Theoretical Stability Derivatives for the X-15 Research Airplane at Supersonic and Hypersonic Speeds Including a Comparison with Wind-tunnel Results

TECHNICAL MEMORANDUM

X-287

Hard copy (HC) #"9* /,ca

Microfiche (MF) L

THEORETICAL STABIIJTY DENVATIVhS JWK. 'YHE X-15 RESEARCH

w lr A COMPARLSON WITH WIND-TUNNEL RESULTS \

By Harold J. Walker and C h e s t e r H. Wolowicz

Flight Research C e n t e r k v l f 0 8 I Edwards, Calif.

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i TABLE OF CONTENTS Page

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . .%.,. . 2

- 4 1 1 DESCRIPTION OF THE AIRPLANE . . . . . . . . . . . . . . . .

= m % ' ..

SCOPE OF THE INVESTIGATION . . . . . . . . . . . . . . F4m I . .

0 W ' I u

* H d

DISCUSSION OF FLOW FIELDS . . . . . . . . . . . . . . . . ESC f a R . 4

H O K Z Z PRESENTATION OF RESULTS . . . . . . . . . . . . . . . . .

& E '

f i C ! J H

ANALYSIS AND DISCUSSION OF LONGITUDINAL DERIVATIVES . . 8

% . .

Lift Characteristics . . . . . . . . . . . . . . . . 8

Wing . . . . . . . . . . . . . . . . . . . . . . . . .

Horizontal tail . . . . . . . . . . . . . . . . . . . . . 11

2 9 : : Fuselage . . . . . . . . . . . . . . . . . . . . . .

11 I

2 4 % '

Airplane . . . . . . . . . . . . . . . . . . . . . . H. u .I . .

v)n I

. . . . . . . . . . . . Pitching-Moment Characteristics

v)*w 1 4 m

. . . . . . . . . . . . . Wing and horizontal tail

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Fuselage . . . . . . . . . . . . . . . . . . . . . . u.+Cr 14

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H 4 4 . .

Airplane . . . . . . . . . . . . . . . . . . . . .

.Q .UQ Longitudinal-Control Characteristics . . . . . . . . . . . . . . .

ANALYSIS AND DISCUSSION OF LATERAL-DIRECTIONAL DERIVATIVES . . . . 20

Sideslip Derivatives . . . . . . . . . . . . . . . . . . . . . . 20

Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Fuselage . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Horizontal tail . . . . . . . . . . . . . . . . . . . . . . . . 24

Vertical tail . . . . . . . . . . . . . . . . . . . . . . . . . 2 6

Airplane . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Derivatives Due to Yawing . . . . . . . . . . . . . . . . . . . . 30

Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Fuselage . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Horizontal and vertical tails . . . . . . . . . . . . . . . . . 31

Airplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Derivatives Due to Rolling . . . . . . . . . . . . . . . . . . . 33

Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Horizontal tail . . . . . . . . . . . . . . . . . . . . . . . . 34

ii

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Page

Vertical tail . . . . . . . . . . . . . . . . . . . . . . . . . 35

Airplane . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Lateral-Directional Control Derivatives . . . . . . . . . . . . . 36

Directional control . . . . . . . . . . . . . . . . . . . . . . 37

Lateral control . . . . . . . . . . . . . . . . . . . . . . . . 38

CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . 38

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APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

REFEFENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4

TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

F I G U F P S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION TECHNICAL MEMORANDUM x-287 THEORETICAL STABILITY DERIVATIVES F O R THE x-15 RESEARCH ALRPLANE AT SUPERSONIC AND HYPERSONIC SPEEDS INCLUDING A COMPARISON W I T H WIND-TUNNEL RF,SCLTS* H By Harold J. Walker and Chester

1 H . Wolow i c z

SUMMARY The s t a b i l i t y and c o n t r o l derivatives f o r t h e X - 1 5 research a i r p l a n e are pre- i n power-off f l i g h t at supersonic and hypersonic Mach numbers sented, both as derived from e x i s t i n g t h e o r e t i c a l methods and a s measured i n various wind-tunnel f a c i l i t i e s . Calculations are made f o r Mach numbers w i t h i n and beyond the estimated f l i g h t envelope and f o r angles of a t t a c k from 00 t o 2 5 O . The r e s u l t s are compared with experimental d a t a i n t h e Mach number range from 2 t o approximately 7 and, f o r t h e s t a t i c deriva- t i v e s , with t h e l i m i t i n g values given by Newtonian theory.

I n general, good approximations of the l o n g i t u d i n a l and l a t e r a l - d i r e c t i o n a l d e r i v a t i v e s are obtained when c a r e f u l a t t e n t i o n i s given t o the shock- and vortex-interference e f f e c t s between t h e various a i r - plane components and t o the increasing n o n l i n e a r i t y of t h e aerodynamic c o e f f i c i e n t s as hypersonic speeds w e approached. The c h a r a c t e r i s t i c s of t h e l i f t i n g surfaces are calculated by t h e modified hypersonic s m a l l - disturbance theory proposed by Van Dyke, and those f o r t h e fuselage, by t h e second-order shock-expansion method. The r e s u l t s of t h e s e methods are subsequently employed i n conjunction with slender-body and l i n e a r The l a t e r a l - theory f o r c a l c u l a t i o n of t h e s t a t i c and r o t a r y d e r i v a t i v e s .

d i r e c t i o n a l d e r i v a t i v e s , although limited t o s m a l l s i d e s l i p angles, are determined f o r combined s i d e s l i p and angle of a t t a c k .

The r e s u l t s of t h e a n a l y s i s indicate t h a t t h e X - 1 5 a i r p l a n e i s s t a t i c a l l y stable i n p i t c h and yaw t o Mach numbers w e l l i n excess of i t s design limits, and t h a t t h e degree o f s t a b i l i t y increases s u b s t a n t i a l l y with increasing angle of a t t a c k at hypersonic speeds. The d i h e d r a l e f f e c t a t t h e s e speeds, on t h e other hand, e x h i b i t s an unstable trend, and t h u s i n d i c a t e s a possible dynamic i n s t a b i l i t y at high mgles of a t t a c k . The calculated longitudinal c h a r a c t e r i s t i c s are f o r t h e most vitle, Unclassified.

p a r t i n close accord w i t h t h e r e s u l t s from wind-tunnel tests. The l a t e r a l and 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 agree w e l l w i t h wind-tunnel data i n t h e lower angle-of-attack range; however, due t o an interference of t h e bow shock wave on t h e lower v e r t i c a l t a i l and o t h e r e f f e c t s not accounted f o r i n the theory, some disagreement i s found a t high angles of a t t a c k .

The r e s u l t s from simple Newtonian theory i n general are s u b s t a n t i a l l y lower than the trends indicated by t h e hypersonic small-disturbance and shock-expansion methods.

H INTRODUCTION An adequate and r e l i a b l e ground simulat-Jn of the f l i g h t character- i s t i c s of hypersonic a i r c r a f t , i n view of t h e wide range of f l i g h t con- d i t i o n s encountered throughout a t y p i c a l design mission, n e c e s s i t a t e s a r a t h e r comprehensive determination of t h e aerodynamic c h a r a c t e r i s t i c s of such vehicles i n t h e e a r l y design stages. Wind-tunnel and b a l l i s t i c - range f a c i l i t i e s normally provide t h e bulk of t h i s information; however, t h e o r e t i c a l methods a r e a l s o employed as a r a t i o n a l basis f o r design of t h e vehicle and as a means f o r extrapolating t h e known c h a r a c t e r i s t i c s t o untested and unexplored regions. Thus, each complements t h e other as a new vehicle configuration proceeds from t h e i n i t i a l design t o t h e f i n a l f l i g h t stage.

The X-15 research a i r p l a n e has been extensively t e s t e d i n various NASA and other wind-tunnel f a c i l i t i e s , employing models which i n many cases are nearly exact r e p l i c a s of t h e f i n a l design configuration ( r e f s . 1 t o 4 ) . A s u b s t a n t i a l amount of d e r i v a t i v e d a t a t h e r e f o r e has been assimilated which encompasses most of t h e o v e r a l l f l i g h t envelope pro- posed f o r t h e X - l c j research program. Although t h e d e r i v a t i v e coverage i s f a i r l y comprehensive i n t h e subsonic and lower supersonic speed it i s incomplete above a Mach number of 3.5 and does not extend ranges, beyond the performance l i m i t estimated t o be i n t h e v i c i n i t y of 6 . 5 .

Theoretical methods, therefore, may be applied t o f i l l t h e remaining gaps and t o extrapolate t h e present r e s u l t s t o Mach numbers beyond 6.5 i n order that t h e c h a r a c t e r i s t i c s of a vehicle of t h i s type may be studied T h i s paper i s undertaken t o supply, i n p a r t , i n an extended speed range.

t h i s needed information through a p p l i c a t i o n of various a v a i l a b l e methods of analyses, and t o assess t h e accuracies and l i m i t a t i o n s of t h e methods by comparison w i t h t h e a v a i l a b l e experimental data.

A brief description of t h e a i r p l a n e i s given i n t h e following sec- t i o n , and a l i s t of symbols used throughout the analyses i s presented i n t h e appendix.

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DESCRIPTION OF THE AIRPLANE The X-15 a i r p l a n e i s a rockst-propelled midwing configuration, employing low-aspec$-rat+o ?:percent-thick wing and h o r i z o n t a l - t a i l sur- faces as i l l u s t r a k e d i n f i g u r e 1. The h o r i z o n t a l t a i l i s swept back and, i n order t o provide s u f f i c i e n t clearance from t h e wing wake at low angles i s mounted a t a dihedral angle of - 1 5 O . To ensure adequate of a t t a c k , d i r e c t i o n a l s t a b i l i t y throughout t h e f l i g h t envelope, l a r g e upper and lower v e r t i c a l t a i l s with loo wedge sections are incorporated. The con- t r o l portion of t h e lower panel i s j e t t i s o n a b l e t o provide ground c l e a r - i s composed of l a r g e i n t e g r a l f u e l and ance during landing. The fuselage liquid-oxygen tanks i n t h e midsections which n e c e s s i t a t e t h e addition of e x t e r n a l triangular-shaped side f a i r i n g s t o house t h e various control systems.

Aerodynamic control i n p i t c h and r o l l i s obtained through symnetric and d i f f e r e n t i a l v a r i a t i o n s of t h e t a i l p l a n e incidence, and i n yaw by r o t a t i o n of t h e outboard panels of the upper and lower v e r t i c a l surfaces.

For maneuvering i n regions of low dynamic pressure, j e t r e a c t i o n controls a r e i n s t s l l e d i n t h e nose of t h e fuselage f o r p i t c h and yaw control and near both wing t i p s f o r r o l l control.

Table I o u t l i n e s t h e geometric c h a r a c t e r i s t i c s of t h e airplane.

SCOPE OF THE INVESTIGATION The airplane disturbances i n general a r e assumed t o be small, t h e r e - f o r e t h e longitudinal and l a t e r a l - d i r e c t i o n a l modes may be t r e a t e d inde- pendently. I n t h e following presentation t h e various derivatives are grouped under t h e two general categories of longitudinal o r l a t e r a l - d i r e c t i o n a l derivatives. These categories, i n t u r n , a r e f u r t h e r sub- divided i n t o s t a t i c , rotary, and control d e r i v a t i v e s . Calculated r e s u l t s are presented f o r each derivative, followed by a b r i e f discussion of t h e A r i g i d airframe i s assumed significance and accuracy of the r e s u l t s .

throughout t h e analysis, and ranges of Mach number from 2 t o 12 and angle of a t t a c k from Oo t o a r e considered.

Power e f f e c t s are The analysis i s r e s t r i c t e d t o power-off f l i g h t .

not n e c e s s a r i l y negligible, however, p a r t i c u l a r l y under conditions where t h e j e t exhaust i s highly underexpanded and extensive pluming may occur.

Some possible e f f e c t s of jet pluming on a i r p l a n e s t a b i l i t y and control are considered i n references 5 and 6.

Results of extensive wind-tunnel tests made with s c a l e models of t h e X-15 provide t h e best available c r i t e r i a f o r judging t h e accuracy of , 0 . 0 . . . ... . . . 0.. 0 .

0 . 0 . e .

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0 . 0 . 0 .

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the t h e o r e t i c a l methods employed. Comparisons t h e r e f o r e a r e made i n each case with data derived from t h e following sources:

Mach number [ F a c i l i t y

Reference I

I a

1 . 4 1 t o 2.01 Langley 4- by 4-foot mpersonic 1 pressure tunnel Ames Unitary Plan tunnel 2 1.55 t o 3.50 ' Langley Unitary Plan tunnel 2.29 t o 4.65 Langley 11-inch hypersonic tunnel H 6.86 1, 4 1, 4 Wind-tunnel data f o r Mach numbers g r e a t e r than approximately 7 are not available at present. Table I1 presents d e t a i l s of the models, which i n a l l cases were nearly exact r e p l i c a s of t h e f i n a l design configuration.

DISCUSSION O F F L O W FIELDS I n t h e following analysis, frequent reference i s made t o various interference e f f e c t s a r i s i n g from the shock waves and flow f i e l d s gen- e r a t e d by t h e various a i r p l a n e components. Since t h e a i r p l a n e s t a b i l i t y and c o n t r o l l a b i l i t y are i n general markedly affected, a b r i e f introductory description of these e f f e c t s preceding t h e d e t a i l e d d e r i v a t i v e a n a l y s i s w i l l , it i s believed, permit a c l e a r e r and more o r d e r l y presentation.

More extensive treatments may be found i n references 7 t o 11.

Interference at high Mach numbers may arise from a number of sources, are t h e fuselage bow wave (including t h e canopy and side- among which f a i r i n g shocks), the shock compression and expansion f i e l d s from t h e wing and t a i l surfaces, the downwash and sidewash induced by t h e wing, and from the v o r t i c e s generated by t h e fuselage. These i n t e r f e r e n c e f i e l d s are i l l u s t r a t e d i n sketches (a) and ( b ) presented on t h e following The shock waves that occur at a Mach number of 6 are a l s o i l l u s - page.

t r a t e d i n f i g u r e 2 i n the form of shadowgraphs of a small f r e e - f l i g h t model t e s t e d a t t h e NASA Ames Research Center.

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WING-COMPR

Side view Sketch (a).

BODY UPWASH

NTERFERENCE

Plan view Sjeicii ("u) .

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The order of magnitude of the dynamic-pressure l o s s from t h e X-15 bow shock i s given i n f i g u r e 3 i n terms of t h e r a t i o of downstream t o free-stream dynamic pressure. The r e s u l t s shown were calculated with t h e aid of Schlieren photographs of t h e bow-wave angles from reference 5 and the shock tables i n r e f e r e n w 12, assuming t h e downstream flow a f t e r passage through the shock wave t o rxpznd i s e n t r o p i c a l l y u n t i l t h e s t a t i c pressure again reaches t h t f r L ’ t i - s t r t ? y i m value. The l o c a l Mach number a t t h i s point, however, i s 1k’ss t h a n the free-stream value, hence t h e lift- curve slopes of t h e downstream surfvces a r e increased. This increase tends t o compensate i n p a r t f o r the l o s s i n dynamic pressure as shown a l s o i n f i g u r e 3 by t h e r a t i o of the product of dynamic pressure and l i f t - c u r v e slope i n the downstream and free-stream regions. This r a t i o , Q, i s applied h e r e a f t e r as a correction f a c t o r f o r t h e designated as l i f t i n g effectiveness of the t a i l surfaces a t low angles of a t t a c k . The it i s observed, crosses t h e wing a t t h e higher Mach numbers bow wave, such t h a t some portions of t h e wing l i e i n t h e region of e s s e n t i a l l y unexpanded flow immediately behind t h e shock, as w e l l as i n t h e highly expanded flow a t t h e fuselage juncture. I n t h e v i c i n i t y of t h e shock i s considerably g r e a t e r than t h e free-stream wave the product value, whereas near t h e body it i s l e s s than t h i s value. A s a r e s u l t , t h e f a c t o r Q f o r t h e wing v a r i e s between values g r e a t e r and less than unity. The average value i s assumed t o be u n i t y .

The wing, as shown i n t h e foregoing sketches, generates shock com- pression and expansion f i e l d s which give rise t o pronounced changes i n l o c a l Mach number, dynamic pressure, and downwash, a l l of which may a l t e r s u b s t a n t i a l l y t h e c h a r a c t e r i s t i c s of t h e t a i l surfaces. The interference of these f i e l d s with t h e horizontal t a i l i s l a r g e l y avoided on t h e X-15 by locating the t a i l surface near t h e extended wing plane but with s u f f i - Large c i e n t dihedral angle t o c l e a r t h e wake a t low angles of a t t a c k .

incidence s e t t i n g s of t h e horizontal t a i l , however, w i l l place some sec- shock f i e l d s , and t h e sta- t i o n s of t h e t a i l within t h e bounds of t h e s e b i l i z e r effectiveness w i l l be correspondingly a l t e r e d depending upon incidence angle, angle of a t t a c k , and Mach number ( r e f . 10). The v e r t i - c a l t a i l s a r e s i m i l a r l y a f f e c t e d by t h e changes i n l o c a l dynamic pressure and Mach number due t o both compression from t h e lower wing surface and expansion from t h e upper surface. A s shown i n a subsequent section, these e f f e c t s a r e of prime importance i n evaluating t h e d i r e c t i o n a l - and 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 at high Mach numbers.

