"Applied Science, Faculty of"@en . "Mechanical Engineering, Department of"@en . "DSpace"@en . "UBCV"@en . "Drummond, Alastair Milne"@en . "2011-11-08T22:02:52Z"@en . "1963"@en . "Master of Applied Science - MASc"@en . "University of British Columbia"@en . "The dynamic stability of a long, slender-bodied vehicle with a flexible fuselage is examined analytically in the supersonic speed regime. The small aspect ratio lifting surfaces are considered to be rigid but dependence of their angles of attack on fuselage flexibility is accounted for. The amplitude of pitching oscillation is restricted to \u00B110\u00B0 about the zero-lift line by the nature of the unsteady, supersonic aerodynamic theory used. The stability problem is formulated by a set of non-linear differential equations with the non-linear contributions arising from both the inertia and the aerodynamic forces. The present analysis accounts for non-linear contributions up to third degree in the rigid body angle of attack. The stability of the short period mode is investigated using Routh-Hurwitz criteria and an expression representing a stiffness criterion for dynamic stability is obtained. The analytical development is so presented as to make it easily applicable to a supersonic, flexible vehicle with or without wings, e.g. a supersonic transport or a missile. Moreover to facilitate the evaluation of the effect of flexibility and non-linearities on dynamic stability, four cases are considered separately:\r\na. Rigid body equations of motion, without non-linear terms\r\nb. Rigid body equations of motion, with non-linear terms\r\nc. Flexible body equations of motion, without non-linear terms\r\nd. Flexible body equations of motion, with non-linear terms.\r\nA numerical example is presented towards the end which investigates the dynamic stability of a flexible, supersonic transport configuration. The conclusions from the example are:\r\n1. The non-linearities can be safely neglected for rigid aircraft, but not for wingless vehicles.\r\n2. Flexibility affects the stability through the lift and pitching moment and also by introducing two more possible equilibrium points.\r\n3. The amount of work involved in finding a solution is markedly increased by the necessity of solution of more characteristic equations of higher degree.\r\n4. The stiffness criterion can be used to adjust the stiffness distribution to one that can make an unstable configuration stable.\r\nThe usefulness of the method is two-fold. For a flexible vehicle with known geometric, mass and elastic properties, the method can predict its dynamic stability. This feature is of considerable importance particularly in the design stage. On the other hand, if an aircraft with known geometry, total mass and centre-of-gravity location proves to be dynamically unstable, then the analysis provides a stiffness criterion by which it can be made stable. The analysis involves a considerable amount of computation and hence seems to be particularly suited for solution by a digital computer."@en . "https://circle.library.ubc.ca/rest/handle/2429/38867?expand=metadata"@en . "ON THE DYNAMIC STABILITY OF FLEXIBLE SUPERSONIC VEHICLES by ALASTAIR MILNE DRUMMOND B.A.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1957 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF . MASTER OF APPLIED SCIENCE i n the Department of Mechanical Engineering We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1963 In presenting th i s thesis i n p a r t i a l fulf i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t free ly avai lable for reference and study. I further agree that per-miss ion. for extensive copying of this thesis f o r . s c h o l a r l y purposes may be granted by the Head of my Department or by his representatives . I t i s understood that copying, or p u b l i -cation of this thesis for f i n a n c i a l gain sha l l not be allowed without my written permission. Department of The Univers i ty of B r i t i s h Columbia, Vancouver 8, Canada. Date ABSTRACT The dynamic s t a b i l i t y of a lo n g , slender-bodied v e h i c l e w i t h a f l e x i b l e fuselage i s examined a n a l y t i c a l l y i n the super-sonic speed regime. The sm a l l aspect r a t i o l i f t i n g surfaces are considered to be r i g i d but dependence of t h e i r angles of at t a c k on fuselage f l e x i b i l i t y i s accounted f o r . The amplitude of p i t c h i n g o s c i l l a t i o n i s r e s t r i c t e d to\u00E2\u0080\u009410\u00C2\u00B0 about the zero-l i f t l i n e by the nature of the unsteady, supersonic aero-dynamic theory used. The s t a b i l i t y problem i s formulated by a set of no n - l i n e a r d i f f e r e n t i a l equations w i t h the non-l i n e a r c o n t r i b u t i o n s a r i s i n g from both the i n e r t i a and the aero-dynamic f o r c e s . The present a n a l y s i s accounts f o r n o n - l i n e a r c o n t r i b u t i o n s up t o t h i r d degree i n the r i g i d body angle of a t t a c k . The s t a b i l i t y of the short period mode i s i n v e s t i -gated using Routh-Hurwitz c r i t e r i a and an expression repre-s e n t i n g a s t i f f n e s s c r i t e r i o n f o r dynamic s t a b i l i t y i s obtained. The a n a l y t i c a l development i s so presented as t o make i t e a s i l y a p p l i c a b l e t o a supersonic, f l e x i b l e v e h i c l e w i t h or without wings, e.g. a supersonic t r a n s p o r t or a mis-s i l e . Moreover t o f a c i l i t a t e the e v a l u a t i o n of the e f f e c t of f l e x i b i l i t y and n o n - l i n e a r i t i e s on dynamic s t a b i l i t y , f o u r cases are considered s e p a r a t e l y : a. R i g i d body equations of motion, without n o n - l i n e a r terms b. R i g i d body equations of motion, w i t h n o n - l i n e a r terms i i i c. F l e x i b l e body equations of motion, without n o n - l i n e a r terms d. F l e x i b l e body equations of motion, w i t h n o n - l i n e a r terms. A numerical example i s presented towards the end which i n v e s t i -gates the dynamic s t a b i l i t y of a f l e x i b l e , supersonic t r a n s p o r t c o n f i g u r a t i o n . The conclusions from the example are: 1. The n o n - l i n e a r i t i e s can be s a f e l y neglected f o r r i g i d a i r c r a f t , but not f o r wingless v e h i c l e s . 2. F l e x i b i l i t y a f f e c t s the s t a b i l i t y through the l i f t and p i t c h i n g moment and a l s o by i n t r o d u c i n g two more p o s s i b l e e q u i l i b r i u m p o i n t s . 3. The amount of work involved i n f i n d i n g a s o l u t i o n i s markedly increased by the n e c e s s i t y of s o l u t i o n of more c h a r a c t e r i s t i c equations of higher degree. 4. The s t i f f n e s s c r i t e r i o n can be used to adjust the s t i f f n e s s d i s t r i b u t i o n to one tha t can make an unstable c o n f i g u r a t i o n s t a b l e . The usefulness of the method i s tw o - f o l d . For a f l e x i b l e v e h i c l e w i t h known geometric, mass and e l a s t i c p r o p e r t i e s , the method can p r e d i c t i t s dynamic s t a b i l i t y . This f e a t u r e i s of considerable importance p a r t i c u l a r l y i n the design stage. On the other hand, i f an a i r c r a f t w i t h known geometry, t o t a l mass and c e n t r e - o f - g r a v i t y l o c a t i o n proves to be dynamically un-s t a b l e , then the a n a l y s i s provides a s t i f f n e s s c r i t e r i o n by which i t can be made s t a b l e . The a n a l y s i s i n v o l v e s a consider-able amount of computation and hence seems to be p a r t i c u l a r l y s u i t e d f o r s o l u t i o n by a d i g i t a l computer. \ ACKNOWLEDGEMENT I would l i k e to express my sin c e r e thanks and appre-c i a t i o n to Dr. V.J. Modi f o r the guidance given to me throughout the prepar a t i o n of t h i s t h e s i s . His help and i n s p i r a t i o n have proved i n v a l u a b l e . Thanks are a l s o due to the Department of Mechanical Engineering f o r f i n a n c i a l support f o r t h i s t h e s i s from funds of the Chair of Aeronautics. TABLE OF CONTENTS PART 1 PAGE I. INTRODUCTION 1 I I . ANALYTICAL FORMULATION OF THE PROBLEM . . . . 5 2 . 1 P r e l i m i n a r y Remarks . 5 2 . 2 Choice, of C o n f i g u r a t i o n ,. 5 2 .3 E s t i m a t i o n of Forces and Moments . . . . 9 2 . 3 . 1 Fuselage . . . 1 2 2 . 3 . 2 Wing and Foreplane . . . . . . . 40 2 .4 E l a s t i c Degrees of Freedom . . . . . . 47 2 . 5 Equations of Motion f o r the Complete Airplane\" . **'*. . . . . . . . . . . 52 2 .6 C o l l e c t i o n of Equations 57 I I I . SOLUTION OF THE EQUATIONS OF MOTION . . . . 63 3 .1 P r e l i m i n a r y Remarks . 63 3 .2 Method of A n a l y s i s . . . . . . . . . 65 3.3 R i g i d Body Cases 70 3.4 E l a s t i c Cases .' . . . . 79 V PART PAGE IV. NUMERICAL EXAMPLE . 96 V. CONCLUDING REMARKS 110 V I . RECOMMENDATIONS FOR FUTURE, RESEARCH . . . . 112 APPENDIX I . COLLECTION OF SHAPE CONSTANTS BIBLIOGRAPHY LIST OF FIGURES FIGURE PAGE 1. Configuration Chosen for Analysis . 7 2. Co-ordinate Systems f o r Equation of Motion . . . 13 3. Moving Co-ordinate System 14 4. Co-ordinate System f o r Disturbed Rigid Body . . 15 5. D e f i n i t i o n of oi and o(r 20 6. C y l i n d r i c a l Body Co-ordinates . 21 7. Pressure Geometry . 25 8. Positive Direction f o r Pitching Moment . . . . 30 9. Resolution of A x i a l and Normal Forces . . . . 35 10. Geometry -of Wing 42 11. Co-ordinate System Relative to Body . . . . . 54 12. Example of a S t a b i l i t y Boundary 69 13. Typical S t i f f n e s s C r i t e r i o n Plot as a Function of Assumed Mode (f)(7) and Xcj 92 14. \"Symmetric\" Bending of Fuselage . . . . . . 93 15. \"Anti-symmetric\" Bending of the Fuselage . . . 95 16. Fuselage Geometry . , 9$ 17. Wing and Foreplane Location . . . . . . . 100 LIST OF TABLES TABLE PAGE I. Numerical Values of the Parameters Used i n the I l l u s t r a t i v e E x a m p l e \u00E2\u0080\u0094 R i g i d Constants 99 I I . Numerical Values of the Parameters Used i n the I l l u s t r a t i v e E x a m p l e \u00E2\u0080\u0094 E l a s t i c Constants 107 LIST OF SYMBOLS* H c o e f f i c i e n t i n viscous c o n t r i b u t i o n to normal f o r c e = 0.49 35 ^ aspect r a t i o 58 . B mass moment of i n e r t i a of a i r c r a f t i n p i t c h about Y a x i s 80 D c o e f f i c i e n t i n viscous c o n t r i b u t i o n to normal \u00E2\u0080\u00A2 for c e - -0.0056 35 (\u00E2\u0080\u0094 c o e f f i c i e n t i n viscous c o n t r i b u t i o n t o normal f o r c e - 0.00003 35 \u00C2\u00A3o c c r o s s - f l o w drag c o e f f i c i e n t = i-^u^t, ^3 Cz g e n e r a l i z e d f o r c e c o e f f i c i e n t , 77 J \u00C2\u00B0 l f\u00C2\u00ABuasb I\u00E2\u0080\u0094 CL* l i f t c o e f f i c i e n t = T4T^ 51 C M * p i t c h i n g moment c o e f f i c i e n t = \u00E2\u0080\u00A2 ^ , , 31 C-M\u00C2\u00A3 p i t c h i n g moment c o e f f i c i e n t at zero angle of att a c k 1 r N C-^ normal for c e c o e f f i c i e n t = l 22 2. \"Too\u00C2\u00AB--' X51> C x* a x i a l f o r c e c o e f f i c i e n t = , *,.n<. 21 4 u*Sb 2 D d i f f e r e n t i a l operator w i t h respect t o non-J . dimensional time \u00E2\u0080\u00A2** 83 D c c r o s s - f l o w drag 33 E Young's Modulus 118 \u00E2\u0080\u0094 E I EI non-dimensional s t i f f n e s s = \u00E2\u0080\u0094 - \u00E2\u0080\u0094 \u00E2\u0080\u0094 ^ 119 7 wing semi-span 5$ v e r t i c a l d e f l e c t i o n of nose i n d>tx) mode x i root chord of l i f t i n g surfaces 58 \u00E2\u0080\u00A2PoO source d i s t r i b u t i o n along fuselage x a x i s 12 a c c e l e r a t i o n due to g r a v i t y 79 ^\u00E2\u0080\u00A2(Xjt) e l a s t i c d e f l e c t i o n i n p o s i t i v e y d i r e c t i o n = L% It) 4 fy(Xjt) non-dimensional e l a s t i c d e f l e c t i o n =.\u00E2\u0080\u00A2-[-_ 23 h r i g i d body displacement i n y d i r e c t i o n 2 I FT 58 6B non-dimensional mass moment of i n e r t i a = \u00E2\u0080\u0094\u00E2\u0080\u0094-\u00E2\u0080\u009483 k reduced frequency parameter = 58 ^ reference l e n g t h = l|- 58 nn mass 80 fly component of u n i t normal v e c t o r to body surface i n y d i r e c t i o n 71 c^? l o c a l s t a t i c pressure 5 ^ 7 \u00C2\u00B0 P - Ro 5 ^. rat e of p i t c h - ^ 8 t time 1 i non-dimensional time = -rv #3 X = = non-dimensional l e n g t h co-ordinate, o r i g i n at vertex 58 X non-dimensional fuselage co-ordinate = \u00E2\u0080\u0094 25 non-dimensional mean foreplane l o c a t i o n = ^ 64 Xur non-dimensional mean wing l o c a t i o n = 59 \u00C2\u00A5(pL0 non-dimensional p o s i t i o n where 4> (X) ~\u00C2\u00B0 j -r-x i i gfc \_ 83 distance aft of l i f t i n g surface vertex in . fraction of c 61 kinematic viscosity 33 p generalized co-ordinate 23 a i r density 5 crCx) doublet distribution along fuselage length 12 0 normalized natural mode shape of fuselage 23 perturbation in velocity potential 1 CO natural frequency 58 x i i i Co-ordinate Systems ^ X y ^ ^ X s s t a t i o n a r y co-ordinates w i t h o r i g i n at fuselage nose 1 X*) y-i.) ZT-L, moving co-ordinates w i t h o r i g i n at fuselage nose 1 y^a, t , moving co-ordinates w i t h r i g i d displacement h i n y, d i r e c t i o n , o r i g i n at fuselage nose 2 X j V ^ ^ t moving co-ordinates w i t h r i g i d and e l a s t i c displacements h and g, o r i g i n at fuselage nose 4 X ; r , 0 c y l i n d r i c a l body co-ordinates, o r i g i n at fuselage nose 10 X body co-ordinate w i t h o r i g i n at centre of g r a v i t y , p o s i t i v e i n the d i r e c t i o n of motion 79 V body co-ordinate normal to the plane of symmetry w i t h o r i g i n at centre of g r a v i t y , p o s i t i v e toward starboard side 79 2? body co-ordinate normal to X i n the plane of sym-metry w i t h o r i g i n at centre of g r a v i t y , p o s i t i v e down 79 Su b s c r i p t s b fuselage base 21 c~ c r o s s - f l o w 33 c<^. centre of g r a v i t y 28 x i v \u00C2\u00A3 equivalent 58 EL e l a s t i c 25 .. -f f oreplane 67 F fuselage 17 t Ltln mode 23 j j t h mode 71 m applicable to general c o e f f i c i e n t s 102 n applicable to general c o e f f i c i e n t s 102 P potential 74 r r i g i d 8 s.s. steady state 39 \u00C2\u00A3j\u00C2\u00BB j ^ singular (equilibrium) point 104 T t o t a l 86 V viscous 36 X,\"'\u00C2\u00AB,t indicate p a r t i a l derivatives 1 UT wing 58 oO free stream 1 Superscripts \u00E2\u0080\u00A2 (dot) d i f f e r e n t i a t i o n with respect to time 6 '(prime) d i f f e r e n t i a t i o n with respect to noted argument 6 ^Mcap) small perturbations about equilibrium point 104 Shape Constants defined i n Appendix I A ; B pressure c o e f f i c i e n t 16 ^ I ~ ^ I O fuselage normal force 49 XV B , -By fuselage a x i a l f o r c e 50 fuselage l i f t f o r c e 54 0, -Dl0 fuselage p i t c h i n g moment 57 wing l i f t 65 Fr wing p i t c h i n g moment 66 Cr, foreplane l i f t 69 foreplane p i t c h i n g moment 70 g e n e r a l i z e d f o r c e 78 cZ, condensed c o e f f i c i e n t s f o r s t a b i l i t y equations 97 V -bin condensed c o e f f i c i e n t s f o r s t a b i l i t y equations 98 condensed c o e f f i c i e n t s f o r s t a b i l i t y equations 99 < s t a r denotes dependence of a,b,c on e q u i l i b r i u m values of \u00C2\u00B0 < r S j . , ^ ^ ^ y . 