An investigation of the dynamic stability and control characteristics for a transport cruising at a mach number of 3

i a : c

AN WESTIGATION OF THE

DYI??mC STABILITY AND CONTROL

CHARACTERISTICS FOR A TRApdS

ISING AT A h4i&CH

W4TlOlAL AfROWAUTlCS AND SPACE ADMiXtSTRATlON 0 WASMlbl.QTBH, I. C. OCTOBER 1964

ERRATA NASA Technical Note D-2483 I ? %( P .

AN LNVESTIGATION OF THE DYNAMIC STABILITY AND CONTROL CHARACTERISTICS FOR A TRANSPORT CRUISING AT A MACH NUMBER OF 3 By Lawrence W. Brown October 1964 Page 17: The value of Cz (tenth item of table I) should have a minus sign added, P Thus,

C z p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -0.124

NASA-Langley, 1967 Issued 6-8-67 TECH LIBRARY KAFB, NU AN INVESTIGATION OF THE DYNAMIC STABILITY AND CONTROL CHARACTERISTICS FOR A TRANSPORT

CRUISING AT A MACH NUMBER OF 3

By Lawrence W. Brown Langley R e s e a r c h C e n t e r Langley Station, Hampton, Va.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For s a l e by the O f f i c e of T e c h n i c a l Services, Department of Commerce, D.C. 20230 -- P r i c e $1.00 Washington,

I

AN INVESTIGATION O F TRE DYNAMIC STABILITY AND C O N T R O L CHARACTERISTICS F O R A TRANSPORT CRUISING AT A MACH NUMBER O F 3 By Lawrence W. Brown Langley Research Center A t h e o r e t i c a l investigation has been made of t h e c h a r a c t e r i s t i c modes of t h e dynamic l a t e r a l s t a b i l i t y of a supersonic-transport configuration, cruising at a Mach number of 3 at a l t i t u d e s of 60,000 f e e t and 7O,OOO f e e t with t r i m angles of attack of 3 . 6 O and 5 . 8 O , respectively. The s t a b i l i t y and flying q u a l i t i e s were studied by using t h e c l a s s i c a l linearized equations of lateral motion, t h e r a t i o of t h e roll angle t o t h e equivalent side velocity, and t h e r a t i o of t h e r o l l angle t o sideslip. The e f f e c t s of t h e cross-control deriva- t i v e s on t h e s t a b i l i t y of t h e configuration w i t h damper augmentation were inves- tigated. I n addition, t h e roll coupling of t h e unaugmented configuration w a s considered.

Results show t h a t t h e interaction of the r o l l and yaw dampers and t h e change i n s t a b i l i t y with a l t i t u d e require a system with variable damper gains t o obtain satisfactory lateral s t a b i l i t y . I n addition, too s m a l l a value of t h e s t a t i c directional derivative may cause large roll-to-sideslip r a t i o s and roll-coupling problems.

INTRODUCTION An investigation of possible configurations f o r t h e supersonic commercial at t h e National Aeronautics and Space transport has been i n progress Administration. This a i r c r a f t w i l l extend commercial f l i g h t s t o Mach numbers of 3 and t o a l t i t u d e s as high as 70,000 f e e t . Considerable a t t e n t i o n has been given t o many different design concepts t o develop a superior cruise vehicle.

Since no generally accepted flying-qualities requirements e x i s t f o r t h e lateral modes of transport type of a i r c r a f t , t h e s t a b i l i t y characteristics presented herein are compared with existing m i l i t a r y specifications f o r l a t e r a l direc- t i o n a l s t a b i l i t y . Preliminary estimates of t h e handling q u a l i t i e s determined from simulator studies reported i n reference 1 indicate t h e necessity f o r fur- t h e r investigation of t h e configurations considered.

A t h e o r e t i c a l investigation w a s undertaken of t h e lateral s t a b i l i t y char- a c t e r i s t i c s of a supersonic-transport configuration incorporating variable- sweep wings f o r which t h e sweep varies with speed and a l t i t u d e u n t i l it reaches This analysis contains cal- 7 5 ' f o r a cruising speed of a Mach number of 3 .

culations of t h e dynamic l a t e r a l - s t a b i l i t y c h a r a c t e r i s t i c s f o r a l t i t u d e s of 60,000 f e e t and 7O,OOO f e e t and a gross weight of 375,000 pounds f o r t h e con- 7'3O sweepback at a Mach number of 3 . Various y a w - and r o l l - f i g u r a t i o n with damper combinations a r e considered. I n these cases, t h e e f f e c t s of omitting o r including t h e cross-control derivatives were studied.

The r e s u l t s a r e presented as p l o t s of t h e reciprocal of t h e t i m e t o damp t o half-amplitude of t h e lateral modes with increasing damper gains and angle of attack. The Dutch r o l l damping of t h e a i r c r a f t augmented with dampers i s presented as a function of t h e r a t i o of r o l l angle t o equivalent side velocity and i s compared with t h e flying-qualities c r i t e r i a . The c r i t i c a l r o l l velocity f o r i n e r t i a l coupling of t h e undamped a i r c r a f t i s a l s o presented.

