Calculated Hovering Helicopter Flight Dynamics with a Circulation Controlled Rotor

NASA TECHNICAL NASA TM 78,443

MEMORANDUM

( Y b S 8 -25-78443) CALCULATED HOVERI86 N78-10002 H E L I C ~ P F E R PLIGAT DYNAMICS 2ITH A CTRCdLhJ!IDB CONThOLLED BilT;@R ( U S A ) 40 p HC

, /$r ~ 3 7 c?;<r, OIA U n c l a s

G3/01 52084

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CALCULATED HOVERING HELICOPTER FLIGHT DYNAMICS WITH A CIRCULATION CONTROLLED ROTOR Wayne Johnson Ames Research Center and Aeromechanics Laboratory U.S. Army Aviation R&D Coni~nand Ames Research Center Moffe t t Field, Calif, 94035 and Inderjit Chopra Anies Research Center Moffett Fieid, Calif. 94035 September i 977 b

CALC'ILATEL) EIOVERZNG MELTCOPTER FLIGHT D Y N A M X C S WLTH A CIRCULATION CONTROLLED ROTOR Wayne Johnson Ames Research Center, NASA and Aeromechanics Laboratory, U.S. Army Aviation R&D Command Ames Research Center Elof f e t t F i e l d , C a l i f . 94035 Tnderj i t Chopra NRC Postdoctoral, Research Associate Ames Research Center, NASA SW4MARY The f l i g h t dynamics of a hovering helicopter with a c i s c u l a t i o n controlled r o t o r a r e analyzed. The influence of the r o t o r blowing c o e f f i c i e n t on the calculated, r o o t s of the longitudinal and l a t e r a l motion is examined fox a range of values of the r o t o r l i f t and the blade f l a p frequenoy. The control c h a r a c t e r i s t i c s of a h e l i c o p t e r with a c i r c u l a t i o n c o n t r o l l e d r o t o r a r e discussed. The g r i n c i p l e f f e c t of the blowing is a reduction i n the r o t o r speed s-tability derivative. Above a c r i t i c a l l e v e l of blowing c o e f f i c i e n t , which d.epends on t,he f l a p frequency and r o t o r l i f t , negative speed s t a b i l i t y is produced and the dynamic charactexistice of the h e l i c o p t e r axe r a d i c a l l y a l t e r e d . The handling q u a l i t i e s of a h e l i c o p t e r with negative speed s t a b i l i t y a r e probably unacc.apt.able without a s t a b i l i t y augmentation system.

INTRO DUCT10 N A c i r c u l a t i o n controlled r o t o r uses blowing a t the blade t r a i l i n g edge t o control the r o t o r blade l i f t , i n place of the geometric p i t c h control of conventional r o t o r s . The blade s e c t i o n l i f t is given by the prod.uct -1-

C

I

I I

I of t h e d y r m i c pxetxure, chord, and l i f t coeff i c l e n t -- L =

V c c (o( ,cP) --

3 2 JI

where the l i f t coefficient, depends now on the blowing c o s f f l c i e n t Cp a a well as on the angle-of-attack, The blowir?g c o e f f i c i e n t is defined as

h V /() 4 ~ a c ) , where hVj is the j e t momentum . With a conventional

I C~ r o t o r a ~ n r t v r b a t i o n of t h e blade inplane v e l o c i t y V influences the lift With a by changing the s e c t i o n dynamic pressure antl angle-of -attack 6 c i r c u l a t i o n controlled r o t o r , inplane v e l o c i t y perturbations influence t h e l i f t alno by changing the s e c t i o n blowing c o e f f i c i e n t :

an& c FP = bcq/ b~,, . Assuming that the j e t

where CJI, = h c d h * momentum is f i x e d , w e have used The a d d i t i o n a l l i f t change due t o inplane v e l o c i t y pertl~xbations with t r a i l i n g I edge blowing w i l l a l t e r the dynamic c h a r a c t e r i s t i c s of the c i r c u l a t i o n controller! r o t o r conipred. t o conventional r o t o r s , One concern is the This r e p o r t influence of t h e blowing on the helicopter f l i g h t dynamics,

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'1 presents the r e s u l t s of an a n a l y s i s of the hovering helicopter f l i g h t dynamj.cs with a c i r c u l a t i o n controlled r o t o r .

EQUATIONS O F 14OTION I Consider a hovering helicopter with the center of g r a v i t y a distance

i

h below the r o t o r hub, and on the s h a f t a x i s . Then the l o n g i t u d . i m l / l a t e r a l d.ynamics decouple from t h e h e l i c o p t e r v e r t i c a l and yaw motions. The degrees

i

of freedom of t h e helicopter r i g i d body mot,ion a r e the longitudinal v e l o c i t y

i 1

, the l a t e r a l v e l o c i t y iF, the p i t c h angle By, and t h e r o l l angle .

9.