Immediately downstream of t h e wing t r a i l i n g edge a small region of upwash may be expected a t high Mach numbers as a r e s u l t of t h e expansion of t h e flow f i e l d from t h e lower wing surface (refs.

8 and 13). Depending upon Mach number, angle of a t t a c k , wing thickness, and proximity of t h e t a i l , t h i s l o c a l upwash could exert a noticeable e f f e c t on t h e l i f t of t h e horizontal tail. I n t h e present application, t h e l a r g e sweep of t h e t a i l i n r e l a t i o n t o t h e wing and t h e extreme slenderness of t h e wing and s t a b i l i z e r p r o f i l e s minimize t h e s e e f f e c t s , and s i g n i f i c a n t upwash e f f e c t s a r e expected only at high angles of a t t a c k i n t h e high Mach number rarlge.

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I n a l i f t i n g a t t i t u d e , the fuselage generates v o r t i c e s along i t s length, similar t o those shown i n t h e sketches, which eventually merge i n t o a p a i r of separated vortex filaments o f f s e t from t h e surface of the fuselage. A s pointed out i n references 10 and 14, these v o r t i c e s o f t e n induce s i z a b l e downwash and sidewash i n the region of t h e t a i l , depending upon angle of a t t a c k and the l o c a t i o n of t h e t a i l surfaces.

Although t h e point of separation from t h e body moves toward the nose w i t h increasing angle of a t t a c k , t h e c r i t e r i a of references 10, 11, and 1-5 indicate t h a t f o r low angles of attack t h i s separation point on t h e X-15 should occur j u s t upstream from the wing leading edges. Shortly t h e r e a f t e r the v o r t i c e s e n t e r t h e expansion f i e l d from t h e wing and are bent i n t h e d i r e c t i o n of l o c a l flow. It i s believed, therefore, t h a t the departure of t h e v o r t i c e s from t h e fuselage i s w e l l below t h e t i p of t h e v e r t i c a l t a i l a t moderate angles of attack (below l5O). With t h e horizon- t a l t a i l located i n a r e l a t i v e l y low position i n t h e p o s i t i v e angle-of- a t t a c k range and w i t h the v e r t i c a l surfaces close t o t h e wing, s m a l l departures w i l l e x e r t r e l a t i v e l y l i t t l e influence on e i t h e r t h e longi- t u d i n a l o r l a t e r a l - d i r e c t i o n a l s t a b i l i t y of the X-15. A t high angles of a t t a c k t h e effectiveness of the upper v e r t i c a l t a i l i s so reduced by t h e wing-expansion f i e l d and t h e horizontal t a i l so far removed t h a t vortex interference again becomes a negligible f a c t o r . A more complete description of t h i s e f f e c t i s given i n a l a t e r section. The e f f e c t s of t h e body v o r t i c e s , therefore, a r e disregarded i n t h e present analysis.

Wing-vortex interference i s confined e s s e n t i a l l y t o the regions inside t h e downstream Mach cones f r o m t h e t i p s , t h e wing leading edges being supersonic i n t h e range of Mach numbers considered. Because of t h e r e l a t i v e proximity of t h e wing and horizontal t a i l , these t i p cones

f o r Mach numbers g r e a t e r than about 4 i n t e r c e p t only minor regions of

Below a t h e h o r i z o n t a l t a i l near t h e t i p s , and hence may be neglected.

Mach number of 4, t h e i r e f f e c t on t h e l o c a l downwash angle a t t h e t a i l should be taken i n t o account.

Sketches (a) and (b) a l s o indicate regions of mutual interference between adjacent components of t h e airplane, such as those of l i f t carry- surfaces onto t h e fuselage, and of body- over from t h e wing and t a i l The d e s c r i p t i o n of t h e s e e f f e c t s induced upwash across the wing span.

i s deferred t o t h e subsequent sections.

PRESENTATION O F RESULTS The longitudinal derivatives a r e r e f e r r e d t o the s t a b i l i t y axes I f \ siiowii i i i rigurt: 4\aj an6 are presented i n t h e next section i n t n e foi- lowing order : ............... e e.. e.

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Figure L i f t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 t o 10

Pitching moment . . . . . . . . . . . . . . . . . . . . . . 9, 1 1 t o 16

Longitudinal control . . . . . . . . . . . . . . . . . . . . 17 t o 20

D a m p i n g i n p i t c h . . . . . . . . . . . . . . . . . . . . . . 2 1 t o 24

The body-axis system i n f i g u r e 4(b) i s used f o r t h e lateral- d i r e c t i o n a l derivatives which are presented as follows: Figure H

S i d e s l i p . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 t o 33

Yawing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 t o 39

Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 t o 45

Directional control . . . . . . . . . . . . . . . . . . . . . 46 t o 48

L a t e r a l control . . . . . . . . . . . . . . . . . . . . . . . 49 and 50

ANALYSIS AND DISCUSSION OF LONGITUDINAL DERIVATIVES The following section presents an analysis and discussion of t h e l i f t , pitching moment, longitudinal control, and pitch-damping character- i s t i c s both as derived from theory and as measured i n t h e wind-tunnel tests previously described.

L i f t C h a r a c t e r i s t i c s The lift f o r t h e complete a i r p l a n e i s calculated by t h e method of reference 16, i n which t h e t o t a l l i f t i s considered i n i t i a l l y t o be t h e sum of the individual l i f t s of the exposed wing and h o r i z o n t a l - t a i l sur- faces and of t h e fuselage, each t r e a t e d as an i s o l a t e d body. Incremental l i f t s are then added which represent corrections f o r the i n t e r f e r e n c e s t h a t arise when t h e components are placed adjacent t o one another i n t h e o v e r a l l configuration. The i n t e r f e r e n c e i s reciprocal, consisting of reflection-plane and upwash e f f e c t s on the wing due t o the presence of the fuselage, and of the carryover l i f t on the fuselage due t o t h e exposed

wing and t a i l panels. Both e f f e c t s , however, are t r e a t e d as wing con- -

t r i b u t i o n s i n accordance with the method of reference 16.

The f o r c e s on t h e h o r i z o n t a l - t a i l surfaces at zero incidence ( c o n t r o l s f i x e d ) are simi- l a r l y derived.

The method i n general has been confirmed experimentally f o r low and moderate angles of a t t a c k a t supersonic Mach numbers, but i t s v a l i d i t y i n t h e hypersonic range has not y e t been established.

Extension of the method t o angles of a t t a c k g r e a t e r than t h e range of t h e present study i s considered i n reference 17.

The procedure of reference 16 when applied t o t h e X - l 5 configuration l e a d s t o t h e following r e l a t i o n s h i p f o r a i r p l a n e l i f t c o e f f i c i e n t

ST cos rT

CL = - sw C'&(Km + KBW> + Q

c ' , - J K ~ + K ~ ~ ) (. - 2) + c LB

S S The K terms are the i n t e r f e r e n c e f a c t o r s which account f o r t h e l i f t of t h e wing and t h e h o r i z o n t a l t a i l i n the presence of t h e body, K m and

Km, and f o r t h e l i f t of t h e body i n the presence of t h e wing and t h e

h o r i z o n t a l t a i l , Km and Km. The c h a r a c t e r i s t i c s , of t h e i n d i v i d u a l components a r e discussed f u r t h e r i n the following sections.

Wing.- The flight envelope f o r the X-15 extends through t h e t r a n s i -

-

t i o n a l range from supersonic t o hypersonic speeds, hence a method of c a l c u l a t i o n s u i t a b l e t o both regimes is desired. The u n i f i e d supersonic- hypersonic small-disturbance theory proposed by Van Dyke i n reference 18 f o r slender configurations appears t o f u l f i l l t h i s need. According t o t h i s method, t h e r e l a t i o n s h i p s developed f o r hypersonic flow about slender shapes i n terms of the hypersonic s i m i l a r i t y parameter (Mach number x flow-deflection angle) are found t o be v a l i d a l s o i n t h e realm of super-

sonic l i n e a r theory if the parameter is simply redefined as i - 1 x

flow-deflection angle. This modification i s a l s o discussed i n reference 19.

For determination of t h e wing l i f t c h a r a c t e r i s t i c s i n t h e present a n a l y s i s , t h e small-disturbance pressure c o e f f i c i e n t s given i n reference 20 f o r w i t h t h e s i m i l a r i t y parameter compression and expansion a r e employed but modified as s t a t e d previously. These c o e f f i c i e n t s , when compared with t h e r e s u l t s of shock-expansion theory, are shown i n reference 2 1 i n t h e unmodified form t o y i e l d accurate estimates of two-dimensional a i r f o i l l i f t c o e f f i c i e n t s a t hypersonic speeds f o r angles of a t t a c k up t o 25'.

When applied t o an i n c l i n e d f l a t plate, as t h e wing and t a i l surfaces are assumed t o be i n t h e present analysis, t h e following r e s u l t i s obtained f o r t h e two-dimensional case by t h i s expression, as a goes t o zero i s found t o reduce t o t h e familiar

-r'

lo

, given by l i n e a r theory. Although t h e i n i t i a l slope

r e s u l t ,

q 2 - L

i s i d e n t i c a l t o t h a t given by l i n e a r theory, t h e v a r i a t i o n of cn with angle of a t t a c k becomes increasingly nonlinear as Mach number i s extended t o the hypersonic range. This progressive change i s i l l u s t r a t e d i n 5 i n which y i s assumed t o b e 1 . 4 . The v a r i a t i o n f o r t h e f i g u r e l i m i t i n g case of i n f i n i t e Mach number reduces t o t h e parabola The r e s u l t s given by simple Newtonian theory a r e a l s o cn = ( 7 + l)u2.

It i s observed t h a t equation ( 2 ) reduces t o included f o r comparison.

t h e Newtonian r e s u l t when M --tm and y -1; t h a t is, cn = 2u . Devia- I

t i o n s o f y from t h e value of 1 . 4 assumed i n t h e present a n a l y s i s , how- ever, a r e believed on t h e b a s i s of t h e r e s u l t s of references 12 and 19 t o be small.

As a means f o r conversion from two-dimensional t o three-dimensional l i f t a t hypersonic speeds, t h e following approximation f o r wing-tip e f f e c t s , based on l i n e a r theory, may be applied C ' (3) C ' N = 'n

\JM2 - 1

I n t h i s expression i s t h e l i f t - c u r v e slope from l i n e a r theory f o r C ' & t h e three-dimensional plan form, as given, f o r example, i n reference 22, and cn i s given by equation ( 2 ) . The l i f t c o e f f i c i e n t f o r t h e i s o l a t e d wing, neglecting t h e s m a l l chordwise f o r c e s due t o s k i n f r i c t i o n and wave drag, therefore becomes

(4)

c'Lw = c" cos

and t h a t f o r t h e wing i n t h e presence of t h e body (based on a r e a S ) ,

'L, sw

( 5 )

ch = (%B + KBW) 4 cn T cos

J M ' - 1

Approximate values f o r t h e i n t e r f e r e n c e terms Km and K B W i n equa- t i o n ( 5 ) a r e given i n reference 16. The l i f t c h a r a c t e r i s t i c s p r e d i c t e d by t h i s equation f o r t h e X-15 wing are shown i n f i g u r e 6 ( a ) . The Newtonian l i m i t ( t h a t is, M = m, y = l), f o r which KWB 3 1 and K B W -10, i s seen am am. a. me m a . m m m ma m e a m m m m o m m a a m m a m a a m m a m m m m m m m a m m m m m m m m m m a a m a m m m m ma 0.0 am t o be s u b s t a n t i a l l y lower than r e s u l t s given by t h e hypersonic s m a l l - disturbance theory, l a r g e l y because of t h e difference i n 7 .

Horizontal t a i l . - The lift c h a r a c t e r i s t i c s of the horizontal t a i l a t zero incidence a r e calculated by the same procedures described f o r t h e wing, b u t with a d d i t i o n a l modifications included f o r dihedral angle ( s e e ref. 23). The fuselage-inducsd upwash at the t a i l plane, however, i s con- sidered t o be n e g l i g i b l e due t o t h e proximity of the wing, and the term qB, corresponding t o KWB i n equation ( 5 ) , i s t h e r e f o r e unity. The win@; downwash parameter d € / d u , as estimated from t h e c h a r t s of reference 24,

i s found t o be negligible beyond a Mach number of approximately 4. The

lift curves f o r t h e horizontal t a i l , based on t h e reference area S and corrected f o r t h e dynamic-pressure loss Q from figure 3, are shown i n f i g u r e 6(b) together w i t h t h e Newtonian l i m i t .

Fuselage.- L i f t from the fuselage as described i n reference 25 i s derived from both i n v i s c i d flow about t h e body and from viscous cross- flow separation. For t n e present case t h e i n v i s c i d l i f t i s believed t o second- be b e t t e r approximated i n the overall Mach number range by the order shock-expansion theory presented i n reference 26, r a t h e r than by t h e slender-body p o t e n t i a l theory employed i n reference 25 The method of reference 26 i s an extension o f the generalized shock-expansion method of reference 27 f o r bodies of revalution a t s m a l l angles of a t t a c k and i s believed applicable f o r Mach numbers intermediate between those of For application the p o t e n t i a l and generalized shock-expansion t h e o r i e s .

t o t h e noncircular cross sections of t h e X-15, t h e r e s u l t s from r e f e r - ence 26 have been multiplied by a f a c t o r equal t o t h e r a t i o of t h e a c t u a l plan-form area t o t h a t of an equivalent body of revolution having t h e same l o c a l cross-sectional areas as t h e present configuration. This approximation f o r t h e i n v i s c i d e f f e c t s (neglecting chordwise f o r c e s ) leads t o t h e r e l a t i o n s h i p i n which i s obtained fro= reference 26 (appendix C ) and Total fuselage plan-form a r e a

% =

Plan-form area of equivalent body of revolution The slopes gii,ren hy the second -order shock-expansion theory, although derived f o r vanishingly small angles of a t t a c k (streamlines approximately ~ 1The u n i f i e d supersonic -hypersonic small -disturbance method described t o slender bodies a t an i n t h e preceding section angle of a t t a c k i n axial f -.

p a r a l l e l t o t h e body meridian l i n e s ) , have been extended through t h e o v e r a l l angle-of-attack range. References 28 and 29 show t h a t , f o r slender bodies having e l l i p t i c cross s e c t i o n s , t h e r a t i o of p o t e n t i a l lift for t h e e l l i p t i c body t o t h a t f o r an equivalent c i r c u l a r body i s equal t o t h e r a t i o of major t o minor axes. This c r i t e r i o n would lead t o values of somewhat higher than those given by (cLB) i n v i s c i d equation (6) .

The l i f t due t o viscous crossflow i s given by t h e following r e l a - H tionship from reference 30 t ( 7 )

The term k, the plan-form a r e a of t h e fuselage, c o n s i s t s i n t h e present

case of t h e forebody area only ( v e r t e x t o wing leading edge approximately), t a i l , i n e f f e c t , block t h e crossflow over t h e remaining since the wing and sections. The term q i s a correction f a c t o r f o r body-fineness r a t i o as i s an average crossflow drag c o e f f i - discussed i n reference 30, and Ed C c i e n t . The latter should be estimated by t h e procedure suggested i n t h e appendix of reference 3 1 using t h e experimental s e c t i o n drag c o e f f i c i e n t s given i n references 30 and 32 t o 34. For s i m p l i c i t y i n t h e present anal- y s i s , c‘ has been assumed t o be constant at 1 . 2 i n t h e o v e r a l l Mach d C number and angle-of-attack ranges. Although confirmed experimentally f o r

Mach numbers up t o approximately 4 ( r e f . 29), t h e v a l i d i t y of t h e pre-

ceding method f o r hypersonic flows i n general has not been e s t a b l i s h e d .

Newtonian theory has been applied i n a simple, approximate manner X-15 fuselage may be represented from t h e v e r t e x by assuming t h a t t h e t o a s t a t i o n immediately rearward of t h e canopy by a c i r c u l a r cone and over the remaining length by a cylinder of constant diamond-shaped cross section similar t o t h a t of t h e combined fuselage and s i d e f a i r i n g s . The r e l a t i o n s h i p s given i n reference 13 then l e a d t o t h e following expression f o r fuselage l i f t c o e f f i c i e n t

cos 2 T~~ s i n 2a + - 2 cos*v s i n

S The r e s u l t s given by equations (6) t o (8) are presented i n f i g u r e 6 ( c ) .