wing span, f t wing mean aerodynamic chord, f t L i f t

lift coefficient, -

qs Rolling moment rolling-moment coefficient, SSb Pitching moment pitching-moment coefficient, qSF d C m

s t a t i c margin, -

dCL Yawing moment yawing-moment coefficient, Ssb Side force side-force coefficient, qs cycles t o damp t o half-amplitude acceleration due t o gravity, 32.2 f t / s e c 2 altitude, f t moment of i n e r t i a about t h e principal body axes, (slugs)(sq f t ) 1x9 Iy, I , roll-damper gain, €ja/$ kl yaw-damper gain, 6r/$ k2 Mach number m a s s , slugs r o l l i n g angular velocity period PV2

dynamic pressure, - , Ib/sq ft

r yawing velocity S wing area, sq f t time t o damp t o half-amplitude, sec t l / 2 time t o double amplitude, sec t 2 v velocity, f t / s e c

V side velocity, - Pv f t / s e c

57.3’ equivalent side velocity, vfi, f t / s e c Ve center-of-gravity position measured from wing pivot point, a f t direc X cg t i o n being positive, f t U angle of attack, degrees, except i n appendix where u i s i n radians angle of sideslip, radians P aileron deflection, radians &a rudder deflection, radians 6 r air density, slugs/cu ft a i r density r a t i o roll, radians angle of angle of yaw, radians Ma MqLu 2 nondimensional pitch parameter, %I2

(- Iy -

nondimensional yaw parameter,

v2

ratio of roll angle to sideslip angle

If1

ratio of roll angle to yaw angle

1 : I

-2v Subscripts: dyn dynamic 0 value at angle of attack of zero U denotes p a r t i a l derivative with respect t o angle of attack ANALYSIS Aircraft Flight Conditions and Characteristics The lateral s t a b i l i t y of t h e susersonic-transport configuration shown i n f i g u r e 1 w a s investigated. The a i r c r a f t motion was represented with reference t o the principal body axis by t h e linearized equations of l a t e r a l motion, as The gross weight w a s assumed t o be 375,000 pounds.

presented i n reference 2.

The flight conditions represented were f o r trimmed l e v e l f l i g h t at a Mach num- b e r of 3 at a l t i t u d e s of 60,000 f e e t and 70,000 feet. The t r i m angle of attack a t 60,000 feet w a s 3.60 and t h a t f o r 70,000 f e e t w a s 5.8O with wing sweep angle of 750. Calculations were made t o determine t h e period and damping, the r a t i o of t h e r o l l angle t o equivalent side velocity

- , and t h e r a t i o of t h e roll

F e I

angle t o s i d e s l i p If1 of t h e o s c i l l a t i n g mode and t h e damping of t h e aperiodic modes. The aircraft-configuration c h a r a c t e r i s t i c s and t h e conditions assumed f o r t h e f l i g h t evaluations a r e presented i n t a b l e I.

For t h i s variable-sweep configuration t h e wing-pivot s t a t i o n w a s considered a f e a s i b l e center-of-gravity location, and pitching and yawing moments w e r e For t h i s center-of- referenced t o t h e longitudinal location of t h e pivot point.

gravity position t h e a i r c r a f t has a positive s t a t i c margin ( C q L = -0.233).

The s t a b i l i t y derivatives were estimated from values computed according t o t h e method of reference 3 and from wind-tunnel data of similar configurations. The contribution of an angle of attack t o t h e s t a b i l i t y derivatives Cnp, Czp, and C is based on wind-tunnel t e s t results, and t h e values are given i n t h e nP appendix. A plot of -Czp and Cn is presented i n figure 2.

P Flying Qualities The problem of establishing flying-qualities requirements f o r the super- sonic transport i s being considered, but as yet no generally accepted require- ments have been established. For t h e purpose of t h i s analysis t h e s a t i s f a c t o r y l a t e r a l - d i r e c t i o n a l c h a r a c t e r i s t i c s a r e accepted as those established f o r m i l i t a r y a i r c r a f t , inasmuch as t h e present configurations should meet many of The Dutch r o l l mode is considered t h e requirements f o r t h e m i l i t a r y a i r c r a f t .

s a t i s f a c t o r y f o r - > 0.24 with a r t i f i c i a l dampers inoperative and f o r c1/2 with a r t i f i c i a l dampers operating, as presented i n reference 4 which

- I > 0.7

c1/2

includes t h e requirements of /&I. I n addition, the c r i t e r i o n from reference 5

~

< 4. Reference 5 s t i p u l a t e s a

is imposed f o r t h e roll-to-sideslip r a t i o s a t i s f a c t o r y c r i t e r i o n f o r t h e roll-to-yaw r a t i o (Ifi < 4). i n which r o l l and yaw motions are defined i n t h e s t a b i l i t y axis system. This c r i t e r i o n i n some investigations has been applied t o r o l l and s i d e s l i p motions, since

lfl= It1

i n t h e s t a b i l i t y axis system. The parameter 1 8 I is e s s e n t i a l l y independent < of t h e axis system, and i n t h i s investigation (based on body axis equations of motion) t h e roll-to-yaw r a t i o c r i t e r i o n of reference 5 could be applied d i r e c t l y as a 1 f I c r i t e r i o n . The c r i t e r i o n f o r t h e damping of t h e r o l l mode w a s

selected from p i l o t s ' opinions as -> 1, and t h e crite,-ion f o r t h e s p i r a l

t l / 2

mode w a s taken from reference 4 as 1 < 0.05. These c r i t e r i a are used f.or

t 2 reference values i n t h i s investigation, but f u r t h e r research t o determine acceptable c r i t e r i a i s required f o r configurations of t h i s type.