For helicopter f l i g h t dynamics analyses it is generally s u f f i c i e n t to consider a q u a s i s t a t i c mod.el f o r the r o t o r hub r e a c t i o n s i n response t o s h a f t motion, I > I c o n t r o l , and. g u s t s . With %he q u a s i s h t i c o r low frequency s o l u t i o n , the ;

- 2-

I

Ci r.

l i I I ,

f r o t o r model rloes n o t odd degrees of freedom to t h e system, n t h e r t h e m t o r i a represented by a oet of s t a b i l i t y d e r i v a t i v e s . C y c l i c blowing c o n t m l

5,, COSY) + d l \ a i n v is coneldered. as w e l l as conventional c y c l i c

dry

p i t o h c o n t r o l 4 8 - a,, COSY + 6 i n 9.

Thus i n l a p l a c e form t h e equations of motion f o r h e l i c o p t e r l o n g i t u d i m l and l a t e r a l dynamics i n hover a r e : * Here s is t h e Laplace v a r i a b l e ; g is t h e a c c e l e r a t i o n ( s e e r e f e r e n c e 1).

due t o gravitjr; anrl uG an!?, vG a r e t h e l o n g i t u d i n a l and l a t e n l g u s t v e l o c i t i e s .

The r o t o r couples t h e hel.icopter longitud,inal anrl l a t e r a l motions, b u t it I s convenient and o f t e n s u f f i c i e n t l y a c c u r a t e t o decouple t h e equations and The drcoupled analyee t h e l o n g i t u d i n a l and l a t e r a l rlynamics s e p m t e l y .

l o n g i t u d i n a l and l a t e r a l equations of motion are:

.

Thesa equations i n v e r t t o

- x M ) s + @;M ; a.nd s i m i l a r l y f o r the

where A = S3 - ( ~ , ~ ' f ' M ~ ) S + (XZq

u l a t e r a l dynamics CHARA CTERISTIC EQUATION The llft incrament due to t r a i l i n g edge blowing is r e l a t i v e l y i n s e n s i t i v e t o the s s c t i o n angle-of-attack perturbatlions about a n unstalled However, the s e c t i o n l i f t change due t o a v e l o c i t y perturbation operating point.

where 504 = X/V 16 the induced angle-of-attack change due t o a s e c t i o n

This l i f t force has a component i n t h e plane of the d.isk v e l o c i t y increase.

Hence the blowing ( the inducod d r a g ) , a s we12 as normal t o the disk plane.

influences only the r o t o r acrodynamic f l a p moments and. hub f o r c e s due t o inplane v e l o c i t y perturbations of t h e hub. The handling q u a l i t i e s of the hovering helicopter l o n g i t u d i m l dynamics a r e determined, primarily by the p i t c h damping d e r i v a t i v e M 9 and the speed s t a b i l i t y d e r i v a t i v e M U ( s i m i l a r l y

the l a t e r a l dynamics depend primarily on the r o l l tsmpiw L P ' and the speed

The p i t c h and r o l l damping produced by the s t a b i l i t y o r cI.iheclra1 e f f e c t 4).

r o t o r a r e due t~ the time l a g of tip-path plane tilt r e l a t i v e to the sh%f-t firming

angular mations of the helLcopter, This t i p p a t h plane tilt produces the i aeredynamic moment necesomy t o p r e c e s ~ the r o t o r disk t o follow the s h a f t motion, 'l'he source of t h e xototl: aeroc~ynam5.c moment i s the f l a p moment due t o f3oppJ,n& v e l o c i t y , i n other words a blade l i f t change due t o an angle-of- a t t a c k perturbationc Thsrefore M and L aze not influenced g r e a t l y by the 9 P t r a i l i n g edge blowing (thfire is a minor e f f e c t s i n c e p i t c h o r r o l l about the I I The helicopter center-of-gravity w i l l produce an inplane hub v e l o c i t y ) .

speed s f a b i l i t y prorluced by the r o t o r is however due t o t h s f l a p moments

I * I

I an? hub f o r c e s which occur dcuring inplane mc~tions of the r o t o r hub, Therefore

?

MU an4 +, a r e d i r e c t l y influenced by the t r a i l i n g edge blowing of the

c i r c u l a t i o n controlled r o t o r , With aero ox low blowing, a forward v e l o c l t~ of t h e h e l i c o p t e r increases the blade lift on the advancing s i d e and decreases i Z , on the r e t r e a t i n g s i d e of the d i s k , The r o t o r responds with a rearward.

t11t of the t i p - p t h plane, which produces a t h r u s t v e c b r tilt a,nd a hub

\

moment, The r e s u l t i n g p i t c h up moment on t h e h e l i c o p t e r I s tho ?peed

!

