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    A U G U S T 1 9 7 5 E N G I N E E R I N G NOTES 68 3

    U.i:"S : 0 . 1oS 0fe1-0.1i-0.2

    -n 3

    G a u s s i a n , T n nnn iPtDis tr ibu t ion, c = 0 . 6 c Uniform Poros i ty

    .

    i i

    = 0. 4/ffN o0.6y/T\k T = 0 0 C losed !// o\\ ^y i j .hd ' ^ _ __---''^ ^C/ph = 0.5

    L o n g i t u d i n a l D i s t a n c e A l o n g T u n n e l C e n t e r l i n e , x / p hF i g . 2 L i f t i n t er f eren ce fo r NASA 64-series a i r f o i l i n t u n n e l s w i t hGaussian an d u n i f o r m d is t r ib u t io n s o f po ro s i t i e s at = Q ) =4.0,

    T h e r e s u l t s of l i f t i n t e r f e r e n c e fo r the NACA-64 a i r fo i lse r ie s i n t u n n e l w a l l s w i t h Gaussian a n d u n i f o r m d i s t r i b u t i o nof porosities are s h o w n in Fig. 2. The comparison betweent h e s e tw o c u r v e s i n d i c a t e s that i m p r o v e m e n t of l i f t i n -t e r f e r e n c e i s f o u n d by u t i l i z i n g t h e Gaussian d i s t r i b u t i o n .S p e c i f i c a l l y , t h e l i f t i n t e r f e r e n c e a t t h e t a i l h a s t h e s am e s igna n d m a g n i t u d e a s t h a t a t t h e w i n g for a win g- t a i l model.T h e p i t c h i n g m o m e n t obtained f rom a u n i f o r m d i s t r i b u t e dp o r o s i t y t u n n e l w h i c h h a s opposite s ign s of i n t e r f e r e n c e bet-w e e n th e w i n g a n d t h e t a i l i s d i f f i c u l t to correct. In a G a u s s i a nd i s t r i b u t i o n of p o r o s i t y , t h e i n t e r f e r e n c e h a s t h e same s igna n d m a g n i t u d e o n t h e w i n g a n d t h e t a i l . T h e on l y co r r e c t ionr e q u i r e d t o t h e model data i s for t u n n e l f l ow a n g u l a r i t y .

    B l o c k a g e I n t e r f e r e n c eFor t he b l o c k a g e i n t e r f e r e n c e c a l c u l a t i o n , t h e m ode l poten-t i a l fo r t h e a i r f o i l t h i c k n e s s i s r e p r e s e n t e d by

    logS i m i l a r l y t o t h e l i f t i n t e r f e r e n c e c a s e, a d i s c re t e d i s t r i b u t i o n ofs o u r c e s i n g u l a r i t i e s i s adopted a sU= 1C

    T h e f o r m u l a t i o n i s s e t u p fo r e a c h s i n g l e d i s c r e t e m o de l p o t e n -t i a l i n t e r m s of t h e a x i a l p e r t u r b a t i o n v e l o c i t y .T h e b l o c k a g e i n t e r f e r e n c e , = Uj/l3hU, is c a l c u l a t e d 8 for ab i c o n v e x a i r f o i l s h o w n i n Fig. 3 for C/,^0.3 a n d 0.5. T h ecomparison demonstrates c l e a r l y that the b l ockage in-t e r f e r e n c e g r a d i e n t fo r u n i f o r m p o r os i t y i s g r e a t l y r e d u c e d byu s i n g t h e G a u s s i a n d i s t r i b u t i o n . I n t h e u n i f o r m p o r o s it y tun-n e l , t h e b l o c k a g e i n t e r f e r e n c e n o t o n l y h a s a r a t h e r l a r g eg r a d i e n t bu t a l s o c h a n g e s s i g n s f r o m t h e l e ad in g edge t o t h et r a i l i n g edge o f a i r fo i l .

