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NASA CP 2085 p t .2 ' NASA Conference Publication 208% c. 1 Part I1 Science and Technology of Low Speed and Motorless Flight Proceedings of a symposium held at NASA Langley Research Center Hampton, Virginia March 29-30, 1979 0099882 NASA Conference Publication 2085 Part 11 Science and Technology of Low Speed and Motorless Flight Perry W . Hanson, Compiler NASA Lungley Research Center Proceedings of a symposium cosponsored by NASA Langley Research Center and the Soaring Society of America, and held at NASA Langley Research Center, Hampton, Virginia March 29-30, 1979 National Aeronautics and Space Administration Scientific and Technical Information Office Page intentionally left blank PREFACE T h i s NASA c o n f e r e n c e p u b l i c a t i o n c o n t a i n s t h e p r o c e e d i n g s o f t h e T h i r d I n t e r n a t i o n a l Symposium o n t h e S c i e n c e and Technology of Low Speed and Motorless F l i g h t h e l d a t t h e NASA Langley Research C e n t e r , Hampton, V i r g i n i a , March 29-30, 1979. The symposium was cosponsored by t h e Langley Research Center (LaRC) and t h e S o a r i n g S o c i e t y of America (SSA). Oran Nicks, Deputy D i r e c t o r of t h e Langley Research C e n t e r , and James Nash-Webber, M a s s a c h u s e t t s I n s t i t u t e o f Technology and p a s t chairman of t h e SSA T e c h n i c a l Board, were g e n e r a l cochairmen. P e r r y Hanson, NASA LaRC, was t h e symposium o r g a n i z e r and t e c h n i c a l program chairman. H e w i t t P h i l l i p s , NASA LaRC ( R e t i r e d ) ; J o s e p h Gera, N S LaRC: and Robert Lamson, Chairman of t h e SSA T e c h n i c a l Board, s e r v e d a s AA chairmen f o r t h e t e c h n i c a l s e s s i o n s . The purpose of t h e Symposium was t o p r o v i d e a forum f o r t h e i n t e r c h a n g e of i n f o r m a t i o n o n r e c e n t p r o g r e s s i n t h e s c i e n c e and t e c h n o l o g i e s a s s o c i a t e d w i t h l o w speed and m o t o r l e s s f l i g h t . Twenty-eight p a p e r s were p r e s e n t e d i n t h e a r e a s of low speed aerodynamics, new m a t e r i a l s a p p l i c a t i o n s and s t r u c t u r a l c o n c e p t s , advanced f l i g h t i n s t r u m e n t a t i o n , s a i l p l a n e o p t i m a l f l i g h t t e c h n i q u e s , and s e l f l a u n c h i n g and u l t r a l i g h t g l i d e r technology'. T h i s NASA c o n f e r e n c e p u b l i c a t i o n c o n t a i n s t h e s e p r e s e n t a t i o n s and a p a p e r , which was n o t p r e s e n t e d , on proposed d e f i n i t i o n s f o r v a r i o u s c a t e g o r i e s of s a i l p l a n e s and g l i d e r s . The use of t r a d e names or m a n u f a c t u r e r ' s names i n t h i s p u b l i c a t i o n does n o t c o n s t i t u t e a n o f f i c i a l endorsement of such p r o d u c t s o r m a n u f a c t u r e r s , e i t h e r e x p r e s s e d or i m p l i e d , by NASA. The included p a p e r s a r e l a r g e l y a s s u b m i t t e d . The p h y s i c a l q u a n t i t i e s , whether i n t h e I n t e r n a t i o n a l System of U n i t s (SI) o r U.S. Customary U n i t s , a r e r e t a i n e d a s s u b m i t t e d by t h e a u t h o r s . Page intentionally left blank CONTENTS PREFACE ................................. PART I* LOW-SPEED AERODYNAMICS iii 1. LOW-SPEED SINGLE-ELEMENT AIRFOIL SYNTHESIS J o h n H. McMasters and M i c h a e l L. H e n d e r s o n .............. SAILPLANE 1 2. A EXPLORATORY INVESTIGATION OF THE EFFECT OF A PLASTIC COATING N O THE PROFILE DRAG OF A PRACTICAL-METAL-CONSTRUCTION N AIRFOIL D a n M. S o m e r s .............................. ............... ................ 33 65 3 . OPTIMUM TAIL PLANE DESIGN FOR SAILPLANES Kay Mayland 4.THEEFFECTOFDISTURBANCESONAWING. Richard Eppler 81 5. GENERATION AND BREAKDOWN OF AERODYNAMIC LIFT: PHYSICAL MECHANISM Wolfgang L i e b e ......................... .............. 93 6. INTRODUCTION TO THE ARCOPTER ARC WING AND THE BERTELSEN EFFECT FOR POSITIVE PITCH STABILITY AND CONTROL W i l l i a m D. B e r t e l s e n 103 7. SOME NEW AIRFOILS Richard Eppler .......................... .................. 1 31 8. A COMPARISON OF THE AERODYNAMIC CHARACTERISTICS OF EIGHT SAILWING AIRFOIL SECTIONS Mark D. Maughmer 9. LENGTH AND BURSTING OF SEPARATION BUBBLES: INTERPRETATION J o h n M. R u s s e l l 155 . . . . . . . . . . . . . .A .PHYSICAL. . . . . . . . . ... 17 7 10. WING SHAPE OPTIMIZATION FOR MAXIMUM CROSS-COUNTRY SPEED, WITHMATHEMATICALPROGRAMMING Gunter Helwig ................... 203 * P a p e r s 1 t o 14 are p r e s e n t e d u n d e r a separate cover. ADVANCED INSTRUMENTATION 1 1 . FURTHER DEVELOPMENTS IN SIMPLE TOTAL ENERGY SENSORS Oran We Nicks 12.HOWACCURATEISNETTO Stephen du Pont ......... 219 ......................... 247 1 3 . THE APPLICATION OF MICROPROCESSOR TECHNOLOGY TO IN-FLIGHT COMPUTATIONS.... P a t r i c i a L. S a w y e r and Dan M. Smers ........................ MOTORSOARERS 267 1 4 . DESIGN OF PROPELLERS FOR MOTORSOARERS E. E u g e n e L a r r a b e e ................ 285 PART I1 OPTIMAL FLIGHT TECHNIQUES 15. MINIMUM ALTITUDE-LOSS SOARING Ig A SPECIFIED VERTICAL WIND DISTRIBUTION B i o n L. P i e r s o n a n d Imao C h e n 1 6 . A STUDY OF COURSE DEVIATIONS DURING CROSS-COUNTRY SOARING S t e v e n M. S l i w a a n d D a v i d J. S l i w a 1 7 . ON GLOBAL OPTIMAL SAILPLANE FLIGHT STRATEGY G. S a n d e r a n d F . X. L i t t ............................ 305 ...... 319 ............. 355 1 8 . BALANCE TRAINING OF THE EQUILIBRIUM ORGAN AND I T S EFFECT ON FLIGHT STRATEGY K. -D. E i k e m e i e r , H e -D, M e l z i g , N. R e i c k e , and W. Schmidt 1 9 . A MONTE CARLO APPROACH TO COMPETITION STRATEGY M i c h a e l P. T e t e r , ......................... ............ 377 389 STRUCTURES AND MATERIALS 20, A GENERAL METHOD FOR THE LAYOUT OF AILERONS AND ELEVATORS OFGLIDERSANDMOTORPLANES.. Manfred H i l l e r 21 ................... 399 . EXPERIMENTAL INVESTIGATION INTO THE FEASIBILITY OF AN "EXTRUDED" WING . . . . . . . . . . . . . . . . . . . . . . . . . . P i e r o Morelli and G i u l i o Romeo 41 9 22. TREATMENT OF THE CONTROL MECHANISMS OF LIGHT AIRPLANES IN THE FLUTTER CLEARANCE PROCESS E l m a r J. B r e i t b a c h ..................... APPLICATION AND 437 23. ADVANCED COMPOSITES I N SAILPLANE STRUCTURES: MECHANICAL PROPERTIES D i e t e r Muser . . . . . . . . . . . . . . . . . . . . . . . 467 ULTRALIGHT SAILPLANES AND HANG GLIDERS 24. THE ULTRALIGHT SAILPLANE J. H. McMasters ....................... 485 25. ANALYTICAL AND SCALE-MODEL RESEARCH AIMED AT IMPROVED HANG GLIDER DESIGN I l a n K r o o and L i - S h i n g C h a n g ........................... ...................... ................ 505 26. IMPROVEMENT OF HANG GLIDER PERFORMANCE BY USE OF ULTRALIGHT ELASTIC WING Jer zy Wolf 523 27. EXPERIMENTAL STUDY OF THE FLIGHT ENVELOPE AND RESEARCH OF SAFETY REQUIREMENTS FOR HANGGLIDERS C l a u d i u s La B u r t h e 537 557 28. WIND TUNNEL TESTS OF FOUR FLEXIBLE WING ULTRALIGHT GLIDERS R o b e r t A. O r m i s t o n ...... CONTEMPORARY SOARING NOMENCLATURE* S. 0. J e n k o *. ................. 591 ATTENDEES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 **Paper n o t presented a t s y m p o s i u m . MINIMUM ALTITUDE-LOSS SOARING I N A SPECIFIED VERTICAL WIND DISTRIBUTION Bion L. P i e r s o n and Imao Chen Iowa S t a t e U n i v e r s i t y SM AY U MR Minimum a l t i t u d e - l o s s f l i g h t of a s a i l p l a n e through a g i v e n v e r t i c a l wind d i s t r i b u t i o n i s d i s c u s s e d . The problem i s posed a s a n o p t i m a l c o n t r o l problem, and s e v e r a l numerical s o l u t i o n s a r e o b t a i n e d f o r a s i n u s o i d a l wind d i s t r i b u t i o n . INTRODUCTION The problem of d e t e r m i n i n g t h e o p t i m a l s a i l p l a n e t r a j e c t o r y through a pres c r i b e d v e r t i c a l wind d i s t r i b u t i o n f o r minimum a l t i t u d e l o s s i s f o r m u l a t e d and s o l v e d a s a n o p t i m a l c o n t r o l problem. The f l i g h t i s assumed t o t a k e p l a c e i n a v e r t i c a l p l a n e over a f i x e d range, and t h e r o t a t i o n a l o r p i t c h dynamics of t h e sailplane a r e neglected. Sailplane l i f t coefficient serves a s the control f u n c t i o n i n t h e n o n l i n e a r point-mass equatYons of motion. For o s c i l l a t o r y v e r t i c a l wind d i s t r i b u t i o n s , t h i s problem b e l o n g s t o t h e c l a s s of "optimal d o l p h i n s o a r i n g " problems. I n q u a l i t a t i v e terms, t h e s e problems e x h i b i t s o l u t i o n s f o r which t h e s a i l p l a n e speed i s d e c r e a s e d i n u p c u r r e n t s t o p r o l o n g t h e a l t i t u d e g a i n and i n c r e a s e d i n downcurrents t o l e s s e n t h e a l t i t u d e l o s s ( r e f . 1 ) . E a r l i e r s o l u t i o n s t o t h e s e problems have assumed e i t h e r p i e c e w i s e - s t a t i c f l i g h t ( e q u i l i b r i u m g l i d e through segments of c o n s t a n t v e r t i c a l wind--see, f o r example, r e f e r e n c e 2 ) o r q u a s i - s t a t i c f l i g h t ( k i n e m a t i c e q u a t i o n s of motion only--see, f o r example, r e f e r e n c e s 3 through 6 ) . Thus, t h e primary d i s t i n g u i s h i n g f e a t u r e of t h i s paper i s t h e u s e of t h e f u l l n o n l i n e a r t r a n s l a t i o n a l e q u a t i o n s of motion and t h e c o r r e s p o n d i n g u s e of a modern o p t i m a l c o n t r o l a l g o r i t h m f o r n u m e r i c a l s o l u t i o n s . A d d i t i o n a l r e s e a r c h on t h e a p p l i c a t i o n of o p t i m a l c o n t r o l t h e o r y t o dynamic s a i l p l a n e performance problems may b e found i n r e f e r e n c e s 7 through 9 . PROBLEM FORMULATION A b r i e f d e r i v a t i o n of t h e e q u a t i o n s of motion used h e r e i s provided i n t h e Appendix. The b a s i c assumptions a r e : f l i g h t i n a v e r t i c a l p l a n e , uniform g r a v i t y a c c e l e r a t i o n g and atmospheric d e n s i t y p, a point-mass s a i l p l a n e of I f t h e v e r t i c a l wind d i s c o n s t a n t mass m, and v e r t i c a l wind of magnitude W. t r i b u t i o n i s f u r t h e r assumed t o b e independent of a l t i t u d e ( = W x ) W (), then t h e right-hand s i d e s of t h e e q u a t i o n s of motion do n o t depend on a l t i t u d e Y. The a l t i t u d e e q u a t i o n (10A) can t h e r e f o r e b e i n c o r p o r a t e d i n t o t h e performance index ( a l t i t u d e l o s s ) J = Y(O) - y(tf) = - ltf - ltf + 0 ( W 0 (- ? ) d t V s i n y)dt and need n o t b e r e g a r d e d a s a d i f f e r e n t i a l c o n s t r a i n t . Furthermore, i t w i l l b e convenient t o r e g a r d t h e range X as t h e independent v a r i a b l e r a t h e r t h a n t h e t i m e t . S i n c e t h e f i n a l range, X ( t f ) = X f , i s t o b e s p e c i f i e d , t h i s change of v a r i a b l e s w i l l r e s u l t i n a f i x e d "end-time" o p t i m a l c o n t r o l problem which i s i n h e r e n t l y e a s i e r t o s o l v e t h a n a v a r i a b l e "end-time" problem. The r a n g e e q u a t i o n (9A) can a l s o b e o m i t t e d from c o n s i d e r a t i o n as a d i f f e r e n t i a l c o n s t r a i n t . It must b e t a c i t l y assumed, however, t h a t t h e o p t i m a l t r a j e c t o r y w i l l n o t i n c l u d e any k i n d of l o o p i n g maneuver which would r e s u l t i n z e r o v a l u e s f o r V c o s y. Using (9A) t h e n , t h e performance i n d e x (1) becomes W +V v sin y cos y and t h e remaining e q u a t i o n s of motion, dV - -- dX dx (11A) and (12A), become I [ P V CDS/(2m) a + [(V cos y j ( d ~ / d ~ ) g l s i n y + 1 / ( v c o s y) (34 dX = ( PVC~S/(2m) - [ c o s y(dW/dX) + g/V] cos y 1/ (V cos y ) (3b) respectively. F i n a l l y , t h e nondimensional q u a n t i t i e s a r e i n t r o d u c e d . The r e s u l t i n g o p t i m a l c o n t r o l problem may b e s t a t e d a s f o l l o w s . Find t h a t c o n t r o l f u n c t i o n u ( x ) , 0 - x - 1, which minimizes t h e augmented < < performance i n d e x s u b j e c t t o t h e second-order dv -= dx dynamic s y s t e m - [rlCg(u)v 2 f (1 + S)s i n y ] / ( v cos y) , V ( ~ = vo ) 2 d x = [17C~(u)v - ( 1 + J ) cos y]/(v 2 c o s y) , y ( 0 ) = yo and s u b j e c t t o t h e t e r m i n a l s t a t e c o n s t r a i n t s *, where CD(u) = al C, (u) = CL = ~ ( 1 ) Yo = 0 C u + max a 2 L + 2 2 a j CL (2 s i n - 1) = and where w(x) i s t h e p r e s c r i b e d wind d i s t r i b u t i o n , ; (dw/dx) ( d x / d t ) = (dw/dx)v c o s y, and i s t h e f i x e d r a n g e . Minimum a l t i t u d e - l o s s e q u i l i b r i u m f g l i d e ( s t i l l a i r ) v a l u e s a r e adopted f o r t h e f i x e d and e q u a l i n i t i a l and t e r m i n a l s t a t e v a l u e s , v and y S e v e r a l a d d i t i o n a l e x p l a n a t o r y comments a r e r e q u i r e d . 0 x 0 . F i r s t , n o t e t h a t minimum ( s t a l l ) and maximum ( f l u t t e r ) s t a t e i n e q u a l i t y c o n s t r a i n t s on t h e a i r s p e e d a r e e n f o r c e d Gsing i n t e g r a l i n t e r i o r p e n a l t y f u n c t i o n s ( r e f . 1 0 ) shown i n terms two and t h r e e , r e s p e c t i v e l y , of e q u a t i o n Thus, a sequence of o p t i m a l c o n t r o l problems (5) - (10) must b e s o l v e d (5) The p e n a l t y c o n s t a n t s a r e f o r s p e c i f i e d p o s i t i v e p e n a l t y c o n s t a n t s K and K2. 1 t h e n i n c r e a s e d between subproblems. The s o l u t i o n o b t a i n e d from e a c h subproblem i s used a s s t a r t i n g d a t a f o r t h e subsequent subproblem. The sequence of subproblems i s t e r m i n a t e d when each p e n a l t y f u n c t i o n v a l u e i s s u f f i c i e n t l y small. . Secondly, i t may b e observed t h a t t h e l i 5 , t c o e f f i c i e n t i s bounded v i a t h e t r a n s f o r m a t i o n ( 9 ) . That i s , f o r any v a l u e of t h e c o n t r o l f u n c t i o n u ( x ) , t h e control inequality constraints -cL < - C,(u) 5 CL (11) max max i s a s p e c i f i e d c o n s t a n t . Also, n o t e t h a t a q u a d r a t i c L d r a g p o l a r ( 8 ) i s used?laX~orea c c u r a t e d r a g p o l a r s may b e used i n s t e a d provided F i n a l l y , i t should t h a t an a n a l y t i c a l r e l a t i o n i s a v a i l a b l e between C and C,. D b e observed t h a t f o r a f i x e d wing l o a d i n g , t h e nondimensional aerodynamic parameter q i n e q u a t i o n (10) i s p r o p o r t i o n a l t o t h e s p e c i f i e d t e r m i n a l r a n g e X f" T h i s i s t h e b a s i c o p t i m a l c o n t r o l problem c o n s i d e r e d h e r e . t h i s problem w i l l a l s o b e p r e s e n t e d l a t e r . A v a r i a t i o n of a r e s a t i s f i e d where C NUMERI C L RESULT S A A l l computations have been performed on coupled IBM 360/65 and I t e l AS/5 computers u s i n g a FORTRAN I V compiler and double p r e c i s i o n a r i t h m e t i c . The n u m e r i c a l i n t e g r a t i o n of t h e r e q u i r e d d i f f e r e n t i a l e q u a t i o n s h a s been performed u s i n g a s t a n d a r d f o u r t h - o r d e r Runge-Kutta method w i t h 100 f i x e d uniform integration steps. The n u m e r i c a l r e s u l t s have been o b t a i n e d f o r t h e c a s e of a s i n u s o i d a l wind distribution and t h e Nimbus I1 open-class s a i l p l a n e u s i n g t h e g r a d i e n t p r o j e c t i o n a l g o r i t h m p r e s e n t e d i n r e f e r e n c e 1 . The s i n u s o i d a l wind d i s t r i b u t i o n (12) i s simply 1 an i d e a l i z e d model of an o s c i l l a t o r y v e r t i c a l wind which s a t i s f i e s a " c o n t i n u i t y " c o n d i t i o n : t h e i n t e g r a l of w(x) over t h e f i x e d r a n g e i s z e r o . The v a l u e s for the coefficients of t h e q u a d r a t i c d r a g p o l a r (8) f o r t h e Nimbus I1 h a v e been o b t a i n e d from a l e a s t - s q u a r e s f i t of d a t a t a k e n from t h e m a n u f a c t u r e r ' s v e l o c i t y p o l a r . For s t a n d a r d s e a l e v e l c o n d i t i o n s and a wing l o a d i n g , mg/S, of (32) (9.81) t h e aerodynamic parameter i n e q u a t i o n (10) i s given by q = 0,01916 X,. A d d i t i o n a l c o n s t a n t d a t a chosen i n c l u d e : = 1.4, ( g ~ f ) % vstallL= 1 8 m / s and (gX )$ vmax = 70 m / s . F i n a l l y , i n q u a l i @ f i v e terms, t h e g r a d i e n t p r o j ect i o n meEhod i s a d i r e c t method i n t h e s e n s e t h a t t h e c o n t r o l f u n c t i o n u(x) i s changed d u r i n g each i t e r a t i o n s o a s t o produce b o t h a d e c r e a s e i n t h e performance i n d e x v a l u e (eq. (5)) and f u l l s a t i s f a c t i o n of t h e t e r m i n a l s t a t e constraints (eqs.(7)). Specified I n i t i a l State I n t h i s c a s e , t h e i n i t i a l and f i n a l s t a t e a r e t o b e h e l d f i x e d and e q u a l . I n particular, t h e values (gXf))i v = 28.1676 m / s and yo = 1 C~ 0 - 0.019106 r a d (14) are t o b e used i n e q u a t i o n s (6) and (7) and correspond t o t h e minimum a l t i t u d e l o s s e q u i l i b r i u m g l i d e c o n d i t i o n s f o r t h e Nimbus I1 i n s t i l l a i r w i t h a d r a g p o l a r g i v e n b y e q u a t i o n s (8) and ( 1 3 ) . The v e r t i c a l wind amplitude i s chosen a s ( g ~ f ) % = 2 m / s , and a f i x e d r a n g e of 1000 m e t e r s i s used. wA The r e s u l t i n g o p t i m a l t r a j e c t o r y and t h e corresponding o p t i m a l l i f t c o e f f i c i e n t d i s t r i b u t i o n are p r e s e n t e d i n f i g u r e s 1 and 2, r e s p e c t i v e l y . The o p t i m a l f l i g h t can b e d i v i d e d i n t o t h r e e s u c c e s s i v e segments: an i n i t i a l cliinb, a maximum C a r c , and a d i v e followed by a s h o r t pull-up. The i n i t i a l climb L i s i n t u i t i v e l y r e a s o n a b l e s i n c e t h e s a i l p l a n e must g a i n a s much a l t i t u d e a s p o s s i b l e w h i l e i n t h e i n i t i a l u p c u r r e n t . The maximum C a r c i s a c o n t i n u a t i o n L of t h e f i r s t phase and l a s t s a s l o n g a s t h e wind i s s t r o n g enough t o s u s t a i n i t . The f o l l o w i n g d i v e i s made t o p a s s through t h e downcurrent a s q u i c k l y a s p o s s i b l e . The f i n a l pull-up i s n e c e s s a r y t o meet t h e t e r m i n a l s t a t e c o n s t r a i n t s ( 7 ) . The s t a l l speed i n e q u a l i t y c o n s t r a i n t w a s a c t i v e f o r t h i s s o l u t i o n , b u t t h e maximum speed c o n s t r a i n t was n o t . The minimum a l t i t u d e l o s s f o r t h i s o p t i m a l t r a j e c t o r y i s o n l y 12.19 m. By comparison, t h e minimum a l t i t u d e l o s s d u r i n g an e q u i l i b r i u m g l i d e i n s t i l l a i r o v e r t h e same 1000 m r a n g e i s 19.11 m. T h i s r e p r e s e n t s a 36% a l t i t u d e l o s s reduct ion. F r e e b u t Equal I n i t i a l and F i n a l S t a t e s Here, t h e i n i t i a l and f i n a l speed and l o c a l f l i g h t p a t h a n g l e v a l u e s a r e n o l o n g e r s p e c i f i e d , b u t t h e r e s p e c t i v e i n i t i a l and f i n a l v a l u e s a r e s t i l l r e q u i r e d t o b e e q u a l . The g r a d i e n t p r o j e c t i o n a l g o r i t h m , a s d e s c r i b e d i n 1 r e f e r e n c e 1 , can accomodate t h e a d d i t i o n of t h e two c o n t r o l p a r a m e t e r s v ( 0 ) and y ( 0 ) r e p r e s e n t i n g v a r i a b l e i n i t i a l s t a t e s . However, t h e p r e s e n c e of t h e s e same two c o n t r o l p a r a m e t e r s i n t h e t e r m i n a l s t a t e c o n s t r a i n t s n e c e s s i t a t e s a f u r t h e r modification t o t h e proj e c t i o n operator equations. S i n c e t h e o p t i m a l t r a j e c t o r i e s a r e nqw b e i n g s e l e c t e d from a l a r g e r c l a s s , a d d i t i o n a l performance g a i n s a r e expected. However, f o r a f i n a l r a n g e of 1000 m, t h e minimum a l t i t u d e l o s s improves o n l y 1.4%: from 12.19 m t o 1 2 . 0 1 m. For comparison purposes, t h e o p t i m a l t r a j e c t o r y i s a l s o shown i n f i g u r e 1. I n t h i s case, t h e optimal t r a j e c t o r y e x h i b i t s a higher a l t i t u d e gain during t h e climb phase, a l o n g e r maximum C a r c and a l e s s e r a l t i t u d e l o s s i n t h e L downcurrent when compared w i t h t h e p r e v i o u s s o l u t i o n . The i n i t i a l (and f i n a l ) a i r s p e e d h a s i n c r e a s e d approximately 1.2 m / s t o 29.384 m / s . E f f e c t s of Wind Amplitude The i n i t i a l and f i p a l s t a t e s a r e a g a i n f r e e b u t e q u a l . For an i n c r e a s e d wind amplitude of (gX )3 w = 5 m / s , a s u b s t a n t i a l improvement is o b t a i n e d a s £ A may b e n o t e d from t h e o p t i m a l t r a j e c t o r y shown i n f i g u r e 3. A n e t a l t i t u d e g a i n of 5.158 m i s now a v a i l a b l e over t h e 1000 m coucse. C l e a r l y , f o r h i g h e r a m p l i t u d e s of an o s c i l l a t i n g v e r t i c a l wind, more e n e r g y can b e e x t r a c t e d from t h e wind t o s u s t a i n cross-country f l i g h t . E f f e c t s of Varying t h e Fixed Range The wind amplitude i s h e l d f i x e d a t 5 m / s , and f r e e b u t e q u a l i n i t i a l a n d f i n a l s t a t e s a r e a g a i n c o n s i d e r e d . Changes i n t h e f i n a l range X a f f e c t o n l y f t h e c o n s t a n t aerodynamic parameter q and t h e c h a r a c t e r i s t i c l e n g t h and t i m e used i n t h e n o n d i m e n s i o n a l i z a t i o n . Varying X i s e q u i v a l e n t t o v a r y i n g t h e f frequency of t h e s i n u s o i d a l wind d i s t r i b u t i o n f o r s u s t a i n e d f l i g h t s . Upon r e d u c i n g X from 1000 m t o 500 m, a r a d i c a l l y d i f f e r e n t o p t i m a l traf j e c t o r y was o b t a i n e d . The corresponding o p t i m a l t r a j e c t o r y and o p t i m a l C L d i s t r i b u t i o n a r e shown i n f i g u r e s 4 and 5 , r e s p e c t i v e l y , and w i l l b e r e f e r r e d t o a s a Type I1 s o l u t i o n . I n t h i s c a s e , a d i v e - f i r s t c l i m b - l a t e r f l i g h t p a t t e r n i s observed. The a v e r a g e speed i s much h i g h e r , and t h e n e t a l t i t u d e g a i n i s c o n s i d e r a b l y h i g h e r t h a n f o r t h e e a r l i e r Type I t r a j e c t o r y . The n e t a l t i t u d e g a i n of 23.10 m exceeds t h a t achieved on t h e p r e v i o u s Type I s o l u t i o n f o r d o u b l e t h e r a n g e . For t h e Type I1 t r a j e c t o r i e s , t h e maximum speed i n e q u a l i t y c o n s t r a i n t i s a c t i v e r a t h e r t h a n t h e s t a l l speed c o n s t r a i n t . By employing t h e r e s u l t s of p r e v i o u s Type I s o l u t i o n s as s t a r t i n g d a t a , i t was p o s s i b l e t o a l s o o b t a i n a Type I s o l u t i o n f o r Xf = 500 m. However, t h i s Type I r e l a t i v e minimum s o l u t i o n y i e l d s a n e t a l t i t u d e l o s s of 4.45 m. Thus, a t l e a s t f o r r e l a t i v e l y s m a l l X and l a r g e W , t h e Type I1 s o l u t i o n i s d e c i d e d l y a m a t t e r oQ c o n j e c t u r e , t h e Type I1 s o l u t i o n s u p e r i o r t o Type I s o l u t i o n s . may c e a s e t o e x i s t f o r s u f f i c i e n t l y l a r g e f i n a l r a n g e s a n d / o r f o r s u f f i c i e n t l y small wind a m p l i t u d e s . Two a d d i t i o n a l s o l u t i o n s were a l s o o b t a i n e d : a Type I s o l u t i o n f o r Xf = 750 m and a Type I1 s o l u t i o n f o r X = 625 m. f 1s E f f e c t of Wing Loading I f t h e nominal wing l o a d i n g , mg/S, i s i n c r e a s e d by 15%, t h e aerodynamic parameter q becomes 0.01666 X ( s e e e q u a t i o n ( 1 0 ) ) . Nothing e l s e i s changed. f I n t h i s c a s e , a Type I s o l u t i o n was o b t a i n e d f o r X = 1000 m, W = 5 m / s and f r e e f A b u t e q u a l i n i t i a l and f i n a l s t a t e s . The r e s u l t i n g o p t i m a l t r a j e c t o r y p r o v i d e s a n e t a l t i t u d e g a i n of 1 . 1 4 m which i s n e a r l y 4 m l e s s t h a n t h e comparable 5.16 m o b t a i n e d e a r l i e r f o r t h e nominal wing l o a d i n g . Both t h e o p t i m a l t r a j e c t o r i e s and t h e o p t i m a l l i f t c o e f f i c i e n t h i s t o r i e s f o r t h e two s o l u t i o n s a r e very s i m i l a r . The key r e s u l t s f o r t h e e i g h t o p t i m a l s o l u t i o n s p r e s e n t e d h e r e a r e summarized ir! t a b l e I . CONCLUSIONS AND DISCUSSION The o p t i m a l c o n t r o l problem t r e a t e d h e r e i s of a t l e a s t moderate d i f f i c u l t y i n view of t h e s t a t e v a r i a b l e i n e q u a l i t y c o n s t r a i n t s p r e s e n t . R e l a t i v e l y few n u m e r i c a l s o l u t i o n s a r e . c u r r e n t l y a v a i l a b l e b e c a u s e of t h e c o n s i d e r a b l e comput a t i o n a l e f f o r t involved. However, s e v e r a l t e n t a t i v e c o n c l u s i o n s emerge from t h e computational r e s u l t s obtained thus f a r . 1) For a s i n u s o i d a l v e r t i c a l wind d i s t r i b u t i o n , which s e r v e s as a simple model of a z e r o range-averaged o s c i l l a t o r y wind, s u b s t a n t i a l a l t i t u d e s a v i n g s a r e a v a i l a b l e when compared w i t h o p t i m a l e q u i l i b r i u m g l i d e s i n s t i l l a i r . The r e l a t i v e advantage i n c r e a s e s f o r h i g h e r wind amplitudes. 2) Equal i n i t i a l and f i n a l s t a t e v e c t o r elements can b e t r e a t e d a s addit i o n a l c o n t r o l parameters i n t h e o p t i m a l c o n t r o l a l g o r i t h m and t h e r e f o r e v a r i e d as p a r t of t h e o p t i m i z a t i o n p r o c e s s . The a d d i t i o n a l a l t i t u d e g a i n s o b t a i n e d i n t h i s c a s e are, however, r a t h e r s m a l l . For r e l a t i v e l y s h o r t r a n g e s and h i g h wind amplitude, i t i s p o s s i b l e t o o b t a i n o p t i m a l t r a j e c t o r i e s which e x h i b i t an unexpected " d i v e f i r s t , climb l a t e r " maneuver sequence. Optimal t r a j e c t o r i e s of t h i s second t y p e i n v o l v e h i g h e r speeds and b e t t e r f i n a l a l t i t u d e g a i n s than t h e u s u a l "dolphin" s t y l e o p t i m a l t r a j e c t o r y . 3) 4) A s expected, an i n c r e a s e i n s a i l p l a n e wing l o a d i n g i n c r e a s e s t h e minimum a l t i t u d e l o s s when o t h e r c o n d i t i o n s a r e h e l d f i x e d . The most s u r p r i s i n g f i n d i n g of t h i s s t u d y i s t h e a p p a r e n t e x i s t e n c e of two d i s t i n c t t y p e s of e x t r e m a l s o l u t i o n s a t l e a s t f o r r e s t r i c t e d r a n g e s of t h e parameters i n v o l v e d . C l e a r l y , t h e Type I1 t r a j e c t o r y d e s e r v e s f u r t h e r r e s e a r c h effort. There i s a l s o a need t o o b t a i n r e s u l t s f o r o t h e r wind d i s t r i b u t i o n s and t o make d e f i n i t i v e comparisons w i t h p r e v i o u s o p t i m i z a t i o n s t u d i e s which do n o t i n c o r p o r a t e t h e f u l l t r a n s l a t i o n a l e q u a t i o n s of motion. The r e l a t e d problem of minimum-time f l i g h t through a given v e r t i c a l wind d i s t r i b u t i o n f o r a s p e c i f i e d a l t i t u d e l o s s i s of perhaps even g c e a t e r i n t e r e s t . Research on t h i s l a t t e r problem i s c u r r e n t l y underway. REFERENCES Reichmann, H., Cross-Country S o a r i n g ( E n g l i s h t r a n s l a t i o n of S t r e c k e n s e g e l f l u : Thompson P u b l i c a t i o n s , S a n t a Monica, C a l i f o r n i a , 1978. Meyer, R., "Dolphin-style May 1978. g l i d i n g , " T e c h n i c a l Soaring, Vol. 5, No. 1, 1-9, Arho, R . , "Optimal d o l p h i n s o a r i n g a s a v a r i a t i o n a l problem," Acta Polytechnics Scandinavica, Mechanical Engineering S e r i e s No. 68, 1972. I r v i n g , F. G., "Cloud-street 1-8, Winter 1973. f l y i n g , " T e c h n i c a l Soaring, Vol. 3, No. 1, Metzger, D. E. and Hedrick, J . K., "Optimal f l i g h t p a t h s f o r s o a r i n g f l i g h t , ' 1 J o u r n a l of A i r c r a f t , Vol. 12, No. 1 , 867-871, December 1975. d e Jong, J . L., "The ' o p t i m a l - r a n g e - v e l o c i ~ y - p o l a r ' , a new t h e o r e t i c a l t o o l f o r t h e o p t i m i z a t i o n of s a i l p l a n e f l i g h t t r a j e c t o r i e s , " Memorandum COSOR 77-28, Department of Mathematics, Eindhoven U n i v e r s i t y of Technology, December 1977. P i e r s o n , B. L., "Maximum a l t i t u d e s a i l p l a n e winch-launch t r a j e c t o r i e s , " A e r o n a u t i c a l Q u a r t e r l y , Vol. 28, No. 2, 75-84, May 1977. P i e r s o n , B. L. and de Jong, J . L . , "Cross-country s a i l p l a n e f l i g h t as a dynamic o p t i m i z a t i o n problem," I n t e r n a t i o n a l J o u r n a l f o r Numerical 1 Methods i n Engineering, Vol. 12, No. 1 , 1743-1759, 1978. Genalo, L. J . and P i e r s o n , B . L., "A s i n g u l a r - a r c approximation t o a dynamic s a i l p l a n e f l i g h t p a t h o p t i m i z a t i o n problem," Engineering O p t i m i z a t i o n , Vol. 3, No. 4, 175-182, 1978. 1 0 . Lasdon, L. S., Waren, A. D . , and Rice, R. K . , "An i n t e r i o r p e n a l t y f u n c t i o n method f o r i n e q u a l i t y c o n s t r a i n e d o p t i m a l c o n t r o l problems," X E T r a n s a c t i o n s on Automatic C o n t r o l , Vol. AC-12, No. 4, 388-395, August 1967 1 . P i e r s o n , B. L . , "Panel f l u t t e r o p t i m i z a t i o n by g r a d i e n t p r o j e c t i o n , " 1 I n t e r n a t i o n a l J o u r n a l f o r Numerical Methods i n Engineering, Vol. 9, NO. 2, 271-296, 1975. APPENDIX: DERIVATION O THE EQUATIONS O MOTION F F The s a i l p l a n e v e l o c i t y v e c t o r , r e l a t i v e t o t h e s u r r o u n d i n g a i r , i s g i v e n by ( s e e f i g u r e 6) where R i s t h e i n e r t i a l v e l o c i t y v e c t o r f o r t h e s a i l p l a n e , and -r T = [0, W(X,Y)] W i s t h e d e t e r m i n i s t i c v e r t i c a l wind d i s t r i b u t i o n . The a n g l e y shown i n f i g u r e 6 i s t h e u s u a l f l i g h t p a t h a n g l e f o r t h e c a s e of no wind. I n t h e i n e r t i a l (X,Y)-coordinate frame, t h e t r a n s l a t i o n a l e q u a t i o n s of motion f o r unpowered f l i g h t i n a uniform g r a v i t y f i e l d may b e w r i t t e n a s where L and 3 a r e t h e u s u a l aerodynamic l i f t and d r a g f o r c e s , r e s p e c t i v e l y , G i s t h e weight f o r c e , and m i s t h e c o n s t a n t s a i l p l a n e mass. Equation (2A) can be rearranged t o y i e l d 4 .. - where g i s t h e c o n s t a n t g r a v i t y a c c e l e r a t i o n . P r i m a r i l y b e c a u s e of t h e d e f i n i t i o n s of l i f t and d r a g , i t i s d e s i r a b l e t o r e w r i t e e q u a t i o n (3A) w i t h r e s p e c t t o a r o t a t i n g ( 5 , ~ ) - c o o r d i n a t e frame d i r e c t e d toward V, and $ , d i ~ e c t e d normal d e f i n e d by t h e u n i t v e c t o r s $ t o V a l o n g t h e l i f t v e c t o r . !&e i n e r t i a l t i m e d e r i v a t i v g of V, r e f e r r e d t o t h i s r o t a t i n g frame, i s t h e n given by where e^ = x 2 Note t h a t any v e c t o r i n t h e p l a n e , s a y A, can b e e x p r e s s e d i? $5 i n t h e r o t a t i n g (<,n)-frame u s i n g t h e f o l l o w i n g r o t a t i o n m a t r i x . < . -. - s i n y cosy Using e q u a t i o n (5A), one o b t a i n s fi = g i ( a w / a x ) The term, g gravity acceleration. + + + ? ( a w / a ~ ) , is o f t e n r e f e r r e d t o a s t h e apparent F i n a l l y , u s i n g e q u a t i o n s (4A) and (6A) and t h e u s u a l e x p r e s s i o n s f o r l i f t and d r a g , e q u a t i o n (3A) can b e w r i t t e n i n s c a l a r form a s f o l l o w s . From e q u a t i o n (lA) and t h e i n v e r s e of e q u a t i o n (5A), t h e k i n e m a t i c e q u a t i o n s may b e w r i t t e n as X = v cos y T h e r e f o r e , e q u a t i o n s (7A) and (8A) become t = - pv 2CDS/(2m) = pVCLS/(2m) - [(V cos y ) (awlax) ( W + +V s i n y ) (aW/aY) + g]sin y (11A) - [ c o s y(aW/aX) + (W/V + sin y ) (aW/aY) (12A) + g/Vlcos Y TABLE I. Summary of Optimal S o l u t i o n s WIND AMPLITUDE ALTITUDE CHANGE TYPE RANGE v(0) = V ( l > m/s y(0) = ~ ( 1 ) rad m m/s . m a f i x g d boundary c o n d i t i o n s wing l o a d i n g i n c r e a s e d 15% RANGE, X, m Figure 1.- Optimal t r a j e c t o r i e s of Type I: WA = 2 m / s , C L ~ , = 1.4. Xf = 1000 m y V s t a l l = 18 m / s , RANGE, X, m Figure 2.- Optimal l i f t c o e f f i c i e n t d i s t r i b u t i o n f o r f i x e d and equal boundary c o n d i t i o n s : WA = 2 m / s , Xf = 1000 m y Vscall = 18 m / s , C, 4 = 1.4. RANGE, Figure 3 X, m WA = 5 m / s , .- Optimal Type I t r a j e c t o r y f o r high wind amplitude: Chx = 1.4. Xf = 1000 m y VStall = 18 m l s , RANGE, F i g u r e 4.Optimal Type I1 t r a j e c t o r y : X, m = 5 m/s, WA Xf = 500 m , , V y = 70 m / s . 31 7 RANGE, X, m Figure 5.- Optimal l i f t c o e f f i c i e n t d i s t r i b u t i o n f o r Type I1 t r a j e c t o r y : = 70 m / s . WA = 5 m / s , Xf ,= 500 m,, , V Figure 6 . - Velocity v e c t o r diagram. A S U Y O C U S DEVIATIONS DURIZiG C O SC U T Y SOARING TD F O RE R S- O NR Steven M. Sliwa Langley Research Center David J . Sliwa University of I l l i n o i s a t Urbana-Champaign ABSTRACT Severa1,models a r e developed f o r studying t h e impact of d e v i a t i o n s from course during cross-country soaring f l i g h t s . Analyses a r e performed a t t h e micro-strategy and macro-strategy l e v e l s . T o ty-pes of lift sources a r e w considered: concentrated thermals and thermal s t r e e t s . The s e n s i t i v i t y of t h e optimum speed s o l u t i o n s t o various model, p i l o t i n g and performance parameters i s evaluated. Guides a r e presented t o provide t h e p i l o t with c r i t e r i o n s f o r making i n - f l i g h t decisions. I n g e n e r a l , course d e v i a t i o n s a r e warranted during weak lift conditions, but a r e l e s s j u s t i f i a b l e with moderate t o strong lift conditions. INTRODUCTION There have been many attempts t o develop optimum p i l o t i n g s t r a t e g i e s f o r t h e v e r t i c a l plane of cross-country soaring ( f o r example, r e f e r e n c e s 1 through 5 ) , which b a s i c a l l y y i e l d an optimal a i r s p e e d given t h e airmass c h a r a c t e r i s t i c s , but l i t t l e has been done with t h e h o r i z o n t a l plane. References 6 through 8 point out t h a t s u b s t a n t i a l departures from t h e optimum speed-to-fly r e s u l t i n small degradations i n achieved speed. I n f a c t , t h e biggest c o n t r i b u t i n g f a c t o r s influencing average speed a r e maximizing t h e achieved rate-of-climb i n l i f t and minimizing t h e atmospheric s i n k r a t e between regions of l i f t . So it seems t h a t cross-country soaring performance i s most influenced by t h e p i l o t ' s decisions made i n t h e h o r i z o n t a l plane. This paper w i l l address i t s e l f t o developing some models r e f l e c t i n g t y p i c a l course d e v i a t i o n d e c i s i o n s a p i l o t i s l i k e l y t o be confronted with during a cross-country soaring f l i g h t . The accompanying a n a l y s i s should provide guidel i n e s f o r t h e p i l o t t o r a t i o n a l l y s e l e c t f l i g h t paths which optimize t h e a n t i c i p a t e d lift conditions. T o t y p e s of convective lift conditions a r e w considered: soaring conditions where t h e regions of lift a r e s m a l l r e l a t i v e t o t h e d i s t a n c e flown ( c i r c l i n g r e q u i r e d ) and conditions where t h e regions of l i f t a r e of t h e order of t h e d i s t a n c e flown (thermal s t r e e t f l y i n g ) . I n a d d i t i o n , two c a t e g o r i e s of models a r e i n v e s t i g a t e d . Micro-strategy models a r e used t o analyze t h e choice of lift along a d e s i r e d course l i n e . Macro-strategy models a r e used f o r studying t h e influence of choosing a course l i n e t o a goal. The a n a l y s i s contained h e r e i n assumes parabolic performance p o l a r s with numerical examples computed f o r parameters t y p i c a l of a modern standard c l a s s s a i l p l a n e . The p i l o t i s assumed t o f l y a t t h e optimal airspeed f o r a l l course choices s i n c e p e r t u r b a t i o n s a r e assumed t o have a minor e f f e c t . Since f i n a l g l i d e s a r e not considered and p o t e n t i a l energy i s conserved, a l l models begin and end at t h e same a l t i t u d e , cloudbase. Furthermore, a l l s o l u t i o n s neglect s u r v i v a b i l i t y , i.e . , t h e y assume t h e p i l o t w i l l complete t h e t a s k no m a t t e r which choices a r e made. F i n a l l y , a l l s i t u a t i o n s assume t h a t t h e p i l o t i s f a r from a ground referenced g o a l and t h a t t h e l i f t i s not ground referenced so t h e i n f l u e n c e of wind can be neglected. LIST O S M O S F Y BL 1 P a r a s i t i c drag f a c t o r , 2v02 (L/Dlmax " Induced drag f a c t o r , D ~ ( L / D ) ~ ~ Distance on course t o lift source g o a l f o r thermal s t r e e t model Distance on course t o lift, source g o a l f o r thermal models Projected d i s t a n c e of a l t e r n a t e lift source onto course l i n e , Fig. 1 F2 Intermediate c a l c u l a t i o n v a r i a b l e s Intermediate c a l c u l a t i o n v a r i a b l e A l t i t u d e gained climbing i n s t r e e t lift A l t i t u d e l o s t c r u i s i n g between s t r e e t s Average rate-of-climb while c i r c l i n g i n thermals Rate-of-climb averaged while c r u i s i n g thermal s t r e e t l i f t d d' f h~~ h~~ i, is K Intermediate c a l c u l a t i o n conshant, defined i n Appendix C , Equation 3 Intermediate c a l c u l a t i o n constant, defined i n Appendix C , Equation 1 0 Equivalent t o maximum g l i d e r a t i o i n s t i l l a i r Distance flown along s t r e e t , Fig. 1 2 Distance t o f l y along s t r e e t f o r optimum time K' (L/D Imax R R* Distance t o f l y along s t r e e t f o r time equal t o not making course deviation Non-dimensionalized d i s t a n c e t o f l y along s t r e e t , break-away point Slope of tangent l i n e Length of second l e g of course d e v i a t i o n , Fig. 1 2 T o t a l d i s t a n c e of a cruise/climb s t r e e t c y c l e Distance of climb phase of a s t r e e t c y c l e Distance of c r u i s e phase of a s t r e e t cycle Value of d e f i n i n g polynomial f o r i t h i t e r a t i o n T o t a l d e v i a t i o n d i s t a n c e of using a s t r e e t p a r a l l e l t o course l i n e , Fig. 9 Distance of i n d i v i d u a l l e g s of course d e v i a t i o n , Fig. s/D T~ 9 9 Deviation d i s t a n c e r a t i o of p a r a l l e l s t r e e t model, Fig. Time t o f l y g l i d e l c l i m b thermal cycle on course Time t o f l y course d e v i a t i o n Airspeed while c r u i s i n g , knots Optimum speed-to-fly between l i f t , knots th - i t e r a t i o n , knots Guess of V* during i Sink r a t e f l y i n g a t an a i r s p e e d of V*, knots T,tn V V* v" i V* s 'at Average v e r t i c a l sinking v e l o c i t y of atmosphere between l i f t , knots Airspeed while climbing i n a s t r e e t , knots Required a i r s p e e d t o c r u i s e i n s t r e e t l i f t and maintain constant a l t i t u d e , knots Airspeed along l e g s D , 2, n r e s p e c t i v e l y V c ~ 'CR VD,VR,Vn v~ 'GS Average ground speed a f t e r a complete glide/climb thermal c y c l e , knots Average ground speed a f t e r a complete glide/climb thermal s t r e e t l i f t c y c l e , knots Airspeed f o r minimum s i n k r a t e , knots Speed a t which (L/D),,~ occurs, knots 'MIN vo vs 'sn Sink r a t e f l y i n g a t a i r s p e e d V , knots Sink r a t e f l y i n g a t a i r s p e e d Vn, knots Geometry l a b e l s f o r course d e v i a t i o n models T o t a l d e v i a t i o n d i s t a n c e , Fig. 1 x2 Deviation d i s t a n c e l e g s , Fig. 1 Deviation d i s t a n c e r a t i o Distance between p a r a l l e l s t r e e t and course l i n e , Fig. 9 Spacing d i s t a n c e r a t i o R a t i o of average rate-of-climb along course d e v i a t i o n on course t o average rate-of-climb W X, Y, Z , x Ratio of average atmospheric s i n k r a t e between lift sources t o average rate-of-climb i n l i f t Ratio of average ground speed on course d e v i a t i o n i n augmented l i f t t o ground speed acheived on course w i t h average lift conditions Angle between thermal s t r e e t and course l i n e Angle of thermal model course d e v i a t i o n PRESENTATION O RESULTS F Thermal Models Micro-Strategy The f i r s t c a s e considered i s d e p i c t e d i n f i g u r e 1. It r e p r e s e n t s a frequent d e c i s i o n confronting t h e p i l o t during cross-country soaring. The p i l o t , a f t e r departing t h e th=al at X a t cloudbase, must. choose between staying on course along p a t h XZ and achieving t h e average rate-of-climb f o r t h a t time of day a t thermal Z o r deviating along XY t o t h e thermal a t Y, which looks a s i f it might y i e l d a higher achieved rate-of-climb. Then t h e p i l o t r e t u r n s t o t h e course a f t e r deviating t o Y by f l y i n g t o thermal Z. Given t h e geometry, t h e question remains how much g r e a t e r must be t h e rate-of-climb a t thermal Y than t h e rate-of-climb a t thermal Z t o y i e l d t h e same time f o r both t h e d i r e c t course and t h e extended r o u t e . Figure 2 shows t h e r e s u l t f o r a s a i l p l a n e r e p r e s e n t a t i v e of t h e standard c l a s s . The