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On the attitude dynamics of slowly spinning axisymmetric satellites under the influence of gravity gradient… Neilson, John Emery 1968

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ON THE ATTITUDE DYNAMICS OF SLOWLY SPINNING AXISYMMETRIC SATELLITES UNDER THE INFLUENCE OF GRAVITY GRADIENT TORQUES by JOHN EMERY NEILSON B.Sc. (ME) University of Manitoba, 1960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t he L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t udy . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rpo se s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumb ia Vancouve r 8, Canada Date 3 0 / ^ C>6> ABSTRACT The dynamics of slowly spinning axisymmetric s a t e l l i t e s under the influence of gravity gradient torque i s investigated using a n a l y t i c a l and. numerical techniques. P a r t i c u l a r emphasis i s on motion near the equilibrium p o s i t i o n i n which the spin axis i s normal to the o r b i t a l plane. The problem i s studied i n increasing orders of d i f f i c u l t y . Phase I deals with the response and s t a b i l i t y of a s i m p l i f i e d model free to l i b r a t e i n r o l l while the more general problem i s treated i n Phase I I . Phase I serves as a proving ground fo r techniques to be used i n subsequent analysis. A closed form solution i s obtained i n terms of e l l i p t i c functions for the autonomous case. In general, f o r non-circular o r b i t s , motion i n the large i s studied using the concept of the invariant solution surface. These surfaces, obtained numerically, reveal the nature of motion i n the large i n terms of the dominant periodic solutions and allow one to determine the l i m i t s of o s c i l l a t o r y motion i n terms of the state parameters. Floquet theory i s employed i n conjunc-t i o n with numerical solutions of the l i n e a r i z e d equations of motion to study s t a b i l i t y i n the small. This technique i s ex-tended to assess the v a r i a t i o n a l s t a b i l i t y of the dominant perio-dic motions i n the large. Phase II investigates a more general model with three degrees of freedom i n attitude motion. The presence of an ignor-able coordinate gives a fourth order, non-autonomous system for an e l l i p t i c t r a j e c t o r y . Motion i n the small i s studied extensive-I l l l y , again using Floquet theory, and s t a b i l i t y charts suitable for design purposes are presented. The invariant surface con-cept i s successfully extended to the study of the autonomous case i n the large. Methods are developed for determining the maximum response to a given disturbance r e s u l t i n g i n a set of charts which are useful i n assessing the e f f e c t s of non-linear-i t i e s and the v a l i d i t y of the analysis i n the small. Procedures are explained for determining periodic solutions of the problem, as well as t h e i r s t a b i l i t y , for a r b i t r a r y e c c e n t r i c i t y . The analysis suggests the p o s s i b i l i t y of attitude i n -s t a b i l i t y during spin-up operations. I t i s shown that stable motion can be established by providing either a p o s i t i v e or neg-ative spin to the s a t e l l i t e with the former preferrable. Given s u f f i c i e n t spin any configuration, even those with an adverse gravity gradient e f f e c t , can be s t a b i l i z e d . E c c e n t r i c i t y affects the attitude motion of a s a t e l l i t e adversely as regions of un-stable motion increase i n size and number with i t . TABLE OF CONTENTS Chapter Page 1 Introduction 1 1.1 Preliminary Remarks . 1 1.2 H i s t o r i c a l Background 3 1.3 Purpose and Scope of the Investigation . . . 5 2 Phase I - R o l l Dynamics of a Spinning S a t e l l i t e . 9 2.1 Preliminary Remarks . . . . . 9 2.2 Formulation of the Problem 10 2.3 Motion i n the Small 16 2.4 Motion i n the Large 21 2.4.1 C i r c u l a r O r b i t a l Motion 25 2.4.2 Non-Circular O r b i t a l Motion (Analytic-a l Approach) 37 2.4.3 Non-Circular O r b i t a l Motion (Numer-i c a l Approach) 59 2.4.4 V a r i a t i o n a l Analysis of Periodic Solutions 67 2.5 Concluding Remarks . 74 3 Phase II - Attitude Dynamics of a Spinning Axisymmetric S a t e l l i t e 7 6 3.1 Preliminary Remarks 76 3.2 Formulation of the Problem 77 3.3 Motion i n the Small 81 3.4 Motion i n the Large 88 3.4.1 C i r c u l a r O r b i t a l Motion (Analytical Approach) 93 3.4.2 C i r c u l a r O r b i t a l Motion (Numerical Approach) 106 V C h a p t e r Page 3 . 4 . 3 N o n - C i r c u l a r O r b i t a l M o t i o n . . . . . . 145 3.5 C o n c l u d i n g Remarks 159 4 C l o s i n g Comments 166 4 . 1 Summary 166 4.2 Recommendations f o r F u t u r e Work 167 B i b l i o g r a p h y 169 A p p e n d i x I 171 LIST OF TABLES T a b l e Page I P r i n c i p a l C r o s s S e c t i o n s 115 LIST OF FIGURES Figure Page 1- 1 Schematic diagram of the proposed program . . . 7 2- 1 Geometry of Phase I model 11 2-2 S t a b i l i t y chart for motion i n the small; e * o 18 2 - 3 S t a b i l i t y chart for motion i n the small; e s . j 22 2 - 4 S t a b i l i t y chart for motion i n the small; e= ."3 23 2-5 S t a b i l i t y chart for motion i n the small; e.= .5 24 2-6 L i b r a t i o n a l response i n a c i r c u l a r o r b i t ; 1= 2. , *3~= - |.5 28 2-7 L i b r a t i o n a l response i n a c i r c u l a r o r b i t ; I* 2 , <sr * a. . 29 2-8 • L i b r a t i o n a l response i n a c i r c u l a r o r b i t ; r • .5 , 0-= -.5 30 2-9 L i b r a t i o n a l response i n a c i r c u l a r o r b i t ; I« .5 , <3T= 21 31 2-10 S t a b i l i t y chart for motion i n the large; e« ©, *; m O 3 4 2-11 V a r i a t i o n of the c r i t i c a l v e l o c i t y with i n e r t i a and spin parameters; S « O . . . . 3 5 2-12 E f f e c t of spin on the s t a b i l i t y bounds; e = O, 1= . 5 36 2-13 Va r i a t i o n of F with ^" and & ; e « . S , I * 2. 40 2-14 Va r i a t i o n of F with €* and 9 ; I« 2, <r= o 4 1 2-15 Comparison between the exact and WKBJ solutions for l i b r a t i o n a l response; ^=0, 8/* 1.5, I* 2; CT- \, e . « . z 4 5 v i i i Figure Page 2-16 Comparison between the exact and WKBJ solutions for l i b r a t i o n a l response; O, 1,1=2., *r=o, e « . e . . . • 46 2-17 Comparison between the exact and perturbation solutions f o r l i b r a t i o n a l response; O, tf.'= 11 I - I, cr- = 2 , e ~ . 2 ! . . . 50 2-18 Comparison between the exact and perturbation solutions for l i b r a t i o n a l response; V « o , 5'e2, 1=2., cr- i , e s . i s 51 2-19 Comparison between the exact, WKBJ and perturbation solutions for l i b r a t i o n a l response; ¥.»0, | I « I, <3~= 2 , e = . 2 52 2-20 Schematic of an invariant solution surface i n state space 55 2-21 Comparison of cross sections a t 6 s-Oof invariant surfaces generated by the WKBJ and numerical analyses; 1 = 2 , CT = - I , e « . 4- 57 2-22 Comparison of cross sections at 9 =» O of invariant surfaces generated by the WKBJ and numerical analyses; I=2. /^~aO > e».5 58 2-23 T y p i c a l example of l i m i t i n g invariant surface; Ie.5, C-a 2, e = .35 6 2 2-24 Typical example of l i m i t i n g invariant surfaces; 1= . 5 , vT = 2 , - . IS 63 2-25 Study of invariant surface cross sections; X « 2 . , < 3 - = i , e s . 3 , © = 0 65 2-26 Study of invariant surface cross sections; I « .5, <r-= a , e s , 3 3 ; ®- O . . . . . . 66 2-27 V a r i a t i o n of c r i t i c a l v e l o c i t y with o r b i t e c c e n t r i c i t y ; O , X s „ 5, a"« 2 . . . . . . 68 2-28 V a r i a t i o n of c r i t i c a l v e l o c i t y with o r b i t e c c e n t r i c i t y ; &•= O, I = 2 , CT* I 69 2-29 V a r i a t i o n a l analysis of periodic solutions; O, I« .5 , OT= 2 72 2-30 V a r i a t i o n a l analysis of periodic solutions; )(,= O , I . 2, I 73 ix Figure Page 3-1 Geometry of Phase II model 78 3-2 S t a b i l i t y chart for motion i n the small; e « o • 86 3-3 S t a b i l i t y chart for motion i n the small; e * . i 89 3-4 S t a b i l i t y chart for motion i n the small; e * .2 90 3-5 S t a b i l i t y chart for motion i n the small; e « . 3 91 3-6 S t a b i l i t y chart for motion i n the small; e > . 4 92 3-7 Zero v e l o c i t y curves; I* 2., <3T« 2. . . . 97 3-8 Zero v e l o c i t y curves; I* 2 , 0 " » - 2 . . . 98 3-9 Zero v e l o c i t y curves; I « 1 .25 , c r * 2. . . . 99 3-10 Zero v e l o c i t y curves; I = . 5 , <x = I . . . 100 3-11 Motion envelope i n — space; I = 2 , CT = 2 , M = - 3 . 6 6 102 3-12 Motion envelope i n & — space; 1 -2 , , <T--2 , H r - 3 . 6 6 103 3-13 Motion envelope i n space; .r» <ar= i , H« - . 5 104 3-14 Motion envelope i n J — space; I* .5 ^T-- I , H » - . 5 105 3-15 Invariant solution surface i n space; 1= 2 , 5 5 - = O, / 3 ; = « ' = Ot 2.-46, *.= .5 . . HO 3-16 Invariant solution surface i n V' — space; 1= «3-= O , ^ . - J . ' - O ^ . ' s 2.-*<Z», V ;=.S . • H I 3-17 Invariant solution surfaces i n /S'— space; I « 2 ( f l r s I, Y.= O, / 5 « - . I, 2 . o o 2 . . 112 3-18 Invariant solution surfaces i n V — space; 1= 2 , <X= |, *.« 0 , ^ = - . I, */= 2-OOg • • 113 X Figure Page 3-19 Type 2 cross sections of invariant surface; I« , cr = O, 5 = o, /3- • 5, *'= 1.503 . . 117 3-20 Type 4 cross sections of invariant surface; I « a , < r * 0 , / 3 . ' « O, .S, X ' = I . S 0 3 . . 118 3-21 Type 2 cross sections of invariant surface; 1= e., <r= o , /2>'- o , », 3.6-45 . . . 119 3-22 Type 4 cross sections of invariant surface; la g, <3-= O, * = O, /C3= 1,^^3.64-3 . . 120 3-23 Invariant surface section studies of types 1 and 3; I « «T= 1 , H » - 3 . . . . . . 122 3-24 Invariant surface section studies of types 1 and 3; I » \91 Q - B \t H «= I . . . . . . 123 3-25 Invariant surface section studies of types 1 and 3; 3 > " 2 , <J- » 1 , H = 5 124 3-26 Invariant surface section studies of types 1 and 3; Is 1.5, T * - 2 , H» 4 125 3-27 Invariant surface section studies - type 1; I s .3, <r *• I , H « - . 9 126 3-28 Maximum /3 response chart; « z^;'* O, - 0» 129 3-29 Maximum Y response chart; » O, .OI 130 3-30 Maximum /3 response chart; — ^ « fc.'e . 5 131 3-31 Maximum X response chart; « /Ss •= fc^s O, ^= . 5 132 3-32 Mapping scheme for locating periodic solutions; G.m O 135 3-33 Periodic solution for a s a t e l l i t e i n a c i r c -u l a r o r b i t ; 1= . 3 , I, O, /^'= - 9 5 , tf.= M 5 3 137 3-34 Periodic solution for a s a t e l l i t e i n a c i r c -u l a r o r b i t ; 1= . 3 , I, /S* O, /3'= .217, . l"7B 138 x i Figure Page 3-35 Va r i a t i o n of t and T r L $ C * ) l with H for fundamental periodic motion; I s a , c r = o 1 4 2 3-36 Var i a t i o n of , " C and ~Tr[_§(rt)2 with H for fundamental periodic motion; 1 * 2 , I . . . . 143 3-37 Va r i a t i o n of /3j,'t. and T r £ < $ < t ) ] with H for fundamental periodic motion; I - 2 , <r« 2 . . . 144 3-38 Scattered intersections of a trajectory with the stroboscopic phase planes at 8 « O : X * . S , «ar» I, e • . I , / $ * * > O , - », : . O 1 147 3-39 Trajectory intersections with the stroboscopic phase planes at 0*O showing banding about a periodic solution; I » . S , XTs \. O. ~ I • • , ; . > ; - . . . . • . . . . . 148 3-40 Trajectory intersections with the stroboscopic phase planes at Q * O showing banding about the o r i g i n ; I s a , I , . l,/3;» O, ' / ^ - ' « I. ft, 8;0 ~.S 149 3-41 • Periodic solution of a s a t e l l i t e i n an e l l i p t i c o r b i t ; I • 2 , <r « I , - . I, *'« O, I . S 9 4 , Vj* - . 9 6 4 , 156 3-42 Va r i a t i o n of i n i t i a l conditions for stable periodic motion with o r b i t e c c e n t r i c i t y ; l a 2 , < T - I 157 3-43 Varia t i o n of i n i t i a l conditions for stable periodic motion with o r b i t e c c e n t r i c i t y ; I s . 5 , * S " S 1 • • 158 3-44 Comparison of s t a b i l i t y regions as determined by several investigators.; & . *3 161 3-45 Relationship between spin parameters X and CS" . . 162 1-1 Incremental change per o r b i t i n o r b i t a l para-meters due to coupled periodic l i b r a t i o n a l motion; 1 = 2 , C T ^ - I , nr» <»- I , r» •=• I 175 1-2 Incremental change per o r b i t i n o r b i t a l para-meters due to coupled periodic l i b r a t i o n a l motion; I* 1 . 5 , ^ = - ! , ^ = I , n = 2. 176 x i i F i g u r e Page 1-3 I n c r e m e n t a l change p e r o r b i t i n o r b i t a l parameters due t o c o u p l e d p e r i o d i c l i b r a t i o n a l m o t i o n ; I = . 5 , *3~ = 4-i m « I, r i« I . . . . . 177 ACKNOWLEDGEMENT The author wishes to express his gratitude to Dr. V.J. Modi for his encouragement and guidance throughout the study and p a r t i c u l a r l y for his help during the c r i t i c a l i n i t i a l stages. Since a large part of t h i s i n v e s t i g a t i o n involved a numer-i c a l approach, the f a c i l i t i e s of the Computer Centre of the University of B r i t i s h Columbia were extensively employed. The use of these f a c i l i t i e s i s g r a t e f u l l y acknowledged. The research f o r t h i s thesis was supported i n part by the Defence Research Board of Canada, Grant number 0201-01. L I S T OF SYMBOLS A L(cr+i) B 31-4-C,,Ca,C3,C* C o n s t a n t s d e f i n e d i n e q u a t i o n (2.45) ?,iCa C o n s t a n t s d e f i n e d i n e q u a t i o n (2.42) D,tD2 C o n s t a n t s d e f i n e d i n e q u a t i o n (3.11) and (3.12) E T o t a l e n e r g y , U F F u n c t i o n d e f i n e d i n e q u a t i o n (2.41) G F u n c t i o n d e f i n e d i n e q u a t i o n (2.39) H H a m i l t o n i a n I I n e r t i a p a r a m e t e r , IX / 1 ^ IXJ^IK P r i n c i p a l moments o f i n e r t i a Kj C o e f f i c i e n t i n e q u a t i o n (2.52) L L a g r a n g i a n Lj C o e f f i c i e n t i n e q u a t i o n (2.53) O C e n t e r o f f o r c e P P e r i c e n t e r R D i s t a n c e between s a t e l l i t e c e n t e r o f mass and c e n t e r o f f o r c e S S a t e l l i t e c e n t e r of mass T K i n e t i c energy U P o t e n t i a l energy V L i a p o u n o v t e s t i n g f u n c t i o n X S t a t e v e c t o r a h e ^ u d - e f b , E lement of c h a r a c t e r i s t i c m a t r i x XV Periodic c o e f f i c i e n t of equations of motion i n state vector form Function defined i n (1.5) Function defined i n (1.9) Constants of motion Modulus of e l l i p t i c functions P r i n c i p a l r a d i i of gyration Constants i n equations (2.29) and (2.30) Spin parameter Number of o s c i l l a t i o n s involved i n periodic motion Mass of the s a t e l l i t e Number of o r b i t s over which motion i s periodic Distance from center of force to a mass element Root of c h a r a c t e r i s t i c equation Time Spinning body coordinates I n e r t i a ! coordinates P r i n c i p a l body coordinates Intermediate body centered coordinates Incremental change i n «S per o r b i t Incremental change i n ^  per o r b i t Normal solution basis Orbit e c c e n t r i c i t y A+Bc©sS Constants defined i n equations (2.28) 1 i normal solution vector $ Solution basis 9 i t h solution vector $j ) t h element of i t h solution vector OLt/StH Spin, yaw and r o l l rotations, respectively tfA Maximum amplitude of r o l l o s c i l l a t i o n TS0,X,,}$j,,--- Perturbation elements of r o l l solution 9 x(i-+eco»e) £ ^ , S 8» I t e r a t i o n changes in/3-, /£•, , and ft; respectively £ Perturbation parameter 0 True anomaly Aj i + W c h a r a c t e r i s t i c m u l t i p l i e r JX G r a v i t a t i o n a l constant p Size parameter O" Spin parameter t Period of l i b r a t i o n a l motion ft Perturbation of the pericenter f & + ? CO L i b r a t i o n a l frequency c*>f Fundamental l i b r a t i o n a l frequency CO X TcOv,,a> 2 Angular v e l o c i t y components i n p r i n c i p a l body coordinates c u 0 Basic l i b r a t i o n a l frequency, cu,,(A> a> perturbation frequencies p X V I 1 S u b s c r i p t s c. C r i t i c a l v a l u e I n i t i a l v a l u e P e r i o d i c s o l u t i o n v V a r i a t i o n f rom p e r i o d i c s o l u t i o n Dots and pr imes i n d i c a t e d i f f e r e n t i a t i o n w i t h r e s p e c t t o 1" and & r e s p e c t i v e l y . 1. INTRODUCTION 1.1 Preliminary Remarks Interest i n the attitude dynamics of r i g i d o r b i t i n g bodies dates back to the eighteenth century when astronomers studied motions of natural s a t e l l i t e s , e.g., lunar l i b r a t i o n s . In recent years and p a r t i c u l a r l y since the launching of the f i r s t a r t i f i c i a l s a t e l l i t e research i n t h i s f i e l d has measurably accelerated. Modern s a t e l l i t e systems, capable of performing sophisticated on-board experiments, usually demand a corresponding degree of sophistication i n attitude control to overcome a v a r i e t y of disturbing influences, e.g., solar r a d i a t i o n pressure, gravity gradient and magnetic f i e l d e f f e c t s , and micrometeorite impacts. Ideally, however, the attitude control of a s a t e l l i t e should be accomplished with the minimum expenditure of energy since space and weight are at a premium aboard instrument packed space vehicles. In many applications where attitude control requirements are not too severe, passive techniques involving no expenditure of stored energy have proven to be adequate. Among the methods belonging to t h i s category, those u t i l i z i n g gravity gradient and/or gyroscopic e f f e c t s are commonly used. The former u t i l i z e s the moment due to the g r a v i t a t i o n a l gradient across a s a t e l l i t e so that i t s long axis ( i . e . , the axis of l e a s t i n e r t i a ) points i n the d i r e c t i o n of the a t t r a c t i n g body. The l a t t e r , i n i t s simplest form, turns the entire 2 s a t e l l i t e into a gyroscope by permitting i t to spin about a suitable axis. When thi s axis coincides with an axis of sym-metry, the system i s i n equilibrium with the spin vector normal to the o r b i t a l plane. For c e r t a i n s a t e l l i t e configurations, the gravity grad-ient torque tends to reinforce spin s t a b i l i z a t i o n . For example, a thin disk or r i n g shaped s a t e l l i t e would be i n a position of stable equilibrium under the influence of the gravity torque i f i t s axis of symmetry were normal to the plane of the o r b i t . On the other hand, the gravity torque may work against spin s t a b i l i z a t i o n as i n the case of a slender, pencil-shaped s a t e l -l i t e i n the i d e n t i c a l o r i e n t a t i o n . The gravity gradient torque i s always present, except for the case of a spherical s a t e l l i t e , and i t s e f f e c t i s p a r t i c u l a r l y s i g n i f i c a n t where the rate of spin i s small. Consequently, an investigation of the behavior of slowly spinning s a t e l l i t e s should take the gravity gradient torque into account. One may ask: Why not simply spin the s a t e l l i t e at a high rate so that gravity e f f e c t s are minimized? While t h i s approach may be permissible i n certain s p e c i f i c applications, high rates of spin are usually not compatible with other design objectives. Moreover, during i n j e c t i o n into o r b i t and subsequent spin-up f o r s t a b i l i z a t i o n , there i s a t r a n s i t o r y period during which g r a v i t a t i o n a l and gyroscopic e f f e c t s are of comparable magnitudes. Thus a study of the attitude dynamics of a slowly spinning s a t e l l i t e i n a g r a v i t a t i o n a l f i e l d should lead to information of considerable p r a c t i c a l s i g n i f i c a n c e . 3 1.2 H i s t o r i c a l Background A survey of the pertinent l i t e r a t u r e suggests that com-pared to gravity gradient s t a b i l i z e d systems the dynamical analysis of slowly spinning s a t e l l i t e s has received l i t t l e attention. Research i n the f i e l d of gravity orientated systems has, u n t i l recently, been confined to the study of the i d e a l i z e d dumbbell s a t e l l i t e configuration. Much of th i s work pertains to a s i m p l i f i e d model of the system with a single degree of freedom i n attitude allowing for planar l i b r a t i o n only. Klemperer ^ obtained the exact solution of th i s system i n terms of e l l i p t i c functions for c i r c u l a r o r b i t a l motion. 