    C o n c l u s i o n sT h e s o l u t i on i s c o m p u t a t i o n a l l y s i m p l i f i e d b y a s s u m i n g ad i s c r e t e d i s t r i b u t i o n of s i n g u l a r i t i e s for a f i n i t e c h o r d a i r f o i l .T h e f o r m o f i n f l u e n c e c o e f f i c i e n t s i s ob t a i n e d f o r a g i v e nw i n d t u n n e l . T h e t a b u l a t e d i n f l u e n c e c o e f f i c i e n t s c a n b e u s e dto c a l c u l a t e i n t e r f e r e n c e f a c t o rs fo r a n y f i n i t e a i r f o i l w i t h ag iv en l o a d in g a n d t h i c k n e s s di s t r i b u t i o n a t a g iv en an g l e o f a t -t a c k .T h e p o t e n t i a l to r e d u c e t u n n e l i n t e r f e r e n c e s on a g iv enf i n i t e a i r f o i l i s d e m o n s t r a t e d b y a s e l e c t e d G a u s s i a nd i s t r i b u t i o n of p o r o s i t y . For a s p e c i f i c a i r f o i l , one of a f a m i l yo f G a u s s i a n d i s t r i b u t i o n m a y b e u s e d f o r i n t e r f e r e n c em i n i m i z a t i o n a l t h o u g h s u p e r i o r p o r o s i t y d i s t r i b u t i o n s m a y b ef o u n d fo r t h e g e n e r a l case.T h e i n t e r f e r e n c e c a l c u l a t i o n fo r t h e p r e v i o u s e x a m p l e s i n -d i c a t e s t h a t a l a r g e g r a d i e n t of p o r o s i t y , e s p e c i a l l y i n t h eneighborhood of t he test model, i s r e q u i r e d to s i g n i f i c a n t l y

    G a u s s i a n Di s t r i bu t i on , c = 0.4 C/|3hj=JX5^

    -2.0 -1.5 -1.0 5 0~~ 0.5 1.0 1.5 2.0Long i t ud ina l Distance A long T unne l Cen ter li ne , x / | 5 hF i g . 3 Blockage in t e r f e r e n c e f o r a b i c o n v e x ai r fo i l w i th 6% t h i c k n e ssrat i o in t u n n e l s w i t h Gaus s ian an d u n i f o r m d i s tr i b u t io n s o f po ro s i t i e sa t 0T(x = o) = 1.5,< / / / * =-1.0.c h a n g e t h e v a l u e of i n t e r f e r e n c e f a c to r s of a t u n n e l . In otherw o r d s , a n y s m a l l a m o u n t of v a r i a t i o n of po ro s i ty doe s n o tg r e a t l y a f fe c t t u n n e l i n t e r f e r e n c e s . F or t h e s u b s o n i c c a s e , t h eb l ockage i n t e r f e r e n c e is u s u a l l y neg l i g ib l e 9 a s compared t o t h el i f t i n t e r f e r e n c e . H e n c e , t h e c o n s i d e r a t i o n of l i f t i n t e r f e r e n c es h o u l d be e m p h a s i z e d i n t h e se l ec t ion of t u n n e l w a l l p o r o s i tyd i s t r i b u t i o n .

    R e f e r e n c e s! L o , C. F . and Ol ive r , R. H., "S u bson i c L i f t I n t e r f e r e n c e in aW i n d T u n n e l wi th Per fora t ed W a l l s , " Journal of Aircraft, V o l . 7,M a y - J u n e 1970, pp. 2 81-283.2 J a c o c k s , J . L., " E v a l u a t i o n of In t e r fe rence Effects on a L i f t i n gModel i n t h e A E D C PW T 4-ft T r an son i c T u n n e l , " AEDC-TR-70-72(AD868290), April 1970, A rnold Engineer ing Development Center ,Arnold Air Forc e S t a t ion , Tu l l ahoma , T e n n .3B i n i o n , T. W. , Jr., "A n Invest iga t ion of Several Slot ted W i n dT u n n e l W a l l Conf igu r a t ions with a High Disc Loading V / S T O LM o d e l / ' A E D C - T R -7 1 - 77 ( A D 7 2 3 2 9 4 ) , M a y 1971,A r n o l dEnginee r i ng Deve lopmen t Cen t e r , Arno ld Air Force Sta t ion ,T u l l a h oma , T e n n .4 L o , C , F. , " W i n d T u n n e l W a l l I n t e r f e r e n c e Reduc t ion by St ream-wise Poros i ty Di s t r i b u t i on , " AIAA Journal , Vol . 10 , A p r i l 1972 , p p .

