Riding and handling qualities of light aircraft: A review and analysis

change in rolling moment coefficient due to change in aileron deflection (found in control forces section) section lift-curve slope due to elevator deflection (this is not to be confused with rolling moment coefficient) Xi - a6R i ng-moment coef f i cient (M/ipU2Sc) pitch ing-moment coefficient due to thrust force pitch ai rfo i I section Ditching moment about the aerodynamic center Lma.c.

coefficient of wing force normal to the airplane reference line yawing-moment coefficient (N/ipU2Sb) mean aerodynamicchord a i l e r o n c h o r d e I e v a t o r c h o r d f l a p c h o r d ,i t y a i r p l a n e c e n t e r o f grav d r a gf o r c e base o f naturalsystem of logarithms or Oswald'sspan e f f i c i e n c y f a c t o r induced-angle span e f f i c i e n c y f a c t o r s t i c k - f o r c e due t o a i l e r o n a c t u a t i o n s t i c k f o r c e t h e e q u i v a l e n t p a r a s i t e a r e a d i s c u s s e d i n t h e CD s e c t i o n r a t i o o f e l e v a t o r d i s p l a c e m e n t t o t h e p r o d u c t o f s t i c k l e n g t h and s t i c k angulardisplacement a c c e l e r a t i o n due t o g r a v i t y (32.2 f t / s e c 2 ) horsepower h e i g h t o f t h e h o r i z o n t a l t a i l a.c. above t h e c.g. ( p o s i t i v e f o r a.c. above c.g. 1 moment o f i n e r t i a a b o u t t h e x - a x i s moment o f i n e r t i a a b o u t t h e y - a x i s moment o f i n e r t i a a b o u t t h e z - a x i s p r o d u c to fi n e r t i a incidence ang I e r a d i u s of g y r a t i o n a b o u t t h e x - a x i s r a d i u s o f g y r a t i o n a b o u t t h e y - a x i s r a d i u s o f g y r a t i o n a b o u t t h e z - a x i s l i f t o r r o l l i n g moment l e n g t ho ff u s e l a g eo r body lengthfromc.g. t o t a i l q u a r t e r c h o r d lengthfromwingquarterchordto t a i l quarterchord t a i l volume f a c t o r lengthfromc.g. t o v e r t i c a l t a i l aerodynamiccenter p'itching-momentaboutthec.g.

pitching-momentaboutthec.g. due t o t h e f u s e l a g e and n a c e l l e s M f us nac aerodynamic moments a b o u t t h e y - a x i s ( p i t c h i n g moment) m mass i ns l u g s N wingforcenormal t o t h e a i r p l a n e r e f e r e n c e l i n e o r y a w i n g moment n normal a c c e l e r a t i o ni ng ' s r o l l i n g v e l o c i t y P .

r a t e o f change o f r o l l i n g v e l o c i t y P

(e) h e l i x r o l l a n g l e

2U dynamicpressure ($pU2) o r p i t c h i n g v e l o c i t y r a t e o f change o f p i t c h i n g v e l o c i t y dynamicpressure a t t h e h o r i z o n t a l t a i l q t dynamic p r e s s u r e a t t h e v e r t i c a l t a i l qv RPM p r o p e l l e r r e v o l u t i o n s p e r m i n u t e Reynolds Number RN r y a w i n g v e l o c i t y .

r r a t e o f change o f y a w i n g v e l o c i t y S w i ng area s e a t r e f e r e n c e p o i n t SRP rudderarea sR e I e v a t o r a r e a Se f l a p a r e a S f bodysidearea SS h o r i zonta I t a i I area 'h S d i s t a n c ei nh a l fc h o r d sw h i c hi s used as a n o n - d i m e n s i o n a l i z i n g f a c t o r T t h r u s t TDPF t a i l damping power f a c t o r ,s e ef i g u r e 72 TDR t a i l damping r a t i o , see f i g u r e 72 1 aT

- - (see Appendix A )

TU m a u t t i m e o r a i r f o i I t h i c k n e s s t t i m e A t t i m e o f t h e downwash a t t h e t a i l p l a n e I ag a i r p l a n e v e l o c i t y U s t a I I speed trim speed UTr i m

w a i r p l a n e w e i g h t

maximum w i d t h of t h e f u s e l a g e o r n a c e l l e s W f X’ d i s t a n c ef r o mc . g .t ow i n gq u a r t e rc h o r d( p o s i t i v e f o r c.g. ahead o f q u a r t e r c h o r d ) - l o n g i t u d i n a ld i s t a n c er e a r w a r df r o mc . g . t o w ng aerodynami c c e n t e r X d i s t a n c e p a r a l l e l t o r e l a t i v ew i n df r o mt h e w ng a.c. t o t h e C . Q .

( p o s i t i v e f o r a.c. ahead o ft h ec . g . 1 d i s t a n c ef r o m body c e n t e r l i n e t o i n b o a r d edge o f a i l e r o n Yi p e r p e n d i c u l a r d i s t a n c e f r o m t h r u s t l i n e t o c . g . ( p o s i t i v e f o r t h r u s t ZT l i n e belowthec.g.1 v e r t i c a ld i s t a n c ef r o mw i n ga . c .t oc . g .( p o s i t i v ef o ra . c .a b o v ec . g . 1 ‘a d i s t a n c ef r o mt h ec e n t e r of p r e s s u r e o f t h e v e r t i c a l t a i l t o t h e a i r c r a f t =V c e n t e r l i n e ( p o s i t i v e f o r v e r t i c a l t a i l above t h ex - a x i s ) d i s t a n c ef r o mb o d yc e n t e r l i n e to q u a r t e r - c h o r d p o i n t o f exposedwing Z W r o o t c h o r d ( p o s i t i v e for t h eq u a r t e r - c h o r dp o i n tb e l o wt h e body c e n t e r I i ne)

r d i h e d r a l a n g l e

A wing sweep a n g l e r e s u l t a n t a n g u l a r v e l o c i t y arlgle o f a t t a c k r a t e of change o f a n g l e of a t t a c k inducedangle o f a t t a c k a n g l e o f a t t a c k f o r z e r o l i f t s t a I I a n g l e o f a t t a c k

e I e v a t o r e f f i c i e n c y f a c t o r I (dCL/dBE1/(dCL/dat) I

l e s i d e s 1 i p a n g f l i g h t p a t h a n g l e from t h e h o r i z o n t a l c o r r e c t i o nf a c t o r f o r induceddrag a i I e r o n d e f I e c t i o n e l e v a t o r d e f l e c t i o n r u d d e r d e f I e c t i o n a i l e r o n d e f l e c t i o n e I e v a t o r d e f I e c t i o n e l e v a t o r a n g l e a t z e r o a i r c r a f t l i f t f l a p d e f l e c t i o n trim t a b d e f l e c t i o n downwash a n g l e change i n downwash angledue t o change i n a n g l e o f a t t a c k .

Dutch Roll damping r a t i o s h o r tp e r i o d damping r a t i o %.p.

rl e f f i c i e n c y f a c t o r for t a i l , qL./q, o r t h e p r o p e l l e r e f f i c i e n c y p i t c h a n g l e r a t e o f change of p i t c h ang l e t a p e r r a t i o ( t i p c h o r d / r o o t chord 1 a i r p l a n e r e l a t i v e d e n s i t y f a c t o r (m/pSb) 3.1416 d e n s i t y s i dewash ang I e d a t c o r r e c t i o nf a c t o rf o ri n d u c e da n g l e or e l e v a t o r e f f e c t i v e n e s s f a c t o r ( - 1 d6 E t i m ec o n s t a n t for r o l l mode r o t I a n g l e second d e r i v a t i v e w i t h r e s p e c t t o t i m e o f t h e roll a n g l e a measure o f t h e r a t i o o f t h e o s c i l l a t o r y c o h p o n e n t o f bankangle t o t h e averagecomponent o f bank a n g l e f o l l o w i n g a r u d d e r - p e d a l - f r e ei m p u l s e a i I e r o n c o n t r o 1 command yaw a n g l e phaseang l e of Dutch Roll component of s i d e s l i p Dutch Rol I n a t u r a l f r e q u e n c y s h o r t p e r i o dn a t u r a 1 frequency S u b s c r i p t : a.c. aerodynamiccenter c.g. c e n t e r of g r a v i t y w i n gq u a r t e r - c h o r d c / 4 h h o r i z o n t a l t a i l 0 t a i I V v e r t i c a l t a i l W wing

STABILITY OERIVATIVES

I t i s customary,asnotedearlier,torepresenttheaerodynamicand t h r u s t f o r c e s o n a n a i r c r a f t b y a T a y l o r e x p a n s i o n a b o u t t h e c o n d i t i o n s for s t e a d y ,l e v e lf l i g h t , Because of t h ea s s u m p t i o nt h a tp e r t u r b a t i o n s from e q u i l i b r i u ma r es m a l l ,o n l yt h el i n e a rt e r n sa r er e t a l n e d Js q u a r e s , p r o d u c tt e r m s ,a n dh i g h e ro r d e rd e r i v a t i v e sa r en e g l e c t e d ,T h er e m a i n i n g d e r i v a t i v e s , c a l l e d s t a b i l i t y d e r i v a t i v e s , b e l n g more numerous t h a n t h e e q u a t i o n s of m o t i o n ,c a nt h e r e f o r e b ee v a l u a t e do n l yt h r o u g hs p e c i a lt e s t s o rt h r o u g ha n a l y s e sw h i c h limit responses t o t h o s e r e s u l t i n g f r o m a s i n g l e v a r i a b l e . The d l s c u s s i o nb e l o wi s o l a t e se a c h of t h e s ed e r i v a t i v e s ,t h e m e t h o df r o mt h el i t e r a t u r er e g a r d e da sm o s ts u i t a b l ef o ri t se v a l u a t i o n , and t y p i c a l v a l u e s f o r l i g h t a i r c r a f t , U s u a l l y , by t h et i m ea na i r p l a n ei ss u b j e c t e d t o a s t a b i l i t y a n a l y s i s , t h ed e s i g nh a sp r o g r e s s e df a r enough t h a t t h e l i f t c o e f f i c i e n t s f o r t h e f l i g h t c o n d i t i o n so fi n t e r e s ta r e known f o rt h ec o m p l e t ea i r p l a n e .S i n c e 2-D ( s e c t i o n )d a t aa r ea v a i l a b l e f o r many a i r f o i l and a i r f o i l f ! a p c o n f i g u r a t i o n s , i t i s d e s i r a b l e t o have a means o f c o n v e r t i n g s e c t i o n d a t a t o three-dimensional d a t af o r a p a r t i c u l a ra i r f o i l .I n Theory of F l i g h t ,R i c h a r d Von Mises (Ref. 6 ) g i v e sa ne s t i m a t eo f l i f t c o e f f i c i e n t , CQ CL =

1 + 2/AR

An example 2-D p l o t of Cg v e r s u sa n g l e of a t t a c k i s shown below f o r t h e 2412 a i r f o i ls e c t i o n .

2.01

1.61 3.1 RN xL06

0 5.7 1 - 2 1 o 8.9 -4l- " 4 -.8 - 1 . 2 8 16 24 32 -32 -24 -16 -8 0

Section Angle of Attack, Q , deg

F i g u r e 1 . 2-0 p l o to fs e c t i o n l i f t c o e f f i c i e n t v e r s u s a n g l e o f a t t a c k f o r 2412 a i r - f o i I s e c t i o n .

The a p p r o x i m a t i o n does n o t i n c l u d e s u c h f a c t o r s a s t a p e r e f f e c t s a n d t i p e f f e c t s .

.- It should be noted that 2-D section data are available for a large number of airfoils in Theory of Wing Sections (Ref. 7 ) ; and the same data are included in TR-824 (Ref. 8 ) .

A refined analysis to determine lift coefficient should include the lift contribution of the tail and, if possible, the lift contribution of the fuse- lage. The interference effects between the wing and the fuselage may be significant and, for this reason, wind tunnel tests or actual flight tests contribute to determining lift coefficient. The tail coefficient, can be included in that for the entire aircraft where ‘It = q+/q:

- St + c

(CL)Airp lane - ‘LW + ‘Lt S, ‘It Lfuselage’

nacel les The tail contribution to the total airplane CL at cruise can be approximated by using the moment equation: Xa Sw

*

or CLt = c - -

‘It -

Lw R t S t The fuselage lift coefficient may be quite difficult to estimate, unless some simplification, such as slender body theory, i s applied. NACA TR-540 (Ref. 9 ) discussed the interference effects between the wing and the fuselage and gives some example lift coefficients for the fuselage at various angles of attack. The table below gives an example of some of trlese experimental data.

a=OO a= 1 Z0 a= 8O a=40 CL CD cL cD

Fuse I ag6 Eng i ne CL CD CL CD

Round None .011 .0062 .005 ,0049 .OOl .0042 .OOO .0041 .008 .0216 .004 .0200 .001 .0191 .OOO .0189 Round Uncow I ed .028 .0115 .017 .0088 .008 .0073 .OOO .0069 Round Cow I ed Rectangular None .OOO .0049 ,005 .0054 .014 .0068 .026 .0097 ; Table 1 . Example lift coefficients for the fuselage at various angles of attack.

* A method for calculating the elevator deflection required for equilibrium

cruise is given in the sample calculations in Appendix G.

The f u s e l a g e c o e f f i c i e n t s i n t h e t a b l e a r e b a s e d o n w i n g a r e a of t h ew i n gf u s e - l a g ea r r a n g e m e n tt e s t e d . Thus, b ye x a m i n i n gt h et a b l ew i t h a knowledge o f t h e a i r p l a n e a n g l e of a t t a c k ,o n ec a ne s t i m a t et h ef u s e l a g ec o n t r i b u t i o n t o t h e t o t a la i r p l a n e l i f t c o e f f i c i e n t .F o rs m a l la n g l e s of a t t a c k , 00 t o 8O, t h e f u s e l a g e l i f t c a np r o b a b l yb en e g l e c t e dw i t h o u t a s i g n i f i c a n t change i n t h e t o t a l a i r p l a n e l i f t c o e f f i c i e n t ,u n l e s st h ef u s e l a g ei sh i g h l y cambered.

Datcom (Ref., 10) g i v e s a method f o r c a l c u l a t i n g C L , of a body based on p o t e n t i a lf l o wt h e o r y( s e e CL,). Thus, CL c a nb ea p p r o x i m a t e db ym u l t i p l y i n g C L , by a of t h e body, w h i c hi s known.

F o rl i g h ta i r p l a n e si nt h ec r u i s i n g mode of f l i g h t , t h e l i f t c o e f f i c i e n t w i l lp r o b a b l yf a l lb e t w e e n0 . 2 5 and 0.45. I nc l i m b i n gf l i g h t , CL will p r o b a b l y be larger”0.5 t o 0 . 9 . F o r t h e l a n d i n g a p p r o a c h , t h e u s u a l CL will p r o b a b l y range from .95 t o 1.18. F o rt h e Cessna 182, t y p i c a lv a l u e sa r e 0.309 a tc r u i s e , 0.719 i nc l i m b i n gf l i g h t , and 1.12 i nt h el a n d i n ga p p r o a c h .

T h ed r a gc o e f f i c i e n t , CD, i s a na e r o d y n a m i cf o r c ec o e f f i c i e n tw h i c hc a n b et h o u g h t o f a s a damping c o e f f i c i e n t . T h ee q u i l i b r i u ma i r p l a n ed r a ge f f e c - t i v e l y damps t h e t h r u s t b y a c t i n g i n t h e o p p o s i t e d i r e c t i o n o f t h e r e l a t i v e wind and i s a l w a y sp o s i t i v ei ns i g n . When consideringperformance o f an a l r - c r a f t , t h e s m a l l e s t p o s s i b l e v a l u e o f CD i s desired;however, i n a i r f r a m e dynamics, CD i s t h e m a i n c o n t r i b u t o r t o . t h e damping of t h e p h u g o i d mode.

Thus, t h el a r g e rt h ev a l u e of CD, t h eb e t t e rt h e damping. Since phugoid damping is n o tc o n s i d e r e d o f m a j o r i m p o r t a n c e i n t h e f l y i n g q u a l i t i e s o f t h e a i r p l a n e , t h e p e r f o r m a n c e r a t h e r t h a n t h e f l y i n g q u a l i t i e s o f t h e a i r c r a f t s h o u l d d i c t a t e t h e d e s i g n v a l u e o f CD.

A p r e l i m i n a r ys t a g e of d r a g e s t i m a t i o n o f a n a i r p l a n e i n anincompress- i b l e f l o w may b ea c c o m p l i s h e db ya d d i n gt h ei n d i v i d u a ld r a g so fs e v e r a l components o ft h ea i r p l a n e .T h i s method i sc a ll e d" d r a g breakdown." The m a j o r i t y of a i r p l a n ed r a gc u r v e sc a nb ee x p r e s s e da s yr. 2 where CDf = z e r o l i f t o r p a r a s i t e d r a g , M a t h e m a t i c a l l y , CDf i s ' t h e i n t e r c e p t o n t h e CD a x i s o f a graph of CD v e r s u s C L ~ , and l/TeAR i st h es l o p e .O n l ya tv e r y low or h i g hv a l u e so fl i f t c o e f f i c i e n t s does t h e p a r a b o l i c a p p r o x i m a t i o n d e v i a t e f r o m t h e a c t u a l d r a g p o l a r .

The f o l l o w i n gp r o c e d u r ei s commonly used t o o b t a i n t h e v a l u e of C D ~ f o r a p a r t i c u l a ra i r p l a n e . The d r a gc o e f f i c i e n to f each component i s based on an area, A , , " p r o p e r "t ot h a t component. For example,oneusuallybasesthedrag c o e f f i c i e n tf o rt h ef u s e l a g eo nt h e maximum f u s e l a g ea r e a . The e q u i v a l e n t p a r a s i t ed r a ga r e a , f , i s thenexpressedas The t o t a l f f o r a na i r p l a n ei sa p p r o x i m a t e l yt h e sum o f CD A , f o r t h e i n d i v i d - ua I components, p I us f i v e o r t e n p e r c e n t f o r mutua I i n t e r y e r e n c e b e t w e e n t h e components.Anotherpercentageerror ( 3 % t o 5%) may a Is0 beadded f o r sma I I protuberances such as handles, hinges, antennas, and c o v e rp l a t e s .F i f t e e np e r c e n t of t h e t o t a l CD,A~ was used t o account for p r o t u b e r a n c e sa n di n t e r f e r e n c e s o nt h e Cessna 182 d i s c u s s e dl a t e ri nt h ep r e s e n ts t u d y . Once t h et o t a l f i s found, CDf f o r t h e a i r p l a n e c a n bedeterminedfrom - f "

cDf s

where S '= wing area.

CL2 . - T o t a l CD i s t h e n cDf -t TeAR

* e c a n b e e s t i m a t e d i n t h e CL s e c t

ion.

a Values of C D , and t h e a r e a s o n w h i c h t h e y a r e b a s e d a r e g i v e n i n t h e t a b l e s b e l o w . T h i s i n f o r m a t i o n was s e l e c t e d from References 11, 12, 13, and 14.

Thewingdrag C D , f o ra i r f o i l sw i t hs t a n d a r dr o u g h n e s sc a nb eo b t a i n e d from a t a b l eg i v e ni nR e f e r e n c e ( 1 5 ) and shown below. The comparison is based onstandardroughness a t RN = 6 x l o 6 ; t h e w i n g C D , i s basedonwingareaand i st a b u l a t e da s C d , . Wing f l a p d e f l e c t i o n may g e n e r a t ea na d d i t i o n a l C D , which should b ea c c o u n t e df o r .

A i r f o i l S e c t i o n

632-41 5 .15 .0098 .20 1 .33 12. -. 070

23016.5 ,165 .0102 .15 1 12. -. 005

.20

23012.0 .12 .0099 .18 1 .22 12. -. 01 5

241 2 .12 .0098 .20 1 .22 14. -. 048

2301 8 .18 .0105 1 -. 005

.10 .05 1 1 .

241 0 .10 .0095 .20 1 .22 -. 050

14.

632-21 5 .15 .0097 .16 1 .26 14. -. 030

641-412 .12 .0097 1 -. 070

.31 .34 12.

642-AZ 1 5 .15 .0102 .20 1.20 14. -. 035

652-4 15 .15 ,0100 .22 1.25 14. -. 065

641 -21 2 .12 .0088 .18 1.18 1 1 . -. 025

T a b l e 2. A i r f o i l s e c t i o n s .

The f u s e l a g e C D , c a nb eo b t a i n e df r o mt h et a b l eb e l o wb yc h o o s i n gt h ef u s e l a g e m o s tn e a r l yl i k et h eo n ei n v e s t i g a t e d .T w e n t yp e rc e n t of C D , can be added t o a c c o u n tf o rt h e canopyandwindshield.Forthe Cessna 182, t h ef a u r t h f u s e l a g e was chosen.

Ref.. Area 0.266 SC 0.062 0.071 SC 0.063 0.116 SC T a b l e 3. Fuselage drag where Sc = maximum c r o s ss e c t i o n a la r e a and L = Fuselagelength.

.-

I II

The empennage CD v a l u e sf o ru n d e f l e c t e dc o n t r o ls u r f a c e sc a nb ef o u n d from Tab I e 4 based on'the t a i I p I anform area where is i s t h e h o r i z o n t a I t a i I s u r f a c e i n c i d e n c e .

Tai I ' D , Lrrangement Description \rea

"-

Tapered f i I l e t s , v e r t i c a l and h o r i z o n t a lt a p e r e ds u r f a c e s is = OO .0043 S t i5 = -40 .0063 Tapered f i l l e t s ,t a i ls u r f a c e s d i t h e n dp l a t e s is = OO ,0058 S t is = -40 .0063 Symmetr i c a I t a p e r e d f i I I e t s i s = 00 .0059 S t d e r t i c a l and h o r i z o n t a lt a i l s u r f a c e s is = OO .0070 S t . = -40 .0058

' s

T.a i I s u r f a c e s w i t h end 3 I a t e s .0058 is = 0 ' S t rapered f i I I e t s , h o r i z o n t a I ta i I s u r f a c e s i s = 0 ' .0039 Sb i s = -40 .0083 i s = 4O - 0 0 6 1 T a b l e 4 . Empennage drag.

The i n f o r m a t i o nr e q u i r e df o rl a n d i n gg e a rd r a gp r e d i c t i o ni sg i v e ni nT a b l e 5.

F o rt h e nose gear, C D ~ i sg i v e n ; however, f o r t h eo t h e rl a n d i n gg e a r ,t h e v a l u eo f f i s g i v e n . The t a b u l a t e dv a l u eo f f i st h ev a l u e f o r bothwheels of t h el a n d i n gg e a ra s s e m b l y .

Remarks 8.50-10 wheels, not faired . . . . . . . .

1.67

8.50-10 wheels,faired . . . . . . . . . . 1.50

8.50-10 wheels, no streamline members . . 3.83

8.50-10 wheels, faired . . . . . . . . . . 0.74

27-in. streamlined wheels, not faired . . . . . . . . . . . . . . .

0.98 27-in. streamlined wheels, not faired . .

0.84

8.50-10 wheels, faired . . . . . . . . . . 0.68

21-in. streamlined wheels, not faired . .

0.53

8.50-10 wheels . . . . . . . . . . . . . . 0.51

8.50-10 wheels, not faired . . . . . . . . 1.52

8.50-10 wheels, faired . . . . . . . . . .

1.02

8.50-10 wheels, not faired . . . . . . . . 1.60

24-in. streamlined wheels, 8 , intersections

filleted . . . . . . . . . . . . . . 0.86

8.50-10 wheels, no fillets . . . . . . . .

1.13

8.50-10 wheels . . . . . . . . . . . . . . 1.05

Low pressure wheels, intersections filleted 0.31 Low pressure wheels, no wheel fairing . .

0.47 Streamlined wheels, round strut, half fork

no fairing . . . . . . . . . . . . . 1.25

For the nose gear Cg = .%.8 based on Nose Gear 7T A , = (wheel diameterl(whee1 width)

I

Table 5, Landing gear drag.

.- The C D , v a l u e sf o ro t h e rc o m p o n e n t s of i n t e r e s tc a nb ee s t i m a t e d by u s i n g t h e t a b l e b e l o w : AREA FOR DRAG COMPONENTS CA LCU LAT I ON cD7T Nace I l e s 1 . abovewing, smal I a i r p lane C r o s s s e c t i o n a r e a .250 2. large leading edge

n a c e l l e ,s m a l la i r p l a n e C r o s s s e c t i o n a r e a . 1 20

3. smal I leadingedge n a c e ll e ,l a r g ea i r p l a n e Cross section area .080 4 . improved nacelle, no c o o l i n g f l o w C r o s s s e c t i o n a r e a .050 5. i m p r o v e d n a c e l l e , t y p i c a l

c o o l i n g a i r f l o w C r o s s s e c t i o n a r e a . l o o

Wing Tanks 1 . c e n t e r e do nt i p Cross s e c t i o na r e a .05-. 07 2. belowwing t i p Cross s e c t i o na r e a .07-. 10 3. inboard below wing Cross s e c t i o n a r e a .15-. 30 W i r e s and S t r u t s 1 . smooth round wires and s t r u t s( p e rf o o t ) F r o n t a I a r e a 1.2-1.3 2. s t a n d a r d a i r c r a f t cab l e F r o n t a I a r e a 1.4-1.7 ( p e rf o o t ) 3. smooth e l l i p t i c a lw i r e F r o n t a I a r e a ( p e rf o o t ) f i n e n e s s r a t i o 2 : l 0.6-0.4 f i n e n e s s r a t i o 4 : l .35 f i n e n e s s r a t i o 8 : l .3-.2 4. standardstream1inedwire .45-. 20 ( p e r f o o t ) F r o n t a l area 5. s q u a r ew i r e( p e rf o o t ) F r o n t a I area .16-. 20 6. s t r e a m l i n e ds t r u t s ( p e r f o o t ) F r o n t a I area .075-0.10 T a b l e 6. A i r p l a n e components.

Forsmoothroundwireofdiameterlessthan 4 inch, assume

twoend f i t t i n g s e q u i v a l e n t t o t h r e e f e e t o f w i r e .

Forsmoothround s t r u t s of d i a m e t e rg r e a t e rt h a n5 / 1 6i n c h , assume twoend f i t t i n g s e q u i v a l e n t t o one foot of s t r u t .

Forsmooth e l l i p t i c a l w i r e , assume twoend f i t t i n g s e q u i v a I e n t t o 10 t o 15 f e e t of w i r e .

Forsquarewire, assume two end f i t t i n g s equ i v a l e n t t o t w o f e e t o f w i r e .

F o rs t r e a m l i n e ds t r u t s , assume two end f i t t i n g s e q u i v a l e n t t o f i v e f e e t of s t r u t i f f a i r e d ,t e nf e e t i f u n f a i r e d .

The t o t a l d r a g c o e f f i c i e n t f o r t h e a i r p l a n e i s t h u s The dragbreakdownmethod was used t o e s t i m a t e a d r a g c o e f f i c i e n t of 0.0311 f o r t h e Cessna 182. D r a gc o e f f i c i e n t sf o ro t h e rl i g h ta i r p l a n e sc a nb ef o u n d u s i n gt h et a b l eb e l o wt a k e n from Reference (14).

YEAR A I RPLAN E TYPE b S W "POWER" u

cD (ft) (ft') ( I b s ) ( h p l ( k t s ) I I 1903 W r i g h t - B r o f h e r s b i p l a n e 40: 5.10 750 12 26 0.074 1945 35 5 79 1500 1 35 110 0.032 P i p e r "Cub" persona I 36 175 2200 140 122 0.032 1950 Cessna rr170rf persona I 1942 M e s s e , r s c h m i t f - I 0 9 f i g h t e r 32 172 6700 1200 330 0.036 0.020 1943 N. A . P-51 "Mustang" f i g h t e r 37 23.5 10000 1380 38,O Table 7. D r a gc o e f f i c i e n t sf o rs e l e c t e dl i g h ta i r p l a n e s I t s h o u l db en o t e dt h a tR e f e r e n c e ( 1 5 ) g i v e sa n o t h e r method o f e s t i m a t i n g d r a g c o e f f i c i e n t w h i c h sums f r i c t i o n d r a g and t h e p r o f i l e d r a g for each o f t h e components.

* e = 1 / ( 1 + 6 1 and a g r a p hf o re s t i m a t i n g ( 1 + 6 ) can be found i nt h e d i s c u s s i o n o f t h e C L , s e c t i o n .

The aerodynamic pitching moment a b o u tt h e c.g. i sd e f i n e da s Cm. I n e q u i l i b r i u m f l i g h t , t h e a e r o d y n a m i c p i t c h i n g moment m u s tb a l a n c et h e moment c o e f f i c i e n t r e s u l t i n g from t h r u s t .A l t h o u g h Cm appears i n t h e l i s t of dimen- s i o n a l s t a b i l i t y d e r i v a t i v e s i n A p p e n d i x B of t h ep r e s e n ts t u d y , it i s n o t u s u a l l yr e f e r r e d t o as a s t a b i l i t y d e r i v a t i v e . The p r i n c i p a l e f f e c t o f Cm i s t o c o n t r i b u t e t o t h e p e r i o d o f t h e p h u g o i d mode.

Severalforcesand moments, such as t a i l , wing, and fuselage l i f t and drag, and moments a b o u tt h ea e r o d y n a m i cc e n t e r so ft h ew i n ga n dt a i i ,c o n t r i b u t e t o t h e p i t c h i n g moment c o e f f i c i e n t ; however, f o r e q u i l i b r i u m ,p o w e r - o f ff l i g h t , t h ee l e v a t o ri sp o s i t i o n e d so t h q t Cm i sz e r o .F o rp o w e r e df l i g h t , Cm i s t h e n e g a t i v e of t h e Cm r e q u i r e d t o b a l a n c et h et h r u s tf o r c e moment, CmT. CmT canbe w r i t t e n a s T ZT - " ' , T q S i where ZT = p e r p e n d i c u l a rd i s t a n c e from t h r u s t l i n e t o c.g., p o s l t l v e f o r t h r u s t l i n e below t h ec . g .

T = t h r u s t The p a r t i c u l a ra i r p l a n ed i s c u s s e dl a t e r was examined w i t hp o w e r - o f f ;t h u s , Cm = 0. From t h ee q u a t i o n above, i t i sa l s oa p p a r e n tt h a t C , i s z e r o when t h e t h r u s t l i n e p a s s e dt h r o u g ht h ec e n t e ro fg r a v i t y ( Z T = 0 ) .

An e q u a t i o n f o r Cr? i s a l s o v e r y h e l p f u l i n d e s i g n i n g or s i z i n ga na i r p l a n e .

be w r i t t e n s i m p l y b y summing theaerodynamic moments o f t h e The equationcan v a r i o u s a i r c r a f t components about the c.g. The e q u a t i o ng i v e nb e l o wi s a modi- f i c a t i o n of t h a t c o n t a i n e d i n TR-927 (Ref. 16). D e f i n i n g "nose-up" as a p o s i - t i v e moment, t h e moments can be summed u s i n gF i g u r e 2. The f o l l o w i n ge q u a t i o n s arebasedonthe a s s u m p t i o nt h a tc o s a = 1 and s i n a = 0.

L

\ i'

F i g u r e 2. Forces and moments i nt h ep l a n e of symmetry.

Mc.g. = L x a + Dza + Ma.c. - L t R t + D t h t +

where I x , i s p o s i t i v e f 0 r . a . c . o f - w i n g ahead of c.g.

Za i s p o s i t i v e f o r a.c. ofwingabovec.g.

h t i s p o s i t i v e f o r t a i l a.c. above c.g.

= d i s t a n c e from c.g. t o t a i I q u a r t e rc h o r d Thus, t h e e q u a t i o n f o r t h e moment c o e f f i c i e n t c a n be w r i t t e n as S i n c e t h e t a i l i s u s u a l l y a s y m m e t r i c a ls e c t i o n , where

at = (aw - iw - E + it)

Thus, I t s h o u l db en o t e dt h a tt h ea b o v ee q u a t i o n does n o t c o n s i d e r t h e f u s e l a g e or p r o p e l l e rp i t c h i n g moment. The f u s e l a g ep i t c h i n g moment can be approximated by i se v a l u a t e di nt h ed i s c u s s i o no ft h e Cma i nt h ep r e s e n ts t u d y .

( C m a ) f se age The e f y e c s o f t h e p r o p e I l e r a r e d i s c u s s e d i n t h e s e c t I o n dea I i ng w i t h power 4 e f f e c t s o n s t a b i l i t y d e r i v a t i v e s .

The t h r u s t c o e f f i c i e n t , . C ~ , l i k e CL i s n o t s t r i c t l y a s t a b i l i t y d e r i v a t i v e b u ti s . n e c e s s a r yi n a dynamicana,lysis. . The c o e f f i c i e n t i s non-dimensional . .

. . , ..

and i s d e f i n e d , a s .

Forthe,Cessna 182 d i s c u s s e dl a t e r , CT = 0.0 s i n c et h ea n a l y s i s was made w i t h power-off; tiowever, f o r power-on c r u i s e CT = .04.

C C and Cm

L"' D" U

The s t a b l l i t y d e r i v a t i v e s C L ~ , C D ~ , and Cmu a r e t h e changes i n l i f t , drag, and p i t c h i n g moment c o e f f i c i e n t s ,r e s p e c t i v e l y ,w i t ha i r s p e e d . These changes a t low speeds a r er e a l l yR e y n o l d s number e f f e c t s andcan u s u a l l y beconsidered t o be zero. The f a c tt h a tt h e ya r ev e r ys m a l l f o r low Mach numbers i sp o i n t e d o u tb yt h ef i g u r eb e l o wt a k e n from Reference (18).

0.3 0.5 0.7 0.3 0.5 0.7 0.9 Mach number Mach number F i g u r e 3. V a r i a t i o no f Cg and Cdowith Mach number.

S i n c ef o rt h ec r u i s ec o n d i t i o n CT CD and C T ~ N CD, and i'n t h e same manner as above CT i s n e g l e c t e d f o r c r u i s i n gf l i g h t .I ts h o u l d b en o t e dt h a tf o r approach an8 I and i ng maneuvers, CT, and CD" may become s i g n i f i c a n t and t h e r e f o r e s h o u l dn o t beneglected.

C L ~ , CDu, Cmu, and C T ~ were chosen t o be z e r o f o r t h e Cessna 182 i n v e s t i g a t e d l a t e r i n t h i s r e p o r t .

The s t a b i l i t y d e r i v a t i v e C b i s t h e change i n l i f t c o e f f i c i e n t w i t h a n g l e of a t t a c k and i s commonly known a st h e l i f t c u r v es l o p e .T h i sd e r i v a t i v ei s a l w a y sp o s i t i v e f o r a n g l e s o f a t t a c kb e l o wt h es t a l l .O r d i n a r i l y ,t h ew i n g accounts f o r 85% t o 90% of t h e t o t a l C b . T h i sd e r i v a t i v ei sv e r yi m p o r t a n t i n e q u i l i b r i u m f l i g h t and i n d y n a m i cc o n d i t i o n s .S i n c ea na i r p l a n ew i t h a h i g h e rv a l u e o f CLC, u s u a l l y has a lowerdrag, a h i g hv a l u e o f C b i s d e s i r a b l e f o r optimum performance. Higher values of C a r en e c e s s a r i l ya s s o c i a t e dw i t h h i g h a s p e c t r a t i o , u n s w e p t w i n g s . T h e d e r i v a % i v e a l s o makes a n i m p o r t a n t c o n t r i b u t i o n t o t h e damping of t h e l o n g i t u d i n a l s h o r t p e r i o d mode.

T h a tp o r t i o no ft h ee f f e c t i v ew i n ga n g l eo fa t t a c ki n d u c e db yw i n g l i f t canbe w r i t t e n from Reference ( 1 9 ) : where I

el = l + . r ' = i n d u c e d - a n g l es p a ne f f i c i e n c yf a c t o r

T = c o r r e c t i o nf a c t o rf o ri n d u c e da n g l e( s e eF i g u r e 4 ) AR = w i n ga s p e c tr a t i o S i n c et h ea n g l e o f a t t a c k f o r z e r o l i f t does n o t v a r y w i t h a s p e c t r a t i o , t h e g e o m e t r i ca n g l e s of a t t a c k f o r t h e same a i r f o i l w i n g o p e r a t i n g a t t h e same CL b u t w i t h t w o d i f f e r e n t a s p e c t r a t i o s a r e r e l a t e d b y Thus, t h ee x p r e s s i o n f o r t h e l i f t c u r v es l o p eo f a w i n gw i t ho n ea s p e c tr a t i o i n t e r m s o f t h e l i f t c u r v e s l o p e a t a n o t h e ra s p e c tr a t i o becomes where dCL/da = change i n ' l i f t c o e f f i c l e n tp e rd e g r e e The e q u a t i o na b o v ei m p l i e st h a tt h e l i f t c u r v es l o p e i s a c o n s t a n t ,w h i c h i s a f a i r l ya c c u r a t ea s s u m p t i o n f o r a n g l e s of a t t a c k up t o 10 o r 12 degrees.

However, o n c et h el i n e a rr a n g e of t h e 2-D l i f t c u r v es l o p ei s exceeded, t h e l i f t c u r v es l o p ec a no n l yb ee v a l u a t e df o r a p a r t i c u l a rv a l u e of a. I f c o n d i - t i o n "2" i s t h a t o f i n f i n i t e a s p e c t r a t i o , t h e l i f t c u r v es l o p ep e rd e g r e e o f a f i n i t e a s p e c t r a t i o c a nb ew r i t t e ni nt e r m so ft h e 2-D a i r f o i l l i f t c u r v e slope, (dCL/da), as 4 1 T y p i c a lv a l u e s f o r CCk), perdegreerange from a,bout 0.110 f o r t h i n a i r f o i l s t o a b o u t 0.115 f o r t h i c k a i r f o i l s a t ReynoldsNumbersgreaterthanabout lo6.

The s l o p e i s somewhat l e s sa tl o w e rR e y n o l d s Numbers and a tt h eh i g h e r( n e a r 1.0) l i f t c o e f f i c i e n t s .F o re x a c t i n g use, r e f e r e n c es h o u l d be made t o c a r e f u l w i n dt u n n e lt e s t s of t h e a i r f o i l b e i n g examined.

A r a p i d a p p r o x i m a t i o n t o t h e e x a c t v a l u e of e l may beachievedbyusing e i t h e ro rb o t h of t h e f i g u r e s b e l o w . F i g u r e 4, t a k e n f r o m R e f e r e n c e (19) hasboth 'I and 6 p l o t t e d f o r v a r i o u s t a p e r r a t i o s a t a n a s p e c t r a t i o o f 6 . 2 8 ( 6 i s used i nt h ed i s c u s s i o no f C o a l . F i g u r e 5 i s a p l o t o f 1 + T and 1 + 6 f o rv a r i o u sa s p e c tr a t i o sa t a t a p e r r a t i o o f 1.0 fromReference (20). Thus, i f a g i v e n d e s i g n has an aspect r a t i o 6 , F i g u r e 4 may be used; i f t h e d e s i g n has a t a p e rr a t i oo f 1.0, F i g u r e 5 may be used. For most l i g h ta i r p l a n e s , F i g u r e 4 s h o u l dg i v ea c c e p t a b l er e s u l t s .

Tip chod'Root chord F i g u r e 4. V a r i a t i o no f T and 6 w i t h t a p e r r a t i o s a t a s p e c t r a t i o of 6 . 2 8 .

AR

F i g u r e 5. V a r i a t i o no f 1 + T

and 1 + 6 w i t ha s p e c t

r a t i o s a t t a p e r r a t i o o f 1.0.

The l i f t c u r v e s l o p e o f p a r t i a l o r f u l l span f l a p p e dw i n g sc a no f t e n be a p p r o x i m a t e ds a t i s f a c t o r i l yo rd i r e c t l yf o u n d by c o n s u l t i n g t h e NACA l i t e r a t u r e .

Many r e p o r t sh a v e been w r i t t e nc o n c e r n i n gv a r i o u sf l a pc o n f i g u r a t i o n so ns e v e r a l a i r f o i l s o r wings. Thus, NASA CR-1485 ( R e f . 2 ) should be v e r y v a l u a b l e , because it l i s t s NACA r e p o r t sd e a l i n gw i t hs p e c i f i cf l a pc o n f i g u r a t i o n s .I n the usual case, CL i sl a r g e rf o rf l a p sd e f l e c t e dt h a nf o rf l a p sr e t r a c t e d .

F o r t h e c r u i s e c o d i t i o n , a n u n d e f l e c t e d f l a p c a n u s u a l l y be c o n s i d e r e d p a r t o ft h ea i r f o i ls e c t i o n ,t h u sr e q u i r i n g no a d d i t i o n a lc a l c u l a t i o n .F o ro t h e r f l i g h t modes, however, it may benecessary t o f i n d C L ~ f o r a g i v e nf l a pd e f l e c - t i o n . The l i f tc u r v es l o p eo ft h ew i n gi sp r o b a b l y known f o r a l l f l i g h t c o n d i t i o n sb yt h et i m e a d y n a m i ca n a l y s i si s made, b u t i f n o t , NASA CR-1485 s h o u l d be usedas a m a j o rr e f e r e n c e f o r i n f o r m a t i o no np r e d i c t i o no fa e r o d y n a m i c c h a r a c t e r i s t i c so ff l a p s .I ft h ea i r p l a n ea n a l y z e d has p a r t i a l span f l a p s i n s t e a d o f f u l l s p a nf l a p s ,t h ea p p r o x i m a t e l i f t c u r v es l o p ec a nb eo b t a i n e d f r o m where S = wing area f = s u b s c r i p t d e n o t i n g f l a p E L

- = mean s e c t i o n v a l u e o f t h e l i f t c u r v e

da s I opebetween t h e r o o t and t i p s e c t i o n L.

Although ~ L C , f o rt h ec o m p l e t ea i r p l a n e ,i n many cases, i s assumed equal t o t h e C b . of t h ew i n ga l o n e , amethod o f p r e d i c t i n g t h e bodyand t a i l C b s h o u l db ea v a i l a b l e . T h em e t h o dp r e s e n t e dh e r ef o ra p p r o x i m a t i n gt h e body c o n t r i b u t i o ni st a k e nf r o mR e f e r e n c e (10).

where

k 2 - k, = apparent mass f a c t o rw h i c hi s a f u n c t i o n o f

f i n e n e s sr a t i o( l e n g t h / m a x i m u mt h i c k n e s s ) vb = t o t a l body volume So = c r o s ss e c t i o n a la r e aa t x .

x . = body s t a t i o n where f l o wc e a s e st o be p o t e n t i a l , t h i s i s a f u n c t i o no f x i , t h e body s t a t i o n where theparameter dSx/dx f i r s t reaches i t s minimum value.

( T h i s s t a t i o n where t h e change i na r e a w i t hr e s p e c tt o x f i r s t reaches i t s lowest valuecan be e s t i m a t e df r o m a s k e t c ho f t h e body. 1 Sx = body c r o s ss e c t i o n a la r e aa t any body s t a t i o n R b = bodylength.

F i g u r e s 6 and 7 r e p r e s e n tg r a p h sf o re s t i m a t i n gb o t hk 2 - k l and xo/Rb.

FINENESS RATIO F i g u r e 6 . Reduced mass f a c t o r .

F i g u r e 7 . Body s t a t i o n where f l o w becomes v i s c o u s .

F o ra ne s t i m a t eo f some v a l u e so f (C~,)Body, T a b l e 1 from TR-540 (Ref. 9 ) i nt h ed i s c u s s i o no f C L canbeconsidered. The t a b l e i s a t a b u l a rf o r m of CL v e r s u s a f o rv a r i o u sf u s e l a g e ,w i n gc o m b i n a t i o n s . The v a l u eo f C L i s based on w i n ga r e a ,a n da l t h o u g ht h ef u s e l a g e sg i v e nd on o tr e a l l yr e p r e s e n tt h ef u s e - l a g e o f a l i g h ta i r p l a n e ,t h e ys e r v ea s an i n d i c a t o rf o rt h ed e g r e eo ff u s e l a g e c o n t r i b u t i o nt o( C b ) T o t a l .

The t a i l a sw e l la st h ef u s e l a g e may make a s i g n i f i c a n t c o n t r i b u t i o n t o C L , . F o r t h e u s u a l a e r o d y n a m i c a n a l y s i s , t h e w i n g C L ~ i su s u a l l yc o n s i d e r e d t o be t h et o t a l C o f t h ea i r f r a m e ; however, f o r a dynamic s t a b i l i t ya n a l y s i s , t h e t o t a l CLO, i s t % e change i n t o t a l C L r e s u l t i n g f r o m a change i n a n g l e o f a t t a c ko n l y . An a i r p l a n e i n f l i g h t must be trimmed a f t e r t h e a n g l e o f a t t a c k i s changed i f t h e new a n g l eo fa t t a c ki st o bemaintained. Thus, t h e trim f o r c e sc h a n g eo v e rt h ea n g l eo fa t t a c kr a n g e ,m a k i n g it i m p o s s i b l e t o measure C L , i n f l i g h t w i t h e v e r y t h i n g e l s e ( i n c l u d i n g trim f o r c e s )c o n s t a n t . * Wind t u n n e lt e s t sc a n be used t o o b t a i n t h e t o t a l C , s i n c et h e model can be r e s t r a i n e d . S i n c e t h e t a i l c o n t r i b u t i o n i s sma 9 I compared t o t h ew i n gc o n - t r i b u t i o n( u s u a l l yl e s st h a n l o % ) , i n many cases, C L o ft h ew i n ga l o n ec a n beused.For t h e a i r p I ane t o beana l y z e d l a t e r i n t a i s s t u d y , t h e w i ng (4.61 I i s used.

cLa The t a i l c o n t r i b u t i o n t o C b can be estimatedas,

T o t a l L i f t = L i f t w i n g + L i f t f u s e l a g e + L i f t t a ; I

Thus, The d e r i v a t i v e of t h e above e x p r e s s i o n i s t a k e n t o y i e l d * A l t h o u g h i m p o s s i b l e t o measure e x a c t l y , C L ~ can be approximated with a gooddegree of accuracy from f l i g h t t e s t r e c o r d s .

S i nce = a , - i, + i t - E , at t h u s , The t a i l c o n t r i b u t i o n c a n t h e r e f o r e be w r i t t e n a s T y p i c a lv a l u e so f C L f o r l i g h ta i r p l a n e sf a l li nt h er a n g e o f 4.0 t o 7.0 per r a d i a n ,d e p e n d i n gc h y e fl y on t h e t y p e o f wingused.

The s t a b i l i t y d e r i v a t i v e C D ~ i s t h e . c h a n g ei nd r a gC o e f f i c i e n tw i t hv a r y i n g a n g l e of ' a t t a c k . : - A b o v e . t h e a n ' g l e of a t t a c k f o r minimum drag - t h e . d r a g c o e f f i c i e n t increases a s t h e , a n g I e o f a t t a c ki n c r e a s e s ;t h u s , ' C di s p o ' s i t t i v ei ns i g n . ' 8 C D a i , i s u a I I y Has .I i t t l e e f f e c t o n t h e s h o r t p e r i o d mo%e and h a s o n l y a . sma I I ' , e f f e c t o n t h e p h u g o i d mode i n t h a t a d e c r e a s e ' i n C D ~ u s u a l l y i n c r e a s e s s t a b i l i t y .

C o , i s made up o fc o n t r i b u t i o n sf r o mt h e wing,fuselage,and t a i l s e c t i o n , w i t ht h ew i n gb e i n gb yf a rt h el a r g e s t . The w i n gd r a gc o e f f i c i e n tc a n be w r i t t e n a s

CD = cdo + (CL*/.rreAR)

where Cdo = p r o f i l e d r a g c o e f f i c i e n t

e = 1 / ( 1 + 6 ) = O s w a l d ' ss p a ne f f i c i e n c yf a c t o r *

Thus, C I J ~ i s wing dCdo ~ C L - "

+ -

CDawing d a TeAR 'h

For small angles o f attack, (dCdo/da) i su s u a l l ys m a l l ; however, for f l i g h t modes other than cruise, (dCdo/da) could have a s i g n i f i c a n tv a l u e . The wing C D ~ , inmostcases,servesas a goodapproximation o f t h e t o t a l C D ~ .

F o rf u s e l a g ea n g l e s o f a t t - a c kl e s st h a n loo, t h ef u s e l a g e C I J ~ can be i g n o r e d ,a sc a nt h et a i l C D ~ , w i t h good accuracy. Wind t u n n e lt e s t ss h o u l d a c t u a l l y be used t o measure a i r p l a n e C D ~ , butbecausethewing CD dominates and s i n c e t h e e q u a t i o n s a r e r e l a t i v e l y i n s e n s i t i v e t o changes i n it i s f e l t t h a t C D ~ can be approximatedbythewingcontribution.Thus, The v a l u e o f C D ~ c a l c u l a t e d f o r t h e Cessna 182 i s 0.126 p e rr a d i a n .

* A g r a p hf o re s t i m a t i n g ( 1 + 8 ) can be f o u n di nt h ed i s c u s s i o n of C L , .

" i s p e r h a p st h em o s ti m p o r t a n td e r i v a t i v er e l a t e dt ol o n g i t u d i n a l y and c o n t r o l ,s i n c e it p r i m a r i l ye s t a b l i s h e st h en a t u r a lf r e q u e n c yo f t h e s h o r t p e r i o d mode and i s a m a j o rf a c t o ri nd e t e r m i n i n gt h er e s p o n s eo f t h ea i r f r a m et oe l e v a t o rm o t i o n s and g u s t s .U s u a l l y , a l a r g en e g a t i v ev a l u e o f C% i sd e s i r e d (-0.5 t o -1.0 f o rl i g h ta i r p l a n e s ) ,b u t i f C i st o ol a r g e , t h e r e q u i r e d e l e v a t o r e f f e c t i v e n e s s may become u n r e a s o n a b l y h i g ? .

F i g u r e 8 , takenfromReference ( 1 11, shows t h a t

N = L c o s ( a - i,) + D s i n ( a - i,)

C = D c o s ( a - i,) - L s i n ( a - iw)

where i = i n c i d e n c ea n g l e t = s u b s c r i p t w h i c h r e f e r s t o t h e t a i l w = s u b s c r i p tw h i c hr e f e r st ot h ew i n g -Airplanereferenceline

$& t it

Wind -.

Direc F i g u r e 8 . Forces and moments i n p l a n e o f s y m m e t r y .

N e g l e c t i n g Cct, the d e r i v a t i v e o f t h e p i t c h i n g moment w i t hr e s p e c t t o C L can be w r i t t e n a s C o n t r i b u t i o n o f w i n g C o n t r i b u t i o n C o n t r i b u t i o n o f o f f u s e l a g e h o r i z o n t a l & nacel les t a i I - where C m a a C , = moment c o e f f i c i e n t abou t aerodyanmi c c e n t e r i zonta I t a i I q u a r t e r c h o r d R t = l e n g t h from c.g. t o h o r

rlt = st/sw

Eva I u a t i n g some o f t h e d e r i v a t i v e s shows t h a t , i f a - i, i ss m a l lt h e n s i n ( a - i,) y ( a - i,) and cos(a - i,) 1.0, C D ( ~ - i w l / C L a i s s m a l l compared

w i t h ( a , - iwl, and dCdo/dCL = 0 t h e n

where e = span e f f i c i e n c y f a c t o r o b t a i n e d f r o m C L ~ AR = w i n g a s p e c t r a t i o Usingtheaboveequations, where C L ~ i sp e rr a d i a n .I ft h e c.g. i s ahead o ft h ea e r o d y n a m i cc e n t e r , X , i sp e g a t l v e (Xa = d i s t a n c e f r o m c . g . t o a . c . 1 . I ft h ec . g .p o s i t i o ni su n d e r thewinga.c., Za i s p o s i t i v e ( z a = v e r t i c a ld i s t a n c ef r o mc . g .t oa . c . 1 .

The a n g l e o f a t t a c k o f t h e t a i l i s a f u n c t i o no ft h ew i n ga n g l e of a t t a c k , aownwash, t a i l i n c i d e n c e , and wing incidence. Thus, a t = a w - ~ + i t - i W .

The f i r s t t e r m of a T a y l o re x p a n s i o n of CNt g i v e s Therefore, Theaboveequationcanbeused t o w r i t e (dCm/dCL)tas: Reference ( 1 9 ) g i v e st h ec h a n g ei n downwash w i t h r e s p e c t t o a as where X = t i p c h o r d / r o o t c h o r d ( u s e 0 . 6 7 f o r e l l i p t i c a l w i n g ) AR = w i n g a s p e c t r a t i o

R’ = d i s t a n c e f r o m w i n g q u a r t e r c h o r d t o h o r i z o n t a l

t a i I q u a r t e r c h o r d .

I f t h e h o r i z o n t a l t a i l i s l o c a t e d k 0 . 5 ~ o r more v e r t i c a l l yf r o mt h ec e n t e r l i n e o f t h e wake, a c o n s t a n t o f 18 i n s t e a do f 20 shouldbeused.

Two methods, explainedbelow, may be used t o e s t i m a t e t h e e f f e c t s of t h e f u s e l a g eo n Cb. The l a t t e r method i s c o n s i d e r e dt h e more a c c u r a t eo ft h e two b u tr e q u i r e s more c a l c u l a t i o n s . The f i r s t method r e q u i r e st h ee v a l u a t i o no f t h e e q u a t i o n Nacel les where = e m p i r i c a lf a c t o r shown i nF i g u r e 9 kf w f = maximum w i d t h o f t h e f u s e l a g e o r n a c e l l e E,, = l e n g t ho ff u s e l a g eo rn a c e l l e 3 . 2 2.8 2 A 2 -0 I . . , kf 1-6 . .

1.2 -8 -4 0 20 40 60 Position of wing 114 root chordonbody, F i g u r e 9 . E m p i r i c a l f a c t o r f o r f u s e l a g e o r n a c e l l e c o n t r i b u t i o n s t o C%.

The above formula, taken from TR-711 (Ref. 21), i s a s i m p l e method f o re s t i m a t i n g t h e e f f e c t o f t h e f u s e l a g e o r n a c e l l e s o n CYa.

Thesecondmethod,takenfromPerkinsand Hage ( R e f . 1 1 1 ,c o n s i d e r st h ef a c t t h a t t h e v a r i a t i o n o f t h e f u s e l a g e l o n g i t u d i n a l p i t c h i n g moment w i t ha n g l e of a t t a c k i s g r e a t l y a f f e c t e d by t h e upwash i n f r o n t o f t h e w i n g and t h e downwash b e h i n d t h e w i n g . T h e w i n g ' s i n d u c e d f l o w has a heavy d e s t a b i l i z i n gi n f l u e n c e o nt h ef u s e l a g e o r n a c e l l es e c t i o n s ahead of t h ew i n g and a s t a b i l i z i n g i n f l u e n c e b e h i n dt h ew i n g . Thus, t h el o c a t i o no ft h ew i n gw i t hr e s p e c tt ot h el o n g i t u d i n a l a x i s i s o f c o n s i d e r a b l e i m p o r t a n c e . M u l t h o p p ( R e f . 2 2 ) p r o p o s e s t h e f o l l o w i n g f o r m u l a t o a c c o u n t f o r t h i s phenomenon: where B = a n g l e of l o c a lf l o w( f r e e s t r e a ma n g l e of a t t a c kp l u st h ea n g l eo ft h ei n d u c e df l o w ahead o f or b e h i n dt h ew i n g ) w f = f u s e l a g ew i d t h = l e n g t h of f u s e l a g e T h ee q u a t i o nc a n be i n t e g r a t e dn u r n e r i c a I l y i n t h e manner i l l u s t r a t e d i n F i g u r e 10.

Segments 1-5 da/da from curve a " from Segment 6 dB/da curve b b - 4 I F i g u r e 10. T y p i c a l l a y o u t for computingfuselage moments.

The i n t e g r a t i o n r e q u i r e s t h a t t h e a v e r a g e v a l u e o f w f 2 (dB/da)Axbefound f o r each segment and t h a tt h ei n d i v i d u a l segment p a r t s be summed. The c u r v ef o r dB/da v e r s u sp o s i t i o n s ahead o ft h ew i n gl e a d i n g edge i np e r c e n tw i n gc h o r da r e g i v e ni nc u r v e( a ) of t h ef o l l o w i n gf i g u r e . For t h es e c t i o ni m m e d i a t e l y ahead o ft h ew i n gl e a d i n g edge, dB/da r i s e s so a b r u p t l yt h a ti n t e g r a t e dv a l u e sa r e g i v e n based on t h el e n g t h of t h i s segment a f t of t h ew i n gc h o r d( c u r v eb ) .

F o rt h e segment a f to ft h ew i n g , it i s assumed t h a t dB/da r i s e s l i n e a r l y f r o m z e r o a t t h e r o o t t r a i l i n g edge t o ( 1 - de/da) a tt h eh o r i z o n t a lt a i la . c . I n t h er e g i o nb e t w e e nt h ew i n gl e a d i n g and t r a i l i n g edge, dB/da i sc o n s i d e r e d zero. I t s h o u l d be remembered t h a tu s i n gt h ep r o c e d u r ea b o v eg i v e s cm p e r r a d i an. a.

0 -4 .8 1.2 1.6 2.0

-

X 1 X 1

- and -

C C F i g u r e 1 1 . F u s e l a g e p i t c h i n g moment c o n t r i b u t i o n t o C m , .

I t may be d e s i r a b l e t o i n c l u d e a f a c t o rw h i c ht a k e si n t oa c c o u n tt h e i n t e r f e r e n c e e f f e c t o f t h e f u s e l a g e o r n a c e l l e onthewing Cm. T h i sa d d i t i o n - a l c o n t r i b u t i o n c a n beapproximatedbytheformulabelow and added t o t h e C h o f t h ew i n g .

dCm - C "

( W ~ . ~ . + WMid. - WT.E.)

da ( 2 9 0 ) ( S I where W L . ~ . , W M ~ ~ . , and WT.E. = w i d t h so ft h ef u s e l a g e a t thewingleading edge, mid- chord,and t r a i l i n g edge, r e s p e c t i v e l y .

Thus, t h e t o t a l a i r p l a n e Cm c a n b e w r i t t e n a s t h e sum o f t h e w i n g c o n t r i b u - t i o n , t h e f u s e l a g e c o n t r i b u t i o n , and t h e t a i l c o n t r i b u t i o n : A t y p i c a lv a l u e for C , , i s t h e o n ea s s o c i a t e dw i t ht h e Cessna 182. For t h e c . g . a t 26% o f t h e mean aerodynamicchord Cmcl = -0.885.Moving t h e c.g.

f o r w a r dp r o d u c e sh i g h e rn e g a t i v ev a l u e sf o r Cb; moving it a f t a c h i e v e sl e s s n e g a t i v ev a l u e s .F o rm o s tl i g h ta i r p l a n e s , Cm p r o b a b l yf a l l si nt h er a n g e o f -0.5 t o -1.0.

C

La

The s t a b i l i t yd e r i v a t i v e C i s t h e change i n l i f t c o e f f i c i e n tw i t ht h e r a t e o f change of a n g l e of a t t a c ? . T h i s d e r i v a t i v e a r i s e s from a t y p e o f " p l u n g i n g "m o t i o na l o n gt h ez - a x i s ,d u r i n gw h i c ht h ea n g l e of p i t c h , 8, remains zero. For low speed f l i g h t ,t h ed e r i v a t i v er e s u l t sp r i m a r i l yf r o mt h ea e r o d y - namic t i m el a ge f f e c ta tt h eh o r i z o n t a lt a i l , and i t ss i g ni sp o s i t i v e . For t h ec o n v e n t i o n a ll i g h ta i r p l a n e ,t h eh o r i z o n t a lt a i li s immersed i nt h e down- wash f i e l d o f t h e wing some d i s t a n c eb e h i n dt h ew i n g . When t h ew i n ga n g l eo f a t t a c k i s changed, t h e downwash f i e l d i s a l s o a l t e r e d ; however, it t a k e s a f i n i t e l e n g t h o f t i m e f o r t h e downwash a l t e r a t i o n s t o r e a c h t h e t a i l , r e s u l t i n g i n a t a i l l i f t w h i c hl a g st h em o t i o no ft h ea i r c r a f t . C @ c a na l s oa r i s e f r o m a e r o e l a s t i c e f f e c t s a t h i g h speed, b u t t h e s e a e r o e l a s t i c c o n t r i b u t i o n s a r e n e g l i g i b l e f o r l i g h t a i r c r a f t .

