UBC Theses and Dissertations

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UBC Theses and Dissertations

A Stability study of gravity oriented satellites Brereton, Robert Cloudesley 1967

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The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY ROBERT CLOUDESLEY BRERETON B, Eng., M c G i l l U n i v e r s i t y , 1959 WEDNESDAY, NOVEMBER '22, 1967, AT 10:30 a.m. ROOM 208, MECHANICAL ENGINEERING ANNEX Ex t e r n a l Examiner: J . Mar Head, Space Mechanics S e c t i o n Defence Research Telecommunications Establishment Ottawa, Ontario of COMMITTEE IN CHARGE Chairman: I . McTo Cowan J.P. Duncan G,V. Parkinson A.C. Soudack V.Jo Modi C.R. H a z e l l E.V. Bohn Research Supervisor: V . J . Modi A STABILITY STUDY OF GRAVITY ORIENTED SATELLITES ABSTRACT The s t a b i l i t y of g r a v i t a t i o n a l gradient o r i e n t e d s a t e l l i t e s i s examined by c o n s i d e r i n g four s i m p l i f i e d models'; The i n v e s t i g a t i o n i s c a r r i e d out numberically and a n a l y t i c a l l y o The techniques employed i n v o l v e considerable computation and hence are p a r t i c u l a r l y s u i t e d to s o l u t i o n by a d i g i t a l computer. The a n a l y s i s of the planar motion of a r i g i d s a t e l l i t e leads to the concept of an i n v a r i a n t surface or i n t e g r a l manifold. Numerical i n t e g r a t i o n of the equation of motion i s employed to determine the mani-f o l d s . I t i s shown that f o r s p e c i f i e d values of the parameters d e s c r i b i n g the s a t e l l i t e , the region i n phase space that i s c o n s i s t e n t w i t h s t a b l e motion cor-responds to the l a r g e s t i n v a r i a n t surface which can be found. I t i s a l s o demonstrated that the manifolds are i n t i m a t e l y connected w i t h p e r i o d i c s o l u t i o n s of the equation of motion and t h i s knowledge permits determin-ing l i m i t s on the parameters so as to ensure s t a b l e motion by a study of the s o l u t i o n of the v a r i a t i o n a l equation. S e v e r a l charts s u i t a b l e f o r design purposes are presented. The planar motion of a s a t e l l i t e c o n t a i n i n g a • damping mechanism i s studied using a s i m p l i f i e d model. I t i s shown that f o r small dampers the motion e v e n t u a l ! becomes nearly i d e n t i c a l w i t h a p e r i o d i c s o l u t i o n of the undamped case. The t h i r d model represents a f l e x i b l e s a t e l l i t e f r e e to deform under the i n f l u e n c e of s o l a r h e a t i n g . An a n a l y s i s of the temperature d i s t r i b u t i o n i n the s t r u c t u r e permits determination of the shape of the s a t e l l i t e s o l e l y i n terms of i t s p o s i t i o n ^ .The r e s u l t -ing equation of motion i s derived and i t i s shown that f l e x i b i l i t y does not g r e a t l y a f f e c t the s t a b i l i t y pro-vided that the f l e x i b l e member i s not too long-The case of an axi-symmetric 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 s al s o considered- I t i s shown that i n t h i s case manifolds a l s o e x i s t although i n some cases apparently ergodic motion can occur. S t a b i l i t y can be guaranteed i f the Hamiltonian i s le s s than a p r e s c r i b e d value„ Values of the Hamiltonian l a r g e r than t h i s may al s o permit s t a b l e motion and i n t h i s case an i n v a r i a n t surface i s always described i n phase space. The s t a b i l i t y of the general motion i s somewhat greater than that f o r the planar motion. Charts are presented g i v i n g the maximum p e r m i s s i b l e disturbances f o r s t a b l e motion. GRADUATE STUDIES F i e l d of Study: Mechanical Engineering Space Dynamics ( I and I I ) Mechanical V i b r a t i o n s High Speed Gas Dynamics Mechanics of R a r i f i e d Gases Non-linear Systems Analogue Computers CAo Brockley G.V. Parkinson G.V. Parkinson A.C. Soudack E.V. Bonn V.J. Modi PUBLICATIONS Modi, V.J., and Brereton, R.C., " L i b r a t i o n A n a l y s i s of a Dumbbell S a t e l l i t e Using the WKBJ Method," Journal of A p p l i e d Mechanics, V o l . 33, No. 3, Sept. 1966, pp. 676-678. Brereton, R.C., and Modi, V.J., "On the S t a b i l i t y of a Dumbbell 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 , " Journal of the Royal A e r o n a u t i c a l S o c i e t y , V o l . 