required rate-of-climb i n t h e thermal a t Y i s p l o t t e d a g a i n s t t h e non-dimensional d e v i a t i o n d i s t a n c e r a t i o f o r a v a r i e t y of average lift conditions assuming t h e p i l o t f l i e s t h e optimum a i r s p e e d , t h e c a l c u l a t i o n of which i s shown i n Appendix A. The curves i n f i g u r e 2 can be t r e a t e d a s time boundaries. P o i n t s t o t h e above and l e f t of a curve i n d i c a t e t h a t a d e v i a t i o n would be f a s t e r t h a n s t a y i n g on course whereas p o i n t s t o t h e bottom and r i g h t r e p r e s e n t c o n d i t i o n s f o r which staying on course would be more p r o f i t a b l e . The importance of d e v i a t i n g f o r minor gains i n l i f t when t h e conditions a r e weak i s shown by examining t h e curve f o r 1 knot average rate-of-climb on course. A 25% course elongation r e q u i r e s a l i t t l e over 2 knots rate-of-climb i n t h e thermal a t Y . I f t h e expected rate-of-climb i n Z were 4 k n o t s (moderate lift c o n d i t i o n s ) , a 25% course d e v i a t i o n r a t i o would need t o have an achieved rate-of-climb b e t t e r than 1 5 knots, t o r e s u l t i n t h e same time t o t h e t o p of t h e thermal a t Z. The implication i s t h a t when lift conditions a r e weak (1-2 knots average rate-of-climb), course d e v i a t i o n s would be advantageous f o r modest g a i n s i n lift. However, f o r moderate t o s t r o n g lift conditions ( 4 knots and above average rate-of-climb), s i z e a b l e g a i n s i n l i f t w i l l permit only minor d e v i a t i o n s from t h e course l i n e . This r e s u l t i s f u r t h e r emphasized i n f i g u r e 3 where t h e d e v i a t i o n d i s t a n c e r a t i o i s p l o t t e d a g a i n s t a non-dimensionalized l i f t r a t i o f o r a number of lift conditions. The weak conditions warrant s u b s t a n t i a l d e v i a t i o n d i s t a n c e r a t i o s even i n non-dimensional form while, i n c o n t r a s t , t h e s t r o n g e r conditions begin t o coincide upon a boundary r e q u i r i n g l a r g e lift r a t i o s f o r any appreciable distance r a t i o . The influence of s a i l p l a n e performance upon t h e p i l o t ' s d e c i s i o n s i s shown i n f i g u r e 4. Rate-of-climb required a t thermal Y i s p l o t t e d a s a function of deviation d i s t a n c e r a t i o f o r t h r e e c l a s s e s of s a i l p l a n e s . S a i l p l a n e A i s t h e standard c l a s s a i r c r a f t considered previously; s a i l p l a n e B r e p r e s e n t s a onedesign s p o r t c l a s s ; and a i r c r a f t C r e p r e s e n t s a s a i l p l a n e i n t h e open c l a s s . It i s r e a d i l y apparent t h a t s a i l p l a n e performance has a minor e f f e c t on t h e p i l o t ' s willingness t o d e v i a t e from course. However, t h e r e i s a t r e n d f o r s a i l p l a n e s of l e s s e r performance t o be w i l l i n g t o make s l i g h t l y g r e a t e r course deviations. The previous curves were developed with an assumed average atmospheric subsidence equal t o 20 percent of t h e rate-of-climb ( r e f e r e n c e 9 ) . As expected, s l i g h t course extensions with t h i s model can be j u s t i f i e d with reduced s i n k r a t e ( f i g u r e 5 ) . However, t h e influence of s i n k r a t e on t h e p i l o t ' s d e c i s i o n t o d e v i a t e from course, assuming t h a t both f l i g h t paths undergo t h e same average sink r a t e , i s negligible. An important v a r i a b l e i n t h e geometry shown i n f i g u r e 1 i s d l / d . It impacts t h e performance of t h e extended course by determining how much of t h e a l t i t u d e t o be regained w i l l be done i n t h e s t r o n g e r thermal at Y. The generalized r e s u l t s f o r d ' / d of .25, .5, and .75 a r e shown i n f i g u r e 6 f o r average l i f t conditions of 2 knots and 6 knots. It i s r e a d i l y apparent from f i g u r e 6 t h a t s u b s t a n t i a l l y l a r g e r course d e v i a t i o n s can be j u s t i f i e d with l a r g e r values of d l / d . The g r e a t e r t h e d i s t a n c e between X and Y f o r a given d e v i a t i o n d i s t a n c e r a t i o , t h e g r e a t e r t h e a l t i t u d e which i s gained i n t h e s t r o n g e r l i f t a t Y , t h e r e b y i n c r e a s i n g t h e achieved speed. The net r e s u l t of t h e foregoing a n a l y s i s i s t h a t t h e d e v i a t i o n angle, Y, should be kept a s small a s p o s s i b l e . This i s e s p e c i a l l y t r u e f o r moderate t o strong l i f t conditions. This r e s u l t i s i n b a s i c agreement with t h e macros t r a t e g y model presented i n r e f e r e n c e 1 0 which i s of s i m i l a r format t o t h e micro-strategy model considered here. It should be noted t h a t t h e preceding r e s u l t s can be d i r e c t l y applied t o a more generalized model including m u l t i p l e g l i d e / c i r c l e cycles along t h e course l i n e segments XZ and This i s t r u e a slong as t h e d e v i a t i o n f l i g h t p a t h includes only one g l i d e / c i r c l e c y c l e along YZ. The reason m u l t i p l e thermals do not a f f e c t t h e a n a l y s i s i s due t o t h e s i m p l i f i c a t i o n t h a t n e t ground speed i s a f u n c t i o n of achieved rate-of-climb, s o t h e time t o reach cloudbase a t t h e end of a segment w i l l be t h e same no matter how many thermals a r e used. E . The r e s u l t s of another micro-strategy a n a l y s i s a r e shown i n f i g u r e 7. Speed r a t i o , achieved ground speed with v e r t i c a l a i r motion between thermals normali z e d by achieved ground speed with no v e r t i c a l a i r motion between thermals, i s p l o t t e d a g a i n s t s i n k r a t i o , which i s t h e r a t i o of average v e r t i c a l a i r motion between l i f t sources t o achieved rate-of-climb i n l i f t f o r a v a r i e t y of l i f t c o n d i t i o n s , Negative s i n k r a t i o s a r e i n d i c a t i v e of what p i l o t s c a l l "reduced sink," i . e . , p o s i t i v e v e r t i c a l a i r motion t o o weak t o y i e l d a p o s i t i v e rate-ofclimb, but s t i l l r e s u l t i n a reduction of t h e r a t e a t which a l t i t u d e i s l o s t . The curves i n f i g u r e 7 a r e continued i n t h e negative s i n k r a t i o d i r e c t i o n u n t i l "zero sink" ( t h e p o i n t a t which t h e net a l t i t u d e l o s s during c r u i s i n g i s z e r o ) i s achieved. Speed r a t i o s g r e a t e r t h a n 1 can be i n t e r p r e t e d a s d e v i a t i o n d i s t a n c e r a t i o s . For example, a speed r a t i o of 1.1 implies t h a t a p i l o t could d e v i a t e from h i s s t r a i g h t l i n e course by 10% and s t i l l have t h e same achieved ground speed f o r a complete g l i d e / c i r c l e cycle. If t h e p i l o t d e v i a t e s from course any l e s s , f o r t h e i n d i c a t e d l i f t and s i n k c o n d i t i o n s , a n e t g a i n i n cross-country speed w i l l r e s u l t . These r e s u l t s r e i t e r a t e t h e n e c e s s i t y f o r minimizing s i n k r a t e by making minor d e v i a t i o n s during inter-thermal c r u i s e t o optimize t h e achieved cross-country performance. Macro-Strategy Macro-strategies involve t h e choice of courses t o a d e s i r e d goal r a t h e r t h a n t h e f l i g h t path s e l e c t i o n t o i n d i v i d u a l sources of lift. Macro-strategies a r e used t o f l y through regions of improved l i f t conditions. So once a macros t r a t e g y i s developed, an undetermined number of micro-strategies a r e used t o f l y t h e p r e s c r i b e d course. The r e s u l t s of t h e thermal macro-strategy model a r e shown i n non-dimensional form i n f i g u r e 8. Speed r a t i o i s p l o t t e d a s a function of lift r a t i o f o r a v a r i e t y of average lift conditions. As before, t h e non-dimensionalized speed r a t i o can be i n t e r p r e t e d a s t h e d e v i a t i o n d i s t a n c e r a t i o boundary required f o r equal time t o reach t h e goal. It i s immediately obvious,by comparing f i g u r e s 3 and 8 , t h a t decisions on t h e macro-strategy l e v e l have a much g r e a t e r impact upon t h e achieved cross-country soaring performance than decisions a t t h e micro-strategy l e v e l . A lift r a t i o of 2.0 y i e l d s more t h a n twice t h e speed r a t i o f o r a l l lift conditions f o r t h e macro-strategy case i n comparision with t h e micro-strategy case. The importance of choosing courses t h a t w i l l pass through more favorable s e c t o r s i s of g r e a t e r importance f o r weak conditions a s opposed t o moderate o r strong thermal conditions. As b e f o r e , although s a i l p l a n e performance and sink between thermals w i l l a f f e c t achieved groundspeed, t h e y have l i t t l e influence upon t h e p i l o t ' s decision of when t o make course deviations. S t r e e t Models Many times t h e p i l o t w i l l have occasion t o u t i l i z e convective lift while c r u i s i n g along course l i n e . This condition where t h e regions o r l i f t make up a s u b s t a n t i a l p o r t i o n of t h e f l i g h t p a t h i s u s u a l l y r e f e r r e d t o a s s t r e e t i n g . Making e f f e c t i v e use of t h e s e l a r g e regions of l i f t u s u a l l y involves speeding up i n s i n k and slowing down i n l i f t . There have been s e v e r a l analyses of t h i s mode of f l y i n g , f o r example, r e f e r e n c e s 2 through 5 and 1 through 1 4 . I n t h i s 1 paper, however, s i m p l i f i e d and conservative c o n t r o l laws have been implemented f o r studying thermal s t r e e t f l y i n g . For t h e most p a r t , t h e p i l o t f l i e s a t t h e speed f o r minimum s i n k r a t e while i n lift u n t i l cloud base i s reached, a t which time t h e p i l o t speeds up and f l i e s so a s t o maintain a l t i t u d e . The p i l o t c r u i s e s between l i f t a s d i c t a t e d by t h e equations of Appendix B. As it t u r n s o u t , t h i s procedure i s not f a r from t h e optimum a s demonstrated i n reference 5. Micro-Strategy The f i r s t model t o be considered i s shown i n f i g u r e 9 . The p i l o t has reached cloud base a t Point W and i s t r y i n g t o g e t t o Point Z. H must e decide i f f l y i n g s t r a i g h t t o Z o r d e v i a t i n g t o use t h e thermal s t r e e t along XY w i l l y i e l d t h e f a s t e s t time t o cloud base a t Point Z. It i s assumed t h a t t h e p i l o t i s capable of achieving an average rate-of-climb h a l f t h e rate-of-climb along XY equal t o 11 obtainable from c i r c l i n g i n thermals on course S = 0.5. Optimal i n t e r - l i f t c r u i s i n g speeds a r e obtained from Appendices A and B. The p i l o t u s e s t h e c o n t r o l law previously mentioned f o r c r u i s i n g i n t h e l i f t along XY. The r e s u l t s a r e shown i n f i g u r e s 1 0 and 1 f o r t h i s model. Boundaries of 1 d e v i a t i o n d i s t a n c e r a t i o , S / D , y i e l d i n g t h e same time t o cloudbase a t Z a r e p l o t t e d a g a i n s t average l i f t conditions f o r a v a r i e t y of s t r e e t l e n g t h r a t i o s , s'/D, i n f i g u r e 10. As a n t i c i p a t e d , t h e geometry of t h e s i t u a t i o n confronting t h e p i l o t . has a much g r e a t e r influence on h i s d e c i s i o n t h a n rate-of-climb, s a i l plane performance o r i n t e r - l i f t s i n k . Obviously, t h e g r e a t e r t h e l e n g t h of - XY (sv) , t h e g r e a t e r t h e d i s t a n c e t h e p i l o t should be w i l l i n g t o t r a n s v e r s e t o improve h i s cross-country soaring performance. Course d e v i a t i o n s f o r weak conditions can be about 10%longer t h a n f o r moderate t o strong conditions. A more convenient way f o r t h e p i l o t t o p i c t u r e how f a r of a course d e v i a t i o n i s warranted i s shown i n f i g u r e 1 . It i s a p l o t of curves showing 1 allowable spacing-distance r a t i o , y/D, a g a i n s t achieved rate-of-climb f o r s t r e e t l e n g t h r a t i o s of 0.2 and 0.8. Spacing d i s t a n c e r a t i o s of about 35% and 45% r e s p e c t i v e l y a r e j u s t i f i e d f o r weak cdnditions while spacing d i s t a n c e r a t i o s of about 25% and 35% a r e allowed f o r moderate t o strong thermal conditions. The second micro-strategy thermal s t r e e t model i s shown i n f i g u r e 12. The p i l o t has j u s t reached cloudbase i n a thermal a t X and d e s i r e s t o reach cloudbase a t t h e thermal a t point Z. He must decide between f l y i n g d i r e c t l y on course o r d e v i a t i n g t o use t h e s t r e e t along XY and t h e n fly&g t o Z. It i s assumed t h a t t h e average v e r t i c a l atmospheric v e l o c i t y along XY i s equivalent t o t h a t which would y i e l d h a l f t h e rate-of-climb from thermalling a t p o i n t s X o r Z. The p i l o t f l i e s along XY a t t h e speed which y i e l d s no n e t a l t i t u d e change and t h e n f l i e s along YZ a t t h e speed-to-fly i n d i c a t e d by t h e methods of Appendix A. - P r i o r t o analyzing t h e model, it i s necessary t o determine t h e optimum method of f l y i n g t h e s t r e e t and t h e s e n s i t i v i t y t o v a r i a t i o n s from t h e optimal procedures. Figure 1 3 i s a s e r i e s of p l o t s showing speed r a t i o , i . e . , t h e speed obtained by d e v i a t i n g t o f l y t h e s t r e e t at angle @ normalized by t h e speed obtained f l y i n g s t r a i g h t ahead i n t h e c l a s s i c a l c i r c l e / g l i d e manner, a s a function of breakaway d i s t a n c e r a t i o , R/D, f o r a v a r i e t y of s t r e e t angles. Speed r a t i o s g r e a t e r t h a n one correspond t o f l i g h t path extensions which would y i e l d a f a s t e r time t o cloudbase a t Z t h a n t h e straight-ahead choice. Figu r e 1 3 shows t h e following: 1) t h e r e a r e many ways t o f l y t h e thermal s t r e e t so a s t o o b t a i n a speed r a t i o g r e a t e r t h a n 1; 2 ) t h e r e e x i s t s , l o r thermal s t r e e t angles l e s s than about 60°, an optimal d i s t a n c e along t h e s t r e e t t o break away and begin f l y i n g toward Z t o optimize speed r a t i o ; 3) t h i s optimum breakaway d i s t a n c e i s highly s e n s i t i v e t o s t r e e t angle and not very s e n s i t i v e t o rate-of- climb; 4 ) t h e g r e a t e s t speed r a t i o s a r e obtained wizh small angles and weak l i f t c o n d i t i o n s ; and, 5 ) optimum speed r a t i o i s highly s e n s i t i v e t o breakaway point f o r weak l i f t and small s t r e e t angles. The breakaway point which y i e l d s equal time t o f l y t h e s t r e e t and t h e s t r a i g h t ahead g l i d e / c i r c l e c y c l e and t h e breakaway point f o r t h e optimumtime by f l y i n g t h e s t r e e t i s a n a l y t i c a l l y derived i n Appendix C . The l o c u s of breakaway p o i n t s f o r equal time ( s t r a i g h t ahead versus deviating along t h e i s t r e e t ) , R 1 / D , and optimum time, ,%*ID,s shown a s a f u n c t i o n of obtainable average rate-of-climb thermalling f o r a v a r i e t y of s t r e e t angles i n f i g u r e 1 4 . The optimum breakaway p o i n t from t h e s t r e e t i s not g r e a t l y a f f e c t e d by average rate-of-climb whereas t h e breakaway point f o r equal time can be extended along t h e s t r e e t s u b s t a n t i a l l y during weaker conditions a s compared with moderate t o strong l i f t conditions. As expected, f i g u r e 1 5 , which shows obtaina b l e speed r q t i o f o r a v a r i e t y of thermal s t r e e t a n g l e s , i n d i c a t e s t h a t t h e l a r g e s t g a i n s i n speed r a t i o from f l y i n g t h e thermal s t r e e t a r e p o s s i b l e with weak conditions and/or small thermal s t r e e t angles. The influence of s t r e e t angle on breakaway p o i n t s f o r optimum time and equal time i s shown i n f i g u r e 16. It i s c l e a r t h a t deviating along a s t r e e t i s not b e n e f i c i a l f o r s t r e e t angles of 60' o r more. I n a d d i t i o n , it can be observed t h a t t h e r e i s a very l a r g e margin between t h e locus of p o i n t s equal time and optimum time, i n d i c a t i n g t h a t t h e p i l o t can choose a l a r g e number of breakaway p o i n t s and s t i l l improve h i s performance. Even s o , it would probably be b e n e f i c i a l f o r t h e p i l o t t o study t h i s 'plot and develop r u l e s of thumb f o r deciding upon t h e optimum breakaway p o i n t given a geometry and l i f t condition. For example, neglect obtainable average rate-of-climb thermalling and j u s t decide by r e f e r e n c e t o s t r e e t angle--15' f l y an RID of 90%; 30' f l y an of 70%; 45' f l y an R I D of 50%; and 60' and g r e a t e r f l y s t r a i g h t ahead ignoring t h e s t r e e t . The magnitude of t h e b e n e f i t s t o be obtained from devia t i n g along s t r e e t s a s a function of s t r e e t angle i s demonstrated i n f i g u r e 17. Y D Macro-Strategy The equations f o r studying t h e e f f e c t of s t r e e t i n g a r e developed i n Appendix B. The macro-strategy model t o be considered i s b a s i c a l l y t h e same a s considered previously except t h a t some p o r t i o n of t h e course d e v i a t i o n i s under t h e influence of convective l i f t . As before, it i s assumed t h a t t h e average v e r t i c a l a i r v e l o c i t y encountered while c r u i s i n g i s equivalent t o h a l f t h e achieved rate-of-climb i n thermals. It i s assumed t h a t a f t e r a long enough s t r e t c h of cloud s t r e e t f l y i n g t h a t t h e net change i n a l t i t u d e i s constrained t o be zero. This i s v a l i d only at t h e macro-strategy l e v e l because t h e p i l o t might be w i l l i n g , i n t h e s h o r t term, t o t o l e r a t e slow l o s e s of a l t i t u d e i n order t o make progress along t h e d e s i r e d course. The required r a t i o of d i s t a n c e flown while climbing t o t o t a l d i s t a n c e covered i s p l o t t e d i n f i g u r e 18 a g a i n s t achieved rate-of-climb i n thermals f o r 3 s a i l p l a n e s . The s p o r t c l a s s s a i l p l a n e r e q u i r e s considerably more of t h e f l i g h t pa3h i n l i f t t h a n t h e o t h e r two c l a s s e s studied. It should a l s o be remembered t h a t t h i s assumes s t a t i c equilibrium f l i g h t and n e g l e c t s t h e performance d i f f e r e n c e s due t o t h e dynamics of p u l l i n g up and pushing over, which should i n c r e a s e t h e d i f f e r e n c e s between c l a s s e s . Some of t h e s e dynamic e f f e c t s have been s t u d i e d previously, f o r example, reference 14. The importance of d e v i a t i n g from course t o be a b l e t o c r u i s e while climbing i s shown i n f i g u r e 19. Speed r a t i o i s shown a s a function of r a t e of-climb achievable by thermalling f o r t h r e e r a t i o s of d i s t a n c e covered while climbing i n thermal s t r e e t s t o t o t a l d i s t a n c e covered. Here it i s assumed, t h a t i n order t o have no net change i n a l t i t u d e a f t e r a long period of t i m e , one of two approaches must be taken: 1) i f t h e r e i s more l i f t a v a i l a b l e t h a n necessary t o maintain a l t i t u d e , t h e excess w i l l be used t o i n c r e a s e speed at cloudbase u n t i l no net change i n a l t i t u d e w i l l occur; o r , 2 ) i f t h e r e i s not enough l i f t a v a i l a b l e t o maintain a l t i t u d e , t h e p i l o t w i l l c i r c l e t o cloudbase a t t h e end of t h e c r u i s e a t t h e average rate-of-climb expected i n thermals a t t h a t time. The f o u r t h curve i s a locus of p o i n t s obtained from f i g w e 18 showing t h e achieved performance i f t h e r a t i o of d i s t a n c e covered climbing t o t o t a l d i s t a n c e covered were a t t h e c o r r e c t value t o y i e l d no n e t a l t i t u d e change from climbing by c r u i s i n g a t t h e speed f o r minimum s i n k and c r u i s i n g between lift a t t h e a p p r o p r i a t e speed-to-fly ( ~ p p e n d i xB). Several assumptions have been made during t h e development of t h e s t r e e t f l y i n g analyses which need t o be considered. The authors have s t u d i e d t h e influence of s a i l p l a n e performance and inter-thermal s i n k and found t h a t , although t h e cross-country performance may be s i g n i f i c a n t l y a f f e c t e d , t h e p i l o t ' s d e c i s i o n i n regards t o non-dimensionalized course d e v i a t i o n s i s not a l t e r e d . The assumption t h a t t h e average v e r t i c a l atmospheric v e l o c i t y encountered while climbing i s 50% t h a t of t h e v e r t i c a l a i r v e l o c i t y obtainable i n thermals at t h e time does i n f l u e n c e various p a r t s of t h e a n a l y s i s . It i s f e l t , however, t h a t t h i s does not have a major ?Impact upon t h e t r e n d s demons t r a t e d i n t h i s paper. Furthermore, neglecting winds i n t h e s e analyses probably would a f f e c t t h e d e c i s i o n s a p i l o t would make during cross-country s t r e e t f l y i n g . Thermal s t r e e t s a r e u s u a l l y f o s t e r e d by g e n t l e winds and t h e i n c l u s i o n of t h i s f a c t o r warrants f u r t h e r research. As exemplified i n reference 1 5 , t h e p i l o t would probably be w i l l i n g t o make f u r t h e r progress a g a i n s t t h e wind i n s t r e e t s t h a n t h e optimum s o l u t i o n s f o r s t i l l a i r r e p o r t e d herein. S M A Y O RESULTS U MR F Several t r e n d s came out of t h e a n a l y s i s of t h e thermal models i n t h i s paper. It i s apparent t h a t d e c i s i o n s t o d e v i a t e from course a r e of much g r e a t e r s i g n i f i c a n c e a t t h e macro-strategy l e v e l t h a n t h e micro-strategy l e v e l . A p i l o t can enhance h i s performance by choosing s e c t o r s of t h e sky t o improve h i s achieved rate-of-climb. A t both t h e micro- and macro-strategy l e v e l s it i s c l e a r t h a t s u b s t a n t i a l d e v i a t i o n s from course may be warranted f o r weak lift whereas when t h e thermal conditions a r e moderate o r s t r o n g , only very minor course d e v i a t i o n s can be j u s t i f i e d . I n a l l cases, cross-country soaring performance can be augmented by making course deviations with t h e smallest p o s s i b l e d e v i a t i o n angles. This i n d i c a t e s t h a t p i l o t s should make course change d e c i s i o n s a s soon a s p o s s i b l e and be w i l l i n g t o ignore l i f t not near t h e course, which i s e s p e c i a l l y t r u e f o r moderate o r strong lift. A l a r g e improvement i n average cross-country speed i s a t t a i n a b l e by c r u i s i n g while climbing, such a s i n s t r e e t i n g conditions. I n t h e s t r e e t models considered, t h e percentage of t h e f l i g h t p a t h i n lift had a b i g i n f l u e n c e upon t h e achieved performance and p i l o t ' s d e c i s i o n c r i t e r i a . I n t h e case of t h e p a r a l l e l s t r e e t micro-strategy model, s t r e e t s with spacing d i s t a n c e r a t i o s of 30% o r l e s s could be j u s t i f i e d t o i n c r e a s e t h e a t t a i n e d cross-country speed. Deviation d i s t a n c e r a t i o s can be extended by about 10% f o r weak c o n d i t i o n s a s compared t o moderate o r strong lift conditions. The study of s t r e e t s at an angle t o t h e course l i n e r e s u l t s i n some i n t e r e s t i n g observations. There e x i s t s an optimum point of breakaway f r o m t h e s t r e e t t o minimize t h e time r e q u i r e d t o reach t h e t o p of t h e next thermal. This breakaway point i s p r i m a r i l y a function of s t r e e t angle. Although t h e optimum augmentation of speed i s highly s e n s i t i v e t o breakaway point f o r weak conditions a t small s t r e e t angles, f o r most combinations of s t r e e t geometry and l i f t c o n d i t i o n s t h e r e e x i s t s a range of p o s s i b l e s o l u t i o n s which y i e l d s a f a s t e r time t h a n t h e s t r a i g h t ahead g l i d e / c i r c l e cycle. It can be shown t h a t cloud s t r e e t s at a n angle g r e a t e r t h a n 60 degrees a r e not b e n e f i c i a l and should not be used t o improve average ground speed. CONCLUDING R M R S E AK Several assumptions have been made which may a f f e c t t h e a p p l i c a b i l i t y of t h e r e s u l t s r e p o r t e d upon herein. A premise f o r a l l t h e cases s t u d i e d was t h a t s u r v i v a b i l i t y i s ignored. Speed was considered a s t h e performance f u n c t i o n t o be optimized whereas i f t h e p i l o t was concerned about not being a b l e t o complete t h e f l i g h t , a l t i t u d e conservation would be of prime importance. A c o n s t r a i n t f o r each e x e r c i s e was t h a t n e t a l t i t u d e l o s s would be zero; hence, t h e r e s u l t s a r e not a p p l i c a b l e t o f i n a l g l i d e s . A p o s s i b l e focus of f u t u r e r e s e a r c h may be t o study t h e impact of course d e v i a t i o n s upon f i n a l g l i d e s . Also, it was a r b i t r a r i l y assumed f o r t h e s t r e e t models t h a t average lift i n a s t r e e t was approximately 50% of t h e l i f t found i n thermals a t t h a t time. This has an obvious impact upon t h e performance gains of d e v i a t i n g t o use s t r e e t s , but g e n e r a l t r e n d s of t h e analyses a r e s t i l l v a l i d . A s i g n i f i c a n t l i m i t a t i o n of t h e approach presented i n t h i s paper i s t h e assumption implied by considering lift a s s o l e l y a i r referenced. This negates t h e i n f l u e n c e of winds f o r reaching ground referenced g o a l s o r l i f t sources. It i s expected t h a t d e c i s i o n s reached during t h e s t r e e t a n a l y s i s w i l l be s h i f t e d i n t o t h e wind. For example, t h e p i l o t w i l l probably want t o make more progress i n t o t h e wind while i n lift t h a n otherwise i n d i c a t e d by t h e breakaway point s o l u t i o n s . Since thermal s t r e e t s a r e u s u a l l y formed i n l i g h t t o medium winds, t h e i n c l u s i o n of winds i n t h e foregoing analyses i s c u r r e n t l y being undertaken by t h e authors. The models developed i n t h i s paper a r e s i m p l i f i e d and general i n n a t u r e . It i s hoped t h a t t h e y o r a l i n e a r superposition of more t h a n one of them a r e r e p r e s e n t a t i v e of t y p i c a l l i f t geometries a p i l o t may encounter during a crosscountry soaring f l i g h t . The r e s u l t s presented i n t h i s paper a r e not meant t o be cockpit a i d s f o r i n t e r p r e t i n g t h e most promising f l i g h t paths. I n s t e a d , t h e y i l l u s t r a t e t h e d e s i r a b i l i t y and i n d i c a t e an approach, f o r a n a l y t i c a l l y studying t y p i c a l course s e l e c t i o n d e c i s i o n s beforehand, enabling t h e p i l o t t o more e f f e c t i v e l y evaluate t h e p o t e n t i a l t r a d e o f f s f o r a r r i v i n g upon a more optimal s o l u t i o n while i n f l i g h t . APPENDIX A O TM M SPEED-TO-FLY CALCULATIONS F R T E M L M D L PI U O HR A O ES To f a c i l i t a t e t h e c a l c u l a t i o n s required i n t h i s paper and i n o t h e r i n v e s t i g a t i o n s ( r e f e r e n c e 1 6 ) , a n a l y t i c a l expressions needed t o be derived f o r ) Although t h e t h e f a m i l i a r inter-thermal speed-t o-fly s o l u t i o n ( r e f e r e n c e 1 defining equations a r e e a s i l y derived and have been presented i n numerous p u b l i c a t i o n s ( f o r example, r e f e r e n c e s 3 and 5 ) , a closed form a n a l y t i c a l s o l u t i o n f o r c a l c u l a t i n g numerical r e s u l t s i s not g e n e r a l l y a v a i l a b l e i n t h e l i t e r a t u r e and i s given below. The g r a p h i c a l i n t e r p r e t a t i o n of t h e r e s u l t s which i s widely used by p i l o t s i s shown i n f i g u r e 20. The f i r s t case considered i s where t h e s a i l p l a n e performance i s known and i s assumed t o be p a r a b o l i c ; t h e average rate-of-climb i n t h e next thermal i s known; t h e r a t i o of s i n k r a t e between thermals t o rate-of-climb i n them i s known; and t h e optimum speed-tof l y between t h e thermals and t h e corresponding average ground speed i s d e s i r e d . The s a i l p l a n e performance r e l a t i o n i s : . where The defining equation can be found from f i g u r e 20 o r by d e r i v a t i o n t o be ) By applying t h e d e f i n i t i o n of s i n k r a t i o , 11, and u t i l i z i n g equations ( ~ 1,(k21, and C~37, equation ( ~ 4becomes t h e following f o u r t h degree polynomial: ) The r o o t o f ' i n t e r e s t was found t o be c a l c u l a t e d by t h e following r e l a t i o n s : The average ground speed f o r a complete g l i d e / c i r c l e cycle i s given by The second case considered i s where t h e s a i l p l a n e performance i s known i n t h e form a s b e f o r e , t h e s i n k r a t i o can be assumed, t h e d e s i r e d average ground speed i s known, and t h e optimum speed-to-fly and t h e r e q u i r e d rate-of-climb given t h e preceding a r e t o be found. The defining equation can be e a s i l y a t t a i n e d from f i g u r e 20 by equating t h e slope of t h e tangent l i n e , t o t h e slope sf t h e s a i l p l a n e p o l a r found by d i f f e r e n t i a t i n g equation [Al) The d e f i n i n g equation f o r t h e optimum s o l u t i o n becomes Use Newton's method f o r estimating r o o t s . Let then, A good i n i t i a l guess f o r Vi W could a r b i t r a r i l y be Vo + 5(1 + TI).; A fair amount of accuracy can be obtained with j u s t f i v e i t e r a t i o n s i n t h i s manner. The required rate-of-climb f o r an average ground speed of VG i s given by t h e following r e l a t i o n : O TM M SPEED-TO-FLY CALCULATIONS F R T E M L STREETS PI U O HR A The defining r e l a t i o n s and a geometric i n t e r p r e t a t i o n ( f i g u r e 21) of t h e optimum speed-to-fly between lif't,given t h e s a i l p l a n e performance, t h e i n t e r lirt s i n k r a t i o , t h e rate-of-climb and t h e speed a t which t h e l i f t i s t r a n s versed ( v C L ) , were presented i n r e f e r e n c e 5. The defining equation i s , 2 Assuming a parabolic p o l a r , equations ( ~ l () ~ ) , and (A3 ) , t h e following f i f t h degree polynomial can be derived Newton's i t e r a t i v e method of egtimating r e a l r o o t s was used t o solve t h e f i r t h degree equation f o r t h e d e s i r e d r o o t . Let then A good value f o r t h e i n i t i a l guess of * Vi might a r b i t r a r i l y be t h e s o l u t i o n t o t h e thermal model problem developed i n Appendix A. A near optimum value f o r t h e climbing v e l o c i t y , VCL, would be t h e speed f o r minimum sink r a t e , 'I* MN The average ground speed f o r a complete c y c l e , a s p i c t u r e d i n f i g u r e 22, i s c a l c u l a t e d a s follows: These equations were derived assuming t h a t t h e n e t a l t i t u d e change a f t e r each cruise/climb c y c l e was zero. Referring t o f i g u r e 22, a r e l a t i o n can be derived t o y i e l d t h e proportion of t h e f l i g h t p a t h which must be under t h e influence of lift t o r e s u l t i n no net a l t i t u d e change a f t e r each c y c l e kCR/hCL= 1) S t a r t i n g with and The following equation i s derived A p l o t of R /R C L f i g u r e 18. a s a function of is f o r t h r e e s a i l p l a n e s i s shown i n I n t h e event t h a t t h e r e i s a l a r g e r proportion of t h e f l i g h t p a t h under t h e influence of l i f t t h a n required f o r no n e t a l t i t u d e change, t h e n t h e p i l o t needs t o c r u i s e a t a v e l o c i t y which g i v e s a s i n k r a t e equal t o t h e v e r t i c a l a i r veloc i t y t o keep from climbing i n t o t h e cloud. This a i r s p e e d can be c a l c u l a t e d a s follows : APPENDIX C CALCULATION O BREAK-AWAY POINTS F O F RM H O RE A C O D STREET AT AN ANGLE TO T E DESIRED C U S LU Using t h e geometry defined i n f i g u r e 1 2 , a r e l a t i o n can be defined t o determine t h e a p p r o p r i a t e breakaway p o i n t s i n terms of s a i l p l a n e performance parameters and atmospheric l i f t c o n d i t i o n s . The f i r s t case considered i s f i n d i n g t h e breakaway p o i n t , t h e d i s t a n c e t o be flown along t h e s t r e e t , 2 , t o y i e l d t h e same time t o t h e t o p of t h e thermal a s Z a s by f l y i n g d i r e c t l y from X t o Z. The time t o f l y along t h e s t r e e t , f l y t o Z and t h e n climb t o cloudbase a t Z i s given by: then Using t h e Law of Cosines n2 = D Squaring equation L C ~ )i e l d s y 2 + a2 - 2 ~ cos 2 + S u b s t i t u t i n g equation ( ~ 5 i)n t o ( ~ 6 gives ) From t h e d e f i n i t i o n of completing e i t h e r r o u t e i n equal time and from t h e assumptions of Appendix A , Vn and V a r e equal s i n c e t h e y a r e both calcuD l a t e d based on t h e thermal a t Z , t h e following can be w r i t t e n : S u b s t i t u t i n g ( ~ 8 i)n t o ( ~ 7 r)e s u l t s i n I f we d e f i n e t h e following c o n s t a n t , ) t h e n equation ( ~ 9 can be solved f o r t h e r o o t s a s follows The second case considered i s t h e s o l u t i o n f o r t h e non-dimensionalized breakaway p o i n t , R* -, D f o r minimum time t o reach t h e t o p of t h e thermal a t Z. S t a r t i n g with equation ( ~ 4 and s u b s t i t u t i n g t h e square r o o t of equation ( ~ 5 ) ) i n t o i t , t h e following function i s obtained: The minimum time s o l u t i o n f o r T i s found by d i f f e r e n t i a t i n g with r e s p e c t t o R n R and s e t t i n g it t o zero. 3 38 d -T dR R n r o = -I + - K (D2 - D cos 0 + t2 - &a cos 0) 4 R 1 (~14) Rearranging ((214) and s u b s t i t u t i n g i n equation following quadratic equation : ( ~ 1 0 ) gives t h e The root of i n t e r e s t from equation ( ~ 1 5 ) is R - = cos 0 D lk v' K12 (1 - cos2 0 ) (1- Kt2 ) REFERENCES McCready , Paul B. : Mozdiniewicz, W.: pp. 26-27. I r v i n g , F. G . : "An Optimal Airspeed S e l e c t o r , " "Cloud S t r e e t Soaring, May 1954. lying," Soaring, Vol. 35, No. 1 , 1971 1 "Cloud S t r e e t Flying," N S CR-2315, 1972, pp. 274-286. AA Gedeon, J o z s e f : "Dynamic Analysis of Dolphin-Style Thermal Cross-Country Flying," Technical Soaring, Vol. 3, No. 1, 1973. Abzug, Malcolm J . : "A Speed Ring f o r Cloud S t r e e t Flying," Soaring, Vol. 1 4 , NO. 1, 1976. Technical I r v i n g , F. G . : he Effect of E r r o r s i n Inter-Thermal Speed on t h e Average Cross-Country Speed," Technical Soaring, Vol. 3, No. 2 , 1973. Schuemann, W i l : h he P r i c e You Pay f o r MacCready speeds," Presented a t t h e 1972 Symposium on Competitive Soaring, Soaring Symposia, Cumberland, Maryland, 1973. T e t e r , Michael P.: "Competition S t r a t e g y and T a c t i c s f o r ~ e g i n n e r s , " Soaring, August, 1975, pp. 26-28. Johnson, Richard H. : "cross-country Soaring," American Soaring Handbook, Soaring Society of America, 1962, Chapter 6, pg. 8 . Alleman, Rudolf T. : " ~ o t e son t h e Dilemma of Deviation from Course ~ i n , Ie 1 Soaring, March, 1962, pp. 12-14. Arho, Risto: "Optimal Dolphin Soaring a s a V a r i a t i o n ~ r o b l e m , " Technical Soaring, Vol. 3, No. 1, 1973. Arho, R i s t o : "Some Notes on Soaring F l i g h t Optimization Theory," Soaring, Vol. 1 4 , No. 2 , 1974. Technical Metzger, Darryl E. and Hedrick, J. Karl: "Optimal F l i g h t Paths f o r Soaring F l i g h t ," Proceedings of Second I n t e r n a t i o n a l Symposium on t h e Technology and Science of Low-Speed and Mot o r l e s s F l i g h t , Cambridge, Mass. , 1974. Gedeon, J o z s e f : h he Influence of S a i l p l a n e Performance and Thermal Strength on Optimal Dolphin-Flight T r a n s i t i o n P i l o t i n g Techniques," Presented a t t h e XV OSTIV Congress, Rayskala, Finland, 1976. Reichman, Helmut; Cross-Country Soaring: A Handbook f o r Performance and Competitive Soaring, Thompson P u b l i c a t i o n s , Santa Monica, C a l i f o r n i a , 1978. Sliwa, Steven M.: "A Computer Simulation of Cross-Country Soaring," Soaring, January 1976. Figure 1. - Micro-strategy thermal model. Average rate-of-climb, h,, knots = 38 Vo = 50 knots " = 0.2 - = 0.5 di 0 1.0 I 1.1 I r I 1.2 ~ 1.3 I I 1.4 I 1.5 I I 1.6 ~ I 1.7 I l 1.8 , I 1.3 , I 210 , Deviation distance r a t i o , Figure 2. - Required rate-of-climb a t Y a s a function of deviation d i s t a n c e r a t i o f o r micro-strategy thermal model. Average rate-of-climb. hZ, knots s = 0.2 0 L i f t ratio, Figure 3. - Deviation d i s t a n c e r a t i o a s a function o f l i f t r a t i o f o r micros t r a t e g y thermal model. Average rate-of-climb, hZ, knots Vo (knots) 50 42 52 0 1 1 1 , 1 , 1 l0 . 1.1 12 . 13 . , ~ ( 1.9 1!5 1!6 1!7 1!8 1!3 2!0 Deviation distance r a t i o , Figure 4. - Required rate-of-climb a t Y versus deviation d i s t a n c e r a t i o i l l u s t r a t i n g impact o f s a i l p l a n e performance f o r micro-strategy thermal model. 342 Average rate-of-climb, h,, knots Sink r a t i o , = 38 Yo = 50 knots $ = 0.5 1.0 11 . 1.2 13 . ! ' . I 15 . 1.6 1.7 1.8 13 . 2.0 Oeviation distance r a t i o , Figure 5 . - Required rate-of-climb a t Y versus deviation d i s t a n c e r a t i o i l l u s t r a t i n g impact of s i n k r , a t i o f o r micro-strat egy thermal model. Average rate-of-climb, hz, knots (LID),, = 38 Yo = 50 knots s = 0.2 t I 0 1.0 1.1 I I 1.2 I I 1.3 I l.q I I 15 . I I I 16 . Oeviation distance r a t i o , $ I I 1.7 I I I l ~ 1.8 13 . 2.0 I Figure 6. - Required rate-of-climb a t Y i l l u s t r a t i n g impact of d l / d versus deviation d i s t a n c e r a t i o f o r micro-strategy thermal model. h , knots .6 -.q 1 1 .5 -.5 -.3 1 -.2 , 1 -.l , 0 1 , .1 1 .2 , ~ . , 3 9 ~ , .5 ~ , ~ , ~ Sink r a t i o , Q Figure 7 . - Speed r a t i o versus s i n k ' r a t i o f o r micro-strategy thermal a n a l y s i s . 1.7 " = 38 Average rate-of-climb, h , knots Vo = 50 knots " = 0.2 1 1 18 . , 20 . 1 , 22 . 1 2.9 1 ~ 26 . 1 2.8 ~ , 310 ~ , L i f t ratio, 5 Figure 8. - Speed r a t i o versus l i f ' t r a t i o f o r macro-strategy thermal model. Figure 9. - Micro-strategy model with thermal s t r e e t p a r a l l e l t o course l i n e . (L/O),ax = 38 Vo = 50 knots Average rate-of-climb, h, knots Figure 10. - Deviation d i s t a n c e r a t i o versus average rate-of-climb thermal s t r e e t micro-strategy model. for parallel 1 .o.9 (L/O)m, . 8 = 38 * a 0 . 7 - v , n = SO knots = 0.2 Ts - 0.5 1 0 I I 2 I I Y I 6 I I I I I I I I I I I I I I I 8 10 12 llt 16 18 20 Average rate-of-climb, h, knots Figure 1 . 1 - Spacing d i s t a n c e r a t i o versus average rate-of-climb thermal s t r e e t micro-strategy model. for parallel Figure 1 2 . - Micro-strategy model with thermal s t r e e t a t an angle with course line. Average r a t e - o f - c l imb, h, knots 0 1 0 1 1 a1 1 1 1 1 , 1 , 1 , 1 , 1 , 1 , 1 1 1 1 1 1 1 1 1 1 1 1 .2 -3 .9 .5 .6 -7 .8 . 9 1.0 1.1 1.2 1.3 1.9 l!5 Breakaway distance r a t i o , -Si. 9. 2 -0 1.8 1.6 Average rate-of-ci imb, h, knots 1.9 0 ,1.2 0 . r CI m 1.0 L u a J l n P .8 .6 Breakaway distance r a t i o , 9. Figure 13. - Speed r a t i o versus breakaway d i s t a n c e r a t i o f o r a thermal s t r e e t a t an angle t o course l i n e . 1.6 1.9 Average rate-of-cl imb, h, knots .q .2 0 0 I I -i l " .2 0.5 I I I I I I I I ~ I I I I I ~ ~ ~ -1 .3 .'-I .5 .6 .7 .8 .9 1.0 II 1.1 1.2 1.3 1.9 l(5 Breakaway distance r a t i o , Breakaway d i Stance r a t i o , L ! Figure 13. - Concluded. Breakaway d i s t a n c e r a t i o f o r equal time, W . r % I 0 1.2- w L 6 w U s c, W . P 1.0- 6 .8.6 -0 1 w m L .I .2 1.6 Optimum breakaway d i s t a n c e r a t i o , a* D (L/DIma, = 38 q = 0.2 Vo = 50 knots h 2 = 0.5 h Average r a t e - o f - c l i m b , h, knots Vo = 50 knots 0 s m ll = 0.2 U) distance r a t i o f o r equal time, $0 Optimum breakaway distance r a t i o , m Average r a t e - o f - c l imb, h , knots Figure 14. - Breakaway d i s t a n c e r a t i o s f o r equal time and optimum time versus average rate-of-climb line. f o r a thermal s t r e e t a t an angle t o course '4 .o Thermal street angle, 4 , degrees Average rate-of-cl imb, h, knots Figure 15. - Speed r a t i o versus rate-of-climb f o r a v a r i e t y of s t r e e t angles f o r a thermal s t r e e t a t an angle t o course l i n e . Average rate-of-cl imb, h, knots 1 --- -- 5 - 10 = 38 Vo = 50 knots .9 0 I 10 n = 0.2 .2 0 I I 20 I I 30 I I 90 I I 50 I I 60 $ 70 80 30 100 Thermal s t r e e t angle, , degrees Figure 16. - Influence of thermal s t r e e t angle upon breakaway d i s t a n c e r a t i o f o r a thermal s t r e e t a t an angle t o course l i n e . : . 5 0 f , l , l , l , l , l , l , , , l , l , , 10 20 30 40 50 60 70 80 30 100 Thermal street angle, 6 , degrees Figure 17. - Speed r a t i o versus thermal s t r e e t angle f o r micro-strategy thermal s t r e e t model. Average rate-of-cl imb, h, knots Figurel8. - Required climb d i s t a n c e r a t i o versus average ratecof-climb f o r thermal s t r e e t macro-strategy a n a l y s i s . 