T h i sd e r i v a t i v ei su n i m p o r t a n ti n a dynamic l o n g i t u d i n a ls t a b i l i t y a n a l y s i s f o r l i g h t a i r p l a n e s , s i n c e i t s e f f e c t i s e s s e n t i a l l y t h e same as i f t h ea i r p l a n e ' s mass o r i n e r t i a were changed about the z-axis. Many s t u d i e s e i t h e r n e g l e c t CG by c a l l i n g I t s e f f e c t s s m a l l or f a i l t o mention i t a t a l l because o f t h e n e g l i g i b l e e f f e c t s .

An e m p i r i c a lm e t h o df o rc a l c u l a t i n g C b canbefoundusingthe same a p p r o a c ht h a ti s used f o r C . Cm& i sm e r e l yt h e moment r e s u l t i n g from t h e t i m e l a g of t h e t a i l l i f t ; t 2 u s , C u canbethought o f a s CG d i v i d e d b y t h e non-dimensional moment arm *. Thus, -C% -c C~ c G = R t / c = Rt The n e g a t i v es i g ni s includedbecause a p o s i t i v e p i t c h i n g moment from t h e t a i i ng I r e q u i r e s a n e g a t i v e l i f t (nose-up i s a p o s i t i v ep i t c h i n g moment). Us t h e e q u a t i o n d e r i v e d f o r C ~ , CG can be w r i t t e n as,

C G =

where k t ' = d i s t a n c eb e t w e e nt h ew i n gq u a r t e rc h o r d and t h eh o r i z o n t a lt a i lq u a r t e rc h o r d .

A t y p i c a l r a n g e of v a l u e s o f C f o rl i g h ta i r p l a n e si s 1.5 t o 3.0. For t h e Cessna 182, CG was found t o b e q . 7 4 .

* t h el e n g t h from t h ec . g .t ot h et a i lq u a r t e rc h o r dd i v i d e db yt h e mean aerodynamic wing chord, ( Q / c ) .

C

D a

The s t a b i l i t y d e r i v a t i v e CDG i s t h e change i n d r a g c o e f f i c i e n t w i t h r a t e of change o f a n g l e o f a t t a c k . , (3% a r i s e sf r o mt h ea e r o d y n a m i cl a g e f f e c t and v a r i o u s" d e a d - w e i g h t "a e r o ea s t i ce f f e c t s .F o ra i r p l a n e si nt h e L'ke speed and w e i g h tr a n g eo fl i g h ta i r p l a n e s ,h o w e v e r ,t h ed r a gv a r i a t i o n due t o b o t ht h e s ee f f e c t si sn e g l i g i b l e .C o n s e q u e n t l y , C D ~ i s t a k e n t o be zero.

"

C

ma

The d e r i v a t i v e Cm6 i s t h e change i n p i t c h i n g moment c o e f f i c i e n t w i t h r e s p e c t t o b , t h e t ime r a t e of change of t h e a n g l e o f a t t a c k . I t i s q u i t e i m p o r t a n ti nl o n g i t u d i n a l dynamics, since it i si n v o l v e di nt h e damping o f t h es h o r tp e r i o d mode. A n e g a t i v ev a l u e of Cm& i n c r e a s e ss h o r tp e r i o d damping; t h u s ,h i g hn e g a t i v ev a l u e sa r ed e s i r a b l e .T h i sd e r i v a t i v ei sa c t u a ! l y caused by a l a g e f f e c t o f t h e downwash a t t h e h o r i z o n t a l t a i l o f t h e a i r c r a f t .

War Report W R L-430 (Ref. 23) i sc o n s i d e r e dt og i v et h eb e s td e r i v a t i o n o f an e m p i r i c a lf o r m u l af o r C a based on t h e l a g o f t h e downwash between t h e t r a i l i n g edge o f t h ew i n ga n dt h eh o r i z o n t a lt a i l . The downwash a t t h e t a i l a t t h e t i m e t dependson t h e a n g l e o f a t t a c k o f t h e w i n g a t t h e t i m e a; t " U where

R i = l e n g t h from t h eq u a r t e rc h o r d of t h ew i n g

c t ot h eq u a r t e rc h o r d of t h eh o r i z o n t a lt a i l

-

" = l a g t i m e o f t h e downwash

U The a n g l e of a t t a c k of t h e " a i I, at, canbe w r i t t e n a s For a g i v e n maneuver, t h e a n g l e o f a t t a c k c a n be w r i t t e n a s a f u n c t i o n o f t i m e , a = f ( t ) . Thus, expanding t h e downwash i n a T a y l o rs e r i e s and i g n o r i n g second o r d e rt e r m s ,t h e downwash canbe w r i t t e n a s a f u n c t i o n o f (t - A t ) a s

E = - " f ( t - A t )

aa where

A t = -

U As a simp1 i f y i n gt e c h n i q u e , f ( t - A t ) can be expanded i n a T a y l o rs e r i e sa s ( A t 1 f A t f ' ( t ) + - f " ( t ) - - (At)3 f"'(t) + ...

2 6 Thus, A q u a n t i t y s i s now i n t r o d u c e d t o non-dimensionalize 6 , where Therefore, can be expressed as * 2U da 2U da a = " = - c ds c 2u d(- t) C and can be expressed as

The downwash can now be w r i t t e n i n t e r m s o f ,s a s

da + ( 2 g)'

E = Ea {a - 2 - d2a - ... 3

2u c 2u d(- t) d(- C C The e q u a t i o n f o r C Y + can now be w r i t t e n ,w i t ht h ea b o v ee q u a t i o ns u b s t i t u t e d f o r t h e downwash, as 2% da 2 q d 2 a -

a t - a, - i, + i t - { a - -- + ( - 1 z- ... 1

C c ds S i n c e a and a , r e f e r t o t h e same a n g l e o f a t t a c k , 2a-i 2 ( T I da a t = a ( l - + it - iw + -- - 9 - ...

c ds Ea 2 S i n c e Cm- r e s u l t s from t h e l a g o f t h e downwash a t t h e t a i l , o n l y t h e t a i l c o n t r i buy ion t o t h e p i t c h i ng moment must be considered when Cm& i s d e r i v e d .

The p a r t o f t h e p i t c h i n g moment c o e f f i c i e n t c o n t r i b u t e d by t h e t a i l canbe w r i t t e n a s where fit = d i s t a n c e f r o m c . g . t o & c h o r d o f t a i I S i n c e Cm i s a f u n c t i o n of t a i la n g l eo fa t t a c k ,t h ee x p r e s s i o nd e r i v e da b o v e

f o r % can be s u b s t i t u t e d i n t o t h e Cm f o r m u l a t o y i e l d

2% 2 fit st da "

Cm = - {a(l - E a ) + Ea - - - ( T - ) d 2 a + ...}

c tis Ea 2 z

cbt O t c sw

The . p a r t i a l d e r i v a t i v e of Cm w i t h r e s p e c t t o da/dscan now beexpressedas S t 2 a i - ( 7 1 rlt SW or I t s h o u l db en o t e dt h a tt h ea b o v ef o r m u l aa g r e e sw i t ht h eo n ed e v e l o p e di n P e r k i n s and Hage (Ref. 111, e x c e p t f o r t h e manner i nw h i c h & i s non- d i m e n s i o n a l i z e d . I t s h o u l da l s o be p o i n t e do u tt h a t C m i can be found from t h e a b o v ee q u a t i o nb ym e r e l yd i f f e r e n t i a t i n gw i t hr e s p e c tt od 2 a / d s 2 .I n g e n e r a l ,h o w e v e r ,a n g u l a ra c c e l e r a t i o n sa r ec o n s i d e r e ds m a l l when l i n e a r i z e d e q u a t i o n s of m o t i o na r eu s e d and t h e r ea r eo n l ys m a l ld i s t u r b a n c e sf r o m e q u i l i b r i u m . Thus, f o r I i g h ta i r p l a n e s , CmE would n o t be o f g r e a ti m p o r t a n c e .

A t y p i c a lr a n g eo f Cm& f o r a l i g h ta i r p l a n e i s -3.0 t o -7.0. F o rt h e Cessna 182, Cm& was c a l c u l a t e da t -5.24.

The s t a b i l i t y d e r i v a t i v e CL,q r e p r e s e n t st h ec h a n g ei na i r p l a n e lift w i t h v a r y i n g p i t c h i n g v e l o c i t y w h i l e The a n g l e of a t t a c k of t h e a i r p l a n e a s a whole r e m a i n sc o n s t a n t .C o n t r i b u t i o n sa r e made b yb o t ht h ew i n g and h o r i z o n t a l t a i l , b u t t h e t a i l Is by f a r t h e more important. The general concensus i s t h a t CL p l a y so n l y a m i n o rp a r ti ne s t i m a t i n gt h el o n g i t u d i n a lr e s p o n s e o f t h e a i r c r a f ? .

Volume V o f Aerodynamic Theory by Durand (Ref. 24) e x p l a i n st h ep h y s i c a l phenomena a s s o c i a t e dw i t h C L ~ . F i g u r e 1'2 shows t h a t an a i r p l a n e f l y i n g w i t h v e l o c i t y U i n a c i r c u l a r f l i g h t p a t h o f r a d i u s R and c e n t e r 0 hasanangular v e l o c i t y , de, such t h a t U = R ( d e / d t la n d a i s c o n s t a n t .

F i g u r e 12. P i t c h i n ga tc o n s t a n t a.

F o r s m a l l p e r t u r b a t i o n s , ( d O / d t ) = q (Appendix A ) . I ft h ea i r p l a n ec . g .i s t r a v e l i n g w i t h v e l o c i t y u w h i l e t h e a i r p l a n e i s r o t a t i n g w i t h a n g u l a r v e l o c i t y q, t h e d i r e c t i o n o f m o t i o n o f any p o i n t o n t h e t a i l , d i s t a n c e Rt b e h i n dt h e c.g., makes a na n g l et a n - l ( q R t / U )w i t ht h ed i r e c t i o no fm o t i o no ft h ec . g .

Provided qkt i sn o tt o ol a r g e compared t o u , t h ee f f e c t i v ei n c i d e n c e of t h e t a i I i si n c r e a s e d byapproximately qRJu r a d i a n s .

The w i n gc o n t r i b u t i o nt o CL can be e x p l a i n e di n much t h e same way. The change i n a n g l e of a t t a c k o f t h e w i n g (measured a t t h e a e r o d y n a m i cc e n t e r )i s xc.g. - Xa.c.

ACX = - q ( I

U where x c a g . = d i s t a n c e t o t h e c.g.

x a a c . = d i s t a n c e t o t h e a.c.

T h i se x p r e s s i o ni n d i c a t e st h a tt h e r e i s a r e d u c t i o ni n l i f t i f t h ec . g . is behind the a.c. and an increase i f t h ec . g .i si nf r o n to ft h e a.c. For a c . g .v e r yn e a rt h e a.c., t h ew i n gc o n t r i b u t i o ni sn e g l i g i b l e ; however, t h e w i n gc o n t r i b u t i o ni n c r e a s e sa st h ed i s t a n c eb e t w e e nt h ec . g . and a.c.

increases. I t i sf e l tt h a t ,f o rl i g h ta i r p l a n e s ,t h ef u s e l a g ec o n t r i b u t i o n t o C L ~ i s s m a l l e r t h a n t h a t o f t h e wing;thus, it i sn e g l e c t e dh e r e .

I . , -.. .. .

A t h e o r e t i c a l d e r i v a t i o n o f CLq canbeobtainedfrom W R L-430 (Ref. 231 b ym o d i f y i n gt h e Cmq d e r i v a t i o n t o be d i s G u s s e dl a t e r . A t t h eh o r i z o n t a l t a i l ,t h ea n g l e o f a t t a c ki si n c r e a s e d R t e / U b yp i t c h i n g . The t o t a l l i f t c o e f f i c i e n ti s ,t h e r e f o r e ,i n c r e a s e db yt h e amount where = t a i l I i f t c u r v es l o p e The f o l l o w i n ge x p r e s s i o ni s used as a n o n - d i m e n s i o n a l i z i n gt e c h n i q u e .

Thus, D i f f e r e n t i a t i n g w i t h r e s p e c t t o (ci)/2U) o r The w i be o b t a i n e d s i m i l a r l y by s u b ‘ s t i t u t i ng t h e d i stance ng ccm t r i b u t i o n can f r o mt h ec . g .t ot h ew i n gq u a r t e rc h o r d * f o r t h ed i s t a n c e R t . Thus, where x’ = distancefromc.g. t o wingquarterchord ( p o s i t i v e f o r c . g . ahead o fq u a r t e rc h o r d , n e g a t i v ef o r c.g.behindquarterchord) CLC, = wing l i f t c u r v es l o p e * A t t h i sp o i n t it is assumed t h a tt h ea . c .o ft h ew i n gi sv e r yn e a rt h e w i n gq u a r t e rc h o r d ;t h e r e f o r e ,t h eq u a r t e rc h o r di s usedasthereference.

The airplane C becomes Lq 8CL - - -

cLq a(c9-j

2u For the Cessna 182 aircraft investigated later in the paper, the c.g. is located very near the quarter chord a n d the wing contribution i s therefore neglected. The value of aircraft C L was calculated to be 3.9.

The s t a b i l i t y d e r i v a t i v e CD i s t h e change i nd r a g of t h ea i r p l a n ew i t h v a r y i n g p i t c h i n g v e l o c i t y w h i l e ?he a n g l e o f a t t a c k o f t h e a i r p l a n e a s a whole r e m a i n sc o n s t a n t .T h i sd e r i v a t i v eh a sc o n t r i b u t i o n s from b o t ht h ew i n ga n d t h ef u s e l a g eb u tb o t hc o n t r i b u t i o n sa r ev e r ys m a l l .I na l lo ft h el i t e r a t u r e f o r s u b s o n i cf l i g h t , CD i si g n o r e d because it i sr e a l l yu n i m p o r t a n ti n a n a l y z i n gf l i g h t dynami2s and very small i nm a g n i t u d e . Qq f o r t h e Cessna 182 i s t a k e n t o be 0.

The change i n p i t c h i n g moment c o e f f i c i e n t due t o a change i n p i t c h i n g v e l o c i t y c o n s t i t u t e s t h e s t a b i l i t y d e r i v a t i v e Cm . If a na i r p l a n eh a s a p o s i - t i v e p i t c h r a t e w i t h a c o n s t a n t a n g l e of a t t a c k ? f l y i n g a curved f I-,igh't p a t h ) , t h e a n g l eo fa t t a c ka tt h et a i li si n c r e a s e d ,t h e r e b ya d d i n g more p o s i t i v e l i f t t o t h e t a i l and c r e a t i n g a moment t o oppose t h ep i t c h i n gm o t i o n .F o rt h i s reason, t h e d e r i v a t i v e i s sometimes r e f e r r e d t o as t h e " p i t c h damping" d e r i v a - t i v e and i su s u a l l yn e g a t i v e . The w i n gc o n t r i b u t i o nt o Cm e i t h e r opposes o ri n c r e a s e st h ep i t c h i n gm o t i o nd e p e n d i n go nt h ec . g .l o 2 a t i o n( s e ed i s c u s s i o n o f C L ~ ) ; however, t h i s i s r e l a t i v e l y i n s i g n i f i c a n t compared t o t h e t a i l c o n t r i b u t i o n . The f u s e l a g ec o n t r i b u t i o ni sa l w a y sn e g l e c t e df o rl i g h ta i r - planes.

T h i sp a r t i c u l a rd e r i v a t i v ei sv e r yi m p o r t a n ti nl o n g i t u d i n a ld y n a m i c s because it p l a y s a m a j o r r o l e i n t h e damping o f t h e s h o r t p e r i o d mode and a m i n o rr o l ei np h u g o i d damping. High negative values of Cm (-10.0 t o -15.0) u s u a l l yi n s u r e good s h o r tp e r i o d damping f o r l i g h t a i r p l a n e s .

' Cmq can be considered a sum o f moments due t o t h e component p a r t s o f C L ~ . Consequently, fit ac,

- a c m -

" - -" I a(%,

a ( : )

t a i I t a i I (The negativesignappearsbecauseCmqtail i s alwaysnegative,while C L 9 t a i I i sa l w a y sp o s i t i v e ) and 2% l x c l acL - " " 'mqw i ng

a a(%)

L U wing wing where

1x.l = d i s t a n c e from c.g . t o w i n gq u a r t e r

chord (a I ways pos i t i v e ) .

The s i g no f Cmqwing i so p p o s i t et h a to f CLqwing; t h u s , / x c / i s used i n s t e a d o f X * . When t h e a.c. i nf r o n to ft h e c.g. C L ~ ~ ~ ~ ~ i sn e g a t i v eb u t Cmqwing i s p o s i t i v e . The t o t a l Cmq i s 2x * S t - - " "r) C 2 c"q where x c = d i s t a n c ef r o mc . g .t ow i n gq u a r t e rc h o r d ( p o s i t i v ef o rc . g . ahead o fq u a r t e rc h o r d , n e g a t i v ef o r c.g.behindquarterchord.

/ x c / = m a g n i t u d eo f x* The v a l u eo f Cm c a l c u l a t e df o rt h e Cessna 182 i s -12.43.

The change i n l i f t c o e f f i c i e n t due t o e l e v a t o r d e f l e c t i o n i s t h e s t a b i l i t y d e r i v a t i v e C L ~ ~ . S i n c e downward d e f l e c t i o n of t h ee l e v a t o ri sd e f i n e da s p o s i t i v e , p r o d u c l n g a p o s i t i v e l i f t , C L ~ i sn o r m a l l yp o s i t i v ei ns i g n . Many e r r o n e o u s l y c o n s i d e r t h i s d e r i v a t i v e t o Ee t h e same a s t h e c h a n g e i n e q u i l i b r i u m l i f t c o e f f i c i e n t w i t h r e s p e c t t o e l e v a t o r d e f l e c t i o n , i n w h i c h t h e e l e v a t o r d e f l e c t i o nc a u s e s a change i n t h e a n g l e of a t t a c k ,r e s u l t i n gi nw i n g and t a i l l i f t c h a n g e s .F o rt h ed e r i v a t i v e C L ~ ~ , t h ee l e v a t o ra n g l ei st h eo n l yq u a n t i t y whichcanchange;thus,theangle o f a t t a c k a s we I I as a I I o t h e r a n g l e s m u s t r e m a i nt h e same. CL6E d o e sn o ta p p e a ri nt h ec h a r a c t e r i s t i ce q u a t i o n of t h e a i r c r a f t , b u t it does appear i nt h en u m e r a t o r o f t h et r a n s f e rf u n c t i o n s ; it t h e r e f o r e a f f e c t s t h e g a i n o f a p a r t i c u l a r t r a n s f e r f u n c t i o n .

For a c o n v e n t i o n a la i r c r a f tw i t ht h eh o r i z o n t a lt a i l mounted an a p p r e c i a b l e d i s t a n c e a f t o f t h e c e n t e r of g r a v i t y , CL6E i ss m a l l ,a p p r o x i m a t e l y 0.1 t o 0.2 p e rr a d i a n .F o rl i g h ta i r p l a n e sw i t hr e l a t i v e l yl a r g et a i l s , C L ~ ~ may takeonvaluesbetween 0.3 and 0.5 p e rr a d i a n .

NACA TR-791 (.Ref. 2 5 ) m e n t i o n st h a t C L ~ i s usua I l y obtainedfromwind t u n n e ld a t a ;h o w e v e r ,a ne m p i r i c a lr e l a t i o nf r o mR e f e r e n c e ( 1 1 ) can be used t o approximate C L ~ ~ : dCLtdat

CLQ = - -

dat d6E where ( d C L t ) / ( d a t l = l i f t c u r v es l o p e o f t h e t a i l and dat/d6E i sp l o t - t e d a s a f u n c t i o no fe l e v a t o ra r e ad i v i d e d by t a i l area ( F i g u r e 131, where S E / S ~ = 1.0 f o r a l I-movable t a i I .

.8 .6 .4 .2 0 .l .2 .3 .4 .S .6 .7 ‘E”t F i g u r e 1 3 . E l e v a t o r e f f e c t i v e n e s s .

The d e r i v a t i v e c a n b ee s t i m a t e dm o r ee x a c t l yb yu s i n gt h et w o - d i m e n s i o n a l d a t aa v a i l a b l ei nR e f e r e n c e( 1 9 ) a n dm e r e l yc o r r e c t i n gf o ra s p e c tr a t i o . The d a t ap r e s e n t e da r ef o rt h e 0009 a i r f o i l and a r eu s e f u l ,t h e r e f o r e ,f o rm o s t l i g h ta i r p l a n e s( F i g u r e s 14 and 1 5 ) . Dommash (Ref. 19) m e n t i o n st h a tt h ed a t a g i v e n f o r t h e NACA 0009 a i r f o i l w i t h a p l a i n f l a p i l l u s t r a t e t h e p r i n c i p l e s i n v o l v e d and a r en o ti n t e n d e da se x a c te n g i n e e r i n gd e s i g nd a t a ; however, t h e d a t as h o u l dg i v er e s u l t sw e l lw i t h i nt h es a t i s f a c t o r yr e q u i r e m e n t s .

L V Y -Test data V 0) Extrapolated r r 0 .02 e+ m 0 -2 .4 .6 .8 1.0 Flap-airfoil chord ratio,cf/c F i g u r e 1 4 . F l a pe f f e c t i v e n e s sp a r a m e t e rv e r s u sf l a p - a i r f o i l c h o r d r a t i o f o r NACA 0009 a i r f o i l w i t h p l a i n , u n s e a l e df l a D and O..OO5c gap.

0 - 1.0 a

-

.w

'0 -8

.-

V

.-

r r v -6 c1

z

- -4

c

.-

M .2 c a a 0 5 10 1s 20 25 30 Flapdeflection c f , deg F i g u r e 15 Change i n l i f t c o e f f i c i e n t ( w i t h c o n s t a n t a n g l e o f a t t a c k ) v e r s u s f l a p d e f l e c t i o n f o rs e v e r a lv a l u e s o f f l a p - a i r f o i lc h o r d r a t i o s . NACA 0009 a i r f o i lw i t hp l a i n , u n s e a l e df l a p and 0 . 0 0 5 ~ gap.

The two-dimensional values of C L ~ found above can be changed t o CQ c o r r e c t e d f o r a s p e c tr a t i ou s i n gt h ec o r r e c F i o np r o c e d u r eg i v e ni nR e f e r e n c e ( f 0 ) f o r flappedwings: s e c t i o nl i f tc u r v ei n c r e m e n t due t o f l a p ( e l e v a t o r ) d e f l e c t i o n( t h i sd e r i v a t i v es h o u l dn o t beconfused w i t h t h e r o l l i n g moment c o e f f i c i e n t a p p e a r i n g i n t h e I a t e r a I dynamics 1.

l i f t c u r v es l o p e of t a i l w i t h o u t f l a p d e f l e c t e d (3-D 1 s e c t i o n l i f t c u r v e s l o p e o f b a s i c a i r f o i l r a t i o o f 3-D f l a p e f f e c t i v e p a r a m e t e r t o t h e 2-D f l a pe f f e c t i v e n e s sp a r a m e t e ro b t a i n e df r o mt h e f i g u r eb e l o wa s a f u n c t i o no fw i n g( t a i l )a s p e c t

r a t i o and t h e t h e o r e t i c a I va I ue o f (a6 )c2. The

t h e o r e t i c a lv a l u ei sa l s og i v e na s a f u n c t i o n of f l a p c h o r d t o a i r f o i l c h o r d .

f l a p - s p a nf a c t o rw h i c h i s = 1.0 f o r e l e v a t o r h o r i z o n t a lt a i ls u r f a c ef o rl i g h ta i r p l a n e s .

For most l i g h ta i r p l a n e s , ( a & ) ~ /(a&) N 1.0 t o 1 . 1 ; C L ~ ~ and C k t can be e s t i m a t e du s i n gt h ep r o c e d u r e s knumer%'ed f o r C k , and Cg6 canbeestimated u s i n g t h eg r a p h s f o r t h e 0009 a i r f o i I. The f o r m u l a f o r CLgE based on wing areacanthenbe wri "ten as o r ,s i n c e (a&) /(cl&)cR i s a p p r o x i m a t e l y 1.05 for most l i g h ta i r c r a f t , CL The v a l u e of c a l c u l a t e d for t h e Cessna 182 is 0.427.

"""__"..__._~."."""..._.I .._.".."_... * " " - . . . " " " - ---..... .... --..-.-------.

The change i n d r a g c o e f f i c i e n t due t o a change i n e l e v a t o r a n g l e i s t h e c o n t r o ls u r f a c es t a b i l i t yd e r i v a t i v e C D ~ . F o ra ne l e v a t o r o f reasonable s i z e , t h e t o t a l a i r p l a n e d r a g does n o t c F i ange a p p r e c i a b l y w i t h e l e v a t o r d e f l e c t i o n . F o r t h i s r e a s o n , C D ~ ~ i s o f t e n n e g l e c t e d . The c h a r a c t e r i s t i c e q u a t i o n o f t h e a i r c r a f t i s n o t a f u n c t i o n o f CD~,-; t h u s , a f f e c t s o n l y t h e g a i n o f t h e p a r t i c u l a r t r a n s f e r suchas u / ~ E .

Probably +he b e s t method o f e s t i m a t i n g C D ~ , i s t o p e r f o r m w i n d t u n n e l t e s t s on a p a r t i c u l a r a i r c r a f t model; however, i f w i n dt u n n e lt e s t i n gi sn o t f e a s i b l e , CD6 can be a p p r o x i m a t e db yu s i n gd a t af r o mw i n dt u n n e lt e s t sw h i c h have a I ready Eeen performed. NACA TR-688 (Ref. 26) d i s c u s s e dt h ea e r o d y n a m i c c h a r a c t e r i s t i c s of s e v e r a lh o r i z o n t a lt a i l s . Using t h e figures beluw and t h e appropriate graphs, can be estimated. Five t a i l shapes from NACA TR-688 were chosen f o ri n c l u s i o ni nt h i sa n a l y s i s . To estimate CD”, the t a i l s u r f a c ew h i c h is m o s tl i k et h eo n ei nq u e s t i o n s h o u l d 6.e use . Th,e numerical v a l u e of CD6 c a nb et a k e nf r o mp l o t so f CD versus CY f o r d i f f e r e n t e l e v a t o r d e f l e c t i o n s F o r e a c h t a i I surface;however,the va I ues o f CQ must be m u l f i p l i e d by S t / S so t h a t w i l l be based on wing area.

I ( 5 ) F i g u r e 17. F i v et a i ls u r f a c e ss e l e c t e df r o m NACA TR-688 (Ref. 2 6 ) .

T a i I Span S t Se Ct T e s t u T e s t

S u r f a c e AR ( i n . ) ( s q . ( s q . ( i n . ) ( f p s ) RN i n . ) i n . 1 1 3.4 155 701 5 2450 45.25 88.0 1 ,960,000 2 3.1 23.6 181 68 7.68 110.0 448,000

""""-

3 4.3 39.4 361 81 9.15 -----

" " " " _

4.3 39.4 361 117 9.15 -----

""""-

5 4.3 39.3 356 1 62 9.06 -----

I I T a b l e 8. D i m e n s i o n a lc h a r a c t e r i s t i c sf o rh o r i z o n t a lt a i l s .

.5 2 A8 A4 .40 3 6 . 3 2 4- c b

0 " .24

M

e .20

Ip .16 .12 .O 8 .04 -4 0 4 8 12 16 20 24 -4 0 4 8 12 16 20 24 Angle of attack (deg) Angle of attack (deg) F i g u r e 19. Drag c o e f f i c i e n t a g a i n s t a n g l e o f F i g u r e 18. Drag c o e f f i c i e n t a t t a c k a t v a r i o u s a g a i n s t a n g l e of e l e v a t o r d e f l e c - a t t a c k a t vari,ous t i o n s f o r t a i t e l e v a t o r d e f l e c - s u r f a c e 2.

t i o n s f o r t a i I 69 s u r f a c e 1.

I

20 - 4 0 4 8 12 16 20 - 4 0 4 8 12 16 Angle of attack (deg) Angle of attack (deg) F i g u r e 20. Drag c o e f f i c i e n t F i g u r e 21. Drag c o e f f i c i e n t a g a i n s t a n g l e of aga i n s t a n g l e o f a t t a c k a t v a r i o u s a t t a c k a t v a r i o u s e l e v a t o r d e f l e c - e l e v a t o r d e f l e c - t i o n s f o r t a i I t i o n s for t a i I s u r f a c e 3.

s u r f a c e 4.

- 4 0 4 8 12 16 20 24 Angle of a t t a c k (deg) F i g u r e 22. Drag c o e f f i c i e n t a g a i n s t a n g l e of a t t a c k a t v a r i o u s e l e v a t o r d e f l e c - t i o n s f o r t a i l s u r f a c e 5.

7a The s t a b i l i t y d e r i v a t i v e Cm6E i s t h e change i n p i t c h i n g moment c o e f f i c i e n t w i t h changes i n e l e v a t o r d e f l e c t i o n , u s u a l l y r e f e r r e d t o a s" e l e v a t o rp o w e r " , o r " e l e v a t o re f f e c t i v e n e s s . " I f Cm6E and t h e maximum d e f l e c t i o n o f t h e e l e v a t o r a r e known, t h e maximum r o t a t i o n moment w h i c h t h e t a i l c a n e x e r t c a n b e e s t i - mated. I t must be remembered t h a t Cm6 i se v a l u a t e dw i t h o u ta l l o w i n gt h e a i r p l a n e t o r o t a t e or a n y o t h e r parame F e r s t o change; t h u s , Cm6E i s r e a l l y t h e moment produced by C L ~ ~ . \ The p r i m a r y f u n c t i o n of t h e e l e v a t o r i s t o c o n t r o l t h e a n g l e o f a t t a c k o f t h ea i r p l a n ei ne q u i l i b r i u mf l i g h t o r i n maneuvering f l i g h t . Depending o nt h e maximum a l l o w a b l ef o r w a r dc e n t e r of g r a v i t y t r a v e l , t h e h o r i z o n t a l t a i l i s designed t o g i v e enough t a i l power i n a l l f l i g h t c o n d i t i o n s t o c o n t r o l t h e a i r - c r a f t . I f t h e needed t a i l o r e l e v a t o rs i z ei se n t i r e l yu n r e a s o n a b l e ,t h ed e s i r e d c.g. t r a v e l may have t o be l i m i t e d .F o ra l lp r a c t i c a lp u r p o s e st h e maximum p r a c t i c a l Cm6E d e t e r m i n e st h e maximum f o r w a r dc e n t e r of g r a v i t y t r a v e l .

S i n c e a p o s i t i v e e l e v a t o r d e f l e c t i o n i s d e f i n e d a s down, a p o s i t i v e e l e v a t o r d e f l e c t i o n g i v e s a n e g a t i v e p i t c h i n g moment c o n t r i b u t i o n , m a k i n g t h e s i g n o f C ! s E n e g a t i v e . A d e s i r a b l ev a l u e of Cm6E c a n n o t be s t a t e di ng e n e r a lf o ra l l a t r c r a f t because each case must be a n a l y z e ds e p a r a t e l y .F o rm o s tl i g h ta i r - c r a f t , however, should have a value between-0.75 and-2.0.

The numericalvalue of CmBE can be o b t a i n e d by m u l t i p l y i n g C L ~ ~ b yt h e d i s t a n c ef r o mt h ec . g . t o t h et a i lq u a r t e rc h o r dd i v i d e db yt h ew i n g mean aerodynamic chord : The negativesignappearsbecause C L ~ i s p o s i t i v e and Cm6E must be n e g a t i v e , as discussed above. For the Cessna 162, Cm6E was c a l c u l a t e dt o be-1.26. I n t h ee x p e r i m e n t a lc a s eu s e df o rt h i ss t u d y ,t h e r ei s a c o n t r i b u t i o n t o Cm6E due t o t h e moments a b o u tt h ea . c .o ft h eh o r i z o n t a lt a i l ; however, t h i s c o n t r i b u t i o ni s so s m a l l i t i su s u a l l yn e g l e c t e d .

The s t a b i l i t y d e r i v a t i v e C B i s t h e change i n s i d e f o r c e causedby a v a r i a t i o ni ns i d e s 1i pa n g l e .d e nt h ea i r f r a m eh a s a p o s i t i v es i d e s l i p , f3, t h e r e l a t i v e w i n d s t r i k e s t h e w i n g ,f u s e l a g e ,a n dv e r t i c a lt a i lo b l i q u e l y from t h e r i g h t , r e s u l t i n g i n a n e g a t i v es i d ef o r c e . The m a j o rc o n t r i b u t i o n t o C comes f r o mt h ev e r t i c a lt a i l ,w i t h a s m a l l e rc o n t r i b u t i o nf r o mt h e fuseyage and a n e a r l yn e g l i g i b l ec o n t r i b u t i o nf r o mt h ew i n g .T h i sd e r i v a t i v e c o n t r i b u t e s t o t h e damping o f t h eD u t c hR o l l mode; t h u s ,l a r g en e g a t i v e v a l u e s of CY@ m i g h t seem d e s i r a b l e .L a r g en e g a t i v ev a l u e so f CYB, however, may c r e a t e a l a r g et i m el a gi nt h ea i r p l a n e ' sr e s p o n s e and cause it t o r e a c t s l u g g i s h l y to t h e p i l o t ' s commands.

I ne s t i m a t i n gv a l u e sf o r CyB, f o r c e - t e s td a t a f o r t h ed e s i g ni nq u e s t i o n should be used, i f p o s s i b l e . A c c o r d i n g t o TR-1098 (Ref. 271, i n t e r f e r e n c e e f f e c t s a r e so l a r g e t h a t a g e n e r a l i z e df o r m u l aw o u l dn o tb ec o m p l e t e l y s a t i s f a c t o r y .I n s t e a d , a method o fc o r r e c t i n gd a t a of a s i m i l a rd e s i g n f o r use i na n a l y z i n gt h ed e s i g ni nq u e s t i o ni s recommended. A more r e c e n t p u b l i c a t i o n , Datcom (Ref. 101, p r o b a b l yg i v e st h em o s ta c c u r a t e method, s i n c ei n t e r f e r e n c ee f f e c t s b a s e do ne x p e r i m e n t a lr e s u l t sa r ei n c l u d e d .

The w i n g c o n t r i b u t i o n t o CyB i s sma I I , o n t h e o r d e r o f a2 ( a n g l e o f a t t a c ki nr a d i a n s ) , so i t sa c c u r a t ee s t i m a t i o n i s n o t v i t a l t o t h e t o t a l Cyb.

For swept wings, TN-1581 (Ref. 2 8 ) g i v e st h ef o l l o w i n gf o r m u l af o r CYB o f t h ew i n g : - 6 t a n A s i n A

. ( p e rr a d i a n )

(Cyg)wing - cL2 T A R ( A R + 4 cos A )

For z e r o sweep ( A = 0'1, (C ) w i n g i s e q u a l t o z e r o . T R - l 0 9 8 ( R e f . 2 7 ) s t a t e s t h a t t h e aboveformula i s n 8 s a t l s f a c t o r y i n p r a c t i c e and t h a t no c o r r e c t i o n of (Cyglwing i s n e c e s s a r yf o rs i m i l a rd e s i g n ss i n c et h i sc o n t r i b u t i o n i s so s m a l l . The method i n Datcom (Ref. 10) g i v e st h ee f f e c t of w i n g d i h e d r a l o n a s f o l l o w s : (CygIwing = -.0001 I I ' l per degree, where r i s i n degrees.

The f u s e l a g e c o n t r i b u t i o n i s g r e a t e r t h a n t h a t o f t h e w i n g and may be estimated by a method adapted from Datcom (Ref. 10). The b a s i cr e l a t i o ni s where Body ReferenceArea = ( f u s e l a g e ~ o l u m e ) ~ ' ~ , K i = w i n g - f u s e l a g ei n t e r f e r e n c ef a c t o r o b t a i n e df r o mt h eg r a p hb e l o w , zw = d i s t a n c e f r o m body c e n t e r l i n e t o q u a r t e r - c h o r d p o i n t o f e x p o s e d w i n gr o o tc h o r d( p o s i t i v e f o r t h e q u a r t e r - c h o r d p o i n t b e I ow t h e body c e n t e r l i n e ) , d = maximum body h e i g h t a t wing-body i n t e r s e c t i o n .

0 -1.0 F i g u r e 23. Values f o r w i n g - f u s e l a g e i n t e r f e r e n c e f a c t o r .

TR-540 (Ref. 9 ) g i v e sv a l u e s of ( C b I f u sa s .0525 p e rr a d i a n f o r round fuse- lages and .1243 p e rr a d i a nf o rr e c t a n g u l a rf u s e l a g e s .O b s e r v a t i o n o f t h e s e 7 3 d a t ai n d i c a t e st h a t , f o r m o s tc o n v e n t i o n a ll i g h ta i r c r a f t ,t h ev a l u e o f ( C y B ) f u s i s s m a l l .

The v e r t i c a l t a i l i s t h e m o s ti m p o r t a n tc o n t r i b u t o r t o CyB, and ( C y g l t a i l i s used i n t h e c a l c u l a t i o n of t a i l e f f e c t s on many o f t h e o t h e r l a t e r a l s t a b i l - i t y d e r i v a t i v e s . TR-1098 (Ref. 27) g i v e st h ef o l l o w i n gf o r m u l a f o r a d a p t i n g i l a r model t o t h ed e s i g nu n d e rc o n s i d e r a t i o n : i qn" r e f e r s t o t h e new C o n f i g u r a t i o n , and t h es u b s c r i p t" d a t a " i- r e f e r st ot h ec o n f i g u r a t i o nf o rw h i c h ( C y g ) t a i ; i sa l r e a d y known. Datcom (Ref.10) presentsprobablythemostcomprehensivemethod o f o b t a i n i n g ( C y B ) t a i I y e t d e v e l o p e d . I n t h i s m e t h o d , t h e f o l l o w i n g f o r m u l a i s shown: The v a l u e o f ( C L ~ ) ~ m u s tb ed e t e r m i n e d ,u s i n gt h ee f f e c t i v ea s p e c tr a t i o o f t h e v e r t i c a l t a i l , t o c o n v e r t t h e t w o - d i m e n s i o n a l l i f t - c u r v e s l o p e t o t h e three-dimensional value. The i m p o r t a n c eo ft h i sa p p r o a c hi ss t r e s s e di n W R L-487 (Ref. 291, i n w h i c hd a t ao ff r e e - f l i g h tt e s t s show t h a t , f o r v e r t i c a l t a i l s o f t h e same a r e a ,w i t ht h ea s p e c t r a t i o increased from 1.0 t o 2.28, e f f e c t i v e n e s si n c r e a s e s 67%. The v a l u eo f k, an e m p i r i c a lf a c t o r , may be o b t a i n e df r o mt h ef o l l o w i n gg r a p h as a f u n c t i o n o f t h e r a t i o o f v e r t i c a l t a i l span t o f u s e l a g ed i a m e t e ri nt h et a i lr e g i o n ,( b v / Z r l ) .

K 0 1 2 3 4 5 6 F i g u r e 2'4. Values f o r k as a f u n c t i o n o f t h e r a t i o o f v e r t i c a l t a i l span t of u s e l a g ed i a m e t e ri nt h et a i lr e g i o n .

The v a l u e of t h ec o m b i n a t i o ns i d e w a s ha n dd y n a m i cp r e s s u r er a t i op a r a m e t e r for z e r o sweep is t h ee m p i r i c a l l y - d e r i v e de x p r e s s i o n , SV ZW

qv - .724 + 1.53 + .4 a - + .009 ( A R ) , ( 1 +"I"

9 W where z w = d i s t a n c e ,p a r a l l e l t o z-axis, from wing r o o t q u a r t e r - c h o r d p o i n t t o f u s e l a g e c e n t e r l i n e , d = maximum d i a m e t e r o f fuselage.

A comDarison of c a l c u l a t e d v a l u e s o f t h i s p a r a m e t e r ( u s i n g t h e aboveformula) w i t h t e s t e d va I u e s i n d i c a t e s t h a tt h ea v e r a g ee r r o ri sl e s st h a nf i v ep e r c e n t .

P u t t i n g t o g e t h e r e a c h o f t h e components, t h ef o l l o w i n gf o r m u l a from Datcom (Ref. 10) f o r t o t a l $6 r e s u I t s : Body ReferenceArea

1 - .OOOI Irl

(cy6 )tots I = - K i

(CLJfus ( S W I n c o r p o r a t i n gt h e aboveformula and t h e c h a r a c t e r i s t i c s o f a t y p i c a l l i g h t a i r p l a n ey i e l d s CyB = -.31 perradian,which seems t o be a t y p i c a lv a l u e .

7 5 The s t a b i l i t y d e r i v a t i v e Cg normally r e f e r r e d t o as t h e " e f f e c t i v e d i h e d r a l d e r i v a t i v e , " is t h e cha E ' ge i n r o l l i n g moment c o e f f i c i e n t caused by v a r i a t i o n i n s i d e s l i p angle. I n popular usage, a " p o s i t i v e d i h e d r a l e f f e c t " means a n e g a t i v e v a l u e o f CQ . During a i r c r a f t s i d e s l i p p i n g , t h e r o l l i n g moment produced i s t h e r e s u l f o f wing d i h e d r a l e f f e c t and t h e moment r e s u l t i n g from t h e v e r t i c a l t a i l c e n t e r of pressure located above t h e e q u i l i b r i u m x-axis.

For most conventional c o n f i g u r a t i o n s , t h e value o f CAB is negative; however, t h i s value can e a s i l y be a d j u s t e d by changing t h e amount o f b u i l t - i n wing d i hedra 1 .

CgB i s q u i t e important t o l a t e r a l s t a b i l i t y , s i n c e it a i d s i n damping both t h e Dutch RoI I mode and t h e s p i r a l mode. For f a v o r a b l e Dutch R o l l damping c h a r a c t e r i s t i c s , small negative values o f CQB a r e desired, b u t f o r improved s p i r a l s t a b i l i t y , l a r g e n e g a t i v e values a r e necessary. A compromise i s t h u s i n order, as i n d i c a t e d i n TN-1094 (Ref. 301, which shows t h a t b e s t general f l i g h t behavior i s obtained when t h e e f f e c t i v e d i h e d r a l a n g l e i s approximately 2 O . * I n a c t u a l p r a c t i c e , C Q ~ is u s u a l l y n o t determined a n a l y t i c a l l y because o f t h e l a r g e e r r o r s i n v o l v e d as compared w i t h f o r c e - t e s t d a t a on t h e design i n question. I f p o s s i b l e , f o r c e - t e s t data should be used t o determine t h e amount o f wing d i h e d r a l , t h u s determining t h e value o f CgB. An a n a l y t i c a l approach f o r s t r a i g h t and l e v e l f l i g h t ( B = - 9 ) i s g i v e n i n Perkins and Hage (Ref. 1 1 ) f o r c a l c u l a t i n g C Q ~ o f t h e t o t a l a i r c r a f t . T h i s formula i s g i v e n below, followed by a d i s c u s s i o n o f methods f o r o b t a i n i n g each component: i s From P e r k i n s and Hage, fhe wing d i h e d r a l c o n t r i b u t i o n fo CQ P

(cQB)w = ( - 1 CQB r -4- ( A C R g ) t i p shape

r

The e f f e c t o f wing t i p shape on t h e v a l u e o f ACQ i s shown below.

B

Maximum Ordinates On Upper Surface ~~~ AC"= --0°02 Maximum Ordinates

*%= O-O

Maximum Ordinates On Lower Surfaces

1 - p A c l p = -0002

F i g u r e 25.

E f f e c t o f winq t i p shape on CgB p e r radian.

- ~~ The v a l u e of ((2% / r ) f o r a p a r t i c u l a r a s p e c t r a t i o and t a p e r r a t i o may be B o b t a i n e d from t h ef o l l o w i n gg r a p h .

0 2 4 6 8 10 Aspect Ratio, AR F l g u r e 26. Values f o r (Cg / r ) f o rv a r i o u sa s p e c t and t a p e rr a t i o s .

A t h e o r e t i c a ls t u d yf o ru n s w e p t ,e l l i p t i c a lw i n g sw i t hz e r od i h e d r a li s p r e s e n t e di n TR-1269 (Ref. 311, p r o d u c i n gt h ef o l l o w i n ge q u a t i o nf o rw i n g c o n t r i b u t i o n t o CgB :

( c ~ ~ ) ~ , = CL[- -- 16 + .05] p e r r a d i a n

IT AR

r=o

A l s op r e s e n t e di s a r e v i s e df o r m u l a ,w h i c hc o n s i d e r sc h a n g e si nt a p e rr a t i o :

( C R g ) w , = CL [- k ( ' 7 1 x + + .05] p e r r a d i a n

AR A

r=o

where k = 1.0 f o r s t r a i g h t wing t i p s k = 1.5 f o rr o u n dw i n gt i p s I ft h ev e r t i c a lt a i li sl o c a t e d a b o v et h el o n g i t u d i n a la x i s ,t h ev e r t i c a l t a i lc o n t r i b u t i o ni se a s i l yc a l c u l a t e df r o mc o m p u t a t i o no ft h en o r m a lf o r c e caused by s i d e s l i p . Thus, where zv = d i s t a n c e f r o m t h e c e n t e r o f p r e s s u r e of t h e v e r t i c a l t a i l t o t h e a i r p l a n e ' s x - a x i s ( p o s i t i v e f o r v e r t i c a l t a i l above t h ex - a x i s ) .

interference effects on the vertical Most studies on this subject consider two interference--both influenced by the tail--wing-fuselage and wing-vertical tail ted in Perki ns and Hage, as shown wing's location. These effects are tabula in Table 9.

Wing-fuselage Wing-vertical Tail

( m g 1 ( ACRg 12

-. 0006 .00016

High Wing 0 0 M i d Wing

.0008 -. 0001 6

Low Wing Table 9. Values for interference effects on the vertical fail per rad i an.

Typi.cal yalues of CR range from -.03 to -.12 per radian.

The s t a b i l i t y d e r i v a t i v e Cn , o f t e nc a l l e dt h e" w e a t h e r c o c k " or s t a t i c d i r e c t i o n a l d e r i v a t i v e , i s t h e c E ange i n yawing moment c o e f f i c i e n t r e s u l t i n g from a change i ns i d e s l i pa n g l e .P h y s l c a l l y , CnB i s t h e r e s u l t of t h ea i r f r a m e s i d e s l i p p i n g , w i t h t h e r e l a t i v e w i n d s t r i k i n g it o b l i q u e l y ,c a u s i n g a yawing moment a b o u tt h ec e n t e r o f g r a v i t y . The v e r t i c a lt a i l ,f u s e l a g e , and wing c o n t r i b u t e t o CnB, w i t ht h ev e r t i c a lt a i lt h ed o m i n a n tf a c t o r .F o rp o s i t i v e s i d e s l i p , t h e v e r t i c a l t a i l c a u s e s a p o s i t i v ey a w i n g moment; thus, CnB i s u s u a l l yp o s i t i v e ,e v e nt h o u g ht h ef u s e l a g ec o n t r i b u t i o ni sn o r m a l l yn e g a t i v e .

The w i n gc o n t r i b u t i o ni su s u a l l yp o s i t i v e ,b u tq u i t es m a l l compared t o t h a t o f t h e v e r t i c a l t a i l andfuselage.

The v a l u e o f CnB d e t e r m i n e sp r i m a r i l yt h eD u t c h Roll n a t u r a lf r e q u e n c y and a f f e c t st h es p i r a ls t a b i l i t y o f t h ea i r c r a f t . I t i sg e n e r a l l ya g r e e dt h a t v a l u e s o f Cn a sh i g ha sp r a c t i c a l l yp o s s i b l ea r ed e s i r a b l ef o r good f l y i n g q u a li t i e s .I a t u e sf o r CnB s h o u l db eo b t a i n e df r o mf o r c e - t e s td a t a f o r t h e model i n q u e s t i o n wherepossible.

A t p r e s e n t , two a n a l y t i c a l methods o f c a l c u l a t i n g t o t a l Cne appearmost complete. The f i r s ti st h a tp r e s e n t e di nP e r k i n s and Hage (Ref.

1 1 ) f o r s t r a i g h t ,l e v e lf l i g h t ( 6 = -$I. The v a l u e of CnB f o rt h et o t a l planecan be o b t a i n e df r o mt h ec o m p o s i t ef o r m u l a , Perk i n s and Hage a l s o g i v e a v a l u e f o r (CnglWas ( C n g I w = ,00006 ( A 0 1 1'2 perdegree For unswept wings, TM-906 (Ref. 3 2 ) g i v e s C,.

(CnBB)W = - p e r r a d i a n

TAR T h i sf o r m u l ai sm o d i f i e di n TN-1581 ( R e f .2 8 )f o r sweep, by t h ef o l l o w i n g r e l a t i o n : 1 t a n A AR A R ~

(COS A - 2 -

(CngIw = C L ~ [ "

8 cos A

4rAR T A R ( A R + 4 cos A )

+ 6 C T I x sin p e r r a d i a n where

-

x = long i t u d i na I d i s t a n c e . r e a r w a r d f r o m c.g. t o wingaerodynamiccenter / A v a l u e o f (CngIfuscanbeobtainedfromPerkinsand Hage by use of The v a l u e o f KB, a ne m p i r i c a lc o n s t a n t ,c a nb eo b t a i n e d from t h e f o l l o w i n g g r a p ha s a f u n c t i o n of f i n e n e s sr a t i o and C.Q. l o c a t i o n . The d i s t a n c e from t h e nose t o t h e c.g. i s d; S, i s t h e body s i d ea r e a . The o t h e rv a r i a b l e sa r e shown below.

F i g u r e 27. E m p i r i c a l c o n s t a n t KB as a f u n c t i o n o ff i n e n e s sr a t i o and C.Q. l o c a t i o n .

Perkinsand Hage a l s op r e s e n tt h ef o l l o w i n gf o r m u l af o rv e r t i c a lt a i l c o n t r i b u t i o n t o C n B : The v a l u e of a , , l i f t - c u r v es l o p e o f v e r t i c a l+ a i l ,i sd e t e r m i n e d by u s i n gt h e e f f e c t i v ea s p e c tr a t i o ,w h i c hi sc a l c u l a t e d bV

Ae = 1.55 -

S V The v a l u e o f a , can now be o b t a i n e df r o mt h ef o l l o w i n gg r a p h p a r t i c u l a r f o r a v a l u eo f Ae. T h i sv a l u ei s based on a c o n v e n t i o n a l , low hor i zon t a I t a i l c o n f i g u r a t i o n ;t h e r e f o r e ,f o rd e s i g n ss i g n i f i c a n t l yd i f f e r e n t, one i s r e f e r r e d t o Datcom (Ref. 10) f o r an a l t e r n a t e method of f i n d i n g Ae.

0 1 2 3 4 5 6 1 Effective aspect ratio vertical tail, Ae F i g u r e 2 8 . Values o fa v as a f u n c t i o n o fv e r t i c a lt a i la s p e c tr a t i o .

The v e r t i c a I t a i I e f f i c i e n c y f a c t o r , vv, i s a l s o needed f o r t h ed e s i g ni n q u e s t i o n . Two i n t e r f e r e n c ef a c t o r sm u s t bedetermined t o c o m p l e t et h ec a l c u - l a t i o n o f t o t a l Cn . The f i r s to ft h e s e , A1Cn , i st h ew i n g - f u s e l a g ei n t e r - f e r e n c e f a c t o r , a p u n c t i o no fw i n gl o c a t i o n . 7 he f o l l o w i n gc h a r ti s used t o determine i t sv a l u e .

WING POSITION A I C n B PER DEGREE High Wing .0002 Mid Wing ,000 1 Low Wing T a b l e 10. Values f o r A1CnB as a f u n c t i o no fw i n gl o c a t i o n .

The second i n t e r f e r e n c ef a c t o r , A2CnB, i s t h e r e s u l t o f s i d e w a s h a t t h e v e r t i c a l t a i l c a u s e d b y w i n g - f u s e l a g e i n t e r f e r e n c e . I t s v a l u e may be determined from t h ef o l l o w i n gg r a p ha s a f u n c t i o n o f w i n g p o s i t i o n and maximum f u s e l a g eh e i g h t .

F i g u r e 2 9 . V a l u e sf o r A2CnB as a f u n c t i o n o f w i n gp o s i t i o n and maximum f u s e l a g eh e i g h t .

Now t h a t each o f t h e componentshasbeendetermined, t h e v a l u e f o r t o t a l CnB, u s i n gt h e method o fP e r k i n s and Hage, can be evaluated. They i n d i c a t e t h a t a CnB 1 .0005 t J " / b ) i s necessary f o r adequate d i r e c t i o n a I s t a b i I i t y .

The second method o f o b t a i n i n gt o t a l C i s p r e s e n t e d i n Datcom (Ref. 10) nB where a w i n g - f u s e l a g ec o r r e c t i o n i s added t o t h e v e r t i c a l t a i l c o n t r i b u t i o n , r e s u l t i n gi n The i n t e r f e r e n c ef a c t o r , K , (perdegree), may b ed e t e r m i n e df r o mt h ef o l l o w i n g graphas a f u n c t i o n o f a i r c r a f t g e o m e t r y andReynolds Number.

SB, = Body Side Area

w = Maximum Body Width

0.001 0.002 0.003 0-004 F i g u r e 30. E m p i r i c a l i n t e r f e r e n c e f a c t o r , Kn, as a f u n c t i o n of a i r c r a f t geometryandReynolds Number.

The v a l u e of (C ltai I may be o b t a i n e d from t h ed i s c u s s i o n of t h e s t a b i l i t y d e r i v a t i v e Cvs t h e p r e s e n t s t u d y .

F o rl i g h ta i r c r a f t ,t y p i c a lv a l u e s of C seem t o range from 0.03 t o "6 0.12 p e r r a d i a n .

The s t a b i l i t y d e r i v a t i v e Cyp i s t h e change i n s i d e f o r c e r e s u l t i n g from r o l l i n g v e l o c i t y , w i t h t h e v e r t l c a l t a i l t h e m a i n c o n t r i b u t o r , eventhough, f o r some c o n f i g u r a t i o n s ,t h ew i n g may make a significant c o n t r i b u t i o n . The r o l l i n g v e l o c i t y , p, c r e a t e s a n e f f e c t i v e a n g l e of a t t a c k on t h e t a i l , which, i nt u r n ,p r o d u c e s a s i d e - f o r c e . The s i g n o f C may b e e i t h e rp o s i t i v eo r n e g a t i v e . I t i s r e l a t i v e l yi n s i g n i f i c a n t and %nmonly neglected.

I n TN-1581 (Ref. 281, a c o m p l e t e l yt h e o r e f i c a la p p r o a c h ,t h ef o l l o w i n g f o r m u l a f o r C of t h e w i n g i s p r e s e n t e d : Y P

AR + cos A tan A

'YP = c L AR + 4 cos A

€or zero sweepback, t h i sf o r m u l ag i v e s C = 0.0, w h i c h a g r e e s w i t h o t h e r t h e o r e t i c a l t r e a t m e n t s . From w i n d t u n n e l t e s Yp s on wings alone, TR-968 (Ref. 33) adds CL/AR t o t h e above f o r m u l at oa c c o u n tf o rw i n gt i ps u c t i o n ,r e s u l t i n gi n T h i s r e d u c e s t o Cy = CL/AR f o r z e r o sweepback,whichdoes n o ta p p e a r sma I I enough t o be cons i Hered neg I i g i b I e.

The w i n gc o n t r i b u t i o ni su s u a l l ym i n o r compared t o t h e v e r t i c a l t a i l c o n t r i b u t i o n . From NASA MEMO 4-1-59t (Ref. 341, u s i n g a t h e o r e t i c a l method o f discrete-horseshoe-vortices, with t h eh o r i z o n t a lt a i li nt h em i d - p o s i t i o n , a v a l u eo f ( C ) t a i li s( - . 8 ) [ b v / ( b w / 2 ) ]f o rt h ev e r t i c a lt a i 1 c o n t r i b u t i o n .

Also, i n TR-1&6 (Ref. 351, a f o r m u l af o rt a i I c o n t r i b u t i o n i s where zv = h e i g h t of v e r t i c a l t a i l c e n t e r of p r e s s u r ea b o v et h el o n g i t u d i n a la x i s (J = s i d e w a s h a n g l e a t t h e v e r t i c a l t a i l The r a t i o (acs/[a(pb/2U)]),, i s t h e a v e r a g ee f f e c to fs i d e w a s h on t h e v e r t i c a l t a i l a n dc a nb eo b t a i n e df r o mt h ef o l l o w i n gg r a p ha s a f u n c t i o n of a n g l e o f a t t a c k and v e r t i c a l t a i l t o semispan r a t i o for a w i n g w i t h a s p e c t r a t i o i n t h e v i c i n i t y of 6.

4 8 12

(Y , deg

F i g u r e 31. i s t i m a t i o no fa v e r a g es i d e w a s ha n g l ea t t h ev e r t i c a lt a i lw i t hw i n ga s p e c tr a t i o equal t o 6.

I n TR-1098 (Ref. 271, t h e r e l a t i o n f o r t a i l c o n t r i b u t i o n i s Z

(Cyp)ta i 1 = 2 e - ( ~ ) a = 0 1 (CyB)ta i I

w bw which seems q u i t es m a l l f o r c r u i s e a t an a n g l eo fa t t a c kn e a rz e r o .

Datcom (Ref. 101 a I so p r e s e n t s a method f o r c a l c u l a t i n g Cyp, b u t , f o r c o n v e n t i o n a ll i g h ta i r c r a f tw i t hz e r ow i n g sweep, lt a p p e a r s t h a t C i s v e r y near zero. The f o l l o w i n gf o r m u l ai st a k e n from t h i s work: YP The followinggraphs,whichhavebeenadaptedfrom Datcom f o r l i g h t a i r c r a f t , show ( C /CL) a s a f u n c t i o no fw i n g sweep and t a p e rr a t i o , and [ ( A C y p ) r / ( C ~ p ) r = o ] a s a f u n yg t i o n o f d i h e d r a l a n g l e .

AC14 30 0 -30 -5 X 1 . 0 " ' " " " ' " " ' " " ! ' " ~ -.4 "2 0 -2 A F i g u r e 3 2 . Values f o r ( C / C L ) as a f u n c t i o no f wing sweep and YP t a p e r r a t i o .

r, Dihedral Angle [deg]

TN-40.66 (Ref. 361, whichuses f l i g h t measurements t o d e t e r m i n e s t a b i l i t y d e r i v a t i v e s ,s t a t e st h a t ,i np r a c t i c e , Cyp may l i e between 0.3 and -0.3. U s i n g t h i s r a n g e o f Cyp, t h e o t h e r s t a b i l i t y d e r i v a t i v e s w e r ec a l c u l a t e d ,w i t h CY showing a s m a l le f f e c to n l yo n CYB and C Thus, it appears t h a t Cyp = 0.6 "P' i s p r o b a b l ya sa c c u r a t e an e s t i m a t ea s i s necessary.

The s t a b i l i t yd e r i v a t i v e Cg, t h er o l l damping d e r i v a t i v e ,i st h e change P’ i n r o l l i n g moment c o e f f i c i e n t due t o v a r i a t i o n i n r o l l i n g v e l o c i t y . F o r a p o s i t i v e r o l I, i s t h e r e s u l t , p r i m a r i l y , o f a ni n c r e a s ei nl i f t on t h e down moving wing nd a d e c r e a s e i n l i f t on t h e up m o v i n gw i n g ,t h u sc r e a t i n g Cg% a moment whichopposesthemotion o f t h e r o l l . T h i s moment i s n e g a t i v e , making CfiP n e g a t i v ei ns i g n . The wing, h o r i z o n t a lt a i l , and v e r t i c a lt a i l c o n t r i b u t e t o CgP, w i t h t h e w i n g t h e d o m i n a n t f a c t o r f o r a i r f r a m e s w i t h c o n v e n t i o n a l - s i z et a i l s . Cgp i st h ep r i n c i p a ld e t e r m i n a n t o f the damping-in- r o l l c h a r a c t e r i s t i c s o f t h e a i r c r a f t .