The equation above shows that blowing reduces s e c t i o n s t a b i l i t y d.erivative (PIU > 0).

b L / & V , and t h a t f o r s u f f i c i e n t l y high blowing coefficienah the l i f t can u a c t u a l l y decrease due t o an inplane v e l o c i t y increase, r e s u l t i n g i n negative speed s t a b i l i t y . I n fitxmrnary, t h e p r i n c i p a l invluenca of blowing on the r o t o r s t a b i l i t y d e r i v a t i v e s is a change i n the speed s t a b i l i t y M U and Lv,

I

which a r e reduced f o r C p > 0. For s u f f i c i e n t l y high blowing, negative

speed s t a b i l i t y is possible ( M ~ z 0 and Lv < 0 correspond t o positive speed

s t a b i l i t y ) . An expression f o r t h e speed s t a b i l i t y d e r i v a t i v e is given i n the appendix.

The c h a r a c t e r i s t i c equation of the h e l i c o p t e r longitudinal dynamics

" I

I The influence of the speed s t a b i l i t y MU on the three r o o t s of t h i s equation is sketched below. With no blowing the helicopter has p o s i t i v e speed

i

s t a b i l i t y , M , > O . The11 the r o o t s of the hover l o n g i t u d i n a l dynamics consist of a s t a b l e r e a l r o o t due to t h e p i t c h damping M and. a mildly unstable 9 ' long period o s c i l l a t o r y mode clue t o the speed s C a b i l i t y . With a low l e v e l

i

t -5-

The r e s u l t is of blowing, the speed s t a b i l i t y derivative Mu is red,uced.

a red'uction i n the damping of .the r e a l r o o t , and, an increase i n the period and t i n e to double-,amplitude of the o s c i l l a b x y mod.@. This change i n the o s c i l l a * t o r y mode is favorable, and the influence on the r e a l r o o t w i l l be With s u f f i c i e n t l y high blowing t h e r o t o r small with a hingeless r o t o r .

w i l l Mve negative a p e d s t a b i l i t y , M U < 0. Then the r o o t s of t h e hover dynamics c o n s i s t of two s t a b l e r e a l roots (ox a s t a b l e o s c i l l a t o x y mode), !

and a n unstable r e a l divergence due t o the speed. s t a b i l i t y . The tLme t o There a r e double-amplitude of thPs unstab3.e mode can be unacceptably s h a r t , i

similar e f f e c t s of the blowing on the hovering helicopter l a t e r a l dynamics. 1

i

I i

I ti

CALCULATElD DYNAMIC CHARACl'ERISTICS I The calculated f l i g h t dynamics of the hovering h e l l c o p t e r with a c i r c u l a t i o n controlled r o t o r w i l l be presented f o r t h e following example: = 0.085; blade Lock numbor r o t o r solidl,ty = 5; mast height h / ~ = 0.3;

i

- 6-

preconb pp

9'1 p i t c h radiua of gyration ( k y / ~ ) ' - 0.10 i r o l l radius

of gymtion ( ~ $ 3 ) ~ 0.04; and a, g r a v i t a t i o n a l constant o f g / ~ % = 0.002 .

A r o t o r speerl of = 27 xad/stsc i~i used t o obtain dimensional xesulCs

0.002, the rotor t i p epeed and r a d i u s m e RR - 180 n/sec

(henco with @/sL% and R Fi 6.7 m). The c i ~ c u l a t i o n controlled. a i r f o i l c h a r a c t e r i s t i c s a t conhtant Mach number were approxiwted by P c p P 8 M F b C p w i t h a a 7 .O1 b For the blade s e c t i o n drag c o e f f i c i e n t , 10.8, and p = 2/3.

cd = 0.012 watj used. The hove2* induced v e l o c i t y was obdained using the mornentwn theory r e s u l t \ h , = k h w, with t h e empirical f a c t o r kh 1.15 .

..

Z t was assumed that the blade s e c t i o n blowing c o e f f i c i e n t varied inversely with the r a d i a l s t a t i o n along the spanr The f l i g h t Cp - C ~ & ( R / ~ ) .

dykunica a r e considered over a mnge of r o t o r l i f t ( k h ~ u s t c o e f f i c i e n t t o

a o l i d i t y r a t i o n of cT/- - 0.05 t o 0.15), as a function of t h e t i p blowing

o o e f f i c i e n t C p C . ~ e v e m l value8 of the f l a p frequency a r e consid.eredi 9 = 1, as $ox tz t e e t e r i n g o r glmbsllad r o t o r : 3 = 1 ,Oki ~epresentat.2,ve of an

offset-hinge a r t i c u l a t e d r o t o r ; and 9 = 1 .@ and 1 . 18, representative of

hingeless r o t o r s .