127 I . INTRODUCTION The c o n s i d e r a t i o n of s t r u c t u r a l f l e x i b i l i t y i n the a n a l -y s i s of the d i s t u r b e d motion of a i r c r a f t i s a r e l a t i v e l y r e -cent development that has been made necessary by the new generation of aerodynamic shapes s u i t a b l e f o r high speed f l i g h t . The i n t e r p r e t a t i o n of high speed f l i g h t i s the super-sonic speed range up t o three or f o u r times the speed of sound. The shape of t h i s c l a s s of a i r c r a f t i s c h a r a c t e r i z e d by a long, slender fuselage and t h i n , small aspect r a t i o wings. B a l l i s t i c m i s s i l e s , some i n t e r c e p t o r a i r c r a f t and projected designs f o r supersonic a i r l i n e r s a l l f i t t h i s p a t t e r n . The trend i s i n -e v i t a b l y toward n e e d l e - l i k e fuselages and very t h i n wings i n order to reduce the drag and hence power and f u e l requirements f o r sustained f l i g h t at supersonic speeds. F l e x i b i l i t y then becomes a s i g n i f i c a n t problem. The p r o p o r t i o n i n g of s t r u c t u r a l weight, stre n g t h and s t i f f n e s s f o r optimum use of c o n s t r u c t i o n m a t e r i a l s r e q u i r e s some c r i t e r i o n which the designer can use as a standard. The problem of the optimum design i s d e f i n i t e l y important, because every unnecessary pound of s t r u c t u r a l weight causes a d i s p r o p o r t i o n a t e increase i n t o t a l a i r c r a f t weight. The e x t r a f u e l and engine c a p a c i t y r e q u i r e d to keep the heavier a i r c r a f t airborne p e n a l i z e the performance unduly. I t must a l s o be noted that f l e x i b i l i t y of the s t r u c t u r e can a l t e r the aerodynamic fo r c e s and moments on the a i r c r a f t causing quite d i f f e r e n t c o n t r o l and s t a b i l i t y c h a r a c t e r i s t i c s . 2 In the past, i t was s u f f i c i e n t t o consider f o r s t a b i l i t y -a n a l y s i s that the a i r c r a f t be r i g i d , because the aerodynamic shapes most s u i t e d t o low speed f l i g h t were so s t i f f t h a t t h e i r d e f l e c t i o n was too small t o be of importance. The s t a b i l i t y a n a l y s i s of such a i r c r a f t was based on a l i n e a r i z e d set of equations of motion, and on the assumption that the aerody-namic f o r c e s and moments were l i n e a r f u n c t i o n s of the a i r c r a f t p o s i t i o n v a r i a b l e s . Only s m a l l p e r t u r b a t i o n s were considered, so t h a t large excursions from the reference f l i g h t c o n d i t i o n could not be allowed, and the s t a b i l i t y c h a r a c t e r i s t i c s of the a i r c r a f t at l a r g e displacements were unknown. The n a t u r a l frequencies of v i b r a t i o n of such r i g i d s t r u c t u r e s were s u f f i -c i e n t l y h i g h , as compared w i t h the frequencies of motion of the e n t i r e a i r f r a m e , so th a t steady aerodynamic theory was qu i t e adequate i n d e s c r i b i n g the d i s t u r b e d motion. For the shapes under c o n s i d e r a t i o n f o r modern a i r c r a f t , i t i s not p o s s i b l e to neglect the f l e x i b i l i t y because of the lower r i g i d i t y inherent i n such shapes. A l s o , the reduced s t i f f n e s s of the s t r u c t u r e causes the n a t u r a l frequencies t o become s u f f i c i e n t l y low that s i g n i f i c a n t coupling can occur between the a i r c r a f t and e l a s t i c degrees of freedom. Unsteady aerodynamic theory must be used t o p r e d i c t the f o r c e s and moments, thus i n c r e a s i n g the complexity of the s t a b i l i t y equa-t i o n s . The complications a r i s i n g from the c o n s i d e r a t i o n of many degrees of freedom l i e i n the s o l u t i o n of a l a r g e number of simultaneous n o n - l i n e a r d i f f e r e n t i a l equations as, i n 3 theory, there can be an i n f i n i t e number of e l a s t i c degrees of freedom. I f l a r g e disturbances are to be allowed, then these equa-t i o n s become n o n - l i n e a r through both the i n e r t i a and the aero-dynamic f o r c e s . The l a t t e r source of n o n - l i n e a r i t y i s accentu-ated by the long slender fuselage r e q u i r e d f o r the supersonic speed range. The exact s o l u t i o n of a l a r g e number of s i m u l t a -neous n o n - l i n e a r d i f f e r e n t i a l equations d e s c r i b i n g the general motion of a completely f l e x i b l e a i r c r a f t w i l l not i n general be obtaina b l e , hence some r e s t r i c t i o n s and approximations must be made i n order to i n i t i a t e the s o l u t i o n of such a complex prob-lem. A s t a r t i n g p o i n t i n the s t a b i l i t y a n a l y s i s of a f l e x i b l e a i r c r a f t i s d e s i r e d which i s reasonably g e n e r a l , not too com-p l i c a t e d , and yet i s a p p l i c a b l e to a r e a l i s t i c s i t u a t i o n . Fur-the r refinements can be made at a l a t e r date a f t e r the e s s e n t i a l f e a t u r e s of the problem have been w e l l understood. R e s t r i c t i o n s on the number of degrees of freedom, amount of n o n - l i n e a r i t y and s i z e of amplitudes during d i s t u r b e d motion can be made so as to reduce the problem to a more amenable form f o r a n a l y s i s without unduly a f f e c t i n g i t s p h y s i c a l r e p r e s e n t a t i o n or the major e f f e c t s of f l e x i b i l i t y . Very l i t t l e published work e x i s t s on the e f f e c t s of s t r u c t u r a l f l e x i b i l i t y on dynamic s t a b i l i t y . A comprehensive statement of the problem i s made i n [ 1 ] , but no experimental data have been uncovered that would be re l e v a n t to the type of body discussed i n t h i s t h e s i s . The work contained h e r e i n i s the f i r s t known attempt to include the f l e x i b i l i t y v a r i a b l e s i n the a n a l y s i s r i g h t from the equation f o r the l o c a l s t a t i c pres-sure. General p r a c t i c e i n the pastwhen t h i s problem was d i s -cussed has been to make allowances f o r f l e x i b i l i t y by the a d d i t i o n of quasi-steady e l a s t i c terms to the r i g i d body s t a -b i l i t y equations..; Thus the importance of t h i s work l i e s i n the rigorous s t a b i l i t y a n a l y s i s of a s i m p l i f i e d model of a g e n e r a l l y f l e x i b l e a i r c r a f t and i n the r e v e l a t i o n of the major e f f e c t s of f l e x i b i l i t y . Furthermore, the f o r m u l a t i o n of a s t i f f n e s s c r i -t e r i o n as i s c a r r i e d out i n I I I should prove extremely h e l p f u l i n the design of modern a i r c r a f t or m i s s i l e s . I I . ANALYTICAL FORMULATION OF THE PROBLEM 2.1 P r e l i m i n a r y Remarks This s e c t i o n deals w i t h the choice of c o n f i g u r a t i o n to be s t u d i e d and w i t h the d e t a i l e d development of the equations to be solved i n I I I . A rigorous point of view has been adopted and when assumptions are made they are c l e a r l y s t a t e d . A cer-t a i n n o t a t i o n problem e x i s t s because of the l a r g e number of v a r i a b l e s considered. I t i s necessary to denote many of the geometric parameters i n a general form under a lumped constant which can be determined when the a c t u a l body data are s u p p l i e d . 2 \u00C2\u00BB 2 Choice of C o n f i g u r a t i o n The usefulness of the r e s u l t s of a t h e o r e t i c a l i n v e s t i -g a t i o n depends to a l a r g e degree on the choice of p h y s i c a l s i t u a t i o n that i s s t u d i e d . The c r i t e r i o n of ease of a n a l y s i s i s of l i m i t e d v a l i d i t y when the r e s u l t s must be a p p l i e d to p r a c t i c a l s i t u a t i o n s . A l s o , the s i t u a t i o n s t u d i e d should be r e p r e s e n t a t i v e of a whole c l a s s of problems, and not so spe-c i a l i z e d as t o be of l i t t l e general use. However, i f the problem i n hand i s of such a nature that the exact s o l u t i o n of the equations i s too l a b o r i o u s , d i f f i c u l t , or even i m p o s s i b l e , then i t may be necessary to represent the a c t u a l p h y s i c a l s i t u a t i o n by a s i m p l i f i e d model, thus s a c r i f i c i n g accuracy to an extent i n order to o b t a i n at l e a s t a g u i d i n g knowledge of the b a s i c p h y s i c a l processes. The above comments were taken i n t o c o n s i d e r a t i o n when the shape shown i n Figure I was chosen f o r a n a l y s i s . The long slender body i s r e p r e s e n t a t i v e of e i t h e r a supersonic a i r l i n e r or of a long-range m i s s i l e ; the a d d i t i o n of the wing and foreplane somewhat s p e c i a l i z e s the c o n f i g u r a t i o n to the supersonic a i r l i n e r case, but omission of the terms a p p l i c a b l e to the wing and foreplane i n the equations y i e l d s the m i s s i l e case. This i s made e a s i e r since i n t e r f e r e n c e e f f e c t s of the wing-body-foreplane combination have been neglected. The reason f o r t h i s i s that the i n t e r f e r e n c e problem has been solved t h e o r e t i c a l l y f o r only a few s p e c i a l cases, and a l s o the i n c l u s i o n of i n t e r f e r e n c e e f f e c t s would increase the complexity of the equations unduly. The arrangement of the h o r i z o n t a l s t a b i l i z i n g surface ahead of the wing (canard c o n f i g u r a t i o n ) was chosen f o r the f o l l o w i n g reasons. The requirements f o r s t a t i c s t a b i l i t y and t r i m are that 1_J3 be negative and.CM* be p o s i t i v e respec t i v e l y . The wing and body each supply a negative Ch* that must be counteracted by the l i f t of the h o r i z o n t a l s t a b i l i z e at a moment arm. For the normal c o n f i g u r a t i o n ( s t a b i l i z e r a f t of the centre of g r a v i t y ) t h i s i s achieved w i t h the s t a b i l i z e r at negative incidence producing a down-load, and v i c e versa f o r the foreplane case. Thus, the l i f t on the 7 Figure 1 C o n f i g u r a t i o n Chosen f o r A n a l y s i s 8 h o r i z o n t a l s t a b i l i z e r necessary f o r t r i m i s p o s i t i v e f o r the canard c o n f i g u r a t i o n . A l s o , i n the lan d i n g phase, the exten-s i o n of f l a p s produces a lar g e nose down increment i n p i t c h -in g moment which n e c e s s i t a t e s an increase i n foreplane l i f t . This allows the a i r c r a f t t o a t t a i n more l i f t and approach touch-down at a lower speed; thus the airborne p o r t i o n of the l a n d i n g distance i s considerably reduced. For super-sonic a i r l i n e r s designed f o r low drag, t h i s c o n t r i b u t e s to a s u b s t a n t i a l improvement i n performance. The fuselage i s considered to be f l e x i b l e , and the wing and foreplane are both r i g i d . Thus e l a s t i c deforma-t i o n s are confined t o the f u s e l a g e , but the e f f e c t s of e l a s t i c d e f l e c t i o n are f e l t through a changed fuselage l i f t and p i t c h i n g moment, as we l l , as through foreplane and wing incidences which depend on the slope of the fuselage d e f l e c -t i o n curve. I t was considered to be beyond the scope of t h i s t h e s i s to account f o r f l e x i b i l i t y of the l i f t i n g s u r f a c e s . The wing of a high speed v e h i c l e n e c e s s a r i l y has a small aspect r a t i o thus g i v i n g high s t i f f n e s s and consequently high n a t u r a l f r e q u e n c i e s . In most c o n f i g u r a t i o n s of prac-t i c a l importance, the fundamental frequency of the l i f t i n g surfaces i s expected to be considerably higher than t h a t of the fuselage i n bending. Furthermore, i t i s the wing t o r s i o n a l mode, r a t h e r than bending mode tha t i s l i k e l y t o 9 be more e f f e c t i v e i n a l t e r i n g the pressure d i s t r i b u t i o n due to f l e x i b i l i t y . Of course, t h i s c o n t r i b u t i o n would be i n a d d i t i o n to that provided by the fuselage f l e x i b i l i t y and can be looked upon as a second order p e r t u r b a t i o n e f f e c t which, i t i s f e l t , can be neglected without a f f e c t i n g the accuracy unduly. As mentioned e a r l i e r , the theme of the present i n v e s t i g a t i o n i s to o b t a i n a fundamental under-standing of the extremely complex problem which could only be accomplished by c o n s i d e r i n g a s i m p l i f i e d yet r e a l i s t i c model of the a c t u a l p h y s i c a l problem. I n c l u s i o n of i t s various minor, yet c o m p l i c a t i n g , f e a t u r e s can be made as a refinement to the a n a l y s i s as developed here. Thus the shape of the a i r c r a f t f o r study i s a com-promise between usefulness and ease of a n a l y s i s . The f l e x i b i l i t y i s confined to one major part of the a i r c r a f t , but the problem has to be s t a r t e d somewhere. There are both c i v i l and m i l i t a r y a p p l i c a t i o n s f o r such a shape. 