2 Schechter attempted, with li m i t e d success, to extend t h i s solution to non-circular o r b i t a l motion by perturbation methods 3 while Baker found periodic solutions of the problem for small 4 o r b i t e c c e n t r i c i t y . Brereton has presented an excellent 5 review of t h i s work. More recently, Zlatousov, et a l and Brereton and Modi ^ employed numerical methods involving the use of stroboscopic phase planes to study motion i n the large for o r b i t s of a r b i t r a r y e c c e n t r i c i t y . The study of the more general problem involving out of 7 plane l i b r a t i o n s was undertaken by Modi and Brereton i n which g a r o t a t i o n a l constraint was involved and by DeBra allowing three degrees of freedom. In both cases numerical methods were used to determine s a t e l l i t e response. In the f i e l d of slowly spinning s a t e l l i t e s , early work has been r e s t r i c t e d to the study of systems undergoing c i r c u l a r 4 9 o r b i t a l motion^ Thomson p r e s e n t e d a s t a b i l i t y c r i t e r i o n i n terms o f s p i n and i n e r t i a p arameters u s i n g l i n e a r i z e d a n a l y s i s . S u b s e q u e n t l y Kane e t a l a p p l i e d t h i s c r i t e r i o n t o o b t a i n a s t a b i l i t y c h a r t i n terms o f t h e s e p a r a m e t e r s . The s t a b i l i t y o f a s p i n n i n g unsymmetric s a t e l l i t e was i n v e s t i g a t e d by Kane and Shi p p y ^ a p p l y i n g F l o q u e t t h e o r y t o a l i n e a r i z e d model o f t h e system. The l a r g e a m p l i t u d e l i b r a t i o n a l m o t i o n o f a s p i n n i n g 12 s a t e l l i t e i n a c i r c u l a r o r b i t was s t u d i e d by P r i n g l e u s i n g a L i a p o u n o v t y p e o f a n a l y s i s . P o s i t i o n s o f e q u i l i b r i u m as w e l l as bounds o f m o t i o n , i . e . s e p a r a t r i c e s , about t h e s e p o s i t i o n s were o b t a i n e d . 13 The a t t e m p t by Kane and Barba t o a n a l y z e m o t i o n i n an e l l i p t i c o r b i t s h o u l d be mentioned h e r e . By a p p l y i n g F l o q u e t t h e o r y t o a l i n e a r i z e d system, t h e y d e v e l o p e d a p r o c e d u r e f o r t e s t i n g t h e s t a b i l i t y , i n t h e s m a l l , o f a s p i n n i n g s a t e l l i t e u n d e r g o i n g o r b i t a l m o t i o n o f a r b i t r a r y e c c e n t r i c i t y . 14 I n a r e c e n t p a p e r , W a l l a c e and M e i r o v i t c h s t u d i e d the same p r o b l e m u s i n g a s y m p t o t i c a n a l y s i s i n c o n j u n c t i o n w i t h L i a p o u n o v 1 s d i r e c t method. A n o r m a l i z e d H a m i l t o n i a n f u n c t i o n was employed as t h e t e s t i n g f u n c t i o n . Much of t h i s work i n v o l v e d the use o f l i n e a r i z e d a n a l y s i s a l t h o u g h p e r t u r b a t i o n methods were used i n an a t t e m p t t o i n v e s t i g a t e t h e e f f e c t s o f f i r s t o r d e r n o n - l i n e a r i t i e s . N o n - l i n e a r s t i f f e n i n g e f f e c t s and r e s o n -ance r e g i o n s were o b s e r v e d ; however, t h e r e i s an element o f doubt about the v a l i d i t y o f t h e s e r e s u l t s s i n c e even the c o n c l u -s i o n s based on t h e l i n e a r i z e d a n a l y s i s were n o t i n g e n e r a l 13 agreement w i t h t h o s e o f Kane and Barba , p a r t i c u l a r l y i n the 5 n e g a t i v e s p i n r e g i m e . 1.3 Purpose and Scope o f t h e I n v e s t i g a t i o n From t h e f o r e g o i n g i t i s c l e a r t h a t , a l t h o u g h r e s e a r c h -e r s have been a c t i v e i n t h i s f i e l d f o r sometime t h e r e i s s t i l l much t o be l e a r n e d about the b e h a v i o r of s p i n n i n g , g r a v i t y i n f l u e n c e d s a t e l l i t e s . The main purpose o f t h i s i n v e s t i g a t i o n i s t o d e v e l o p t e c h n i q u e s by w h i c h one may e x p l o r e r e g i o n s o f parameter space t h a t a r e o f p a r t i c u l a r i n t e r e s t d u r i n g the f e a s i b i l i t y s t u d y and d e s i g n s t a g e of a s a t e l l i t e program. I t i s a l s o i n t e n d e d here t o a p p l y t h e s e t e c h n i q u e s t o a number o f s p e c i f i c s i t u a t i o n s so as t o o b t a i n a fundamental u n d e r s t a n d i n g o f the dynamics o f s l o w l y s p i n n i n g s a t e l l i t e s . Due t o t h e complex n a t u r e o f t h e problem, i t was judged a d v i s a b l e t o approach i t i n two s t a g e s . Phase I s t u d i e s a s i m p l i f i e d system c o n s i s t i n g o f a s p i n n i n g s a t e l l i t e f r e e t o l i b r a t e i n r o l l . The model would s e r v e as a t e s t bed f o r methods o f a n a l y s i s . Such a system, a l t h o u g h i t f a i l s t o r e p -r e s e n t the p h y s i c a l s i t u a t i o n a c c u r a t e l y , does p o s s e s s s e v e r a l p r o p e r t i e s o f a more g e n e r a l model i n c o r p o r a t i n g t h r e e degrees o f freedom. T h i s approach r o u g h l y p a r a l l e l s t h a t t a k e n by i n v e s t i g a t o r s i n s t u d y i n g the a t t i t u d e dynamics o f a dumbbell s a t e l l i t e where t h e e a r l y work r e s t r i c t e d l i b r a t i o n a l m o tion t o the p l a n e o f the o r b i t . Phase I I o f t h e a n a l y s i s t r e a t s a more g e n e r a l s i t u a t i o n i n v o l v i n g a model w i t h t h r e e degrees o f freedom i n a t t i t u d e , v i z . , r o l l , yaw and s p i n . I n t h i s c a s e , a t t e m p t s have been 6 made to generalize and extend those methods which proved most f r u i t f u l i n Phase I. In each phase of the study, i t was considered appropriate to approach the problem i n stages representing increasing degrees of complexity. In the beginning, a l i n e a r i z e d model of the system i s used to study motion i n the small. Since prac-t i c a l applications such as communications s a t e l l i t e s usually demand considerable pointing accuracy, a knowledge of the be-havior i n the small would be of great value. Subsequently, studies of motion i n the large are undertaken since gravity torques and spin coupling e f f e c t s are inherently non-linear. A further subdivision of the problem i s possible. I t i s shown that to a high degree of accuracy o r b i t a l motion, i . e . , motion of the center of mass, and l i b r a t i o n a l motion may be considered to be uncoupled. Thus, the equations governing a t t i -tude dynamics are autonomous for c i r c u l a r o r b i t a l motion while those for e l l i p t i c a l o r b i t s are non-autonomous. A considerable s i m p l i f i c a t i o n i s , therefore, r e a l i z e d by i n i t i a l l y r e s t r i c t i n g the study to those systems i n which o r b i t a l motion i s c i r c u l a r . This also helps provide a firm basis for subsequent studies involving non-circular o r b i t a l motion. Figure 1-1 schematically i l l u s t r a t e s the proposed method of attack. In each phase the investigation begins with the simplest model based upon a l i n e a r , autonomous representation and progresses through to the most complex system governed by non-linear, non-autonomous equations of motion. It i s f e l t that t h i s approach, by the nature of the escalating complexities Attitude Dynamics of Spinning Satellites Phase 1 Roll Freedom Phase II Yaw& Roll Freedom Motion in the Small [Linear] Motion in the Large [Non-Linear] Motion in the Small [Linear] Motion in the Large [Non-Linear] Circular Orbit Circular Orbit Circular Orbit Circular Orbit F i g u r e 1-1 Schematic diagram of the proposed program 8 o f t h e v a r i o u s models, p r o v i d e s a c o h e r e n t program t o e x p l o r e t h e s u b j e c t . 2. PHASE I-ROLL DYNAMICS OF A SPINNING AXISYMMETRIC SATELLITE 2.1 Preliminary Remarks This chapter investigates the attitude dynamics of a r i g i d , axisymmetric, spinning s a t e l l i t e free to l i b r a t e i n r o l l . The e f f e c t s of o r b i t e c c e n t r i c i t y and the gravity gradient torque are included i n the analysis. L i b r a t i o n a l motion i s studied both i n the small ( i . e . l i n e a r i z e d analysis) and i n the large. P a r t i c u l a r attention i s paid to the development of tech-niques suitable for the investi g a t i o n of the more general prob-lem of a s a t e l l i t e free to l i b r a t e i n both yaw and r o l l . The analysis of s t a b i l i t y i n the small i s performed using a numeric-a l method based on Floquet theory which i s subsequently adapted to the v a r i a t i o n a l analysis of a large amplitude periodic motion. A closed form solution i n terms of e l l i p t i c function i s obtained for the p a r t i c u l a r case of c i r c u l a r o r b i t a l motion. S u i t a b i l i t y of several approximate methods i s investigated for cases involving o r b i t a l motion of ar b i t r a r y e c c e n t r i c i t y . The WKBJ solution i s shown to lead to the concept of invariant surfaces i n three dimensional state space. This concept, an extension of the stroboscopic phase plane method, i s u t i l i z e d i n the numerical analysis of the problem to obtain l i m i t i n g surfaces for stable (non-tumbling) motion. 10 2.2 F o r m u l a t i o n of t h e P r o b l e m C o n s i d e r a r i g i d , a x i s y m m e t r i c , s p i n n i n g s a t e l l i t e w i t h c e n t e r o f mass a t S l i b r a t i n g about the l o c a l h o r i z o n t a l w h i l e moving i n o r b i t about t h e c e n t e r o f f o r c e a t 0 ( F i g u r e 2-1). The c o o r d i n a t e s R and 0 d e f i n e the p o s i t i o n o f the c e n t e r o f mass w i t h r e s p e c t t o the i n e r t i a l frame X 0,^ 0 / Z 0 . L e t xp, Z p r e p r e s e n t t h e p r i n c i p a l body c o o r d i n a t e s w i t h o r i g i n a t the c e n t e r o f mass ( S ) and t h e c o o r d i n a t e X P c o i n c i d i n g w i t h the a x i s o f symmetry. L e t X6, ^ & , Z s be a n o t h e r s e t of o r t h o g o n a l c o o r d i n a t e s w i t h o r i g i n a t S b u t o r i e n t a t e d such t h a t X s i s normal t o t h e o r b i t a l p l a n e and ^ s l i e s a l o n g the e x t e n s i o n o f t h e r a d i u s v e c t o r R . The a n g u l a r o r i e n t a t i o n o f t h e s a t e l l i t e i s s p e c i f i e d by t h e E u l e r a n g l e s H and Oi r e l a t i v e t o the non-i n e r t i a l frame Xs, ^ s , Z s . The f i r s t r o t a t i o n , V , about the l o c a l h o r i z o n t a l i s r e f e r r e d t o as r o l l w h i l e the second r o t a t i o n , OC , about th e a x i s o f symmetry r e p r e s e n t s the s p i n o f the s a t e l l i t e . Owing t o t h e symmetry o f t h e s a t e l l i t e , f o r m u l a t i o n o f t h e p r o b l e m i s most e a s i l y a c c o m p l i s h e d u s i n g the c o o r d i n a t e frame X, y, Z ( F i g u r e 2-1) i n w h i c h the s a t e l l i t e s p i n s w i t h a n g u l a r v e l o c i t y oc about i t s a x i s o f symmetry l y i n g a l o n g the X a x i s . The k i n e t i c and p o t e n t i a l e n e r g i e s o f the system may be w r i t t e n as (2.1) F i g u r e 2-1 Geometry o f Phase I model (2.2) where the a n g u l a r v e l o c i t i e s a r e g i v e n by Ct>x » & + © C O S ft oo^ss - 6 sintf (2.3) and r-R[|+{WT+(*HiM *^^ HI* (2.4) E x p a n d i n g e q u a t i o n (2.4) u s i n g the b i n o m i a l theorem g i v e s + 0 > L _ U ( . (2.5) S i n c e the coordinates X, ^  , «• were chosen with o r i g i n S a n d the X axis along the axis of symmetry, the following r e l a t i o n s hold: J^x dm s = J~ J m s » dm s « O (2.6) •"Us m« and (2.7) dm<= — U s i n g e q u a t i o n s ( 2 . 5 ) , (2.6) and ( 2 . 7 ) , n e g l e c t i n g O ^ J L J ^ and n o t i n g t h a t I * \ J = \J I Z g i v e s U = - ^ - ^ * ( J T 1 ) ( , - 3 s i ^ ) and hence the L a g r a n g i a n f u n c t i o n , L— , can be w r i t t e n (2.8) as 1 / ¥+ © ' s i * - ? * )I •+ / U m s -+- >g I x * 2 (2.9) As b o t h OC and 0 a r e c y c l i c c o o r d i n a t e s , t h e r e a r e two f i r s t i n t e g r a l s o f m o t i o n g i v e n by = J_ c^L = JL ( « f + 6 c o s X ) (2.10) 14 and h Q m * L m,l I Sirf^ yjfi -+ C O S OC (2.11) where and h e are constants of motion. Using Lagrangian formulation the equations of motion for the remaining coordinates, v i z . , R and 16 , can be written as R - R § 2 + 3 M I * / 1 - 0/ l - 3 s i nK )= O (2.12) and -+• ( I~0 ^ 8 % 3 S i n Ycos + Ia&sin)$= O . (2.13) These non-linear, non-autonomous, coupled equations of motion do not possess any known closed form solution. Some s i m p l i f i c a t i o n of these equations can be achieved by neglecting the e f f e c t s of l i b r a t i o n a l motion on o r b i t a l motion. An analysis, using the method of v a r i a t i o n of parameters, i s per-formed i n Appendix I which v e r i f i e s t h i s approach. Neglecting attitude motion e f f e c t s on the o r b i t a l motion leads to the c l a s s i c a l equations for a p a r t i c l e undergoing 15 c e n t r a l f a c e m o t i o n , and R - + - O. (2.14) These e q u a t i o n s , i n t u r n , y i e l d the K e p l e r i a n r e l a t i o n f o r R and 0 , R •» //^c ( I + e c o s f i ) . (2.15) Note t h a t h<*/ w h i c h i s a c o n s t a n t o f m o t i o n , i s a measure of s p i n . C o n s e q u e n t l y a d i m e n s i o n l e s s s p i n p a r a m e t e r , CT" , d e -f i n e d as oc m& Vic 0=0 may be used t o e l i m i n a t e the c y c l i c c o o r d i n a t e OC . R e c o g n i z i n g t h a t jd_ « be <j <=Jt* F T de R A d e 2 the g o v e r n i n g e q u a t i o n of m o t i o n can be w r i t t e n as V"-/g<gsine>S * [731--4. - ecose\ A \ |+ecose/ L\ i-*-ecos& / I l+ecoseyj (2.16) (2.17) I t should be noted that, i n equation (2.17), the spin parameter i s based upon B « 0 such that ^ " - . I f i n i t i a l conditions other than these are chosen, the spin parameter i s related to ot' by <sT" -Analysis of the non-linear, non-autonomous equation governing motion, i . e . equation (2.17), involving three system parameters, T , and , forms the subject of t h i s study. 2.3 Motion i n the Small Investigation of motion i n the small i s an important phase i n the general study of any dynamical systems. In most s a t e l l i t e applications only small amplitude l i b r a t i o n a l motion i s permissible. Consequently, the question of the s t a b i l i t y of the l i n e a r i z e d system i s of considerable s i g n i f i c a n c e . L i n e a r i z a t i o n of the equation governing attitude motion (i.e. , equation 2.17) r e s u l t s i n f— 2 ^ s i n 0 ? y ' -+• ^31-A-ecose L i + e c o s © J I i + e c o s e I(<5-+ I ) / ' + e \ ( * = O . (2.18) \ l + e c o s & ; J I t should be noted that although equation (2.18) i s l i n -ear, i t i s not i n general autonomous since the independent 17 v a r i a b l e 6 i s p r e s e n t e x p l i c i t l y . L e t us c o n s i d e r t h e s p e c i a l case where & * O , i . e . , c i r c -u l a r o r b i t a l m o t i o n . I n t h i s c a s e , e q u a t i o n (2.18) becomes autonomous and s t a b i l i t y i n t h e s m a l l i s d e t e r m i n e d by the s i g n o f t h e c o e f f i c i e n t o f 8 . The s t a b i l i t y c r i t e r i o n f o r t h i s s i t u a t i o n i s s i m p l y , > 0, s t a b l e 31- A •+ I (cr+i) ^  - °> c r i t i c a l (2.19) < O , u n s t a b l e . E x p r e s s e d i n terms o f a c r i t i c a l s p i n parameter t h e s e r e l a t i o n s become r > CTk 9 s t a b l e CT J = <3I , c r i t i c a l , <T_ = 4- (2.20) < CT^ . , u n s t a b l e . A c h a r t showing the r e g i o n s o f s t a b l e and u n s t a b l e con-f i g u r a t i o n s i n r ,CP-space i s p r e s e n t e d i n F i g u r e 2 -2 . The n e x t l o g i c a l s t e p would be t o ex t e n d t h e a n a l y s i s t o cases i n v o l v i n g n o n - c i r c u l a r o r b i t a l m o t i o n . S i n c e t he c o e f f i c i e n t s i n e q u a t i o n (2.18) a r e p e r i o d i c i n 9 ( p e r i o d 2.TT ) , F l o q u e t t h e o r y may be employed t o s t u d y s t a b i l i t y . 15 t h F l o q u e t t h e o r y a s s e r t s t h a t f o r an " n 1 o r d e r system governed by a s e t o f l i n e a r , homogeneous, d i f f e r e n t i a l e q u a t i o n s h a v i n g c o e f f i c i e n t s o f a common p e r i o d i c i t y ( s a y f ) i n the inde p e n d e n t v a r i a b l e , t h e r e e x i s t s a b a s i s c o n s i s t i n g o f n normal solution vectors. Further, from the theory of l i n e a r equations, we know that any solution can be constructed from a l i n e a r combination of basis solutions. Normal solutions have the property that © (1> t ) « \\ fi> ( t ) where the con-stants A ; are referred t o vas c h a r a c t e r i s t i c m u l t i p l i e r s . Here T i s used as the independent variable and the super-s c r i p t i denotes s p e c i f i c solution vectors. For a second order system, such as the one with which we are dealing, the s t a b i l i t y c r i t e r i o n can be expressed as follows, , , i £ *>•.•••* * i 2- » stable \ A;| j (2.21) > I, i * l , 2 ^ u n s t a b l e . In order to construct a basis of normal solutions, i t i s necessary that some basis, say - « t ) , of l i n e a r l y independent solutions be known. Since must also be a solution due to the peri o d i c nature of the c o e f f i c i e n t s of the governing equations, one can express $ ( t V t ) as a li n e a r combination of the basis solutions, $'( t) . Thus both © ( t ) and©'(tVt)can be written i n terms of $(t") , and a r e l a t i o n for the character-i s t i c m u l t i p l i e r s A'« i n terms of the combinative constants re-l a t i n g $'<tVt) to f ' ( t ) can be obtained as follows: For * T ( t V t ) - ± ^ $J(T) , 20 Here the s u b s c r i p t j denotes a s p e c i f i c element w i t h i n a s o l u t i o n v e c t o r . I n p a r t i c u l a r , i f . $j ( o ^ = t < S ; J , we see t h a t fc> * <S(T) s i n c e = ZI b «f J( o) . J J j » I J Thus t h e p r o c e d u r e i s c l e a r ; f i r s t o b t a i n $ ( t ) from e q u a l t o t h e i d e n t i t y m a t r i x and second, f i n d the e i g e n -v a l u e s o f $ ( t ) , t h e s e b e i n g the c h a r a c t e r i s t i c m u l t i p l i e r s o f t h e system. The d i f f i c u l t y l i e s i n the e x e c u t i o n o f the f i r s t s t e p . N u m e r i c a l s o l u t i o n o f the g o v e r n i n g d i f f e r e n t i a l e q u a t i o n s o f f e r s a r e l a t i v e l y easy and a c c u r a t e means t o t h i s end. The c r i t e r i o n f o r s t a b i l i t y can be f u r t h e r s i m p l i f i e d i n t h i s case s i n c e i t may be shown t h a t the p r o d u c t o f the c h a r a c -t e r i s t i c m u l t i p l i e r s i s u n i t y . T h i s f o l l o w s from a c o n s i d e r -a t i o n o f t h e Wronskian and i t s d e r i v a t i v e s y i e l d i n g the r e l a t i o n , f t A , - exp( f rzcnctndt ) where the f u n c t i o n s C\\ a r e the p e r i o d i c c o e f f i c i e n t s o f the governing equations arranged i n state vector form, i . e . , % \ = 21 C ; j ( t ) X j , i * 1 , 2 , r» . For the case J - ' i n hand, ZTr X,^» exp( \(2£L&iix£)c}e) « | . r v J l i + e c o « 6 / ' (2.24) Thus i t i s seen that the c h a r a c t e r i s t i c m u l t i p l i e r s -lie on eit h e r the unit c i r c l e or on the r e a l axis. In the former case, the system i s stable and the trace of the matrix l i e s between — "2. and 2. . In the l a t t e r case, the system i s unstable and the trace of $ ( 2 . IT) i s either greater than+2. or less than—2 . . Thus the s t a b i l i t y c r i t e r i o n may be stated i n terms of the trace of $ ( 2 t r ) as, l T r[$(aTT) ^ 2 , stable (2.25) unstable. Using the above c r i t e r i o n and integrating equation (2.18) numerically using the i n i t i a l conditions X j ** ' I , = O and "Xj n O , 8;' « I for various values of I , QT and © , the s t a b i l i t y charts shown i n Figures 2-3, 2-4 and 2-5 were obtained. 