For a given value of the r o t o r l i f t , t h e c o l l e c t i v e p i t c h decreases vaxlation of 8 as the blowing increases. Figure i shows the with C , , f o r the present example, a t several l e v e l s of c+. A h e l i c o p t e r wikh a c i r c u l a t i o n controlled r o t o r would l i k e l y be operated by using the blowing t o conCrol cT/v-, f o r a fixed value of geometric c o l l e o t i v e p i t c h . Biguce 2.

compares the s o l u t i o n f o r the coupled hover dynamics with the resu1.t;~ of separate s o l u t i o n s of the decoupled. squatiovrs f o r the longitudinal and l a t e r a l % dynamics, The decoupled equations a r e lower o r d e r , hence e a s i e r solve and e a s i e r t o i n t e r p r e t . From f i g u r e 2 i C 3 . s conclud.ed t h a t the s o l u t i o n I of the decoupled equations c o n b i n s the Tunrl.aments1 c h a r a c t e r i s t i c s of the hovering h e l i c o p t e r f l i g h t dynamics. I n particula'r, the behavior of the

i

r o o t s a t high vailues of the blowing c o e f f i c i e n t a r e c o r r e c t l y obtained.

I

Figma 3 shows the influence of t r a i l i n g edge Slowing on the calculated longitud tlnal f l i g h t dynamj.cs i n hover. The r e a l and imaginary

p r t a of the xoot8 aaro p l o t t e d as o, function of t h e t i p b 1 0 ~ i n g coeffic%ent C t h e right,-hand axe. show the corresponding time to double- o r half-amplitude anif period of the mode. F i ~ u r e s 30 and 3b $ive t h e r o o t s for the axticulatec'.

1.09) r e s p e c t i v e l y , a t several

rotor: ( 9 i) and the hingelens r o t o r ( 4

vnlueo of the r o t o r l i f t . Figure 3 c gives the r o o t s at cT/w - 0.10 f o r

several, valuee of the f l a p frequency, F i g w e 1~ ahows t h e influence of blowing on the, P a t e r a l dynmicn, The vaxiation of the r o o t s a s t h e b l o w i n ~ c o e f f i c i e n t increases follows the behavior sketched iri t h e xoot LOCUS above .

I Consider fip;uxe 3a, A s C p L i n c r e a ~ e s from zero, the damping of the s t a b l e r e a l r o o t tlecreasec while the damping and time J;o double-amplitude of the A t a value of C p k ~ h i c h depends on t h e unstable o ~ c i l l a t o r y mocle increase.

r o t o r l i f t , the two complex roots reach the o r i g i n , and f o r still higher For thin case Cpt booome t,wo r e a l r o o t s , one a t a b l e and one unstable.

of 9 1,0, $he two f i h b l e reaJ. r o o t s .then meet on the r e a l a x i s With a hingoloss m t o r ( f i g u r s 3b) I axeak off t o form a complex conjugate p i x .

the behavior is similar, except t h a t the s t a b l e x e a l xoot due b the p i t c h damping has much l a r g e r magnitude, and so does not combine with the o t h e r s t a b l e r e a l r o o t a t high C p k . Tncleed t h i s p i t c h r o o t is o f f the s a a l e of f i g u r e s j b ant1 g c , even f o r the offset-hinge a r t i c u l a t e d r o t o r ( 9 = 1.04).

The influence of blowing on the r o o t s is similar fox the l a t e r a l dynamics.

With a quasis-katic r o t o r morlal, the e f f e c t s of unsteady aerodynamics The r e s u l t s shown can be accounted f a r by uaing a lift deficiency function, i n f i g u r e s 3 and It were o b b i a e d uslng C = I f o r the l i f t deficiency function.

I

However, the reduction of the r o t o r semdynanic hub moments i n hover due t o Pigurc 5 shows the l i f C deficiency the unsteady wake e f f e c t s can be very l a r g e , Figures function v a r i a t i o n with f o r the present example (aeo the appendix).

I

6 and 7 presen.t; t h e resu2ts f o r the helicoptex longitudinal. and l a t e r a l I The influence of dynamics includSng t h i s l i f t Ceficiency function (C *< 1) .

I i +.railing edge blowing on the calculated dynamics is not: s i g n i f i c a n t l y changed. by includ.ing the l i f t deficiency funci;ion.

The p o i n t a% which the compXex z o o t ~ r e a c h t h e o r i g i n , and form a p 3 i r of real r o o t s , comespont!~ t o zero apead s f ~ b i l i t y (MU 0 o r Lv - 0 ) .

Examining C i g w e c 3 t o 7 , it i n conclutled that t h e value of C p k f o r z e r o apeeb n t a b i l i t y i n c r o a ~ e s with C ~ V 9, is n o t sensitive t o t h e lift d e f i c i e n c y f u n c t i o n , and is t h e same f o r tho Xonlr;itudinal and l a t e r a l motion&, Note t h a t t h e unstabla xenl r o o t i n i t i a l l y I n c r e a s e s very q v i c k l y a f t e r this c r i t i c a l Cpk is reached, e s p e c i a l l y with low l o a d i n g o r a n a c f i c u l a t c d raLor. Therefore t h e blowing c o a f f i c i e n t can n o t be i n c r e a s e d much above t h i s p o i n t b e f o r e t h e time t;o double-amplitude bocomes unacceptably I I small.