70, Dec. 1966, pp. 1098-1102. Brereton, R.C., and Modi, V.J., " S t a b i l i t y of the Planar L i b r a t i o n a l Motion 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 , " Proceedings of the XVII 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 Congress, Madrid, 19 67. Modi, V.J., and Brereton, R.C., " S t a b i l i t y Boundaries f o r Planar L i b r a t i o n s of a Long F l e x i b l e S a t e l l i t e , Paper No. 67-126, AIAA 5th Aerospace Sciences Meet-in g , New York, New York, Jan. 23-26, 1967. Modi, V.J., and Brereton, R.C "The S t a b i l i t y A n a l y s i s of Coupled L i b r a t i o n a l Motion of an Axi-Symmetric 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 , " Proceedings of the  XV I I I 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 Congress, Belgrad ( i n press) . A STABILITY STUDY OF GRAVITY ORIENTED SATELLITES by ROBERT CLOUDESLEY BRERETON B. Eng., McGill University, 1959 A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILMENT OF FOR THE DEGREE 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 September, 1967 In presenting t h i s thesis i n p a r t i a l fulfilment of the require-ments f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l -able f o r reference and study. I further agree that permission f o r extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his repre-sentatives. It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. R.C. B r e re ton Department of Mechan i ca l Eng inee r i ng The University of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT The s t a b i l i t y o f g r a v i t a t i o n a l g r a d i e n t o r i e n t e d s a t e l l i t e s i s examined by c o n s i d e r i n g f o u r s i m p l i f i e d models. The i n v e s t i g a t i o n i s c a r r i e d out n u m e r i c a l l y and a n a l y t i c a l l y . The t e c h n i q u e s employed i n v o l v e c o n s i d e r a b l e c o mputation and hence a r e p a r t i c u l a r l y s u i t e d t o s o l u t i o n by a d i g i t a l computer. The a n a l y s i s o f t h e p l a n a r m otion o f a r i g i d s a t e l l i t e l e a d s t o t h e concept o f an i n v a r i a n t s u r f a c e o r i n t e g r a l m a n i f o l d . N u m e r i c a l , i n t e g r a t i o n o f t h e e q u a t i o n o f motion i s employed t o determine t h e m a n i f o l d s . I t i s shown t h a t f o r s p e c i f i e d v a l u e s o f t h e parameters d e s c r i b i n g t h e s a t e l l i t e , t h e r e g i o n i n phase space t h a t i s c o n s i s t e n t w i t h s t a b l e m otion c o r r e s p o n d s t o t h e l a r g e s t i n v a r i a n t s u r f a c e w hich can be found. I t i s a l s o demonstrated t h a t t h e mani-f o l d s a r e i n t i m a t e l y connected w i t h p e r i o d i c s o l u t i o n s o f th e e q u a t i o n o f mo t i o n and t h i s knowledge p e r m i t s d e t e r m i n -i n g l i m i t s on t h e parameters so as t o ensure s t a b l e motion by a s t u d y o f t h e s o l u t i o n o f t h e v a r i a t i o n a l e q u a t i o n . S e v e r a l c h a r t s s u i t a b l e f o r d e s i g n purposes a r e p r e s e n t e d . The p l a n a r m otion o f a s a t e l l i t e c o n t a i n i n g a damping mechanism i s s t u d i e d u s i n g a s i m p l i f i e d model. I t i s shown t h a t f o r s m a l l dampers t h e motion e v e n t u a l l y becomes n e a r l y i d e n t i c a l w i t h a p e r i o d i c s o l u t i o n o f t h e undamped case. The t h i r d model r e p r e s e n t s a f l e x i b l e s a t e l l i t e f r e e t o deform under t h e i n f l u e n c e o f s o l a r h e a t i n g . An a n a l y s i s o f the t e m p e r a t u r e d i s t r i b u t i o n i n t h e s t r u c t u r e p e r m i t s d e t e r m i n a t i o n o f t h e shape o f t h e s a t e l l i t e s o l e l y i n terms o f i t s p o s i t i o n . The r e s u l t i n g e q u a t i o n o f motion i s d e r i v e d and i t i s shown t h a t f l e x i b i l i t y does not g r e a t l y a f f e c t t h e s t a b i l i t y p r o v i d e d t h a t the f l e x i b l e member i s not t o o l o n g . The case o f an a x i - s y m m e t r i c 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 s a l s o c o n s i d e r e d . I t i s shown t h a t i n t h i s case m a n i f o l d s a l s o e x i s t a l t h o u g h a t tim e s a p p a r e n t l y e r g o d i c motion can o c c u r . S t a b i l i t y can be gu a r a n t e e d i f the H a m i l -t o n i a n i s l e s s t h a n a p r e s c r i b e d v a l u e . V a l u e s o f the H a m i l -t o n i a n l a r g e r t h a n t h i s may a l s o permit s t a b l e motion and i n t h i s case an i n v a r i a n t s u r f a c e i s always d e s c r i b e d i n phase space. The s t a b i l i t y o f the g e n e r a l motion i s somewhat g r e a t e r t h a n t h a t f o r t h e p l a n a r motion. C h a r t s a r e p r e s e n t e d g i v i n g t h e maximum p e r m i s s i b l e d i s t u r b a n c e s f o r s t a b l e motion. TABLE OF CONTENTS Chapter 1 I n t r o d u c t i o n . . . . . . . . . . . . 1 .1 P r e l i m i n a r y Remarks . . . . . . . 1 .2 G r a v i t y - G r a d i e n t S t a b i l i z e d S a t e l l i t e s 1.3 Purpose and Scope o f I n v e s t i g a t i o n . 2 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 R i g i d S a t e l l i t e 2 . 1 F o r m u l a t i o n o f t h e Problem . . . . 2 . 2 Simple E x a c t S o l u t i o n s . . . . . . 2 . 2 . 1 C i r c u l a r O r b i t (e = 0) . . . 2 . 2 . 2 P e r i o d i c S o l u t i o n s U s i n g t h e Method o f Harmonic Ba l a n c e 2 . 2 . 3 N u m e r i c a l D e t e r m i n a t i o n o f P e r i o d i c S o l u t i o n s 2 . 3 2 . 4 2 . 5 Approximate S o l u t i o n s . . . . . 2 . 3 . 1 WKBJ Method . . . . . . 2 . 3 . 