mple x c oLjug a t e s.A s _ in creases th e pe ri odi ci ty o f th e system in c r ease s , a nd the roots start mo ving towa rd n / rev ( o r n + _ / rev) l ine s re m ain in g co m p l e x co n jugate pairs th oug h ). At s ome critical _ (the poi n t Y ' , on the locu s ) the loci reach I m ; k = n / rev, and the n for sti ll larger / _ t h e f req uency re m ai n s fi x e d while t h e real p art of o ne root _s decreased an d th at of th e o ther is i n crease d This b eh a v i or should be co m pared w i th th at of t w o root s of a co nst a n t coeff i c ie nt sy s te m which s tart out a s comple x co nj u gates, m eet at the re al a xi s, a u d th e n p ro ceed in opposite di re ctio n s along th e real axi s . The ex is tenc e of periodic coefficie n ts i n th e eq u ations of m ot i o n ge ne r al izes thi s beha vi or so that it ca _ occur at an y Imk = n / rev or n + _ / re v, not just Imk= O. The pro pert y of the solution th at al lows th is beha vi or is the fact th at the eigenvalue s _l (t) are the m selves pe riodic.

For a sin gl e degree of freedom, second order syste m , let xR be the so l ut ion obtained from integrating the equation with initi al condition s _ (0) = 1, x(O) = O; and let Xp be the so lution with initial conditi on s _ (0) = O, x(O) ffi O. Then the ro o ts _ 'e are given by the quaudrztte equation -105- Appendix_ H. Solution of the Secular Equation The me th od of multiple time scale s often leads to an ordinary differenti al eq u ati on of the form

- + (a + i d) / 3 + (b + Ic ) _ = 0

where _ i s a complex quantity, and th e constants a, b, c, and d are re a l.

Je tting D2=d 2-(b 2+c 2) =d 2-[(b=|c)[ 2 it m ay be verified that the solution of th e above differenti al equation is V2>0: fl - e -a _ [A(d - D + l(b + ic _ )e ID _ j • A( d + D + i{b + ic))e "I D o l wher e A i s a compI _x co n stant D2 = _ : _ ffi• "a @ [A(( 6 + iCo + ie)) _ + I) + B(d + i(b+ ic) ) ] wh e re A ar _ l B ar e real cc a s' _ uts " D _ ,< 0: j 3 ffie "a @ [A(d + tD + i(b D _

J

i i " lc)) e .l _ ] I +B _ d - I D + i(b + ! wlmx _ A and B ar e real emstan _

' _

-1 0 6- The limiting ca se b = c = 0 give s D = d so the solution is = A e-(a + i d)¢ where A is a co m plex constant # The region of decreased stability, i.e., the region where th e real part of the eigenvalue becomes mo re po s itive ( th e critical re gi on), is given by D2<0. The boundary of th e critical re gion b J D2 = 0. One root in th e critic al region beco m es less st able, but th e other be co m es more stable. Furt _ er- m o re , in s ide th e critical re gi on (D2<0) the re i s a change in the re al _ art of the root but no chang e in th e imaginary pa _ i.e., the frequency; while outside the re f _ on (D2>0) the ll i s a change du e to D in the f re qu en cy, but no change du e to D in the re al part. This be l _ vior follows th at expected of the t lgenv _ luc s of periodic sy s te ms ( _ ee Appendix I). I n deed, D2 i s a measure of the relative effects of the _ and _ te rms in the diffe re ntial equati on ; th e fo rme r u s ually comes from th e constant co e fflcfent s in th e equation of moti on and the latt e r fro m the pe ri odic co e flici e_ t s . I / -107- Re[erences 1 Si s si n gh, G.J., "Dy n am ic s o f Ro tors O pe r ati n g at Hi gh A d * ¢a nce R at i o s, " Journal of the A m e r ic a n Helicopter Society, Vo l 13, No 3, Jul y 1 96 8 ' 2 P e te rs , D.A., a nd Ho h ene m ser, K . H . , "Applicatio n o f the F loqu et Tra n sitio n Matrix to P r ob l e ms of Lifti n g R otor S tability," Jo ur nal of the American Helicopter Society, Vo l 16, No. 2 , Apri l 1971 3 Cole, J.D., Perturbation Methods in A ppl ied Mathematics, Blaisdell P ub l ishing C o mpany, Wal th am, M a s sachusetts, 1968 4 P e a rs o n, C.E., "On a Dif f erent ial Eq uation of Boun da r y L a yer T ype, " Journal of Mathemativs an d. physics , Vol 47, No 2 , June 1968 _ "