I lPi&ure 8 shows t h e v a r i a t i o n of t h e apeod s t a b i l i t y d e r i v a t i v e M U w i t h the t i p bl.owing c o e f f i c i e n t G p k and t h e x c t o r lift C ~ V . F i g u r e 9 VS. C F L p l a n e , as a showa tho nero speed ~ t a b i l i t y bouneary on t h e C ~ U - function of flap frequency, The inf l uence of t h e Z l f t Bef Lciency f unokinn $ = 1.09

i n n e g l i ~ i b l e f o r t h e - 1.0 boundary, and amall f o r t h e

bounflaxy. P'or a high enough blowing c o e f f i c i e n t , o r a small enough r o t o r l i f t , the h e l i c o p t e r w i l l have negative speed n t a b i l i t y and hence a n u n s t a b l e r e a l r o o t Tor t h e hover2 dynamics. It'igures 10 and 11 g i v e t h e correnponding r e s u l t s w i t h a p i t c h / f l n p coupling of Kp = 0.5; t h e r e is a small unfcvorable I n f l u e n c e o f p i t c h / f l a p coupling. The speed s t a b i l $ . t y d e r i v a t i v e s depend pxlmarily on t h e r o t o r aeradymmic f l a p moment due t o tlhe hub i n p l a n e

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v e l o c i t y , M An e s t i m a t e of t h e boundary f o r zero speed s t a b i l i t y can

P'

be obtained from My. 0, which is a l s o shown i n f i g u r e s 9 and ii ( s e e t h e appendix).

With a,n a r t i c u Z a t e d r o t o r tho c r i t i c a l blowing f o r zero E s p e e d s t a b i l i t y is s i g n i f i c a n t l y lower because t h e direct; inplano hub f o r c e due +to hub v e l o c i t y ( H ~ ) decreases f a s t e r w i t h Cpt t h a n My Boes. With I hingelesfi r o t o r s t h e i n f l u e n c e cf H is much Xess than t h a t o f M s o I

r r '

t h e c r i t e r i o n M = 0 gives a b e t t e r e s t i n a t e of the boundary.

P

I

t o cyclic Ths rasponre o f tho lrolicoptsr l o n g i t ~ r d i n a l v e l o c i t $ p i t c h o r blowing control haa two s s r o a , a, complax conJu@te wA3: of x e l a t i v a l y l a q e magnitude. The p i t c h angle QI, ha8 n oinp;le r c a l sox0 of very amall mngnltu4e, Ths zsroo o f tho l a t e r a l reuponce ZA3 c o n t r o l E ~ X ' Q a&mkl&r. The soroo of the r e a p m e to oyalic blowin@; brs a r e iden.f;ical t ; c 4 tho zeros o f the reaponae to c y c l i c p i t c h B,( i f ( x ~ ~ M ~ , ) / ( x ~ , N ~ ~ ) 1, which is

exactly o a t i s f i s d with $ - 1.0, and is catinfled within i O $ f o r 3 1

Calculations of tho zero6 fox the numerical example t,oneidered above confixm that t h e zeroc f o r c y c l i c blowing atid c y c l i c giLch a z e elrnentbally irJentScal. Hence it 2 5 3 only tho pole ccnfigwntion which Snfluencos the helicopter response t o c s n l r o l .

A conventional helicopter has d mild1y unstable Long period v a c i l l a t i o n i n hover, bu.t %he motion citn be ~ t a b i l i z e d by the p i l o t with r a t e plus

prcportional foedbaclc of t h e p i t c h motsdn (GIs = - K ~ ( Z S + 1) &$ , and

s i m i l a r l y f o r the l a t e r a l dynamics. Tho roo b locua fox such feedback is

The eg/eIi reoponne hon a s i n g l e zero a t t h e o r i g i n , and

skctched below.

+he lead adds another a t G a t-ic With a hingelearn r o t o r t h e control power and damping of the r o t o r a r e high, and. the lead tt: required is not long. A moderate l e v e l of blowing w i l l improve the h e l i c o p t e r c o n t ~ o l c h a r a c t e ~ i s t i c s by reducing the speed a t a b i l i t j , Blowing l a r g e enough . t o produce negatlve speed cta.bility r a d i c a l l y a l t e r s the pole configuration

hawevor, Xn t M o C&GO, RG altcttchad btllow, rate plus p r o p r t l o m l TeredbCk

- K ~ { V Z G * i) @,,I improveo t h e d y m n f c

of tho p i t c h a t t i t d a ( 8 , c r w a c t r r l r , t i c o by in~roaoeng tho t h e to doublo s n p l i t u d e o f tho r o n l tlivergenco, but it 4oeo not complotoly s t o b i l i s e the nysten 20@Edl0~8 To obtain a n b b l o syatam, Zeadback of the of the valuct of %ha Xend 'C; v e l o c i t y perturbation could bo added t o counter t h s nel(ative a p o d s t a b i l i t y , ha + I)$, * ~~g~ ) , and t k n r n t o plus proportional foedbnck of p l t c h used ( b I ?