2 P r i n c i p l e o f Harmonic Ba l a n c e 2 . 3 . 3 P e r t u r b a t i o n o f P e r i o d i c S o l u t i o n s . . . . . 0 . Phase Space and I n v a r i a n t S u r f a c e s . Accuracy o f t h e Method . . . . . 2 . 6 The S i g n i f i c a n c e o f P e r i o d i c S o l u t i o n s 2 . 6 . 1 The R e l a t i o n s h i p Between Mani-f o l d s and P e r i o d i c S o l u t i o n s 2 . 6 . 2 D e t e r m i n a t i o n o f a Complete Set of P e r i o d i c S o l u t i o n s . 2 . 7 2 . 6 . 3 The Degree o f S t a b i l i t y C o n c l u d i n g Remarks . . . . Page 1 1 5 12 14 14 19 19 24 29 35 35 47 56 71 92 97 97 99 105 109 V Chapter Page 3 P l a n a r L i b r a t i o n s o f a Damped S a t e l l i t e . . . 118 3.1 F o r m u l a t i o n o f t h e Problem 118 3.2 N u m e r i c a l R e s u l t s . . . . . . . . 123 3-3 C o n c l u s i o n s . . . . 132 4 P l a n a r L i b r a t i o n s o f a Long F l e x i b l e S a t e l l i t e 134 4.1 P r e l i m i n a r y Remarks . . . . . . . . -134 4.2 F o r m u l a t i o n o f t h e Problem 136 4.3 Thermal A n a l y s i s o f t h e Boom 145 4.4 S o l u t i o n o f t h e Heat Balance E q u a t i o n . . 151 4.5 Thermal D e f l e c t i o n o f t h e Boom . . . . 156 4.6 S t a b i l i t y A n a l y s i s . . . . . . . . 166 4.7 C o n c l u d i n g Remarks 182 5 Two D i m e n s i o n a l M o t i o n o f an Axi-Symmetric S a t e l l i t e . I 8 4 $.1 I n t r o d u c t o r y Remarks I84 5.2 F o r m u l a t i o n o f t h e Problem 185 5.3 The H a m i l t o n i a n and Z e r o - V e l o c i t y Curves . 192 5.4 Phase Space and T r a j e c t o r i e s 194 5.5 Symmetry P r o p e r t i e s 197 5.6 Numerical Results . 198 5.7 Concluding Remarks 210 6 Concluding Remarks 215 6.1 General Conclusions 215 6.2 Recommendations f o r F u t u r e Work . . . . 216 B i b l i o g r a p h y . . . . . . . . . 219 LIST OF TABLES Ta b l e Page I R e p r e s e n t a t i v e 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 h a r a c t e r i s t i c s o o o » « « » « . 5 I I V a r i a t i o n o f R e s u l t s Obtained by I n t e g r a t i n g t h e E q u a t i o n s o f M o t i o n With S e v e r a l Step S X Z 6 S » a o o o o o • o o • » • • 93 I I I C h a r a c t e r i s t i c s o f R e p r e s e n t a t i v e STEM C o n f i g u r a t i o n s . . . . . . . . . . . 135 LIST OF FIGURES F i g u r e Page 1-1 Magnitude o f f o r c e s a c t i n g on a r e p r e s e n t a t i v e S3.t)6lllt6 o o • o • e o • * • • • • 6 1- 2 Models o f m u l t i - b o d y s a t e l l i t e s . . . . . 11 2- 1 Geometry o f p l a n a r m otion o f a r i g i d s a t e l l i t e 15 2-2 Phase p l a n e t r a j e c t o r i e s d e s c r i b i n g t h e s o l u t i o n when e = 0 20 2-3 L i m i t i n g phase p l a n e t r a j e c t o r i e s f o r e = 0 . 22 2-4 I n i t i a l a n g u l a r v e l o c i t i e s r e q u i r e d t o produce s p e c i f i e d p e r i o d i c s o l u t i o n s . . . . 23 2-5 P e r i o d i c s o l u t i o n s as f u n c t i o n s o f e c c e n t r i c i t y (K^ = 1, n = 1) 31 2-6 P e r i o d i c s o l u t i o n s as f u n c t i o n s o f e c c e n t r i c i t y (K^ = 0,3, n = 1) 32 2-7 I n i t i a l d e r i v a t i v e r e q u i r e d t o produce s o l u t i o n s w i t h p e r i o d o f 2Tt . . . . . . 33 2-8 T y p i c a l v a r i a t i o n s o f t h e e r r o r found i n t h e n u m e r i c a l d e t e r m i n a t i o n o f p e r i o d i c s o l u t i o n s . 34 2-$ P e r i o d i c s o l u t i o n s w i t h p e r i o d o f 4K 2-10 I n i t i a l d e r i v a t i v e r e q u i r e d t o produce p e r i o d i c s o l u t i o n s w i t h p e r i o d o f 4Jt . . . 37 2-11 The v a r i a t i o n o f F w i t h o r b i t a n g l e and o r b i t e c c e n t r i c i t y (K^ = 1) . . . . . . . . 39 2-12 Comparison o f t h e ex a c t s o l u t i o n o f t h e e q u a t i o n o f motion w i t h t h a t d e t e r m i n e d by t h e WKBJ method and t h e approximate WKBJ method (K^ = 1 , e = 0 . 1 ) o . . . . . . . . . 44 v i i i F i g u r e 2-13 2-14 2-15 Comparison o f t h e ex a c t s o l u t i o n o f t h e e q u a t i o n o f motion w i t h t h a t d e t e r m i n e d by the WKBJ method and t h e approximate WKBJ method (K. 1, e = 0.3) Comparison o f t h e exact s o l u t i o n o f t h e e q u a t i o n o f motion w i t h t h a t d e t e r m i n e d by the approximate WKBJ method over e i g h t o r b i t s (K^ ~~ 1 j 6 0 © 3) • • • • • ° ° * • D e t e r m i n a t i o n o f the f i r s t two terms o f t h e s i n e s e r i e s s o l u t i o n 2-16 V a l u e s o f t h e i n i t i a l d e r i v a t i v e r e q u i r e d t o produce s o l u t i o n s w i t h p e r i o d o f 271 as determ i n e d by the f i r s t two terms o f t h e s i n e s e r i e s s o l u t i o n . 2-17 Comparison o f e x a c t p e r i o d i c s o l u t i o n s w i t h two and t h r e e term s i n e s e r i e s s o l u t i o n s 2-18 V a l u e s o f Kj and e whic h l e a d t o v a r i a t i o n a l l y s t a b l e p e r i o d i c s o l u t i o n s o f p e r i o d 2TC . 