The b a s i c w i n g c o n t r i b u t i o n t o Cgp may be foundfromthegraphbelowas a f u n c t i o n o f a s p e c ta n dt a p e rr a t i of o r a w i n gw i t hz e r o o r small sweep, a l i f t c o e f f i c i e n t o f z e r o , and l i f t c u r v es l o p eo f 2 ~ . For swept wings, the r e a d e ri sr e f e r r e dt o TR-1098 (Ref. 271, from which t h i s f i g u r e was adapted.

2 4 6 8 10 Aspect Ratio, AR F i g u r e Wing c o n t r i b u t i o n t o Cg f o rw i n g 34.

w i t h z e r o o r sma I I sweeg.

The p r e s e n ts t u d y assumes a l i n e a r l i f t c u r v es l o p ew i t h no c o r r e c t i o n f o r n o n - l i n e a r i t i e si nl i f tc o e f f i c i e n t .I no r d e rt o u s eF i g u r e3 4t oc a l c u l a t e wing Cgp f o r l i f t c u r v e s l o p e s o t h e r t h a n ZIT, s e v e r a l means o f c o r r e c t i n g t h e data must be considered. TN-1839 ( R e f .3 7 )p r e s e n t st h ef o l l o w i n gm e t h o d : which, f o rz e r o sweep, reduces t o The e f f e c to fw i n gd i h e d r a lo n Cg i s cons idered by TN-1732 (Ref. 38) i n a c o r r e c t i o n f o r w i ng Cg ; however, s i k e most l i g h t a i r c r a f t h a v ed i h e d r a l o f sevendegrees o r I ess, t R i s c o r r e c t i o n a p p e a r s q u i t es m a l l . The d r a gc o n t r i - b u t i o n t o wing Cg i sg i v e ni n TN-1924 (Ref. 39 1, f o r s w e p tw i n g sw i t he l l i p t i c - c h o r dd i s t r i b u t i o R , by t h ef o l l o w i n gf o r m u l a : whichreduces, f o r z e r o sweep, t o t h e f o l l o w i n g r e l a t i o n : where CD = e x p e r i m e n t a l l y d e t e r m i n e d d r a g c o e f f i c i e n t .

Thus, f o r an a i r c r a f t w i t h a l i n e a r l i f t c u r v es l o p e ,s m a l ld i h e d r a l , and z e r o sweep, t h ef o l l o w i n gf o r m u l af o rw i n g Cg r e s u l t s : P

AR + 4

(cEp)w i ng = [(Cgp)ao = 2 7 r 1 [( 27r

A -

( a o ) w ) A R + By c o n s i d e r i n gt h eh o r i z o n t a lt a i la sa ni s o l a t e da i r f o i l ,t h eb a s i cv a l u e f o r (Cg ) h may a l s o bedetermined from F i g u r e 34 as a f u n c t i o n o f h o r i z o n t a l t a i I a s F e c tr a t i o ,t a p e rr a t i o , and amount o f sweep. T h i sp r o c e d u r ei st h e same as t h a tf o rc a l c u l a t i n gt h ew i n gc o n t r i b u t i o n ;t h u s , (Cg )hmust be P c o r r e c t e df o rl i f tc u r v es l o p eo t h e rt h a n 2 7 r . The v a l u eo b t a l n e di ss c a l e d down by t h e method o f TR-1098 (Ref. 2 7 ) : The f a c t o r 0.5 i si n c l u d e d t o a c c o u n t f o r t h e f l o w r o t a t i o n a t t h e t a i l caused by thewinq; A i s t h e amount o f sweep o f t h e h o r i z o n t a l t a i l . The h o r i z o n t a l t a i l i s assumed t o have n e g l i g i b l e d i h e d r a l .

The v e r t i c a I t a i I c o n t r i b u t i o n t o C t i sa l s og i v e n i n TR-1098 (Ref. 2 7 ) : - p ( C R p ) V Ot- where zv = h e i g h t o f c e n t e r o f p r e s s u r e o f v e r t i c a l t a i l a b o v e t h e x - a x i s ( d i f f e r e n t f o r e a c h a n g l e o f a t t a c k ) .

. . . . . . . .. . .. . ... , . . _.

. . . ....., ... .... ".. .. . . I 1 1 . . 1 1 1 1.1 I.I. I"..," _I. rn ..- 11-11 , .,.", , II ""- ...".. , _.,,.

From these formulas, it is obvious that the vertical tail contribution is negligible at low angles of attack. A much more elaborate formula for estimating vertical tail contribution is given in TN-2587 (Ref. 401, but because of the relative unimportance of this contribution to total Cgp, that method is not included.

In summary, for an aircraft with zero sweep a va I ue f o r tota I Cg may

P

be obtained by simply adding the various contributions as follows:

AR + 4 1

(Ctp)tota I = [(Cgp)ao = ZT]

- 8 CD

[ ( - A R +

An adaptation from TN-1309 (Ref. 4 1 ) for various sideslip angles is

(CRp)tota I = [(cgp)+ota I J B = 00 cos2 B *

Typical val ues of Cg range from -.25 per radian to -.60 per radian.

P

C"

P

_The stab ility derivativeC?p is the change in yawing moment caused by

r o l I ing, with the wing and vertlca I tai I the main contributors. For a

positive roll, the produced yawing moment is a result of the unsymmetrical lift distribution causing increased drag on the left wing and decreased drag on the right wing and is, therefore, negative. The vertical tail contri- bution may be either positive or negative, depending on tail geometry, angle of attack, and sidewash from the wing. Dutch Roll damping is influenced by Cnp in that the larger its negative value, the less Dutch Roll damping.

therefore, a positive C i s desired.

"P

Using an elliptical lift distribution, Perkins and Hage (Ref. 1 1 ) give the following formula for Cnp: The results of wind tunnel testing on a wing with AR = 5.16, h = 1.0, and 00,wing sweep is reported in TR-968 (Ref. 3 3 ) . The resulting formula for (Cn Iw, by a curve fit, i s P (CnpIw = -.043 CL - .0044 for 0 < CL < 1.05. This report shows that, for large CL ( i n the stall region), and large positive values are obtained. This same trend Cnp reverses signs, occurs at smallerCL when sweepback is encountered. Below is a formula for with sweepback, from TN-1581 (Ref. 28).

use

AR + 4

(Cnp)w = CL 1 + 6 ( 1 + -

AR + 4 cos A AR 12

The ratio (Cnp/CL)A = 00 as a function o f aspect ratio and taper ratio may be obtained from the following figure.

- 2 4 6 8 10 12 14 16 A s p e c t R a t i o , A R Figure 3 5 . Values o f (Cn /CL) for zero sweep.

P 9 1 TN-2587 (Ref. 401 explains, through the following formula, how the vertical tail contribution to C is influenced by sidewash variations:

"P

Values for ao7/[3(pb/2U)], effect of wing sidewash, as a function of (h /b 1, may be obtained from the fol lowing graph, reproduced from TN-2332 (Ref. t 4 2 r ! 2 f o r

wings with aspect ratios in the vicinity of six, where he is the distance from

the wing centerline to the center of pressure of the vertical tail (positive above the wing centerline).

"0 .1 . 2 .3 .4 .5

Figure 3 6 . Effect of wing sidewash on vertical tail.

The e f f e c t o f fuselagesidewash,ao2/[a(pb/2U)], may be d e t e r m i n e df r o mt h e f o l l o w i n gf o r m u l aa d a p t e d from TN-2587 (Ref. 40): where

Ah zv - ( z v c o s a - R , s i n a )

”

-

b W b W F o r z e r o a n g l e o f a t t a c k , t h i s s idewash f a c t o rr e d u c e st oz e r o and i s q u i t e sma I I f o r lowang l e s o f a t t a c k .W i t ht h e s ef a c t o r s , a v a l u e of (Cn Iv can be determined and added t o t h e w i n g c o n t r i b u t i o n t o y i e l d t o t a l Cn P P ’

(Cnp)tot- 1 = (Cn ) W + (Cn ) V

P P T y p i c a lv a l u e sf o r Cn range from -.01 p e rr a d i a nt o - . l o p e rr a d i a n .

P The s t a b i l i t y d e r i v a t i v e Cyr i s t h e change i n s i d e f o r c e r e s u l t i n g f r o m a change i n y a w i n gv e l o c i t y . As t h ea i r f r a m eu n d e r g o e s a p o s i t i v e yaw, an e f f e c t i v ep o s i t i v es i d ef o r c ed e v e l o p so nt h ev e r t i c a lt a i l ,w h i c hi st h e d o m i n a n tc o n t r i b u t o r t o Cyr. S i n c et h i sf o r c ei sn o r m a l l ys m a l l , Cyr u s u a l l y has a s m a l lp o s i t l v ev a l u e .

The w i n gc o n t r i b u t i o ni sn o r m a l l yn e g l i g i b l e ,a si n d i c a t e d by TN-1669 (Ref. 431, which i s t h e r e s u l t o f w i n dt u n n e lt e s t so nr e c t a n g u l a rw i n g sw i t h z e r o sweep and a s p e c tr a t i oo f 5.16. The f o l l o w i n gf o r m u l ai sp r o d u c e db y c u r v e - f i t t i n g t h e d a t a o f t h i s r e p o r t , w h i c h g i v e s (CyrIwing as a f u n c t i o n of l i f t c o e f f i c i e n t f o r an a n g l eo fa t t a c kb e l o wt h es t a l lr e g i o n : ( C 1 = ,143 CL - .05.

Y r The t a i l c o n t r i b u t i o n may be e s t i m a t e db yt h ef o r m u l ab e l o wf r o m TR-1098 (Ref. 2 7 ) w h i c hg i v e s( C y r ) t a i 1a s a f u n c t i o no ft h ev a l u e sf o r( C Y b ) t a i l o r ( C n g ) t a i I d i s c u s s e di nc o n n e c t i o nw i t ht h es t a b i l i t yd e r i v a t i v e Cyg I n t h ep r e s e n ts t u d y : F o rw i n dt u n n e lt e s t s o f a model o s c i l l a t i n g i n yaw, TR-1130 (Ref. 44) i n d i c a t e s t h a t much l a r g e r v a l u e s f o r Cyr a r e o b t a i n e d t h a n f r o m o t h e r t e s t i n g methods. These r e s u l t sa r ep r e s e n t e dg r a p h i c a l l yi nF i g u r e 37, i n c l u d i n g fuselage, wing, and t a i le f f e c t s . The f u s e f a g ee f f e c t sa r er e l a t i v e l yl a r g e and n e g a t i v ei ns i g n .

-0- Fuselage "C+ Fuselage + T a i l

"0- f r s t l a l t + Wing +Tail

0 2 4 6 8

Angle o fA t t a c k , a , deg

F i g u r e 37. Values f o r f u s e l a g e , w i n g , and t a i l c o n t r i b u t i o n s t o C y , .

TN-4066 ( R e f .3 6 1 ,w h i c hu t i l i z e sf l i g h tt e s td a t at oc a l c u l a t es t a b i I i t y d e r i v a t i v e s , i n d i c a t e s t h a t Cy, couldrangefrom-0.3 t o 0 . 3 w i t h o u t show i ng any s i g n i f i c a n te f f e c t so nt h eo t h e rs f a b i l i t yd e r i v a t i v e s .T h i sh e l p s d e m o n s t r a t et h ei n s i g n i f i c a n c e of Cy, t o t h e l a t e r a l s t a b i l i t y o f l i g h t a i r c r a f t . , The change i n r o l l i n g moment due t o v a r i a t i o n i n y a w i n g v e l o c i t y c o n s t i - t u t e s t h e s t a b i l i t y d e r i v a t i v e Cgr. The w i n gp r o v i d e st h em a j o rc o n t r i b u t i o n , w i t h t h e v e r t i c a l t a i l h a v i n g a m l n o re f f e c t . When t h e r e i s a p o s i t i v e yaw r a t e , t h e l e f t w i n g moves f a s t e r t h a n t h e r i g h t wing,producingmore l i f t on t h el e f tw i n g and, consequently, a p o s i t i v e r o l l i n g moment. The t a i l c o n t r i - b u t i o n may be e i t h e r p o s i t i v e o r n e g a t i v e , d e p e n d i n g o n t a i l g e o m e t r y and a n g l e o f a t t a c k o f t h ea i r p l a n e .A l t h o u g h C t r has l i t t l e e f f e c t o n D u t c h Roll damping, it i sq u i t ei m p o r t a n t t o t h es p i r a l mode. F o rs p i r a ls t a b i l i t y , it Is d e s i r a b l e t h a t Ckr beassmall a p o s i t i v e number a sp o s s i b l e .

The w i ng c o n t r i b u t i o n t o Ckr i sp r e s e n t e di n TR-589 (Ref. 4 5 ) as a f u n c t i o n of l i f t c o e f f i c i e n t b yt h ef o l l o w i n gf o r m u l a s : (Ckr’w i ng = C L / ~ f o r r e c t a n g u l a r l i f t d i s t r i b u t i o n , o r (Cgr)wing = CL/4 f o r an e l l i p t i c a l l i f t d i s t r i b u t i o n .

I n TN-1669 (Ref. 431, t h e r e s u l t s of w i n d t u n n e l t e s t s o f a NACA 0012 a i r f o i l o f a s p e c t r a t i o 5.16 and z e r o sweep i n d i c a t e t h a t (Ckr),,,ing = C L / ~ a g r e e sw i t h e x p e r i m e n t a ld a t af o rl i f tc o e f f i c i e n t sb e l o wt h es t a l lr e g i m e . The wing c o n t r i b u t i o n t o Cgr as a f u n c t i o n o f a s p e c t r a t i o , t a p e r r a t i o , and sweep a n g l ei sp r e s e n t e di nF i g u r e 38, adapted from Datcom (Ref. 10).

.1 . 2 . 3 .4 F i g u r e 38. Wing c o n t r i b u t i o nt o Cgr a s a f u n c t i o no fa s p e c t r a t i o , t a p e r r a t i o , and sweep angle.

. . . . . . , .

T h em i n o rc o n t r i b u t i o n of t h e v e r t i c a l t a i l t o Cgr may be c a l c u l a t e d f r o m t h e f o r m u l a b e l o w from TN-I 984 (Ref.46) : By a d d i n gt h ew i n ga n dv e r t i c a lt a i lc o n t r i b u t i o n s ,t o t a l Cgr may be determined.

Typ.icalvalues o f C k , range from .04 p e rr a d i a nt o .12 p e rr a d i a n .

Cnr, commonly known a st h e yaw damping d e r i v a t i v e , i s t h ec h a n g ei ny a w i n g moment due t o v a r i a t i o ni ny a w i n gv e l o c i t y . As t h ea i r f r a m eu n d e r g o e s a p o s i - t i v e r, a yawing moment w h i c ho p p o s e st h em o t i o ni sp r o d u c e d .T h i s moment c o n s i s t s o f c o n t r i b u t i o n s from t h ew i n g ,f u s e l a g e , and v e r t i c a l t a i l , a l l o f w h i c ha r en e g a t i v ei ns i g n . TN-1080 (Ref. 47) s t a t e st h a tt h ev e r t i c a lt a i l c o n t r i b u t e sa b o u t 70% t o 90% o f Cnr f o rc o n v e n t i o n a ld e s i g n s . The d e r i v a t i v e Cnr i s t h e m a i n c o n t r i b u t o r t o t h e damping o ft h eD u t c h Roll mode and a l s o p l a y s a s i g n i f i c a n tr o l ei nd e t e r m i n i n gs p i r a ls t a b i l i t y ,m a k i n g i t v i t a l t o l a t e r a ls t a b i l i t y .F o rb e s te f f e c t si n each of t h e s e modes, l a r g en e g a t i v e v a l u e s o f Cnr a r e d e s i r e d .

The wing c o n t r i b u t i o n t o Cnras a f u n c t i o n o f l i f t and d r a g c o e f f i c i e n t s i s g i v e n i n TR-1098 ( R e f .2 7 )b yt h ef o l l o w i n gr e l a t i o n : O r i g i n a l l yp r e s e n t e di n TN-1581 (Ref. 281, t h i sf o r m u l ai st h er e s u l to fs i m p l e sweep t h e o r yw i t hs t r i pi n t e g r a t i o n ;c o n s e q u e n t l y , [(ACnr),/CL2] and [ ( A C ~ , ) ~ / C D ~ ] a r ef u n c t i o n s of sweep, t a p e rr a t i o , and a s p e c tr a t i o .F o rw i n g s w i t hz e r o sweep a n da s p e c tr a t i og r e a t e rt h a nf i v e , however, t h e s et e r m sa r e c o n s t a n t a t t h e f o l l o w i n g a p p r o x i m a t e v a l u e s : (ACrlr) 1 = -0.020 , CL2 and (AC, 12 r = -0.30 .

co, W i t h t h e s e r e s t r i c t i o n s , t h e aboveformulareduces t o The g r a p ht h a tf o l l o w sf r o m TN-1669 (Ref. 43) shows, for a r e c t a n g u l a r wing, thecloseagreementbetweentheabovetheory and e x p e r i m e n t a ld a t af o rw i n g c o n t r i b u t i o n t o Cn,-.

---Theory 43- Experimental

T

0 .2 .4 .8 1.0 1.2 LiftCoefficient, CL F i g u r e 39. R e s u l t s o f e x p e r i m e n t a l d a t a o n w i n g c o n t r i b u t i o n t o Cnr.

T h i s g r a p h p o i n t s o u t t h e r e l a t i v e i n s i g n i f i c a n c e o f w i n g c o n t r i b u t i o n t o Cnr.

The t a i l c o n t r i b u t i o n t o Cnr i sp r e s e n t e di n TR-1098 (Ref. 271, as f o l lows: B l a k e l o c k( R e f . 18) t r e a t st h i sc o n t r i b u t i o n as, The v e r t i c a l t a i l e f f i c i e n c y f a c t o r , qv, compensates f o rt h ei n t e r f e r e n c e between t h ef u s e l a g e and t h ev e r t i c a lt a i l . When t h e r ei s no i n t e r f e r e n c e , nv i s equal t o one. Total Cnr i st h e nd e t e r m i n e db ya d d i n gt h ew i n g and v e r - t i c a lt a i lc o n t r i b u t i o n s ,s i n c et h ef u s e l a g ec o n t r i b u t i o ni sn e g l i g i b l e .

Anotherapproach t o c a l c u l a t i n g t o t a l Cnr i s f r e e - o s c i l l a t i o n t e s t s , a s presented i n W R L-387 (Ref. 48). These t e s t s on a m i d - w i n ga i r p l a n ei n c l u d e e f f e c t s o f w i n g ,f u s e l a g e ,a n dv e r t i c a lt a i lo n Cnr. The w i n gc o n t r i b u t i o n i s c a l c u l a t e d a s which, f o r a n a s p e c t r a t i o o f s i x and t a p e r r a t i o of ope, seems t o a g r e ec l o s e l y w i t ht h ep r e v i o u sf o r m u l af o rw i n gc o n t r i b u t i o n . WR L-387 (Ref. 48) also

L

.. . - . .... .. . . . . . . .-. . . . . . . .. ". -. . .. - .. . " - . .

s t a t e s t h a t t h e f u s e l a g e c o n t r i b u t i o n o f t e n r a n g e s frorrc -.003 t o -.006, small enough t o be neglected. The v e r t i c a lt a i lc o n t r i b u t i o nf r o mR e f e r e n c e (48) i s Combiningthesetwoformulas,theresultistheempiricalexpressionbelow, showing Cnr f o r a conventional,mid-wingairplane: T y p i c a lv a l u e so f Cnr for g e n e r a l a v i a t i o n a i r c r a f t range from-.05per r a d i a n t o -.14 p e rr a d i a n .

C

The s t a b i l i t y d e r i v a t i v e CY^^ i s t h e change i n s i d e f o r c e c o e f f i c l e n t w i t h v a r i a t i o ni na i l e r o nd e f l e c t i o n .F o rm o s tc o n v e n t i o n a ll i g h ta i r c r a f t ,t h i s d e r i v a t i v e i s zero; however, f o r an a i r f r a m ew i t h low a s p e c t r a t i o and h i g h l y sweptwings, it may have a J a l u eo t h e rt h a nz e r o .

I

The s t a b i l i t yd e r i v a t i v e C Q ~ known a st h ea i l e r o ne f f e c t i v e n e s so r A' " a i l e r o n power", i s t h e v a r i a t i o n i n r o l l i n g moment c o e f f i c i e n t w i t h change i na i l e r o nd e f l e c t i o n .S i n c el e f ta i l e r o n down i sd e f i n e da sp o s i t i v e , a p o s i t i v ed e f l e c t i o np r o d u c e s a r o l l i n g moment t o t h er i g h t ,w h i c h i s a l s o p o s i t i v e ,m a k i n g Cg6 p o s i t i v e .D e s i r a b l ev a l u e so f C Q ~ ~ a r eg i v e ni nt e r m s o f t h e w i n g t i p he1 1 8 angle(pb/2U) f o r a f u l I a i l e r o nd e f l e c t i o n .F o r I i g h t a i r c r a f t ,t h i sr a n g ei sn o r m a l l yf r o m 0.07 t o 0.08 r a d i a n s .I nP e r k i n s and Hage (Ref. 1 1 1 , s t r i pi n t e g r a t i o nc a n be used t o e v a l u a t e C Q ~ ~ as, b 4 i F i g u r e 40. I l l u s t r a t i o no fs t r i pi n t e g r a t i o n .

I no r d e r t o i n t e g r a t e ,t h ec h o r d , c, as a f u n c t i o no ft h es p a n w i s ed i s t a n c e , y, must be known. F o rs t r a i g h t ,t a p e r e dw i n g s ,t h i sr e l a t i o n s h i pi s where C R = r o o tc h o r d .

The v a l u e o f T may be o b t a i n e df r o mt h ef o l l o w i n gg r a p ha s a f u n c t i o n o f a i l e r o n c h o r d t o w i n gc h o r dr a t i o .

F i g u r e 4 1 . V a l u e sf o r T as a f u n c t i o no fa i l e r o n chord t o w i ng c h o r d r a t i o .

The method o fs t r i pi n t e g r a t i o np r e s e n t e d above i s seldom used i n p r a c t i c e because o f l a r g ee r r o r si n c u r r e di na s s u m i n g a d i s c o n t i n u o u s l i f t d i s t r i b u t i o n .

I n r e a l i t y , t h e l i f t d i s t r i b u t i o n a d j u s t s b t h e a i l e r o n d e f l e c t i o n q u i c k l y b u t s m o o t h l y .T h i sb e i n gt h ec a s e ,t h ev a l u eo f CQ i s n o r m a l l y f o u n d f r o m t h e spanwise load d i s t r i b u t i o nd a t aa sp r e s e n t e di n ?R-635 (Ref. 49). These dataarereproducedbelow f o r ( C J ? , ~ ~ / T ) as a f u n c t i o n o f t h e e x t e n t o f t h e u n i t a n t i s y m m e t r i c a la n g l eo fa t t a c k .

.2 .4 . -6 1.0 Extent of unit antisymmetrical angle of attack; %b/2 F i g u r e 42. V a l u e sf o r ( C Q ~ / T ) as a f u n c t i o n o f t h e e x t e n t of u n i t a n t i s y m m e t r i c a l a n g l e o f a t t a c k .

"" ~ .. .. . .. - . .

. ..

A v a l u eo f ( C Q ~ / T ) i so b t a i n e df r o mF i g u r e 42 by f i r s t u s i n g t h e d i s t a n c e f r o m t h e body c e n t e r l i n e t o t h e o u t b o a r d edge of t h e a i l e r o n d i v i d e d b y t h e wingsemispan t o g e t a v a l u e ,f r o mw h i c hi ss u b t r a c t e d a v a l u eo b t a i n e d by u s i n gt h ed i s t a n c ef r o mt h e body c e n t e r l i n e t o t h e i n b o a r d edge o f t h e a i l e r o n dividedbythewingsemispan.Thevalue of ( C R ~ ~ / T ) from t h eg r a p hi s m u l t i p l i e d by a v a l u e o f T f r o mt h ep r e c e d i n gg r a p h t o g i v e C!?,6A p e rr a d i a n .

T y p i c a lv a l u e so f C J ? , ~ ~ rangefrom 0.1 t o 0.25 p e rr a d i a n .

The s t a b i l i t y d e r i v a t i v e Cn6A, t h e change i n yawing moment c o e f f i c i e n t w i t h v a r i a t i o n in a i l e r o n d e f l e c t i o n , r e s u l t s from t h e d i f f e r e n c e betweendrag o nt h e up and down a i l e r o n s .S i n c e a p o s i t i v ed e f l e c t i o n i s w i t h t h e a i l e r o n onthe. l e f t wing down, Cn6A i su s u a l l yn e g a t i v e ,e v e nt h o u g h it i s h e a v i l y d e p e n d e n to nt h ep o s i t i o na n ds i z e o f t h e a i l e r o n s and t h e a n g l e o f a t t a c k o f t h ea i r f r a m e . A n e g a t i v ev a l u e f o r Cn6A i s known as"adverse yaw c o e f f i - c i e n t due t o a i l e r o n s " because it i s t h e r e s u l t o f i n i t i a l yawing o f t h e a i r - frame i n a d i r e c t i o no p p o s i t et h a td e s i r e d f o r a t u r n . Thus, t h ed e s i r e d v a l u e o f C i s e i t h e rz e r o or a v e r y s m a l l p o s i t i v e v a l u e .

"A To compute a v a l u e f o r Cn6A, a f o r m u l a ,a d a p t e dp a r t i a l l y from Datcom (Ref. 101, i s Here, 6~ i s a g a i n p o s i t i v e f o r l e f t a i l e r o n down and r i g h t a i l e r o n up, w i t h K b e i n g an e m p i r i c a lf a c t o ro b t a i n e df r o mt h ef o l l o w i n gg r a p h .

s p a n w i s ed i s t a n c ef r o mc e n t e r l i n et ot h ei n b o a r d edge o ft h ec o n t r o ls u r f a c e 'i ="- - semi span bw/2 K -25 -75 1 . 0 F i g u r e 43. E m p i r i c a lf a c t o r K as a f u n c t i o n of T I f o r t a p e r r a t i o = 0.5.

" 3 -.1 K " 1 .25 .5 1 .o F i g u r e 44. E m p i r i c a lf a c t o r K as a f u n c t i o no f Q f o r t a p e r r a t i o = 1.0.

T y p i c a lv a l u e so f Cn6A range from -0.004 t o -0.09 p e rr a d i a n .

The s t a b i l i t y d e r i v a t i v e CysR i s t h e change i n s i d e f o r c e r e s u l t i n g f r o m r u d d e r d e f I e c t i o n . F o r a p o s i t I v e r u d d e r d e f I e c t ion, o r r u d d e r t o w a r d t h e l e f t wing, a p o s i t i v e s i d e f o r c e r e s u l t s ; hence, t h ev a l u e of CysR i s p o s i t i v e .

Foran a i r p l a n e . w i t h o u t a u t o p i l o - f , t h e e f f e c t of CYsR i s r e l a t i v e l y unim- p o r t a n t t o l a t e r a l s t a b i l i t y and o f t e n i s assumed equal t o zero.

i s s e t f o r t h : I n E t k i n (Ref. 501, t h ef o l l o w i n gf o r m u l a f o r e s t i m a t i n g where a , = l i f t c u r v e s l o p e o f t h e v e r t i c a l t a i l ( c a l c u l a t e da s shown i n d i s c u s s i o n o f Cng), T = a f u n c t i o n o f r u d d e r a r e a t o v e r t i c a l t a i l a r e a r a t i o a s f o u n d f r o m t h e g r a p h below.

F i g u r e 4 5 . V a l u e s f o r T as a f u n c t i o n o f r u d d e r a r e a t o v e r t i c a l t a i l a r e ar a t i o .

T y p i c a lv a l u e s of CYsR rangefrom .12 p e rr a d i a nt o .24 p e rr a d i a n .

The s t a b i l i t y d e r i v a t i v e Cg,6R i s t h e v a r i a t i o n i n r o l l i n g moment c o e f f i c i e n t w i t h change i nr u d d e rd e f l e c t i o n . Because t h er u d d e ri sn o r m a l l yl o c a t e da b o v e t h ex - a x i s , a p o s i t i v e r u d d e r d e f l e c t i o n ( r u d d e r t o t h e l e f t ) c a u s e s a p o s i t i v e r o l l i n g moment, making C Q ~ p o s i t i v e .T h i sv a l u e may p o s s i b l y be n e g a t i v ef o r an u n u s u a la i r f r a m ec o n f i g u r a t i o no r an abnormal angle of attack.Forconven- t i o n a ll i g h ta i r c r a f t ,t h i sd e r i v a t i v ei so fo n l ym i n o ri m p o r t a n c e and i s u s u a l l yn e g l e c t e d .

The f o l l o w i n gf o r m u l aa d a p t e df r o mE t k i n( R e f . 50) may be used t o determine CQR : where zv = d i s t a n c ef r o mt h ex - a x i st oa e r o d y a n m i c c e n t e r o f t h e v e r t i c a I t a i I The v a l u e o f T may be obtainedfromthegraphbelowas a f u n c t i o n o f r u d d e r a r e a t o v e r t i c a I t a i I a r e a r a t i o .

.6 I - - 4 . 2 0 .1 .2 .3 .4 .5 .6 .7 F i g u r e 46 V a l u e s f o r T as a f u n c t i o n o f r u d d e r a r e a t o v e r t i c a I t a i I a r e a r a t i o .

The s t a b i l i t y d e r i v a t i v e Cn6R i s t h e v a r i a t i o n i n yawing moment c o e f f i c i e n t w i t h a change i nr u d d e rd e f l e c t i o n .A l s o known a st h e" r u d d e r power," t h i s d e r i v a t i v ei sn e g a t i v e ,s i n c e a p o s i t i v er u d d e rd e f l e c t i o nt o w a r dt h el e f t w i n gc r e a t e s a n e g a t i v ey a w i n g moment.

P e r k i n s and Hage (Ref. 1 1 1 g i v e s v Rv C = - a v ~ - - n V

"6R s w bw

The v a l u e of T may be o b t a i n e df r o mt h ef o l l o w i n gg r a p hf o r a p a r t i c u l a r r u d d e r area t o v e r t i c a l t a i I a r e a r a t i o .

F i g u r e 4 7 . Values f o r T as a f u n c t i o no f rudderarea t o v e r t i c a l t a i I a r e ar a t i o .

The v a l u e o f Cn6R i s n o r m a l l y o n t h e o r d e r o f - . 0 6 p e r r a d i a n b u t may v a r yg r e a t l y ,d e p e n d i n go nt h ea i r f r a m ec o n f i g u r a t i o n . Power has a g r e a te f f e c t on Cn6R, as seen i nt h ef o l l o w i n gg r a p h from TR-781 (Ref. 5 1 ) f o r a s i n g l e e n g i n ep l a n ew i t hp r o p e l l e rr o t a t i n g t o t h e r i g h t , f l y i n g a t a speed o f 1 1 1 m i l e sp e rh o u r .

-04 -0 2 Cnlfr: -.0012 Normal Power Cn "02 : "0023 Full Power '"6 r Flap+Gear Down ' I I -.04- both per degree

-20 - 10 10 20

b o

F i g u r e 48 . E f f e c t o f a i r c r a f t poweron Cn6R.

POWER EFFECTS

A l t h o u g ht h ep r i m a r yf u n c t i o n of a p r o p u l s i o ns y s t e m i s t o o v e r c o m ea i r p l a n e d r a g ,t h el o c a t i o n of t h es y s t e mw i t hr e s p e c t t o aerodynamicsurfaces may i n - f l u e n c e some aerodynamic parameters and, u l t i m a t e l y , a i r c r a f t s t a b i l i t y . Thus, aswingloadingsincrease,morepronouncedpowereffectsare t o beexpected.

The l i t e r a t u r ei n d i c a t e st h a ta n a l y t i c a ld e t e r m i n a t i o n of power e f f e c t s f o r l i g h t a i r c r a f t i s , a t b e s t , anapproximation. Many r e f e r e n c e ss u g g e s to n l yt h e o r i g i n of a p a r t i c u l a r e f f e c t and e s t i m a t e o n l y i t s o r d e r of magnitude. The f o l l o w i n g k i l l d i s c u s s t h e s i g n i f i c a n t r e p o r t s d e a l i n g w i t h power, t h ec h a r a c t - e r i s t i c changes t o b ee x p e c t e di nl o n g i t u d i n a la n dl a t e r a ls t a b i l i t y due t o p o w e ra p p l i c a t i o n , and t h ea n a l y t i c a lp r o c e d u r eg i v e ni n Datcom (Ref. IO) f o r e s t i m a t i n g t h e e f f e c t s of p o w e ro nl o n g i t u d i n a ls t a b i l i t y .

Power e f f e c t s f o r l i g h t a i r c r a f t c a nu s u a l l yb ec l a s s i f i e da s I ) d i r e c t p r o - p e l l e r e f f e c t s and 2 ) s l i p s t r e a m e f f e c t s , w i t h more a c c u r a t e e s t i m a t i o n p o s s i b l e f o r p r o p e l l e rt h a n f o r s l i p s t r e a me f f e c t s .S i n c et h ee n g i n ei sd i r e c t l yi n f r o n t o f t h e h o r i z o n t a l and v e r t i c a l t a i l s , t h e s l i p s t r e a m e f f e c t s o n s i n g l e e n g i n e a i r c r a f t may bemore d i f f i c u l t t o p r e d i c t t h a n t h o s e o n t w i n or m u l t i - e n g i n ea i r c r a f t ,e s p e c i a l l yi nt h el a t e r a l mode. The l a c k of documentation on ways t o p r e d i c t a n a l y t i c a l l y t h e e f f e c t s o f power on l a t e r a l s t a b i l i t y b e a r s o u t t h i s c o n c l u s i o n .

The need f o r h i g h e r - p o w e r e da i r c r a f td u r i n gW o r l d War I I l e d t o c o n s i d e r - a b l ei n t e r e s ti ne f f e c t s of power on a i r c r a f ts t a b i l i t y . W R L-710 (Ref. 102) i n d i c a t e s t h a t power has a s i g n i f i c a n ti n f l u e n c eo nb o t ht h el o n g i t u d i n a l and l a t e r a ls t a b i l i t y and c o n t r o lc h a r a c t e r i s t i c s .U s i n gt h i sr e p o r t , one concludes t h a t power a p p l i e dt os i n g l e - e n g i n e ,l o w - w i n gm o d e l sd e c r e a s e sl o n g i t u d i n a l s t a b i l i t y and e f f e c t i v ed i h e d r a l .D i r e c t i o n a ls t a b i l i t y and r u d d e r and e l e v a - t o re f f e c t i v e n e s s w e r eu s u a l l yi n c r e a s e db ya p p l i c a t i o n s of power. A t t h e same t i m e ,t h er e p o r tw a r n st h a t power-onwindtunneltestsshouldbeused t o p r e d i c t a c t u a l f l i g h t s t a b i l i t y a n dc o n t r o l ,l e s tt h er e s e a r c h e r be m i s l e db y t h e o r e t i ca I data.

NACA TR-690 (Ref.103) i s a r e p o r t of t h e e f f e c t s of p r o p e l l e ro p e r a t i o n o nt h ef l o wa b o u te i g h tw i n dt u n n e lm o d e l s . A t a b u l a t i o n o f downwash angles, t h ed y n a m i cp r e s s u r e s a t t h e t a i l , and t h ep i t c h i n g - m o m e n tc o n t r i b u t i o no ft h e p r o p e l l e r and t h e w i n g i s p r e s e n t e d . TR-941 ( R e f . 1 0 4 ) c o r r e l a t e s some p e r t i - n e n te x p e r i m e n t a ld a t ao np o w e re f f e c t so b t a i n e dd u r i n gt h e War Years. I t g i v e ss e m i e m p i r i c a lp r o c e d u r e s t o p r e d i c t power-on l o n g i t u d i n a ls t a b i l i t yc h a r - a c t e r i s t i c sw i t hf l a p su n d e f l e c t e d , a n da g r e e sw e l lw i t he x p e r i m e n t a ld a t a .

I n two Technical Notes, 1339 (Ref. 105) and 1379 ( R e f . 1061, Hagerman d i s - c u s s e st h ee f f e c t so fp o w e ro nt h el o n g i t u d i n a l and l a t e r a l s t a b i l i t y of a s i n g l e - e n g i n e , h i g h - w i n g a i r p l a n e m o d e l . He f o u n d t h a t , f o r l o n g i t u d i n a ls t a - b i l i t y , power g r e a t l yi n c r e a s e dt h e l i f t increments and t h e t a i l - o f f l i f t c u r v es l o p ew h i l e ,i ng e n e r a l , it d e c r e a s e d t h e s t a b i l i t y o f t h e model f o r a l l t h r e ef l a pc o n f i g u r a t i o n st e s t e d .F o rl a t e r a ls t a b i l i t y ,a p p l i c a t i o n of power hadno e f f e c t o n t h e e f f e c t i v e d i h e d r a l , w i t h t h e f l a p n e u t r a l ; however, w i t h b o t hs i n g l ea n dd o u b l es l o t t e df l a p sd e f l e c t e d ,p o w e ri n c r e a s e dt h ee f f e c t i v e d i h e d r a l . The d i r e c t i o n a ls t a b i l i t y of t h ee n t i r e model was i n c r e a s e d e x c e p t w i t hf l a p sn e u t r a la t low l i f t c o e f f i c i e n t s .R u d d e re f f e c t i v e n e s s was decreased w i t hf l a p sn e u t r a la n dd o u b l es l o t t e df l a p sd e f l e c t e d and i n c r e a s e dw i t hs i n g l e s l o t t e df l a p sd e f l e c t e d .T r i mc h a n g e sc a u s e db yp o w e rw e r es m a l l ,i n d i c a t i n g good c o n t r o l . TN 1327 (Ref. 107) o nl a t e r a ls t a b i l i t y ,w r i t t e na b o u tt h e same t i m ea s Hagerman's work, r e v e a l st h a tp o w e rd e c r e a s e dt h ed i h e d r a le f f e c tr e - g a r d l e s s o f t h ef l a pc o n d i t i o n ,i n c r e a s e dt h ed i r e c t i o n a ls t a b i l i t y ,a n di n - c r e a s e do v e r a l ll a t e r a ls t a b i l i t ya s lift c o e f f i c i e n t was increased.

The p r o b l e m of power e f f e c t s a l s o j u s t i f i e d t h e i n v e s t i g a t i o n u n d e r t a k e n i n TN I474 (Ref. 1081, whichsought t o o f f - s e t power e f f e c t s 'by u s i n ga n unsymmet- r i c a l t a i l o nt h es i n g l e - e n g i n ea i r p l a n e .A l t h o u g ht h et e s t sa n da n a l y s e s showed t h a t e x t r e m e asymmetry i n t h e h o r i z o n t a I t a i I i nd i c a t e d a r e d u c t i o n i n power e f f e c t so nt h el o n g i t u d i n a ls t a b i l i t y ,t h e" p r a c t i c a l "a r r a n g e m e n tt e s t e d d i dn o t show m a r k e di m p r o v e m e n t .T h r e ey e a r sa f t e rt h ea s y m m e t r i ci n v e s t i g a - t i o n , a d y n a m i cf r e e - f l i g h ts t u d y o f d y n a m i cl o n g i t u d i n a ls t a b i l i t ya si n f l u - enced by s t a t i c s t a b i l i t y measured i nw i n d - t u n n e lf o r c et e s t su n d e rc o n d i t i o n s of c o n s t a n t t h r u s t a n d c o n s t a n t p o w e r was undertaken. (Ref. 109) The r e s u l t s a g r e e dw i t hp r e v i o u ss t u d i e st h a tt h el o n g i t u d i n a l" s t e a d i n e s s " of a i r p l a n e s i s a f f e c t e d t o a much g r e a t e r e x t e n t bychanges i n c o n s t a n t - t h r u s t s t a t i c m a r g i n t h a n by changes i nc o n s t a n t - p o w e rs t a t i cm a r g i n .

TN D-3726 (Ref. 1101, a r e c e n tr e p o r td i r e c t l yr e l a t e dt ol i g h ta i r c r a f t , d i s c u s s e st h ee f f e c t so fp o w e r on t h el a n d i n gc o n f i g u r a t i o ns t i c k - f i x e da n d s t i ' c k - f r e e s t a t i c l o n g i t u d i n a l s t a b i I i t y f o r t h e a i r c r a f t ( i n c l u d i n g b o t h t w i n and s i n g l ee n g i n ea i r c r a f t )w i t ht h em o s tp r o n o u n c e dp o w e re f f e c t s . When t h e power was c y c l e d from approach t o maxi mum a t an a i r s p e e d o f 80 k n o t s t h e p i l o t had t o push w i t h a f o r c eo fa p p r o x i m a t e l ye i g h t pounds t o c o u n t e r t h e r e s u l t i n g nose-up p i t c h .T h i sc h a r a c t e r i s t i ca l s op r e s e n t e d a problem when thepower was b e i n g r e d u c e d i n t h e l a n d i n g p h a s e . F o r l i g h t a i r c r a f t , p o w e r e f f e c t s c a u s e d b yp r o p e ll e r s I i p s t r e a m can become q u i t el a r g e . A t a speed o f I IO knots, ap- p r o x i m a t e l y IO degrees of r u d d e r and 90 pounds o f f o r c e was r e q u i r e d t o m a i n t a i n heading when changingpowerfrom maximum t o i d l e .T h i s was c o n s i d e r e de x c e s s i v e by t h e p i l o t .

Anotherrecentpaper(Ref. I I I ) on l i g h t a i r c r a f t g i v e s methods f o ra n a l y z - i n g power-on s t a t i cl o n g i t u d i n a ls t a b i l i t y . The m e t h o d sa r es i m il a r t ot h o s e g i v e ni n Datcom (Ref. I O ) ; however, t h ep a p e ra l s od e s c r i b e s how power e f f e c t s c a nb ea n a l y z e du s i n ga i r p l a n es t i c kf o r c e ,e l e v a t o rd e f l e c t i o n , and n e u t r a l p o i n t s . The method u t i l i z e s a p o i n t - t o - p o i n tc a l c u l a t i o nt e c h n i q u e f o r each speed or l o a df a c t o rv a r i a t i o n from trim.

I n 1969 and 1970, NASA i n v e s t i g a t e dl o n g i t u d i n a la n dl a t e r a ls t a b il i t yc h a r - a c t e r i s t i c s of b o t h a l i g h t s i n g l e e n g i n e and a l i g h t t w i n e n g i n e a i r c r a f t i n a f u l I s c a l e t u n n e l ( R e f s . 112 and 113). These i n v e s t i g a t i o n sw e r ec a r r i e do u t f o r severa I c o n d i t i o n s of power, as i n d ic a t e db yt h ee x t e n s i v ed a t ai ne a c hr e - p o r t . These two r e p o r t s may be used e i t h e r t o o b t a i n a r o u g he s t i m a t eo r t o h e l pv e r i f yt h ee x a c t n e s so fa n a l y t i c a lp r o c e d u r e sf o re s t i m a t i n gp o w e re f f e c t s on a i r c r a f t s i m i l a r t o t h o s e i n v e s t i g a t e d .

I i t e r a t u r e m e n t i o n e d above may g i v e t h e g e n e r a l t r e n d s f o r power A l t h o u g ht h e e f f e c t s . ,a n a l y t i c a lt e c h n i q u e sa r eo f t e nh a r d t o fi,nd, cumbersome t o use, and i n a c c u r a t e . F o r t h e p r e s e n t , Datcom (Ref. IO> p r o b a b l yg i v e st h eb e s ta n a l y t i - c a lp r o c e d u r e s f o r p r e d i c t i n g p o w e re f f e c t s . They a r ee x t e n s i o n s from t h o s e of P e r k i n s and Hage (Ref. I l l , among o t h e r s( R e f s . 114 and 1151, w i t h improvements added where p o s s i b l e .D a t c o m ' sm e t h o d so fe s t i m a t i n gp o w e re f f e c t so n l i f t and 1 1 1 I p i t c h i n g moment v a r i a t i o nw i t ha n g l eo fa t t a c ka r es u m m a r i z e db e l o w .C u r v e so f b o t h ACL and ACm versus a c a nb ep l o t t e d f o r s e v e r a l f l i g h t c o n d i t i o n s u s i n g theseprocedures;thus, C L ~ and Cma due t o powercanbeevaluated.Scrutiny of t h e p r e s e n t d i s c u s s i o n p o i n t s o u t t h a t t h e e s t i m a t i o n o f p o w e r e f f e c t s d e a l s o n l yw i t hs t a t i cl o n g i t u d i n a ls t a b i l i t y . I t would seem t h a t an a p p r o x i m a t i o n o f power e f f e c t s f o r t h ed y n a m i cd e r i v a t i v e sc o u l d b ea c h i e v e db ye s t i m a t i n g t h e change i n TI& w i t h power,computed a n a l y t i c a l l y i n a s i m i l a r manner t o t h a t g i v e n below, and u s i n gt h i sc o r r e c t e dv a l u e . Some o ft h em e t h o d sg i v e nc a na l - so be used t o e s t i m a t e l a t e r a l s t a b i l i t y d e r i v a t i v e changes w i t h power. F o r example,(C~,)p i s analagous t o (Cy Ip o r ( - C y g I Pf o rc r u i s i n gf l i g h t and c o u l d

be used t o e s t i m a t e o r (Cy6 Y p. When approached from the engineering

v i e w p o i n t ,s i m i l a r methods c o u l da l s o be used i nd e t e r m i n i n gs l i p s t r e a me f f e c t s on t h e v e r t i c a l t a i 1 .

A g a i n t h e r e a d e r s h o u l d remember t h a t ,a tb e s t ,a n a l y t i c a lp r e d i c t i o no f power e f f e c t s i s c r u d e .

L i f t Increment Due t o Propellet-Thrust ( A C L I T = nCT s i n a T where n = number o f eng i nes c t = t h r u s ta x i sa n g l eo fa t t a c kt of r e es t r e a mi nd e g r e e s T L i f t Increment Due t o ProDe I f e r NormalForce nN S

(acLlN = 2 cos a = n f ( c 1 ( a p ) ( $j cos a ( p e r r a d i a n )

T T N" p W P qsW where aP = a n g l eb e t w e e nl o c a la ir s t r e a m and t h r u s t i n degrees Sp = p r o p e l I e r d i s c a r e a f = p r o p e l l e r i n f l o w f a c t o r ( C 1 = p r o p e l l e rn o r m a l - f o r c ed e r i v a t i v ea t T ' = 0 p e rr a d i a n Na p C where a0 = w i n ga n g l e of a t t a c k f o r z e r o I i f t i n degrees

a €

A= wing upwash d e r i v a t i v eg i v e ni nF i g u r e 49.

a"

The f a c t o r f a c c o u n t s f o r t h e i n c r e a s e i n ve l o c i t y a t t h e p r o p e I l e r p I ane due t ot h ei n d u c e df l o wo ft h ep r o p e l l e r and i s g i v e n i n F i g u r e 50 as a f u n c t i o n of 'w 'T 8R P wherr R = p r o p e ll e rr a d i u si nf e e t .

P

( C 1 = [ ( C 1 3 [ 1 + 0 . 8 ( - KN -

11 ( p e r r a d i a n )

Na p Na KN=80.7 80.7 b b b ( p e r

where K , , , = 262 (8). 3Rp + 262 (8) + b I ade)

. 6Rp

35 (8). 9R

P P

P P

w i t h b = b l a d ew i d t h 'i nf e e t ' P

C(C 3 = p r o p e l l e r n o r m a l - f o r c e d e r i v a t i v e a t CT

Na K =80.7 a n d g i v e n i n F i g u r e 51 as a f u n c t i o n of N B , t h en o m i n a l b l a d e a n g l e a t 0.75 rad- i a n s ; t h i s b l a d e a n g l e s h o u I d b e ob- t a i n e d f r o m a performanceengineer o r from a power p I a n t e n g i n e e r .

0 2 4 6 8 10 12 1 4 16 1 8 20 22 SWC~/8R2 F i g u r e 50. P r o p e l l e r i n f l o w f a c t o r .

0 I d 20- 30. 40. sd 60' Nominal Blade Angle-@ at.75 radius F i g u r e 5 I . P r o p e ll e rn o r m a lf o r c ep a r a m e t e r .

L i f t ' Increment Due t o a Change i n S I i pstrearn'Dynarni c P r e s s u r e on t h e W i n g S .

where K1 = c o r r e c t i o np a r a m e t e rf o ra d d i t i o n a lw i n g l i f t due t o power and g i v e n i n F i g u r e 52, where ARi = e f f e c t i v ea s p e c tr a t i o of wing immersed i nt h es l i p s t r e a m , A i = 2Rp/ci , c i = average chord of w i ng immersed i n s I i pstream.

-

Ans - g = r a t i o of change i n dynamicpressure i n

p r o p e l l e r s l i p s t r e a m t o f r e e s t r e a m dynamicpressure and g i v e n by SwcT Ans = 7 ( p e r e n g i n e ) .

*RP S i = wingarea immersed i nt h es l i p s t r e a mi ns q u a r e f e e t .

F i g u r e 52. C o r r e l a t i o n p a r a m e t e r f o r a d d i t i o n a lw i n gl i f t due t o prope I l e r power.

I16 L j f t Due -fo t h e Change i n Ang le of _Attack -l.ndu-qed by. the Prope I l e r Flow Fie1 d S .

“E where Aa = 1- - aa a E U

a = aT - -(a -a 1

P aa w o The p r o p e l l e r downwash d e r i v a t i v e i s g i v e n by where C and C2 a r ep r e s e n t e di nt h ef i g u r eb e l o w and i n one o ft h ep r e v i o u ss e c t i o n s .

( C 1 I S g i v e n N a p 0 2 4 6 8 10 12 14 16 18 20

sw or/8 +

F i g u r e 53. F a c t o r sf o rd e t e r m i n i n g downwash due t op r o p e l l e r s .

Lift Increment Due to the Horizontal Tail

(AC = -(ACmIt (~t C

L t

where (ACm), = total change in pitching moment of the

horizontal tail due to power and can be calculated using

(ACmIt = (AC, 1 + (ACm IE

t q t given in the next section--pitching moment variations with power.

Total Lift Increment Due to Power Pitching Moment Increment Due to the Offset of the Thrust Axis from Oriain of Axis ZT (ACmIT = C - T c Pitching Moment Due to the Propeller Normai Force I h l (Acm)N = ( * ‘ L ) N c cos a

P P T

where x = distance from the intersection of the propel ler plane with the thrust axis to the wing quarter chord.

(ACLIN is evaluated in a previous section.

D Pitching Moment Due to the Change in the Lift of the Wing Caused bv Power Effect X

(AC m L = - [ , A , L Ans + (ACLlaw] $-

where x = distance parallel to x-axis from wing quarter- W chord to aerodynamic center o f wing area im- mersed in the sli~stream. in feet (AC 1 and (AC ) are eva’luated in a previous section.

L Ans aw P i t c h i n g Moment ". ~ ~ Due t o Change i n Dynamic Pressure Acting on the H o r i z o n t a I Ta i I s a t t

- -

(AC I = -C -

L t 9 sw =

where 7 i s p r e s e n t e d i n t h e f i g u r e b e l o w .

p r e s s u r er a t i o F i g u r e 54. E f f e c t of p r o p e l l e r power on dynamic a t t h e h o r i z o n t a l t a i l .

P i t c h i n g Moment Due t o t h e Change i n Angle of A t t a c k of t h e H o r i z o n t a l T a i I St q t

(ACm 1 E = CL AE - -(-)

Sw c q power a t t where A E i s g i v e n i n F i g u r e 55 f o r s i n g l e e n g i n e a i r p l a n e s and F i g u r e 56 f o r mu1 t i - e n g i n e a i r p l a n e s . o f f can be obtained from a formu I a i n t h e CL s e c t i o n .

Aq t where -is e v a l u a t e d i n F i g u r e 5 4 above.

q 1 4 O 12O 10" y.

8 ' f

t

6 O Y) 4"

e = Downwash

O0 0 . 4 .8 12 F i g u r e 55. I n c r e m e n t i n downwash due t op r o p e l l e r power f o rs i n g l e - e n g i n ea i r p l a n e s .

Figure 56. Increment in downwash due to propeller power for multi-engine airplanes.

Total Change in Pitching Moment Due to the Propeller Power Effects

(ACm)power = (ACm.)T + (AC I + (ACmIL f (ACmtIq

N P

+ (ACmt)E

CONTROL FORCES AND DEFLECTIONS

P i t c h C o n t r o l As i n d i c a t e dp r e v i o u s l y ,t h es t a t e o f k n o w l e d g er e g a r d i n gd e s i r a b l ea i r - c r a f t h a n d l i n g q u a l i t i e s up t o 1948 i s c o d i f i e d i n NACA TR-927 (Ref. 16) and i n t h e t e x t by P e r k i n s and Hage (Ref. 1 1 ) . These works have served those of t h e p r e s e n tg e n e r a t i o no fa e r o n a u t i c a le n g i n e e r sw i t h o u ta c c e s s t o l a r g er e s e a r c h anddevelopmentbudgets v i r t u a l l y a s h o l y writ, and t h e h a n d l i n g q u a l i t i e s o f m o s t a i r c r a f t o f l e s s t h a n 10,000 pounds g r o s sw e i g h t now f l y i n g r e f l e c t t h i s t e c h n o l o g y . I t i st h e r e f o r ed e s i r a b l et o examine t h ep a r a m e t e r sc i t e di nt h e s e works, t h e i rv a l u e s , and t h ef a c t o r sw h i c hp r o d u c et h e mi n some d e t a i l a s a f o u n d a t i o nf o rr e c e n ta d v a n c e si nu n d e r s t a n d i n g and new r e s u l t s .

Foran a i r c r a f tw h i c he m p l o y s a trim t a b t o s e t l o n g i t u d i n a l f l i g h t v e l o c i t y , it can be shown t h a t a t anygiventrimmedspeedthe i n c r e m e n t a l s t i c k f o r c e v a r i a t i o n w i t h speed i s g i v e n by P e r k i n s and Hage (Ref. 1 1 ) as where G = r a t i o e l e v a t o r d i s p l a c e m e n t t o t h e p r o d u c t o f s t i c k l e n g t h and s t i c ka n g u l a rd i s p l a c e m e n t , and S i n c e and t h e n A p u l lf o r c e i sn e g a t i v e . I t i s seen t h a tf o r a g i v e na i r c r a f t ,t h es t i c kf o r c e g r a d i e n t a t t rim depends upon t h e trim speed, t h ew e i g h t , and t h e c . g .l o c a t i o n .

S h o r t of m a j o rg e o m e t r i cm o d i f i c a t i o n s ,t h ef o r c eg r a d i e n t s f o r a g i v e n speed, weight, and c.g. l o c a t i o nc a n be changed by m o d i f y i n g G and t h eh i n g e moment parameters, C h , and Chg. The e f f e c to fe l e v a t o rb a l a n c ep o i n t ,e l e v a t o r nose shape, e l e v a t o r t r a i l i n g - e d g e shape, mass b a l a n c i n g and s e a l i n g o n Cha and Ch6 a r et r e a t e de x t e n s i v e l ' yI n TR-868 (Ref, 521, F i g u r e s 57, 5 8 , 5 9 , 6 0 , and 61 i l l u s t r a t i n g t h e s e e f f e c t s a r e t a k e n from t h i s work, A g u i d e f o r u s l n gt h e s e c h a r t sf o l . l o w sF i . q u r e 61, A0 .36 .32 .2a .24 b F l 4

4, .2a

.16 .12 .oa .w F i g u r e 5 7 . C h a r t sf o rd e t e r m i n i n gn u m e r i c a lv a l u e s o fo v e r h a r gf a c t o r F1 and o f nose-shape f a c t o r F2 f r o mg e o m e t r i cc o n s t a n t s o f b a l a n c e da i l e r o n s .

'I

# Nose shape factor, F2 Nose Section showing nose shape I type k c b -I c o r r e l a t i o n o f pIain,overhang,and Figure 58a. Various nose shapes considered in F r i se ba I ances and correspond i ng e x p r e s s i o n s f o r nose-shape f a c t o r .

1 24 I ? - .60 .so .4 0

-

'b

-

-

Ca .30 .2 0 . l o .6 0

.s 0

.4 0

-

'b

-

-

3 0 .20 .10 0 .04 .O 8 12 .16 0 .04 .OS .12 .16

4 F 2 4 F 2

F i g u r e 58b. C h a r t s f o r e s t i m a t i n gt h er e q u i r e dl e n g t h s of overhangs having nose shapes. L e t t e r s A,B,C, and D r e f e r t o t h e c o r r e s - v a r i o u s shapes o f F i g u r e 58a.

pond ingnose 0 4 8 12 16 20 24 28 32 36 40 Trailing-edge angle, 4, deg.

F i g u r e 5 9 . Hinge moment parameters o f p l a i n 2-0 a i l e r o n sw i t hv a r i o u sc h o r d s and v a l u e s of 9. Gaps s e a l e d ;c a / c = a i l e r o nt ow i n gc h o r dr a t i o .

1 26 .OO!

.004 .003 .OOi ch, .001 ,001 -001 -.007 -.005 -.003 -.001 .001 .003 .005 9.6' A I RPLANECONSTANTS

Wing span, b , f t . . . . . . . . . . . . . . . . 43

Wing area, S , sq f t . . . . . . . . . . . . . 308

A s p e c t r a t i o , AR. . . . . . . . . . . . . . . 6.0

T a p e r r a t i o , X . . . . . . . . . . . . . . . .5

Root a i r f o i ls e c t i o n . . . . . . . . . . NACA 23015

T i pa i r f o i ls e c t i o n . . . . . . . . . . NACA 23009

A i r p l a n e w e i g h t , I b s . . . . . . . . . . . . 12,000

S t i c k length, f t . . . . . . . . . . . . . . . 2.33

F i g u r e 60, E f f e c t s of aerodynamic balances on aileron hinge-moment para- m e t e r se s t i m a t e d from c o r r e l a t i o n s . c- = 0.25.

I 0 .002 .004 .006 .008 . 0 1 0 . 0 1 2 . 0 1 4 . 0 1 6 . 0 1 8 .020 0 .o 2 .04 .06 .OS .12 .14 .16 .lS .20 A+2 F i g u r e 6 1 a . E f f e c t o f s e a l e d i n t e r n a l b a l a n c e s o nt h e hinge-momentparameters of c o n t r o l s u r f a c e s . M=0.2 or less.

4 8 12 16 20 24 28 32 Trailing-edge angle (deg) F i g u r e 61b. E f f e c t o f gap on hinge-moment v a r i a t i o n w i t h t r a i l i n g edge angle.

NACA 0009 a i r f o i l ; 2-D model, 2 =0.30.

C The g e n e r a ie x p r e s s i o n sf o rc h a n g e si nt h eh i n g e moment c h a r a c t e r i s t i c s which resu I t f rom geometric changes a r e

AC = & [O.O17F, + 5.05 x A@] 9

ha AR+2 and

- - AR [o. l o F 1F2

- + 5.7 X A@] .

Ach6 AR+2 The t r a i l i n g edge a n g l ed a t aa r e f o r b a l a n c e sw i t h a gap o f .005E.