    547-550. .5S e a r s , W . R ., "Se l f -Cor r ec t i ng W i n d T u n n e l s , " R e p t . RK-5070-A-2, J u l y 1973, Ca l span Corp . , Buffa lo , N . Y .6 B a r o n t i , P ., Ferr i , A ., a n d W e e k s , T., "An a ly s i s of W a l lModifica t ion in a T r an son i c W a l l T u n n e l , " R ep t . TR-181, Feb. 1973,Advanced Technology L a b s , W e s t b u r y , N . Y .7E r i ck son , J.C. Jr., a n d Ne rn i , J. P., "A N u m er i c a l D emon -s t ra t ion of t he Establ i shmen t of Unconf i ned-Flow Cond i t ions i n aSelf-Correc t ing W i n d T u n n e l , " Re p t . RK-5070-A-1, Nov. 1973,C a l s p a n C o r p . , Buf f a lo , N . Y .8 Ladd, T. G. , "L i f t In t e r f e rence Fac tor a n d Blockage Fac tors inSome Per fora t ed Wind T u n n e l s , " Ms t h e s i s , Dec.1972, Dept . . ofMa t h ema t i c s , Unive rs i ty of Tennes see , Knoxvi l l e , T e n n .9P i n dzo l a , M . and Lo, C . F. , " B o u n d a r y In t e r f e rence a t Subson icSpeeds i n W i n d T u n n e l s with Ven t i l a t ed W a l ls , " AEDC-TR-69-47 ,M a y 1969, Arnold Enginee r i ng Deve lopmen t Cen t e r , Arno ld Ai rForce Sta t ion , Tu l lahoma, Tenn.

    E f f e c t o f C o m b i n e d R o l l R a t e a n dSides l ip A n g l e o n A i r c r a f tF l i g h t S tab i l i tyRober t F. Stenge l*Th e Analy t ic Sciences Corporat ion, Reading, Mass.

    I n t r o d u c t i o nIT has l o n g b e e n k n o w n that rapid ro l l rates can de-s tabi l ize t h e motions of a i r c r a f t . Th e i n t r o d u c t i o n of h i g h -Received Decem be r 2 , 1974. This r es earch was conducted a t TheC ha r l e s S ta rk Drape r L abo ra to ry under NASA C o n t r a c t NAS9-10268.Index categories: Aircr af t Handl ing. S tabi l i ty , a n d Con t ro l ;Naviga t ion , Con t ro l , a n d G u i d a n c e T h e o r y; Spacecr af t At t i t udeD yn ami c s a n d Con t ro l .*Member of the Techn i c a l Staff . Assoc i a t e Fellow A I A A .

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    684 J. AIRCRAFT VOL. 12, NO. 8