S t a b i l i t y Augment at i on For t h e purpose of t h i s analysis, t h e a i r c r a f t s t a b i l i t y w a s considered t o be augmented by t h e inclusion of auxiliary dampers, which provided control- When t h e air- surface deflection proportional t o r o l l i n g and yawing v e l o c i t i e s .

c r a f t i s augmented with auxiliary dampers, t h e cross-control effectiveness may considered i n t h e analysis. Dampers have a destabilizing e f f e c t and should be were added t o t h e basic configuration and were included as increments of damping i n Cz and Cn,, and t h e cross-control moments w e r e added as increments i n P C% and C z r by t h e use of t h e Pollowing equations: where k l and kg a r e t h e roll- and yaw-damper gains Sa/$ and Sr/$, respectively.

REULTS AND DISCUSSION Unaugmented Configuration w i t h angle of Since some of t h e aerodynamic parameters vary appreciably attack, as indicated i n f i g u r e 2, t h e basic configuration w a s analyzed t o deter- 9 mine t h e e f f e c t s of angle of attack on t h e l a t e r a l modes. Calculations were

made f o r angle of attack varying from Oo t o loo, and account was taken of t h e

v a r i a t i o n i n t h e s t a t i c derivatives C z p and CnP and the rotary derivative

. Figure 3 shows t h e variation i n t h e damping of the l a t e r a l modes and the

cnP period of t h e Dutch r o l l mode with angle of attack f o r a l t i t u d e s of 60,000 feet and 70,000 f e e t . A t an a l t i t u d e of 60,000 feet, t h e damping of t h e r o l l mode decreases rapidly as a increases. Calculations show that a t approximately The a = 8 ' t h e r o l l and s p i r a l modes merge i n t o a long-period oscillation.

period and damping of t h e Dutch r o l l mode increases gradually; t h i s increase i s a t t r i b u t e d t o a t r a n s f e r of damping from t h e r o l l mode t o t h e Dutch r o l l mode with an increase i n a .

With an increase i n the a l t i t u d e t o 70,000 f e e t t h e air density decreases The period of t h e Dutch r o l l mode and causes a decrease i n t h e t o t a l damping.

increases. The rate of change of damping of each mode w i t h a maintains about t h e same trend as a t 60,000 f e e t . The foregoing analysis indicates that, t o f u l f i l l t h e damping requirements and t o establish s a t i s f a c t o r y s t a b i l i t y , augmen- t a t i o n of t h e basic configuration i s necessary throughout the angle-of-attack range.

Effects of R o l l and Yaw Dampers Without Cross-Control Moments m Effects of r o l l damper.- Figure 4(a) shows t h e variation i n t h e damping of t h e l a t e r a l modes with an auxiliary r o l l damper k l with ensa = 0. For an a l t i t u d e of 60,000 f e e t and a t r i m angle of attack of 3.6', t h e damping of t h e r o l l mode increases w i t h kl and a t t a i n s a s a t i s f a c t o r y value f o r kl 2 0.52, as shown by t h e s a t i s f a c t o r y roll-mode boundary. "he damping of t h e Dutch r o l l mode increases with because of t h e t r a n s f e r of damping from the r o l l mode k l t o t h e Dutch r o l l mode, as indicated previously, and becomes s a t i s f a c t o r y f o r k l 1 . 0.20, as shown by t h e satisfactory Dutch r o l l mode boundary. The damping of the s p i r a l mode decreases as k l increases but remains s t a b l e throughout t h e k l range.

When t h e a l t i t u d e i s increased t o 70,000 feet ( f i g . &(a)), the diminution of t h e air density causes t h e t o t a l damping of t h e system t o diminish, and an increase i n the angle of attack f o r t r i m ( a = 5.80) causes t h e r o l l damper t o a c t as a p a r t i a l yaw damper. A s k l increases, t h e damping of t h e r o l l mode t h e flying-quality c r i t e - increases gradually but i s not sufficient t o s a t i s f y A s a result of an increase i n rion.

a, t h e damping of t h e Dutch r o l l mode has almost t h e same r a t e of increase as t h e damping of t h e r o l l mode and becomes s a t i s f a c t o r y f o r k l h 0.19. The damping of t h e s p i r a l mode diminishes as k l increases but remains satisfactory throughout t h e k l range.

Effect of yaw damper.- The variation i n t h e damping of t h e lateral modes with an auxiliary yaw damper k2 with Cz8 = 0 shows much t h e same trend as r d t h a t due t o an auxiliary r o l l damper. (See f i g . 4 ( b ) . ) A t an a l t i t u d e of 60,000 f e e t ( a = 3.60) t h e damping of t h e r o l l mode increases as increases.

k2 The damping of t h e Dutch r o l l mode, as expected, increases with k2 and a t t a i n s i a satisfactory value f o r k2>= 0.15. The damping of t h e s p i r a l mode remains satisfactory as it increases with t h e yaw-damper gain.

a = 5 . 8 O A t an a l t i t u d e of 7O,OOO f e e t f o r which ( f i g . k(b)), t h e increased a f o r t r i m causes t h e yaw dnmper t o have more e f f e c t on t h e r o l l mode than on t h e Dutch r o l l mode. A s k2 increases, t h e rate of increase i n t h e damping of t h e r o l l mode i s greater than t h e rate of increase i n t h e damping of t h e Dutch r o l l mode. I n fact, at approximately a = 80, k2 no longer a f f e c t s t h e Dutch r o l l damping. The damping of t h e r o l l mode i s not sufficient, however, with any of t h e values of k2 used i n t h i s analysis. The damping of t h e Dutch roll mode becomes satisfactory f o r The damping of t h e k2 2 0.55.