3.02.8- Required climb distance r a t i o , R (L/DIma, = 38 R~~ V, /.25 = 50 knots rl = 0.2 1.21 .o I /' I 0 ' ; 2 I I 3 I I I 5 ~ ' 6 I ' 7 I ' 8 I ' 9 1 8 10 1 11 b I 1 12 t1 I 13 I I 1'4 ' l 15 Average r a t e - o f - c l imb, h, knots Figure19 - Speede gr ay t model. average rate-of-climb i o versus strat f o r thermal s t r e e t macro- Figure 20. - S a i l p l a n e polar thermal soaring . showing optimum speed-to-fly constructions f o r Figure 21. - S a i l p l a n e p o l a r showing optimum speed-to-fly thermal s t r e e t soaring. constructions f o r Figure 22. - Flight p r o f i l e of a glide/climb cycle f o r thermal s t r e e t soaring- 353 Page intentionally left blank ON GLOBAL OPTIMAL SAILPLANE FLIGHT STRATEGY G. Sander and F. X. L i t t U n i v e r s i t y o f ~ i s g e ,Belgium S M AY U MR The p r e s e n t paper c o n c e n t r a t e s on t h e d e r i v a t i o n and i n t e p r e t a t i o n of t h e necessary c o n d i t i o n s t h a t a s a i l p l a n e cross-country f l i g h t has t o s a t i s f y t o achieve t h e maximum g l o b a l f l i g h t speed. Simple r u l e s a r e obtained f o r t w o s p e c i f i c m e t e o r o l o g i c a l models. The f i r s t one uses concentrated l i f t s of var ious s t r e n g t h s and unequal d i s t a n c e . The second one t a k e s i n t o account f i n i t e , non-uniform spa= amplitudes f o r t h e l i f t s and allows, t h e r e f o r e , f o r dolphins t y l e f l i g h t . I n both models, a l t i t u d e c o n s t r a i n t s c o n s i s t i n g of upper and lower l i m i t s a r e shown to be e s s e n t i a l t o model r e a l i s t i c problems. Numerical examples i l l u s t r a t e t h e d i f f e r e n c e w i t h e x i s t i n g techniques based on l o c a l optimality conditions. INTRODUCTION The problems a s s o c i a t e d w i t h t h e o p t i m i z a t i o n of s a i l p l a n e f l i g h t p a t h s t o achieve maximum cross-country speeds have r e c e n t l y received s p e c i a l a t t e n t i o n i n t h e l i t e r a t u r e . This has been s t i m u l a t e d by t h e modern competitive s o a r i n g which c o n s i s t s almost e x c l u s i v e l y i n r a c i n g and by t h e development of high performance s a i l p l a n e s allowing f o r new, h i g h l y e f f i c i e n t f l i g h t techniques. S t a r t i n g w i t h t h e now c l a s s i c a l MacCready [ 1 1 r e s u l t s , most of t h e i n v e s t i g a t i o n s have been concerned e s s e n t i a l l y with l o c a l o p t i m i z a t i o n problems, t h a t is, f i n d i n g t h e optimum f l i g h t s t r a t e g y f o r v a r i o u s s p e c i f i c s i t u a t i o n s encountered i n a s h o r t s e c t i o n of a f l i g h t [l t o 101. I n r e c e n t papers [2, 4, 5, 8 1 t h e optimum speeds t o f l y i n a v a r i e t y of atmospheric v e r t i c a l v e l o c i t y d i s t r i b u t i o n s have been determined from t h e b a s i c assumption t h a t t h e corresponding f l i g h t segments had t o be crossed with zero n e t a l t i t u d e l o s s . Conditions under which a t r a n s i t i o n from t h e c i r c l i n g mode of climb t o t h e dolphin o r e s s i n g modes has t o be decided have Although such r e s u l t s y i e l d extremely valuable g u i d e l i n e s been examined [4 1 f o r s e l e c t i n g a f l i g h t s t r a t e g y , they o n l y o p t i m i z e t h e speed over a l i m i t e d p o r t i o n of t h e t o t a l f l i g h t . . I t is w e l l known, however, i n o p t i m i z a t i o n t h e o r y t h a t a s u c c e s s i o n of l o c a l l y optimum s o l u t i o n s does n o t , i n g e n e r a l , l e a d t o a g l o b a l l y optimum r e s u l t [ill. It is worth p o i n t i n g o u t t h a t a g l o b a l l y optimum f l i g h t s t r a t e g y can only be determined i f t h e assumption is made t h a t t h e d i s t r i b u t i o n of atmospheric v e l o c i t i e s over t h e whole f l i g h t path is known i n advance and is independent of t i m e . Although t h i s is never achieved i n p r a c t i c e , it is f e l t t h a t t h e d e r i v a t i o n of g l o b a l o p t i m a l i t y c o n d i t i o n s a l l w s f o r a new i n s i g h t i n t o t h e s a i l p l a n e f l i g h t technique by g i v i n g a p o s t e r i o r i t h e decis i o n s t h a t t h e p i l o t should have taken and t h e i n f l u e n c e of f a c t o r s t h a t have been up t o now n e g l e c t e d i n t h e a n a l y s i s , such a s t h e e f f e c t of the unequal d i s t r i b u t i o n and s t r e n g t h of t h e l i f t s and t h e e f f e c t of minimum and maximum a l t i t u d e l i m i t a t i o n s . Such a l t i t u d e c o n s t r a i n t s r e v e a l an e s s e n t i a l i n g r e d i e n t i n t h e formulation. Their n e c e s s i t y appears a s follows. I f they a r e absent and i f t h e l i f t s a r e of unequal s t r e n g t h , t h e g l o b a l l y optimum s o l u t i o n t u r n s o u t t o be t r i v i a l and c o n s i s t s of a g l i d e u n t i l t h e maximum l i f t e x i s t i n g along t h e p a t h is reached and a climb t o an a l t i t u d e t h a t allows completion of t h e t a s k , t h e speed on both segments corresponding to t h e MacCready s e t t i n g f o r t h a t s t r o n g e s t l i f t [ I 21 . The p r e s e n t paper provides simple r u l e s f o r g l o b a l o p t i m a l i t y f o r t w o simple atmospheric models. These appear to be i n agreement w i t h t h e techniques i n t u i t i v e l y used by good competition p i l o t s . I n both atmospheric models used i n t h e following, t h e h o r i z o n t a l (wind) v e l o c i t y of t h e a i r mass is assumed t o be e i t h e r zero o r t o be taken i n t o account by an a p p r o p r i a t e m o d i f i c a t i o n of t h e p o l a r equation. The v e r t i c a l v e l o c i t y ( l i f t ) of t h e a i r mass c i is defined by t h e so-called n e t t o value. I t is c o n s t a n t i n t h e v e r t i c a l d i r e c t i o n between t h e a l t i t u d e l i m i t s . The f l i g h t p a t h is supposed t o be c o n s t i t u t e d by a succession of segments of v a r i a b l e l e n g t h s i n which t h e a i r mass e x h i b i t s v e r t i c a l v e l o c i t i e s ci which a r e c o n s t a n t along a given segment but v a r y from one t o t h e o t h e r . The a l t i t u d e c o n s t r a i n t s c o n s i s t i n c o n s t a n t upper and lower l i m i t s denoted h and h. Note t h a t v a r i a b l e a l t i t u d e l i m i t s could be e a s i l y incorporated. For siGplici t y t h e lower a l t i t u d e l i m i t is taken a s z e r o (h = 0 ) . Concentrated L i f t Model I n a f i r s t model, t h e l e n g t h s of t h e segments where a p o s i t i v e v e r t i c a l v e l o c i t y is encountered a r e supposed t o be n e g l i g i b l e , t h a t is, t h e l i f t s a r e considered a s concentrated. The a i r mass between t h e l i f t s is supposed t o be s t a b l e . Climbing is t h e r e f o r e achieved o n l y by c i r c l i n g a t f i x e d l o c a t i o n s corresponding t o t h e l i f t s . I f a l i f t is not used, its c r o s s i n g is supposed n o t t o a f f e c t t h e g l i d e and t h e dynamical a s p e c t s of t h e t r a n s i t i o n from g l i d i n g t o c i r c l i n g a r e n o t considered. The model is i l l u s t r a t e d i n f i g u r e 1 which is drawn i n t h e v e r t i c a l plane. The s i g n s of t h e v e l o c i t i e s a r e taken according t o t h e p o s i t i v e s i g n of t h e axes . The f l i g h t c o n s i s t s of a s u c c e s s i o n of climbs i n t h e s e l e c t e d l i f t s f o l lowed by a g l i d e a t c o n s t a n t speed which p o s s i b l y c r o s s e s discarded l i f t s . The p i l o t c o n t r o l s t h e s e l e c t i o n of t h e l i f t s where he decides t o g a i n a l t i t u d e , t h e amount of a l t i t u d e gained, and t h e speed t o f l y between t h e s e l e c t e d l i f t s . During t h e climbs t h e v e r t i c a l speed of t h e s a i l p l a n e is simply taken a s t h e a l g e b r a i c s m of t h e a i r mass v e r t i c a l v e l o c i t y and t h e minimum s i n k i n g speed u of t h e g l i d e r . The i n c r e a s e of s i n k i n g speed due t o v a r i a t i o n s i n bank a n g l e is not considered e x p l i c i t e l y and should be i n c o r p o r a t e d i n t h e polar d e f i n i t i o n . The v a r i a t i o n s i n a l t i t u d e s a r e g i v e n by where t h e s i n k i n g speed w ( v i ) is g i v e n by t h e p o l a r e q u a t i o n . q u a d r a t i c approximation has been used f o r numerical examples The c l a s s i c a l The t i m e s p e n t a t each s t e p c o n s i s t s of t h e sum of t h e t i m e used i n climbing and t h e t r a n s i t i o n t i m e between l i f t s The achieved r a t e of c l i m b i s t h e sum of t h e a i r mass v e l o c i t y C i and t h e g l i d e r minimum s i n k i n g speed y,. The c o n s t r a i n t s on t h e a l t i t u d e and a l t i t u d e g a i n s a r e e x p r e s s e d by A control constraint Ahi 2 0 b = O (4) I n i t i a l and t e r m i n a l c o n s t r a i n t s , say Altitude constraints a t each s t e p h2n=0 h2i+l 5 6 ; h2i > 0 = i = O,l,...,n-1 (6) The mathematical problem can be t r e a t e d by t h e c l a s s i c a l d i s c r e t e o p t i m a l cont r o l t h e o r y [ 111 and c o n s i s t s of f i n d i n g t h e sequence (s) Ahor vOI Ahl # vl # , vn-1 which s a t i s f y t h e r e l a t i o n s (11, ( 4 ) , ( 5 ) , and (6) and minimizes t h e t o t a l f l i g h t time ... D i s t r i b u t e d L i f t Model I n t h i s second model, t h e l e n g t h of t h e segments is always f i n i t e and nonvanishing. Between l i f t s t h e a i r mass may p r e s e n t negative v e r t i c a l v e l o c i t i e s s o t h a t any d e s i r e d a i r mass v e r t i c a l balance can be achieved. The model is i l l u s t r a t e d i n f i g u r e 2. An important d i f f e r e n c e with t h e preceding model is brought by t h e p o s s i b i l i t y of c r o s s i n g a l i f t i n g segment a t a h o r i z o n t a l speed v i less than t h e speed corresponding t o t h e minimum s i n k i n g speed wm. This is achieved by using t h e e q u i v a l e n t polar i l l u s t r a t e d i n f i g u r e 3 and a l r e a d y For h o r i z o n t a l speeds l e s s than t h a t correused i n o t h e r s i m i l a r works [41 sponding t o t h e minimum s i n k i n g speed, t h e s i n k i n g speed remains c o n s t a n t . This appears t o be s u f f i c i e n t l y a c c u r a t e t o s i m u l a t e t h e t r a n s i t i o n from pure dolphin f l i g h t t o e s s i n g o r c i r c l i n g o r a combination of e q u i v a l e n t manoeuvers achieved t o c r o s s a l i f t i n g a r e a i n t h e t i m e r e q u i r e d t o g a i n a c e r t a i n amount of a l t i tude. Note t h a t t h e same approximation is used a s t h e b a s i s f o r speed c o n t r o l i n some modern i n s t r u m e n t a t i o n . I n t h e numerical a p p l i c a t i o n s , t h e q u a d r a t i c approximation (2) remains a p p l i c a b l e a t speeds higher than v(wm). . The v a r i a t i o n s i n a l t i t u d e a r e governed by I f V i 2 v(wm), then w = wm = Constant and t h u s t h e a l t i t u d e g a i n is e n t i r e l y c o n t r o l l e d by t h e e q u i v a l e n t h o r i z o n t a l speed v i . The a l t i t u d e g a i n no longer appears a s an e x p l i c i t c o n t r o l v a r i a b l e . The t i m e s p e n t i n each segment is given by while t h e a l t i t u d e c o n s t r a i n t s read A t i n i t i a l and t e r m i n a l h , = O hn=O points A t each s t e p O 5 h i S h - i = 1,2,. ..,n-1 (11) The mathematical problem c o n s i s t s of f i n d i n g t h e sequence(s) vO, v1, vn,l s a t i s f y i n g t h e r e l a t i o n s (8) , (10) , and (1 I-) and minimizing t h e t o t a l flight t i m e ..., NECESSARY OPTIMALITY CONDITIONS The f i r s t - o r d e r necessary c o n d i t i o n s f o r o p t i m a l i t y can be deduced by t h e c l a s s i c a l methods of d i s c r e t e optimal c o n t r o l [ 111 Such methods have been used i n previous work [ 4, 61 A d e t a i l e d t r e a t m e n t can be found i n [ 12, 131 f o r t h e t w o atmospheric models presented here, a s w e l l a s f o r c e r t a i n more comp l e x s i t u a t i o n s . The conclusions a r e summarized a s follows. . . Concentrated L i f t Model The Hamiltonian t u r n s o u t t o be I t has t o be maximized w i t h r e s p e c t to Ahi o r v i f o r each ad j o i n t v a r i a b l e s p i have t o s a t i s f y t h e r e l a t i o n s i. The s o - c a l l e d where t h e X i a r e Lagrange m u l t i p l i e r s . From those c o n d i t i o n s , it can be shown El21 t h a t a reduced s e t of a d j o i n t v a r i a b l e s rlor n l , nn-lr $,, , $ 1 and of Lagrange m u l t i p l i e r s P I , ~ 2 , , - 1 can be derived which, i n an o p t i m a l s o l u t i o n , have t o s a t i s f y t h e f o l l a w i n g a n d i t i o n s ... ... ..., D i s t r i b u t e d L i f t Model The o p t i m a l s o l u t i o n ( h i , v i ) must be such t h a t t h e Hamiltonian is maximized w i t h r e s p e c t t o t o s a t i s f y the relations vi f o r each i. The a d j o i n t v a r i a b l e s pi have Xi (2) (hi - 6) = 0 (11 (2) where X i and X i a r e Lagrange m u l t i p l i e r s . a b l e s and Lagrange m u l t i p l i e r s Again a set of reduced v a r i - allows t o p r e s e n t t h e o p t i m a l i t y c o n d i t i o n s i n a more s u i t a b l e form which t u r n s o u t t o be [ 131 (2) p i (hi -h) = 0 (2) Pi 6 O Vi -- w(vi) - c i dv: *( v i ) + 'bi = O PHYSICAL INTERPRETATION The i n t e r p r e t a t i o n of t h e two s e t s of o p t i m a l i t y c o n d i t i o n s f o l l o w s s i m i l a r l i n e s . A f i r s t c o n c l u s i o n i s drawn frcan e q u a t i o n (19) o r (25) which governs t h e speed t o f l y i n a segment. Note t h a t e q u a t i o n (25) reduces t o e q u a t i o n (19) i f t h e a i r mass v e r t i c a l v e l o c i t y c i is z e r o , a s assumed i n t h e f i r s t model. From f i g u r e 3, t h e reduced a d j o i n t v a r i a b l e $ a p p e a r s t o correspond t o a c l a s s i c a l MacCready s e t t i n g and indeed e q u a t i o n (25) a p p e a r s i n most o t h e r works In the following, the notation M S (ci) denotes the C on o p t i m i z a t i o n [ 4, 5, 81 s e t t i n g c o r r e s p o n d i n g t o an a i r mass v e l o c i t y ci a s d e f i n e d by e q u a t i o n ( 2 5 ) . The n e x t i n t e r p r e t a t i o n concerns t h e Lagrange m u l t i p l i e r s i n equation ( I S ) , . ( 1 6 ) , or ( 2 4 ) . These m u l t i p l i e r s a r e z e r o i n e n t e r i n g (p2i \ or i (1 1) o r leav- a Segment i f t h e LAL (lower a l t i t u d e l i m i t ) o r UAL (upper is n o t reached. C From e q u a t i o n (20) o r (26) t h e important c o n c l u s i o n is drawn t h a t t h e M S c a n n o t change from a segment t o t h e n e x t one u n l e s s e i t h e r t h e U L o r t h e LAL A has been r e a c h e d , t h a t i s , i f one of t h e Lagrange m u l t i p l i e r s P becomes negat i v e . I f , and o n l y i f , t h e U L is reached, t h e n t h e M S may be reduced. ConA C v e r s e l y , t h e M S may be i n c r e a s e d o n l y i f t h e LAL is touched. To proceed f u r C t h e r with t h e i n t e r p r e t a t i o n r e q u i r e s d i s t i n g u i s h i n g between t h e two models. Concentrated L i f t Model The reduced a d j o i n t v a r i a b l e s rli a p p e a r s from e q u a t i o n (17) a s i n d i c a t o r s of whether t h e l i f t s C i may be used (rli = 0) t o c l i m b o r n o t (rli > 0 ) . With t h e s e r e s u l t s i n mind, it becomes e a s y t o deduce from e q u a t i o n s (1 8 ) , (20) , and (21) t h e l o g i c f o r d e r i v i n g i t e r a t i v e l y t h e optimum s o l u t i o n . Consider t h e beginning of t h e f l i g h t a t a l o c a t i o n where a l i f t A e x i s t s . Denote by B t h e f i r s t l i f t s t r o n g e r than A a l o n g t h e f l i g h t path. The following i t e r a t i v e r e a s o n i n g can be made. I f B c a n be reached from t h e UAL i n A w i t h a M S (A) , t h a t is, a M S corresponding t o t h e l i f t A, t h e n one has t o climb C C i n A j u s t enough t o r e a c h B a t LAL w i t h a MCS(A). I f B cannot be reached w i t h MCS(A) even when climbing a t UAL i n A, t h e n c l i m b t o UAL i n A and look f o r t h e n e x t best l i f t , denoted C, between A and B. E v i d e n t l y C < B and C 5 A. Try t o r e a c h B a t LAL w i t h a M S i n f e r i o r to M S (A) b u t s u p e r i o r t o M S (C) C C C If this is i m p o s s i b l e because C cannot be reached, then r e s t a r t t h e r e a s o n i n g w i t h A unchanged and B r e p l a c e d by C; i f t h i s is i m p o s s i b l e because B c a n n o t be reached, then t a k e a M S (C) t o r e a c h C and r e s t a r t t h e r e a s o n i n g w i t h B unchanged and C A r e p l a c e d by C. . Consider now t h e c a s e where a l i f t A is s t r o n g e r than a l l remaining ones on t h e f l i g h t path. Denote by B t h e s t r o n g e s t of t h e remaining l i f t s , Unless i n t h e s p e c i a l c a s e where t h e t a s k could be ended from A with a MCS(A) without climbing up t o t h e UAL, one n e c e s s a r i l y has t o climb up t o t h e UAL and t a k e a M S s u p e r i o r t o t h e M S (B) b u t e q u a l o r i n f e r i o r to t h e M S (A) C C C I f the task cannot be ended, reach B with a MCS(B) and climb i n it up t o t h e UAL. Repeat t h e reasoning i n B f o r t h e n e x t s t r o n g e s t l i f t . I f B cannot be reached from A w i t h M S (B) , look f o r t h e best l i f t between A and B, denoted C, and t r y i f C B can be reached a t L L w i t h a M S between M S (B) and M S (C) A C C C I f necessary C climb i n C i f t h e M S is equal MCS(C). I f it is impossible t o reach C a t LAE w i t h a M S (C) look f o r t h e best l i f t , say D , between A and C and r e p e a t t h e C C I f it s t i l l does not work, t h e reasoning. I f t h e r e is no l i f t , t r y M S ( 0 ) f l i g h t is e v i d e n t l y impossible. . . . A combination of t h e two r e a s o n i n g s proposed above f o r i n c r e a s i n g o r decreasing l i f t s allcws t h e c o n s t r u c t i o n of t h e optimal s o l u t i o n . Note t h a t it is n o t n e c e s s a r i l y unique. I t is worth p o i n t i n g o u t t h a t t h e optimal s o l u t i o n l e a d s always when going from a l i f t A t o a l i f t B t o use a M S corresponding t o t h e weaker of t h e t w o C l i f t s . I f A > B t h e U L has t o be taken i n A while B has t o be reached a t LAL A i f A < B. S i m i l a r conclusions have been obtained independently by a v a r i a t i o n a approach i n [ 141 . D i s t r i b u t e d L i f t Model In t h i s model, t h e d e c i s i o n t o g a i n a l t i t u d e i n a l i f t is d i c t a t e d by equa t i o n (27). I f $i > wm + c i t h e speed v i is l a r g e r than v ~ ( w , ) and theref o r e is uniquely determined by equation ( 2 5 ) . The l i f t has then t o be c r o s s e d a t t h e corresponding speed. This appears to be a pure dolphin mode. I f , and o n l y i f , $i = Wm + c i t h e d e c i s i o n of climbing may be taken a s t h e speed v i becomes e q u a l t o or smaller than t h e speed v(wm). Indeed, due to t h e form use v i cannot be computed by equation (25) f o r t h e e q u i v a l e n t p o l a r , t h e speed Its value is d i c t a t e d by t h e need t o g a i n a c e r t a i n amount of a l t i t u d e i n t h a t l i f t , given by . I f t h e speed v i is i n f e r i o r to v i (wm), Ahi is l a r g e r than t h e value o b t a i n e d i n pure dolphin mode which implies t h a t some s o r t of manoeuver l i k e e s s i n g and/or c i r c l i n g is achieved w h i l e f l y i n g through t h e segment. Except f o r t h e process of g a i n i n g a l t i t u d e i n a non-dolphin mode, t h e conc l u s i o n s reached f o r t h e preceding model a r e s t i l l v a l i d . From equation (26) t h e M S may n o t be changed u n l e s s t h e a l t i t u d e l i m i t s a r e reached. I t may C i n c r e a s e o n l y i f t h e LAL i s touched and may decrease o n l y when reaching t h e UAI The process f o r c o n s t r u c t i n g numerically an optimum s o l u t i o n is a s follows. S t a r t by t r y i n g t o use t h e M S of t h e b e s t e x i s t i n g l i f t , s a y A, f o r a l l C t h e segments, I f t h e L L is n o t reached, i n c r e a s e t h e s e t t i n g u n t i l e i t h e r A the t a s k can be ended or t h e L L is reached. A t t h a t p o i n t , t h e M S may be A C increased. I f t h e M S corresponding t o t h e b e s t l i f t a l l w s reaching t h e L L b e f o r e C A t h e segment where it occurs, look f o r t h e s t r o n g e s t l i f t between t h e p r e s e n t A C p o i n t and A. Denote it by B. Then t r y t o reach A a t t h e L L w i t h a M S (B) I f t h i s is n o t p o s s i b l e , climbing i n B is allowed. I f B cannot be reached with a MCS(B), look f o r t h e best l i f t between t h e p r e s e n t p o i n t and B and r e p e a t t h e reasoning a s necessary, keeping i n mind t h e r u l e s t h a t allow climbing i n a l i f t and t h o s e t h a t a l l w changing t h e MCS. . NUMERICAL EXAMPLES Concentrated L i f t Model A s a simple example, a 300 k f l i g h t is schematized i n f i g u r e 4. m The l i f t s a r e e q u i d i s t a n t (10 km) f o r s i m p l i c i t y although it is by no means implied i n t h e preceding r u l e s f o r o p t i m a l i t y . The l i f t s t r e n g t h s a r e i n d i c a t e d i n m/sec along t h e y-axis. They i n c r e a s e p r o g r e s s i v e l y during t h e f l i g h t , then decrease, but a r e i n g e n e r a l unequal. The a l t i t u d e l i m i t s a r e 0 and 1000 m. The s a i l plane polar is approximated by and corresponds t o a d r y open c l a s s s h i p . The o p t i m a l s t r a t e g y f o r t h a t l i f t d i s t r i b u t i o n and a l t i t u d e c o n s t r a i n t s is i l l u s t r a t e d i n f i g u r e 4. The M S f o r C each g l i d e is i n d i c a t e d . I t follows a s a simple and s y s t e m a t i c a p p l i c a t i o n of t h e r u l e s f o r o p t i m a l i t y e s t a b l i s h e d above. Note t h a t t h e f l i g h t s t r a t e g y consists i n h i t t i n g s y s t e m a t i c a l l y t h e a l t i t u d e l i m i t s , except a t 110 km and 170 km where t h e a l t i t u d e j u s t necessary f o r reaching t h e n e x t best l i f t a t L L is A gained. Note a l s o t h a t t h e M S is not always e q u a l t o t h e s t r e n g t h of one of C the t w o l i f t s d e f i n i n g t h e g l i d e . F i n a l l y , note t h a t t h i s example j u s t i f i e s t h e p r a c t i c a l r u l e of f l y i n g "low" when t h e l i f t s a r e improving and keeping "high" when they a r e d e t e r i o r a t i n g . The c r u i s i n g speed f o r t h a t f l i g h t is 81 01 km/h. . As a test f o r t h e s e n s i t i v i t y of t h e speed w i t h r e s p e c t t o t h e MCS, t h e same problem has been solved with t h e a d d i t i o n a l c o n s t r a i n t s of keeping t h e same M S which implies t h e same speed between any t w o s e l e c t e d l i f t s . The optimal C M S f o r t h a t s i t u a t i o n has been obtained i n [131 i n t h e form C where L is t h e t o t a l l e n g t h of t h e f l i g h t (300 km) and ci + wm is t h e achieved r a t e of c l i m b i n each s e l e c t e d l i f t . Note t h a t t h e s e l e c t i o n of t h e l i f t s is h i g h l y dependent upon t h e a l t i t u d e l i m i t s , I n t h e p r e s e n t example IlJ = 1.23 m/sec and t h e corresponding c o n s t a n t speed between t h e t h e r m a l s is 133 km/h. The new c o n s t r a i n e d o p t i m a l speed becomes 79.51 km/h which d i f f e r s by o n l y 1.85% froan t h e e x a c t optimum. Although some r e s t r i c t i o n s have t o be mentioned (see [131) concerning t h e a p p l i c a b i l i t y o f e q u a t i o n (29) i n a g e n e r a l c a s e , it i n d i c a t e s c l e a r l y t h a t i n an atmosphere corresponding t o t h e p r e s e n t model, t h e M S is much less important than t h e c o r r e c t s e l e c t i o n of t h e l i f t s . C T h i s is a g a i n w e l l known f o r many c o m p e t i t i o n p i l o t s [ 151 . D i s t r i b u t e d L i f t Model The f l i g h t p o l a r has been approximated by which c o r r e s p o n d s a l s o t o a d r y open c l a s s g l i d e r and t o t h e model used i n [41 and [ 51 Three d i s t r i b u t i o n s of l i f t s have been s e l e c t e d and a r e p r e s e n t e d i n f i g u r e s 5, 6, and 7 and denoted f l i g h t s I , 11, and 111. These f l i g h t s a r e a l l 200 km long and correspond t o d i f f e r e n t weather c o n d i t i o n s . I n f l i g h t I t h e l i f t s a r e r e l a t i v e l y c o n c e n t r a t e d e x c e p t a t two p l a c e s and t h e i r s t r e n g t h s a r e r a t h e r d i f f e r e n t f r m each o t h e r . The l e n g t h of t h e l i f t i n g zones r e p r e s e n t s 36% of t h e t o t a l which is r a t h e r c r i t i c a l f o r t h e t r a n s i t i o n £ram t h e r m a l i n g t o d o l p h i n i n g 141. The a i r mass b a l a n c e is p o s i t i v e , t h a t is, t h e average over t h e d i s t a n c e of t h e a i r mass ( n e t t o ) v e r t i c a l v e l o c i t i e s y i e l d s 0.39 m/sec. I n f l i g h t I1 t h e l i f t i n g zones r e p r e s e n t 31% of t h e d i s t a n c e and t h e i r s t r e n g t h : a r e much more s i m i l a r t o each o t h e r . The l i f t i n g and s i n k i n g a r e a s e x a c t l y balance each o t h e r ; t h a t is, n o t o n l y t h e average v e r t i c a l v e l o c i t y is z e r o , b u t t h e a i r mass is o r g a n i z e d i n a s u c c e s s i o n of c e l l s which a r e 20 t o 40 km l o n g where t h e e x a c t a i r mass balance is a l s o achieved. T h i s a l l o w s f o r u s i n g t h e classical r u l e s f o r l o c a l o p t i m a l i t y [41 i n c r o s s i n g t h e s e cells and compares w i t h t h e g l o b a l l y o p t i m a l s o l u t i o n . I n f l i g h t I11 t h e l i f t s a r e weaker and t h e i r s t r e n g t h s s t i l l c l o s e r t o each o t h e r . The l i f t i n g zone r e p r e s e n t s 49% of t h e t o t a l . The a i r mass balance y i e l d s 0.236 m/sec and t h e l i f t i n g zones are a g a i n o r g a n i z e d i n c e l l s i n which a p p r o x i m a t e l y t h e same a i r mass balance is maintained. For each of t h e s e atmospheric m o d e l s t h r e e upper a l t i t u d e l i m i t s have been c o n s i d e r e d = 1000 m, 1500 m, and 2000 m. The LAL has been k e p t a t h = 0 which is e v i d e n t l y n o t n e c e s s a r i l y t h e ground l e v e l . . The n u m e r i c a l l y o b t a i n e d o p t i m a l s o l u t i o n s a r e i l l u s t r a t e d i n t a b l e s I, The s a t i s f a c t i o n of t h e o p t i m a l i t y c o n d i t i o n s d e s c r i b e d above a r e e a s i l y v e r i f i e d . The l i f t s i n which g a i n i n g a l t i t u d e i n c i r c l i n g o r e s s i n g a r e i n d i c a t e d a s w e l l a s t h e c o r r e s p o n d i n g e q u i v a l e n t h o r i z o n t a l speed which is then s m a l l e r t h a n the speed of minimum s i n k v(wm) = 20.52 m/sec. I n t h e o t h e r segments, c r o s s e d i n d o l p h i n mode, t h e optimum M S is given. Note t h a t % = 0.47 m/sec. C The i n f l u e n c e of t h e a l t i t u d e l i m i t s is i l l u s t r a t e d by t h e f o l l o w i n g t a b l e : 11, and I11 i n d i g i t a l form and i n f i g u r e s 8 , 9, and 10 i n g r a p h i c . a l form. . Ah = 1000 m Flight I F l i g h t I1 F l i g h t I11 v = 94.5 v = 73.76 v = 85.87 Ah = 1500 m Ah = 2000 m Ah = Unlimited 97 -94 81 . 2 87.98 100.19 83.10 88.16 100.57 84,20 88.16 - where Ah is t h e allowed a l t i t u d e range and v is t h e optimum average speed i n km/h. The a p p l i c a t i o n of v a r i o u s non-globally o p t i m a l f l i g h t s t r a t e g i e s based on t h e use of e x i s t i n g r u l e s f o r o p t i m i z i n g t h e speed i n each i n d i v i d u a l m e t e o r o l o g i c a l c e l l [4, 5 r e s u l t e d i n average speed from 5 t o 15% i n f e r i o r 1 depending on t h e allowed a l t i t u d e l i m i t s and on t h e v a r i o u s c o n d i t i o n s s e l e c t e d i n applying t h e s e r u l e s . CONCLUSIONS Simple r u l e s f o r o b t a i n i n g n u m e r i c a l l y t h e optimum f l i g h t s t r a t e g y i n two m e t e o r o l o g i c a l models have been o b t a i n e d . Their a p p l i c a t i o n s r e v e a l t h a t t h e a l t i t u d e l i m i t s imposed f o r t h e f l i g h t may have, a s known from e x p e r i e n c e , a much more s i g n i f i c a n t i n f l u e n c e on t h e achieved speed t h a n t h e s e l e c t i o n of MCS. A d d i t i o n a l i n v e s t i g a t i o n is r e q u i r e d t o dete'rmine t h e r e l a t i o n beween t h e var io u s p o s s i b l e weather p r o f i l e s and t h e optimum f l i g h t s t r a t e g i e s a s w e l l a s a l t i t u d e l i m i t s i n f l u e n c e . The i n f l i g h t r e c o r d i n g o f such atmospheric p r o f i l e s over r a t h e r l o n g d i s t a n c e s would a l l o w f o r s t u d y i n g s y s t e m a t i c a l l y t h e optimum s o l u t i o n corresponding t o a number of c l a s s i c a l s i t u a t i o n s . SYMBOLS A,B,C f l i g h t polar c o e f f i c i e n t s a i r mass n e t t o v e r t i c a l v e l o c i t y altitude upper a l t i t u d e l i m i t Hamiltonian f l i g h t segment l e n g t h adjoint variable elapsed time t o t a l f l i g h t time h o r i z o n t a l speed s a i l p l a n e s i n k i n g speed minimum s i n k i n g speed A j r r l jr!Ji Qi Lagrange m u l t i p l i e r s C reduced a d j o i n t v a r i a b l e = M S index of a f l i g h t segment upper (lower) a l t i t u d e l i m i t s MacCready s e t t i n g -i UAL, LAL MS C REFEm as Soaring, Apr. 1954. 1. MacCready, P.:: Optimum Speed S e l e c t o r . 2. Reichmann, H , : Zum Problem d e r Fahr t o p t i m i e r u n g i m S t r e c k e n s e g e l f l u g . D i s s . Univ. K a r l s r u h e , I n t B e r i c h t no. 76/2. . 3. Reichmann, H. : Strecken-Segelflug. Motor Buch Ver l a g , S t u t t g a r t , 1976. 4. Metzger, D.; and Hedrick, J.: Optimum F l i g h t P a t h s f o r S o a r i n g F l i g h t s . J. of A i r c r a f t , v o l . 12, no. 11, Nov. 1975. 5. Arho, R,: Same Notes on S o a r i n g F l i g h t O p t i m i z a t i o n Theory. v o l e IV, no. 2, 1977. 6. I r v i n g , F.: Cloud-Street F l y i n g . Tech, S o a r i n g , Tech. S o a r i n g , v o l . 111, no, 1 , 1976. 7. Gedeon, J,: Dynamic A n a l y s i s of Dolphin S t y l e Thermal C r o s s Country F l i g h t . Tech, S o a r i n g , v o l e 111, no. 1 , 1976. 8. Arho, R.: Optimal Dolphin S o a r i n g as a V a r i a t i o n a l Problem. vol. 111, no. 1 , 1976. Tech. S o a r i n g , 9. B o h l i , H.: Optimale Dolphinfluggeschwindigkeit auf S t r e c k e n s e g e l f l u g . Aero Revue, no, 8 , Aug. 1971. 10. P i r ker , H.: Sane C m p u t e r C a l c u l a t i o n s on Optimum Water-Bollest of S a i l planes. 1 5 t h OSTIV Congress, Rayskala, F i n l a n d , 1976. 11. Cannon, M.; Cullum, C.; and Polak, E.: Theory o f Optimal C o n t r o l and McGraw H i l l , 1970. Mathematical Programming. 12. L i t t , F. X.; and S a n d e r , G.: Optimal S t r a t e g y i n a Given Space D i s t r i b u t i o n o f L i f t s With Minimum and Maximum A l t i t u d e C o n s t r a i n t s . Report SART 78/03, S e r v i c e de R g g u l a t i o n e t Automatique , Univ. ~ i g g e ,Belgium, June 1978. A l s o p r e s e n t e d a t XVI OSTIV Congress i n ~ h ^ a t e a u r o u x ,France. 13. L i t t , F. X.; and S a n d e r , G.: n i q u e s . Report SART 79/02, ~ i s g e ,Belgium, Feb. 1979. G l o b a l O p t i m i z a t i o n o f S a i l p l a n e F l i g h t TechS e r v i c e d e ~ 6 g u l a t i o ne t Automatique, Univ. 14. d e Jong, J. L.: D e o p t i m a l e MacCready-ring i n s t e l l i n g e n by inachtname van v e r t i c a l e begrenzingen van d e b r u i k b a r e thermiek hoogte. COSOR R-78-05, Tech. Hogeschool Eindhoven, Holland, A p r i l 1978. 15. Schuemann, W.: The P r i c e You Pay f o r MacCready Speeds. C m p e t i t i v e S o a r i n g , 1972, U.S.A. Proc. Syrnp. on T A B L E I.- O P T I M A L S O L U T I O N F O R F L I G H T I bout = a l t i t u d e at the end o f the s e g m e n t (meters) Mode = D for dolphin C for climbing with v i MCS, v i = m l s e c c vi(wm) TABLE 11.- OPTIMAL SOLUTTON FOR FLIGHT I1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Speed 0.53 0.53 1.03 1.03 2.03 1.03 1.03 1.03 1.03 1.03 1.03 2.03 0.53 0.53 0.53 0.53 0.53 1.53 0.78 541 0 617 0 1000 443 237 181 895 278 0 1000 729 144 890 781 0 1000 0 20.5 38.4 8.3 38.4 10.1 41.7 38.4 30.8 7.2 38.4 41.7 10.1 38.4 34.8 10.6 30.8 34.8 7.6 36.7 D D C D C D D D C D D C 'D D C D D C D 0.53 0.53 1.03 1.03 2.03 2.03 2.03 2.03 2.03 2.03 2.03 2.03 1.53 1.53 1.53 1.53 1.53 1.53 1.53 541 20.5 0 38.4 617 8.3 0 38.4 2000 5.0 1398 47.6 1165 44.7 1014 38.4 1149 30.8 450 44.7 149 47.6 2000 5.5 1711 44.7 1055 41.7 1215 30.8 107438.4 200 41.7 1092 8.6 041.7 8 3 . 1 0 kmlh D D C D C D D D D D D C D D D D D C D 7 3 . 7 6 kmlh bout = altitude at the end of t h e segment (meters) Mode = D f o r d o l p h i n C for climbing w i t h v i MCS, v i = m / s e c < vi(w,) T A B L E 111.- O P T I M A L S O L U T I O N F O R F L I G H T 1 1 1 bout = altitude at the end of the segment (meters) Mode = D for dolphin f o r climbing w i t h v i C 1 < vi(wm) MCS, v. = m / s e c F i g u r e 1.- Concentrated l i f t model. F i g u r e 2.- D i s t r i b u t e d l i f t model. Figure 3.- The equivalent polar. CONCENTRATED LIFT MODEL distances (km) Figure 4.- Optimum flight strategy. DISTRIBUTED LIFT MODEL -1.5 flight segment no. ci (mlsec) li (km) 11 12 3.5 -1.5 5 5 13 -1 5 1 1 .5 16 2 5 2 0 19.5 17 -.5 15 5 3 4 1.5 0 2.5 .5 19.5 1 18 .5 5 19 2.5 10 20 4.5 5 G 0 14 21 -1 10 7 .5 5 22 -.5 1 0 8 1 5 23 2 5 9 -.5 5 1 0 2 5 25 14 .5 10 15 1 10 24 0 15 c = Zcili - -33 m. -L sec LIFTING ZONES -.5 10 = 36 % Figure 5,- Lift distribution for flight I, DISTRIBUTED LIFT MODEL 25 LIFTING ZONES Figure 6.- Lift distribution for flight 11. = 31 % 373 DISTRIBUTED LIFT MODEL -2 -2 E= a= -236m i sec L = 49% LIFTING ZONES Figure 7.- Lift distribution for flight 1x1. 8 climbing with vi < vi(wm) Figure 8.- Optimal solution for flight I. @ climbing with vi is given by figure 10.) Due to the variation process, the coupler force fg has completely changed its behaviour (see figure 12). However, the maximum values can be considerably reduced by prescribing maximum limiting values of the coupler forces as secondary conditions, as shown in figure 13. In this case, the accuracy of the required assignment is reduced, but still sufficient. During flight, the input-output displacement propagates from the stick to the aileron, whereas, looking to the forces, the aerodynamic forces on the aileron are the input-loads, which have to be balanced by the pilot through the stick. In this case, the application of the variation method gets more complicated, because the transfer mechanism has to be investigated in both directions ) (see ref. 4 . Figure 14 shows an example, where critical values of the coupler force in the seventh four-bar mechanism have been reduced. The input-load at the aileron has been the constant moment 2 = (23.34,0,0) [~m] . CONCLUSIONS The layout of spatial transfer-mechanisms consisting of series-connected spatial four-bar mechanisms can be performed more effectively if the graphical techniques are replaced by a numerical method. Within this method, the individual four-bar mechanisms are treated analytically as transfer elements. By variation of selected parameters, the input-output behaviour of the train can be adjusted to prescribed values. The acting loads in the mechanism can be limited, considering maximum values of the loads as secondary conditions during the variation process. The effectiveness of the proposed method has been demonstrated at the layout of the driving mechanism for the aileron of a glider. REFERENCES 1. Beyer, R.: Technische Kinematik. Ambrosius-Verlag, Leipzig, 1931. 2. Hall, A. S.: Kinematics and Linkage Design. Prentice-Hall, Englewood Cliffs, N.J., 1961. 3. Eppler, R.; Hiller, M.: Numerische Behandlung von Gelenkvierecken mit vorgeschriebenem Ausgang. DLR-Forschungsbericht 75-132, pp. 53-66, 1975. g 4. Hiller, M. : Ein Verfahren zur optimalen ~ u s l e ~ u nvon mechanischen Ruderantrieben bei Segelflugzeugen und Motorsportflugzeugen. Zeitschrift fiir Flugwissenschaften und Weltraumforschung (ZFW), May 1979. TABLE 1.- I N I T I A L DATA OF AILERON D R I V I N G MECHANISM OF THE EXPERIMENTAL GLIDER, f s - 2 9 f our-bar mechanism -1 Cml 1 u. 2 - . [ml 1 0.1250 0.0400 -0.0450 0.0000 -0.0200 -0.88 1 0 0.0000 -0.0960 -0.3820 -0.0060 -0.2430 -0.1490 s -1 . [m] 1 W. 1 0.0000 0.0430 0.0150 0.0800 0.0000 0.0000 0.0000 0.0610 -0.0210 0.0000 0.0400 -0.0430 0.0000 -0.0190 0.2000 0.0000 -0.3875 -0.0400 1.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 -0.0062 0.0596 0.0000 0.0650 -0.0100 0.0000 0.0580 0.0060 -0.0800 0.0000 0.0000 0.0000 -0.387 5 -0.0400 1.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0590 -0.1000 2 3 4 5 0.0000 -0.0690 -0.0400 0.0900 -0.0070 -0.0665 0.0000 0.5900 -0.1000 0.0000 1.0000 0.0000 .0600 -0.0570 0.1120 0.1150 -0.0510 -0.0090 1 -0.0080 O.OOOO 0.1350 -0.0250 0.0590 -0.0150 0.0000 1.0000 0.0000 -1.0000 0.0000 0.0000 6 7 - 0.0000 0.0420 -0.0270 0.0020 -0.0170 0.0470 0.0220 -0.0490 0.0120 0.0400 0.0000 -0.0020 -1.0000 0.0000 0.0000 0.0810 0.0000 -0.0030 -0.0030 0.0000 -0.0835 0.0000 1.0000 0.0000 0.0000 0.0290 -0.0900 -0.0480 0.0220 0.0510 0.5860 -0.0200 -0.0640 0.0500 0.0400 -0.0740 0.0000 -0.0125 0.0480 0.0500 0.0000 -0.0015 -0.0480 0.0280 0.0510 0.0000 -0.0405 0.0045 1.0000 0.0000 0.0000 -0.0030 0.0000 -0.083 5 0.0000 -1.0000 0.0000 -1.0000 0.0000 0.0000 8 . 9 10 TABLE 2.- INPUT/OUTPUT ANGULAR RELATIONSHIPS FOR STICK AND AILERON OF f s - 2 9 GLIDER input-angle at stick output-angle at aileron 8, Lo] 1°1 -22.1 -12.7 14.1 26.4 Vlo -25.0 -12.6 8.9 15.0 Figure 1: The spatial four-bar mechanism. ~ Figure 2: The spatial transfer mechanism. $wj-1= uj Figure 3: Coupler force. Figure 4: Correction step 1 for secondary condition.. Figure 5: Correction step 2 for secondary condition. Figure 6: Input-output behaviour of four-bar mechanisms 1 to 8. Figure 414 7: Input-output behaviour of four-bar mechanisms 9 and 10. Figure 8: Sensitivity of the cranks r 9 and -9 s Figure 9: Sensitivity of the cranks r -10 and s -0 1 . Figure 10: Output-angle y 9 before (--- ) and after ( - ) variation. Figure 11: Aileron deflection y l o 41 6 before (--- ) and after ( - ) variation. Figure 1 2 : Coupler f o r c e f b e f o r e (-- - -1 and a f t e r ( - ) variation. Figure 13: Limited c o u p l e r f o r c e f9 Figure 14: Coupler force f 7 before (--- ) and after ( - ) variation. EXPERIMENTAL INVESTIGATION I N T O THE FUSIBILITY P i e r o Morelli and Giulio Romeo P o l i t e c n i c o d i Torino, I t a l y S M AY U MR Research work i n t h e P o l i t e c n i c o d i Torino and r e a l i z a t i o n s ( f a b r i c a t i o n s ) of extruded aluminium a l l o y s t r u c t u r e s during t h e p a s t y e a r s i s b r i e f l y reviewed. The design c r i t e r i a and t h e r e a l i z a t i o n of t h e main s t r u c t u r e of a s a i l p l a n e wing made of a few extruded p r o f i l e s l o n g i t u d i n a l l y connected one t o t h e o t h e r a r e then i l l u s t r a t e d . S t r u c t u r a l t e s t s r e c e n t l y c a r r i e d out a r e r e p o r t e d upon. INTRODUCTION Early r e s e a r c h work and t h e f i r s t r e a l i z a t i o n s on t h e M-300 s a i l p l a n e prototypes were reported upon i n r e f e r e n c e 1. Figure 1 i l l u s t r a t e s t h e c r o s s sect i o n of t h e M-300 extruded s t r u c t u r e s : f i r s t and second r e a l i z a t i o n of t h e a i l e r o n s ( a , b ) , t a i l p l a n e ( c ) , wing s p a r ( d ) . An aluminium a l l o y AlMgSi ~ ~ 1 (A.A. 6 0 6 3 - ~ 6 )was employed f o r t h e e x t r u s i o n except f o r t h e s p a r , which was of A.A. 7075. In more recent years t h e same s t r u c t u r a l concept was adopted by t h e firm Caproni Vizzola Costruzioni Aeronautiche, manufacturer of t h e two-seater s a i l plane C a l i f A-21s ( r e f . 2 and 3 ) . Figure 2 i l l u s t r a t e s some of t h e p a r t s of t h i s g l i d e r which were r e a l i z e d by extrusion using t h e same aluminium a l l o y mentioned above: a i r b r a k e ( a ) , f l a p ( b ) , a i l e r o n ( c ) , e l e v a t o r ( d ) , and l e a d i n g edge of t h e wing c e n t r a l p a r t ( e ) . The a i l e r o n and e l e v a t o r extruded p r o f i l e s incorpor a t e t h e hinge ( A ) . I n t h e a i l e r o n l e a d i n g edge lodging i s provided ( B ) f o r t h e counterweight, uniformly d i s t r i b u t e d along t h e span f o r s t a t i c and dynamic balance. In t h e M-300 and Calif extruded s t r u c t u r e s t h e o r i g i n a l wall thickness of 1 . 8 t o 2.0 mm was reduced t o design values of .5 t o .8 rnm by chemical m i l l i n g of t h e o u t e r s u r f a c e . A l l t h e s e s t r u c t u r e s a r e b a s i c a l l y r i b l e s s . They proved l i g h t and l a r g e l y adequate i n s t r e n g t h and s t i f f n e s s . 6 One of t h e M-300 prototypes i s s t i l l a c t i v e . The Calif two-seater has been s e r i e s produced with t h e extruded p a r t s mentioned here s i n c e 1975; except t h e extruded a i r b r a k e which was introduced i n 1978. Advantages and l i m i t a t i o n s of t h e extruded s t r u c t u r e s were discussed i n r e f e r e n c e s 1, 2, 3 . They a r e b r i e f l y summarized h e r e . Main advantages a r e : 1. Reduction of manhours r e q u i r e d t o r e a l i z e t h e s t r u c t u r e , mainly d u r i n g t h e assembling s t a g e . 2. Reduction of c o s t i n a s e r i e s p r o d u c t i o n when t h e c o s t of t h e expensive ext r u s i o n d i e s can be d i s t r i b u t e d over a h i g h number of p i e c e s . 3. C o r r e c t r e p r o d u c t i o n of s e c t i o n contours w i t h consequent aerodynamic b e n e f i t . The p r a c t i c a l l i m i t a t i o n s a r e p r i n c i p a l l y t h e following: 1. The extruded p r o f i l e has n e c e s s a r i l y a c o n s t a n t c r o s s s e c t i o n . Through s u i t a b l e mechanical and chemical o p e r a t i o n s , however, it i s p o s s i b l e t o achieve a c e r t a i n degree of c r o s s s e c t i o n v a r i a t i o n a l o n g t h e beam a x i s . 