    p e r f o r m a n c e a i r c r a f t fo l lowing W o rl d W a r II de mon s t r a t e dth e p r ob l e m , mot iva t i ng t he c l a s s i c ana lys i s o f P h i l l i p s in1 9 48 .l This a n a ly s i s h a s b e e n e x t e n de d , a s r e por t ed in Ref s . 2 -4 , wi t h p r i m a r y a t t e n t i on p a i d to t h e c a s e of zero s ides l ipa n g l e . In s u c h a n i n s t a n c e , th e r e s on a n c e be tween l o n g i t ud i n a la n d l a t e r a l -d i r e c t iona l mot ion i s coup led by ro l l r a t e s of thes a me o r de r a s t h e n a t u r a l f r equenc i e s of the s hor t pe r iod andD u t c h r o ll modes . Ef f ec t s of c o u p l i n g i n d u c e d by s ides l ip wi t hzero a n g u l a r r a t e s a re r e po r t e d in Ref s . 5 and 6 , wh i c h in -dic a te t h a t s ides l ip a lone i s only mild ly des tab i l iz ing;h o w e v e r , n on z e r o s ides l ip ang l e c a n l ead to adver se coup l i nga t m u c h lowe r ro l l r a t e s t h a n w o u ld be p red i c t ed by rol lr e s o n a n c e alone.T he s e combined des t ab i l i z ing ef fect s of rol l r a t e a n ds ide s l i p a n g l e h a v e b e e n e x a mi n e d u s i n g a l i n e a r s imu l a t i on ofth e coup l ed mot ion s of a Space Shut t l e - l ik e c on f ig u r a t i on .Var i a t ions i n t h e e igenva lue s fo r a f l i gh t condi t ion nomina l lyi den t i f i ed by a n g l e of a t t a ck (a ) =3 3 . 2 9 a n d M a c h n u m b e r(M)=4.9 a r e e x a min e d wi th i l l u s t r a t ions of the separate e f-fect s of body-axis roll r a t e (p B) and yaw r a t e (r B), s tab i l i ty-ax i s ro l l r a t e (/ ? s) , dynamic p r e s s u r e (#), a n d s ide s l ip a ng l e(0).N u m e r i c a l R e s u l t s

    Th e combined di f f e r en t i a l e q ua t i on s of the l o n g i t ud i n a la n d l a t e r a l - d i r e c t i o n a l mot ions o f t h e veh i c l e a re 12th order,wi t h s t a t e var i ab le s r e p r e s e n t i n g a n g u l a r or ien ta t ion a n dr a t e s , a s well a s t r a n s l a t i on a l p o s i t ion and veloci ty. The e igen-v a l u e s o f t h e l o n g i t u d i n a l a n d l a t e r a l -d i r e c t iona l modes a recoup led i f t h e r e f e r ence va lue s o f ro l l r a t e , ya w r a t e , p i t c hrate, or s ides l i p ang le a r e n on z e r o . A ll 12 e igenva lue s ha v ebeen c o m p u t e d for a f l ight condi t ion r e p r e s e n t i n g t h e SpaceShu t t l e ' s r e t u r n from o r b i t , a l thoug h a t t e n t i on will be di r e c t -e d to the sho r t p e r iod a n d D u t c h rol l complex p a i r s . A s ar e su l t of s t e ep e n t r y a n g l e , t h e p h u g o id mode h a s degene r a t edin to two r e a l e i g e n va l u e s , r e p r e s e n t i n g a veloci ty mode a n d af l ight pa t h mode i n t h e c a s e cons ide red h e r e . Th e rol l a n ds p i r a l roo t s h a v e coa lesced in a s t ab l e * lateral phugoid"m o d e , wi th e igenva lue s whose rea l a n d ima g i n a r y p a r t s a re-0.008 a n d 0.007, r e s p e c t i v e l y / T h e h e i g h t , r a n g e , c ros s -r a n g e , a n d y a w a t t i t ud e mode s a r e e s sen t i a l ly p u r e in -t eg r a t ion s o f tr a n s l a t i on a l a nd a n gu l a r rates. Th e attitude a n dve loc i t y a re n o m i n a l l y 144,700 f t a nd 5305 fp s , l e a d i n g to q= 62 p s f a n d M =4.9 (ba s ed on the 1962 U.S. Standa rd At-m o s p h e r e ) . This l eads to the fo l lowing s h o r t period a nd D u t c hro l l e ig enva lu e s : s h o r t period: (-0.009385, 0.6289) ( rea la n d i m a g i n a r y p a r t s ) ; D u t c h roll: (-0.008078, 0.9887)( rea l a n d imag ina ry parts). Both modes are seen to be l ight lyd a m p e d , a n d t h e l a t e r a l -d i r e c t iona l mode i s n e a r l y 60% f a s t e rt h a n t h e l o n g i t u d i n a l m o d e .Se pa r a t e Effec t s of Body-Ax is Rol l an d Y aw R a t e sA n g u l a r ro t a t i o n a b ou t t h e b ody ro l l ax is a lone or a b o u tt h e body-yaw-ax i s alone w o u ld lead to nontrivial changes in aa n d 0; h e n c e , l i n e a r mo d e l i n g of a s t e ady r a t e a b o u t e i the rax i s i s r e s t r i c t ed . I t i s in t e r e s t i n g , neve r the l e s s , to e xa min e th eprog r e s s ion of the s hor t pe r iod a n d D u t c h r ol l roots fo r t h e s etwo ca se s .Tab l e 1 p r e s e n t s t h e ef f ec t s of body-axis ro l l r a t e s be tween 0and 32 d e g / s e c . Th e r e f e r ence v a l u e o f /3 is 0. I t can be seen