2-19 V a l u e s o f K. and e which l e a d t o v a r i a t i o n a l l y s t a b l e p e r i o d i c s o l u t i o n s of p e r i o d 47t • 2-20 V a r i a t i o n o f parameter a w i t h o r b i t e c c e n t r i c i t y (Kj = 1) f o r s o l u t i o n s o f p e r i o d 27T . . 2-21 V a r i a t i o n o f parameter a w i t h o r b i t e c c e n t r i c i t y f o r s o l u t i o n s o f p e r i o d LK (K» 3 - ) O O O O O O O O « 0 « * < 1 2-22 Schematic view o f an i n v a r i a n t s u r f a c e Page 45 46 - i K i "~* I t 0 O 0 o o © © c . . . . . 50 - i i K i 1 0©9* * * • * • * 51 - i i i K. l 0o'7° ° ° • • * * © © © © o 5 2 - i v K. - 0 . 5 53 l -V K i = 0 1 54 - v i K i ™~ 0 © 1 © e » o e © * . . . . . 55 57 5S 69 70 72 73 77 i x F i g u r e 2-23 2-24 2-25 - i - i i A s p e c i f i c s o l u t i o n w h i c h i l l u s t r a t e s t h e symmetry p r o p e r t i e s o f t h e 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 o f an i n v a r i a n t s u r f a c e a t v a r i o u s o r b i t a n g l e s (K^ = 0.7, e = 0.2) T y p i c a l i n v a r i a n t s u r f a c e = X«j 6 = 0«25 « • • • *> • • • K. = 0.7, e = 0.2 . . 2-26 T y p i c a l i n v a r i a n t s u r f a c e w i t h " i s l a n d s " 2-27 Range o f v a l u e s o f t h e d e r i v a t i v e when ^ = 9 = 0 f o r s t a b l e m otion - l - i i - i i i - i v -v - v i K± = 1.0 K. = 0 . 9 K. l 0.7 K ± = 0.5 K. = 0.3 K± = 0.1 2-28 Comparison o f t h e 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 u s i n g d i f f e r e n t i n t e g r a t i o n s t e p s i z e s - i N o n - l i m i t i n g s u r f a c e s . - i i L i m i t i n g s u r f a c e s . . . . 2-29 I n v a r i a n t s u r f a c e i l l u s t r a t i n g t h e appearance o f s t a b l e and 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 i n t h e s t r o b o s c o p i c phase p l a n e 2 - 3 0 The t r a n s f o r m a t i o n o f l i n e s o f c o n s t a n t ip when th e e q u a t i o n o f motion i s i n t e g r a t e d over 271 ( K^ — 1 , e — 0 ) O . » O . . . O . B I 2-31 The t r a n s f o r m a t i o n o f l i n e s o f c o n s t a n t <p when t h e e q u a t i o n o f motion i s i n t e g r a t e d over 2K 2-32 D e t e r m i n a t i o n o f a complete s e t o f f i x e d p o i n t s o f t h e t r a n s f o r m a t i o n ( K ^ = l , e = 0 ) Page 78 80 81 82 85 86 87 88 89 90 91 94 95 100 102 103 104 X F i g u r e 2-33 2-34 2-35 2-36 The t r a n s f o r m a t i o n of l i n e s o f c o n s t a n t <p when t h e e q u a t i o n o f motion i s i n t e g r a t e d o ver 27t ( ~ - L j 6 — 0 o l ) o O O 9 • O O • • » The t r a n s f o r m a t i o n o f l i n e s o f c o n s t a n t ¥ when th e e q u a t i o n o f mo t i o n i s i n t e g r a t e d over 2K (K« G"~0»1) O o O O • o a • • • D e t e r m i n a t i o n o f a complete s e t o f f i x e d p o i n t s o f t h e t r a n s f o r m a t i o n (K^ = 1, e = 0.1) Maximum momentum change r e q u i r e d t o d e s t a b i l i z e a s a t e l l i t e 3-1 3 - 2 3-3 3 - 4 3 - 5 3 - 6 3 - 7 3 - 8 4 - 1 4 - 2 S o l u t i o n o f e q u a t i o n o f motion i l l u s t r a t i n g t h e e f f e c t o f t h e damper . S t r o b o s c o p i c phase p l a n e o f the s o l u t i o n i l l u s t r a t e d i n F i g u r e 3 - 2 T y p i c a l s t r o b o s c o p i c phase p l a n e o f damped Sclt @!X.X i t 6 o o o o o o » o o » » o T y p i c a l s t r o b o s c o p i c phase pla n e o f damped S cl t @X *L j. t 6 o e o o o o t o o o o * L i m i t c y c l e s (K^ = 1 . , e = 0 . 1 ) . . . . L i m i t c y c l e s (K^ = 1 . , e = 0 . 2 ) . . . . L i m i t c y c l e s (K^ = 1 . , e = 0 . 3 ) . . . . Geometry o f motion o f f l e x i b l e s a t e l l i t e Assumed c r o s s - s e c t i o n o f s a t e l l i t e boom . Page 106 107 108 - i K i " 3- O 0 O O O O • O 0 s • * . . 110 - i i K. l *"•" 0*9 ° • • • • • * • * * . . I l l - i i i K i ~~ 0 o 7 ° * * ° • • • * • • . . 112 - i v K i ~" 0 « 5 ° ° ° • • ° • • • • . . 113 - V K ± = 0.3 . . 114 - v i K i = 0.1 . . . . . . 115 Geometry o f motion o f a damped s a t e l l i t e . . 119 125 126 127 128 129 130 131 137 146 XI Figure Page 4 - 3 Heat balance f o r an element of the s a t e l l i t e 4 - 4 Geometry of radiant heat transfer i n the i n t e r i o r of the s a t e l l i t e boom . . . . . . 149 4 - 5 Thermal defl e c t i o n of the s a t e l l i t e boom . . . 157 4 - 6 Shape of thermally deflected s a t e l l i t e boom . . l 6 l 4 - 7 I l l u s t r a t i o n of the p r i n c i p a l axes of the deflected s a t e l l i t e . . . . 162 4 - 8 Maximum i n e r t i a variations as functions of boom length 163 4 - 9 Relative i n e r t i a v a r i a t i o n as a function of the angle between the sun and the x-axis . . . . 164 4 - 1 0 Typical invariant surface (e = 0 . 2 , c< = 0 ° , l ) * • ° ° o o • • o © © » • • l6*7 4-11 Variation of the cross-section of a l i m i t i n g invariant surface with orbit angle (e = 0 . 2 , (X = 0 , L*= 1) . . . 169 4-12 Variation of the cross-section of a l i m i t i n g invariant surface with orbit angle (e = 0 . 2 , CK - 3 0 ° , L* = 1) . . . . . . . . . . 170 4-13 Range of values of the derivative when f = 0 = 0 f o r stable motion (L*= 1) — 1 — O o o o o o . o . a o . . . 171 — IX 30 o « . . o o © © o o . . . 