2. The maximum l i n e a r dimension and n e t a r e a of t h e p r o f i l e s e c t i o n a r e l i m i t e d by t h e power of t h e a v a i l a b l e e x t r u s i o n p r e s s . 3. The d i f f i c u l t y of e x t r u d i n g i n c r e a s e s w i t h h i g h s t r e n g t h aluminium a l l o y such a s 2024 o r 7075. 4. A minimum w a l l t h i c k n e s s i s imposed by t h e e x t r u s i o n p r o c e s s , which i s somet i m e s e x c e s s i v e i n r e l a t i o n t o t h e s t r e n g t h and w e i g h t / s t r e n g t h r a t i o r e quired. A wide f i e l d of p o s s i b l e a p p l i c a t i o n s seem t o e x i s t notwithstanding t h e s e l i m i t a t i o n s , p a r t i c u l a r l y f o r g l i d e r s and l i g h t powered a i r c r a f t . A g l i d e r h a s been conceived, which i s s i m i l a r t o t h e M-300 from which it i s d e r i v e d and i s s u i t a b l e f o r a wide u s e of extruded s t r u c t u r e s , whose l o c a t i o n s a r e i n d i c a t e d by t h e shadowed a r e a s i n f i g u r e 3. The r e a l i z a t i o n of t h e c e n t r a l p a r t of t h e wing of t h i s g l i d e r i s t h e aim of t h e r e s e a r c h work s t a r t e d a few y e a r s ago a t t h e P o l i t e c n i c o d i Torino, a f t e r t h e completion of t h e f i r s t s t a g e which l e d t o t h e r e a l i z a t i o n of t h e above des c r i b e d M-300 extruded p a r t s . THE DESIGN O AN "EXTRUDED" WING F The wing i l l u s t r a t e d i n f i g u r e 3 i s 15 m span w i t h r e c t a n g u l a r - t r a p e z o i d planform, t h e c e n t r a l r e c t a n g u l a r p a r t b e i n g extended over 9 m. It i s a t h r e e - p i e c e wing: t h e c e n t r a l p a r t i s a f l a t s i n g l e p i e c e connected t o t h e f u s e l a g e by a b p o i n t attachment; t h e o u t e r t r a p e z o i d p a n e l s a r e a t t a c h e d t o t h e ends of t h e c e n t r a l p a r t and g i v e t h e wing t h e r e q u i r e d d i h e d r a l a n g l e . The c e n t r a l p a r t i s conceived a s a combination of extruded. p r o f i l e s : a p o s s i b l e t y p i c a l c r o s s s e c t i o n i s i l l u s t r a t e d i n f i g u r e 4 ( a i r f o i l FX 67-K- l 7 0 / l i ) , which i s p u r e l y i n d i c a t i v e of t h e b a s i c i d e a . Corresponding t o t h e a i r f o i l maximum t h i c k n e s s a box s t r u c t u r e can b e s e e n which c a r r i e s p r a c t i c a l l y a l l bending l o a d s and a good p o r t i o n of t h e s h e a r / t o r s i o n l o a d s . The o t h e r two t h i n w a l l e d e x t r u d e d p r o f i l e s a r e r i v e t e d t o t h e c e n t r a l box and c o n t r i b u t e t o t h e s h e a r / t o r s i o n s t r e n g t h and s t i f f n e s s of t h e whole s t r u c t u r e . The e x t r u d e d p r o f i l e a t t h e t r a i l i n g edge i s a f l a p . The wing s t r u c t u r e i s i n t e n d e d as b a s i c a l l y r i b l e s s , a s f a r a s t e s t s w i l l confirm t h a t r i b s can b e e l i m i n a t e d . I n o r d e r t o p r o v i d e t h e c e n t r a l box w i t h t h e r e q u i r e d bending s t r e n g t h and s t i f f n e s s under t h e p r e s c r i b e d l o a d i n g c o n d i t i o n s ( a c c o r d i n g t o t h e OSTIV A i r w o r t h i n e s s Requirements, r e f . 4 ) a c e l l u l a r s t r u c t u r e was adopted f o r t h e d o r s a l and v e n t r a l p a n e l s . T h i s m-dlti-cell s t r u c t u r e w a s t e n t a t i v e l y d e s i g n e d t o prev e n t g e n e r a l and l o c a l e l a s t i c i n s t a b i l i t y . The c e n t r a l box i s made of two p r o f i l e s j o i n e d by r i v e t i n g t h e two halfwebs a l o n g t h e span (A.A. 6 0 6 1 - ~ 6 ) . The l a r g e bending d e f o r m a t i o n , t y p i c a l of a h i g h a s p e c t r a t i o s a i l p l a n e wing, combined w i t h t h e absence of r i b s makes t h e problem of r e s i s t i n g t h e " c r u s h i n g " l o a d s a b a s i c one. One of t h e main o b j e c t i v e s of t h e t e s t i n g program i s t o a s c e r t a i n how f a r t h e webs a l b n e a r e c a p a b l e of w i t h s t a n d i n g t h e crushing loads. The c e n t r a l box s e c t i o n i s reduced a l o n g t h e span by c h e m i c a l l y e t c h i n g t h e o u t e r s u r f a c e of each of t h e two p r o f i l e s s o t h a t t h e o r i g i n a l w a l l t h i c k n e s s of t h e s k i n p a n e l s i s decreased from 2.3 mm down t o .8 mm a t a spanwise s t a t i o n about 2.65 m from t h e wing c e n t e r l i n e . T h i s t h i c k n e s s i s t h e n k e p t cons t a n t o v e r t h e r e s t o f t h e wing. F i g u r e 5 shows t h e r e d u c t i o n of s k i n t h i c k n e s s a l o n g t h e span i n two possi b l e ways. A s t e p r e d u c t i o n (above) o r a continuous t a p e r i n g (below) can b e r e a l i z e d , t h e l a t t e r r e q u i r i n g , however, a d d i t i o n a l equipment f o r chemical m i l l i n g a t v a r i a b l e t i m e of immersion. F i g u r e s 6 and 7 show t h e c e n t r a l box c r o s s s e c t i o n s a t t h e wing r o o t and a t a spanwise s t a t i o n from 2.65 m on. EXPERIMENTAL P O R M AND RESULTS RGA S e v e r a l problems a r e t o b e f a c e d i n t h e r e a l i z a t i o n o f a wing as d e s c r i b e d i n t h e preceding paragraph. A p r e l i m i n a r y e x p e r i m e n t a l i n v e s t i g a t i o n was c o n s i d e r e d n e c e s s a r y i n o r d e r t o check t h e f o l l o w i n g p o i n t s : 1. The c a p a b i l i t y of t h e c e l l u l a r p a n e l s t o w i t h s t a n d t h e h i g h d e s i g n c o m ~ r e s s i o n 421 s t r e s s e s without i n c u r r i n g l o c a l i n s t a b i l i t y phenomena a t low l o a d f a c t o r s . 2. The c a p a b i l i t y of t h e box s t r u c t u r e t o withstand t h e design bending moments and, i n p a r t i c u l a r , t h e crushing loads due t o bending deformation. 3. The c a p a b i l i t y t o o b t a i n t h e c e n t r a l box p r o f i l e s by e x t r u s i o n of a s u i t a b l e material a t an acceptable degree of accuracy and reasonable c o s t . 4. The f e a s i b i l i t y of s a t i s f a c t o r y chemical m i l l i n g i n r e l a t i o n t o t h e part i c u l a r aluminium a l l o y adopted f o r t h e e x t r u s i o n . With r e f e r e n c e t o p o i n t s 1 and 2, it was decided t o check t h e g e n e r a l des i g n of t h e c e n t r a l box s t r u c t u r e , and of t h e c e l l u l a r panels i n p a r t i c u l a r , by r e a l i z i n g a "simulated1' extruded s t r u c t u r e and submitting it t o pure compression and pure bending t e s t s . The c r o s s s e c t i o n of t h e simulated s t r u c t u r e i n f i g u r e 8 shows i t s conventional c o n s t r u c t i o n through Z-stringers and metal s h e e t , both of d u r a l , connected by r i v e t s . Notwithstanding t h e d i f f e r e n c e i n m a t e r i a l and some geometrical f e a t u r e s t h e s e t e s t s gave some v a l u a b l e i n d i c a t i o n s ( r e f . 2 ) s o t h a t t h e r e a l i z a t i o n of t h e expensive e x t r u s i o n d i e s could be undertaken with reasonable confidence. The two extruded p r o f i l e s were t h e n obtained, having t h e c r o s s s e c t i o n shown i n f i g u r e 6. S e v e r a l attempts were necessary, w i t h modification of t h e d i e , b e f o r e an a c c e p t a b l e degree of accuracy of t h e s e c t i b n contour was achieved. The aluminium a l l o y employed on t h e f i r s t extruded p r o f i l e s was not s a t i s f a c t o r y (inadequate values of t h e y i e l d and r u p t u r e s t r e s s ) . A d i f f e r e n t alumini u m a l l o y was t h e n used of higher s t r e n g t h b u t , perhaps, r a t h e r poor p l a s t i c characteristics. It should be remarked h e r e t h a t , i n our I t a l i a n s i t u a t i o n , t h e choice of m a t e r i a l s f o r e x t r u s i o n i s extremely l i m i t e d . I n f a c t , s i n c e our f a c t o r i e s a r e not f u r n i s h e r s of t h e a i r c r a f t i n d u s t r y , t h e supply of a small q u a n t i t y of e x t r u s i o n s such a s r e q u i r e d f o r r e s e a r c h can only be made of a m a t e r i a l of c u r r e n t u s e , i . e . having r a t h e r low s t r e n g t h c h a r a c t e r i s t i c s . Pure bending t e s t s were planned and c a r r i e d out on s e v e r a l specimen,1000 mrn long, of t h e r e a l extruded s t r u c t u r e using t h e bending t e s t machine of t h e I s t i t u t o d i P r o g e t t o d i Aeromobili - P o l i t e c n i c o d i Torino. Figure 9 shows t h e t e s t i n g equipment. Figures 10 and 1 show t h e d e f l e c t i o n 1 curves measured onspecimenswith w a l l t h i c k n e s s of 2.3 and .8 mm, i . e . having t h e c r o s s s e c t i o n s i l l u s t r a t e d i n f i g u r e 6 and 7 , r e s p e c t i v e l y . Figure 12 shows t h e t y p i c a l f a i l u r e i n coapression due t o bending which occurred on one of t h e t = . 8 mm specimen. The r e s u l t s of t h e s e t e s t s were encouraging, although of s t i l l l i m i t e d v a l i - d i t y f o r two main reasons: 1. Since t h e ends a rather short ing influence. crushing l o a d s of t h e specimen a r e r i g i d l y a t t a c h e d t o t h e t e s t machine, only c e n t r a l p o r t i o n of t h e s t r u c t u r e i s f r e e from t h e i r r e s t r a i n Therefore, t h e c a p a b i l i t y of t h e s t r u c t u r e t o withstand t h e cannot be f u l l y evaluated. 2. Shear i s not p r e s e n t . T e s t i n g on a f u l l s c a l e s t r u c t u r e was t h e r e f o r e planned. A t e s t s t r u c t u r e was prepared corresponding t o t h e c e n t r a l box of t h e r e c t a n g u l a r p a r t of t h e s a i l p l a n e wing i l l u s t r a t e d i n f i g u r e 3. The span of t h i s t e s t specimen was 7.67 m, l e s s t h a n t h e 9.0 m span of t h e wing r e c t a n g u l a r p a r t , due t o l i m i t a t i o n s of t h e a v a i l a b l e equipment f o r chemi c a l milling. The s k i n t h i c k n e s s was reduced spanwise through chemical m i l l i n g by .3 mm s t e p s from 2.3 down t o .8 mm a s shown on t h e upper p a r t of f i g . 5 . Two extens i o n s were added a t both ends of t h e s t r u c t u r e t o a l l o w t h e a p p l i c a t i o n of conc e n t r a t e d l o a d s corresponding t o t h e a c t u a l d i s t r i b u t e d load c a r r i e d by t h e o v t e r p o r t i o n s of t h e wing ( s e e f i g u r e 1 3 ) . The spanwise wing l i f t and mass d i s t r i b u t i o n s were evaluated and t h e n r e p l a c e d b y t e n c o n c e n t r a t e d l o a d s , g i v i n g a good approximation of t h e bending moment and shear d i s t r i b u t i o n ( s e e f i g u r e 1 3 ) . Figure n=8. The incremental l o a d was 2,413 N corresponding t o a u n i t y l o a d f a c t o r increment. The u l t i m a t e l o a d was 24,074 N corresponding t o a r a t h e r high u l t i m a t e l o a d f a c t o r of 9.975. The s t r u c t u r a l f a i l u r e occurred a t a l o a d f a c t o r n=8.72. 1 4 shows t h e s t r u c t u r e under t h e load corresponding t o load f a c t o r As shown by f i g u r e 1 5 , t h e d o r s a l c e l l u l a r panel between t h e f i t t i n g s , sirnul a t i n g t h e wing-fuselage attachments, c o l l a p s e d under t h e combined e f f e c t s of compression and crushing l o a d s . I n t h i s a r e a both webs were l a r g e l y c u t out t o allow t h e connection of t h e f i t t i n g s t o t h e s t r u c t u r e . Figure 1 6 shows t h e d e f l e c t i o n curves of t h e whole s t r u c t u r e a t l o a d f a c t o r s of 2, 4 , 6 and 8 . It can be seen t h a t , a t high load f a c t o r s , t h e d e f l e c t i o n of t h e l e f t wing becomes a l i t t l e higher i f compared with t h e o t h e r wing. This i s presumably due t o t h e growing e l a s t i c buckling of t h e d o r s a l panel caused by t h e l a r g e cut-outs of t h e wing c e n t r a l p a r t where t h e f a i l u r e f i n a l l y occurred ( f i g . 1 5 ) . The d e f l e c t i o n s a t d i f f e r e n t s t a t i o n s a r e p l o t t e d v e r s u s l o a d f a c t o r i n f i g u r e 17. S t r a i n gage measurements showed: a ) a s l i g h t e l a s t i c buckling of b o t h webs i n t h e i r l o n g i t u d i n a l l y compressed p a r t a t l o a d f a c t o r s above n= 4 ; b ) no b u c k l i n g whatever o f t h e d o r s a l p a n e l s a l o n g t h e span; and c ) a maximum'local normal s t r e s s of 235 N/mm2 a t n=8 on b o t h d o r s a l and v e n t r a l p a n e l s . CONCLUSIONS The f a i l u r e under bending having occurred a t a v e r y h i g h l o a d f a c t o r (~8.72) and i n t h e c e n t r a l p a r t of t h e s t r u c t u r e where t h e webs can be e a s i l y r e i n f o r c e d , t h e r e s u l t of t h i s f i r s t s t a t i c t e s t can be considered s u c c e s s f u l . There i s a r e a s o n a b l e confidence t h a t , a f t e r reinforcement of t h e web cut-outs i n t h e c e n t r a l p o r t i o n of t h e s t r u c t u r e , t h e r e s i d u a l l o a d f a c t o r increment d ~ 9 . 9 7 5 - 8 . 7 2 0 = 1.255 w i l l be a t t a i n e d . Although t o r s i o n s t a t i c s t r e n g t h and f a t i g u e l i f e a r e t o be demonstrated b e f o r e a s t r u c t u r e of t h i s t y p e can be a s s e s s e d t o b e adequate f o r a s a i l p l a n e wing, t h e r e s u l t of t h e a c t u a l shear/bending t e s t should probably be considered of b a s i c importance a s it p r a c t i c a l l y demonstrates t h e f e a s i b i l i t y of a r i b l e s s s t r u c t u r e made of a few extruded p r o f i l e s l o n g i t u d i n a l l y connected one t o t h e other. REFERENCES 1. Morelli, P.: Extruded Light Alloy Aircraft Structures. Proceedings of the 1st International Symposium on the Technology and Science of Motorless Flight. M.I.T., Cambridge, U.S.A., October 1972. 2. Romeo, G.: Realizzazione per estrusione di strutture aeronautiche - Ricerca su struttura alare estrusa. Prestampa AIDAA n.60, IV Congress0 Nazionale dell'Associazione Italiana di Aeronautica e Astronautics, Milano, September 1977. 3. Romeo, G.: Progress on Extruded Structures. 16th OSTIV Congress,Chateauroux, France, July 1978. 4. OSTIV Airw0rthine.s~ Requirements for Sailplanes, September 1976. FIG.1- M-300 EXTRUDED STRUCTURES FIG.2 - CALIF A-21s EXTRUDED STRUCTURES FIG.3 DESIGNED FOR WIDE USE OF EXTRUDED STRUCTURES - GLIDER FIG. 4 - TYPICAL CROSS SECTION OF AN "EXTRUDED" WING FIG.5 - SKIN THICKNESS REDUCTION ALONG THE SPAN FIG.6 - CENTRAL BOX ROOT SECTION FIG.7 - CENTRAL BOX OUTER SECTION FIG.8 - CROSS SECTION OF "SIMULATED" STUCTURE FIG.9 - EXTRUDED SPECIMEN TESTED AT BENDING MACHINE mm 76 5DEFLECTION 4 - I . A @ Mb=31800 Mb=26150 Mb=21200 Mb=17650 Mb=12350 Nm Nm Nm Nm Nm 3 2 ,,, :/, - ,", -*-----, ,, ,' ' N - 1 -- /,y, ~9,' 0 $ ,, ' ,- FIG. 1 0 mm - - DEFLECTION, CURVES OF THE EXTRUDED SPECIMEN t=2.3mm UNDER BENDING TEST 8- . Mb=I Nm 7400 M b=I Nm 5200 Mb=12900 Nm Mb=10150 Nm DEFLECTION 4 - FIG. 11 - DEFLECTION CURVES OF THE EXTRUDED SPECIMEN t = . 8 m m UNDER BENDING TEST FIG.12 - BENDING FAILURE OF THE tz.8 EXTRUDED SPECIMEN I BENDING MOMENT -.- S H EAR --- FIG.13 -- LOAD DISTRIBUTION ON TEST STRUCTURE FIG.14- STRUCTURE AT LOAD FACTOR n = 8 FIG.15 - STRUCTURE FAILURE /' b ' d DEFLECTION FIG.16 - DEFLECTION CURVES LOA D FACTOR I 100 I 200 I ,300 I 400 I 500 mm I DEFLECTION FIG.17 - DEFLECTION VS. LOAD FACTOR TREATMENT O T E CONTROL MECHANISMS O LIGHT AIRPLANES F H F I N THE FLUTTER CLEARANCE PROCESS Elmar J. Breitbach* Langley Research Center S M AY U MR Recently, it h a s become more and more e v i d e n t t h a t many d i f f i c u l t i e s encountered i n t h e course of a i r c r a f t f l u t t e r a n a l y s e s can be t r a c e d t o s t r o n g l o c a l i z e d n o n l i n e a r i t i e s i n t h e c o n t r o l mechanisms. To cope w i t h t h e s e problems, more r e l i a b l e mathematical models paying s p e c i a l a t t e n t i o n t o c o n t r o l system n o n l i n e a r i t i e s may be e s t a b l i s h e d by means of modified ground v i b r a t i o n t e s t procedures i n combination with s u i t a b l y adapted modal s y n t h e s i s approaches. Three d i f f e r e n t concepts a r e presented i n d e t a i l . INTRODUCTION A t f i r s t glance t h e f l u t t e r c l e a r a n c e of soaring and l i g h t a i r p l a n e s does n o t seem to r a i s e any s e r i o u s problems which cannot be solved by means of today's a e r o e l a s t i c t o o l s . This is t r u e even f o r t h e determination of t h e unsteady aerodynamic loads a s long a s c a s e s with l a r g e a s p e c t r a t i o s a t compar a b l y low speeds a r e considered. The elastodynamical c h a r a c t e r i s t i c s can be determined by using common experimental o r a n a l y t i c a l methods i f s t r u c t u r a l l i n e a r i t y can be assumed t o be a proper approximation. However, a s experience has shown, t h e c o n t r o l mechanisms of l i g h t a i r p l a n e s 1 a r e g e n e r a l l y nonlinear t o such a l a r g e e x t e n t t h a t s e t t i n g up a dependable mathematical model r e q u i r e s s p e c i a l a t t e n t i o n , including m o d i f i c a t i o n s t o s t a n d a r d l i n e a r i z e d procedures. I n t h e f i r s t p a r t of t h i s paper some of t h e most f r e q u e n t l y o c c u r r i n g types of control-system n o n l i n e a r i t i e s a r e d e s c r i b e d . To g e t an i d e a of t h e i n f l u e n c e of some t y p i c a l n o n l i n e a r i t i e s on t h e a e r o e l a s t i c s t a b i l i t y t h e r e s u l t s o f wind t u n n e l f l u t t e r t e s t s on a nonlinear wing a i l e r o n model a r e presented. After t h a t , it is shown i n d e . t a i l how t h e a e r o e l a s t i c e q u a t i o n s of l i g h t a i r p l a n e s with l o c a l i z e d n o n l i n e a r i t i e s may be formulated by using v a r i o u s s u i t a b l y modif i e d ground v i b r a t i o n t e s t (GVT) procedures a l l based on t h e well-known modal s y n t h e s i s approach. The shortcomings a s w e l l a s t h e u s e f u l n e s s of t h e d i f f e r e n t concepts a r e discussed. *NRC-NASA Senior Resident Research Associate. l ~ i g h a i r p l a n e s a s sued i n t h i s paper include both powered and unpowered t v e h i c l e s where t h e power t o t h e f l i g h t c o n t r o l system is s u p p l i e d by t h e p i l o t without e l e c t r i c a l o r h y d r a u l i c boost through a system of c a b l e s , p u l l e y s , pushrods, b e l l c r a n k s , o r o t h e r mechanical linkages. SYMBOLS a, b hinge a x i s coordinates of c o n t r o l s u r f a c e s and t a b s , r e s p e c t i v e l y of geometrical displacements ~ ~ B I C mass, damping, and s t i f f n e s s matrices, r e s p e c t i v e l y , defined i n terms AA,AB,AC matrices of mass, damping, and s t i f f n e s s changes, r e s p e c t i v e l y , defined i n terms of geometrical displacements e q u i v a l e n t l i n e a r s t i f f n e s s of a nonlinear f o r c e d e f l e c t i o n diagram, defined i n equations (1 ) and ( 2 ) center-of-gravity respectively coordinates of c o n t r o l s u r f a c e s and t a b s , Ce e,f F g h Iv,Itv f o r c e o r moment a c t i n g on a c o n t r o l s u r f a c e o r t a b column matrix of c o n s t r a i n t f u n c t i o n s gg bending d e f l e c t i o n of t h e quarter-chord l i n e of l i f t i n g s u r f a c e mass moments of i n e r t i a per span u n i t of c o n t r o l s u r f a c e and t a b , r e s p e c t i v e l y , r e f e r r e d t o t h e c e n t e r of g r a v i t y axes of i n e r t i a I R ~ , I R ~ , I mass moments of i n e r t i a of c o n t r o l s u r f a c e r e f e r r e d t o the main R~ R m~ span width coordinate control-surface mass mass per u n i t span of c o n t r o l s u r f a c e and t a b , r e s p e c t i v e l y generalized mass, damping, and s t i f f n e s s matrices, r e s p e c t i v e l y generalized matrices taking i n t o account mass, damping, and s t i f f n e s s changes, r e s p e c t i v e l y column matrix of e x t e r n a l f o r c e s column matrices of generalized coordinates column matrix of generalized f o r c e s mv,mtv MIDIK AM,AD,AK P 9rP Q t T time i n e r t i a energy column matrix of geometrical d e f l e c t i o n s s t i f f n e s s energy u U 438 V W f l i g h t speed damping energy transformation matrices, defined i n equations (53) and (55) r o t a t i o n about t h e quarter-chord l i n e of l i f t i n g s u r f a c e control-surface r o t a t i o n about t h e hinge l i n e t a b r o t a t i o n about t a b hinge l i n e r o t a t i o n of a c o n t r o l s u r f a c e r e f e r r e d t o i t s c e n t e r of g r a v i t y damping loss angle column matrix of Lagrange's m u l t i p l i e r s Xg diagonal matrix of the square values of t h e normal c i r c u l a r frequencies modal matrices c i r c u l a r frequency u n i t y matrix zero matrix imaginary u n i t x IY 01 13 Y rl 0 X A @ IY w I 0 j = Subscripts: A,B,C,R,v,t substructures indices c o n s t r a i n t index, normal mode index indices referring t o a c o n s t r a i n t s and E R r Or€ k = 1 , 2, . . ., 0 independent coordinates NL L index r e f e r r i n g t o nonlinear p r o p e r t i e s index r e f e r r i n g t o l i n e a r p r o p e r t i e s Superscripts: T transposed matrix indices referring t o substructures r e a l , imaginary p a r t A AIB 1 and B I n GENERRt REMARKS Sources of Control-System Nonlinearities Aeroelastic investigations are usually carried out on the basis of simplified linearized mathematical models. In many cases this approach has been adequate to ensure sufficient flutter safety margins for light airplanes. However, in the last few years, it has become evident that disregarding nonlinear phenomena can lead to hazardously misleading results. For example, it is shown in reference 1 that so-called concentrated or localized nonlinearities in control systems have a significant effect on the flutter behavior. Nonlinearities of this kind may be produced by such things as (1 ) Backlash in the joints and linkage elements (2) Solid friction in control-cable and pushrod ducts as well as in the hinge bearings (3j Kinematic limitation of the control-surface stroke (4) Application of special spring tab systems provided for pilot handling relief The most critical parts of a control mechanism where localized nonlinearities may arise are shown schematically in ,figure 1 . An aeroelastic investigation may become even more complicated if it is necessary to account for items such as the following: (1) Preload changes due to maneuver loads and specially trimmed flight attitudes (2) Changes in friction and backlash over an airplane's lifetime (3) Additional mass, stiffness, and damping forces randomly activated by the pilot Coping with all these difficulties requires special measures throughout the flutter clearance process. First, the ground vibration test (GVT) used to determine the elastodynamical coefficients of the flutter equations has to be modified so that a consistent and superpositionable set of orthogonal, or well-defined nonorthogonal, normal modes can be measured. In reference 2 a proposed experimental approach employs a high frequency auxiliary excitation superimposed upon the much lower sinusoidal excitation to be tuned to the several normal frequencies. Thus, "slip-stick" effects and related nonlinearities in the control mechanisms can be minimized. The method requires additional test and control devices capable of exciting all controls simultaneously. Of course, the simplest solution appears to be to build control-surface mechanisms without either friction or backlash. However, aside from a consider- a b l e i n c r e a s e i n manufacturing c o s t s , t h e r e is no guarantee t h a t such an i d e a l c o n d i t i o n could be kept unchanged f o r t h e l i f e t i m e of an a i r p l a n e . Moreover, a f r i c t i o n l e s s c o n t r o l system i s n o t n e c e s s a r i l y e q u i v a l e n t t o b e t t e r handling q u a l i t i e s , because f r i c t i o n h e l p s g i v e t h e p i l o t t h e " f e e l " of f l y i n g t h e a i r plane, From an e x p e r i m e n t a l i s t ' s s t a n d p o i n t , t h e r e a r e some simpler, b u t e f f e c t i v e , methods using s p e c i a l modal coupling and modal s u p e r p o s i t i o n approaches. A d e t a i l e d p r e s e n t a t i o n of some of t h e s e methods is given i n t h e subsequent sect i o n s of t h i s paper. They w i l l be r e f e r r e d t o a s Concepts I , 11, and 111. I l l u s t r a t i v e Examples of Control-System N o n l i n e a r i t i e s To g e t a r e a l i s t i c impression of control-mechanism n o n l i n e a r i t i e s , t h e f o r c e d e f l e c t i o n diagrams F(B) of t h e rudder and a i l e r o n system (antisymmetr i c a l and symmetrical c a s e ) of a s o a r i n g a i r p l a n e (ASW-15, A. S c h l e i c h e r , Poppenhausen. W. Germany) a r e shown i n f i g u r e s 2 ( a ) , 3 ( a ) , and 4 ( a ) Using t h e p r i n c i p l e of t h e e n e r g e t i c equivalence ( r e f s . 1 and 3) t h e s t i f f n e s s and damping p r o p e r t i e s of a nonlinear f o r c e d e f l e c t i o n diagram can be approximated by t h e s o - c a l l e d e q u i v a l e n t complex s t i f f n e s s : . The c o e f f i c i e n t s Ce (B) and Ce t i v e l y , can be c a l c u l a t e d from 1 II (B) , r e p r e s e n t i n g s t i f f n e s s and damping, respec- ce(B) = - I lTB . $ = O 1 2T l F(B c0.s 4, -Bw s i n 4) cos 4 d4 Ce(B) = .rrB - 3 $=O F(B cos 4 , -$w s i n 0) s i n 4 d4 1J where t h e c i r c u l a r frequency w = 2 l ~ f (where f is frequency i n h e r t z ) and t h e i n t e g r a t i o n v a r i a b l e 4 = w t . Damping can a l s o be expressed by t h e l o s s angle The functions C (8) and Ce(8) corresponding to the force deflection diagrams , of figures 2 (a), 3 (a), and 4(a) are plotted in 2 (b), 3 (b), and 4 (b), respectively. Figure 3(b) shows that the antisymmetric aileron hinge stiffness in the range of the normal aileron stroke varies between 390 N-m and 44 N-m. Because of the stiffness variation, the normal frequency of the antisymmetrical aileron vibration (wing assumed to be fixed) varies over a wide range, between 2 4 Hz . and 7 4 Hz. At least two other antisymmetric normal modes lie in this frequency . range and are consequently characterized by highly amplitude-dependent portions of aileron vibrations. Similar effects can also be observed for the symmetric aileron mode and for the rudder vibration. The effects of strong nonlinearities on the flutter behavior have been demonstrated in some wind-tunnel tests carried out on a nonlinear wing-aileron mode. in the low-speed wind tunnel of DFVLR ~Gttingen. The nonlinear flutter boundaries for a backlash-type and for a spring-tab-type aileron hinge stiffness are . shown in figure 5 Unlike the flutter boundaries of linear systems, both curves are characterized by a considerable dependence of the critical flutter speed on the aileron amplitude. Thus, the flutter boundary of the spring-tab-type system / / . The backlash-type system shows varies between V = 12.5 m s and V = 24 m s a flutter boundary variation between V = 13.5 m/s and V = 20 m/s. More detailed information, especially about the geometric and elastodynamic data of the wing-aileron model, is presented in reference 1. MATHEMATICAL MODELING USING MODAL SYNrTHE SIS CONCEPTS As mentioned previously, the determination of the elastodynamic characteristics by means of GVT can be affected severely by localized nonlinearities in the control mechanisms. It will be shown in the following discussion that the uncertainties resulting from these nonlinear effects can be avoided by applying experimental-analytical concepts based on the well-known modal synthesis approach. Each of these concepts can be used to set up the aeroelastic equations of the actual airplane including all control-mechanism nonlinearities. The nonlinear force deflection diagrams of the different controls can be determined by static or dynamic tests. Three different concepts will be presented. as follows: They may be briefly described I R Concept I Measurement of a set of orthogonal normal modes with the con: trol surfaces rigidly clamped; separate determination of the controlsurface normal modes with the rest of the airplane rigidly fixed. Concept 11: GVT on a configuration artificially linearized by replacing the nonlinear control-mechanism elements by linear and lightly damped dunmy devices; thus, a set of orthogonal normal modes for the entire system is available. Concept 111: Measurement of a set of orthogonal normal modes with the control surfaces removed; separate determination of the normal modes of the control surfaces in uncoupled condition. Concept I The governing equations of motion of an aeroelastic system, formulated in terms of physical coordinates, can be written in matrix notation as follows: where A B C u P mass matrix damping matrix stiffness matrix column matrix of the physical displacements; and u and second derivatives with respect to time t are first column matrix of external forces, for instance, unsteady aerodynamic forces are nonlinear because of It is obvious that parts of the matrices B and C the localized nonlinearities of the controls. Controls without tabs.- If the GVT is carried out with the controls rigidly clamped to the adjacent structure, a set of n largely linear normal modes @Ar can be measured and combined in the modal matrix The modes satisfy the orthogonality condition where MA KA diagonal matrix of the generalized masses MA, 2 diagonal matrix of the generalized stiffnesses K u = wArMAr AA diagonal matrix of the square values of the circular normal frequencies WA, (not necessarily diagonal) is defined by The generalized damping matrix DA Next, assuming that the control surfaces are rigid in the frequency range of interest, a number of additional control-surface rotation modes with the adjacent main structure at rest can be determined and combined in the modal matrix The physical displacements of the complete structure are related to the generalized coordinates by where the column matrix of the generalized coordinates qr and qv is and The basic idea of this modal superposition is outlined in figure 6, Substituting equation (9) into equation (4) and premultiplying it by QT yields where The m a t r i c e s MA, KA, and DA measured i n a GVT a r e d e f i n e d i n t h e equat i o n s ( 6 ) and ( 7 ) . The d i a g o n a l m a t r i c e s MB, DB, and KB c o n t a i n t h e genera l i z e d masses, damping v a l u e s , and s t i f f n e s s e s o f t h e c o n t r o l - s u r f a c e r o t a t i o n modes. I n t h e c a s e of nonlinear hinge s t i f f n e s s and damping, t h e m a t r i x elements of K g and DB a r e where cAv(8) and Cev(8) can be determined from e q u a t i o n ( 2 ) The term By, denotes t h e c o n t r o l r o t a t i o n i n t h e a c t i o n l i n e o f t h e c o n t r o l a c t u a t o r f o r c e . The m a t r i x MB can be determined by c a l c u l a t i o n o r measurement t a k i n g i n t o account n o t o n l y t h e control-surface mass b u t a l s o t h e moving mass of such a t t a c h e d hardware a s pushrods, c a b l e s , and c o n t r o l s t i c k . The elements of t h e coupling m a t r i x n . can be found by i n t e g r a t i o n over s u r f a c e s SgV of t h e c o n t r o l s where t h e following terms correspond t o t h e vth c o n t r o l with t a b locked t o t h e control mv IV mass of t h e c o n t r o l s u r f a c e p e r u n i t span mass moment of i n e r t i a per u n i t span r e f e r r e d t o t h e c e n t e r of g r a v i t y d i s t a n c e between c e n t e r of g r a v i t y and hinge a x i s ( s e e f i g . 7) d i s t a n c e between hinge a x i s and t h e quarte'r-chord p o i n t ( s e e f i g . 7) ev av A l l t h e s e d a t a a s w e l l a s t h e amplitudes f u n c t i o n s of t h e span c o o r d i n a t e R. h,, a,, and By ( s e e f i g . 7) a r e I n c a s e of an i d e a l locking of t h e con- trols, n e i t h e r hinge s t i f f n e s s f o r c e s nor h i n g e damping f o r c e s a r e g e n e r a t e d i n Hence, t h e normal modes @Are Extension t o c o n t r o l s w i t h tabs.- The above procedure c a n e a s i l y be extended t o systems with c o n t r o l s and t a b s ( s p r i n g t a b s , t r i m t a b s , o r g e a r e d t a b s ) by i n t r o d u c i n g t h e t a b movement a s a s e p a r a t e d e g r e e of freedom. For t h i s s p e c i a l case t h e main GVT c o n f i g u r a t i o n is c h a r a c t e r i z e d by c o n t r o l s locked t o t h e a d j a c e n t a i r p l a n e s t r u c t u r e and t a b s locked t o t h e c o n t r o l s . T h i s l e a d s t o t h e same s e t of normal modes QAr a s d e f i n e d i n e q u a t i o n ( 5 ) . Furthermore, t h e d e g r e e s of freedam of t h e c o n t r o l s a r e s e p a r a t e l y determined w i t h t h e main s t r u c t u r e a t r e s t and w i t h t a b s locked t o t h e c o n t r o l s . The r e s u l t i n g normal modes a r e i d e n t i c a l t o t h e ones d e f i n e d by e q u a t i o n ( 8 ) . F i n a l l y , i n a t h i r d s t e p t h e t a b modes QCv a r e determined w i t h both t h e main s t r u c t u r e and t h e c o n t r o l s a t rest. T h i s concept is s c h e m a t i c a l l y i l l u s t r a t e d i n f i g u r e 8. I n accordance w i t h t h i s , u c a n be e x p r e s s e d a s a s e r i e s expans i o n of t h e normal mode sets OA, QBI and QC where Replacement of u i n e q u a t i o n ( 4 ) by e q u a t i o n (18) and p r e m u l t i p l i c a t i o n by aT l e a d s t o an e q u a t i o n s i m i l a r t o e q u a t i o n ( 1 2 ) . Because of t h e a d d i t i o n a l t a b d e g r e e s of f r e e d m t h e m a t r i c e s M, D, K, and Q have t h e extended form The m a t r i c e s MA, MB, MAB = KA, Kg, DA, and DB a r e i d e n t i c a l t o The m a t r i c e s t h e m a t r i c e s d e f i n e d i n e q u a t i o n s (6) , (7) , (14) , (1 5) , and (16) KC and DC can be determined i n t h e same way a s KB and DB by measuring t h e n o n l i n e a r f o r c e d e f l e c t i o n diagrams of t h e t a b s and u s i n g e q u a t i o n (2) t o calculate . The term Yva d e n o t e s t h e t a b r o t a t i o n i n t h e l i n e where t h e f o r c e a c t i n g on the t a b is a p p l i e d . The m a t r i x Mp can be determined by t e s t o r c a l c u l a t i o n . The elements of t h e c o u p l i n g m a t r i x c a n be found by i n t e g r a t i o n over t h e t a b s u r f a c e Sa where t h e f o l l o w i n g terms correspond t o t h e v t h t a b ( p a r t of t h e Vth c o n t r o l ) m t ~ mass of t h e t a b per u n i t span Itv f mass moment of i n e r t i a p e r u n i t span r e f e r r e d t o t h e c e n t e r of g r a v i t y d i s t a n c e between t h e t a b h i n g e a x i s a n d . t h e t a b c e n t e r of g r a v i t y ( s e e f i g . 7) d i s t a n c e between t h e t a b h i n g e a x i s and the c o n t r o l h i n g e a x i s ( s e e f i g . 7) v h a,, and yv ( s e e The q u a n t i t i e s I t v , mt-, f v , and h a s w e l l a s h,, The elements of t h e f i g . 7) a r e f u n c t i o n s of t h e t a b span c o o r d i n a t e k t . coupling matrix between t h e c o n t r o l s u r f a c e s and t h e a p p e r t a i n i n g t a b s can be c a l c u l a t e d by i n t e g r a t i o n over t h e t a b s u r f a c e S m Provided that the normal modes @A, can be measured with ideally locked control and tab hinges, neither hinge damping forces nor hinge stiffness forces are generated in @Ar. This leads to Concept I1 As described in references 1, 6, and 7, the replacement of the control nonlinearities by artificial linear stiffnesses results i n a modified linearized . test configuration represented in matrix notation by which is formulated in terms of physical displacements. The governing dynamic equations of the unchanged nonlinear system can be written in the same form as equation ( 4 ) by subdividing the matrices A, B, and C as follows: The term A h L - AAL represents the difference in the mass distribution between the artificial linear system and the real nonlinear system; ABL and AcL define the damping and stiffness properties of the artificial linear elements; and kNL describe the damping and stiffness properties of the replaced nonlinear elements. Development of the arbitrary displacement vector u in a series expansion of the measured normal modes QLr of the linearized system yields Inserting this modal transformation into equation ( 4 ) , premultiplying by QLT, and taking into account equation ( 2 8 ) results in generalized equations of motion in the same form as equation ( 1 2 ) , but with the mass, damping, and stiffness matrices now defined as The m a t r i c e s MLI Furthermore , DL, and KL a r e measured i n a GVT on t h e l i n e a r i z e d system. For s i m p l i c i t y , c o n s i d e r o n l y one c o n t r o l s u r f a c e . For t h e v t h c o n t r o l s u r f a c e , t h e modal m a t r i x QL d e g e n e r a t e s t o t h e row m a t r i x and ABNL - ABL and ACNL - ACL d e g e n e r a t e t o t h e 1 x 1 matrices 1 Il where t h e n o n l i n e a r s t i f f n e s s and damping v a l u e s Ce(B) and Ce(B) can be determined a g a i n by a p p l y i n g e q u a t i o n (2) t o t h e measured n o n l i n e a r f o r c e d e f l e c t i o n diagram. The damping and s t i f f n e s s m a t r i c e s BL and CL, respect i v e l y , of t h e a r t i f i c i a l l i n e a r element can be measured by means of s i m p l e tests. The m a t r i x AMNL - AML c a n a l s o be c a l c u l a t e d by u s i n g t h e modal m a t r i x a s d e f i n e d i n e q u a t i o n ( 3 2 ) , provided t h e two p a r t s of t h e 1 x 1 m a t r i x &NL ' A A ~ c a n be d e f i n e d a s moments of i n e r t i a by r e f e r r i n g t h e removed mass of t h e n o n l i n e a r system, a s w e l l a s t h e a d d i t i o n a l mass r e s u l t i n g from t h e a r t i f i c i a l l i n e a r i z a t i o n , t o t h e hinge a n g l e B. Concept I I I The a e r o e l a s t i c e q u a t i o n s of an a i r p l a n e can a l s o be e s t a b l i s h e d by means of both a set of normal modes measured i n a GVT with c o n t r o l s removed and r i g i d body and some e l a s t i c normal modes of t h e s e v e r a l c o n t r o l s ( s e e f i g . 