2.4 Motion i n the Large Ideally a s a t e l l i t e should be stable i n the small and should be able to withstand large disturbances without tumbling. gure 2 - 5 S t a b i l i t y c h a r t f o r motion i n the small; e = .5 25 During large amplitude motions, the non-linear terms i n equation (2.17) become s i g n i f i c a n t and, hence, the l i n e a r i z e d analysis may lead to erroneous conclusions. I t i s , therefore, necessary to study the question of s t a b i l i t y i n the large. In th i s sense, o s c i l l a t o r y motion w i l l be referred to as stable and non-oscillatory or tumbling motion as unstable. Motion i n the large i s treated here i n four parts: c i r -cular o r b i t a l motion, non-circular o r b i t a l motion (analytical approach), non-circular o r b i t a l motion (numerical approach) and, f i n a l l y , the v a r i a t i o n a l analysis of periodic motion. 2.4.1 C i r c u l a r O r b i t a l Motion For c i r c u l a r o r b i t a l motion ( « S = o ) , the governing equation becomes autonomous: V ( A -* Beosfc) S i n X » O (2.26) where A = I ( cr-*-1) and B a 3 1 - 4 - . Multiplying equation (2.26) by 2 ^ and integrating with respect to 8, gives the f i r s t i n t e g r a l (&')- aAeostf B s in 2If- 2.E. (2.27) Here E i s a constant of motion and a measure of the t o t a l l i b r a t i o n a l energy of the system. 26 S u b s t i t u t i n g U • A •+• Bcos # i n t o e q u a t i o n (2.27) l e a d s t o V+( y-H)( u-H)(u-4)(u-u A) where u, =* - \/(A +Bf- B(K;T (2.28) U 2 -u 3= A-B V f A + B f - BfffiT A+B f o r tf(o) a O and S ( o ) « X] . 16 E q u a t i o n (2.28) can be i n t e g r a t e d t o g i v e two d i s t i n c t s o l u t i o n s c o r r e s p o n d i n g to r e a l and i m a g i n a r y v a l u e s o f u, and U 3 . i ) i f o , and o 3 a re r e a l , & - c o s ' ) k,+ k?_sr?(ksei-kJk)< \ ka+k^sn^kp + Klk) i i ) i f u , and U 3 a re i m a g i n a r y , (2.29) 5 = cos'J U -t- erf ( k a e + k j k ) / \ i + c „ H K 0 4 k | k ) 7 (2.30) I t may be pointed out that the modulus k and the constants are dependent upon the parameters I and c as well as the i n i t i a l v e l o c i t y H\ . Moreover, the appropriate r e l a t i o n s defining them depend upon the r e l a t i v e magnitudes of U , , u a , U 3 and u,* . Hence, application of the solutions involves a consid-erable amount of computation which tends to reduce th e i r e f f e c t -iveness . Ty p i c a l s a t e l l i t e response obtained using the exact closed form solutions presented here i s shown i n Figures 2-6 to 2-9 i n c l u s i v e . I t i s apparent that the r e s u l t i n g l i b r a t i o n a l motion can be stable or unstable depending on the magnitude of the impulsive disturbance. E f f e c t of n o n - l i n e a r i t i e s i s also e v i -dent through a strong dependence of the l i b r a t i o n frequency on i n i t i a l condition. In studying the question of s t a b i l i t y i n the large, the f i r s t i n t e g r a l obtained previously, equation (2.27), offers a more d i r e c t approach. I t can be written as T + U = E (2.3i) where and (J= - A C O S B S i n # • For maximum or minimum values of the p o t e n t i a l energy, d U / d * = o . ( A + B c o s ^ ) s i n ^ O (2.32) and i f | A / B | <'U Tf« ±- cos" '(-A/B). Thus U(o)=-A, U(±tr)= A and i f l A / B U I, U(* cos (-A/B))- (A% BV2 B. Now i f either U ( ± T T ) or U(±cos"'(~A/B)) i s greater than U(o) , constant t o t a l energy curves e x i s t i n the 0,0" phase plane which enclose the o r i g i n . Thus motion i n the large can be considered stable ( i . e . , o s c i l l a t o r y ) i f A > 0 or i f A / B ^ l and B^O , i . e . , B> 1 A 1 Further i f B ^ | A) , U(±cos (-A/B))^ U( -ft) since Noting that a separatrix between o s c i l l a t o r y and non-oscillatory motion passes through the points of maximum pot e n t i a l energy, the r e l a t i o n s f o r s t a b i l i t y bounds i n the phase plane are zAcosfl-BsirfiUaA; A ^ o , A ^ B 0^ = < (2.33) EAcosX-Bsirf^ + ^iB" . , B>|A|. B In p a r t i c u l a r f o r X; =. O , the c r i t i c a l or maximum i n i t i a l v e l o c i t y , Kc • for stable motion i s CAA ; A ^ o , A> B ((A+BZ/B; B>|A|. Thus for large amplitude stable motion, . O , for B i O A>) ' ( 2 3 5 ) - B, f o r B > O . In terms o f the system parameters I and CT these c o n d i t i o n s are (2.36) The c o r r e s p o n d i n g e x p r e s s i o n s f o r the c r i t i c a l v a l u e o f 8 a r e 2Vl (cr-H) J o r crV-2(l-2.)/l K ~ I (2.37) fa--*;); I>^ /3 and Cr< e ( l - z V l . F i g u r e 2-10 summarizes these f i n d i n g s i n the form o f a c h a r t showing the n e c e s s a r y c o n d i t i o n s f o r s t a b i l i t y i n the l a r g e f o r e « 0 and 8;«.0 . A c a r p e t p l o t of Vc v e r s u s I and CT" i s shown i n F i g u r e 2 -11 . In a more g e n e r a l s i t u a t i o n i n v o l v i n g a r b i t r a r y i n i t i a l c o n d i t i o n s , s t a b i l i t y i n the l a r g e can be d e t e r m i n e d u t i l i z i n g e q u a t i o n s (2.33) i n the f o r m , 2l(<r-n)(»^cos«)-(3l--4.,)sinX-, I<<4/3,or>-l o r ^•)*= J I>4/3,T>a(I-2)/l Z ICcr* \)cosX -(3l-4.)sin X + I2(cr-*if-»(3l-4.)a. 31-4 crc^cr 4 2(I - 2)/I. (2.38) These r e l a t i o n s define the s t a b i l i t y bounds i n the phase plane. Figure 2-12 i l l u s t r a t e s the e f f e c t of the spin parameter on s t a b i l i t y for a s p e c i f i c value of the i n e r t i a parameter I . 2.4.2 Non-Circular O r b i t a l Motion (Analytical Approach) For the general case where 6 ^ 0 there are no known exact solutions to equations (2.17). Further, i f such a solu-t i o n did e x i s t , i t s int e r p r e t a t i o n would l i k e l y be d i f f i c u l t . Moreover, i n t h i s case there i s no f i r s t i n t e g r a l of motion and consequently no d i r e c t method of establishing s t a b i l i t y bounds as there was i n the preceding section. Numerical techniques i n such a s i t u a t i o n can be used to advantage. On the other hand, approximate methods can provide information suitable f or preliminary design purposes. In order to be useful approximate analyses should y i e l d solutions which are: a) acceptably accurate b) simple i n form to f a c i l i t a t e i n t e r p r e t a t i o n , (i) WKBJ method 1 7 To apply the WKBJ method i t i s necessary to l i n e a r i z e the equation of motion and remove the f i r s t derivative term. Con-sider the l i n e a r i z e d system, i.e., equation (2.18). The term containing 16' may be eliminated by introducing the transforma-A t i o n }f = *6 ( I c o s 6) giving 9 " + G Z ( e ) % = 0 (2.39) where G*(8) = 2>1-A- + I(sr-»t) ( i + e f . 38 E q u a t i o n (2.39) i s now i n a form amenable t o t r e a t m e n t by the WKBJ method p r o v i d e d F « I (2.40) where 2G 3 A \ G V U s i n g the d e f i n i t i o n o f G ( 6 ) , the f u n c t i o n p may be w r i t t e n as 1 6 kecosd 3 1 - 4 ( 3 1 - 4 - Ktr+rtO + e ? I e c o s 8 2V3 e c o s e 31-4- + zl(<T+\)(\ + ef  I +ecos 8 (3r~4 + I(cr+»)(iH-eyr V i + e c o s e / + e g s i n 2 8 4 ( l + e c o s 8 ) 6 1 - e Hr 6>I(q"^iK»-»-et  i •¥ e cos e ( 3 1 - 4 + Iia±JlQ±«f V \ i + e c o s e / (2.41) Typical plots of t h i s function are shown i n Figures 2-13 and 2-14 which indicate the a p p l i c a b i l i t y of the method for a wide range of values of I , CT and «B . Rigorous application of the WKBJ method yi e l d s the solution 8 = G / g Tc. Sing + £Lcosg1 (2.42) 1+ecosBL J where ? = j ~ G ( e ) d e and the constants C, and are determined by the i n i t i a l conditions. Although equation (2.42) appears to be simple, i t s evaluation, i n general, i s not straightforward. Substitution for Ca can create d i f f i -c u l t i e s . Though not imperative, i t i s convenient to simplify the solution by introducing the assumption of small o r b i t e c c e n t r i c i t y . This i s j u s t i f i a b l e , as for most situations of p r a c t i c a l importance, Q i s indeed small. I t i s , therefore, use-f u l to develop an approximate WKBJ solution involving only f i r s t order terms i n e c c e n t r i c i t y . Writing (2.43) G = o>e[l-+ e I(cr-f0(l~^cose)/6o5 where UC>\ - I ( A) - 4 - , equation (2.41) can be reduced to F = ecosef 3 l - 4 - + zI(cr-n)l + CKe*) . (2.44) A-cot L J Thus the condition of a p p l i c a b i l i t y , equation (2.40), for small values of e i s related to the basic frequency of l i b r a t i o n a l motion, OJo . Except for situations involving extremely low frequency motion (uJ0 O ) , the method i s applicable. Substituting from (2.43) into equation (2.42) and dropping OCe 2 } / the following solution i s obtained for )S(o)=.0 and * * ( c ) + C c o s e ) s i n | c o 0 ( c 3 e - » - c ^ s i n e ) ^ . Here C, * tf[ f I + e (I (zcr+ I7>-sj] , C 3 = I + e. Kcr+l) = - e/X(z.«r-*-s } - 2 V cola / (2.45) and The nature of the motion i s now cle a r . Both the amplitude and the frequency of motion are modulated for e 0 , More-over, there are f i r s t order e f f e c t s of e c c e n t r i c i t y on both the mean amplitude and frequency. I t i s possible to extract f u r -ther information about the motion by examining the c o e f f i c i e n t s of equation (2.45). For example, the maximum amplitude i s re-lated to K' , I , <r and e by *" ^bi^S^/^^- (2-46) Furthermore, the mean or fundamental frequency of motion i s given by i . e . , O J p * (l ( ^ 4 - l ) ( l +e ) + 3l - 4)/(l(\r - + 4 ) - 4 , ) / ? (2.47) While the WKBJ solution s a t i s f i e s the requirement of f a c i l i t y i n in t e r p r e t a t i o n , there are li m i t a t i o n s i n i t s app l i c a t i o n due to the s i m p l i f i c a t i o n s introduced. Since the solution i s based upon the l i n e a r i z e d system and terms of 0(e 2) have been neglected, i t may not agree c l o s e l y with the exact solution i f either the amplitude of l i b r a t i o n a l motion or the o r b i t a l e c c e n t r i c i t y i s large. Furthermore, i n order to y i e l d 44 meaningful r e s u l t s G00 > O , i . e . , <T > A ( l - l ) / r . To assess the accuracy of the method, the WKBJ solution given by equation (2.45) was compared with the "exact" numerical solu t i o n of the governing equation, (2.17). Typical comparisons are shown i n Figures 2-15 and 2-16. In general, agreement i s quite good considering the amplitude of motion and the magnitude of G. , however, i t i s clear that studies of s t a b i l i t y i n the large would require a more precise solution. I t may be pointed out that the res u l t s obtained using the WKBJ method are i d e n t i c a l to those given by the method of Kr y l o f f and Bogoliuboff involving l i n e a r i z a t i o n with respect to e and 8 . ( i i ) Perturbation analysis In order to overcome the shortcomings of the WKBJ solu-t i o n , i t i s necessary to include the non-linear e f f e c t s . While i n p r i n c i p a l t h i s can be done u t i l i z i n g equation (2.45) as a generating solution, i t s unwieldly form together with the fac t the non-linear terms i n the governing equation are trigonometric r e s u l t s i n considerable d i f f i c u l t i e s . The usefulness of such an approach i s , therefore, somewhat questionable. An alternate approach would be to represent the non-l i n e a r trigonometric terms i n series form d i r e c t l y and work from a simpler generating solution. Consider equation (2.17) i n the form — — Numerical WKBJ 0 180 360 540 720 0,deg. F i g u r e 2-15 Comparison between the e x a c t and WKBJ s o l u t i o n s f o r l i b r a t i o n a l r e s p onse; Y; i.S, 1*2, c=» 1, .2 - 9 0 0 180 F i g u r e 2-16 360 0 , deg. 540 720 Comparison between the e x a c t and WKBJ s o l u t i o n s f o r l i b r a t i o n a l r e s p o n s e ; "6,-0, %'a I ,I=2,CTs o,e= .5 * % [ l ( c r - + A ) - 4 - ] X - e j" - | 2*s in8 3(l - l ) c o s 0 - 2 K a - + i > ( l - cos - r OCez)~ o . (2.48) Here £ i s a perturbation parameter, equal to unity, assigned to the o r b i t a l e c c e n t r i c i t y , & , and to a l l non-l i n e a r terms a r i s i n g from the series expansion of the trigono-metric functions. Neglecting and writing 0 = 0 o + C O i and uJ~ — c o ^ -+• £ t j f gives V + a/*„= 0 which for tf(o) = O, U'(o) « fy' (2.49) y i e l d s = jfj s i n ( c o B ) CO (2.50) 48 and (2.51) s i n ( ( c o ^ l ) e ) j H- ^ 2c^>- 3(1-0- 2 I(<STH-I)J» Jmax X. K£fej+l)sin ( to9)-Sin ( (2j+06ue)| ( 2 .52 : where \ represents the number of non-linear terms included i n the series representation of the trigonometric functions and 49 ; jmax . . The elimination of secular terms requires that r^= ^ e - i - 2i u(ii\ (2.53) where <j + ' ) ! j l C l e a r l y the p r a c t i c a l usefulness of t h i s solution i s l i m i t e d unless one r e s t r i c t s the number of terms i n the series representations of the trigonometric functions. Even i n the case of J m a K m ' / i . e . , when terms are neglected, the r e s u l t i n g solution i s considerably more complex than the WKBJ solution as i t contains terms of the form €in(i*/ 0) , S i n ( 3co8 ) , and sin((cJ-t) 0 ) where a quadratic i s involved i n defining U>a . Before attempting to compare t h i s solution with the WKBJ method, an i n v e s t i g a t i o n was performed with the aim of e s t a b l i s h i n g the best compromise between accuracy and s i m p l i c i t y of form. This was done by comparing the responses as obtained using j m a x t i I , * Z , 3 and A- against accurate numerical solutions. T y p i c a l examples are shown i n Figures 2-17 and 2-18. On the basis of these t e s t s , i t was decided to select the perturbation solution based upon j , ^ ^ — 2 . to compare with the WKBJ method using an accurate numerical standard. The example given i n Figure 2-19 indicates that, i n general, l i t t l e , i f any, improve-ment i n accuracy was attained over the simpler WKBJ method. 1.2 8 Numerical Perturbation ( j m a x = l ) Perturbation (jmax>1) -1.2 180° F i g u r e 2-17 360 6 540 720 Comparison between the e x a c t and p e r t u r b a t i o n s o l u t i o n s f o r l i b r a t i o n a l r e s p o n s e ; X,.=0,fc/:s 1,1*I ,va2,es& o 1.2 Numerical, Perturbation (jmax=1) -- Perturbation (j m a x=2), Perturbation(jmax>2) -1.2 180° 360° 6 540° 720° F i g u r e 2-18 Comparison between the e x a c t and p e r t u r b a t i o n s o l u t i o n s f o r l i b r a t i o n a l response;^aO,Sl.«2,I»Z>«'tB|<e« .16 •1.2 .8 Numerical -1.2 WKBJ ——- Perturbation (j m a x = 2) _ i _ 180° 360° @ 540° F i g u r e 2-19 Comparison between the e x a c t , WKBJ and p e r t u r b a t i o n s o l u t i o n s f o r l i b r a t i o n a l 720° response t o 53 In order to answer the question of s t a b i l i t y i n the large, i t i s necessary to have a solution accurate for large amplitude motion, p a r t i c u l a r l y i n the neighbourhood of 1S=-7T. As indicated by Figure 2-18 the c o r r e l a t i o n between the best ^ Jmax ~ "3 ) perturbation and the numerical solutions i s r e l a t i v e l y poor even for X as small as one radian. Thus i t seems f a i r to conclude that, i n order to a t t a i n s u f f i c i e n t accuracy f o r an a n a l y t i c a l study of s t a b i l i t y i n the large, a highly sophisticated and complex perturbation procedure i s required. This, i n turn, would make such a study so involved that i t would defeat the purpose of searching for an a n a l y t i c a l s o l u t i o n . ( i i i ) Invariant surface concept Returning to the WKBJ solution, v i z . equation (2.45), and d i f f e r e n t i a t i n g i t with respect to Q gives (c i^c z c o s e)(c 3-tc 4 c o s e ) c o e * The argument to„(C 8 -+• O s i n © ) appearing i n 'o » 3 ' A-equations (2.45) and (2.54) can be eliminated giving 54 cos 0 ) LO0 * [ I - ( ft / ( C, H - C 2 c o s 0 ) f ] / 2 - ( C 2 s m 9 / ( C ,+ C 2 C 0 S 9 ) ) . (2.55) Equation (2.55) r e l a t i n g the state variables ft, X and & suggests the existence of a time invariant solution surface i n a three dimensional state space. Since & enters t h i s r e l a t i o n by way of trigonometric functions periodic i n 2. TT, the state space may be truncated to cover any i n t e r v a l of 2. ft i n 0 without loss of generality. In t h i s analysis the i n t e r v a l i s used. A schematic diagram of an invariant solution surface i s shown i n Figure 2-20. A traj e c t o r y eminating from point 1 f o l -lows the surface to point 2 at 0 * 2ft . Continuation of the tra j e c t o r y beyond t h i s point can be represented by a succession of t r a j e c t o r i e s eminating at 6= O having the same values of ft and 8 as the preceding tr a j e c t o r y had when terminated at S =5 2\ Tr . This procedure when repeated over a large number of or b i t s should define a surface referred to as an invariant surface or i n t e g r a l manifold. Moreover, for any spec i f i e d value F i g u r e 2-20 Schematic o f an i n v a r i a n t s o l u t i o n s u r f a c e i n s t a t e space of 0 , i t should be possible to generate a cross section of the i n v a r i a n t surface simply by using the values of 8 and tf obtained from a solu t i o n . Hence a meaningful comparison between methods should be possible i n terms of the cross sec-tions generated by them. I t should be pointed out that the existence of an invar-i a n t surface implies ordered intersections of the t r a j e c t o r i e s with a given plane r e s u l t i n g i n a well defined cross section. On the other hand, an ergodic d i s t r i b u t i o n of points of i n t e r -section would suggest the absence of such a surface. The concept of an invariant surface i n state space i s 18 not new. Theoretical work by Moser and recent numerical experiments by Henon and Heiles and by Jeffreys 2 ^ have demonstrated the existence and usefulness of such surfaces. 4 6 Brereton ' showed, numerically, that s i m i l a r surfaces can be generated f o r a s p e c i f i c , conservative, non-linear, non-autono-mous system with periodic c o e f f i c i e n t s . To v e r i f y the a p p l i -cation of t h i s concept to the problem i n hand, an attempt was made to generate cross sections at © = 0 using an accurate numerical solution. Typical examples shown i n Figures 2-21 and 2-22 c l e a r l y suggest the existence of invariant solution surfaces for t h i s system. The figures also compare cross sec-tions obtained using the WKBJ solution with those obtained numerically. I t i s i n t e r e s t i n g to note that f or 2. ,CT« O and €*- .S there i s good agreement between the a n a l y t i c a l and numerical methods (Figure 2-22) despite r e l a t i v e l y poor agreement i n dynamical response (Figure 2-16). Thus the disagreement i n 100 0-75 -0-50 0-25 0 0 gure 2-21 0 2 5 0 5 0 6 , rad. Comparison of cross sections at 0 « 0 c invariant surfaces generated by the WKBJ and numerical analyses; w = — I , 58 2 0 0 r - • 1 — 1 WKBJ 100 X , rad. F i g u r e 2-22 Comparison o f c r o s s s e c t i o n s a t © « 0 o f i n v a r i a n t s u r f a c e s g e n e r a t e d by the WKBJ and n u m e r i c a l a n a l y s e s ; C T = i O , e » . 3 59 phase between the responses predicted by the two methods does not seem to a f f e c t the geometry of the surface for r e l a t i v e l y small amplitude motion. The p r i n c i p a l l i m i t a t i o n of the WKBJ method appears to ari s e as a r e s u l t of l i n e a r i z i n g the equation of motion. Differences between approximate and "exact" r e s u l t s grow as the amplitude of motion gets larger. Nonetheless, the WKBJ method y i e l d s r e s u l t s of s u f f i c i e n t accuracy f o r preliminary design purposes. 