Airplane.- The combined r e s u l t s from equations (1) t o ( 8 ) , r e p r e - senting t h e l i f t c h a r a c t e r i s t i c s f o r t h e complete a i r p l a n e (untrimmed), 0 0 0 0 0 . 0 0 9.0 0 * . o 0 .

0 . 0 .

0 .

L

0 0 . 0 0 . . 0

0 . 0 . I).

0 . 0.0 0 . 0 0 0 0 0 .

figure 7 f o r the Mach number range from 2 t o 12 and f o r the a r e shown i n Newtonian l i m i t . The component buildup i s f u r t h e r i l l u s t r a t e d i n f i g u r e 8, showing t h e e f f e c t of Mach number on l i f t - c u r v e slope a t s e v e r a l angles of a t t a c k . These f i g u r e s a r e seen t o r e f l e c t the increasing n o n l i n e a r i t y which c h a r a c t e r i z e s the t r a n s i t i o n from supersonic t o hypersonic Mach numbers. Thus t h e l i f t - c u r v e slope a t high angles of a t t a c k i s seen i n f i g u r e 8 t o diminish r e l a t i v e l y l i t t l e with increasing Mach number as compared t o t h e f a m i l i a r l o s s e s associated w i t h small angles.

The c a l c u l a t e d r e s u l t s a r e compared w i t h wind-tunnel d a t a from references 1 t o 3 i n f i g u r e s 9 and 10 f o r s e v e r a l Mach numbers from 2.01 Although t h e wind- t o 6.86 and f o r angles of a t t a c k from 0 ' t o 2 5 O .

t u n n e l r e s u l t s appear t o be s l i g h t l y underestimated, t h e general agree- ment i s good. Some of t h e apparent discrepancy i s due t o an i r r e g u l a r v a r i a t i o n of the z e r o - l i f t i n t e r c e p t s among t h e various data. The Newtonian l i m i t s shown i n f i g u r e 10, due t o t h e absence of the various y , interference e f f e c t s among t h e components and t h e reduced value of a r e considerably lower than t h e trends indicated by t h e other methods.

Pitching-Moment C h a r a c t e r i s t i c s The pitching-moment c h a r a c t e r i s t i c s f o r t h e a i r p l a n e are r e a d i l y determined from t h e values of l i f t c o e f f i c i e n t presented i n figures 6 The buildup t o 8 and t h e center-of-pressure charts given i n reference 16.

of the moments due t o t h e various components about a center-of-gravity l o c a t i o n at 20 percent of t h e mean aerodynamic chord (based on area S) proceeds as follows : Wing and h o r i z o n t a l t a i l . - The moment arm f o r t h e l i f t of t h e wing i n t h e presence of the body d i f f e r s i n general from t h a t f o r the l i f t induced by the wing on t h e body, w i t h the d i f f e r e n c e depending p r i m a r i l y The moments from t h e two sources on Mach number and fuselage diameter.

t h e r e f o r e must be determined separately; however, f o r consistency with t h e foregoing l i f t calculations both e f f e c t s are charged t o t h e wing.

are The c h a r a c t e r i s t i c s f o r the horizontal t a i l (at zero incidence) determined i n l i k e manner, although t h e moment arms f o r t h e various i n t e r f e r e n c e e f f e c t s , due t o t h e absence of fuselage afterbody, are e s s e n t i a l l y equal. The following expression f o r t h e combined wing and t a i l i n t h e presence of t h e fuselage is obtained The r e s u l t s given by t h i s equation a r e presented i n f i g u r e s l l ( a ) and l l ( b ) together with t h e Newtonian l i m i t s .

Fuselage.- The center of pressure f o r t h e l i f t due t o i n v i s c i d flow about t h e fuselage i s calculated by t h e second-order shock-expansion method presented i n appendix C of reference 26, acd t h a t due t o viscous crossflow, by t h e procedure described i n t h e appendix of reference 31.

The former i s found t o vary s l i g h t l y with Mach number and t h e l a t t e r t o be e s s e n t i a l l y constant. The moment c o e f f i c i e n t f o r t h e fuselage may be expressed as H and, as shown i n f i g u r e l l ( c ) , i s d e s t a b i l i z i n g .

Airplane. - Figures 1 2 and 1-3 present t h e 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 both t h e t a i l - o n (zero incidence) and t a i l - o f f configurations as calculated from equations (9) and (10). The gradual departure from l i n e a r i t y as Mach number i s increased from supersonic t o hypersonic l e v e l s i s again evident i n t h e s e f i g u r e s . A t high angles of a t t a c k t h e s t a b i l i t y , l i k e t h e a i r p l a n e l i f t c o e f f i c i e n t , declines r e l a t i v e l y l i t t l e with increasing Mach number. These trends a r e a l s o apparent i n t h e 14 f o r several angles of a t t a c k . Newtonian buildup presented i n f i g u r e theory, since it underestimates t h e l i f t of t h e wing and horizontal t a i l ( f i g . 7), a l s o underestimates t h e s t a b i l i t y as shown i n f i g u r e s 12 and 13.

Figure 9 shows t h a t t h e t h e o r e t i c a l methods a r e generally i n close accord with t h e experimental data, although t h e s t a b i l i t y contribution from t h e horizontal t a i l at a Mach number of 6.86 ( f i g . 9 ( e ) ) , appears t o be underestimated t o some extent. The discrepancy i s a possible con- sequence of the upwash i n t h e expanding flow downstream from t h e wing

t r a i l i n g edge as described i n references 8 and 13 - an e f f e c t which has

been neglected i n t h e present analyses. Also shown i n f i g u r e 9 a r e t h e l i f t curves f o r trimmed l e v e l f l i g h t based on t h e foregoing calculated p i t ching-moment c h a r a c t e r i s t i c s .

Further comparisons between experiment and theory a r e presented i n f i g u r e s l’j and 16. These f i g u r e s i n d i c a t e t h a t s t a t i c s t a b i l i t y a t small p o s i t i v e angles of a t t a c k w i l l not become marginal u n t i l Mach numbers well i n excess of t h e design l i m i t a r e reached. A s noted previously, t h e Newtonian limits i n f i g u r e 15 d i f f e r s u b s t a n t i a l l y from t h e apparent trends of t h e hypersonic small-disturbance and shock-expansion methods.

3T

Longitudinal-Control Characteristics The methods of reference 16 enable r a p i d estimates t o be made of t h e l i f t v a r i a t i o n s due t o incidence as well as angle of a t t a c k f o r wing-body combinations. When applied t o t h e X-15 s t a b i l i z e r , t h e following rela- t i o n s h i p s a r e obtained f o r t h e l i f t and moment increments due t o a change of incidence angle (a = Constant) The term E",,,, because of t h e nonlinear character of t h e flow, should - be determined with t h e a i d of equations (2) and (3) f o r t h e combined angles of a t t a c k and incidence as measured i n a plane perpendicular t o t h e surface. If subscript P i s used t o designate angles measured i n t h e plane perpendicular t o t h e plane of t h e tail,2 then the normal-force increment due t o incidence i s i n which

J

The f a c t o r s % and kgT account f o r t h e mutual interference between

t h e s t a b i l i z e r and fuselage f o r incidence v a r i a t i o n s i n a manner analogous t o Km and KBT f o r angle-of-attack v a r i a t i o n s . I n trimmed f l i g h t , however, a and iT are normally of opposite sign and t h e magnitude of

t h e combined angle a + iT i s generally s m a l l . Assuming t h a t t h e terms

i n equations (11) and (12) are u n i t y and t h a t cos a + LT' zr?d cos iT

i I

*The incidence of the X - 1 5 s t a b i l i z e r a c t u a l l y is varied by r o t a t i o n about an axis i n t h e plane of t h e surface r a t h e r than an axis normal t o t h e v e r t i c a l plane of symmetry (see f i g . 1).

............... . . 0.. r .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

t h e tangents i n equation (14) are equal t o t h e radian values of t h e angles, t h e following approximate r e l a t i o n s h i p s f o r small incidence angles are obtained dC

%

- - - C L i T ai, I n these equations, t h e c a l c u l a t i o n of from equation (2) may be simplified by adoption of t h e following n o t a t i o n

r r+il

where

The r e s u l t s given by equations (13) t o (17) are shown i n f i g u r e 17

f o r several combined angles, a + i T ' The s t a b i l i z e r effectiveness, similar t o t h e l i f t c h a r a c t e r i s t i c s described earlier, increases sub- s t a n t i a l l y with angle of a t t a c k a t hypersonic speeds. Experimental d a t a f o r s m a l l incidence angles are not a v a i l a b l e t o confirm t h e t r e n d s shown i n f i g u r e 17.

References 3 and 4 present lift and moment d a t a f o r l a r g e incidence angles t h a t may be compared with the moment increments predicted by Results f o r Mach numbers from 2.29 t o 6.86 are presented equation (12).

i n f i g u r e 18 i n terms of angle of a t t a c k a t constant incidence s e t t i n g (-20° and l5O) and i n f i g u r e 19 i n terms of incidence s e t t i n g a t constant angle of a t t a c k ( O O ) . Both f i g u r e s show f a i r agreement a t the lower Mach numbers. A t t h e higher Mach numbers, however, t h e s t a b i l i z e r e f f e c t i v e - ness i n f i g u r e 18, appears t o be underestimated a t high angles of a t t a c k and f o r negative incidences somewhat overestimated i n t h e lower range.

The discrepancy a t t h e high angles of a t t a c k i s undoubtedly due t o t h e combined influence of dynamic pressure, Mach number, and downwash i n t h e compression f i e l d from t h e lower wing surface b r i e f l y described i n a preceding section. These shock e f f e c t s , as discussed i n reference 10, occur i n varying degree depending upon Mach number, incidence s e t t i n g , and angle of a t t a c k . The l a r g e sweep and d i h e d r a l angles of t h e sta- b l i z e r , however, preclude any r e l a t i v e l y simple procedure f o r estimating t h e s e e f f e c t s .

Further evidence of the various interference e f f e c t s a t l a r g e i n c i - dence s e t t i n g s i s found i n t h e pitching-moment c h a r a c t e r i s t i c s presented i n f i g u r e 20 f o r four test Mach numbers. The comparisons between theory and experiment f o r a Mach number o f 2.29 ( f i g . 2 0 ( a ) ) show some disagree- ment primarily i n the magnitude of the moment increments a t l a r g e nega-

t i v e incidences - a probable e f f e c t of both t h e w a k e from t h e wing and

t h e l a r g e abrupt d i s c o n t i n u i t y between t h e fuselage s i d e - f a i r i n g and inboard end of t h e s t a b i l i z e r . A t a Mach number of 2.98 ( f i g . 20(b)) t h e r e i s , i n a d d i t i o n t o the e f f e c t s o f wing wake and fuselage f a i r i n g , some evidence of the leading edges of t h e s t a b i l i z e r dipping i n t o the compression f i e l d from the wing at l i f t c o e f f i c i e n t s above 0.4. The immediate r e s u l t of t h i s interference i s a sharp increase i n trim l i f t c o e f f i c i e n t and an apparent reduction i n a i r p l a n e s t a b i l i t y . This t r e n d becomes more and more pronounced a s t h e Mach number i s r a i s e d t o 4.65 and 6.86 as shown i n f i g u r e s 20(c) and 20(d), respectively.

Damping i n Pitch The buildup procedure of reference 16 i s employed a l s o f o r calcula- t o p i t c h i n g r a t e about t h e t i o n of the pitching-moment coefficient due and t o steady v e r t i c a l a c c e l e r a t i o n of t h e center center of g r a v i t y

%

It i s assumed t h a t the previously determined i n t e r f e r - of g r a v i t y CmaL.

ence terms (K f a c t o r s ) f o r angle-of-attack v a r i a t i o n s are a l s o a p p l i - cable t o t h e cases of steady pitching rate and v e r t i c a l a c c e l e r a t i o n .

The Newtonian l i m i t s , which are generally i n disagreement w i t h t h e r e s u l t s from t h e shock-expansion and small-disturbance t h e o r i e s , are omitted.

.. ... . 0.. . 0 . .. . . . . . e 0 .

0 . . a . a . . 4 .

3. .. 0 . . .

0 . e . . . . . .. ..

Wing.- References 35 and 36 provide c h a r t s based on l i n e a r theory from which estimates of C, and C f o r t h e i s o l a t e d wing may be 9 % obtained. Preliminary inspection of t h e charts shows t h a t f o r t h e assumed center-of-gravity location, t h e wing damping e f f e c t s are r e l a t i v e l y s m a l l .

Following t h e notation used i n t h e l i f t calculations, t h e following expressions f o r C f o r t h e wing (including interference) and

cmq 9%

are obtained i n which t h e terms are obtained from references 35 and ( C t m . ) \ a w and 36.

It i s expected t h a t both q u a n t i t i e s would e x h i b i t a nonlinear v a r i a t i o n with l o c a l angle of a t t a c k at hypersonic speeds and should therefore be modified accordingly. I n t h e present a n a l y s i s t h e r e s u l t s f o r a given angle of a t t a c k were adjusted by t h e r a t i o from l i n e a r theory of given by equation ( 2 ) t o t h a t represented by t h e product C N The r e s u l t s from equations (18) and (19) a r e presented i n u.

(CNU)CL,O .~ f i g u r e s 2 1 and 22 f o r several angles of a t t a c k .

Fuselage.- The damping d e r i v a t i v e s f o r t h e fuselage may be approxi- mated through application of t h e r e l a t i v e l y simple r e s u l t s derived from slender-body theory i n a manner similar t o t h a t described i n reference 16.

It i s found t h a t , although slender-body theory alone does not accurately p r e d i c t t h e c h a r a c t e r i s t i c s of nonslender configurations, t h e r a t i o of slender-body derivatives may be employed with reasonable accuracy i n t h e following manner body body The s t a t i c d e r i v a t i v e i n these expressions has been previously determined by more precise methods ( r e f . 26). The slender-body r a t i o s may be r e a d i l y derived from t h e relationships developed i n references 25, 37, and 38, giving L 2 ' B Volume xo

-4(F) s S , L ( 7 - ; )

I .

- - -.

,lender body where xo/L and xc/L are the center-of-gravity and area-centroid loca- t i o n s r e l a t i v e t o t h e o v e r a l l body length, and where t h e term "volume" designates t h e a c t u a l volume of t h e fuselage including t h e s i d e f a i r i n g s .

Figures 21 and 22 show t h e damping derivatives f o r t h e X - 1 5 fuselage as estimated by t h e previous relationships t o be nearly constant i n t h e Mach number range from 2 t o 12.

Horizontal t a i l . - The damping contributions from t h e horizontal t a i l are determined by t h e method of reference 39 from which t h e following approximate equations are derived

18.. \ m

represent t h e average upwash induced by The terms

$[*) and

9cw Both are s m a l l , becoming negligible i n t h e wing a t t h e t a i l location.

t h e hypersonic range. The t a i l - p l a n e l i f t - c u r v e slope i s nonlinear, as .

shown i n t h e preceding sections.

Results a r e presented i n f i g u r e s 21 and 22.

Airplane.- The fuselage i s seen i n f i g u r e s 2 1 and 22 t o be t h e pre- dominant component i n t h e buildup of t h e a i r p l a n e damping c h a r a c t e r i s t i c s , with the tail-damping contribution becoming increasingly s i g n i f i c a n t as angle of a t t a c k i s increased. The e f f e c t s of v e r t i c a l t r a n s l a t i o n are seen i n f i g u r e 22 t o be q u i t e s m a l l and may be neglected at Mach numbers

The calculated damping c o e f f i c i e n t (Cmq + C'%) f o r the a i r p l a n e

above 4.

H with t h e t a i l on and o f f i s summarized i n figure 23.

Wind-tunnel data from reference 2 are compared with t h e calculated r e s u l t s a t Mach numbers up t o 3.5 i n f i g u r e 24 f o r both t h e t a i l - o n and t a i l - o f f configurations. The calculated damping with t h e t a i l o f f , f o r t h e most p a r t , i s l e s s than t h e experimental r e s u l t , i n d i c a t i n g possibly t h a t the fuselage moments may be underestimated. With t h e t a i l on, some- what b e t t e r agreement i s obtained where t h e t a i l incidence i s held con- s t a n t near t h e zero s e t t i n g . The l a r g e negative incidences employed at t h e higher angles of attack, it i s noted, tend t o reduce t h e o v e r a l l airplane damping due t o t h e nonlinearity of the t a i l - p l a n e l i f t charac- t e r i s t i c s . Inclusion of these e f f e c t s i n t h e c a l c u l a t i o n s would reduce t h e differences noted.