10) d e t e r mined experimentally or by f a i r l y simple c a l c u l a t i o n s . The e q u a t i o n s of motion of t h e coupled system can be set up by means of Lagrange's e q u a t i o n s where The m a t r i c e s AK and AD i n equation (35) t a k e i n t o account t h e e l a s t i c coupling between c o n t r o l s u r f a c e s and main s t r u c t u r e by means of t h e r e a l hinge s t i f f n e s s and hinge damping elements. The term on t h e r i g h t s i d e of equation (34.) i s formulated i n terms of Lagrange's undetermined m u l t i p l i e r s lk which c o r r e spond t o a number of a c o n s t r a i n t c o n d i t i o n s They e x p r e s s c o m p a t i b i l i t y i n t h o s e coupling p o i n t s , where t h e c o n t r o l s can be assumed t o be r i g i d l y f i x e d t o t h e main s t r u c t u r e . Application of equation (34) t o e q u a t i o n s (35) and (36) y i e l d s where t h e elements of t h e r x R matrix yT are Confining the further derivation to the coupling of only two systems, A and B (main structure and control surface) results in the following generalized mass, stiffness, and damping matrices of the uncoupled system: where the submatrices are The matrices Air Bi, and Ci describe mass, damping, and stiffness of the subsystems A and B in terms of geometrical coordinates; @ is the modal . matrix of subsystem i The elements of the diagonal matrices Mi and Ki and of the damping matrices Di, which are not necessarily diagonal, can be determined by GVT or, as in the case of the controls, by calculation, also. According to reference 7 the generalized coupling matrices AK can be written as follows: and AD When the main structure and control surface are coupled by one single ccnnplex hinge stiffness Ce in the action line of the control force, we obtain , n+2 I : a A B , and aar are defined in figure 9 For the special . The angles of rotation a case of coupling two systems A and B the compatibility condition for a physical degrees of freedom can be expressed by the constraints A UJ, gJ,= A - UJ, = B 0 If A B and UJ, are expressed in a series of the normal modes of the systems and B, then UR or in matrix notation The aeroelastic equations of motion are defined now by the n~ + ng = m generalized coordinates. Due to the a constraints there remains a number of s = m - a independent generalized coordinates in terms of which the aeroelastic equations have to be formulated. To do this, the term yT in equation (37) has to be rearranged rowwise so that where Yo is a nonsingular a x a matrix. The matrices M, K, D, AH, and & I with respect to both their columns and rows and the column matrix q have to be rearranged in the same sense. The rearranged equations can be written as where The new structure of the matrices MI D, KI following equation using as an example - - - AD, and AK is shown in the Thus, A can be determined as follows + ([KO& %o] + L a o s I moo]) G) From equations ( 4 5 ) and ( 4 6 ) it follows that Inserting equation ( 5 0 ) into the first s rows of equation ( 4 7 ) and taking into account equation ( 5 1 ) results in the following equation where It can easily be shown that equation ( 5 2 ) can be transformed to the more convenient equation w i t h t h e u n i t y m a t r i x I. I t s h o u l d be mentioned t h a t a n o n s i n g u l a r m a t r i x Ya c a n be determined o p t i m a l l y by a p p l y i n g common mathematical t o o l s f o r t h e d e t e r m i n a t i o n of t h e l i n e a r independence of a g i v e n number of v e c t o r s , a s d e s c r i b e d , f o r example, i n r e f e r e n c e 8. These methods a r e a l s o a p p l i c a b l e t o c a s e s w i t h t h e number of c o n s t r a i n t s h i g h e r t h a n t h e rank of m a t r i x Yo. P r a c t i c a l a p p l i c a t i o n s t o s t r u c t u r a l dynamics problems a r e p r e s e n t e d i n r e f e r e n c e 9. I t is obvious t h a t t h e unsteady aerodynamic f o r c e s c a n n o t immediately b e c a l c u l a t e d on t h e b a s i s of t h e s e p a r a t e normal mode sets of t h e s e v e r a l subHowever, t h i s problem c a n s t r u c t u r e s (main s t r u c t u r e and c o n t r o l s u r f a c e s ) e a s i l y be s o l v e d a s follows: . (1) Couple t h e c o n t r o l s t o t h e main s t r u c t u r e u s i n g t h e above d e s c r i b e d procedure. I n doing s o , t h e a c t u a l n o n l i n e a r s t i f f n e s s e s Ce a r e r e p l a c e d by l i n e a r s t i f f n e s s e s chosen t o be a n average r e p r e s e n t a t i v e of t h e n o n l i n e a r ones. ( 2 ) C a l c u l a t e t h e normal mode c h a r a c t e r i s t i c s of t h i s l i n e a r l y coupled system and c a l c u l a t e t h e unsteady aerodynamic f o r c e s based on t h i s s e t of normal modes. (3) I n t h e c a s e of hinge s t i f f n e s s v a r i a t i o n s o r n o n l i n e a r f l u t t e r c a l c u l a t i o n s , t h e combination of c o n c e p t s I11 and I1 d e s c r i b e d s u b s e q u e n t l y may be used. Combined A p p l i c a t i o n of Concepts I , 11, and I11 A d e t a i l e d examination of t h e p o s s i b i l i t i e s o f f e r e d by t h e t h r e e c o n c e p t s makes it obvious t h a t sometimes t h e i r combined a p p l i c a t i o n may be v e r y b e n e f i c i a l . Four p o s s i b l e v a r i a t i o n s c a n be o u t l i n e d as f o l l o w s : Cambination of Concept I11 and Concept 11: (1) Apply Concept 111, t a k i n g i n t o a c c o u n t l i n e a r and l i g h t l y damped h i n g e c o u p l i n g elements. ( 2 ) C a l c u l a t e t h e normal mode c h a r a c t e r i s t i c s of t h e l i n e a r l y coupled s y st e m . (3) Vary t h e l i n e a r c o u p l i n g e l e m e n t s o r i n t r o d u c e t h e n o n l i n e a r c o u p l i n g e l e m e n t s by means of Concept 11. Combination o f Concept I11 and Concept I: (1) Apply Concept I11 w i t h a completely r i g i d c o u p l i n g i n c l u d i n g t h e cont r o l h i n g e d e g r e e s o f freedom r e s u l t i n g i n a c o n f i g u r a t i o n w i t h r i g i d l y locked controls. (2) Take i n t o a c c o u n t t h e c o n t r o l d e g r e e s of freedom a c c o r d i n g to Concept I by adding a s e p a r a t e set of c o n t r o l normal modes w i t h t h e main s t r u c t u r e a t r e s t . Combination o f Concept I1 and Concept I: (1) T e s t t h e a i r c r a f t s t r u c t u r e w i t h c o n t r o l s removed as a b a s i c con£ i g u r a t i o n . (2) E s t a b l i s h a n a l y t i c a l l y a second c o n f i g u r a t i o n w i t h t h e c o n t r o l s r i g i d l y locked t o t h e main s t r u c t u r e by a p p l y i n g Concept 11. This c a n be a c h i e v e d by adding modal mass c o u p l i n g m a t r i c e s 1IM t o t h e e q u a t i o n s of motion of t h e b a s i c c o n f i g u r a t i o n s i m i l a r to t h o s e d e f i n e d i n e q u a t i o n (31 ) . When a s i n g l e c o n t r o l s u r f a c e is c o n s i d e r e d t h e c o e f f i c i e n t s of t h e mass c o u p l i n g m a t r i x AM can be w r i t t e n a s where The column m a t r i x %, r e p r e s e n t s t h e t r a n s l a t i o n a l and r o t a t i o n a l d i s p l a c e ments a t t h e c o u p l i n g p o i n t of t h e main s t r u c t u r e i n r e l a t i o n t o t h e XYZ a x i s system (see f i g . 10) I f t h e c e n t e r o f g r a v i t y of t h e c o n t r o l l i e s o u t s i d e t h e (x,, ys, zS) = (sXI , sZ) t h e i n e r t i a m a t r i x AAR c a n be 0 , coupling p o i n t , w r i t t e n i n t h e form . where m R mass of t h e c o n t r o l s u r f a c e r e l a t i o n t o its c e n t e r of g r a v i t y IRxrIRy,IRz mass moments of i n e r t i a of t h e c o n t r o l s u r f a c e i n (3) Take i n t o account t h e c o n t r o l d e g r e e s of freedom a c c o r d i n g t o Concept I by adding a s e p a r a t e set of c o n t r o l normal modes w i t h t h e main s t r u c t u r e a t r e s t . Conbination of Concept I1 and Concept 111: (1) T e s t t h e a i r c r a f t s t r u c t u r e with r i g i d c o n t r o l dummies i n locked cond i t i o n as a b a s i c c o n f i g u r a t i o n . The r i g i d dummies a r e used t o determine a b e t t e r basic set of normal mode shapes r e p r e s e n t i n g t h e dynamic deformations of t h e coupled system t h a n c a n be determined i n the test c o n f i g u r a t i o n w i t h removed c o n t r o l s . T h i s procedure can b e s t be d e s c r i b e d a s convergence a c c e l e r a t i o n by means of i n t e r f a c e l o a d i n g . (2) E s t a b l i s h a n a l y t i c a l l y a second c o n f i g u r a t i o n with t h e dummy c o n t r o l s removed. T h i s c a n be achieved i n accordance w i t h Concept I1 by s u b t r a c t i n g a I modal mass c o u p l i n g m a t r i x & a s d e f i n e d i n e q u a t i o n (56) £ran t h e e q u a t i o n s of motion of t h e b a s i c c o n f i g u r a t i o n . (3) Apply Concept I11 c o u p l i n g t h e e l a s t i c c o n t r o l s t o t h e main s t r u c t u r e . COMPARATIVE CONSIDERATIONS The c o n c e p t s p r e s e n t e d o f f e r a number of p o s s i b i l i t i e s to i n c o r p o r a t e t h e c o n t r o l systems of l i g h t a i r p l a n e s , which i n g e n e r a l a r e a f f e c t e d by s t r o n g conc e n t r a t e d n o n l i n e a r i t i e s , i n t o t h e f l u t t e r a n a l y s i s . S p e c i a l emphasis is p l a c e d on t h e mathematical modeling of t h e e l a s t o m e c h a n i c a l system based on GVT. I t is obvious t h a t a f i n a l e v a l u a t i o n of t h e a p p l i c a b i l i t y and a c c u r a c y of t h e d i f f e r e n t c o n c e p t s is r a t h e r d i f f i c u l t because, up t o t h e p r e s e n t t i m e , o n l y Concept I h a s been a p p l i e d t o some e x t e n t t o r e a l a i r p l a n e s t r u c t u r e s . Only l i t t l e e x p e r i e n c e w i t h t h e o t h e r c o n c e p t s is a t hand. Thus, Concept I1 has r e c e n t l y been employed i n t h e c o u r s e of t h e f l u t t e r c l e a r a n c e p r o c e s s of t h e s o a r i n g a i r p l a n e ASW-15. F l u t t e r c a l c u l a t i o n s based on t h i s c o n c e p t p r e d i c t e d That r e s u l t w a s v e r i f i e d by f l i g h t f l u t t e r t a i l f l u t t e r a t about 200 km/hr. t e s t s , where t h e a i r p l a n e showed n o n l i n e a r f l u t t e r i n a speed range from 175 t o 220 km/hr, s t a r t i n g with c a n p a r a b l y s m a l l amplitudes a t 175 km/hr and i n c r e a s i n g t o v e r y high amplitudes f a r beyond t h e r e g u l a r rudder s t r o k e a t h i g h e r speeds. T h i s behavior i n concurrence w i t h s u b s t a n t i a l a l t e r a t i o n s of t h e f l u t t e r modes is symptomatic of h i g h l y n o n l i n e a r f l u t t e r c a s e s . A more d e t a i l e d c o n s i d e r a t i o n of t h i s s p e c i a l problem exceeds t h e s u b j e c t of t h i s paper and s h o u l d be r e s e r v e d f o r f u r t h e r i n v e s t i g a t i o n s . I t is a l s o worth mentioning t h a t t h e ground v i b r a t i o n test c a r r i e d o u t i n accordance w i t h t h i s c o n c e p t took f a r less test t i m e t h a n a normal t e s t on t h e unchanged s t r u c t u r e ( r e d u c t i o n of about 80%) . The first comparative investigation of the Concepts I, 11, and 111 has been the special concern of reference 1 0 , where results are reported for a simple plate-type wing-aileron model with largely linear elastodynamical properties. Although this model cannot be considered representative in all respects of the elastodynamical behavior of real airplanes, it seems to be opportune to use the results of this investigation together with the present experience with the Concepts I and I1 as a basis for a preliminary assessment concerning the advantages and the weak points of different methods. For this purpose a selected number of criteria is used taking into consideration several requirements such as (1) Test effort required (2) Numerical effort required (3) General applicability (4) Physical consistency Table 1 shows in a condensed form how the criteria are met by the several concepts. CONCLUDING REMARKS It has been known for many years that the flutter clearance of light airplanes can be highly afflicted by uncertainties stemming from strong localized nonlinearities in the control mechanisms. It is shown that the establishment of more reliable and accurate mathematical models for the flutter analysis requires modified ground vibration test procedures combined with suitably adapted modal synthesis approaches. Three basic concepts with several variations have been described in detail. They offer a diverse choice of tools for carrying out both approximately linearized and nonlinear flutter investigations. A canparative consideration has been made as to the capacity as well as the drawbacks of the different concepts. However, because of lack of practical experience with Concepts I1 and 111, it is not possible at present to make a conclusive evaluation. REFERENCES 1. Breitbach, E : Effects of Structural Non-Linearities on Aircraft Vibration . and Flutter. AGARD Report No. 665, Sept. 1977. R; . 2 Dat, R.; ~rgtout, . and Lafont, J. M : Eassais de Vibration d'une Struc. No. ture Camportant du Frottement Sec. La Rech. ~e)ros~atiale, 3, 1975, pp. 169-174. 3. Bogoljubow, N. N. ; and Mitropolski, J. A: . Theorie der nichtlinearen Schwingungen. Asp.totische Methoden in der Akademie-Verlag, Berlin, 1965. 4. Scanlan, R H ; and Rosenbaum, R : Introduction to the Study of Aircraft . . . Vibration and Flutter. MacMillan Co., 1951. 5 ~Gssner, G.; and Gollnitz, H : Theorie und Methode der Flatterrechnung . H. . von Flugzeugen unter Benutzung des Standschwingungsversuchs. AVAForschungsbericht 64-01, 1964. 6. ~ussner, . G.; and Breitbach, E: Bestimmung der Korrekturglieder der H . Bewegungsgleichungen bei Aenderungen eines elastomachanischen Systems. AVA-Report 69 J 01, 1969. 7. Breitbach, E : Investigation of,SpacecraftVibrations by Means of the Modal . Synthesis Approach. ESA-SP-121, Oct. 1976, pp. 1-7. 8. Courant, R.; and Hilbert, I : Methoden der mathematischen Physik I. Heidel. berger ~aschenbi.icher, Springer-Verlag, 1968. . . 9. Walton, W C ; and Steeves, E C: A New Matrix Theorem and Its Application . . for Establishing Independent Coordinates for Complex Dynamical Systems With Constraints. NASA TR R-326, 1969. H. 10. ~Gners, : ~erkksichtigungder Ruderfreiheitsgrade im Flatternachweis von Flugzeugen. DFVLR-Report IB 253-78 J 07, 1978. TABLE 1.- COMPARATIVE CONSIDERATION OF CONCEPTS I, 11, AND I11 r Criterion Test Concept I Concept I1 Concept I11 Medium removal o f t h e controls effort locking t h e sevMedium replacement of P r e p a r a t i o n time Low therealhingestiffe r a l c o n t r o l s a n d tabs n e s s e s by l i n e a r ones T e s t time - - - no a n g l e Very low measurements Low - angle measurement Low/medium a n g l e measureLow i n the hinge s t i f f n e s s ment i n coupling p o i n t s , several substructures points - - T e s t equipment r e g u l a r number Medium more a c c e l e r ometers i n t h e hinge o f accelerometers, no a n g l e measurements s t i f f n e s s points Higher accuracy angle required measurement - - Medium/high h i g h number o f acclerometers a t coupling points, several substructures Higher accuracy r e q u i r e d a n g l e measurement - Measuring accuracy Regular accuracy sufficient - I c a l c u l a t i o n of i n t e g r a t i o n Low.- very simple d e t e r - Medium Numerical ~ l a s t o d y n a m i c a l Low/medium normal modes o f coupled over mass d i s t r i b u t i o n mination of coupling equations effort matrices system necessary of c o n t r o l s Low most normal modes Unsteady aerowithout hinge a n g l e dynamic f o r c e s - - - Low/medium a l l normal modes a f f e c t e d w i t h hinge a n g l e s - Low/medium - a l l normal modes a f f e c t e d w i t h hinge a n g l e s P h y s i c a l Convergence consistency and general ' Type of applicontrols cation Good for controls statically Good can be improved by Excellent i n t e r f a c e loading, s t a t without resonances i n indeterminate coupling i c a l l y indeterminate t h e frequency range of included interest coupling included - - - Nonlinearities conNo r e s t r i c t i o n s Restricted t o controls without resonances i n t r o l s can be very f l e x i b l e and l a r g e t h e frequency range o f i n t e r e s t , small s i z e - No r e s t r i c t i o n s controls can be very f l e x i b l e and large - low No r e s t r i c t i o n s number o f nonlinear coupling terms - No r e s t r i c t i o n s , b u t higher No r e s t r i c t i o n s , b u t higher number of nonnumber of nonlinear coul i n e a r coupling terms p l i n g terms I AILERON CONTROL Figure 1.- Schematical sketch of t h e c o n t r o l system of a l i g h t a i r p l a n e . RUDDER HINGE ANGLE, Pt deg ( a ) Force d e f l e c t i o n diagram. RUDDER HINGE ST1FFNESS, CJP)' 1 DAMPING DECREMENT, RUDDER HINGE ANGLE, P. deg 021 Hinge s t i f f n e s s and damping v e r s u s hinge a n g l e . F i g u r e 2.- Force d e f l e c t i o n diagram and s t i f f n e s s and damping f o r the rudder system of a s o a r i n g a i r p l a n e . AILERON HINGE MOMENT. N-rn ANT ISYMMETR ICAL CASE Car Force d e f l e c t i o n diagram. A ILERON H INGE STIFFNESS, 2 c;cP), N-rn 1 I DAMP ING DAMPING DECREMENT, 0.2 0.1 I 1 0 10 20 30 ANTISYMMETR ICAL CASE AILERON HINGE ANGLE, a deg @I Hjinge s t i f f n e a s and damping versus hinge angle. Fkgure 3 . - Force d e f l e c t i o n diagram and s t i f f n e s s and damping f o r t h e a i l e r o n system of a soaring a i r p l a n e . Antisymmetrical case. AILERON HINGE MOMENT, 0 N -m I / I SYMMETRICAL CASE AILERON HINGE ANGLE, P, deg kt) Force d e f l e c t i o n diagram. A lLERON H l NGE STIFFNESS, Ck(B), N-m A l U R O N HINGE ANGLE. P. deg (bl Hinge s t i f f n e s s and damping versus hinge angle. Figure 4.- Force d e f l e c t i o n diagram and s t i f f n e s s and damping f o r the a i l e r o n system of a soaring a i r p l a n e . Symmetrical case. r A lLERON 5 HINGE ANGLE, BACKLASH P, deg \ \ \ SPRING TAB 10 20 FLUTTER SPEED, VF, rnl sec I 30 Figure 5.- Measured f l u t t e r boundary of a nonlinear model. NORMAL MODE WITH CONTROL SURFACE LOCKED 'B" NORMAL MODE WITH LIFTING SURFACE AT REST AND CONTROL SURFACE FREE -\ Figure 6 LI ARB ITRARY DEFLECTION .- Modal ~ ~ ~ e r ~ o s iaccording t o concept I. t i o n Controls without t a b s . -- I ,- C. G. OF CONTROL- Figure 7.- Lifting surface with control surface and tab. -- - @A' mBv v NORMAL MODE W l T H CS AND T LOCKED NORMAL MODE WlTH CS FREE T LOCKED. AND LS AT REST NORMAL MODE WITH CS LOCKED, T FREE, AND LS AT REST ARB ITRARY DEFLECTION u LS: LIFTING SURFACE CS: CONTROL SURFACE T: TAB Ffgure 8.- Modal superposition according to concept I. Controls with tabs. SUBSYSTEM A SUBSYSTEM B HINGE STIFFNESS COUPLED SYSTEM Figure 9.- Modal coupling of a wing control surface system according,to concept 111. U PC COUPLING POINT CG CENTER OF GRAVITY Rlgure 10.- Modal coupling according to a combination of concepts- I and 11. A V N E COMPOSITES I N SAILPLANE STRUCTURES: D A CD APPLICATION AND EJECHANICAL PROPERTIES D i e t e r Muser Research Center St u t t g a r t Deutsche Forschungs- und V e r s u c h s a n s t a l t f 6r Euf t- und Raumfahrt e.V. Advanced Composites i n S a i l p l a n e s mean t h e use of carbon and aramid f f b e r s i n an epoxy matrix. Weight s a v i n g s a r e i n t h e range of 8 to 18% i n compari s o n with g l a s s f i b e r s t r u c t u r e s . The l a m i n a t e s w i l l be produced by hand-layup techniques and a l l m a t e r i a l t e s t s shown here have been done with t h e s e mater i a l s . These v a l u e s may be used f o r c a l c u l a t i o n of s t r e n g t h and s t i f f n e s s a s w e l l a s f o r comparison of t h e m a t e r i a l s t o g e t a weight-optimum c o n s t r u c t i o n . Proposals f o r material-optimum c o n s t r u c t i o n a r e mentioned. TEWICAL HISTORY The f i r s t f i b e r - r e i n f o r c e d g l i d e r , a Phoenix developed by P r o f . Eppler, made its maiden f l i g h t i n 1957. Now, more than 4000 g l i d e r s with g l a s s - f i b e r r e i n f o r c e d s t r u c t u r e s a r e i n t h e a i r a l l over t h e world, I n c r e a s i n g t h e wing loading p e r m i t t e d i n c r e a s e s i n maximum speed, but s t r u c t u r a l demands i n c r e a s e d t h e weight a l s o . A l a r g e span enabled t h e c o n s t r u c t o r s to b u i l d p l a n e s with l i f t t o drag r a t i o s of about 50 (ASW 17: 48.5, Nimbus 2: 49) and s i n k i n g speeds of 0.50 m/s (1.64 f t / s ) But it was n o t p o s s i b l e to r e a l i z e wing spans with more than 22 meters without a very s o f t wing s t r u c t u r e . This was p o s s i b l e when carbon f i b e r s were used i n t h e c e n t e r wing s e c t i o n of t h e Akaflieg Braunschweig SB 10 i n 1972 ( f i g . 1 ) . With a maximum wing span of 29 meters, t h i s g l i d e r has t h e b e s t g l i d e r a t i o of 53 and a s i n k i n g speed of 0.41 m / s (1.35 f t / s ) , But t h e p r i c e of carbon f i b e r s was very high a t t h i s time and s o t h i s m a t e r i a l was used o n l y i n another p r o t o t y p e , t h e Akaflieg S t u t t g a r t fs-29 i n 1975, To r e a l i z e t h e o l d dream t o vary t h e span during f l i g h t , it was a b s o l u t e l y necessary t o use carbon f i b e r s i n t h e o u t e r moving p a r t of t h e wing and i n t h e s p a r of t h e inner wing s e c t i o n , When t h e Akaflieg Braunschweig b u i l t t h e f i r s t all-carbon g l i d e r i n 1977/78, they used carbon f i b e r s t o reduce weight and t o s t i f f e n t h e wing, s o t h a t a l l f l a p s move o n l y very s l i g h t l y and t h e p i l o t is a b l e to hand l e them. And t h i s was t h e year when carbon f i b e r s were used i n a l a r g e r volume i n d i f f e r e n t t y p e s of commercial g l i d e r s , . WEIGHT SAVINGS Weight and s t i f f n e s s problems occur e s p e c i a l l y , and s o it is n o t s u r p r i s ing t h a t most of t h e new f l a p g l i d e r s use carbon f i b e r s i n t h e spar. The wings of some of t h e of t e n - b u i l t g l i d e r s a r e shown i n f i g u r e 2. A l l t h e p l a n e s use a s p a r with carbon-fiber-reinforced epoxy and t h e weight s a v i n g s a r e i n t h e range of about 11 t o 14%. When carbon f a b r i c is a l s o used i n s t e a d of some g l a s s f i b e r f a b r i c l a y e r s , weight s a v i n g s i n c r e a s e up t o 17.4% compared w i t h t h e f u l l y equipped wing or up t o 24.3% compared with t h e wing s t r u c t u r e i t s e l f . I n t h e m a t t e r of f u s e l a g e s , weight saving r a t e s a r e lower ( f i g . 3 ) , because t h e r e is a higher weight p e r c e n t of c o n t r o l s and of t h e l a n d i n g gear. When carbon is o n l y used i n f u s e l a g e s t r i n g e r s , weight s a v i n g s a r e about 8%. I f some g l a s s l a y e r s a r e replaced by aramid o r carbon f a b r i c , t h e range w i l l i n c r e a s e t o about 15%. But t h e s e v a l u e s a r e n o t t h e maximum weight savings which can be r e a l ized. Looking a t s p e c i f i c t e n s i o n s t r e n g t h of r e i n f o r c e d epoxy l a m i n a t e s i n f i g u r e 4, mass r e d u c t i o n s of 50% by s u b s t i t u t i o n of aramid f i b e r s and of 40% by s u b s t i t u t i o n of carbon f i b e r s a r e p o s s i b l e , when b a r e s t r u c t u r e s a r e considered. MATERIAL PROPERTIES A l l m a t e r i a l p r o p e r t i e s shown i n t h e following f i g u r e s a r e test r e s u l t s of hand-laminated systems. Most of t h e tests have been undertaken a t room temperature and normal outdoor humidity. e Resins were of t h e epoxy type, such a s ~ u t g e r s - ~ a k e l i tL02/SL or L20/SL or CIBA XB 2878. These r e s i n systems a r e normally cured f o r g l i d e r purpose a t room temperature f o r 24 hours and postcured a t 60° C (1 40° F) f o r 15 t o 20 hours. They have shown b e t t e r i n t e r f a c e c h a r a c t e r i s t i c s with carbon and aramid f i b e r s and a l s o higher temperature s t a b i l i t y than t h e o l d e r S h e l l Epikote systems. The f i b e r types a r e mentioned i n each f i g u r e . The carbon i s u s u a l l y untwisted T300 B produced by TORAY. F a b r i c t y p e s which have been used have t h e following c h a r a c t e r i s t i c s : Car bon-UD : Carbon f a b r i c : A r amid-UD: Aramid f a b r i c : G l ass-UD: Glass fabric: TORAY Interglas Interglas Interglas Interglas Interglas 2002 03040 98616 9861 2 92145 92125 13 0 200 170 170 220 276 g/m2 g/m2 l i n e n g/m2 g/m2 t w i l l g/m2 g/m2 t w i l l M a t e r i a l tests have been done i n a l o t of d i f f e r e n t works ( r e f s . 1 t o 5 ) . But a l l laminates have been prepared under t h e same c o n d i t i o n s and have been t e s t e d a t t h e same test f a c i l i t i e s . To use advanced composites - i.e., carbon- and a r a m i d - f i b e r - r e i n f o r c e d epoxy l a m i n a t e s - i n s p a r f l a n g e s f o r g l i d e r s and l i g h t w e i g h t p l a n e s , t e n s i l e s t r e n g t h and modulus a r e t h e most important c h a r a c t e r i s t i c s t o c o n s i d e r . Figu r e 5 shows a s m a l l advantage of Kevlar 49 compared w i t h carbon and E-glass e s p e c i a l l y when UD-laminates a r e i n t e n d e d t o be used f o r a wet l a m i n a t i o n proc e s s . For t o r s i o n s h e l l s , f a b r i c s under d i a g o n a l o r i e n t a t i o n a r e normally used. T h e r e f o r e Kevlar and carbon have t h e same q u a l i t i e s . But a s s p a r f l a n g e s may a l s o be loaded under compression, aramid f i b e r s a r e n o t u s a b l e f o r t h i s purpose. Because of i t s c h a i n l i k e molecular s t r u c t u r e , t h i s material. has o n l y about 20% of t e n s i o n s t r e n g t h c a p a c i t y under compression load ( f i g . 6) . I n a l l h i g h l y l o a d e d s t r u c t u r e s t h e s h e l l s a r e a l s o c a r r y i n g l o a d s . To c a l c u l a t e t h e l o a d d i s t r i b u t i o n between t h e s h e l l and s p a r , it i s n e c e s s a r y t o know t h e e l a s t i c moduli of t h e m a t e r i a l s used ( f i g . 7 ) . A c o n v e n t i o n a l s t r u c t u r e has a carbon s p a r , l a m i n a t e d w i t h r o v i n g s or UD-tapes and a +45O r e i n f o r c e d s h e l l . So t h e very s t i f f s p a r w i l l c a r r y most of t h e bending l o a d s , w h i l e t h e s h e l l w i t h o n l y 10% s t i f f n e s s i n carbon o r 3 t o 4% i n aramid o r g l a s s f i b e r f a b r i c w i l l c a r r y o n l y a s m a l l p a r t of t h e bending f o r c e s . T h i s is v a l i d o n l y when t h e l a m i n a t e a r e a s of t h e s p a r and t h e s h e l l a r e i n t h e same range. Due t o t h e higher allowed s t r e s s e s i n carbon compared w i t h g l a s s , t h e c r o s s s e c t i o n s of s p a r s d e c r e a s e w h i l e t h e s h e l l a r e a remains c o n s t a n t . So t h e l o a d - c a r r y i n g r a t i o is pushed t o t h e s i d e of t h e s h e l l and t h e wing s t i f f n e s s i n c r e a s e s . On t h e o t h e r hand, s h e a r moduli of 45O l a m i n a t e s a r e h i g h e r t h a n t h o s e of O0 o r 90° l a m i n a t e s ( f i g . 8 ) . A s t h e s h e a r a r e a of t h e s h e l l is much h i g h e r t h a n t h e a r e a of t h e s p a r , most of t h e t o r s i o n and s h e a r l o a d s a r e c a r r i e d by the shell. F i g u r e 9 shows t h e s h e a r s t r e n g t h of epoxy l a m i n a t e s found by t u b e - t o r s i o n t e s t s . T h i s t e s t method g e n e r a t e s t h e h i g h e s t s h e a r v a l u e s , a s t h e r e is no problem w i t h f o r c e i n t r o d u c t i o n i n t o t e s t specimens. Carbon l a m i n a t e s with +45O f i b e r o r i e n t a t i o n show t h e h i g h e s t v a l u e s compared w i t h aramid o r g l a s s f i b e r s . Woven m a t e r i a l s a l s o produce h i g h e r v a l u e s t h a n nonwoven u n i d i r e c t i o n a l l a y e r s o r i e n t e d under +45O. These l a y e r s a r e b e t t e r t o handle and t o o r i e n t . I n t e r l a m i n a r s h e a r s t r e n g t h ( f i g . 1 0 ) of carbon l a m i n a t e s is h i g h e r t h a n i n g l a s s o r aramid f i b e r l a m i n a t e s . The epoxy r e s i n s used most i n combination with aramid and carbon f i b e r s i n Germany a r e t h e R u t g e r s - B a k e l i t e L20 and CIBA XB 2878. There a r e o n l y s m a l l d i f f e r e n c e s i n m a t e r i a l s t r e n g t h , n o t o n l y i n i n t e r l a m i n a r s h e a r s t r e n g t h , s o t h a t t h e s e r e s i n s may be s u b s t i t u t e d one f o r t h e o t h e r . Both r e s i n s have f u l f i l l e d t h e a i r w o r t h i n e s s requirements i s s u e d by t h e Luf tfahrtbundesamt. I t i s n o t n e c e s s a r y t o use o n l y l a m i n a t e a n g l e s of oO, 0°/900 o r +45O, which a r e based on p r o d u c t i o n e x p e r i e n c e s t o s a v e m a t e r i a l and time d u r i n g f a b r i c a t i o n . When d i f f e r e n t angle-ply l a m i n a t e s a r e used, t h e t e n s i l e modul u s can be c a l c u l a t e d a s shown i n f i g u r e 11 f o r UD-tapes i n a symmetric laminate. For g l i d e r s , temperatures of 54O C (12g0 F) i n s t r u c t u r e s with a white s u r f a c e a r e normally n o t exceeded. But t h e c o e f f i c i e n t s of thermal expansion should be considered ( f i g . 1 2 ) . Additional s t r e s s e s may occur i n some m a t e r i a l combinations. T h i s is a l s o v a l i d when carbon is bonded t o aluminium o r s t e e l . I n t h i s matter t h e r e must be a l s o a n t i c o r r o s i o n c o a t i n g s t o provide c o r r o s i o n p r o t e c t i o n without any adhesive system. S t a i n l e s s s t e e l s should be used i n t h i s case, A s shown before, aramid f i b e r s a r e n o t very u s e f u l f o r primary s t r u c t u r e s . E s p e c i a l l y when weight s a v i n g s a r e necessary i n some p a r t s of p l a n e s , aramid f i b e r s i n combination with carbon f i b e r s can be used t o i n c r e a s e t h e impact resistivity. The low impact energy of pure carbon ( f i g . 1 3) can be improved by combin a t i o n with aramid f i b e r s ( f i g . 1 4 ) . The h i g h e s t g a i n s can be reached w i t h a 36% carbon f i b e r weight r a t i o i n an aramid-carbon-hybrid laminate ( r e f . 4 ) , where carbon is t h e s u r f a c e m a t e r i a l . Such a m a t e r i a l combination may be used i n f u s e l a g e s , e s p e c i a l l y i n t h e cabin a r e a , t o provide l a r g e impact r e s i s t a n c e i n c a s e of an a c c i d e n t . I f such hybrid laminates should be s u b j e c t e d t o high l o a d i n g s t o o , Poisson% r a t i o of t h e combined m a t e r i a l s must be considered ( f i g . 1 5 ) . I n c a s e of l a r g e d i f f e r e n c e s i n P o i s s o n ' s r a t i o , secondary s t r e s s e s perpendicu l a r t o t h e loading d i r e c t i o n w i l l be generated. The i n v e s t i g a t i o n of f a t i g u e u s u a l l y ends a t 1o6 t o 1 o7 load c y c l e s . I n case of t h e hand-laminated, room-temperature-cured epoxy laminates normally used, t h e r e a r e o n l y l i m i t e d v a l i d test r e s u l t s a v a i l a b l e . The published r e s u l t s a r e normally v a l i d f o r prepreg systems ( f i g . 1 6 ) . Larger d i f f e r e n c e s between prepreg r e s i n systems and room-temperature-curing systems a t o p e r a t i o n temperatures of p l a n e s a r e n o t expected and t h e t e s t r e s u l t s can be extrapol a t e d t o t h e s e laminates. F a t i g u e s t r e n g t h of carbon epoxy (about 600 N/mm2) i s much higher than of g l a s s f i b e r epoxy (about 200 ~ / m r n ~ ) . But more tests have t o be run with t h e new r e s i n systems, because t h e normally used S h e l l Epikote/Laromin has poorer q u a l i t y i n combination with carbon f i b e r s . S p e c i a l tests on wing s p a r s have been c a r r i e d o u t with d i f f e r e n t f i b e r r e s i n systems ( f i g . 17 and r e f . 5 ) . A loading spectrum of v a r i o u s o p e r a t i o n l o a d s has been run with about 6 m i l l i o n load c y c l e s o r 9000 hours f l i g h t s i m u l a t i o n f o r g l a s s f i b e r s p a r s . A s t h e l i f e t i m e of f i b e r - r e i n f o r c e d g l i d e r s is higher than expected, an i n c r e a s e d program f o r carbon s p a r s with a s a f e l i f e s i m u l a t i o n of 12 000 hours has been run. Residual s t r e n g t h s of d i f f e r e n t s p a r s i n d i c a t e t h e s a f e l i f e value of 600 N/rmn2 a t t h e maximum demanded o p e r a t i o n temperature of 54O C (12g0 F) ( r e f s . 6, 7 ) . A new problem appears when carbon f i b e r s a r e used i n a i r p l a n e s t r u c t u r e s . Lightning damage may occur t o unprotected carbon-fiber-reinforced p l a s t i c (CFRP) up t o t o t a l f a i l u r e of a 6-mm laminate i n an a r e a of 80 mm diameter, The whole corresponding t o a s t r i k e of 200 kA ( f i g . 18 and r e f s . 8 t o 10) carbon-reinforced a r e a must be p r o t e c t e d with an aluminium mesh. The weight Damage i s reduced t o g a i n is small, because mesh weight i s o n l y 100 g/m2. f a i l u r e of t h e s u r f a c e l a y e r s , . D i f f e r e n t a p p l i c a t i o n s combining a l l m a t e r i a l q u a l i t i e s a r e p o s s i b l e . For fuselage tubes, f i b e r winding technology i s p o s s i b l e and has a l r e a d y been t e s t e d ( f i g s . 1 9 , 20, and r e f . 1 1 ) . For wing s t r u c t u r e s a combination of carbon s p a r s , carbon t o r s i o n a l s h e l l , and aramid t r a i l i n g edge box may be t h e weight optimal s t r u c t u r e ( f i g . 2 1 ) . I n t h e c o c k p i t region hybrid s h e l l s of aramid and carbon f a b r i c may f u l f i l l t h e accident requirements, while the carbon s p a r s c a r r y most of t h e bending loads. Comparing p r i c e s , a decrease is s t i l l observed and a more severe decrease is expected when automotive i n d u s t r i e s s t a r t using t h e s e f i b e r s o r new product ion technologies a r e developed. Also new manufacturing methods, such a s winding or prepreg a p p l i c a t i o n , have t o be introduced t o t h e s a i l p l a n e industry t o make t h e new m a t e r i a l s cost-competitive with t h e "oldn g l a s s f i b e r . ABBREVIATIONS cm GFRP SFRP carbon-fiber-reinforced glass-fiber-reinforced plastic plastic plastic synthetic-fiber-reinforced REFERENCES von Gewebe 1. Hald, H. : Versuche zur ~ e s t i g k e i t IB 454-79/4 (1 979) . - Laminaten. DFVLR S t u t t g a r t , 2. Schmid, T.: ~ i e g e f e s t i g k e i t e nvon Gewebe I B 454-79/5 (1 979) . - Laminaten. DFVLR S t u t t g a r t , 3. Kensche, Chr. ; Muser , D. : Neuere Werkstoffentwicklungen und F e r t i g u n g s t e c h n i k e n f G den S e g e l f lugzeugbau. DFVLR S t u t t g a r t , IB 454-77/17 r (1 977) . 4. Hector, B.: Untersuchungen symmetrischer Mischlaminate. Studienarbeit I n s t i t u t f u r Flugzeugbau, U n i v e r s i t h t S t u t t g a r t , May 1976. 5. Hinz, B.: S t a t i s c h e und dynamische Biegeversuche an CFK-Holmen m i t kastenfijrmigem Q u e r s c h n i t t . DFVLR S t u t t g a r t , IB 454-78/2 (1 978) . 6. LBA: R i c h t l i n i e n zur ~ u h r u n gd e s F e s t i g k e i t s n a c h w e i s e s f u r B a u t e i l e a u s K o h l e n s t o f f a s e r v e r s t a r k t e n K u n s t s t o f f e n von Segelflugzeugen und Motorseglern Luftf ahr tbundesamt (LBA) Braunschweig, March 1978. . 7. N i e d e r s t a d t , G K r i t e r i e n und B e i s p i e l e f iir d i e Wahl von Ver bundwerks t o f f e n m i t F a s e r n hoher s p e z i f i s c h e r E i g e n s c n a f t e n . In: Kohlenstoff- und a r a m i d f a s e r v e r s t k k t e n K u n s t s t o f f en, VDI-Verlag, ~ i i s s e l d o r f1 977. 8. Molly, J. P,: Blitzschutzuntersuchungen und Auslegung d e s B l i t z s c h u t z s y s t e m s fiir d a s GROWIAN R o t o r b l a t t . DFVLR S t u t t g a r t , IB 454-78/11 (1 978). .: - 9. Skouby, C. D.: R e l a t i v e Behaviour o f Graphite/Epoxy and Aluminium i n a L i g h t n i n g Environment. Annual Book o f SAMPE, Volume 23, 1977. 10. Hendricks, C. L.: L i g h t n i n g P r o t e c t i o n Techniques f o r Graphite/Epoxy A i r c r a f t S t r u c t u r e s . Annual Book o f SAMPE, Volume 23, 1977. 11. S p e t h , J. F. : ~aserverbund-RurnpfrGhre. S t u d i e n a r b e i t I n s t i t u t fGr Flugzeugbau, ~ n i v e r s i t a tS t u t t g a r t , J a n u a r y 1976. TYPE SPAN m WEIGHT N 1. FLIGHT FIBER SB 10 29 58 00 1972 SlGRl Figure 1 fs-29 SB 1 1 13 - 19 15 2600 197 8 VARI OU S 3700 1975 T 300 .- CFRP i n prototypes. 980 860 SlingsbyT59 I 12(Z 10.9 1375 1225 ASW 17 950 1150 COMPLETE WING 820 620 STRUCTURE 650 5LO 17.1 Nimbus 2 2L.3 16.9 Mini-Nimbus 700 600 V J [%I I , PIK20D CFRP GFRP WEIGHT [ N l WEIGHT SAVING Figure 2 - Material substitution in sailplane wings. . 1250 1150 8 25 C ASW 17 940 830 11.7 20 CIS M~ni-N~rnbus 1020 905 11,2 50 C ASW 19 SB 11 930 790 15,O 1 CIS LS3 GFRP CISFRP WEIGHT PROD WEIGHT WEIGHTSAVING UNl T S TYPE I N1 [%I Figure 3.- Material substitution in fuselages. KILOMETER 126 SPECIFIC TEhS ILE STRENGTH 100 80 60 40 20 ORIENTATION TYPE FIBER 0 0 90 0/90 ROV liD UD FAB T - 300 @ 0 9 0 0/90 ROV UD UD FA6 KEVLAR 49 r! @ SO O/YO R@V UD UD FAB E - GLASS Figure 4 . - S p e c i f i c tension strength of epoxy laminates. TENSIOM STRENGTH 800 RESIN: ROTGERS L2O 200 ORIENTATION TYPE FIr3ER C 0 9 0 0/90 0 0 9 0 @/90 0 ROV 0 90 @/90 ROV UD UC FAB T - 306 ROV UD UD FPB KEVLAR - 49 UD UD FABRIC E - GLASS Figure 5.- Tension strength of epoxy laminates. TENS ION AND COWPRESSION STRENGTH ORIENTATION TYPE FIBER 0 0 0 D ROV U FAB T 300 0 0 0 ROV U FA6 D KEVLAR 49 0 0 0 4 5 RGV U FABRIC D E - GLBSS Figure 6 . - Tension and canpression strength. YOUNG'S MODULUS BY TE?ISIOU 75 ORIENTATION TYPE FIBER 0 0 90 O/90 45 ROV U U FP,B D D T - 300 0 0 9@ 0/90 q ! ~ 0 0 90 0/90 45 RPV U U FAB D D ROV U U FABRIC D D E - GLASS KEVLPR 49 Figure 7.- Young's modulus of epoxy laminates. SHEAR MODULUS TUBE - SPECIMENS UNDEF TORSION-TEST F i g u r e 8.- Shear modulus of epoxy l a m i n a t e s . SHEAR STRENGTH RESIN: ROTGERS L 20 120 80 40 ORIENTATION TYPE FIBER 0 45 45 0/9@ 0 45 FABRIC , G 45 0 UD UD FABRIC T - 300 FABRIC liD E - GLASS KEVLPR 49 F i g u r e 9.- Shear s t r e n g t h of epoxy l a m i n a t e s . 60 N/ M M ~ INTERLAMINAR SHEAR STRENGTH 50 40 30 2G 1C E - GLASS FABRIC ' 0 450 KEVLP.K 49 - FABRIC GO T 3CO - FPBRIC 45O c0 45O Figure 10.- Interlaminar shear strength. TENSILE MODULUS 102 8C Figure 1 1 . - T e n s i l e modulus of angle-ply laminate. COEFFICIENT OF THERP'IAL mO C 40 LAMINATE 0RIENTP.TION Figure 12.- C o e f f i c i e n t o f thermal expansion. 200 IMPACT ENERGY 150 ern E-GLASS FAERIC W4 49-1 I LR FP3AIC s* T 3C0 B FABFIC 11 G 1111 1111 S C ROVING Figure 1 3 . - Impact energy of epoxy laminates. 1i.iPP.CT ENERGY 15C FIBER: CFRP: T 30O SFRP: KEVLAR 49 RESIN: ROTGERS L 02 100 Tkr O; M 0 20 80 WEIGhT-RATIO CFRP 4C 60 60 4C 8G 20 lop 0 Ofo "10 'VIEIGHT-RP.TIO SFRP 10C Figure 14.- Impact energy of hybrid laminates. ORIENTATION 0 10 20 3C 4C 50 b0 70 80 90 Figure 15.- P o i s s o n ' s r a t i o of epoxy laminates. KILOMETER SPECIFIC STRENGTH 3 4 5 6 7 LOG 8 N LOAC CYCLES Figure 1 6 . - S p e c i f i c f a t i g u e strength of O0 laminates. RESIDUAL STRENGTH I N TEKSION SPAR SPAR BOX LOADING GEVI CE - 4 , 9 2 6 , COO 9 0 0 . GCO FIEER: RESIN: T 3CC EPIKOTE T 300 S I G R I !F XE 2378 XB 2 8 7 8 TOTAL LOAD CYCLES : 6 POC, COO - CK 1 5 YEARS SAFE L I F E TAKE GbST LOAGING SPECTROX OR 9 . 0 0 0 HOURS FLIGHT SIMULP,TION Figure 1 7 . - Dynamic t e s t s on CFRP g l i d e r spar box. F i g u r e 18.- Lightning damage of an u n p r o t e c t e d c a r b o n - r e i n f o r c e d wing s t r u c t u r e . F i g u r e 19.- Carbon f i b e r winding of a f u s e l a g e tube with h y b r i d s t r u c t u r e . Figure 20.- Hybrid fuselage tube under bending load. 1 LEADING EDGE ShELL TORSICN LGAD? $5 CARBOP Ff,BFiIC -SCWDFICF SPAR CARBON ROVIEES @R LC-TAFE 1 STRIPGFF CCRFCN EOVIBC OK UD-TAPE WELL CP.RDC!l-ARI\,EIG-HYBRID 2 2 SHELL 3 4 iiEE +4> CARBOF! Ff.ERIC - SFMd"ICH 5 STPIPGFP CF,I?BOb RCVIPI: OK UD-TPPE TRAILING ECCE +4>/G-YO I\RFflIU FAEEIC-Slif:LiiICii Figure 21.- P r s p s a b s for lightweight structures, Page intentionally left blank THE ULTRALIGHT SAILPLANE J. H. McMasters Boei ng Commerci a1 Airplane Company Seattle, Washington S M AY U MR The increasing cost of traditional soaring has lead to a search f o r less expensive alternatives. During the past decade, the r i s e in the popularity of hang gl i ding, together w i t h advances made in other branches of ul t r a l i g h t weight a i r c r a f t design (e.g , human powered a i r c r a f t ) , has demonstrated the possibility of development of a "new" category of soaring device - the " u l t r a l i g h t sailplane." A presently envisioned, the u l t r a l i g h t sailplane i s s intermediate in size, cost and performance between current hang gl i ders (defined here as a "sailplane" having a foot launch/landing capability) and the lower end of the traditional sailplane spectrum (as represented by the Schweizer 1-26, "Duster" and "Woodstock"). In the design of an u l t r a l i g h t sailplane, safety, low cost and operational simplicity are emphasized a t the expense of absolute performance. The present paper presents an overview of the design requirements f o r an u l t r a l i g h t sailplane. I t i s concluded that by a judicious combination of the technologies of hang gliding, human powered f l i g h t , conventional soaring and motor gliding, an operational 1y and economically viable class of u l t r a l i g h t , self-1 aunching sailplanes can be developed. INTRODUCTION . The purpose of the present paper i s t o summarize and place i n context the technical design trade-offs, performance potential and operational characteristics of a category of u l t r a l i g h t sailplanes which would combine several desi reable characteristics of present hang gliders, sailplanes and motorgliders into a viable, low-cost a1 ternative or supplement to a l l three. There are few modern examples of the u l t r a l i g h t sailplane envisioned here, and a central purpose of t h i s paper i s to establish the existence of an "ecological niche" for such devices. The remarkable r i s e i n the popularity of hang gliding during the past decade has paralleled an increase i n both cost and regulation of traditional sport aviation (powered and unpowered). This has lead to a rebirth i n interest in a range of ultra-light weight sport a i r c r a f t . The wretched safety record and generally low performance (by modern sailplane standards) of hang gliders has resulted in substantial controversy within organizations 1 ike the Soaring Society of America (SSA) regarding the wisdom and desi reabi 1i t y of associating themselves in any way with the vital new sport of " u l t r a l i g h t soaring.'' To many participants in traditional soaring, the term " u l t r a l i g h t sailplane" i s taken as synonymous with the explitive "hang glider," which conjures visions of wretched wood and fabric (or bamboo and p l a s t i c ) anachronisms. T h i s lack of discrimination among the possible types of ul tra-light weight soaring devices i s unfortunate and i s as wrong-headed as considering "soaring" t o be synonymous w i t h fiberglass racing sailplanes and contest flying. Despite i t s obvious l i a b i l i t i e s , hang gliding has several a t t r a c t i v e features, not l e a s t of which are low cost and simplicity (both in construction and i n operation). In view of i t s advantages, and a surprisingly benign regulatory environment, hang gliding has gone its own way, largely oblivious t o the outcries of i t s c r i t i c s . Progress has been rapid and separate organizations have been formed to provide goals and a measure of self regualtion. A t present, hang gliding i s represented by the U Hang Glider Association S (USHGA), i t s British counterpart, the BHGA, and, within the Federation Aeronautique Internationale (FAI), by the Commission International du Voile1 Li bre (CIVL). Several authorities (including the FAA) have attempted to define the term hang glider and identify i t as only one element of a larger "ultralight" matrix. Attempts to rigorously define classes of vehicles whose development i s a t a rudimentary stage are often inadequate and frequently degenerate into a s o r t of pointless legal exercise. Regarding the problem of "disassociating" the hang glider from other types of soaring device, i t must be acknowledged t h a t a l l b u t the crudest of modern hang gliders are capable of soaring under favorable conditions, and there appears to be no satisfactory way to ignore these devices when discussing the broad spectrum of possible soaring activities. Despite the d i f f i c u l t y of formulating adequate general defintions, following simple morphology i s considered adequate f o r purposes of subsequent discussion: Hang Glider the the - An airplane whose dominant mode of f l i g h t i s gliding or soaring, wherein the p i l o t s legs serve as the primary 1i n c h i n g and/or 1andi ng gear .