2 \u00C2\u00AB3 E s t i m a t i o n of Forces and Moments The f o r c e and moment c o n t r i b u t i o n s of the f u s e l a g e , wing and foreplane are c a l c u l a t e d s e p a r a t e l y , and the sum i s taken to be the r e s u l t a n t f o r the body as a whole. No i n t e r f e r e n c e e f f e c t s are considered but unsteady f l o w i s assumed. This i s necessary [2] when co n s i d e r i n g aero-e l a s t i c e f f e c t s even at low frequencies of v i b r a t i o n since unsteady flow e f f e c t s manifest themselves i n the damping. The l i f t and moment c o n t r i b u t i o n s from the fuselage are considered to come from two sources, p o t e n t i a l and viscous f l o w , which are added to give the t o t a l f o r the f u s e l a g e . The viscous f l o w l i f t and p i t c h i n g moment a r i s e from the separation of the flow from the f u s e l a g e . This i s essen-t i a l l y a steady f l o w phenomenon, and i t has been suggested [ 3 ] , w i t h some experimental v e r i f i c a t i o n , t h a t the unsteady flow delays t h i s s e p a r a t i o n causing a reduced viscous l i f t and moment. No q u a n t i t a t i v e r e s u l t s of t h i s delay have been uncovered. Slender-body theory [4] was chosen f o r the c a l c u -l a t i o n of the p o t e n t i a l f l o w - f o r c e s . I t i s agreed t h a t t h i s theory i s not the most accurate (Van Dyke's 2nd order theory i s much more accurate [ 5 ] ) , but i t i s simple enough to use i n the s t a b i l i t y a n a l y s i s without the excessive complications that would a r i s e from the i n c l u s i o n of e l a s -t i c e f f e c t s i n Van Dyke's method. Another s i m p l i f i c a t i o n i s that l i f t and moment are independent of Mach Number, This puts a r e s t r i c t i o n on the k i n d of body to which the theory can be a p p l i e d , but modern aerodynamic shapes f i t t h i s r e s t r i c t i o n without d i f f i c u l t y . Long, slender bodies are very common. This t h e s i s i s concerned w i t h o b t a i n i n g a good estimate of the e f f e c t s of fuselage f l e x i b i l i t y on the s t a b i l i t y of a c e r t a i n shape of body, and not w i t h the most accurate e s t i m a t i o n of the l i f t and p i t c h i n g moment. Some controversy e x i s t s among experts about t h i s question of accuracy ([5] and [6]), but slender-body theory i s considered to be s u f f i c i e n t l y accurate f o r the purposes of t h i s t h e s i s . I t would be i n c o n s i s t e n t i n the d i s c u s s i o n of wing l i f t t o use a more accurate p o t e n t i a l f l o w theory f o r l i f t and p i t c h i n g moment. Therefore, unsteady sle n d e r -body theory was chosen again. There i s no viscous l i f t t o consider i n t h i s case. The l i f t of the wing i s a l i n e a r f u n c t i o n of angle of a t t a c k f o r a much l a r g e r range of incidence than f o r the f u s e l a g e . The p h y s i c a l character of the f l o w i s q u i t e d i f f e r e n t ; as the wing aspect r a t i o tends to zero, the e f f e c t s of v i s c o s i t y become more s i g n i f i c a n t because the f l o w separates at the wing l e a d i n g edge causing the s i t u a t i o n analogous to the fuselage case. In the f o l l o w i n g pages a d e t a i l e d development of the l i f t and p i t c h i n g moment c o n t r i b u t i o n s from the fuse-l a g e , wing and foreplane i s given. The e l a s t i c equations of motion are then discussed f o l l o w e d by a statement of the r i g i d body equations of motion w i t h s p e c i a l i z a t i o n to the present problem. F i n a l l y , a l l the equations are c o l l e c t e d , combined and placed i n the most convenient form f o r s t a b i l i t y study. 2 . 3 . 1 Fuselage The aerodynamic f o r c e s and moment of ,the fuselage due to p o t e n t i a l flow are obtained from a small p e r t u r -ba t i o n approximation to the v e l o c i t y p o t e n t i a l . This p e r t u r b a t i o n must s a t i s f y the wave equation [4] with reference to the s t a t i o n a r y co-ordinate system. This equation can be transformed i n t o the case of a uniform steady stream by a of the references axes. f a m i l i a r aerodynamic steady t r a n s l a t i o n \u00E2\u0080\u00A2..Let the co-ordinate transformation between the moving system ( x 2 ,ye. ,z 2 , t z ) . a n d the s t a t i o n a r y system ( x 3 , y 3 ,z 3 ,t3) be, - X 3 = X t r Utr t Then the wave equation becomes where ^ Z i n d i c a t e s the operator ^7*\"is to be a p p l i e d t o the {xz ,y2 ,z t) system. Note that t h i s equation reduces to the f a m i l i a r P r a n d t l - G l a u e r t equation < \Jtz > Figure 2 Co-ordinate Systems f o r Equation of Motion when the f l o w i s ' steady as viewed from the moving co-ordinate system.... J \u00E2\u0080\u00A2 ....: ... The wave equation can be r e w r i t t e n i n the form Considerable s i m p l i f i c a t i o n , r e s u l t s when the source of disturbance p o t e n t i a l i s a slender body. For slender bodies performing slow motions of small amplitude i t i s found [6] t h a t the cross flow p e r t u r b a t i o n s are much l a r g e r than the a x i a l disturbances such t h a t r y ^ i . > \u00E2\u0080\u00A2 ^ t j . and T ^ I X ^ can be neglected thus g i v i n g the c l a s -s i c a l Laplace's Equation as the equation of motion. The speed of sound and Mach Number are thus e l i m i n a t e d from the problem. I t i s con-venient to work w i t h co-ordinates f i x e d to the body as the statement of the c o n d i t i o n of no flow of a i r through the body i s s i m p l i f i e d . Figure 3 Moving Co-ordinate System The s o l u t i o n of Laplace's Equation leads to the pressure r e l a t i o n I t can be shown [4] that t h i s form of the pressure c o e f f i c i e n t should be used i n preference to A-R =1 - 2 \u00E2\u0080\u0094 2. (fey as the term V i s shown to be comparable i n magnitude w i t h 2 ffi*- near the surface and cannot be neglected. U . Next the e x p r e s s i o n , f o r the l o c a l s t a t i c pressure on the surface of a plunging, p i t c h i n g , s l e n d e r , e l a s t i c body i s developed i n considerable d e t a i l . , Let the co-ordinate system (x^^y^ fzz) f i x e d to the r i g i d body w i t h o r i g i n at the nose be, d i s p l a c e d by a s m a l l distance h along the y 2 a x i s and r o t a t e d through a s m a l l angle Q about the z z a x i s so as to occupy the new p o s i t i o n ,y, ,z/ ) as shown i n Figure 4. 02 Figure 4 Co-ordinate System f o r Disturbed R i g i d Body 16 To a second order InQ , the r e s u l t s of the t r a n s -formation are: X-2_= X, cose 4-y, sine y 2 . - y; c ^ s ^ - x , s/>> & -t- h (3) Rearranging (3) i n t o a more convenient form g i v e s : X\", = X 2 \u00C2\u00A3 c s & - y t s / n & y , = y x c o s e 4- X v S i n <9 - in t , = t z Now l e t the system x, ,y, ,z, , undergo a small e l a s t i c d e f l e c t i o n g ( x , t ) , and l e t the new c o n f i g u r a -t i o n of the system be denoted by (x,y,z,t) then / = X j . c o s B -yzs/n 0 y- y^co^e+K^)nG-h-gs Using equations (5) the pressure r e l a t i o n (2) can be transformed i n t o the new co-ordinate system as described below. The disturbance p o t e n t i a l ^ i s a f u n c t i o n of the s p a t i a l co-ordinates x 2 , y 2 , z 2 and time t 2 . In t u r n x 2 , y 2 ,z 2 ,t are f u n c t i o n s of x,y,z,t by equations (5). The co-ordinates x,y,z are a l s o f u n c t i o n s of t , hence (4) (5) where ^ - C o s \u00C2\u00A9 33 - O 3 x i T h e r e f o r e , f*^* \?%+\*y{e-p') S i m i l a r l y , f y , = i j P = ^ P^Y 4 - ^ ^ + where 3y' = c o s 0 j?x = - s /h e A l S O , % s ^ 4 - ^ < L * 4 .^ j f ^ ^ = / 5 ^ * = 0 . sL/ = I t i s necessary when c a l c u l a t i n g t o account f o r changes i n if from the time v a r i a t i o n of ( x , y , z ) . For t h i s i t i s necessary t o have x z , y z i n terms of x,y which can be obtained from (5) as: = X c o s & +ys/n & 18 Now, _ *}\u00C2\u00A32p +^p^x +.a)& \u00E2\u0080\u00A2hX-L(cose)d -f)^) -h cose(x cose -r-y s/h<9)j-( ti+p) = e\ x- sih&( h+^)j - (ti -hp) 2? = O S u b s t i t u t i n g these equations i n t o the expression f o r ^t-L above y i e l d s : Here, V * ^ s s m a l l . Therefore, to second order i n T* Q, and t h e i r d e r i v a t i v e s , 19 The t h i r d term, _L_ (i^^ +\u00E2\u0080\u00A2 f ^ ) , i n the pressure r e l a -t i o n i s s t i l l t o be m o d i f i e d . Vx^ and 7 ^ were obtained b e f o r e . F i n a l l y the s u b s t i t u t i o n of , V^-z , i n t o (2) y i e l d s the pressure r e l a t i o n i n the moving c o - o r d i -nate system f i x e d to the e l a s t i c body as: (7) The q u a n t i t i e s h and g r e p r e s e n t s m a l l r i g i d and e l a s t i c displacements of the body i n the p o s i t i v e y d i r e c t i o n r e s p e c t i v e l y . The v e c t o r sum of - ( h + g) and U i s the v e l o c i t y of the f r e e stream r e l a t i v e to the x a x i s of the body. T h i s r e l a t i v e v e l o c i t y i s a p p r o x i -m a t e l y ^ . Since h and g are s m a l l , the angle between the t r u e r e l a t i v e v e l o c i t y v e c t o r and U i s - i ( ^ + ^ J . T h i s angle of a t t a c k i s d e f i n e d as the angle between the x a x i s of the body and the r e l a t i v e v e l o c i t y and i s approximately equal t o oi = e - & - i _ 20 i . e . 0 can be considered as a geometric angle of attack and - Q ( & + $ ) a s an induced angle of a t tack. This r e l a t i o n -ship i s shown i n Figure 5. (8) Figure 5 D e f i n i t i o n ofo) as i t s i m p l i f i e s the s p e c i f i -c a t i o n of boundary conditions and subsequent determination of the v e l o c i t y p o t e n t i a l . The p o l a r co-ordinates are defined i n Figure 6 . Figure 6 C y l i n d r i c a l Body Co-ordinates 22 Transformation between the co-ordinate system i s defined by the r e l a t i o n s x = x y = rcos\u00C2\u00A9 z = rsin\u00C2\u00A9 Therefore , = ^ +dtf\u00C2\u00A3v +ZJ> 2jb - & co% 9 f The boundary c o n d i t i o n f o r a slender body of r e v o l u t i o n i n c y l i n d r i c a l body co-ordinates x, r,@ i s ( f t ) p B R = UdB -[U&r-f'-fo) +fx]cos0 (12) This boundary c o n d i t i o n i s equi v a l e n t to the statement that no fl o w through the body surface i s pos-s i b l e , i . e . , the flow i s tangent to the body s u r f a c e . 23 The equation for the disturbance p o t e n t i a l ^ 1 i s f- fo +QQSe[u(\u00C2\u00ABr f'-fa) 1-fxJ (13) The complete description of equation (13) i s quite beyond the scope of t h i s t h e s i s , but some i n -sight into i t s meaning should be given. The complete solution f or the disturbance poten-t i a l applicable to a pointed, smooth, slender body of revolution i s [6]: LPfx r , 0 ) ^ - ^ ^ ttcx>)Ccsh x-x, c/X| -co$0~X \ fx -x , )^y ,Wxi o For small values of ( i r , the equation becomes Lp(X. r , & ) = - \u00C2\u00B1 ^ f#x.)In 2.(y-yQdv, - cos\u00C2\u00AE Q-(X) where \"frx) ^ US'60 The term i s a source d i s t r i b u t i o n along the x ax i s , and 0~(x) i s a doublet d i s t r i b u t i o n also along the x a x i s . Note the role of(3r. As the Mach Number increases, a more slender body i s required to f i t the approximations of the theory. Evaluating the in t e g r a l i n the above equation gives the f i n a l expression f o r as given i n equation (13). Referring to the o r i g i n a l equation of motion (1), 24 in the region close to the body, i t is found [4] that ^tt \u00C2\u00BB $tt \u00C2\u00BB a n c * ^2* a r e s m a H a n d hence can be neglected, y i e l d i n g the equation of motion as It i s now possible by use of (13) to obtain the pressure equation in terms of the geometric and position variables. ^ -' ^ Q S ^ ^ U f r r - f - ^ + f X ^ (14) Also f o r r=R , S - T T R * ' , S =ZTTRCTR ; therefore, the pressure r e l a t i o n at the body surface becomes: -zj? cos a [*r - Cf^kjj (is) i . e . A_\u00C2\u00A3 =- Plcos0 + B(l-4-<>ir>~i\u00C2\u00AE) (16)* Equation (16) can be integrated to obtain the a x i a l and normal forces. Figure 7 shows the geometrical * Collective constants A, B, C, , C t, . . . , D, , D 2, . . . are defined in Appendix I. 25 Figure 7 Aerodynamic Forces and Moment 26 nature of the i n t e g r a t i o n s . X P = f o r c e i n p o s i t i v e x d i r e c t i o n Lo I J@-.o d y J N p = normal f o r c e i n p o s i t i v e y d i r e c t i o n . rX-L C r.<\u00C2\u00A3)=-2.TT ^ T - 0 , , \u00E2\u0080\u009E '6> = 0 Equations (17) and (18) contain s e v e r a l i n t e -g r a l s of the form f ^(R)^ , R =RriO which are f u n c t i o n s of the body geometry only and are most e a s i l y solved s e p a r a t e l y . The i n t e g r a t i o n i s aided by n o t i n g that at the nose, R = 0 X - 0 while at the a f t extremity of the body R = Rb X = L Note a l s o = T T R 2 6 0 (19) 27 cU 1 Io Jo 2 2TT 2.TT (20) \"TT TT Furthermore, (17) and (18) can be separated i n t o \" r i g i d \" and \" e l a s t i c \" p a r t s . The i n t e g r a t i o n of (17) and ( 1 8 ) can be performed w i t h the a i d of (19) and (20). The t o t a l f o r c e s are put i n c o e f f i c i e n t form by the use of the d e f i n i t i o n s r.* - * F . r . , = U . A / f .. . The f i n a l r e s u l t s are: '*rtC(?'w**gd*] (2i) + r ^ / ^ ^ (22) Equations (21) and (22) s t i l l i n v o l v e i n t e g r a l s r e p r e s e n t i n g e l a s t i c c o n t r i b u t i o n s which can be eval u -ated once the form of g ( x ; t ) i s known. Let the e l a s t i c 28 d e f l e c t i o n g ( x , t ) be denoted by (23) where fyt?) i s the normal ized n a t u r a l s mode, and i s the CtJl g e n e r a l i z e d c o - o r d i n a t e . Note t h a t S/^J d e s c r i b e s the amount of the t o t a l d e f l e c t i o n g(5c^t) c o n t r i b u t e d by the n a t u r a l mode. W i t h t h i s , (T/v/g. can be w r i t t e n a s : C ^ B L F - ( \u00C2\u00A3 s r / J * j \"'\"J00) The c o e f f i c i e n t s a , b , c , depend o n l y on the fuse lage geometry and the d e f l e c t i o n mode shape which i n t u r n depends on the mass and s t i f f n e s s d i s t r i b u t i o n s of the s t r u c t u r e . Thus CHE. i s obta ined i n terms of the normal co-ord ina te s whose c o e f f i c i e n t s can be determined when the body i s g i v e n . The d e t e r m i n a t i o n of the can be a l a b o r i o u s problem, but many approximate methods are a v a i l a b l e to c a l c u l a t e them [ 3 ] . S u b s t i t u t i o n of (23) i n t o (21) and (22) y i e l d s : 29 The e l a s t i c deformations are considered t o be s m a l l j so second and higher order terms i n and i t s d e r i v a t i v e s can be neglected. Therefore, the sum of the r i g i d and e l a s t i c c o n t r i b u t i o n s g i v i n g a x i a l and normal f o r c e s become: The p i t c h i n g moment caused by the p o t e n t i a l flow around the fuselage can be obtained i n a s i m i l a r manner. For o r i g i n at the nose and c e n t r e - o f - g r a v i t y l o c a t i o n XCJ , the p i t c h i n g moment about the centre of g r a v i t y i s : M ^ ^ L J ^ d ^ Q ^ ^ d V (2d) 30 M 1 ->x Figure 8 P o s i t i v e D i r e c t i o n f o r P i t c h i n g Moment J p L J \u00C2\u00B0 ._ Moment of Volume (30) the p i t c h i n g moment c o e f f i c i e n t about the centre of g r a v i t y due t o fuselage f o r c e s can be w r i t t e n as: h < S F USbJ Out u( U^b) ^ > J ^ ? \u00C2\u00A3 ^ -hx^C^ (31) < S u b s t i t u t i n g f r o m (27) and re-arranging terms i n the above equation y i e l d : Equations (26), (27) and (32) complete the poten-t i a l f low c o n t r i b u t i o n s to the t o t a l a x i a l and normal f o r c e s and p i t c h i n g moment of the fuselage. For sm a l l angles of a t t a c k these equations are s u i t a b l y accurate to describe the l i f t and p i t c h i n g moment; i n t h i s case \u00C2\u00A3 x * i s n e g l i g i b l y small and ^-NF \u00E2\u0080\u00A2 equals Ci* . However, f o r a steady motion, i f the angle of atta c k r i s e s much beyond two degrees, s i g n i f i c a n t e r r o r s a r i s e because of the assumption that the e f f e c t s of v i s c o s i t y are n e g l i g i b l e . K e l l y [7] has presented a new method based on tha t of A l l e n [ 8 ] , but w i t h refinementSjto account f o r viscous e f f e c t s . The bas i c p r i n c i p l e upon which the method i s based i s tha t the normal f o r c e of a body can be evaluated by adding a \"viscous f o r c e \" t o the \" p o t e n t i a l f o r c e \" . 