ANALYSIS AND DISCUSSION OF LATERAL-DIRECTIONAL DERIVATIVES The s i d e s l i p and t h e r o t a r y and control d e r i v a t i v e s f o r yaw and r o l l a r e considered i n t h e following section. The p r i n c i p a l interference flow f i e l d s a f f e c t i n g these d e r i v a t i v e s have been b r i e f l y described i n an e a r l i e r section, however more d e t a i l e d information may be found i n refer- ences 1 4 , 40, and 41. I n general, t h e procedures, assumptions, and nomenclature employed are s i m i l a r t o those f o r t h e longitudinal deriva- t i v e s . It is assumed, i n addition, t h a t t h e s i d e s l i p angles are small and t h e incidence of t h e horizontal t a i l i s zero.

The Newtonian limits are determined f o r t h e s i d e s l i p d e r i v a t i v e s only.

S i d e s l i p Derivatives The procedure of reference 16, when applied t o t h e case of steady s i d e s l i p leads t o the following b a s i c r e l a t i o n s h i p s f o r t h e side-force, yawing-moment, and rolling-moment c o e f f i c i e n t s due t o t h e wing, fuselage, horizontal t a i l , and t h e upper and lower segments of t h e v e r t i c a l t a i l

a

= ‘fh) s a w a + $(‘yB) inviscid a p

ac

ST s i n r T NT

+ & s (%B + KBT)(l g ) F

B viscous ( 2 7 ) These equations a r e t h e basic forms from which t h e s i d e s l i p d e r i v a t i v e s f o r t h e individual components and f o r t h e complete a i r p l a n e are derived i n t h e following subsections. The r o l l i n g moments due t o t h e fuselage are n e g l i g i b l e and have been omitted from equation (28). A s i n t h e e a r l i e r analyses, t h e interference l i f t s between t h e fuselage and various are combined w i t h the l i f t f o r t h e adjacent surface, l i f t i n g surfaces and, where appropriate, t h e nonlinear r e l a t i o n s h i p s given by equations (2) and ( 3 ) a r e introduced. The results as applied t o t h e X-15 are presented i n figures 25 t o 29 f o r angles of a t t a c k of Oo, 8O, 1 6 O , and 24’.

Wing.- The terms i n equations (26) and (27)

-

o r i g i n a t e primarily from edge-suction f o r c e s e x i s t i n g along t h e wing t i p s .

rm u e s e e f f e c t s are fsuzd f r m the r e s u l t s of reference 42 t o be extremely s m a l l i n comparison with those due t o t h e fuselage and v e r t i c a l tail, and, t h e r e f o r e , are neglected. The r e s u l t s of reference 43 i n d i c a t e t h a t t h e

term r+) f o r t h e wing alone i s a l s o very small. For t h e wing i n

w

t h e presence of the body, however, it i s demonstrated i n reference 44 t h a t a s u b s t a n t i a l r o l l i n g moment w i l l occur due t o cross-coupling e f f e c t s of t h e sidewash v e l o c i t i e s t h a t a r i s e when t h e wing-body com- bination i s displaced both i n s i d e s l i p and angle of a t t a c k . According

t o t h e method of reference 44 as applied t o bodies of c i r c u l a r cross

section, t h i s moment i s given approximately by the r e l a t i o n s h i p B or This equation i s applied t o t h e X-15 fuselage by replacing t h e l a t t e r with an equivalent c i r c u l a r cylinder having a cross-sectional area equal t o t h e a c t u a l cross-sectional a r e a at the wing-body juncture (including t h e side f a i r i n g s ) . For hypersonic Mach numbers the nonlinear v a r i a t i o n of t h e term C ' N ~ with angle of a t t a c k should not be overlooked. The ( 3 0 ) a r e presented i n f i g u r e 27 which shows t h e r e s u l t s from equation cross-coupling e f f e c t t o be s i g n i f i c a n t at high angles of a t t a c k .

Fuselage.- Equations (6) and (7) f o r the force c o e f f i c i e n t s due t o inviscid and viscous crossflow, when converted t o s i d e force i n combined s i d e s l i p and angle of a t t a c k , transform t o t h e following o v e r a l l expression The term i s t h e angle of a t t a c k as measured i n t h e plane containing cp t h e free-stream-velocity vector and t h e body a x i s , and p i s t h e angle between t h i s plane and t h e X-Y plane i n f i g u r e 4 ( b ) . The term RT corresponds t o i n equation (6) and v a r i e s from 1.4 f o r displacements i n pitch ( p = 0) t o approximately 0.9 f o r displacements i n yaw (a = 0 ) .

Correspondingly, t h e term AT ( t h e counterpart of h) v a r i e s from 0.60

t o 0.485. It can be shown t h a t such t h a t t h e d e r i v a t i v e may be approximated as and AT t o be constant. The assuming t h e terms Rv, 7, 'dC, angle cp i s r e l a t e d t o u and P by the equation --

s i n cp = {syn'u + sin 2 p - s i n 2 a s i n 2 p

(34) which, f o r r e l a t i v e l y s m a l l angles, simplifies t o the form The r e s u l t a n t values f o r t h e various f a c t o r s assumed t o be constant i n equation (33) may be determined f o r combined u and p by resolving t h e known magnitudes f o r u = 0 and P = 0 i n t o components proportional a P

. This procedure i s not exact, however, since

t o and

1 - Jd+p2

the resultant-force vector, as pointed out i n reference 14, does not necessarily l i e i n t h e plane containing t h e v e l o c i t y vector and t h e body a x i s . The various c o e f f i c i e n t s f o r the i n v i s c i d and viscous crossflow are determined by t h e methods discussed previously i n t h e longitu- terms f o r s i d e s l i p alone d i n a l analysis. I n the calculation of (%)B (a = O o ) , however, t h e X - 1 5 fuselage and canopy have been approximated as by a cone-cylinder combination rather than an ogive and cylinder, assumed previously f o r angle of attack.

The yawing-moment derivative for t h e fuselage a t combined angles of a t t a c k and s i d e s l i p may then be obtained from t h e r e l a t i o n s h i p i n which t h e moment arm Poi- t h e viscous crossflow should be determined of t h e iliitial w i t h due consideration given t o the downstream location point of crossflow separation along t h e body a x i s as discussed i n r e f e r - ence 31. The wing, it i s noted, has l i t t l e o r no blanketing e f f e c t on t h e crossflow i n s i d e s l i p .

The side-force and yawing-moment d e r i v a t i v e s f o r t h e X - 1 5 fuselage are shown i n f i g u r e s 25 and 26.

The e f f e c t of Mach number i s s m a l l , but an increase i n angle of a t t a c k i s accompanied by a gradual increase of both derivatives.

The Newtonian l i m i t s are derived from the r e s u l t s of reference 13 i n t h e manner described previously f o r t h e longitudinal c h a r a c t e r i s t i c s H (eq. (8)). The following equation i s obtained f o r t h e nose cone and 1 c y l i n d r i c a l afterbody sections 4 ' N C 2 SAB

= 2 - C O S TNC C O S U + 2 - U C O S V

(37) S S i n which Sm i s the s i d e area of t h e afterbody section. The r e s u l t s from equation (37) a r e presented i n figures 25 and 26.

Horizontal t a i l . - The s i d e force from the horizontal t a i l arises s o l e l y from dihedral angle, and i n d e r i v a t i v e form i s given by The term C ' i s proportional t o an e f f e c t i v e angle of a t t a c k ae of NT t h e t a i l defined as follows f o r t h e combined angles of a t t a c k and s i d e - s l i p ( r e f . 45, appendix B) s i n fi t a n r T

t a n a cos B -

(39) cos a For s m a U angles of s i d e s l i p and equation (38) becomes i s given by equation (17). The upwash e f f e c t due t o

where (' INUIT

fuselage crossflow i s assumed t o be negligible a t t h e t a i l , hence t h e

term KTB becomes unity. Furthermore, t h e fuselage may be regarded as

having zero d i h e d r a l angle, and thus t h e term KBT i s zero.

The yawing-moment d e r i v a t i v e f o r t h e h o r i z o n t a l t a i l i s r e a d i l y obtained as H where % has been determined previously (see eq. ( 9 ) ) .

The sidewash cross-coupling e f f e c t s mentioned previously i n regard t o t h e wing-body r o l l i n g moments are assumed t o occur a l s o i n t h e v i c i n i t y of t h e fuselage and t h e horizontal- and v e r t i c a l - t a i l surfaces. Following the method of reference 44, the rolling-moment d e r i v a t i v e f o r t h e h o r i - zontal t a i l becomes For s m a l l s i d e s l i p angles, t h e expression i s obtained with the a i d of equation (40).

The contributions of t h e horizontal t a i l t o t h e o v e r a l l s i d e s l i p d e r i v a t i v e s are g e n e r a l l y q u i t e small, as can be seen i n f i g u r e s 25 t o 27.

The simple Newtonian approximations f o r t h e t a i l - p l a n e d e r i v a t i v e s are given by . .

* E (45) 1 s T 2

= -4 - s i n r?

C t

= 4 P ) ' T - - s i n2 rp

b S Figures 25 t o 27 show t h a t t h e Newtonian r e s u l t s f o r t h e horizontal t a i l are not s i g n i f i c a n t .

Vertical t a i l . - For improved s t a b i l i t y at high Mach numbers, 1 0 ' wedge-type sections were chosen f o r both t h e upper and lower t a i l The normal-force c o e f f i c i e n t s f o r these surfaces may be d e t e r - surfaces.

mined f i r s t by applying t h e r e s u l t s of l i n e a r theory (ref. 22), assuming the surfaces t o be f l a t p l a t e s , and then by correcting t h e f l a t - p l a t e values by the r a t i o of t h e l i f t - c u r v e slopes f o r t h e wedge and f l a t p l a t e given in reference 46. The terms i n equation (26) (cNu)L and (cNu)lJ a r e considered t o have been modified i n t h i s manner.

The i n t e r f e r e n c e c o e f f i c i e n t s of reference 16 a r e introduced, but t h e hypersonic nonlinear e f f e c t s a r e neglected f o r s m a l l s i d e s l i p angles.

A s noted i n t h e descrip- t i o n of the interference flow f i e l d s , t h e e f f e c t of t h e displacements of t h e wing and fuselage v o r t i c e s r e l a t i v e t o t h e t a i l surfaces is believed t o be s l i g h t a t s m a l l s i d e s l i p angles; consequently, t h e term i n da/dp equation (26) w i l l be neglected.

The side-force d e r i v a t i v e f o r t h e v e r t i - c a l t a i l s therefore becomes 8 . 0.. . . . .a 0 . . 0.. 0 0.. 0 .

0 8 . 0 . . 0 . . . 0 . 0 . 0 .

0 . 0 . . 0 * . . 0 0 . - * . I - -

I n general, t h e various terms i n equation (46) f o r t h e upper and lower surfaces d i f f e r . P a r t i c u l a r l y noteworthy a r e t h e l a r g e differences i n Q t h a t occur with increasing angle of attack. For s m a l l angles, t h e magnitudes shown i n f i g u r e 3 may be applied equally t o both surfaces. A t moderate and high angles of a t t a c k , however, t h e upper t a i l i s l a r g e l y contained within t h e shock-expansion f i e l d from t h e upper wing and body surfaces and t h e lower t a i l within the compression f i e l d from t h e lower surfaces. Consequently, Q f o r the upper t a i l diminishes while t h a t f o r t h e lower t a i l increases rapidly as shown i n f i g u r e 28. The v a r i a - t i o n s of Q shown i n t h i s f i g u r e were estimated from t h e c h a r t s of r e f - erence 1 2 f o r supersonic flow past wedges and f o r Prandtl-Meyer expansions.

The yawing-moment c h a r a c t e r i s t i c s f o r t h e v e r t i c a l t a i l are r e a d i l y found from equation (46) and t h e pmviously determined moment arms.

Rolling moments from t h e v e r t i c a l t a i l a r e generated by differences i n geometry and dynamic pressure between t h e upper and lower surfaces as w e l l as by t h e previously mentioned cross coupling of sidewash v e l o c i t i e s f o r t h e f u s e l a g e - t a i l combination. The following r e l a t i o n s h i p s from reference 44 a r e used f o r calculation of t h e i r combined e f f e c t J Values f o r t h e c o e f f i c i e n t s Km, KuB, and Kq and f o r t h e moment arms

(2y.Ju)

are determined from t h e c h a r t s given i n reference 44.

Newtonian theory provides t h e following simple expressions f o r t h e assuming t h e s i d e s l i p angles t o be less v e r t i c a l - t a i l c h a r a c t e r i s t i c s , than t h e semivertex angle of t h e wedge p r o f i l e s ( T ~ = 5') ............... .......

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

............

- - - e -

\ sL (%JL = -8- S TV

-

Figures 25 t o 27 i l l u s t r a t e the dominating e f f e c t t h a t t h e v e r t i c a l t a i l s have on t h e s i d e s l i p c h a r a c t e r i s t i c s of t h e X-15. P a r t i c u l a r l y remarkable are t h e , r a p i d gain i n effectiveness of t h e lower surface and t h e loss i n effectiveness of t h e upper surface as angle of a t t a c k i s increased. A s a r e s u l t , t h e d i r e c t i o n a l s t a b i l i t y at hypersonic speeds remains at a high l e v e l , and t h e net contribution of t h e v e r t i c a l t a i l s t o t h e dihedral e f f e c t changes from negative t o p o s i t i v e magnitudes as angle of a t t a c k i s increased. Newtonian theory, as shown i n the figures, does not p r e d i c t these c h a r a c t e r i s t i c s .

Airplane.- The s i d e s l i p c h a r a c t e r i s t i c s f o r t h e a i r p l a n e a t angles of attack of Oo, 8 O , 16O, and 24O, as b u i l t up from the foregoing component e f f e c t s , a r e presented i n f i g u r e 29 f o r both the v e r t i c a l t a i l on and o f f .

The airplane, it i s noted, i s predicted t o be d i r e c t i o n a l l y stable w e l l i n excess o f i t s estimated l i m i t speed, but t h e calculated dihedral e f f e c t

, due t o t h e pronounced asymmetry i n effectiveness of t h e upper and

c z P lower v e r t i c a l tails, changes r a p i d l y from near zero t o r e l a t i v e l y l a r g e positive magnitudes as angle of a t t a c k i s increased. This p o s i t i v e t r e n d i s generally undesirable from the standpoint of t h e Dutch roll i n C z P s t a b i l i t y due, as discussed i n reference 47, t o i t s i n d i r e c t e f f e c t of as angle of a t t a c k i s reducing t h e d i r e c t i o n a l s t a b i l i t y of t h e a i r p l a n e increased. Thus, from a simplified a n a l y s i s of t h e l a t e r a l - d i r e c t i o n a l of motion t h e e f f e c t i v e d i r e c t i o n a l - s t a b i l i t y paxameter equations i s evolved which shows t h a t i n s t a b i l i t y i n Dutch roll i s promoted by a positive t r e n d i n at high angles of a t t a c k . It i s a l s o noteworthy

czB

t h a t the interference of t h e wing and fuselage v o r t i c e s on t h e s t a b i l i t y contribution from t h e upper v e r t i c a l t a i l a t p o s i t i v e angles of a t t a c k , due t o t h e l o s s i n effectiveness of the surface, i s of l i t t l e consequence as hypersonic speeds are a t t a i n e d .

Figures 30 t o 32 present a comparison of t h e c a l c u l a t e d s i d e s l i p d e r i v a t i v e s with those obtained i n t h e wind-tunnel tests of references 1 Generally fair agreement i s noted i n these figures f o r t h e v e r t i c a l - t o 4.

t a i l - o f f r e s u l t s , however t h e v e r t i c a l - t a i l contribution ( d i f f e r e n c e between t h e t a i l - o n and t a i l - o f f values) appears t o have been o v e r e s t i - mated a t t h e higher Bngles of a t t a c k f o r Mach numbers of 2.01, 4.68, and 6.86. The discrepancies a t a Mach number of 2.01 are believed t o be associated w i t h detachment of t h e wing leading-edge shock waves as high angles of a t t a c k are approached (see f i g . 4 of ref. 12). With shock detachment a s u b s t a n t i a l weakening of the wing compression f i e l d , and hence a l s o t h e t a i l contributions, would be expected t o occur, although l i t t l e can be predicted by present methods of t h e mixed-flow character- i s t i c s downstream from detached shocks (see r e f . 48). The discrepancies a t t h e two highest Mach numbers r e s u l t i n p a r t from the extension of t h e lower v e r t i c a l surface through t h e wing-fuselage compression f i e l d i n t o t h e reduced dynamic pressure of t h e f r e e stream a t t h e higher angles of a t t a c k - an e f f e c t not taken i n t o account i n t h e foregoing c a l c u l a t i o n s .