2.4.3 Non-Circular O r b i t a l Motion (Numerical Approach) Fundamentally the problem of obtaining a numerical solu t i o n to a system governed by a set of ordinary d i f f e r e n t i a l equations i s comparatively easy as several techniques are av a i l a b l e . Interpretation of these r e s u l t s , on the other hand, i s often d i f f i c u l t . I t i s here that the invariant surface con-cept introduced previously proves to be most useful. As shown i n the foregoing, the WKBJ method applied to the l i n e a r i z e d system gives r i s e to invariant solution surfaces i n state space. In addition, i t was seen that for e s O, the energy r e l a t i o n , equation (2.27), defines curves i n the ft, ft-phase plane which are, i n e f f e c t , equivalent to uniform cross sections of solution surfaces i n ft,X,9 - state space. Thus the existence of invariant surfaces for both large amplitude motion i n a c i r c u l a r o r b i t and for small amplitude motion i n an e l l i p t i c o r b i t has been demonstrated. As well, the numerical cross sec-t i o n tests discussed i n the previous section strongly suggest the existence of such surfaces under more general conditions. 60 15 In e f f e c t , t h i s i s the basis of Minorsky's method involving a stroboscopic phase plane which, i n essence, i s just a 0 s constant cross section of state space. Minorsky asserts that a stationary point i n the strobo-scopic phase plane i s associated with a periodic solution. This i s evident i n t, 9 — state space as for e 0 , a pe r i o d i c solution must be of period "2.n TT , where n i s a f i n i t e integer. Thus the solution i s represented by a f i n i t e number of t r a j e c t o r i e s i n the truncated state space whose cross sec-tions are a f i n i t e number of stationary points. Furthermore, i n the stroboscopic phase plane, periodic solutions appear as singular points of an ordinary phase plane. For the case under consideration, i t can be shown, by Floquet theory, that periodic solutions appear as e i t h e r centers (considered stable) or saddles. I t i s c l e a r , therefore, that closed cross sections about stable periodic solutions give r i s e to tube-like invariant surfaces surrounding the periodic motion. Since the governing equation (2.17) s a t i s f i e s the L i p s h i t z condition for uniqueness, invariant surfaces cannot i n t e r s e c t as t h i s would imply non-unique solutions. In p r i n c i p l e , begin-ning with a stable periodic solution, a succession of nested inva r i a n t surfaces can be generated by progressively choosing i n i t i a l conditions exterior to the preceding surface. Floquet theory does not predict a l i m i t to t h i s exercise but since equation (2.17) i s non-linear, a breakdown should eventually occur. For example, tumbling motion cannot generate a closed surface about an o s c i l l a t o r y periodic solution. Thus i t would seem a p p r o p r i a t e t o d e v i s e a s e r i e s o f " n u m e r i c a l e x p e r i m e n t s " t o d e t e r m i n e the outermost o r l i m i t i n g i n v a r i a n t s u r f a c e as a f u n c t i o n o f the system p a r a m e t e r s , I , O* and & . In p r a c t i c e , l i m i t i n g s u r f a c e s can be o b t a i n e d to a l m o s t any d e s i r e d degree o f a c c u r a c y by n u m e r i c a l l y i n t e g r a t i n g the g o v e r n i n g e q u a t i o n and a s s e s s i n g a r e s u l t i n g c r o s s s e c t i o n , f o r example, see F i g u r e s 2-21 and 2-22. In most o f the cases i n -v e s t i g a t e d , the e v e n t u a l breakdown o c c u r r e d d r a m a t i c a l l y w i t h an a b r u p t f r a g m e n t a t i o n o f the c r o s s s e c t i o n i n d i c a t i n g tumb-l i n g o r u n s t a b l e o s c i l l a t o r y m o t i o n . On a few o c c a s i o n s u n c e r t a i n t y e x i s t e d near the l i m i t i n g s u r f a c e where f r a g m e n t a t -i o n o f the c r o s s s e c t i o n o c c u r r e d b u t t u m b l i n g was e i t h e r n o t o b s e r v e d w i t h i n the p e r i o d o f i n t e g r a t i o n , g e n e r a l l y 30 o r b i t s , o r o n l y o b s e r v e d b e l a t e d l y . In g e n e r a l , however, the e n v e l o p e o f s t a b l e m o t i o n i n the l a r g e was d e f i n e d t o a h i g h l y s a t i s f a c -t o r y degree o f a c c u r a c y by the outermost unfragmented s u r f a c e . I s o m e t r i c v iews o f t y p i c a l l i m i t i n g i n v a r i a n t s u r f a c e s , r e p r e s e n t i n g the bounds of s t a b l e m o t i o n , a re shown i n F i g u r e s 2-23 and 2-24. Note t h a t f o r the case o f I « . _» , 2> and S - T . I6» , t h r e e such s u r f a c e s are d e f i n e d : a major e n v e l o p e o f s t a b l e m o t i o n (mainland) and two s u b s i d u a r y e n v e l o p e s a s s o -c i a t e d w i t h p e r i o d i c s o l u t i o n s , s u b s e q u e n t l y r e f e r r e d t o as i s l a n d s . Such secondary s u r f a c e s wrap around the m a i n l a n d i n c l o s e p r o x i m i t y t o i t and r e p r e s e n t s m a l l zones o f s t a b i l i t y o f l i t t l e p r a c t i c a l s i g n i f i c a n c e . A s i d e from the a c t u a l e n v e l o p e o f m o t i o n , the c h a r a c t e r -i s t i c s o f a system are b e t t e r r e p r e s e n t e d i n the s t r o b o s c o p i c 64 phase plane or 0 cross section rather than i n state space. The terms mainland and i s l a n d may be seen to take on more p r a c t i c a l s i g n i f i c a n c e i n t h i s context. Generally, mainlands have large cross sections centered on the o r i g i n while islands have smaller cross sections and are peripheral to the mainland. Since a l l t r a j e c t o r i e s span the i n t e r v a l ( O,2.i0 i n 9 , i t i s immaterial which cross section i s chosen for a study of system c h a r a c t e r i s t i c s . I t i s , however, advisable to choose a cross section which w i l l provide maximum information for a given amount of integration. The cross section at 8= O was selected because, as shown by the governing equation, i t i s symmetric with respect to both tf and tf r e s u l t i n g i n a consider-able saving i n computer time. Using t h i s concept of cross sectioning, studies of the c h a r a c t e r i s t i c s of large amplitude motion were re a d i l y accomp-l i s h e d . T y p i c a l examples are shown i n Figures 2-25 and 2-26. The r o l e of periodic solutions as backbones or spines of inv a r i a n t surfaces i s evident. The sphere of influence of any given p e r i o d i c solution i s apparent as well. This suggests that many of the s a l i e n t features of large amplitude motion can be ascertained through the v a r i a t i o n a l analysis of relevant periodic solutions. This i s analogous to the study of singular points i n an autonomous system. Yet another useful extension of the invariant surface concept i s the development of a s t a b i l i t y chart using some spe-c i f i c intercept of the l i m i t i n g surfaces as a measure of sta-b i l i t y . This r e s u l t s i n a considerable condensation of data. F i g u r e 2-25 Study o f i n v a r i a n t s u r f a c e c r o s s s e c t i o n s ;I« 2 , «•« I ,e » 3 F i g u r e 2-26 Study o f i n v a r i a n t s u r f a c e c r o s s s e c t i o n s ;Io.5,«»2,e«.33 I t i s c l e a r , however, that any point on a l i m i t i n g invariant surface i s s u f f i c i e n t to define the e n t i r e envelope of stable motion i n state space. Figures 2-27 and 2-28 are examples of such charts u t i l i z i n g (tf'J a t X « 0 » O a s a measure of s t a b i l i t y showing the d e s t a b i l i z i n g e f f e c t s of o r b i t e c c e n t r i c i t y . The ragged nature of the s t a b i l i t y bound with sp i k e - l i k e features can be a t t r i b u t e d to the emergence of periodic solutions and r e l a t e d islands from the mainland l i m i t i n g surface. This again underlines the need to pursue the analysis of periodic solutions I t should be emphasized at t h i s time that, although the cross sectioning concept i s r e l a t i v e l y simple and y i e l d s much in s i g h t into the nature of motion i n the large, the numerical character of t h i s approach requires a considerable amount of computer time. For example, a t y p i c a l cross section generated over 30 o r b i t s requires nearly one minute on an IBM 7044 compute On occasion, double p r e c i s i o n arithmetic and as many as 720 inte gration steps per o r b i t were required to give accurate r e s u l t s . Since t h i s phase of the analysis was primarily intended for the development and t e s t i n g of techniques, a large scale parametric study was considered unwarranted. 2.4.4 V a r i a t i o n a l Analysis of Periodic Solutions In the foregoing the need to perform a v a r i a t i o n a l analy s i s of p e r i o d i c motion became apparent. To obtain the v a r i -a t i o n a l equation for periodic motion, l e t X= 8 p c»v , where )S p i s the periodic solution and tfv represents a small perturbation. Substituting into equation (2.17) and l i n e a r i z i n g with respect to ^ v y i e l d s : F i g u r e 2-27 V a r i a t i o n o f c r i t i c a l v e l o c i t y w i t h o r b i t e c c e n t r i c i t y ; If;« O, I=.5, C - 2 . F i g u r e 2-28 V a r i a t i o n o f c r i t i c a l v e l o c i t y w i t h o r b i t e c c e n t r i c i t y ; tfj«e O , I s 2, CT * - I 70 ft" - \ 2esine ( £ + } I 3l - -4 . -eco ie)» I 4 -e c o s 6 / 2 fcostf p - s i n * . ) + I ( q - 4 i l f l + e ) 7 \ I + ecose/ C O S 6 B ( 6 V — O. (2.56) Xv - oS i n c e ftp must be p e r i o d i c i n 2 . n T T , where n i s an i n t e g e r , F l o q u e t t h e o r y can be a p p l i e d as the c o e f f i c i e n t s o f e q u a t i o n (2.56) have a common p e r i o d namely t h a t o f the p e r i o d i c s o l u t i o n . F u r t h e r m o r e , as i n the s t u d y o f m o t i o n i n the s m a l l , the p r o d u c t o f the c h a r a c t e r i s t i c m u l t i p l i e r s i s u n i t y b e c a u s e , once a g a i n , e q u a t i o n (2.24) a p p l i e s . T h i s c o n s t i t u t e s a p r o o f of the s ta tement made p r e v i o u s l y t h a t p e r i o d i c s o l u t i o n s behave as e i t h e r c e n t e r s o r s a d d l e s i n the s t r o b o s c o p i c phase p l a n e s i n c e the c h a r a c t e r i s t i c m u l t i p l i e r s l i e e i t h e r on the u n i t c i r c l e o r on the r e a l a x i s . I n o r d e r t o u t i l i z e t h i s a n a l y s i s the f o l l o w i n g p r o c e d u r e was a d o p t e d . F i r s t , i n i t i a l c o n d i t i o n space was scanned i n the a r e a o f i n t e r e s t f o r p o t e n t i a l p e r i o d i c s o l u t i o n s . T h i s was done by i n t e g r a t i n g e q u a t i o n (2.17) n u m e r i c a l l y o v e r a number o f o r b i t s and o b s e r v i n g the s t a t e of the system a t the end of each o r b i t . When a p e r i o d i c s o l u t i o n was b r a c k e t e d i n t h i s manner, an i n t e r a t i v e t e c h n i q u e based on the v a r i a b l e s e c a n t method was used t o match i n i t i a l and f i n a l c o n d i t i o n s . W i t h t h i s , e q u a t i o n (2.56) c o u l d be i n t e g r a t e d s i m u l t a n e o u s l y w i t h e q u a t i o n (2.17) t o o b t a i n $ ( - 2 L n T r ) f rom $(o)=. 1 . The c r i t e r i o n f o r the v a r i a t i o n a l s t a b i l i t y o f any p a r t i c u l a r s o l u -t i o n i s t h e n Tr > 2, u n s t a b l e (2.57) ^ 2, s t a b l e . T h i s method was a p p l i e d t o s t u d y the s t a b i l i t y c h a r t s p r e s e n t e d i n F i g u r e s 2-27 and 2-2 8. R e s u l t s o f the a n a l y s i s are shown i n F i g u r e s 2-29 and 2-30. The n o t a t i o n m/n i s used t o i n d i c a t e p e r i o d i c m o t i o n o f m o s c i l l a t i o n s i n o r b i t s . As p o i n t e d o u t b e f o r e , the appearance o f s p i k e s i s a s s o c i a t e d w i t h s t a b l e p e r i o d i c s o l u t i o n s s e p a r a t i n g f rom the m a i n l a n d . On the o t h e r h a n d , i t s h o u l d be emphasized t h a t u n s t a b l e p e r i o d i c s o l u t i o n s have l i t t l e e f f e c t on the s t a b i l i t y b o u n d a r y . T e r m i n -a t i o n o f the s p i k e s o r i s l a n d s a t c e r t a i n d e f i n i t e v a l u e s of o r b i t e c c e n t r i c i t y as p r e d i c t e d by the v a r i a t i o n a l a n a l y s i s agrees w e l l w i t h the r e s u l t s o b t a i n e d by c r o s s s e c t i o n i n g . In f a c t , the v a r i a t i o n a l a n a l y s i s would be e x p e c t e d to be more a c c u r a t e i n t h i s r e g a r d as l e s s c o m p u t a t i o n i s i n v o l v e d and the c r i t e r i o n (2.57) i s f a r l e s s s u b j e c t i v e t h a n an assessment o f c r o s s s e c t i o n d a t a . I t s h o u l d be s t r e s s e d here t h a t a l t h o u g h t h i s type o f v a r i a t i o n a l a p p r o a c h appears t o resemble an a n a l y s i s o f s i n g u -l a r i t i e s o f a c o n s e r v a t i v e autonomous s y s t e m , i t does n o t , i n f a c t , o f f e r a " q u i c k l o o k " a t the f e a t u r e s o f m o t i o n i n the i r Variationally Stable Variationally Unstable Stability Bound 1 -2 3 e -4 -5 F i g u r e 2-29 V a r i a t i o n a l a n a l y s i s o f p e r i o d i c s o l u t i o n s ; O , I».5,*T 0 .1 2 -3 e .4 -5 F i g u r e 2-30 V a r i a t i o n a l a n a l y s i s o f p e r i o d i c s o l u t i o n s ; O, 1= _., C3~= I 74 large since p e r i o d i c solutions must f i r s t be found. Despite t h i s d e f iciency, however, the analysis i s useful i n seeking b i f u r c a t i o n values of certa i n parameters. 2.5 Concluding Remarks As the purpose of t h i s investigation was to serve as a tes t i n g ground for Phase I I , i t would be appropriate to emphas-ize the important aspects of the analyses and the conclusions based on them: (i) L i b r a t i o n a l motion of a spinning s a t e l l i t e i n r o l l i s coupled to the o r b i t a l motion. Fortunately, the coupling e f f e c t s are small and can be neglected. In the r e s u l t i n g non-l i n e a r , non-autonomous equations of motion, o r b i t a l character-i s t i c s are s p e c i f i e d by a single parameter(e). ( i i ) S t a b i l i t y i n the small can be successfully treated using a numerical approach involving Floquet theory. The method y i e l d s s t a b i l i t y charts delineating stable and unstable regions i n parameter space. ( i i i ) For the p a r t i c u l a r case of e « 0 , the f i r s t integ-r a l of motion (2.27) as well as a closed form solution are a v a i l -able. The f i r s t i n t e g r a l proves to be of greater use i n anal-yzing motion i n the large as the closed form solution i s d i f f i -c u l t to in t e r p r e t . . (iv) Approximate a n a l y t i c a l solutions have limited value as they are not s u f f i c i e n t l y accurate to assess s t a b i l i t y i n the large. Among the methods developed, the WKBJ solution y i e l d s r e s u l t s of greatest value. For small o s c i l l a t i o n s , i t predicts the amplitude and frequency with s u f f i c i e n t accuracy 75 to be useful i n preliminary design. (v) Bounded (o s c i l l a t o r y ) motion was found to generate closed i n v a r i a n t solution surfaces i n ft , 0 --state space. The envelopes of motion can be determined e a s i l y and accurately using numerical techniques. This approach to the study of motion i n the large was found to be most useful and d e f i n i t i v e . (vi) Stable periodic solutions form backbones or spines of i n v a r i a n t surfaces. V a r i a t i o n a l analysis using Floquet theory establishes the s t a b i l i t y of periodic motion and hence, the s a l i e n t features of system behavior. A comment should be made at t h i s stage on the use of Liapounov's d i r e c t method f o r determining s t a b i l i t y i n the large. The method involves determination of a suitable testing function V f o r which no d e f i n i t e procedure i s available when the system i s non-linear and non-autonomous. The very existence of closed i n v a r i a n t solution surfaces, however, implies neutral s t a b i l i t y , i . e . dV/_lt"» O . Thus the testing function i s a constant when the motion i s bounded and, i n f a c t , could be defined by the r e l a t i o n describing the invariant surface. The complicated nature of l i m i t i n g surfaces suggests, however, that the analyt-i c a l form of the Liapounov function would be highly complex thus v i r t u a l l y r u l i n g out any p o s s i b i l i t y of obtaining i t by t r i a l and error. On the other hand, numerical methods y i e l d the desired information with r e l a t i v e ease. 3. PHASE II - ATTITUDE DYNAMICS OF A SPINNING AXISYMMETRIC SATELLITE 3.1 Preliminary Remarks This chapter investigates the attitude dynamics of a r i g i d , axisymmetric s a t e l l i t e free to l i b r a t e i n both r o l l and yaw. The e f f e c t s of o r b i t e c c e n t r i c i t y and the gravity gradient torque are included i n the study. As indicated i n the i n t r o -ductory chapter most of the research i n thi s f i e l d involves a li n e a r or quasi-linear representation of the system. In t h i s analysis, l i b r a t i o n a l motion i s studied both i n the small and i n the large. The pattern established i n the previous chapter i s f o l -lowed with extensive use being made of the numerical methods employed i n that study, i . e . , the invariant surface and cross sectioning concepts and Floquet theory i n the v a r i a t i o n a l anal-y s i s of both motion i n the small and periodic motion. S t a b i l i t y charts are presented which are of considerably greater d e t a i l than those found i n current l i t e r a t u r e . Through the use of the Hamiltonian function,a v a r i a t i o n of the invariant surface concept developed i n Chapter 2 i s used to investigate motion i n the large for the case of & «• © . The concept of p r i n c i p a l cross sections i s introduced enabling the maximum re-sponse of the system to be ascertained given an a r b i t r a r y d i s t u r -bance. Methods are given for determining the i n i t i a l conditions for and the v a r i a t i o n a l s t a b i l i t y of periodic solutions for either c i r c u l a r or e l l i p t i c a l o r b i t a l motion. 3.2 Formulation of the Problem Consider a s a t e l l i t e with center of mass S moving i n a Keplerian o r b i t about the center of force at O (Figure 3-1). As before, the coordinates R and 0 define the position of the center of mass with respect to an i n e r t i a l frame X „ , ^ 0 , 2T0 . Let X p , ^ p , Z P represent the p r i n c i p a l body co-ordinates with o r i g i n at S and the coordinate X p coinciding with the axis of symmetry. Further, l e t X s >^/s>^s ^ e another set of orthogonal coordinates with o r i g i n at S but or-ientated such that the X s axis i s normal to the o r b i t a l plane and ^ s l i e s along the extension of the radius vector R . The coordinates ft, /-3 and OL are modified Euler angles defining the attitude of the s a t e l l i t e r e l a t i v e to the n o n - i n e r t i a l frame X S ; >^s , _ s . The f i r s t r o t a t i o n , ft' , about the l o c a l h o r i z o n t a l , Z . s axis, i s referred to as r o l l ; the second rota-t i o n , about the ^ axis represents yaw while the t h i r d rota-t i o n , OC , about the axis of symmetry i s the spin of the s a t e l l i t e Assuming l i b r a t i o n a l motion to have a n e g l i g i b l e e f f e c t on the motion of the center of mass, a Lagrangian formulation need only involve those terms d i r e c t l y related to attitude motion. This approach i s deemed j u s t i f i a b l e i n the l i g h t of the r e s u l t s of the analysis of Appendix I and i s consistent with the 9 11 12 13 1 methods employed by other researchers i n t h i s f i e l d . ' ' ' ' Again owing to the symmetry of the s a t e l l i t e , the co-ordinate frame X , ^ , Z (Figure 3-1) i s used i n which the s a t e l -l i t e spins with angular v e l o c i t y oc about the X axis. The l i b r a t i o n a l k i n e t i c energy of the s a t e l l i t e can be F i g u r e 3-1 Geometry o f Phase I I model w r i t t e n as T = 1 |l x c4-> I 3 < ^ + ^ x ^ J (3-D where the a n g u l a r v e l o c i t i e s are g i v e n by • * CO* =. c& - "8 s i n / 3 © c o s / 3 c o s t f co^ = /3 - 9 s i n 8 co z = 9 s i n / 5 c o s # . (3.2) F o l l o w i n g the method used p r e v i o u s l y the p o t e n t i a l energy can be w r i t t e n as (3.3) 2F?H I r N o t i n g t h a t I » I x / I ^ I x / I z t the L a g r a n g i a n f u n c t i o n f o r the system becomes e l L >U ( I - l ) ( / - 3 s i n 8 c o s / 3 ) . (3.4) J 80 As b e f o r e , oc i s a c y c l i c c o o r d i n a t e w i t h the f o l l o w i n g f i r s t i n t e g r a l d e f i n i n g OL : oL - 0 siry_» B c©s/_»cos&). (3.5) The s p i n p a r a m e t e r , O " , i s a g a i n d e f i n e d t o be e 1 x 6 (3.6) 0« O C h a n g i n g the i n d e p e n d e n t v a r i a b l e from 1* t o d t h r o u g h the use o f e q u a t i o n s (2.14) and (2.15) and making use o f the s p i n p a r a m e t e r , the g o v e r n i n g e q u a t i o n s f o r l i b r a t i o n a l m o t i o n become: /£>- ft 'cos ft •+ T f a r - nV \ -» _> \l ft ' c . o ^ A * . gin / 4 c o O ( \ l»-«-ecose;l / -(ft*cos/$-*-sin/£cosft) (cos/3cos)f- o"'sin/_0 — 3 (T-l) sim/3cos/_3 sin* "ft -I •+• ecosB / sine \ (/_>'— s m X ) s= o , VI + ecos0 / (3.7) ftcos/lb -2/3'2T ' s in/_> + a / ^ ' c o s / S c o s f t -81 l l + e c o t 8 / 3(1-0 c o s / g . sitn ^ c o s ) i — / 2 e s i n 9 \ » I+ecose \i-t-ecos6/ H - s i n / S c o s 1$) = O . (3.8) This fourth order, non-linear, non-autonomous system i s c l e a r l y more complex than that analyzed i n the previous chapter. As a n a l y t i c a l methods i n such a formidable s i t u a t i o n are not l i k e l y to be successful, the numerical approach i s used extensively i n the analysis of t h i s system. 