32 The viscous f o r c e a r i s e s from the s e p a r a t i o n of the c r o s s - f l o w from the body; i t i s analagous t o the drag of a h i g h l y yawed c y l i n d e r . The viscous c o n t r i b u -t i o n to the c r o s s - f o r c e from a c y l i n d r i c a l element of length dx can be w r i t t e n as where T i s the body ra d i u s at the p o i n t X , Uc i s the c r o s s - f l o w v e l o c i t y = (Js/'nc<, O ^ i s the drag c o e f f i c i e n t of a c i r c u l a r c y l i n d e r at Reynold's number =2jl^r Therefore, d F = 2 r C ^ , ^ i ^ (;z5/^o< o i x ( 3 4 ) where tYJ equals r e d u c t i o n i n C ( f r o m two-dimensional data) due t o the f i n i t e l e n g t h of the c y l i n d e r . K e l l y ' s contribution t o the theory i s that of i n t e r p r e t i n g the q u a n t i t i e s r f j andCo^ . According to K e l l y , A l l e n was aware th a t \u00C2\u00A3T D^ should not be the steady-state drag co-e f f i c i e n t which was used, but should be r e l a t e d to the t r a n s i e n t e f f e c t found by Schwabe [ 9 ] . Schwabe measured the drag c o e f f i c i e n t f o r a c i r c u l a r c y l i n d e r moving crosswise i n a f l u i d when s t a r t e d i m p u l s i v e l y from r e s t . K e l l y approximated the experimental curve by P)-*($)l5no( +B*ffltanz*+C*(xytGr)s'c< ( 3 5 ) stopping at the f i f t h power because of a change i n flow c o n d i t i o n s at l a r g e angles. 33 From (34), the normal f o r c e and pitching\"moment due to v i s c o s i t y are the above equations can be w r i t t e n as: K e l l y suggests that/?/* 1/, and tha t a boundary l a y e r cor-r e c t i o n should be a p p l i e d . The l a t t e r has been neg-l e c t e d f o r the sake of s i m p l i c i t y . However, A l l e n ' s assumption t h a t the c r o s s - f l o w depends only on the cro s s - f l o w Reynold's Number and not at a l l on the a x i a l f l o w leads to s i g n i f i c a n t e r r o r s . ( I f a cr o s s - f l o w encounters a tu r b u l e n t a x i a l boundary l a y e r the proper cro s s - f l o w drag c o e f f i c i e n t should be the one a p p r o p r i -ate to t u r b u l e n t f l o w . The same a p p l i e s f o r a laminar a x i a l boundary l a y e r . ) I f the a x i a l boundary l a y e r i s 34 laminar, \u00C2\u00A3j> = /\u00E2\u0080\u009E2 , and i f i t i s t u r b u l e n t , ^ T V - -03S~. S.s, -.. -S.S. The c o e f f i c i e n t s A*^ B*and ( f a r e 0.49, -O.OO56 and 0.00003 r e s p e c t i v e l y . Note that t h i s viscous f o r c e i s steady, as no time d e r i v a t i v e s of angle of a t t a c k appear. U n t i l now, the angle of attackc< i n the equations has not been made p r e c i s e ; the i n c l u s i o n of e l a s t i c d e f l e c t i o n s makes S u b s t i t u t i o n of (38) and (40) i n t o (36) and (37) and n o t i n g (39) y i e l d s : For angles of a t t a c k up to ten degrees, only the f i r s t term, the term i n o^ .\"3, i s of importance [7] and the terms i n o( and o< can be neglected. With t h i s approximation the viscous c o n t r i b u t i o n s become: (43) 35 (44) This approximation i s v a l i d f o r \u00C2\u00B1 ^ n<5T I f t h i s l i m i t i s exceeded, the complete polynomial must be used to obtain adequate p r e c i s i o n . In summary, equation (26) gives the t o t a l a x i a l f o r c e , equations (27) together w i t h (43) give the t o t a l normal f o r c e , and equations (32) plus (44) give the t o t a l p i t c h i n g moment about the centre of g r a v i t y . I t only remains to f i n d the l i f t of the fuselage from the normal and the a x i a l f o r c e s . (45) Figure 9 R e s o l u t i o n of A x i a l and Normal Forces Representing cosc7 + SL^L %'L^jf^%-)7R0c/xJ (26) This can now be rewritten i n a s i m p l i f i e d form as: + *<.teXH)+B>(rb)* ,50)* The combination of (49), (50) and (46) results i n : * Shape constants A, to A,0 , B, to B y are defined in Appendix I. 38 i . e . Q y v / W ^ r ^ ^ \"^^fe;i^-^ 7^r^^ (52) Combining the c o e f f i c i e n t s of the same v a r i a b l e gives C ^ . ft, \u00C2\u00ABr ^4ArL -B, -R e - d e f i n i t i o n of the above \"shape -constants\" i s d e s i r -able f o r conciseness. The r e s u l t of t h i s operation i s : f ^fb&bZt 4 / (54)* Equations (32) plus (44) y i e l d the t o t a l p i t c h i n g moment * C, to C/7 are defined i n Appendix I. 39 c o e f f i c i e n t f o r the body about the centre of g r a v i t y : - v ^ ) - * - % - j r j -*t$(Wj-*3 m o -T- J ^ . tf>5T-y^) /? o/*J (55) A f t e r c o l l e c t i n g the appropriate terms, re-arranging and d e f i n i n g \"shape constants\", equation (55) becomes: C M ; = - l \u00C2\u00AB r ( / - Zy-gj -2*rfr ( \%}-W-*J -2.3.2 Wing and Foreplane To o b t a i n a c o n s i s t e n t degree of accuracy slender-body theory i s again used to evaluate the aerodynamic for c e s a c t i n g on the wing and f o r e p l a n e . Both the wing and f o r e p l a n e , i n g e n e r a l , can have any plan form w i t h d i f f e r e n t aspect r a t i o s and areas. The lower the aspect r a t i o the b e t t e r i s the approximation from slender-body-theory. I t i s noted [1] t h a t a more s i g n i f i c a n t para-meter than geometric aspect r a t i o PR i s the q u a n t i t y P r e v i o u s l y , f o r the f u s e l a g e , (3r was assumed small to a r r i v e at an approximate simple s o l u t i o n f o r the d i s -turbance p o t e n t i a l a p p l i c a b l e t o slender bodies. Thus, i n t h i s context, the r o l e of [i r i s taken by ^fR . As the f l i g h t speed i n c r e a s e s , the PR must decrease to keep the wing s u f f i c i e n t l y s lender. I t i s p o s s i b l e t h a t a low aspect r a t i o wing may be considered slender at M = 1.5, but not at M = 3.5. The o r i g i n a l slender-body approximations a p p l i c a b l e * D( to D / c are defined i n Appendix I . to low aspect r a t i o wings were made by Jones [10] f o r the case of steady p o t e n t i a l f l o w . His r e s u l t s were extended to unsteady p o t e n t i a l f l o w by M i l e s [11]. The r e s t r i c t i o n s that must be a p p l i e d to the slender-body r e s u l t s i n unsteady f l o w are that \u00C2\u00A3 , fe\u00C2\u00A3 , ftS , hhg , be very much l e s s than u n i t y , where % = dimension l e s s wingspan = wingspan k = reduced frequency = M = Mach Number. The l i f t i n g surfaces are assumed to be r i g i d ; how-ever the incidence of both the wing and foreplane are dependent on fuselage f l e x i b i l i t y . The f o l l o w i n g develop-ment i s a p p l i c a b l e to both the wing and f o r e p l a n e . Slender wing-body combinations can be analyzed [1] by only a s l i g h t i m o d i f i c a t i o n of the method d e s c r i b e d , i . e . by r e p l a c i n g the semi-span b(x) i n combination w i t h a body of radius R(x) by an equivalent semi-span '/2 b^x-j = + Bias - RVXOT Z x' being a non-dimensional l e n g t h . The treatment of the wing-body combination accounts f o r i n t e r f e r e n c e e f f e c t s between the wing and body as w e l l as the i n d i v i d u a l con-t r i b u t i o n s of the wing and body. However, i n t e r f e r e n c e e f f e c t s are neglected i n t h i s work. The geometry of the wing i s shown i n Figure 10. 42 As shown by M i l e s , t'^)^TTf^U^(^^lk^b^^(^ (58) where b(x') = semi-span x' = non-dimensional len g t h based on the wing root chord = k = reduced frequency. Since the wing i s r i g i d , where X^- i s the mean wing l o c a t i o n and Cj,(*ur) i s the f u s e -lage e l a s t i c d e f l e c t i o n at the wing l o c a t i o n . 43 Moreover, assuming the planform to be t r i a n g u l a r , 6fx1) -x' tan r ; tanp^bO) S 4-where f\"7 = semi-vertex angle of the wing planform. Therefore, t ! ^ x ' J =77 f U^tJr[^^) dkx1) + ik (59) i . e . , L^= C L ^ U ) d ^ JO - I T f (JZ(xt)dl>lxt)+jk T ^ M l d x 1 Jo Noting t h a t C * = gives C * = Sur- r r / ^ ^ c K ^ i +1 L k ) ( 6 0 ) ^ Sh 2 ^ J / The p i t c h i n g moment about the p o i n t ^ ^ a f t of the v e r t e x i n percentage of wing root chord ( p o s i t i v e nose-up) i s : M r ^ [ C ^ ' ^ i A - - ^ ^ 1 (61) 'O 'C.C'W'fJJ^w^ W) iWs Ti k ^ ( J ^ ' / j o/x' (62 ) U t i l i z i n g the same s u b s t i t u t i o n s as i n the i n t e -g r a t i o n f o r the l i f t f o r c e , d e f i n i n g C M * - u r-, and noting t h a t ^ r = ^ . , equation (62).reduces t o : 44 C M * = | ^ \u00C2\u00AB r 7 T ^ R ^ t K u r [ ( ^ - | ) + ^ ( i ^ \u00C2\u00BB r ^ ) \ (63) The incidence of the foreplane i s d i f f e r e n t from that of the wing because of the fuselage f l e x i b i l i t y . The f o r e -plane of area S_p i s mounted at the mean dis t a n c e x_p a f t of the nose of the f u s e l a g e , hence the foreplane incidence i s given by h fb^'7*) 7 ^ ( 6 4 ) The term g(x^) i s the e l a s t i c d e f l e c t i o n at the mean l o c a t i o n of the foreplane planform. S u b s t i t u t i o n of the f o l l o w i n g r e l a t i o n s o< = cX 0 U # , . c u)~E-GK ~ o\ 0 L CO \u00C2\u00A3. = CCOoi * lk LM L_ S i m i l a r r e l a t i o n s being v a l i d f o r q and g i n the expres-s i o n f o r wing l i f t (60) y i e l d s : 45 i . e . , \u00E2\u0080\u00A23 (65)* S i m i l a r l y &M^=&^^M&urj\u00C2\u00B0'\u00C2\u00ABr-^j i.e.', ^ M^r = C K ^ ( 6 6 ) * The l i f t and moment c o e f f i c i e n t s f o r the f o r e p l a n e are found by s u b s t i t u t i n g ( 6 4 ) i n t o ( 6 5 ) and ( 6 6 ) , and m u l t i p l y i n g them by \u00C2\u00A7f and ^\u00C2\u00A3^\u00C2\u00A3 r e s p e c t i v e l y : Sur-^ur C L f \"It IR*tr*tu L^'^] '(f^ ( 6 7 ) * E, to E 7 , and F, to F 7 are d e f i n e d i n Appendix I. 46 These aerodynamic c o e f f i c i e n t s are based on the fuselage base area and fuselage reference l e n g t h L . Note that the wing i s operating i n the downwash f i e l d of the f o r e p l a n e . A rough check from a simple theory [12] shows c l e a r l y that f o r a small f o r e p l a n e , \u00C2\u00A7^^.0.25, the Sle-wing e f f i c i e n c y i s greater than 95% i f both are mounted on the fuselage c e n t r e - l i n e . The e f f i c i e n c y i s even greater i f the foreplane i s i n a \"high\" p o s i t i o n and the wing i n a \"low\" p o s i t i o n w i t h respect to the c e n t r e - l i n e . A l s o , the wake from the foreplane i s probably r o l l e d up by the time i t reaches the wing because of the long f u s e l a g e . There-f o r e , t h i s e f f e c t can be s a f e l y neglected since i t i s of second order of magnitude f o r a small f o r e p l a n e . As before the r e l a t i o n g - 4{^%'^) i s sub-s t i t u t e d i n t o the foreplane l i f t and moment equations (6?) and (68): * G, to G 7, and H; t o H ? are defined i n Appendix I . 2 . 4 E l a s t i c Degrees of Freedom For the s t a b i l i t y a n a l y s i s of a r i g i d a i r c r a f t the previous a n a l y s i s would apply i f ^ and i t s d e r i v a t i v e s were a l l set equal to zero. However, i f ^ , ^ . , and ^ are not zero, a d d i t i o n a l equations of motion to determine them must be found. The i n s e r t i o n of equation ( 2 3 ) , namely that ^-Z J: , was performed i n a somewhat a r b i t r a r y manner, but there i s a good reason f o r t h a t choice of f u n c t i o n to represent the e l a s t i c d e f l e c t i o n g. I f a body i s set i n f r e e v i b r a t i o n , then i t w i l l v i b r a t e i n i t s n a t u r a l modes of v i b r a t i o n which depend only on the mass and s t i f f n e s s d i s t r i b u t i o n s of the body. There i s an i n f i n i t e number of these mode shapes, and * G, t o G 7, and H, to H 7 are defined i n Appendix I . 48 the d e f l e c t i o n of a point i n the body i s the sum of a cer-t a i n m u l t i p l e of a l l the mode shapes at any time, the propor-t i o n of each mode changing w i t h time. Thus, the f^/t.) i s a measure of how much of the t o t a l d e f l e c t i o n i s cont r i b u t e d by the / r h mode (fi,, In the theory of l i n e a r algebra of an n - dimensional vector space, ^>^, \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 . ,^ are an ortho-normal co-ordinate system, an ordered b a s i s , f o r a v e c t o r space. I f g l i e s w i t h -i n the space generated by (p,; $ 2 ;.. m)0^ , then ^ , % n are the co-ordinates of g r e l a t i v e to (^P\i07_r^i 4h) \u00E2\u0080\u00A2 The term ortho-normal means tha t each C/X/y) i s ortho-gonal to every other and that the magnitude of each b a s i s v e c t o r cannot be gr e a t e r than one. Thus g i s a l i n e a r com-b i n a t i o n of a l l the mode shapes. The important property of the n a t u r a l mode i s i t s o r t h o g o n a l i t y . This r e s u l t s i n no i n e r t i a c o upling between the e l a s t i c and r i g i d - b o d y degrees of freedom enabling a much simpler mathematical a n a l y s i s of the problem. The modes of motion are independent of one another i n s o f a r as e l a s t i c and i n e r t i a f o r c e s are concerned but are coupled through the aerodynamic f o r c e s . F u r t h e r , since the r e s u l t a n t l i n e a r and angular moments of each mode i s zero, there i s no i n f l u e n c e of e l a s t i c i t y on the E u l e r equations of motion [13]. These equations remain unchanged i f the small time v a r i a t i o n of 49 the moments of i n e r t i a of the body are neglected. B i s p l i n g h o f f has a very c l e a r d e r i v a t i o n of the e l a s t i c equations of motion [3]. Because of the o r t h o g o n a l i t y , i t i s p o s s i b l e to w r i t e the equation of motion f o r one mode, say t h e j , without d i s t u r -b i ng any other mode. The equations are: where Mj = ge n e r a l i z e d mass i n mode, = normalized n a t u r a l mode of v i b r a t i o n of the j f u s e l a g e , or the normalized e i g e n f u n c t i o n of d i f f e r e n t i a l equation of f r e e v i b r a t i o n , mass d i s t r i b u t i o n of the fu s e l a g e , = gl Kn(7) ^j(i)cs normal co-o r d i n a t e , tAlj = frequency of v i b r a t i o n of the j ^ mode, or LUj\" i s the eigenvalue of d i f f e r e n t i a l equa-t i o n of f r e e v i b r a t i o n , JSj = g e n e r a l i z e d f o r c e i n j mode due to aero-dynamic pressure p ( x ) , = ~ fJ[f*Ap(t,0)rt/'$M^tfJr ( 7 2 ) and ^v= component i n the y d i r e c t i o n of u n i t vector normal to the body s u r f a c e . 