One f i r s t s e l e c t s a nose shape from F i g u r e 58 and s e c u r e st h e r e f r o m a v a l u e o f F 2 ' . T h i s i n v o l v e s , i n a d d i t i o n , a s e l e c t i o n o f c b / c q . The symbols Mg, MB, Mc, MD, ME, and M F r e f e r t o moments a b o u tt h eh i n g ea x t so ft h ep r o - f i l e a r e a s o f exposedoverhangbalances of t y p e s c o r r e s p o n d i n g t o t h e sub- s c r i p t s 0, 6, C, and so f o r t h . T h eb a l a n c ep r o f i l ea r e ai sd e f i n e da st h e t o t a lp r o f i l ea r e a o f t h e a i r f o i l ahead o ft h eh i n g ea x i s . F2=F2' f o r nose shapes 0, A, 6, D, and G. F2 can a l s o be found from F i g u r e 57. The same f i g u r e i s used t o d e t e r m i n e F1. One t h e nc a l c u l a t e s F1F2' and uses F i g u r e 61b o r t h ee q u a t i o n sa b o v e to c h e c kw h e t h e rt h ei n i t i a ls e l e c t i o no fn o s e shape andnoseoverhang will g i v e t h e d e s i r e d m o d i f i c a t i o n i n h i n g e moment c h a r a c t - e r i s t i c s .A d d i t i o n a lm o d i f i c a t i o n si nc h and Cb c a n t h e n b e o b t a i n e d from t r a i l i n g e d g ea n g l ev a r i a t i o n sa si n d i c a t g di nF t a u r e s 59 and 61a o r t h e equationsabove.

F o rs e a l e db a l a n c e s ,t h ee q u a t i o n sa r e and Theincrementsrepresentedbytheseequationsare t o be added t o t h e h i n g e moment c o e f f i c i e n t sp r o d u c e d by c o n t r o ls u r f a c e sw i t h noareaforward of t h eh i n g el i n e .T h er e s u l t s of c a l c u l a t i o n s f o r t h eb a s i ch i n g e moment c o e f f i c i e n t s of t h i n ,s i m p l es h a p e s shown i n TR-868 (Ref. 5 2 ) canberepre- sentedby Ca Ca Cha - ,0003 c - < 0.4 , C and I t shouldbenoted,however, t h a tt h e s ev a l u e sw i l lb ea l t e r e d t o some ex- t e n t by t h e c o n d i t i o n of t h e flow o v e rt h eb a s i cw i n g( c o n d i t i o n of boundary layer,presence of t i p v o r t i c e s , e t c . ) and by t h e p a r t i c u l a r shape of t h e c o n t r o ls u r f a c e .F o rt h i sr e a s o n ,p r e c i s ev a l u e sa r eu s u a l l yo b t a i n e de x - p e r i m e n t a l l y .F u r t h e rd e t a i l s may be found i nR e f e r e n c e 52.

One f i n a lc o n s i d e r a t i o ns h o u l d be mentioned here: To reduce drag, i n e r t i a l l y - i n d u c e dc o n t r o ls u r f a c ed e f l e c t i o n ,t h ep o s s i b i l i t y of f l u t t e r , and e f f e c t s of c o n t r o ls u r f a c ed r o o p , and t h ec o n t r o ls u r f a c ea c t u a t i o n f o r c e ,t h ec o n t r o ls u r f a c e i s a l s o mass b a l a n c e da b o u tt h eh i n g el i n e .T h i s will o f t e nr e q u i r et h a tl e a dw e i g h t s beplaced i n t h e nose of t h ea e r o - dynamicbalance or a t t a c h e d bylongarms t o t h eh i n g ea x i s .

R e p o r t 927 ( R e f .1 6 )i l l u s t r a t e ss u i t a b l ev a l u e s f o r Cha and Ch6 f o r a t y p i c a l example. T h i s i s r e p r o d u c e d a s F i g u r e 62. I n g e n e r a l , o n e d e s i r e s t o keep t h ev a l u e so f Cha and Ch6 s m a l l .T h i s will a l s o keep t h e s t i c k f o r c e s t o reasonablevalues f o r h i g h speeds o r heavyweights.

Some of t h e means f o r a l . t e r i n g t h e e f f e c t i v e G m e c h a n i c a l l y( s p r i n g s and bob w e i g h t s )a r ed i s c u s s e db yP e r k i n sa n d Hage (Ref. 1 1 ) . F u l l y powered con- t r o l systems, of course,canuseother means i f necessary t o p r o d u c et h e d e s i r e d f e e l . (See, f o r example, TN D-632 (Ref. 531.1 F o r m e r l y ,c o n s i d e r a b l ee f f o r t was e x p e n d e dt o w a r ds p e c i f y i n ga c c e p t a b l e values f o r dFs/dU, b u tr e c e n t l ys p e c i f i c a t i o nw r i t e r s( b o t hi n FAR p a r t 23 (Ref. 54) and i n MIL-F-8785B (Ref. 4) 1 h a v ea s k e do n l yt h a t dFs/dU b es t a b l e P a t a l l f l i g h t c o n d i t i o n s , e.g., t h a ti n c r e a s i n gp u l lf o r c e s and a f t m o t i o n o f t h e e l e v a t o r c o n t r o l be r e q u i r e dt o , , m a i n t a i nl o w e rt h a n trim airspeed.

There i s no r e q u i r e m e n tt h a t dF,/dU be l i n e a r . * I n a d d i t i o n t o c a l l i n g for stab1.eforce-speedgradients,both FAR p a r t 23 and MIL-F-8785B s p e c i f y The maximum o u t - o f - t r i m f o r c e s a n a i r c r a f t s h o u l d r e q u i r e i n a v a r i e t y o f f l i g h t c o n d i t i o n s . The t a b l eb e l o w summarizes t h e s e maximum f o r c e s and c o n d i t i o n s .

Stable ngim Neutral stick-free stability C Positive values of C h i . not used hat because of unstable short- period -.w5 oscillations with stick free stability for static margin of .05c Unstable region

I

-. 010 -

-.015 -.010 -. 005 0 .005

F i g u r e 62. Boundary between s t a b l e and u n s t a b l ev a l u e s o f c h and C f o r t h e example g i v e n i n TR-927. hb, U n s t a b l e s i d e of boundaries i nd i c a t e d by c r o s s - h a t c h i n g .

* Indeed, dFg/dU i s d i r e c t l y p r o p o r t i o n a l t o U away from trim. Other

c a u s e s o f n o n - l i n e a r f o r c e v a r i a t i o n s i n c l u d e t h e f a c t t h a t a l l t h e d e r - i v a t i v e si nt h ee q u a t i o na r ee v a l u a t e da ts p e c i f i cp o i n t so n l y . A t h i g h v a l u e s of CL, f o r example, i s l e s s t h a n for CL near 0. Thus, one s h o u l d t a k e c a r e t h a t t h e v a t i v e v a l u e s used f o r c a l c u l a t i o n s a r e t h o s ev a l i df o rt h er a n g eo fa n g l e sb e i n gc o n s i d e r e d .

1.5 U s t a l lw i t h power o f f and gear and f l a p s 10 I b s FAR 23.145 down a t most forwardc.g.

Any c r u i s e speed between 1.3Us and Umax; Approach speeds 40 I b s FAR 23.175 between 1.1Us and 1.8Us Maximum f o r c ef o rp r o - l o n g e d a p p l i c a t i o n ; 10 I b s FAR 23.143 T e m p o r a r y a p p l i c a t i o n S t i c k 60 I b s Whee I 75 I b s Ta ke-of f 20 I b s p u l l t o MI L-F-8785B 10 Ibs push § 3.2.3.3.2 Land i ng Must be a p u l l MI L-F-8785B f o r c e o f n o t more § 3.2.3.4.1 t h a n 3 5 I b s ~~ D i v e s w i t h a i r c r a f t < 50 Ibs push

1 s t i c k

t r i m m e d f o r l e v e l f l i g h t < 10 I b s p u l l < 75 I b s push < 15, I bs pu I I ] M t L-F-87855 5 3.2.3.5 w i t h trim a t d i v e e n t r y < 10 fbs s t i c k < 20 f b s wheel 1 1 . S p e c i f i c a t i o nr e q u i r e m e n t s f o r maximum o u t - o f - t r i mf o r c e s .

T a b l e T h eg e n e r a le x p r e s s i o nf o rs t i c kf o r c ei sg i v e ni nP e r k i n s and Hage (Ref. 1 1 ) as where a, = a i r c r a f t a n g l e o f a t t a c k f o r z e r o l i f t , iw = w i n g i n c i d e n c e a n g l e , i e = t a i lp l a n ei n c i d e n c ea n g l e , Cho = r e s i d u a lh i n g e moment c o e f f i c i e n t ( h i n g e moment n o t c a u s e d b y d e f l e c t i o n or a n g l e o f a t t a c k ) , &eo = e l e v a t o r a n g l e a t z e r o a i r c r a f t l i f t , 8, = trim t a bd e f l e c t i o n .

N o t et h a tf o rt r i m m e df l i g h t , 6 t i s such t h a t Fs = 0. F o ra i r c r a f tw i t h i r r e v e r s i b l el o n g i t u d i n a lc o n t r o ls y s t e m s ,F si st h ef o r c ew h i c ht h ec o n t r o l systemmustapply t o t h e e l e v a t o r o r s t a b i l a t o r , b u t t h e f o r c e w h i c h t h e p i l o t f e e l si sd e t e r m i n e d by c o n t r o ls y s t e md e s i g n .

The o n l ym e n t i o nf o u n di nt h el i t e r a t u r e f o r d e s i r a b l ec o n t r o ld e f l e c t i o n g r a d i e n t si st h er e q u i r e m e n ti n MIL-F-8785B 93.2.2.2.2 (Ref. 4) t h a t t h e p i l o t a p p l yn o tl e s st h a n 5 I b sf o r c e f o r e a c hi n c ho fs t i c kt r a v e l .

A n y t h i n gw h i c hc h a n g e st h ef l o wf i e l di nt h en e i g h b o r h o o do ft h eh o r i z o n t a l c o n t r o ls u r f a c ec a nh a v ea ne f f e c to ni t sa c t u a t i n gf o r c e s . Such changes can be t h e r e s u l t o f a p p l i c a t i o n s o f p o w e r( w h i c hr e s u l ti ni n c r e a s e si n nt and c o n t r i b u t i o n s t o t h e downwash f i e l d f r o m t h e p r o p e l l e r s l i p s t r e a m ) or a l t e r a t i o n s i n t h e l i f t c o n f i g u r a t i o n , a sw i t hd e f l e c t i o no ff l a p s( w h i c ha l s or e s u l ti n a l t e r e d downwash f i e l d s ) . The change i n w i n gp i t c h i n g moment accompanying t h e d e f l e c t i o n o f f l a p s v a r i e s t h e t r i m m i n g moment w h i c h t h e t a i l i s r e q u i r e d t o g e n e r a t e a n d t h u s c h a n g e s t h e s t i c k f o r c e o r t h e trim t a b s e t t i n g r e q u i r e d .

A d d i t i o n a l moments a s s o c i a t e d w i t h t h e a p p l i c a t i o n o f powerdevelop when t h e t h r u s t does n o ta c tt h r o u g ht h ec . 9 . and because p r o p e l l e r sg e n e r a t ei n - p l a n e f o r c e si n upwash f i e l d s . Some o ft h e s ee f f e c t st e n dt or e d u c es t i c kf o r c e s and g r a d i e n t s ;o t h e r st e n d t o i n c r e a s e them; none o f themareeasy t o e s t i m a t e a c c u r a t e l yf r o mt h e o r e t i c a lc o n s i d e r a t i o n sa l o n e . Wind t u n n e lo rf l i g h tt e s t i n g i sn e c e s s a r yf o ra c c u r a t ee v a l u a t i o n .F o rm o s ta i r c r a f t ,t h ea p p l i c a t i o n o f power r e s u l t s i n a more p o s i t i v e v a l u e o f C m , and a r e d u c t i o ni nc o n t r o lf o r c e s .

The d i s c u s s i o ni nP e r k i n sa n d Hage (Ref. 1 1 ) enablesone t o a r r i v e a t some v e r ya p p r o x i m a t eq u a n t i t a t i v ev a l u e s f o r t h e s ee f f e c t s .T h ep r o b l e m o f u n d e s i r a b l e trim c h a n g e sw i t ha p p l i c a t i o n of power or e x t e n s i o n o f g e a r a n d / o r f l a p s i s a p p a r e n t l y a f a i r l y common o n e s i n c e s e v e r a l o f t h e l i g h t a i r c r a f t t e s t e d f o r TN D-3726 (Ref. 5 5 ) e v i d e n c e ds u b s t a n t i a ls h i f t si ns t i c kf o r c ew i t ht h ea p p l i - c a t i o n o f power o r t h e e x t e n s i o n o f f l a p s o r gear.

I n a d d i t i o n t o t h e f o r c e s r e q u i r e d t o change speed, t h e f o r c e s needed t o change f l i g h t d i r e c t i o n c o n t r i b u t e s i g n i f i c a n t l y t o t h e p i l o t ' s i m p r e s s i o n o f t h ev e h i c l e ' sh a n d l i n gq u a l i t i e s .I nt h el o n g i t u d i n a l mode, t h e s ea r eu s u a l l y s t a t e da st h es t i c kf o r c ep e r "g". The f o r c e sr e q u i r e dt o change d i r e c t i o na r eh i g h e rt h a n f-hose r e q u i r e d t o i n i t i a h a change i n speedalone a t t h e same dynamicpressurebecausean a d d i t i o n a l trim moment i s r e q u i r e d t o p i t c h t h e a i r c r a f t t o a na n g l eo fa t t a c k s u f f i c i e n t t o g e n e r a t et h ea d d i t i o n a l l i f t needed t o c u r v e t h e f l i g h t p a t h and because r o t a t i o n( u p - p i t c h i n g )i n d u c e s a p o s l t i v e a n g l e o f a t t a c k component i n t h ef l o wa b o u tt h et a i lp l a n e ,w h i c ho n em u s tc o u n t e rw i t h a compensatory e l e v a t o r d e f l e c t i o n .

To f i n d t h e e l e v a t o r s t i c k f o r c e f o r a n a r b i t r a r y maneuver, it i s necessary t os o l v et h eg e n e r a le q u a t i o n sf o rt h en o r m a la c c e l e r a t i o np r o d u c e d by t h a t e l e v a t o r m o t i o n w h i c h r e s u l t s f r o m t h e s p e c i f i e d a p p l i c a t i o n o f s t i c k f o r c e .

F o rs i m p l i c i t y and s t a n d a r d i z a t i o ni na n a l y s i s and i n f l i g h t t e s t s , however, maneuversdesigned t o e v a l u a t e s t i c k f o r c e p e r g a r e l i m i t e d t o s t e a d yp u l l - u p s i n t h e x z p l a n e o f i n e r t i a l space and t o s t e a d yt u r n si nt h ex yp l a n eo f i n e r t i a l space. The general expressions f o r s t i c kp e r g i nt h e s ec i r c u m s t a n c e s a r e - GntSeCe(W/S)Chg + " " -

( > P u I I -ups

chg C Sw '6 d6e cm6 N o t e t h a t t h e s t i c k f o r c e g r a d i e n t s w i l l be l i n e a ri np u l l - u p s so longas G, Cma, C b , Cha, Chg, (dat/d6,), (de/da), C k t , and n t r e t a i nt h e same v a l u e s a sf o rn = l .N o t ea l s ot h a ta st u r n st i g h t e n ,t h es t i c kf o r c ep e r g approaches t h e same v a l u ea s f o r p u l l - u p s .

On t h eq u e s t i o n of l i n e a r i t y , MIL-F-8785B (Ref. 4 ) s p e c i f i e s t h a t t h e l o c a lv a l u eo fd F s / d ns h a l ln o td i f f e r by more t h a n 50% from i t s averagevalue, The f o r c e l i m i t s i n maneuvers a r e s t a t e d d i f f e r e n t l y f o r s t i c k and wheel con- t r o l s , t h e r a t i o n a l e b e i n g t h a t s t i c k c o n t r o l l e r s a r e e a s i e r t o m a n i p u l a t e p r e - c i s e l y a t low f o r c e s and wheel c o n t r o l l e r s p e r m i t l a r g e r f o r c e s t o be a p p l i e d .

Not u n e x p e c t e d l y ,t h ef o r c el i m i t sa r es t a t e di nt e r m s of t h e limit l o a df a c t o r w h i c ht h es t r u c t u r ec a ns u s t a i n . One wouldnotwish t o be a b l e t o exceed t h i s l o a df a c t o rw i t h a v e r yl i g h tf o r c e ;n o r ,o nt h eo t h e r hand, wouldonewishthe s t i c k f o r c e s t o be so g r e a t t h a t t h e maneuver c a p a b i l i t i e s o f t h e c r a f t c o u l d * ch6 i s o f t e n i n c r e a s e d b y IO$ t o a c c o u n t r o u g h l y f o r the fuselage damping, n o tb ea c h i e v e d .F o rt h e limit l o a df a c t o ro f 3.8 t y p i c a lo fm o s tl i g h ta i r - c r a f t , t h e s p e c i f i c a t i o n s t i p u l a t e s t h a t f o r s t i c k c o n t r o l l e r s t h e maximum s t i c k f o r c e p e r g s h a l l be no more t h a n 28 I b s p e r g n o rl e s st h a n 20 I b sp e r g. The minimum g r a d i e n ti s 7.5 Ibs p e rg .

W i t h wheel c o n - - r o l l e r s , t h e maximum v a l u e s a r e a t l e a s t 42.8 I b sp e r g b u tn o t more t h a n 120 I b sp e rg . The minimum v a l u ei s 16 I b sp e r g.

llThe'-.term g r a d i e n t d o e s n o t i n c l u d e t h a t p o r t i o n of t h e f o r c e v e r s u s n c u r v ew i t h i nt h ep r e l o a d e db r e a k o u tf o r c e or f r i c t i o n band." These, accord.ing t o paragraph 3.5.2.1 (Ref.4)shouldbebetween1/2and 3 I b s f o r a s t i c k and 4 I bs for a wheel.

FAR p a r t 23 ( R e f .5 4 )p l a c e sn on u m e r i c a ll i m i t so nt h ee l e v a t o rs t i c k f o r c e i n m a n e u v e r i n g f l i g h t o t h e r t h a n t o say(23.143) t h a t t h e f o r c e o n t h e p i t c hc o n t r o ls h o u l dn e v e re x c e e d6 0 Ibs ( s t i c k ) or 75 I b s( w h e e l )d u r i n g tem- p o r a r ya p p l i c a t i o n so r 10 I b sd u r i n gp r o l o n g e da p p l i c a t i o n s . The p o p u l a r f l y i n g j o u r n a l s g e n e r a l l y do notreportsuchdataperhapsbecause i t s a c q u i - s i t i o n r e q u i r e s t h e i n s t a l l a t i o n o f c o n s i d e r a b l e i n s t r u m e n t a t i o n o r b e c a u s e t h e o m i s s i o n o f q u a n t i t a t i v e s t a n d a r d s i n t h e FAR'S suggests t o some e i t h e r a l a c k o f s i g n i f i c a n c e o r a l a c ko fp o p u l a ra p p r e c i a t i o nf o rt h em e a n i n go f q u a n t i t a t i v ev a l u e si nd e s c r i b i n gh a n d l i n g .S i n c et h e r ea r e few r e p o r t s d e a l i n gw i t hl i g h ta i r c r a f ta v a i l a b l ee l s e w h e r e , it i s d i f f i c u l t t o d e t e r m i n e t h ef o r c eg r a d i e n t s now common i n l i g h t a i r c r a f t .

Two sources, however, are helpful. TND-3726 ( R e f .5 5 )r e p o r t st h a t ,f o r some o ft h el i g h ta i r c r a f ti n v e s t i g a t e d ,s t i c kf o r c eg r a d i e n t sv a r i e db e t w e e n 8 and 17 I b sp e r g f o r speeds of l e s st h a n 100 k n o t s . When compared w i t ht h e r e q u i r e m e n t so f MIL-F-8785B (Ref. 41, t h e s e a i r c r a f t have g r a d i e n t sl o w e rt h a n d e s i r e d . On t h eo t h e r hand, t e s t s on a l i g h t a i r c r a f t a d a p t e d f o r m i l i t a r y use (Ref.56) showed c o m p l i a n c ew i t ht h es p e c i f i c a t i o na ta l lf l i g h tc o n d i t i o n s and a i r c r a f tl o a d i n g s . Based o nt h e s el i m i t e ds a m p l i n g s ,o n ew o u l de x p e c tt o f i n d a w i d ev a r i a t i o ni nt h eh a n d l i n gd u r i n g maneuvers e x h i b i t e db yc o n t e m p o r - a r y I i g h t a i r c r a f t .

One a d d i t i o n a la r e ag i v e np r o m i n a n c e by MIL-F-87858 b u t n o t r e f e r r e d t o elsewhere i n q u a n t i t a t i v e t e r m s i s t h e phase r e l a t i o n b e t w e e nt h ea p p l i c a t i o n o f c o n t r o l f o r c e and t h em o t i o no ft h ec o c k p i tc o n t r o lo nt h ea e r o d y n a m i cc o n - t r o l s u r f a c e . P a r a g r a p h 3 . 5 . 3 . 1 r e q u i r e s t h a t c o n t r o l d e f l e c t i o n s h o u l d n o t l e a dt h ea p p l i c a t i o no fc o n t r o lf o r c e .P a r a g r a p h3 . 5 . 3s p e c i f i e st h a tt h e c o n t r o ls u r f a c ed e f l e c t i o ns h a l ln o tl a gt h ec o c k p i tc o n t r o lf o r c e s by m r e t h a n 30° i n p h a s ea n g l ef o ra p p l i c a t i o nf r e q u e n c i e se q u a lo rl e s st h a n W n SP ' I nc o n t r o ls y s t e m s where t h ea e r o d y n a m i cs u r f a c e sa r ea c t u a t e d by r i g i d l i n k a g e sf r o mt h e wheel o rs t i c kt h e s er e q u i r e m e n t so fc o u r s ea r ea l w a y sm e t .

If, however, t h es y s t e mc o n t a i n se l a s t i ce l e m e n t s ,l i n k a g ef o r c eb o o s t e r s ,o r r e m o t e l yc o n t r o l l e da c t u a t o r st h i s may n o t be t h ec a s e . The c h a r a c t e r i s t i c s o fs u c hs y s t e m sw i l lr e q u i r ei n v e s t i g a t i o n and p o s s i b l e a l t e r a t i o n t o i n s u r e c o m p l i a n c ew i t ht h es p e c i f i c a t i o n .R e f e r e n c e 56 p r e s e n t sr e s u l t st a k e nw i t h t y p i c a lf o r c e and a n g u l a rp o s i t i o nt r a n s d u c e r sw h i c hc o u l d be used a l o n gw i t h o s c i l l o g r a p h i c o r t a p e r e c o r d e r s t o o b t a i n d a t a s u i t a b l e f o r a n a l y s i s o f t h e controlsystemphaseresponses.

1 35 Rol I C o n t r o l I f o n e r e v i e w s t h e l i t e r a t u r e d e a l i n g w i t h t h e h a n d l i n g q u a l i t i e s of l i g h t a i r c r a f t i n t h e r o l l mode c h r o n o l o g i c a l l y ,h e becomes aware of a s u b t l e s h i f t of e m p h a s i sf r o mc o n c e r np r i m a r i l yw i t ha c h i e v e m e n t of a g i v e n r o l l i n g r a t e a t low speeds and l i m i t a t i o n of a i l e r o n f o r c e s a t h i g h speeds t o concern w i t h t o t a l p i l o t w o r kl o a d( i n c l u d i n gr u d d e ra n de l e v a t o rc o o r d i n a t i o nr e - q u i r e d ) d u r i n g r o l l s and t o t h e a n g l e a t t a i n e d i n a s p e c i f i e d p e r i o d of t i m e .

T h i s i s n o t s u r p r i s i n g when one r e c a l l s t h a t d u r i n g t h e e r a o f W o r l d War I I good r o l l i n gp e r f o r m a n c ei nf i g h t e ra i r c r a f t( w e i g h i n gg e n e r a l l yl e s st h a n 10,000 Ibs, a t l e a s t a t t h e b e g i n n i n g of t h ew a r ) was e s s e n t i a l t o s u r v i v a l i n dog f i g h t s . As knowledge of how t o a c h i e v et h i sp e r f o r m a n c ea e r o d y n a m i c - a l l y grew and t h e c o n t r o l f o r c e s a t h i g h speedwerereducedbypowerboost systems, a t t e n t i o n c o u l d b e d e v o t e d t o f a c t o r s a f f e c t i n g s a f e t y and p r e c i s i o n r a t h e rt h a ns i m p l es u r v i v a l . Hence, t h e 1969 r e v i s i o no f MIL-F-8785B ( R e f . 4 ) d e v o t e s c o n s i d e r a b l e a t t e n t i o n t o l i m i t i n g t h e f o r c e s t h e p i l o t m u s t a p p l y t o r u d d e r and e l e v a t o rd u r i n ga i l e r o na p p l i c a t i o n .I ta l s o changed t h er e q u i r e - ment f o r t h e a t t a i n m e n t of a giver; pb/2U t o t h ea t t a i n m e n t of a g i v e n bank a n g l e i n a f i x e dt i m eo nt h ep r e m i s et h a ts u c h a r e q u i r e m e n t was more mean- i n g f u li ne s t a b l i s h i n gc o l l i s i o na v o i d a n c ec a p a b i l i t i e sa n d was e q u a l l y s u i t e d t o e s t a b l i s h i n g t h e d e s i r a b i l i t y o f o t h e r a r e a s of r o l l i n g performance.

The s h i f t i n areas of c o n c e r nw i t ht i m ea l s o had i t s a n a l o g i n t h e ana- l y t i c a lt e c h n i q u e se m p l o y e d and t h ep a r a m e t e rv a l u e si d e n t i f i e d . Whereas one wouldbegin a d i s c u s s i o no fr o l lh a n d l i n gw i t hc o n s i d e r a t i o n of t h o s ec o n t r o l f o r c e s ,c o n t r o ld e f l e c t i o n s ,a n dt h e i ry r a d i e n t sw h i c ha f f e c tt h ep i l o t ' s o p i n i o n of an a i r c r a f t ' sr o l lh a n d l i n gq u a l i t i e s , it w i l l b er e c o g n i z e dt h a t s u c ht h i n g sa st h ep h a s er e l a t i o n s h i p sb e t w e e nc o n t r o la p p l i c a t i o n and a i r c r a f t response,thepresence of s p u r i o u sr e s p o n s e sw h i c hr e q u i r ec o n t r o li n p u tt o c o u n t e r ,a n dt h ep r e c i s i o nw i t hw h i c hd e s i r e dm a n e u v e r sc a nb ee x e c u t e da l s o c o n t r i b u t e s i g n i f i c a n t l y t o h i so v e r a l !i m p r e s s i o n of h a n d l i n gc h a r a c t e r i s t i c s , T h e s ef a c t o r sp l u st h ei n e v i t a b l ec o u p l i n g of l a t e r a l and d i r e c t i o n a l modes and t h e f a c t t h a t a e r o d y n a m i c a l l y t h e a i r c r a f t h a s no i n h e r e n tb a n ko r i e n t a - t i o n and t h u s no s t a t i c r o l l s t a b i l i t y i n t h e c o n v e n t i o n a l sense, make it necessary t o employ a moregeneralapproach t o r o l l h a n d l i n g a n a l y s i s t h a n t h es i m p ! e rv i e ws u i t a b l ef o rt h et r e a t m e n to fp i t c hh a n d l i n g . The d i s c u s s i o n w h i c hf o l l o w sb e g i n s ,t h e r e f o r e ,w i t ht h es i m p l e ,o n e - d i m e n s i o n a lv i e w and moves on t o d e t a i l m e t h o d sw h i c ht h es p e c i f i c a t i o nw r i t e r sh a v eu s e di na n e f f o r t t o q u a n t i f y o t h e r p h a s e so fa c c e p t a b l er o l lh a n d l i n g .

The g e n e r a l ,l i n e a rt r e a t m e n to fs t e a d y ,o n e - d i m e n s i o n a lr o l lh a s been a v a i l a b l es i n c et h em i d - 1 9 4 0 I s . (See References I I and 16, f o r e x a m p l e . ) W i t h t h e a s s u m p t i o n t h a t a l l d e r i v a t i v e s r e m a i n c o n s t a n t i r r e s p e c t i v e o f a i l e r o n d e f l e c t i o n , speed, a l t i t u d e , and r o l l i n g r a t e , and t h a t t h e t i m e r e q u i r e d t o a t t a i n t h e maximum r o l l i n g r a t e f o r a g i v e na i l e r o nd e f l e c t i o nc a nb ec o n s i d e r e d 1 36 zero*,onecan w r i t e f o r t h e bankangle as a f u n c t i o n of a i l e r o n d e f l e c t i o n and * One-dimensional r o l l i n g m o t i o n i s d e s c r i b e d b y t h e e q u a t i o n ..

whose s o l u t i o n f o r a s t e pa i l e r o ni n p u ti s I nt h i se q u a t i o n , q = 1/2 pU . S i n c et h ed e n o m i n a t o ro ft h e second t e r m ** (equal t o ( 1 / - c R ) 1 g e n e r a l l yh a s a v a l u eo f -10 o r more ( n e g a t i v e ) , it i s o b v i o u st h a tf o rt i m e si ne x c e s s of a b o u t 1.0 s e ct h e second t e r mc o n t r i b u t e s v e r y l i t t l e t o t h e b a n ka n g l ea t t a i n e d . MIL-F-8785B, §3.3.1.2 r e q u i r e st h a t t h e maximum v a l u eo f TR f o r l i g h t a i r c r a f t i s 1.0. §3.3.1.4 f u r t h e rs t a t e s t h a t t h e r o l l i n g mode and t h e s p i r a l mode s h a l ln o tc o u p l e ,i e .t h et i m ec o n - s t a n t s have t h e same value, t o p r o d u c ea no s c i l l a t o r y mode. F o r a s i m u l a t o r s t u d y of t h i s case, t h er e a d e ri sr e f e r r e d t o TND-5466 (Ref. 5 7 ) .

F o rt h ec o m p l e t et h r e e - d i m e n s i o n a lt r e a t m e n to fr o l l i n gr e s p o n s et o a i l e r o n d e f l e c t i o n , t h e a p p e n d i c e s t o t h ep r e s e n ts t u d ys h o u l d beexamined.

** See Appendix D f o r a d i s c u s s i o n of t h e s t a b i l i t y d e r i v a t i v e s m o s t i m p o r t a n ti nd e t e r m i n gt h ev a l u e of t h e r o l l i n g mode t i m ec o n s t a n t , TR.

pub SacaG f o r t h e bank a n g l e a s a f u n c t i o n o f c o n t r o l f o r c e . i s t h e t o t a l ( u p p l u s down, assumed t o be e q u a la n de q u a l l ye f f e c t i v e )a i l e r o nd e f l e c t i o n .y ’i s t h es p a n w i s el o c a t i o n o f t h e a i l e r o n c e n t r o i d , I t i s seen t h a t a t low speed t h er o l l i n gp e r f o r m a n c ei n c r e a s e sw i t hi n c r e a s i n g speed u n t i l t h e f o r c e l i m i t s arereached; a t t h i s p o i n t t h e b a n ka n g l ep o s s i b l ep e ru n i tt i m ed e c r e a s e s w i t h i n c r e a s i n g s p e e d . The r a t i o (Cg6/CEp) i s a f u n c t i o n of t a p e rr a t i o , a i l e r o n - t o - w i n gc h o r dr a t i o ,a n dp e rc e n t of span denoted t oa i l e r o n s .I n a d d i t i o n t o making t h i s r a t i o a s l a r g e as p r a c t i c a b l e (maximum p o s s i b l ev a l u e i s a b o u t 1.51, o n e d e s i r e s t o a d j u s t t o be a p p r o x i m a t e l y 0.5 so a s t o i n c r e a s e t h e r o l l c a p a b i l i t y a t h i g h speeds.

C a r e f u l a t t e n t i o n t o a i l e r o n g e o m e t r y i s t h e r e f o r e n e c e s s a r y t o o b t a i n t h e p r o p e r r a t i o o f cha t o chg. TR-868 (Ref. 5 2 ) i s an e x c e ll e n ts o u r c eo fe x - p e r i m e n t a ld a t ao ng e o m e t r i ce f f e c t s and designmethods.

I t i s a p p a r e n tf r o mt h ef o r e g o i n gt h a t i f o n e m i n i m i z e s f r i c t i o n and e l a s t i c i t y i n t h e a i l e r o n l i n k a g e , t h e f a c t o r s c o n t r i b u t i n g t o wheel o r s t i c k f o r c e and t o r o l l i n gp e r f o r m a n c ea r ew e l l known. Even n o n - l i n e a r i t i e si nt h e d e r i v a t i v e sc h g , cha, e t c . ,w h i l er e q u i r i n gt h a tt e d i o u sc o m p u t a i - i o n sb e made t o o b t a i n +(t), a r ew e l l documented f o r a l a r g e number o f c o n f i g u r a t i o n s . Thus it i s a f a i r l ys t r a i g h t f o r w a r dm a t t e rt ot r a n s l a t ec o n t r o lf o r c e ,c o n t r o l de- f l e c t i o n , and r o l l i n g p e r f o r m a n c el i m i t a t i o n si n t oh a r d w a r es p e c i f i c a t i o n s .

I t remainsthen t o s t a t e t h e a p p r o p r i a t e v a l u e s f o r t h e s e q u a l i t i e s .

The maximum f o r c e g i v e n i n FAR p a r t 2 3 (Ref. 5 4 ) and MIL-F-8785B (Ref. 4 ) seem, f o r t h e most, t o havebeenselectedbyexperience from many y e a r s of p i l o t comments; nevertheless,comparison of t h e s e r e s u l t s w i t h a v a i l a b l e an- t h r o p o l o g i c a ld a t a (See d i s c u s s i o n of Human F a c t o r s i n t h i s w o r k ) f o r t h e com- f o r t a b l ea p p l i c a t i o n of l a t e r a lf o r c e s shows good agreement. The t a b l eb e l o w g i v e s a summary of t h e s p e c i f i c a t i o n r e q u i r e m e n t s .

-__ -" REQU I REMENT REFERENCE - __T_" "~ , . .

Temporary: Stick 601 Wheel 73# (Max) FAR 23. I43 P r o I onged I O# I S t i c k Whee I Theminimum f o r c e f o r maximum M 1 L-F-87858 r o I I i ng performance sha I I n o t 9 3.3.4.2 b el e s st h a nt h eb r e a k o u tf o r c e p l u s 1/4 o f t h e abovevalues.

C o n t r o lc e n t e r i n ga n db r e a k o u t f o r c e ss h a l ln o t bemorethan 53.5.2. I 2# ( s t i c k ) o r 3# ( w h e e l )n o r l e s s t h a n I /2 #.

R o l l i n g p e r f o r m a n c e : a i r c r a f t mustreach a 60' b a n ka n g l ei n 53.3.4.14 1.7 sec i n c r u i s e and a 30° bank i n 1.3 sec f o ra p p r o a c h N o tm o r et h a n 5# ( s t i c k ) o r I O # ( w h e e l )s h o u l db er e q u i r e d t o a c h i e v e a 45O bank w i t hr u d d e r 53.3.2.6 f r e e and t h ea i l e r o n st r i m m e d f o r w i n g si nl e v e lf l i g h t .

"_

+80° f o r a Not morethan 60° of wheel c o m p l e t e l y 53.3.4.4 r o t a t i o n ,i ne i t h e rd i r e c t i o n -._C rnechan i ca I system T h e r es h a l lb en oo b j e c t i o n a b l e n o n - l i n e a r i t i e s i n v a r i a t i o n of 53.3.4.3 r o l l i n g r e s p o n s e t o wheel m o t i o n .

i C o n t r o l s u r f a c e r e s p o n s e s h a l l i n o t l a g c o c k p i t c o n t r o l f o r c e i n p u t bymore t h a n 30° phase 53.5.3 a n g l e f o r f r e q u e n c i e sl e s s t h a n 1/-rR o r Wnd, whichever i s I a r g e r .

C o c k p i tc o n t r o ld e f l e c t i o n s h a l ln o tl e a dc o c k p i tc o n t r o l 53.5.3. I f o r c e .

I a t e r a I T a b l e 12. S p e c i f i c a t i o nr e q u i r e m e n t sf o rc o m f o r t a b l ea p p l i c a t i o no f f o r c e s .

Therequirementsquoted i n t h e p r e c e d i n g t a b l e a r e t h o s e r e s t r i c t e d t o C l a s s I ( s m a l ll i g h ta i r p l a n e ss u c ha sl i g h tu t i l i t y ,p r i m a r yt r a i n e r ,a n d l i g h t o b s e r v a t i o n up t o a b o u t 12,000 I b sg r o s sw e i g h t ) ;c a t e g o r y B w h i c hi n - cludesclimb,cruise,anddescentandcategory C w h i c hi n c l u d e st a k e - o f f , approach, and landing; and Level I, i e ,h a v i n gf l y i n gq u a l i t i e sc l e a r l y adequate f o r t h e M i s s i o n F l i g h t Phase. Thisapproach was s e l e c t e d f o r t h e present-workbecause it was d e s i r e d t o show t h e p r e s e n t s t a t e of understanding of what i s r e q u i r e d t o o b t a i n s a t i s f a c t o r y h a n d l i n g q u a l i t i e s .

The r e q u i r e m e n t o n t h e t i m e c o n s t a n t of t h e r o l l i n g mode was g i v e n p r e - v i o u s l ya sn o tl e s st h a n 1.0 sec. Examination of t h ee q u a t i o n of m o t i o n c i t e d p r e v i o u s l y w i l l a l s o show t h a t t h e a i r c r a f t w i l l r e s p o n dw e l lt os i n u - s o i d a la i l e r o ni n p u t s up t o f r e q u e n c i e s o f P rad/sec, 2u T x x where q = 1/2 pu .

T h i si sg e n e r a l l y above t h er a n g ea tw h i c ht h ep i l o tc a nt r a c k . Thus, i f t h e phase lag i nt h ec o n t r o ls y s t e mi sl e s st h a n 30' up t o t h i sf r e q u e n c y , +he p i l o t w i l l f i n d t h e a i r c r a f t a b l e t o g e n e r a t e r o l l r a t e s c o r r e s p o n d i n g t o wheel p o s i t i o n v i r t u a l l y a s r a p i d l y a s hecan t u r n t h e wheel o r move t h e s t i c k .

i sn o t e w o r t h yt h a ti np r e p a r i n gt h i ss p e c i f i c a t i o n ,i t sw r i t e r s knave q u a n t i t a t i v el i m i t so n The r o l I ingperformance of w h i c h t h e a i r c r a f t m u s t be capable, The maximum and minimum f o r c e s needed t o produce maximum r o l l i n g v e l o c i t y , P e r m i s s i b l eb r e a k o u t and c e n t e r i n gf o r c e s , The maximum c o n t r o l d e f l e c t i o n .

These, c o u p l e dw i t ht h ef r e q u e n c yr e s p o n s er e q u i r e m e n t sm e n t i o n e de a r l i e r and t h e b a no no b j e c t i o n a b l en o n - l i n e a r i t i e s ,p r e s e n t a complete and q u a n t i t a t i v e d e s c r i p t i o n o f t h e f a c t o r s c o n t r i b u t i n g t o t h e p i l o t ' s s a t i s f a c t i o n w i t h r o l l mode hand I i ng.

The usual methods of p r o d u c i n g a r o l l i n g m o t i o n i s t h e d i f f e r e n t i a l de- f l e c t i o no fo u t b o a r dp o r t i o n so ft r a i l i n g edge o ft h e wing, o r a i l e r o n s , I f onewishes t o r o l l t o t h e r i g h t , he d e f l e c t s t h e r i g h t a i l e r o n up t o reduce l i f t on t h a t wing and d e f l e c t s t h e l e f t a i l e r o n down t o i n c r e a s e l i f t on t h a t w i n g .U s u a l l y ,s u c ho p e r a t i o ni s a p o r t i o n of a c o o r d i n a t e dt u r nt ot h er i g h t .

The d r a go nt h er i g h tw i n gi sd e c r e a s e d and t h ed r a go nt h el e f tw i n gi n c r e a s e d by t h i sa i l e r o nd e f l e c t i o n . The r e s u l t i n gy a w i n g moment ( c a l l e da d v e r s ey a w ) , i f unbalancedby a r u d d e rd e f l e c t i o n ,r e d u c e st h ee f f e c t i v er o l l i n gv e l o c i t y b y m o v i n g t h e r i g h t w i n g a t a h i g h e r f o r w a r d v e l o c i t y t h a n t h e l e f t w i n g .

The l i f t on t h e r i g h t w i n g i s t h e r e f o r e g r e a t e r ( a n d l e s s o n t h e l e f t ) t h a n wouldbethecase i f t h e a d v e r s e yaw w e r en o tp r e s e n t .

Adverse yaw i s a l s o undesirablebecause i f unopposed it e x c i t e s t h e l i g h t l y damped Dutch Roll o s c i l l a t i o n . The p i l o tt h u sf i n d sh i m s e l fp r o - d u c i n go f t e nd i s c o n c e r t i n gy a w i n gm o t i o n su n i n t e n t i o n a l l y . The s p e c i f i c a t i o n r e q u i r e m e n t sl i m i t i n gi t se x t e n ti n c l u d e 53.3.2.5 ( R e f .4 )w h i c hp r o v i d e st h a t no more t h a n 50 I b s o f r u d d e rf o r c e be r e q u i r e d t o make a c o o r d i n a t e d (B=O) t u r n a t c r u i s e speeds o r 100 I b s i n t h e a p p r o a c h c o n f i g u r a t i o n f o r h i g hp e r - formance f i g h t e r a i r c r a f t . T h e r e i s a l s o 53.3.2.4, w h i c hp r o v i d e st h a ta d - v e r s es i d e s l i ps h a l ln o te x c e e d IOo and t h a tp r o v e r s es i d e s l i ps h a l ln o te x - ceed 3 O d u r i n g f I i g h t p h a s ec a t e g o r i e s B and C ( c r u i s e , c I imb, descent,and t a k e - o f f and l a n d i n g ) . TND-3726 ( R e f .5 5 )s u g g e s t st h a tp i l o t sf i n d a s i d e - s l i p a n g l e o f 10' i n response t o a r u d d e r - l o c k e d a i l e r o n d e f l e c t i o n t h e max- imum a c c e p t a b l e . Some o f t h e a i r c r a f t t e s t e d f o r t h i s r e p o r t e x h i b i t e d s i d e - s l i pa n g l e si ne x c e s so f 13'. Reference 56 r e p o r t st h a t a c o n v e r t e dl i g h t a i r c r a f t c a r r y i n g m i l i t a r y s t o r e s u n d e r t h e w i n g r e a c h e d s i d e s l i p a n g l e s i n excess o f 10' a n dr e q u i r e d 125 I b so fr u d d e rf o r c et op e r f o r m a c o o r d i n a t e d t u r n a t low speed. The m i l i t a r ys t o r e s w e r er e s p o n s i b l ef o r a 5 O i n c r e a s ei n s i d e s l i p a n g l e .

I nt h ea b s e n c eo f a r u d d e rf o r c et oc o u n t e rt h ea d v e r s ey a w i n g moment, t h e a i r c r a f t w i l l d e v e l o p s u f f i c i e n t s i d e s l i p t o p e r m i t a balancingyawing moment due t o s i d e s l i p t o be produced, Since a l l l i g h t a i r c r a f t employ some d i h e d r a lt oa i di nm a i n t a i n i n gt h ew i n g s 'l e v e ld u r i n gn o r m a lf l i g h t * and s i n c e t h i s s t a b i l i z i n g r o l l i n g moment due t o g i d e s l i p a c t s t o r e d u c e t h e r o l l r a t ea c h i e v e db y a g i v e na i l e r o nd e f l e c t i o n , i t i si m p o r t a n tf o r good r o l l i n g r e s p o n s et h a ta d v e r s e yaw be h e l d t o a minimum. 53.3.6.3.2 o f MIL-F-8785B ( R e f . 4 ) s t a t e st h a tt h ep o s i t i v ed i h e d r a le f f e c ts h a l ln o tb e so g r e a t t h a t morethan75% of t h e r o l l c o n t r o l power a v a i l a b l e t o t h e p i l o t , and no more t h a n 10 I b so fa i l e r o ns t i c kf o r c eo r 20 I b s of wheel f o r c ea r er e q u i r e df o r s i d e s l i pa n g l e sw h i c hm i g h t beexperienced i ns e r v i c ed e p l o y m e n t , The comments of AFFDL-TR-69-72 (Ref. 4 ) r e l a t i v e t o s i d e s l i pr e q u i r e m e n t s d e v e l o p e df o rt h es p e c i f i c a t i o n seem p a r t i c u l a r l yp e r t i n e n t : I' t h ep r i m a r y s o u r c eo fd a t af r o mw h i c ht h es i d e s l i pr e q u i r e m e n te v o l v e d( S e e A F F D L - T R - ~ ~ - ~ ~ ) a r e t h o s e t e s t s f o r w h i c h t h e r a t i o o f b a n k a n g l e t o s i d e s l i p a n g l e d u r i n g D u t c h Rol I equaledabout1.5. "The p i l o t comments a s s o c i a t e dw i t ht h e s ec o n - f i g u r a t i o n s i n d i c a t e d t h a t p i l o t ' s d i f f i c u l t i e s w e r ea l m o s te x c l u s i v e l ya s - s o c i a t e d w i t h s i d e s l i p , r a t h e r t h a n w i t h b a n k a n g l e t r a c k i n g a s was t h e c a s e f o r l($/f31dE6 c o n f i g u r a t i o n s .A n a l y s i so ft h ed a t ar e v e a l e dt h a tt h e amount o f s i d e s l i p a p i l o t w i l l a c c e p t or t o l e r a t e i s a s t r o n gf u n c t i o n of t h e phase * FAR p a r t 23 ( R e f . 5 4 ) s t a t e s t h a t ' ' t h e s t a t i c l a t e r a l s t a b i l i t y , a s shown b yt h et e n d e n c y t o r a i s e t h e lowwing i n a s l i p , m u s t b e p o s i t i v e f o r a n yl a n d i n gg e a r and f l a p p o s i t i o n . " a n g l eo ft h eD u t c h Roll component of s i d e s l i p , When t h ep h a s ea n g l ei ss u c h t h a t B i s p r i m a r i l y a d v e r s e , t h e p i l o t c a n t o l e r a t e q u i t e a b i t of s i d e s l i p .

On t h e o t h e r hand, when t h ep h a s i n gi sp r i m a r i l yp r o v e r s e ,t h ep i l o tc a no n l y t o l e r a t e a small amount of s i d e s l i p because o f d i f f i c u l t y of c o o r d i n a t i o n .

"There i s more t oc o o r d i n a t i o n , however, t h a nw h e t h e rt h es i d e s l i pi s a d v e r s e o r p r o v e r s e ;t h es o u r c e andphasing of t h ed i s t u r b i n gy a w i n g moment a l s os i g n i f i c a n t l ya f f e c tt h ec o o r d i n a t i o np r o b l e m . If t h ey a w i n g moment i s c a u s e db ya i l e r o na n d i s i n t h e a d v e r s e sense, t h e n i n o r d e r t o c o o r d i n a t e t h e p i l o t m u s t p h a s e e i t h e r r i g h t r u d d e r w i t h r i g h t a i l e r o n o r l e f t r u d d e r w i t hl e f ta i l e r o n .S i n c ep i l o t sf i n dt h i st e c h n i q u en a t u r a l ,t h e yc a ng e n e r - a l l yc o o r d i n a t ew e l le v e n i f t h ey a w i n g moment i sl a r g e .I fo nt h eo t h e r hand, t h ey a w i n g moment i s i n t h e p r o v e r s e sense or i s caused by r o l lr a t e ,c o o r d i n - a t i o n i s f a r more d i f f i c u l t . F o rp r o v e r s ey a w - d u e - t o - a i l e r o nt h ep i l o tm u s t c r o s sc o n t r o l ; and f o re i t h e ra d v e r s eo rp r o v e r s ey a w - d u e - t o - r o l l - r a t e ,r e - q u i r e d r u d d e r i n p u t s m u s t be p r o p o r t i o n a lt or o l lr a t e .P i l o t sf i n dt h e s e t e c h n i q u e su n n a t u r a l and d i f f i c u l tt op e r f o r m .S i n c ey a w i n g moments may a l s o beintroducedby yaw r a t e , it can be seen that,dependingonthemagnitude andsense o ft h ev a r i o u sy a w i n g moments, c o o r d i n a t i o n may e i t h e r be easy o r e x t r e m e l yd i f f i c u l t .I fc o o r d i n a t i o ni ss u f f i c i e n t l yd i f f i c u l tt h a tp i l o t s cannot be expected t o c o o r d i n a t e r o u t i n e l y , t h e f l y i n g q u a l i t y r e q u i r e m e n t s m u s tr e s t r i c tr u d d e r - p e d a l sf r e eu n w a n t e dm o t i o n st o a s i z e a c c e p t a b l e t o p i l o t s . " " A n a l y s i s f u r t h e r r e v e a l e d t h a t it was n o t so much t h ea b s o l u t em a g n i t u d e o f t h e s i d e s l i p t h a t b o t h e r e d t h e p i l o t , b u t r a t h e r t h e maximum changeoccuring i ns i d e s l i p . The l a t t e r was a b e t t e r measure o ft h e amount o fc o o r d i n a t i o n r e q u i r e d . Thus t h e d a t a . . . w e r e p l o t t e d ... a s t h e maximum change i n s i d e s l i p o c c u r i n gd u r i n g a r u d d e r - p e d a l s - f i x e dr o l l i n g maneuver, ABmax, v e r s u st h ep h a s e a n g l e of t h eD u t c h Roll component o f s i d e s l i p $6, The phase angle, I ) @ , i s a measure o f t h e sense o f t h e i n i t i a l s i d e s l i p response,whetheradverse o r p r o - v e r s e ,w h i l e ABmax i s a measure o f t h ea m p l i t u d e of t h e s i d e s l i p g e n e r a t e d .

B o t ht h es e n s ea n da m p l i t u d ea f f e c tt h ec o o r d i n a t i o np r o b l e m . " F o rs m a l la i l e r o nc o n t r o l commands (93.3.2.4.11, t h e amount o fa l l o w a b l e s i d e s l i pi sg i v e ni nt h ef i g u r eb e l o w .

F i g u r e 63. A l l o w a b l es i d e s l i pf o rs m a l la i l e r o nc o n t r o l commands, T h i sr e q u i r e m e n ta p p l i e df o rs t e pa i l e r o nc o n t r o l commands up t o t h e magnitudewhichcauses a 60' b a n ka n g l ec h a n g ei n two seconds o r 2 l T seconds, whichever i s g r e a t e r .

The d i f f e r e n c e i n a l l o w a b l e ABmax w i t h Jlg " i s a l m o s t t o t a l l y due t o t h e d i f f e r e n c ei na b i l i t yt oc o o r d i n a t ed u r i n gt u r ne n t r i e s and e x i t s , $6 i s a d i r e c t i n d i c a t o r of t h e d i f f i c u l t y a p i l o t w i l l e x p e r i e n c e i n c o o r d i n a t i n g a t u r n e n t r y , F o r -180°2 $6 2 -270°, n o r m a lc o o r d i n a t i o n may be e f f e c t e d ....

As $B v a r i e s from -270' t o -360' c o o r d i n a t i o n becomes i n c r e a s i n g l y d i f f i c u l t , and i n t h e r a n g e -360' I $6 I -90' c r o s s c o n t r o l l i n g i s r e q u i r e d t o e f f e c t c o o r d i n a t i o n .S i n c ep i l o t s do n o tn o r m a l l yc r o s sc o n t r o l and, i f they must, h a v e g r e a t d i f f i c u l t y i n d o i n g so f o r -360' 5 Jlg I -go', o s c i l l a t i o n s i n s i d e s l i p e i t h e r go unchecked o r a r e a m p l i f i e d b y t h e p i l o t ' s e f f o r t s t o co- o r d i n a t ew i t hr u d d e rp e d a l s . " To e x t e n dr o l l - s i d e s l i pc o u p l i n gr e q u i r e m e n t s t o l a r g e rc o n t r o ld e f l e c - t i o n s ,§ 3 . 3 . 2 . 3s t a t e st h a tt h ev a l u eo ft h ep a r a m e t e r

9 , + 93 - 292

- - 9osc"average

9, + 93 + 293 '

i n t h e b a n ka n g l et i m eh i s t o r y where $1, $ 2 , $3 representsuccessivepeaks ($2 i s a minimum peak), s h a l l be w i t h i n t h e l i m i t s shown o n t h e f i g u r e b e l o w .

1.0 -40' -80' -120' -160' -200' -240' -28W-320"360.

when p leads @ by 45' t o 225' I I I I I I I I I -220'-260'-300'-340'-20'-60. -100. -140" -180'

fie when p leads Q by 225" through 360'to 45"

Figure 64. Limits for extending roll-sideslip coupling requirements to larger control deflections.

Similar requirements designed to insure control precision in the roll mode are given i n §3.3.2.2.1, which requires thatPoSc/P~vbe as shown in the figure below.

- 0.6 - - OA

* 43-

n!

\

v, 92- n a1 L

L

I I I I I I I 1 I

-40' -80' -1zo' 160. -200'-240'-280' - 320' -360'

#@ (see Fig. 64)

I 1 I I I I I I I

-220' -260' -300' -340' -20' -60' -100' - 140'-180'

$p (see Fig. 64) F i g u r e 6 5 . Requirements f o r i n s u r i n gc o n t r o lp r e c i s i o ni nt h er o l l mode.

T h i sr e q u i r e m e n ta p p l i e df o rs t e pa i l e r o nc o n t r o l commands up t o t h e magni- tudewhichcauses a 60° bank a n g l e change i n 3.4Tr seconds,

w n r,-- 2

d - 'd

F o rl a r g e rr o l lr a t e s , § 3 . 3 . 2 . 2 s t a t e st h a tf o l l o w i n g a r u d d e r - p e d a l s - f r e e s t e pa i l e r o nc o n t r o l command, t h e r o l l r a t e a t t h e f i r s t minimum f o l l o w i n g t h e f i r s t p e a ks h a l lb e of t h e same s i g n and no l e s st h a n 60% ( c a t e g o r i e s A and C ) o r 25% ( c a t e g o r yB )o ft h er o l lr a t e o f t h e f i r s t peak.

A d d i t i o n a le v i d e n c e of t h e b e a r i n g w h i c h r o l l i n g moment due t o s i d e s l i p hason p i l o t t s o p i n i o n of a n a i r c r a f t ' s h a n d l i n g q u a l i t i e s i s p r o v i d e d b y .:43.3.2.1 w h i c hs t a t e st h a tr o l la c c e l e r a t i o n ,r a t e , and displacement responses t os i d eg u s t ss h a l lb ei n v e s t i g a t e df o ra i r p l a n e sw i t hl a r g er o l l i n g moments due t o s i d e s l i p , As mentioned a t t h e b e g i n n i n g of t h i s s e c t i o n , m o d e r nc o n c e r nw i t ha i r - c r a f t h a n d l i n g c h a r a c t e r i s t i c s h a sc e n t e r e do np r e c i s i o n of responseand t o t a l p i l o t work load. The paragraphs o f MIL-F-8785B (Ref. 4 ) c i t e d above d e m o n s t r a t et h i sc o n c e r nc l e a r l y . TND-3726 (Ref. 551, c i t i n g t h e f a c t N t h a t t h e s t a b i l i t y and c o n t r o l c h a r a c t e r i s t i c s o f t h e a i r p l a n e s ( t e s t e d ) a r e . , . .. . . - - . . . ... .. . . . . . . . ... . .. .. . . . - g e n e r a l ! ys a t i s f a c t o r yb u td e t e r i o r a t ew i t hr e d u c e d speed, i n c r e a s e d power, a f tc e n t e r - f - g r a v i t yl o c a t i o na n dc h a n g e s t o t h el a n d i n gc o n f i g u r a t i o n , . .

and a r e c r i t i c a l l y degraded i n t u r b u l e n t a i r " and t h a t " i n g e n e r a l , e a c h s t a b i l i t y and c o n t r o ld e g r a d a t i o ni sr e l a t i v e l ys m a l l "s u g g e s tt h a t" r e d u c e d p i l o t r a t i n g r e s u l t s from t h e combined e f f e c t s of a l l d e t e r i o r a t i o n s " and t h a t it i sV - h e r e f o r en e c e s s a r y t o c o n s i d e r t h e c o m b i n e d e f f e c t o f a l l t h e s t a b i l i t y and c o n t r o l c h a r a c t e r i s t i c s i n o r d e r t o p r o p e r l y a s s e s s t h e p i l o t r a t i n g s . t ' The a u t h o r s of TND-3726 (Ref. 55) s u g g e s tt h a tt h i sb e done by e v a l u a t i n g a work l o a df a c t o rd e f i n e da s w o r kl o a df a c t o r = The i n t e g r a l sa r et h ea r e a su n d e r t h e c u r v e s of t h e t h r e e p i l o t - a p p l i e d f o r c e t i m e h i s t o r i e s t a k e n o v e r a 3 0 s e c p e r i o d w i t h t h e a i r c r a f t o r i q i n - a l l y i n a t r i m m e ds t a t e . "The s a t i s f a c t o r ya i r p l a n ep r e s e n t s a w o r kl o a d f a c t o r o f a p p r o x i m a t e l y 300; whereas t h eu n s a t i s f a c t o r ya i r p l a n ea p p r o a c h e s a f a c t o r o f 5 0 0 . "C l e a r l y , a r e q u i r e m e n t of t h i s t y p e will u l t i m a t e l yb e made p a r t o f t h e f l y i n g q u a l i t y s p e c i f i c a t i o n s , Many of t h e q u a n t i t a t i v e t e s t s from w h i c h t h e s p e c i f i c a t i o n w r i t e r s gainedguidancewereperformed i n s i m u l a t o r s o r i n v a r i a b l e s t a b i l i t y a i r - p l a n e s . R e s u l t s r e p o r t e d i n TND-746 (Ref. 58) and TND-779 ( R e f . 5 9 ) f o r example, showed t h ei m p o r t a n c e of t h eD u t c hR o l lf r e q u e n c y and damping i n e s t a b l i s h i n gt h el i m i t i n gb e h a v i o rw h i c ht h ep i l o tc o u l dc o n t r o l . TND-221 ( R e f . 6 0 ) r e p o r t s p i l o t s r e q u i r e 0.23 sec t o b e g i n t o move t h e s t i c k l a t e r - a l l y a n d 0 . 3 3 s e c l o n g i t u d i n a l l y . TND-173 ( R e f .6 1 )r e p o r t sr e s u l t so f a s t u d yt op r o v i d ea r t i f i c i a ll o n g i t u d i n a ls t a b i l i t y . TND-1782 (Ref. 62) p r o v i d e sd a t a on t h et r a n s f e rf u n c t i o n of human p i l o t s . I t s h o u l d be noted, however, t h a t suchdatacannotbeacceptedwithout some q u a l i f i c a t i o n : human p i l o t s d i s p l a y a v e r yh i g hd e g r e e o f a d a p t a b i l i t y t o d e v i c e s designed t o measure t h e i rr e s p o n s ec a p a b i l i t i e s and t h u s e x h i b i t d i f f e r e n t t r a n s f e r f u n c t i o n sw i t hi n c r e a s i n gt i m e o r a st h ec o n t r o lt a s ki sv a r i e d .

S e v e r a lv e r yr e c e n tf u l l - s c a l ew i n dt u n n e ls t u d i e so nt h es t a b i l i t y and c o n t r o l c h a r a c t e r i s t i c s of r e p r e s e n t a t i v e l i g h t a i r c r a f t a r e r e p o r t e d i n References 63, 64, and 6 5 .R e s u l t so f a p r o g r a mo fa r t i f i c i a ls t a b i l i t y augmentation andworkloadreductionon a p o p u l a rl i g h tt w i ni sp r e s e n t e d i nc o n s i d e r a b l ee n g i n e e r i n gd e t a i li nR e f e r e n c e6 6 ,T h i sp r o g r a mi sn o t a b l e i n b e g i n n i n g w i t h a c o m p e t e n ta n a l y t i c a la t t a c ko nt h ep r o b l e m( a n di n de- t a i l i n gt h en u m e r i c a lc a l c u l a t i o n s ) ,i nd e v i s i n ga ni n g e n i o u sr o l l - y a w c o u p l e re s p e c i a l l ys u i t e d t o t h i s t y p e a i r c r a f t , and i n p r e s e n t i n g d e t a i l s , i n c l u d i n g t r a n s f e r f u n c t i o n s , of t h e f l i g h t hardware, F i n a l l y , f o r a v e r yi n t e r e s t i n gr e v i e w of t h ec h r o n o l o g yo fm a n r s un- d e r s t a n d i n g of a i r c r a f t s t a b i l i t y and c o n t r o l , t h e r e a d e r i s r e f e r r e d t o t h e 1970 von Ka"rrnSn L e c t u r eg i v e n by Dean P e r k i n s( R e f ,6 7 ) .