3.3 Motion i n the Small Following the pattern established i n the preceding phase, a study of the l i n e a r system i s f i r s t attempted. L i n e a r i z a t i o n of the governing equations of motion, (3.7) and (3.8), with respect to the coordinates & and ft res u l t s i n the following: 2 - H e r ^ f U e t] K' \ l+ecoseiJ (3.9) Li + ecose J L \ l + e c o s 9 / 82 / 3 ' + i - I ( ^ \ ) | L L I-I • + - e c o s 9 tf 4- | Z c s i n S . i + e c o s ©, (3.10) F o r the autonomous case o f £ a O f i . e . , c i r c u l a r o r b i t a l m o t i o n , e q u a t i o n s (3.9) and (3.10) reduce t o : where - ( D f - l ) / 3 - ( D . - 2 ) * ' tf" = ( D , - . e ) / a ' - (D.-4- Q ) tf D,= I(<s-+i), (3.11) (3.12) D = 3 1 - 4 > ( - 4 - ^ Q < 2 ) . The c h a r a c t e r i s t i c e q u a t i o n i s g i v e n by Det - S I - (Q - l ) -s o o o o \ o -(q-2) -s 1 = o \ o (p-2) -(p+q) -s / o r 83 4 s -*• [(D,-2j + (q + r4) +( D |_,)]s' jjD, -1)(D, - D2)j = o . (3.i3) For s t a b i l i t y , 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 must be of the same sign. Beginning with the C O e f f i -fe cient of S , f o r s t a b i l i t y i.e., D,c= l± Ll-(3-hD2)J . (3.14) (3.15) For D,e to be meaningful, i t must be r e a l . Thus ( l - ( 3 t l ^ ) ) ^ O o r D^2,i.e.n2/3. Hence the s t a b i l i t y re-quirement involving the c o e f f i c i e n t of S a i s met i f f i) I >s 2 / 3 o, i » K a/3 ^ ^ » < r < ( l - « 3 l ) / i - i Cb> >(lWa-3lVl-|. Turning to the c o e f f i c i e n t of S ° , s t a b i l i t y requires that (D.-OD.-D* > o, (3.17) 84 S o l v i n g (3.17) f o r the c r i t i c a l v a l u e s o f D, , y i e l d s (3.18) i . e . , ' c ^ o r 1 - a . D,_ « i (3.19) T h u s , the s t a b i l i t y r e q u i r e m e n t i n v o l v i n g the c o e f f i c -i e n t o f S ° i s met i f f i ) <r>(l-l) /I i ) I ^ | and < o r b) cr < - 4 ( | - I ) / I o r (3.20) i i ) 1 < \ and a) <r < ( l - l ) / I b) o r <r>4(\-I)/I. F u r t h e r m o r e , i t i s c l e a r t h a t , s i n c e the c h a r a c t e r i s t i c e q u a t i o n (3.13) i s a q u a d r a t i c i n S 2 , the o r i g i n w i l l be s t a b l e i f f i t s r o o t s are n e g a t i v e and r e a l . O t h e r w i s e some of the c h a r a c t e r i s t i c r o o t s w i l l have p o s i t i v e r e a l p a r t s . T h u s , f o r s t a b i l i t y y e t a n o t h e r r e q u i r e m e n t must be met, v i z . , 2 [ D ' - 4 Q + 5] D 2-r 85 [Dr-4 .p 3 +eD*-eD,+9] > o . { 3 . 2 D S o l v i n g (3.21) f o r the c r i t i c a l v a l u e s o f D2 r e s u l t s i n Cfcc- - [D ? -4D, + 5]^2\/-C?-v-5D?-eD (+4.. 0 . 2 2 ) F o r r e a l and hence m e a n i n g f u l v a l u e s of D2c * - D ' + 5 D ' - 8 R + 4 > O. (3.23) As the z e r o s o f t h i s e x p r e s s i o n o c c u r a t I , 2. and 2., i t i s c l e a r t h a t t h e r e i s a bound on D2 i f f D,4l i.e.,0"^ I n t h a t c a s e , e q u a t i o n (3.22) can be used t o d e f i n e the r e g i o n o f s t a b i l i t y i n parameter s p a c e . U t i l i z i n g the r e l a t i o n s ( 3 . 1 6 ) , (3.20) and ( 3 . 2 2 ) , i t i s p o s s i b l e t o produce a s t a b i l i t y c h a r t i n v o l v i n g the parameters I and CT ( F i g u r e 3 - 2 ) . T h i s c h a r t when compared w i t h F i g u r e 2-2 i l l u s t r a t e s the g r e a t e r c o m p l e x i t y o f t h i s phase of the s t u d y . T u r n i n g now t o the more g e n e r a l s i t u a t i o n o f O , i t i s seen t h a t the g o v e r n i n g e q u a t i o n s , (3.9) and ( 3 . 1 0 ) , have p e r i o d i c c o e f f i c i e n t s and h e n c e , can be s t u d i e d u s i n g F l o q u e t t h e o r y . R e c a l l i n g t h a t the method i n v o l v e s the d e t e r m i n a t i o n of a f i n a l c o n d i t i o n m a t r i x f rom an i n i t i a l c o n d i t i o n m a t r i x e q u a l t o the i d e n t i t y m a t r i x w i t h the subsequent e v a l u -00 F i g u r e 3-2 S t a b i l i t y c h a r t f o r m o t i o n i n the s m a l l ; 6 = O 87 a t i o n o f the c o r r e s p o n d i n g e i g e n v a l u e s o r c h a r a c t e r i s t i c m u l t i -p l i e r s , A; , t h e s t a b i l i t y c r i t e r i o n can be e x p r e s s e d as f o l l o w s : A, ^ | . s t a b l e (3.24) i=il2>a,4 / > | • u n s t a b l e . A l t h o u g h i t can be shown, by e q u a t i o n ( 2 . 2 3 ) , t h a t 7TA; as I , the t r a c e o f $ ( 2 T r ) cannot be used as the s o l e measure o f s t a b i l i t y as the c h a r a c t e r i s t i c m u l t i p l i e r s a re n o t , i n t h i s c a s e , r e s t r i c t e d t o be on the u n i t c i r c l e or the r e a l a x i s . I t i s c l e a r , however, t h a t i f , the system i s u n s t a b l e . Thus the s t a b i l i t y c r i t e r i o n becomes, ^ 4 . u n s t a b l e Tr[$(atr)]| |AJ ^ h s t a b l e 4 4 ' J i , , w | A | > I * u n s t a b l e . (3.25) U s i n g the above c r i t e r i o n and i n t e g r a t i n g e q u a t i o n s (3.9) and (3.10) n u m e r i c a l l y t o g e n e r a t e $(zfr) f rom the f o l l o w i n g i n i t i a l c o n d i t i o n s : 88 i> /SMcOsi, /3'(o)- tt(o)-t'(o) « p i i } /3'(o)=l, /3(o)» *(o)* *'(o)= o i i i ) o ( O ) = I, y_Wo)« /_»'(0) a 8 '(o) = O iv) 8"(o) = I, /3(o)=/2>Yo)=8(o)= o the s t a b i l i t y charts shown i n Figures 3-3, 3-4, 3-5 and 3-6 were obtained. Although these charts d i f f e r considerably i n d e t a i l from t h e i r counterparts obtained i n Phase I (Figures 2-3, 2-4 and 2-5), they have some s i m i l a r i t y i n basic structure. Furthermore, the usefulness of Phase I i n developing suitable analyses i s demonstrated here by the r e l a t i v e ease with which these, r e s u l t s were obtained. 3.4 Motion i n the Large U n t i l t h i s stage of the analysis, motion was rea d i l y c l a s s i f i e d as stable or unstable. Motion i n the small never presents a problem i n t h i s regard. In Phase I, motion i n the large was seen to be either o s c i l l a t o r y or non-oscillatory. With the i n c l u s i o n of the/2> degree of freedom, the s t a b i l i t y of motion i n the large i s d i f f i c u l t to define. Rather, for p r a c t i c a l purposes, the response of the system may or may not be acceptable and would then be referred to as stable or unstable. For example, a configuration may be deemed acceptable i f the axis of symmetry does not deviate from the normal to the o r b i t a l plane by more than some prescribed angle. Since the matter of s t a b i l -i t y i s not well defined i n the case, no attempt i s made here to impose an a r b i t r a r y standard. This, however, does make i t d i f f i c u l t to condense r e s u l t s i n a manner s i m i l a r to that i n Phase I. pertains to c i r c u l a r o r b i t a l motion and the l a t t e r to e l l i p t i c a l t r a j e c t o r i e s . In each case, the analysis involves the extension of the numerical, techniques which proved useful i n Phase I. 3.4.1 C i r c u l a r O r b i t a l Motion (Analytical Approach) As i n Phase I, the r e s t r i c t i o n of e » O reduces the governing equations to autonomous form: Motion i n the large i s treated i n two parts: The f i r s t /3"-*- [l(cr+i)cos/3 - 2 c o s z / 6 c o s tf] tf'-[| + (3l-4.)sin*tf -U'f) sin/3 cos/d-t-(3.26) tf'cos/3 + [a c o s / 3 c o s tf - I(or+ 0 - 2tf sJn/3]/*>+ 3l-4.)cos/5]sintfcostf + I(cr+i)stn tf * O . (3.27) Prospects of finding a closed form solution to these equations are dim indeed. However, as seen i n Phase I, considerable i n -sight into the problem can be obtained by examining the f i r s t i n t e g r a l . Noting that, for S *= O f x £ C / R = © and the Lagran-gian of the system can be expressed as ( i c o s / 6 + 9&m/&aosl)+(l-\)ff(\-'3>c.o&5siY?'6) (3.28) where & = constant. Since _L/Z>t= o , dH/d+= O , i . e . the Hamiltonian i s a f i r s t i n t e g r a l or constant of motion. By d e f i n i t i o n , H = <* + x2> + c K ft - L . (3.29) &<k Thus, within a m u l t i p l i c a t i v e constant,the Hamiltonian can be written as (*'Tco€4> + CO^/&COSZ%+3(l-\)<ZOS?/*><siY?t - (3.30) Furthermore, from equations (3.5) and ( 3 . 6 ) , the variable OC; can be eliminated by introducing the spin parameter C giving H = - _ I 1) cos/3cosK -+• [(5')2-+-3(1-0 - ( 3 l - 4 ) c o s * S ] c o s 2 ^ . (3.31) The above e q u a t i o n can be r e w r i t t e n as C ^ ' f + ( Jffcos*/S> = H + a l ( c r - n ) c o s / 3 c o s * - [ 3 ( I - l ) - ( 3 I - 4 . ) c r f t f 3 c o s / d . (3.32) F o r r e a l m o t i o n , (3.33) F o r the c r i t i c a l s i t u a t i o n where the e q u a l i t y i s s a t i s f i e d , the r e l a t i o n y i e l d s z e r o v e l o c i t y c u r v e s r e p r e s e n t i n g the a b s o l u t e bounds o f m o t i o n i n / 3 , tf- s p a c e . S o l v i n g f o r ^ > i n terms of tf g i v e s /*> « cos ' 1 I (cr+l)costf 3 ( 1 - 0 - ( 3 1 - 4 ) cos^S I ( c r - u l c o s 8 H s7 L 3 ( I - l ) - ( 3 l - 4 ) c o s e t f / 3(I-|)-(3l-4)cos*o| 13.34) I t i s e v i d e n t t h a t zero v e l o c i t y c u r v e s are s y m m e t r i c a l w i t h r e s p e c t t o b o t h / h and tf . In a d d i t i o n , due to the p e r i o d i c n a t u r e of the COS f u n c t i o n and the manner i n w h i c h tf e n t e r s the e q u a t i o n ( 3 . 3 4 ) , z e r o v e l o c i t y c u r v e s are not o n l y p e r i o d i c i n 2 TT i n b o t h / £ > and tf but p o s s e s s symmetry about the p o i n t TttTf . Thus i t i s s u f f i c i e n t t o p r e s e n t these c u r v e s i n the 96 region covering the i n t e r v a l (o> Tr ) i n ft and (o,_Tr*) i n /2> , I t may be pointed out that i f the argument of cos"' exceeds unity i n absolute value or i s complex, the expression does not give a r e a l value of/£> i n d i c a t i n g the absence of zero v e l o c i t y . Zero v e l o c i t y curves f o r several configurations (l,<S") over a va r i e t y of i n i t i a l conditions ( H ) are shown i n Figures 3-7 to 3-10. Interpretation of these plots i s r e l a t i v e l y simple. Real motion i s possible only on the side of decreasing H . Of p a r t i c u l a r i n t e r e s t are the curves about the o r i g i n enclosing a region of r e a l motion. From equation (3.34) i t follows that such curves e x i s t i f f ^">— I. However, even for 5" > - I , the o r i g i n does not represent the minimum r e a l i z a b l e H i f I »^ I . In these cases the zero v e l o c i t y curves, near the o r i g i n but not necessarily enclosing i t , cover r e l a t i v e l y large regions as shown i n Figure 3-10. I t should be emphasized at t h i s point that although curves cl o s i n g around the o r i g i n y i e l d the maximum possible amp-li t u d e of/2> and IS motion f o r a given value of H , they cannot predict the actual maximum amplitude of motion for a given disturbance. Furthermore, curves which do not enclose the o r i g i n do not necessarily imply large amplitude motion but merely i n -dicate i t s possible occurance. Note that for a given value of H f the disturbance i s not uniquely determined, i . e . , there are an i n f i n i t e number of values of the state parameters which s a t i s f y equation (3.31). I t i s possible to obtain some further information from the Hamiltonian. Since the Hamiltonian i s a constant of motion, 0° 30° 60° 90° Y F i g u r e 3-7 Zero v e l o c i t y c u r v e s ; 1*2 , CT = 2 98 F i g u r e 3-8 Zero v e l o c i t y c u r v e s ; I = 2 , CT = - 2 101 i t can be u t i l i z e d to eliminate any one of the four state para-meters and thus three dimensional state space can be used to describe the response of the system. Within the bounds imposed by the zero v e l o c i t y curves, r e l a t i o n s defining surfaces i n /3, ^ jft or "ft,ft t /_3 space can be obtained by equating the ignored v e l o c i t y to zero. Thus motion i s bounded by COS 6 -[ 3 ( I - I ) - ( 31-4- ) . e o « ] cos*/3 i n / - » , / 2 $ ' space and by ( X'f = ^ H + 21 I (<r-t-1 ) c o s / i c o s f t -[ 3 ( I - l ) - ( 3 l - 4 ) cos^Jcos^J/coS?/S i n * , f t space. (3.35) (3.36) Examples of such surfaces are shown i n Figures 3-11 to 3-14. They represent envelopes of possible motion i n state space for a given value of the Hamiltonian. Note that the zero v e l -o c i t y curves are merely cross sections of these surfaces show-ing the extent of maximum possible motion. I t should be emphasized that the actual motion of a system i s dependent upon the i n i t i a l conditions and not merely the value of the Hamiltonian. Thus, i n order to e s t a b l i s h i t s c h a r a c t e r i s t i c s , such as amplitude and frequency, i t i s necessary 106 to solve the governing equations(3.26) and (3.27). The concept of an invariant surface obtained numerically i s used to study the system i n the following section. 3.4.2 C i r c u l a r O r b i t a l Motion (Numerical Approach) From equation (3.31) i t i s seen that although any one of the state elements can be expressed i n terms of the other three and the Hamiltonian, there i s an ambiguity as to i t s sign. As pointed out by Henon and Heiles ^ , i t i s necessary to d e l i n -eate between these two p o s s i b i l i t i e s to u t i l i z e the invariant surface concept i n the study of such a system. Hence, two surfaces must be used to describe the state of the system; one for p o s i t i v e values of the eliminated state element and the other f o r negative values. That i s to say, once a state para-meter i s chosen for elimination, i t s sign should be used to de-termine the portions of the trajectory pertaining to a given surface. For example, i f Y i s eliminated, the system i s des-cribed i n /3,/~> j tf state space which may or may not be bounded by a closed envelope of possible motion (Figures 3-11 and 3-13). I t should be pointed out that for f i n i t e values of H , the angular v e l o c i t i e s /£> and tf are bounded but not necessarily the angular displacements. As pointed out before, motion of the representative point occurs i n two state spaces; one for situations where tf >0 and the other for tf ^ O . For a given i n i t i a l condition, the tra j e c t o r y defined b y / ^ , / 3 and tf l i e s i n a s p e c i f i c state space depending upon the value of tf; . Switching points between the 107 spaces occur when ft «= O . This means that the traj e c t o r y termin-ates i n a given space when i t meets the envelope of possible motion and then continues i n the other space. A p a r a l l e l may be drawn between the state space ft or ft, ft',/3 and the one used i n Phase l ( f t ; f t ' , 0 ) . They are si m i l a r as each involves a phase plane stretched into a t h i r d dimension by a coordinate. Furthermore, i n each case a repre-sentative point moves from one switching point to the next i n a given space i n such a manner that the stretching coordinate, i . e . , / - ^ ^ o r 0, i s either monotonically increasing or decrea-sing with time. This was evident i n Phase I as © was always p o s i t i v e . I t also holds for t h i s phase of the study as the cases corresponding to p o s i t i v e and negative values of the eliminated v e l o c i t y are treated separately. This property of a monotonically changing state element f a c i l i t a t e s the interpre-t a t i o n of cross sections of the invariant surface. I t should be emphasized that such monotonic behavior of the stretching coordinate i s not available i n eithe r ft or ft, ft',/3' space. For t h i s reason, the discussion of invariant surfaces i s r e s t r i c t e d to those i n / - 5 , ft and tf, t' f/3 state spaces, referred to as/3 and ft space respectively. Consider now the symmetry properties of invariant sur-faces in/-3 and V spaces. The governing equations of motion are invariant during the transformation — 0, ft- ft and /_2 = -/3 Since t h i s also implies that = - X and - / £ , i t i s clear that: / , i) for /» of the same sign, symmetry exists about the 108 tf axis i n tf space. i i ) for tf of opposite sign, there i s symmetry about the / 3 ) J - plane. S i m i l a r l y with the transformation 0~-6 ; ) f = - t f a n d / ^ s / ^ , the equations of motion are unchanged. Since t h i s implies that tf'- 5' and /3'=-/£>', i t follows that: i i i ) f or tf of the same sign, symmetry exists about t h e / 3 axis i n space. iv) f o r /^ of opposite sign, there i s symmetry about the X ,/5-plane. I t should be noted that the above symmetry properties only imply the existence of an image solution but t h i s does not necessarily mean that the image surface i s coincident with that generated by the given so l u t i o n . Coincidence of these surfaces can be guaranteed only i f i n i t i a l conditions are chosen to l i e along an axis of symmetry. Turning now to the problem of invariant surface gener-ation; due to symmetry properties ( i i ) and (iv), i t i s s u f f i c i e n t to discuss only those surfaces pertaining totf>Oin /2> space and t o / 3 > O i n space. Thus the problem i s s i m i l a r to that en-countered i n Phase I except that the stretching coordinate, tf o r /3 , i s not the independent var i a b l e . Furthermore, i n t h i s case switching points are unknown i n terms of the independant v a r i a b l e . In Phase I, cross sections of the invariant surface were e a s i l y obtained since the numerical integration of the equation of motion was performed using a f i x e d step-size i n & , the stretching coordinate. Here, however, to obtain cross sections in/3 ox }f space, i t was necessary to develop an int e r p o l a t i o n scheme so that the state of the system could be- ascertained for any given value of the stretching coordinate. This was achieved 21 by using the Adams Bashforth predictor-corrector method i n conjunction with a polynomial f i t to the past history of the state coordinates and t h e i r d e r i v a t i v e s . Having f i t t e d the numerical solution with polynomials i n B , the state of the system could r e a d i l y be computed using Newton Raphson i t e r a t i o n . The mechanics of generating invariant surfaces i s now cl e a r . At a number of preselected values of the stretching coordinate, cross sections were obtained by determining the state of the system each time the trajectory intersected the section plane. The symmetry properties discussed above were u t i l i z e d to minimize computational e f f o r t . I t should be noted that during the integration of the equations of motion no attempt was made to reduce the order of the system using the Hamiltonian. Rather, the Hamiltonian, which i s a constant of motion, was computed along with cross section data and was used as a check on the o v e r a l l accuracy of the method. Typical examples of surfaces generated i n t h i s manner are shown i n Figures 3-15 to 3-18. Note the presence of open cross sections. These openings are analogous to the ends of the invariant surfaces obtained i n Phase I (Figures 2-23 and 2-24). As against a common value of the stretching coordinate for switching points, here they occur over a range of values of the stretching coordinate. As mentioned previously, switching Y -1.5 v. P F i g u r e 3-15 I n v a r i a n t s o l u t i o n s u r f a c e i n /^ >- space 1*2, <r= 0 , 4 = K'=0,/3±2.AG,\=.S F i g u r e 3-16 I n v a r i a n t s o l u t i o n s u r f a c e i n 0 — s p a c e 1 = 2, cr= o , / £ = & L « 0 , / 3 » _ . 