    that th e p r i n c i p a l e f f e c t of body-axis ro l l rate i s to dec rea set h e d a m p i n g a n d i n c r e a s e t h e n a t u r a l f r equenc i e s of bothmodes ; h o w e v e r , ne i the r mode is dr iven to in s t ab i l i ty , a s t h erea l p a r t s of both modes r ema in n ega t ive .Th e ef fect of body-axis ya w r a t e is p r e s e n t e d in Tab l e 2 .T he r e i s a s imi l a r ef f ec t o n t h e D u t c h r o ll n a t u r a l f r equency ,bu t t h i s mode's damping i s i nc r ea s ed . The sho r t period

    mode ' s f r equency i s r e l a t ive ly una f f e c t ed , bu t i t s d a m p i n gbecomes nega t ive a t a y a w r a t e of 29 deg/sec, i.e., th e root'sr e a l part becomes pos i t ive .E f f e c t o f S t a b i l i t y - A x i s R o l l R a t e

    If the veh i c l e is ro ll ed abou t the ve loc i ty vec to r with zero /3 ,th e body c e n t e r l i n e desc r ibes a cone wi th ha l f - ang l e e q u a l toa , a n d t h e a n g u l a r r a t e i s t r a n s f o rme d i n t o body-axis rol l a n dya w r a t e s a spB=pscosa

    = 0 deg/sec

    (1 )(2 )

    32deg/sec

    -0.6 0.2F i g . 1 E x a m p l e o f i n e r t i a l c o u p l i n g e f f e c t s o n a i r c r a f t d y n a m i cs t a b i l i t y .

    Table 2 Ef f e c t o f b o d y - a x i s ya w rate o n t h e r o o t s o f t h e s h o r t p e r io da n d d u t c h r o l l m o d e s

    Body-axis ya w r a t e ,deg/ sec

    041632

    Shor t pe r iode i g e n v a l u e s ,rad/sec-0.009385, 0.6287-0.009186, 0.6280-0.005666, 0 . 6 223+ 0.000928, 0.6201

    D u t c h r o ll e i g e n v a l u e s ,r a d / s e c-0.00878, 0.9887-0.008136, 0.9921-0.009244, 1.043-0.01041, 1.183

    T a b l e 1 E f f e c t o f b o d y - a x i s r o l l ra te o n t h e roots o f t h e s h o r t p e r io da n d D u t c h r o l l m o d e s

    B o d y - a x i s ro l l r a t e ,d eg / s e c

    041632

    S h o r t p e r i o de igenva lue s ,rad/sec

    -0.009385, 0.6287-0.009502, 0.6300-0.009351, 0.6507-0.009169, 0.7271

    D u t ch r ol l e igenva lue s ,rad/sec

    -0.008078, 0.9887-0.008077, 0.9922- 0.007970, 1.042 --0.007734, 1.178

    T a b l e 3 E f f e c t o f s i d e s l i p a n g l e o n t h e r o o t s o f t h e s h o r t p e r i o d a n dD u t c h r o l l m o d e s ,ps = 0

    Sidesl ip angle ,de g

    Shor t pe r iode i g e n v a l u e s ,rad/sec Du t c h rol l e igen va l u e s ,rad/sec

    0 -0.009385, 0.6287 -0.008078, 0.98872 -0.009385, 0 . 6 28 7 -0.008075, 0.98864 -0.009385, 0.6287 -0.008068, 0.9883