, s p i r a l mode obtains an i n i t i a l increase with t h e increase i n a at k2 = 0 and continues t o increase with k2 at almost t h e same rate as t h e damping of t h e Dutch r o l l mode.

Effect of yaw and r o l l dampers.- For t h e purpose of determining t h e damper gains necessary f o r s a t i s f a c t o r y damping of t h e lateral modes and of b e t t e r evaluating t h e e f f e c t s of t h e cross-control moment, values of k l were selected and calculations were m a d e when w a s varied and when t h e cross-control k2 and C were neglected. Figure 5(a) shows t h e damping of t h e moments '%a '6r l a t e r a l modes f o r h = 60,000 f e e t and a = 3 . 6 O f o r two values of k l (0.35 and 0.50) and with k2 varying.from 0 t o 0.5. For a value of k l = 0.35, t h e r o l l mode does not a t t a i n a satisfactory damping within t h e k2 range. The k2 = 0, as a r e s u l t of t h e damping of t h e Dutch r o l l mode i s satisfactory f o r roll-damper effects,and increases with k2. Increasing t h e roll-damper gain t o k l = 0.50 causes t h e damping of t h e r o l l mode t o a t t a i n a satisfactory value at k2 2 0.165. The damping of t h e Dutch r o l l mode increases as a result of an increase i n k l and continues t o increase with k2 s o t h a t a satisfac- t o r y value i s maintained. The damping of t h e s p i r a l mode decreases as k l increases but maintains a satisfactory value as k2 increases.

Because of t h e decrease i n t h e t o t a l damping with an increase i n t h e a l t i - tude and t h e change i n c e r t a i n aerodynamic parameters with angle of attack, t h e roll-damper gain w a s increased with an increase i n t h e a l t i t u d e t o 70,000 feet and a t r i m angle of attack of 5.8'. With roll-damper gains of 0.70 and 0.90 and with 5(b)), t h e damping of t h e r o l l mode varying from 0 t o 1.0 ( f i g .

k2 increases but does not a t t a i n a s a t i s f a c t o r y value within t h e k2 range. The damping of t h e Dutch r o l l mode i s quite s a t i s f a c t o r y f o r k2 = 0 because of t h e amount of damping a t t r i b u t e d t o k l and the increases with k2 at almost t h e same r a t e as t h e r o l l mode. !Be damping of t h e s p i r a l mode decreases w i t h increase i n kl and increases with k2 so t h a t a s a t i s f a c t o r y value i s t maintained.

i Effects of Cross-Control Moments The possible e f f e c t s of the cross-control moments which occur when auxil- i a r y dampers are added have been discussed i n reference 2 and indicate t h e destabilizing e f f e c t s t h a t might occur i n t h e Dutch r o l l and s p i r a l modes. I n view of these effects, t h e cross-control moments C and Czgr were intro- %a duced i n t o t h e analysis t o determine t h e e f f e c t s they would have on t h e stabil- i t y . For the p a r t i c u l a r configuration and f l i g h t conditions of t h i s investiga- = -0.00464 and C 2 = 0.0056.

t i o n ( t a b l e I), these values were 6 r Aileron cross-cgntrol effects.- For an a l t i t u d e of 60,000 f e e t and an angle of attack of 3.60, a co6rparison. of f i g u r e s 4(a) and 6(a) indicates that t h e aileron cross-control moment Cns causes a t r a n s f e r of damping from the a Dutch r o l l and s p i r a l modes t o t h e r o l l mode. The damping of t h e Dutch r o l l and s p i r a l modes decreases as kl increases, and t h e daarping of t h e roll mode increases as k l increases.

A t an a l t i t u d e of 70,000 feet and an angle of attack of 5.8O, t h e aileron cross-coatrol moment becomes l e s s effective because of the coupling e f f e c t s of t h e r o l l and Dutch r o l l modes w i t h an increase i n angle of attack. Figure 6(a) shows that damping of t h e r o l l mode at h = 70,000 feet is s a t i s f a c t o r y f o r

kl >, 0.8. The damping of t h e Dutch r o l l mode, although decreased because of

undergoes a slight increase as kl increases. However, a t both a l t i t u d e s considered.

t h e Dutch r o l l damping remains unsatisfactory f o r t h e range of kl There i s l i t t l e o r no e f f e c t on t h e damping of t h e s p i r a l mode, which remains satisfactory, decreasing as k l increases.

Rudder cross-control effects.- A comparison of figures 4(b) and 6(b) shows that t h e rudder cross-control moment C z causes a comparatively s l i g h t B r tendency t o r e d i s t r i b u t e the damping t o t h e Dutch r o l l and s p i r a l modes as k2 increases. Because of t h e coupling e f f e c t s of t h e r o l l and Dutch r o l l modes w i t h an increase i n angle of attack, a t an a l t i t u d e of and h = 70,000 feet a = 5 . 8 O t h e damping of t h e r o l l and Dutch r o l l modes increases at almost t h e same r a t e with an increase i n The variation i n t h e damping due t o k2.

Cz8, is very small compared with the variation due t o Cn6a; therefore, values of k2 l a r g e r than kl a r e indicated t o o f f s e t t h e e f f e c t s of C .