4 G , * F « . 5 F i g u r e 3-18 I n v a r i a n t s o l u t i o n s u r f a c e s i n 8 - s p a c e ; U z , cr= * = o , 1 ; *> a . o o a p o i n t s o c c u r a t b o u n d i n g s u r f a c e s s i m i l a r t o t h o s e shown i n F i g u r e s 3-11 t o 3-14. I t i s c l e a r , t h e r e f o r e , t h a t i n v a r i a n t s u r f a c e s a r e , i n f a c t , t e r m i n a t e d by the a s s o c i a t e d bounding s u r f a c e o r e n v e l o p e . In Phase I o f the s t u d y , much economy of e f f o r t was r e a l i z e d by d e t e r m i n i n g o n l y the r e l e v a n t c r o s s s e c t i o n s r a t h e r than the e n t i r e s u r f a c e . C r o s s s e c t i o n s c l e a r l y r e v e a l e d the s t r u c t u r e o f the n e s t i n g s u r f a c e s and the r o l e t h a t the p e r i o d i c s o l u t i o n s p l a y e d i n d e t e r m i n i n g these s t r u c t u r e s . H a v i n g c h o s e n / ^ and ft space f o r t h i s s t u d y , the i n v a r -i a n t s u r f a c e s a s s o c i a t e d w i t h t h e s e spaces are o f a s i m i l a r t y p e w i t h the s t r e t c h i n g c o o r d i n a t e b e h a v i n g i n a monotonic f a s h i o n . C o n s e q u e n t l y , i t s h o u l d be e x p e c t e d t h a t 0. c o n s t a n t c r o s s s e c t i o n s i n a n d ft space r e s p e c t i v e l y y i e l d s i m i l a r i n f o r -m a t i o n t o t h a t o b t a i n e d f o r 6 «=• c o n s t a n t s e c t i o n s i n Phase I . The d i f f i c u l t y h e r e l i e s i n the q u e s t i o n of where t o s e c t i o n the s u r f a c e . In Phase I o f the s t u d y , p o s i t i o n o f the c r o s s s e c t i o n was i m m a t e r i a l as the s w i t c h i n g p o i n t s were known a p r i o r i , i . e . , a t 9 •= O and 2 TT . On the o t h e r h a n d , i n t h i s case s w i t c h i n g p o i n t s a re n o t known a p r i o r i . In absence o f such knowledge i t would seem n e c e s s a r y t o use s e v e r a l s e c t i o n s t o o b t a i n a complete p i c t u r e o f m o t i o n i n the l a r g e . However, s i n c e i n most p r a c t i c a l a p p l i c a t i o n s the p r i m a r y i n t e r e s t i s i n the m o t i o n about the o r i g i n , i . e . , near /£> - ft - O , s e c t i o n s were t a k e n a t ft - O i n /3' space and / _ £ * O i n ft' s p a c e . F u r t h e r m o r e , a d d i t i o n a l i n f o r -m a t i o n can be o b t a i n e d by s e c t i o n i n g the s u r f a c e a t /3 » 0 i n /h space and a t ft = O i n ft s p a c e . These s e c t i o n s g i v e the maximum 115 amplitude of motion i n t h e a n d tf modes. The foregoing cross sections are c o l l e c t i v e l y referred to as the p r i n c i p a l cross sections. A summary of the properties and uses of these sections i s presented i n the following table: TABLE I P r i n c i p a l Cross Sections Type Zero Element Relevant Space Use 1 tf' tf mode study 2 A' amplitude 3 tf {/S,/2>\ tf) mode study 4 tf' tf'(M;/3) tf amplitude A mention should be made of the two problems encountered at t h i s stage. The f i r s t involves the d i f f i c u l t y imposed by the si n g u l a r i t y i n the tf equation (3.27). At |/2>| * fr/2 ± ZnTT , where n i s an integer, i t i s seen that tf i s i n f i n i t e . This d i f f i c u l t y i s inherent with the use of Euler angles and while i t can be overcome by the use of a d i f f e r e n t set of Euler angles, i n addition to the set already used, i t would seem p r a c t i c a l to simply l i m i t the study to motion of |/^| 7X / . This i s j u s t i f i a b l e as the motion of larger amplitude i s of l i t t l e p r a c t i c a l i n t e r e s t . The second d i f f i c u l t y arises when apparent breakdown of the invariant surfaces occurs. In Phase I, breakdown represented 116 unstable or tumbling motion. In thi s case, however, the concept of s t a b i l i t y i s undefined. Nonetheless, breakdown was observed to occur, and occasionally even when the associated bounding surface was closed. I t was not clear whether the breakdown was r e a l , i . e . , i n d i c a t i n g ergodic behavior of the system, or i f i t was simply the r e s u l t of the l i m i t a t i o n s of the numerical approach. Its occurance was always associated with r e l a t i v e l y large amp-lit u d e motion and consequently the uncertainty of t h i s data was of l i t t l e importance. The study of the maximum amplitude of motion for a given system was performed using cross sections of types 2 and 4. Unfortunately, symmetry properties could not guarantee that a given cross section, say i n % space with /_5 > O , would have i t s mirror image generated by the same trajectory for /—* ^  O . I t was, therefore,necessary to consider the sections for both v e l o c i t y conditions to esta b l i s h the maximum response. Figures 3-19 and 3-20 show sections of types 2 and 4 respectively for the s p e c i f i e d values of the system parameters and i n i t i a l conditions. These sections reveal that the ensuing motion i s bounded ( - 0.5 < /2> < C».5 0.76 < o*< 0.~7& ) . S i m i l a r l y Figures 3-21 and 3-22 show, for d i f f e r e n t values of the i n i t i a l conditions, the motion to be bounded by — I . O and — M © 5 < C & I. I 8 5 . Note that while i n the former case, bounded motion i s guaranteed by the closed zero v e l o c i t y curve, i n the l a t t e r , bounded motion occurs despite the fact that the envelope as derived from the Hamiltonian i s open. Thus the point made e a r l i e r , that bounding surfaces indicate only 118 .3 P red. .2 .1 L o - 1 . 0 Prod I .2 -.5 -1.0 -.5 F i g u r e 3-20 "< r P'>o Zero Velocity Curve Cross Section Yrod. 1.0 1.5 2.0 f r p'<o ± ± 0 Y r Q d .5 1.0 1.5 2.0 Type 4 c r o s s s e c t i o n s of i n v a r i a n t s u r f a c e ; <r-o,/3/= V - o ^ = . 5 , I . S 0 3 120 i — i 1 r T 1 1 1 i r P'<o F i g u r e 3-22 Type 4 c r o s s s e c t i o n s o f i n v a r i a n t s u r f a c e ; .121 the r e g i o n o f p o s s i b l e m o t i o n , i s c l e a r . C o n s i d e r now the p r o b l e m of s t u d y i n g the s t r u c t u r e of the i n v a r i a n t s u r f a c e s u s i n g c r o s s s e c t i o n s o f types 1 and 3. As i n Phase I , i t i s e v i d e n t t h a t u n i q u e n e s s a s s u r e s t h a t s u r -f a c e s do not i n t e r s e c t . As w e l l , i t i s c l e a r t h a t p e r i o d i c s o l u -t i o n s appear as s t a t i o n a r y p o i n t s i n c r o s s s e c t i o n and t h a t s e c -t i o n s o f s u r f a c e s c l o s i n g about such p o i n t s a re n e s t e d . C o n s e -q u e n t l y , r e l a t i v e l y few, a p p r o p r i a t e l y c h o s e n , s o l u t i o n s are r e q u i r e d t o g i v e c o n s i d e r a b l e i n s i g h t i n t o the n a t u r e of mot ion i n the l a r g e . T y p i c a l examples of c r o s s s e c t i o n s t u d i e s are shown i n F i g u r e s 3-23 t o 3-27 . In each case s e v e r a l s o l u t i o n s h a v i n g the same v a l u e s o f I , and H a re s e c t i o n e d t o show the s t r u c t u r a l p r o p e r t i e s o f the s o l u t i o n s u r f a c e s . In F i g u r e s 3-23 t o 3-26, c r o s s s e c t i o n s o f b o t h t y p e s , i . e . , types 1 and 3, are shown. F i g u r e 3-27, on the o t h e r h a n d , c o n t a i n s type 1 c r o s s s e c t i o n s f o r and /5'<0 showing the a lmost com-p l e t e l a c k o f symmetry i n the b e h a v i o r of the system Cj " " s a I , H=- .9 ) . In t h i s case the m a j o r i t y o f s o l u t i o n s were such t h a t the c o o r d i n a t e tf d i d not change s i g n a n d , f o r these s o l u t i o n s , the c o r r e s p o n d i n g type 3 s e c t i o n s were n o n - e x i s t e n t . These s t u d i e s show the e x i s t e n c e o f s t a t i o n a r y p o i n t s which appear as e i t h e r c e n t e r s o r s a d d l e s as d i d p e r i o d i c 122 123 T T 3 L r —- Motion Bound Cross Section Typel; p>0 I n v a r i a n t s u r f a c e s e c t i o n s t u d i e s of t y p e s 1 and 3 ; I = " 2 , <T = 1 , H = I 124 I n v a r i a n t s u r f a c e s e c t i o n s t u d i e s o f t y p e s 1 and 3; 1= 2 , <T= I , H - 5 8 T T 7 1 Typel; P>0 6i_ r \ Motion Bound Cross Section / I -1.5 -1.0 F i g u r e 3-26 0 p rod. I n v a r i a n t s u r f a c e s e c t i o n s t u d i e s o f t y p e s 1 and 3; 1= |.5,CT = -a, H -A -3 -2 F i g u r e 3-27 ' 1 I , .,4 ,. 11 LJ—•_ - 1 0 Yrod. ' I n v a r i a n t surface s e c t i o n I = .5, O* » I , H B - , 9 ad  i t i  s t u d i e s - t y p e 1; . 9 127 solutions i n Phase I. When tested for s t a b i l i t y using Floquet theory, the former proved to be v a r i a t i o n a l l y stable while the l a t t e r were found to be unstable. Referring to Figures 3-23, 3-24 and 3-25, i t i s seen that varying H for a given configuration materially affects the i n i t i a l conditions required for periodic motion. Hence a study of periodic solutions, s i m i l a r to that conducted i n Phase I but using H rather than e as the varied parameter, i s useful. In t h i s manner i t i s possible to r e l a t e v a r i a t i o n a l s t a b i l i t y , i n i t i a l conditions, and the period of the motion to the Hamiltonian. I t may also be pointed out that, for closed envelopes of possible motion, cross section studies suggested the e x i s t -ence of several periodic solutions for a given value of H (Figures 3-23 and 3-27). However, for p o s i t i v e values of the Hamiltonian ( i . e . , open envelopes of possible motion) only one of these appeared to p e r s i s t (Figures 3-24 and 3-25). This ten-dency was observed i n a l l the cases studied. The periodic solu-t i o n e x i s t i n g over the largest range of the Hamiltonian i s referred to as "fundamental" i n the subsequent discussion. Figure 3-23 shows sections of invariant surfaces, for a variety of i n i t i a l conditions, bounded by a closed envelope. As pointed out before, for H > 0 , the zero v e l o c i t y curves are open. Even i n these cases, closed cross sections can e x i s t as i l l u s t r a t e d i n Figures 3-24 and 3-25. S i m i l a r l y , for the case of I 1.5, = - 2 . 0 studied i n Figure 3-26 i n which no closed envelope e x i s t s , the analysis of motion i n the small 128 c o r r e c t l y p r e d i c t e d the n a t u r e o f the response to s m a l l d i s t u r -b a n c e s . F i g u r e 3-27 s t u d i e s a c o n f i g u r a t i o n w h i c h was found u n s t a b l e i n the s m a l l d e s p i t e the f a c t t h a t e n v e l o p e s o f mot ion as d e t e r m i n e d by s u i t a b l y s m a l l v a l u e s o f the H a m i l t o n i a n are c l o s e d i n d i c a t i n g m o t i o n o f f i n i t e a m p l i t u d e . These c r o s s s e c t i o n s show t h a t the system i s c h a r a c t e r i z e d by an u n s t a b l e o r i g i n b u t m o t i o n remains f i n i t e , p r o v i d e d the H a m i l t o n i a n i s s m a l l , due t o the i n f l u e n c e o f s t a b l e p e r i o d i c s o l u t i o n s n e a r b y . N o n e t h e l e s s , the a n a l y s i s of m o t i o n i n the s m a l l c o r r e c t l y p r e -d i c t e d the u n s t a b l e n a t u r e o f the o r i g i n as t r u l y s m a l l a m p l i -tude m o t i o n o f t h i s system i s n o t p o s s i b l e . C r o s s s e c t i o n s t u d i e s were c a r r i e d out f o r a v a r i e t y o f system parameters b u t f o r c o n c i s e n e s s o n l y r e p r e s e n t a t i v e r e s u l t s are p r e s e n t e d h e r e . A u s e f u l a p p l i c a t i o n of the c r o s s s e c t i o n i n g method s h o u l d be mentioned a t t h i s t i m e . C r o s s s e c t i o n s o f types 2 and 4 were shown t o g i v e the maximum a m p l i t u d e o f the m o t i o n . As an a c c u r a t e e s t i m a t e o f the maximum response u s u a l l y i n v o l v e s r e l a t i v e l y l i t t l e c o m p u t a t i o n a l e f f o r t , a p a r a m e t r i c s tudy can be u n d e r t a k e n q u i t e r e a d i l y . I t would be u s e f u l t o o b t a i n r e s p o n s e c h a r t s , s i m i l a r to the s t a b i l i t y c h a r t s o b t a i n e d by the l i n e a r i z e d a n a l y s i s , showing the maximum response f o r a g i v e n d i s t u r b a n c e as a f u n c t i o n of the system parameters I and C T ( F i g u r e s 3-28 to 3 -31 ) . F i g u r e s 3-28 and 3-29 p e r t a i n t o systems exposed t o a d i s t u r b a n c e c h a r a c t e r i z e d by the i n i t i a l c o n d i t i o n s : 133 and tf/ « O . O I . S i m i l a r l y , Figures 3-30 and 3-31 pertain to systems exposed to a larger i n i t i a l disturbance: and tf/ » O. S . I t i s i n t e r e s t i n g to compare Figures 3-28 and 3-29 to the s t a b i l i t y chart shown i n Figure 3-2. While d e t a i l s d i f f e r , there i s a remarkable agreement i n the fundamental regions of s t a b i l i t y and i n s t a b i l i t y i n d i c a t i n g the v a l i d i t y of the l i n e a r -ized analysis. The primary purpose of the second set of response charts (Figures 3-30 and 3-31) i s to demonstrate the e f f e c t s of the inherent n o n - l i n e a r i t i e s upon system response. I t should be noted that the contour lev e l s shown i n these charts are for responses f i f t y times greater than the lowermost leve l s shown i n Figures 3-28 and 3-29. Noting that the disturbance for the charts shown i n Figures 3-30 and 3-31 i s f i f t y times greater than that used for the preceeding charts, a comparison of the corresponding contours shows the "hardening" or "softening" e f f e c t of the n o n - l i n e a r i t i e s . For example, a large discrepancy i n po s i t i o n i s seen between the 0.02 contour i n Figure 3-28 and the 1.0 contour i n Figure 3-30, p a r t i c u l a r l y i n the region of ( 3 " > •O and 1 ^ 1 . This behavior may also be noted i n the corres-ponding tf response charts in d i c a t i n g that a non-linear s t i f f e n -ing e f f e c t i s present i n t h i s region of parameter space. 134-C o n s i d e r now the q u e s t i o n o f p e r i o d i c m o t i o n . By s t u d y -i n g a s e r i e s o f c r o s s s e c t i o n s , s i m i l a r to those shown i n F i g u r e s 3-23 t o 3-27, i t i s a p p a r e n t t h a t some form o f p e r i o d i c m o t i o n e x i s t s f o r most system c o n f i g u r a t i o n s . As p e r i o d i c s o l u t i o n s c o n t i n u e t o d e t e r m i n e the c h a r a c t e r o f i n v a r i a n t s u r -f a c e s as i n Phase I , t h e i r s t u d y i n r e l a t i o n t o the parameter H i s i m p o r t a n t . F o r a c i r c u l a r o r b i t the e q u a t i o n s o f mot ion are a u t o n -omous and hence , p e r i o d i c m o t i o n can be s p e c i f i e d i n terms o f an i n f i n i t e v a r i e t y o f i n i t i a l c o n d i t i o n s . In most c a s e s , however, the e x a m i n a t i o n o f c r o s s s e c t i o n d a t a r e v e a l e d the e x i s t e n c e of p e r i o d i c m o t i o n f o r i n i t i a l c o n d i t i o n s o f the f o r m : o , O, ft/** o o r / £ _ 0 / ft;* O , / ^ O A scheme f o r d e t e r m i n i n g the n o n - z e r o i n i t i a l c o n d i t i o n s i n terms of the r e l e v a n t system parameters i s now d e v e l o p e d . C o n s i d e r a % - p l a n e as shown i n F i g u r e 3 - 3 2 . F o r g i v e n I , , H and a s p e c i f i e d v a l u e o f the c o r r e s p o n d i n g can be d e t e r m i n e d u s i n g the e x p r e s s i o n f o r the H a m i l t o n i a n w i t h /£x =• « O . U s i n g these i n i t i a l c o n d i t i o n s a v a l u e o f ft can be d e t e r m i n e d by i n t e g r a t i n g the g o v e r n i n g e q u a t i o n s u n t i l / - ^ e x e c u t e s some p r e s c r i b e d number o f o s c i l l a t i o n s . R e p e a t i n g the p r o c e s s f o r a range o f v a l u e s o f / £ > \ r e s u l t s i n a p l o t shown i n F i g u r e 3-32 where the i n i t i a l c o n d i t i o n s r e p r e -s e n t e d by the / £ \ a x i s a re mapped i n t o f i n a l c o n d i t i o n space u t i l i z i n g the e q u a t i o n s o f m o t i o n . C l e a r l y , i n t e r s e c t i o n s o f the 135 F i g u r e 3-32 Mapping scheme f o r l o c a t i n g p e r i o d i c s o l u t i o n s ; — O 136 r e s u l t a n t mapping w i t h the a x i s r e p r e s e n t i n i t i a l c o n d i t i o n s f o r w h i c h , a f t e r some s p e c i f i e d number o f / 3 o s c i l l a t i o n s , a t least/2> and tf a t t a i n t h e i r i n i t i a l v a l u e s . Note t h a t / 5 = O whenever completes an o s c i l l a t i o n . A l l t h a t i s n e c e s s a r y then i s t o d e t e r m i n e whether and tf are a l s o matched t o t h e i r i n i t i a l v a l u e s . I f s o , p e r i o d i c m o t i o n has been found and i t s p e r i o d i s e q u a l t o the t ime t a k e n f o r t o execute the s p e c i f i e d number o f o s c i l l a t i o n s . I f / £ > o r tf are not matched w i t h i n i t i a l v a l u e s , p e r i o d i c m o t i o n i s s t i l l i n d i c a t e d but w i l l o c c u r o v e r a d i f f e r e n t number o f / £ > o s c i l l a t i o n s . Thus a method of d e t e r m i n i n g p e r i o d i c s o l u t i o n s i s e s t a b l i s h e d . A t f i r s t s i g h t the r e s u l t a n t mapping would appear t o be a s i n g l e v a l u e d and c o n t i n u o u s f u n c t i o n o f /Q\ . I f such were the c a s e , an i t e r a t i v e approach such as the v a r i a b l e s e c a n t method c o u l d be employed w i t h o u t d i f f i c u l t y t o determine the i n i t i a l c o n d i t i o n s f o r p e r i o d i c m o t i o n . In f a c t , such mappings a r e n o t , i n g e n e r a l , c o n t i n u o u s because the number o f / £ > o s c i l l a t i o n s d e t e r m i n e s the f i n a l c o n d i t i o n s . I f an o s c i l -l a t i o n i s d e f i n e d t o o c c u r a f t e r two s u c c e s s i v e s i g n changes of / £ > , the a d d i t i o n o r d e l e t i o n o f a s m a l l r i p p l e i n the r e -sponse can l e a d t o a sudden change i n the number of o s c i l l a t i o n s o v e r a g i v e n i n t e r v a l o f t i m e . T h i s i n t u r n would g i v e r i s e to a d i s c o n t i n u i t y i n the mapping . A p p a r e n t i n t e r s e c t i o n s of the mapping w i t h the a x i s due t o such d i s c o n t i n u i t i e s g i v i n g e r r o n e o u s i n i t i a l c o n d i t i o n s can be a v o i d e d by c h e c k i n g f o r a d i s c o n t i n u i t y i n the p e r i o d of m o t i o n . T y p i c a l examples o f p e r i o d i c mot ion found u s i n g the 138 0° 360° 720° Q 1080° 1440° 1800° F i g u r e 3-34 P e r i o d i c s o l u t i o n f o r a s a t e l l i t e i n a c i r c u l a r o r b i t ; I».5,<T= I,/^ .