Altcmativaly, i n t e p l fee0back of pitoh could be u ~ e d an uo2l to producs

a s&%blo ayaten (St$ a - K ~ ( (%I"*i + t t 'c~~E) BF). Such control lava

I

would likoLy ba too comp1,lex $or tho p i l o t to c ~ n f o r h b l y .blynALe le. I n any I coca, an ~ u t o n n t i o c t x b i l i t y rrugnsnCtion ay 6tem i a c l e a r l y c~eoimlrle with ' I .

a b e negztive opescl s t a b i l i t y . L i t t l e work ban been done on the hand.ling q,ualities of helicopter6 wi$h n q a t i v o speed s t a b i l i t y , Limited data i n reference 2 i n d i c a t e s thaL Lha k n r i l i n g qmXitlies axe probably u m c c e p k b l e if M,, 16 188s than about -0.01 (m-liec)ul, and c l e a r l y it is BeliimbLe ?o U have ~ s i 2 i v e speed s t a b i l i t y ( M , 3 0 ) .

1 1

ETJCLUDIflG RENARKS With s conventional r o t o r , an increase I n the blade inplane v e l o c i t y ixlcreases the dynamic pressure and increasbd the angle-of-attack ID^ the s e c t i o n , thereby increasing the l i f t , With a c i r c u l a t i o n controlled r o t o r , -11- ORIGINAL PAGE 1s ON POOH. QZiAWTY I

an inplane velool%y tynncreso sofucefi the blowing c o o f f i c l a n t Cpt, and thuo tanda t o xetluco t h e l i f t , If tha blowlng 18 high enough t b t there i s a ncst rsJuaLion an t h s blade l i f t f o r an inplsno veloc%.ty i n c r s a o s , fhrj ilynnrric behavior of tho r o t o r l a m d i m l l y al%extlO, For the hovcsrine; Sl5ght k?ynnmica, t h e m ID & c r i t i c a l , blowing l s ~ v e l , above wl~ich tho r o t o r proilucdo a nog;l%lvo speed s b b l l i t y derivntlva, Bla hnndlinlz; qualities af a halicopter with neC;EL.tiva cpood fitabilj ty ara probably unaccsptable without a c t n b i l i t y augmentnt;ion cyfitem,

IICVEilZNG IiELTCGpZIER SPEED STABILITY DERTVATIVE The equations of notion f o r the helicopter r i g i d body degrees of freedom, r o t o r f l a p motion, hub reactions, and inflow perturbation were obtained Srom the aexoelastic a n a l y s i s of xcfexence 3 , For the r o t o r 1 ) only the f l a p motion was considered, o p e c i f i c a l l y the t i p - p t h plane tilt

degrees of freedon pl, an4 PIS. FOXthe f l a p node shape r i g i d body

r o t a t i o n with no hinae o f f s e t waa asstuned i n order t o simplify the aerodynamic I a n d i n e r t i a l coefficienta. Hingeless r o t o r s were modelled by r e t a i n i n g I ~ r b i t r a r y f l a p frequency 4. Per:twbations of the r o t o r induced v e l o c t t y , l t n e a x l y varyit16 over the r o t o x d.isk, were includeA to account f o r t h e r o t o r unsteady aerotlynamicc , The influence of these inflow perturbations * p r i n c l p l l y takes the form of a l i f t deficiency function C multiplying the aerodynamic coefficients. A q u a s i s t a t i c s o l u t i o n f o r the r o t o r f l a p response was used, The flapping v e l o c i t y and a c c e l e r a t i o n terms were dropped and then the f l a p equations were solved f o r the Lip-path plane tilt response, The flapping v e l o c i t y terns were dropped, from the hub r e a c t i o n s , :Fa), t h e s o l u t i o n s f o r @, and P I S s u b s t i t u t e d . From the r e s u l t i n g e x p n s i u n of t h e r o t o r low frequency response i n terms of the pexturbation h0d.y motion, the s t a b i l i t y derivatives f o r the f l i g h t dynamics analysiu were identified..

The r e s u l t f o r t h e speed s t a b i l i t y derivative MU is as follows.

where q 2 - 1 N* i;P + K~ U c/8 and %he l i f t deficiency function ( frcn reference 3) is 8 2 2 The l a t e r a l speed Oerivatkw is Ly = -(ky / k , ) M ~ . The parameter6 required i n t h i a expression a r e :

i3 a c c e l e r a t i o n due t o g r a v i t y 1

p i t c h radius of gyration k~ r o l l rad,ius of gyxation

Ir *

n r o t o r angular v e l o c i t y

r o t o r t h r u s t c o e f f i c i e n t C~ 'P" r o t o r solidit,y r a t i o a blade s e c t i o n l i f t - c u r v e slope Q f l a p frequency (per r e v , i n the r o t a t i n g frame) ?f blade Lock 11wnber

I

h r o t o r mast height R r o t o r rad.ius he induced v e l o c i t y

0. blad,e coning angle

'?he momentum theory value f o r t h e induced v e l o c i t y i n hover is 'L;' kh ~ 7 2 , where kh is a n empirical constant.

