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On the dynamic stability of flexible supersonic vehicles Drummond, Alastair Milne 1963

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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—10° 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 — 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 — 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 • for c e - -0.0056 35 (— 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 £o 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 ° l f«uasb I— 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 = • ^  , , 31 C-M£ 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«--' 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 •** 83 D c c r o s s - f l o w drag 33 E Young's Modulus 118 — E I EI non-dimensional s t i f f n e s s = — - — — ^ 119 7<o5bU L * Numbers i n r i g h t hand margin r e f e r t o the equation where the symbol f i r s t occurs. X F viscous f o r c e 33 3r g e n e r a l i z e d f o r c e 71 X area moment of i n e r t i a of fuselage i n bending 116* L* l i f t f o r c e i n negative Z d i r e c t i o n 45 L* reference l i f t 33 L~ l e n g t h of fuselage 17 M Mach Number = ^ 5$ M p i t c h i n g moment 2& M«V. moment of volume of fuselage 20 Mj' g e n e r a l i z e d mass i n mode 71 A/j = a / 'QKtx)yt V 7 ) ^* 9 4 Jo J N 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 18 c o e f f i c i e n t of i n c h a r a c t e r i s t i c equation 103 t e s t f u n c t i o n f o r s t a b i l i t y 103 R fuselage radius 17 R non-dimensional fuselage r a d i u s = ^  25 R* Routh's Disc r i m i n a n t 103 S area 21 Vo/. volume of fuselage 20 V.MX volume moment of i n e r t i a of fuselage 20 (J. v e l o c i t y of centre of g r a v i t y i n X d i r e c t i o n 1 W v e l o c i t y of centre of g r a v i t y i n 2 d i r e c t i o n 79 X aerodynamic 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 17 <£. v e l o c i t y of sound 1 t> 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 •PoO 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 ^•(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 =.•-[-_ 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 = ——-—83 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 ° 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 = — 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 ¥(pL0 non-dimensional p o s i t i o n where 4> (X) ~° j -r-x i i <X angle of attack - angle between x axis and velocity U from zero l i f t l ine = Q - £ - ^ - « «  8 (3 = N / M M ' 1 ^ reference climb angle - measured from horizontal 83 P semi-vertex angle of l i f t i n g surfaces 59 £ non-dimensional wing span = J^, 58 y 7 " Laplace operator (^+ayi.-^.) in Cartesian co-ordinates 1 /Tj fract ional reduction in C^c from two-dimensional tests 34 O angle of pitch 0 2 ©*- total angle of pitch including reference climb angle - V +• 9 . 80 7\ coefficient to be determined from characteristic equation 106 IT") yU. non-dimensional mass = j> 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 £ 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 £j» j ^  singular (equilibrium) point 104 T t o t a l 86 V viscous 36 X,"'«,t indicate p a r t i a l derivatives 1 UT wing 58 oO free stream 1 Superscripts • (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 ° < 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 » 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 «3 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 •..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 • ....: ...  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 . > • ^ 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 — 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 £ 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 © 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, _ *}£2p +^p^x +.a<P^y f ^ f a i (6) = yz(-s/h£>)& •hX-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^^ +• 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<andO< r Subs t i tu t ion from (8) in to (7) and re-organizing y ie lds (9) Note the e l a s t i c contributions ^ and ^ modify only the l i n e a r , steady-flow terms i n the pressure equation. Now th a t the pressure r e l a t i o n has been transformed i n t o C a r t e s i a n co-ordinates f i x e d t o the body, the general case of an a r b i t r a r y body undergoing plunging and p i t c h i n g can be handled. However, i t i s convenient, f o r the study of slender bodies of r e v o l u t i o n t o use a c y l i n d r i c a l body co-ordinate system (x,r,'fi>) 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© z = rsin© Therefore , = ^ +dtf£v +ZJ> 2jb - & co% 9 f <fz s/h 0 and 4^ = S<P£x +.