~~ Yaw C o n t r o I The uses of t h e yaw c o n t r o l i n a i r c r a f t i n c l u d e , a c c o r d i n g t o AFFDL-TR-69r72 (Ref. 4 1 , To perform a cross-windIandingYTeitheremploy (a 1 a s t e a d yr u d d e r - p e d a l - i n d u c e ds i d e s l i p o r e l s e a decrabmaneuver, To augment r o l l r a t e anywhere w i t h i n +he f l i g h t enve I ope, To r a i s e a wing when t h e p i l o t i s busy w i t h h i s hands, such a s when t a k i n g a c l e a r a n c e .

F o r t r a c k i ng .

For wing-overs ... t o o b t a i n a r a p i d c h a n g e i n

h e a d i n go rb a n ka n g l e .

For c I ose-format ion f I y i ng .

To l o s e a l t i t u d e a s i n a f o r w a r d s i d e s l i p o r t o improve v i s i b i I i t y .

To c o u n t e ry a w i n g moments f r o m p r o p e l l e r t o r q u e s , s p e e dc h a n g e ,a s y m m e t r i ct h r u s t ,s t o r e s ,e t c .

To t a x i .

The r e a d e r w i I I n o t e t h a t f r o m a hand I i n g s t a n d p o i n t t h e s e u s e s a r e o f two t y p e s :p r i m a r yc o n t r o lo fm o t i o na b o u tt h e yaw a x i s a n dp r e c i s i o no fc o n t r o l i nc o u p l e dl a t e r a l - d i r e c t i o n a lm o t i o n s .P r i m a r yc o n t r o la b o u tt h e yaw a x i s i s much l i k ep r i m a r yc o n t r o la b o u tt h ep i t c ha x i s :t h ea i r c r a f tc a n be pro- v i d e dw i t hi n h e r e n ts t a t i cs t a b i l i t y ,t h ef o r c e se x p e r i e n c e d by t h e p i l o t a r e a good measure o f t h e o u t - o f - t r i m c o n d i t i o n , and most of t h e l o n g i t u d i n a l s t a b i I i t y parametershave d i r e c t i o n a I ana logs. Thus, o n e c a n w r i t e - dF r = -G(g)qvSrcr [.I + Ch [ $ ) I , dB a "6 r and Wil-h t h eu s u a la s s u m p t i o n o f c o n s t a n c yi nt h ev a l u e s o f t h e s t a b i l i t y d e r - i v a t i v e s , t h e s e r e l a t i o n s p r o v i d e a means t o compare t h e p r i m a r y d i r e c t i o n a l h a n d l i n gp a r a m e t e r s o f proposed a i r c r a f t w i t h t h e s p e c i f i c a t i o n r e q u i r e m e n t s q u o t e db e l o w .N o t et h a t t o keep t h ev a r i a t i o ni nf o r c er e q u i r e dw i t hc h a n g e s i n s p e e dr e a s o n a b l ea n dt h u sm i n i m i z et h e need for r e t r i m m i n g it i sn e c e s s a r y t o make Chcl andchgassmallaspossible. Such a s t e p w i l l a l s o a i d i n k e e p i n gt h ef o r c er e q u i r e d t o p r o d u c eg i v e ns i d e s l i p s( a n dt h u sa i di nr o l l i n g maneuvers) w i t h i nr e a s o n .

P r i m a r y yaw c o n t r o li nt h ep r e s e n tc o n n o t a t i o nc o n s i s t s of t h o s ef u n c t i o n s w h i c hc a nb ep r o v i d e do n l yb yr u d d e rd e f l e c t i o n .I n c l u d e di nt h i sl i s tw o u l d be items such as (a), (e), (g), (h), and ( i ) . The p r e c i s i o n - o f - c o n t r o l i t e m s i nt h el i s ti n d i c a t et h a tf o rt h e s ec i r c u m s t a n c e st h er u d d e ri su s e dp r i n c i - p a l l y t o complement a i l e r o nc o n t r o l ,t oi m p r o v ei t sp r e c i s i o n by c o u n t e r i n g adverseyawing moments (Cngr or Cn 1 o rt op r o v i d es m a l la d d i t i o n a lf a v o r a b l e P r o l l i n g moments ( C E r ) . T h e t a b l e b e l o w shows how t h ed e s i r et op e r m i tt h e s e uses was t r a n s l a t e di n t oq u a n t i t a t i v er e q u i r e m e n t s .

REQU I REMENT Max imum r u d d e r f o r c e 150# FAR 23.143 temporary 20# steady Rudder f o r c e <50# t o c o u n t e r s i d e s l i p MIL-F-8785B i n r o l l s .

Coordinatedturnwhichreaches 45O o f b a n k s h o u l d r e q u i r e l e s s t h a n 40# o f § 3 . 3 . 2 . 5 r u d d e r .

50# o fr u d d e rf o r c em u s ti n d u c e a r o l l 93.3.4.5 r a t e o f 3O/sec.

I f t h e a i r c r a f t i s t r i m m e d w i t h s y m m e t r i c power it must be ab l e t o changespeed?30% §3.3,5.1 w i t h o u tr e q u i r i n gm o r et h a n 100# o fr u d d e r f o r c e .

No more t h a n 100# of r u d d e rf o r c em u s t be 93.3.5.1.1 necessaryforasymmetricloading.

(continuedonnextpage) . " REFERENCE REQUIREMENT ~ .- ~~ ~~ The r a t i o B/6, mustbe e s s e n t i a l l y l i n e a r t o +15O w i t h t h e s l o p e p e r m i t t e d t o be s m a l l e r f o r 8 > 1 5 O b u t s t i l l p o s i t i v e . The 53.3.6. I r a t i o B / F r mustbe e s s e n t i a l t y l i n e a r t o +IOo. F r may be l e s sf o rl a r g e rv a l u e s b u tn e v e rz e r o .

-~ ~- ~~ F o r t h e a i r c r a f t t r i m m e d f o r @=O f l i g h t t h e s er e q u i r e m e n t sa p p l ya ts i d e s l i pa n g l e s produced o rl i m i t e d by ( a ) f u l l r u d d e r p e d a l d e f l e c t i o n 53.3.6 (b 1 250H r u d d e rp e d a lf o r c e ( c ) maximum a i l e r o nc o n t r o lo r s u r f a c e d e f l e c t i o n .

- C o n t r o l c e n t e r i n g and b r e a k o u tf o r c e ss h a l l §3.5.2. I be between I # and 7#.

C o n t r o l s u r f a c e r e s p o n s e s h a l l n o t l a g c o c k p i tc o n t r o lf o r c ei n p u t by more §3.5.3 t h a n 30° ( p h a s ea n g l e )f o rf r e q u e n c i e s equal t oo rl e s st h a nI / T R .

~~ .. - . .- ~. ~ " ~~ ~ C o c k p i tc o n t r o ld e f l e c t i o ns h a l ln o t §3.5.3. I l e a dc o n t r o lf o r c e .

I t s h a l l be p o s s i b l et ot a k eo f f and land w i t h normal p i l o t s k i l l i n 90' cross winds 93.3.7 f r o m e i t h e r s i d e w i t h v e l o c i t i e s up t o 20 knots. Rudder forces shall not exceed loo#.

Rudder c o n t r o l power s h a l l be adequate t o m a i n t a i nw i n g sl e v e l and s i d e s l i pz e r o w i t h o u t r e t r i m m i n g t h r o u g h o u t d i v e s and 93.3.8 p u l l u p s . I n t h es e r v i c ef l i g h te n v e l o p e , s h a l ln o te x c e e d 180#.

I t s h a l l be p o s s i b l e t o t a x i a t anyangle 53.3.7.3 t o a 35-knotwind.

Rudder f o r c e s s h a l l notexceed #I80 t o m a i n t a i n a s t r a i g h t p a t h i n t h e e v e n t 53.3.9. I of sudden loss o f t h r u s t d u r i n g t a k e - o f f , ~ " " ~~~ ' a b l e 13. Yaw c o n t r o l r e q u i r e r n e n t s .

C u r r e n t l i g h t a i r c r a f t o f t e n r e q u i r e f a i r l y l a r g e ("185 I b s )r u d d e r f o r c e s i n 15O s t e a d ys i d e s l i p s( R e f . 56) o r a s a r e s u l t of t h e a p p l i c a t i o n of power--up t o 90 I b s changebetween i d l e powerand maximum power a t a g i v e n a i r s p e e d ( R e f , 5 5 ) . P i l o t sc o n s i d e r e dt h el a t t e ru n s a t i s f a c t o r ya l - though it would s a t i s f yt h es p e c i f i c a t i o n .N o t et h a tb r a k ef o r c e si na u t o - m o b i l e si ne x c e s s of 100 I b sa r eg e n e r a l l yr e g a r d e da su n d e s i r a b l ew h i l e f o r c e si ne x c e s s of 200 I b sa r er e g a r d e da su n a c c e p t a b l e ,I nt h i sr e g a r d , t h e FAR maximum r u d d e rf o r c er e q u i r e m e n ta p p e a r s t o b e f a r morereasonable t h a n t h a t of MIL-F-87858 (Ref. 4 1 , p a r t i c u l a r l y c o n s i d e r i n g t h a t d i m i n u a t i v e women s h o u l db ea b l e t o o p e r a t e a l i g h ta i r p l a n ec o m f o r t a b l y . The s p e c i f i - c a t i o n s a r e a l s o n o t a b l e i n o m i t t i n g a n y q u a n t i t a t i v e l i m i t a t i o n o n r u d d e r p e d a lt r a v e l .W h i l et h i si sa n o t h e ra s p e c t of t h ep r e s e n tu n s a t i s f a c t o r y s t a t e of d i r e c t i o n a lc o n t r o lr e q u i r e m e n t s ,t h ec o n d i t i o nl i k e l y stems from t h e f a c t t h a t m o s t d i r e c t i o n a l c o n t r o l d u r i n g f l i g h t i s now accomplished t h r o u g h a i l e r o n m a n i p u l a t i o n w h i c h t h e p i l o t c a n p e r f o r m v e r y p r e c i s e l y and f o rw h i c ht h e force-deflection-response r e l a t i o n s a r e p r e s c r i b e d i n g r e a t d e t a i I .

I t i s f e l t by s e v e r a lp i l o t sc o n s u l t e d by t h ea u t h o r st h a +t h ef o r c e l i m i t s q u o t e d i n t h e m i l i t a r y s p e c i f i c a t i o n s s h o u l d be a p p l i e d o n l y t o t h e f a s t e r and h e a v i e ra i r c r a f t of t h eg e n e r a la v i a t i o nc l a s s .I t was t h e i r c o n t e n t i o nt h a to n es h o u l dr e d u c et h e s el i m i t sp r o g r e s s i v e l ya s maximum speed and w e i g h ta r er e d u c e d so t h a t for t h e l i g h t e s t a n d s l o w e s t a i r c r a f t i n t h e c l a s s t h e maximum f o r c e s w i l l beno g r e a t e rt h a n 1/3 o r 1/2 t h e s p e c i f i c a t i o nv a l u e s .T h er e a s o n i n ga p p a r e n t l yf o l l o w st h eu s u a l human e x p e c t a t i o nt h a tl a r g e rv e h i c l e sr e q u i r el a r g e rf o r c e st oc o n t r o l them.

T h a tt h i si sn o tn e c e s s a r y , however, i s e v i d e n t from t h ep o p u l a ra c c e p t a n c e o f power s t e e r i n g andpowerbrakesonautomobiles.Withthesedevicesthe f o r c e sa r e made more or l e s si n d e p e n d e n to fc a rs i z e . I t i s t o be e x p e c t e d ,t h e r e f o r e ,t h a ta s" f l y - b y - w i r e "c o n t r o ls y s t e m sa r ei n s t a l l e d more w i d e l y i n a i r c r a f t a t r e n dt o w a r ds t a n d a r d i z a t i o n of c o n t r o lf e e l , r i d i n g q u a l i t i e s , and h a n d l i n g q u a l i t i e s f o r a i r c r a f t of a l l s i z e s , speeds, and w e i g h t s will develop.

The discussionabovehas assumed t h a t t h e a i r mass throughwhichan a i r c r a f t i s f l y i n g i s u n i f o r m l y s t a t i o n a r y w i t h r e s p e c t t o i n e r t i a l space.

Nature,however, i s seldom so accomodating,and many a i r c r a f t e x p e r i e n c e s e v e r e d e t e r i o r a t i o n i n h a n d l i n g q u a l i t i ' e s d u r i n g f l i g h t i n t u r b u l e n t a i r (Ref. 5 5 ) . A n a l y s i sr e p o r t e di nR e f e r e n c e 4 I n d i c a t e st h a t a f i r s t o r d e r t r e a t m e n t of theproblemcanbeperformed.Gustsareconsidered t o be i s o - t r o p i c away f r o mt h eg r o u n d ,G u s tv e l o c i t yv a r i a t i o nw i t hs p a t i a lf r e q u e n c y f o r t w o m o d e l s i s used t o o b t a i n g u s t components o f u ( x ) , v ( x ) , andw(x,y).

From these,onecancalculategustcomponents for a, ( 3 , p, q, and r and, s u b s e q u e n t l y ,t h ea l t e r e dv a l u e so ft h es t a b i l i t yd e r i v a t i v e s . I t i s seen t h e r e f o r et h a ta l la i r f r a m e dynamic modes c a nb ee x c i t e di ng u s t s . I f t h e s e a r ei n s u f f i c i e n t l y damped, i f t h el a t e r a l - d i r e c t i o n a lc o u p l i n gi se x c e s s i v e , o r i f t h e s t a t i c s t a b i l i t y i s m a r g i n a l , t h e p i l o t ' s work load will i n c r e a s e markedly i f he a t t e m p t s t o m a i n t a i n a r e a s o n a b l yp r e c i s ec o u r s e or comfor- t a b l er i d e .O u t s t a n d i n gh a n d l i n gq u a l i t i e si ns t i l la i ra r ec o n s e q u e n t l y a n e c e s s a r yp r e r e q u i s i t et oa c c e p t a b l eh a n d l i n gi nt u r b u l e n t a i r .

INERTIAL CHARACTERISTICS

T h e a i r p l a n e d e s i g n e r i s d i r e c t e d t o t h e s e c t i o n of t h i s s t u d y d e a l i n g w i t h s p i n e n t r y for a d i s c u s s i o n of dynamic response r e s u l t i n g from changes i n moments of i n e r t i a ;t e c h n i q u e s f o r e s t i m a t i n g moments of i n e r t i aa r ed e t a i l e dh e r e .

When a n a i r p l a n e i s r o t a t e d a b o u t i t s c e n t e r of g r a v i t y , t h e r e s u l t i n g t o r q u e , I ? , i s equal t o t h e p r o d u c t o f t h e moment of i n e r t i a a b o u t t h e c.g.

and t h ea n g u l a ra c c e l e r a t i o n ( r = I d d d t ) .S i n c et h et o r q u ei sa p p l i e db y a c o n t r o ls u r f a c e ,t h ea n g u l a ra c c e l e r a t i o nc a n befound f o r a p a r t i c u l a r c o n t r o ld e f l e c t i o n by d i v i d i n gt h et o r q u e by t h e moment o fi n e r t i a . The t h r e ea n g u l a rd e g r e e s of freedom f o r t h e a i r p l a n e a r e p i t c h , r o l l , and yaw; thus, it i s necessary t o know t h e moments o fi n e r t i aa b o u tt h e x, y, and z a i r p l a n e body axes. One p r o d u c t - o f - i n e r t i at e r m , Ixz, a l s oa p p e a r si nt h e e q u a t i o n so fm o t i o n( s e eA p p e n d i x A ) , b u t t h i s t e r m i s u s u a l l y s m a l l , and, f o r many a n a l y s e s ,i sc o n s i d e r e dz e r o ,i n d i c a t i n gt h a ta l lt h ea x e sa r e p r i n c i p a l a x e s .

Moments o f i n e r t i a canbeobtainedbyexperimentalmeasurements o rc a n be e s t i m a t e df r o ma i r p l a n e mass and g e o m e t r i cc h a r a c t e r i s t i c s . The usual method f o r d e t e r m i n i n ge x p e r i m e n t a l l yt h e moments o f i n e r t i a i s a pendulum method.

The a p p l i c a t i o n o f t h e pendulum method t ot h ee x p e r i m e n t a ld e t e r m i n a t i o n of moments o fi n e r t i ao fa i r p l a n e si sd i s c u s s e di n TR-467 ( R e f . 6 8 ) . The moments o f i n e r t i a a b o u t t h e x and y a x e sa r ef o u n db ys w i n g i n gt h ea i r p l a n e as a compound pendulum, whereasthe moment o f i n e r t i a a b o u t t h e z a x i s i s d e t e r m i n e db yo s c i l l a t i n gt h ea i r p l a n ea s a b i f i l a r - t o r s i o n a l pendulum. The d i f f e r e n t i a le q u a t i o nw h i c hd e s c r i b e st h e pendulum motion i s o f t h e form,

d 2 8 + be = 0 where I = measured moment o f i n e r t i a

I F

b = c o n s t a n td e p e n d i n go nt h ew e i g h t and dimensions of thependulum 8 = angulardisplacement.

The p e r i o dc a nb ew r i t t e na s T = Z ' I T / / ~ , and t h e moment o f i n e r t i a canbe s o l v e d by s o l v i n gf o r I. For each o f t h ec a s e sm e n t i o n e da b o v e ,t h et r u e moments o fi n e r t i aa r ed e t e r m i n e d by c o r r e c t i n g t h e measured moments o f i n e r t i a f o r ( 1 ) t h eb u o y a n c yo ft h es t r u c t u r e , ( 2 ) t h e a i r e n t r a p p e d w i t h i n t h e s t r u c t u r e , and ( 3 ) t h ea d d i t i o n a l mass e f f e c t . T h e s e t h r e e f a c t o r s c a u s e an a p p a r e n ta d d i t i o n a l moment o fi n e r t i a ,w h i c hi se v a l u a t e do nt h eb a s i so f ( 1 ) t h e a i r p l a n e s i z e and shape normal t o t h e d i r e c t i o n of m o t i o n and ( 2 ) t h e r e s u l t so ft e s t so ft h ea d d i t i o n a l mass e f f e c to ff l a tp l a t e s .R e f e r e n c e6 8 should be used t o e v a l u a t et h er e q u i r e dc o r r e c t i o n s . The a d d i t i o n a l mass e f f e c t (moment o fi n e r t i ai n f l u e n c e d by t h es u r r o u n d i n g medium) r e s u l t s from t h e f a c t t h a t t h e p e r i o d o f t h e p e n d u l u m ' s v i b r a t i n g i n a i r i s t o some e x t e n td e p e n d e n t o nt h e momentum i m p a r t e db yi t sm o t i o nt h r o u g ht h ea i r .T h e momentum imparted t o t h e body i sp r o p o r t i o n a l t o t h e momentum o f t h e b o d y ;t h u s ,t h ee q u i v a l e n t a d d i t i o n a l mass may be used.

The p r e c i s i o no ft h ep e n d u l u m method t o e s t i m a t e t r u e moments of i n e r t i a i sa p p r o x i m a t e l y k2.5 p e r c e n t , t 1 . 3 percent,and kO.8 p e r c e n t f o r t h ex , y, and z a x e sr e s p e c t i v e l y .S e v e r a lt y p e so fa i r p l a n e s of g r o s sw e i g h tl e s st h a n 10,000 pounds have been t e s t e d a t t h e NACA l a b o r a t o r i e s ,t h er e s u l t s of which werecompiledandpublishedas NACA TN-780 ( R e f . 6 9 ) whichsupersedes TN-375.

O n l y a few of t h e a i r p l a n e s t e s t e d a r e c h a r a c t e r i s t i c of t h e l i g h t a i r p l a n e d e s i g n s of t o d a y .T h i sr e p o r t , however, i s v a l u a b l ei no b t a i n i n ga na p p r o x i - mate number f o r t h e moments of i n e r t i a and t h er a d i i of gyration. Agard Report 2 2 4( R e f .1 0 1 )a l s od i s c u s s e sm e t h o d so fo b t a i n i n gt h e moments of i n e r t i a by t h es p r i n go s c i l l a t i o n method. Schematic representation of t y p i c a l methods f o r d e t e r m i n i n g( w i t h i n 5% or b e t t e r ) t h e r o l l i n g and p i t c h i n g moments o f i n e r t i a a r eg i v e na sw e l la so t h e rr e f e r e n c e sw h i c h may be h e l p f u li nu s i n gt h e s e methods.

Because o f t h ee q u i p m e n tr e q u i r e d t o measure t h e moment of i n e r t i a by t h e pendulummethod,probablythemostpopularmethodutilizestheweightand lo- c a t i o no f component p a r t so u t l i n e di n TN-575 ( R e f .7 0 ) .T h i si s a step-by-step, t a b u l a t e d methodwhich y i e l d s t h e moments of i n e r t i a a b o u t t h e t h r e e a x e s , t h e p r o d u c t s of i n e r t i a , and t h ec e n t e r of g r a v i t yl o c a t i o n s . I t i s b e l i e v e dt h a t t h e moments o f i n e r t i a c a nb ee s t i m a t e dw i t h i n 10% by t h i s method and can be a p p l i e d by c o m p l e t i n gT a b l e 14, w h i c hi se x p l a i n e db e l o w .

The f i r s t s t e p i n t h e p r o c e d u r e i s t o d e f i n e a s e t of t h r e em u t u a l l y p e r p e n d i c u l a rr e f e r e n c ep l a n e s , x’z: y’z: and x’y’, as shown i n t h e f i g u r e below.

\ -y 1 2‘

\ R e f e r e n c ep l a n e

/

k X I 21 R e f e r e n c e p l a n e \

\ 1 - y l R e f e r e n c e a x i s

F i g u r e 66. S e t of t h r e em u t u a l l yp e r p e n d i c u l a rr e f e r e n c ep l a n e s .

The p l a n e of symmetry i s c h o s e na st h ex * z *r e f e r e n c ep l a n e ,s i n c e it c o n t a i n s t h ec . g . The y * z *r e f e r e n c ep l a n ei su s u a l l ys e ta tt h en o s e of t h ea i r p l a n e , w h i . l e t h e x * y * r e f e r e n c e p l a n e c a n b e e s t a b l i s h e d a t t h e t o p o r t h e b o t t o m .

These r e f e r e n c e p l a n e s d e f i n e t h e o r i g i n o f t h e c o o r d i n a t e a x e s .

The t a b l ew h i c hm u s t becompleted i s shown belowandexplainedcolumn bycolumn.

( 2 ) (31 ( 5 ) ( 6 ) ( 7 1 ( 8 1 ( 9 )

-

X N X N N N 3 3 3 X r u 3 5 t Tota I s Table 14. Sample t a b l ef o ro b t a i n i n gi n e r t i a lc h a r a c t e r i s t i c s .

Column ( 1 ) Elements o ft h ea i r p l a n ew h i c ha r ec o n s i d e r e d( f l a p s , wheels, baggage doors, engine, seats, p i l o t ,f u e l , e t c . 1.

Column ( 2 ) Weight of each i n d i v i d u a l e l e m e n t .

Columns (31, (41, & ( 5 ) D i s t a n c ef r o mt h ex * ,y * and z * axes t o t h ee l e m e n t . I t i si m p o r t a n tt on o t et h a t some o f t h e s e d i s t a n c e s may be e i t h e r pos’i t i v e o r n e g a t i v e .

Columns ( 6 ) 8, ( 7 ) Moments c o n t r i b u t e d by each element. Since the a i r p l a n e i s symmetricaboutthe x * z * plane, wy = 0; t h e r e f o r e , it does notappear.

Columns (81, ( 9 1 , & (10) The p r o d u c to ft h es q u a r e so ft h ei n d i - v i d u a ld i s t a n c e s and t h ei t e mw e i g h t .

Columns (111, (121, & ( 1 3 ) The e s t i m a t i o no ft h e moments o fi n e r - t i a o f t h e l a r g e r i t e m s a b o u t t h e i r own c e n t e r of grav i t y .

Column (14) The t o t a lp r o d u c to f‘ i n e r t i a , Ixz.

The i n f o r m a t i o n i n t h e above columns can be used t o f i n d and zc. 1 ( 1 ) c.g. Location--The x and z c.g. l o c a t i o n s ( x c.g.

c a nb ef o u n db yd i v i d i n gt h et o t a l so fc o l u m n s ( 6 ) and ( ? j r e s p e c t i v e l y by t h et o t a lw e i g h t .

( 2 ) Moments o fI n e r t i a - - T h et o t a l moments of i n e r t i a of t h e a i r - p l a n e a b o u t t h e t h r e e r e f e r e n c e a x e s a r e

Ix*x* = Cwy2 + C W Z ~ + Z A I x x

I y ’y ’ = Cwx2 + Cwy2 + Z A I y y

= Cwx2 + Cwy2 + C A I z z

I, ’z The moments of i n e r t i a a b o u t t h e c.g. c a nb ew r i t t e na s

- 2

I y y - Iy’y’ - W(Xc.g.

+ zc.g. 2,

I , , = I x y - w(zc,g.2)

I , , = - W(XC. g . 2, where W = t o t a l a i r p l a n e w e i g h t ( 3 ) P r o d u c t of I n e r t i a - - T h e p r o d u c t of i n e r t i a , Ixz, a b o u tt h e c.g.canbefoundbytheformula, I,, = cwxz - W ( X c a g . + zc.g.

Even for t h el a r g ei t e m s ,t h ep r o d u c t s of i n e r t i a a b o u t t h e i r own c.g.can u s u a l l y b e n e g l e c t e d . A l s o , f o r l i g h ta i r c r a f t ,t h ep r o d u c t so fi n e r t i a for t h ec o m p l e t ea i r c r a f ta r eu s u a l l ys m a l lo rn e g l i g i b l e . Thus, f o r a f i r s t approximation,assumingtheproducts o f i n e r t i a t o be z e r o i s f a i r l y a c c u r a t e .

I f g r e a t e ra c c u r a c yi sr e q u i r e d ,t h ef o r m u l aa b o v ec a n be used.

Some t y p i c a lv a l u e so f moments o fi n e r t i ac a nb e seen f o r l i g h t a i r c r a f t b ye x a m i n i n gt h et a b l eb e l o w : A i r p l a n e Number I x x I Y Y I z z Weight of ( i b s . 1 Eng i nes ( s 1 ug-f t2 ( s I u g - f t 2 1 ( s I u g - f t 2 1 902 1335 1922 2650 94 1 1479 21 10 1495 2207 2878 8884 1939 11,001 5,000 64,811 17,300 64,543 21 00 766 1275 1805 T a b l e 15. T y p i c a ll i g h ta i r c r a f t moments of i n e r t i a .

STALL

The flow o v e r a body i s s a i d t o s t a l l when t h ep r e s s u r eg r a d i e n t becomes so u n f a v o r a b l e t h a t t h e v e l o c i t y a t t h e s u r f a c e i s z e r o and t h em a i n s t r e a m i s no longer attached t o t h e body. When s t a l lo c c u r so na i r f o i l s ,t h e r ei sb o t h a loss o f l i f t and an i n c r e a s ei nd r a g .I ng e n e r a l ,s t a l lc a n be d e f i n e da s t h a t flow c o n d i t i o nw h i c hf o l l o w st h ef i r s tl i f t - c u r v e peak. TN-2502 (Ref. 71) i n d i c a t e st h r e ec l a s s i f i c a t i o n sf o rs t a l la t low speeds: 1 ) t r a i l i n g - e d g e s t a l l , 2 ) leading-edge s t a l l , and 3 ) t h i n a i r f o i l s t a l l .

T r a i l i n g - e d g es t a l li sp r e c e d e d by t h e movement o f t h e p o i n t o f t u r b u l e n t b o u n d a r y - l a y e rs e p a r a t i o nf o r w a r d from t h e t r a i l i n g edge w i t hi n c r e a s i n ga n g l e o f a t t a c k ( u s u a l l y c h a r a c t e r i s t i c o f a i r f o i l s o f 15% t h i c k n e s s or m o r e ) .T h i s t y p e o f s t a l l i s d i s t i n g u i s h e d by a g r a d u a l ,c o n t i n u o u sf o r c e and moment v a r i - a t i o n w i t h a w e l l - r o u n d e dl i f t - c u r v e peak.

The leading-edge s t a l l i s an a b r u p tf l o ws e p a r a t i o no ft h el a m i n a r bound- a r yl a y e rn e a rt h el e a d i n g edge, g e n e r a l l yw i t h o u ts u b s e q u e n tf l o wr e a t t a c h m e n t ( u s u a l l yt y p i c a l o f a i r f o i l s 9% t o 15% t h i c k ) . L i t t l e o r no change i n l i f t - c u r v e> l o p es h o u l d b ee x p e c t e dp r i o r t o maximum l i f t andanabrupt,often s u b s t a n t i a l ,d e c r e a s ei nl i f ts h o u l do c c u ra f t e r maximum l i f ti sa t t a i n e d . A c o m b i n e d l e a d i n g - a n d t r a i l i n g - e d g e s t a l l i s p o s s i b l e .

The t h i n a i r f o i l s t a l l i s p r e c e d e db yf l o ws e p a r a t i o nf r o mt h el e a d i n g edge, w i t ht h er e a t t a c h m e n tp o i n tm o v i n gp r o g r e s s i v e l yd o w n s t r e a mw i t hi n c r e a s i n g a n g l eo fa t t a c k .T h i st h i na i r f o i ls t a l l has a rounded l i f t - c u r v e peak, gen- e r a l l y preceded by a d i s c o n t i n u o u sf o r c e and moment v a r i a t i o n f o r a i r f o i l s w i t h a rounded leading edge.

The s t a l l sm e n t i o n e d above a r e shown i nt h ef i g u r eb e l o w .

Angle of Attack -

F i g u r e6 7 .C h a r a c t e r i s t i cs t a l lt y p e s .

N o t e v e r y a i r f o i l c a n be c l a s s i f i e du n i q u e l yi n t o a g i v e ns t a l lc a t e g o r y ,n o r i s e a c h t y p e o f s t a I I I i m i t e d t o a c e r t a i n r a n g e o f t h i c k n e s s r a t i o s .

S i n c e s t a l l c h a r a c t e r i s t i c s p l a y a m a j o rr o l ei nt h ed e s i g n o f any a i r - plane, it i s i m p o r t a n t t o know a t t h e o u t s e t a t whatangle of a t t a c k t h e s t a l l occurs,dependingontheweight, speed, and maximum l i f t c o e f f i c i e n t o f t h e a i r p l a n e . The a b r u p ts t a l l has proven t o be q u i t e dangerous because the p i l o t r e c e i v e s l i t t l e o r n ow a r n i n gb e f o r ea t t a i n i n g a c r i t i c a la t t i t u d e .A l s o , i f s t a l lo c c u r sd u r i n gl a n d , i n go rt a k e - o f f ,t h ep i l o t has l i t t l e o r no a l t i t u d e i nw h i c h t o r e c o v e r . Many p a r a m e t e r sa f f e c tt h es t a l lo f a body or wing: t a p e r ,a s p e c t ,a n dt h i c k n e s sr a t i o s ;R e y n o l d s number; camber; washout; and l e a d i n g - e d g es h a p e .B e f o r ec o n s i d e r i n gt h ep a r a m e t e r sv h k h a f f e c ts t a l l , some mentionshould be made o f t h er e q u i r e m e n t s f o r good s t a l lw a r n i n g andrecovery.

4 ) d e f i n e ss t a l lw a r n i n gr e q u i r e m e n t s , T e c h n i c a lR e p o r t AFFDL-TR-69-72 (Ref.

s t a l lc h a r a c t e r i s t i c s , and s t a l lr e c o v e r yr e q u i r e m e n t sf o rl i g h tm i l i t a r y a i r c r a f t .

Sta I I Warning I a p r o a c hs h a l l be accompanied by an e a s i l y The s t a p e r c e p t b l e w a r n i n g .

Acceptab I e warn i ng f o r a l l t y p e s o f s t a l l s c o n s i s t s o f s h a k i n g o f t h e c o c k p i tc o n t r o l s , bu f f e t i n go rs h a k i n g of t h ea i r p l a n e ,o r a c o m b i n a t i o no f these. The o n s e t o f t h i sw a r n i n gs h o u l do c c u rw i t h i nt h er a n g e ss p e c i f i e d by t h et w ot a b l e sb e l o w Warning speed f o r s t a l l s a t 1g normal t o t h ef l i g h tp a t h . Warning o n s e t f o r s t a l l s a t l g normal t o t h e f l i g h t p a t h s h a l l o c c u r be- I tween t h e f o l l o w i n g l i m i t s : Minimum S t a l l Warning Maximum Sta I I Warning F I i g h t Phase Speed Speed t Approach Higher o f 1.05Us o r H i g h e r o f 1.1OUs or

Us + 5 k n o t s Us + 10 k n o t s

AI I O t h e r H i g h e r o f 1.05Us o r H i g h e r o f 1.15Us o r

Us + 5 k n o t s Us + 15 k n o t s

T a b l e 16. Warning speed f o r s t a l l s a t l g normal t o t h e f l i g h t p a t h .

- Warning range f o r a c c e l e r a t e ds t a l l s .O n s e to fs t a l lw a r n i n gs h a l l o c c u ro u t s i d et h eO p e r a t i o n a lF l i g h tE n v e l o p ea s s o c i a t e dw i t ht h e A i r p l a n e Normal S t a t e and w i t h i n t h e f o l l o w i n g a n g l e - o f - a t t a c k ranges: F I i g h t Minimum S t a l l Warning Maximum S t a l l Warning Phase Angle o f A t t a c k Angle o f A t t a c k

a0 + 0.82 ( a , - a,) a . + 0.90 (a, - a,)

Approach

AI I O t h e r a . + 0.75 (as - a,) a0 + 0.90 ( a , - ao)

where as i s t h e s t a l l a n g l e o f a t t a c k and a, i s t h e a n g l e o f a t t a c k f o r z e r o I i f t .

T a b l e 17. Warning range f o r a c c e l e r a t e ds t a ti s .

S t a l l a n g l e o f a t t a c k i s t h e a n g l e o f a t t a c k a t c o n s t a n t speed f o r t h e c o n f i g u - r a t i o n ,w e i g h t , a n dc . g .p o s i t i o nw h i c hi st h el o w e s to ft h ef o l l o w i n g : ( a > The a n g l e o f a t t a c k f o r t h eh i g h e s ts t e a d y l o a df a c t o r ,n o r m a l t o t h e f l i g h t p a t h , t h a t canbeobtained a t a g i v e n Mach number; ( b ) The a n g l eo fa t t a c kf o r a g i v e n speed o r Mach number a t w h i c h u n c o n t r o l l a b l e p i t c h - i n g ,r o l l i n g ,o ry a w i n go c c u r s( i . e . , l o s s o fc o n t r o la b o u t a s i n g l e a x i s ) ; ( c l A n g l eo fa t t a c kf o r a g i v e n speed o r Mach number, a t w h i c hi n t o l e r a b l eb u f f e t i n gi s encountered.

The i n c r e a s ei nb u f f e t i n gi n t e n s i t yw i t hf u r t h e ri n c r e a s ei na n g l eo fa t t a c k s h o u l db es u f f i c i e n t l y marked t o be n o t e db yt h ep i l o t .T h i sw a r n i n g may be p r o v i d e d a r t i f i c i a l l y o n l y i f it can be shown t h a tn a t u r a ls t a l lw a r n i n gi sn o t f e a s i b l e .

S t a l lC h a r a c t e r i s t i c s I nu n a c c e l e r a t e ds t a l l s , +he a i r p l a n es h a l n o t e x h i b i tu n c o n t r o l l a b l er o l l i n g , yawing, o r downward p i t c h i n g a t t h e s t a l l i n e x c e s s o f 200.

t e d or a c c e l - I t i s d e s i r e d t h a t no p i t c h - u pt e n d e n c i e so c c u ri n u n a c c e l e r a t c h may be a c c e p t a b l e e r a t e ds t a l l s .I nu n a c c e l e r a t e ds t a l l s ,m i l d nose-up p i f n oe l e v a t o rc o n t r o lf o r c er e v e r s a lo c c u r s and i f nodangerous,unrecoverable, o r o b j e c t i o n a b l e f l i g h t c o n d i t i o n s r e s u l t . A m i l d nose-up tendency may be a c c e p t a b l ei na c c e l e r a t e ds t a l l s i f t h eo p e r a t i o n a le f f e c t i v e n e s s of t h e a i r - p l a n e i s n o t compromised and ( a ) The a i r p l a n e has adequate s t a l l warning; ( b )E l e v a t o re f f e c t i v e n e s si ss u c ht h a t it i s p o s s i b l e t o s t o p t h e p i t c h - u p p r o m p t l y a n dr e d u c et h ea n g l e o f a t t a c k ; ( c ) A t n op o i n td u r i n gt h es t a l l ,s t a l l ap- proach, o r r e c o v e r y d o e sa n yp o r t i o no f t h ea i r p l a n e exceed s t r u c t u r a l limit loads.

Theserequirementsapply t o a l l s t a l l s r e s u l t i n g f r o m r a t e s o f speed r e d u c t i o n up t o f o u rk n o t sp e rs e c o n d (4.6 m i l e sp e rh o u r ,p e rs e c o n d ) .S t a l lc h a r a c - t e r i s t i c sa r eu n a c c e p t a b l e i f a s p i n i s l i k e l y t o r e s u l t .

S t a l lP r e v e n t i o n and Recovery I t s h a l l be p o s s i b l e t o p r e v e n t t h e c o m p l e t e s t a l l bymoderateuse o f t h e c o n t r o l s a t t h e o n s e t o f t h e s t a l lw a r n i n g .I ts h a l l be p o s s i b l e t o r e c o v e r f r o m a c o m p l e t es t a l l by use o ft h ee l e v a t o r ,a i l e r o n s , and r u d d e rc o n t r o l sw i t hr e a s o n a b l ef o r c e s , and t o r e g a i nl e v e lf l i g h tw i t h o u te x c e s s i v e loss o f a l t i - t u d e o r b u i l d - u p of a i r s p e e d .T h r o t t l e ss h a l lr e m a i n f i x e d u n t i l speed has begun t o i n c r e a s e when anangle of a t t a c kb e l o wt h es t a l lh a s been regained. In the s t r a i g h t - f l i g h t s t a l l s w i t h t he a i r p l a n et r i m m e d a t a speed n o tg r e a t e rt h a n 1.4 Us and w i t h a speed r e - d u c t i o n r a t e o f a t l e a s t 4.0 k n o t sp e r second, e l e - v a t o r c o n t r o l powersha I I be s u f f i c i e n t t o r e c o v e r f r o m a n y a t t a i n a b l e a n g l e o f a t t a c k . On m u l t i e n g i n e a i r c r a f t , it s h a l l be p o s s i b l e t o r e c o v e r s a f e I y from s t a l I s w i t h t h e c r i t i c a l e n g i n ei n o p e r a t i v e .

T h i sr e q u i r e m e n ta p p l i e sw i t ht h er e m a i n i n ge n g i n e s up t o t h r u s t f o r l e v e l f l i g h t a t 1.4 US, b u tt h e s ee n g i n e s may be t h r o t t l e db a c kd u r i n gr e c o v e r y .

Although some l i g h t a i r c r a f t may h a v es p e c i a ls t a l l i n gp r o b l e m s ,s u c ha s n a c e l l e s s t a l l i n g a t h i g h speedswhere theycanproduce a s i g n i f i c a n t l i f t , c a u s i n g t h e a i r p l a n e t o a t t a i n d a n g e r o u s a t t i t u d e s , t h e s t a l l p r o b l e m i s u s u a l l ys o l v e d i f t h e w i n g has good s t a l lc h a r a c t e r i s t i c s . NACA TR-703 (Ref. 72) i s a s e t of d e s i g nc h a r t sp r e p a r e d t o show t h ee f f e c t so fw i n gg e o m e t r yo nt h e s t a l l i n g c h a r a c t e r i s t i c s o f taperedwings; a summary o f t h e s e e f f e c t s i s p r e s e n t e di nT a b l e 18.

I

I

I GEOMETR I C PROPERTY/W I NG EFFECTS ON STALL

Taper I n c r e a s i n g t a p e r t e n d s t o move t h e s t a l l i n g p o i n t p r o g r e s s i v e l y o u t b o a r d and t o decrease t h e s t a l l i n g m a r g i n o f most of t h er e m a i n i n g wing.

A s p e c tR a t i o An i n c r e a s ei na s p e c tr a t i ot e n d s t o f l a t t e n t h e s e c t i o n l i f t d i s t r i b u t i o n ; up t o an as- p e c t r a t i o of 18, t h e e f f e c t o n t h e s t a l l i n g p o i n ti sr e l a t i v e l ys m a l l .

Th i ckness An i n c r e a s ei nr o o tt h i c k n e s sr a t i o beyond 0.15 c a u s e s t h e s t a l l i n g p o i n t t o move i n - board,except f o r t h e l o w e s t va I ues of Reynolds numbers t e s t e d , and t e n d s t o r e - duce t h e r a t e a t w h i c h t h e s e c t i o n l i f t and maximum s e c t i o n l i f t d i v e r g ei n b o a r d o f t h e i n i t i a l s t a l l i n g p o i n t .

Camber I n c r e a s i n g camber l i n e a r l y f r o m r o o t t o t i p ( 4 % ) appears t o be u s e f u l t o good s t a l l c h a r a c t e r i s t i c s o n l y f o r low t a p e rr a t i o s .

Washout Although washout i se x p e n s i v ei nr e g a r dt o d r a gc o n s i d e r a t i o n s , i t o f f e r s an e f f e c t i v e means o f i m p r o v i n gs t a l l i n gc h a r a c t e r i s t i c s .

Washout becomes more e f f e c t i v e asReynolds number i s increased.

Reyno I ds Number An i n c r e a s e i n R e y n o l d s number t e n d s t o move t h ei n i t i a ls t a l l i n gp o i n ti n b o a r d .

I

Sharp Leading Edge A sharp leading edge reduces the maximum s e c t i o n l i f t c o e f f i c i e n t so t h a t s t a l l i n g takesplaceinboardwherethesharpedge i s l o c a t e d .

T a b l e 18. E f f e c t so fw i n gg e o m e t r yo ns t a l l c h a r a c t e r i s t i c s o f t a p e r e d w i n g s .

Thus, f o r good s t a l lc h a r a c t e r i s t i c s ,w i n g sw i t h low t a p e rr a t i o s ,m o d e r a t e t h i c k n e s s( 1 0 % t o 15%), low t o moderateReynolds numbers, lo t o 3O washout, and a sharpleadingedge a t t h e r o o t seem t h e m o s td e s i r a b l e .

W R L-145 (Ref. 73) p o i n t so u tt h a ts u r f a c er o u g h n e s sa n dp r o p e l l e re f f e c t s s h o u l da l s o b ec o n s i d e r e da sp a r a m e t e r sa f f e c t i n ga i r p l a n es t a l l . The a i r - p l a n e st e s t e d f o r t h e r e p o r t were m i l i t a r y a i r c r a f t o f t h e m i d 1940's. Because of t h e armament r e q u i r e d , t h e a i r p l a n e s wereequippedwith numerousaccess d o o r s ,i n s p e c t i o np l a t e s , and numerous o t h e rf e a t u r e st h a tt e n d t o make t h e wing extremely rough and allow a i r leakage through it. I t was f o u n d , t h a t a wingwhich was f a i r e d and sealedgave a h i g h e r C L ~ ~ ~ t h e na nu n f a i r e dw i n g , e v e nt h o u g hb o t hw i n g ss t a l l e da ta p p r o x i m a t e l yt h e same a n g l e o f a t t a c k .

T h es t u d ya l s oi n d i c a t e dt h a tp r o p e l l e ro p e r a t i o ng e n e r a l l yi n c r e a s e st h e s e v e r i t y o.f t h es t a l l ,e s p e c i a l l yo ns i n g l e - e n g i n ea i r p l a n e s . The r o t a t i o n w i t h i n t h e s l i p s t r e a m i n c r e a s e s t h e e f f e c t i v e a n g l e o f a t t a c k of t h e w i n g s e c t i o nb e h i n dt h eu p - g o i n gp r o p e l l e rb l a d e s and d e c r e a s e st h ee f f e c t i v ea n g l e o f a t t a c ko ft h ew i n gs e c t i o nb e h i n dt h e down-going p r o p e l l e rb l a d e s . An a s y m m e t r i c a ls t a l lp a t t e r ni st h u sp r o d u c e d ,l e a d i n gt o a sometlme severe rot I .

WR L-296 (Ref. 74) i n d i c a t e st h a ta i r p l a n e so ft h e 1930's s o l v e dt h e p r o b l e m o f s t a l l by p r o v i d i n g a d e f i n i t ew a r n i n g of t h ea p p r o a c h i n gs t a l lt h r o u g h backward movement, p o s i t i o n , and f o r c e s on the control column. Monoplanes u s u a l l y had l i t t l e o r no t a p e r w i t h " i n e f f i c i e n t " w i n g - f u s e l a g e j u n c t u r e s , c a u s i n g a g r a d u a l l yd e v e l o p i n gs t a l l ,b e g i n n i n ga tm i d s p a n . Thus, t h es t a l l e d c o n d i t i o nd e v e l o p e dp r o g r e s s i v e l ya f t e r a r e a s o n a b l yd e f i n i t ew a r n i n g ;a l s o , l a t e r a lc o n t r o l was o f t e nm a i n t a i n e d up t o o r beyond t h e s t a l l . Many o f t h e a i r p l a n e sd e s i g n e di nt h e 1940's d e p i c t e dt r e n d st o w a r dh i g h e rw i n gl o a d i n g s andlandingspeeds;lessemphasis was p l a c e do ns t a l l i n gt e n d e n c i e s ,t h u s l e a d i n gt oa i r p l a n e sw i t hp o o rs t a l l i n gc h a r a c t e r i s t i c s . WR L-296 a l s o i n d i c a t e st h a ts h a r pl e a d i n g edges a t t h e w i n gr o o t may i m p r o v ep o o rs t a l l t e n d e n c i e s . A n o t h e r p r o p o s e d s o l u t i o n i s l i m i t a t i o n of l o n g i t u d i n a l c o n t r o l t o p r e v e n tt h ew i n gf r o mr e a c h i n g maximum l i f t . To be e f f e c t i v e , a warning m u s t o c c u r a t a n a n g l e o f a t t a c k c o n s i d e r a b l y b e l o w t h a t o f maximum l i f t becausegusts or i n e r t i a e f f e c t s may m o m e n t a r i l y c a r r y t h e a i r p l a n e beyond t h ew a r n i n ga t t i t u d e . An i n v e s t i g a t i o n was made i n WR L-296 (Ref. 74) o f a " s t a l l - c o n t r o lf l a p . " The b a s i cw i n gt e s t e d was a 23012 s e c t i o nw l t h a 60% c h o r df l a p . The idea was t od e f l e c tt h es t a l l - c o n t r o lf l a p so t h a t t h e m o d i f i e da i r f o i lw o u l dh a v e a shape s i m i l a r t o t h e 4412 a i r f o i l , whichhas good s t a l lc h a r a c t e r i s t i c s because o f i t s r e l a t i v e l y f l a t l i f t - c u r v e peak.

T'he f l a p was c o n s i d e r e da e r o d y n a m i c a l l ys a t i s f a c t o r yw i t h or w i t h o u th i g h l i f t devices, even though it may be veryexpensive t o i n s t a l l . The graph b e l o wi n d i c a t e st h ee f f e c to ft h ef l a pd e f l e c t i o no nI i f t - c u r v e shape.

3 . 0 -.2 F i g u r e 68. I n v e s t i g a t i o no f a s t a l lc o n t r o l f l a pi n W R L-296 (Ref. 7 4 ) .

NACA TN-1868 (Ref. 7 5 ) i s a s t u d y made t o c o r r e l a t e p i l o t s ' o p i n i o n s o f t h es t a l lw a r n i n gp r o p e r t i e so f 16 a i r p l a n e s ,r a n g i n g from s i n g l e - e n g i n ef i g h t e r s t o four-engine bombers, w i t h a number of q u a n t i t a t i v e f a c t o r s o b t a i n e d from t i m e ,h i s t o r yf l i g h tr e c o r d s f o r speeds near s t a l l . The a l t i t u d et e s tr a n g e was 4,000 t o 12,000 f e e t , w i t h s t a l l s a t t a i n e d i n s t r a i g h t f l i g h t by g r a d u a l l y a p p r o a c h i n gt h es t a l lw i t ht h e normal a c c e l e r a t i o nf a c t o ra sc l o s et ou n i t ya s p o s s i b l e . The t e s t si n d i c a t et h a t ,i ng e n e r a l ,t h es t a l !w a r n i n gi sc o n s i d e r e d s a t i s f a c t o r y by t h e p i l o t s when c h a r a c t e r i z e db y any of t h e f o l l o w i n g q u a l i t i e s : ( a )A i r p l a n eb u f f e t i n ga t speeds 3 t o 15 mph above s t a l l i n g speedand o f a magnitude t o p r o d u c ei n c r e m e n t a li n d i c a t e dv a l u e s of normal a c c e l e r a t i o nf a c t o rf r o m 0.04 t o 0.22; ( b lP r e l i m i n a r yc o n t r o l l a b l er o l l i n gm o t i o n f r o m 0.04 t o 0.06 radianspersecondoc- c u r i n g anywhere w i t h i n a range o f 2 t o 12 mph above t h e s t a l l i n g speed; ( c ) A t l e a s t 2.75 i n c h e s r e a r w a r d t r a v e l o f t h e c o n t r o l s t i c k d u r i n g t h e 15 mph speedrangeimmediatelyprecedingthe s t a l 1 .

The t w og r a p h sb e l o wi n d i c a t et h es a t i s f a c t o r y and u n s a t i s f a c t o r yr a n g e so f b o t hn o r m a la c c e l e r a t i o ni n c r e m e n t s ,F i g u r e 6 9 , and rearward movement of t h e c o n t r o ls t i c k ,F i g u r e 70, v e r s u s s p e e d a b o v e t h e s t a l l . Each l i n er e p r e s e n t s a s e p a r a t et e s to f one o ft h e 16 a i r p l a n e s . A , i s t h e normal a c c e l e r a t i o n increment.

0 4 8 12 16 20 24 28 Speed above stall (rnph) F i g u r e 6 9 . C o r r e l a t i o n of p i l o to p i n i o no fs t a l lw a r n i n gw i t h a i r p l a n e b u f f e t a t v a r i o u s speedsabove s t a l l .

Speed above stall, mph F i g u r e 70. C o r r e l a t i o n of p i l o t o p i n i o no fs t a l lw a r n i n gw i t h r e a r w a r d movement o f c o n t r o l s t i c k i n t h e speed r a n g ei m m e d i a t e l ya b o v es t a l l .

I n a summary o f v a r i o u ss t a lI - w a r n , i n gd e v i c e s , TN-2676 (Ref. 76) p o i n t s t o t h e fact t h a t a w a r n i n g l i g h t , s t a l [-warning d i a l ,r u d d e rs h a k e r ,u r . s t i c k shaker may benecessary for any a i r c r a f t w i t h v e r y p o o r s t a l l - w a r n i n g c h a r a c - t e r i s t i c s . Such i n d i c a t o r sc a np o s s i b l yb ea c t u a t e d bysuchdevices as a l e a d i n g - e d g eo r i f i c e ,l e a d i n g - e d g et a b ,s p o i l e ra n dp i t o t - s t a t i ct u b e ,t r a i l i n g - edge p i t o t s t a t i c t u b e , o r t r a i i i n g - e d g e vane. L i g h ta i r p l a n e sw i t h o u th y d r a u - l i c c o n t r o ls y s t e m s will p r o b a b l yn e v e r need t o usethesedevices.

TN-2923 (Ref. 7 7 ) d e s c r i b e st h et e s t i n g of a low-wing, l i g h t a i r p l a n e model d u r i n gt h es t a l l and I n t ot h ei n c i p i e n ts p i n . A more d e t a i l e dr e v i e wo ft h i s r e p o r ti sp r e s e n t e db e l o wi nt h ed i s c u s s i o n of s p i ne n t r y .

I t should be n o t e dt h a ti ns e e k i n gt oa n a l y z et h em o t i o n so fa i r c r a f tn e a r s t a l l ,t h eu s u a la s s u m p t i o nt h a tt h el o n g i t u d i n a ls t a b i l i t yd e r i v a t i v e sa r e c o n s t a n t si s n o l o n g e r t r u e . The d e r i v a t i v e sa r ev e r ys t r o n gf u n c t i o n s of a.

Also, downwash andsidewash f i e l d s a r e v e r y s t r o n g and u n s t a b l eu n d e rt h e s e c o n d i t i o n s . Dynamic p r e s s u r e may v a r yc o n s i d e r a b l ya l o n gt h e span. F i n a l l y , a deep s t a l l i s u s u a l l y accompanied by s u f f i c i e n t l y l a r g e m o t i o n s t h a t one can nolonger assume w i t ha c c u r a c yt h a tp r o d u c t s and squares of p e r t u b a t i o n v e l o c i t i e s c a nb e c n e g l e c t e d .F o rt h e s er e a s o n s ,e x t r e m ec a r em u s tb eu s e di na t t a c k i n g t h i s p r o b l e m a n a l y t i c a l l y .

SPIN ENTRY

The c a p a c i t y f o r a t t a i n i n g a s p i n n i n g a t t i t u d e w h i l e p e r f o r m i n g o r d i n a r y f l i g h t maneuvers and t h e f a c t t h a t many l i g h t a i r c r a f t a c c i d e n t s h a v e been a t t r i b u t e d i n t h e l a s t few y e a r s t o s t a l I / s p i np r o b l e m sh a sl e d t o e x t e n s i v e researchby NACA/NASA c o n c e r n i n gs p i ne n t r ya n dr e c o v e r y .M o r et h a n 25 NACA/NASA r e p o r t sd e a lw i t hs p i na p p l i c a b l e t o l i g h ta i r c r a f td e s i g n . Much of t h i s l i t e r a t u r e was w r i t t e n i n t h e l a t e 1940’s and e a r l y 1950’s because t h eh i g h e rw i n g and f u s e l a g el o a d i n g s of more modern a i r c r a f t a r e n o t c h a r - a c t e r i s t i c of t h e m a j o r i t y of t h e g e n e r a l a v i a t i o n a i r c r a f t .

A d e v e l o p e ds p i ni su s u a l l yc o n s i d e r e d t o be a m o t i o ni nw h i c ha na i r - p l a n e i n f l i g h t , a t some a n g l e of a t t a c kb e t w e e nt h es t a l la n d 90°, descends r a p i d l y t o w a r d t h e e a r t h w h i l e r o t a t i n g , w i t h t h e w i n g s n e a r l y p e r p e n d i c u l a r t o a v e r t i c a l o r n e a r - v e r t i c a la x i s .S p e c i a lc o n s i d e r a t i o nm u s tb eg i v e n t o s p i n si nt h ed e s i g ns t a g e ,s i n c ec o n t r o l se f f e c t i v ei nn o r m a lf l i g h tm i g h t be i n a d e q u a t ef o rr e c o v e r yf r o mt h es p i n . The same f a c t o r sw h i c h may cause d i f f i c u l t y i n a t t a i n i n g a s p i na l s o may impede r e c o v e r yf r o m it, o n c ea t t a i n e d .

B e s i d e st h ep r o b l e m of p r o v i d i n ga d e q u a t ec o n t r o l s f o r r e c o v e r y ,t h e r ei s a l s ot h ep r o b l e m o f p i l o t d i s o r i e n t a t i o n r e s u l t i n g from t h ed e v e l o p e ds p i n ; t h u s ,p r e v e n t i n gt h es p i no rr e c o v e r i n gd u r i n gt h ei n c i p i e n tp h a s e( t h em o t i o n between t h e i n i t i a l s t a l l and t h ed e v e l o p e ds p i n )i se s s e n t i a l .C i v i l Air R e g u l a t i o n s r e q u i r e t h a t t h e p i l o t of a p e r s o n a l - o w n e ra i r p l a n er e c o v e rf r o m a o n e - t u r ns p i n upon r e l e a s eo fc o n t r o l s by t h e p i l o t ; t h i s may mean t h a t c o n t r o l s bedesigned t o f l o a t a g a i n s t t h e s p i n .

O f t h e numerous s i g n i f i c a n tr e p o r t sr e v i e w e d f o r t h i ss t u d y ,t w oa r e c o n s i d e r e do fD r i m a r vi n t e r e s t - - A i r D l a n eS D i n n i n ab v James S . Bowman (Ref. 78) and S t a t u so fS p i nR e s e a r c h f o r RecentAirplane6esignsby A. I.Neihouse, e t a l (Ref. 7 9 ) . The l a t t e r d i s c u s s e d s D i n - t u n n e l t e s t i n a and i t sc o r r e l a t i o n u ” w i t h a c t u a I f I i g h t t e s t s , t h e i n f I uence of a i r p l a n e g e o m e t r y on s p i n n i n g , and a summary of t h es p i n n i n gc h a r a c t e r i s t i c s of 21 model a i r p l a n e s ,i n c l u d i n g model and f u l ls c a l ec o r r e l a t i o n .A i r p l a n eS p i n n i n gi s a s t a t e - o f - t h e - a r t summary o f s p i n p r e d i c t i o n and a l l e v i a t i o n .

E x p e r i e n c eh a si n d i c a t e dt h a ts p i n s and s p i nr e c o v e r i e so fa i r p l a n e s c a nb ei n v e s t i g a t e ds a f e l y and a t a c o m p a r a t i v e l ym o d e r a t ec o s tu s i n gs m a l l dynamic models i n a s p i nt u n n e l . * I t i s i m p o r t a n t t h a t t h e model be designed so t h a t g e o m e t r i c a l l y s i m i l a r p a t h s of m o t i o nb e t w e e nt h e model and t h e a i r p l a n ea r ea t t a i n e di nt h es p i n ;t h i si sa c c o m p l i s h e db yh o l d i n gt h ef o r c e , mass, a n dt i m er a t i oc o n s t a n ta sw e l la st h er a t i o o f l i n e a rd i m e n s i o n si nt h e model d e s i g n . I t i s hand-launched with a s p i n n i n gm o t i o ni n t ot h es p i n t u n n e l , and t h e v e r t i c a l speed of t h e column o f a i r i s a d j u s t e d t o m a i n t a i n a s p i n n i n ga t t i t u d e of t h e model a t a p a r t i c u l a rh e i g h ti nt h et u n n e l . I t i s assumed t h a i ,f o rm o s ts p i n s ,t h ep i l o tw o u l dp r o b a b l yh a v et h ea i r p l a n e * a v e r t i c a lw i n dt u n n e lc o n t r o l l e db y a p r o p e l l e rw i t hv e r yf a s tr e s p o n s e t i m e c o n t r o l ss e ta p p r o x i m a t e l ya t" n o r m a ls p i n n i n gc o n t r o lc o n f i g u r a t i o n " - - t h a t i s ,s t i c kf u l l back, and l a t e r a l l yn e u t r a 1 , r u d d e rf u l lw i t ht h es p i n .A f t e r t h es p i ni se s t a b l i s h e d ,t h e mode! c o n t r o ls u r f a c e s( r u d d e r ,e l e v a t o r , and a i l e r o n s )a r ed e f l e c t e d t o attemptrecovery.Experiencehas shown t h a t t h e c r i t e r i o n f o r s a t i s f a c t o r yr e c o v e r y f o r model t e s t s i s r e c o v e r yw i t h i n 28 t u r n s of t h e model a f t e rt h ec o n t r o ls u r f a c ed e f l e c t i o n s , Based on t h i s a n a l y s i s , when t h er e c o v e r yi nt h es p i nt u n n e lr e q u i r e sm o r et h a nt h i s number of t u r n s , t h e c o n t r o l s a r e n o t s u f f i c i e n t l y e f f e c t i v e and t h ec o r r e s p o n d i n g a i r p l a n ep r o b a b l yw o u l dh a v eu n s a t i s f a c t o r yr e c o v e r yc h a r a c t e r i s t i c s . One p o o rr e c o v e r yo u to fs e v e r a lr e c o v e r ya t t e m p t si su s u a l l yc o n s i d e r e da s u n d e s i r a b l ea sc o n s i s t e n t l yp o o rr e c o v e r i e s . The p h i l o s o p h yi s t o assume t h a t a proposeddesign i s i n a d e q u a t ef o rs p i nr e c o v e r yu n l e s s i t can be p r o v e ns a t i s f a c t o r y .