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    A U G U S T 1975 ENGINEERING NOTES 68 5This m o t i o n i s realizable, a l t h o u g h th e g r a v i t a t i o n a l effec to n t h e m o t i o n i s a f u n c t i o n o f th e ro l l angle, < / > . Rol l orien-tation e f f e c t s a r e i m p o r t a n t i n t h e d e t e r m i n a t i o n o f t h e l o n g

    period m o d e s but do not a f f e c t the a t t i t u d e modes ap-p r e c i a b l y . In the c u r r e n t case, w i t h cos a = 0.836 and sin a= 0.549, s t ab i l i t y - ax i s ro l l r a t e h a s t h e e f f e c t s h o wn in Fig. 1.I n c r e a s i n g r o l l r a t e s t ab i l i ze s th e sho r t p e r iod mode a n dde s t ab i l i ze s th e D u t c h rol l mo d e ; th e l a t t e r b e c o me s u n s t a b l ea t a ro l l r a t e of 14.5 deg/sec. T h e s e r e s u l t s a s s u m e n o c o n t ro l -loop c lo s u r e s . A u g m e n t e d d a m p i n g wo u l d m a t e r i a l l y a l t e rt h e s e n u m e r i c a l r e s u l t s , a l t h o u g h t h e t r e n d towa rd dec r e a s eds t ab i l i t y wi t h i n c r e a s i n g ro l l r a t e s t i l l w o u l d o c c u r .E f f e c t o f D y n a m i c P r e s s u r e

    M a i n t a i n i n g ve loc i t y m a g n i t u d e w h i l e v a r y i n g a l t i t u d ec a u s e s s ign i f i c an t c ha n g e s in q a n d mi n o r c h a n g e s in M. Someef f ec t s of a 20,000-f t a l t i t ud e sh i f t a re c o n s i d e r e d he r e .R e d u c i n g th e a l t i t u d e by 20,000 ft i n c r e a s e s q to 148 p s f , andit i n c r e a s e s t h e s t ab i l i t y of the s h o r t p e r i o d a n d D u t c h r o l lmo d e s . Th e ro l l r a t e a t w h i ch i n s t ab i l i t y o c c u r s is about 17deg/sec. Conve r s e ly , s t ab i l i t y i s r educed wi t h i n c r e a s i n ga l t i t u d e . W i t h a 20 ,000- f t i n c r e a s e , q i s r e d u c e d t o 27 p s f , a n dt h e D u t c h r o l l m o d e is s l i gh t ly u n s t a b l e , even w i t h zero ro l lr a t e ( t h e roots a re l oc a ted a t 0.0007506, 0.6568). In -c r e a s i n g th e a l t i t u d e by 10,000 ft r e s u l t s in < ? = 4 0 p s f . Th eD u t c h r o l l i s s t a b l e a t t h i s p o i n t (wi th ze ro rol l rate), bu t i tbecomes u n s t a b l e w h e n ps = S d e g / s e c . It is c on c l ude d t h a tth e s t ab i l i t y -ax i s rol l r a t e a t w h i c h i n s t ab i l i t y f i r s t appears i n -c r e a s e s a s q i n c r e a s e s .E f f e c t o f S i d e s l i p A n g l e

    As s h o wn b y T a b l e 3 , t h e d i rec t e f f e c t of n o n z e ro s ide s l i pang l e on t h e e i g e n v a l u e s i s not l a rge ; h o w e v e r , nonze ro @c a u s e s i n s t ab i l i t y to oc cur a t very low rol l r a t e s . T he e n l a r ge dc i r c l e in Fig. 1 i l l u s t r a t e s t h e e f f e c t of a 0 of 2 on t h ep r o g r e s s i o n of the roots wi t h s t ab i l i t y - ax i s ro l l rate. T h ecros sover to i n s t ab i l i t y o c c u r s w h e n /?5 = 5 deg/sec. A n i n -t e r e s t i n g r e s u l t i s t h a t th e s h o r t p e r i od mode, n o t t h e D u t c hro l l mo d e , becomes u n s t a b l e a s a r e s u l t of t he ro l l rate. F u r -t h e r e x a m i n a t i o n i n d i c a t e s t h a t th e s h o r t p e r i o d mo d e i sde s t ab i l i zed whenps a n d /3 h a v e t h e s a m e s ign , bu t t h e D u t c hrol l mo d e i s de s t ab i l i zed when ps a n d /3 h a v e opposite s i g n .The r e a r e i n s u f f i c i e n t da t a to ind ic a t e w h e t h e r or not t h i s i s agene ra l r e su l t ; however , i t i s r e a s on a b l e to e xpe c t d i f f e r i ngb e h a v i o r w h e n t h e veh i c l e i s s ide - s l ippe d "into" or "out-of"th e ro l l i ng mo t i o n . D e c r e a s i n g /3 to 1 i n c r e a s e s th e c ro s soverps to 10 d e g / s e c , w h i l e i n c r e a s i n g 1 3 t o 4 d e c r e a s e s th e ro l lr a t e fo r i n s t ab i l i t y to 3 deg/sec.C o n c l u s i o n s