^P^/ ^ S f 5 2 r = rs i 'n0+<£r^osg? Thus % = ^ x tf+Vf^VS+^fe (10) These r e l a t i o n s (10) are i n s e r t e d i n t o equation (9) to give the equation f o r the pressure d i f f e r e n c e Ap. 'D^-A^+r^') (11> 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(«r 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 , & ) = - ± ^ f#x.)In 2.(y-yQdv, - cos® 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 » $tt » 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_£ =- Plcos0 + B(l-4-<>ir>~i®) (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.<£)=-2.TT ^ T - 0 , , „ '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 - ( £ 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 ° <ix 3Jodx -* = -L(ldNcx) VcJv + LFea M ( 2 9 ) Noting t h a t : / X ^ R ^ J x Volume Moment of I n e r t i a y* = o " T P [ X^RdTscix . L Si>._ 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 ^ ? £ ^ -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 £ x * i s n e g l i g i b l y small and ^-NF • 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 £T 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, £j> = /„2 , 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 ± ^ 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 cosc<r and sino< r by the f i r s t two 36 terms of the i n f i n i t e s e r i e s ( v a l i d f o r o<r4:\0 ) g i v e s : the l i f t c o e f f i c i e n t can be w r i t t e n as: C^.C^^-oir-j-Cx+tUr-ob?) (46) 3 No terms of order higher than o<r are r e t a i n e d ; the f o l -lowing few pages complete the development of the l i f t and p i t c h i n g moment c o e f f i c i e n t s . Equation (27) together w i t h (43) y i e l d s the t o t a l CNp: C N p = ^ P s s A * L"jj4f" " °< r&J£tfm xJr-3*r%:fcfk^xdij + + %-£f^*-)2K^xJ (47) The rearrangement of (47) g i v e s : CNF-(Z)^+(,)fL ^ ( ^ ) ^r(C^L - «rfs(^ss Ifj&flWrJvJ-oilfr^^catftjttyT)7c/F) -(48) 37 The quantities i n brackets are constants f o r a given shape; hence, equation (4#) can be written as: + fl7 +flg % +- Af & +Alo g. 43 ( 4 9 ) * The expression f o r the a x i a l force c o e f f i c i e n t was obtained before, which was: ^ Jo C dx " U U Jo ed>7 + 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, «r ^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 « 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 £ , fe£ , 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<Xur ( fel>(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 * = | ^ « r 7 T ^ R ^ t K u r [ ( ^ - | ) + ^ ( i ^ » 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<z-^ r ° k- COL-> U # , . c u)~E-GK ~ o\ 0 L CO £. = 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 . , •3 (65)* S i m i l a r l y &M^=&^^M&urj°<u-C'#*r-%) +°<ur^(l>'«r-^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 §f and ^£^£ 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 , §^^.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, ^>^, • • • . ,^ 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) • 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/'$M^tfJr ( 7 2 ) and ^v= component i n the y d i r e c t i o n of u n i t vector normal to the body s u r f a c e . 50 The x and z components of the g e n e r a l i z e d f o r c e would have to be considered i f f l e x i b i l i t y of the s t r u c t u r e were allowed i n the x and z d i r e c t i o n s . The only c o n t r i b u t i o n to i s from the aerodynamic pressure. Considering the p o t e n t i a l f l o w , equation (16) leads t o : f' fz7r - — l-= - L3* ^jj7- J J j~ f)(?) cos Or Bm(i- Rcos0($) 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 °fJpy 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©; 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* =• 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 •o !