A d e v e l o p e ds p i ni n v o l v e s a balanceofaerodynamic and i n e r t i a l moments and f o r c e s ;t h u s ,t h ee f f e c t i v e n e s so f any c o n t r o li np r o m o t i n g o r i n t e r - m i n a t i n g t h e s p i n depends n o to n l yo nt h ea e r o d y n a m i c moments and f o r c e s p r o d u c e db yt h ec o n t r o lb u ta l s oo nt h ei n e r t i a lc h a r a c t e r i s t i c so ft h e a i r p l a n e . A s p i na b o u ta n ya x i si ns p a c ec a n be c o n s i d e r e da s a r o t a t i o n a l m o t i o na b o u ta n ya x i st h r o u g ht h ec e n t e ro fg r a v i t y . The e q u a t i o n sf o rt h e moments a c t i n g i n a s p i n *a r e .

I y y - Izz

u 2 z c , +

= 2pkX q r I X X

. U 2 Izz - I x x

4 = - 2 cm,b + 'P 2 1 - 1ky =Y Y The a i r p l a n e s p i n n i n g a t t i t u d e ( s t e e p o r f l a t ) and i t s r a t e o f r o t a t i o n depend p r i m a r i l yo nt h ey a w i n g and p i t c h i n g moment c h a r a c t e r i s t i c s o f t h e a i r p l a n e . Low damping i n yaw a ts p i n n i n ga t t i t u d e s o r h i g ha u t o r o t a t i v ey a w i n g moments lead t of l a t( h i g h a), f a s tr o t a t i n g( h i g h S 2 ) s p i n s . Some i n t e r e s t i n g f a c t s can be p o i n t e do u tb ya p p r o x i m a t i n gt h ep i t c h i n g - m o m e n te q u a t i o no b t a i n e d i ne q u a t i n gt h ea e r o d y n a m i c and i n e r t i a l p i t c h i n g moments: Q 2 = - Myaer o

$(Izz - Ixx) s i n 2a

Remembering t h a t a p o s i t i v e p i t c h i n g moment i s nose up, i t can be seen t h a t a nose-down ( n e g a t i v e )p i t c h i n g moment may nose t h e a i r p l a n e down b u t l e a d t o a h i g h e rr a t e o f r o t a t i o n and may, i nf a c t ,f l a t t e nt h es p i n .F o rg i v e nd i r e c - t i o n a l and l a t e r a lc h a r a c t e r i s t i c s ,t h ep i t c h i n g moment c a ni n f l u e n c et h e m o t i o n so t h a t i t may v a r yf r o m a h i g h r o t a t i v e s p i n t o a low r o t a t i v es p i n .

The e f f e c t o f any c o n t r o li nb r i n g i n ga b o u ts p i nr e c o v e r y depends upon t h e momi3nts t h a t t h e c o n t r o l p r o v i d e s and upon t h e e f f e c t i v e n e s s o f t h o s e * assuming t h e x, y, and z a x e sa r ep r i n c i p a la x e sa n dt h a te n g i n ee f f e c t can be i-gnored .

moments i n p r o d u c i n g a change i n a n g u l a r v e l o c i t y and t h u s i n u p s e t t i n g t h e s p i ne q u i l i b r i u m .E x p e r i e n c eh a s shown t h a t t h e b e s t way t o a l l e v i a t e t h e s p i n n i n gm o t i o ni sp r o v i d e a yawing moment a b o u tt h e z body a x i s t o oppose t h e s p i n r o t a t i o n ; -thus, t h er u d d e ri su s u a l l yc o n s i d e r e dt h em o s ti m p o r t a n t c o n t r o l ,e s p e c i a l l y f o r l i g h ta i r p l a n e s . I t a l s oa p p e a r st h a te l e v a t o re f f e c - t i v e n e s s and a i l e r o n e f f e c t i v e n e s s , i n t h e f i n a l a n a l y s i s , depend upon t h e i r a b i l i t y t o a l t e r t h e y a w i n g moment of t h es p i n . * Thus, it would seem t h a t t h e m o s t e f f e c t i v e way t o i n f l u e n c e t h e s p i n and t o b r i n g a b o u t r e c o v e r y i s t o o b t a i n a yawing moment by a p p l y i n g a moment a b o u ta na x i sw h i c he x p e r i e n c e s t h el e a s tr e s i s t a n c e' t 0 . a change i na n g u l a rv e l o c i t y . * *F o r example, t h em o s t p r o f i c i e n t way t o o b t a i n an a n t i s p i n y a w i n g moment f o r r e c o v e r y may be t o r o l l t h e a i r p l a n e i n such a d i r e c t i o n t h a t a g y r o s c o p i c y a w i n g moment t o oppose t h es p i ni so b t a i n e d .S i m i l a r l y , i f mass i sh e a v i l yc o n c e n t r a t e di nt h ew i n g s , movement o f e l e v a t o r s downward may p r o v i d e t h e m o s t e f f e c t i v e means of a p p l y i n g an a n t i s p i ny a w i n g moment. Thus, because of i n e r t i a lc o u p l i - n g of a r o t a t i n g body, if t h e moment a b o u to n ea x i si s changed, t h e moments a b o u t t h e o t h e r axesarealsochanged.

By i n s p e c t i n g t h e e q u a t i o n g i v e n e a r l i e r f o r P , it can be seen t h a t t h e r u d d e ri st h em o s ti m p o r t a n tc o n t r o l when IXX - Iyy = 0 because defined as m/pSb) and k za r er e l a t i v e l ys m a l l , and Cn i s a f u n c t i o n of r u d d e rd e f l e c t i o n .

F o r modern, h i g h speed f i g h t e r s and r e s e a r c ha i r p l a n e s ,l a r g en e g a t i v ev a l u e s o f Ixx - I y yp r e d o m i n a t e ,s i n c et h e mass i sh e a v i l yc o n c e n t r a t e di nt h ef u s e - l a g e .F o rt h e s ea i r p l a n e s , it would be extremely important t o make t h ei n e r t i a t e r m sa n t i s p i n( n e g a t i v ef o rr i g h ts p i n ) f o r recovery. This can be accom- p l i s h e d by c o n t r o l l i n g t h e a l g e b r a i c s i g n o f t h e p i t c h i n g v e l o c i t y , e.g., by t i l t i n g t h e i n n e r w i n g ( r i g h t w i n g i n a r i g h t s p i n ) down r e l a t i v e t o t h es p i n a x i s . T h i s t i l t i n g o f t h e w i n g downward makes p i t c h i n g v e l o c i t y p o s i t i v e ( q = 52 s i n + f o r low v a l u e so f I X x ) and g i v e sr i s e t o a c r o s s - c o u p l e de f f e c t , w h i c ha c t si n a d i r e c t i o nt h a tt e r m i n a t e st h es p i n n i n g .L i g h ta i r p l a n e s , however, f a l l , f o r t h e m o s tp a r t ,i n t ot h ec a t e g o r y o f a i r p l a n e sd e s i g n e di n t h el a t e 1930's and e a r l y 1940's. These l i g h ta i r p l a n e s have r e l a t i v e l ys m a l l changes i nt h ei n e r t i at e r m sw h i c hc o n t r i b u t et o i-, i n d i c a t i n gt h a tt h er u d d e r s h o u l db et h em o s ti m p o r t a n tc o n t r o l .

The p r i n c i p a lf a c t o r si ns p i n n i n ga r e mass d i s t r i b u t i o n , by f a r t h e most i m p o r t a n ts i n g l ep a r a m e t e r , and t a i l d e s i g n ,p a r t i c u l a r l yi m p o r t a n tf o rc o n - d i t i o n s of z e r o or n e a r - z e r o l o a d i n g ( s m a l l I , , - IYY). By knowing the mass d i s t r i b u t i o n and t a i ld e s i g n , it i sp o s s i b l e ,I n many cases, t o p r e d i c t w h e t h e ra na i r p l a n eh a ss a t i s f a c t o r ys p i n - r e c o v e r yc h a r a c t e r i s t i c s . The mass d i s t r i b u t i o n of a i r p l a n e sc a nb eg r o u p e di n t ot h r e eg e n e r a ll o a d i n gc a t e g o r i e s , as shown on t h e f o l l o w i n g page.

* I t s h o u l d b e p o i n t e d o u t t h a t i f s p o i l e r s a r e used instead o f a i l e r o n s , t h es p o i l e r sa r eg e n e r a l l yi n e f f e c t i v ei nt h ed e v e l o p e ds p i nb e c a u s eo ft h e a r e as h i e l d e di nt h es p i n n i n ga t t i t u d e .

** a moment a b o u tt h ea x i sw i t ht h el e a s t amount o fi n e r t i a .

A i l e r o n s W i t h R u d d e r A g a i n s t E l e v a t o r s D o w n Followed B y P l u s Plus R u d d e rA g a i n s t E l e v a t o r s D o w n R u d d e r A g a i n s t Fuselage Heavy Z e r ol o a d i n g Wings Heavy l o a d i n g Loading F i g u r e 71. P r i m a r y r e c o v e r y c o n t r o l s a s determinedby mass d i s t r i b u t i o n .

The t y p eo fl o a d i n g shown o n t h e r i g h t i s c l a s s i f i e d a s wing-heavyloading, i n w h i c ht h er o l l moment o f i n e r t i a i s g r e a t e r t h a n t h e p i t c h moment o f i n e r t i a .

The a i r p l a n eo nt h el e f ti sa n example offuselage-heavyloading,inwhich t h e r o l l moment o fi n e r t i ai sl e s st h a nt h ep i t c h moment o fi n e r t i a . The a i r - p l a n e i n t h e c e n t e r h a s r o l l and p i t c h moments of i n e r t i aw h i c ha r ea b o u t equal; t h i sc o n d i t i o ni sr e f e r r e dt oa sz e r ol o a d i n g .

The l o a d i n g o f t h e a i r p l a n e c a n d i c t a t e w h a t c o n t r o l s a r e r e q u i r e d f o r r e c o v e r y ,a se x p l a i n e di n WR L-168, A M a s s - D i s t r i b u t i o n C r i t e r i o n f o r P r e d i c t i n g t h eE f f e c to fC o n t r o lM a n i p u l a t i o no nt h eR e c o v e r yf r o m a S p i n( R e f .8 0 ) .

D e f l e c t i n gt h er u d d e ra g a i n s tt h es p i ni sa l w a y s recommended, b u t , f o r s a t i s - f a c t o r yr e c o v e r y ,d e f l e c t i o n of o t h e rc o n t r o l si s s o m e t i m e s r e q u i r e d . F o r t h e caseofwing-heavyloading, down e l e v a t o ri st h ep r i m a r yr e c o v e r yc o n t r o l , w h i l ea i l e r o n sa g a i n s ts h o u l da l s ob eb e n e f i c i a l ;f o rf u s e l a g e - h e a v yl o a d i n g , t h ea i l e r o ni st h ep r i m a r yr e c o v e r yc o n t r o l .I nt h el a t t e r c a s e ,t h ea i l e r o n needs t o be d e f l e c t e d w i t h t h e s p i n - - f o r example, s t i c k r i g h t f o r a s p i n t o t h er i g h t .P r e d i c t i n gw h a tt h ee f f e c t s will be f o rt h ez e r ol o a d i n gi s d i f f i c u l t ;b u t ,a l m o s ti n v a r i a b l y ,t h ep r o p e rr e c o v e r yp r o c e d u r ei s t o move +he r u d d e ra g a i n s tt h es p i n and, a s h o r t t i m e l a t e r , move t h e e l e v a t o r down.

The a b o v er e c o v e r yt e c h n i q u e s seem t o i n d i c a t e t h a t s p i n r e c o v e r y i s s i m p l e , b u t it i s sometimeshard t o c l a s s a g i v e n a i r p l a n e i n o n eo ft h et h r e e c a t e g o r i e sa b o v e a n d , o n c e c l a s s e d , t h e c o n t r o l s s t i l l may o r may notproduce enough a n t i s p i n moment t o a c h i e v e r e c o v e r y .

T a i ld e s i g ni sa l s oa ni m p o r t a n tf a c t o ri nd e s i g n i n ga na i r p l a n et or e c o v e r from s p i n s .S i n c em o s tl i g h tp l a n e sf a l li n t ot h ez e r ol o a d i n gc a t e g o r y ,t h e r u d d e ri so fp r i m a r yi m p o r t a n c et o a good d e s i g n .I n a s p i n ,t h e r ei s a dead a i r r e g i o n o v e r much o f t h e v e r t i c a l t a i l causedbythe wake o f t h e s t a l l e d h o r i z o n t a lt a i l .F o r o p t i m u mr u d d e re f f e c t i v e n e s s ,p a r to ft h er u d d e rm u s t be o u t s i d et h i ss t a l l e d wake. A n o t h e rf a c t o rw h i c ha f f e c t st a i ld e s i g n from t h es p i ns t a n d p o i n ti st h a tt h e r es h o u l d be a s u b s t a n t i a l amount o f f i x e d a r e ab e n e a t ht h eh o r i z o n t a lt a i lt op r o v i d e damping o ft h es p i n n i n gm o t i o n .

A c r i t e r i o n f o r good t a i l d e s i g n was d e t e r m i n e di nt h em i d d l e 1940's i s o f s p i n - t u n n e (TN 1045, Ref. 81 and TN 1329, Ref.82)onthebas I t e s t s i s c a t l e d t h e t a i I damping w i t ha b o u t 100 d i f f e r e n td e s i g n s .T h i sc r i t e r i o n Dower f a c t o r (TDPFI. a measure of t h e damping p r o v i dedby t h e f i x e d a r e a , - .

beneaf-h t h e h o r i z o n t a l t a i l and t h ec o n t r o lp o w e rp r o v i d e d b y t h eu n s h i e l d e d p a r t of t h er u d d e r . The t a i l - d a m p i n g power f a c t o rc a nb ec a l c u l a t e da s shown below i n p a r t s a , b, and c of F i g u r e 72.

Relative Wind /

F u l l - l e n g t h r u d d e r ; a assumed to be 45’ P a r t a To c.g. o f

-

A i r p l a n e - + l l I To

-

A i r R e l a t i v e W i n d 1 P a r t i a l - l e n g t hr u d d e r ; a a s s u m e d t o be 45.

TDR c 0.019 P a r t b

To c.g. of u”

-

Airplane

R e l a t i v e W i n d /

P a r t i a l - l e n g t h r u d d e r ; a assumed t o be 30‘ TDR 3 0 019 P a r t c F i g u r e 72. C a l c u l a t i o no ft a i l - d a m p i n g power f a c t o r .

The t a i l damping power f a c t o rr e q u i r e d t o i n s u r es a t i s f a c t o r yr e c o v e r y i sg i v e ni nt h ef i g u r eb e l o w .

Airplane Density m l S b

’’ Air Density P

1600”x10-6

-

-

-

Tai I Damping

-

Power 8 0 0

-

F a c t o r

-

-

I 0 -

- 2 0 0 0 - 800 - 4 0 0 0 400 n ld4

F u s e l a g e W i n g Heavy Heavy Mass Distribution, Clxx-lyy)/mb2 F i g u r e 73. T a i l d e s i g n r e q u i r e m e n t s .

1 70 A l l t h e d a t a a r e i n t h e a r e a o f z e r o or near-zeroloading,wheretherudder i s a p r i m a r yr e c o v e r yc o n t r o l and t h e t a i l d e s i g n i s o f p a r t i c u l a r i m p o r t a n c e .

As can be seen from t h eg r a p hi nF i g u r e 68, o n l y a l i m i t e d p a r t o f t h e t o t a l e x i s t i n gr a n g e of mass d i s t r i b u t i o n sa p p l i e s . The p l o t shows boundaries i n d i c a t i n g t h e minimum v a l u e s o f t h e t a i l - d a m p i n g power f a c t o r r e q u i r e d t o i n s u r es a t i , s t a c t o r yr e c o v e r y . The h a t c h e ds i d eo ft h eb o u n d a r i e si st h e u n s a t i s f a c t o r ys i d e . The s o l i dl i n e sa r ef o rr e c o v e r yb yr u d d e ra l - o n e , and t h eb r o k e nl i n e shows t h eb o u n d a r yf o rr e c o v e r yb yr u d d e r and e l e v a t o r . The b o u n d a r i e sa r ep r e s e n t e di nt e r m so ft h er e l a t i v ed e n s i t yf a c t o r p. The v a l u eo f 1 . 1 = 6 i sr e p r e s e n t a t i v eo fl i g h t ,s i n g l e - e n g i n e ,p e r s o n a l - o w n e r a i r p l a n e s , w h i l e p = 35 i s r e p r e s e n t a t i v e o f t h a t f o r e x e c u t i v e j e t s .

TN-1329, T a iI - D e s i g nR e q u i r e m e n t sf o rS a t i s f a c t o r yS p i nR e c o v e r yf o r Personal-Owner-TypeLightAirplanes,(Ref. 8 2 ) i s a r e p o r td e a l i n gs p e c i f i c a l l y w i t hl i g h ta i r p l a n ed e s i g n .I t sc o n c l u s i o n sa r e based on t e s t s of 60 models i nt h eL a n g l e ys p i nt u n n e l s .R e s u l t so ft h i s ' i n v e s t i g a t i o na r e shown i nt h e p l o t below.

Region for satisfactory recovery by rudder reversal alone versa1 of hoth rudder and elevator

I n e r t i a Yawing- moment Parameter, - Ixx-lYY

mb2 F i g u r e 74. V e r t i c a l - t a i l d e s i g n r e q u i r e m e n t s f o r p e r s o n a l - o w n e r - t y p ea i r p l a n e s .

An i m p o r t a n t p l o t w h i c h i n d i c a t e s o t h e r i n f l u e n c e s of mass d i s t r i b u t i o n o n o p t i m u mc o n t r o l movement f o r r e c o v e r y from s p i n i s presentedbelow.

" W e i g h t a l o n g W e i g h t along- fuselage wing F i g u r e 7 5 . I n f l u e n c e of mass d i s t r i b u t i o n onoptimumcontrol movement f o r r e c o v e r y from s p i n .

The u n s a t i s f a c t o r yr e g i o ne x i s t s because t h e d i f f e r e n c e i n i n e r t i a i s low.

As t h ed i f f e r e n c ei n c r e a s e se i t h e r way t h e c o n t r o l s u r f a c e ( e l e v a t o r i n t h e case o fp o s i t i v ei n c r e a s e ,a i l e r o ni nt h ec a s eo fn e g a t i v ei n c r e a s e )g a i n s e f f e c t i v e n e s s . Based o nt h ei n e r t i a s ,t h er e v e r s a l of a i l e r o ne f f e c ts h o u l d o c c u r a t I , , - I y y . = 0; however, due t o t h ea e r o d y n a m i ce f f e c t s ,t h er e v e r s a l of a i l e r o n e f f e c t I S s h i f t e d from 0 t o[ ( I x x - I y y ) / m b 2 1 X IO4 = -50. Thus, i n t h i s v i c i n i t y , a i l e r o n s w i t h t h e s p i n ( s t i c k r i g h t i n a r i g h t s p i n ) g e n e r a l l y l o o s e t h e i r f a v o r a b l e e f f e c t and become adverse; f o r a i l e r o n s a g a i n s t t h e s p i n , t h ec o n v e r s ei st r u e .T h i sr e s u l t , it i sb e l i e v e d ,i sp r i m a r i l y a r e s u l to f a secondary e f f e c t a s s o c i a t e d w i t h p o s i t i v e Cng o f t h e a i r p l a n e and a r e s u l t i n g r e l a t i v ep r o s p i ni n c r e m e n ti ny a w i n g moment because o ft h ei n c r e m e n ti ni n w a r d s i d e s l i pt h a ti n v a r i a b l yo c c u r s when a i l e r o n sa r es e tw i t ht h es p i n .A n o t h e r importantgraph, shown below, i s a summary of t h em o s ti m p o r t a n tf a c t o r si n s p i n n i n g and i n d i c a t e st h ep r e s e n ts t a t e - o f - t h e - a r t .

I RECOVERY CONTROL ' R u d d e r E l e v a t o r

A i l e r o n s W i t h \ A g a i n s t 1 Down

PI us dFollowed# Plus

Rudder Against ;r' BY f Rudder

#Elevator Against $Down 1.

1600 '% 1 4 6

-

-

TAIL 1200

-

DAMPING POWER

-

FACTOR

-

p=35

-

- 15

0 - 4 I I I I -2000 -800 -400 0 400 X Fuselage Wing Heavy Heavy M A S S D I S T R I B U T I O N , ( Ixx-lyy)/mb2 F i g u r e 76. Summary of t h em o s ti m p o r t a n tf a c t o r si ns p i n n i n g .

The f i g u r ea b o v ea g r e e sw i t hc u r r e n tl i t e r a t u r e ; however, it was p r e v i o u s l y c o n s i d e r e di nr u d d e r - e l e v a t o rr e c o v e r yp r o c e d u r e st h a tr u d d e ra p p l i c a t i o n s h o u l dl e a de l e v a t o ra p p l i c a t i o n .F o ra ne r e c ts p i n down e l e v a t o ri n c r e a s e s t h e s h i e l d i n g o f t h e r u d d e r more t h a n up e l e v a t o r ;t h u s ,m a i n t a i n i n ge l e v a t o r up p r o v i d e s maximum r u d d e re f f e c t i v e n e s sd u r i n gr u d d e rr e v e r s a l . I t s h o u l d b en o t e dt h a t , f o r t h ee x t r e m el o a d i n g so nb o t he n d so ft h es c a l e ,n oc r i t e r - i a havebeendeveloped f o r p r e d i c t i n g t h e e f f e c t i v e n e s s of t h e c o n t r o l s f o r s a t i s f a c t o r yr e c o v e r y . The s a f e s t way t o i n s u r e good s p i nr e c o v e r yi s by s p i nt u n n e l or f l i g h t t e s t i n g .

The p r e s c r i b e dm e t h o d s f o r s p i nr e c o v e r yh a v e been g i v e na b o v ef o rt h e p a r t i c u l a r mass c o n f i g u r a t i o nd e s i r e d ,b u t a r e c o v e r y may s t i l l behard t o a c h i e v e .I nf l i g h t , an a i r p l a n ee n t e r s a s p i nf o l l o w i n g r o l l - o f f j u s t above t h e s t a l l i n g a n g l e o f a t t a c k a f t e r b e i n g b r o u g h t up fromlowerangles of a t t a c k . I tu s u a l l yt a k e sa na i r p l a n e two t o f i v e t u r n s t o a t t a i n a f u l l y d e v e l o p e ds p i na f t e rs t a r t i n gt h ei n c i p i e n t - s p i nm o t i o n ,w i t ht h e number o f t u r n s d e p e n d i n g u p o n c o n f i g u r a t i o n a n d c o n t r o l t e c h n i q u e . One i m p o r t a n t f a c ts h o u l db e remembered; r e c o v e r i e sa r eg e n e r a l l ya c h i e v e d much more r e a d i l y when a t t e m p t e dd u r i n gt h ei n c i p i e n tp h a s e of t h e s p i n t h a n when a t t e m p t e d a f t e rt h es p i n becomes f u l l y developed. Thus, some c o n s i d e r a t i o ns h o u l db e g i v e n t o t h e t e c h n i q u e s of n o t i c i n g a s p i n e n t r y a t t i t u d e and t o ways of a v o i d i n g t h e f u l l y d e v e l o p e d s p i n .

TN-2352 (Ref. 8 3 ) i s a s p i n - t u n n e li n v e s t i g a t i o n of a low-wing p e r s o n a l - owner a i r c r a f t w h i c h was conducted t o p r o v i d ed e s i g ni n f o r m a t i o n for p r o p o r t i o n - ingpersonal-owner o r l i a i s o n a i r p l a n e s f o r s a t i s f a c t o r y r e c o v e r y f r o m s p i n s and f o r s p i np r o o f i n g . The i n v e s t i g a t i o n was i n t e n d e d t o b ee x t e n s i v e enough t o d e t e r m i n e t h e c o n f i g u r a t i o n s m o s t l i k e l y t o meet t h es p i n - r e c o v e r yr e q u i r e - m e n t sg i v e ni nP a r t 3 of The C i v i l A i r Regulation(Ref.54)andsummarizedbelow.

F o r an a i r p l a n el i c e n s e di nt h en o r m a lc a t e g o r y : ( 1 ) A I S - t u r nr e c o v e r ya f t e r a I - t u r ns p i nb yr e l e a s i n gc o n t r o l s ( c o n t r o l s a s s i s t e d t o t h e e x t e n t n e c e s s a r y t o overcome f r i c t i o n ) ( 2 )" U n c o n t r o l l a b l es p i n "c h e c k - - a i r p l a n ec a p a b l e of r e c o v e r i n g from a I - t u r n s p i n w i t h a i l e r o n s a t n e u t r a l b y f i r s t c o m p l e t e l y r e v e r s i n g e l e v a t o r and then, i f necessary, f u l l yr e v e r s i n gt h er u d d e r .

F o ra i r p l a n e sl i c e n s e di nt h ea c r o b a t i cc a t e g o r y : ( 1 ) A 4 - t u r nr e c o v e r ya f t e r 6 t u r n s of t h es p i n by r e l e a s i n gc o n t r o l s ( 2 ) Recovery from a 6 - t u r ns p i ni n 1 % a d d i t i o n a lt u r n sa f t e rn e u t r a l - i z a t i o n o f r u d d e ra n de l e v a t o r ,a i l e r o n sa tn e u t r a l ( 3 ) " U n c o n t r o l l a b l es p i n "c h e c k - - a i r p l a n ec a p a b l eo fr e c o v e r i n gf r o m a 6 - t u r ns p i nw i t ha i l e r o n sa tn e u t r a lb yf i r s tc o m p l e t e l yr e v e r s i n g e l e v a t o r and then, i f n e c e s s a r y ,f u l l yr e v e r s i n gt h er u d d e r ( 4 ) Recovery from " a b n o r m a ls p i n s " - - a2 - t u r nr e c o v e r ya f t e r 6 t u r n s o f t h e s p i n w i t h a i l e r o n s i n i t i a l l y e i t h e r f u l l w i t h or f u l l a g a i n s t t h e s p i n by n e u t r a l i z i n g a i l e r o n s and f u l l y r e v e r s i n g rudder and e l e v a t o r ( 5 ) A I S - t u r n r e c o v e r y from a I - t u r ns p i n by n e u t r a l i z a t i o no fr u d d e r and e l e v a t o r w i t h f l a p s and landinggearextended.

An e x t e n s i v e amount o f t e s t i n g o f t h i s l i g h t a i r p l a n e i n d i c a t e d t h a t s a t i s f a c - t o r yr e c o v e r yc a nb er e a d i l yo b t a i n e de v e n i f t h et a i l - d a m p i n g power f a c t o r i sn o tv e r yg r e a t ,p r o v i d e dt h er e c o v e r yt e c h n i q u eu s e di sf u l lr a p i dr u d d e r r e v e r s a lf o l l o w e da p p r o x i m a t e l y % t u r n l a t e r by f o r w a r d movement of t h e s t i c k .

The r e s u l t s a l s o i n d i c a t e d t h a t f o r r e c o v e r yb ym e r e l yn e u t r a l i z i n gb o t h c o n t r o l s ,e s p e c i a l l y f o r r e a r w a r dc . g .p o s i t i o n s ,h i g hv a l u e s of t a i l - d a m p i n g power f a c t o r may have an adverse e f f e c t upon r e c o v e r i e s . I t was f o u n dt h a t d i f f e r e n tw i n gp l a n f o r m sh a dl i t t l ee f f e c t on t h e model s p i n and r e c o v e r y c h a r a c t e r i s t i c s . I t was c o n c l u d e dt h a tu n l e s st h er u d d e rc a nb ed e s i g n e dt o f l o a ta g a i n s tt h es p i n ,r e c o v e r y from a s p i n by r e l e a s i n gc o n t r o l sm i g h tb e d i f f i c u l t u n l e s s t h e e l e v a t o r c a n be made t o f l o a t a t d e f l e c t i o n s f a r t h e r down t h a nn e u t r a l . ! t was a l s oc o n c l u d e dt h a to t h e rr e q u i r e m e n t s for recoveryby v a r i o u s movements o ft h ec o n t r o l sa ss p e c i f i e di nt h ea f o r e m e n t i o n e dr e g u l a - t i o n sc o u l dp r o b a b l yb em e tf o rt h ev a r i o u s model c o n f i g u r a t i o n s and mass d i s t r i b u t i o n s i n v e s t i g a t e d by m a i n t a i n i n g t h e c e n t e r o f g r a v i t y a t a f o r w a r d p o s i t i o n and u t i l i z i n g a h i g ht a i l - d a m p i n g power f a c t o r .F o rf u s e l a g eh e a v y l o a d i n g and low TDPF a p r e m a t u r ef o r w a r ds t i c k movement may r e t a r dr e c o v e r y .

Mass changes were s i g n i f i c a n t a t low TDPF b u t n o t a t h i g h TDPF. I t may be of i n t e r e s t t o m e n t i o n t h a t t h e model spun w i t h a t o t a l a n g u l a r v e l o c i t y o f approximately0.35 t o 0 . 5r e v o l u t i o n sp e rs e c o n d .

An i n v e s t i g a t i o n o f t h e e f f e c t o f c e n t e r o f g r a v i t y l o c a t i o n o n t h e s p i n n i n g c h a r a c t e r i s t i c s of a low-wing monoplane model i s g i v e n i n TR-672 (Ref. 8 4 ) .

M o v i n gt h ec . g .f o r w a r ds t e e p e n st h es p i n ,i n c r e a s e s Qb/2U, and improves r e c o v e r y ;s i m i l a r l y ,m o v i n gt h ec . g .b a c kt e n d st of l a t t e nt h es p i n ,d e c r e a s e Qb/2U, and r e t a r d r e c o v e r y . A r e p o r ti nt h e same s e r i e s o f i n v e s t i g a t i o n s , TR-691 (Ref. 851, g i v e st h ee f f e c t s of a i r p l a n er e l a t i v ed e n s i t y . The f i n d i n g s i n t h i s r e p o r t a r e t h a t , i n most c a s e s ,a ni n c r e a s ei nr e l a t i v ed e n s i t y p r o d u c e sf l a t t e rs p i n s ,h i g h e rv e l o c i t i e s ,l o w e rv a l u e s of Qb/2U, andslower r e c o v e r i es.

NACA TN-570 (Ref. 8 6 ) i s d i r e c t l y a p p l i c a b l e t o l i g h t a i r p l a n e d e s i g n .

T h i s p a r t i c u l a r r e p o r t i n v e s t i g a t e s t h e e f f e c t of d i f f e r e n t t a i l a r r a n g e m e n t s o nt h es p i n n i n gc h a r a c t e r i s t i c s of a low-wing monoplane model. Results o f t h i s i h v e s t i g a t i o ni n d i c a t et h a t a r e d u c t i o n i n t a i l l e n g t h r e s u l t s i n s p i n s w i t h h i g h e r a n g l e s o f a t t a c k , h i g h e r v a l u e s of Qb/2U, a n ds l o w e rr a t e s of descent.

Recoveries from t h e s p i n seem t o depend c r i t i c a l l y upon t h e e x a c t l o c a t i o n o ft h ev e r t i c a ls u r f a c e s .I ti sa l s oc o n c l u d e dt h a t , b ym a k i n gc e r t a i n reasonably small changes i n t h e t a i l arrangement, a l lt h es p i n n i n gc h a r a c t - e r i s t i c s , e x c e p t t h e amount o fs i d e s l i p ,c a n bechangedthroughwideranges.

T h i sa g a i np o i n t so u t , t h ei m p o r t a n c eo ft a i ld e s i g ni nl i g h ta i r p l a n es p i n r e c o v e r y .

TN-608 ( R e f .8 7 )i sa n o t h e ri n v e s t i g a t i o ni nw h i c h a s e r i e so fm o d e l s were t e s t e di nt h es p i nt u n n e l .I t was found t h a tr e c t a n g u l a r and f a i r e dt i p sg i v e t h es t e e p e s ts p i n sa n df l a p st e n dt or e t a r dr e c o v e r y ;f o rc o n t r o l sw i t ht h e s p i n , t a i l B ( b e l o w )g i v e ss t e e p e rs p i n st h a nt a i l A, w i t hg e n e r a l l ys a t i s f a c - t o r y r e c o v e r y f o r e i t h e r t a i l , w h i l e t a i l C g e n e r a l l yg i v e ss l o w e rr e c o v e r i e s .

Empennage Arrangements Investigated F i g u r e 7 7 . E f f e c t so fv a r i o u sa i r c r a f tt a i l so ns p i nr e c o v e r y .

The e f f e c t o f s u c h d e v i c e s a s a n t i s p i n f i l l e t s a n dd o r s a lf i n so ns p i n r e c o v e r yc h a r a c t e r i s t i c sc a n be found i n TN-1779 (Ref. 8 8 ) . Data from 21 d i f f e r e n t modelswereused t o d e t e r m i n e t h e a c t i o n o f f i l l e t s i n damping of s p i n r o t a t i o n , and 30 modelswereinvestigated f o r t h e e f f e c t of d o r s a lf i n s ; The e f f e c t i v e n e s s of a n t i s p i n f i l l e t s f o r s p i nr e c o v e r ya p p e a r t o depend p r i m a r i l y upon t h e f a c t t h a t t h e f u s e l a g e a r e a b e l o w t h e f i l l e t s becomes e f f e c t i v e i n damping t h es p i nr o t a t i o n . Whether or n o t t h e f i l l e t s improve t h e r e c o v e r y c h a r a c t e r i s t i c s of a g i v e n d e s i g n i s s t i l l a f u n c t i o n o f t h e t a i l - d a m p i n g power f a c t o r of t h ed e s i g n and t h e mass d i s t r i b u t i o n .D o r s a l f i n s g e n e r a l l y have l i t t l e e f f e c t o n s p i n and r e c o v e r yc h a r a c t e r i s t i c s .

TN-1801 (Ref. 89) should be mentioned because it i s a ;pin i n v e s t i g a t i & o f a t w i n - t a i l l i g h t a i r p l a n e model w i t hl i n k e d and u n l i n k e da i l e r o nc o n t r o l s .

I t was f o u n dt h a t when t h er u d d e r s and a i l e r o n s a r e l i n k e d f o r t w o - c o n t r o l o p e r a t i o n ,t h e model g e n e r a l l y does n o ts p i n . The s p i n so b t a i n e di nt h i s s t u d yw e r es t e e p ,a n dt h et e s t sr e s u l t e di ns a t i s f a c t o r yr e c o v e r y .

TN-2923 (Ref. 77) i sa n o t h e rr e p o r ti nw h i c h a l i g h ta i r p l a n e was t e s t e d . The motion of a p e r s o n a l - o w n e ro rl i a i s o na i r p l a n et h r o u g ht h e i n c i p i e n ts p i n was analyzed. I t was f o u n dt h a t ,a f t e rt h ei n i t i a ls t a l l and i m m e d l a t e l ya f t e rt h e model becomes u n s t a l l e d ,t h er a t e so f yaw and p i t c h a r er e l a t i v e l ys m a l l , and t h er a t e so fr o l lb e g i nt od e c r e a s e .T h e r ea l s oi s l i t t l e loss o f a l t i t u d e up t o t h i s t i m e and t h e r e s u l t s i n d i c a t e t h a t , even t h o u g ht h ea i r p l a n e may be i n v e r t e d , a t i m e s o o n a f t e r t h e r o l l - o f f has s t a r t e d a p p e a r s t o be a d e s i r a b l e t i m e t o a t t e m p t t o t e r m i n a t e t h e m o t i o n .

I t i s f e l t t h a t t h e m o t i o n s a t t a i n e d i n t h i s i n v e s t i g a t i o n a r e i n d i c a t i v e o f t h em o t i o n so f a low-wing, l i g h ta i r p l a n ei nt h ei n c i p i e n ts p i n .A l t h o u g h e v a l u a t i o n of t h ep a r a m e t e r si ni n c i p i e n t - s p i nm o t i o ni sn o tt r e a t e d s p e c i f i c a l l yi nt h ee q u a t i o n so fm o t i o np r e s e n t e d i n Appendix A o f t h e p r e s e n t s t u d y ,t h e s ep a r a m e t e r sc o u l db eo b t a i n e db yp r o p e ra p p l i c a t i o n of t h et e c h - n i q u e su s e di nt h a td e r i v a t i o n .B e l o wa r e some example p l o t so fa n g u l a r displacements and a n g u l a r v e l o c i t i e s v e r s u s t i m e f o r t h e i n c i p i e n t s p i n caused b y a r i g h t r o l l - o f f shown i nF i g u r e s 78 and 79.

450 1 1 1 1 1 \ 1 I l l 1 0 1 2 3 4 5 6 7 8 Time, sec.

c - 2 o b

Q) Repeat - -- - \

I

-

-

-25 \ I M \ / P

I I l l 1 1 1 1 1 1 1 1 1 1 I I

-30 0 1 2 3 4 5 6 7 8 Time, sec.

30r

-5 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~ 0 1 2 3 4 5 6 7 8 Time, scc.

Figure 78. Angular displacements versus time f o r the 'Encipient s p i n caused by a r i g h t r o l l - o f f .

0 '

3 '1 Original--

=

m

L ReDeat- - - - - n

0 1 2 3 4 5 6 8 T i m e , sec.

2-

- Original Original

Repeat - - - - Repeat - - - -

1-

I I I I I I I i I I I I I I I I I I I I I I I i I I I I I I I I

-1 -1 0 1 2 3 4 5 6 7 Time, sec.

3 . 0 -

Original -

-

2.5

Repeat -- ---

-

2.0

-

1.5

-

1.0

-

.5

C'l

I I I I I I I I 1 I I I I I

" 5' 0 1 2 3 4 5 6 7 8 l i m e , sec.

F i g u r e 79. A n g u l a rv e l o c i t i e sv e r s u st i m ef o rt h ei n c i p i e n ts p i n caused by a r i g h t r o t I-off .

HUMAN FACTORS

For maximum s a f e t y t o an a i r c r a f t and i t s o c c u p a n t s , t h e p i l o t m u s t be a b l e t o c o n t r o l a i r c r a f t m o t i o n d u r i n g a l l f l i g h t c o n d i t i o n s . One p o s s i b l e means of a t t a i n i n g t h i s o b j e c t i v e i s t o d e s i g n t h e a i r c r a f t c o n t r o l s f o r t h e "average" man; however, AFSC DH 1-3 (Ref. 9 0 ) i n d i c a t e st h a tl e s st h a no n e p e rc e n to ft h ep o p u l a t i o ni s" a v e r a g e "i nt h ef i v ed i m e n s i o n sc o n s i d e r e d . Thus, d e s i g n i n gf o rt h e" a v e r a g e " man appears unsound. The concept o f" d e s i g n I i m i t s " o f f e r s a more r e a l i . s t i ca p p r o a c h .W i t ht h i si d e ai n mind, t h ef o l l o w i n g d i s c u s s i o nc e n t e r so nt h ep i l o t l sc o m f o r tf r o m a "design limit" v i e w p o i n t .

U s i n ga n t h r o p o l o g i c a ld a t a ,c o m f o r tl i m i t sf o rp i l o t so fg e n e r a la v i a t i o na i r - c r a f t a r e examined.

S e t t i n g t h e p r o p e r minimum f o r c er e d u c e st h el i k e l i h o o do fa c c i d e n t a l a c t i v a t i o n o f a c o n t r o l ,e s p e c i a l l yt h o s ec o n t r o l so nw h i c ht h ep i l o tm u s t c o n t i n u o u s l y keep h i s hands o rf e e t . An upper l i m i t i s needed t oi n s u r et h a t r e q u i r e dc o n t r o lf o r c e s do n o te x c e e dt h ep i l o t l sc a p a b i l i t i e s .S e v e r a l g e n e r a l l z a t i o n sc o n c e r n i n gf o r c ea p p l i c a t i o n st oc o n t r o ld e v i c e sa r eg i v e ni n AFSC DH 1-3 ( R e f . 9 0 ) : a )F o r c ea p p l i c a t i o ni se q u a l l ya c c u r a t ef o r hands a nd f e e t ; b ) Contro I s c e n t e r e d i n f r o n t o f t h e o p e r a t o r maximum f o r c e a p p l i c a t i o n ; p e r m i t c ) C o n t r o f o r c eg r e a t e rt h a n 30 t o 40 Ibs I applied byhand o r g r e a t e r t h a n 60 I bs by f o o ti sf a t i g u i n g ; d ) The p r e f e r r e d hand and a r m a r eg e n e r a l l y 10% s t r o n g e rt h a nt h en o n - p r e f e r r e d .

The c a p a b i l i t yw h i c h 9 5 $ , o r t h e f i f t h p e r c e n t i l e , o f a p o p u l a t i o nc a n be expected t oe x e r ti sc o n s i d e r e dt h es t a n d a r d .

From AFSC DH 1-3, t h et a b l eb e l o w shows arm s t r e n g t h f o r d i f f e r e n t a n g l e s o f e l b o wf l e x i o n .

Arm Strength in SEATED ELBOW FLEX I ON OUT DOWN UP PULL PUSH I N R L R L L R L R L R ””” 1 80° 14 8 20 13 17 13 14 9 52 50 50 42 15 8 20 15 20 18 18 15 56 42 42 30 1 5 0 ° 15 10 22 20 26 21 24 17 42 34 36 26 1200 36 22 37 32 900 20 17 16 10 18 16 26 21 17 12 20 17 20 18 20 15 24 26 34 22 60’ T a b l e 19. A r m s t r e n g t hf o rd i f f e r e n ta n g l e so fe l b o wf l e x i o n .

The maximum f o r c e e x e r t e d o n a n a i r c r a f t c o n t r o l s t i c k b y t h e r i g h t arm o f male Air F o r c ep e r s o n n e li nt h es i t t i n gp o s i t i o n ,a c c o r d i n gt o Morgan (Ref. 911, i s r e p r o d u c e di nT a b l e 20.

RIGHT ARM ON AIRCRAFT CONTROL STICK (POUNDS) ~ ISTANCE I N INCHES FROM R I GHT PULL LEFT MIDPLANE PUSH SRP* 12 26 24 34 8 ( l e f t ) 28 31 31 44 ( l e f t ) 18 26 34 30 23 9 0 44 ( r i g h t ) 34 39 26 37 39 26 8 ( r i g h t ) 18 33 23 8 ( l e f t ) 12 1/2 49 22 16 8 ( r i g h t ) 43 39 20 25 8 ( l e f t ) 23 43 54 24 15 1/2 55 24 13 8 ( r i g h t ) 53 45 16 22 8 ( l e f t ) 36 64 56 8 15 18 3/4 0 58 22 14 8 ( r i g h t ) 70 29 51 1 1 8 ( l e f t ) 62 14 0 54 23 3/4 58 20 12 8 ( r i g h t ) *Forward; c o n t r o li s 13 1/2 inches above seat r e f e r e n c e po i n t .

Table 20. Maximum f o r c ee x e r t e do na na i r c r a f t c o n t r o l s t i c k b y t h e r i g h t arm.

From t h i s same r e f e r e n c e , s i m i l a r i n f o r m a t i o n f o r amount o f f o r c e o n a n a i r - c r a f t c o n t r o l wheel i s g i v e n i n t h e t a b l e below.

1 3 : R I G H T ARM ON AIRCRAFT CONTROL WHEEL (POUNDS) D I STANCE IN INCHES CONTROL FORWARD P O S I T I O N PUSH P U L L L E F T R I GHT FROM SRP* 900 ( l e f t ) 32 23 23 27 45O ( I e f t ) 48 40 21 24 10 3/4 0 52 44 26 20 450 ( r i g h t ) 40 39 31 24 80° ( r i g h t ) 19 15 18 21 13 1/4 90' ( l e f t ) 32 33 26 21 90' ( r i g h t ) 25 31 25 19 goo ( I e f t ) 32 42 27 19 15 3/4 0 61 66 27 27 90° ( r i g h t ) 32 49 29 20 900 ( l e f t ) 37 60 22 27 19 0 64 73 25 30 90' ( r i g h t ) 33 61 33 22 900 ( l e f t ) 82 73 21 26 23 1/4 0 105 77 20 35 90° ( r i g h t ) 49 74 26 22 *Wheel g r i p s 18 inches above SRP and 15 i n c h e sa p a r t .

Table 21. Amount o ff o r c ee x e r t e do na na i r c r a f tc o n t r o lw h e e l .

1 82 Sinceeach o f t h e s e t e s t s g i v e s t h e maximum f o r c e f o r t h e r i g h t arm of amale, it must be remembered t h a t t h e s t a n d a r d l e f t arm, which i s u s u a l l y weaker, and t h e p o s s i b i l i t y o f f e m a l e p i l o t s f o r g e n e r a l a v i a t i o n a i r c r a f t w o u l d d i c t a t el o w e r maximum f o r c e s . These l i m i t s on wheel f o r c e s need n o t be so s t r i n g e n t i f a w o r s tc a s ea n a l y s i s shows them t o be i m p r a c t i c a l , s i n c e t h e p i l o t c o u l d useboth hands i f necessary. When p o s s i b l e ,u s e o f two hands on t h e wheel s h o u l db ea v o i d e do nl a n d i n g ,a st h ep i l o t may need h i s r i g h t hand t o p e r f o r mo t h e rt a s k s . Damon (Ref. 92) g i v e st h el e f tr o t a t i o no ft h e wheel a sa p p r o x i m a t e l y 25 I b s and r i g h t r o t a t i o n asabout 30 Ibs.

The maximum f o r c e t h a t c a n be e x e r t e d i n e x t e n s i o n o f t h e leg a t t h e h i p and knee f o r 17 t e s t c o n d i t i o n s o n m a l e B r i t i s h c i v i I i a n s i n t h e s i t t i ng p o s i t i o ni sg i v e ni n Morgan (Ref. 91).

T e s t C o n d i t i o n s Avg .

f o r c e A B C D ( I b ) I 0 0 63 0 90

"""""""" "_

0 0 89 0 113 0 0 156 0 135 0 5 559 0 164 0 6 73 0 94 0 8 87 0 93 0 10 77 0 80 0 10 59 0 90 0 10 27 0 0 135 0 10 346 0 165 0 15 227 0 149 0 15 845 0 160 0 15 530 0 169 0 16 31 9 0 129 0 17 272 0 117 0 17 684 0 151 0 33 184 0 106 Table 22. Maximum f o r c ee x e r t e di ne x t e n s i o n o f t h e l e g a t t h e h i p and knee.

Morgan (Ref. 91) a l s oc o n s i d e r st h e maximum f o r c et h a t can be e x e r t e di ne x t e n - s i o no ft h ea n k l e ,c o r r e s p o n d i n g t o f o o tp e d a lo p e r a t i o n( T a b l e 231, by male Air F o r c ep e r s o n n e lf o r 18 t e s tc o n d i t i o n s .

1 83 I T e s t C o n d i t i o n s P e r c e n t i I e s ( I b 1 A E F G 5 t h 13 10 35t 37 14 1 3 1 0 37 54 13 30 37 25 13 30 37 64 38; 13 50 37 24 35$ 13 50 37 48 13 10 39 15 35; 1 3 1 0 39 37 13 30 39 26 35; 13 30 39 60 38; 13 50 39 22 1 3 50 38; 39 50 13 10 4 1 18 35; 13 1 0 41 35 13 30 41 32 35; 13 30 41 50 13 50 41 23 35; 13 50 41 50 Tab l e 23. Maximum f o r c ee x e r t e di ne x t e n s i o n of t h ea n k l e .

The f o l l o w i n g t a b l e ,a d a p t e d from MIL-F-8785B (Ref. 41, g i v e sf o r c el i m i t sf o r t h e e I e v a t o r , a i l e r o n s , and rudder.

CONTROL MAXIMUM ( I b s ) MINIMUM ( I b s ) E I e v a t o r S t i c k c o n t r o l l e r s 28.0 3.0 Wheel c o n t r o l l e r s 120.0 6.0 A i I erons S t i c k c o n t r o l l e r s 20.0 5.5 Wheel c o n t r o l l e r s 40.0 10.5 Rudder Peda I s f o r s h o r t d u r a t i o n 100.0 f o r s t e a d y c o o r d i n a t e d t u r n s 40.0 T a b l e 24. Force l i m i t s f o rt h ee l e v a t o r ,a i l e r o n s , and rudder.

1 84 Bureau o fA e r o n a u t i c sR e p o r t AE-61-4-11 (Ref. 9 3 ) d i s c u s s e sf r i c t i o n f o r c e s i n t h e c o n t r o l s y s t e m a s t h e y a f f e c t t h e p i l o t ' s t r a c k i n g a c c u r a c y and recommends t h a t f r i c t i o n f o r c e s f o r hand c o n t r o l si ne x c e s s o f t h r e e pounds beavoided,sincetheydonotimproveperformancebut do i n c r e a s e p i l o t f a t i g u e .

A t t h e same t i m e , t h i s r e p o r t a l s o recommends t h a t no hand c o n t r o l r e q u i r e l e s st h a n two pounds o f f o r c e and no pedal movement r e q u i r el e s st h a ns e v e n pounds o f f o r c e f o r i n c r e a s e d p i l o t t r a c k i n g a c c u r a c y .

I t i s n o t c o n s i d e r e d n e c e s s a r y t o s e t s t a n d a r d s f o r s w i t c h e s a n d d i a l s , s i n c et h e y will most l i k e l y n o t r e q u i r e limit f o r c e s ; however, Woodson and C o n o v e r( R e f .9 4 )s u g g e s tt h a t ,f o ri n c r e a s e de f f i c i e n c y ,r o t a r y knob diameters rangefromone-half t o two inches andhave a maximum r e s i s t a n c e of one pound or less.

O f t e n ,t h ef o r c et ob e overcome i s usedas a feedbackcue;thus, it i.c necessary t o reproduce a p r e v i o u s l ye x p e r i e n c e df o r c e and a s s o c i a t ew i t h i t a c e r t a i nr e a c t i o n o f t h e a i r c r a f t . The a b i l i t y t o reproduce a g i v e nf o r c e ,a s s t a t e db y McCormick (Ref. 951, v a r i e s w i t h t y p e o f c o n t r o l and amount o f f o r c e t o be e x e r t e d ,a s shown i nt h eg r a p hb e l o w . The c o n t r o l st e s t e dw e r eo ft h e p r e s s u r et y p e , so v a r i o u s amounts o fp r e s s u r ec o u l d be a p p l i e d w i t h l i t t l e o r no displacement, making amount o fd i s p l a c e m e n tc o n s t a n t . The d e v i c e st e s t e d were a s t i c k , an a i r c r a f t - t y p e wheel,and a r u d d e r - l i ke peda I . The d i f f e r e n c e between t h ea c t u a lf o r c er e p r o d u c t i o n and t h ed e s i r e dr e p r o d u c t i o n was expressed i n I imens ( t h es t a n d a r dd e v i a t i o nd i v i d e d - b yt h es t a n d a r dp r e s s u r e ) .T h i s f i g u r ei n d i c a t e st h a t ,f o rp r e s s u r e so ff i v e pounds o rl e s s ,t h ee r r o r si n r e p r o d u c i n gt h ed e s i r e df o r c e sa r ep r o p o r t i o n a l l yg r e a t e r .F o rf i v et ot e n pounds, t h ee r r o r sa r e somewhat less,but s t i l l g r e a t e r t h a n t h o s e o f f o r c e s t e n t o f o r t y pounds; p r e s s u r e sg r e a t e rt h a nf o r t y pounds, o v e rl o n gp e r i o d so f t i m e , a r e a p t t o c a u s e p i l o t f a t i g u e .

- stick control

- - - - - wheel control

----o---- pedal control J 5 10 15 20 25 30 35 Pressure,pounds F i g u r e 80. R e s u l t s of d a t a o n r e p r o d u c l n g c o n t r o l f o r c e s .

1 85

L

O f c o n s i d e r a b l ei m p o r t a n c ei nt h ed e s i g n of any c o n t r o ls y s t e ma r et h e l i m i t a t i o n s o n t h e p i l o t ' s r e s p o n s e t i m e , o r t h e speed, c o n s i d e r i n g t h e e f f e c t of load, a t w h i c ht h ep i l o tc a na c t u a t et h ec o n t r o l s .O r l a n s k y( R e f . 9 6 ) r e p o r t sa ne x p e r i m e n t t o d e t e r m i n e t h e maximum r a t e a t w h i c h p i l o t s canpush or p u l l a c o n t r o ls t i c ka st h el o a dp e ru n i td i s p l a c e m e n t changes. From t h i s and o t h e rs t u d i e s ,h ec o n c l u d e st h a t , f o r a 3 5 - l bl o a d ,t h e maximum r a t e of s t i c k movement i s about 50 in./sec,pushing a s t i c k i s n e a r l y 25% f a s t e r t h a n p u l l i n g , a n d t h e r a t e of c o n t r o ls t i c km o t i o nd e c r e a s e sa st h el o a do nt h e s t i c ki n c r e a s e s . The Handbook of Human Engineering Data (Ref. 9 7 ) i n d i c a t e s t h a t , as t h ed i s t a n c e f o r a p o s i t i o n i n g movement i n c r e a s e s ,t h eo p e r a t o ri n c r e a s e s h i s speed of movement; t h et i m er e q u i r e df o rt h er e s p o n s e does n o ti n c r e a s ea s much as would be expected. For example, data from t h i s handbook i n d i c a t et h a t t h e t i m e f o r t o t a l movement i n c r e a s e s 15% when t h ed i s t a n c ei sd o u b l e d and 25% i f t h ed i s t a n c ei st r i p l e d . Also n o t e di st h e loss o f t i m ei nm a k i n g a p o s i t i o n movement when t h e p i l o t mustchange d i r e c t i o n s w i t h t h e c o n t r o l , when about 15% t o 24% o f t h e movement t i m e i s i n v o l v e d i n s t o p p i n g t h e movement i n one d i r e c t i o n and b e g i n n i n g i t i n a n o t h e r d i r e c t i o n . C o n t i n u o u s c u r v e d m o t i o n s , t h e r e f o r e ,a r ed e s i r e do v e rm o t i o n sw i t hs h a r pd i r e c t i o n a lc h a n g e s .

The p i l o t ' s r e s p o n s et i m ei st h e sum of h i s r e a c t i o n t i m e and h i s movement t i m e . R e a c t i o n t i m e , a s d e f i n e d i n AFSC DH 1-3 (Ref. 901, i st h ep e r i o d between t h eo n s e to ft h es i g n a l t o r e s p o n da n dt h eb e g i n n i n g of t h ea c t u a lr e s p o n s e .

Among t h e f a c t o r s a f f e c t i n g r e a c t i o n t i m e a r e t y p e o f s i g n a l , m o t i o n u n i t r e s p o n d i n g ,p r e c i s i o no ft h er e s p o n s e , age and sex of t h er e s p o n d e r ,p r e p a r a t i o n f o r t h er e s p o n s e ,p r a c t i c ef o rc o m p l e xr e s p o n s e s , and a m b i e n tc o n d i t i o n s .

AFSC DH 1-3 r e p o r t s t h a t hand response i s 20% f a s t e rt h a nf o o tr e s p o n s e , and t h ep r e f e r r e dl i m bi sa b o u tt h r e ep e r c e n tf a s t e rt h a nt h en o n - p r e f e r r e d .

The Handbook of Human EngineeringData(Ref. 9 7 ) g i v e s mean r e a c t i o nt i m e s for s i m p l e movements r a n q i n qf r o ma b o u t 0.22 sec t o 0.3 sec.Theconclusion from t h e s e d a t a i s t h a t t h e a u d i t o r y s y s t e m r e a c t s f a s t e r t h a n t h e v i sua I system.

The f o r e g o i n gi n f o r m a t i o nh a s been i n c l u d e dt oi n d i c a t et h ea n t h r o p o l o g i c a l b a s i sf o rs a t i s f a c t o r yh a n d l i n g :t h ea c t u a t i o nf o r c el e v e l s ,l i m bd i s p l a c e - ments, and phase r e l a t i o n s h i p sw i t hw h i c h a p i l o ti sc o m f o r t a b l e .I ti st h e n t h ed e s i g n e r ' st a s k t o p r o v i d e s a t i s f a c t o r y a i r c r a f t r e s p o n s e u s i n g t h e s e a n t h r o p o l o g i c a ld a t a t o d e s c r i b et h ei n p u tt ot h ea i r c r a f tc o n t r o l system. The r e a d e rw i l ln o t et h a tt h ep r i m a r ye m p h a s i s of t h i s r e p o r t i s on i n s u r i n gs a t i s - f a c t o r ya i r c r a f tr e s p o n s e .W h i l e many f u t u r el i g h ta i r c r a f tw i l lr e t a i n en- t i r e l y manual c o n t r o ls y s t e m sw h i c hr e q u i r et h ed e s i g n e r t o make c e r t a i n com- p r o m i s e sb e t w e e nw h a tf o r c e s ,d i s p l a c e m e n t s ,e t c .h ew o u l dl i k e t o p r e s e n tt h e p i l o t w i t h t h e responses t h a tt h e s ei n p u t sc a np r o d u c e , some f u t u r e l i g h t a i r - c r a f t will employ a r t i f i c i a l f e e l s y s t e m sw h i c hc a np r e s e n tt h ep i l o tw i t h whathewould l i k e w h i l e a t t h e same t i m ep r o v i d i n gs a t i s f a c t o r yr e s p o n s e s .

When thesystem i s capable of o p t i m i z i n gb o t ht h e s ef a c e t s it t h e n becomes i m p o r t a n tt oi n s u r et h a ta n t h r o p o l o g i c a lr e q u i r e m e n t sa r ec o n s i d e r e di nd e t a i l .

DESIGN FOR DESIRABLE RIDING QUALITIES

I n t h e p a s t t h e p r i m a r y d e s i g n and s p e c i f i c a t i o n e f f o r P s f o r a i r c r a f t have r i g h t l y been concerned w i t hi n s u r i n gs a f eo p e r a t i o n and s u c c e s s f u l com- p l e t i o n of t h em i s s i o n .P r o g r e s si nt h e s ea r e a s now appears t o have reached t h e p o i n t t h a t some a t t e n t i o n may b e d i r e c t e d t o w a r d p r o v i d i n g t h e p i l o t w i t h a c o m f o r t a b l er i d ea sw . e l l . A s i g n i f i c a n t p o r t i o n of what t h e p i l o t d e s c r i b e s a s r i d i n g q u a l i t i e s depends upon t h ed e s i g n of h i s s e a t and o t h e r r e s t r a i n t s .

N o i s e - i n d u c e d v i b r a t i o n s a l s o c o n t r i b u t e t o t h e p i l o t ' s g r o s s i m p r e s s i o n of t h er i d e .F o rt h ep r e s e n td i s c u s s i o n , however, c o n s i d e r a t i o nw i l lb er e - s t r i c t e d t o t h o s e a s p e c t s o f r i d e w h i c h c a n be a l t e r e d by theaerodynamicde- s i g no ft h ea i r c r a f t . Thus t h ec o n c e r nw i l l be d e s i r a b l ev a l u e s of l i n e a r and a n g u l a ra c c e l e r a t i o n sa s s o c i a t e dw i t ht h ea i r f r a m ed y n a m i c s .