172 — DL X X 0^  — 60 . . . O O O A . . . . . . 173 — i v (X = 90° • • 174 4-14 Range of values of the derivative when ifJ = & = 0 f o r stable motion (L* = 2) - i tX = 0° . . . . . . . . 175 - i i = 30° . . . . . . . . . . . . . 176 - i i i = 60° . . . . . . . . . . . . . 177 -xv W = x i i F i g u r e Page 4-15 Range o f v a l u e o f t h e d e r i v a t i v e , when <// = 6 = 0 f o r l o n g term s t a b i l i t y — 1 Xj'"" ! 1 O O 9 • O 9 9 9 9 9 • °" 1 1 ly 2 o O O O • 9 9 9 9 9 9 Geometry f o r t h e two d i m e n s i o n a l motion o f a Z e r o - v e l o c i t y c u r v e s f o r an a x i - s y m m e t r i c s a t e l l i t e in. a c i r c u l a r o r b i t (K^ = 1) I n v a r i a n t s u r f a c e r e s u l t i n g from t h e f i r s t c l a s s o f s o l u t i o n s . . . . . . . . C r o s s - s e c t i o n of a s u r f a c e - . s i m i l a r t o t h a t p r e s e n t e d ^ i n ' F i g u r e 5-3 when d> = 0. The c r o s s - s e c t i o n <p = 0 i n phase space i l l u s t r a t i n g t h e e r g o d i c n a t u r e o f the second c l a s s o f s o l u t i o n s . . . . . . . . 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 The c r o s s - s e c t i o n tp = 0 i n phase space i l l u s t r a t i n g t h e t r a n s i t i o n from a l a r g e s i m p l e " m a i n l a n d " t o an e r g o d i c t r a j e c t o r y v i a a number o f " i s l a n d s " . . . . . . . . . . T y p i c a l i n v a r i a n t s u r f a c e when C^ > - 1 . L i m i t i n g i n v a r i a n t s u r f a c e . - i CH " "•0o5* ° ° ° O O 0 • - i i CH - 0 O o o o o O _ 9 9 9 - i i i CH '• 0 • 5 ° ° ° ° 9 9 9 9 5-9 S o l u t i o n o f t h e e q u a t i o n s o f motion f o r ; s p e c i f i c i n i t i a l c o n d i t i o n s , i l l u s t r a t i n g t h e q u a s i - p e r i o d i c n a t u r e o f t h e motion 5-10 A l l o w a b l e v a r i a t i o n s i n t h e a n g u l a r v e l o c i t i e s w hich may be imposed on an a x i - s y m m e t r i c s a t e l l i t e when i n a s p e c i f i e d o r i e n t a t i o n D 1 0 *""" O o o o o o o o e o o o e - i i Ci = ± 15° •"111 ^ ~~* ~~ Zf- 5 ° ° <• ° * © Q « O • O 180 181 186 195 199 201 202 203 205 206 207 208 209 211 212 213 ACKNOWLEDGEMENT The a u t h o r w i s h e s t o e x p r e s s h i s t h a n k s t o Dr. V . J . Modi f o r t h e gui d a n c e g i v e n throughout t h e p r e p a r a t i o n o f t h e t h e s i s . H i s h e l p and encourage-ment have been i n v a l u a b l e . The n u m e r i c a l work i n t h i s t h e s i s was c a r r i e d out i n t h e Computing C e n t r e o f the U n i v e r s i t y o f B r i t i s h Columbia. The use o f t h e s e : , f a c i l i t i e s i s g r a t e f u l l y acknowledged. LIST OF SYMBOLS A C o e f f i c i e n t i n e q u a t i o n (2.100) Aa Aerodynamic r e f e r e n c e a r e a Ae Area from w h i c h r a d i a t i o n i s e m i t t e d A, A m p l i t u d e o f t h e i * ^ normal mode o f t h e 1 s a t e l l i t e boom Area on wh i c h r a d i a t i o n i s i n c i d e n t Am n m t h F o u r i e r c o e f f i c i e n t i n t h e s o l u t i o n o f '> e q u a t i o n (2.14) o f p e r i o d 2n n P r o j e c t e d a r e a o f t h e s a t e l l i t e A£ A r b i t r a r y c o e f f i c i e n t i n t h e s o l u t i o n o f t h e 5 v a r i a t i o n a l e q u a t i o n (2.64) 33 C o e f f i c i e n t i n e q u a t i o n (2.100) f3g A r b i t r a r y c o e f f i c i e n t i n t h e s o l u t i o n o f t h e v a r i a t i o n a l e q u a t i o n (2.64) G C o e f f i c i e n t i n e q u a t i o n (2.100) C j , n C o e f f i c i e n t i n e q u a t i o n (4.59) C c Constant o f i n t e g r a t i o n i n e q u a t i o n (2.16) CL| C onstant v a l u e o f the H a m i l t o n i a n , e q u a t i o n (5.26) ^•PWV" C o e f f i c i e n t i n e q u a t i o n (2.23) Cp Aerodynamic p r e s s u r e c o e f f i c i e n t CS)r\ n F o u r i e r c o e f f i c i e n t e x p r e s s i n g t h e s o l a r heat i n p u t t o t h e s a t e l l i t e boom CyC^C^C^ A r b i t r a r y c o n s t a n t s X) C o e f f i c i e n t I n e q u a t i o n (2.100) Parameter d e t e r m i n i n g t h e n a t u r e o f a c o n i c s e c t i o n £ Modulus o f e l a s t i c i t y p 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 . 4 1 ) & D i s s i p a t i o n f u n c t i o n Ff G e n e r a l i z e d f o r c e a c t i n g on the i * " * 1 normal mode o f t h e s a t e l l i t e boom L o n g i t u d i n a l f o r c e i n t h e s a t e l l i t e boom p F F u n c t i o n s d e f i n e d i n e q u a t i o n ( 5 . 2 9 ) Q 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 . 4 0 ) H~f H a m i l t o n i a n f u n c t i o n $4" M a g n e t i c f i e l d s t r e n g t h I Large moment o f i n e r t i a o f an a x i - s y m m e t r i c 1 s a t e l l i t e J I n e r t i a o f t h e c r o s s - s e c t i o n o f t h e s a t e l l i t e b boom T I n e r t i a o f t h e i mode o f v i b r a t i o n o f t h e 1 s a t e l l i t e boom, d e f i n e d i n e q u a t i o n (4 . 