The improvements gained by inclusion of t h i s e f f e c t are i l l u s t r a t e d i n figure 33. Since methods f o r predicting t h e p o s i t i o n and shape of wing- body shock f i e l d s at high angles o f a t t a c k appear t o be unavailable at t h i s t i m e , t h e proportions of the lower surface within and beyond the compression f i e l d were estimated from s c h l i e r e n photographs obtained

from references 3 and 4 and unpublished sources. The d i r e c t i o n a l sta-

b i l i t y as a r e s u l t of t h i s change i s somewhat reduced at t h e higher angles of a t t a c k , but t h e d i h e d r a l e f f e c t i s a l s o less unstable. The general agreement between theory and experiment, although improved i n some areas, i s s t i l l not completely s a t i s f a c t o r y , i n d i c a t i n g t h a t o t h e r e f f e c t s not accounted f o r are present.

It should be observed a l s o that large negative incidence s e t t i n g s of t h e h o r i z o n t a l t a i l i n which t h e leading edges penetrate t h e wing compression f i e l d w i l l , due t o t h e expanding flow around t h e leading edge, cause noticeable reductions i n effectiveness of the lower v e r t i c a l surface a t high angles of a t t a c k . Although t h e complex geometry of t h e X-15 precludes any ready c a l c u l a t i o n o f t h i s i n t e r f e r e n c e e f f e c t , it would be expected that t h e dynamic d i r e c t i o n a l s t a b i l i t y of t h e a i r p l a n e (eq. (49)) may be improved when it i s trimmed f o r l e v e l f l i g h t at high angles of a t t a c k . T h i s possible improvement, which would arise primarily from t h e reduced unstable t r e n d i n CzB, i s indicated I n t h e r e s u l t s of preliminary wind-tunnel tests a t high angles of a t t a c k .

do not The r e s u l t s from Newtonian theory, a s shown i n f i g u r e 29, i n general agree w i t h t h e l i m i t i n g values approached by the o t h e r methods ~lt. high Mach numbers. The Newtonian approximation therefore, s i n c e it 1s omitted appears t o be generally inadequate for t h e s t a t i c d e r i v a t i v e s , from t h e remaining analyses of t h e r o t a r y and c o n t r o l d e r i v a t i v e s .

Derivatives Due t o Yawing .I The derivations of t h e side-force, yawing-moment, and r o l l i n g - moment c o e f f i c i e n t s due t o steady yawing r a t e follow c l o s e l y those due t o s i d e s l i p i n t h e preceding section, and equations (26) t o (28) comprise t h e b a s i c relationships from which t h e a i r p l a n e yawing d e r i v a t i v e s may be As before, t h e fuselage r o l l i n g moments a r e i n s i g n i f i c a n t and deduced.

Results a r e presented i n f i g u r e s 34 t o 39.

a r e therefore disregarded.

Wing.- Wing e f f e c t s due t o yawing a r e caused by suction forces along

-

H t h e subsonic edges and by spanwise v a r i a t i o n s of velocity and Mach number.

E s t i m a t e s of these e f f e c t s a r e presented i n reference 42, which i n d i c a t e s t h a t the wing force and moment c o e f f i c i e n t s i n t h e present application \ fuselage and v e r t i c a l - t a i l a r e quite s m a l l compared t o thosy f A due t o t h e Ir surfaces. The derivatives ( % ) w and r$jW therefore are neglected.

The rolling-moment derivative (2)w, has been r e t a i n e d as shown i n

f i g u r e 36. The l a t e r a l - a c c e l e r a t i o n d e r i v a t i v e s ( b terms) f o r t h e wing a r e a l s o found t o be negligible ( r e f . 49).

Fuselage.- Slender-body theory i s used t o estimate t h e side-force and yawing-moment c o e f f i c i e n t s due t o yawing i n t h e manner described e a r l i e r i n t h e pitch-damping calculations, t h a t i s

(Cnr)B = (" np ) b) cn slender body

a r e given by equations (33) and (36), respec- where (Cyg) B and ('"B)B The various Slender-body terms a r e derived from t h e r e s u l t s of t i v e l y .

references 37 and 2 5 , giving ( s e e a l s o eqs. (22) and (23))

5T

- -

..

S B

slender body -2 -

S

( . )

'"p slender body Volume - sb - SB

slender body -2 - S

- - ( E ) ( % ) b C

"a slender body The r e s u l t s from equations (50) and ( 5 l ) ' a r e presented i n f i g u r e s 34 and 35. The d e r i v a t i v e (Crib),, l i k e ( C m 6 ) , , i s noted i n f i g u r e 35 t o be s m a l l .

Horizontal and v e r t i c a l t a i l s . - The yawing d e r i v a t i v e s f o r t h e t a i l surfaces may be approximated by t h e r e l a t i v e l y simple approach proposed i n reference 9, i n which t h e yawing derivatives a r e r e l a t e d d i r e c t l y t o those f o r s i d e s l i p i n t h e following way 2 :(e), Assuming t h e yaw angles t o be small, p may be expressed as b 2 V whence and where the dimension 2 i s t h e moment arm of e i t h e r t h e horizontal o r 4 v e r t i c a l t a i l . Figures 34 t o 36 show t h a t t h e magnitudes of these quan- 6 t i t i e s are negligible f o r t h e horizontal t a i l and f o r t h e upper v e r t i c a l t a i l a t high angles of a t t a c k . The lower v e r t i c a l t a i l , on t h e o t h e r hand, exhibits s u b s t a n t i a l damping e f f e c t s .

The e f f e c t of sidewash l a g a t t h e v e r t i c a l t a i l i s analyzed i n it i s shown t h a t t h e d e r i v a t i v e s are r e l a t e d reference 50, i n which t o t h e rate of change of sidewash angle with angle of s i d e s l i p as follows The sidewash v a r i a t i o n s at t h e t a i l , however, a r e not s i g n i f i c a n t , and t h e derivatives therefore have been dropped.

Airplane.- The calculated damping-in-yaw c h a r a c t e r i s t i c s f o r t h e airplane a r e summarized i n f i g u r e 37, i n which t h e general l e v e l of The damping is observed t o increase with increasing angle of a t t a c k .

asymmetry i n loading on the v e r t i c a l surfaces introduces negative r o l l i n g moments which a l s o increase with angle of a t t a c k . These t r e n d s are com- pared with t h e r e s u l t s from wind-tunnel tests ( r e f . 2) i n figures 38

and 39. The predicted l e v e l s of yaw damping (Cnr - Cni) appear t o be i n

good agreement with t h e t e s t r e s u l t s except, perhaps, i n t h e higher angle-of-attack range a t t h e lower Mach numbers. The r o l l i n g d e r i v a t i v e s on t h e other hand, do not agree i n e i t h e r magnitude or trend

(Czr -

except near zero angle of a t t a c k , although some improvement would be t a i l from t h e gained i f t h e p a r t i a l emergence of t h e lower v e r t i c a l wing-body compression f i e l d were taken i n t o account as previously d i s - cussed i n r e l a t i o n t o t h e a i r p l a n e dihedral e f f e c t .

Derivatives Due t o Rolling The a i r p l a n e force and moment c o e f f i c i e n t s due t o steady r o l l i n g v e l o c i t y , as determined by summing t h e various component and i n t e r f e r e n c e c h a r a c t e r i s t i c s , a r e presented i n f i g u r e s 40 t o 45.

The fuselage, when considered as an i s o l a t e d body, i s assumed t o have zero loading; however, H i n r e l a t i o n t o t h e adjacent components i t s presence must be taken i n t o 1 account. The b a s i c r e l a t i o n s h i p s f o r steady r o l l i n g a r e (57)

= ( c z ) + (czp)T + ( c z ) p L,U

IP P W Wing.- Side forces and yawing moments on a wing with supersonic leading edges a r e generated s o l e l y by t i p suction f o r c e s as noted i n reference 51. For high t a p e r t h i s e f f e c t i s generally s m a l l but may

become noticeable at high angles of a t t a c k . The terms (2) and

I n \ f o r t h e X - 1 5 wing i n equations (55) and (56), as determined from

(2jW

the r e s u l t s of reference 51, are shown i n figures 40 and 41 t o be q u i t e s i g n i f i c a n t a t high angles of attack.

The damping i n r o l l f o r the wing i s t h e predominant d e r i v a t i v e i n t h i s group. Its m&gni+,udemay be r e a d i l y determined from t h e r e s u l t s of l i n e a r theory i n reference 22, but should be corrected f o r wing-body i n t e r f e r e n c e and f o r the nonlinear v a r i a t i o n s with angle of a t t a c k at hypersonic speeds. The slender-body r e s u l t s of reference 52 may be applied as a correction f o r wing-body i n t e r f e r e n c e , and equation (17) as an adjustment f o r nonlinearity. Then i s obtained from reference 22. (See a l s o ref. 53 f o r

where ( c ‘4 w

wing-body combinations. ) Figure 42 shows t h a t t h e damping i n r o l l f o r t h e wing, as predicted by equation ( 5 8 ) , should increase s u b s t a n t i a l l y with angle of a t t a c k at high Mach numbers.

Horizontal t a i l . - The geometric dihedral, as w e l l as t i p suction forces, introduces small side forces and yawing moments due t o r o l l i n g of t h e horizontal t a i l . Their magnitudes, however, are reduced somewhat H by the downwash from t h e wing as described i n reference 9. When corrected f o r downwash e f f e c t s , t h e c o e f f i c i e n t f o r t h e side force due t o t i p SUC- 4 t i o n , a r a t h e r s m a l l e f f e c t compared with t h a t due t o dihedral, may be 6 approximated as

= Q

S

[ (“P) T ] t i p suction

2v

wherein values f o r ( % ) T may be obtained from reference 51. A rough

estimate of t h e downwash parameter may be deduced from t h e r e s u l t s f o r a similar configuration i n reference 9. The side-force c o e f f i c i e n t r e s u l t i n g from dihedral i s proportional t o t h e loading on t h e t a i l due t o roll, and may be calculated with s u f f i c i e n t accuracy by t h e expression

= Q

[(“p) 4 dihedral

slender body Interference between t h e fuselage and t a i l i s taken i n t o account by introducing t h e slender-body term from reference 52 f o r cruciform t a i l surfaces. The term d i d may be calculated w i t h the a i d of s t r i p

T I (*3

theory, as suggested i n references 54 and 55, t h a t is, i s given by equation (17).

and (cNU)T The corresponding yawing-moment c o e f f i c i e n t s a r e obtained by multi- plying t h e r e s u l t s from equations (59) and (60) by t h e moment arm Z T ~ .

The side-force and yawing-moment r e s u l t s are shown i n f i g u r e s 40 and 41.

The r o l l damping due t o t h e horizontal t a i l i s c a l c u l a t e d i n t h e same manner as t h e wing damping (eq. ( 5 8 ) ) , but must be a d d i t i o n a l l y corrected f o r wing downwash e f f e c t s . Thus slender body S f T i n t h i s where (C zp)T i s obtained from reference 22. The a r e a equation i s t h e t o t a l a r e a including the portion enclosed w i t h i n t h e fuselage, and t h e slender-body terms are f o r cruciform configurations A s shown i n f i g u r e 42 t h e t a i l - p l a n e damping i s r e l a t i v e l y ( r e f . 52).

s m a l l .

V e r t i c a l t a i l . - The side-force c o e f f i c i e n t s due t o r o l l i n g of t h e upper and lower v e r t i c a l t a i l s a r e of opposite sign and may be determined by expressions similar t o equation (&), t h a t is, slender body The sidewash parameter similar t o t h e downwash parameter f o r t h e horizontal t a i l , has been determined f o r a similar configuration i n reference 9. (For subsonic leading edges see r e f . 56.) The f a c t o r V,/V i s introduced because of t h e r e l a t i v e l y l a r g e v a r i a t i o n s i n local-stream velocity t h a t occur at t h e higher angles of a t t a c k . This f a c t o r may be r e a d i l y estimated from t h e c h a r t s of reference 12. Although t h e two sur- faces tend t o n u l l i f y one another at low angles of a t t a c k , t h e i r e f f e c t s H become highly unsymmetrical as t h e higher angles are approached, as shown

i n figures 40 and 41. The corresponding yawing-moment c o e f f i c i e n t s - are

xL u

e a s i l y obtained from t h e previously determined moment arms A.

b Equation (62) may be r e a d i l y adapted t o c a l c u l a t i o n of t h e damping- moment derivatives f o r t h e v e r t i c a l surfaces. The downwash parameter should be replaced by t h a t f o r sidewash, V,/V and t h e velocity r a t i o may be introduced f o r g r e a t e r accuracy. The v e r t i c a l - t a i l damping, however, i s not consequential, as shown i n f i g u r e 42.

Airplane.- The predicted r o l l i n g d e r i v a t i v e s C Yp and cnp f o r

the airplane a r e given i n f i g u r e 43, i n which t h e magnitudes of both derivatives are shown i n general t o be i n s i g n i f i c a n t at low angles of The theo- attack, but t o grow r a p i d l y with increasing angle of a t t a c k .

r e t i c a l damping i n r o l l i s a l s o seen i n t h e same f i g u r e t o increase sub- s t a n t i a l l y with angle of a t t a c k . For Mach numbers up t o 3.5 t h e s e t r e n d s a r e confirmed experimentally t o some extent as shown i n f i g u r e s 44 and 45, but the agreement i s q u a l i t a t i v e a t b e s t . The wide s c a t t e r i n t h e data, notably i n t h e t a i l - o f f r e s u l t s , leaves t h e comparison somewhat i n doubt as t o which source, experiment o r theory, i s the more accurate.

Lateral-Directional Control Derivatives For aerodynamic d i r e c t i o n a l c o n t r o l t h e X - 1 5 i s equipped with a l l - movable t i p - c o n t r o l surfaces which b a s i c a l l y comprise t h e outer segments of t h e upper and lower v e r t i c a l s t a b i l i z e r s . The surface d e f l e c t i o n s a r e limited t o +7A0. L a t e r a l control, provided through d i f f e r e n t i a l c incidence of t h e horizontal s t a b i l i z e r s , has an angular range of k7- f o r each panel except at s e t t i n g s near l i m i t i n g values of s t a b i l i z e r deflection where t h e d i f f e r e n t i a l incidence a v a i l a b l e decreases. The two systems a r e aerodynamically coupled i n t h e sense t h a t d i r e c t i o n a l control a l s o induces lateral moments, and vice versa. The effectiveness of each type of c o n t r o l and the cross-coupling moments may be r e a d i l y determined from t h e r e s u l t s of t h e preceding analyses.

D i r e c t i o n a l control.- The s i d e force due t o t i p - c o n t r o l d e f l e c t i o n may be estimated i n a r e l a t i v e l y simple manner by an extension of the methods of reference 16. The t i p control and adjacent stabilizer-body combination may, as an approximation, be regarded as a wing and body combination, thereby permitting ready use of t h e c h a r t s of reference 16 f o r determination of t h e mutual interference e f f e c t s due t o incidence.

T h i s procedure l e a d s t o t h e expressions .

L sRZ

(CYER), = QL - S FRzB + kBRz) ('"U)RZ

I

i n which t h e various k terms are analogous t o the i n t e r f e r e n c e f a c t o r s contained i n equation (11) f o r t h e h o r i z o n t a l - s t a b i l i z e r e f f e c t i v e n e s s .

may be determined from

A s discussed earlier, ("&)Ru and (' "a)Rz

the r e s u l t s of l i n e a r theory i n reference 22 corrected f o r the e f f e c t s of t h e wedge-type p r o f i l e s as recommended i n reference 46. The dynamic- pressure terms QL and Qu are given i n f i g u r e 28. The corresponding yawing-moment c o e f f i c i e n t s a r e then found by multiplying equation (64) by the appropriate c o n t r o l moment arms (Z/b).

Rolling moments due t o tip-control d e f l e c t i o n s arise from d i f f e r - ences both i n geometry and i n dynamic pressure Q a t high angles of a t t a c k . Their magnitudes a r e proportional t o t h e d i s t a n c e s from t h e axis of r o t a t i o n t o t h e l a t e r a l centers of pressure which, i n the present a n a l y s i s , a r e assumed t o be the centroids of the control-surface areas.

It follows that The d i r e c t i o n a l - c o n t r o l derivatives predicted by equations (64) and ( 6 5 ) f o r t h e upper and lower surfaces a c t i n g both independently arid Figure 46 shows again i n combination are presented i n figures 46 and 47.

the r e l a t i v e influence of t h e previously described shock-expansion and compression f i e l d s from the wing on the r e l a t i v e e f f e c t i v e n e s s of t h e two surfaces.

The n e t r e s u l t s shown i n f i g u r e 47 indicate a r e l a t i v e l y high l e v e l of c o n t r o l l a b i l i t y f o r t h e a n t i c i p a t e d f l i g h t p r o f i l e s of t h e airplane, but a l s o a s u b s t a n t i a l cross-coupling e f f e c t a t high angles of attack.