-Uj-O,/^ . F.217, XB',7S 139 above scheme are shown i n F i g u r e s 3-33 and 3-34. They i l l u s -t r a t e the complex n a t u r e of the s o l u t i o n s o f t h i s s y s t e m . T h u s , i t i s e v i d e n t t h a t h i g h l y s o p h i s t i c a t e d a n a l y t i c a l t e c h n i q u e s would be r e q u i r e d t o a c c u r a t e l y d e s c r i b e system r e s p o n s e . As i n Phase I , the s t a b i l i t y o f p e r i o d i c s o l u t i o n s i s s t u d i e d u s i n g v a r i a t i o n a l a n a l y s i s . In t h i s c a s e , t h e . v a r i a t i o n a l e q u a t i o n s can be w r i t t e n as and I ( < r + i ) c o s/3pCostf p - t - ( 4 c o s / 4 p c o s X p - I ( < S - - H ) * cos tfp -«- I(cr-»- i) s m / 3 p s i n Xp- 2 K pcos/6 P * sin tfv - ^  K^ "-*-1) cos/^pt 2 ( tfpSin/^ COS /4 p - C O S / f ) p C O s ) l p ) j tfv (3.37) jVan/2Py;' + 2 / 5 ; u; + t a n g o s aP) + (3 1-4) tan/3 psin tfpcosO/3v •+ fIfe"+0sec/5p+ 140 (31 - 4 ) c os 2 XP - I (cr + 0 cos K psec/3 P^ V + [ 2 / ^ ; t a n / 3 p ^ * ; . < 3- 3 7> I t i s c l e a r t h a t the above f o u r t h o r d e r l i n e a r system has p e r i o d i c a l l y v a r y i n g c o e f f i c i e n t s o f a common p e r i o d . T h i s p e r i o d , say 1" , i s the same as t h a t o f the p e r i o d i c m o t i o n s i n c e the c o e f f i c i e n t s a re f u n c t i o n s of the s t a t e e l e m e n t s . T h u s , F l o q u e t t h e o r y i s a p p l i c a b l e t o t h i s p r o b l e m . Because the system i s o f f o u r t h o r d e r , s t a b i l i t y i n f o r m a t i o n would n o r m a l l y be i n the form o f f o u r c h a r a c t e r i s t i c m u l t i p l i e r s . I t i s not n e c e s s a r y , however, i n t h i s case to deduce the v a l u e s o f these m u l t i p l i e r s t o a s s e s s s t a b i l i t y . S i n c e the system i s autonomous, one o f the c h a r a c t e r i s t i c m u l t i p l i e r s i s e q u a l to u n i t y as the d e r i v a t i v e o f the p e r i o d i c s o l u t i o n s a t i s f i e s the v a r i a t i o n a l e q u a t i o n s . F u r t h e r , i t has been demonstra ted t h a t p r o v i d e d such systems have a f i r s t i n t e g r a l o f m o t i o n , a second m u l t i p l i e r has u n i t v a l u e . M o r e o v e r , as the system i s c o n s e r v a t i v e , i t s s t a t e space r e p r e s e n t a t i o n i n terms o f H a m i l t o n i a n v a r i a b l e s i s volume p r e s e r v i n g , i . e . , i t p o s s e s s e s an i n t e g r a l i n v a r i a n t . Under such c i r c u m s t a n c e s , i t may be shown t h a t TT A = I . T h u s , i 1 i f A,= Az= I , then I . Hence, the two f r e e c h a r a c t e r i s t i c m u l t i p l i e r s , A 3 and A^ , which determine the s t a b i l i t y o f the s o l u t i o n must l i e on the u n i t c i r c l e or the r e a l a x i s i n the complex p l a n e . A s t a b i l i t y c r i t e r i o n can t h e r e -141 f o r e be based upon the sum of the f o u r c h a r a c t e r i s t i c m u l t i p l i -e r s , v i z . , > o r ^ O ' u n s t a b l e Z X; | (3.39) ^ ^ and >^  Q. s t a b l e . The a p p l i c a t i o n o f F l o q u e t t h e o r y i n v o l v e s the com-p u t a t i o n o f a f i n a l c o n d i t i o n m a t r i x $ ( t ) and the s u b s e -quent e v a l u a t i o n o f i t s e i g e n v a l u e s . S i n c e the t r a c e o f a m a t r i x i s i n v a r i a n t under o r t h o g o n a l t r a n s f o r m a t i o n , i t i s c l e a r t h a t T r L $ ( T ) 1 = _L A; . Thus the s t a b i l i t y o f p e r i o d i c m o t i o n can be d e t e r m i n e d by the f o l l o w i n g c r i t e r i o n : > 2.' u n s t a b l e T r [ § ( r ) ] - 2 ' " (3.40) ^ 2 - s t a b l e U t i l i z i n g t h i s method, s t u d i e s were made o f the f u n d a -m e n t a l p e r i o d i c s o l u t i o n s a s s o c i a t e d w i t h v a r i o u s c o n f i g u r a t i o n s . In a l l c a s e s , the fundamenta l p e r i o d i c mot ion remained v a r i a t i o n -a l l y s t a b l e even f o r l a r g e d i s t u r b a n c e s , i . e . , l a r g e v a l u e s o f the H a m i l t o n i a n . Examples shown i n F i g u r e s 3-35 t o 3-37 r e l a t e the p e r i o d of m o t i o n , t , and the i n i t i a l c o n d i t i o n e l e m e n t , / £ > „ to the H a m i l t o n i a n . Of c o u r s e , the r e m a i n i n g elements of the i n i t i a l c o n d i t i o n v e c t o r a re g i v e n by /£>S = «• O a n < ^ 8 ; — X^/ 142 T 1 — -1 1 1 1 Figure 3-35 Variation of /2>. , T and T r with H for fundamental periodic motion; I»2.,cr=0 1 4 4 F i g u r e 3-37 V a r i a t i o n o f / 3 . , " t and Tr Q$ (T) 3 w i t h H f o r fundamental p e r i o d i c m o t i o n ; I=2.,a"='2, 3 .4 . 3 N o n - C i r c u l a r O r b i t a l M o t i o n To t h i s s t a g e o f the s t u d y , the system was c o n s t r a i n e d i n one way o r a n o t h e r . F o r example, i n Phase I, o n l y a r o l l degree o f f reedom was a l l o w e d which p e r m i t t e d e x t e n s i v e a n a l -y s i s u t i l i z i n g t h r e e d i m e n s i o n a l s t a t e s p a c e . In the p r e c e d i n g s e c t i o n where e was r e s t r i c t e d t o z e r o v a l u e , s i m i l a r t e c h n i q u e were a p p l i c a b l e as the s t a t e o f the system c o u l d a g a i n be d e s -c r i b e d i n t h r e e d i m e n s i o n s . The g e n e r a l i z a t i o n a l l o w i n g f o r n o n - z e r o o r b i t e c c e n t r i c i t y w i t h b o t h r o l l and yaw degrees o f freedom r e q u i r e s f i v e d i m e n s i o n s /_3 } "ft, ft', 0) to s p e c i f y the s t a t e o f the s y s t e m . R e d u c t i o n o f t h i s system i s n o t p o s s i b l e as no known f i r s t i n t e g r a l s e x i s t . Note t h a t the H a m i l t o n i a n i s no l o n g e r a c o n s t a n t o f m o t i o n as W h i l e i t i s r e a s o n a b l e t o e x p e c t i n v a r i a n t s o l u t i o n s u r f a c e s , more p r e c i s e l y h y p e r s u r f a c e s , t o e x i s t i n t h i s s t a t e s p a c e , i t i s c l e a r t h a t t h e i r n u m e r i c a l g e n e r a t i o n would i n v o l v e a g r e a t d e a l o f c o m p u t a t i o n . As w e l l , i n t e r p r e t a t i o n and r e p r e -s e n t a t i o n o f these s u r f a c e s would p r e s e n t f o r m i d a b l e p r o b l e m s . F o r example , a c r o s s s e c t i o n t a k e n a t 5 » c o n s t a n t of such a s u r f a c e would be a t h r e e d i m e n s i o n a l r e g i o n i n f o u r d i m e n s i o n a l (/_3,/2>^ ft^ ft' ) s p a c e . Thus i t would seem t h a t the i n v a r i a n t s u r f a c e and c r o s s s e c t i o n i n g c o n c e p t s which p r o v e d so s u c c e s s f u l i n the f o r e g o i n g are o f l i t t l e p r a c t i c a l v a l u e i n t h i s c a s e . There i s , however, a s l i m chance t h a t something w o r t h -w h i l e can be a c c o m p l i s h e d u s i n g t h i s method. I n s t e a d o f at tempt i n g to v i s u a l i z e the f o u r d i m e n s i o n a l space f o r a 0 — c o n s t a n t c r o s s s e c t i o n , c o n s i d e r two s t r o b o s c o p i c phase p l a n e s , and ft . ft . In t h i s way the p r o b l e m of r e p r e s e n t i n g n u m e r i c a l 146 d a t a would be e l i m i n a t e d a l t h o u g h the r e s u l t i n g s e c t i o n s c o u l d n o t be e x p e c t e d to be as d e f i n i t i v e as those o b t a i n e d p r e v i o u s l y . In f a c t , a t b e s t , the r e s u l t i n g s c a t t e r o f d a t a would e x h i b i t some degree o f o r d e r i n g which c o u l d be used t o e s t a b l i s h the n a t u r e o f the m o t i o n . F o r example, q u a s i - p e r i o d i c m o t i o n would be e x p e c t e d to form banded r i n g s about the p e r i o d i c c o n d i t i o n s . These bands would be analogous t o the c l o s e d c u r v e s about s t a t i o n a r y p o i n t s p e r t a i n i n g to p e r i o d i c mot ion i n two d i m e n s i o n -a l s e c t i o n s . By i n t e g r a t i n g the e q u a t i o n s o f m o t i o n (3.7) and ( 3 . 8 ) , and s e c t i o n i n g a t p e r i c e n t e r , i . e . , © =• O , a t tempts were made t o u t i l i z e t h i s scheme. In p r a c t i c a l l y a l l i n s t a n c e s , however, the r e s u l t i n g s e c t i o n d a t a was w e l l s c a t t e r e d and showed l i t t l e o r no e v i d e n c e o f o r d e r i n g ( F i g u r e 3 -38 ) . I t was o n l y i n a r a r e s i t u a t i o n t h a t some degree o f o r d e r i n g i n c r o s s s e c t i o n d a t a was o b s e r v e d . Examples o f b a n d i n g about a n o n - t r i v i a l as w e l l as a n u l l p e r i o d i c s o l u t i o n a re shown i n F i g u r e s 3-39 and 3-40, r e -s p e c t i v e l y . The f a i l u r e , i n t h i s c a s e , o f the i n v a r i a n t s u r f a c e t e c h n i q u e o r even a m o d i f i e d form t h e r e o f i s u n f o r t u n a t e . C l e a r -l y , n u m e r i c a l s t u d i e s of s o l u t i o n s u r f a c e s t r u c t u r e s are impos-s i b l e a n d , f u r t h e r , g e n e r a t i o n o f response c h a r t s s i m i l a r to those o b t a i n e d f o r the autonomous case ( F i g u r e s 3-28 to 3-31) would be e x t r e m e l y l a b o r i o u s . T h u s , d e s p i t e i t s g r e a t v a l u e , a p a r a m e t r i c s t u d y o f t h i s form was not u n d e r t a k e n f o r t h i s c a s e . The n e x t l o g i c a l s t e p would be to l o c a t e and v a r i a t i o n -a l l y a n a l y z e p e r i o d i c s o l u t i o n s . One p r o c e d u r e would be to s t a r t w i t h known p e r i o d i c mot ion f o r A « O and extend i t , u s i n g 1 I 1 • 1 I o Initial Conditions • • • • .8 • • • • • • -P 0 • • • • • • • • • • • • --.8 • • • . 0 • --1.6 i i « i i -1.5 1.6 -1.0 -5 0 P rod. 5 1.0 1. i 1 i 1 • i • .8 • • • • • • • • • • • • • Y • 0 0 • • • -.8 • • • • • • • -1.6 i i • i i i -2.4 -1.6 -.8 0 Y rad. 8 1.6 2, F i g u r e 3-38 S c a t t e r e d i n t e r s e c t i o n s o f a t r a j e c t o r y w i t h the s t r o b o s c o p i c phase p l a n e s a t 6s O, I=.5, .8 148 .4 1 1 i i r — ° Initial Conditions — • • • • .° • • • • • • • • • • • • • • -, „ - 1 1 1 j - i -.16 -.08 p r o d . .08 .16 .24 .32 I 1 T — 1 1 .2 • • • r • • • • • • 0 • • • • -• • • • • -.2 --.4 l I 1 1 1 -1.24 -1.19 -1.14 Y r a d-1.09 -1.04 -.99 -.9 Figure 3-39 Trajectory intersections with the stroboscopic phase planes at 0 = O showing banding about a periodic solution; I = <r« l , _ - . i , / 3 i . S.*0,/S«.5,)f.»-| F i g u r e 3-40 T r a j e c t o r y i n t e r s e c t i o n s w i t h the s t r o b o s c o p i c phase p l a n e s a t 0 = O showing banding about the o r i g i n ; 1 = 2, <r= l,e = - 8 j * * o , / S { * 1 . 9 , *U - .5 some i t e r a t i v e scheme, t o v a l u e s o f n o n - z e r o e c c e n t r i c i t y . W h i l e n o t p r o v i d i n g a complete p i c t u r e o f the dynamics o f the s y s t e m , such an a p p r o a c h , i f s u c c e s s f u l , would be a s t e p i n the r i g h t d i r e c t i o n . Replacing/2> by /dp-** / d v and tf by tfp-*- tfv? the g o v e r n i n g e q u a t i o n s y i e l d t h e f o l l o w i n g v a r i a t i o n a l r e l a -t i o n s : . i \ l+ecos8/ cos L ) -r / 3 ( 1 - 1 ) s i n g t f r + c o s a o r - (frf\* ( c o s ^ p - s i n / ^ P ) — A tfp s i n / 5 P c o s ^ c o s 8P^ * / 3 -»• f 2 e s i n 6 ? ^ tl<q--,)/ 1-t-e t » L I + e c o s & ) C \ U e c o s e / S ' m / f > p S i n tfp-t- g . / 3 ( 1 -0 - » \ s i n x 3 p c o s / ^ p « \ l + e c o s e / S i n t f p C - o s 5 p - 2 e s m 9 c o s tfp — tfp'sin tfp* {•+ e c o s 9 / I -v e t 2 tfpSin/3j c o s / 3 p + ( c o s 2 / 5 p -\ I <e cos6/ " S m / ^ + | ) c o S ? p j l f y (3.41) 151 and c o s /2>p« ] 3 ( r - Q s i n /_»>psiVi ftpC o s ftp+ C l + e c o s 0 2 c s i n 8 / C o s / - » p C o S o * p - UpSih/_> pj n-I.<*« c o s © \ I & 5p c o s / ^ p s i n/3p| ftp c o s tfP ( 2/^p-s i n ftp)j^ /^ >v+ ^ l ^ e . q J -+-2 ftp eini./3p — 2 c o s / 3 p c o S tfp^ /^v + f ~ I ( c r - n ) f U e f c o ^ ftp- / 3 ( I - i ) - i l t \ I n - e c o s S y \ I + e c o s e / c o s / 5 p ( c o s 2 , f t p - sin*"ftp) — s m tfP^ sm/<3p* l + e c o s 8 CO«»/3p -h e / ^ S l V l / S p ^ ^ . (3.42) Despite the complex form of the above equations, i t i s seen that a l l of the c o e f f i c i e n t s vary p e r i o d i c a l l y with a per-iod which i s some multiple of 2.1V . Thus, Floquet theory i s again applicable although, i n th i s case, i t w i l l be necessary to evaluate the four c h a r a c t e r i s t i c m u l t i p l i e r s to assess s t a b i l i t y . The fundamenta l p r o b l e m here then i s n o t i n a s s e s s i n g the v a r i a t i o n a l s t a b i l i t y o f p e r i o d i c mot ion b u t i s i n l o c a t i n g the p e r i o d i c m o t i o n i t s e l f . In e s s e n c e , t h i s means f i n d i n g an i n i t i a l s t a t e o f the sys tem, as d e f i n e d by /S{} , tf; and tf; , such t h a t a f t e r a s p e c i f i e d number o f o r b i t s , the s t a t e matches the i n i t i a l s t a t e . In p r e v i o u s s t u d i e s o f p e r i o d i c m o t i o n , the p r o b l e m i n v o l v e d the matching of fewer s t a t e e l e m e n t s . In those c a s e s , the v a r i a b l e s e c a n t method was a p p l i e d to o n l y one e l e m e n t . O b v i o u s l y a more s o p h i s t i c a t e d i t e r a t i o n scheme i s r e q u i r e d i n t h i s c a s e . C o n s i d e r the p o s s i b i l i t y o f u t i l i z i n g the v a r i a t i o n a l e q u a t i o n s i n such an i t e r a t i o n scheme. R e c a l l t h a t i n u s i n g t h e s e e q u a t i o n s , an i n i t i a l c o n d i t i o n matrix^ e q u a l to the i d e n -t i t y m a t r i x Is i n t e g r a t e d to y i e l d a f i n a l c o n d i t i o n m a t r i x , C o n s i d e r o n l y the f i r s t column o f the i d e n t i t y m a t r i x as an i n i t i a l c o n d i t i o n v e c t o r f o r the v a r i a t i o n e q u a t i o n s , i . e . , A ' * \ $ (o). I n t e g r a t i n g o v e r a g i v e n p e r i o d t would r e s u l t , i n g e n e r a l , i n a new s t a t e v e c t o r of the v a r i a t i o n a l e q u a t i o n : 153 Note t h a t the e lements o f t h i s v e c t o r r e p r e s e n t the change i n the s t a t e e lements and tf f o r a u n i t change i n /£.?; , As t h e system i s l i n e a r and /&\t,\ — 1 7 154 Hence, iz2 d/3;' cM.' Aft, i i oft; a*- ^ f t ; Recognizing t h i s , an i t e r a t i o n scheme for locating periodic motion can be developed which e s s e n t i a l l y amounts to a fourth order Newton Raphson approach. Let ^/?>sy/^>X) ft; ; ft-, ) be an estimate of the required i n i t i a l state giving periodic motion i n 2 n Tr . Integration of the equations of motion, together with the v a r i a t i o n a l equations, w i l l , i n general, r e s u l t i n a state d i f f e r i n g from the i n i t i a l state of the system. Let ^ /3} / 3 ) ft^ X ) represent t h i s state. To approach a matching of these states, the following l i n e a r system must be solved for the changes or corrections made to the i n i t i a l conditions: /3,-t- ^  = /3 + ^ £ + ,^+ ^3 S,-iZiS*< / * > ' + M . ' 5*+ i^ . ' & , - v^2Ls f +^; sv . a/*> X A/5,' oft, ax; 5fi <^ *, (3 .44) In more c o n c i s e n o t a t i o n , these can be w r i t t e n as 6y« ^ $ ( e h i v ) - 1 j / ' If 9 (3 .45) where ^ > , y ^ y a n a - ^ t f ' r e p r e s e n t the c o r r e c t i o n s to the i n i t i a l s t a t e e lements t o reduce the matching e r r o r . I t may be p o i n t e d out t h a t t h i s i t e r a t i o n scheme i s p a r t i c u l a r l y c o m p a t i b l e w i t h v a r i a t i o n a l a n a l y s i s s i n c e $ (t) d e t e r m i n e d here can a l s o be used t o a s s e s s s t a b i l i t y by F l o q u e t t h e o r y . Hence, once an a c c e p t a b l e match i s a t t a i n e d , the e i g e n v a l u e s o f $ , i . e . , the c h a r a c t e r i s t i c m u l t i p l i e r s A-, o f the v a r i -a t i o n a l sys tem, can be computed. The c r i t e r i o n f o r s t a b i l i t y o f the p e r i o d i c m o t i o n can be s t a t e d as i « 1,2,3,4-^ | u n s t a b l e ^ I t s t a b l e . (3 .46) 159 A t y p i c a l example o f p e r i o d i c m o t i o n found by t h i s i t e r a t i o n scheme i s shown i n F i g u r e 3-41 . Examples o f the use o f t h i s method i n t r a c i n g p e r i o d i c m o t i o n t h r o u g h v a r i o u s v a l u e s o f € a r e shown i n F i g u r e s 3-42 and 3-43. F i g u r e 3-42 i l l u s t r a t e s the e f f e c t o f <5 upon the i n i t i a l c o n d i t i o n s f o r t h r e e s t a b l e t y p e s o f p e r i o d i c m o t i o n w i t h I» *2. and <S"= I . I t s h o u l d be noted t h a t o t h e r p e r i o d i c s o l u t i o n s were a l s o found f o r t h i s c a s e . These s o l u t i o n s appeared t o be r e l a t e d to those shown i n t h a t they had the same v a r i a t i o n a l c h a r a c t e r i s t i c s , i . e . A^s were i d e n t i c a l , and were d e r i v e d from the same fundamental p e r i o d i c s o l u t i o n s a t •» O . The r e s u l t s shown sugges t t h a t , due t o the m u l t i p l i c i t y o f p e r i o d i c s o l u t i o n s , the sphere o f i n f l u e n c e o f any one o f them i s s m a l l compared t o t h a t o b s e r v e d i n Phase I . F i g u r e 3-43 t r a c e s s t a b l e p e r i o d i c m o t i o n f o r the case of I« .5 and CJ" = I i n w h i c h the X m o t i o n o c c u r s near tf « I r a t h e r than near tf = O . 3.5 C o n c l u d i n g Remarks As s t a t e d i n the b e g i n n i n g , the aim of t h i s phase o f the work i s t o e x t e n d the n u m e r i c a l t e c h n i q u e s d e v e l o p e d i n Phase I . To a l a r g e degree t h i s a p p r o a c h p r o v e d to be s u c c e s s f u l . O n l y i n the s t u d y o f the non-autonomous system i n the l a r g e were d i f f i c u l t i e s e n c o u n t e r e d . Even i n t h a t c a s e , i t was p o s s i b l e to l o c a t e and v a r i a t i o n a l l y a n a l y z e p e r i o d i c m o t i o n . S t a b i l i t y i n the s m a l l was s t u d i e d n u m e r i c a l l y u s i n g F l o q u e t t h e o r y . R e s u l t s i n terms o f s t a b i l i t y c h a r t s were o b t a i n -ed o v e r a wide range o f parameters which s h o u l d p r o v e u s e f u l i n 160 system d e s i g n . I t i s i m p o r t a n t t o note t h a t w h i l e these r e s u l t s a r e based upon a l i n e a r i z e d a n a l y s i s , they were s u b s t a n t i a t e d by the studies o f m o t i o n i n the l a r g e . T h i s l i n e a r a n a l y s i s 9 13 p a r a l l e l s t h o s e o f Thomson fo r , £ « 0 and Kane and Barba f o r &\0 but the r e s u l t s o b t a i n e d here a re i n c o n s i d e r a b l y g r e a t e r d e t a i l . F igure 3 - 4 4 compares re su l t s f o r the case o f ' e * . 3 . A l s o shown i n t h i s f i g u r e i s the s t a b i l i t y boundary as de termined u s i n g L i a p o u n o v ' s d i r e c t method a p p l i e d t o the l i n e a r i z e d system.