I) The r o t o r aeroCynamic c o e f f i c i e n t s My and Hp a r e respectively the f l a p moment and blade drag force due t o inplane velocity of the hub (see I Jt is only through these aerodynamic c o e f f i c i e n t s t h a t the reference 3).

I To evaluate My an6 H p , t r a i l i n g edge blowing influences the f l i g h t dynamics.

i t was assumed t h a t the blade has constant chord and l i n e a r twist ( qw),

I

and that the blowing c o e f f i c i e n t varied along t h e blade span a s Cp = Cyt ( ~ / r ) .

The c i r c u l a t i o n controlled a i r f o i l c h a r a c t e r i s t i c s were approximated by P

c = a o ( + b C The r e s u l t s f o r M

X I ' *

and Hr

i

- 14-

.I

C

T I

and f o r ths blade coning and c o l l e c t i v e p i t c h angles : where f p is the hub precone angle.

The s p e d s t a b i l i t y d e r i v a t i v e depends primarily on the r o t o r aerodynamic c o e f f i c i e n t My. Hence an estimate of t h e boundary f o r 5ero speed s t a b i l i t y can be obtained from M y = 0, which gives o r with a - 7.0, b = 10.8, p = 2/3, and kh = 1.15:

1 S o c k e l , Edward, S t : ~ b i l L t y and Control of Airplanes and H e l i c o p t e r s , Acaclemic ~ ' ~ R G s , New York, 1964 :! K e l l y , James R , , and, Gmxen, John E ' , , F r , , "Stuc?y of t h e Cptimwn Values of S e v e r a l Pailmetarc Affecting Lona;itudirtzl Handling Q u a l i t i e s of YWL A i r c r a f t , " NASA TN D-4624, J u l y 1960 3 JoF;ison, Wayne, "Aeroelastic Analysis f o x Rotorcraf t i n F l i g h t or I n a Wind Tunnel, " N A S A TN D - R S l R , J u l y 1977

I i I 1

20 - I ' I - . i

! i *

i

I $ * I !

4 . * . .*,I i - 7 * , ..

I i t

9 7 5 a . 1 i

1 * I i i 1.l i; llf 1 4 1 * .

t I 4 ;

i: 4

, ! i(

j j

t 3 I i 1 J / 1 t i

I i

!

$ 3 1 i 8

1- r

1 ' , , 1 * . , . 1 : 1 , ;; 1

-10 L t I L

I , I i . ! . ,

4 I -

I I

, . * + .I. ' .. r -. .* - I . . .,. . . r

L . 1 Figure 1 Rotor blade c o l l e c t i v e p i t c h as a function of r o t o r loading and blowing c o e f f i c i e n t .

I I

I

-17-

Ir. I *

P - I

i

- " -----L; A

-d

LL *

- .L =-: ;L .. ._ .*-&::L-_L*_--A .

8 " d - - -- ----- L a L - -- - . .-"-. -0-- - -

(b) Ylap frequency = I. (9 Figure 2 Concluded.

aec

(a) Flap frequency 9 = 1.0

Calculated roots of the hover longitudinal dynamics Figure 3 a? & function of the t i p blowing c o e f f i c i e n t ,

I * I (b) Flap frequency -? = 1.09

I i

i r e 3 Continued.

- ORIGINAL PAGE I8 OF POOIE QUALITY

I

I I ( c ) L i f t C ~ / P = 0.10

i

E'i.gure 3 Concluded..

i

I -22-,

t

I 'i

i

9 A " . . _ " - *,, < - --< - - - 0 - A .

sec I (a) Flap frequency 9 = 1.0 Calculated roots of the hover l a t e r a l dynamics as a function Figure 4 of the t i p blowing coefficient.

i

oac aoc

( a ) Lift c p - = 0.10

0 . 0 ~ ) + Figure 5 Lift deficiency f uilction for rotor hub moments ( o .

Bec T s e c

(a) Flap frequency 3 = 1.0

Calculated r o o t s of t h e hover l o n g i t u d i n a l dynamics as a f u n c t i o n F i g u r e 6 o f t h e t i p blowing c o e f f i c i e n t , i n c l u d i n g t h e lift d e f i c i e n c y function.

* 0.02 , c p '+@. I S I .

-- :0,10 -

1 . : --.I--- 0.05

-

[je ), II 3.

,,- -

-4- e-'-

44- , - -

5.

sec /@*

10 ao ,0 z :/ /- -

L 0 i,, v I 20' I\

\-

h% >\ 10

'

--*--

\

--

< 5 t & ---sr.---- 1 ..

s e c * 1 . 1 1 & I . . . . A . . a .