2a.. '

if Fig. 1 . Root loci for varying _ , based on the small _ results (to order / _2) ; V = I and / _ = 0 and O. I " P Fig . 2. Root loci for varying t _ , based on the small _ results (to order _2 ); fixed for each locus (R eX = - y / 16 for _ = 0)

-.4

I mX: .99

,, , ,,, • ,,,, _ ' _ I ',

0 . 5

, F i g. 3. Lin e s of c onstant Im P , and R e_ ,, based on the smal l # r e sults ( te order _2 ); v = I, Kp " 0; -----Im p , ---- Re _ , w circled values of Im _ in dicate areas in which Im P , is constant.

1 i

/ Fig. 4 . L [_ es of constant Ira ) , and Re ) ,. based on the small _ results " (to order _ x 2 ); V = 1, Kp = O.1; --.---IM P , , - " -'- " " -" Re) , , circled values of Im ) , Lndicate areas in which Im _ is constant.

8 5

T 6 - !

_35

(None f or = , = I)

,. Cri tica l reg i on f or v = I .OIq

0 .5

Fig. 5. Lin e s of constant ImX and Re _ ,, based on th e small _ results • (to order _ 2); _ = 1, Kp = -0.1; ------Im;k, ....

Re _ , circled v al ues of Im; _ indicate areas in whi c h Im P , is " constant.

|

/ i

0 I 2 !

/

Fig. 6. The ratio (-M _ / M _ ) , which governs the effect of KR and Kp

t

I for s m all _ . .

.!

/ / 7 ""

l

, u . 2

• Fig . 7 . The averages of the aerodynamic coeff i cients , which g ive the roo t s for small y. ,

- I 0

R e X

Io I - L 8 I _ ' o r I mX = 3 / 2

I,_1 '.

21-_, " _-"_, _,,,,,,,,,.,..,..,.:...,_,__

" I _ _f j' /'/ . N eor Im X = ! / 2 .... ,. ".,.. - .. .

1 ...... _ _ - ...... l .... l t , i_ii, . -'"_ .- 1

0 I 2 3 4 5

/ = ¥ _ . 9 . The Kp vs. _ b oundaries for stability in the center of the • _ critical region, based on the mall Y re s u l ts (to order 7 1 ; ' for roots n ear ImX _ =I / _ , I , u d 3 / 2 .

! ,

'1

, !

.875

w t

3-

/

Z-

Negative dampin g

on r et r eatings id e

J t /, kR _ ' // e a o ii II I I I I IIII I I I I II I I I I I I II I I

' I I- I "- -

- i

0 . 641 t 2 3

i . th e rotor _=_ : -p _ - x R_ )<o (e n t e r =_ o tL _1=_ _ c u o), Fi g . l I. F l ap rata _ db _ required fo r m l _ atlve diu n_pln g over part of Stab l e __ _ -_1 - Unstable ' w

~ 25

Fig. 12. Sketch of the chara c teristic behavior of the critical regio n botmdaries and s tability botmdaHea for large / _ ; ----- boundary of region in which Ira, Is fixed at n / rev or n + _ / rev; ....

{ boun dary of region in which the real part of ooe root is positive, i.e.,unstable.

IM_ -- A I _ - Cm m h °