- U t r a l ight Sailplane l Any "lightweight" (by Schweizer 1-26 standards) sailplane capable of steady controlled f l i g h t a t a w (zero wind) minimum speed be1 o 15 m/s (-30kt). SOURCES While few modern examples of the sort of u l t r a l i g h t sailplane to be discussed here e x i s t , i t s possible development must draw heavily on the wealth of data and experience gained i n other branches of low-speed and motor less f 1i ght. Prior to discussing the prospects f o r synthesizing t h i s information into a "new" whole, i t i s advisable to indicate some sources of such information. A definitive technical history of soaring, charting i t s evolution from the notions of Cayley and Rayleigh, through the experiments of Lilienthal and the Wrights, the early experience a t the Wasserkcppe, and the fundamental transition which occurred as ridge soaring gave way to cross-country thermal and wave soaring, has yet to be written. Along the way, the classic u l t r a l i g h t sailplane (perhaps epitomized by the Darmstadt D28 "Mi ndspiel") was discarded as competition sailplane performance rose to i t s present dramatic levels. Serious hang gliding died with Lilienthal. The brief summary of t h i s evolution presented by Zacher ( r e f . 1) remains the single best semi-technical source of information on developments up to the advent of the current range of f i berg1 ass sai 1planes , and a popularized overview has been presented by D i ggi ns ( r e f . 2). Modern sail pl ane developments are covered extensi vely in w the various journals devoted in whole or part t o soaring (e.g., Soaring, Sailplane and Gliding, AeroRevue, Technical Soaring). Possible future trends have been discussed recently in references 3 through 5. The history and technology of hang gliding has been documented in several sources ( r e f s . 6, 7, 8) and an excellent survey a r t i c l e by MacCready ( r e f . 9) describes technical and operational trends f o r a range of unpowered hang glider type vechicles. Developments in t h i s branch of u l t r a l i g h t aviation have been very rapid and the interested reader should consult publications specifically devoted to t h i s sport (e.g., Hang Glider, nee' Ground Skimmer; Glider Rider). Perhaps the most important development in "hang gliding" since publication of ref. 9 has been the rapid r i s e of powered (self launching) hang gliders (ref .7), both rigid and f l e x i b l e winged. Good sources of information on related areas of u l t r a l i g h t a i r c r a f t development (e. g., human powered a i r c r a f t ) are contained i n refs. 10 through 12. Specific background information f o r the present paper has been published i n references 3, 8, 13 through 16. To place the subsequent discussior; in quantitative perspective, the characateristics of tweleve u l t r a l i g h t a i r c r a f t and small sailplanes are presented in Table 1. PRELIMINARY ANAY LSIS In order to discuss the specific design requirements for an " u l t r a l i g h t sailplane" which could represent a true a1 ternative to e i t h e r the traditional sailplane or the modern hang glider, i t i s necessary to examine the possible performance ranges of existing low-speed " a i r c r a f t . " For t h i s purpose i t i s instructive t o examine the vari ation of maximum aerodynamic efficiency (maximum lift-drag r a t i o ) with the f l i g h t speed a t which these values are Such a plot, with the achieved f o r a i r c r a f t operating below 40m/s (-80kt). apparent (approximate) bounds of the f e a s i b l e indicated, i s shown in Figure 1. Lift-drag r a t i o by i t s e l f is not an adequate index of soaring performance, and Figure 2 has therefore been prepared to show the approximate ranges of minimum sink r a t e as functions of horizontal speed for some of the same categories of device shown i n Figure 1. The foot launching capability limitation on cross-country speed for hang gliders i s clearly shown in Figure 2. Figures 1 and 2 show that there exists a rather large void area between the performance ranges of current hang gl i ders and s a i lpl anes. This i s presumably the performance range or "ecological niche'" of the u l t r a l i g h t sailplane. While Figures 1 and 2 provide few clues to the size-performance-cost trade-offs i n ultra1 i g h t design, they remain instructive of the general nature of the performance spectrum to be investigated. As in Nature, i f a vacant niche' e x i s t s , and good reasons f o r f i l l i n g i t exist, i t will be f i l l e d - by new genera or species as necessary. "Good" reasons for f i l l i n g the u l t r a l i g h t niche' can be readily identified on the basis of an analysis of cost and operational penalties of traditional soaring and the performance limitations of hang gliders. Detailed cost-perf ormance comparisons f o r sai 1planes a r e always controversi a1 , and a f u l l discussion of the many factors involved i s f a r beyond the scope of the present paper. However, two brief a r t i c l e s by Sharp (ref. 17) and Bell ( r e f . 18) present interesting insights into the problem of the spiralling cost of traditional soaring, and allow one to make the following observations: 1. I n i t i a1 equipment cost (airframe, instruments, t r a i 1er) i s a substantial portion of the cost of soaring and probably looms largest to the average p i l o t contemplating a f i r s t purchase. There i s a direct relation (with possible substantial s c a t t e r around the mean) between sai 1plane cost, empty weight and performance increase. Bell ' s analysis ( r e f . 18) supports the i n t u i t i v e conclusion that the cost-performance relation i s non-1 i near, w i t h cost increasing ever more rapidly with increasing performance. Over several years of utilization, the overall cost per hour of soaring dominates the cost consciousness of the enthusiast. These costs are stongly influenced ( f o r those who neither crash nor travel frequently to national contests) by: a. b. 2. 3. The requirements f o r aero towing (either i t s direct cost or the problem of a v a i l a b i l i t y 1imiting sailplane u t i l i z a t i o n ) . Factors associated tie-down, travel ). with fixed base operations (hangaring, There are obvious options and alternatives to the above. Homebuilding can reduce airframe costs substantially. However, many 1ower cost/performance sailplanes f o r which plans or k i t s are presently available suffer from a level of structural complexity which limits t h e i r appeal to homebuilders due t o the large amount of construction time involved. Further, these a i r c r a f t , once built, remain traditional sailplanes carrying the f u l l burden of operating costs associated w i t h any performance level sailplane (fiberglass or otherwise). In principle, motor gliders (or self 1aunching sailplanes) could reduce direct operating costs (e.g., towing, outlandings), and increase u t i l i z a t i o n . Motor g l i d i n g has n o t y e t become a popular a l t e r n a t i v e i n t h i s country, due t o a number o f f a c t o r s besides t h e p h i l o s o p h i c a l d i f f i c u l t i e s o f I f soaring performance mating an engine t o a "motorless" soaring machine. comparable t o unpowered equivalents i s sought (e.g., PIK 20E, Motor Nimbus), equipment c o s t becomes v e r y high. I f s i m p l i c i t y o r c o s t r e d u c t i o n i s sought i n a conventional s a i l p l a n e weight vehicle, power requirements become excessive and/or p e r f ormance d e t e r i o r a t e s dramati c a l ly. A l l t o o f r e q u e n t l y , a device resembling t h e mooncalf o f f - s p r i n g o f a d a l l i a n c e between a Piper "Cherokee" and a Ka6 r e s u l t s . F i n a l l y , commerical motor g l i d e r development has been plagued f o r decades by t h e problem o f a v a i l a b i l i t y o f low-cost, r e l i a b l e , l i g h t - w e i g h t , l i c e n s a b l e engines. Current hang g l i d i n g (powered o r unpowered) may p r o v i d e an a1t e r n a t i v e t o s a i l p l a n e s o a r i n g f o r some, b u t many more conservative i n d i v i d u a l s are p u t - o f f by t h e s a f e t y record o f t h e sport, t h e apparent f l i m s i n e s s o f t h e equipment, and t h e l a c k o f s u i t a b l e i n s t r u c t i o n o r f l y i n g s i t e s i n t h e i r area. Extremely 1i g h t weight s t r u c t u r e s and u l t r a - l o w speeds are i n t r i n s i c c h a r a c t e r i s t i c s o f t h e hang g l i d e r , t h e r e s u l t i n g compromises i n performance and crash p r o t e c t i o n made i n exchange f o r t h e freedom and low cost o f b a s i c f o o t launched hang g l i d i n g being c h e e r f u l l y accepted by i t s proponents. To many accustomed t o 1-26 d u r a b i l i t y and performance, hang g l i d i n g i s no a1t e r n a t i v e a t a l l . The l a t e s t b o l d e x t r a p o l a t i o n i n hang g l i d i n g i n v o l v e s f i t t i n g "anything a i r w o r t h y " w i t h a "chainsaw" engine. This development has caused v e r y s e r i o u s concern, even among many o f those who have been s t o u t advocates , o f basic hang gliding. The sometimes crude, o f t e n unenlightened "cut-and-try" n a t u r e o f some o f these r e t r o f i t s t o marginal o r i n a p p r o p r i a t e airframes seems a sure r o u t e t o d i s a s t e r . The obvious appeal i s undeniable, however. It should a l s o be noted here t h a t several designs f o r "low-cost" s a i l p l a n e s have r e c e n t l y appeared. Only two o f these, however, ( t h e powered v e r s i o n o f "Monerai" and t h e American "Eaglet", c f . Table 1) s e r i o u s l y address b o t h t h e problems o f reducing a i r f r a m e c o s t (through reduced s i z e and complexity) and o p e r a t i n g c o s t (by i n c o r p o r a t i n g a s e l f 1 aunching capabi 1i t y ) Both t h e "Eaglet" and the "Monerai" remain r e l a t i v e l y s o p h i s t i c a t e d by contemplated u l t r a l i g h t standards and t h e i r appeal as a t r u e a l t e r n a t i v e t o conventional s a i l p l a n e s remains t o be f u l l y demonstrated. . I n view o f t h e preceeding discussion, i t appears t h a t t h e e c o l o g i c a l n i c h e ' f o r a safe, u l t r a l i g h t , low-cost s a i l p l a n e indeed e x i s t s and i s n o t adequately f i l l e d by o t h e r a v a i l a b l e types of soaring equipment. As hang g l i d i n g matures and t h e cost o f t r a d i t i o n a l s o a r i n g continues t o increase, i t seems u n l i k e l y t h a t overlap between t h e two s p o r t s w i l l occur ( t h u s l e a v i n g t h e u l t r a l i g h t n i c h e 1 i n t a c t ) , and t h e requirements f o r t h e u l t r a l i g h t a l t e r n a t i v e w i l l increase. If t h e f a v o r a b l e prognosis f o r t h e u l t r a l i g h t s a i l p l a n e i s v a l i d , why do so few examples o f t h i s type o f machine e x i s t a t present? The reasons f o r the llvacancyll i n the u l t r a l i g h t sailplane niche1 have several historical roots, b u t i t may be conjectured that basically i t s time has not yet come (or returned). Soaring in the U (unlike Europe) is not a major S branch of sport avi a t i on. Potent i a1 domest i c manufacturers of conventional sailplanes are faced w i t h a limited market and the huge expense of co.mplying with existing airworthiness c e r t i f i c a t i o n requirements. Ultralights l i k e the "Windspiel" became "obsolete" i n the early 19301s, and domestic sailplane designers have remained enthralled w i t h the chall enges of developing high performance racing sailpl anes (or more affordable imitations) ever since. The low p r i o r i t y of soaring due t o i t s limited commercial potential has also resulted i n a 1ack of the research necessary t o maintain a strong modern data base from which designs can compete e f f i c i e n t l y w i t h European (largely German) manufacturers. A racing sailplane performance and cost have spiralled upward together, an s alternative presented i t s e l f on the extreme low end of the soaring spectrum i n the form of a rebirth in i n t e r e s t i n hang gliding. Here, a t l e a s t , no technology gap existed between domestic and foreign manufacturers. In the absence of any direct government regulations on hang g l i d i n g , this t u r n from the sublime to the ridiculous has flourished. Hang gliding development has brought with i t a whole new s e t of challenges to designers, and remarkable progress has been made very largely on a cut-and-try basis. A developments s on both ends of the soaring spectrum mature and s t a b i l i z e , the time may again become ripe to turn attention to the middle range of u l t r a l i g h t sailplanes, and a class of machines as different from the "Windspiel" as the Rogallo i s f rom the box k i t e may emerge. Regardless of the route future soaring developments take, i t appears that there i s a valid place f o r an u l t r a l i g h t sailplane i n the overall scheme. I t can be argued t h a t both the "Eaglet" and "Monerai" are commendable half measures of what may eventually be possible, and a large gap s t i l l remains between these machines and the 1-26 on one side and the motorized Mitchell Wing ( r e f . 19) on the other. The technology e x i s t s to design a good u l t r a l i g h t and the l a s t stumbling block t o i t s early realization appears t o be lack of a definite goal f o r i t s development. Ann Welch's a r t i c l e ( r e f . 15), advocating establishment of an internationally recognized "Ultralight Class1' (100 kg empty weight limit) f o r record and competition purposes, discusses what may be wanted, provided the rules are not too confining, and the resulting machines represent clear a1 ternatives to e i t h e r present hang gliders or pseudo-racing sailplanes. AN ULTRALIGHT SAILPLANE On the basis of the preceding discussion i t i s now possible to define in more detail the concept and design requirements of a I1typical" u l t r a l i g h t sailplane. Concept This l i g h t weight (empty weight l e s s than about 1300 N--300 1bs. ) s a i l p l a n e i s intended f o r l o c a l and 1 i r n i t e d cross-country soaring. The a i r c r a f t may be s u i t a b l e f o r home c o n s t r u c t i o n from a l i m i t e d number o f p r e f a b r i c a t e d components. Launching i s t o be by means o f o t h e r than aero towing (e.g., bungee, w i nch o r s e l f -1 aunched by an a i r r e s t a r t a b l e engine). Design P r i o r i t i e s I n order o f importance: 1. S a f e t y (benevolent launch and f 1i g h t c h a r a c t e r i s t i c s , no unusual demands on p i 1o t s k i 11, adequate strucutral strength and c o n t r o l a b i l i t y over the e n t i r e f l i g h t envelope, crash p r o t e c t i o n f o r the p i l o t ) . S i m p l i c i t y ( i n b o t h c o n s t r u c t i o n and operation). "Low c o s t " ( i n both c o n s t r u c t i o n and o p e r a t i o n ) . Performance (adequate m i 1d thermal soaring p e n e t r a t i o n i n t o winds up t o 15 m/s- 30 k t ) . Additional Constraints capabi 1ity, adequate 2. 3. 4. 1 . 2. 3. The machine should be t r a n s p o r t a b l e on n o t h i n g more e l a b o r a t e than a simple boat type t r a i l e r , towed by a compact car. The machine should break down i n t o components which a l l o w convenient storage a t the owner's residence. There should be minimum requirements f o r , or l i m i t a t i o n due t o , s p e c i a l launching s i t e s (e.g., a h i l l o f s u f f i c i e n t slope and h e i g h t ) . No comp1 e t e l y adequate airworthiness standards (U. S. or i n t e r n a t i o n a l ) p r e s e n t l y e x i s t f o r t h i s category o f u l t r a l i g h t aircraft. Until such standards are formulated, the OSTIV A i r w o r t h i n e s s Standards f o r S a i l p l a n e s should be used as a guide. 4. A t e n t a t i v e concept f o r t h e t y p e o f machine which might meet these requirements i s shown i n F i g u r e 3, together w i t h an e x i s t i n g " f i r s t g e n e r a t i on" version. TECHNICAL CONSIDERATIONS A detailed technical discussion of u l t r a l i g h t design trade-offs i s beyond the space limitations of the present paper. Such a study is i n preparation by the author, and, in the inter i m , some additional technical references up-dating those i n ref. 8 are presented. Although absolute performance i s not the primary design goal of the u l t r a l i g h t sailplane, i t remains necessary t o examine careful l y several areas of performance compromi se involved in meeting primary design objectives (e.g,, safety, low-cost, simplicity). Aerodynamic Requirements In general, sailplane aerodynamic prel irninary design optimization i s performed assuming a "glider" operating i n r e c t i l i n e a r f l i g h t , w i t h central emphasis placed on the achievement of a high lift-to-drag r a t i o ( m i n i m u m glide angle) a t a "desired" forward speed. Around t h i s pivot point in the performance polar, low sink rates a t both low ( f o r climb) and high (cross-country) speeds are juggled until a satisfactory "racer" has been defined. If thermal soaring i s envisioned, only towards the end of the analysis i s sink r a t e in a banked t u r n seriously considered. I t has recently been argued by Eppler ( r e f . 20) and Irving ( r e f , 21) t h a t emphasis on analysis of the r e c t i l i n e a r portion of the glide may lead to non-optimum .sizing (selection of w i n g area and aspect r a t i o ) of 15m span sailplanes which must both thermal e f f i c i e n t l y and achieve good h i g h speed performance. Under a variety of conditions ( b a l l a s t levels and thermal models assumed), a racing sailplane optimized for minimum sink r a t e in a turn and a high forward speed in the region around 2-3 m/s r a t e of sink should have a somewhat lower than customary aspect ratio. In the 15 m examples considered, t h i s means larger area. In these examples, absolute r e c t i l i n e a r L/D suffers somewhat, b u t average cross-country speed ( i n the MacCready sense) increases, For somewhat different reasons, the u l t r a l i g h t sailplane presents the same two-poi n t optimizati on problem confronted by the classic thermal soaring racer, with r e c t i l i n e a r maximum L/D being of importance only insofar as i t r e f l e c t s m i n i m u m sink rate ( a t an arbitrary bank angle) and high speed (penetration) capability. High speed penetration capability i s basically a safety objective, and only secondarily a desireable performance objective i n the u l t r a l i g h t . Minimum sink r a t e i s the fundamental performance objective. An indication of banked turn performance i s shown i n Figure 4. Unfortunately, the banked t u r n , minimum sink r a t e , optimum sizing problem i s a great deal more complex than the simple r e c t i l i n e a r f l i g h t problem, For further discussions, the papers by Marsden (ref 22, 23) and Cone ( r e f . 24), i n addition to those by Eppler ( r e f . 20) and Irving ( r e f . 21), should be consulted. Any serious u l t r a l i g h t design must also consult the report by Shenstone and Scott-Hall ( r e f . 25). While u l t r a l i g h t detailed aerodynamic wing design should follow conventional sailplane practice (although aspect r a t i o s may be substantially lower), the selection of suitable a i r f o i l sections presents a major problem due to the general lack of experimental data f o r appropriate sections optimized a t sufficiently low Reynolds number. Existing data i s surveyed in refs, 26 through 34. The a i r f o i l selection and design problem i s further complicated by the strong coupling between high-lift/low-drag aerodynamic and simple, light weight wing structural requirements. Modern 1aminar sailpl ane sections are generally inapplicable to an airplane wherein the structure i s unlikely t o support laminar flow much beyond 30-40% of the wing chord. In t h i s light, the experience w i t h sailplanes of 1930-40 vintage ( r e f s . 25, 27) provide a f a r better guide to a i r f o i l selection and performance than do those of the 1970's. Aerodynamically, the u l t r a l i g h t i s an excel1 ent candidate f o r a f u l l y f 1apped wing (preferably involving flaps with a high degree of Fowler motion). Unfortunately, t h i s desirable feature directly conflicts w i t h the simplicity requirement, and cannot be advised f o r early generations of such a i r c r a f t . Further data on t h i s topic can be found i n references 35 through 39. Aerodynamic Constraints The basic f i r s t order equations of sailplane motion (cf. refs. 22, 24, 40) show that both minimum sink r a t e and maximum L/D are (for equal weight vehicles) most powerfully influenced by w i n g span. High speed performance i s 1argely one of prof i l e/parasi t e (viscous dependent) drag which increases as the square of the f l i g h t speed. Further, whereas weight and/or, wing loading increase helps high speed performance, i t seriously erodes minimum sink performance. Overall, then, f o r a racing sa'ilplane the trend should be towards large span ( t o regain low-speed performance) and high wing loading and In extreme aerodynamic "cleanliness" t o maximize h i g h speed performance. addition, a better match between desired 1ow- and high-speed performance can be had by use of flaps. This simplistic view ignores important aspects of the low-speed thermalling (banked t u r n ) mode, however; these e f f e c t s may be particularly important in attempts to transfer the above recipe t o an u l t r a l i g h t . Increasing span leads to increasing wing weight. Drag cleanup and flaps are contrary to structural' simplicity. Most important i s the "low-speed t u r n problem" which puts the u l t r a l i g h t in closer kinship w i t h the vulture and the HPA than the racirig sailplane, a n d may ultimately establish a practical upper bound on wing span, just as aeroelastic e f f e c t s a t high speed ultimately limit the span of high-performance sailpl anes. In a steady turn, the radius i s a purely kinematic function proportional t o the square of the sailplane's velocity and the reciprocal of the tangent of the bank angle. For an u l t r a l i g h t type vehicle (vulture, hang g l i d e r ) , the normal thermalling speed may be decreased to the point where the wing span becomes a significant percentage of the turn radius. A discussed in refs. s 41-43, t h i s situation results i n substantial gradients of velocity (and hence dynamic pressure and Reynolds number) across the span, and a corresponding distortion of the untrimmed span loading accompanied by an outboard s h i f t i n the center of l i f t which tends to steepen the turn. To counteract t h i s overbanking tendency, powerful trimming devices ( a i 1erons and rudder) and dihedral are required, and/or the bank angle or wing span must be limited. Regardless of other precautions, the depressed Reynolds number over the inboard semi-span during a t u r n may aggravate any tendency towards t i p s t a l l w i t h the danger of a subsequent spin. The vulture's solution to t h i s problem ( r e f . 24) i s worth noting, since i t represents a marvelous example of the coupling between structural strength/stiffness, high-lift aerodynamics and minimization of trim drag. Structures and Weight Surprisingly l i t t l e good information on u l t r a l i g h t structural techniques exists. The best sources r e l a t e t o human powered a i r c r a f t , the structures of which are generally complex. O a l l the aspects of u l t r a l i g h t development, f structural weight reduction and si)nplification are in most need of major e f f o r t . The author's favorite sources on these topics are references 44 through 48. Launching Provision of an alternative to aero towing for launching an u l t r a l i g h t sailplane i s central to the operational simplicity concept. The success of the motorized hang glider makes the notion of a motorized self-launching u l t r a l i g h t sailplane an a t t r a c t i v e idea. The key to success here l i e s i n avail abi 1i t y of re1 iable "engines" (internal combustion or otherwise). I t should also be noted that either the canard or the flying wing configuration seem natural f o r a powered u l t r a l i g h t due to the ease of low drag integration of an engine into the design. Some information on suitable engine/propeller combinations are contained in references 49 and 50. CONCLUSIONS An evaluation of the requirements f o r an inexpensive alternative t o conventional soaring has shown that an "ecological niche"' apparently exists f o r an u l t r a l i g h t sailplane intermediate i n performance and weight between modern hang gliders and traditional sport sailplanes. There appear to be no serious constraints on the ecomonic or operational v i a b i l i t y of such a device. Four factors appear to be central to progress towards i t s early realization: 1. Development of simple structural techniques for minimum time and cost construction of wings of adequate aerodynamic quality, strength and stiffness. Availability of reliable, light weight, low powered and low-cost engines to provide a self 1 aunchi ng capabi 1i t y , and/or development of "minimum" non-aero tow launching methods. Establishment of suitable goals f o r u l t r a l i g h t sailplane performance and design (e.g., national or international recognition of an Ultralight Class f o r record or competition purposes). Clarification of the unpowered), u l t r a l i g h t Whether regulated by airworthiness standards relationship of hang gliding (powered or sailpl anes and government (FAA) regulation. the government or not, a suitable s e t of f o r "ultralights" needs t o be developed. 2. 3. 4. As a f i n a l thought, i t can be argued t h a t the single most important factor which made the modern hang glider renaissance flourish as i t has was the structural and aerodynamic model presented a t the outset by the Rogallo w i n g . The u t t e r simplicity of t h i s concept completely outweighed i t s very modest performance. As i t turned out, this performance was quite good enough to launch a new sport and i t s supporting industry. The great progress i n hang gliding since a few visionaries began diving off sand dunes i n bamboo and pl a s t i c monstrosities has been accompl ished very 1 argely by cut-and-try, further t r i b u t e to the basic simplicity of the i n i t i a l concept. I t now remains f o r some individual to make the same sort of creative leap which could usher i n the modern u l t r a l i g h t sailplane. REFERENCES 1. Zacher, H., "The Shape of High Performance Sailplane Technical Development," The World's Sailplanes, Vol. 11, edited by B S. Shenstone . and K. G. ilki ins on, OSTIV Publication,196-3, 2. Dwiggins, D., Silent Wings, NY: Gossest and Dunlap, 1970. 3. McMasters, J. H, and Nash-Webber, J., "Some Technical Extrapolations," Soaring, January 1977, pp. 21-28. 4. Carmichael, B. H., "On Predicting the Future of Soaring Flight," Soaring,. April 1977, pp. 25-30. 5. Brain, V., "The Next 25 Years 14-19. - An Overview," Soaring, January 1977, pp. 6. McMasters, J. H. and Cole, C. J., "Ecoflight Is Here," Lifestyle No. 3, February 1973. (Available from - Mother - News P.O. Box 70, The Earth Hendersonvi 11 e, NC 28739. -9 7. Gustafson, D., "Powered Hang Gliders and Such," Sport Aviation, August 1978, pp. 10-14. 8 McMasters, J. H., "Some Opportunities for . Aeronautics," Soaring, June 1975, pp. 22-26. Progress in Ultralight 9. MacCready, P. B. Jr., "Developments in Ultralight Gliding," Soaring, June 1976, pp. 23-29. 10. McMasters, J. H. and Palmer, G. M., "At the Threshold of Man Powered Flight," Aeronautics and Astronautics, September 1977, pp. 60-70. of 11. Reay, D. A., The History - Man-Powered Flight, NY: Pergamon Press, 1977. 12. McMasters, J.H. and McLean, J. D., "The Formulation Flight of Human Powered Aircraft Across the English Channel in the Spring," presented at the XVI OSTIV Congress, Chateauroux, France, July 1978. (See also Soaring, June 1978, pp. 16-22). 13. McMasters, J. H., "Four Ultralight 14-15. question^,^' Soaring, July 1977, pp. 14. McMasters, J. H., "Ultralight Survey Results," Soaring, December 1977, pp. 27-30. 15. Welch, A., "The Orphans," Soaring, July 1978, p. 14. 16. Worthington, G., "Wheels vs. Foot Launching," Soaring, July 1978, p. 15. 17. Sharp, T., "The Myth of Cheap Soaring," Soaring, August 1977, p. 27. 18. Bell, D., 31. "A Look a t the Used Sailplane Market," Soaring, January 1978, p. 19. Patmont, S., "Free a t Last," Soaring, August 1978, pp. 12-13. 20 Eppler, R., "The Optimum Design and Wing Section of a 15m Glider Without Flaps," Sailplane - Gliding, June/July 1977, pp. 110-117. and 21. Irving, F., "Computer Analysis of the Performance of 15m Sailplanes," i n Motorless Flight Research - 1972, J. L. Nash - Webber ed., N S CR 2315, AA November 1973. 22. Marsden, D. J., "Circling Performance of Sailplanes," OSTIV Pub. XI, 1970. 23. Marsden, D. J . , "Gemini - A Variable Geometry Sailplane," AIAA Paper 74-1035, September 1974. (Proc. 2nd Int Symposi um Low-Speed and Motorless F l i g h t ) . . 24. Cone, C. D. J r . , "The Design of Sailplanes f o r O~timum Thermal Soaring Performance," N S TN D-2052, January 1964, AA 25. Shenstone, B. S. and Scott-Hall S., T 780, 1937. M "Glider Development in Germany," NACA 26. McMasters, J. H., "Low-Speed Airfoil Bibliography," Tech. Soaring, Vol. 3, No. 4, Fall 1974, pp.40-42. 27. Riegels, F. W., A i r f o i l Sections, London: Butterworth, 1961. as 28. Jacobs, E. N. and Sherman, A., "Airfoil Section C h a r a c t e r i s t i c s Affected by Variations of the Reynold Number," NACA TR 586, 1937. 29. Liebeck, R e H., "Design of Subsonic A i r f o i l s f o r High L i f t , " - - of Jour. A i r c r a f t , September 1978, pp. 547-561. 30. Henderson, M. L . , "Inverse Boundary-Layer Technique f o r A i r f o i l Design," Proceed. of the N S AA Advanced Technology Airfoi 1 Research (ATAR) Conference, N S CP-2045, March 1978. AA 31. Wortman, F. X., "Airfoil Design f o r Man-Powered A i r c r a f t , " Proceed. of the 2nd Royal Aero. Soc. MPA Group Symposium, February 1977. (Roy. Aero. Soc., 4 Hami 1ton Pl ace, London WV OBQ, England) I 32. Eppler, R., "Turbulent A i r f o i l s f o r General Aviation," February 1978, pp. 93-99. 33. H i scocks, R. D., " A i r f o i 1 s," -- o f Jour. Aircraft, Soaring, November-December 1951. 34. McGhee, R. J. and Beasley, W. D., " E f f e c t s o f Thickness on t h e Aerodynamic C h a r a c t e r i s t i c s o f an I n i t i a l Low-Speed Family o f A i r f o i l s f o r General Avi a t i on A p p l i c a t i o n s , " NASA TMX-72843, December 1976. 35. Smith, A. M. 0. "High L i f t Aerodynamics," June 1975, pp. 501-530. Jour. A i r c r a f t , Vol. 12, No. 6, "Low-Speed Aerodynamic C h a r a c t e r i s t i c s 36. Bingham, G. J. and Noonan, K. W., o f NACA 6716 and NACA 4416 A i r f o i l s w i t h 35% Chord S i n g l e - S l o t t e d Flaps," NASA TMX 2623, 1974. 37. Wortmann, F. Publ. XI). X., "The Sailplane," Aero-Revue, June 1971. (Also OSTIV 38. Wenzinger, C. J. and H a r r i s , T. A., "Wind Tunnel I n v e s t i g a t i o n o f an NACA 23012 A i r f o i l w i t h Various Arrangements o f S l o t t e d Flaps," NACA T 664, R 1939. "combination o f A i l e r o n and F l a p D e f l e c t i o n on Minimum Induced 39. F e i f e l , W., Drag R o l l Control," presented a t X V I OSTIV Congress, Chateauroux, France, J u l y 1978. 40. Falk, T. J. and Matteson, F. H., Handbook Vol. 9, LA: SSA, 1971. S a i l p l a n e Aerodynamics, American Soaring Tech. Soaring, 41. McMasters, J. H., "The Prospects f o r Man-Powered F l i g h t , " . Vol. 1 No. 2, October 1971. 42. McMasters, J. H., "The O p t i m i z a t i o n o f Kremer Competition Man Powered A i r c r a f t , " Proceed. o f t h e 2nd I n t e r n a t . Sym, on t h e Tech. and Sci. o f Low-Speed and Motor l e s s F l i g h t , September 1974. 43. P h i l l i p s , W. H., "Analysis and Experimental Studies o f t h e C o n t r o l o f Hang Gliders," AIAA Paper 74-1030, September 1974. "Dominant Factors i n L i g h t Weight Design," 44. Czerwinski, W., Space J., January 1967, pp. 9-22. "Man-Powered. F l i g h t , 45. Czerwinski, W., 70-879, J u l y 1970. 46. Wimpenny, Aircraft," I t s Purpose and Future," Considerations in Can. Aero. AIAA Paper Man-Powered -J Aero _a J. C., 9 " S t r u c t u r a l Design May 1975, pp. 198-207. 47. Pazmany, L. e t . a1 , "Potenti a1 S t r u c t u r a l Materials and Design Concepts f o r Light Airplanes,' NASA CR-1285, March 1969. 48. Stender, W e , " S a i l p l a n e Weight Estimation," OSTIV P u b l i c a t i o n , June 1969. 49. "Dyad 280, A Self-Launch Engine," Soaring, December 1977, pp. 31-34. 50. Rautio, J., "15-Meter December 1978, p. 3. Motorglider Performance Calculations," Soaring, . 51. Rogers, B e , "1974 S a i l pl ane Directory," Soaring, August 1974. 52. Hoose, S. ( e d ) , "1978 S a i l p l a n e D i r e c t o r y Suppl iment," 1978, pp. 21-44. 53. Simons, M. "The D38 Windspiel ,I1 Soaring, August Soaring, February 1972, pp. 35-36. 54. Zacher, H. "Flugmessungen mit Segelf 1ugzeugen von 12 b i s 13m Spannwei t e , " OSTIV P u b l i c a t i o n IX (1965, p t . 2 ) . 55. Maupin, J. "Woodstock," Soaring, May 1978, pp. 38-39. 1978, p. 3. 56. Hall, S., "The Vector I , " Soaring, ~ u g ' u s t 1975. 57. Foreman, J . M., 58. Haig, L., "Monerai," Soaring, June 1978, pp. 14-15. Also, Soaring, June "American Eaglet," Soaring, March 1976, pp. 16-19. 59. Worthington, G., "Getting Acquainted With the American Eaglet," Soaring, March 1978, pp. 10-14. 60. M i t c h e l l , D., 61. Allen, W. 56-57. A., "The Mitchell Wing," Soaring, June 1977, pp. 26, 40. "Hoist on Your Own Canard," Pop. Mech., Sept. 1978, pp. 62. P r i c e , C. B., ed., " USHGA Hang Glider Directory," Ground Skimmer (now Hang Gl i d e r ) , December 1975. 63. McMasters, J. H., "An MA P r e d i c t i o n , " Soaring, April 1977, p, 3. P 64. Johnson, D., et. a1 "The 1977 Superships - A Consumer Report," G l i d e r N Rider, May 1977, pp. 27-30, (P. 0. Box 6009, Chattanooga, T 37401). ., Table 1. Wing Span m (ft) Schweizer 1-26 Darmstadt D28b " W i ndspiel" Maupin "Woodstock" Hal 1 "Vector I" Monett "Monerai " Haig/American IiEag1 e t " Mars ke "Monarch Hi1 1 "Superf 1o a t e r " M i t c h e l l Wing Avi a f ib r e Canard 2FL Bennett "Phoenix 8" Nihon U. "Stork Bti Wing Area mZ(ft2) Aircraft Characteristics Weights Empty Loaded* N (lb) N (Ib) Wing* Estimate Loading L/D ~ / m z ( p s f ) max. Ref. 51 Aspect Ratio 1 53, 54 , 52, 55 52, 56 52, 57 52, 58, 59 52 52, 62 52, 60 61 62, 64 12, 63 "Weight assumes p i l o t p l u s equipment weight o f 800 N - 180 I b . Partial I ~r > :: . :. . : , ; .:.-.!'. . ....... i g ht : ; : ;; ;: ....... , , .:.....-. ,. : , . :..., :.. ..,. ..... .q$j Zeppel ins . . Motor Gl i ders 0 Speed For Maximum Aerodynamic Effi ci ency FIGURE 1. - ~'(~1s) APPROXIMATE BOUNDARIES OF THE FEASIBLE/ECONOMICAL LOW-SPEED FLIGHT SPECTRUM L / ~ 5 -. \ FIGURE 2. S I N K RATE PERFORMANCE COMPARISON KLAUS HILL I' SUPERFLOATER" High L i f t A i r f o i y s Optimum Wing for ~il i r c ng Drag Reduction Through S treaml in i ng Optional H ~ e Launching f l Option FIGURE 3. Low Weight Simp1 e S t r u c t u r e Q LOW cost TYPICAL ULTRALIGHT SAILPLANES H o r i z o n t a l Speed - V (m/s) S o a r i n g B i r d Vul t u r e 4 - Bank Angle i m io3 / cos v2 9 t a n cp ' 9 @ V$2 v0/cos 1/2# R = * @ Z m i n m 2.5 m/s T u r n Radius FIGURE 4. - 0 R (m) TYPICAL GLIDER TURN PERFORMANCE ANALYTICAL AND SCALE M D L RESEARCH AIMED A IMPROVED HANG GLIDER DESIGN OE T I l a n Kroo and Li-Shing Chang Stanford University SM AY U MR A program of r e s e a r c h on t h e aerodynamics, a e r o e l a s t i c i t y , and s t a b i l i t y of hang g l i d e r s h a s r e c e n t l y begun a t S t a n f o r d U n i v e r s i t y w i t h s u p p o r t from NASA. The r e s e a r c h c o n s i s t s of a t h e o r e t i c a l a n a l y s i s which a t t e m p t s t o p r e d i c t aerodynamic c h a r a c t e r i s t i c s u s i n g l i f t i n g s u r f a c e t h e o r y and f i n i t e - e l e m e n t s t r u c t u r a l a n a l y s i s a s w e l l a s an e x p e r i m e n t a l i n v e s t i g a t i o n u s i n g 1 / 5 - s c a l e e l a s t i c a l l y s i m i l a r models i n t h e NASA Ames 2m x 3m ( 7 ' x 1 0 7 ) wind t u n n e l . Experimental d a t a w i l l b e compared w i t h t h e o r e t i c a l r e s u l t s i n t h e development of a computer program which may b e used i n t h e d e s i g n and e v a l u a t i o n of u l t r a light gliders. T h i s p a p e r d e s c r i b e s t h e g o a l s and g e n e r a l procedures of t h e i n v e s t i g a t i o n begun i n J a n u a r y 1979. INTRODUCTION I n r e c e n t y e a r s t h e performance and v a r i e t y of hang g l i d e r d e s i g n s have i n c r e a s e d d r a m a t i c a l l y . F l i g h t c o n d i t i o n s and demands t h a t a r e p l a c e d on hang g l i d e r s a r e v e r y d i f f e r e n t from t h o s e encountered by o l d e r d e s i g n s . Whereas l i f t - t o - d r a g r a t i o s of 3 were common n o t long ago, some p r e s e n t d e s i g n s a c h i e v e g l i d e r a t i o s of c l o s e t o 10 and have been flown c r o s s c o u n t r y f o r 160km (100mi) a t a l t i t u d e s a s h i g h a s 6000 m (19,000 f t . ) (Ref. 1 ) . I n a d d i t i o n t o ( o f t e n t u r b u l e n t ) thermal f l y i n g , i n c r e a s e d c o n t r o l l a b i l i t y h a s made l i m i t e d a e r o b a t i c maneuvers p o s s i b l e . S e v e r a l y e a r s ago t h e r e s u l t s of N S wind t u n n e l s t u d i e s AA of t h e Rogallo wing (Ref. 2-7) i n t h e 1960's could b e used t o o b t a i n some i d e a of t h e c h a r a c t e r i s t i c s of new d e s i g n s . Although n o t a l l f l i g h t regimes and r e l e v a n t p a r a m e t e r s were thoroughly i n v e s t i g a t e d , t h e d a t a t h a t d i d e x i s t proved u s e f u l . The hang g l i d e r h a s evolved, however, t o t h e p o i n t t h a t t h e s e o r i g i n a l i n v e s t i g a t i o n s can no l o n g e r b e a p p l i e d . The f l i g h t c h a r a c t e r i s t i c s of modem hang g l i d e r s (Ref. 8) w i t h spans extending t o 3 1 m ( 3 6 f t . ) , a s p e c t r a t i o s from 5 t o 7.6 and s a i l s y i t h low b i l l o w and sweep, cannot b e e s t i m a t e d from t h e s e d a t a f o r t h e h i g h b i l l o w ( 4 - 5 d e g r e e s ) , low a s p e c t r a t i o (2.5) "standards". I n f o r m a t i o n on t h e aerodynamic c h a r a c t e r i s t i c s of p r e s e n t d e s i g n s i s almost e n t i r e l y q u a l i t a t i v e , deduced from l i m i t e d f l i g h t t e s t s of new d e s i g n s . Many problems t h a t have been encountered might have been prevented had such d a t a been a v a i l a b l e . Pitch-down d i v e r g e n c e a t low a n g l e s of a t t a c k c o n t i n u e s t o b e an i m p o r t a n t problem. T h i r t y p e r c e n t of f a t a l i t i e s i n 1976 involved f u l l - l u f f i n g d i v e s from a l t i t u d e s i n e x c e s s of 60 m (200 f t .) ( ~ e f 9) a l t h o u g h r e c o v e r y i s t h e o r e t i c a l l y p o s s i b l e i n l e s s t h a n 15m (50 f t . ) (Ref. 1 0 ) . S t a t i s t i c s from hang g l i d i n g a c c i d e n t s i n 1977 and 1978 show t h a t , d e s p i t e a more . thorough t e s t i n g program pursued by t h e i n d u s t r y i n t h e l a s t few y e a r s , such i n s t a b i l i t i e s are a l l t o o common even up t o t h e p r e s e n t t i m e . Work was begun i n January 1979 on a program of r e s e a r c h aimed a t p r o v i d i n g q u a n t i t a t i v e t o o l s f o r u s e i n t h e d e s i g n and e v a l u a t i o n of modern hang g l i d e r s . The i n v e s t i g a t i o n c o n s i s t s of two c o n c u r r e n t and c l o s e l y i n t e g r a t e d phases: 1 ) B a s i c f o r c e and moment measurements w i l l b e made on s c a l e models i n AA one of t h e 2 m x 3 m wind t u n n e l s a t N S Ames Research Center. Models a r e b e i n g c o n s t r u c t e d t h a t w i l l reproduce t h e g e o m e t r i c , e l a s t i c , and aerodynamic p r o p e r t i e s of a r e p r e s e n t a t i v e c l a s s of modern g l i d e r . 