    Th e e f f e c t s of s ide s l i p a n g l e a n d a n g u l a r r a t e s h a v e beend e m o n s t r a t e d for a high-a f l igh t cond i t ion u s i n g fu l ly c o u p l e dl i n e a r e q u a t i o n s of m o t i o n . I t i s s how n t h a t t h e s t ab i l i t y of t h ef ree mot ion of the v eh i c l e i s s en s i t i v e to ro l l r a t e a n d t h a t t h i ss e n s i t i v i t y i s m ag n i f i ed by n o n z e r o s ide s l i p a n g l e . T he s er e s u l t s s u g g e s t t h a t fu l ly c o u p l e d l i n e a r i zed e q u a t i o n s c a n b eof v a l u e n ot on ly for the s t u d y of Space S h u t t l e s t a b i l i t y a n dcon t ro l , bu t for a b e t t e r u n d e r s t a n d i n g of pos t - s t a l l gy ra t ion s ,i n c i p i e n t s p i n , a n d d e p a r t u r e p r e v e n t i o n f o r h ig h -p e r f o r m a n c e a i r c r a f t .R e f e r e n c e s

    Phillips, W . H ., "Effec t of Steady Rol l ing on L ong i t ud i n a l a n dD i r e c t i o n a l S t a b i l i t y , " TN 1627, Ju ne 1948 , N A C A .2 A b z u g , M . J ., "Effec t s of Cer t a in S t eady M ot ions on Sma l l -D i s t u r b a n c e A i r p l a n e M o t io n s , " Journa l of the Aeronaut ica l Scien-ces , Vol. 21, Nov. 1954, pp. 749-762.3B y u s h g e n s , G.S. a n d S t ud ne v , R . V. , " D y n a m i c s of t he Spat i a lM o t i o n of an Ai r c r a f t , " TT F-555, A p r i l 1969, N ASA .

    4 E t k i n , B ., Dynamics o f Atmospheric Flight , W i l ey , N e w Y o r k ,1972.5 P or t e r , R . F . and Loomis , J.P., "Examination of an A er o d y n am icC o u p l i n g Phenomenon," Jou rn a l of Aircraf t , Vol . 2 , Nov . - De c .1965, pp. 553-556.6Johnston, D.E. a n d Hogge, J.R., "The Eff ec t of No n - S y m m e t r i cF l igh t on Ai r c r a f t High A n g l e o f A t t a ck H an d l i n g Q u a l i t i e s a n dD e p a r t u r e C h a r a c t e r i s t i c s , " A I A A Paper 74-792 , New York A u g1974.

    S l i d e - V a l v e - C o n t r o l l e dV e c t o r i n g N o z z l eJ.A. C . K e n t f i e l d *Universi ty of Calgary,Calgary, Alberta, Canada