50 The x and z components of the g e n e r a l i z e d f o r c e would have to be considered i f f l e x i b i l i t y of the s t r u c t u r e were allowed i n the x and z d i r e c t i o n s . The only c o n t r i b u t i o n to i s from the aerodynamic pressure. Considering the p o t e n t i a l f l o w , equation (16) leads t o : f' fz7r - \u00E2\u0080\u0094 l-= - L3* ^jj7- J J j~ f)(?) cos Or Bm(i- Rcos0($) m 2 JQ J The rearrangement of (73) g i v e s : (74) Again, the o r t h o g o n a l i t y of the n a t u r a l modes allows a s i m p l i f i c a t i o n because equals zero i f L&J and equals i f i = j . 51 Therefore (74) becomes: Jjp z 0 0 I Jo V- ^^//o dX 7 4 t^ -jf A TT i f 4 j c ^ 7 > (75) I t i s p o s s i b l e t o extend the above development to the case of the viscous normal f o r c e even though no pres-sure appears e x p l i c i t l y i n the equations, but of course the c r o s s - f l o w drag i s a r e s u l t of a c e r t a i n pressure d i s -t r i b u t i o n . Equation (41) presents i n the d e s i r e d form; r e c a l l i n g that powers of o{ greater than three and products \u00C2\u00B0fJpy a n d i t s d e r i v a t i v e s are neglected leads t o : \" 3*?%'QfotfWxci*\ (76) The t o t a l g e n e r a l i z e d f o r e e ^ j i n the j mode i s the sum of (75) and ( 7 6 ) . D e f i n i n g the g e n e r a l i z e d f o r c e c o e f f i c i e n t as C z , ^ ? J (77) and adding equations (75) and (76) t o obtain the t o t a l 52 2.5 Equations of Motion of Complete A i r c r a f t The general approach to the s t a b i l i t y study of a r i g i d a i r c r a f t i s w e l l e s t a b l i s h e d [13], hence only the important f e a t u r e s of the a n a l y s i s are reviewed here. A r i g i d body of general shape i n space has s i x degrees of freedom, namely, three i n t r a n s l a t i o n and three i n r o t a t i o n about the t r a n s l a t i o n a l d i r e c t i o n s . However, * K, t o K/0 are defined i n Appendix I , i f a plane of symmetry e x i s t s i n the body, and i f there are no gyroscopic e f f e c t s , . t h e n \"pure symmetric motion\" i s pos-s i b l e . This i s the case f o r the a i r c r a f t under d i s c u s s i o n . The s p e c i a l type of motion means that the a i r c r a f t has degrees of freedom i n the l i f t and drag d i r e c t i o n s and can r o t a t e about an a x i s perpendicular to both those d i r e c t i o n s . A f u r t h e r r e s t r i c t i o n i s introduced by s t a t i n g that the speed U i s approximately constant. Even a c c e l e r a t i o n s o f i \" 2 0 g introduce n e g l i g i b l e f o r c e s at speeds away from M = 1 [ 6 ] . I f a l l three degrees of freedom are considered, the r i g i d - b o d y modes u s u a l l y break up i n t o a \" s h o r t - p e r i o d \" mode whereU i s s e n s i b l y con-stant and angle of atta c k v a r i e s , and a \"phugoid\" mode of long p e r i o d where U v a r i e s and angle of attack i s n e a r l y constant. The p i l o t can u s u a l l y c o r r e c t f o r small i n s t a b i l i t i e s i n the phugoid mode, but he cannot do t h i s i n the s h o r t - p e r i o d mode. Therefore, only the \" l i f t f o r c e \" and \" p i t c h i n g moment\" equations are considered here, w i t h a l l anti-symmetric v a r i a b l e s i d e n t i -c a l l y zero and speed changes n e a r l y so. A l s o , over a small speed range, the for c e s and moments are considered t o be constant [13]. The co-ordinate system f o r the s t a b i l i t y equations i s shown i n Figure 11. S t a b i l i t y axes are chosen w i t h o r i g i n at the centre of. g r a v i t y . OX p o i n t s i n the d i r e c t i o n of motion of the a i r p l a n e i n a reference c o n d i t i o n of steady symmetric f l i g h t , OY i s p o s i t i v e to starboard and perpendicular to the plane of 54 y X Figure 11 Co-ordinate System R e l a t i v e to Body P o s i t i v e D i r e c t i o n s Are Indicated symmetry, and 02 is. perpendicular to OX i n the plane of sym-metry p o s i t i v e downward. Under these c o n d i t i o n s the equations of motion a r e , Equation (79) represents e q u i l i b r i u m of aerodynamic and i n e r t i a f o r c e s i n the Z d i r e c t i o n and (80) r e f e r s to moment e q u i l i b r i u m about the QY a x i s . These equations are dimensional and exact. L and M are aerodynamic l i f t and p i t c h i n g moment r e s p e c t i v e l y ; mgcosfi* i s the component of weight i n the Z d i r e c t i o n ; U i s the v e l o c i t y of the centre of g r a v i t y i n the X d i r e c t i o n ; Wis the (79) (80) 55 v e l o c i t y of the centre of g r a v i t y i n the Z;ydirection; d, equals\u00C2\u00A9; and B represents the mass moment of i n e r t i a i n p i t c h . The q u a n t i t y Q represents the induced angle of attack and corresponds tot&fioV. The r e p r e s e n t a t i o n of o^by the f i r s t two terms of the i n f i n i t e s e r i e s i s con-s i s t e n t w i t h the previous a n a l y s i s , where only terms i n o/^were r e t a i n e d . U 3 (81) = UC*r)(l +*r) (g2) The term m^COS3* represents the weight component i n the Z d i r e c t i o n ; i f the a i r c r a f t reference c o n d i t i o n (steady s t a t e ) i s taken at some climb angle ^, then po^ cos 0* =\u00E2\u0080\u00A2 fD^ ^cosYcvs & ~siV) $s in @J L r t i s the reference l i f t and L i s the t o t a l aerodynamic l i f t . L* = mq cos y \u00E2\u0080\u00A2o !7 S u b s t i t u t i o n of these r e l a t i o n s i n t o equation (79) and not i n g equation (82) leads to 56 = -L* -hLgCosG -\u00C2\u00A30 lantisinB \u00C2\u00AB -L*\u00C2\u00B1L?0(l-f )-L*0fari(9-f) This r e q u i r e s the angle of a t t a c k i n the expression f o r C ^ to be measured from the z e r o - l i f t l i n e . I f the reference c o n d i t i o n i s the one corresponding t o ^=Q, then the equation s i m p l i f i e s t o - L* j-mcj(h\u00C2\u00A3z) =rr\U[*r(n-\u00C2\u00B0\u00C2\u00A9D% +o-*sNj) + + rCn-Nj') +0^ . + o ^ & \u00C2\u00A3 j faw) = 0. (99)* Equations (97), (98), (99)'are rearranged i n t o the most convenient form f o r s o l u t i o n : <*V (a\ +r<*r2) + (02+2*4 +(0g +2>u) D9(fc3 -2u) +\u00E2\u0080\u00A2 * j j'fra -W/^) +\u00C2\u00ABrbd(aiDe +#<(,b%>^h^^ts^'O (100) + 5 JAo+V* 2> D ^ f ^ + i ^ ^ + l ^ ' f t a ) (101) * &, to a / g , b, to b,y and c, to c/^ are defined i n Appendix I . ^ f c K 5 ^ ) +b\u00C2\u00AB r c 3 ) + D ^ r ^ ) +\u00E2\u0080\u00A2 *$i\(\u00C2\u00ABo~Njky)Kii\u00C2\u00ABr\ f D f y fa. TtiEjfaz-Nj') ( 1 0 2 ) The general c o e f f i c i e n t i n equations (100), (101) and (102) i s of the form P e r m i t t i n g cxV to vary up to ten degrees, and the i n -c l u s i o n of the e f f e c t s of v i s c o s i t y modify the \" l i n e a r co-e f f i c i e n t \" l?n by the a d d i t i o n of the term ky^dr\". A l s o , s i n c e account was taken of unsteady flow e f f e c t s a term i n fr^j' e x i s t s i n each of equations (100), (101) and (102) i n a d d i t i o n to the term Ny DaJ^, . I t w i l l be shown i n a n a l y s i s of these equations that considerable complication r e s u l t s i n the s t a -b i l i t y a n a l y s i s by the i n c l u s i o n of t h i s unsteady term. The convenience of t h i s method of arranging the three equations of motion l i e s i n the ease of o b t a i n i n g the l i n e a r from the n o n - l i n e a r equations and i n the symmetry of the co-e f f i c i e n t s . The s e c t i o n on s o l u t i o n of the equations can now be discussed as a separate e n t i t y , without s p e c i a l i z i n g i t to e i t h e r the m i s s i l e or the a i r l i n e r case. The geometry of the a i r l i n e r and the main parameters i n v o l v e d i n i t s l o n g i t u d i n a l s t a b i l i t y have been l e f t i n a very general form. This completes the a n a l y t i c a l f o r m u l a t i o n of the prob-lem. The next s e c t i o n i s devoted to the s o l u t i o n of equations (100) , (101) and (102) . I I I . SOLUTION OF EQUATIONS 3.1 P r e l i m i n a r y Remarks Se c t i o n I I was devoted to the mathematical f o r m u l a t i o n of the p h y s i c a l problem, i . e . the d e r i v a t i o n of equations of motion of the e l a s t i c s t r u c t u r e undergoing plunging and p i t c h i n g motion. In t h i s s e c t i o n an attempt i s made t o solve these equations. The equations are n o n - l i n e a r , o r d i n a r y , d i f -f e r e n t i a l equations which must be solved simultaneously. I t may be pointed out that the e l a s t i c degrees of freedom can be represented e x a c t l y only by an i n f i n i t e number of equations. However, the c o n s i d e r a t i o n of more than one e l a s t i c degree of freedom causes immense a l g e b r a i c problems i n the attempt t o handle the equations a n a l y t i c a l l y , and ther e f o r e only one mode of v i b r a t i o n i s considered here. The extension of the method to more than one e l a s t i c degree of freedom (mode) i s obvious, but i t i s f e l t t hat t h i s should be reserved f o r a numerical r a t h e r than a n a l y t i c a l approach. In general, the problems a s s o c i a t e d w i t h a i r c r a f t dynamics f a l l i n t o two c a t e g o r i e s . 1. response problems, 2. s t a b i l i t y problems. I n v e s t i g a t i o n of the response of the e l a s t i c a i r c r a f t to d i f -f e r e n t e x c i t a t i o n s or the design of a s u i t a b l e c o n t r o l system to o f f - s e t any undesirable e f f e c t s of f l e x i b i l i t y are the prob-64 lems that belong to the f i r s t category. An a n a l y t i c a l approach to the response of a many-degree-of-freedom system i s a formid-able undertaking, and the expectation of achieving exact solu-tions w i l l , i n general, not be r e a l i z e d . Approximate solutions are the rule, except i n some s p e c i a l cases [14]. Much s k i l l and ingenuity are required to obtain even these approximate solutions, which are considered preferable: to no solution. The method of Varia t i o n of Parameters [14] can often y i e l d an approximate solution, however, the evaluation of the integrals involved may not be possible i n every case. The study of a i r c r a f t s t a b i l i t y as affected by fuselage f l e x i b i l i t y belongs to the second group of problems which i s the area under consideration here. The term \" s t a b i l i t y \" needs some amplification i n the case of a non-linear system. The usual (linear) sense of s t a b i l i t y i s that a system w i l l have convergent amplitude when disturbed from an equilibrium condition. However, non-linear systems can possess \" l i m i t cycles\" where the amplitude does not converge to the equilibrium point, and yet i t does not diverge to destroy or saturate the system. In t h i s sense, the system i s stable, but not i n the sense of l i n e a r systems. For the non-linear system, large disturbances leading to large ampli-tudes , may cause the system to leave the region of influence of an equilibrium point to go to a l i m i t cycle, to the region of another equilibrium point, or diverge to i n f i n i t y . Also, i f the system i s i n a l i m i t cycle, a disturbance may send the 65 system out of the o r i g i n a l c y c l e i n t o a d i f f e r e n t one, or the system may r e t u r n t o i t s o r i g i n a l l i m i t c y c l e . In the l i g h t of the above d i s c u s s i o n , two kinds of s t a b i l i t y f o r n o n - l i n e a r systems may be d e f i n e d : 1. Asymptotic s t a b i l i t y . I f the d i f f e r e n c e s between the amplitudes of the d i s t u r b e d motion and the o r i g i n a l undis-turbed motion u l t i m a t e l y r e t u r n to zero, the system i s a s y m p t o t i c a l l y s t a b l e . 2. O r b i t a l s t a b i l i t y . I f a small disturbance i s a p p l i e d to a system i n a steady s t a t e o s c i l l a t i o n ( l i m i t c y c l e ) , and i f the system returns to the o r i g i n a l l i m i t c y c l e , i t i s r e f e r r e d t o as o r b i t a l l y s t a b l e . In the present a n a l y s i s only asymptotic s t a b i l i t y i s considered w i t h the r e s t r i c t i o n that u s u a l l y goes w i t h asymptotic s t a b i l i t y , that the disturbance i s s u f f i c i e n t l y small to keep the system w i t h i n the region of i n f l u e n c e of the point of e q u i l i b r i u m i n question. Asymptotic s t a b i l i t y i s d e s i r a b l e from the a i r c r a f t point of view because a non-o s c i l l a t o r y steady s t a t e e x i s t s . A i r l i n e r s and m i s s i l e s de-s i r e t h i s f o r passenger comfort and guidance r e s p e c t i v e l y . 3.2 Method of A n a l y s i s The general approach to the problem i s best given as a s e r i e s of operations to be a p p l i e d to the system of equations derived before. 