The r e s u l t s of theory and experiment a r e compared i n f i g u r e 48 i n which generally f a i r agreement i s indicated, although some improvement a t t h e higher angles of a t t a c k and Mach numbers would be expected i f t h e emersion of t h e lower control surface from t h e wing-body compression f i e l d were taken i n t o account (see e a r l i e r discussion of d i h e d r a l e f f e c t ) € Lateral control.- The l a t e r a l - c o n t r o l derivatives s t e m d i r e c t l y from 1 t h e relationships developed f o r t h e pitch-control effectiveness. Referring I

t o equation (15) and noting t h a t each panel i s deflected one-half of t h e c

t o t a l d i f f e r e n t i a l deflection, t h e following r e l a t i o n s h i p s are obtained The l a t e r a l centers of pressure a r e assumed t o be located at t h e area centroids of t h e exposed panels, and t h e moment arm h i s determined on t h i s basis.

The calculated l a t e r a l - c o n t r o l d e r i v a t i v e s a r e given i n f i g u r e 49 which, as a n t i c i p a t e d from t h e r e s u l t s of t h e earlier longitudinal- control analysis, shows an increasing degree of control e f f e c t i v e n e s s with increasing angle of a t t a c k . A noticeable cross-coupling e f f e c t i s found a l s o f o r t h i s control. Figure 50 shows t h a t i n some cases theory and experiment f o r undetermined reasons do not agree w e l l at high angles of attack.

CONCLUDING REMARKS I n t h e foregoing a n a l y t i c a l study a number of a v a i l a b l e t h e o r e t i c a l methods were employed f o r c a l c u l a t i o n of the longitudinal and lateral- d i r e c t i o n a l s t a b i l i t y and control d e r i v a ‘ves f o r the X-15 a i r p l a n e at

d

T

Mach numbers ranging from 2 t o 12 and a n i i e s of a t t a c k as high as 25'.

The a n a l y t i c a l r e s u l t s were compared with e x i s t i n g wind-tunnel data f o r Mach numbers between 2 and 7, and thus enabled an o v e r a l l assessment t o be made of t h e accuracy and a p p l i c a b i l i t y of t h e methods used. The analysis as a whole provided valuable i n s i g h t s i n t o t h e significance of t h e various shock-interference phenomena a f f e c t i n g t h e a i r p l a n e as hyper- sonic speeds a r e approached, and t o the r e l a t i v e importance of t h e i n d i - vidual a i r p l a n e components i n systematic buildups of t h e o v e r a l l deriva- t i v e c h a r a c t e r i s t i c s . I n addition, the analysis w a s extended t o speeds w e l l beyond t h e estimated l i m i t f o r the X-15 so t h e d e r i v a t i v e s f o r more advanced versions of t h i s type vehicle could be examined.

I n general, s a t i s f a c t o r y agreement with wind-tunnel r e s u l t s was obtained. Notable exceptions, however, were found i n t h e s t a b i l i z e r e f f e c t i v e n e s s and several of t h e 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 at high angles of a t t a c k where shock interference on the horizontal and v e r t i c a l t a i l s could not be r e a d i l y calculated.

P a r t i c u l a r l y noteworthy f o r the X-15 configuration a r e t h e increases i n longitudinal and d i r e c t i o n a l s t a t i c s t a b i l i t y as angle of a t t a c k i s increased at hypersonic speeds. These c h a r a c t e r i s t i c s , however, a r e accompanied by an unstable trend i n dihedral e f f e c t . Pronounced cross coupling of t h e d i r e c t i o n a l - and l a t e r a l - c o n t r o l moments are a l s o noted a t high angles of a t t a c k and Mach numbers. I n general, t h e calculations i n d i c a t e t h a t both s t a b i l i t y and c o n t r o l l a b i l i t y a r e maintained w e l l beyond t h e estimated l i m i t speed.

The l i m i t i n g values predicted by Newtonian theory f o r t h e s t a t i c d e r i v a t i v e s a r e found i n general t o be lower than the t r e n d s shown by shock- t h e u n i f i e d supersonic-hypersonic small-disturbance theory, expansion theory, and other methods employed i n t h e analysis.

F l i g h t Research Center, National Aeronautics and Space Administration, Edwards, Calif., March 1, 1960.

t LIST OF SYMBOLS I n t h e following l i s t of symbols, t h e aerodynamic c o e f f i c i e n t s , when used without a superscript, a r e based on the dimensions of t h e wing with leading and t r a i l i n g edges extended t o the plane of symmetry of t h e airplane, and, when primed, on t h e dimensions of t h e i s o l a t e d surface o r body. H

plan-form area of fuselage forebody from vertex t o wing 4

leading edges 6 projected area of fuselage on plane normal t o crossflow a t combined a and p f o r portions of fuselage a f f e c t e d by crossflow b (refers t o wing without subscript) o v e r a l l span Drag

drag c o e f f i c i e n t , -

CD q , s l i f t c o e f f i c i e n t , Lift CL L S l i f t - c u r v e slope i n downstream flow based on l o c a l Mach number l i f t - c u r v e slope i n free-stream flow Rolling moment r o l l i n g -moment c o e f f i c i e n t , 6 , S b H pitching-moment coefficient, Pitching Cm

iasc'

Normal force normal-force coefficient, C" GmS Yawing moment yawing-moment coefficient, Cn 6 , S b H

*

effective directional-stability parameter (eq. (49) )

CnP -

cni - -

a(&) 2v

normal-force coefficient for two-dimensional flat plate Cn pressure coefficient, cP

Local pressure - Free-stream static pressure

o m e m m a m me ma- m m m ma e m m m m m m e a m a a m m a m m m m a m a e m m m m m e a m

e e m mea m m m a - - -

b Side force side-force c o e f f i c i e n t , CY G , S H

-

C mean aerodynamic chord (refers t o reference a r e a S when used without subscript ) average crossflow drag c o e f f i c i e n t f o r fuselage as calcu- l a t e d by method of reference 3 1

s i m i l a r i t y parameter, (M2 - 1 ) 1 ' 2 a

H " moment arm from fuselage center l i n e t o center of pressure h of horizontal t a i l measured i n plane of t a i l moment of i n e r t i a about X - a x i s , slug-ft2 ( f i g . 4 ( b ) ) I X e e e e .me e e e e a e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e a e e e e e e e e e e e w e e e e e e e e e e e e e e e e e e e e e a e

- i= a

-

p - - . -

i”=

moment of i n e r t i a about Z-axis, s l u g - f t 2 ( f i g . 4 ( b ) ) I Z i angle of incidence incidence of h o r i z o n t a l t a i l measured i n plane of symmetry i T r e l a t i v e t o fuselage center l i n e , p o s i t i v e f o r upward r o t a t i o n of leading edge d i f f e r e n t i a l incidence of h o r i z o n t a l - t a i l panels i ‘T H K r a t i o of l i f t due t o angle of a t t a c k of a component i n presence of an adjacent component t o l i f t of i s o l a t e d wing o r t a i l surface (see ref. 16 and s u b s c r i p t s ) f a c t o r representing coupling of sidewash v e l o c i t i e s due t o a and p (see ref. 44) k r a t i o of l i f t due t o angle of incidence of a component i n presence of an adjacent component t o l i f t of i s o l a t e d wing o r t a i l surface L o v e r a l l length of fuselage M Mach number r a t e of roll P p i t c h i n g r a t e 1 2 free-stream dynamic pressure, p , V , dynamic pressure of downstream flow where l o c a l s t a t i c pressure behind t h e bow shock wave i s equal t o f r e e - stream s t a t i c pressure r a t i o of t o t a l fuselage plan-form a r e a t o plan-form area R, of equivalent body of revolution having same l o c a l cross-sectional a r e a s ( s e e eq. ( 6 ) ) e. .e 0 e..) e e.. e * e 0 e . e . e .

e e e e e e e e * e e . * e . e . 45 e. e. e e e e.. e.

r a t i o of projected area of fuselage on plane normal t o fuselage crossflow a t combined a and p t o area of equivalent body of revolution having t h e same l o c a l cross-sectional area (see eq. (31) ) r rate of yaw S reference area equal t o area of wing with leading and t r a i l i n g edges extended t o plane of symmetry plan form and side,area of fuselage afterbody f o r normal- SAB and side-force considerations, respectively (see Newtonian theory, eqs. (8) and (37)) fuselage f r o n t a l area sB base a r e a of nose cone approximating fuselage nose section sNC (see Newtonian method, eq. (8)) area of directional-control surface SR area of exposed h o r i z o n t a l - t a i l surfaces ST area of horizontal t a i l with leading and t r a i l i n g edges S ' T extended t o fuselage center l i n e area of exposed wing panels local-stream v e l o c i t y f r e e -stream velo c it y V s i d e v e l o c i t y W v e r t i c a l velocity coordinate axes (see f i g . 4)

x,y>z

-

longitudinal distance from center of g r a v i t y t o center X of pressure of component l i f t measured i n d i r e c t i o n of fuselage center l i n e area longitudinal distance from vertex t o centroid of xC of fuselage measured i n d i r e c t i o n of fuselage center l i n e e. 0.0 0 e.. . ..

.. .. 0 .

O o 0 4 6 0 0 0 0 0 0 .

0 0 ... 0 . . 0 . ..

l o n g i t u d i n a l distance from vertex of fuselage t o center XO of g r a v i t y measured i n d i r e c t i o n of fuselage center l i n e

-

l a t e r a l distance from fuselage center l i n e t o center of Y pressure Z v e r t i c a l distance from fuselage center l i n e t o center of pressure angle of a t t a c k , deg , H U constant time r a t e of change of angle of a t t a c k due t o 4 U v e r t i c a l acceleration (plunging motion) 6 e f f e c t i v e angle of a t t a c k of h o r i z o n t a l t a i l at combined ue a and P ( s e e eq. ( 3 9 h de@; angle of s i d e s l i p , deg time r a t e of change of s i d e s l i p angle due t o constant l a t e r a l a c c e l e r a t i o n dihedral angle of h o r i z o n t a l t a i l measured from X-Y plane, p o s i t i v e when r o t a t e d upward r a t i o of s p e c i f i c heat a t constant pressure t o s p e c i f i c Y heat at constant volume d e f l e c t i o n angle of d i r e c t i o n a l - c o n t r o l surfaces, p o s i t i v e f o r r o t a t i o n of leading edge t o r i g h t , deg downwash angle a t t a i l , deg semiapex angle of wing o r t a i l surface, deg r a t i o of drag c o e f f i c i e n t of a c i r c u l a r cylinder of f i n i t e length t o t h a t f o r cylinder of i n f i n i t e length (see r e f . 30) p o t e n t i a l function f o r constant r a t e of p i t c h (see r e f . 39) Tip chord t a p e r r a t i o of h o r i z o n t a l - t a i l panels, Root chord angle between plane containing v e l o c i t y vector and fuse- lage center l i n e and t h e normal t o t h e plane of symmetry at combined a and p (see eqs. (31) and (32))

7T

. angle of i n c l i n a t i o n of fuselage s i d e f a i r i n g s ( s e e

V Newtonian method, eq. (8)) free-stream density, slugs/cu f t sidewash angle at t a i l , deg semivertex angle of nose cone (see Newtonian method eq. ( 8 ) ) semivertex angle of v e r t i c a l - t a i l wedge s e c t i o n s ( s e e rV f i g . 1) angle between free-stream v e l o c i t y and fuselage center cp l i n e at combined a and f3 p o t e n t i a l function f o r constant angle of a t t a c k (see

sla

r e f . 39) average upwash at t a i l due t o steady p i t c h i n g rate of wing (see r e f . 39)

A(-- 4%

average upwash at t a i l due t o steady plunging motion of '= wing (see r e f . 39) Subscripts: average av B fuselage BL fuselage i n presence of lower v e r t i c a l t a i l fuselage i n presence of lower d i r e c t i o n a l - c o n t r o l panel BR1 fuselage i n presence of upper d i r e c t i o n a l - c o n t r o l panel BRU fuselage i n presence of h o r i z o n t a l t a i l BT BU fuselage i n presence of upper v e r t i c a l t a i l BW fuselage i n presence of wing L lower v e r t i c a l t a i l (exposed) lower vertical tail in presence of fuselage LB designates that angle is measured in plane perpendicular P to the plane of the horizontal tail lower directional-control panel lower directional-control panel in presence of fuselage ZB upper directional-control panel RU H upper directional-control panel in presence of fuselage RUB T horizontal tail (exposed) horizontal tail in presence of fuselage TB upper vertical tail (exposed) U upper vertical tail in presence of fuselage BU W wing (exposed) W B wing in presence of fuselage

U quantity due to angle of attack or sideslip (ref. 44)

quantity due to combined angles of attack and sideslip T (ref. 44) REFERBNCES 1. Driver, Cornelius: Effect of Forebody Strakes on the Aerodynamic Characteristics in Pitch and Sideslip of a Hypersonic Airplane NASA Configuration at Mach Numbers of 1.41, 2.01, and 6.86.

TM x-116, 1959.

2. Tunnell, Phillips J., and Latham, Eldon A . : The Static and Dynamic- Rotary Stability Derivatives of a Model of the X-15 Research Air- 1.53 to 3.50. NASA MEMO 12-23-5&, plane at Mach Numbers From

3. Franklin, Arthur E., and Lust, Robert M . : Investigation of the Aero-

dynamic Characteristics of a 0.067-Scale Model of the X-15 Airplane NASA (Configuration 3) at Mach Numbers of 2.29, 2 . 9 8 , and 4.65.

TM X-38, 1959.

Static Longitudinal, 4. Penland, Jim A., and Fetterman, David E., Jr.: Directional, and Lateral Stability and Control Data-ata Mach Number of 6 . 8 3 of the Final Configuration of the X-13 Research Airplane. NASA TM X-236, 1960.

5. Dryer, Murray, and North, Warren J . : Preliminary Analysis of the

Effect of Flow Separation Due to Rocket Jet Pluming on Aircraft Dynamic Stability During Atmospheric Exit. NASA MEMO 4-22-393,

1959 9

Effects of Jet Exhausts 6. Wolowicz, Chester H., and Rediess, Herman A.: on Flight-Determined Longitudinal and Lateral Dynamic Stability NACA Characteristics of the Douglas D-538-11 Research Airplane.

RM H57W9, 1937.

Supersonic Wave Interference Affecting Stability.

7. Love, Eugene S.: NACA TN 4358, 1958.

8 . U l m a n n , Edward F., and Ridyard, Herbert W.: Flow-Field Effects on Static Stability and Control at High Supersonic Mach Numbers.

NACA RM L55Llga, 1956.

9. Margolis, Kenneth, and Bobbitt, Percy J.: Theoretical Calculations of the Stability Derivatives at Supersonic Speeds for a High-speed Airplane Configuration. NACA RM L53G17, 1953.

The Effects of Body Vortices and the Wing Shock- 1 0 . Nielsen, Jack N . : Expansion Fieid oii the Pitch-V;, Characteristics of Supersonic Airplanes. NACA RM A57L23, 1958.

.. 0 . 0 .

0 . 0 . . 0 . C c e o * e o b .

O b .e. 0 b .r O b 0 - - _.

11. Nielsen, Jack N., and Kaattari, George E . : The Effects of Vortex and Shock-Expansion Fields on Pitch and Yaw Instabilities of Supersonic Airplanes. Preprint No. 743, Inst. Aero. Sci., 1957.

12. Ames Research Staff: Equations, Tables, and Charts for Compressible

Flow. NACA Rep. 1135, 1953. (Supersedes NACA TN 1428.)

Estimation of Static Longitudinal Stability of 13. Dugan, Duane W.: Aircraft Configurations at High Mach Numbers and at Angles of Attack Between Oo and f1800. NASA MEMO 1-17-5gA, 1959.

H Estimation of Directional Stability Deriva- 14. Kaattari, George E.: NASA MEMO l2-l-58A, tives at Moderate Angles and Supersonic Speeds.

15. Jorgensen, Leland H., and Perkins, Edward W . : Investigation of Some Wake Vortex Characteristics of an Inclined Ogive-Cylinder Body at (Supersedes NACA RM A55E31.)

Mach Number 2. NACA Rep. 1371, 1958.

Lift 16. Pitts, William C., Neilsen, Jack N., and Kaattari, George E . : and Center of Pressure of Wing-Body-Tail Combinations at Subsonic, Transonic, and Supersonic Speeds. NACA Rep. 1307, 1957.

17. Spahr, J. Richard: Longitudinal Aerodynamic Characteristics to Large Angles of Attack of a Cruciform Missile Configuration at a Mach Number of 2 . NACA RMA9H27, 19%.

1 8 . Van Dyke, Milton D.: A Study of Hypersonic Small-Disturbance Theory.

NACA Rep. 1194, 1954. (Supersedes NACA TN 3173.)