    I n t r o d u c t i o nSO M E t i m e ago, a b r i e f d e s c r i p t i o n w a s g iven ofa v e n t r a l - t y p e vec to r i ng nozz l e w i t h t h e f low t h r o u g h t h ev e n t r a l a p e r t u r e , c u t in t h e w a l l of t he j e t p i p e , con t ro l l ed by as l ide-valve of cu rved c ros s - s e c t ion co n f o r m i n g to t h a t of theje t p i p e . ' M o r e r e c e n t l y , ven t r a l nozz l e s wi th s l ide -va lv e c o n -t rol of t he v e n t r a l or i f i c e were des c r i bed i n f u r t h e r d e t a i l . 2Th e p r i m e a d v a n t a g e s of s l ide-valve c o n t r o l of the v e n t r a lflow a r e a m i n i m a l i n c r e m e n t in e n g i n e c r o s s - s e c t i o n a l a r e a ,c o m p a r e d wi t h t h a t of a n o n v e c t o r i n g e n g i n e a n d s i m p l i c i t y ;t h e r e i s on ly on e m a j o r m o v i n g p a r t , t h e s l ide -va lv e ,a s s o c i a t e d wi t h t h e v e n t r a l or i f i ce . T h e a d v a n t a g e s o f v e n t r a la s di s t i n c t f rom o t h e r t yp e s of v e c t o r i n g n o z z l e h av e beend i s c u s s e d , i n r e l a t i o n t o a p a r t i c u l a r m i s s i o n s t u d y , by G i l l . 3T h e v e n t r a l - t y p e vec to r i ng nozz l e w i t h s l ide-valve c o n t r o l a p -p e a r s to be s u i t a b l e on ly for e n g i n e s w h i c h a r e n o t a u g m e n t e di n t h e l i f t m o d e . Lif t mode a u g m e n t a t i o n c r e a t e s s e v e r ecoo l i ng p ro b l e ms w i t h t h e s l i d e-va lve and f low g u i d e - v an e s .T h i s No t e p r e s e n t s r e c e n t l y o b t a i n e d e x p e r i m e n t a l d a t ar e l a t i n g t o t h e a e r o d y n a m i c p e r f o r m a n c e , i n c l u d i n g t h e vec -t o r i n g c a p a b i l i t y , of a s l i d e-va lve-con t ro l l ed v e n t r a l o u t f l o wsys t em. A m a j o r f a c t o r c o m p l i c a t i n g t h e d e s i gn o f a l l v e n t r a lvec to r i ng nozz l e s i s t h e need to p rov ide vec to r c o n t r o l w i t h th ee n t i r e e n g i n e flow p a s s i n g t h r o u g h t h e v e n t r a l o p e n i n g .

    T h r u s t V e c t o r C o n t r o lI t i s p a r t i c u l a r l y i m p o r t a n t , w i t h ven t r a l nozz l e s , t o a r r a n g efo r a m e a s u r e of t h r u s t v ec t o r c o n t r o l i n t h e l i f t m o d e w i t h t h e

    f u l l e n g i n e f low p a s s i n g t h r o u g h t h e v e n t r a l a p e r t u r e . V ec t o rc o n t r o l , i n a n e l e m e n t a r y v e n t r a l nozz l e , so le ly by d i f f e r e n t i a lva r i a t ion of t h e a f t a nd v en t r a l e x i t a r e a s , r e s u l t s , d u e t o t h egeomet ry o f s u c h a n a r r a n g e m e n t , i n a f u n d a m e n t a lt h r u s t r e d u c t i o n . W i t h t h e a x e s of t he two f lows a t r i g h ta n g l e s , t h i s r e d u c t i o n h a s a m a x i m u m v a l u e o f j u s t u n d e r 3 0%w h e n the f low def l ec t i on a n g l e 8 is 4 5 ; for 0 = 0 and 6 =90 th e r e d u c t i o n i s ze ro . T h e s e p o i n t s a re a p p a r e n t f rom Fig. 1f rom c o m p a r i s o n o f t h e c u r v e s f o r t h e e l em en t a r y a n d i d e a lc a s e s . Figu r e 1 was e s t a b l i s h e d f rom s i m p l e r e s o l u t i o n of thet h r u s t f o r c e s r e s u l t i n g f rom ou t f l ow t h r o u g h t h e p r o p u l s i o na n d v e n t r a l a p e r t u r e s o f h y p o t h e t i c a l v en t r a l v e c t o r i n g noz-zles .Fu l l v e c to r c o n t r o l over t h e r a n g e 60