2) A computer program, based on t h e b e s t a v a i l a b l e a n a l y t i c t o o l s from p o t e n t i a l aerodynamics and f i n i t e - e l e m e n t s t r u c t u r a l methods, f o r p r e d i c t i n g t h e measured a i r l o a d s w i t h s t a t i c a e r o e l a s t i c c o r r e c t i o n s i s b e i n g developed. A f t e r r e f i n e m e n t by comparison w i t h t h e t e s t s , t h i s program w i l l b e promulgated f o r t h e a n a l y s i s of f u t u r e g l i d e r designs. A s t h i s r e s e a r c h i s t o b e conducted o v e r t h e n e x t two y e a r s , t h i s paper d e s c r i b e s t h e g o a l s and g e n e r a l approach of t h e p r o j e c t w i t h r e s u l t s t o b e published a t a l a t e r date. WIND TUNNEL TESTS 'Models Planned wind t u n n e l t e s t s c o n s i s t of measurements of t h e b a s i c f o r c e s and moments on a group of 115-scale models a t Reynold's numbers v e r y c l o s e t o t h e f u l l s c a l e value. Although t h e r e e x i s t today a wide v a r i e t y of hang g l i d e r d e s i g n s and i t i s no l o n g e r p o s s i b l e t o t e s t a "standard" c o n f i g u r a t i o n and u s e t h e r e s u l t s t o p r e d i c t u n i v e r s a l c h a r a c t e r i s t i c s f o r t h e s e a i r c r a f t , s u f f i c i e n t s i m i l a r i t y does e x i s t s o t h a t c e r t a i n c h a r a c t e r i s t i c s may b e determined from t e s t s on a l i m i t e d number of models and a p p l i e d t o many o t h e r d e s i g n s w i t h s i m i l a r f e a t u r e s . I n t h i s way, good approximations t o t h e p r o p e r t i e s of such g l i d e r s may b e o b t a i n e d from t e s t s on a s m a l l group w i t h d i f f e r e n t , b u t c a r e f u l l y s e l e c t e d , g e o m e t r i e s . The models s e l e c t e d span a wide r a n g e of g l i d e r t y p e s , from t h e o l d e r Rogallot y p e "standards" t o more r e c e n t " i n t e r m e d i a t e " and h i g h performance d e s i g n s . (See T a b l e 1 ) . The e f f e c t on o v e r a l l aerodynamic c h a r a c t e r i s t i c s of v a r i o u s wing t i p g e o m e t r i e s , s a i l planforms, and camber and t w i s t d i s t r i b u t i o n s common t o many g l i d e r s w i l l b e determined from t e s t s on t h i s group of models. The importance of e l a s t i c s c a l i n g h a s been demonstrated r e c e n t l y (Ref. 1 1 ) . The f l i g h t c h a r a c t e r i s t i c s of g l i d e r s a r e s e e n t o v a r y c o n s i d e r a b l y w i t h changes i n l o a d i n g . T h i s i s caused by t h e f l e x i b i l i t y of t h e frame and d e f o r mation of t h e s a i l of t h e s e u l t r a l i g h t g l i d e r s . For t h i s r e a s o n , i t i s import a n t t h a t s c a l e models b e c o n s t r u c t e d i n such a way as t o remain g e o m e t r i c a l l y s i m i l a r t o f u l l s i z e g l i d e r s under corresponding l o a d s . Another key assumption u n d e r l y i n g t h e d e s i g n of f l e x i b l e models i s t h e a t t a i n m e n t of f u l l - s c a l e Reynolds' number, Re. T h i s i s because r a t h e r complex separated-flow e f f e c t s a r e a n t i c i p a t e d a t t h e l a r g e r v a l u e s of a and B S i n c e a v a i l a b l e wind t u n n e l s o p e r a t e a t e s s e n t i a l l y s e a - l e v e l c o n d i t i o n s , i t f o l l o w s t h a t any r e s u l t a n t f o r c e , e x p e r i e n c e d by t h e model must e q u a l Fm' t h e corresponding Ff a t f u l l s c a l e . (Mach number e f f e c t s a r e n e g l i g i b l e a t t h e s e "microsonic speeds. ") Force e q u a l i t y can b e reasoned from t h e f a c t t h a t : . where t h e product of speed and t y p i c a l l e n g t h , s c a l e s . With a i r d e n s i t i e s : VR, must b e t h e same a t b o t h and f o r c e s p r o p o r t i o n a l t o 'v2R2 , t h e n : The combination of e q u a l f o r c e and e q u a l s t r a i n requirements l e a d t o d i f f i c u l t i e s i n t h e c o n s t r u c t i o n of e l a s t i c a l l y s c a l e d models. Consider t h a t b o t h t h e model and f u l l s c a l e g l i d e r s a r e c o n s t r u c t e d of t u b e s and c a b l e s of approxim a t e l y c i r c u l a r c r o s s s e c t i o n s of r a d i u s r , s u p p o r t i n g t h e f a b r i c s a i l s . S i n c e r s h o u l d b e p r o p o r t i o n a l t o R f o r aerodynamic s i m i l a r i t y , t h e s t r a i n s i n FrR/EI o r t o F r 2 / E I , w i t h EI t h e f a m i l i a r these tubes a r e proportional t o bending r i g i d i t y . The s e v e r e requirement on model c o n s t r u c t i o n i s t o e n s u r e For a t y p i c a l g l i d e r , assembled of thin-walled aluminum t u b e s , t h i s q u a n t i t y i s of t h e o r d e r N (4 x 1b.-l). I f t h e same c o n s t r u c t i o n and m a t e r i a l were employed on t h e model, one would g e t (6) ill = ( m ($)£ T h i s f a c t o r of 25 a t o n e - f i f t h s c a l e i s q u i t e u n a c c e p t a b l e . S i n c e weight i s n o t b e l i e v e d t o b e a v e r y s i g n i f i c a n t f a c t o r , t h e s i t u a t i o n can b e a l l e v i a t e d by going t o s o l i d - s e c t i o n c y l i n d e r s of s t i f f e r m a t e r i a l on t h e model. Models c o n s t r u c t e d of s t e e l i n t h i s manner approach t h e d e s i r e d s t i f f n e s s : It a p p e a r s t h a t t h e requirement of e q u a l s t r a i n s , t h e r e f o r e , can o n l y b e m e t by some r e l a x a t i o n of t h e Reynolds number requirement. The f o l l o w i n g v a l u e s correspond t o t h e model c o n s t r u c t i o n above: 507 T h i s d i f f e r e n c e i s n o t l a r g e and can b e reduced f u r t h e r w i t h t h e u s e of t u b e s and c a b l e s of s l i g h t l y l a r g e r t h a n s c a l e d r a d i i . E s p e c i a l l y f o r newer hang g l i d e r d e s i g n s w i t h low b i l l o w , i t i s i m p o r t a n t t o d u p l i c a t e t h e s t r e t c h i n g of t h e f a b r i c s a i l a s w e l l a s t h e bending of frame elements. T h i s requirement may b e s e e n approximately a s f o l l o w s . Requiring e q u a l s t r a i n s i n t h e model and f u l l - s c a l e g l i d e r s a i l s , f o r geometric s i m i l a r i t y i m p l i e s t h a t E = E m = E ~ , - 0 E - p(x)dx E dA - p(x)dx E t dx = F - EtR and from above we have l e t i s t h e same a t b o t h s c a l e s . Now s i n c e 1 Em = - R 5 f T h i s can b e achieved w i t h F = 113.4 F we r e q u i r e t h a t (Et) = 1 . 8 ( E t ) m f m f' t h e a p p r o p r i a t e c h o i c e of Dacron f a b r i c . Values of E t f o r Dacron s a i l s a r e may b e g i v e n i n r e f e r e n c e 24, from which i t can b e seen t h a t t h e p r o p e r (Et), achieved w i t h two l a y e r s of m a t e r k a l s l i g h t l y l i g h t e r t h a n t h a t used on full-scale gliders. Data Reduction From measurements of t h e b a s i c f o r c e s and moments on t h e s e models, t h e f o l l o w i n g performance c o e f f i c i e n t s and s t a t i c s t a b i l i t y d e r i v a t i v e s can b e calculated. -45' through Data w i l l b e o b t a i n e d g e n e r a l l y a t a n g l e s of a t t a c k , a from 0 s t a l l and a t s i d e s l i p a n g l e s B to 20 T e s t s w i l l b e conducted a t v a r i o u s p r e s s u r e s t o o b t a i n d a t a on t h e presumably s i g n i f i c a n t v a r i a t i o n of t h e s e q u a n t i t i e s w i t h t h e dynamic p r e s s u r e , q ( e l a s t i c e f f e c t s ) . T e s t r e s u l t s w i l l b e c o r r e c t e d f o r j e t blockage and w a l l e f f e c t s . + . Much of t h i s d a t a could b e used immediately f o r d e s i g n purposes w i t h l i t t l i n t e r m e d i a t e m a n i p u l a t i o n . With t h e u s e of d a t a on p i t c h i n g moment c o e f f i c i e n t t h e l o n g i t u d i n a l e q u a t i o n s of motion may b e n u m e r i c a l l y i n t e g r a t e d t o show t h e e f f e c t i v e n e s s of "weight-shift" c o n t r o l , i n c l u d i n g r e q u i r e d b a r p r e s s u r e s ( s t i c f o r c e s ) , under v a r i o u s f l i g h t c o n d i t i o n s . S t a l l , d i v e r e c o v e r y , and o t h e r a s p e c t s of l o n g i t u d i n a l motion w i l l b e analyzed. A s i m i l a r a n a l y s i s f o r l a t e r a motion, t a k i n g account of t h e u n u s u a l l y l a r g e coupling between l o n g i t u d i n a l and l a t e r a l modes a s s o c i a t e d w i t h hang g l i d e r s , w i l l a l s o b e c a r r i e d o u t . A t t h e p r e s e n t t i m e , t h e f i r s t wind t u n n e l model i s being c o n s t r u c t e d a t t h e machine shop f a c i l i t i e s a t S t a n f o r d . The frame w i l l have two p o s s i b l e n o s e a n g l e s and by a t t a c h i n g d i f f e r e n t s a i l s many c o n f i g u r a t i o n s may b e t e s t e d . Although a f i n a l l i s t of c o n f i g u r a t i o n s t o b e t e s t e d h a s n o t y e t been d e t e r mined, a t e n t a t i v e group of t e s t models i s d e s c r i b e d i n t a b l e 1. THEORETICAL ANALYSIS I n c o n j u n c t i o n w i t h t h e t e s t i n g program, t h e o r e t i c a l aerodynamic and a e r o e l a s t i c methodology i s b e i n g a p p l i e d toward t h e development of a computer program which w i l l u n d e r t a k e t o p r e d i c t some o r a l l of t h e q u a n t i t i e s measured i n t h e e x p e r i m e n t a l p o r t i o n of t h e p r o j e c t . S e v e r a l t h e o r e t i c a l t r e a t m e n t s of "parawing" aerodynamics were p u b l i s h e d L i f t i n g s u r f a c e t h e o r y was used t o p r e d i c t i n t h e 1960's (e.g. Ref. 12-14). l i f t and moment of v a r i o u s parawing c o n f i g u r a t i o n s w i t h t h e assumption of a p a r t i c u l a r mode shape ( g e n e r a l l y t a k e n t o b e a p o r t i o n of a r i g h t c i r c u l a r cone). Induced and p r o f i l e d r a g s and t h e e f f e c t s of r i g i d l e a d i n g edges were t r e a t e d . Recent e x p e r i m e n t a l work (Ref.11) h a s shown, however, t h a t changes i n s a i l shape w i t h a n g l e of a t t a c k and dynamic p r e s s u r e a r e extremely i m p o r t a n t , e s p e c i a l l y f o r c u r r e n t hang g l i d e r d e s i g n s . Thus, n o t o n l y i s t h e assumption of c o n i c a l canopy shape no l o n g e r v a l i d , b u t no r i g i d a n a l y t i c assumption of mode shape c a n b e used. The approach t a k e n i n t h e p r e s e n t a n a l y s i s c o n s i s t s of two major p a r t s : 1 ) The d e t e r m i n a t i o n of a i r l o a d s f o r a ' p r e s c r i b e d mode shape, and 2) The f l e x i b l e s t r u c t u r a l r e s p o n s e t o t h i s c a l c u l a t e d l o a d i n g , r e s u l t i n g i n a new approximation f o r canopy shape. The i t e r a t e d procedure, shown s c h e m a t i c a l l y i n F i g . 1 , i s used t o o b t a i n a s o l u t i o n f o r p r e s s u r e d i s t r i b u t i o n w i t h o u t t h e need f o r s p e c i f y i n g t h e e x a c t s a i l shape i n i t i a l l y . From t h e s e p r e d i c t e d a i r l o a d s , f o r c e and moment c o e f f i c i e n t s may b e c a l c u l a t e d and compared w i t h e x p e r i m e n t a l r e s u l t s . Aerodynamics L i n e a r i z e d , s t e a d y , l i f t i n g - s u r f a c e t h e o r y f o r i n c o m p r e s s i b l e flow i s used i n t h e p r e d i c t i o n of aerodynamic l o a d s on t h e g l i d e r . Under such c o n d i t i o n s v2$ = 0 which t h e flow over t h e g l i d e r s a t i s f i e s L a p l a c e ' s e q u a t i o n : may b e s o l v e d w i t h t h e u s e of v o r t e x - l a t t i c e o r k e r n e l - f u n c t i o n methods. The approach t a k e n h e r e u t i l i z e s t h e former method d e s c r i b e d by Woodward and Rubbert (Refs. 15,16) w i t h a code by N a t h a n (Ref. 17) used a t S t a n f o r d ' s computing f a c i l i t i e s . The s a i l i s d i v i d e d i n t o f i n i t e elements a s shown i n F i g . 2. Each element i s i d e a l i z e d a s a f l a t p a n e l of c o n s t a n t d o u b l e t s t r e n g t h , K , d e f i n e d a s t h e d i s c o n t i n u i t y i n p o t e n t i a l between t h e upper and lower s u r f a c e s , A s shown i n t h e appendix, t h i s l e a d s t o t h e f o l l o w i n g e x p r e s s i o n f o r t h e v e l o c i t y induced a t p o i n t s o u t s i d e t h i s s u r f a c e where [C] i s t h e aerodynamic i n f l u e n c e m a t r i x d e s c r i b e d i n t h e appendix. The d o u b l e t s t r e n g t h f o r each p a n e l i s chosen s o t h a t t h e flow a t t h e s u r f a c e of t h e g l i d e r i s tangent (zero s a i l p o r o s i t y ) . This condition is s a t i s f i e d i f t h e normal v e l o c i t y induced by t h e system of d o u b l e t s j u s t c a n c e l s t h e f r e e - s t r e a m normal v e l o c i t y : S i n c e t h e s u r f a c e normals and i n f l u e n c e n a t r i x may b e computed from t h e assumed s a i l geometry and s i n c e t h e f r e e - s t r e a m v e l o c i t y i s g i v e n , t h e v a l u e of K can b e c a l c u l a t e d over t h e s u r f a c e . Once K i s known, t h e v o r t i c i t y on t h e s u r f a c e i s g i v e n by: and t h e l o a d i n g : These p r e s s u r e s a r e t h e n used t o c a l c u l a t e t h e desir.ed f o r c e and moment c o e f f i c i e n t s a c c o r d i n g t o s t a n d a r d d e f i n i t i o n s ( c f . Ref. 18). The procedure i s summarized i n F i g . 3 . F i g s . 4-6 show t h e p r e l i m i n a r y r e s u l t s of t h i s t h e o r y a p p l i e d t o some s i m p l e planforms f o r which e x p e r i m e n t a l d a t a i s a v a i l a b l e . (Ref. 1 9 , 2 0 ) . Agreement i s c l o s e a l t h o u g h e f f e c t s of l e a d i n g edges and d e v i a t i o n from c o n i c a l geometry a r e n o t considered. It s h o u l d b e noted t h a t t h e s e r e s u l t s a r e t h e p r e d i c t i o n s of t h e aerodynamic p o r t i o n of t h e program o n l y . A r i g i d mode shape i s assumed and s o agreement w i t h experiment can o n l y b e expected a t i n t e r m e d i a t e a The combination of t h i s p o r t i o n of t h e program and t h e s t r u c t u r a l a n a l y s i s d e s c r i b e d below i s p r e s e n t l y underway and r e s u l t s a r e n o t y e t a v a i l a b l e . . T h i s a n a l y s i s does n o t i n c l u d e t h e e f f e c t of p i l o t , c a b l e o r frame i n t e r f e r e n c e . It a p p l i e s o n l y t o u n s e p a r a t e d flow and does n o t i n c l u d e v i s c o u s e f f e c t s . C o r r e c t i o n s t o t h e f i r s t s t a g e of t h e a n a l y s i s , t a k i n g t h e s e e f f e c t s i n t o a c c o u n t , a r e b e i n g s t u d i e d and can, h o p e f u l l y , b e implemented i n l a t e r work. S t r u c t u r a l Analysis The s a i l and frame of a hang g l i d e r c o n s t i t u t e a r a t h e r f l e x i b l e s t r u c t u r e assumed t o b e i n a s t a t e of q u a s i - s t a t i c e q u i l i b r i u m . Tension members, a x i a l l y loaded beams, bending members, and membrane s u r f a c e s a r e a l l i n v o l v e d , w i t h c l e a r l y d e f i n e d modal connections. It i s e v i d e n t t h a t t h e f i n i t e - e l e m e n t method of s t a t i c , s t r u c t u r a l a n a l y s i s i s t h e o n l y f e a s i b l e way of r e p r e s e n t i n g and b a l a n c i n g t h e complete system of i n t e r n a l and e x t e r n a l l o a d s . The approach t a k e n h e r e i n v o l v e s a n a n a l y s i s of t h e g l i d e r frame by c l a s s i c a l methods and modelling of t h e s a i l a s a membrane w i t h v e r y s m a l l f l e x u r a l r i g i d i t y . The procedure i s diagrammed i n F i g . 7 . An i n c r e m e n t a l l o a d i n g t e c h n i q u e a s d e s c r i b e d by Turner e t a l . (Ref. 21) i s used t o p r e d i c t t h e r e s p o n s e of t h e e n t i r e s t r u c t u r e t o t h e g i v e n a p p l i e d l o a d . The p r e s s u r e d i s t r i b u t i o n g i v e n by t h e aerodynamic p o r t i o n of t h e a n a l y s i s i s broken down i n t o s m a l l increments and t h e change i n shape due t o t h i s i n c r e m e n t a l l o a d i s c a l c u l a t e d . T h i s i s done by e x p r e s s i n g t h e p r e s s u r e , Api, o v e r each p a n e l i n terms of e q u i v a l e n t n o d a l l o a d s , Fi, and c a l c u l a t i n g t h e d i s p l a c e m e n t , D i , of t h e nodes by t h e r e l a t i o n : IF} = [S] (D} Here, [S] i s a s t i f f n e s s m a t r i x , made up of a l i n e a r , e l a s t i c p a r t [Se] which a c c o u n t s f o r s a i l s t r e t c h i n g ( d e s p i t e t h e a n i s o t r o p i c s t r e t c h i n g b e h a v i o r of t e x t i l e m a t e r i a l s , t h e g l i d e r s a i l i s assumed i s o t r o p i c f o r t h e e a r l y s t a g e ISg] which depends of t h e i n v e s t i g a t i o n ) and a non-linear geometric p a r t on t h e geometry and i n i t i a l t e n s i o n . The a d d i t i o n of t h i s geometric s t i f f n e s s t o t h e conventional s t i f f n e s s matrix allows t h e non-linear strain-displacement r e l a t i o n s a s s o c i a t e d w i t h t h i s l a r g e displacement problem t o b e i n c o r p o r a t e d i n an approximate manner. A method d e s c r i b e d by A r g y r i s (Ref. 22) i s a d a p t e d h e r e t o g e n e r a t e t h e geometric s t i f f n e s s m a t r i x . T h i s method assumes a l i n e a r s t r a i n - d i s p l a c e m e n t r e l a t i o n s h i p w i t h i n t h e elements and i s c o n s i d e r a b l y s i m p l e r t h a n c o n v e n t i o n a l t e c h n i q u e s which r e q u i r e c a l c u l a t i o n of t h e s t r a i n energy ( c f . Ref. 23). A t each s t e p t h e geometric s t i f f n e s s m a t r i x i s updated and n o d a l f o r c e s and i n c r e m e n t a l d i s p l a c e m e n t s c a l c u l a t e d . A f t e r t h e step-by-step p r o c e s s i s completed, t h e i n c r e m e n t a l d i s p l a c e m e n t s a r e summed t o o b t a i n a new mode shape which i s t h e n used a s i n p u t t o t h e aerodynamic program f o r a n o t h e r i t e r a t i o n . A code based on t h i s approach h a s been developed and i s p r e s e n t l y b e i n g checked by comparison w i t h test c a s e s f o r which a n a l y t i c s o l u t i o n s a r e possible. P r e l i m i n a r y work i n d i c a t e s agreement t o w i t h i n a few p e r c e n t i n displacement a l t h o u g h f u r t h e r work i s needed t o a s s u r e convergence i n some cases. R e s u l t s from t h e e x p e r i m e n t a l p o r t i o n of t h e i n v e s t i g a t i o n w i l l be used t o e s t a b l i s h t h e t h e o r e t i c a l r e s u l t s v range of v a l i d i t y and w i l l g u i d e e f f o r t s t o i n c o r p o r a t e t h e e f f e c t s of v i s c o s i t y , i n t e r f e r e n c e , leading-edge s u c t i o n , and o t h e r phenomena i n t h e a n a l y t i c a l p o r t i o n of t h e r e s e a r c h . CONCLUDING REMARKS The t h e o r y p r e s e n t e d h e r e i s i n t e n d e d t o p r o v i d e a g e n e r a l i d e a of some of t h e methods t o b e used i n t h i s i n v e s t i g a t i o n . Much work i s r e q u i r e d b e f o r e t h e a n a l y s i s can p r o p e r l y t a k e account of t h e complex aerodynamic and aeroe l a s t i c e f f e c t s a s s o c i a t e d w i t h modern hang g l i d e r s . A t t h e time of t h i s w r i t i n g , t h e aerodynamic and s t r u c t u r a l r o u t i n e s have n o t been combined a l t h o u g h i t i s expected t h a t t h i s w i l l accomplished s h o r t l y . Wind-tunnel models a r e p r e s e n t l y being f a b r i c a t e d f o r t e s t s t o b e conducted l a t e r t h i s y e a r . R e s u l t s from b o t h t h e t h e o r e t i c a l and e x p e r i m e n t a l p a r t s of t h i s r e s e a r c h w i l l b e publ i s h e d a s t h e y become a v a i l a b l e . APPENDIX Aerodynamic I n f l u e n c e M a t r i x C a l c u l a t i o n Expressing t h e v e l o c i t y p e r t u r b a t i o n p o t e n t i a l , $ ( P ) , a t a p o i n t P, i n terms of t h e v a l u e of $ and i t s normal d e r i v a t i v e - , on t h e f l u i d boundary a$ an by Green's theorem: where r i s t h e d i s t a n c e between P and P ' , a p o i n t on t h e boundary, S. I f t h e s a i l i s t a k e n t o b e a 2-dimensional s u r f a c e , t h e n o r d e r t h a t t h e flow b e t a n g e n t t o t h e s u r f a c e , n = U - nR and, i n so; and If K i s assumed c o n s t a n t over each of t h e p a n e l s S i , then: The aerodynamic i n f l u e n c e c o e f f i c i e n t of t h e r e g i o n thus defined as: Ci = j S j on t h e p o i n t ' i is - /v sj & (+) d s ; Expressing t h i s v e l o c i t y a t s e v e r a l p o i n t s i n m a t r i x n o t a t i o n : REFERENCES 1. 2. "Hang G l i d i n g Magazine" (Formerly "Ground Skimmer"), Dec. 1973-Feb. 1979, a Monthly p u b l i c a t i o n of t h e U n i t e s S t a t e s Hang G l i d e r A s s o c i a t i o n . Johnson, Joseph L., Jr., "Low-Speed Wind-Tunnel I n v e s t i g a t i o n t o Determine AA t h e F l i g h t C h a r a c t e r i s t i c s of a Model of a Parawing l J t i l i t y v e h i c l e , " N S T D-1255, 1962. N Johnson, Joseph L., J r . , and H a s s e l l , James L., J r . , " F u l l - S c a l e Wind-Tunnel I n v e s t i g a t i o n of a Flexible-Wing Manned T e s t Vehicle," N S T D-1946, 1963. AA N Naeseth, Roger L . , and G a i n e r , Thomas G . , "Low-Speed I n v e s t i g a t i o n of t h e E f f e c t s of Wing Sweep on t h e Aerodynamic C h a r a c t e r i s t i c s of Parawings Having Equal-Length Leading Edges and Keel," N S T N D-1957, 1963. AA Bugg, Frank M., " E f f e c t s of Aspect R a t i o and Canopy Shape on Low-Speed AA Aerodynamic C h a r a c t e r i s t i c s of 50' Swept Parawings," N S T N D-2922, J u l y 1965. Croom, Delwin, R., Naeseth, Roger L., and Sleeman, William C . , J r . , " E f f e c t s of Canopy Shape on Low-Speed Aerodynamic C h a r a c t e r i s t i c s of a 550 Swept Parawing w i t h Large-Diameter Leading Edges," TN D-2551, Dec. 1964. Johnson, Joseph L., J r . , "Low-Speed F o r c e and F l i g h t I n v e s t i g a t i o n s of a AA N Model of a Modified Parawing U t i l i t y V e h i c l e , " N S T D-2492, March 1965. 3. 4. 5. 6. 7. 8. Hang G l i d e r Design C a t a l o g , Ground Skimmer, December 1975. "Accident Summaries", Hang Gliding, 1974-1978. 9. W i l l s , R.V., 10. 11. J o n e s , R.T., "Dynamics of U l t r a l i g h t A i r c r a f t : Dive Recovery of Hang G l i d e r s , " May 1977, NASA T X-73229, M D i s c u s s i o n w i t h Robert Ormiston, NASA Ames Research C e n t e r , r e g a r d i n g r e s u l t s of f u l l - s c a l e tests i n s e t t l i n g chamber of #2 7 ' x 10' wind t u n n e l a t Ames, J u n e 1977. " I n v e s t i g a t i o n of Mendenhall, M.R., Spangler, S.B., and N i e l s e n , J . N . , Methods f o r P r e d i c t i n g t h e Aerodynamic C h a r a c t e r i s t i c s of Two-Lobed Parawings," NASA CR-1166, S e p t . 1968. Polhamus, Edward C . , and Naeseth, Roger L., "Experimental and T h e o r e t i c a l S t u d i e s of t h e E f f e c t s of Camber and Twist on t h e Aerodynamic C h a r a c t e r i s t i c s of Parawings Having Nominal Aspect R a t i o s of 3 and 6," NASA T D-972, N 1963. N i e l s e n , J a c k N . , and B u r n e l l , J a c k A , , " T h e o r e t i c a l Aerodynamics of F l e x i b l e Wings a t Low Speeds: Engineering Method f o r E s t i m a t i n g Parawing Performance, F i n a l R e p o r t , Feb. - Nov. 1965," VIDYA-209, Dec. 1965. Woodward, F.A., "Analysis and Deqign of Wing-Body Combinations a t Subsonic and S u p e r s o n i c Speeds," J. A i r c r a f t , 5:528-34. Rubbert, P.E., " T h e o r e t i c a l C h a r a c t e r i s t i c s of A r b i t r a r y Wings by a NonP l a n a r Vortex L a t t i c e Method," Boeing Co., Rep. D-6-9244, 1964. Nathman, J . K . , "Delta Wings i n I n c o m p r e s s i b l e Flow," Meeting, 1977. AIAA 1 3 t h Annual 12. 13. 14. 15. 16. 17. 18. 19. Ashley, H., Engineering A n a l y s i s of F l i g h t V e h i c l e s , Addison-Wesley P u b l i s h i n g Co., Reading, Mass,, 1974, ( S e c t s . 3.1, 3.2, 8.2 e t c ) . T u r n e l l , J . A . , and N i e l s e n , J . N . , "Aerodynamics of F l e x i b l e Wings a t Low Speeds, P a r t I V -- Experimental Program and Comparison w i t h Theory," VIDYA Report 172, Feb. 1965. From unpublished s t u d y c i t e d i n Ref. 12. T u r n e r , M . J . , D i l l , E.H., M a r t i n , H.C., and Melosh, R . J . , " ~ a r g eD e f l e c t i o n s of S t r u c t u r e s S u b j e c t e d t o Heating and E x t e r n a l ~ o a d s , " J. Aerospace S c i . , 27:97-102, 127, 1960. A r g y r i s , J . H . , Kesley, S . , and Kamel, H . , M a t r i x Methods of S t r u c t u r a l A n a l y s i s , " AGARDograph 72, Pergamon P r e s s , Oxford, England, 1964. Prezeminiecki, J.S., New York, 1968. Theory of M a t r i x S t r u c t u r a l A n a l y s i s , McGraw-Hill, 20. 21. 22. 23. 24. Ormiston, Robert A . , " T h e o r e t i c a l and Experimental Aerodynamics of an E l a s t i c Sailwing," Ph.D. T h e s i s , Oct. 1969, Dept. of Aerospace and Mechanical S c i e n c e s , P r i n c e t o n U n i v e r s i t y , P r i n c e t o n , N . J . LIST OF SYMBOLS Rotation Aerodynamic i n f l u e n c e m a t r i x ( s e e appendix) Displacement of p a n e l nodes E l a s t i c i t y constant Bending r i g i d i t y Force Doublet s t r e n g t h Typical length U n i t v e c t o r normal t o s u r f a c e P o i n t on s u r f a c e of s a i l Pressure Loading on s a i l p e r u n i t l e n g t h Dynamic p r e s s u r e Reynolds number S a i l thickness Stiffness matrix Fluid velocity Free-stream v e l o c i t y Angle of a t t a c k Angle of s i d e s l i p Vorticity Strain Fluid density Stress Velocity perturbation p o t e n t i a l L i f t coefficient C~ Drag c o e f f i c i e n t P i t c h i n g moment c o e f f i c i e n t (based on k e e l l e n g t h and r e f e r r e d t o t h e c /2 point) m ' C r m a S l o p e of p i t c h i n g moment c u r v e w i t h r e s p e c t t o a E f f e c t i v e d i h e d r a l ( r o l l i n g moment c o e f f i c i e n t due t o yaw) Yawing moment c o e f f i c i e n t due t o s i d e s l i p C B f3 C Subscripts elastic f u l l scale geometric i n d i c i e s r e f e r t o individual panels lower s u r f a c e model normal component upper s u r f a c e s High sweep, low aspect ratio, "standard" High sweep, medium aspect ratio, Z0 billow "intermediate" High performance medium sweep (35O) zero tip chord I 5 ' Same as / 3 with 4 High performance low billow fixed minimum twist with "floating" High performance low billow high twist I Same as / 6 with decreased twist I Same as / 6 without geometric dihedral Same as 16 with "keel 1 at root chord Similar to /I6 with low taper planform, low twist, reflex For comparison with more recent designs and previous wind tunnel studies Comparison with standard and high performance designs; effects of "billow" Washout not fixed by tips Effect of sweep on Effect of this common tip geometry on C Features common to many contemporary hang gliders Effect of twist on performance and stability Dihedral effects on lateral stability and control Reflex effects on longilateral control Common to some of the highest performance gliders. * Some configurations can be changed with minor model modifications, which results in the need for only 6 airframes for the 10 configurations listed. MODE SHAPE I CALCULATE PRESSURES PROM AERODYNAMICS I YES a CALCULATE MODE SHAPE BY ITERATIVE STRUCTURES I CALCULATE C , CL, m AND OTHER PARAMETERS OF INTEREST 1 F i g u r e 1.- B a s i c s t r u c t u r e of l o a d - p r e d i c t i o n program. F i g u r e 2.- Finite-element r e p r e s e n t a t i o n of hang g l i d e r s a i l . P O I N GEOMETRY FROM STRUCTURES PROGRAM FORMULATE LC], AERODYNAMIC INFLUENCE MATRIX COMPUTE NORMAL COMPONENT OF FREE STREAM SOLVE FOR DOUBLET STRENGTH CALCUIATE TOTAL VELOCITY FIELD DISTRIBUTION F i g u r e 3. - Algorithm f o r aerodynamic a n a l y s i s . F i g u r e 4.- R e s u l t s of aerodynamic program. I- 0 - REFERENCE 19 Billov AR 2 A = 63' - - 2.z0 F i g u r e 5. - R e s u l t s of aerodynamic program. AR = A = Billov = l.oO 3.8 3s0 F i g u r e 6. - R e s u l t s of aerodynamic program. I INPUT ASSUMED MODE SHAPE AND PRESSURE DISTRIBZPPION 1 PRESSURE AND EOUIVALENT NODAL FORCES 1 COMPUTE LINEAR AND GEOMETRIC STIFFNESS I I SOLVE FOR INCREMEWAL DISPIACEMENT, TEST FOR COMPLETIO ROCEDURE ANALYSIS F i g u r e 7. - Computational procedure - deflection analysis. Page intentionally left blank IMPROVEMENT O HANG GLIDER PERFORMANCE F BY USE O ULTRALIGHT ELASTIC WING F J e r z y Wolf Aviation I n s t i t u t e Warsaw, Poland SM AY U MR The problem of t h e l a t e r a l c o n t r o l l a b i l i t y of t h e hang g l i d e r by t h e p i l o t ' s weight s h i f t i s c o n s i d e r e d . The i n f l u e n c e of t h e span and t h e t o r s i o n a l e l a s t i c i t y of t h e wing i s determined. It i s s t a t e d t h a t an u l t r a l i g h t e l a s t i c wing of a new k i n d developed by t h e a u t h o r i s most s u i t a b l e f o r good c o n t r o l . The wing a l s o h a s o t h e r advantageous p r o p e r t i e s . INTRODUCTION The main problem a f f e c t i n g t h e development of u l t r a l i g h t g l i d i n g i s t h e d e c r e a s e of t h e c o n t r o l e f f e c t i v e n e s s of t h e p i l o t ' s weight s h i f t when t h e wing span i n c r e a s e s . However, i n c r e a s i n g t h e span and consequently t h e a s p e c t r a t i o i s t h e o n l y way t o improve t h e l i f t - d r a g r a t i o (LID). The i m p o r t a n t e f f e c t o f t h e a s p e c t r a t i o on t h e LIDf o r a d e f i n i t e t y p e of e x t e r n a l s k e l e t o n of t h e u l t r a l i g h t wing c a n b e shown as i n d i c a t e d i n f i g u r e 1 r e f . 1 ) Areas A, B , and C i n d i c a t e t h e c a u s e s of t h e d i m i n i s h i n g of LID. F i g u r e 1 shows t h a t t h e induced d r a g A i s t h e main p r i c e of l i f t p r o d u c t i o n and c a n b e d i m h i s h e d mainly by i n c r e a s i n g span. Changing t h e unadvantageous tria n g u l a r wing planform of t h e e a r l y f l e x i b l e wings improves i t t o some d e g r e e and t h e a p p l i c a t i o n of f i n a l w i n g l e t s makes i t p o s s i b l e t o improve i t even more. Area B on f i g u r e 1 i l l u s t r a t e s t h e i n f l u e n c e of t h e wing p r o f i l e e f f e c t i v e n e s s on t h e hang g l i d e r LID, which i s n o t v e r y s e n s i t i v e t o p r o f i l e shape above a n a s p e c t r a t i o of 5. F i n a l l y a r e a C , t h e s k e l e t o n d r a g , c o n s t i t u t e s t h e main f i e l d of t h e d e s i g n e r ' s a c t i v i t y . It i s v e r y i n t e r e s t i n g t h a t f o r a l l wings w i t h e x t e r n a l s k e l e t o n ( w i t h e x t e r n a l s p a r s and s t r u t s ( a ) , w i t h e x t e r n a l s p a r s and c a b l e s ( b ) , and w i t h e x t e r n a l c a b l e s o n l y ( c ) ) , a n optimum a s p e c t r a t i o always e x i s t s . The maximum of LID c a n b e e x p l a i n e d by t h e c o n s i d e r a b l e d r a g i n c r e a s e , which f o r some a s p e c t r a t i o s exceeds t h e d e c r e a s e of induced d r a g . It h a s been shown i n f i g u r e 1 a l s o t h a t t h e optimum a s p e c t r a t i o can b e a c o n s i d e r a b l e one f o r u l t r a l i g h t wings. It enhances a p p l i c a t i o n of wings w i t h e n l a r g e d spans. A d i f f i c u l t y w i t h h i g h e r a s p e c t r a t i o s i s t h a t t h e l a t e r a l c o n t r o l of s i m p l e hang g l i d e r s by t h e p i l o t ' s body s h i f t o n l y i s worsened. ANALYSIS O LATERAL CONTROL F To a n a l y z e t h i s c h a l l e n g i n g problem, t h e time t o bank t h e wing 60' w a s c a l c u l a t e d (from +30° t o -30') as shown i n f i g u r e 2. F i r s t a completely s t i f f wing was c o n s i d e r e d , f o r which t h e i n e r t i a f o r c e s were n e g l e c t e d . Next a wing completely e l a s t i c i n t o r s i o n was c o n s i d e r e d , f o r which a l l t h e l a t e r a l aerodynamic moments were n e g l e c t e d . It was a s o f t wing, l o n g i t u d i n a l l y s t a b i l i z e d aerodynamically, w i t h t h e r o l l moment of i n e r t i a f o r c e s o n l y c o n s i d e r e d . I n t h e f i r s t c a s e t h e r e s p o n s e s on t h e c o n t r o l f o r c e moment were aerodynamic f o r c e s and i n t h e second c a s e s o l e l y t h e i n e r t i a f o r c e s . These two c a s e s c a n be regarded a s boundary l i m i t s on t h e r o l l r a t e s of a l l r e a l wings of hang gliders. For t h e f i r s t c a s e t h e f o l l o w i n g r e l a t i o n was found: where C~ l i f t coefficient a n g l e of i n c i d e n c e , d e g bank a n g l e i n f i g u r e 2, deg wing s p a n , m l i f t f o r c e (L = W1 p i l o t w e i g h t , daN g l i d e r w e i g h t , daN mean body s h i f t of t h e p i l o t , rn f l i g h t speed, m/sec t i m e t o bank from -30° a 'J l R L + W2), daN W1 W2 r v t t o +30°, sec and f o r t h e second c a s e : where m i s t h e g l i d e r mass assumed t o b e u n i f o r m l y d i s t r i b u t e d spanwise. Furthermore i t w a s assumed t h a t t h i s m a s s grows l i n e a r a s a f u n c t i o n of t h e span a c c o r d i n g t o t h e formula, where g E a r t h ' s a c c e l e r a t i o n , m/sec 2 W2, R* wing span, m y of hang g l i d e r weighing daN For t h e c a l c u l a t e d p r a c t i c a l examples t h e same v a l u e s were assumed: W1 = 75 daN, W2 = 25 daN, r = 0,75 m y = 60° and f u r t h e r m o r e dCL/da = 0,06, + v = 8 mlsec, C = 0,7, L R* = 12 m. The r e s u l t s of t h e c a l c u l a t i o n a r e shown i n f i g u r e 3. They concern two i d e a l boundary c a s e s 1 and 5 and t h r e e known t y p e s of hang g l i d e r s 2, 3 , and 4. P a r t i c u l a r c u r v e s concern t h e f o l l o w i n g t y p e s of u l t r a l i g h t wings: 1 - s t i f f wing 2 - Rogallo wing w i t h f l e x i b l e canopy c h a r a c t e r i z e d by l i m i t e d washout of t h e wing s a i l w i n g o r Rogallo h y b r i d wing of i n c r e a s e d washout s a i l w i n g o r h y b r i d Rogallo wing w i t h a u t o m a t i c a l l y changing s a i l b i l l o w and washout e l a s t i c wing of maximum a b r i t r a r y washout 3 4 5 I n f i g u r e 3 , t h r e e r a n g e s of bank t i m e f o r t h e mean body s h i f t r = 0,75 m of t h e p i l o t weighing 75 daN are shown. The f i r s t r a n g e of t from 0 t o 2 s e c i s t h e s a f e r a n g e of good m a n o e u v r a b i l i t y of t h e hang g l i d e r . It corresponds t o p r a c t i c a l o b s e r v a t i o n s of g l i d e r s and BCAR, s e c t i o n K, f o r t h e l i g h t a i r p l a n e s ( r e f . 2 ) . The second r a n g e of t = 2 t o 4 s e c i s , under some weather c o n d i t i o n s , a n a c c e p t a b l e r a n g e of s u f f i c i e n t m a n o e u v r a b i l i t y . The t h i r d r a n g e , t g r e a t e r t h a n 4 s e c , i s dangerous f o r hang g l i d e r s and can b e a c c e p t e d o n l y i n p a r t i c u l a r c a s e s as f o r man-powered a i r p l a n e s a t wind speed l e s s t h a n 2 m/sec. I n f i g u r e 3 , t h e e s t i m a t e d bank time of t h e h i s t o r i c a l L i l i e n t h a l ' s g l i d e r s of 7 m span i s i n d i c a t e d by a c i r c l e . They were c o n t r o l l e d l e s s e f f e c t i v e l y t h a n contemporary hang g l i d e r s . T h e i r bank t i m e s of 7 s e c were w i t h i n a n u n s a f e range. That e x p l a i n s t h e h a l f - c e n t u r y of s t a g n a t i o n i n development of t h a t form of g l i d i n g . I t s r e v i v a l was p o s s i b l e when t h e v a l u e of r = 0,2 m was i n c r e a s e d t o n e a r l y 0,7 m when t h e h a r n e s s f o r t h e p i l o t was i n v e n t e d . The bank times i n d i c a t e d i n f i g u r e 3 concern a c o n s i d e r a b l y low f l i g h t speed v = 8 m/sec, and i t i s known t h a t t h e aerodynamic c o n t r o l e f f e c t i v e n e s s d i m i n i s h e s w i t h t h e a i r speed. However t h i s bad p r o p e r t y does n o t o c c u r i n t h e c a s e of hang g l i d e r s c o n t r o l l e d by weight s h i f t , a s was expressed by formulas (1) and ( 2 ) . T h i s problem c a n b e p r e s e n t e d c l e a r l y by t a k i n g i n t o account t h a t f o r t h e formula (1) and f o r t h e weight c o n t r o l t h e r e l a t i o n CL 1 l / v 2 i s v a l i d . . Next f o r t h e formula (2) and aerodynamic c o n t r o l (when t h e i n e r t i a f o r c e s are t h e o n l y r e s p o n s e on t h e c o n t r o l f o r c e ) , t h e c o n t r o l moment r W 1 % r v 2 a p p l i e s . Then we o b t a i n r e l a t i o n s shown i n f i g u r e 4. T h i s t a b l e shows v e r y unadvantageous c h a r a c t e r i s t i c s ( t % 1 / v ) of t h e aerodynamic c o n t r o l f o r low speed f l y i n g d e v i c e s o p e r a t i n g n e a r s t a l l and b e i n g i n t e n d e d t o o p e r a t e l i k e a p a r a c h u t e . On t h e o t h e r hand t h e weight c o n t r o l h a s s u i t a b l e c h a r a c t e r i s t i c s a t low s p e e d s and improves when t h e speed d i m i n i s h e s ( t % v ) . It even c a n be independent of t h e speed ( t # f ( v ) ) i n t h e c a s e of t h e t o r s i o n a l l y v e r y e l a s t i c wing under c o n s i d e r a t i o n . Of c o u r s e , t h i s r e l a t i o n s h i p remains v a l i d i f t h e wing i s s t a b l e d u r i n g s t a l l o r , i n o t h e r words, i f t h e s e p a r a t i o n i s symmetrical. DEVELOPMENT O 2-77 HANG GLIDER F The development of a n u l t r a l i g h t wing of t h i s k i n d w a s v e r y troublesome and t o o k t h e a u t h o r about 1 0 y e a r s . I n i t i a l l y t h e work concerned a wing w i t h a c a b l e l e a d i n g edge ( r e f . 3 ) s t r e t c h e d by means of a p u l l e y and a s p r i n g o r r u b b e r rope expanded a l o n g t h e s p a r t u b e of t h e s k e l e t o n . These experiments showed advantageous f e a t u r e s of t h e u l t r a l i g h t f o l d a b l e wing w i t h t h e canopy f i x e d a t one p o i n t of t h e t i p t o t h e wing s p a r and having a hinged end r i b . The r i b h i n g i n g on t h e c a b l e o r on t h e t u b e c a n change t h e a n g l e of a t t a c k of t h e wing l i p . The t o r s i o n a l e l a s t i c i t y a l l o w s s e l f - a d j u s t m e n t of t h e wing t o t h e f l i g h t c o n d i t i o n s and good l a t e r a l c o n t r o l by weight s h i f t o n l y . Theref o r e i t was decided t o d e s i g n t h e e x p e r i m e n t a l hang g l i d e r 2-77 w i t h a cons i d e r a b l e s p a n of 1 2 m, a r e c t a n g u l a r wing planform, and a s i n g l e c e n t r a l v e r t i c a l s t a b i l i z e r ( r e f . 4). T h i s s i m p l e f l y i n g p l a n k arrangement was chosen a s a r e s u l t of t h e a u t h o r ' s own wide experiments and of a n a n a l y s i s of p o s i t i v e swept f l y i n g wings. I t s g e n e r a l p r o p e r t i e s a r e u n s t a b l e s t a l l f o r l a r g e r a s p e c t r a t i o s and bad d i v e r e c o v e r y of f l e x i b l e wings w i t h s o f t t i p s and no p r o f i l e d c e n t r a l r i b . These p r o p e r t i e s c r e a t e l i m i t s of a narrow speed r a n g e due t o u n s a f e c h a r a c t e r i s t i c s i n t u r b u l e n t wind c o n d i t i o n s . It was found t h a t t h e g r e a t e s t chance of e l i m i n a t i n g t h e s e u n d e s i r a b l e p r o p e r t i e s i s by a p p l i c a t i o n of a n arrangement w i t h s l i g h t l y n e g a t i v e sweep of t h e wing. I t i s j u s t t h e arrangement of t h e hang g l i d e r w i t h r e a s o n a b l e a p p l i c a t i o n of a n e l a s t i c wing c h a r a c t e r i z e d by one p o i n t c o n n e c t i o n of t h e s a i l t i p s t o t h e s k e l e t o n , and by t o r s i o n a l e l a s t i c i t y of t h e wing p l a n e . The hang g l i d e r 2-77 was d e s i g n e d a c c o r d i n g t o t h e g e n e r a l r u l e , " f i r s t s a f e t y and l a t e r t h e performance." The second more s o p h i s t i c a t e d r u l e w a s "do n o t c o u n t e r a c t t h e d e f o r m a t i o n b u t o r g a n i z e and e x p l o i t i t f o r s a f e t y and performance purposes." According t o t h i s second r u l e t h e wing bends and t w i s t s c o n s i d e r a b l y around t h e l e a d i n g edge which a c t s as a spanwise hinge. The f i r s t v a r i a n t of 2-77 t e s t e d i n 1977 had t h e c a b l e l e a d i n g edge and e x t e r n a l s p a r ( f i g . 5). Its s t a b i l i t y and c o n t r o l w a s e x c e l l e n t and t h e o n l y drawback was t e a r i n g of t h e canopy a s r e s u l t of c o n t a c t w i t h t h e w i r e s , when t h e g l i d e r was s t a n d i n g windward n o s e down on t h e ground. T h i s drawback was s o s i g n i f i c a n t t h a t a f t e r 1 5 minutes of wind p r e s s i n g on t h e wing t h e s a i l had t o be repaired. T h i s d e f e c t l e d t o a m o d i f i c a t i o n of t h e c o n s t r u c t i o n by i n s e r t i n g a s p a r t u b e i n t o t h e s a i l . Furthermore t h e s p a r w a s s u p p o r t e d by o n l y t h r e e w i r e s s o s i t u a t e d t h a t t h e s a i l would n o t touch t h e w i r e under any c o n d i t i o n s . I n t h e second v a r i a n t of 2-77 a double membrane a i r f o i l ( d a r k i n t h e p i c t u r e s ) f o r 50% of t h e chord w a s used w i t h duraluminium s h e e t p r o f i l e s s i m i l a r t o t h o s e i n the f i r s t variant. T h i s second v a r i a n t of 2-77 ( f i g . 6) had a n e x t r a o r d i n a r i l y wide speed r a n g e and a v e r y s o f t and s t a b l e s t a l l . The g l i d e r was g e n e r a l l y f a s t , cons i d e r i n g t h e area of 20 m2. T h i s w a s t h e r e s u l t of r e l a t i v e l y f l a t s e l f s t a b l e p r o f i l e s of t h e same k i n d as t h o s e used i n s i n g l e membrane v e r s i o n . The g l i d e r w a s v e r y s t a b l e i n t u r b u l e n t winds and i t s l o n g i t u d i n a l and l a t e r a l cont r o l w a s good. It p a r t i c i p a t e d i n hang g l i d e r c o m p e t i t i o n i n t h e ZakopaneT a t r a mountains i n 1978. A f t e r numerous f l i g h t s t h e n e x t m o d i f i c a t i o n of t h e wing ( f i g . 