As s t a t e d e a r l i e r it i s t h e s e changes i n v e l o c i t y w h i c h t h e p i l o t f e e l s as imposed f o r c e so nh i s b o d ya n dw h i c hh ei n t e r p r e t sa sm a j o rc o n t r i b u t o r s t o t h er i d i n gq u a l i t i e s of h i sa i r c r a f t .U n f o r t u n a t e l y , no s u b s t a n t i v ed i s - c u s s i o n o f t h e r e l a t i o n between t h em a g n i t u d e and f r e q u e n c yo ft h e s ea c c e l e r - a t i o n s and t h ea c c e p t a b i l i t yo ft h er i d e was f o u n di nt h el i t e r a t u r e . The followingarguments,however,lead t o c r i t e r i a w h i c h may f i n d u t i l i t y .

T h e r e a r e f i v e c h a r a c t e r i s t i c m o t i o n s a s s o c i a t e d w i t h r i g i d a i r c r a f t .

The s p i r a l mode has a v e r yl o n gt i m ec o n s t a n t and i sa p e r i o d i c . I t i st h e r e - f o r e u n l i k e l y t o i n d u c es i g n i f i c a n ta c c e l e r a t i o n si n n o r m a lo p e r a t i o no rt o o c c u ra t a r a t ew h i c hw i l l be uncomfortable. The r o l l mode, a l s oa p e r i o d i c , i sv e r yh e a v i l y damped and i ss e n e r a l l yn o t sensed by t h ep i l o t .T h i sl e a v e s t h e t h r e e o s c i l l a t o r y modes a st h es o u r c eo fr i d ed i s c o m f o r t s .

A-34 Consider f i r s tt h el o n g i t u d i n a lc a s e . From t h e second e q u a t i o no f it i s e v i d e n t t h a t t o a f i r s t o r d e r

a , - - w - Uoq = ZUu + Zww + Z ~ G ~ G

The n o t a t i o n dG i s usedhere t o i n d i c a t e an e f f e c t i v ea e r o d y n a m i ci n p u t , - s i m i l a r t o a f l a p o ne l e v a t o rd e f l e c t i o n , due t o a v e r t i c a lg u s t , w. Thus, S i n c e t h i s f o r m i s s i m i l a r t o t h a tr e s u l t i n g from a s i m p l ec o n t r o li n p u t it may beused t o d e s c r i b e p i l o t i n d u c e do s c i l l a t i o n sa sw e l la sg u s ti n d u c e d o s c i l l a t i o n s . O n l y t h e va l u e of Z6G and 6~ must be changed. Following t h i s t a c k , t h e n T h i s may b ee v a l u a t e dt h r o u g ht h eu s eo fE q u a t i o n s C-4 and C-7.

The r e s u l t i n g Bode p l o t has two w e l l - d e f i n e dp e a k sc o r r e s p o n d i n gt ot h e Phugoid mode and t h e s h o r t p e r i o d mode. S i n c et h ep e a ka c c e l e r a t i o n sa r e r e a l l y t h e v a l u e s of i n t e r e s t , it i sh e l p f u lt od e v e l o pa p p r o x i m a t ef o r m s f o r t h ea m p l i t u d e of a,/bE c o r r e s p o n d i n gt ot h e s et w op e a k s . The numerator o ft h et r a n s f e rf u n c t i o nc o n t a i n sf o u rz e r o s :o n ea tt h eo r i g i n , one j u s t t o t h e r i g h t o f it, one f a r t o t h e r i g h t o f t h e o r i g i n andone f a r t o t h e l e f t o f t h e o r i g i n . The l a t t e rt w oh a v en oi n f l u e n c eo nt h ep h u g o i d mode and v e r y l i t t l e o nt h es h o r tp e r i o d mode.

The d e n o m i n a t o rh a st h et w os e c o n do r d e rf a c t o r sc o r r e s p o n d i n gt ot h e p h u g o i da n ds h o r tp e r i o d modes. I t w i l l be r e c a l l e dt h a tt h ep h u g o i df a c t o r 25 s s2 + p + 1 7 w n n P P reduces t o 2Cp when w = wn and can be approximated by W n 2/wnp2 when SP w = wnsp. The s h o r tp e r i o f jf a c t o r is about I .O when w = wp and 2cSp when w = wsp. F o rt h e s e two c o n d i t i o n s t h e t r a n s f e r f u n c t i o n becomes K Talwn 2 a

" = - P

& E 25P When t h eg a i n and t i m ec o n s t a n t ,T a l ,a r ee v a l u a t e da sg i v e n byRef. 17 one has w 2 p u 4 'L n a o !

" -

I + -

z - * cm

6E o !

where t h e damping - r a t i o r e f e r s t o t h e mode b e i n gc o n s i d e r e d .I nt e r m so ft h e g u s tv e l o c i t y , w, t h i s can be w r i t t e n The q u a n t i t i e si np a r e n t h e s e sa r er e l a t i v e l yc o n s t a n tw i t h speed i n t h e r a n g eo v e rw h i c hl i g h ta i r c r a f to p e r a t e .S i n c e w w = c o n s t a n t n n P SP and w

n - uo

SP 1 88 t h e n

aZ . . . W u0/s

z ssp i sa p p r o x i m a t e l yi n d e p e n d e n t o f s p e e dw h i l e sp i n c r e a s e sw i t h Uo .

N o t et h a ti n c r e a s i n gt h ew i n gl o a d i n g is f a v o r a b l ef o ri m p r o v e dr i d e w h i l er e d u c i n go s c i l l a t o r y damping r e s u l t s i n a p o o r e rr i d e . Two a d d i t i o n a l comments a r ei no r d e rr e g a r d i n gt h i sr e l a t i o n : ( 1 ) w ' h e r ei st h ea m p l i t u d e o f an o s c i l l a t o r y g u s t h a v i n g a f r e q u e n c y of e i t h e r Unp or wnssp. a Z will be l e s s f o r o s c i l l a t o r yg u s t s of any o t h e rf r e q u e n c y . ( 2 ) The equations from w h i c ht h er e l a t i o n was d e r i v e d assume n os t e a d yp i t c h i n gv e l o c i t y . Hence o n ec a n n o tf i n dt h ev a l u e of az/6E i n a s t e a d yp u l l - u pf r o mt h i sr e l a t i o n .

T y p i c a l l y , t h e damping o ft h ep h u g o i d mode i s a b o u t 1/10 t h a t o f t h e s h o r tp e r i o d mode w h i l e i t s f r e q u e n c y i s a b o u t 1/20 t h a t o f t h e s h o r t p e r i o d mode. T h i s means t h a tt h ea c c e l e r a t i o n sa s s o c i a t e dw i t ht h ep h u g o i d mode c a nb et e nt i m e sa sl a r g ea st h o s ea s s o c i a t e dw i t ht h es h o r tp e r i o d mode. On t h e o t h e r hand, s i n c ed e s i r a b l ev a l u e so fs h o r tp e r i o d mode frequencyrange betweenabout I and 6 r a d i a n s / s e cf o re f f e c t i v eh a n d l i n g ,t h ep e r i o d of t i m e r e q u i r e d f o r t h ep h u g o i da c c e l e r a t i o n t o b u i l d up t o i t s peak i s o nt h eo r d e r of 5 t o 30 s e c o n d s .D u r i n gt h i sp e r i o do ft i m et h ep i l o th a st h eo p p o r t u n i t y t o t a k ec o r r e c t i v ea c t i o n .A u t o p i l o t sa l s o damp t h i sm o t i o ne f f e c t i v e l y .I n any case, a p i l o ti sn o tl i k e l yt oa s s o c i a t ep h u g o i di n d u c e da c c e l e r a t i o n s w i t hr i d eb u tr a t h e rw i t hh a n d l i n g .I ti se v i d e n t , however, t h a tb o t hr i d e and h a n d l i n ga n dt h e r e f o r es a f e t yc a n be improved s i g n i f i c a n t l y by i n c r e a s i n g t h e damping.

The phugoid mode i s a l s o accompanied by a l o n g i t u d i n a la c c e l e r a t i o nw h i c h has a magnitudeabout 1/10 t o 1/6 as l a r g ea sa z .S i n c ee i t h e rt h ep i l o to r an a u t o p i l o t will a t t e m p t t o s u p p r e s s a z a t wnp, it seems r e a s o n a b l e t o i g n o r e ax. Note t h a tv a r i a t i o n si n ax a r ee s s e n t i a l l yz e r oa t wnsp. Other than pos- s i b l e nausea r e s u l t i n g from t h el o n gp e r i o ds w a y i n gm o t i o na n dt h ep i l o tf a - t i g u ei n c u r r e di nc o n t r o l l i n g it, t h ep h u g o i do s c i l l a t i o nc a np r o b a b l y be i g - nored. I t a p p e a r st h e r e f o r et h a to n e may t a k e t h e f o l l o w i n g a s t h e r i d e c r i t e r i a i n t h e x-zplane: between I and 6 rad/sec.

w"sp A s i m i l a ra r g u m e n tc a nb e made f o r t h e l a t e r a l a c c e l e r a t i o n w i t h t h e r e s u l t t h a t * The a n t h r o p o l o g i c a lb a s i s f o r s u i t a b l e v a l u e s i st r e a t e dl a t e r .

...". . . ."

. .

where ..

I n t e r m s of a l a t e r a l g u s t w i t h o s c i I l a t i o n amp1 i t u d e v , t h i s becomes Psuo 2 Wd

-

a = v K a '--

4w <d

Y Y where Ka ' i n d i c a t e s t h a t Cn6R and C Y Q havebeenreplacedby cn6G and C Y ~ G .

W d i s d i r e c t l y p r o p o r t i o n a l t o Uo whl l e 5d i sa p p r o x i m a t e l yc o n s t a n t . The c r i t e r i a f o r s i d e a c c e l e r a t i o n a r e , t h e r e f o r e , Psuo 2 Wd a = ; K ~ ' " - - - 4w Sd Y Y w between I and 6 rad/sec.

d w i t h t h e maximum v a l u e f o r a y t o be s p e c i f i e d . ay of c o u r s er e f e r so n l yt op e r - t u b a t i o n s f r o m s t r a i g h t l i n e f l i g h t .

A d d i t i o n a l componentsmustbeadded t o t h e a c c e l e r a t i o n s t o a c c o u n t f o r s t e a d yr o t a t i o n .F o re x a m p l e ,i n a s t e a d yt u r nw i t h 0 = 0 t h ea c c e l e r o m e t e r i n d i c a t i o n sa r e A = U R - g s i t ~ @ ~ Y 0 0 A , = QoUo - COS @o .

I ft h et u r ni sc o o r d i n a t e d , R = - s i n 0 , , o u A = O Y l-cos2@o g

Qo = - uo cos@o '

and A z - 9 .

z cos @ The a c c e l e r a t i o n s f e l t b y t h e p i l o t a r e t h o s e v a l u e s w h i c h d i f f e r f r o m A , = g and Ay = 0. Thus i n a c o o r d i n a t e dt u r nt h ec o m f o r tl i m i ti sd e t e r m i n e db y A Z = g (cos 1 @o - 1) .

McFarland (Ref. 9 8 ) suggests a comfort limit of IO f t / s e c f o r I i near a c c e l e r a t i o n s .A c c o r d i n g t o t h i sc r i t e r i o nc o o r d i n a t e dt u r n s n c l e a r a i r shouldnotexceedbankangles o f 40°. I f t h e a i r i s t u r b u l e n t t h e a l l o w a b l e bankangleapparentlywouldbereduced.

While a p i l o t may t o l e r a t e t h e s e a c c e l e r a t i o n l e v e l s d u r i n g t u r n i n g f l i g h t becausetheyareunavoidableandareimposed f o r r e l a t i v e l y s h o r t p e r i o d s of t i m e , it does n o ta p p e a rr e a s o n a b l et h a t a 180 I b . p i l o t w o u l d f i n d n e a r l y 60 Ibs. of f o r c ea p p l i e ds i n u s o i d a l l ya l o n ga n y of h i s p r i n c i p a l axes f o r an e x t e n d e dp e r i o d of t i m ec o m f o r t a b l e . I t i sa l s or e a s o n a b l e t o e x p e c tt h a tb e c a u s e of h i s c o n s t r u c t i o n , a p i l o t will b em o r es e n s i t i v e t o l a t e r a lf o r c e st h a n t o v e r t i c a lf o r c e s .W h i l en os u b s t a n t i v ei n f o r m a t i o ni s a v a i l a b l e t o s u p p o r tt h ea u t h o r ' sq u a l i t a t i v ee x p e r i e n c e , it i s suggested t h a t because of t h e s e c o n s i d e r a t i o n s a = 4 f t / s e c Z and a = 2 f t / s e c Y may be more s u i t a b l e a c c e l e r a t i o n l i m i t s f o r t h e d u t c h r o l l and s h o r tp e r i o d m o t i o n st h a nt h a to f f e r e d byMcFarland.Sincesuchaccelerationsaresubstan- t i a l l y b e l o wt h o s ei m p o s e db ys t e a d yt u r n s ,t h ep i l o tp r o b a b l yw i l ln o tb ea s s e n s i t i v e t o t h e md u r i n gt u r n sa sa to t h e rt i m e s . The 40° b a n ka n g l el i m i t may t h e r e f o r eb ea c c e p t a b l ed u r i n gt u r n s ,p r o v i d e dt h e ay and a Zl i m i t sq u o t e d above a r e met d u r i n g s t r a i g h t - l i n e f l i g h t .

I n t e r e s t i n g l y enough, McFarland suggests a l i m i t o f 5 O bankangle a t low a l t i t u d e s and 25O bankangle a t h i g h a l t i t u d e s f o r t i l t a n g l e sw i t hw h i c h passengers would be comfortable. I f t h e 2 5 O t i l t o c c u r r e di n a s t e a d yt u r n , i t wou I d correspond t o a 3 . 2 f t / s e c 2 a c c e l e r a t i o n i ncrement a I ong t h e z- d i r e c t i o n .

McFarland was a l s oc o n c e r n e dw i t ht o l e r a b l el e v e l s of a n g u l a ra c c e l e r a t i o n .

However, it can be shown t h a t t h e g a i n o f i s a b o u t t i m e s t h e g a i n of I t a p p e a r s ,t h e r e f o r e ,t h a ta n g u l a ra c c e l e r a t i o n sa r e of no s i g n i f i c a n c e i f az and ay a r em a i n t a i n e da tt h el e v e l si n d i c a t e d . .

O t h e rd a t a of i n t e r e s t t o t h e r i d i n g q u a l i t y d i s c u s s i o n a r e p r e s e n t e d i n F i g u r e 81. T h i s shows a summary o f p i l o t comments o n t h e s h o r t p e r i o d mode hand I i ng qua l i t i e s o f a j e t f i g h t e r . N o t e t h a t good h a n d l i n g c h a r a c t e r i s t i c s l yi n s u r e good r i d i n g qua l i t i e s a c c o r d i n g t o t h e c r i t e r i a d e v e l o p e d w i I I v i r t u a l here.

a

. 7 . 3 .l . 2 3 .I 5 .6 .7 .8 .9 1 . 0 Damping Ratio F i g u r e 81. R e s u l t s of p i l o to p i n i o nr a t i n g s o nt h e h a n d l i n g q u a l i t i e s o f a j e t f i g h t e r (Ref. 9 9 ) .

One f i n a l comment r e g a r d i n g t h e r i d i n g q u a l i t i e s o f an a i r c r a f t i n g u s t y a i r may be made. R i g i d a i r c r a f t e x h i b i t f a i r l y r a p i d a t t e n u a t i o n o f normal a c c e l e r a t i o nr e s p o n s e f o r w>wsP. For example, a tf r e q u e n c i e si nt h er a n g e where human i n t e r n a lo r g a nr e s o n a n c e sa r ee x c i t e d ( - 4 2 rad/sec), a Z i sl e s s t h a n 10% a s much as a t wsp f o r a g i v e na m p l i t u d eo s c i l l a t o r yg u s t .O n l y i f t h e a i r c r a f t has a p o o r l y damped f u s e l a g eb e n d i n g mode or wingbending mode a tt h e s ef r e q u e n c i e sw o u l do n ee x p e c tt h e r e t o be s i g n i f i c a n t s t r u c t u r a l shake r e s u l t i n g i n p i l o t d i s c o m f o r t .

THE

EFFECT OF VARIATIONS IN STABILITY DERIVATIVES

ON THE

MOTIONS OF A TYPICAL LIGHT AIRCRAFT

INTRODUCTION

S p e c i f i c a t i o n o f a i r c r a f t geometry, mass d i s t r i b u t i o n , and c o n t r o ls y s t e m c h a r a c t e r i s t i c s t o y i e l d g i v e n r i d i n g and h a n d l i n g q u a l i t i e s i s u n f o r t u n a t e l y an i n t e r a t i v ep r o c e d u r e and t h e r e f o r e a l a b o r i o u sp r o c e s s . One c a nc o n s t r u c t t h en e c e s s a r yt r a n s f e rf u n c t i o n s ,b u tt h e r ei sn ou n i q u em e t h o d t o a s s i g n numericalvalues t o p a r t i c u l a r s t a b i l i t y d e r i v a t i v e s w h i c h assumes t h a t t h e s e v a l u e sw i l lb e a rt h eo f t e nn e c e s s a r yi n t e r r e l a t i o nw i t ho n ea n o t h e rn o r correspond t op h y s i c a l l yr e a l i z a b l eg e o m e t r y o r mass d i s t r i b u t i o n s . The approach employed here proceeded through several phases. $he f i r s t was t o t a k e a n e x i s t i n g l i g h t a i r c r a f t , i n t h i s c a s e a Cessna 182 , compute i t s s t a b i l i t y d e r i v a t i v e s , s u b s t i t u t e i n t h e t r a n s f e r f u n c t i o n , and e x t r a c t t h e r o o t s . The d e r i v a t i v e s w e r et h e nv a r i e di n d i v i d u a l l yt od e t e r m i n et h e sen- s i t i v i t y o f t h el o c u s o f r o o t s t o changes i n t h a t p a r t i c u l a r p a r a m e t e r .

T h i sp r o c e d u r ea l s op e r m i t so n et od e t e r m i n ea p p r o x i m a t e l yt h er a n g e of v a l u e s w h i c ht h ep a r t i c u l a rd e r i v a t i v e may have f o r s a t i s f a c t o r yp e r f o r m a n c e . I t i s onlyanapproximaterangebecause some d e r i v a t i v e s c a n n o t p h y s i c a l l y be v a r i e di n d e p e n d e n t l yo fo t h e r s .T h i sa s p e c t o f t h ep r o c e d u r e and i t s s i g n i - f i c a n c ew i l l become c l e a ri nt h es u b s e q u e n td i s c u s s i o n . Once reasonable v a l u e sa r eo b t a i n e df o rt h ed e s i r a b l ev a l u e s of t h e s t a b i l i t y d e r i v a t i v e s , p a r t i c u l a r l y t h o s e w h i c h have a s t r o n g e f f e c t o n movement o f t h er o o t s ,o n e thenproceeds t o d e t e r m i n et h eg e o m e t r i c and mass d i s t r i b u t i o n s w h i c h w i l l producethesevaluesandare a t t h e same t i m es e l f - c o n s i s t e n t .T h i s phase will be e l a b o r a t e dl a t e r .

I n p r e p a r i n g t h e f i g u r e s f o r t h e s t a b i l i t y d e r i v a t i v e v a r i a t i o n , t h e d e r i v a t i v e s c h o s e n f o r e x a m i n a t i o n w e r e g e n e r a l l y w i t h i n !IUS o r minusone o r d e ro fm a g n i t u d eo ft h o s ec a l c u l a t e df o rt h e Cessna 182 , a tc r u i s e .

I n c l u d e d w i t h t h e f i g u r e s f o r t h e l o c u s o f r o o t s due t o a v a r i a t i o n o f a s i n g l e s t a b i l i t y d e r i v a t i v e a r e t a b l e s w h I c hi n d i c a t e how t h e g a i n o f each p a r t i c u l a r t r a n s f e r f u n c t i o n v a r i e s a s a f u n c t i o n o f t h e s t a b i l i t y d e r i v a t i v e .

T a b l e 16 t h r o u g hT a b l e 31 andTable 35 t h r o u g hT a b l e4 6a r et a b u l a t i o n s o f t h en u m e r a t o rr o o t sf o rt h el o n g i t u d i n a l and l a t e r a ls t a b i l i t yd e r i v a t i v e v a r i a t i o n s ,r e s p e c t i v e l y .I n c l u d e da tt h e end o ft h i ss e c t i o na r e a s e r i e s o f s i x Bode p l o t s( F i g u r e s 90 t h r o u g h 9 5 ) i l l u s t r a t i n g t h e m o t i o n s o f t h e a i r c r a f ti n response t o c o n t r o ls u r f a c es t e pi n p u t s . The numerical values used t o p r e p a r et h e s eg r a p h sa r et h o s e f o r a Cessna 182 a t c r u i s e .

* The p r i n c i p a l g e o m e t r i c d i m e n s i o n s f o r t h e Cessna 182 a r e shown i n F i g u r e s 61a and 61b. The l o n g i t u d i n a l and l a t e r a l derivatives used f o r t h ea n a l y s i sa r et a b u l a t e di nT a b l e s 1 4 ~ and 14b.

** The v a l u e s w e r e c a l c u l a t e d b y t h e m e t h o d s p r e s e n t e d i n e a r l i e r s e c t i o n s of t h i s r e p o r t . They compare f a v o r a b l yw i t ht h o s ec a l c u l a t e db y Cessna a c c o r d i n g t o a personalcommunication.

I 8 80 L I S uwD.Io 2aOoLrn 1 2 IN NACA 2412 WINO AIRFOIL ROOT NACA 0001.5 TIP NACA O W 8 VERT. TAIL AIRFOIL ROOT NACA OOOO TIP M I C A OW6 W R Z . TAIL AIRFUL ROOT ANGLES OF INCIDENCE t l o 30' Y I N WHO-ROOT cwm -1. 30' N I N --TIP CHORD - 3 O t l S ' N I N STABILIZER DIHEDRAL + l o 4 4 ' MIN WINO MOMENTS OF INERTIA 941 S L U O - F T ~ I P I 3 4 6 S L U O - F T ~ IY Y I967 SLUO-FT2 I n 329.37

7-I

THRUST 225.441 I 1 \ I SJ.00 I- -1 F i g u r e 82b. Three View Drawing Continued - . . . . . . . .

CL 0.309 CD 0.031 1 Cm 0.0 CT 0.0 0.0 -0.3086 cLU GYB 0.0 -0.089 cDU

c%3

0 . 0 0.06455 CmU c"B C -0.0373 CTU 0 . 0 YP 4.61/rad. -0.4708 c L a cgP 0.126/rad. -0.0292 cDct cnP -0.885/rad. 0.2103 r cmct 0.0958 CL& 1.74/rad.

cgr- -0.09924 CD& 0.0 'nr Cm& -5.24/rad.

'Y6R 0.187 3.9/rad. 0.0147 cg6 R cLq 0.0 -0.0658 C

cDq n6R

-12.43/rad.

P 0.00205 S I u g s / f t .

cmq 0.427/rad. U 219.0 f t . / s e c .

Y 00

CD6E 0.0596/rad.

Cm6 E - 1 .28/rad.

T a b l e 25b. L a t e r a l S t a b i l i t y P 0.00205 s I u g s / f t . d e r i v a t i v e s( p e rr a d i a n ) .

U 2 1 9 . 0 f t . / s e c .

Y O0

T a b l e 25a. L o n g i t u d i n a l s t a b i l i t y d e r i v a t i ves.

LONGITUDINAL VARIATIONS

F i g u r e s 83a and 83b show t h e e f f e c t of CL v a r i a t i o n s o n t h e l o n g i t u d i n a l dynamlcs. I t w i l l be seen t h a t t h e r e i s l i t t l e change i nt h el o c a t i o n of t h e s h o r tp e r i o dr o o t s f o r a l l u s u a l v a l u e s of CL. The phugoid mode, however, i s s i g n i f i c a n t l y a l t e r e d by changes i n CL. A t h i g h C L ' S , b o t ht h ef r e q u e n c y and damping r a t i oa r ei n c r e a s e d . A t C L ' S n e a rz e r o ,t h er o o t s become r e a l w i t h o n eg o i n gu n s t a b l e .T h u so n ew o u l de x p e c td i f f i c u l t y i p m a i n t a i n i n g speed s t a b i l i t y i n a shallow, high-speed dive.

The e f f o r t t o p r o v i d e low d r a g f o r good performanceleads t o a n e u t r a l l y damped p h u g o i d , a s p o i n t e d o u t i n F i g u r e s 84a and 84b. S t a b i l i t ya u g m e n t a t i o n i st h e r e f o r er e q u i r e d i f one i s t o o b t a i nb o t h low ( Q O ) drag and good r i d i n g and h a n d l i n g( l o ww o r kl o a d )q u a l i t i e s .

Cm and CT mustbeconsideredsimultaneouslybecause Cm p r o v i d e s t h e aerodynamic moment t oc o u n t e rt h e moment produced by t h et h r u s t .I ng l i d i n g f l i g h t , Cm = 0. F i g u r e s 85a and 86a show t h a tt h es h o r tp e r i o d mode i sn o t a f f e c t e d by changes i n e i t h e r Cm o r CT. F i g u r e s 85b and 86b show t h e e f f e c t o nt h ep h u g o i d mode o f a l t e r i n g Cm and CT r e s p e c t i v e l y .A d d i n gp o w e rw i l l make CT p o s i t i v e and Cm n e g a t i v e .R e f e r e n c et ot h ef i g u r e sw i l l show t h a t making Cm n e g a t i v e w i l l c a u s e t h e p h u g o i d r o o t s t o s p l i t a l o n g t h e r e a l a x i s w i t h o n eg o i n gu n s t a b l e . On t h eo t h e r hand making CT p o s i t i v e r e s u l t s i n an u n s t a b l ep h u g o i do s c i l l a t i o n . Thus w h i l e i t i sn o tp o s s i b l e t o c o n c l u d e from t h e s ef i g u r e sa l o n et h ed e t a i l e da i r p l a n eb e h a v i o r when power i s added (because Cm and CT c a n n o tb ev a r i e di n d e p e n d e n t l yi nf l i g h tb u to n l yt h r o u g hd e s i g n changes such as t h el o c a t i o n of t h ee n g i n et h r u s tl i n e ) , i t i so b v i o u st h a t t h e a p p l i c a t i o n o f power i s d e s t a b i l i z i n g .

F i g u r e s 87a and 87b i n d i c a t et h ev a r i a t i o n si nl o n g i t u d i n a ld y n a m i c s produced by changing C L ~ . I n c r e a s i n g C L i s seen t o r e d u c e t h e f r e q u e n c y and z r .

t oi n c r e a s et h e damping of t h es h o r tp e r l o d mode. A s u f f i c i e n t l yl a r g ev a l u e of C L ~ s u g g e s ta p e r i o d i cs h o r tp e r i o dr o o t s .T h et i m ef o rt h ep h u g o i dt o damp t o h a l f a m p l i t u d e i s l i t t l e a f f e c t e d bychanging C L ~ ( F i g u r e8 7 b ) ,b u t t h eo s c i l l a t i o nf r e q u e n c yi s a d i r e c tf u n c t i o no f C L ~ . A l lu s u a lv a l u e s of C L ~ a r e t h e r e f o r e a c c e p t a b l e f o r s a t i s f a c t o r y a i r c r a f t r i d i n g q u a l i t i e s .

From F i g u r e 88a, it i s seen t h a tt h es h o r tp e r i o d mode i s v i r t u a l l : !

i n s e n s i t i v e t o moderate changes i n CDa. I n c r e a s i n gv a l u e so f C D ~ d e s t a b i l i z e s t h e p h u g o i d mode ( F i g u r e8 8 b ) .I n s t a b i l i t yi s most l i k e l yt oo c c u ri nt h e a p p r o a c hc o n f i g u r a t i o nw h e r e C D ~ i sg r e a t e s t . The maximum a c c e p t a b l ev a l u e i sa b o u tt w i c et h ev a l u ec a l c u l a t e d f o r t h e Cessna 182.

F i g u r e s 89a and 89b show t h e movement o f t h e s h o r t p e r i o d andphugoid r o o t sr e s p e c t i v e l y due t o a v a r i a t i o ni n Cma. F o rm o s tl i g h ta i r p l a n e sa s u s u a l l y l o a d e d , Cma will p r o b a b l yl i e between -.3 and -1.5. I nt h i s range, t h e t i m e t o damp t h e s h o r t p e r i o d o s c i l l a t i o n t o h a l f - a m p l i t u d ei si n d e p e n d e n t of t h e v a l u e of Cma, w h i l et h ef r e q u e n c yi n c r e a s e sa s Cma assumes g r e a t e r n e g a t i v e v-al-ues. Cma w i t hg r e a t e rn e g a t i v ev a l u e st h a n -1.5 g i v e ss h o r t p e r i o d mode f r e q u e n c i e sh i g h e rt h a nt h e 5-6 r a d / s e cl i m i td e s i r e d .V a l u e s o f Cma g r e a t e rt h a ta b o u t -0.23 c a u s e t h e s h o r t p e r i o d o s c i l l a t i o n t o d i s a p p e a r and become t w oa p e r i o d i c modes.

R e a s o n a b l ev a r i a t i o n si n Cma ( F i g u r e8 9 b )a l s oh a v el i t t l ee f f e c to nt h e t i m e t o damp t h ep h u g o i do s c i l l a t i o n .T h ep h u g o i df r e q u e n c yi n c r e a s e sa s Cma becomes morenegative,whilevaluesmorethanabout -0.05 c a u s et h e phugoid t o d e g e n e r a t ei n t oa p e r i o d i cm o t i o n s . I t i s seen t h e r e f o r e t h a t t h e p r o p e rv a l u eo f Cma i s u s u a l l yd e t e r m i n e db yr e q u i r e m e n t s f o r a c c e p t a b l es h o r t p e r i o d mode c h a r a c t e r i s t i c s . A n o t a b l ee x c e p t i o n may be mentioned, however.

A r e c e n tp a p e r( R e f . 116) d i s c u s s e dr e s u l t so b t a i n e dw i t h a l i g h t a i r c r a f t m o d i f i e d t o e v a l u a t e t h e p i l o t a c c e p t a n c e of v a r i o u s s h o r t p e r i o d mode damping r a t i o s . The damping r a t i o was made u n i t y( i . e .b o t hr o o t sl i eo nt h e n e g a t i v er e a la x i s )b yi n c r e a s i n g Cma (making i t l e s s n e g a t i v e ) . One would e x p e c t t h a t t h e p i l o t w o u l d b e p l e a s e d w i t h t h i s c o n d i t i o n s i n c e it means t h a t t h e a i r c r a f t w o u l d t r a c k r a p i d e l e v a t o r commands w i t h o u t o s c i l l a t i n g .

I n s t e a d ,t h ep i l o tf o u n d it q u i t eo b j e c t i o n a b l e ,s a y i n gt h a t it was d i f f i c u l t t o m a i n t a i n a i r s p e e d s t a b i l i t y . T h ep a p e rt h e r e f o r e recommends t h a t damping r a t i o sa p p r o a c h i n gu n i t yn o t be used., The f a l a c yi nt h e argument i si m m e d i a t e l yo b v i o u s when one examines t h e e f f e c t s of n e a rz e r o Cma v a l u e so nt h ep h u g o i d mode. The r o o t s become a p e r i o d i cw i t ho n eg o i n g u n s t a b l ea s Cma i n c r e a s e s . O f c o u r s et h ep i l o tw o u l df i n dt h i su n d e s i r a b l e .

I t i s t h e r e f o r e i m p o r t a n t t h a t u n i t y damping r a t i o f o r t h e s h o r t p e r i o d mode

not be a c h i e v e db yi n c r e a s i n g Cma a l o n e . As discussed below the most

s a t i s f a c t o r ys i n g l e means o fo b t a i n i n gt h i sb e h a v i o r( u n i t y damping r a t i o ) i s t o make Cmq more n e g a t i v e .I n c r e a s i n gt h et a i ll e n g t hi st h em o s te f f e c t i v e geometricchangeonecan make t o a c c o m p l i s ht h i si n c r e a s ei s damping.

Because o ft h ev e r ys i g n i f i c a n tl e n g t h e n i n gr e q u i r e d , however, it i sd e s i r a b l e t o combine t h i sw i t hs m a l lr e a r w a r ds h i f ti no p e r a t i n gc . g .( e f f e c t i v e l y a s m a l l i n c r e a s e i n Cma).

F i g u r e s 90a, 90b, 91a, and 91b show t h a tt h er o o t sa r er e l a t i v e l y i n s e n s i t i v e t o changes i n CL& and CQ&. Changing CL& by an o r d e ro fm a g n i t u d e f r o mi t so r i g i n a lv a l u e moves t h er o o t so n l ys l i g h t l yf o rb o t ht h es h o r t p e r i o d and phugoid modes. CD& i su s u a l l yc o n s i d e r e dt o be z e r of o rm o s t l i g h t a i r c r a f t . I f a v a l u ew e r ec a l c u l a t e d , it would be a tl e a s t an o r d e r o fm a g n i t u d es m a l l e rt h a n C&, a tl e a s t as small as .177. Thus, f o r a l l n o r m a lv a l u e so f CD&, b o t hp h u g o i d and s h o r tp e r i o dr o o t sr e m a i ni nv i r t u a l l y t h e same l o c a t i o n .

F i g u r e s 92a and 92b i n d i c a t e t h a t Cm& i s q u i t e i m p o r t a n t i n t l l e s h o r t p e r i o d mode b u tu n i m p o r t a n ti nt h ep h u g o i d mode. M o r en e g a t i v ev a l u e so f Cm& d e c r e a s et h ef r e q u e n c i e s and i n c r e a s et h e damping. High negative values can even lead t o t w oh i g h l y damped a p e r i o d i c modes. V a l u e sg r e a t e rt h a nz e r o should be avoided. In the normal range of v a l u e s ,a ne r r o ri n Cm& o f 20% could cause a 5% e r r o ri nt h ec a l c u l a t e df r e q u e n c yo fo s c i l l a t i o n . The phugoid mode r o o t sa r er e l a t i v e l yu n a f f e c t e d ,e v e n when Cm&, i s changed by an o r d e r o f magnitude.

F i g u r e s 93a, 93b, 94a, and 94b i n d i c a t et h a tb o t ht h es h o r tp e r i o d mode and t h ep h u g o i d mode a r e r e l a t i v e l y i n s e n s i t i v e t o v a r i a t i o n s i n CLq and C D ~ .

I t i s n o t e w o r t h yt h a t a v a l u eo f C L ~ = 0 w o u l dg i v ea l m o s tt h e same c h a r a c t e r - i s t i c s a st h ev a l u ec a l c u l a t e d .

L

Cmq i s q u i t e i m p o r t a n t t o thefrequencyanddamping of t h e s h o r t p e r i o d mode b u t r e l a t i v e l y u n i m p o r t a n t t o t h o s e of t h ep h u g o i d mode. The s h o r t p e r i o d roots f o r v a r i o u sv a l u e s o f Cmq a r e s i m i l a r i n b e h a v i o u r t o t h e r o o t s f o r v a r i o u sv a l u e s of Cm&.(see above). As Cmq becomes m o r en e g a t i v e ,t h e frequencydecreasesandthedampingincreases f o r t h e s h o r t . p e r i o d mode ( F i g u r e9 5 a ) .F o rt h en o r m a lr a n g e o f Cmq values, the damping i s more sensi- t i v e t o changes i n Cm t h a nt h ef r e q u e n c y of o s c i l l a t i o n .F o rv e r yn e g a t i v e va I ues of Cmq, a p e r i o a i c m o t i o n s a r e a c h i e v e d . V a r i a t i o n s of Cmq have a Imos-t no e f f e c to nt h ep h u g o i d damping. The frequency o ft h ep h u g o i d mode decreases as Cmq becomes more negative, as can be observed from Figure95b. Cmq, of course, i sa l w a y sn e g a t i v e .

Theabove r e s u l t sa r ed i s c u s s e db e l o w from t h e v i e w p o i n t of a c c e p t a b l e r i d i n g q u a l i t i e s , s i n c e t h e p r o p e r r a n g e o f t h e more i m p o r t a n t s t a b i l i t y d e r i v a t i v e si ss i g n i f i c a n ti na c h i e v i n gt h e s eq u a l i t i e s . T h ei m p o r t a n td e r i - v a t i v e s w h i c h a f f e c t t h e s h o r t p e r i o d mode appear t o be C L ~ , Cma, Cm&, and Cmq. I t would be d e s i r a b l e t o have t h es h o r tp e r i o d damp t o o n e - h a l fa m p l i t u d e i n one second o rl e s s ,w i t h a damping r a t i o g r e a t e r t h a n 0.6 and t o have t h e f r e q u e n c yl e s st h a nf i v er a d i a n sp e rs e c o n d f o r a c c e p t a b l er i d i n gq u a l i t i e s and l e s st h a nf o u rr a d i a n sp e rs e c o n df o r good r i d i n gq u a l i t i e s . Any r e a s o n a b l ev a l u e of C L ~ w i l l p r o v i d e a damping r a t i o g r e a t e r t h a n 0.6, a frequehcy less t h a n 4.25rad/secand a t i m e f o r damping t o o n e - h a l fa m p l i t u d e of l e s st h a n 0.25 seconds. Cma values between -.23 and -1.0 g i v ea c c e p t a b l e s h o r t p e r i o d r i d i n g q u a l i t i e s w i t h a f r e q u e n c yo fl e s st h a n5 . 0r a d / s e ca n d a damping r a t i og r e a t e rt h a n 0.6. I t i s d e s i r a b l e t o have Cm& l i e between -4.0 and -17.0; t h e more n e g a t i v et h ev a l u e ,t h el o w e rt h ef r e q u e n c y and h i g h e rt h e damping. Cmq should have values between -6.0 and -29.0 f o r good r i d . i n g q u a l i t i e s .

The frequencyanddamping c r i t e r i am e n t i o n e da b o v ep r o b a b . l ys h o u l db e r e g a r d e da sa p p l y i n go n l y t o a i r c r a f t w i t h o u t a r t i f i c i a l s t a b i l i t y augmenta- t i o n . The geometric changes necessary t o i m p r o v et h er i d i n g and h a n d l i n g q u a l i t i e s f u r t h e r ( i . e . t o o b t a i n u n i t y damping r a t i o f o r an undamped n a t u r a l f r e q u e n c yo f 6 r a d i a n sp e rs e c o n d )s u f f i c i e n t l yd e g r a d ea i r c r a f tp e r f o r m a n c e and payload capacity as t o make t h e s e c h a n g e s u n a t t r a c t i v e . I m p r o v e m e n t s i n r i d i n g and h a n d l i n gq u a l i t i e s , however, can be o b t a i n e d w i t h s t a b i l i t y augmen- t a t i o n w i t h o u t s a c r i f i c i n g e i t h e r p e r f o r m a n c e o r p a y l o a d .

Thephugoid mode m u s ta l s ob ec o n s i d e r e d when d i s c u s s i n g r i d i n g q u a l i t i e s .

T h i s mode shouldhave a damping r a t i o o f a t l e a s t 0.04, w i t h a maximum f r e - quency o fa b o u t 0.3 rad/sec. I f t h e p e r i o d i s o v e rt h r e et of o u r seconds, it c a nb et r a c k e db yt h ep i l o ta l t h o u g ht h ep i l o tw o r kl o a dw i l l behigh.

T h ep h u g o i df r e q u e n c yi sr e a l l yn o tv e r yc r i t i c a la sl o n ga s i t does n o t a p p r o a c ht h es h o r tp e r i o dr a n g e .T h ed e r i v a t i v e s Cm and CT c o n t r i b u t eo n l y t ot h ep h u g o i d mode a n ds h o u l db ec o n s i d e r e dt o g e t h e r .M o r ep o s i t i v ev a l u e s o f CT y i e l du n s t a b l ep h u g o i dr o o t s ;f o rt h ep a r t i c u l a ra i r p l a n ea n a l y z e d , Cm becomes more n e g a t i v ea s CT became more p o s i t i v ew h i c hl e a d st o anincrease i np h u g o i df r e q u e n c y and a decrease in damping. CL has t h e most e f f e c t on t h ef r e q u e n c yo ft h ep h u g o i d mode, w h i l e CD m a i n l ya f f e c t s damping. For t h e normal range of CL v a l u e s ( . 1 - 1 . 5 ) , t h ef r e q u e n c i e s may t a k e on v a l u e s as h i g ha s 0.3 t o 0.4rad/sec;however,asthefrequencyincreases,the damping a l s oi n c r e a s e s .T h el a r g e rt h ev a l u eo f CD, t h eb e t t e rt h er i d i n g q u a l i t i e s ; h o w e v e r ,p e r f o r m a n c er e q u i r e m e n t sm u s td i c t a t et h ev a l u eo f CD.

Values of Cma between -0.01 and-2.5appear t o g i v e s a t i s f a c t o r y p h u g o i d r e s p o n s eb ya f f e c t i n gt h ef r e q u e n c yw h i l ek e e p i n gt h ed a m p i n ge s s e n t i a l l y c o n s t a n t . Cmq has a s m a l le f f e c to nt h ep h u g o i df r e q u e n c y ,b u tv a l u e s between 0.0 and -30 g i v e d e s i r a b l e p h u g o i d r i d i n g q u a l i t i e s .

I n t h e a b o v ed i s c u s s i o n , a range o f d e s i r a b l e o r a c c e p t a b l ev a l u e s of t h e i m p o r t a n t s t a b i l i t y d e r i v a t i v e s hasbeengiven f o r t h ep h u g o i da n ds h o r t p e r i o dr i d i n gq u a l i t i e s . I t should be emphasized t h a tt h ed e r i v a t i v e sh a v e beendiscussedas i f t h e yw e r ei n d e p e n d e n to ft h eo t h e rd e r i v a t i v e s ,w h i l e i nr e a l i t yt h e ya r en o t .I nt h et a b l e below, t h e m o s ti m p o r t a n tl o n g i t u d i n a l s t a b i l i t y d e r i v a t i v e s a r e g i v e n w i t h t h e r a n g e of v a l u e sw h i c hs h o u l dr e s u l t i nt h ed e s i r a b l eh a n d l i n gq u a l i t i e si n d i c a t e d above. I t s h o u l d b e remembered t h a t eventhoughvalues of Cmq between 0.0 and -30.0 appearacceptable, i f t h ev a l u e sa r ev e r yn e a rz e r ot h e n it w i l l p r o b a b l y b ei m p o s s i b l e t o o b t a i n a c c e p t a b l ev a l u e s f o r o t h e r s t a b i l i t y d e r i v a t i v e s .

S t a b i l i t y D e r i v a t i v e A c c e p t a b l e Ranqe 0.03" t o 1 .O -0.23 t o -1.0 -4.0 t o -17.0 0 . 0 t o -30.0 range f o r l o n g i t u d i n a ls t a b i l i t y d e r i v a t i ves.

T a b l e 26. A c c e p t a b l e S i n c et h el o n g i t u d i n a ls t a b i l i t yd e r i v a t i v e sc a n n o tr e a l l y be v a r i e d independently, it was f e l t t h a t t h e movement of t h e r o o t s due t o t h e changes i nt h ea i r p l a n e ' sg e o m e t r yw o u l db e more i n f o r m a t i v et h a nj u s t a v a r i a t i o n o f t h el o n g i t u d i n a ls t a b i l i t yd e r i v a t i v e s .I nt h ec a s e o f l o n g i t u d i n a l dynamics, t h e movement o f t h e r o o t s was c a l c u l a t e df o rv a r i o u sc . g .l o c a - t i o n s , h o r i z o n t a l t a i l a r e a s , t a i l l e n g t h s , and t a i le f f i c i e n c i e s .I ts h o u l d be n o t e dt h a tt h e s ev a r i a t i o n sw e r e made w i t h o u t c h a n g i n g t h e o r i g i n a l i n e r t i ac h a r a c t e r i s t i c s of t h ea i r p l a n e . As an example o f how i n e r t i a changes would e f f e c tt h eg e o m e t r i cv a r i a t i o n s ,t h es h o r tp e r i o dp l o to f Rt i n d i c a t e s t h er o o t sw i t h and w i t h o u tt h ei n e r t i a changes. I t s h o u l da l s ob ep o i n t e d o u t t h a t t h e t a i l a r e a was v a r i e di ns u c h a way t h a t t h e t a i l a s p e c t r a t i o was h e l dc o n s t a n t .T h ef i g u r e sa n dt a b l e sb e l o wc a nb eu s e dt ot r a c kt h e r o o t s .

T h ep h u g o i da n ds h o r tp e r i o df r e q u e n c i e sd e c r e a s e da st h ec . g . was moved a f t , u n t i l a p e r i o d i c modes w e r eo b t a i n e df o rb o t h( F i g u r e 96 a n dT a b l e4 4 ) .

The s h o r tp e r i o d damping decreased and thephugoiddampingremainedalmost c o n s t a n ta st h e C.Q. moved a f t UP t o 42.5% m.a.c. The s h o r tp e r i o dr o o t s

. a t 45% i l e w i t h t h e c . g

were a p e r i o d i c w i t h t h e c . g . a t 42.5% m.a-.c., wh * L o w e rv a l u e so f CD will r e q u i r e a r t i f i c i a l damping t o meet t h e phugo i d damping c r i t e r i o n .

20 1 m.a.c., t h ep h u g o i d roots w e r ea p e r i o d i cw i t ho n eu n s t a b l er o o t .F o r a c . g .l o c a t i o na t 47.5%, t h e r e was a p h u g o i d - s h o r tp e r i o dc o u p l i n ga sw e l l a so n ev e r ys t a b l es h o r tp e r i o dr o o t and an u n s t a b l ep h u g o i dr o o t .

The v a r i a t i o n s i n h o r i z o n t a l t a i l a r e a w i t h t a i l a s p e c t r a t i o h e l d c o n s t a n t showed t h a t t h e s h o r t p e r i o d dampingincreasedandthefrequency decreased f o rt a i la r e a sg r e a t e rt h a nt h a t of t h e Cessna 182. T a i la r e a sl e s s t h a nt h eo r i g i n a la r e ag a v el o w e rf r e q u e n c i e sa n dl o w e rs h o r tp e r i o d damping ( F i g u r e9 7 a ) . The phugoid damping was v i r t u a l l yu n a f f e c t e d by t h ev a r i a t i o n s i nt a i la r e aw h i l et h en a t u r a lf r e q u e n c y showed onlysmallchanges.

F o r t h e v a r i a t i o n i n t a i l l e n g t h w i t h o u t i n e r t i a e f f e c t s , t h e s h o r t p e r i o d damping increased and the frequency decreased for t a i ll e n g t h s g r e a t e rt h a nt h a to ft h e Cessna 182 ( F i g u r e9 8 a ) .T a i ll e n g t h sl e s st h a nt h e o r i g i n a l l e n g t h g a v e l o w e r f r e q u e n c i e s w i t h l o w e r s h o r t p e r i o d d a m p i n g . Two a p e r i o d i cr o o t sa r eo b t a i n e df o rt a i ll e n g t h ss l i g h t l yb e l o w 45% o f t h e o r i g i n a ll e n g t h . The t r a j e c t o r yo ft h es h o r tp e r i o dr o o t sc o n s i d e r i n g i n e r t i a changes was s i m i l a r t o t h a tc a l c u l a t e dw i t h o u tc o n s i d e r i n gi n e r t i a changes b u te x h i b i t e dh i g h e r damping r a t i o sf o rl o n g e rt a i ll e n g t h s and l o w e rd a m p i n gr a t i o sf o rs h o r t e rt a i ll e n g t h s . The phugoid damping was a l m o s t u n a f f e c t e d b y v a r i a t i o n i n t a i I l e n g t h f r o m 200% t o 45% o f t h e o r i g i na I value, as seen i nF i g u r e 98b. The frequency did decrease as t a i ll e n g t h d e c r e a s e d ,b u tn ou n r e a s o n a b l ef r e q u e n c i e sw e r eo b t a i n e d f o r t h er a n g e of t a i ll e n g t hv a r i a t i o n s . As may b ee x p e c t e d ,t h ev a r i a t i o no ft a i le f f i c i e n c y , qt, gave r e s u l t ss i m i l a r t o t h o s ef o rt a i ll e n g t hv a r i a t i o n( F i g u r e s 99a and 99b).

R

LATERAL VARIATIONS

F i g u r e 100 shows t h e e f f e c t of CYB v a r i a t i o n so nl a t e r a ld y n a m i c s .I n F i g u r e 100a, t h eD u t c h R o l l n a t u r a lf r e q u e n c ya p p e a r sn e a r l yu n a f f e c t e db y changes i n CY@; however, Dutch Roll dampingincreasesslowly f o r more n e g a t i v e v a l u e s of cy^. F o rp r a c t i c a la i r f r a m ec o n f i g u r a t i o n s , cy^ i sn e v e rp o s i t i v e .

F.igures 100b and 1OOc show t h e n e a r l y n e g l i g i b l e e f f e c t of cy^ o n t h e s p i r a l and P o l l modes, r e s p e c t i v e l y .I ng e n e r a l ,t h et h r e e modes seem r e l a t i v e l y i n s e n s i t i v e t o changes i n CY6.

The e f f e c t o f CQ v a r i a t i o n o nt h eD u t c h Roll mode i s seen i n F i g u r e l o l a , w h e r el a r g en e g a t i v ev a l u e s of CQ d e c r e a s et h e dampingandincreasethe n a t u r a lf r e q u e n c y , and p o s i t i v ev a l u e si m p r o v e damping and decrease t h e natural frequency; however, f o r t y p i c a lv a l u e s of CRB, t h eD u t c hR o l l mode i so n l ys l i g h t l ya f f e c t e d . The s p i r a l mode, a sp o i n t e do u ti nF i g u r e 101b, i s more s t a b l e f o r l a r g en e g a t i v ev a l u e so f Cj$ b u tl e s ss t a b l ef o rb o t h s m a l ln e g a t i v ev a l u e sa n da n yp o s i t i v ev a l u e s .T h u sa na i r c r a f t may be made more s p i r a l l y s t a b l e by i n c r e a s i n gt h ew i n gd i h e d r a l ; however, t h e r e i s a l i m i t t o t h e amount o f d i h e d r a l because of t h ea d v e r s ee f f e c t of l a r g e nega- t i v ev a l u e so f C@ o nt h eD u t c hR o l l mode. The r o l l mode ( F i g u r e1 0 1 c )i s a l s o more s t a b l ef o rl a r g e rn e g a t i v ev a l u e so f CQ, b u t s u b s t a n t i a l changes a r en e c e s s a r yf o rt h ee f f e c t t o be v e r yn o t i c e a b l e .

The Dutch R o l l f r e q u e n c yi sh i g h l ys e n s i t i v et o changes i n CnB, F i g u r e 102a, w i t hl a r g ep o s i t i v ev a l u e sc a u s i n gv e r yh i g hf r e q u e n c i e s ;s m a l ln e g a t i v e v a l u e s ,w h i c ha r ep o s s i b l ef o ra i r f r a m e sw i t hs m a l lv e r t i c a lt a i l s ,p r o d u c e an unstable system. Dutch R o l l damping i sr e l a t i v e l yu n a f f e c t e d by changes i n Cng. The s p i r a l mode ( F i g u r e 102b) i sm o d e r a t e l ys e n s i t i v et o changes i n Cng; it becomes more s t a b l ea s CnB g r o w ss m a l l e r and becomes n e g a t i v e . The r o l l mode ( F i g u r e1 0 2 ~ )i su n a f f e c t e d by changes i n CnB.

V a r i a t i o ni n Cy has no e f f e c t on any o ft h el a t e r a l modes, a si n d i c a t e d i nF i g u r e s 103a, 103&, and 103c.

F i g u r e 104a shows t h a tt h eD u t c h R o l l mode i s o n l y s l i g h t l y a f f e c t e d by v a r i a t i o ni n CgP. The s p i r a l mode i nF i g u r e 104b i s somewhat more s e n s i t i v e ; l e s sn e g a t i v ev a l u e so f Cgp i n c r e a s e t h e s t a b i l i t y ; p o s i t i v e v a l u e s o f CgP a r en o tp h y s i c a l l yp o s s i b l e .O b v i o u s l y ,t h er o l l mode i nF i g u r e 104c i s h i g h l ys e n s i t i v e t o changes i n CkP, t h ed a m p i n g - i n - r o l ld e r i v a t i v e ,a s it becomes more s t a b l e f o r i n c r e a s i n g l y n e g a t i v e v a l u e s o f Cf, F o r v e r y s m a l l P ' n e g a t i v ev a l u e sa n df o rp o s i t i v ev a l u e s ,t h e r o l l and s p i r a l modes c o u p l e t o p r o d u c et h er o l l - s p i r a lo s c i l l a t o r y mode. MIL-F-8785B, ( R e f . 41, s p e c i - f i c a l l y p r o h i b i t s t h i s .

F i g u r e 105a shows t h ee f f e c to nt h eD u t c h R o l l mode of v a r i a t i o n s i n Cnp.

L a r g en e g a t i v ev a l u e sc a u s ea ni n c r e a s ei nn a t u r a lf r e q u e n c y ,w h e r e a sp o s i t i v e valuesdecreasethefrequencyanddampinguntilthesystem becomes u n s t a b l e .

The s p i r a l mode ( F i g u r e 105b) i sv i r t u a l l yu n a f f e c t e d by changes i n Cnp, and t h e r o l l mode ( F i g u r e1 0 5 ~ ) becomes l e s ss t a b l e f o r l a r g e , -n e g a t i v ev a l u e s of Cnp. 203 V a r i a t i o n i n Cyr h a sa l m o s t no e f f e c t o n any of t h e l a t e r a l modes, as F i g u r e s 106a, 106b, and 106c i n d i c a t e .

L a r g e rp o s i t i v ev a l u e s of C R r ( F i g u r e 1 0 7 a )i n c r e a s et h eD u t c h Roll damping, b u th a v el i t t l ee f f e c to nt h en a t u r a lf r e q u e n c y .M o r ep o s i t i v e values,though, seem t o g i v ea nu n s t a b l es p i r a l mode ( F i g u r e 107b) and a I ess s t a b l e rol I mode ( F i g u r e 1 0 7 ~ ) .

F i g u r e1 0 8 shows t h e e f f e c t of v a r i a t i o n s i n Cnr o n a l l t h r e e modes.

F o rl a r g e rn e g a t i v ev a l u e s of Cnr, t h eD u t c h Roll becomes two a p e r i o d i c modes, w i t h one root m o v i n gt o w a r dt h e r o l l mode r o o t a n da n o t h e rr o o ta p p r o a c h i n g t h es p i r a l root. These roots meet and, a t one p o i n t , two o s c i l l a t o r y modes e x i s t .T h i ss i t u a t i o n , however, may n o t b e p h y s i c a l l y o b t a i n a b l e . F o r l e s s n e g a t i v ev a l u e s of Cnr, t h eD u t c h Roll damping decreases, b u t t h e r o l l and s p i r a l modes a r e l i t t l e a f f e c t e d .

T h e p r e v i o u s r e s u l t s a r e examinedbelow from t h e v i e w p o i n t o f a c c e p t a b l e r i d i n gq u a l i t i e s .C o n s i d e r a t i o ni sg i v e ns p e c i f i c a l l yt ot h ea l l o w a b l er a n g e of s e v e r a li m p o r t a n ts t a b i l i t yd e r i v a t i v e sg o v e r n i n gt h e s eq u a l i t i e s . Two s t a b i l i t y d e r i v a t i v e s , CnB andCnr, seem t o have t h e m o s t e f f e c t o n t h e D u t c h RoI.1 mode. U s i n g a minimum <d (Dutch Roll damping r a t i o ) of t h el a r g e r of .19 and .35/wnd and l i m i t i n gt h en a t u r a lf r e q u e n c y t o a range from o n er a d i a n persecond t o a b o u t f i v e r a d i a n s p e r second, a r a n g eo fv a l u e s f o r t h e s e t w od e r i v a t i v e s was determined. CnB, w h i c h d e t e r m i n e s t h e n a t u r a l f r e q u e n c y , s h o u l dl i e between about .01 and about .15 per radian. A l s o Cnr, t h e yaw- damping d e r i v a t i v e , was found t o have a minimum v a l u e of -.45 p e rr a d i a n t o s a t i s f y t h e r e q u i r e m e n t f o r minimumandand a maximum v a l u e of -.09 p e r a tl e a s t .19. r a d i a n t o g i v e a damping r a t i o of The s t a b i l i t y d e r i v a t i v e Cg,g seems t o have t h e m o s t e f f e c t o n t h e s p i r a l mode. F o r b e s t r i d i n g q u a l i t i e s , a tl e a s t a n e u t r a l l ys t a b l es p i r a l mode i s d e s i r e d . To a t t a i nt h i s , a v a l u e o f Cg,g more n e g a t i v et h a n -.05 i s needed.

T h i s may be accomplished,physica I l y , b yi n c r e a s i n gt h e amount of a i r c r a f t w i n gd i h e d r a l ,b u tl a r g en e g a t i v e v a l u e s o f Cgg a d v e r s e l y a f f e c t t h e D u t c h frequencyanddecreasingthedamping,thus, a R o l l mode b yi n c r e a s i n gt h e Cg,g o f a b o u t -0.4 p e r r a d i a n i sc o n s i d e r e dt h em o s tn e g a t i v ev a l u ep e r m i s s i b l e .

The s t a b i ! i t y o f t h e r o l l mode i sd e t e r m i n e dp r i m a r i l y by Cg, , t h e dampi n g - in - r o l I d e r i v a t i v e . F o r a ~ / T R l e s st h a na b o u t 0.7, wh icR seems d e s i r a b l e f o r e a s yh a n d l i n g ,t h ev a l u eo f CgP m u s tb el e s st h a n -.04 p e r r a d i a n ;h o w e v e r ,t o ol a r g e a n e g a t i v ev a l u eo f Cg, may c a u s et h ea i r c r a f t P t o r e a c t s l u g g i s h l y t o t h e p i l o t ' s commands, and one of t h er e q u i r e m e n t s f o r e a s y m a n e u v e r a b i l i t y i s t h a t t h e a i r c r a f t be a b l e t o r o l l a c e r t a i n number o fd e g r e e si n a f i n i t e amount of t i m e .

The f o l l o w i n gt a b l el i s t st h ei m p o r t a n tl a t e r a ls t a b i l i t yd e r i v a t i v e s w i t h t h e i r s u g g e s t e d r a n g e o f v a l u e s f o r d e s i r a b l e h a n d l i n g q u a l i t i e s .

- __ - ~~

S t a b i l i t y D e r i v a t i v e s A c c e p t a b l e Range " ~ ~" ~ ( p e r r a d i an) .01 t o .15 c"B -0.45 t o -0.09 'nr -0.40 t o -0.05 QB l e s st h a n- 0 . 0 4 QP T a b l e 27. Acceptable range f o r l a t e r a ls t a b i l i t yd e r i v a t i v e s .

S i n c e t h e l a t e r a l s t a b i l i t y d e r i v a t i v e s c a n n o t p h y s i c a l l y b e v a r i e d independently, it seems necessary t o d i s c u s st h e movement o f t h e r o o t s o f t h e c h a r a c t e r i s t i ce q u a t i o n due t o a v a r i a t i o n of t h ea i r p l a n e ' sg e o m e t r y .F o r t h el a t e r a l dynamics case, t h e l e n g t h t o t h e v e r t i c a l t a i l , Rv, t h ea r e ao f t h e v e r t i c a l t a i l , Sv, t h ew i n gd i h e d r a la n g l e , and a c o m b i n a t i o no fv e r t i c a l t a i l areaandwingdihedralwerevaried.