%a Rudaer and aileron cross-control effects.- For t h e purpose of evaluating t h e selected roll-damper gains f o r satisfactory damping of t h e lateral modes, t h e cross-control moments w e r e included i n t h e calculations and t h e yaw-damper gain w a s varied from 0 t o 1.0. For an a l t i t u d e of 60,000 feet with a r o l l - ( f i g . 7(a)), t h e damping of t h e roll mode i s satis- damper gain of kl = 0.35 factory f o r k2 = 0, because of t h e influence of Cns and increases with k2.

a' but does increase The damping of t h e Dutch r o l l mode decreases because of C nga with k2, and it a t t a i n s a s a t i s f a c t o r y value f o r k2>= 0.25. If t h e damper k l = 0.50, t h e damping of t h e r o l l mode remains satisfac- gain i s increased t o t o r y and increases with k2. The damping of t h e Dutch r o l l and s p i r a l modes decreases with an increase i n k l but increases with k2, and t h e Dutch r o l l k2 > 0.30.

mode a t t a i n s satisfactory damping f o r 7( b ) ) and a roll-dmper gain of For an a l t i t u d e of 70,000 f e e t ( f i g .

kl = 0.70, t h e damping of t h e r o l l mode a t t a i n s a s a t i s f a c t o r y value f o r

k2 2 0.30. satisfactory f o r

The damping of t h e Dutch roll mode i s k2 2 0.435.

t h e damping of t h e r o l l mode When t h e damper gain i s increased t o k l = 0.90, i s satisfactory throughout t h e k2 range. The damping of t h e Dutch r o l l mode, a moderate increase because of because of t h e coupling effects, now shows k l

and a t t a i n s a satisfactory value f o r k2 2 0.26 with k l = 0.9. It should be

noted t h a t t h e rates of increase of t h e damping of both t h e roll and Dutch roll modes are almost equal.

of A n g l e of Attack on Augmented Configuration Effects Some of t h e differences t h a t occur at the d i f f e r e n t a l t i t u d e s are related t o t h e change i n t h e angle of attack and can be seen i n a p l o t of t h e damping t h e configuration of t h e l a t e r a l modes f o r t h e augmented configuration (i.e., a. For damper gains of with both roll and yaw dampers) as a function of k l = 0.35 and k2 = 0.50 and with cross-control derivatives included ( f i g . 8), t h e damping of .the r o l l mode, having satisfactory damping a t a = Oo, decreases as a increases. The damping of t h e Dutch r o l l mode and s p i r a l modes i s a l s o s a t i s f a c t o r y at a = 0 ' and increases with a.

k2 = 0.70, t h e damping of t h e If t h e yaw-damper gain i s increased so t h a t a = 4O, then r o l l mode i s i n i t i a l l y t h e same at a = Oo, decreases u n t i l because of t h e coupling e f f e c t s of t h e Dutch r o l l mode increases as a increases. The damping of t h e Dutch r o l l mode undergoes an i n i t i a l increase at a = 0 ' but begins t o decrease a t approximately a = 4'. A t a > go damping and Q = 0.50.

i s l e s s than t h e damping f o r t h e condition when k l = 0.35 i s a l s o due t o t h e coupling of t h e Dutch r o l l mode.

This condition a t a > go The damping of t h e s p i r a l mode undergoes an i n i t i a l increase and continues t o increase with a. When t h e roll-damper gain i s increased so t h a t kl = 0.50 and k2 = 0.50, t h e damping of t h e r o l l mode increases a t a = Oo but decreases

- - - I

rapidly as a increases. The damping of t h e Dutch r o l l knd s p i r a l modes decreases s l i g h t l y a t a = 0 ' but increases rapidly as a increases because of the coupling of t h e r o l l modes.

Later&-Directional Oscillation The Dutch r o l l c r i t e r i o n of reference 4 i s expressed i n terms of damping

, whereas it i s suggested i n reference 5 t h a t required as a function of

1 4

large values of a r e intolerable, regardless of t h e damping. These c r i - t e r i a are inconsistent f o r high-altitude conditions, and f o r t h e purpose of t h i s analysis both c r i t e r i a are considered, inasmuch as it i s recognized t h a t

1 : I

i n the s t a b i l i t y axis system approximately equals i n t h e body axis system.

"he boundaries f o r t h e s a t i s f a c t o r y f l y i n g q u a l i t i e s of reference 4 a r e indi-

. The roll-to-sideslip r a t i o s cated i n a plot of - as a function of

c1/2 3 which a r e l i s t e d f o r comparison with t h e acceptable c r i t e r i o n of reference

requires t h a t < 4. A s shown i n figure 9, t h e basic configuration

(kl = k2 = 0 ) and t h e configuration with t h e damper combinations which have been accepted as having s a t i s f a c t o r y damping a r e i n a region which provides t o l e r a b l e o r s a t i s f a c t o r y Dutch r o l l o s c i l l a t i o n f o r dampers inoperative and f o r dampers operating at 60,000 f e e t and 70,000 f e e t . (Note t h a t t h e region labeled "tolerable" i s considered s a t i s f a c t o r y f o r operation without dampers.) The I$ I as tabulated f o r these conditions a r e considered unsatisfactory, however,

since they do not meet t h e c r i t e r i o n t h a t I < 4. Since t h e various damper

I

combinations which would tend t o reduce t h e values of I f I would be considered

r a t h e r large, t h e approximate expression of t h i s r a t i o as presented i n refer- ence 6 w a s examined and an increase i n C w a s considered. When (CnP), i s

nP

increased t o 0.25 t o give CnB = 0.1722 at h = 60,000 f e e t and C = 0.1247 a t h = 7O,OOO f e e t ( f i g . l o ) , t h e basic configuration and t h e damper combina- t i o n s s t i l l maintain s a t i s f a c t o r y l a t e r a l - d i r e c t i o n a l o s c i l l a t i o n s at both alti- tudes and have acceptable roll-to-sideslip r a t i o s R o l l Coupling of t h e Undamped Configuration For configurations with highly swept w i n g s and low s t a t i c s t a b i l i t y , i n e r t i a l coupling i s often a problem, and hence a preliminary analysis of t h e r o l l - coupling c h a r a c t e r i s t i c s of t h i s configuration w a s undertaken. C r i t i c a l r o l l i n g 1 2