,

-

I ,. .

I

1 - I*$

!

" 1 0 .

- 0 . 0 2 , !. . . . . . . rrr 1 I . "4 I * + , , 6 , x ,.

I I 0 . 0 2 ~ . ' I , , , .. # , 4 % ' . 1) 0.. , I 15

Im 1

.

;,,. .... 1 ....-.%.,T. . " . a - , , ,

- 20'

, . 1 . . . . . . . .

(b) F l a p frequency $ = 1.09 F i g u r e 6 Continued.

s e c s e c ( a ) F l a p frequency $ " 1.0 F i g u r e 7 Calculated r o o t s of t h e hover l a t e r a l dynamies as a f u n c t i o n o f t h e t i p blowing c o e f f i c i e n t , i n c l u d i n g t h e lift d-eficiency f u n c t i i o n ,

t

O R m AL Y AGE 1s

I -30-

OF POOR Q U A L Z ~

t Z sec sec sec Figure 7 Concluded,

(a) Flap frequency 3 = 1.0 (negligible e f f e c t of the l i f t deficiency function).

Figure 8 Speed stabi1it.y derivative MU as a function of C,/V and C Pt

I

i I

-33-

b

I I i

(b) Flap frequency a= 1.09 (with and without lift d e f i c i e n c y f u n c t i o n ) .

F i g u r e 8 Zonclur3,ed.

7 , s I negative speed

stpbility - *

. . . . .. . ,+, . j. ,-. -

Fipure 9 Boundary for zero speed, ~tability ( Q = 0.085) a

I (a) Flay frequency 3 = 1.0 (negligible e f f e c t of the lift deficiency function) I I Pinure 10 Speed s t a b i l i t y d e r i v a t i v e M u as a function of C ~ W - and C p a , n a K = n - c : I ORIG.tNAL PAGE IS -36- OF POOR QUALITY

C

r \ --- - -

(b) Plap frequency $ = 1.03 (with and without lift d e f i c i e n c y f u n c t i o n ) Figure 10 Concluded, -37- O R W A L P A W OF POOR Q u A L ~

I - *..

a,i5-

- * . 8

0.20 I L t , r I CT : + oagativo qfpt;ed . . ., , stability i o*o!i - ", ;..

I .

.

. : .

, . I * . I * ] * 8 s 1 ! .. : i .

. b ' . ; ....,.

.* ; 7

t . . ! * t ' I . ! .

; .:.. ; ., . : I ; ' I I . I t I, I 1 1 . , . a 4 * r t * . b i I I ,0,01 f' 4 , . . s . . 1 . . * 9 4 0 d o 2 0403 , O°KYk / .,,:,0&5' :

i C=,*

L . , . L

a . 6 - z. ; b i . u 4 p i .

I

. , * t " I

" * I 4 * . ' . 1

* ; i t . . , I 'I Figure ii Boundary for zero speed. s t a b i l i t y , with ppit~hjflap coupling O,5 ( P = 0.085).

K~

NASA Tbi-18,443 A Trtlil md Suhtttlr ULCUIATET) IIOVERTNG fiEI~XGOI'TEIi, FIaXCIIT UY NAtT CS WETH A CZRCUT,nTZON GQNT~BI~LlST) ROTOR *NRC P o ~ t d o c t o r a l Research Asaocintc, Amca Rcscnrelt Center, NASA, Tha f l i a h t dynaniaa o E a hoveriag heZicsptar w i t h a c i r c u l a t i o n c o n t r o l l e d r o t o r a r e analyzed, The InfJuenco oE t h e r o t o r blowing coef- f i c i e n t : sn t h e saZculatcd r o n t s o f t h e l o n g i t u d i n a l auld l a t e r a l motion i s cxnminod f o r a rnngc of v a l u e s o f t h e r o t o r l i f t : and the blade f l a p frequency. The c o n t r o l c h a r a c t o r i s t i c s o f o h a l i c o p t o r w i t h n c i r c u l a t i o n Thc p r i n c i p a l e f f e c t of t h o blowing i b a c o n t r o l l e d r o t o r a r c d i s c u s s e d , r e d u c t i o n i n the rocor speed s t a b i l i t y d e r l v a t i v c . Above a c r i t i c a l Level of blowing coefficient, which depends on t h e f l a p frequency and r o t o r L i f t , ncgatdvc epced s t a b i l i t y 25 produced and t h e dynamic c h a r a c t e r i s t i c s of the h e l l c o p t e r a r c r a d i c a l l y a l t e r e d , Tho lzandling q u a l i t i e s o f a h e l i c o p t e r with n e g a t i v e speed s t a b i l i t y a r c probably unacceptable without a s t a b i l i t y augmentation system.

18, Distribution Statomont 17. Key Words (Sugpwtad by Authorl~)) C i r c u l a t i o n c o n t r o l l e d r o t o r

H e l i c o p t e r Eligh t dynamics I

STAR Category - 0 1

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