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 -£0 lantisinB « -L*±L?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£z) =rr\U[*r(n-°<?)-fr] (83) I*" because i n the h o r i z o n t a l f l i g h t , L0 must equal the a i r -c r a f t weight. Equation (83) i s dimensional and a con-s i d e r a b l e convenience r e s u l t s i f non-dimensional equa-t i o n s are used. This i s achieved by i n t r o d u c i n g the f o l l o w i n g s u b s t i t u t i o n s i n (83) : = I DB t t * = non-dimensional time ( a i r s e c s ) =» L. U t* L 57 Dividing the above equation by L-^JJ^S^ gives: The pitching moment equation (82) i s non-dimensionalized in a s i m i l a r manner by putting B - t'B £ S t L.3 Dividing throughout by 1 f U\ L and denoting t * = •=• , 2 U It now remains to c o l l e c t the C^s andC^+'s from the fuselage, foreplane and wing, and insert them i n (84) and (85). The next section i s devoted to t h i s task. 2,6 C o l l e c t i o n of Equations In the present analysis a l l of the aerodynamic co-e f f i c i e n t s and non-dimensional variables have been based on fuselage base area S b and length L. Normal practice has been to non-dimensionalize the equations on the basis of wing area and some ch a r a c t e r i s t i c reference length, usually the chord. The reason f o r t h i s departure i s to make the equations as presented here r e a d i l y a p p l i c a b l e e i t h e r to the supersonic a i r l i n e r case or the m i s s i l e case, the l a t t e r being achieved by merely d e l e t i n g the wing and foreplane c o n t r i b u t i o n s from the equations,, Since i n t e r f e r e n c e e f f e c t s have been neglected, the t o t a l l i f t and p i t c h i n g moment of the a i r c r a f t can be w r i t t e n as the sum of the c o n t r i b u t i o n s of the fuselage wing and fore p l a n e . In c o e f f i c i e n t form, the t o t a l l i f t and p i t c h i n g moment are: C£T--CL%+^^1* [66) C-h* ^Ch+F + c h ^ <^H* -(x^-fyCi^'C^-x^C-l*^ (87) The, equation of motion f o r the a i r c r a f t i n the i n -d i r e c t i o n was given by equation (84). The fuselage l i f t , wing l i f t and foreplane l i f t are obtained from equations (54), (65) and (69) r e s p e c t i v e l y . S u b s t i t u t i n g these c o e f f i c i e n t s i n t o (84) and making use of (86) g i v e s : 59 The equation of motion for the a i r c r a f t i n p i t c h i s given by equation (85). The aerodynamic contr ibut ions of the fuselage, wing and foreplane are given by equations (57), (66) and (70) r e spec t ive ly . Subs t i tu t ing these coef f ic ien t s into (85) and noting (87) y i e l d s : 2 4 tf& = <X r[p, i F+ff, -(7ey7^n - f % -xV) E ^ I X r [ b * ¥ F, + W2 -+ ^6 - f X g - x ^ - f c y -XMTJ^] 4Efgjp, 0^^ 7 + Ff Y x ^ < r ; - ( 7 y - x ^ W ) The equation for the general e l a s t i c degree of freedom (71) i s s t i l l i n a dimensional form. Introducing the reduced frequency parameter fc=and d i v i d i n g throughout byi-^uSfcL , g ives: Mjf/ +. fay') hj % j ~ 5t -: t (90) Now, M j=L xjf<JfWnn^^ (91) Also J ^ j - ^ b ^ . . Therefore (91) becomes, -^T^Z^U ^ ^ ( l V - i i ^ u ^ L ^ J (92) L e t t i n g = 2- f ^ /o J equation (93) takes the f i n a l form Introducing the r e l a t i o n f o r from (78) g i v e s : For the a n a l y s i s of these equations of motion (88), (89) and (96) a f u r t h e r a b b r e v i a t i o n of the c o e f f i c i e n t s i s d e s i r a b l e i n order that the p h y s i c a l nature of the s o l u t i o n may not be obscured by the use of a l a r g e number of c o e f f i c i e n t s . In a d d i t i o n , i f the equations are r e -arranged, a symmetry e x i s t s that i s o l a t e s the non-l i n e a r i t i e s i n a concise form. Equations (88), (89), (96) can be r e w r i t t e n as 61 o<r(a,) +Dofr^fa+VO +Dfi^3"3A) +tP©fc«0 Mpfa)* o(?D9(a6) + f <*V t>©D% +o<r D © <fcrn) f 6^/5) « 0 . (97) * + f / b ( 0 ) + B ^ ^ i ) + ^ f ; ^ ( X ) ^ ? y ^ ) + 0 ( r ^ / M = O (98)* o< r Cc,) 4lVrrc 2 ) + T)etfc3) -UPefc*) f V ^ r ) +%fat>-*sNj) + + rCn-Nj') +0^ . + o ^ & £ 