I

velocities, as presented i n reference 7, were calculated f o r three center-of- gravity positions (-6, 0, and 6 f e e t with respect t o t h e wing-pivot location) and f o r a l t i t u d e s of 60,000 and 7O,OOO feet and are presented i n figures 1 1 and 12. The e f f e c t of t h i s change w a s t o change t h e values of and Cnpy as shown i n t h e appendix. For these calculations, t h e longitudfnal and l a t e r a l modes of o s c i l l a t i o n were a r b i t r a r i l y assumed t o have zero damping.

t For a center-of-gravity location of x = 0 a t an a l t i t u d e of 60,000 f e e t cg ( f i g . 11) t h e roll-coupling c h a r a c t e r i s t i c s a r e considered unsatisfactory i n t

and %2 > 0.9. Locating t h e

1.68 < p C 2.97 where q2 C 0.8

t h e range xcg = 6 f e e t center-of-gravity position aft of t h e wing-pivot point t o with r o l l r a t e s being decreases t h e upper boundary of t h e s a t i s f a c t o r y range,

1.43 < p C 2.21. Locating t h e center of gravity forward of

unsatisfactory f o r t h e wing-pivot point t o xcg = -6 f e e t increases t h e upper boundary of t h e 1.89 < p < 3.41.

s a t i s f a c t o r y range, with r o l l r a t e s being unsatisfactory f o r When t h e a l t i t u d e is increased t o 70,000 feet ( f i g . 12), t h e upper bound- a r i e s of r o l l r a t e s f o r s a t i s f a c t o r y roll-coupling c h a r a c t e r i s t i c s are g r e a t l y xcg = 0, r o l l r a t e s are unsatis- reduced. For a center-of-gravity location of factory i n the range When 0.95 C p < 2.26 where v2 C 0.8 and %2 > 0.9.

t h e center-of-gravity position i s located forward of t h e wing-pivot point t o 1.17 C p C 2.69.

xcg = -6 feet, t h e r o l l rates a r e unsatisfactory f o r It is evident t h a t the roll-coupling c h a r a c t e r i s t i c s are not t h e best t o be desired i n t h i s configuration f o r t h e center-of-gravity locations presented.

If consideration i s given t o an increase i n t h e s t a t i c d i r e c t i o n a l derivative

as a means of obtaining more favorable values of I gl, t h i s increase could a l s o

be a contributing f a c t o r i n obtaining b e t t e r roll-coupling characteristics. If an increase i n t h e s t a t i c d i r e c t i o n a l derivative, such as t h a t i n t h e section at an e n t i t l e d "Lateral-Directional Oscillation, 'I i s considered (CnP = 0.1722 a l t i t u d e of 60,000 f e e t ) , t h e c r i t i c a l r o l l i n g v e l o c i t i e s a r e s i g n i f i c a n t l y For a center-of-gravity increased and t h e unstable r o l l range is decreased.

location of xcg = 0, t h e r o l l rates f o r s a t i s f a c t o r y roll-coupling character- i s t i c s a r e unsatisfactory f o r 2.2l C p C 2.87. When t h e center-of-gravity xcg = -6 feet, t h e r o l l position is located forward of t h e wing-pivot point t o For an a l t i t u d e of 70,000 f e e t rates are unsatisfactory f o r 2.38 C p < 3.41.

and f o r = 0.1247, r o l l r a t e s are unsatisfactory i n t h e range CnP

1.48 < p C 2.26 and i n t h e range f o r a center-of-gravity location of

xcg = 0

1.63 c p < 2.68 f o r a center-of-gravity location of xcg = -6 f e e t . It

should be recognized t h a t t h i s analysis of t h e roll-coupling c h a r a c t e r i s t i c s i s a limited one. More detailed analysis should include t h e e f f e c t of damping on t h e c r i t i c a l velocity and should determine t h e t r a n s i e n t motions t o be encoun- t e r e d i n r o l l i n g maneuvers.

CONCLUDING RESIARKS A n investigation w a s made of t h e c h a r a c t e r i s t i c modes of t h e l a t e r a l s t a b i l i t y f o r a variable-sweep-wing supersonic-transport configuration, cruising at a Mach number of 3 at a l t i t u d e s of 60,000 and 7O,OOO f e e t with t r i m angles of attack of 3 . 6 O and 5 . 8 O , respectively. Calculations were based on the con- Y figuration at a gross weight of 375,000 pounds w i t h a wing sweep angle of 75'.

Results show t h a t augmentati.on by roll and yaw dampers i s necessary f o r $ s a t i s f a c t o r y lateral s t a b i l i t y . With an increase i n the altitude, the t o t a l aerodynamic damping of t h e system decreases and large increments i n t h e damper gains a r e required. A t t h e higher angle of attack required f o r f l i g h t at an a l t i t u d e of 7O,OOO f e e t , coupling e x i s t s between t h e r o l l and Dutch r o l l modes t o t h e extent t h a t both r o l l damping and yaw damping have nearly equal influence roll and Dutch r o l l modes. Results a l s o show t h a t s m a l l values of the on t h e s t a t i c - d i r e c t i o n a l derivative causes unsatisfactory l a t e r a l - d i r e c t i o n a l o s c i l l a - t i o n s and contributes t o roll-coupling problems. An increase i n the s t a t i c - d i r e c t i o n a l derivative could improve these conditions.