19. Ivey, H. Reese, and Cline, Charles W.: Effect of Heat-Capacity Lag on the Flow Through Oblique Shock Waves. M C A TN 2196, 1950.

The Dynamics and Thermodynamics of Compressible 20. Shapiro, Ascher H . : Fluid Flow. Vol. 11. The Ronald Press Co., 1954.

21. Raymond, Joseph L . : Thin Airfoils in Hypersonic Flow With Strong Rep. P-1189, the RAND Corp., Dec. 19, 1957.

Shocks.

22. Harmon, Sidney M . , and Jeffreys, Isabella: Theoretical Lift and Damping in Roll of Thin Wings With Arbitrary Sweep and Taper at Supersonic Speeds. Supersonic Leading and Trailing Edges. NACA TN 2114, 1950.

23. Purser, Paul E., and Campbell, John P . : Experimental Verification of a Simplified Vee-Tail Theory and Analysis of Available Data on Complete Models With Vee Tails. NACA Rep. 823, 1945.

Charts 24. Haefeli, Rudolph C . , Mirels, Harold, and Cummings, John L.: f o r Estimating Downwash Behind Rectangular, Trapezoidal, and T r i - angular Wings a t Supersonic Speeds. NACA TN 2141, 1950.

25. Allen, H. J u l i a n , and Perkins, Edward W . : A Study of E f f e c t s of Viscosity on Flow Over Slender Inclined Bodies of Revolution.

NACA Rep. 1048, 1951.

A Second-Order Shock- 26. Syvertson, Clarence A., and Dennis, David H.: Expansion Method Applicable t o Bodies of Revolution Near Zero L i f t .

NACA Rep. 1328, 1957. (Supersedes NACA TN 3527.)

27. Eggers, A . J., Jr., and Savin, Raymond C.: A Unified Two-Dimensional A-pproach t o t h e Calculation of Three-Dimensional Hypersonic Flows, With Application t o Bodies of Revolution. NACA Rep. 1249, 1955.

(Supersedes NACA TN 2811.)

E l l i p t i c Cones Alone and With Wings a t Super- 28. Jorgensen, Leland H.: sonic Speeds. NACA Rep. 1376, 1958. (Supersedes NACA TN 4045.)

29. Jorgensen, Leland H.: Inclined Bodies of Various Cross Sections a t Supersonic Speeds. NASA MEMO 10-3-58A, 1958.

Estimation of the Forces and Moments Acting on 30. Allen, H. J u l i a n : Inclined Bodies of Revolution of High Fineness Ratio. NACA RM ~9126, 1949.

31. Perkins, Edward W., and Jorgensen, Leland H.: Comparison of Experi- mental and Theoretical Normal-Force D i s t r i b u t i o n s (Including Reynolds Number E f f e c t s ) on an Ogive-Cylinder Body a t Mach Num- b e r 1.98. NACA TN 3716, 1956.

32. Delany, Noel K., and Sorensen, Norman E.: Low-Speed Drag of Cylin- ders of Various Shapes. NACA TN 3038, 1933.

33. Gowen, F o r r e s t E., and Perkins, Edward W.: Drag of Circular Cylin- d e r s f o r a Wide Range of Reynolds Numbers and Mach Numbers.

NACA TN 2360, 1933.

An Experimental Investi- 34. Carlson, Harry W., and Gapcynski, John P.: g a t i o n a t a Mach Number of 2.01 of t h e E f f e c t s of Body Cross-Section Shape on t h e Aerodynamic Characteristics of Bodies and Wing-Body Combinations. NACA RM L53E23, 1955.

35. Martin, John C., Margolis, Kenneth, and J e f f r e y s , I s a b e l l a : Calcu- L i f t and Pitching Moments Due t o Angle of Attack and l a t i o n of Steady Pitching Velocity a t Supersonic Speeds f o r Thin Sweptback Tapered Wings With Streamwise Tips and Supersonic Leading and

36. Harmon, Sidney M . : Stability Derivatives at Supersonic Speeds of

Thin Rectangular Wings With Diagonals Ahead of Tip Mach Lines.

(Supersedes NACA TN 1706.)

NACA Rep. 925, 1949.

37. Henderson, Arthur, Jr.: Pitching-Moment Derivatives C and

%

at Supersonic Speeds for a Slender-Delta-Wing and Slender-Body Combination and Approximate Solutions for Broad-Delta-Wing and Slender-Body Combinations. NACA TN 2553, 1951.

Experimental 38. Tobak, Murray, Reese, David E., and Beam, Benjamin H . : H NACA RM ~50~26, 1970.

Damping in Pitch of 45O Triangular Wings.

A 39. Martin, John C., Diederich, Margaret S . , and Bobbitt, Percy J . : Theoretical Investigation of the Aerodynamics of Wing-Tail Combi- nations Performing Time-Dependent Motions at Supersonic Speeds.

NACA TN 3072, 1954.

40. Goodwin, Frederick K., and Kaattari, George E . : Estimation of

Directional Stability Derivatives at Small Angles and Subsonic and Supersonic Speeds. NASA MEMO 12-2-38A, 1958.

Aerodynamic Loads on 41. Spahr, J. Richard, and Polhamus, Edward C . : NACA RM A57E21, Tails at High Angles of Attack and Sideslip.

42. Jones, Arthur L., and Alksne, Alberta: A Summary of Lateral- Stability Derivatives Calculated for Wing Plan Forms in Supersonic Flow. NACA Rep. 1052, 1951.

43. Jones, Arthur L., Spreiter, John R., and Alksne, Alberta: The

Rolling Moment Due to Sideslip of Triangular, Trapezoidal, and Related Plan Forms in Supersonic Flow. NACA TN 1700, 1948.

44. Spahr, J. Richard: Contribution of the Wing Panels to the Forces

and Moments of Supersonic Wing-Body Combinations at Combined Angles. NACA TN 4146, 1958.

45. Boatright, William B . : Experimental Investigation of Effects of

Wing Plan Form and Dihedral Angle on Sideslip Derivatives of Sweptback-Wing-Body Combinations at Supersonic Speeds. NACA RM ~ 5 8 ~ 0 8 , 1958.

46. McLellan, Charles H . : A Method for Increasing the Effectiveness

of Stabilizing Surfaces at High Supersonic Mach Numbers. NACA RM L54F21, 1954.

47. Moul, Martin T., and Paulson, John W.: Dynamic Lateral Behavior of High-Performance Aircraft. NACA RM ~ 5 8 ~ 1 6 , 1958.

48. P i t t s , W i l l i a m C.: Force, Moment, and Pressure-Distribution Charac- t e r i s t i c s of Rectangular Wings a t High Angles of Attack and Super- sonic Speeds. NACA RM A55K09, 1956.

49. Margolis, Kenneth, and Bobbitt, Percy J.: Theoretical Calculations of the Pressures, Forces, and Moments at Supersonic Speeds Due t o Various Lateral Motions Acting on Thin Isolated Vertical Tails.

NACA Rep. 1268, 1956.

50. Fisher, Lewis R . , and Fletcher, Herman S.: Effect of Lag of Side- wash on the Vertical-Tail Contribution t o Oscillatory Damping i n Y a w of Airplane Models.

NACA TN 3356, 1955.

51. Harmon, Sidney M., and Martin, John C.: Theoretical Calculations of the Lateral Force and Yawing Moment Due t o Rolling a t Super- sonic Speeds f o r Sweptback Tapered Wings With Streamwise Tips.

Supersonic Leading Edges. NACA TN 2156, 1950.

52. Adams, Gaynor J., and Dugan, Duane W.: Theoretical Damping i n Roll and Rolling Moment Due t o Differential W i n g Incidence for Slender NACA Rep. 1088, 1952.

Cruciform Wings and Wing-Body Combinations.

53. Tucker, Warren A., and Piland, Robert 0.: Estimation of the Damping NACA i n Roll of Supersonic-Leading-Edge Wing-Body Combinations.

TN 2151, 1950.

Rolling Effectiveness of 9. Strass, H. K u r t , and Marley, Edward T.: All-Movable Wings a t Small Angles of Incidence a t Mach Numbers From 0.6 t o 1.6. NACA RM L5IH03, 1951.

55. Purser, Paul E.: An Approximation t o the Effect of Geometric Dihedral on the Rolling Moment Due t o Sideslip f o r Wings a t Tran- sonic and Supersonic Speeds. NACA RM L52BOl, 1952.

Linearized Lifting-Surface and Lifting-Line 56. Bobbitt, Percy J.: Evaluations of Sidewash Behind Rolling Triangular Wings a t Super- sonic Speeds. NACA TN 3609, 1956.

- - o m e . . m . e .

.. e.. c eo 0 .

TABLE I AIRPLANE GEOMETRIC CHARACTERISTICS Wing (extended t o body center l i n e ) : Area, s q f t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Aspect r a t i o . . . . . . . . . . . . . . . . . . . . . . . . . 2.50

. . . .

Taper r a t i o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.20

Mean aerodynamic chord, i n . . . . . . . . . . . .

. . . . . . . . . . . 123.23

Sweep of leading edge, deg . . . . . . . . . . . . . . . . . . . . . . 36.75

Span, f t . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.36

. . . .

Root chord, i n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.89

h Tip chord, i n . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . 35.78

I Dihedral angle, deg . . . . . . . . . . . . . . . 0 . . . . . . .

. . . . I -

C

Incidence angle, deg . . . . . . . . . . . . . . 0

. . . . . . . . . . .

C

Twist, deg . . . . . . . . . . . . . . . . . . . 0

. . . . . . . . . . .

Airfoil section . . . . . . . . . . . . . . . . . . . . . NACA 66005 (modified)

Fuselage s t a t i o n f o r 20-percent mean aerodynamic

. . . . . . . . . . . 339.19

chord, i n . . . . . . . . . . . . . . . . . . .

Wing s t a t i o n f o r 20-percent mean aerodynamic

. . . . . . . . . . . 52.17

chord, i n . . . . . . . . . . . . . . . . . . .

Flap area, s q f t . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.48

Flap travel, deg . . . . . . . . . . . . . . . . 40

. . . . . . . . . . .

Wing (exposed) :

A r e a , s q f t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Aspect r a t i o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.27

Taper r a t i o

Root chord, i n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.95

Tip chord, i n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.78

Horizontal t a i l (exposed) :

Area, s q f t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.76

Aspect r a t i o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.81

Taper r a t i o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.21

Mean aerodynamic chord, i n . . . . . . . . . . . . . . . . . . . . . . . 60.07

Sweep of quarter-chord l i n e , deg . . . . . . . . . . . . . . . . . . . 45

span, overall, ft . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.64

Root chord, i n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.27

Tip chord, in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.28

Dihedral angle, deg . . . . . . . . . . . . . . . . . . . . . . . . . . -15

A i r f o i l section . . . . . . . . . . . . . . . . . . . . . NACA 66005 (modified)

Fuselage s t a t i o n f o r 50-percent h o r i z o n t a l - t a i l mean

aerodynamic chord, i n . . . . . . . . . . . . . . . . . . . . . . . . 537.52

Span station f o r 50-percent h o r i z o n t a l - t a i l mean

aerodynamic chord, from fuselage, i n . . . . . . . . . . . . . . . . . 26.96

T a i l arm, 20-percent wing mean aerodynamic chord t o 50-percent h o r i z o n t a l - t a i l mean aerodynamic chord, i n . . . . . . . . 198.33

Incidence range, normal t o plane of symmetry, deg -

Pitch control . . . . . . . . . . . . . . . . . . . . . . . . .35 down, 15 U P

Roll control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.5

I T

0 0 *.* 0 0 ...............

0 . 4 . 0 . . e . .

:, 0 . 0 0 . 0 .

0 . 0 .. 0 0 55

..........

TABU I . . Concluded

AIRPLANE GEOMETRIC CHARACTERISTICS Vertical t a i l (upper. exposed) :

Area. s q f t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.8

. 1.03

Aspect r a t i o . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Taper r a t i o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.74

Mean aerodynamic chord. i n . . . . . . . . . . . . . . . . . . . . . . . 107.5

Sweep of leading edge. deg . . . . . . . . . . . . . . . . . . . . .

Span (exposed) . i n . . . . . . . . . . . . . . . . . . . . . . . . . .

. 122.5

Root chord. i n . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Tipchord. i n . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* 90.75

A i r f o i l section . . . . . . . . . . . . . . . . . . . . . . . . . . . loo wedge

I Fuselage s t a t i o n f o r 50-percent v e r t i c a l - t a i l mean X

. 520.25

aerodynamic chord. i n . . . . . . . . . . . . . . . . . . . . . . .

Span s t a t i o n f o r 50-'percent v e r t i c a l - t a i l mean

aerodynamic chord. from fuselage. i n . . . . . . . . . . . . . . . . . 26.15

T a i l arm. 20-percent wing mean aerodynamic chord t o

50-percent vertical-tail mean aerodynamic chord. i n . . . . . . . . . 181.06

Movable area. outboard panel. sq f t . . . . . . . . . . . . . . . . . . 26.5

. k7.5

Angular t r a v e l of movable area. deg . . . . . . . . . . . . . . . . .

Vertical t a i l (lower. exposed) :

Area. s q f t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.2

. . . . . . . . . 0.785

Aspect r a t i o . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . 0.79

Taper r a t i o . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . 109.2

Mean aerodynamic chord. i n . . . . . . . . . . . . . .

. . . . . . . . . 30

Sweep of leading edge. deg . . . . . . . . . . . . .

. . . . . . . . . 44

Span. exposed. i n . . . . . . . . . . . . . . . . . .

Root chord. i n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4

. . . . . . . . . 96

Tip chord. i n . . . . . . . . . . . . . . . . . . . .

. . . . . . . . loo wedge

A i r f o i l section . . . . . . . . . . . . . . . . . . .

Fuselage s t a t i o n f o r 50-percent v e r t i c a l - t a i l mean

aerodynamic chord. i n . . . . . . . . . . . . . . . . . . . . . . . . 519.4

Span s t a t i o n f o r 50-percent v e r t i c a l - t a i l mean

. . . . . . . . . 21.15

aerodynamic chord from fuselage. i n . . . . . . . .

T a i l arm. 20-percent wing mean aerodynamic chord t o

50-percent v e r t i c a l - t a i l mean aerodynamic chord. in . . . . . . . . 180.21

Movable (jettisonable) area. s q f t . . . . . . . . . . . . . . . . . . 19.9

. . . . . . . . . k7.5

Angular t r a v e l of movable area. deg . . . . . . . . .

Fuselage :

Length. high-speed nose. f t . . . . . . . . . . . . . . . . . . . . . . 49.17

Length. low-speed nose. less boom. f t . . . . . . . . . . . . . . . . . 50.16

Width. including side fairings. station 346 t o s t a t i o n 411. i n . . . . . 88.0

Height. s t a t i o n 186 t o station 530. in . . . . . . . . . . . . . . . . . 56.0

Maximum cross-sectional area. s q f t . . . . . . . . . . . . . . . . . . 21.4

Fineness ratio. average . . . . . . . . . . . . . . . . . . . . . . . . 9.4

k e e apex angle. deg . . . . . . . . . . . . . . . . . . . . . . . . . 31.0

Speed brakes (upper and lower): Location hinge line. fuselage station. i n 5% . . . . . . . . . . . . . . .

Side area. each. sq ft . . . . . . . . . . . . . . . . . . . . . . . . 4.88

Angular travel. fuselage center line. deg . . . . . . . . . . . . . . . 4 1

. .

N V N CI u u d 0 0 i n cu N N '" d 0 ;t cu -t 0 - M a m a c, 0

c, s a

5 a !

's f f f (u f T i (u N 0 ..M 0 0 c, 0 c, c, c, d $ 0 0 0 4 ?I

Y

2 3 3 : v) d K\ f tl 0 f c, c, v) d M ;f 0 d x c, d r4 d d R

. 50016 I

I U 22.36 I l l W

Figure 1 . - Three-view drawing of t h e X-15 airplane. All dimensions in

f e e t .

L-60 -281 Figure 2.- Shadowgraphs of a f r e e - f l i g h t model of t h e X-15 a i r p l a n e at a Mach number of 6 taken i n t h e Ames supersonic f r e e - f l i g h t wind tunnel.

1 . 0

I

’. 9

b 8 .7

i-

.6

L

I .5 8 12 M Figure 3.- Estimated losses in dynamic pressure and lift effectiveness for surfaces in the flow downstream from the fuselage bow-shock wave.

..

"1

L a - \

J '

1 . 0 -9 .8 \D - 7 I X .6 .5 . 4 .2 . 1 Figure 5.- Calculated normal-force curves f o r two-dimensional f l a t p l a t e .

a m h b

m N If N N u3 d bo a U N r( Q

9T

(u (u h h UJ 0 d rl .d cd -4J rl 2 d cd \D -4J d f c r;' N X .d k a ,

x"

n P W -3 (u UJ d

d

rl M d cd cue d -4J d rl cd -P c co N .d k n cd f W CLa ’ per deg .08 .06 C L . , ’ ,04 per deg .02 n M Figure 8.- Buildup of calculated lift curves for the X-15 airplane at iT = 0’.

angles of attack of Oo, 8 O , 16O, and 2 4 O .