66 Step 1 - Determine (the e q u i l i b r i u m c o n d i t i o n s or singu-l a r points i n the phase plane of the system. Set a l l the d e r i v a t i v e s equal to zero and solve the r e s u l t i n g set of s t a t i c a l g e b r a i c equations f o r the e q u i l i b r i u m values of o(y) . Knowing the designer can adjust the mass and s t i f f n e s s d i s t r i b u t i o n s of the s t r u c t u r e c o n s i s t e n t w i t h tyL(x) . This may t u r n out to be very easy f o r him to accomplish but a c r i t e r i o n 70 i s d e s i r a b l e . The e x t r a c t i o n of a s t i f f n e s s c r i t e r i o n from the equa-t i o n s PQ =0 and R*=0 i s discussed l a t e r i n the a n a l y s i s . The method of a n a l y s i s given above a p p l i e s to non-l i n e a r equations i n general, and i t can be a p p l i e d t o l i n e a r equations i n the degenerate case when a l l the n o n - l i n e a r i t i e s are zero. This i s important because i t i s i n s t r u c t i v e to carry out the a n a l y s i s of the s t a b i l i t y of the f l e x i b l e a i r -c r a f t i n stages, r a t h e r than to proceed immediately to the most complex problem. There are f o u r \" c l a s s e s \" of problem contained i n equations (100), (101) and (102): 1. R i g i d body cases a. l i n e a r equations of motion b. non-linear equations of motion 2. E l a s t i c body eases a. l i n e a r equations of motion b. n o n - l i n e a r equations of motion. These cases and t h e i r s o l u t i o n s w i l l be discussed i n order to obtain as much information as p o s s i b l e from the equations developed i n I I . 3.3 R i g i d Body Cases The case of a r i g i d body w i t h l i n e a r equations i s t r e a t e d by using equations (100), (101) and (102), s e t t i n g ^ and i t s d e r i v a t i v e s equal t o zero, and n e g l e c t i n g a l l higher 71 order terms. I t may be pointed out that the e l a s t i c equations of motion reduce to the set of equations normally taken f o r the r i g i d body a n a l y s i s when^=#. A l l d e r i v a t i v e s are set equal to zero i n accordance w i t h the approach to the a n a l y s i s described before. The r e s u l t i s : \u00C2\u00B0<%(b)=0 (103) The f i r s t equation shows that'the l i f t equals the weight i n steady l e v e l f l i g h t , angle of attack being meas-ured from zero l i f t c o n d i t i o n s to s u i t the o r i g i n a l non-l i n e a r r e l a t i o n of witho< r, and that the p i t c h i n g mom-ent i s zero (the a i r c r a f t i s trimmed). The method of a n a l y s i s when d e a l i n g with l i n e a r equations i s t o consider (Xi-as a p e r t u r b a t i o n about the reference l i f t angle of a t t a c k ; t h i s leads to the con c l u s i o n that the e q u i l i b r i u m point i s the angle of attack corresponding to steady l e v e l f l i g h t , which i s not a s u r p r i s i n g r e s u l t . I f the assump-t i o n that Ci* v a r i e d l i n e a r l y w i t h P ( r were t o be made, then the equation would be v a l i d , independent of the s i z e of the angle of attack r e quired to achieve C\* . Thus ACL[ corresponds to $\u00C2\u00A3ir*. The a d d i t i o n of the term ' t f ^ r ' t o xa' makes i t necessary to measure Oir from the z e r o - l i f t l i n e because the s i z e of i s dependent on the s i z e of (XR . 72 Note that Q i s indeterminate, i n d i c a t i n g an arbitrary-reference point f o r angle of p i t c h . Considering small perturbations about the e q u i l i b r i u m value of 0(r , the f o l l o w i n g r e l a t i o n s are obtained: o ?i/ a n c* A 4. The root ?\ = 0 i n d i c a t e s that\u00C2\u00A9 i s indeterminate. The Routh-Hurwitz c r i t e r i o n as a p p l i e d to the quadratic equation ( a n d ^ ) s t a t e s t h a t the c o e f f i c i e n t s of /)'and ?{ must be of the same s i g n . The q u a n t i t i e s ^ and c.3 are the r e l a t i v e mass and i n e r t i a c o e f f i c i e n t s r e s p e c t i v e l y . Both of them are p o s i t i v e and very l a r g e compared to the a Ts and b's. Hence, an approximate c h a r a c t e r i s t i c equation can be obtained which i s much simpler and s t i l l y i e l d s accurate r o o t s . D i v i d i n g throughout by - 4 / u d g and d i s c a r d i n g s m a l l terms: The s t a b i l i t y c r i t e r i a then become: i , < 0 ,2* > ^ (107) In the context of l i n e a r equations, \u00C2\u00A3>,<0 means th a t \"2-\u00C2\u00A3L must be negative. This i s the s t a t i c s t a b i l i t y c r i -t e r i o n which i n d i c a t e s t h a t a system must be s t a t i c a l l y s t a b l e i f i t i s t o be dynamically s t a b l e . This i s a necessary but not s u f f i c i e n t c o n d i t i o n to ensure dynamic s t a b i l i t y . The other r e l a t i o n , ~^ , > <^ r e q u i r e s more i n v e s t i g a t i o n . A l l the q u a n t i t i e s i n the equation depend on r e l a t i v e magnitudes of the c o n t r i b u t i o n s of the wing, foreplane and fuselage to C L * a n d C M * , and on the mass to 74 i n e r t i a r a t i o of the a i r c r a f t . I t is-'very d i f f i c u l t t o discuss t h i s i n general terms because the major v a r i a b l e s of the design of the a i r c r a f t are i m p l i c i t l y contained i n the equations. I t i s s u f f i c i e n t to say at t h i s point that the c e n t r e - o f - g r a v i t y l o c a t i o n \"Xcj i s \"the most important s i n g l e f a c t o r ; a numerical example of a s p e c i f i c a i r c r a f t i s given i n Part IV where the r e l a t i v e magnitudes of the various c o n t r i b u t i o n s can be appreciated i n b e t t e r p e r s p e c t i v e . This would normally complete the s t a b i l i t y a n a l y s i s of the sho r t - p e r i o d mode. The non- l i n e a r equations of motion a p p l i c a b l e t o the r i g i d body are: o(r(^-h^r) T D * r ( Q +be(bz) + &&(kr*i&)stO ( 1 0 8 ) S e t t i n g a l l d e r i v a t i v e s equal to zero, 0 10\u00C2\u00B0 should be accepted w i t h extreme care. This i s f u r t h e r s u b s t a n t i a t e d i n the Numerical Example. Note that ^o- ,n~ ' A D*r=D<3r S u b s t i t u t i n g these approximations i n t o (108) g i v e s : + ^ rsj ( i, + k y o< rSj.)|j = O (110) The q u a n t i t i e s i n the \u00C2\u00A3 j are i d e n t i c a l l y zero from the d e f i n i t i o n of c>J +\u00E2\u0080\u00A2 ^ [ > 3 ] t D X 0 [ k 4 - l 4 j = \u00C2\u00A3 2 (111) This i s a l i n e a r p a i r of equations i n o(r and 0 . I t i s not p o s s i b l e to make the assumption from the outset that^/c i s so la r g e as to dwarf the other terms, because b^ . can be qu i t e small independent of the a i r c r a f t mass parameter/^.. Compari-son of equations ( 1 1 1 ) and ( 1 0 5 ) shows the change i n the co-e f f i c i e n t s due to the i n c l u s i o n of the n o n - l i n e a r i t y . Expansion of the determinant of the c o e f f i c i e n t s y i e l d s : - tj] - ^ [ ^ 3 ~2*)ks -tf^jj +?l j ' b 3[*i W t ^^\pi-^)LS'^L/]-^f^j + 7.hlfn * \u00C2\u00A3 ) ( 1 1 2 ) A major e f f e c t of the n o n - l i n e a r i t i e s i s to make necessary the s o l u t i o n of a cubic equation, r a t h e r than a quadratic equation as was p o s s i b l e i n the l i n e a r case; Q i s no longer indeterminate and the reference f o r Q must be defined. The n o n - l i n e a r i t i e s have not increased the degree of the c h a r a c t e r i s t i c equation but do r e q u i r e the s o l u t i o n of the higher degree equation. In p r i n c i p l e t h i s does not increase the complexity of the problem, but i n p r a c t i c e i t r e q u i r e s a s i g n i f i c a n t increase i n labour. Some s i m p l i f i c a t i o n of ( 1 1 2 ) i s p o s s i b l e since ^ t ^ t f g , 2 * a \u00C2\u00BB i > 4 > 3 s A \u00C2\u00BB A z , \u00E2\u0080\u00A2 Incorporating t h i s 78 approximation i n t o (112) g i v e s : j^(ai ^ b , ) - 2 4>, ^)~t2r,j\u00E2\u0080\u00A2\u00C2\u00BB-2b(J_'= O (113) The s t a b i l i t y c r i t e r i a f o r a cubic c h a r a c t e r i s t i c equation P37? + P^ T?- f P, 7\ + P0 \u00C2\u00BB 0 ( P3 < O) are P3 ;' P*j Po *C C and (P2F! -P 3 Po)<0 (114) Since b, must be negative f o r s t a t i c s t a b i l i t y , the c o n d i t i o n f ^ Q i s always s a t i s f i e d f o r p o s i t i v e values of the square r o o t . The other c o n d i t i o n s may or may not be s a t i s f i e d depending on the r e l a t i v e magnitude of the terms. The cubic c h a r a c t e r i s t i c equation has one more s t a b i l i t y c r i t e r i o n compared to the quadratic case. One e f f e c t of a l t i t u d e on s t a b i l i t y can be seen from examination of . The c o n d i t i o n i s that P-i<0 Therefore, \u00E2\u0080\u0094 i^^'s ( i s ' ^ - ^ f ^ ^ - ^ ^ ^ O B o t h ^ and Lg are i n v e r s e l y p r o p o r t i o n a l t o the a i r d e n s i t y 79 which decreases as a l t i t u d e i n c r e a s e s . Thus, as the a l t i t u d e of f l i g h t changes the s t a b i l i t y c o n d i t i o n may or may not be s a t i s f i e d depending on the signs and magnitudes of the terms i n v o l v e d . 3.4 E l a s t i c Cases As has been noted p r e v i o u s l y i n the a n a l y s i s of the l i n e a r - r i g i d and n o n - l i n e a r - r i g i d cases, i t i s exceedingly d i f f i c u l t to make general conclusions about s t a b i l i t y from the c h a r a c t e r i s t i c equations because of the lar g e number of co-e f f i c i e n t s whose r e l a t i v e magnitudes are important. The c o n t r i b u t i o n s of the various components of the a i r c r a f t t o C ^ y and to Cft* may be e i t h e r s t a b i l i z i n g or d e s t a b i l i z i n g , and the t o t a l e f f e c t can be to produce e i t h e r a s t a b l e or unstable a i r -c r a f t . The i n c l u s i o n of the e l a s t i c degree of freedom i n t o the l i n e a r equations results i n a q u i n t i c c h a r a c t e r i s t i c equation, again w i t h A-Cbeing one of the r o o t s . Thus one must solve a q u a r t i c r a t h e r than a quadratic equation. The c h a r a c t e r i s t i c equation i s of the form The Routh-Hurwitz c r i t e r i a f o r s t a b i l i t y as mentioned before are th a t and (115) 80 These are the necessary and s u f f i c i e n t c o n d i t i o n s f o r s t a b i l i t y , and the f a c t that must be~~yO i s a derived r e s u l t . To obtain the c h a r a c t e r i s t i c equation f o r the l i n e a r -e l a s t i c case i t i s necessary to expand a three-by-three deter-minant (see pages Gl , 6 2 ) , c o l l e c t the terms w i t h l i k e powers of A to f i n d Pn , and c a l c u l a t e the value of Routh's D i s c r i m i -nant. This i s a formidable e x e r c i s e i n alge b r a . The n o n - l i n e a r terms make i t necessary to solve the f u l l q u i n t i c equation. The s t a b i l i t y c r i t e r i a f o r a character-i s t i c equation of the form are that Ps} P 4 } , P0 > \u00C2\u00B0 (P4P3 - P s R r ) > 0 and LP 4 ^ - ^ P ^ [ ( P f ^ - P r/l ) P 2 - ( ^ - ^ H ) I \ ] > \u00C2\u00A3 ( 1 1 6 ) The derived c o n d i t i o n s that P^ and P, must a l s o be p o s i -t i v e f o l l o w from the l a s t two equations of ( 1 1 6 ) . The general theory of Routh s t a t e s t h a t f o r a polynomial i n ?\ of degree n there w i l l be n-H t e s t func-t i o n s , designated Qn\u00C2\u00BB which must a l l be p o s i t i v e f o r s t a b i l i t y . I f the a i r c r a f t i s s t a b l e , and some design parameter i s v a r i e d to cause i n s t a b i l i t y , two cases are of i n t e r e s t : i f only P Q changes s i g n from p o s i t i v e to negative, then one r e a l root changes s i g n from negative to p o s i t i v e i n d i c a t i n g one divergence; i f only Qn-/ 81 changes s i g n from p o s i t i v e to negative, then the r e a l part of one complex p a i r of roots changes from negative to p o s i t i v e , i n d i c a t i n g one divergent o s c i l l a t i o n . Thus PQ =0 and Qn_( =0 represent boundaries between s t a b i l i t y and s t a t i c i n s t a b i l i t y , and s t a b i l i t y and divergent o s c i l l a t i o n r e s p e c t i v e l y . These s t a b i l i t y boundaries f o r a q u i n t i c c h a r a c t e r i s t i c equation are given by P . - O and [PnFi-PsP{)[(:P+P3-P5-rV)P2-('R,^-Ps-R)PH7 =0 (117) Two p o s s i b l e l i n e s of approach e x i s t f o r the d i s -cussion of the s t a b i l i t y of a f l e x i b l e a i r c r a f t using the e l a s t i c equations of motion. The f i r s t approach i s to consider an a i r c r a f t whose complete geometry, mass and s t i f f n e s s d i s t r i b u t i o n s are known. A l l the c o e f f i c i e n t s i n the equations of motion can be evaluated and the c h a r a c t e r i s t i c equation can be obtained. The Routh-Hurwitz c r i t e r i o n can then be a p p l i e d to check the s t a b i l i t y . S t a b i l i t y boundaries could be c a l c u l a t e d on the basis of two important design parameters. The equations and s t a b i l i t y c r i t e r i a described p r e v i o u s l y can be a p p l i e d d i r e c t l y t o t h i s problem. This a n a l y s i s i s c a r r i e d out i n d e t a i l i n the Numerical Example considered. The second approach i s t o consider an a i r c r a f t of known geometry, t o t a l mass and c e n t r e - o f - g r a v i t y l o c a t i o n . 