Langley Research Center, National Aeronautics and Space Administration, Langley Station, Hampton, Va., June 22, 1964.

APPENDIX STABILITY DERIVATIVES AFFECTED BY ANGLE OF ATTACK AJXD CENTER-OF-GRAVITY POSITION The s t a b i l i t y derivatives affected by angle of attack are: and Those derivatives affected by the center-of-gravity position are: X Cnp = and X cg Gma = CmcLCL, + CL, 7 C The following values a r e used t o solve the preceding equations:

(Czp) = -0.063

(Cnp>, = 0.023

(cZp) = - 0 ~ 2 9 5

U ( c 4 u = -Oo108 (cnp) = -0.177 Cr, = 1.550

(cnp) = -1.238

C = -0.233 a q L Cyp = -0.028 REFERENCES 1. White, Maurice D., Vomaske, Richard F., McNeill, Walter E., and Cooper, A Preliminary Study of Handling-Qualities Requirements of George E.: Supersonic Transports i n High-speed Cruising Flight Using Piloted Simulators. NASA TN D-1888, 1963.

2. Brown, Lawrence W.: A Theoretical Investigation of the Effect of Cross- Control Derivatives on the S t a b i l i t y Characteristics of Airplanes Designed f o r Flight at High Mach Numbers.

NASA TN D-1223, 1962.

3 . Margolis, Kenneth, and Bobbitt, Percy J. : Theoretical Calculations of t h e S t a b i l i t y Derivatives at Supersonic Speeds f o r a High-speed Airplane Con- figuration. NACA RM L53G17, 1953.

4. Anon.: Flying Qualities of Piloted Airplanes. Military Specification mi-F-8785 ( A S ) , Sept . 1, 1954; Amendment-4, Apr . 17, 1959.

5. W i l l i a m s , Walter C., and Phillips, W i l l i a m H.: Some Recent Research on the Handling Qualities of Airplanes. NACA RM H55L29aY 1956.

6. Sherman, Windsor L.: A Theoretical Investigation of t h e Dynamic Lateral S t a b i l i t y of Three Possible Airplane Configurations f o r Flight at a Mach Number of 3.0. NASA MEMO 5-15-59L, 1959.

7. Phillips, W i l l i a m H.: Effect of Steady Rolling on Longitudinal and Direc- t i o n a l Stability. NACA TN 1627, 1948.

TABU 1.- CONFIGURATION CHARACTERISTICS AND FLCGHT CONDITIONS USED FOR FLYING-QUALITIES EVALUATION b , f t . . . . . e . . . .

-

C , f t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i Ix, (slugs) ( s q f t ) . . . . . . . . . . . . . . . . . . . . . . . . .

1,484, ooo Iy, (slugs) ( s q f t ) . . . . . . . . . . . . . . . . . . . . . . . . .

11,784,000 i Iz, (slugs) ( s q f t ) . . . . . . . . . . . . . . . . . . . . . . . . .

13,112, ooo my slugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11,650 s, s q f t . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4,040 V, f t / s e c . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2,920 C k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1-55

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -0.124

czP czr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.1018

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C

-0 0055 2b

C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.0056

'6r C"CL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-0.233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-1.045 Cn, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-0.453 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-0.00464 cnsr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-0.028 cyp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-0.347 .

cy8, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-0.028

h, f t . . . . . . . . . . . . . . . . . . . . . . . . 60,000

70, ooo . . . . . . . . . . . . . . . . . . .

q, l b / s q f t . . 953 590 . . . . . . . . . . . . . . . . . . . . . . .

a, deg 3.6 5 - 8

p, siug/cu f t . . . . . . . . . . . . . . . . . . . . 0.900223

0.000138

cZp . . . . . . . . . . . . . . . . . . . . . . . . -0.0815

-0.0929 . . . . . . . . . . . . . . . . . . . . . . . .

0.0517 0 9992

cnp . . . . . . . . . . . . . . . . . . . . . . . . 0 . 0 1 6 ~ 0.0121

(a) Drawings of configuration.

Figure 1 . - Profile and plan views of variable-sweep-wing supersonic-transport configuration used in this investigation.

f- - -c (b) Photographs of configuration.

Figure 1.- Concluded.

.

I -

_L

L

cc

- 1 1 - *

8 I O 12

Figure 2.- Effect of angle of attack on s t a b i l i t y derivatives CzP and Cn P' c a- Rol I - - - - - - - Dutch roll --- Spiral h, ft 0 60,000 70,000 .5 . 4 v) W .3 L Q) Q

-12 N .2

.I 0 !o I Figure 3 . - Effect of angle of attack on period and damping of lateral modes.

I .4 1.2

I .o

3 .8 L a J Q c

- 1 : .6

.4 .2 (a) Effects of roll-damper gains.

Figure 4.- Damping of l a t e r a l modes f o r no cross-control moments.

Mode

------

Rol I --__--- Dutch roll

---

1.2

- - - Spiral

B mrrrrrrDutch roll, 60,000 ft

1.0 ------

. - - . A nnn C I - Dutch rol I, 70,000 f t h, ft 0 60,000 A 70,000 k p , sec (b) Effects of yaw-damper gains.

Figure 4.- Concluded.