^" The example c l e a r l y shows the c o n t r i b u t i o n of t h i s i n v e s t i g a t i o n i n p r e s e n t i n g more d e t a i l e d and a c c u r a t e r e s u l t s . I t s h o u l d be n o t e d t h a t p r e v i o u s r e s e a r c h i n t h i s f i e l d u t i l i z e d a d i f f e r e n t s p i n p a r a m e t e r , w h i c h i s r e l a t e d t o CJ" and © i n the manner i l l u -s t r a t e d i n F i g u r e 3 - 4 5 . In e s s e n c e , t h i s parameter i s a measure of the average s p i n r a t e over the e n t i r e o r b i t r a t h e r than the i n s t a n t a n e o u s s p i n r a t e a t the p e r i c e n t e r P . The s p e c i a l case of e « 0 r e c e i v e d c o n s i d e r a b l e a t t e n t i o n . M o t i o n i n the l a r g e was s u c c e s s f u l l y t r e a t e d u s i n g the i n v a r i a n t s u r f a c e c o n c e p t . P r i n c i p a l c r o s s s e c t i o n s o f s o l u t i o n s u r f a c e s , t o g e t h e r w i t h the e n v e l o p e s o f p o s s i b l e m o t i o n o b t a i n e d from the H a m i l t o n i a n , r e p r e s e n t a c o n s i d e r a b l e e x t e n s i o n of the p r e -12 l i m i n a r y work by P r i n g l e i n t h i s f i e l d . I n v a r i a n t s u r f a c e methods p r o v e d to be most f r u i t f u l as not o n l y the n a t u r e o f the d y n a m i c a l r e s p o n s e b u t the a m p l i t u d e o f m o t i o n i n b o t h jS> and X modes o f o s c i l l a t i o n was e a s i l y a s c e r t a i n e d . The u t i l i z a t i o n o f the i n v a r i a n t s u r f a c e t e c h n i q u e to o b t a i n response c h a r t s , F i g u r e s 3-28 to 3-31, appears t o be the f i r s t a t tempt t o r e l a t e maximum r e s p o n s e to a s p e c i f i e d d i s t u r -161 -1 - 2 1 I Figure 3-44 Comparison of s t a b i l i t y regions as determined by several investigators; ©-..3 •0 .2 .4 .6 .8 iO e Figure 3-45 Relationship between spin parameters 1 and 163 bance over a wide range of system parameters. As pointed out e a r l i e r , these charts are of considerable value since they not only e s t a b l i s h the extent to which the l i n e a r i z e d analysis can be applied but the nature and magnitude of non-linear e f f e c t s . Cross section studies gave considerable insight 1 into the behavior of periodic solutions. Although studies of closed envelopes indicated the presence of several periodic solutions, only the fundamental continued to e x i s t at higher values of the Hamiltonian. . ' Analyses of periodic motion were performed for both c i r c u l a r and non-circular o r b i t a l motion. I t e r a t i v e techniques for locating such motion based upon numerical integration of the governing equations proved to be successful. The numerical methods involved i n the a p p l i c a t i o n of Floquet theory to the fourth order v a r i a t i o n a l system can be derived d i r e c t l y from those used i n Phase I. V a r i a t i o n a l analysis of the fundamental periodic solution showed i t to be stable for values of H as large as 41 (Figure 3 - 3 6 ) . This method of locating and v a r i -a t i o n a l l y analyzing periodic motion f o r both the autonomous and non-autonomous cases represents a considerable advance i n the f i e l d . As f a r as the a n a l y t i c a l studies are concerned, zero v e l o c i t y curves and.the associated envelopes of motion based upon the f i r s t i n t e g r a l , i . e . , the Hamiltonian, were obtained for the case of fi«0. The use of Liapounov's d i r e c t method was not considered appropriate, as explained i n Phase I. In addition, there i s a problem of defining s t a b i l i t y i n the large. 164 Although the study of large amplitude motion i s not extensive, i t substantiates the r e s u l t s of the l i n e a r i z e d anal-y s i s presented in. Figures 3-2 to 3-6. Conclusions based on these charts can be summarized as follows: i) Except f o r the case of I « I , at least one spin regime exists i n which motion i n the small i s unstable regardless of o r b i t e c c e n t r i c i t y . Thus the spin-up operation of a s a t e l l i t e could lead to attitude i n s t a b i l i t y . i i ) In contrast to the findings of Phase I pertaining to i n s t a b i l i t y with negative «T , two fundamentally stable regions were observed i n parameter space, one i n the p o s i t i v e and the other i n the negative spin regime. In general, the former would be preferred for operation since larger and/or more frequent unstable regions occur i n the l a t t e r . i i i ) Long, slender s a t e l l i t e s , i . e . , I I , while inherently less stable than th i n disk-type configurations due to the adverse gravity gradient e f f e c t , can be e f f e c t i v e l y s t a b i l i z e d given s u f f i c i e n t spin. iv) E f f e c t s of non-circular o r b i t a l motion are numerous. Orbit e c c e n t r i c i t y plays a s i g n i f i c a n t role i n determining both the size and the number of unstable regions. Using Jl instead of as a measure of spin, i t i s seen (Figures 3-2 to 3-6 and 3-45) that increasing the value of e c c e n t r i c i t y adversely affects the attitude motion of a s a t e l l i t e as regions of unstable motion increase i n both size and number. Tracings of periodic solutions with o r b i t e c c e n t r i c i t y have also revealed that such motion i s altered considerably even for numerically small values of © 165 The a n a l y s i s of a t t i t u d e s t a b i l i t y i n the s m a l l , t o -g e t h e r w i t h the s t u d i e s o f the autonomous system i n the l a r g e , p r o v i d e i n f o r m a t i o n u s e f u l f o r the p r a c t i c a l d e s i g n o f a s a t e l l i t e . The a n a l y s i s o f the non-autonomous system i n the l a r g e p r o v e d t o be l e s s s u c c e s s f u l i n t h i s r e g a r d ; however, some p r o g r e s s was made i n the s t u d y of p e r i o d i c m o t i o n . T h u s , i t seems f a i r t o c o n c l u d e t h a t the t e c h n i q u e s and r e s u l t s p r e -s e n t e d i n t h i s c h a p t e r p r o v i d e a sound b a s i s f o r the d e s i g n o f s p i n n i n g a x i s y m m e t r i c s a t e l l i t e s w i t h the p o s s i b l e e x c e p t i o n of systems i n w h i c h l i b r a t i o n a l m o t i o n as w e l l as o r b i t e c c e n t r i c -i t y a re l a r g e . 4. CLOSING COMMENTS 4 .1 Summary As n o t e d a t the o u t s e t , the p r i m a r y aim o f the i n v e s -t i g a t i o n was t o d e v e l o p g e n e r a l methods o f a n a l y s i s by w h i c h the dynamics o f any p a r t i c u l a r c o n f i g u r a t i o n c o u l d be s t u d i e d . As the emphasis was on the development of t e c h n i q u e s r a t h e r than the g e n e r a t i o n o f n u m e r i c a l r e s u l t s , no a t tempt was made to p r e s e n t mass ive d a t a , p a r t i c u l a r l y i n those a reas where the approach i n v o l v e d a c o n s i d e r a b l e e x p e n d i t u r e o f computer t i m e . S t u d i e s o f m o t i o n i n the s m a l l , however , a re q u i t e e x t e n s i v e and d e t a i l e d s i n c e they are f e l t t o be of c o n s i d e r a b l e g e n e r a l i n t e r e s t . W h i l e c o n c e p t u a l l y the d e t e r m i n a t i o n o f a n u m e r i c a l s o l u t i o n t o a system o f d i f f e r e n t i a l e q u a t i o n s i s not d i f f i c u l t , r e s p o n s e c u r v e s o b t a i n e d i n t h i s manner n o r m a l l y p r o v i d e r e l a -t i v e l y l i t t l e i n s i g h t i n t o the n a t u r e of m o t i o n . In t h i s s t u d y , the p r o b l e m o f i n t e r p r e t a t i o n o f n u m e r i c a l d a t a was l a r g e l y overcome t h r o u g h the u t i l i z a t i o n o f i n v a r i a n t s u r f a c e s . The g e n e r a t i o n o f such s u r f a c e s and t h e i r c r o s s s e c t i o n s demonstra ted a most u s e f u l method of u t i l i z i n g n u m e r i c a l t e c h n i q u e s . F o r the a n a l y s i s o f s t a b i l i t y i n the s m a l l , F l o q u e t t h e o r y i n c o n j u n c t i o n w i t h n u m e r i c a l methods p r o v e d to be u s e f u l . T h i s t e c h n i q u e was ex tended t o s t u d y the s t a b i l i t y o f p e r i o d i c s o l u t i o n s u s i n g v a r i a t i o n a l methods. I t s h o u l d be emphasized t h a t the a n a l y s i s i n Phase I r e l a t e s t o a s i m p l i f i e d model o f the r e a l s y s t e m . Hence, i t s 1 6 7 findings merely indicate the approximate behavior of the physic-a l configuration. In t h i s sense, the analysis served i t s purpose quite well as the more general analysis usually found the r e s u l t s i n d i c a t i v e of the correct trend and generally con-servative i n t h e i r predictions. 4.2 Recommendations fo r Future Work P o s s i b i l i t i e s for extension of the work presented are numerous. Application of the methods developed to the study of a s p e c i f i c design would be p a r t i c u l a r l y i n t e r e s t i n g . If t h i s were done i n addition to a detailed simulation, an assessment could be made of the a p p l i c a b i l i t y of the techniques to p r a c t i c a l problems involving such additional factors as mass asymmetries, solar r a d i a t i o n pressure and s a t e l l i t e f l e x i b i l i t y . Although the use of t h i s method for a general parametric study of the problem involves considerable computation, studies of a l i m i t e d range of parameter space could be u s e f u l l y under-taken.' For example, the analysis of motion i n the small, using Floquet theory, could be extended to cover a wider range of o r b i t e c c e n t r i c i t y and/or spin rate. Moreover, a d i f f e r e n t form of s t a b i l i t y chart with I f i x e d and e and <ar as parameters should prove useful to the designer because, while the geometry of an o r b i t i n g s a t e l l i t e i s usually f i x e d , C and © are more amenable to change. A useful contribution could also be made i n the exten-sion of the maximum response charts. For example, plots for e x c i t a t i o n would be of i n t e r e s t . As mentioned previously, response charts for S ^ O would e n t a i l a massive computa-168 t i o n a l e f f o r t b u t would be o f g r e a t v a l u e . A s i d e f rom these r a t h e r o b v i o u s p r o p o s a l s , f u t u r e r e -s e a r c h e r s i n t h i s f i e l d s h o u l d i n v e s t i g a t e more e l a b o r a t e m o d e l s . A s t u d y o f the e f f e c t s o f s m a l l mass asymmetries would be u s e -f u l as s u c h would u n d o u b t e d l y be p r e s e n t i n any p h y s i c a l s y s t e m . 22 A r e c e n t paper by M e x r o v i t c h has s t r e s s e d the e f f e c t s o f h i g h e r - o r d e r i n e r t i a l i n t e g r a l s on the a t t i t u d e dynamics o f c e r t a i n g r a v i t y g r a d i e n t s a t e l l i t e c o n f i g u r a t i o n s . In the case o f a s p i n n i n g s a t e l l i t e , such i n t e g r a l s may be s i g n i f i c a n t f o r I e q u a l t o o r n e a r l y e q u a l t o u n i t y . Thus an i n v e s t i g a t i o n as t o the magnitude o f such e f f e c t s would be o f v a l u e . An i m p o r t a n t e x t e n s i o n s u g g e s t e d by t h i s work would be the i n v e s t i g a t i o n o f a t t i t u d e b e h a v i o r d u r i n g t r a n s i t i o n t h r o u g h the u n s t a b l e s p i n r e g i m e s . Such a s t u d y would n e c e s s a r i l y have t o take i n t o a c c o u n t the h i s t o r y o f s p i n r a t e as w e l l as the d i s -t u r b a n c e s i n t r o d u c e d d u r i n g the s p i n - u p o p e r a t i o n . T h u s , the mechanism employed f o r a l t e r i n g s p i n r a t e would e n t e r the p i c t u r e . BIBLIOGRAPHY 1 K l e m p e r , W . B . , " S a t e l l i t e L i b r a t i o n s o f L a r g e A m p l i t u d e , " ARS J o u r n a l , V o l . 30, No. 1, J a n . 1960, p p . 123-124. 2 S c h e c h t e r , Hans , B . , " D u m b b e l l L i b r a t i o n s i n E l l i p t i c O r b i t s , " AIAA J o u r n a l , V o l . 2, N o . 6, June 1964, p p . 1000-1003. 3 B a k e r , R . M . L . , J r . , " L i b r a t i o n s on a S l i g h t l y E c c e n t r i c O r b i t , " ARS J o u r n a l , V o l . 30, No. 1, J a n . 1960, p p . 124-126. 4 B r e r e t o n , R . C . , " A S t a b i l i t y Study of G r a v i t y O r i e n t e d S a t -e l i t e s , " P h . D . d i s s e r t a t i o n , U n i v e r s i t y o f B r i t i s h C o l u m b i a , November 1967. 5 Z l a t o u s o v , V . A . , O k h o t s i m s k y , D . E . , S a r g h e v , V . A . , and T o r z h e v s k y , A . P . , " I n v e s t i g a t i o n o f a S a t e l l i t e O s c i l l a -t i o n s i n the P l a n e of an E l l i p t i c O r b i t , " P r o c e e d i n g s  o f the X l t h I n t e r n a t i o n a l Congress o f A p p l i e d M e c h a n i c s , G b r t h e r , H e n r y , e d . , S p r i n g e r - V e r l o g , B e r l i n , 1964, p p . 436-439. 6 B r e r e t o n , R . C . , and M o d i , V . J . , " S t a b i l i t y o f the P l a n a r L i b r a t i o n a l M o t i o n o f a S a t e l l i t e i n an E l l i p t i c O r b i t , " P r o c e e d i n g s o f the X V I I t h I n t e r n a t i o n a l A s t r o n a u t i c a l  C o n g r e s s , Gordon and B r e a c h I n c . , New Y o r k , 1967, p p . 179-192. 7 M o d i , V . J . , and B r e r e t o n , R . C . , " S t a b i l i t y of a Dumbbell S a t e l l i t e i n a C i r c u l a r O r b i t D u r i n g C o u p l e d L i b r a t i o n a l M o t i o n , " P r o c e e d i n g s o f the X V I I I t h I n t e r n a t i o n a l A s t r o n - a u t i c a l C o n g r e s s , Pergamon P r e s s , L o n d o n , 1968, p p . 109-120. 8 De B r a , D . , "The L a r g e A t t i t u d e M o t i o n s and S t a b i l i t y , Due to G r a v i t y , of a S a t e l l i t e w i t h P a s s i v e Damping i n an O r b i t o f A r b i t r a r y E c c e n t r i c i t y about an O b l a t e B o d y , " P h . D . d i s s e r t a t i o n , S t a n f o r d U n i v e r s i t y , June 1962. 9 Thomson, W . J . , " S p i n S t a b i l i z a t i o n o f A t t i t u d e A g a i n s t G r a v i t y T o r q u e s , " The J o u r n a l of the A s t r o n a u t i c a l S c i e n c e s , V o l . 9, N o . 1, J a n . 1962, p p . 31-33 . 10 Kane, T . R . , M a r s h , E . L . , and W i l s o n , W . G . , " L e t t e r to the E d i t o r , " The J o u r n a l o f the A s t r o n a u t i c a l S c i e n c e s , V o l . 9, N o . 1, J a n . 1962, p p . 108-109. 170 11 Kane , T . R . , and S n i p p y , D . J . , " A t t i t u d e S t a b i l i t y o f a S p i n -n i n g U n s y m m e t r i c a l S a t e l l i t e i n a C i r c u l a r O r b i t , " The  J o u r n a l o f the A s t r o n a u t i c a l S c i e n c e s , V o l . 10, N o . 4, W i n t e r 1963, p p . 114-119. ~ 12 P r i n g l e , R . , J r . , "Bounds o f L i b r a t i o n s o f a S y m m e t r i c a l S a t e l l i t e , " AIAA J o u r n a l , V o l . 2, N o . 5, May 1964, p p . 908-912. 13 Kane , T . R . , and B a r b a , P . M . , " A t t i t u d e S t a b i l i t y of a S p i n n i n g . S a t e l l i t e i n an E l l i p t i c O r b i t , " J o u r n a l o f A p p l i e d M e c h a n i c s , June 1966, p p . 402-405. 14 W a l l a c e , F . B . , J r . , and M e i r o v i t c h , L . , " A t t i t u d e I n s t a b i l i t y Regions o f a S p i n n i n g Symmetric S a t e l l i t e i n an E l l i p t i c O r b i t , " AIAA J o u r n a l , V o l . 5, N o . 9, S e p t . 1967, p p . 1642-1650. : 15 M i n o r s k y , N i c h o l a s , N o n l i n e a r O s c i l l a t i o n s , D. Van N o s t r a n d C o . I n c . , P r i n c e t o n , 1962, p p . 127-133, 390-415. 16 F r a n k l i n , P . , Methods o f Advanced C a l c u l u s , M c G r a w - H i l l Book C o . , I n c . , New Y o r k , 1944, p p . 285-290. 17 Cunningham, W . J . , I n t r o d u c t i o n t o N o n l i n e a r A n a l y s i s , McGraw-H i l l Book C o . , I n c . , New Y o r k , 1958, p p . 250-257. 18 M o s e r , J u r g e n , " P e t u r b a t i o n T h e o r y f o r A l m o s t P e r i o d i c S o l u t i o n s fo-r Undamped N o n l i n e a r D i f f e r e n t i a l E q u a t i o n s , " i n t e r n a t l o n a 1  Symposium on N o n l i n e a r D i f f e r e n t i a l E q u a t i o n s and N o n l i n e a r  M e c h a n i c s , e d i t e d by L a S a l l e , J . P . , and L e f s c h e t z , S . , Academ-i c P r e s s , New Y o r k , 1963, p p . 71-79 . 19 He*non/ M . , and H e i l e s , C , "The A p p l i c a b i l i t y o f the T h i r d I n t e g r a l o f M o t i o n : Some N u m e r i c a l E x p e r i m e n t s , " A s t r o n o m i c a l  J o u r n a l , V o l . 69, N o . 1, F e b . 1964, p p . 73-79 . 20 J e f f r e y s , W . H . , "Some D y n a m i c a l Systems o f Two Degrees of Freedom i n C e l e s t i a l M e c h a n i c s , " A s t r o n o m i c a l J o u r n a l , V o l . 71, N o . 5, June 1966, p p . 306-313. 21 Hamming, R . W . , N u m e r i c a l Methods f o r S c i e n t i s t s and E n g i n e e r s , M c G r a w - H i l l Book C o . , I n c . , New Y o r k , 1962, p p . 183-222. 22 M e i r o v i t c h , L . , "On the E f f e c t s o f H i g h e r - O r d e r I n e r t i a I n -t e g r a l s on the A t t i t u d e S t a b i l i t y o f E a r t h - P o i n t i n g S a t e l l i t e s , " The J o u r n a l of the A s t r o n a u t i c a l S c i e n c e s , V o l . 15, N o . 1, J a n . - F e b . 1968, p p . 14-18. APPENDIX I A n a l y s i s o f the E f f e c t s o f L i b r a t i o n a l M o t i o n on O r b i t a l M o t i o n One o f the i m p o r t a n t s t e p s i n f o r m u l a t i n g problems i n s a t e l l i t e a t t i t u d e dynamics i s the assumption t h a t l i b r a t i o n a l m o t i o n has a n e g l i g i b l e e f f e c t upon o r b i t a l m o t i o n . T h i s e n a b l e s the e q u a t i o n s g o v e r n i n g a t t i t u d e and o r b i t a l m o t i o n to be d e c o u p l e d . The method of v a r i a t i o n o f parameters i s employed t o e s t a b l i s h the v a l i d i t y o f t h i s c o n c e p t f o r the p r o b l e m i n h a n d . E q u a t i o n s (2.11) and (2.12) can be r e w r i t t e n as (1.1) R - RB%^ -r £ ^ 2>JUL k '( l-Q( I- 3 s i n * ) ? - o a . 2 ; where £ has u n i t v a l u e and i s used t o denote s m a l l t e r m s . C o m b i n i n g e q u a t i o n s (1.1) and (1.2) and n e g l e c t i n g terms o f OCe?) y i e l d s R H T T . ( R J / 3 a O . (1.3) I t i s c o n v e n i e n t a t t h i s s tage to i n t r o d u c e a change i n v a r i a b l e s . R e p l a c i n g R by V « 1 — JU and c h a n g i n g the i n d e pendent v a r i a b l e f rom T to © u s i n g the r e l a t i o n g i v e s h» v " + V - £ - f = O (1.5) where -f f = | . ^ «)( I - 3 s i n 2 * ) ( v v - + ^ J 4- 2 k* s i n b^v+^j(v ' )* + ^2 k*sin tfcostf ^ v - f ^ j " ^ c S i n t f j t f ' v ' . As shown by the method of K r y l o f f and B o g o l i u b o f f , e q u a t i o n (1.5) has a s o l u t i o n o f the form 173 V « V o C O f t ( 0 + * ) a V c C O S ^ (1.6) where b o t h Vo and & v a r y w i t h 9 . In terms o f more m e a n i n g f u l v a r i a b l e s , the s o l u t i o n (1.6) may be w r i t t e n as h ^ ( l + e c o s ( 9 + ^ ) ) (1.7) where e » V 0 h 6 / ^ U . . T a k i n g the a t t i t u d e m o t i o n and hence £ t o be p e r i o d i c and c o n s i d e r i n g ^ t o be s m a l l ^ ^= O i n i t i a l l y ) , g i v e s f sin ^  M ' 4 , o and ( I - 8 ) where U 4 > ) and d » / d f are approximated by and 0 r e s p e c t i v e l y . L e t " a " be a parameter e q u i v a l e n t to the semi-major a x i s o f a K e p l e r i a n o r b i t d e f i n e d as a s h e / ^ u ( l - c ) . I n t r o d u c -i n g a s i z e parameter j^ssk^/SL e q u a t i o n s (1.8) can be r e -w r i t t e n as 174 average \\ —Gj L J and average \ I - ^ 7 e t J ^ 0 (1.9) where <j=: ( i + ecos i^) }J|( l - lX l -3s in e }n -r 2 sin - 2 e s i n X c o s f t s i n 0 ©]jf? 4-(l + ecos^)* i 3 I l + ^i)( i+e) 2 /( l-e1 Z cos* e sin % sin 0 s|]L .^ From the f o r e g o i n g , s i n c e O ^ < I ( n o r m a l l y . 1 0 ) , i t i s a p p a r e n t t h a t f o r most s i t u a t i o n s changes i n 6 and & are e s s e n t i a l l y p r o p o r t i o n a l t o p 2. F u r t h e r , i t i s e v i d e n t t h a t a l t h o u g h a n a l y t i c a l s o l u t i o n s of e q u a t i o n s (1.9) a re not p o s s i b l e , the i n t e g r a l terms i n these e x p r e s s i o n s s h o u l d not be l a r g e s i n c e the i n t e g r a n d s c o n s i s t o f t r i g o n o m e t r i c f u n c t i o n s F i g u r e 1-2 I n c r e m e n t a l change per o r b i t i n o r b i t a l p a r a -meters due t o c o u p l e d p e r i o d i c l i b r a t i o n a l m o t i o n ; I- 1.5, cr as - | f rr\ * I , n = 2. 178 w i t h r e l a t i v e l y s m a l l c o e f f i c i e n t s . T h u s , a t l e a s t q u a l i t a t i v e -l y , e q u a t i o n (1.9) i n d i c a t e s t h a t changes i n e and $ due to a t t i t u d e m o t i o n o f the s a t e l l i t e a re s m a l l . N u m e r i c a l s o l u t i o n s o f (1.9) and ( 2 . 1 7 ) , the e q u a t i o n g o v e r n i n g l i b r a t i o n a l m o t i o n , are p o s s i b l e . R e p r e s e n t a t i v e examples f o r l a r g e a m p l i t u d e , p e r i o d i c , l i b r a t i o n a l mot ion are shown i n F i g u r e s 1 -1 , 1-2 and 1-3 . These r e s u l t s i n d i c a t e t h a t the e f f e c t o f s a t e l l i t e l i b r a t i o n s on the mot ion of the c e n t e r o f mass i s n e g l i g i b l e and t h a t d e c o u p l i n g of the e q u a t i o n s o f m o t i o n i s i n d e e d v a l i d . 

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