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) +• * j j'fra -W/^) +«rbd(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«<r^O 4 - D 6 > r c 3 ) + D ^ r ^ ) +• *$i\(«o~Njky)Kii«r\ 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<rs., ©s. , , where the s u b s c r i p t Sj represents the e q u i l i b r i u m value a s s o c i a t e d w i t h the j s i n g u l a r p o i n t . Non-linear systems w i l l , i n ge n e r a l , have more than one s i n g u l a r p o i n t . Step 2 - Inv e s t i g a t e the e f f e c t s of small changes i n cK r , 0 , and ^  n e a r e r , 0 S . , and j ^ . , by p l a c i n g J J v l si and s u b s t i t u t i n g these r e l a t i o n s i n t o the o r i g i n a l n o n - l i n e a r A A A equations. Here, c*r , 0 , and !~ are sm a l l changes away from the s i n g u l a r i t y . This leads to a set of l i n e a r equations i n terms of perturbations^,©, and ^ . The c o e f f i c i e n t s of the p e r t u r -b a t i o n parameters w i l l depend on p(r 4., 6^ ., and^ s.. , Due to assumptions commonly as s o c i a t e d w i t h p e r t u r b a t i o n theory, the r e s u l t i n g equations are l i n e a r . Step 3 - Find the c h a r a c t e r i s t i c equation of the set of equations i n o\r , Q, and € at each s i n g u l a r i t y . This i s ob-ta i n e d by assuming a s o l u t i o n of the form X = X D ^ where X Q i s a constant determined by the i n i t i a l c o n d i t i o n s and 7s* i s a co-e f f i c i e n t to be determined. This assumption of p e r i o d i c motion 67 i s v a l i d i n the reg i o n of i n f l u e n c e of the s i n g u l a r i t y f o r small disturbances. When these s o l u t i o n s are s u b s t i t u t e d i n t o the l i n e a r equations of o<f , Q, and £ the determinant equation f o r a n o n - t r i v i a l s o l u t i o n r e s u l t s which, when expanded, y i e l d s an a l g e b r a i c polynomial equation i n ? \ . Step 4 - F i n d the signs of the r e a l parts of the roots of the c h a r a c t e r i s t i c equation corresponding to each singu-l a r i t y . These signs must be negative f o r asymptotic s t a b i l i t y . A p p l i c a t i o n of the Routh-Hurwitz c r i t e r i o n [13] t o the co-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 y i e l d s i n f o r m a t i o n about the s t a b i l i t y . U n fortunately, t h i s c r i t e r i o n merely gives an answer to the question, " I s the system s t a b l e or not?" I t does not t e l l the designer how s t a b l e the system i s . For t h i s information a Nyquist c r i t e r i o n i s more s u i t a b l e [13]. However, i f data on f l e x i b i l i t y can be supplie d to a designer as a c r i t e r i o n beyond which i n s t a b i l i t y w i l l occur, he i s i n a much b e t t e r p o s i t i o n to attempt a r a t i o n a l design of the s t r u c -t u r e . This " s t a b i l i t y c o n s t r a i n t " on fuselage f l e x i b i l i t y would be only one of the many competing c o n s t r a i n t s t h a t the designer would have to apply i n a c t u a l l y a r r i v i n g at the l i m i t i n g c r i t e r i o n . With reference t o Step 4, the Routh-Hurwitz c r i t e r i o n f o r s t a b i l i t y can be a p p l i e d d i r e c t l y to 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. The roots themselves need not be evaluated, thus c o n s t i t u t i n g a lar g e saving i n e f f o r t f o r an equation of l a r g e degree. However, s e v e r a l l a r g e determinants 68 must be evaluated. The case where a q u a r t i c equation r e s u l t s w i l l be discussed here because t h i s a p p l i e s to the present problem w i t h the f i r s t e l a s t i c mode includ e d . The degree i s r a i s e d by two f o r every a d d i t i o n a l e l a s t i c mode. Consider the c h a r a c t e r i s t i c equation, 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 no root to have a p o s i t i v e r e a l part ( i n s t a b i l i t y ) are that each of a s e r i e s of t e s t f u n c t i o n s should be p o s i t i v e . This requires Therefore must a l s o be p o s i t i v e . The quan t i t y R(P^P2-P*77")-Tf T£ i s commonly known as Routh's Discriminant and i s given the symbol R*. I f Exchanges 