F a I n I I '?

I 'u I rl I I rl W rl t o a d N rl a J f I . 1 1 1 1 1.0 . a 0 . . . .om a .

In . .

\D 4J c f u II I o \ I .

-I

I I rl N k X

T

I

0 . 0.. . 0.. . 0 . 0 . . . . 0.. 0 .

.08 .06 ‘b’ .04 per deg .02 (a) Horizontal t a i l on. iT = 0’.

-08 1 1 1 Calculated i 06 c.ol, per deg b02 C 2 4 6 8 10 12 M ( b ) Horczontal t a i l o f f .

Figure 10.- Comparison of c a l c u l a t e d and experimental l i f t - c u r v e slopes of t h e X-15 a i r p l a n e with t h e h o r i z o n t a l t a i l on and o f f .

. OT

-.l - . z C myr - . 3 -.4 -.5 (a) Wing.

C nlr (b) Horizontal tail.

.4 . 3 C Y3 .2 .1 0 4 8 12 16 20 24 28 a. deg ( c ) Fuseiage.

Figure 1 .- Calcu ated pitching-moment characteristics of the wing and

the horizontal tail. iT = 0'.

In I k k rl d aJ c, rl c d c,

E

N k n P W II H .d .F: rl d a J c, rl a J c, r : N d k

-

aJ W .e e e e e me. e * e e * e e e.. e.

.. .. . e o . e . e . .

_ _ e e me - - e em - - e :e * : * -

e . e . e . .

e. ..e e e e .01 %.

-.01 per deg - .02 -.03 .01 $ 7 -.01 p e r deg -.02 -.03 .01 -.M -.03 . ai %.

- .01 per deg -.02 - . o j Figure 14.- Buildup of calculated pitching-moment characteristics for the X - 1 5 airplane at angles of attack of Oo, 8 O , 16O, and 24'.

.

iT = 00.

.01 %, per deg -.C1

-. 02

-,03 (a) Horizontal tail on. iT = 0'.

. 0 1 ( ! % ? -.01 per deg -.02 Calculated --- Wind tunnel ( r e f s . 1, 3)

I I I I i l

-.03 0 2 4 6 8 10 12 M (b) Horizontal tail off.

Figure 15.- Comparison of calculated and experimental values of

cma

for the X - 1 5 airplane with the horizontal tail on and off.

.2 -.4 -.6 (a) Horizontal t a i l on. i T = 0 ' .

. 2 C -;2 %L

-. 4

--- Wind t u n n e l (refs. 1, 3 ) -.6 0 2 4 6 8 10 12 M (b) Horizontal t a i l o f f .

Figure 16.- Comparison of t h e c a l c u l a t e d and experimental s t a t i c margins f o r t h e X - 1 5 a i r p l a n e with the h o r i z o n t a l t a i l on and off.

.012 %iTj per deg .004

-. 020

- .016

- .012

- .008

- .004

Figure 17.- Calculated effectiveness of the horizontal stabilizer of the X-15 airplane.

............... . . 0.. 0 .

(a) M = 2.29. (b) M = 2.98.

a, de8 10 B

-. 4 -.2 0 . 2 .4

-.4 -.2 0 .2 .4 6% A% ( c ) M = 4.65. ( d ) M = 6.86.

Figure 18.- Comparison of t h e c a l c u l a t e d and experimental s t a b i l i z e r effectiveness of t h e X-15 a i r p l a n e f o r incidence s e t t i n g s of 1 5 ' (leading edge up) and -20° (leading edge down) a t various Mach numbers.

1 l T - 5 .4 .3 .2 . 1 -.l -.2 Figure 19.- Comparison of t h e calculated and experimental s t a b i l i z e r c h a r a c t e r i s t i c s f o r t h e X-15 airplane f o r s e v e r a l Mach numbers a t zero angle of a t t a c k .

* * a a . a * a a. a. a a a * a * a.

a.

a * a * a * * * * a * a a . a .

. a a * . . a * a a a . . a a * * * - - - - - . .

a . a . . a . a a * * . a a * a * a. * a * a a a .a a. *-

i - ._

80 - i.'

In t i I X e,

L 5

I G-4 rn o o m n f 9 ? v1 f 9 9 f .

rl rl I c ! ?

a 0 0 0 0 .

C. 0 0 0 0 0 0 0 0 0 .

0 0 . 0 . 0 .

0 0 0 0 0 . C 0 .

0 0 0 0 0 . 0 .

8 1 0 0 O b 0.0 0 0 0 0 0 0 0.0 0 .

..

I L n I f I m f 0 f N N 9 “9 Y ‘u I rl vi c 9 l - -a -~ Horizontal tail off -6 C %' per r a d i a n -4 -2 -a a cm 9 q per r a d i a n -4 -2 -a -6 Cmq' -4 per r a d i a n -2 -8 -6 Crn 9 -4 per r a d i a n -2 2 4 5 6 7 0 9 10 11 12 M Figure 21.- Buildup of the calculated damping characteristics due to pitching rate for the X-15 airplane at angles of attack of Oo, 8O, 16O, and 2 4 ' . IT = 0'.

c”dr, 0 per radian -2 u l d I X W Y , p e r radian -2 c %, Q per r a d i a n -2

%’

per r a d i a n Figure 22.- Buildup of the calculated damping characteristics due to vertical acceleration for the x-15 airplane.

iT = 0’.

0 . .., e * e . . 0 . .e . . . e14 .e

-8 -6 cmq + %I&, -4 per radian -2 iT = 0'.

(a) Horizontal t a i l on.

-6 -4

cmq + c t l l ( - y

per radian -2 2 4 6 8 10 12 M (b) Horizontal t a i l o f f .

Figure 23.- Calculated damping-in-pitch d e r i v a t i v e s f o r t h e X-15 a i r p l a n e .

L2T a z t d I X ..

.. ..e . ... . 0 . 0 . . 0 . 0.. 0 .

DE Figure 25.- Buildup of the calculated side-force characteristics of the iT = Oo. X-15 airplane at angles of attack of Oo, 80, 160, and 2 4 O .

. e . . . e.. . . e . .. ..

.. .. .. . 0 . ..e ..

,012 .

. 00s per deg .OO?

-. 004 ,012 . OOE ha. Per deg .a34 - ,004 .01P ..

.008 % , , per deg . o a - .004 .012 .om k8, Per deg .004 - .004 5 6 7 a 9 10 11 12 M Figure 26.- Buildup of the calculated directional-stability character- istics of the X-15 airplane at angles of attack of Oo, 8O, 16O, and 24'. iT = Oo.

** 0.. . ..e . .e 0 . . . . 0.. 0 .

e . 0 . . e 0 . . 0 . . 0 . .

0 , e . . ... e 0 . . . a . . .

0 . e . . * . .

e. 0 . 0 . c . 0 . 0 .

- ,002 deg ,002 .OO'l - .004 :: I-

+

- .002 deg ,002 .OO,l Figure 27.- Buildup of t h e calculated dihedral e f f e c t f o r t h e X-15 air- plane a t angles of a t t a c k of Oo, 8 O , 16O, and 2 4 O . iT = 0'.

.

1.2 .8 .4

I I I

7-F

2 4 6 8 10 12 M Figure 28.- Effect of wing-body shock f i e l d s on l i f t effectiveness of t h e upper and lower v e r t i c a l t a i l s .

0 . e.* . ... . 0 . *. . . . ..e 0 .

0 . 0 . 0 . . 0 . . 0 . . 0 . .

0 . 0 . . 0.c c 0 . 0 . ....

0 . 0 . 0 . . .

0 . 0.. . . 0 .

.

- .032 -.0‘24 dee - .016 - .008 .012

,008 e

,004 C dee “B ’ - .004 - .008 - .002 C L p . per deg .002 1 1 12 Figure 29.- Summary of the calculated s i d e s l i p d e r i v a t i v e s for t h e X-13 airplane with v e r t i c a l t a i l on and off at angles of a t t a c k of Oo, Bo, 16O, and 24’.

iT = 0 ’ .

d (a) M = 2.0. (b) M = 2.29.

v3 rl I X n ( c ) M = 3.0. (d) M = 4.68.

Vertlcal Vertical tail on t a i l o f f meom -__ Reference 1

--- Reference 5

- - __ Reference 4

I!

( e ) M = 6.86.

Figure 30.- Comparison of t h e calculated and experimental side-force d e r i v a t i v e s f o r t h e X - 1 5 airplane with t h e v e r t i c a l t a i l on and o f f .

iT = Oo.

c (b) M = 2.29.

per d w (c) M = 3.0. (d) M = 4.68.

(e) M = 6.86.

Figure 3 1 . - Comparison of the calculated and experimental directional- stability derivatives for the X-15 airplane with the vertical tail iT = 0'.

on and o f f .

13T , 0, deg ( a ) M = 2.0. (b) M = 2.29.

u 3 r i I X a. deg a ,002 0 -.002 ,002 0 -.002 = 3.0.

Vertical Vertical tail on tail Off m 0 r Y

-- -__

Reference 1

- -- RefPrpnce i

_ _ _ _ _ _ _ _ _ Reference 2

..------

--__ Reference 4 .002 0 -.002 C L g j per deg ( e ) M = 6.86.

Figure 32.- Comparison of the calculated and experimental dihedral e f f e c t f o r t h e X-15 a i r p l a n e with the v e r t i c a l t a i l on and o f f . iT = 0'.

- Calculated, N l y i n wing-badg cnnpression f i e l d _ - Calculated, w ~ r t h l l y imremed i n wing-body compnsssion fleld - - -- md-tunnel data

M - 4.68

(a) Side-force derivative.

(b) Yawing-moment derivative.

0 . 4 . 8 zE 10

p

.wz 0 -. ' ,002 0 - ( c ) Rolling-moment d e r i v a t i v e .

Figure 3 3 . - Comparison of experimental s i d e s l i p d e r i v a t i v e s at Mach num- bers of 4.65 and 6.86 with those c a l c u l a t e d f o r t h e lower v e r t i c a l t a i l f u l l y and p a r t i a l l y immersed i n wing-body compression f i e l d .

. 5 .8 - CYB.

radian -.8 -1.6 . 8 -.8 CYr - CYB.

per radian -1.6 -2.4 .

1.6 .a CYr - CYi.

Fer radian -.8 -1.6 -2.4 -3.2 5 9 10 11 12 4 6 7 8 M y.

Figure 34.- Buildup of t h e calculated side-force d e r i v a t i v e due t o yawing rate f o r t h e X-15 a i r p l a n e . i T = Oo.

.4 - . 4 C " - c y r per radian -.a -1.2 c - c "r %' per radian .

M Figure 35.- Buildup of the calculated damping-in-yaw derivative for the iT = 0'.

x-15 airplane at angles of attack of 00, 80, 160, and 2 4 O .

.1 c t r -.l per radian - . 2 - . 3 .1 u l f C l r - C l i > r: X -.1 per radian - . 2 _ i ./ .1 C l r - C l i , -.l per radian - . 2

I

-.3 3 4 5 6 7 8 9 10 11 12 M Figure 36.- Buildup of the calculated rolling-moment derivative due to yawing rate for the X-15 airplane. iT = Oo.

. . . . . . .

0 . 0 . . ... .

1 0 0 * * .. , . * * . . . * * * . ..

-.8 CYr - C Y i I per radian -1.6 -2. J+ - . 8 CYr - CYb' -1.6 per radian -9.4 -3.2 -.4 - . D Cnr - cni I per radlbii -1.2 -1.6 -2.0 .1 C l , - C $ .

per radlan -.1 -.l - C L b , radian -.2 5 h 7 8 9 10 11 12 Figure 37.- Summary of the c a l c u l a t e d d e r i v a t i v e s due t o yawing rate f o r t h e X-15 a i r p l a n e w i t h v e r t i c a l t a i l on and off at angles of a t t a c k o f O o , 8 O , 1 6 O , and 24'.

iT = 00.

.

n (b) M = 2.5.

( c ) M = 3.0.

C, - C ., per radian r "B (d) M = 3.5.

Figure 38.- Comparison of t h e calculated and experimental damping-in- yaw d e r i v a t i v e for t h e X-15 airplane with v e r t i c a l t a i l on and off.

* a * * a * * a * * .a . * * a a.

* e a * a . * a . * a * * * a * e 0 . . . * * * * * * . a a , deg 2.0.

(b) M = 2 . 5 .

(c) M = 3.0.

I I I I I I I r i .6 .4 . z 0- -.2 Cir - C l b , per radian ( d ) M = 3.5.

Figure 39.- Comparison of c a l c u l a t e d and experimental rolling-moment derivative due t o yawing r a t e for t h e X - 1 5 a i r p l a n e with v e r t i c a l t a i l on and o f f .

1 4 T

.24 .16 .08 radar PE -.08 .24 \D 4.

.16 r: X per radiar .08 - .08 .24 .15 radlar .08 -.08 yay "p.,

Figure 40.- Buildup of t h e calculated side-force derivative due t o r o l l

r a t e for t h e X-15 airplane a t angles of a t t a c k of 00, 8 O , 160, and 2 4 O . iT = 00.

.'

.04 radian %' -.04 - .08 V e r t i c a l t a i l on Vertical t a l l off - .12 .04

F

r c- 0 (3\ radlar c"P' - .04 - . Ot - .13 . 0 4 r a d i x - .04 -.08 - . 1 2 .04 -.04 radlar - .08 -.12 M Figure 41. - Buildup of t h e c a l c u l a t e d yawing-moment d e r i v a t i v e due t o r o l l rate of t h e X-15 a i r p l a n e .

iT = 00.

I per -.08 C L , per radian P -.16 -.24 -.32 .

-.08 radian -.16 -.24 -.32 M Figure 42.- Buildup of calculated damping-in-roll d e r i v a t i v e of a i r p l a n e a t angles of a t t a c k of 00, 80, 160, and 24'.

iT = 0'.

.24 .16 oer radiar .08 .24 -- 8 .i6 %' per radian .08 -.04 radian -.oe - . 1 2 - .04 radiar -.08 -.12 - .08 radian - . i b - .24 - . .12 M Figure 4 3 . - Summary of t h e c a l c u l a t e d d e r i v a t i v e s due t o r o l l rate f o r t h e X-15 a i r p l a n e w i t h v e r t i c a l t a i l on and off a t angles of a t t a c k of 00, 8 O , 16O, and 24'. iT = 0 ' .

am m m m ma ma m m m m m m am m m m m m m m m m m a a m m a L m m m m m a m m m m m m m a a. deg a (b) M = 2.5.

a, deg ( c ) M = 3.0.

( d ) M = 3.5.

Figure 44.- Comparison of t h e calculated and experimental yawing-moment d e r i v a t i v e due t o r o l l r a t e f o r the X-15 a i r p l a n e with v e r t i c a l t a i l on and off a t several Mach numbers.

ma aam a m m m m a ma a m m ma a m a m m a a m m m m o m . m a m a m e a a m . a * a a m a m a (a) M = 2.0.

(b) M = 2 '5.

I C n, deg (c) M = 3.0.

0 -.1 - . 2 -.3 -.4 (d) M = 3.5.

Figure 45.- Comparison of the calculated and experimental damping-in- roll derivative for the X - 1 5 airplane w i t h vertical t a i l on and off at several Mach numbers.

n 0 0 . 0 . 0 e.. 0 0.0 0 0 0 . e .

0 0 0 0 0 0 0 0 0 0 0 0 log 0 , . 0 0 e .

0. 0 0 0 0.. 0 0 ' y 6 ~ ~ per deg 'I' X .

.001 "6R' per deg

- 001

0 2 4 6 8 10 12 M

Figure 46. - Calculated side-force, yawing-moment, and rolling-moment

derivatives for upper and lower directional-control surfaces of the X-15 airplane.

.012 .om $6 ’ per 2eg .004 t %RJ per deg .001 C 2%’ p e r deg

- .001

2 4 6 8 10 12 M Figure 47.- Calculated d i r e c t i o n a l - c o n t r o l d e r i v a t i v e s f o r t h e X-15 a i r p l a n e .

15T

1 I 4- X M d U .

/ ,002

’. 002

Cn,tT9 .001 per deg 4 6 8 10 12 M Figure 49.- Calculated l a t e r a l - c o n t r o l d e r i v a t i v e s f o r t h e X-15 a i r p l a n e .

....... ................

. . . . . . . . . . . . . . . .

.... .........

:.. : . *

:.*ri : : : Q) k k P a k fi d d V \D f r: X a k fi

-

d .

V ' : M a" k n E-

-

d V N d d M 2l a d NASA - Langley Field, Va.

H-146