2 0 ) I . I n e r t i a o f a s a t e l l i t e w i t h a r i g i d boom ^xxJ^vy^zz. . . . P r i n c i p a l moments o f i n e r t i a o f the s a t e l l i t e T I n e r t i a v a r i a t i o n parameter = I,. \ r wax' K , E l l i p t i c i n t e g r a l o f t h e f i r s t k i n d Parameter d e s c r i b i n g t h e i n e r t i a o f t h e s a t e l l i t e damper, e q u a t i o n (3.6) j\j I n e r t i a parameter = (lXx~ ^zz)/\y K K., I n e r t i a v a r i a t i o n p a r a m e t e r s , d e f i n e d i n *' * '" e q u a t i o n (4.02) l _ Length o f the s a t e l l i t e boom Xj L a g r a n g i a n f u n c t i o n 1_ D i m e n s i o n l e s s l e n g t h o f t h e s a t e l l i t e boom = 9ft R e s i d u a l magnetic moment o f t h e s a t e l l i t e Frr\,Y\ C o e f f i c i e n t i n e q u a t i o n ( 2 . 2 5 ) x v i Q u a n t i t y o f heat Radius o f c u r v a t u r e o f t h e s a t e l l i t e boom K i n e t i c energy Temperature o f the s a t e l l i t e boom The n t h F o u r i e r c o e f f i c i e n t d e s c r i b i n g t h e tem p e r a t u r e d i s t r i b u t i o n i n t h e s a t e l l i t e boom Ref e r e n c e t e m p e r a t u r e V i b r a t i o n a l k i n e t i c energy P o t e n t i a l energy E l a s t i c p o t e n t i a l energy G r a v i t a t i o n a l p o t e n t i a l energy Wronskian d e t e r m i n a n t Shape f u n c t i o n o f t h e i normal mode o f t h e s a t e l l i t e boom D i m e n s i o n l e s s damper d i s p l a c e m e n t V a l u e o f t h e f i r s t s o l u t i o n of the v a r i a t i o n a l e q u a t i o n when Q = 27Cn Diameter o f t h e s a t e l l i t e boom Va l u e o f t h e d e r i v a t i v e o f t h e f i r s t s o l u t i o n o f t h e v a r i a t i o n a l e q u a t i o n when © = 2Trn W a l l t h i c k n e s s o f t h e s a t e l l i t e boom Val u e o f t h e second s o l u t i o n o f t h e v a r i a t i o n a l e q u a t i o n when 0 = 2%n S p e c i f i c heat o f the m a t e r i a l i n t h e s a t e l l i t e boom Damping c o n s t a n t Speed o f l i g h t V a l u e o f t h e d e r i v a t i v e o f t h e second s o l u t i o n o f t h e v a r i a t i o n a l e q u a t i o n when Q = 2 ^ n XVI1 S . O r b i t e c c e n t r i c i t y -ft , Step s i z e employed i n n u m e r i c a l i n t e g r a t i o n •/? A n g u l a r momentum o f t h e s a t e l l i t e boom as seen i n r o t a t i n g c o - o r d i n a t e s $ A n g u l a r momentum per u n i t mass o f t h e s a t e l l i t e about t h e c e n t r e o f f o r c e t ., I n t e g e r If .. Frequency e i g e n v a l u e o f t h e s a t e l l i t e boom K .. Thermal c o n d u c t i v i t y o f t h e m a t e r i a l i n the s a t e l l i t e boom kj . S p r i n g c o n s t a n t o f t h e damper . L e n g t h of an element o f the s a t e l l i t e boom JLfeL R e f e r e n c e l e n g t h o f an element o f the s a t e l l i t e ' boom ^Xftyi^z, D i r e c t i o n c o s i n e s o f t h e l o c a l v e r t i c a l i n t h e p r i n c i p a l c o - o r d i n a t e s /)#• X. - J Thermal r e f e r e n c e l e n g t h o f t h e s a t e l l i t e , d e f i n e d i n e q u a t i o n (4.76) D i s t a n c e between t h e l i n e o f a c t i o n o f t h e aerodynamic f o r c e and t h e c e n t r e o f mass faH^ D i s t a n c e between the l i n e o f a c t i o n o f t h e f o r c e due t o r a d i a t i o n p r e s s u r e and t h e c e n t r e o f mass Yf) . I n t e g e r Mass o f the s a t e l l i t e y/)j . Mass o f t h e damper W Mass p e n u n i t l e n g t h o f t h e s a t e l l i t e boom T ^ ^ t f j I n t e g e r s y% I n t e g e r pf G e n e r a l i z e d momentum c o n j u g a t e t o t h e co-o r d i n a t e q^ XV111 p r R a d i a t i o n p r e s s u r e .Momentum c o n j u g a t e t o t h e X c o - o r d i n a t e Momentum c o n j u g a t e t o t h e 0 c o - o r d i n a t e Momentum c o n j u g a t e t o t h e tf> c o - o r d i n a t e Cjate ...Rate a t which heat i s absorbed per u n i t a r e a Rate o f heat i n p u t from e x t e r n a l s o u r c e s per u n i t a r e a G e n e r a l i z e d c o - o r d i n a t e Rate a t whi c h heat i s i n c i d e n t on u n i t a r e a Rate o f heat i n p u t from i n t e r n a l s o u r c e s per u n i t a r e a °jrtf Rate a t which heat i s r e f l e c t e d p e r u n i t a r e a Rate o f heat i n p u t f rom t h e sun per u n i t a r e a r Radius /I D i s t a n c e between t h e c e n t r e o f f o r c e and an element of mass S D i s t a n c e a l o n g t h e s a t e l l i t e boom t Time V V e l o c i t y . P r i n c i p a l body c o - o r d i n a t e s ^ity^* I n t e r m e d i a t e body c o - o r d i n a t e s w i t h o r i g i n a t th e c e n t r e o f mass p r i o r t o t h e m o d i f i e d XutfLrti, E u l e r i a n r o t a t i o n s <^> , <f> , A r e s p e c t i v e l y ^. Damper o f f - s e t d i s t a n c e ^, , Damper d i s p l a c e m e n t , Aerodynamic t o r q u e P b Moment i n t h e s a t e l l i t e boom P 3 Moment due t o g r a v i t a t i o n a l g r a d i e n t Pyrt Magnetic t o r q u e x i x R a d i a t i o n t o r q u e F u n c t i o n d e s c r i b i n g t h e e r r o r i n t h e n u m e r i c a l l y d e t e r m i n e d p e r i o d i c s o l u t i o n s Mean r o t a t i o n a n g l e o f t h e p e r t u r b a t i o n s o l u t i o n V a r i a b l e d e f i n e d i n e q u a t i o n ( 2 . 