82 An allowable l i m i t on f l e x i b i l i t y f o r s t a b i l i t y can be deter-mined from the e l a s t i c equations of motion by co n s i d e r i n g the s t a b i l i t y boundaries as l i m i t s f o r the f l e x i b i l i t y . This l i m i t on f l e x i b i l i t y can be i n t e r p r e t e d as an allowable s t i f f n e s s d i s t r i b u t i o n along the fuselage length- A means of making an unstable a i r c r a f t s t a b l e i s now a v a i l a b l e . A method f o r ob-t a i n i n g t h i s allowable f l e x i b i l i t y i s developed next. Examination of the equations (100), (101) and (102) shows c o e f f i c i e n t s of the form where f and g are fun c t i o n s of mode shape and geometry, occur-r i n g i n the e l a s t i c c o n t r i b u t i o n s t o , C^\u00E2\u0080\u009EandC^ . The c o e f f i c i e n t the mass and s t i f f n e s s d i s t r i b u t i o n s of the fuselage are known, i t i s p o s s i b l e to c a l c u l a t e the n a t u r a l frequencies and n a t u r a l mode shapes of the fu s e l a g e , the 4>y(*) and UJj . The a n a l y s i s as presented i s concerned w i t h only one e l a s t i c degree of freedom, and t h i s degree of freedom i s the one corresponding to the lowest n a t u r a l frequency of the fu s e l a g e , i . e . the fundamental bending mode. I f the mass d i s t r i b u t i o n ^6t'(x) and n a t u r a l f r e -quency were known, the only \" e l a s t i c unknown\" i n the equations would be (fi(x) , and i t would be p o s s i b l e to consider the s t a b i l -i t y boundaries PQ =0 and Qn_, =0 as equations t o determine a 'o contains the non-dimensional mass d i s t r i b u t i o n ^ - ' f > c ) . Now, i f \" l i m i t i n g \" 0(7). I t i s true that (j)C7) would be the only unknown, but i t i s extremely d o u b t f u l that the s o l u t i o n f o r the f u n c t i o n 0(Z) could be obtained e x p l i c i t l y because of the complexity of the equations Pc =0 and Q p H =0. Note th a t each of the s t a b i l i t y boundaries would give an independent c o n d i t i o n f o r (f>(ll) and that there i s no reason whatsoever f o r the same s o l u t i o n to be obtained by both equations. This i s a d i r e c t consequence of the d i f f e r e n t p h y s i c a l c o n d i t i o n s of i n s t a b i l i t y t o which the boundaries are a p p l i c a b l e . However, i f both n a t u r a l frequency and the mass d i s t r i -b u tion were known, most assuredly the mode shape and s t i f f n e s s d i s t r i b u t i o n would be known because the l a s t named q u a n t i t y would have had to be a v a i l a b l e i n order t o obta i n the n a t u r a l frequency. On the whole, that l i n e of approach would not be too f r u i t f u l . Another way does e x i s t which y i e l d s a valuable r e s u l t c o n s i s t e n t w i t h the approximations made i n r e s t r i c t i n g the a n a l y s i s to a s i n g l e , e l a s t i c degree-of-freedom. I f the mode shape ${7) were to be assumed then a l l the c o e f f i c i e n t s i n the equations of motion could be evaluated except N, and Nk*. Note th a t the terms N and Nk*\" are as s o c i a t e d w i t h D*\"jp , and ^ r e s p e c t i v e l y . Also i t may be pointed out tha t N equals Expansion of the determinant of the co-e f f i c i e n t s and formation of the c h a r a c t e r i s t i c equation, y i e l d s 84 the c o e f f i c i e n t s P n as complicated functions of N and Nk3\"; however, PQ w i l l always contain the term Nk*\" i n a manner where i t can be isolated on the s t a b i l i t y boundary Pe =0. This i s possible because with (ficx) given, every other term can be obtained. This i s one of the s i g n i f i c a n t results of the analysis because by use of the Rayleigh Method i t i s possible to solve f o r a l i m i t i n g s t i f f n e s s d i s t r i b u t i o n of the fuselage on the basis of longitudinal s t a b i l i t y . that the maxima of the potential and k i n e t i c energies of a body in simple harmonic motion can be equated. This leads to the expression f o r fundamental c i r c u l a r frequency as: It i s convenient to non-dimensionalize the equation by the following substitutions The Rayleigh Method i s based on the energy p r i n c i p l e (118) 6U= k U L ETC?) = BT The non-dimensional frequency i s given by (119) Equation (119) i s u t i l i z e d to obtain the equation 85 t h e r e f o r e , A/k 2=2/'EXW (120) s u b s t i t u t i o n of the r i g h t hand side of equation (120) s t i f f n e s s c r i t e r i o n . The next few pages are devoted to the d e t a i l e d development of the c r i t e r i o n i n the general case. The s t a b i l i t y boundary can be obtained i n the general case of the non - l i n e a r equations by the method described p r e v i o u s l y . The v a r i a t i o n a l equations i n cpr , A A 0, and i3 -t- b ( fe l 3 bi bio =0 ( 1 2 5 ) 87 The c o e f f i c i e n t s are w r i t t e n as two-by-two determinants f o r convenience. The a d d i t i o n of e l a s t i c i t y t o the problem has i n c r e a s e d the p o s s i b l e e q u i l i b r i u m p o i n t s by two, a g a i n sym-m e t r i c a l l y disposed about the r e f e r e n c e f l i g h t c o n d i t i o n . However the e q u i l i b r i u m values now depend upon the f a c t o r Nk''*, and every c o e f f i c i e n t i n the equations (121), (122), (123) i s a complicated f u n c t i o n of the s t i f f n e s s E I f o r which a s o l u t i o n was d e s i r e d . Thus the e x t r a c t i o n of a c r i t e r i o n f o r f u s e l a g e s t i f f n e s s w i t h n o n - l i n e a r i t i e s i n c l u d e d must be obtained from a polynomial equation i n Nk^ . T h i s i n c r e a s e s the amount of work in v o l v e d but the method i s s t i l l v a l i d . The c h a r a c t e r i s t i c equation obtained a t the g e n e r a l e q u i l i b r i u m p o i n t from equations (121), (122) and (123) i s of the form Ps*5\" + P 4 t f + P 3 ? ? + P z 7 i , , + - P , * + P o \u00C2\u00AB 0 - (126) where the c o e f f i c i e n t s P-. to P^ , are d e f i n e d below. P4 = a, -a. /ICijC,?/ ' c/*jft\u00C2\u00BB-N)lj I fri- ll) -h (127)* (i2a>* * The s t a r r e d symbols are d e f i n e d i n Appendix I. 88 U c \u00C2\u00B1 >c*\ \c^)Cca'N)\) ticket -h 4-4-4-J bi*1; bo. bx, b (0 a, t i j Cb 4 - 2 ^ 8 ) ^ 4 -(129)* 1 /o b*5b - a bzj b\u00E2\u0080\u009E* /b,* b,i '\u00E2\u0080\u00A2\u00E2\u0080\u00A2zCu bi j i>io Ms b,VWs)// Jb*>3 4 ; C 3 '/ J (130^ * The s t a r r e d symbols are d e f i n e d i n Appendix I. 89 IS wo (132)* These c o e f f i c i e n t s are a p p l i c a b l e to the l i n e a r case as w e l l by p l a c i n g &\r<; - Sk,; -%y\rD. The r e s u l t i n g charac-t e r i s t i c equation i s a l s o a q u i n t i c but Ty^O i s one of the roots The s t a r on some symbols denotes dependence of the c o e f f i c i e n t on the e q u i l i b r i u m values of cx>* \u00E2\u0080\u00A2 , Q$\, and . This n o t a t i o n i s employed f o r conciseness. For the J l i n e a r case equations ( 1 2 7 ) to ( 1 3 2 ) are s t i l l a p p l i c a b l e i f &r$\ , &s' , and are put equal to zero i n the s t a r r e d symbols. I t may be pointed out t h a t f o r the l i n e a r d i s -. *-cussion, &\q-0. The equations f o r the s t a b l e region f o r the l i n e a r and non-linear cases can now be w r i t t e n i n f u l l d e t a i l as f o l l o w s : (133) 90 The terms i n ^ j are i d e n t i c a l l y zero from equation (125), t h e r e f o r e the above equation reduces t o ?0 = -4ataQ^.M, b^ , h>i0 C13 ){/C/e-//kX)\ o \u00C2\u00B0 b\" (134) S o l v i n g equation (134) f o r Nk* y i e l d s : K M T - 3 |br,bli I i t f i t / o I |tl3,b; (135) S u b s t i t u t i n g the r e s u l t of (120) i n (135), the s t i f f n e s s c r i t e r i o n f o r the n o n - l i n e a r case i s obtained: 10 rl 3 I b ^ \u00C2\u00B0 ' 3 I |br . b ' o J ,j>,3 k, 2 j E r ^ f f l fr))c/)T^ ^ r 5 j - / ^ ^ 3 / -ho(r5j j Cf, c,o l+Psj U>U For the l i n e a r case, F^=0( a s a ^ = \u00C2\u00B0) > a n d t h e r e -f o r e equation (132) i s no longer a v a i l a b l e t o ob t a i n the s t a b i l i t y boundary. I t i s necessary to consider equation (131) as the s t a b i l i t y boundary a p p l i c a b l e to the case: (136) Therefore, i f F|><2, a, a;+*,o (137) f o r s t a b i l i t y . S o l v i n g f o r Nk2\" and s u b s t i t u t i n g from equation (120) g i v e s : <2(km(f\u00C2\u00AEp7^ a, \c^%\-fa-&)lt&l\u00C2\u00ABi\u00C2\u00BB/$kl ( 1 3 8 , Thus i t i s p o s s i b l e to solve f o r a l i m i t i n g s t i f f -ness d i s t r i b u t i o n t h a t would give a s t a b l e system. A l l the terms on the r i g h t hand side of equations (136) and (138) are known once the a l t i t u d e , geometry, and weight of the body are known. The fuselage f i r s t bending mode must be assumed as w e l l as the centre of g r a v i t y l o c a t i o n but i t i s not necessary to know the mass d i s t r i b u t i o n i n d e t a i l . The procedure f o r using t h i s s t i f f n e s s c r i t e r i o n i n the p r a c t i c a l design of an a i r c r a f t would be to c a l c u l a t e the r i g h t hand side of (136) or (138) f o r d i f f e r e n t values of Xc^ and <$CZ) so as to obtain a s e r i e s of p l o t s as shown i n Figure 13. The mode shape y i e l d i n g the lowest value of the i n t e g r a l at a given c e n t r e - o f - g r a v i t y l o c a t i o n then gives the best approximation. The r e s u l t s of equations (136) and (138) are p o s s i b l e because the n a t u r a l mode shape (p(x) i s chosen, and the accuracy of the r e s u l t s depends on the choice of f u n c t i o n to 9 2 represent \"=0 a/)a/(&r(f)\")'=0 ( 1 3 9 ) The l a s t c o n d i t i o n i s a p p l i c a b l e to the f r e e v i b r a -t i o n of the fu s e l a g e , and i t s t a t e s that the bending-moment and shear are both zero at each end. The fuselage can be considered to be a \" f r e e - f r e e \" beam (neither end f i x e d ) of non-uniform cross s e c t i o n . The f i r s t \"symmetric\" bending mode w i l l i n general have two nodes and must have zero curvature {(f)\"-0) at e i t h e r end. This i s shown i n Figure 14. Figure 14 \"Symmetric\" Bending of Fuselage 94 If the centre of gravity were half-way along the fuselage, a simple mode shape s a t i s f y i n g the conditions would be The constant 'c' i s important because i t determines the position of the nodes. However, i n general, the centre of gravity would not be at , so equation (140) must be modified. 0 '- $ 1-^ati-Cn , * ^ ' (141) where Xfli-Q represents the distance a f t of the fuselage nose where 0/f>()~O. This type of function requires the shape constants that include (frC9) to be calculated i n two parts, namely: % \u00C2\u00B1 o (/) and (ft are the same when the centre of gravity i s at Anti-symmetric bending i s also possible. This i s ' i l l u s t r a t e d i n Figure 15. The conditions that ^ \u00C2\u00A3 - 0.5 98 The wing i s so p o s i t i o n e d that i t s t r a i l i n g edge co-i n c i d e s w i t h the fuselage base The foreplane t r a i l i n g edge i s taken to be 40 f t . a f t of the fuselage nose (Figure 17) C D r =0.35 ( I f the a x i a l f l o w boundary l a y e r i s laminar, a d i f f e r e n t value f o r c t J c s s should be used, u s u a l l y 1.2). R = 0.04-X \u00E2\u0080\u0094 ^-^T?f^J o \u00C2\u00AB\u00C2\u00A3 _ _ _ _ _ _ >J \u00E2\u0080\u0094>-\u00E2\u0080\u0094>- X Figure 16 Fuselage Geometry The fuselage length L i s chosen on the b a s i s of the condition,l?an<* r <5\". I f =Xr equals 10\u00C2\u00B0 , t a n c * r equals 0.176. T h e n i must be l e s s than 28.4. The fuselage base radius R b i s chosen as 10 f e e t l e a d i n g t o the c o n d i t i o n that L must be l e s s than 284 f e e t . With t h i s i n f o r m a t i o n , the various parameters appearing i n the a n a l y s i s can e a s i l y be computed. These values are l i s t e d i n Table I . 99 TABLE I NUMERICAL VALUES OF THE PARAMETERS USED IN THE ILLUSTRATIVE EXAMPLE - RIGID CONSTANTS Parameter Value Parameter Value Parameter Value lw ( f t . ) 56 - 0.410 c 9 0.126 l f ( f t . ) 17 .7 K - 0.136 D, - 0.34 125 - 1.172 ; D a - 0.016 0.75 - 0.584 D3 - 0.373 B slug-f -S 2.02-x. 107 F3 Q.030 \u00C2\u00B04 - 0.070 i s 4.62 F4 0.015 *s 2.88 V o l . f t ! 6.25 x 103 G * 1.57 a i 19.27 M.V. f t \" . 1.25 x-106 G z 1.03.8 a 2 5.912 V.M.I.ftf 2.605 x IO 8 G3 0.523 a3 2.628 0.496 G4 0.346 a+ 0.336 - f u r 0.776 H, - 0.037 a ^ - 34.2 0.089 Ha - 0.025 a& 0.84 A, 2 H3 - 6.019 a7 - 0.063 2 H4 - 0.012 a8 - 0.080 A3 0.126 C, 2 0.126 A. 0.159 c 2 2 ' - 2.177 A 5 34.2 C3 0.126 - 0.829 B, - 1 C4 0.159 - 1.064 B 2 - 1.840 34.2 - 0.300 Bo - 0.12 0.84 2.88 E/ 15.7 - 0.063 E 2 5.23 Cg - 0.080 100 L=zsd s~ Figure 17 Wing and Foreplane Location The expressions f o r the r i g i d body l i f t and p i t c h i n g moment coef f i c i e n t s , C L * T and \u00C2\u00A3 h * r e s p e c t i v e l y , can now be w r i t t e n as + o3t>o) +0(^(2,%?) a. R i g i d A i r c r a f t w i t h L i n e a r Equations of Motion From equations (106) the c h a r a c t e r i s t i c equation f o r the 101 r i g i d a i r c r a f t excluding n o n - l i n e a r i t i e s i s : or The Routh-Hurwitz c r i t e r i a f o r a quadratic character-i s t i c equation are s a t i s f i e d , namely, th a t Pfl , P, and (P( P 2 ) are greater than zero., The r i g i d a i r c r a f t i s therefore s t a b l e , The roots of.the equation are h ^ O and'\z=~ OJ431 \u00C2\u00B1L(o.473) I t should be noted t h a t the roots as shown are f o r the non-dimensional system. The period of the motion represented by the roots can be obtained as f o l l o w s : U IOOO Therefore, i n ordin a r y time > ) j 2. - -1.145-1(3.78$). The time to h a l f amplitude \" Q ' f f = 0.602 seconds. 1.145 - \u00E2\u0080\u00A2 -The per i o d of the o s c i l l a t i o n i s 3^3^ SS !\u00C2\u00BB66 seconds. This i s q u i t e small compared to the peri o d of a t y p i c a l phugoid mode of 200 seconds. As was noted i n the t e x t , equation (106) can be s i m p l i -f i e d by d i s c a r d i n g terms th a t are of much lower order of magni-102 tude compared t o ^ and i g . In that case, the ch a r a c t e r i s t i c equation i s modified to be: i . e . ?) [ t f + h(o.2B2) 4- (0.23S)j-O The accuracy of this approximation can be tested by calcu l a t i n g the percentage difference between the results f o r the two cha r a c t e r i s t i c equations. The roots of the equation are: 7 \ 3 ' 0 ) \ 2 = -0.14-1 \u00C2\u00B1l(OA6,S) The time to half amplitude - 0.612 seconds. The period of the o s c i l l a t i o n i s 1.688 seconds. The percent error i n time to half amplitude i s 1.66$ and i n the period, 1.69$. Thus, the approximation has only a n e g l i -gible effect on the accuracy. The equilibrium angle of attack i s : ot\u00C2\u00BB - \A8\u00C2\u00B0 This i s the angle of attack required f o r straight and l e v e l f l i g h t . b. Rigid A i r c r a f t with 1 Non-Linear Equations of Motion It was indicated previously (page75\") that the non-l i n e a r equations of motion give r i s e to two more equilibrium values of the variables and that these values might be very-large. The values can be obtained from the solution of i . e . , since i n steady l e v e l f l i g h t , @=otr . Urshz = 0.756 * r S ( x =\u00C2\u00B149.8\u00C2\u00B0 For the present case, with the r e s t r i c t i o n thato( r^10\u00C2\u00B0, these values have no physical s i g n i f i c a n c e . Moving the centre of gravity affects both b( and b^ . . The equilibrium value of the angle of attack corresponding to the reference condition of straight and l e v e l f l i g h t i s obtained from -3 53 33 A l l of the c o e f f i c i e n t s are r e a l and po s i t i v e ; therefore, there i s only one r e a l value of wVj^ that s a t i s f i e s the equation which i s : ^rS-3 e 1\u00C2\u00BB^7\u00C2\u00B0 Comparison of t h i s value with 1.48\u00C2\u00B0 obtained f o r the li n e a r case shows that there i s j u s t i f i c a t i o n f o r considering the aerodynamic forces to be l i n e a r f o r a small perturbation analysis of the s t a b i l i t y of a r i g i d a i r c r a f t . However, f o r 104 a wingless m i s s i l e , the term a f i s markedly reduced and the non-l i n e a r terms become much more important. For t h i s case, a, = 3 . 