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 causing one divergence ( s t a t i c i n s t a b i l i t y ) . I f R* changes from p o s i t i v e to n egative, one divergent o s c i l l a t i o n appears i n the s o l u t i o n (dynamic i n s t a b i l i t y ) . Thus, Po=0 and R*=0 define boundaries between s t a b i l i t y and i n s t a b i l i t y . These boundaries must be determined f o r each s i n g u l a r i t y S j . I f j=3, three q u a r t i c equations must be analyzed; i t i s not guaranteed that i f the a i r c r a f t i s s t a b l e at one s i n g u l a r i t y , i t w i l l be s t a b l e at a l l s i n g u l a r p o i n t s because the c o e f f i c i e n t s depend on the e q u i l i b -rium values of<Xr,&, a n d . The usual method of d i s p l a y i n g the s t a b i l i t y boundaries 69 i s to r e l a t e two of the important s t a b i l i t y d e r i v a t i v e s at R*=0. An example of t h i s i s shown below. STABLE Figure 12 Example of a S t a b i l i t y Boundary Examination of the equations of motion shows th a t they involve the e l a s t i c co-ordinate and n a t u r a l mode shape (p(?) . The mode shape appears i n the c o e f f i c i e n t s of the character-i s t i c equations, and i t i s completely determined i f 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 are known. These data a l s o y i e l d at the same time the n a t u r a l frequency c o r r e s -ponding t o the n a t u r a l mode. B i s p l i n g h o f f [3] gives methods f o r f i n d i n g these mode shapes and f r e q u e n c i e s . Thus, the equations P c=0 and R*=0 can be thought of as equations f o r 0LCx) where the s u b s c r i p t L means a " l i m i t i n g " value of 4>(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 : °<%(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 $£ir*. 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<r= o<rSl 4-oir (104) 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 (100) and (101) y i e l d s : S e t t i n g o(r-o/ro<z ^9~B0<2. g i v e s , f o r n o n - t r i v i a l s o l u t i o n , Expansion of the above determinant r e s u l t s i n the c h a r a c t e r i s t i c equation of the set of simultaneous d i f -f e r e n t i a l equations, which i s : (105) (106) 73 There are three roots to t h i s equation, ^ i=0> ?i/ a n c* A 4. The root ?\ = 0 i n d i c a t e s that© 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, £>,<0 means th a t "2-£L 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<r(l{ + ) -O The term b( must be negative f o r s t a t i c s t a b i l i t y ; t h e r e f o r e , b<- must be p o s i t i v e f o r r e a l e q u i l i b r i u m values o f c * . r t o e x i s t . This i s p o s s i b l e only i f Xcc, y ~b- . L S t \ 2 J 3 J t, = D, + F j +H, -CxcyZ+)Cr, S/'J-Zur)E, A l l the b a s i c l o n g i t u d i n a l s t a b i l i t y design parameters are contained i n b ( , hence once the p h y s i c a l p r o p e r t i e s of the c o n f i g u r a t i o n under i n v e s t i g a t i o n are known, the conclusion concerning the s t a t i c s t a b i l i t y can e a s i l y be made. I t may-be pointed out tha t i f X<^ i s j u s t b a r e l y greater than 2/3, then b^ - becomes very small and values of c*r s as predicted byc*r S ( 7= — j - 1 become very l a r g e , which i s beyond the scope of the theory to be estimated a c c u r a t e l y . The conclusion i s that the e f f e c t of no n - l i n e a r elements i s to allow e q u i l i b r i u m values of the v a r i a b l e s to e x i s t , and a l s o that very l a r g e values are p o s s i b l e . Values of | <*rs] > 10° 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<rs,^= +y i l " j , and th e r e f o r e the e q u i l i b r i u m points are symmetrical about o(r-0. A * Now i f small perturbations o<r,c9, about the e q u i l i b r i u m points are allowed, then A A <Xr - <X +- o(r G s &s, f B 6 = £*sz+-<9 ( 1 0 9 ) 76 A p p l i c a t i o n of the usual approximations of small perturbations y i e l d s : , ~>- ,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 £ 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><rs. , @ s j . S u b s t i t u t i o n of the values of o(r^^/7£ , 0-,t = cJo and ftf */2[< / - d A * / * ^ - ^ i , i n t o equations (110) y i e l d s : cA r[-z/) (] r Drf r[>J +• ^ [ > 3 ] t D X 0 [ k 4 - l 4 j = £ 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 * £ ) ( 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 » i > 4 > 3 s A » A z , • Incorporating t h i s 78 approximation i n t o (112) g i v e s : j^(ai ^ b , ) - 2 4>, ^)~t2r,j•»-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 » 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, — 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 > ° (P4P3 - P s R r ) > 0 and LP 4 ^ - ^ P ^ [ ( P f ^ - P r/l ) P 2 - ( ^ - ^ H ) I \ ] > £ ( 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» 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^„andC^ . 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 <e f o r the general e q u i l i b r i u m point are obtained from equations (100), (101) and (102): i n t o the s t a b i l i t y boundary PQ =0 y i e l d s the d e s i r e d 4 '0(121) (122) 86 ( 1 2 3 ) For n o n - t r i v i a l s o l u t i o n of equations ( 1 2 1 ) , ( 1 2 2 ) and ( 1 2 3 ) , the determinant equation i n the g e n e r a l non-l i n e a r case y i e l d s a q u i n t i c c h a r a c t e r i s t i c equation. I t i s p o s s i b l e t o analyze a q u a r t i c equation i f the e q u i l i b -rium c o n d i t i o n i s the r e f e r e n c e case of s t r a i g h t and l e v e l f l i g h t . G e n e r a l l y f i v e p o s s i b l e cases f o r the PQ =0 s t a b i l i t y boundary e x i s t . The d e r i v a t i v e s are set equal to zero and the f o l l o w i n g equations are s o l v e d f o r the e q u i l i b r i u m p o i n t s : ( 1 2 4 ) The s o l u t i o n f o r the e q u i l i b r i u m v a l u e s of o<r S | i s obtained from the f o l l o w i n g q u i n t i c equation: A bs t>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 « 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»-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 ± >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„* /b,* b,i '••zCu 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>* • , 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 °<r<:\hr> 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 ^ ° ' 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 ^ = °) > 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®p7^ a, \c^%\-fa-&)lt&l«i»/$kl ( 1 3 8 , Thus i t i s p o s s i b l e to solve f o r a l i m i t i n g s t i f f -ness d i s t r i b u t i o n t h a t would give a s t a b l e system. A l l the terms on the r i g h t hand side of equations (136) and (138) are known once the a l t i t u d e , geometry, and weight of the body are known. The fuselage f i r s t bending mode must be assumed as w e l l as the centre of g r a v i t y l o c a t i o n but i t i s not necessary to know the mass d i s t r i b u t i o n i n d e t a i l . The procedure f o r using t h i s s t i f f n e s s c r i t e r i o n i n the p r a c t i c a l design of an a i r c r a f t would be to c a l c u l a t e the r i g h t hand side of (136) or (138) f o r d i f f e r e n t values of Xc^ and <$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 <p(y:). Fortunately, the accuracy of the Rayleigh Method i s not very sensitive to the choice of mode shape, so that even a crude choice would give results of reasonable accuracy. The Rayleigh Method always over-estimates the natural frequency, and i t i s not possible by use of (136) or (138) to underestimate the s t i f f n e s s required f o r s t a b i l i t y . The errors of the approximation are thus conservative. Of course, i f the chosen mode shape happens to coincide with the actual mode, then the s t i f f n e s s calculated by (136) or (138) is a true l i m i t . To ensure that the natural mode shapes w i l l be ortho-gonal when more than one e l a s t i c degree of freedom i s con-sidered, some r e s t r i c t i o n s muct be placed on the boundary c o n d i t i o n s a p p l i c a b l e to the s t r u c t u r e i n f r e e v i b r a t i o n . Any-one of four p a i r s of boundary c o n d i t i o n s must be s a t i s f i e d at each end of the fuselage f o r mode o r t h o g o n a l i t y t o e x i s t . These co n d i t i o n s are: £T(j>"=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: % ± 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 <fi" equal zero at either end must s t i l l be s a t i s f i e d , but now there i s only one node located at the point where ^  equals zero. Considerable s k i l l i s required to pick the best mode. (142) 95 X Figure 15 "Anti-Symmetric" Bending of the Fuselage If the natural modes are known f o r a structure which i s similar, or the s t a t i c d e f l e c t i o n mode of the structure i s known, then either of these modes would be acceptable as a s t a r t . Several polynomial methods are described by Bisplinghoff [3] and Timoshenko [15]. Any function that s a t i s f i e s the boundary conditions of at both ends of the fuselage i s a suitable choice. However, Timoshenko [15] states that "only i n some s p e c i a l cases . . . can the exact forms" of the natural modes "be determined i n terms of known functions" since the s t i f f n e s s and mass d i s t r i -butions are unlikely to be available i n terms of known func-t i o n s . Numerical integration i s often the only recourse. This concludes the discussion of the s t a b i l i t y of the e l a s t i c a i r c r a f t . A Numerical Example i s considered now where the general r e s u l t s obtained so f a r are applied to a s p e c i f i c case. IV. NUMERICAL EXAMPLE In the a n a l y t i c a l development presented i n the previous s e c t i o n s , a l a r g e number of equations were deriv e d and many groups of terms were c o l l e c t e d i n the form of shape constants. These shape constants are known f o r an a i r c r a f t of given geometry and e l a s t i c p r o p e r t i e s . The s t a b i l i t y c r i t e r i a were obtained i n terms of shape constants and other v a r i a b l e s which l e d t o two d i s t i n c t approaches t o the problem depending on the actual' "situation-: 1. The a n a l y s i s developed here can be used to p r e d i c t the s t a b i l i t y of the f l e x i b l e , supersonic c o n f i g u r a t i o n under design i f i t s geometry, mass d i s t r i b u t i o n , s t i f f n e s s and v i b r a t i o n c h a r a c t e r i s t i c s are known from t h e o r e t i c a l i n v e s t i -gations and s t r u c t u r a l t e s t s . 2. T o r a system w i t h known geometric p r o p e r t i e s , 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 , the s t i f f n e s s d i s t r i b u t i o n f o r dynamic s t a b i l i t y can be p r e d i c t e d by use of the s t i f f n e s s c r i t e r i o n . I t i s f e l t t h a t a b e t t e r a p p r e c i a t i o n of the t h e o r e t i c a l development can be gained by the i n v e s t i g a t i o n of a t y p i c a l case. Therefore, a numerical example i s given which considers a l i k e l y c o n f i g u r a t i o n f o r a supersonic t r a n s p o r t and s t u d i e s systemmatically the nature of i t s s t a b i l i t y . The v a l i d i t y of some of the assumptions made i n the a n a l y s i s i s a l s o v e r i f i e d . 97 I t should be pointed out t h a t the numerical values s e l e c t e d f o r the d i f f e r e n t parameters are r e p r e s e n t a t i v e of a supersonic a i r l i n e r i n magnitude only. The optimum design i s not of i n t e r e s t here because the purpose of t h i s i n v e s t i g a t i o n i s to assign c o r r e c t orders of magnitude to the c o e f f i c i e n t s of the equations and demonstrate the method of s o l u t i o n n u m e r i c a l l y . Let the a i r p l a n e c o n f i g u r a t i o n under i n v e s t i g a t i o n have the f o l l o w i n g c h a r a c t e r i s t i c s : Weight = 277,000 l b s . A l t i t u d e = 30,000 f t . , = .00089 s l u g s / f t . 3 Speed = M = 2 or 2000 f.p . s . Wing Aspect Ratio = fiR^- = 1 Foreplane Aspect R a t i o - £fy - 1 Radius of body base = Rb = 10 f t . Fuselage l e n g t h - L - 250 f t . S*r/Sb = 10 3f/S b = 1 E f f e c t i v e thickness of fuselage s t r u c t u r e = 0.1 f t . Mass d i s t r i b u t i o n along the l o n g i t u d i n a l a x i s =^C7) = Fuselage s t i f f n e s s d i s t r i b u t i o n along the l o n g i t u d i n a l a x i s - EjrlW ^ i ^ K ) ^ -/£ \ *"T£R 'KNEW/ .. 4. - V ... 