7) was u n d e r t a k e n i n o r d e r t o improve i t s LIDabove 1 0 which i s poss i b l e f o r t h e s t r u c t u r a l arrangement used and a n a s p e c t r a t i o of 7. For t h i s purpose, new more e f f e c t i v e s p e c i a l p r o f i l e s were developed and t h e planform of t h e wing was s l i g h t l y changed. During v e r y many t e s t f l i g h t s , sometimes of 1 0 minutes d u r a t i o n , t h e g l i d e r demonstrated a v e r y low minimum speed of 20 km/hr and a c o n s i d e r a b l e l i f t c o e f f i c i e n t ( n e a r l y 2 ) . Determination of maximum speed w a s more d i f f i c u l t , b u t speeds of 80 km/hr were reached w i t h o u t any problem. The m o d i f i c a t i o n s and t h e test f l i g h t s a r e c o n t i n u i n g . The main t a s k i s t o improve t h e L I D t o t h e p o s s i b l e n e a r l y 1 5 w h i l e m a i n t a i n i n g t h e hang g l i d e r ' s s a f e t y by good 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 . The s a f e t y achieved i s due t o such p r o p e r t i e s a s - p o s s i b i l i t y of s t a b l e and c o n t r o l l a b l e s t a l l and p a r a c h u t i n g from any altitude i m p o s s i b i l i t y of s l i p p i n g t h e wing and asymmetrical s t a l l i m p o s s i b i l i t y of s p i n c o n t r o l l a b l e d i v i n g and e a s y r e c o v e r y from d i v e v e r y wide speed r a n g e and i t s s a f e b o u n d a r i e s (very i m p o r t a n t under s t r o n g wind t u r b u l e n t c o n d i t i o n s ) p o s s i b i l i t y of immediate t r a n s i t i o n from d i v e t o p a r a c h u t i n g on t h e same s t r a i g h t l i n e t r a j e c t o r y , l o s i n g o n l y a dozen m e t e r s of a l t i t u d e - - The l a s t of t h e s e p r o p e r t i e s i s a n e x t r a o r d i n a r y one and d e s e r v e s some words. It was known t h a t f o r t h e d e f i n i t e geometry of t h e g l i d e r t h e r e i s one speed p o l a r f o r t h e s t e a d y f l i g h t . But t h e s p r i n g wing of 2-77 i s v e r y e l a s t i c i n t o r s i o n and t h e r e f o r e i t s v e l o c i t y p o l a r i s t h e envelope o f a n i n f i n i t e numb e r of p o l a r s f o r d i f f e r e n t t w i s t a n g l e s of t h e wing. T h i s i s shown i n f i g u r e 8 which e x p l a i n s t h e r e a s o n s f o r t h e wide speed r a n g e of 2-77. On t h e r e s u l t i n g p o l a r , f o r t h e g r e a t r a n g e of t h e t r a j e c t o r y i n c l i n a t i o n a n g l e , t h e two p o i n t s A and B can be found f o r which t h e g l i d e a n g l e i s t h e same. However, t h e s p e e d s of d i v i n g and p a r a c h u t i n g d i f f e r . For t h e hang g l i d e r of f i x e d geometry, c o n s i d e r a b l e sweep o r c o n v e n t i o n a l h o r i z o n t a l t a i l s t a b i l i z e r , a q u i c k move from t h e s t a t e A t o B on t h e s t r a i g h t l i n e t r a j e c t o r y AB i s p r a c t i c a l l y i m p o s s i b l e and o c c u r s d u r i n g p u l l up manoeuvre o r a s l a c k s t a l l a l o n g t h e c u r v e AB. Large span f l e x i b l e wings w i t h c o n s i d e r a b l e l e a d i n g edge sweep and a n e g l i g i b l e t o r s i o n a l e l a s t i c i t y of t h e s a i l w i t h t h e u n e l a s t i c f l e x i b l e canopy s t r e s s e d between t h e k e e l and l e a d i n g edge t u b e s behave s i m i l a r l y . A completely d i f f e r e n t s i t u a t i o n o c c u r s when t h e hang g l i d e r h a s a n e l a s t i c wing, h a s no h o r i z o n t a l t a i l s u r f a c e o r sweep, and h a s a low moment of i n e r t i a i n p i t c h . Then a sudden t r a n s i t i o n from ? o i n t A t o B on t h e s t r a i g h t l i n e t r a j e c t o r y i s p o s s i b l e a t a s u f f i c i e n t l y l a r g e and f a s t i n c r e a s e of t h e i n c i d e n c e a n g l e . Of c o u r s e a moderate b u t n o t t o o slow i n c r e a s e of i n c i d e n c e a n g l e normally r e s u l t s i n dynamic climbing. A t a slow i n c r e a s e of i n c i d e n c e a n g l e t h e g l i d e r mushes a c c o r d i n g t o t h e c u r v e AB. The dynamic s t a l l and t h e manoeuvre of l a n d i n g i n a d i f f i c u l t s i t u a t i o n as d e s c r i b e d and e x p l a i n e d above i s g e n e r a l l y s i m p l e . However, t e c h n i c a l l y t h e problem i s more complicated because t h e t o r s i o n a l e l a s t i c i t y and t h e time of manoeuvre have t o b e s u i t a b l e . These f a c t o r s cause t h e d e v i a t i o n of t h e r e a l t r a j e c t o r y from t h e s t r a i g h t l i n e AB. B r i e f l y , t h e c o n t r o l f o r c e s and manoeuvre t i m e a s s o c i a t e d w i t h i n s u f f i c i e n t e l a s t i c i t y exceed t h e p h y s i c a l c a p a b i l i t i e s of t h e p i l o t . On t h e o t h e r hand t o o much e l a s t i c i t y h i n d e r s dynamic climbing and c a u s e s pancaking of t h e g l i d e r . These problems and o t h e r s a r e t h e s u b j e c t of f u r t h e r r e s e a r c h and t e s t s of 2-77 (which h a s made about 400 f l i g h t s t o d a t e ) . Moreover, 2-77 a c t u a l l y e n a b l e s s h o r t and p r e c i s e l a n d i n g s behind o b s t a c l e s u s i n g t h e whole wing a r e a as a powerful aerodynamic brake. The a c t u a l d a t a of 2-77 ( f i g . 9) a r e Weight, 25 daN Span, 1 2 m Length, 5.5 m A r e a , 1 9 m2 Speed r a n g e , 20 t o 90 km/hr Lift-to-drag r a t i o , 12 Profiles, special, self-stable Maximum chord, 1 . 8 m Minimum chord, 1 . 5 m The hang g l i d e r 2-77, which was n o t d e s c r i b e d h e r e t e c h n i c a l l y , i n c l u d e s some e s s e n t i a l p a t e n t e d improvements. The g l i d e r based on t h e a p p l i c a t i o n of t h e u l t r a l i g h t e l a s t i c wing i s c a p a b l e of performing t h e dynamic s t a l l l a n d i n g p r o c e s s a t t a i n a b l e u n t i l now p r a c t i c a l l y o n l y by b i r d s . The u l t r a l i g h t e l a s t i c wing c a n b e used f o r t h e p r a c t i c a l i n v e s t i g a t i o n of t h e new unconventional l a n d i n g t e c h n i q u e , and f o r t h e development of t h e h i g h performance d e p l o y a b l e f l y i n g d e v i c e s ( f o r example, hang g l i d e r s of t h e c l a s s 2 of FAI-CIVL r e g u l a t i o n ) . T h i s wing c a n b e based on t h e a p p l i c a t i o n of t h e c a b l e o r t u b e l e a d i n g edge arrangement. Its a c t u a l and p o s s i b l e f u t u r e l i f t d r a g r a t i o i s compared i n f i g u r e 1 0 w i t h t h a t of o t h e r u l t r a l i g h t wing t y p e s . Because of t h e p o s s i b i l i t y of h i g h LID, i t i s v e r y s u i t a b l e f o r o s c i l l a t i n g wing p r o p u l s i o n of hang g l i d e r s ( r e f . 5) and h a s been p r a c t i c a l l y proved and t r i e d by t h e a u t h o r i n 1976-1977 by u s e of a n e l a s t i c p i l o t h a r n e s s and f o o t straps. REFERENCES 1. Wolf, J.: Dlaczego sprezyste skrzydlo (Why the Elastic Wing). Lotnicza i Astronautyczna, Nr 10, Nr 11, 1977. 2. British Civil Airworthiness Requirements, Section K, 1974. 3. Wolf, J.: The Stretched-Membrane Sailwing. Technical Soaring, No. 4, April 1973. 4. Wolf, J.: 2-77 lotnia nowego typu (2-77 Szkrzydlata Polska, Nr 39, 1978. Technika - A New Kind of Hang Glider). 5. Wolf, J.: The Technological Prospects for Oscillating Wing Propulsion of Ultralight Gliders. AIAA Paper No. 74-1028, presented at AIAA/MIT/SSA International Symposium on the Technology and Science of Low Speed and Motorless Flight, 1974. 10 aspect ratio 1 5 F i g u r e 1.- I n f l u e n c e of t h e induced d r a g ( a r e a A ) , p r o f i l e drag ( a r e a B ) , and s k e l e t o n d r a g ( a r e a C ) on t h e l i f t l d r a g of t h e u l t r a l i g h t wing w i t h e x t e r n a l s k e l e t o n having s p a r s and s t r u t s ( a ) , s p a r s and c a b l e s ( b ) , and only cables ( c ) . F i g u r e 2.- Considered bank a n g l e s of t h e hang g l i d e r . wing soan iro/ Bank angle 60° ~i1ienthal.s gliders unsafe region 0 2 4 6 8 10 bank time t/eec/ F i g u r e 3.- The bank times t f o r t h e hang g l i d e r s c o n t r o l l e d by mean body s h i f t r = 0,75 m of t h e p i l o t weighing W2 = 75 daN as a f u n c t i o n of wing s p a n R. Control type aerodynamic a weight shift . d a , 9 .k 1 t-- v twv n - R 2 E : M -d o -v m (d 1 t-- v t f f/v/ rl a , F i g u r e 4.- C o r r e l a t i o n of t h e bank time t w i t h f l i g h t speed v. F i g u r e 5.- Experimental hang g l i d e r 2-77 ( f i r s t v a r i a n t ) w i t h c a b l e l e a d i n g edge e l a s t i c wing. F i g u r e 6.- Tube l e a d i n g edge hang g l i d e r 2-77 (second v a r i a n t ) demonstrates t h e c o n s i d e r a b l e range of t h e wing t w i s t . F i g u r e 7.- Tube l e a d i n g edge hang g l i d e r 2-77 (third variant) i n flight. t h e p o i n t s of F i g u r e 8.- V e l o c i t y p o l a r of e l a s t i c wing hang g l i d e r . A,B p o l a r curve f o r d i v i n g and p a r a c h u t i n g on t h e same f l i g h t p a t h i n c l i n a t i o n ; C - t h e p o i n t of maximum speed; D - t h e p o i n t of minimum speed; E - t h e p o i n t of t h e maximum v e r t i c a l p a r a c h u t i n g ; F - t h e p o i n t of maximum LID; y - t h e range of t h e f l i g h t p a t h i n c l i n a t i o n a n g l e s f o r dynamic p a r a c h u t i n g ; vmax-vmin - t h e r a n g e of f l i g h t speed. The veloci t y p o l a r s f o r v a r i o u s wing twxst a r e shown by dashed l i n e s . - Figure 9.- Sketch of the third variant of 2-77. 0 5 aspect r a t i o 10 F i g u r e 10.- L i f t l d r a g as a f u n c t i o n of a s p e c t r a t i o f o r v a r i o u s u l t r a l i g h t wings. EXPERIMENTAL STUDY O F THE FLIGHT ENVELOPE AND RESEARCH O F SAFETY REQUIRE~V~ENTS FOR HANG-GLIDERS by Claudius LA BURTHE Head of Aircraft Section Aerospace Mechanics Division Systems Department Office National dlEtudes et de Recherches Adrospatiales (ONERA) 92320 Chdtillon (France) ABSTRACT Hang gliding was born a a popular sport in France in the 70's. After a period of observation, French Officials s decided that hang gliders were no longer to be considered a toys, but a a new kind of aircraft. Then, French Governs s ment funded a two years'research contract at ONERA on the safety of hang-gliders, in an attempt to set up the most adequate acceptance rules. S , 8 x 16 meters wind-tunnel of Chalais-Meudon near Parig, was used for two series of full scale tests, with 15 1 different gliders, including two-seaters, and most of them with a dummy pilot. A six component instrumentation provided lots of aerodynamic data. Flow visualization was used and showed quite unexpected air flov!s. The calculated basic performances were checked in real flight by the author, with some of the same gliders a s used in the tunnel. The flight mechanics computations were then completed, providing both the flight envelopes w i t , all sorts of limits and a fairly precise idea of the influence of several parameters, such a pilot's weight, wing settings, aeros elasticity, etc... The particular problem of luffing dives was thoroughly analysed, and two kinds of causes were exhibited in both the rules of luffing and aeroelastic effects. The general analysis of longitudinal stability showed a strong link with fabric tension, a expected through Nielsen's and Thwaites' theory. Fabric tension strongly depen. s ding upon aeroelasticity, that parameter was found to be the most effective design one for positive stability. Lateral stability was found to be very similar in all gliders except perhaps the cylindro-conical. The loss of stability happens in roll at low angle of attack, whereas i t happens in yaw at high angle. Turning performance was a bit surprising, with a common maximum value of approximately 55' of bank angle for a steady turn. Structure calculations began on the basis of an isostatic technique which did not succeed because the leadingedges, keel, and cross-spar were separated. Then, a linear finite elements technique was used and gave very adequate results for normal loadings, since the comparison with both flight and ground tests was very satisfactory. The prediction of ultimate loadings and breaking of the structure is less precise, and would possibly require a non-linear computation because of the bendings. During the research, all reports about significant casualties happening in France were analysed at ONERA and were of great help in the direction of the study. The conclusions of the research are, first that none of the normal aeronautical requirements would apply to the is case of hang-gliders. One good example would be the stall, wn~ch the base of a good half of a normal aircraft certification. A hang glider would possibly require the half of the certificator's attention on its maximum diving speed. As far a certification means are concerned, it i s intended to make an aerodynamic-test-vehicle which would be devoted s only to development and stability checks. A structural acceptance could be delivered on the basis of a calculation, plus ground-testing, using the ONERA method. But probably the most important impact of the research in terms of hang-gliders flight safety was the dissemination of this information to French instructorsand pilots. SYMBOLS A 0A angle of attack drag coefficient drag coefficient at d lift coefficient Cp C3 0 = ~(linearized) CL (linearized),lift gradient CL derivative : dcr / d ~ CLP rolling moment due to sideslip-coefficient C , pitching moment coefficient C lvlo pitching moment coefficient at o( = ~(linearized) C derivative cj C M /doc (linearized) C MA pitching moment due to sideslip-coefficient CN ) yawing moment due to sideslip-coefficient force exerted by the pilot on the control bar ( F > 0 corresponds to a nose-up action) center of gravity of the vehicle aerodynamic chord (length of the keel) F C! 0 O,X,Y,z L / ~ fineness ratio center of the glider (at the crossing of keel and cross-par) wing axes resulting aerodynamic force on the glider relative air velocity R V V s t o l stalling speed ~ 3 height of center of gravity,wing axis (see fig.) angle of attack (in degree) corresponding to maximum d dg L/D OO( d corresponding to the kink point on Cqacurve corresponding to onset of luffing if d decreases corresponding to minimum sink speed corresponding to maximum of sideslip dm;, dr AS& = $ A E ~ d r = d r p v m (minimum flying speed) - dLw,+ o ( ~ angle between wing-axis 0 2 and pilot strap (see fig.) ( d > o corresponds to a nose-up action) aspect ratio A aircraft in trim with control bar free ( F = 0) luffing limit maneuvering limit (max length of the pilot's arms) force limit (25% of pilot's weight) loss of roll control loss of yaw control P T b -3INTRODUCTION In France, hang gliding started to be a popular sport in 1973, when a national association (FFVL) was born. There were some hundreds of people flying, almost all claiming to be instructors! As usual, some dramatic accidents focused everyones attention on hang gliding, and fairly soon, many flying places became very crowded. Some of them were closed because of the problems created by the people watching and their motor-cars. But the aeronautical authorities were reluctant to consider them a real aircraft, and preferred initially to classify them a beach s s games, in order not to have to certify them. After two years, it was clear that a new kind of aircraft was flying French skies, and something had to be done about i t s flying safety. The DGAC (equiv. to F.A.A.) funded a two years'research at ONERA about the flying envelope of ultralight hang-gliders, and requested advice for future specifications. In order to avoid difficult similarity problems due to the slackness of fabric, it was decided to go through scale 1 tests in S1 Meudon wind-tunnel. The gliders used covered different shapes from the standard Rogallo to the Fledgling 1. Somewhat unexpected results were obtained, and it was decided to check the main performances in flight, which was done successfully. Then, the flight mechanics computations were completed, and highlighted some very interesting and specific features of these vehicles. A t the same time structural calculations were undertaken, and constantly cross-checked with in-flight and ground-test measurements. But the determination of handling, performance and structure specifications remains difficult because of the numerous non-linearitiesencountered in the problem, and the difficulty of defining adequate demonstrations for the manufacturers. Wind-tunnel testing of a sail-wing mock-up raises difficult scale effect questions. Therefore ONERA decided to use S1 Meudon, which allows scale 1 tests of hang-gliders, thanks to its 1 6 x 8 m elliptic facility. Nevertheless, the study is not necessarily free of Reynolds problems, a the paragliders' flying speeds places their Reynolds number in s the range of 1to 8 million. This could explain a good part of the scattering found in the tunnel results. Two series of one month tests were performed with 15 different gliders covering the shapes shown on figure 1. STARDARD SWALLOWTAIL SHAPE C Y L ~ ~ D COHICAL RO. CARARD ALBATR055 or DRAGOn PHOEUIX66 AUSTRALIAII SHAPE FLEDGLIRG I Fig. 1 - Survey of the shapes of gliders used. The mounting is basically made of a tetrahedral tubing (fig. 2), fixed on three vertical masts, through three dynamometric rings. The glider is fixed by means of clutches : a) at its "center", on the top of the tetrahedron, b) at the control bar on both front struts. The rear mast ends with a screw-jack which provides adjustment of the angle-of-attack.~ whole of the mounting can rotate about The ~ ~ ~ ~ a vertical axis for sideslip setting. All tests were made under static conditions, and all measurements had to be strongly filtered because of the effects of wire and fabric vibrations. Flow visualization revealed quite unexpected air flows, in that : (- 20'). ~ ~ ~ - no wing-tip vortex was found around cruise A.O.A. - a fairly high vorticing activity was found in the center-part of the wing, in spite of sweep angles (<45O) well below the admitted 52' for a vortex flow to be organised over the minimum value of - Fig. 2 - Wind tunnel arrangement. wing. This is almost certainly due to wing twist, which is surprisingly always near to 20°, thus preventing early separation. Fig. 3 a) and b ) show the results of visualizations respectively made with tufts and smoke, in the tunnel and in flight at all AOAs. Fig. 4 indicates the general flow around the wing at cruise angle of attack. 0(=10' 0(=18' rnaxL/= 0(=22' min sink 0(=30' 0(=40" stall Fig. 3 - Flow visualization with tufts (a) and smoke () 6. Possible bur5tin g Fig. 4 - Flow visualization, cruise A. O.A. Two important consequences have to be mentioned. The aerodynamic loading vs. wing-span is less severe than s s expected through a two-dimensional theory. The flow above described remains a long a the shape of the fabric is 5 ' self-adapting to the angle of attack, i.e. between luffing angle and approximately 2. The latter characteristic pro- 0 ' vides unique capabilities to Rogallo wings in that their flying envelope is significantly increased (by an angle of 1 or more) with regard to a normal "rigid" aircraft. Fig. 5 shows the flying envelopes infered from the following definition : the usable angles-of-attack AEdare limited by luffing: d luff and stall: dT . + a ; "l;ff Standard Swallowtail Cylindro con. Canard Albatross Phoenix 6B Australian Fledgling L D I rnax max min sink UD 7 9 9 1 12 1 <6 -5 20 23 18 24 1 6 18 24 12.5 4.3 5.5 4.9 41 . 5.5 5.9 "6 O 7 . / 23 24 24 32 1. 95 2 1 27 14 K 30 30 31 ( a<-a as." K ) .\/ 39 36 30 36 27 32 32 20 AE~O a0 ( ~ da~uff) - 27 29 31 - 9 6 -1 0 3 1 - 32 27 24 24 26 > 26 24 25 Fig. 5 - Key A.O.Asused in defining the flight envelopes. Under these conditions, one could expect to find numerous non-linearities in the aerodynamic data. In fact, s there are many, but curiously, the lift coefficient remains pretty linear against H (Fig. 6)a long as the fabric is free of luffing and far from stall conditions, which means able to adapt its own shape to the proposed angle of attack. The local linearity allows drawing a graph of CLd against aspect ratio h for a l l the gliders in the study 3, 1 (Fig. 7).Then it is possible to compare data of different origins : Fig. 8 and refs. [2, 4,5,6 . But CL is the only coefficient t o behave so, and unfortunately the non linearities of the pitching moment CM are very strong. Fig. 6 shows typical results obtained at constant wind speed in the tunnel. But these do not represent the actual conditions of flying, because the variations of speed induce variable loads on the aluminium structure, which is very flexible . Consequently, the shapes of the wings, mainly the billow, are modified, up to the point where it was found essential to make tunnel tests at diffeient speeds (precisely 3 speeds in the range of 8 to 20 m/s or 18 to 45 m.p.h.1. Fig. 6 shows one example of the necessary interpolation. The impact will be analysed in the discussion of longitudinal stability. Fig. 6 - Typical tunnel results. No Wcngor Ref. ' ~ o h 2.83 2.88 3 29 2.8 3.84 5.37 5.0 5.5 5.2 5.7 5.6 5.88 2.8 Simulatnon 1 2 3 4 5 1 Standard No 2 Standard No 3 Standard No Swallowtall No Swallowta81 No Canard 1 2 6 7 8 9 10 11 12 13 14 15 16 17 18 1 Cylcndro-con.No 2 Cyllndro-con No Albatross Phoenlx 68 Aurtrallan Fledgltng Malavard 11) Polhamur 121 Cylmdrlcal Contcal Naereth 131 Mendenhall [4] Pflugrhauptl5J ENSAG [6] 22 . 2.4 2.25 26 . 2.6 3 3.3 3.2 2.7 28 3.2 31 3 2.12 2.24 2.8 2.9 22 . Scale 1 W T. tests IONERA) Camputatton * . T -he<, 3 3 2.82 W.T. scale glider LID < 17; for a sailplane L/D - 17. Both criteria (normal soaring condition and LID specification) appear to be reasonably equivalent. a manned ULTRALIGHT G.LIDER(ULG) (includes hang glider) or SAILPLANE(ULS): flying device as described previously having a wing loading w 5 10 kg/m2 (approximately 2 lb/ft2) a manned £1 ing device as described preLIGHT GLIDER(LG) or SAILPLANE(LS): viously having a wing loading w = 10 to 25 kg/m (approximately 2 to 5.1 lb/ft2 ) GLIDER or SAILPLANE: a manned flying device as described previously having a wing loading w = 25 to 40 kg/m2 (approximately 5.1 to 8 lb/ft2) S During the development of gliders and sailplanes over several decades the wing loading increased noticeably. What appeared to be a "normal" wing loading some 35 years ago is considered "light" today. In view of the increased interest in hang gliders, human-powered aircraft, and other similar, vastly improved sailplanes under development which due to the energy shortage, may well be the only means of soaring in the future, an attempt is made to define these aircraft. Since the wing loading is one of the factors governing the plane's performance, the above specified ranges are in order. AUXILIARY POWERED GLIDER(APG) or SAILPLANE(APS); LIGHT AUXILIARY POWERED GLIDER(LAPG) or SAILPLANE(LAPS); ULTRALIGHT AUXILIARY POWERED GLIDER (ULAPG) or SAILPLANE(ULAPS): a manned flying device as described previously having an auxiliary engine to take off and to overfly with power any severe downdraft 9 kg/HP areas which would otherwise result in a landing; power loading p (approximately 20 lb/HP) Since the beginning of soaring, attempts have been made to overcome the two inherent disadvantages of a sailplane: inability to take off with initial climb and to overfly large areas of sink without landing. Various kinds of propulsion were and are being installed as an auxiliary source of power which preferably would not decrease the sailplane's performance during the soaring phase of flight. The above definition should cover any auxiliary power installation regardless of whether the available power is sufficient for takeoff and initial climb or sustention of level flight only. The expression "self-launching sailplane" (SLS) for an auxiliary powered sailplane (APS) should not be used because it suggests an ultralight (hang) glider or sailplane which can be launched by the pilot's feet (i.e., without any mechanical power); it is also not consistent with the decades old concept of an APS, described above. Another expression, "motorglider", denoting an auxiliary powered sailplane (APS) appears to be inappropriate for several reasons. Most likely it is an old translation of the German word "motorgleiter" by people whose technical and linguistic knowledge was rather poor. It is an accepted view here (USA) that there is a difference between the two words "motor" (electric) and I1 engine" (combustion). The bridge between the two kinds of energy conversion devices is rocket propulsion: it can be called either a rocket motor or a rocket engine. Furthermore, it should be noted that even Germans apparently have preferred for some time the term "MOTORSEGLER". Unfortunately there is no comparable, elegant translation available in English. a glider or a sailplane converted into a POWERED GLIDER(PG) or SAILPLANE(PS): powered aircraft; the engine is essential for flying operation On occasion a glider or a sailplane is converted into a powered aircraft by installing an engine which produces a substantially higher power than required for flying an auxiliary powered glider or sailplane. Thus soaring flight becomes an exception in the usual flight operation of a powered glider or sailplane. One, but not the only such example, is the Schweizer SGS 2-32 sailplane which has been used in various development, research, and promotional projects. In some extreme cases the power installation and other modifications made were of such extent that the identity of the original sailplane almost vanished. HUMAN POWERED AIRCRAFT(HPA): a manned flying device powered only by human efforts This definition covers any heavier than air, manned flying device, which by its nature is an ultralight sailplane of high performance. CONCLUDING REMARKS One would expect that in view of substantial technological developments resulting in outstanding performance of today's sailplanes appropriate terminology would be widely in use. Apparently this is not the case. This paper presents proposed nomenclature as a beginning effort to improve the present unsatisfactory condition. It should also serve as a guide for comparable improvements in other languages. Page intentionally left blank ATTENDEES BILL ABEYOUNIS N S Langley Research C e n t e r AA S Hampton, Va , U A W. S. BLANCHARD, JR. N S Langley Research C e n t e r (Ret ) AA Hampton, Va UA S . ., . E U R AGUSTO DAD ~ssociacgo r a s i l e i r a d e V& B S. Paulo, B r a z i l STEPJ.3EN L. AICHOLZ Media, Pa., USA HOLT ASHLEY Stanford University Stanford, Calif., U A S a Vela, FORREST BLOSSCM Soaring S o c i e t y of America, U A S KENN ALLEN BLUMENSTOCK Columbia P l a s t i c s . , Inc., Columbus, Md., U A S RAY BORST N. Huntingdon, Pa., UA S D N L D. BAALS O AD The George Washington U n i v e r s i t y J o i n t I n s t i t u t e f o r Advancement of F l i g h t S c i e n c e s , Hampton, Va. UA S CHARLES LINN BOTZKO Elmore, Ohio, U A S O E STEVE B W N Bethlehem Stee1 Corp. N a s h v i l l e , Tenn., USA , BN B DN C E AEOH Danville, C a l i f . , PAUL B K R AE Williamsburg, Va., UA S ELMAR BREITBACH D ~ R Gottingen, W. Germany , BRUCE BROSI S Boston, Mass., U A JOE ALLEN BROWNLEE Mississippi State University Mississippi State, M i s s . , U A S GIFFORD BULL Miss i s s i p p i S t a t e U n i v e r s i t y Mississippi State, M i s s . , U A S JAMES BURLEY N S Langley Research Center AA UA S Hampton, Va. UA S RICHARD W. BARNWELL N S Langley Research C e n t e r AA Hampton, Va. USA PATRICK J. BEATTY Bedfordview, S. A f r i c a GERD W. BERCHTOLD T e c h n i c a l U n i v e r s i t y of Munich Munich, W. Germany WILLIAM D. BERTELSEN B e r t e l s e n , Inc., S Neponset, I l l . U A , , ROBERT CALLAGHAN Ather t o n , C a l i f . , UA S WILLIAM R. BERTELSEN Ber t e l s e n , I n c . , S Neponset, I l l . , U A KEVIN B A K LC Durham C o l l e g e O n t a r i o , Canada JOHN CAMPBELL The George Washington U n i v e r s i t y J o i n t I n s t i t u t e f o r Advancement of F l i g h t Sciences S Hampton, Va. U A , WALTER H. CARNAHAN R o c h e s t e r , N.Y., USA ROBERT A. CHAMPINE NASA L a n g l e y R e s e a r c h C e n t e r ( R e t .) H a m p t o n , Va. , USA ROBERT EDWIN CHENEY C h e n e y Models J a c k s o n , N.H., USA ALEJANDRO CHITTY North Carolina S t a t e University R a l e i g h , N.C., USA DONALD L. CIFFONE, NASA Ames R e s e a r c h C e n t e r M f f e t t F i e l d , C a l i f . , USA o LES CLANTON S a n t a Monica, C a l i f . , DANIEL J. D e VRIES K e n t w o o d , M i c h . , USA JOHN DeYOUNG K e n t r o n I n t e r n a t i o n a l , Inc. H a m p t o n , Va., USA MICHEL J. DOUTRELOUX P l a n t e n en M o r e t u s Antwerp, Belgium ED DULLAGHAN N a v a l E d u c a t i o n and T r a i n i n g C e n t e r N o r f o l k , Va., USA JOHN V. DUNCAN S p a r t a n s b u r g , S .C. JEROME DUPREY Durham C o l l e g e O n t a r i o , Canada LEON EDLING Kent, C o n n . , USA USA DAVID W. CODER D a v i d W. T a y l o r N a v a l S h i p R e s e a r c h and D e v e l o p m e n t C e n t e r B e t h e s d a , Md., USA KENNETH N. COLE NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA WILLIAM Be COMPMN I11 NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA W I L L I AM CONARD T i d e w a t e r S o a r i n g S o c i e t y , USA , USA KLAUS EIKEMEIER DFVLR, B r a u n s c h w e i g , W. G e r m a n y FRAUKE ELBER T i d e w a t e r S o a r i n g S o c i e t y , USA WOLF ELBER NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va USA ., DONALD ELLIOTT C o l u m b u s , O h i o , USA E. ENEVOLDSON NASA Hugh L. D r y d e n F l i g h t Research Center E d w a r d s , C a l i f . , USA RICHARD EPPLER U n i v e r s i t y of S t u t t g a r t S t u t t g a r t , W. G e r m a n y WILLIAM V. FELLER NASA L a n g l e y R e s e a r c h C e n t e r ( R e t . ) H a m p t o n , Va., USA JAMES DAVID CONNERTON C o l u m b i a P l a s t i c s , Inc. C o l u m b i a , M d . , USA E. J. CROSS M i s s i s s i p p i State U n i v e r s i t y Mississippi S t a t e , M i s s . , USA K. MICHAEL DAY A m e r i c a n S t a n d a r d Co. D e a r b o r n , M i c h . , USA JAMES C. FERRIS NASA L a n g l e y R e s e a r c h C e n t e r USA H a m p t o n , Va. , THOMAS E. HEAD Severna P a r k , Md., USA GEORGE W. FISHER C h a r l o t t e , N.C., USA HENRY LEE FISHER M o r r i s v i l l e , N.C., DEBORAH M. FLAD T o u g h k e n a m o n , Pa., RICHARD F . HELLBAUM NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA ROBERT DALE HELTON NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA HEINRICH G. HELWIG Rensselaer Polytechnic I n s t i t u t e T r o y , N.Y., USA USA JAMES R. HENRY Soaring A s s o c i a t i o n of C a n a d a USA USA MARINA FORD Lookout Mountain, Tenn., ROBERT E. FORD Birmingham, Ala., USA MANFRED H. HILLER U n i v e r s i t y of S t u t t g a r t S t u t t g a r t , W Germany . WILLIAM TODD HODGES Army S t r u c t u r e s L a b o r a t o r y NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA HAYWOOD HOLDER U.S. Army H a m p t o n , Va., USA REGINALD M. HOLUlWAY NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , V a , USA WILLIAM F. FOSHAG A e r o p h y s i c s Co. W a s h i n g t o n , D.C., USA JEAN M. FOSTER NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va. , USA SAMUEL A. FRANCIS M a r i o n , M a s s . , USA ROBERT CLAYTON GAIRNS Soaring A s s o c i a t i o n of C a n a d a H. DOUGLAS GARNER NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA JOSEPH GERA NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA SAM GREENHALGH Naval A i r Development C e n t e r W a r m i n s t e r , P a . , USA . JAMES HOTELLING R a l e i g h , N.C., USA EDWARD BRUCE JACKSON N o r t h C a r o l i n a State U n i v e r s i t y R a l e i g h , N.C., USA PETER F JACOBS NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA HARRY A. JAMES Teledyne Ryan A e r o n a u t i c a l San D i e g o , ' C a l i f USA . JAMES B. HALLISSY NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA PERRY W HANSON . NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va , USA ., . CHRISTIAN JANSSEN Corning Glassworks C o r n i n g , N.Y., USA SAM0 ONJEGIN JENKO AMTECH Services M a n s f i e l d , O h i o , USA DAVID G. JONES A s p e n , C o l o . , USA TOM C. JONES T i m o n i u m , Md., EUGENE E. LARRABEE M a s s a c h u s e t t s I n s t i t u t e of T e c h n o l o g y C a m b r i d g e , M a s s . , USA DAVID A. LEEDOM Santa B a r b a r a , C a l i f . , AUSTIN W. LEFTWICH R i c h m o n d , Va., USA W LIEBE . T e c h n i c a l U n i v e r s i t y of B e r l i n B e r l i n , W Germany . F'RANCOIS-XAVIER F I T T U n i v e r s i t y of L i e g e ~ i & ~ e l ,g i urn B ROBERT WARREN LONG M o o r e H a r o n , F l a . , USA PAUL B. MacCREADY A e r o V i r o n m e n t , Inc. P a s a d e n a , C a l i f . , USA JOHN MacKAY McGill University Quebec, Canada JOSEF MANDLA S o a r i n g A s s o c i a t i o n of C a n a d a HENRY L. MARKISON L a n s i n g , M i c h . , USA DAVID JOHN MARSDEN U n i v e r s i t y of A l b e r t a Alberta C a n a d a USA USA ROCKY H. W. JOWES Newpor t News, Va. , USA THOMAS C. KELLY NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA WILLIAM L. KING Sport F l i g h t , Inc., G a i t h e r s b u r g , Md., USA ROBERT KNAUFF U.S. A i r F o r c e Hamp ton, V a USA ., ILAN KROO Stanford U n i v e r s i t y Stanford, C a l i f . , USA ANDREW M. KUBIAK Westland, Mich., USA RICHARD KUHN NASA L a n g l e y R e s e a r c h C e n t e r ( R e t . ) USA Hampton, Va ., , CLAUDIUS LaBURTHE ONERA, C h a t i l l o n , F r a n c e WAYNE A. LaDENDORF DEVRY T e c h n i c a l I n s t i t u t e C h i c a g o , Ill., USA DOUG LAMONT GLEN L. MARTIN K e n t r o n I n t e r n a t i o n a l , Inc. H a m p t o n , Va. USA , PETER MASAK Soaring A s s o c i a t i o n of C a n a d a MARK D. MAUGHMER Princeton University P r i n c e t o n , N.J., USA Soaring Society of A m e r i c a , USA ROBERT T. LAMSON Soaring Society of A m e r i c a , USA KAY MAYLAND Technische Hochschule Darmstadt W. G e r m a n y CATHERINE F . AND JERRY McADEN Lor ton, Va USA ORAN W. NICKS NASA L a n g l e y R e s e a r c h C e n t e r Hampton, Va USA ., ., W. BARRY NIXON Princeton University P r i n c e t o n , N.J., USA MM N WALDO o [ A NASA L a n g l e y R e s e a r h C e n t e r H a m p t o n , Va., USA ALBERT M. McCARTY Naval A i r Development C e n t e r W a r m i n s t e r , P a . , USA HUGH McCAY H a m p t o n , Va., USA LACH OHMAN B r y a n , O h i o , USA ESTER M. AND JACK R. McGONIGLE McMurry, P a . , USA JOHN H. McMASTERS B o e i n g C o m m e r c i a l A i r p l a n e Co. Seattle, W a s h . , USA GREGORY R. MOLENAAR H a n g G l i d e r s I n t e r n a t i o n a l Co. B e t h e s d a , Md., USA LAWRENCE C. MONTOYA NASA H u g h L. D r y d e n F l i g h t Research Center E d w a r d s , C a l i f . , USA MARTIN MOORE A p e x , N.C., USA P I E R 0 MORELLI Politecnico di Torino T o r ino, I t a l y RUDOLF MUELLED Soaring A s s o c i a t i o n of C a n a d a DIETER AND HROSWITEI C. MUSER DFVLR, S t u t t g a r t , W G e r m a n y . HENRY T. NAGAMATSU R e n s s e l a e r Polytechnic I n s t i t u t e T r o y , N.Y., USA JAMES L. NASH-WEBBER M a s s a c h u s e t t s I n s t i t u t e of T e c h n o l o g y C a m b r i d g e , M a s s . , USA ALLEN ORMSBEE U n i v e r s i t y of I l l i n o i s U r b a n a , Ill., USA CARL ARTHUR OSOJNAK B i r m i n g h a m , Mich., USA BERNARD PAIEWONSKY M c C l e a n , Va., USA STANLEY PELKOWSKJ T i d e w a t e r Soaring Society, USA J I M ALEX PENLAND NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA WERNER PFENNINGER NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va. USA , J I M PETERS B a l t i m o r e , Md., USA WILLIAM H. P H I L L I P S NASA L a n g l e y R e s e a r c h C e n t e r ( R e t. ) H a m p t o n , V a , USA . P. KENNETH PIERPONT NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA BION LEE PIERSON Iowa State U n i v e r s i t y Ames, I o w a , USA V I C POWELL National Aeronautic Association W a s h i n g t o n , D.C., USA DECIO PULLIN ~ s s o c i a c a o r a s i l e i r a de V o o a V e l a B S. P a u l o , B r a z i l A G I U L I O ROMEO Politecnico d i T o r i n o Torino, Italy ROBERT A. ROSE B i h r l e Applied Research Hampton, Va USA ., THOMAS H. PURCELL F l i g h t Dynamics, Inc R a l e i g h , N.C., USA ., LAWRENCE C. ROSS1 S a l i s b u r y , Md. , USA JOHN F , ROURKE I JR. NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , V a . , USA MURRAY I. ROZANSKY MIRCO, H o p e w e l l , N . J . , ROBERT RUELLE U n i v e r s i t y of I l l i n o i s U r b a n a , I l l . , USA JOHN M. RUSSELL M a s s a c h u s e t t s I n s t i t u t e of T e c h n o l o g y C a m b r i d g e , M a s s . , USA GUY J. SANDER , U n i v e r s i t y of L i e g e Liege, Belgium LOYAL WADE SAVARIA W o n d e r V a l l e y S o a r i n g School Fresno, Calif USA . BALLARD QUASS NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , V a . , USA JAMES C. REDDING W e b s t e r , N.Y., USA WILMER H. REED, I11 NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA MICHAEL M REISMAN . C h a t t e n o o g a , Tenn., USA USA RONALD E. R W N o r t h C a r o l i n a State U n i v e r s i t y R a l e i g h , N.C., USA KURT REUPKE Grumman A e r o s p a c e C o r p . B e t h p a g e , N.Y., USA WILLIAM K. RICKSON M i l l b r a e , C a l i f . , USA DONALD R. R I L E Y NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , Va., USA DAVID ROBSON B a l t i m o r e , Md., ., P A T R I C I A LYNN SAWYER T i d e w a t e r Soaring Society, USA FREDERIC H. SCHMID M e t a i r i e , L a . , USA HARRIS M. SCHURMEIER NASA J e t P r o p u l s i o n L a b o r a t o r y P a s a d e n a , C a l i f . , USA L E S E. SCHWEIZER Schweizer A i r c r a f t Corp. E l m i r a , N .Y., USA GEORGE SEIRMARCO T i d e w a t e r S o a r i n g Society, USA USA FRANCIS M. ROGALLO K i t t y Hawk, N.C., USA NELSON M. ROGERS W r i g h t P a t t e r s o n A i r Force B a s e O h i o , USA GASTON R e SERVANT H a n g G l i d e r s I n t e r n a t i o n a l Co. B e t h e s d a , Md., USA WILLIAM G R I E R SEWALL NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , V a . , USA EDGAR D. SEYMOUR G l i d e r P i l o t s G r o u n d School R o c h e s t e r , N .Y. USA A. J. SMITH Soaring Society of A m e r i c a , USA BERNALD S. SMITH N a t i o n a l Soaring M u s e u m and E l m i r a , N.Y., Soaring Society of A m e r i c a , USA GRANT M SMITH . P o p l a r L a k e , I l l . , USA PAUL M. SMITH LTV C o r p . H a m p t o n , V a . USA , DANA SEYMOYR T o r on to, C a n a d a NELSON AND REIDUN SHAPTER M c L e a n , V a . , USA TIMOTHY M. SHEARS G r a n d R a p i d s , Mich., , ROBERT E . SMITH Pascagoula, Miss. STANLEY W. SMITH N e w a r k , D e l , USA , USA USA . YOUNG T . SHEN D a v i d W. T a y l o r N a v a l S h i p R e s e a r c h and D e v e l o p m e n t C e n t e r B e t h e s d a , Md., USA S. S I D D I Q I U n i v e r s i t y of I l l i n o i s U r b a n a , I l l . , USA JAROSLAW S O B I E S K I NASA L a n g l e y R e s e a r c h C e n t e r Hampton, Va USA ., DAN MICHAEL SOMERS NASA L a n g l e y R e s e a r c h C e n t e r USA Hampton, Va. , DAVID J. S I E G F R I E D New B r i t a i n , Pa., USA ROBERT A. SIMONDS LTV C o r p . H a m p t o n , V a , USA CHARLES M. SOUTHALL, P o q u o s o n , V a , USA . I11 CHRISTOPHER STARBUCK Wildwood, Ga. USA , . HERBERT A. STOKELY V i r g i n i a B e a c h , Va., USA DAVID J. SLIWA U n i v e r s i t y of I l l i n o i s U r b a n a , I l l , USA KENNETH J. SLIWA H a r r i s H i l l Soaring C o r p . E l m i r a , N.Y., USA SHIRLEY A. SLIWA N a t i o n a l Soaring M u s e u m E l m i r a , N.Y., USA STEVEN M. SLIWA NASA L a n g l e y R e s e a r c h C e n t e r H a m p t o n , V a . , USA F M Y D J. AND FRANCES SWEET Soaring S o c i e t y of A m e r i c a , USA ADAM SWETNICK G a i t h e r s b u r g , Md., USA KATHLEEN K. TAYLOR Brookhaven N a t i o n a l Laboratory U p t o n , N.Y., USA WALTER TAYLOR Newpor t News, V a ., USA M. P. TETER Corning Glassworks Corning, N.Y., UA S R. VICTOR TURRIZIAOI Kentron I n t e r n a t i o n a l Inc. Hampton, Va., USA VADYM V. UTGOFF U.S. Naval Academy Annapolis, Md., U A S BRIAN UTLEY Soaring S o c i e t y of America, U A S OTTO W G E A NR T e c h n i c a l U n i v e r s i t y of Munich Munich, W. Germany MICHAEL WATERS Sport F l i g h t , U A S PETE8 WAY Massachusetts I n s t i t u t e of Technology Cambridge, Mass., U A S Q E TON E W A E UN E VR Asheboro, N.C., USA DAVID V. WEBBER Chads Ford, Pa., . USA TOM WILLIAMS North C a r o l i n a S t a t e U n i v e r s i t y R a l e i g h , N.C., U A S JERZY S. WLF Aviation I n s t i t u t e War saw, Poland L R Y EARL W O S AR OD Hydrospeed O n t a r i o , Canada R Y YOUNG A Aero Club A l b a t r o s s Somerville, N.J., UA S O N BO D JAMES W. Y U G L O N S Langley Research Center AA Hampton, Va , U A S . 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. N S CP-2085, P a r t 1 AA 1 4. Title and Subtitle 5. Report Date THE SCIENCE AND TECHNOLOGY OF M W SPEED AND MOTORLESS FLIGHT 7. Author(s) June 1979 6. Performing Organization Code 8. Performing Organ~zationReport No. P e r r y W. Hanson, ccmpiler , L-12973 10. Work Unit No. 9. Performing Organization Name and Address NASA Langley Research Center Hampton, VA 23665 12. Sponsoring Agency Name and Address 505-02-23-0 1 11. Contract or Grant No. 13. Type of Report and Period Covered N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i m Washington, DC 20546 15. Supplementary Notes Conference P u b l i c a t i o n 14. Sponsoring Agency Code 16. Abstract The T h i r d I n t e r n a t i o n a l Symposium on t h e S c i e n c e and Technology of Low Speed and M o t o r l e s s F l i g h t was h e l d a t t h e NASA, Langley Research C e n t e r , March 29-30, 1979. The N S Langley Research C e n t e r sponsored t h e symposium i n c o o p e r a t i o n w i t h t h e AA S o a r i n g S o c i e t y o f America (SSA) The symposium provided a forum f o r t h e i n t e r change of i n f o r m a t i o n on r e c e n t p r o g r e s s i n t h e s c i e n c e and t e c h n o l o g i e s a s s o c i a t e d w i t h low speed and m o t o r l e s s f l i g h t . T h i s c o n f e r e n c e p u b l i c a t i o n i n c l u d e s 28 p a p e r s p r e s e n t e d a t t h e symposium and 1 a d d i t i o n a l paper. The p a p e r s d e a l w i t h low speed aerodynamics, new m a t e r i a l s a p p l i c a t i o n s and s t r u c t u r a l c o n c e p t s , advanced f l i g h t i n s t r u m e n t a t i o n , s a i l p l a n e o p t i m a l f l i g h t t e c h n i q u e s , motor s o a r e r s , and u l t r a l i g h t s a i l p l a n e s and hang g l i d e r s . . 17. Key Words (Su gested by Authorjs)) Low speed aerodynamics Optimum d e s i gn Low d r a g a i r f o i l s Advanced i n s t r u m e n t a t i o n Structural analysis Composite materials Ultralight sailplanes Hang g l i d e r s 18. Distribution Statement Unclassified - Unlimited S u b j e c t Category 01 21. No. of Pages 22. Price* 19. Security Classif. (of this report) 20. Security Classif. (of this page) Unclassified Unclassified 305 $11.75 * For sale by the National Technical Information Service, Springfield, V ~ r g i n ~22161 a NASA-Langl ey, 1979