3 8 ) Complementary s o l u t i o n s o f e q u a t i o n ( 2 . 3 9 ) P a r t i c u l a r i n t e g r a l o f e q u a t i o n ( 2 . 3 9 ) Angle around s a t e l l i t e boom as measured from t h e s u b - s o l a r p o i n t S o l a r a s p e c t a n g l e A b s o r p t i v i t y o f s a t e l l i t e boom m a t e r i a l t o s o l a r energy C o e f f i c i e n t o f t h e r m a l e x p a n s i o n o f boom m a t e r i a l S p e c i f i c v a l u e o f 0( y + 0 - (X C h a r a c t e r i s t i c r o o t P e r t u r b a t i o n Complementary s o l u t i o n s o f v a r i a t i o n a l e q u a t i o n E m i s s i v i t y o f boom m a t e r i a l P e r t u r b a t i o n i n ip1 D i s t a n c e between elements a l o n g s a t e l l i t e boom .Co-ordinate o f d e f l e c t e d boom P o s i t i o n a n g l e o f s a t e l l i t e i n i t s o r b i t as measured from p e r i c e n t r e S p e c i f i e d v a l u e o f S e - % XX P e r i o d o f a p e r i o d i c s o l u t i o n C o - o r d i n a t e d e f i n i n g t h e a n g l e o f s p i n o f a s a t e l l i t e J Angles used i n t h e r a d i a t i o n a n a l y s i s o f t h e i n t e r i o r o f t h e s a t e l l i t e boom ( F i g u r e 4-4) G r a v i t a t i o n a l f i e l d parameter Angle employed i n t h e r a d i a t i o n a n a l y s i s o f the i n t e r i o r o f t h e s a t e l l i t e boom.(Figure 4-4) 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 (2.45) P e r t u r b a t i o n i n ^ C o - o r d i n a t e o f d e f l e c t e d boom Atmospheric d e n s i t y D e n s i t y o f boom m a t e r i a l S t e f a n - B o l t z m a n n c o n s t a n t S t r e s s i n boom m a t e r i a l Damper t i m e c o n s t a n t t h Time c o n s t a n t o f the n c o e f f i c i e n t i n t h e F o u r i e r a n a l y s i s o f t h e temp e r a t u r e d i s t r i b u t i o n o f t h e s a t e l l i t e boom O r b i t a l p e r i o d D i m e n s i o n l e s s damper t i m e c o n s t a n t , e q u a t i o n (3.14) L i b r a t i o n a l a n g l e normal t o the o r b i t a l p l a n e Angle subtended by an element o f t h e s a t e l l i t e boom o f l e n g t h ^ b Angle between t h e sun and t h e a x i s o f an element o f t h e s a t e l l i t e boom • L i b r a t i o n a l a n g l e i n t h e o r b i t a l p l a n e V a l u e o f (/> a t S = 27f P e r i o d i c s o l u t i o n o f e q u a t i o n (2.14) w i t h p e r i o d 27T n XXI % f - 7T/2 (j)' t// Cos 0 (p^lpi Complementary s o l u t i o n o f e q u a t i o n (2 . 3 7 ) (^»* -. P a r t i c u l a r i n t e g r a l o f e q u a t i o n (2 . 3 7 ) Maximum a l l o w a b l e s i z e o f i m p u l s e f o r s t a b i l i t y 03 S o l i d a n g l e 60j N a t u r a l f r e q u e n c y o f t h e damper co. ,. N a t u r a l f r e q u e n c y o f t h e i normal mode o f t h e L s a t e l l i t e boom 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 (2.45) Mr;6^/**^- A n g u l a r v e l o c i t i e s about t h e p r i n c i p a l axes (J& O r b i t a l f r e q u e n c y = 2 7t/t© cJ* D i m e n s i o n l e s s damper n a t u r a l f r e q u e n c y S u b s c r i p t s "f F i n a l v a l u e I I n t e g e r m a / Maximum Y\ V a l u e a t & = 27Tn steely Steady s t a t e o I n i t i a l c o n d i t i o n s Dots and primes 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 t and ©> r e s p e c t i v e l y 1. INTRODUCTION 1.1 P r e l i m i n a r y Remarks The m o t i o n of a space v e h i c l e i n v o l v e s two d y n a m i c a l a s p e c t s o f i n t e r e s t , namely, th e a n a l y s e s o f i t s t r a j e c t o r y and o f i t s o r i e n t a t i o n . The f o r m e r , g e n e r a l l y r e f e r r e d t o as t h e o r b i t a l m o t i o n , i s concerned w i t h t h e d e t e r m i n a t i o n o f t h e motion of t h e mass c e n t r e and may be thought o f as an e x t e n s i o n o f c l a s s i c a l c e l e s t i a l mechanics. On t h e o t h e r hand, th e motion o f a s a t e l l i t e about i t s own c e n t r e o f mass i s c a l l e d l i b r a t i o n . There a r e s i t u a t i o n s o f p r a c t i c a l i m p o r t a n c e where i t i s d e s i r a b l e t o m a i n t a i n a s a t e l l i t e i n a f i x e d o r i e n t a -t i o n r e l a t i v e t o t h e e a r t h . , For example, p r o p e r f u n c t i o n -i n g o f communication s a t e l l i t e s w i t h d i r e c t i o n a l antennae o r of weather s a t e l l i t e s s c a n n i n g c l o u d c o v e r r e q u i r e s a t t i t u d e c o n t r o l . U n f o r t u n a t e l y , t h e 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 , even though p o s i t i o n e d c o r r e c t l y i n t h e b e g i n n i n g , d e v i a t e s i n t i m e under th e i n f l u e n c e o f e x t e r n a l d i s t u r b -a n c e s , e.g. m i c r o m e t e o r i t e i m p a c t s , s o l a r r a d i a t i o n p r e s s u r e , g r a v i t a t i o n a l and magnetic f i e l d i n t e r a c t i o n s . T h i s l e a d s t o u n d e s i r a b l e l i b r a t i o n a l