0 1/ V a r i a t i o n of fuselage r a d i u s along the l o n g i t u d i n a l — _ _(.£• a x i s : R = R^x (Figure 16) ^ > ^ £ - 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 — ^-^T?f^J o «£ _ _ _ _ _ _ >J —>-—>- 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° , 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 °4 - 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 £ h * r e s p e c t i v e l y , can now be w r i t t e n as + o<? (24*) + otrbe(0,V4) +<^r &9(-0*0613) + + o<?botr(-0.0796) +o<r(0e)'z('0,lzS'6) + &9(-o>3t>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 ±L(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$). The time to h a l f amplitude " Q ' f f = 0.602 seconds. 1.145 - •  -The per i o d of the o s c i l l a t i o n i s 3^3^ SS !»66 seconds. This i s q u i t e small compared to the peri o d of a t y p i c a l phugoid mode of 200 seconds. As was noted i n the t e x t , equation (106) can be s i m p l i -f i e d by d i s c a r d i n g terms th a t are of much lower order of magni-102 tude compared t o ^ and i g . In that case, the ch a r a c t e r i s t i c equation i s modified to be: i . e . ?) [ t f + h(o.2B2) 4- (0.23S)j-O The accuracy of this approximation can be tested by calcu l a t i n g the percentage difference between the results f o r the two cha r a c t e r i s t i c equations. The roots of the equation are: 7 \ 3 ' 0 ) \ 2 = -0.14-1 ±l(OA6,S) The time to half amplitude - 0.612 seconds. The period of the o s c i l l a t i o n i s 1.688 seconds. The percent error i n time to half amplitude i s 1.66$ 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» - \A8° 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 =±49.8° For the present case, with the r e s t r i c t i o n thato( r^10°, 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»^7° Comparison of t h i s value with 1.48° 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° ' 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<*£ +_4-<-«r! - C?«4_-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 ° c. E l a s t i c A i r c r a f t ! • 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 $, 7^L0 and , are chosen to minimize the TO n a t u r a l frequency i n accordance w i t h the Rayleigh Method. In a d d i t i o n , using the c o n d i t i o n of zero momentum and adopting a t r i a l and e r r o r approach gives values of the unknown parameters as _ , ^ ,^ = O.ZS" 106 With these v a l u e s , the c o l l e c t i v e e l a s t i c constants can be evaluated. They are l i s t e d i n Table I I . The e q u i l i b r i u m values of o(r and^ are obtained from equations (124). ^s<t?y)+$Zj(((\tr»#) ^ 3 ^ J ) -0 (124) Therefore - ( C , ^r^c^^^j) Sj" (Oo-Nk*) f C | 3 p A ^ _ , S u b s t i t u t i o n of the above r e l a t i o n i n t o the f i r s t of (124) y i e l d s ^ - 1,4*6° =• 0>ooS'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 °? 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 « - 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„ 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°, 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 +<X?b&\C£] ±*?-$-G\C-?J ¥cArhoi^[CBJ -h CM* - o<r [ p | j- Ft 4 r l , - (7^-x„r)E, -(7cj -Zp)6i] + D«xrjj>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 ^ • - * ^ - ^ . _ _ ) t)z - -z f \ -Vcq - m/ \ D 7 --6£DCSS Zfat?)7(7cj-x)<J7 D 5 - - _ p z ^J 0'£<% fW4(7-7c£ R^Jr^l^^J'^-y^^dyj * 'TL^^j^£4t'(7)4(7 -7^ R<^d7 ^^£<pJC7jH7-x'c6)^^ D l o = ^ r y ) 2 ^ y-7^J ltd7 1 Sb _ _b 2- 3 •Sb 2 «J S_ 2 3? 4 s k z G-- = S _ IT f iT^ -x^) S_ 2 ^ 3 - b 2 J ^ 3 -^ 7 = -Spiral £<p.<-Zt) ^ - TTtR; (± jus -J-Jy<^ - y) 5_ L — /v = Sur-turTr&w-fL J<«r -I) ^ _ _ ^ 3 -~ S 7 7 7 ^ (J7«r " J / ^ S_ _ ^ ^ = - S«r^rr4?ur (4.4>J (*ur))(J7ur - h) 1 St y0 _y J 7 S t s S Jo J J • • = C, / £ / -r-Cr, ^7 r ^ 7 •G, *z *3 61, - C,i f ^ , *n -c„ ^/z. " CfT +£7 +6r7 r 1 Cr* 2 e k -• D, + F, +M-,-(X£j -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_ } £>y # r e * not-a-Mo<^e>L.ie.c/. hi-L = Bio+F7 +H7 - (Xcq -Yur) Ey - (Ycg -Xj) fr7 Co- •I* -Id •Ir, 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 , • 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. 

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