5 7 and the r i g i d - l i n e a r e q u i l i b r i u m angle of a t t a c k i s <*rs - 7 .96\u00C2\u00B0 ' 3. . I f n o n - l i n e a r i t i e s are i n c l u d e d , the a p p l i c a b l e equation i s 3,57 Ur* +0.243<*\u00C2\u00A3 +_4-<-\u00C2\u00ABr! - C?\u00C2\u00AB4_-6 _ 3 _-j - 3 This shows th a t n o n - l i n e a r terms c o n t r i b u t e substan-t i a l l y i n the a n a l y s i s of wingless v e h i c l e s as the r e s u l t of ne g l e c t i n g n o n - l i n e a r terms i s to.over-estimate steady s t a t e l i f t by a s i g n i f i c a n t amount. percent e r r o r = 7 ' 9 ^ ^ 9 4 x 100$ = 1 4 . 7 \u00C2\u00B0 c. E l a s t i c A i r c r a f t ! \u00E2\u0080\u00A2 The r i g i d body shape constants were presented i n Table I. Now, i n a d d i t i o n , the e l a s t i c constants must be c a l c u l a t e d . For given s t i f f n e s s and mass d i s t r i b u t i o n s , the mode shape i n general can be obtained by a numerical a n a l y s i s , matrix i t e r a -t i o n or the Stodola Method [ 3 ] , and i n c e r t a i n s p e c i a l cases from the exact s o l u t i o n of the d i f f e r e n t i a l equation, where J i s the mode shape, when normalized gives (f). E I , m and 105 J are, i n general, a l l f u n c t i o n s of X. Thus, determination of the mode shape f o r the fuselage i n f r e e - f r e e v i b r a t i o n presents no problem. The determination i s not the purpose of t h i s Example, so the mode shape i s assumed to be of the form u-z*L0) J } Note that t h i s form of expression s a t i s f i e s the end co n d i t i o n s . In p r a c t i c e , any f u n c t i o n s a t i s f y i n g the boundary cond i t i o n s i s a v a i l a b l e as an approximate p o s s i b l e choice. In Part I I I , a procedure f o r choosing the best mode was given . For conciseness, l e t the s e l e c t e d mode be the best mode ob-tai n e d a f t e r going through t h a t process of t r i a l . The mode i s normalized by the amplitude of the nose. The e l a s t i c constants must be i n t e g r a t e d i n two p a r t s , namely, from O/o X^tQ, and X^jt^ 7o / . The two parameters i n the expression f or $, 7^L0 and , are chosen to minimize the TO n a t u r a l frequency i n accordance w i t h the Rayleigh Method. In a d d i t i o n , using the c o n d i t i o n of zero momentum and adopting a t r i a l and e r r o r approach gives values of the unknown parameters as _ , ^ ,^ = O.ZS\" 106 With these v a l u e s , the c o l l e c t i v e e l a s t i c constants can be evaluated. They are l i s t e d i n Table I I . The e q u i l i b r i u m values of o(r and^ are obtained from equations (124). ^sooS'6B This corresponds t o the reference values of the v a r i -ables and i n d i c a t e s t h a t the i n c l u s i o n of f l e x i b i l i t y i n t o the non-linear equations does a l t e r the steady s t a t e values, but not by a s i g n i f i c a n t amount. The equations 107 TABLE I I NUMERICAL VALUES OF THE PARAMETERS USED IN THE ILLUSTRATIVE EXAMPLE - ELASTIC CONSTANTS Para-meter Value Para-meter Value Para-meter Value Para-meter Value K, 1.835 1.267 c,_ - 0.424 c_ 0.921 K 2 3.68 D, 0.652 2.222 \u00C2\u00B0? 3.68 K3 0.921 D,o P.492 Ci4- -132.5 c+ 0.16 K 4 0.160 *s 9.80 C,s 0.212 cs 44.0 *S 1.55 E. - 6.694 - 1.27 c/o 1.55 K 6 - 2.21 E 7 - 3.275 0.934 cri - 2.21 K 7 - 0.356 - 0.729 N 230 C / 2 - 0.356 44.0 h 0.368 (EI), 6.31 -86.6 h - 86.6 F 7 0.365 k 0.205 -106.3 -106.3 G_ - 0.399 a i o 1.348 A* - 0.716 G 6 - 1.39 a \u00C2\u00AB - 9.434 A 7 -132 G 7 - 0.021 - 3.72 As - 4.472 H_- 0.094 a )3 2.222 A, - 1.35 - 0.049 a ) t + -132.5 Aio - 0.424 H 7 0.001 a/s 0.212 B* - 0.702 0.040 a i u - 1.27. B 5 : 1.85 2.54 0.934 B 6 1.27 0.626 b i o 3.539 B 7 - 0.934 - 0.624 b\u00E2\u0080\u009E 1.717 - 28.33 O/o - 4.472 b,z 0.766 D 7 35.9 - 1.35 c i 1.835 10$ are used t o f i n d the other p o s s i b l e e q u i l i b r i u m angles. J These are p h y s i c a l l y impossible v a l u e s . Thus, the non-l i n e a r i t i e s can be s a f e l y neglected f o r t h i s supersonic a i r -l i n e r c o n f i g u r a t i o n . A small p e r t u r b a t i o n a n a l y s i s about the reference f l i g h t c o n d i t i o n i s adequate f o r the s t a b i l i t y a n a l y s i s . Equations (127) to (132) give expressions f o r the c o e f f i c i e n t s of the c h a r a c t e r i s t i c equation. These can be evaluated f o r the l i n e a r case as: ? s - 6.79 x 10^ P 4 = 1.736 x 10s\" P 3 = 1.516 x 10* P 2 = 0.0132 x l(f ?t = 0.02666 x I d 5 \" P- = 0 The c o e f f i c i e n t s P.- , ? 4 , P_ , and R, are gre a t e r than zero, but R* i s negative (-4.86 x 10^, i n d i c a t i n g an i n s t a b i l -i t y i n accordance w i t h the Routh-Hurwitz c r i t e r i a f o r s t a b i l i t y . The same c o n c l u s i o n can be a r r i v e d at by making use of the s t i f f n e s s c r i t e r i o n derived i n the a n a l y t i c a l development (138). This can serve as a check o r , more p r o f i t a b l y , i t can 109 save a s i g n i f i c a n t amount of computation involved i n evaluation of P 5 , P^ , . ,.P by correctly predicting the i n s t a b i l i t y as shown below: For s t a b i l i t y , \ Eimfofajfclx ^ 1^15\" In the present case , f(B t)^((p'rfjf'dx =^'8S o To make the a i r c r a f t stable, two methods of a l t e r i n g EI77) are possible: 1. at constant ( E X)^ , adjust i t s dependence onX, 2. f o r the same dependence on/, change (pT)^ . The c r i t e r i o n i s s a t i s f i e d by making E X ^ =3.81. This i s achieved by changing the fuselage thickness from .1 f t . to 1 inch. The differences of numbers of the same order of magni^ tude somewhat affect the accuracy. However, f o r diffe r e n t mode shapes t h i s need not be the case. V. CONCLUDING REMARKS From the a n a l y t i c a l study presented here, f o l l o w i n g con-c l u s i o n s can be made concerning the e f f e c t s of n o n - l i n e a r terms i n the equations of motion and the f l e x i b i l i t y of the fuselage: 1. Existence of e q u i l i b r i u m values of the v a r i a b l e s other than the reference f l i g h t c o n d i t i o n i s p o s s i b l e . This r e q u i r e s p e r t u r b a t i o n a n a l y s i s of the s t a b i l i t y t o be c a r r i e d out at more than one e q u i l i b r i u m p o i n t . 2. Presence of n o n - l i n e a r terms r e q u i r e the s o l u t i o n of the higher degree c h a r a c t e r i s t i c equation. 3. The n o n - l i n e a r terms have n e g l i g i b l e e f f e c t i n the s t a b i l i t y study of a r i g i d a i r p l a n e thus making i t p o s s i b l e to neglect them as i n the conventional a n a l y s i s ; but i n the case of wingless v e h i c l e s (e.g. m i s s i l e s ) they do c o n t r i b u t e sub-s t a n t i a l l y and hence must be r e t a i n e d . 4. The i n f l u e n c e of f l e x i b i l i t y appears i n the a n a l y s i s at s e v e r a l p l a c e s . I t a l t e r s the pressure d i s t r i b u t i o n on the body and hence the l i f t and p i t c h i n g moment. This i n t u r n adds two more e q u i l i b r i u m p o i n t s and r a i s e s the degree of the c h a r a c t e r i s t i c equation by two f o r every e l a s t i c mode considered. 5. F l e x i b i l i t y can have s u b s t a n t i a l e f f e c t on the s t a b i l i t y of a high-speed slender-bodies v e h i c l e as shown i n the numerical example. For the p a r t i c u l a r c o n f i g u r a t i o n con-s i d e r e d , a s t a b l e \" r i g i d \" v e h i c l e turned out unstable when the I l l f l e x i b i l i t y was inclu d e d . 6. The a n a l y s i s presents two u s e f u l approaches to the s t a b i l i t y problem. From the f i r s t approach, the s t a b i l i t y of a known a i r c r a f t could be analyzed by the Routh-Hurwitz c r i t e r -i a w i t h the equations presented here. The second approach leads t o the e x t r a c t i o n of a l i m i t i n g s t i f f n e s s d i s t r i b u t i o n on the b a s i s of s t a b i l i t y boundaries and assumed fuselage bending mode. The use of Rayleigh's method assures small and conser-v a t i v e e r r o r . 7. The theory as developed i s a p p l i c a b l e t o the s t a b i l -i t y study of e i t h e r a supersonic a i r l i n e r or m i s s i l e . The e f f e c t i v e n e s s of the a n a l y s i s i s increased considerably by making the c o n f i g u r a t i o n and l o c a t i o n of the wing and f o r e -plane q u i t e general. 8. The lar g e amount of computation work involved makes the method p a r t i c u l a r l y s u i t e d t o a d i g i t a l computer. VI. RECOMMENDATIONS FOR FUTURE RESEARCH The study of f l e x i b i l i t y effects in aerodynamics i s a r e l a t i v e l y new development that has gained importance due to the constant increase i n speed of both c i v i l and m i l i t a r y vehicles. I t , therefore, presents tremendous room f o r both a n a l y t i c a l and experimental investigations. In the l i g h t of the f a c i l i t i e s available i n the Department investigations which are mainly a n a l y t i c a l i n nature are mentioned here. 1. The present analysis i s r e s t r i c t e d to a. pitching motion alone, b. amplitude of motion up to 10\u00C2\u00B0, the l i m i t a t i o n being introduced by the nature of the aerodynamic theory used and the degree of non-linearity included, c. r i g i d l i f t i n g surfaces. An attempt at removing these r e s t r i c t i o n s should l o g i c a l l y form the f i r s t step i n future investigations. 2. The s t i f f n e s s c r i t e r i o n obtained in the present analysis i s i n the in t e g r a l form. The mode shape required f o r the analysis of a f l e x i b l e body i s usually obtained by a matrix i t e r a t i o n process. This suggests that the solution of the present problem ( i f amenable to matrix formulation) may be handled most e f f e c t i v e l y by numerical methods. 3. Attention should be given not only to the motion i n pure degrees of freedom but also to t h e i r coupling e f f e c t s . 113 4. The c l a s s of problems r e f e r r e d to as response problems should be i n v e s t i g a t e d by s u b j e c t i n g the c o n f i g u r a t i o n under study t o d i f f e r e n t f o r c i n g , f u n c t i o n s w i t h the u l t i m a t e aim of o b t a i n i n g the response of an e l a s t i c v e h i c l e t o a random e x c i t a t i o n . 5. The study of response problems using an analogue computer may provide u s e f u l i n f o r m a t i o n which may be d i f f i c u l t , i f not impossible, t o o b t a i n a n a l y t i c a l l y . APPENDIX I . COLLECTION OF SHAPE CONSTANTS Throughout t h i s t h e s i s use was made of \"shape constants\" that depended only on the geometry and e l a s t i c p r o p e r t i e s of the a i r c r a f t and hence are constants once the c o n f i g u r a t i o n i s s p e c i f i e d . These shape constants are l i s t e d here f o r easy reference. Also the expressions f o r the t o t a l aerodynamic co-e f f i c i e n t s Cu*_, C M*, as w e l l as C-^ . , the g e n e r a l i z e d f o r c e co-e f f i c i e n t , are r e w r i t t e n f o r convenience. + ^[Csl +i+-Fz -htt3-- (7cj -Fur) E?, -(Zj -7^)6^J+ DG[bH+\4z+F3 -(X^-TUT) F_j -{7rj-7p) Cr^j 4-^r^jOhJ f ^ 4 H ^ Y ^ - ^ E r V y ^ - ^ ^ / ^ D ^ - [ D ^ 4 / 4 ^ -- * rM4DeOrJ+-D*r^ 4- tfSj [>-J -h^[kj +c< r z 5 j i k i j t>^j i h v j A 4 . - -.Vol-/) r *1 - -mfrt*>^<*+&.4a>4-f*?l -> A X - 3 %h Jo 1 dT 1_ 7_ 4 X . L i r ' ^ ^ ' ^ ^ ^ 2 _ ? c_7 ' s_ Jo L _7 ^ \u00E2\u0080\u00A2 - * ^ - ^ . _ _ ) t)z - -z f \ -Vcq - m/ \ D 7 --6\u00C2\u00A3DCSS Zfat?)7(7cj-x)J (*ur))(J7ur - h) 1 St y0 _y J 7 S t s S Jo J J \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 = C, / \u00C2\u00A3 / -r-Cr, ^7 r ^ 7 \u00E2\u0080\u00A2G, *z *3 61, - C,i f ^ , *n -c\u00E2\u0080\u009E ^/z. \" CfT +\u00C2\u00A37 +6r7 r 1 Cr* 2 e k -\u00E2\u0080\u00A2 D, + F, +M-,-(X\u00C2\u00A3j -Xur) B, ~(7y -^)G-, = D 2 + F 3 4 M 2 -(Tcj ~7ur) B3-(xkj -7j) - D 4 * ^ f -(Vc4 -x LAS') fc4-(T^j -*4)Cr4_ } \u00C2\u00A3>y # r e * not-a-Mo<^e>L.ie.c/. hi-L = Bio+F7 +H7 - (Xcq -Yur) Ey - (Ycg -Xj) fr7 Co- \u00E2\u0080\u00A2I* -Id \u00E2\u0080\u00A2Ir, Cm &ra, not a^Z/occifhcJ. J J J tfio* - ft 10 \"^ft^^ij BIBLIOGRAPHY 1 A e r o e l a s t i c i t y i n S t a b i l i t y arid Control, J.B. Rea Company Inc., WADC Technical Report 55-173, March, 1957, pp. 220,223. 2 Miles, W.J., \"Unsteady Flow Theory i n Dynamic S t a b i l i t y , \" Journal of the Aeronautical Sciences, January, 1950. 3 Bisplinghoff, R.L., Ashley, H., Halfman, R.L., Aero-e l a s t i c i t y , Addison-Wesley, 1955, pp. 420, 111, 132, 139, 154. 4 Sears, W.R., General Theory of High Speed Aerodynamics, High Speed Aerodynamics and Jet Propulsion, Vol. VI, Princeton University Press, 1954, pp. 64, 452, 278. 5 Van Dyke, \" F i r s t - and Second-Order Theory of Supersonic Flow Past Bodies of Revolution,\" Journal of the Aeronautical Sciences, March, 1951, p. 161. 6 Donovan, A.F., Lawrence, H.R., Aerodynamic Components of A i r c r a f t at High Speeds, High Speed Aerodynamics and Jet Propulsion, V o l . VII, Princeton University Press, 1957, PP. 274, 278, 238, 245. 7 Kelly, \"The Estimation of Normal Force, Drag and Pitching Moment Coe f f i c i e n t s f o r Blunt Based Bodies of Revolu-t i o n at Large Angles of Attack,\" Journal of the Aeronautical Sciences, August, 1954, p. 549. 8 A l l e n and Perkins, A Study of Ef f e c t s of V i s c o s i t y on Flow Over Slender Inclined Bodies of Revolution, NACA Report 1048, 1951. 9 Schwabe, M. , Pressure D i s t r i b u t i o n i n Non-Uniform Two-Diaiensional Flow, NACA TM 1039, 1943. 10 Jones, R.T., Properties of Low Aspect Ratio Pointed Wings at Speeds Above and Below the Speed of Sound, NACA Report 835, 1946. 11 M i l e s , J.W., \"On Non-Steady Motion of\"Slender Bodies,\" A e r o n a u t i c a l Q u a r t e r l y , 2, 1950, p. 183. 12 N i e l s o n , M i s s i l e Aerodynamics, McGraw-Hill, I960. 13 E t k i n , B., Dynamics of F l i g h t S t a b i l i t y and C o n t r o l , Wiley, 1959, PP. 138, 94, 152, 193, 371. 14 Cunningham, W.J., I n t r o d u c t i o n t o Nonlinear A n a l y s i s , \u00E2\u0080\u00A2 McGraw-Hill, 1958, p. 63. 15 Timoshenko, S., V i b r a t i o n Problems i n Engineering, D. Van Nostrand Go. Inc., 1937, pp. 386, 388. "@en . "Thesis/Dissertation"@en . "10.14288/1.0105611"@en . "eng"@en . "Mechanical Engineering"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "On the dynamic stability of flexible supersonic vehicles"@en . "Text"@en . "http://hdl.handle.net/2429/38867"@en .