N w D . 6 __ Mode Rol I I .4

-------

Dutch roll

---

Spiral Criterion kl A - Roll 1.2 0.35

B - Dutch roll

0.50

I .o

I a , L .8( a , Q ..

.4 .I .2 .3 .5 .6 k 2 , sec ( a ) Effects of yaw-damper gain at h = 60,000 f e e t and a t a = 3.6'.

Figure 5.- Damping of l a t e r a l modes with no cross-control moments f o r given values of roll-damper gains.

Rol I

--- - -_ -

Dutch roll --- Spiral

'-1-

Criterion A m Roll 0 0.70 C7-rrrrrr Dutch roll 0.90 ( b ) Effects of yaw-damper gains at h = 70,000 f e e t and at a = 5.8'.

Figure 5.- Concluded.

(a) Effects of roll-damper gains.

Figure 6.- Damping of l a t e r a l modes including cross-control effects.

Rol I I .2

. - --------

- -- ---__

Dutch roll

---

Spira I

---

-----

Criterion A 7 i Roll I .o I - - - - - .

B - r r r r r r r r Dutch roll, 60,000 f t

A Tnn-

C - Dutch roll, 70,000 ft . - - - - - h, f t 0 60,000

-------

70,000 7- I --A

k2, sec (b) Effects of yaw-damper gains.

Figure 6. - Concluded.

A ~ Mode Rol I

-------

Dutch r o l l --- Spi ra I Criterion k l A m Roll 0.35 B - r r r r r r r r Dutch roll, 60,000 ft 0.50 3 . 4 .6 . 7 .2 . 5 k2, sec ( a ) Effects of yaw-damper gains a t h = 60,000 f e e t and a t a = 3.6'.

Figure 7.- Damping of l a t e r a l modes f o r r o l l and yaw dampers including cross-control moments.

1.4 1.2 --a

I .o

-- .8 L (u Q

-------

Dutch roll Spiral

-1 c 2 .6

Criterion A-RolI

c- Dutch roll, 70,000 f t 0.90

. 4 .2 I .I .2 .3 .4 .5 .6 .7 .8 .9

I .o

k,, sec (b) Effects of yaw-damper gains at h = 70,000 feet and at a = 5.8'.

Figure 7.- Concluded.

llll11l1l1l11l I I I

I . 6

i

I . 4

L

1.2

x

: 1 . 0 ---I- I /

# A L W

.- ‘ 1

a Mode c Rol I

-1 c 2 .8 ------ - -

Dutch roll

---

Spiral k l k2 Criterion 0 0.35 0.50 0.50 0.50 A . - r r r r r r r r R o l l B r r r r r r r r r Dutch roll, 60,000 f t 0 0.35 0.70 . 6

i

.4 < r ; .2 7 / 2 4

6 a 1 0

12 14 Figure 8.- Effects of angle of attack on damping of l a t e r a l modes at an a l t i t u d e of 60,000 f e e t with roll and yaw dampers.

1 1 1 1 1 I I I 60,000 f t 70,000 f t -- .

5 - 0 0.00 0.00 4.9 0.0 0.0 6.0 0 0.35 0.50 4.7 l 0 0.7 0.7 5.6 0 0,9 0.7 5.4 -- . 0 0.50 0.50 4.5 0.35 0.90 4.8 0 0.7 0.9 5 . 6 , , d 0.90 0.50 4.0 N

- 1 2

. O Figure 9.- Lateral-directional oscillation f o r undamped and damped configurations at altitudes of 60,000 and 70,000 feet.

w P w Iu 60,000 f t

7 0 , O O O f i ~ , 1 1 1

k l k 2 1 $ 1

k l k 2 5 0 0.00 0.0 3.3 0 0.0 0.0 3.9 0 0.35 0.5 3.2 0 0.7 0.7 3.7 0.5 3.1 -- 1 0 0.50

----

0 0.9 0.7 3.6 0.35 0.7 3.2 0 0.7 0.9 3.7

I

/’ Sat isfactory / ;--/ ---- - 2 2 - 2 - - - - .

00 .I .2 . 3 4 .5 . 6 . 7 .a .9 1.0

Figure 10.- Lateral-directional oscillation for undamped and damped configurations with increase in Cnp at altitudes of 60,000 and 70,000 feet.

- \ \ \ \ \ \ \ -. Roll rate, \ radians/sec \ 2.

a 1.20

\ 0 1.43 \ \ 0 1.50 -.

\ A 1.68 \ 0 1.89 \ +.

2.00 \ t l 2.21 \ 2. 0 2.87 \ Q 3.41 \ \ 7- \ \ \ ~~ \ \ \ \ Uns rot I ~~

/

/ . 2 4 . 6 -8 I .o 1 . 2 wicl Figure ll.- Roll-coupling s t a b i l i t y boundary f o r three center-of-gravity positions of undamped configuration a t an altitude o f 60,000 f e e t and a t an angle of attack of 6O.

I I Center of gravity Roll rate, radians/ sec o 0.55 0 0.66

P

n 0.80 6 0.95 0 1.00

/

Ll 1.17 0 1.74 0 2.00 A 2.26 0 2.69

PO

-6

/

/

Unstable roll rates \ \ \ \ \ I .2 . 4 . 6

v

Figure 12.- Roll-coupling s t a b i l i t y boundary f o r three center-of-gravity positions of undamped configuration at an d t i t u d e of 70,000 f e e t and a t an angle of attack of 5.8'.

NASA-Langley, 1964 L3719 “The aeronautical and space activities of the United States shall be

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