The e f f e c to nt h eD u t c hR o l l( F i g u r e 109a) o fi n c r e a s i n g Rv i s t o i n - c r e a s eb o t hf r e q u e n c y and damping of t h i s mode. T h er e a s o nf o rt h i sc a nb e g a t h e r e db ye x a m i n i n gt h ee f f e c to nD u t c hR o l lc h a r a c t e r i s t i c s of CnB and Cnr v a r i a t i o n s .I n c r e a s i n g Rv makes Cng more p o s i t i v e .T h i sh a s a d i r e c t i n f l u e n c eo nt h ef r e q u e n c y of t h eD u t c hR o l l .I n c r e a s i n g Rv a l s o makes C n r m o r e n e g a t i v e . F o r s m a l l i n c r e a s e s i n t h e m a g n i t u d e o f Cnr, t h e damping of t h eD u t c hR o l l is i n c r e a s e db u tt h ep e r i o di su n a f f e c t e d .F i n a l l y , f o r l a r g en e g a t i v ev a l u e s of Cnr, t h ep e r i o d of t h eD u t c hR o l li n c r e a s e s and t h e damping a l s oi n c r e a s e s .F o r Rv up t o 150% of t h eo r i g i n a l , Cng seems t o d o m i n a t ea st h ef r e q u e n c yr i s e sq u i t er a p i d l y ,b u tf o r Rv g r e a t e rt h a n 150% t h ef r e q u e n c yl e v e l so f f and t h es y s t e m becomes more h i g h l y damped as Cnr becomes dominant. For decreasing Rv, b o t ht h ef r e q u e n c y and damping decrease u n t i l , a t a b o u t 35% o ft h eo r i g i n a lt a i ll e n g t h ,t h es y s t e m becomes u n s t a b l e .

The s p i r a l mode ( F i g u r e1 0 9 b ) becomes more s t a b l e f o r b o t hi n c r e a s i n g and d e c r e a s i n g Rv. T h i si sp o s s i b l es i n c ei nt h ee x p r e s s i o n , CgP Cng/(CgB Cnr - Cng Cfir)CyP, w h i c hg o v e r n st h es p i r a lr o o t ,C n r , CnB, and Cgr a l l change w i t h Rv v a r i a t i o n . The r o l l mode ( F i g u r e1 0 9 ~ )i su n a f f e c t e db yc h a n g e si n v e r t i c a lt a i ll e n g t hs i n c e Cgp i sn o td e p e n d e n t on Rv. The moment of i n e r t i a IZZ was h e l d c o n s t a n t f o r o n e s e t of Rv v a r i a t i o n s and a l l o w e d t o change f o r t h e second s e t o f Rv v a r i a t i o n s . The r e s u l t so fb o t ha r e shown i nF i g u r e 109 and i n e r t i a c h a n g e sp r o d u c eo n l ys m a l ld e f l e c t i o n si nt h ec u r v e sw i t ht h e g e n e r a lt r e n d sr e m a i n i n gt h e same.

F i g u r e 110a shows t h e e f f e c t of v e r t i c a l t a i l a r e a v a r i a t i o n o n t h e Dutch Roll mode. I n c r e a s i n g S v g i v e sb o t h a h i g h e r f r e q u e n c y a n d a s l i g h t l y more damped s y s t e ms i n c e CnB and Cnr a r e a g a i n s u b s t a n t i a l l y a f f e c t e d b y changes i nv e r t i c a lt a i la r e a . (Cnr becomes l e s s n e g a t i v e a s S v decreases; f o r t h e same c o n d i t i o n s CnB may g ot h r o u g hz e r o and become n e g a t i v e ) w i t h a v a l u e of a b o u t 48% of t h e o r i g i n a l t a i l a r e a , t h e s y s t e m becomes u n s t a b l e .

The s p i r a l mode ( F i g u r e 110b) becomes more s t a b l ef o rd e c r e a s i n gv e r t i c a l t a i l a r e a ,s i n c et h i sp r o d u c e ss m a l lp o s i t i v e o r e v e nn e g a t i v ev a l u e s of Cne (Figure 102b). The r o l l mode ( F i g u r e 1 1 0 ~ ) r e m a i n s u n a f f e c t e d s i n c e CR i s P 20 5 only slightly affected by changes in vertical tai I area.

The effect of variations in wing dihedral on the Dutch Roll mode is modest as shown in Figure llla. However, the spiral mode (Figure l l l b ) is heavily dependent on wing dihedral, since this determines the value of C ~ Q B , the "effective dihedral" derivative. Positive dihedral angle gives a more stable spiral mode and negative dihedral causes the mode to become unstable. The roll mode (Figure lllc) i s essentially constant for wing dihedral variation.

The combination of changing both vertical tail area and wing dihedral is shown in Figure 112. The Dutch Roll mode (Figure 112a) is the same as that for SV variation alone since dihedral angle has little effect on this mode.

However, the spiral mode (Figure 112b1, which previously became less stable for increasing SV, now can be made more stable by an increase in the dihedral angle. Thus the Dutch Roll and spiral modes may both be improved by simultaneously increasing vertical tail area and dihedral angle. The roll mode (Figure 112c) remains unaffected.

Short Psriod Mode Roots for the CL Variation I r n I r n " 5 " 5 / r O .3093 "4 "4 " 3 " 3 " 2 " 2 " 1 " 1 I 1 1 I 1 1 I I I I I I I I I I Re I I -5 -4 -3 -2 - 1 -5 -4 -3 -2 - 1 1 2 " " -1 -1 CLValues listed CLValues listed beside roots beside roots

-- --

-2 -2 "-3 "-3 " -4 " -4 " - 5 " - 5 F i g u r e 83a F i g u r e 83a Phugoid Mode Roots for theCLVariation I r n .6 .5 A .3 . 7 5 1 .2 .3093 .1 .1 1 c.

" W 1 ;r Re .1 .2 . 3

-. 1

.I CLValues I isted .3093( beside roots - . 2 -3

- .4

-. 5

-.6 F i g u r e 83b NUMERATOR ROOTS U I STAB I L I TY DERIVATIVE REAL I MAG I NARY REAL REAL

-0.5 - 0.71701 0.0 -26.9358 I

2 I .6049

-0. I - I .29165

0.0 -2 I .2794 I 15. I774

0. I - 2.12606 0.0 - I 7.28866

I I .34820 0.3093

- 7.68330 0.0 - 7.93156

6.84425

0.75 - 5.92058 k15.08719

I .58786

I .5 - 6.66843

k26.45906 0.56020

3.0 - 9.03239

k40.574 I 7 0.24139 4.0 -10.68138 k47.65224 0. I7489 4.5 - I I .51 191 k50.80265 0.15371

5.0 - I 2.34472 k53.75583

0. I371 I -0.5 -0.28084 0.0 0.24735 -195.51762 -0. I -0. I3526 0 . 0 0. I0320 -195.46607 0. I -0.0 I509 k0. 1 I668 -195.44030 0.3093 -0.0 I 472 k0.20640 -195.41333 0.75 -0.0 I 393 f0.32196 -195.35655 I .5 -0.0 I259 k0. 045569 -195.25991 3.0 -0.0099 1 k0. 64493 -195.06664 4.0 -0.008 I 2 -10.74498 -194.93780 4.5 -0.00723 k0. 79032 -194.87339 5.0 -0.00633 k0. 83322 -194.80897

e

-0.5 -0.09750 0.0 -9.99285 -0. I -0.03463 0.0 -2.0562 I 0. I -0.03088 0 . 0 -2.06020 0.3093 -0.04605 0.0 -2.04529 0.75 -0. I4779 0.0

- I .94407

I .5 -0.768 I 7 0 . 0 - I .3246 I

3.0 - I .04730 k I .70249

4.0 - I ,04790 k2.45078

4.5 - I ,04820

k2.81025 5.0 - I .0485 I k3. I6440 T a b l e 2 8 N u m e r a t o r r o o t s f o r CL v a r i a t i o n s .

Short Period Mode Roots for CDVariation Im I I - 5 - 4 -3 -2 -1 - 1 CD Values listed beside roots - 2 -3 - 4 - 5 F i g u r e 84a Phugoid Mode Roots for the Cg Variation Re CDValues listed beside roots

t --.20

F i g u r e 84b 21 1 NUMERATOR ROOTS U S T A B I L I T Y R E A L I MAG I NARY R E A L R E A L DER I VAT I VE -7. I7723 0.0 -8.39084 6.82668 -0.03 0.031 I -7.68330 0.0 -7.93 I56 6.84425 -7.907 I9 51.21744 6.91800 0 . 3 -7.92566 kl .32588 6.93144 0.35 -7.94408 51.42569 6.94479 0.4 0.5 -7.98077 *I .60575 6.971 18 -0.03 0.01478 10.20640 -195.41334 0.031 I -0.0 I472 W . 20640 -195.41333 -0.14171 +O. I5078 -195.41328 0 . 3 0.35 -0. I6542 *O. I2430 -195.41327 -0.18914 k0.08393 -!95.4!326 0.4 -0.12191 -195.41324 0.5 -0.35238 0.0 -0.03 0.01315 0.0 -2.0 I60 I -2.04529 0.031 I -0.04605 0.0 -0.30099 0.0 -2.17126 0 . 3 0.35 -0.34860 0.0 -2.19476 -0.39622 0.0 -2.2 I826 0.4 0.5 -0,49148 0.0 -2.26524 Table 24 Numerator r o o t s f o r C v a r i a t i o n s .

D Short Period Mode Roots for the Cm Variation rn 0 . 0 -.5 0 . 5 - e d e I -5 -4 -3 -2 -1 1 2 -1 C, Values listed -2 beside roots -3 -4 -5 Figure 8% 21 3 PhugoidMode Roots forthe C, Variation Im O.( -.25 -.I L I A I Re -. I I

-. 4 73 7 2 -.l

Cm Values listed beside roots . I 5 .3 F i 21 4 NUMERATOR ROOTS U STAB1 L l T Y D E R I V A T I V E REAL I MAG I NARY REAL REAL -0.5 -7.68330 0.0 -7.93156 6.84425 -0.25 -7.68330 0.0 -7.93156 6.84425 -0. I -7.68330 0.0 -7.93 I56 6.84425 0.0 -7.68330 0.0 -7.93156 6.84425 9.15 -7.68330 0.0 -7.93156 6.84425 0.30 -7.68330 0.0 -7.93 I56 6.84425 0.50 -7.68330 0.0 -7.93156 6.84425 -0.5 -0.00400 +O. I4062 -195.43477 -0.25 -0,00936 +O. I7668 -195.42405 -0. I -0.0 I 257 *O. I9507 -195.41762 0.0 -0.0 I472 k0.20640 -195.41333 0.15 -0.0 I793 k0. 22227 -195.40690 0.30 -0.021 I5 k0 .23704 -195.40047 0.50 -0.02544 k0. 25535 -195.39189 -0.5 -0.01557 0.0 -2.05353 -0.25 -0.03078 0.0 -2.04944 -0. I -0.03993 0.0 -2.04696 0.0 -0.04605 0.0 -2.04529 0.15 -0.05524 0.0 -2.04277 0.30 -0.06445 0.0 -2.04022 0.50 -0.07677 0.0 -2.03679 Table 3 0 . Numerator r o o t s for C , variations.

21 5 Short Period Mode Roots for t h e h r i a t i o n I m " 0.0 - 5 0 . 5

I . 1 I

" I I I I I Re -5 -4 -3 -2 -1 1 2 -1 CT Values listed beside roots -2 -3 .)

-4 2 5 0 . 5 0 . 0 -5 Figure 86a

r

PhugoM Mode Roots for the CT Variatlon D . 2 0 0 . 0 -.2

-.lo

- - . O S . S .5 I 1 I

. 1 ".+

1 I 1

-25 720 - . i s -.lo -.05 ,os .lo . l s a 2 0 .25 . 3 O *35

" -a5 CT Valuer listed beside roots

- -.lo

O D "20 F i g u r e 86b NUMERATOR ROOTS U STAB I L I TY I MAG I NARY REAL REAL DERIVATIVE REAL -7.68330 0.0 -7.93 I56 6.84425 -0.5 -7.68330 0.0 -7.93 I56 -0.2 6.84425, -7.68330 0.0 -7.93 I 56 6.84425 0.0 0.2 -7.68330 0.0 -7.93 I 5 6 6.84425 -7.68330 0.0 -7.93 ! 56 6.84425 0.4 0.5 -7.68330 0.0 -7.93 I56 6.84425 AI2 -0.09450 -195.41718 -0.5 -0.40952 0.0 k0. I7076 -195.41487 -0.2 -0. I0940 0.0 -0.0 I472 +O. 20640 -195.41333 k0. I9522 0.2 0.07997 -195.41180 -195.41027 0.4 0. ! 7466 k0.12527 0. I7346 0.0 0.26937 -195.40950 0.5

e

-0.5 -0.52 I96 0.0 -2.04256 -2.04437 -0.2 -0.23624 0.0 -0.04605 0.0 -2.04529 0.0 0. I4399 0.0 -2.04605 0.2 0.33390 -2.04669 0.4 0.0 -2.04697 0.5 0.42882 0.0 T a b l e 31. N u m e r a t o r r o o t s for 5 v a r i a t i o n s .

Short Period Mode Roots for t h e C L 2 r i a t i o n In 4.608 -4.0 20.0 S I 1 S I I I I I I I I I I -7 -6 -5 - 4 -3 -2 -1 -8 -1 -2 -3 -4 "5 CL,Values listed beside roots F i g u r e 87a 21 9 Phugoid Mode Roots for the C L ~ Variation / Re 20.0 F i g u r e 8 7 b NUMERATOR ROOTS U S T A B I L I TY REAL R E A L D E R I V A T I V E R E A L I MAG I NARY

3.8569 I Q. 24363 - 12.4401 0

- 4.0

0.48723 0 . 0 4.46793 - I I .56084

0.0

-7.68330 - 7.93156

4.608 6.84425 0.0

- 8.98685 k4. I0263 7.60946

8.0 8.22106

12.0 - IO. 23230 k5.68619

- I I . 10984 k6.45655 8.56664 15.0 9.01289

20.0 - I 2.50753 k7.331 19

A a -0.0 I472 k0.20640 -195.41333 - 4.0 0.0 -0.0 I 472 k0. 20640 -195.41333 4.608 -0.0 I472 k0. 20640 -195.41333 8.0 -0.0 I472 k0. 20640 -195.41333 k0.20640 -195.41333 12.0 -0.0 I472 15.0 -0.0 I 472 k0. 20640 -195.41333 -0.0 I472 k0.20640 -195.41333 20.0

e

0.0 2.00513

- 4.0 -0.01518

0.04672 a. I5901

0.0 4.608 -0.04605 0.0 -2.04529 -0.03914 0.0 -3.66043 8.0 12.0 -0.036 I 9 0.0 -5.55989 -0.03503 0.0 -6.98343 15.0 20.0 -0.03389 0.0 -9.35520 T a b l e 3 2 . N u m e r a t o r roots for qCc v a r i a t i o n s .

22 1 .

Short Period Mode Roots for the CgaVariation “ 5 “ 5 ao -.5 01.5 - 0 4 - 0 4 “3 “3 “ 2 “ 2 “ 1 “ 1 I I I I I 1 1 Re I I I

1 I * -

-4 -3 - 2 -; - 5 1 2 1 2 “-1 “-1 CgaValueslisted CgaValueslisted beside roots beside roots

--- 2 --- 2

“-3 “-3 ““4 ““4 a0 - 5 0 15 “ - 5 “ - 5 F i g u r e 88a Phugoid Mode Roots tor the Cpa Vartation In

-

m . 2 0 -J256 -5 ) 1.5 '

- . l 5

-

-.lo

--.OS I a I a I

I , rn Re

.os .10 .10 .O 5

cDa Values listed

- - -.os

beside roots -.5 -.20 p i.gu r e 88b NUMERATOR ROOTS U STAB1 LITY REAL I MAG I NARY REAL REAL DERIVATIVE -0.5 -5.98845 k18.51477 I .I0150

0. t256 -7.68330 0.0 - 7.93156 6.84425

I .5 -0.43878 0.0 -32.75372 29.04580 ACI -0.5 -0.01472 +O .20640 -195.41333 -0.0 I 472 k0.20640 0. I256 -195.41333 -0.01472 +O. 20640 -195.41333 I .5

e

-0.5 -0.09055 0.0 -2.00078 0. I256 -0.04605 0.0 -2.04529 0.04537 0.0 -2. I367 I I .5 T a b l e 33. N u m e r a t o r r o o t s f o r C,, v a r i a t i o n s .

c1 Short. &riod Mode Roots for the *Variation

- 2 5

- 1.5

-3 -. 3 CnG,Values listed beside roots -.6 -L5 p.i.gure. 8% PhugoidModeRoots for the Cm,Variation Re

~ m , Values listed

beside roots

p i\gu re 89b

NUMERATOR ROOTS U STAB I L I TY DERIVATIVE REAL I MAG I NARY REAL REAL

-2.5 -6.9 I423 - +4.94 608 5.05789

- I .5 -7.46700 k2.95479 6. I6344

-0.8852 -7.68330 0.0 - 7.93156 6.84425

-0.6 -5.99482 - 9.93315

0.0 7. I5734

-0.3 -5.330 I 2 0.0 - I 0.92447 7.48395

-0.25 -5.24383 0.0 - I I .06490 7.53808

-0.2 -5. I6223 0.0 - I I .20053 7.59212

-0.08 -4.98270 0.0

- I I .50937 7.72143

0.0 -0.02 -4.90034 - I I .65618 7.78587 -0.01 -4.88704 0.0 - I 1.68021 7.79660 0.0 -4.87385 0.0 - I I .70412 7.80732 0 . 0 - I I . 93702 7.91432 0 . I -4.74795 A a -2.5 -0.0 I 472 k0.20640 -195.41333 -0.0 I 472 k0.20640 -195.41333 - I .5 -0.8852 -0.0 I472 +O. 20640 -195.41333 -0.0 I472 +O. 20640 -195.41333 -0.6 -0.3 -0.0 I472 +O .20640 -195.41333 -0.25 -0.0 I472 +O. 20640 -195.41333 -0.2 -0.0 I472 +O. 20640 -195.41333 -0.08 -0.0 I472 -195.41333 - +O. 20640 -0.0 I472 +O. 20640 -195.41333 -0.02 -0.0 I 472 -195.41333 -0.01 - TO. 20640 0.0 -0.0 I 472 k 0 . 20640 -195.41333 -0.0 I472 +O. 20640 -195.41333 0. I

e

-0.05433 0 . 0 - I .78232 -2.5 0.0

- I .5 -0.04893 - I .94544

0.0 -2.04529 -0.8852 -0.04605 -0.6 -0.04480 0.0 -2.09151 -0.04356 0.0 -2. I4008 -0.3 -0.25 -0.04335 0.0 -2.14816 0.0 -0.2 -0.04315 -2. I5625 -0.04268 0.0 -2. I7565 -0.08 -0.04244 0.0 -0.02 -2. I8535 -0.04240 0.0 -2. I8697 -0.01 0.0 -0.04237 0.0 -2. I8858 -0.04 I98 0.0 -2.20474 0 . I Tab l e 34. Numerator roots f o r C, v a r i a t i o n s .

a - .. " Short Period Mode Roots for the CLfariation Im 4 4 9 9 -5 -5 4 4 -3 -3

- -

-1 -1 n 1 1 I i Re

- = I n I I t

- i r = - Re

"5 - 4 -3 -2 "f "5 - 4 -3 -2 1 2 1 2 "f " " 1 . " " 1 .

ChValues listed ChValues listed beside roots beside roots " -2 " -2 " - 3 " - 3 " -4 " -4 " - 5 " - 5 I' Phugoid Mode Roots for the Cb%rlation m .20 .15

.io

.O 5

710 -d5

-.a

Cdalues listed beside roots -.lo -.15 -2.00 ".20 NUMERATOR ROOTS U STAB1 L I P / REAL REAL DERIVATIVE REAL I MAG I NARY

- 2.0 -7.81212 f l .36416 6.7637 I

-7.68330 0.0 - 7.93156 6.84425

I .7419 20.0 -4.97289 0.0 -10.61842 7.21373

- 2.0 -0.0 I472 -195.41333

t-0.20640 I .7419 -0.0 I472 t-0.20640 -195.41333 20.0 -0.0 I 472 20.20640 -195.41333

e

- 2.0 -0.04604 0.0 -2.08659

-0.04605 0.0 -2.04529 I .7419 20.0 -0.04608 0.0 - I .86499 T a b l e 35. N u m e r a t o r r o o t s f o r G. v a r i a t i o n s .

c1 n

- Re

1 I 1 2 -5 -4 -3 -2 - 1 -I CD& Values listed - 2 beside roots -3 - 4

- 1-00 1.0

-5 F i g u r e 91a 23 1 . . .

Phugoid Mode Roots for tbc C M Variation

1 0 -10 .os I I I - R e -30 .os .10 -.os -.os

c Values listed

DL beside roots -u) 1.0 - . 2 0 Figure 91b NUMERATOR ROOTS U STAB I L I TY D E R I V A T I V E REAL I MAG I NARY REAL REAL

- I .o -4.34161 0.0 - 16.685 19 5.55105

0.0 -7.68330

0.0 - 7.93156 6.84425

I .o -5.29188 k4.46484

9.03719 ACi

- I .o -0.0 I 472

+O. 20640 -195.41333 0 . 0 -0.0 I472 * O .20640 -195.41333

I .o -0.0 I472

+O. 20640 -195.41333

e

- I .o -0.0460 I -2.04686

0.0 0.0 -0.04605 0.0 -2.04529

I .o -0.04608 0.0 -2.0437 I

T a b l e 36. Numerator roots f o r C v a r i a t i o n s .

D ; Short Period Mode Roots for the CmVariation Im I

I - Re

-10 -9 -8 -7 -1 1 2 - 1 -2 -3 -4

hd, Values listed

beside roots -5 -6 F i g u r e 92a m .20 I I

" =

110 -.OS 905 .lo C,, Values listed -905 b e s i d e roots -.lo .

-. 15

-.20

p i:gu r e 92h

NUMERATOR ROOTS U STAB I L I TY REAL REAL DER1 VAT I VE REAL I MAG I NARY -50.0 -3.90188 0.0 -24. I I478 4.43293 -4.97207 0.0 -20.0 -14.47813 5.79415

-15.0 -5.34358 - I 2.77020 6. I I235

0.0 -10.0 -5.9201 5 0.0 -10.89391 6.46725

-7.68330 - 7.93156

- 5.237 0 . 0 6.84425

k2.36959 0.0 -7. I7309 7.30864 5.0 -6.59500 +3. I5130 7.80707 ACi -50.0 -0.01472 + O . 20640 -195.41333 -20.0 -0.0 I472 k0.20640 -195.41333 -15.0 -0.0 I472 +O. 20640 -195.41333 k0. 20640 -10.0 -0.0 I472 -195.41333

- 5.237 -0.0 I 472 +O. 20640 -195.41333

0.0 -0.0 I472 + O . 20640 -195.41333 5.0 '-0.01472 k0.20640 -195.41333 -0.04594 0.0 -2.22363 -50.0 -20.0 -0.0460 I 0 . 0 -2. I0095 - ! 5 . 0 -0.04602 0.0 -2.08 I77 -2.06293 -10.0 -0.04603 0 . 0

- 5.237 -0.04605 0.0 -2.04529

0.0 -0.04606 0.0 -2.02623 5.0 -0.04607 0.0 -2.00835 T a b l e 37. N u m e r a t o r r o o t s f o r C , . v a r i a t i o n s .

0 : I I I I I I I I __t__tc Re I I - 4 -3 -2 -1 - 5 . 1 2 -1 C Values listed Lq -2 beside roots -3 -4 -5 pi,gure 9.3d Phugoid Mode Roots for the CLqVariation Im -20 .10 .os

-

7 10 - . O 5 CL Values listed 4 5 beside roots

F i:gu r.e 9.3h

NUMERATOR ROOTS U STAB1 LlTY DERIVATIVE REAL I MAG I NARY REAL REAL -7.73020 ? I .83643 6.60700

- 5.0

3.9168 -7.68330 0.0 - 7.93156 6.84425

20.0 -5.52374 0.0 - I 0.37532 7.27783

50.0 -4.2 I075 0.0 - 12.23076 8.09894

- 5.0 -0.0 I 475

*0.20171 -204.55258 3.9168 -0.0 1472 k0.20640 -195.41333 20.0 -0.0 I465 +O. 2 I575 -178.92893 50.0 -0.0 1449 k0.237 I 8 -148,18066

e

- 5.0 -0.04605 0.0 -2.04529 -0.04605 0.0 3.9168 -2.04529 20.0 -0.04605 0.0 -2.04529 -0.04605 0.0 50.0 -2.04529 T a b l e 38. Numerator roots f o r E , v a r i a t i o n s .

Short Period Mode Roots for the C h Variation in -1.0 11) I I I 1

I I I I - Re

I . 1 2 - 5 -4 -3 - 2 -1 -1 C Values listed Dq -2 beside roots -3 -4

- 1.0 1 . 0

- 5

p i,gure 94a Phugoid Mode Roots for the C b Variation tm ' .a0 m . 1 0 5 -A5 C Values listed Dq beside roots -1 F i g u r e 94b 24 1 . " ... .....I., I, NUMERATOR ROOTS U STAB I L I TY REAL I MAG I NARY REAL REAL DER1 VAT I VE

-4.88199 0.0 - I 6.37834 5.21643

- I .o

0.0 -7.68330 0.0 - 7.93156 6.84425

I .o -5.50557 k3.67822 9.51381

- I .o -0.0 I395 %. 20645 -195.41487

0.0 -0.0 I472 3 , 2 0 6 4 0 -195.41333 I .o -0.0 I 549 %. 20634 -195.41179

e

-0.04605 0 . 0 -2.04529

- I .o

-0.04605 0.0 -2.04529 0.0 -2.04529

I .o -0.04605 0.0

T a b l e 39. N u m e r a t o r r o o t s f o r CD v a r i a t i o n s .

Short Period Mode Roots for the Cmq Variatkn

- 12.43/

I

Re -10 -9 -8 - 6 - S - 4 -3 - 2 - 1 1 2

I

\

t -l

Values listed Cm, beside roots t - 5 Figure 95a Phugoid Mode Roots for the +Variation Im

-

-a5 ,Re I 1

-.lo -.os

.os ,10

% Values listed

t -mi

beside roots F i g u r e 95b NUMERATOR ROOTS U STAB I L I TY REAL REAL DERIVATIVE REAL I MAG I NARY -4.67832 0.0 -21.12196 4.22104 -50.0

-5.26804 0.0 - 14.83064 5.33863

-30.0 0 . 0 -14.51478 5.40867 -29.0 -5.3 I 297 5.48035 -5.36047 0.0 -14.19800 -28.0 0 . 0 -I I .58404 6. I1845 -20.0 -5.88484

-7.68330 0 . 0 - 7.93156 6.84425

- 12.4337

Q.38 1.45 7.69596

- 5.0 -6.96597

8.35687 0.0 -6.44403 k2.89559 -5.96067 k3.21405 9.09494 5.0 9.91393 10.0 -5.5 I777 53.4096 I ACi -208.22433

.O -0.01350 +o. 20000

-50 +O .20333 -201.40389 -30.0 -0.01413 -0.01416 +O .20350 -201.06287 -29.0 -0.01 4 I 9 +O .20368 -200.72185 -28.0

- I 97.99364

-20 .o -0 .O t 446 +O. 20506

+O .20640 -195.41333 - I 2.4337 -0.01472

- 192.87823

-0.0 I498 fO .20774 - 5.0 -0.01516 +O -20866 -191.17308

0 .o

- 189.46792

-0.0 I 534 +O. 20958 5.0

- 187.76275

10.0 -0. o i 552 +0.21053

e

-50.0 -0.04605 0.0 -2.04529 -30.0 -0.04605 0.0 -2.04529 -0.04605 0.0 -2.04529 -29.0 -28 .O -0.04605 0.0 -2.04529 -0.04605 0.0 -2.04529 -20.0 0.0 -2.04529

- 12.4337 -0.04605

-0.04605 0.0 -2.04529

- 5.0

-0.04605 0.0 -2 04529 0.0 0.0 -2 04529 5.0 -0.04605 -0.04605 0.0 -2 04529 10.0 ons .

Table 4 8 N u m e r a t o r r o o t s f o r C v a r i a t i

L

NUMERATOR ROOTS U STAB I L I TY DERIVATIVE REAL I MAG I NARY REAL REAL -7.585 I 1 k2.06610 7.74177 -0.5 -0. I -7.67989 & I .57952 7.35208 0.0 -7.70387 +I .42618 7.25523 0.1 -7.72796 +I .25 I 3 5 7. I5859 0.25 -7.76429 G.92196 7.01404 0.4268 -7.68330 0 . 0 - 7.93156 6.84425 0.75 -6.59867 0.0 - 9.17542 6.53544

I .o -6.220 I 5 0.0 - 9.67839 6.29786

2.0 -5.22593 0.0 - ! I . 17913 5.35625

5.0 -3. I3507 0.0 - 14.73205 2.47395

Act -0.5 -0.0 I 495 G.21 144 158.94513

-0.0 I 484 a. 20920 81 I .6855l

-0. I 0.0 -0.0 I482 S. 20868 0 . I -0.0 I479 S.20816 -820.16606 0.25 -0.0 I476 S. 20734 -330.61069 -0.0 I472 S. 20640 -195.41333 0.4268 0.75 -0.0 I465 %. 20472 -113.03072

-0.0 I460 a. 20345 - 85.83329

I .o

2.0 -0.0 I442 3. I9859 - 45.03736

5.0 -0.0 I 406 S. I8585 - 20.56030

e

-0.5 -0.0440 I 0.0 -2.30446 -0.04482 -2. I939 I -0. I 0.0 -0.04504 0.0 -2. ! 6597 0.0 -2. I3790 0 . I -0.04526 0.0 0.25 -0.0456 I 0 . 0 -2.09556 -0.04605 0.0 -2.04529 0.4268

-0.0469 I 0 . 0 - I .95233

0.75

-0.04765 0.0 - I .87948

I .o

2.0 -0.05 I 4 2 0.0 - I .57923 -0.56084 5.0 -0.09587 0.0 Table 41. Numerator roots f o r v a r i a t i o n s .

&E NUMERATOR ROOTS U STAB I L I TY REAL I MAG.1 NARY REAL REAL DER I VAT I VE k 8.05539 -3.58753 -0. I -2.0983 I

- I .62690 k IO. 80785 -4.16199

-0.05 -4.9259 I I 36.99857 0.0 -7.68330 0.0 -7.93 I56 6.84425 0.0596 k 2.63517 4.82778 0. I -6.67433 -4.76 I77 k 3.95572 I .29735 0.5 -195.52065 -0.0 I 590 k0.20626 -0. I k0. 20630 -195.48703 -0.05 -0.0 I553 -0.01516 k0.20635 -195.45341 0.0 k0.20640 -195.41333 0.0596 -0.0 I 472 -0.0 I442 k0.20644 -195.38617 0. I -195.11722 0.5 -0.01 145 +O. 20676

e

0 . 0 -2.05 I90 -0. I -0.03843 -2.04983 -0.0408 I 0 . 0 -0.05 -2.04776 0.0 -0.04320 0.0 -2.04529 -0.04605 0 . 0 0.0596 -2.0436 I 0. I -0.04798 0.0 -2.02679 -0.0673 I 0.0 0.5 Table 42. Numerator roots f o r C, v a r i a t i o n s .

& E NUMERATOR ROOTS U STAB I L 1 TY REAL I MAG I NARY REAL REAL DERIVATIVE -5,258 I5

-15.0 0.0 -33. I889 I 29.6762 I

-5.30278 -10.0 0.0 -27.27432 23.80632 -5.44662 -19.44515

- 5.0 0.0 l6.lZI06

- 3.0 -5.66806 0.0 -15.01042 I I .9078 I

- 7.93156

- I .283 -7.68330 0.0 6.84425

- 0.75 -6.66992 5.51054 4.56927

-3.963 I7 k4.21546 - 0.84424

0.0

0.5 -2.55070 k6.96412 - 3.66917

1 . 0 -2.25392 k9. I0942 - 4.26274

A a

-15.0 -0.0 I 522 + O . 2085 I -2239.746

-0.01519 * O . 20839 - 1494.563

-10.0 -0.01512 +-0 .20807 - 749.381 I3

- 5.0

- 3.0 -0.01503 +O. 20767 - 451.30835

-0.0 I472 k0. 20640

- I .283 - 195.41333

-0.0 I 434 +-0. 20486 - 115.97763

- 0.75

-0.00003 -4.23938 0.01042 0.0 0 . 0

-0.0 I 682 k0.21467 70.32293

0.5

I .o -0 .O I602 kO.21 158 I 44.83956

-0.04267 0.0 - 2. I5785

-15.0

-10.0 -0.04282 0 . 0 - 2. I 5263

-0.04327

- 5.0 0 . 0 - 2. I3696

-0.04388 - 2. I 1600

- 3.0 0.0

-0.04605 - 2.04529

- I .283

0.0

-0.04899 0.0 - 1.95623

- 0.75

0.0 0.01042 0.0 - I 5.26480

0.5 -0.03486 0.0 - 2.47223

- 2.32224

I .o -0.03832

T a b l e 43. N u m e r a t o r r o o t s f o r C v a r i a t i o n s .

msE Sort h r i o d and mtgoid Roots kr c.g. V a n a t i i -1 -2 -3 F i g u r e 96 c.g. LOCATION

SHORT PERIOD ROOTS SHORT PERIOD / PHUGOID PHUGO I D ROOTS

( % m.a.c.1

Rea I lmag i nary Rea I Rea I I magi nary Rea I I magi nary Rea I

- 4.6345 k7.5094

0.0 -0,0136 k0. 1865

26.4 - 4.0848 k4.3679 -0.0136 k0. I801

42.5 - 5.3421 -2. I568

-0.0182 +O. I I75

45.0 - 6.1725

- I . 1979 -0.0763 0.02000

47.0 - 6.6547 -0.4796 -0.3432 0. I3957

47. I - 6.6764 -0,40098 k0.06724 0.14501

47.5 - 6.7615 -0.36069 k0.19146 0. I6725

- 6.9626

48.5 -0.26908 k0.30056 0.22966 -0.15946 k0.34295

50.0 - 7.2381 0.3566 I

52.5 - 7.6453

-0.06470 k0.31623 0.69487

- 8.0056

55.0 -0.03672 k0.28517 I . I I858

100.0 - I I .5233

-0.01703 k0.23765 7.3668 I Table 44. Short Period a n d Phugoid Roots for c.g. Variations.

Short Period Mode Roots for the St Variation n ARt =constant __f_c Re -1 St values listed beside roots as percent of -2 original value -3 -4 -5 -6 Figure 97a 25 1 Phugoid Mode Roots for the St Variation n 2 0 1 0

ARt = constant

St values listed beside roots as percent of original value 5 0 200 f o Figure 976 Short Period Mode Roots for the It Variation m

I I I I - Re

-8 -7 -6 - 5 -4 -3 -1 1 -1 It values listed beside mots as percent of original value -2 2 5 - 3 -4 -5 F i g u r e 98a Phugoid Mode Roots forthe It Variation m 2 0 " -05 It values listed beside roots as percent of original value

200 :J

Figure 98b

Short Psrlod Mode Roots for the vt Ylrlatlon

. I I I

I I I I I I 7 6 4 3 2 1 vt values listed beside roots F i g u r e 99a

Phugold Mode Rootsfor t h e I+ Variation

bn 9 values listed t beside roots -05

t

- - 9.10

-

- 4 5

-- 720

Figure 99b

% Values listed

beside roots

t -lp

t"

F i g u r e 1OOa Spiral Mode Roots for the Variation

%

-

-.oos 3 . 0 Q4 9 0 8 -2.0 -LO I L. I ~n a , I ,Re "

I - I I

I I -p20 w.015 7 0 I O :005 ,005 C values listed ye beside roots F i g u r e 100b Roll Mode Roots for the Variation Im

9 3

-10 3 . 0 I * a I I I I Re I I I 40 I

- 1 4 -13 -12 -11

F i g u r e 100c I NUMERATOR ROOTS STAB1 LlTY I MAG I NARY REAL REAL DERIVATIVE REAL

0.0 - I 2.58832 -115.40198

-5.0 .02272

.02272 0.0 - I 2.58832 -115.40198

-2.0

- 12.58832 -115.4Ol98

- I .o .02272 0.0

.02272 0.0 - I 2.58832 -115.40198

- ,308

.02272 0 . 0 - I 2.58832 -115.40198

.6

0.0 - 12.58832 -115.40198

1.5 .02272 -115.40198 .02272 0.0 - 12.58832 2.5

- I 2.58832 -115.40198

3.0 .02272 0 . 0 0.0 -6.8 I839 9.22719 -5.0 0.0 0.0 0.0 -5.8646 I 9.69719 -2.0 9.86678

- I .o 0.0 0 . 0 -5.55962

9.86759 0.0 0.0 -5.29036

- .308

0 . 0 -5.08575 IO. I5225 0.6 0 . 0 I 0.32069 0 . 0 0.0 -4.82705 I .5 0 . 0 -4.54640 10.51463 2.5 0 . 0 -4.40880 10.61431 3.0 0.0 0 . 0

- I 2.65862

-2. I I867 0.0 - .I3344

-5.0

f .33074 - I 2.65543

-2.0 - .41643

-12.65501

- .I7978 f .50051

- I .o

- 1 2.65493

- .01387 k .52896

- .308

f .49324 - 12.65086

0.6 . I9895

- I 2.64990

.4 I 204 k ,33633 I .5

- 12.64890

0.0 I .0204 I 2.5 .27717

- 1 2.04842

.2 I422 0 . 0 I .32018 3.0 T a b l e 4 5 . N u m e r a t o r r o o t s f o r C v a r i a t i o n s .

y B ” 5

’ rnm

I a 1 m 1 I -7 4 -s -4 C, Values listed

P

beside mats “ - 5 Figure 101a Spiral Mode Roots for the C Variation

lp

-1.0 +a9

-

-Q5 CI values listed

e

beside roots Figure 101b Roll Mode Roots for the C Variation Im

t 3

“1

- 1.0 7089 0 . 5 1.0 1 . 5

I I n I t - t rn I 1

- w

I I “ m - I L I I *Re

-14 - 13 -12 -11 -10 -9

--

-1 Figure 101c NUMERATOR ROOTS B STAB1 LlTY REAL REAL I MAG I NARY DERIVATIVE REAL 0.0 -12.588 -I 15.42

-I .o .02272

0.0 -12.588 -I 15.42

- .089 .02272

- I 15.42 0.0 -12.588 .5 .02272 - I 15.42

I .o .02272 0.0 -12.588

0.0 - 12.588 - I 15.42

I .25 .02272 -12.588 -I 15.42 I .5 .02272 0.0 31.9475 0.0

- I .o -21.8216 0.0

9.8676 0.0

- .089 - 5.2904 0.0

0.0 .5 .54328 k19.03747 0.0

- .96334 k26.74 I

I .o

0.0

- I .71665 529.8279

I .25 0 . 0

- 2.4700 k32.6067

I .5

- I .o . I6499 + I .8909 -13.0132

- .089 - .01387 k .52896 - 12.65493

.5 - I .5533 0.0 I .268l I -12.3981

-2.2648 0.0 I .74875 -12.16725

I .o

-2.57468 0.0 I .9370 - I 2.04559

I .25 -2.8686 I 0.0 2. I0452 - I I .91919 I .5 Table 46. Numerator roots for CQ v a r i a t i o n s .

B Dutch Roll Mode Roots for the %Variation Im .6 .5 .4 - 2 5 .I!

.064t C C Values listed

"P

beside roots .064 6 .2 5 Figure 102a Spiral Mode Roots for the Variation Im

c"p

A

- * 7 1 0 - . O s - . o 2 1 I 1 m I c. I c . 1 I 0.0 AM6 I I 1 1 w 1 " 1 - 1 I U I Re - 7 -6 -5 -4 -3 -2 -1 *?bo 1 Cn, values listed

t -l

beside roots F i g u r e 102b Roll Mode Roots for the Cn Variation

P Im

A

" 1 I I I I I Re I # @ 1 I - 5 - 4 -3 - 2 - 1

-

"1 F i g u r e 102c NUMERATOR ROOTS STAB1 LlTy REAL I MAG I NARY REAL DERIVATIVE REAL

- 12.588

-I 15.402

- .05 .02272 0.0

-12.588 -I 15.402

- .02 .02272 0.0

- 12.588 - I 15.402

.02272 0.0 0.0

- 12.588 - I 15.402

,06455 .02272 0.0 -12.588 - I 15.402 .02272 0.0 . I 5

- 12.588 - I 15.402

0.0 .25 .02272

0.0 - 12.588 - I 15.402

.4 .02272

0.0 - 12.588 - I 15.402

.5 .02272

- 12.588 - I 15.402

.6 .02272 0.0 -6.4556 0.0

- .05 I I .0918 0.0

-6. I7427 0.0

- .02 10.81052 0.0

-5.98 I44 0.0 10.61768 0.0 0.0 -5.29036 0.0 .06455 9.86759 0.0 -4.3599 0.0 8.9962 0.0 .I5 -3.01 16 0.0 7.6478 0.0 .25 .28833 0.0 4.3478 0.0 .4 0.0 2.3181 I 3-3.47434 .5 0.0 2.3181 I 3-5.31620 .6 0.0

- 12.66270

3- .60700

- .05 - ,08494

- 12.6598

t- .5886

- .02 - .0664 I

- 12.65785

2 .5756

0.0 - .05405

- 12.6549

-.01387 3- .52896 0.06455

- 12.64358

.03877 3- .45654 0. I5

- I 2.63429

. I0076 3- ,34167

0.25

- 12.62066

.00538 0.0 .38233 0.4 -12.61 I77

-. 12719 0.0 .63929

0.5

- 12.60305

0.6 -. I9755 0.0 .834 I9

Table 47. Numerator roots for Cn v a r i a t i o n s .

B

Dutch Roll Mode Roots for the CypVariation Im

-

- 3 . 5

-

-3.Q

-

-25

-

-m

" l . 5 --LO "0.5 -95 CypValues listed beside roots -lD -15

I

-2B -25 -3.0

7.0 w5

-35

I

Figure 103a Spiral Mode Roots for the Cyp Variation

:"

w . 0 0 5 -z5 zo I I I I I 8 I " R e -.020 -,OlS -.a373 -.o 10 ,oos .005 Cyp values listed - 4 0 5 beside roots

F i gure 103b

Roll Mode Roots for the CrpVariatlon

C "

T'

- 7.5 - . a 3 7 3 7.0

1 - I r 1

1 -

I W - I- I

I ' Re

- 14 - 13 -12 - 1 1

F i g u r e 103c I NUMERATOR ROOTS

B

STAB1 L I N DERIVATIVE REAL I MAG 1 NARY REAL REAL -7.5

0.02177 0.0 -15.624 - 98.470

- .0373

0.02272 0.0 -12.588 -I 15.402 7.0 0.02373 0.0 -10.552 -133.746 -7.5 0.0 0.0 -5.2904 9.8676

- .0373 0.0 0.0

-5.2904 9.8676 7.0 0.0 0.0 -5.2904 9.8676 J,

-7.5 0.25191 a. 45598

-13.187

- .0373 -0.0 I387 S. 52896 -12.655

7.0 -0.29057 a. 45964 -12.102

Tab I e 48 . N u m e r a t o r r o o t s f o r Cy v a r i a t i o n s .

P

Dutch Roll Mode Roots for the C Variation ' P CI values listed P beside roots

-

" l D

-

" l s

-"

-

" 2 5 Figure 104a

0.0 -.os -. 2 -.4707 -4.0

I 1 I I a I I t - I c Re

I I - I I I - I I w I - v u

I 716 714 “12 710 ~ 0 8 -.06 -.04 -*02 -1.0 .02 t-,02 Clp values listed beside roots F i g u r e 104b Roll Mode Rootsfor the ClpVariation Im “10 -3.0 -2.0 -1 . o -4707 -.2 -.os

I - I I L . I 1 - I - I . P

I Re I I w I I V I I I - - I * ” I .02

- 90 -80 - 70 - 60 -50 = 40 - 30 - 20 -10 10

-

“10 F i g u r e 104c NUMERATOR ROOTS STAB1 LlTY DERIVATIVE REAL

IMAG I NARY

REAL REAL -4.0 .00270 0.0 -107.822 -I 15.135 -3.0 .00360 0.0

- 79.796

-I 16.713 -2.0 .00539 0.0

- 53.132

- I 16.934

-I .o

. 0 I 076

0.0

- 26.599 -I 17.025

- .4708

.02272 0.0

- 12.588 - I 17.032

- .2

.05256 0.0

- 5.4426 -I 17.066

- .05

. I7927

0.0

- I .5962 -I 17.0724

0.0 .49080 0.0

- .58346

- I 17.0743 .02 I. I5326 0.0

- .3527 I

- I 17.0769 -4.0 -5.29036 0.0 9.8676 0.0 -3.0 -5.29036 0.0 9.8676 0.0 -2.0 -5.29036 0.0 9.8676 0.0

- I .o -5.29036

0.0 9.8676 0.0

- .4708 -5.29036

0.0 9.8676 0.0

- .2 -5.29036

0.0 9.8676 0.0

- .05 -5.29036

0.0 9.8676 0.0 0.0 -5.29036 0.0 9.8676 0.0 .02 -5.29036 0.0 9.8676 0.0

- .03070

-4.0 k .I8119

- 105.960

- .03039

-3.0 k .20996

- 79,514

- .02969

-2.0 k .25798

- 53.068

-I .o - .02673 + .36562

- 26.628

- .4708 - .01387 k .52896

- 12.6549

,03660

- .2 k .79875

- 5.59728

- .05 .3534 I k I .20658

- 2.26388

0.0 .70033 + I .30297

- I .63539

.02 I .00127 ? I . I7834

- I .45731

Table 49. N u m e r a t o r r o o t s f o r C A v a r i a t i o n s .

P 27 1 Dutch Roll Mode Roots for the Cnp Variation m Cnp values listed beside roots I I s 1 -2 -1 -2 -3 -4 -5 F i g u r e 105a Spiral M o d s h o t s for tbo Cnp Varlrrtlelr 0.25 0.15 -.0292 -05 -1.0

- I I I * I I n ~. I

I * I I I U I w I W W

, ,Re

-.06 -.OS -.04 - ,03 -802 -.Ol

.01

t - . O 1

Cnp values listed beside roots F i g u r e 105b

Roll Mode Roots for the CllpVarlatian

t

. .

NUMERATOR ROOTS STAB1 LIP(

DERIVATIVE REAL I MAG 1 NARY REAL REAL

- I .o . 0 I 584 0.0

-I 9.1345 -I 10.497

- .5 .O 1 877 0.0 -15.6501 - 1 13.984

- .02923 .02272 0.0 - I 2.58832

- I 15.402 .I5 .02472 0.0 - I I .4622 I - I 1 8 . I 7 8 .25 .02599 0.0 -10.84717 - I 18.794

.30 .02667 0.0 - IO. 54233 - I 19.099

.4 I .0283 I 0.0 - 9.87778 -I 19.766

cp

- I .o 9.86759 0.0 -5.29036 0.0

- .5 9.86759 0.0 -5.29036 0.0

- .02923 9.86759 0.0 -5.29036 0.0

.I5 9.86759 0.0 -5.29036 0.0 9.86759 0.0 .25 -5.29036 0.0 .30 9.86759 0.0 -5.29036 0.0 .4 I 9.86759 0.0 -5.29036 0.0

1cI

- I .o .04698 k. 43785 - 18.45365

- .5 .02029 k. 48044 - I 5.47606

- .02923 -.01387 k. 52896 -12.655

.I5 - .0337 I k.55521 - I I ,56659

.25 -. 04523 k .56965 -10.95871

-.05145 f .57728 - IO. 65385

.30

.4 I - ,06638 k.59510 - 9.98065

Table 50. Numerator r o o t sf o r Cn v a r i a t i o n s .

P

Dutch Roll Mode Roots for the %Varktion

,2103 Cy, values listed beside roots

- 4.0

Figurc 27 5 Spiral Mode Roots forthe CyrVarlation Im 4.0 0.2103 -4.0 I n e I I

" 8 v r I I , .L Re

-.020 -.015 -.010 -.m

,005 Cyr values listed beside roots F i g u r e 106b Roll Mode Roots for the Cy,Variatlon I ' I I

I '

i I

t"

Figure 1Q6c NUMERATOR ROOTS B STAB I L I TY REAL I MAG I NARY REAL REAL D E R I V A T I V E

. 0 I 953 0.0 -12.589 -136. I67

-4.0

.02272 0.0 - 12..588 - I 15.402

.2103

- 12.5767 - 99.875

.02666 0.0 4.0 0.0 0.0 -5.8982 10.5344 -4.0 0.0 0.0 -5.2904 9.8676 .2103 0.0 0.0 -4.8256 9.4618 4.0 -.Of387 k . 52896 -12.655 -4.0

+. 52896 -12.655

. 2 I03 -.01387

+. 52896 -12.655

4.0 -.01387 Table 51. Numerator r o o t sf o r Cy, variations.

Dutch Roll Roots for the qrVariation n I

. I I

-1 - 4 -3 - 2 Clr values listed -1 b e s i d e roots -2 -4 Figure 107a Short Mode Roots for the C,rV.*latkm

A h

- 3 . 0 - 2.0 0 .o .0959

2,o 4 0 I a r I I Re

* I -

I 1 - \. 1 - -

1 -

-15 - 1 . 0 -a5

Qs ip Clr values listed beside roots "-Qs F i g u r e 107b Roll Mode Roots for the Clr Variation F i g u r e 107c !

NUMERATOR ROOTS STAB1 L I N REAL I MAG I NARY REAL REAL DERIVATIVE

- I .8946 0.0 -10.261 I7 - I 16.459

-5.0

- I .03898 0.0 - I I .2754 -116.30101

-3.0

-2.0 - .66903 0.0 - I I .72497 -I 16.0221

- .00695 - 12.5467 I - I 15.8325

0 . 0 0.0 0.0959 0.02272 -12.588 - I 15.402 0.0

2.0 0.57860 0.0 - 13.29262 - I 1 4 . 9 0 1

4.0 I . I0784 0.0 - 13,98292 - I 14.740

5.0 I .35548 0.0 -14.31 I36 - I 14.659

7.0 I .82262 0.0 - I 4.94063 - I 14.497

-5.0 -289.2 I 0.0 0. I2108 0.0 .24276 -3.0 -174.06 0.0 0.0 -2.0 -I 16.57 .39352 0.0 0.0

0.0 - 7.75308 0.0 6.85029 0.0

0.0959 - 5.29036 0.0 9.8676 0.0

2.0 - .52488 0.0 114.8989 0.0

- .29323 0.0 229.9438 0.0

4.0

- .24677 0.0 287.5357 0.0

5.0

- . I9355

402.759 0.0 7.0 0.0

-5.0 -.01387 2 . 52896 - 12.65493

-3 .O -.01387 f ,52896 -12.65493

-2.0 -.01387 f . 52896 - 12.65493

0.0 -.01387 2.52896 -12.65493

.0959 -.01387 2 . 52896 - I 2.65493

-12.65493 2.0 -.01387 2 . 52896 -.01387 f .52896 -12.65493 4.0

5.0 -.01387 f .52896 - I 2.65493

-.01387 f .52896 - 12.65493

7.0 Table 52. Numerator roots for Cg v a r i a t i o n s .

r Roots for the Cnr Variatkn F i g u r e 108 NUMERATOR ROOTS

B

STAB1 L I T " REAL I MAG I NARY REAL REAL DERIVATIVE

- 128.568

- I .o -. 03548 0.0 - I 2.4639

- .02947 - I 2.4762 - I 27.287

- .9 0.0

-.01077 0.0 -12.51447 -123.444

- .6

- I 22.803

- .55 -. 00755 0.0 -12.521 I 1

-122. I63

-. 00430 0.0 - 12,5278

- .5

.00229 0.0 -12.5415 -120.881

- .4

.O I586 0.0 - 12.5697 - I 18.317

- .2

- I 15.402 .02272 0.0 -12.588

- .09924

-12.589 - I 14.650

- .07 .02496 0.0

.02638 0.0 -12.592 - 1 14.394

- .05

,02996 0.0 -12.599 - 1 13.752 0.0

@

4.93162 - I I .748 0.0

- I .o

5.28858 -10.840 0.0

- - 9

0.0

6.6698 - 8.3875

- .6

0.0

- -55 6.94212 - 8.0225

- 7.6709 0.0

- .5 7.2278

7.8399 - 7.0084 0.0

-. 4

0.0

9.2274 - 5.8466

- .2

0.0

- .09924 9.8676 - 5.2904

0.0

- .07 IO. 2420 - 5.2042

10.4055 - 5.1 128 0.0

- .05

0.0

10.823 - 4.893

0.0

4 )

- I 2.65493

-.01387 f .52896

- 1 .o

f . 52896 - 12.65493

- . 9 -.01387

f ,52896 - 12.65493

- .6 -.01387

-.01387 k. 52896 -12.65493

- .55

- 12.65493

- .O I387 f .52896

- .5 k. 52896 - I 2.65493

- .4 -.01387

-.01387 f .52896 - I 2.65493 - .2

-.01387 2.52896 - 12.65493

- .09924 - 12.65493 -.01387 &. 52896

- .07

- 12.65493

-.01387 f . 52896

- .05

- 12.65493

-.01387 k. 52896 0.0 Tab l e 53 . Numerator roots for $, v a r i a t i o n s .

r NUMERATOR ROOTS B STAB I L I TY DERIVATIVE REAL I MAG I NARY REAL REAL

- 1 .o .02418 0.0

-12.55031 20.22192

- .75 .02377 0.0

- 12.55567 27.3650

- .5 .02333 0.0 -12.56214

4 I .64652

- .25 .02272

0.0 - 12.57005 84.47888

0.0 .02297 0.0

- 12.58048

.I874 .02272

0.0 - 12.59078 -115.40684

.25 .02264 0.0 -12.59431

- 86.7928 I

.5 .02232

0.0 -12.61223 - 43.95034

-75 .02202 0.0 - 12.63853

- 29.64899

I .oo .02 172 0.0

-12.68051 - 22.46934

0.0

- I .o 0 . 0 -6.57089 7.74879

- .75 0 . 0 0.0 -6.26998 8. I6388 0.0 0 . 0

- .5 -5.98623 8.5961 I

- .25 0.0 0.0

-5.7191 I 9.04498 0 . 0 0 . 0 0.0 -5.46802 9.50987

. 187 0 . 0 0.0 -5.2899 I

9.86846 .25 0.0 0.0 -5.23228 9.99012 0 . 0 0.0 .5 -5.01 I 1 6 IO. 48499 0.0 .75 0.0 -4.80392 IO. 99373

I .oo 0.0 0 . 0

-4.60975 I I .51555

YJ

- 1 .o -.31014 f .4297 -12.61691

- .75 -. 24760 f .46833

- 12.62560

-. 18515 k.49616

- .5 -12.6341 I

- .25 - . I2278 f.51497 - 12.64245

0 . 0

-. 06049 f . 52576 - 12.65062

.I874 -.01386 k. 52892 - 12.65664

.25 .00171 f .52906 - 12.65863

.5 .06384 f .52503 - I 2.66649

. I2589

.75 f.51353 - I 2.67420

. I8787

I .oo f . 49407 - 12.68 I 7 6

T a b l e 54 . Numeratorroots f o r

v a r i a t i o n s .

5 6.R

NUMERATOR ROOTS

B

STAB I L I TY

REAL I MAG I NARY REAL REAL

DERIVATIVE

- .48468 -168.74511 25.38497

-3.0 0.0 -153.71383

-2.0 - .57868 0.0 15.71 175

- 1.17612 0.0 -136.61884 4.47829

- I .o

- I 2.32283 0.0 -115.75927 .03009

0.000 I -115.40684 .02272

.o I475 - I 2.59078 0.0

- 13.24437 0.0 -114.55102 .00605

.05

-15.16965 0.0 - I 12.0589 - .03437

.I5

-102.37878 - . 12576

-22.91592 0.0 .50

-39.3933 0.0 - 83.20187 - .I9315

I .o

-58.6349 ' 4 I .37965 - .25419

2.0

- .28250

-55.98877 +62.391 19 3.0

- .72027 0.0

-3.0 +3. I5419

- .72763 +3. I7672 0.0

-2.0 0.0

- .74972 k3.24328

-I .o

0.0 892.35499 0.0

0.0001 - IO. 29436

- 5.28991 0.0 9.86846 0.0

.o I475

- 2.69752 0.0 3.05337 0.0

.05

- . 4 1 1 0 6 + 1 ,96457 0.0

; 1 5

- .6172 k2.81787 0.0

.50 0.0

- .66137 k2.967 I2

I .o

- ,68346 k3.03875 0.0 2.0

- .69082 k3.06222 0.0

3.0

*

0.0 22.92565 - 2.50023

2.51434 -3.0

3.29649 0.0 10.61332 - 2.78665

-2.0

I .70459 k3.08103 - 4. I0259

- I .o

- I 2.49040

- .01042 + .58161

. 000 I

* .52892 - I 2.65664

- ,01386

.o I475

- .02153 f .38078 - 13.05783

.05

- .48245 .40343 - 14.20363

.I5

.94997 - I 8.26858

- I .09987

.50 - I ,42295 I .23465 -24. I3854

I .o

- I .68133 I .47382 -35.93603 2.0

- I .79385 I .58414 -47.75055

3.0 Table 55. Numeratorrootsfor C variations.

%

NUMERATOR ROOTS B STAB I L I TY REAL I MAG I NARY D E R I V A T I V E REAL REAL

- I 2.47042 .026 I7

-. 12 0.0 -209.71058

- 12.49848 0.0 .02532

-.IO -174.92893 -.08 -12.54195 0.0 .02407 -140.13146

- .0658 - 12.59078 0.0

.02272 -115.40684

-.04 - I 2.78700 0.0 .O I796 - 70.374.82

-.02 - 13,49442 0.0 .00657 - 34.90327

-. 005 - I I . 14076 f4.97905 -. 04503

,005 -10.91735 0.0 .2 1055 5.75643

- I I .38982 0.0 .09625

.01 15.03131 .02 - I I .761 14 0.0 .05942 32.81567 I 0.0 15.66940 0.0

-. 12 -6.5992

-. IO -6.20569 0.0 13.61845 0.0

-.08 -5.7 I878 0 . 0 I 1.4741 I 0.0

-. 0658 -5.2899 I 0.0 9.86846 0.0

- 4 . I9447 0.0 6.63494 0.0 -.04 -.02 -2.62335 0 . 0 3.40639 0.0 0.0

-. 005 - .23002 * 2.27629

.005 - .64437 f 3.77708 0 . 0

- .85155 f4.32203 0.0

.o I

.02 - I .2659 I k5.22012 0 . 0

-. 12 -. 0335 f .55252 - 12.57804

- .0287 I f .54692 -12.59717

-.IO

- .08 -.02155 * .53837 -12.62581

- ,0658 -.01386 f .52892 - I 2.65664

- .04 .O I380 * .4925 I - 12.76809

,08232 * .37954

-.02 - I 3.04833

- .005 -.28175 I . I6989 -14.63102

.005 -. 78637 f .85217 - 9.87907

.01 - .37960 = * .79577 - 1 I 626536

-.21060 * ,7084 I - I I .88973

.02 Table 56. Numerator roots for C;, v a r i a t i o n s .

6R Dutch Roll Mode Roots for I,Variatkn

I m

a -

0 -

I 1 lo 1 -

Ru

I I - 3 - 2 -1 1 I, V~I- 11~t.d beside mots as percent of original v a l u e s A-5 F i g u r e 109a A - with inertia effects 0 -without inertia effects I, values listed beside roots as percent of original value F i g u r e 109b Roll Mode Roots for Iv Variation all values

I . 1

> R 8 I " I I -13 -1 2 -1 1 F i g u r e 109c Dutch Roll Mode Rootsfor S,, Variation with Camtant AR I I i I -3 -2 -1 50!

-1 Sv Values listed beside roots as

I

percent of original Value -2 '4 F i g u r e 110a Spiral Mode Roots for S,Variation 50 75 1 0 0

I n a a I I * Re

-

I - F I I I I

-.30 - .2S - .20 - . I s -.lo - .05 0.0

S, vakes listed beside roots as percent of original v a l u e s F i g u r e 110b Roll Mode Rootsfor S, Variation I m

t