motion which must be c o n t r o l l e d f o r the s u c c e s s f u l o p e r a t i o n o f t h e s a t e l l i t e . S e v e r a l methods of a t t i t u d e c o n t r o l a r e a v a i l a b l e . They may be c l a s s i f i e d as a c t i v e o r p a s s i v e t e c h n i q u e s . A c t i v e s t a b i l i z a t i o n i n v o l v e s t h e e x p e n d i t u r e o f 2 energy w h i c h i s a v e r y e x p e n s i v e commodity aboard an i n s t r u -ment packed s p a c e c r a f t . The main advantage o f t h i s t e c h -n i q u e i s i t s a b i l i t y t o m a i n t a i n t h e s p e c i f i e d o r i e n t a t i o n w i t h almost any d e s i r e d degree o f a c c u r a c y . P a s s i v e s t a b i l i z a t i o n t e c h n i q u e s , which use no power, can p r o v i d e t h e n e c e s s a r y a t t i t u d e c o n t r o l i f t h e o r i e n t a -t i o n r e q u i r e m e n t s a r e not t o o s e v e r e . S t a b i l i z a t i o n i s o b t a i n e d by employing t h e n o n - u n i f o r m i t i e s o f t h e e n v i r o n -ment i n c o n j u n c t i o n w i t h t h e p h y s i c a l p r o p e r t i e s o f t h e s a t e l l i t e . . The s i g n i f i c a n t f o r c e s a v a i l a b l e f o r p a s s i v e s t a b i l i z a t i o n o f a s p a c e c r a f t a r i s e from g r a v i t a t i o n a l , 1 2 s o l a r , m a g n e t i c , and aerodynamic e f f e c t s . ' The g r a v i t a t i o n a l moment a r i s e s because o f t h e l o c a l v a r i a t i o n o f t h e g r a v i t a t i o n a l a c c e l e r a t i o n w i t h i n the s a t e l l i t e . I t tends t o make t h e " l o n g " a x i s ( the a x i s o f minimum moment o f i n e r t i a ) o f the s p a c e c r a f t p o i n t i n t h e l o c a l v e r t i c a l d i r e c t i o n . There i s no d i s c r i m i n a t i o n between "up" and "down." The maximum g r a v i t y - g r a d i e n t t o r q u e i s g i v e n by The e l e c t r o m a g n e t i c r a d i a t i o n from t h e sun c a r r i e s w i t h i t momentum and hence when absorbed or r e f l e c t e d e x e r t s a p r e s s u r e . A s a t e l l i t e w i t h a l a r g e s u r f a c e a r e a p l a c e d a s y m m e t r i c a l l y w i t h r e s p e c t t o the c e n t r e o f mass w i l l e x p e r i e n c e a moment which may be u t i l i z e d t o e s t a b l i s h a (1.1) 3 p r e f e r r e d o r i e n t a t i o n . The r a d i a t i o n p r e s s u r e i n t h e v i c i n i t y o f the e a r t h i s g i v e n by f>r = = 9.7 Xf0'3 lb/ft* ( 1 : 2 ) so t h a t the maximum r a d i a t i o n p r e s s u r e t o r q u e i s 17 = frK A<lfm (1.3) The e a r t h ' s magnetic f i e l d extends f o r some d i s t a n c e i n t o space. W i t h i n t e n e a r t h - r a d i i i t i s r e l a t i v e l y s t a b l e and v a r i e s a p p r o x i m a t e l y as the i n v e r s e t h i r d power o f t h e d i s t a n c e . At l a r g e r d i s t a n c e s i t i n t e r a c t s w i t h t h e s o l a r 3 wind and becomes q u i t e unsteady. The e a r t h ' s magnetic f i e l d can i n t e r a c t w i t h a s p a c e c r a f t g i v i n g r i s e t o a moment i n t h r e e d i s t i n c t ways. R o t a r y motion o f t h e c o n d u c t i n g m a t e r i a l i n t h e space-c r a f t i n d u c e s eddy c u r r e n t s which d i s s i p a t e energy. The e f f e c t i s t o p r o v i d e a moment which opposes t h e motion. R o t a t i o n w i t h r e s p e c t t o t h e e a r t h ' s magnetic f i e l d o f f e r r o m a g n e t i c m a t e r i a l s p r e s e n t i n t h e s a t e l l i t e r e s u l t s i n h y s t e r e s i s l o s s e s and hence i n a damping moment. The i n t e r a c t i o n between t h e r e s i d u a l magnetic moment o f t h e s p a c e c r a f t w i t h t h e e a r t h ' s f i e l d a l s o produces a moment. In c o n t r a s t t o the p r e v i o u s two c a s e s , t h i s i n t e r a c t i o n i s c o n s e r v a t i v e . The maximum t o r q u e due t o t h e magnetic moment i s r; = 7 . 3 8 x , o a a f t The problem i s f u r t h e r c o m p l i c a t e d by t h e f a c t t h a t t h e magnitude and d i r e c t i o n o f the e a r t h ' s f i e l d change w i t h t h e p o s i t i o n o f t h e s a t e l l i t e i n i t s o r b i t . Under c e r t a i n c o n d i t i o n s , aerodynamic f o r c e s may p r o v i d e an e f f e c t i v e means o f s t a b i l i z a t i o n w i t h r e s p e c t t o th e v e l o c i t y v e c t o r . U n f o r t u n a t e l y t h e s e f o r c e s cause the s a t e l l i t e t o r e - e n t e r t h e e a r t h ' s atmosphere t h u s l i m i t i n g t h e a p p l i c a t i o n o f the t e c h n i q u e t o a s h o r t i n t e r v a l . The maximum aerodynamic t o r q u e i s g i v e n by (1 K i n g - H e l e has d i s c u s s e d t h e d e t e r m i n a t i o n o f jO a i n c o n s i d e r -a b l e d e t a i l . The v a r i a t i o n o f the moments g i v e n by e q u a t i o n s ( 1 ) , (3)> (4) and (5) w i t h a l t i t u d e f o r t h e r e p r e s e n t a t i v e 5 c o n f i g u r a t i o n ^ d e s c r i b e d i n T a b l e 1 i s shown i n F i g u r e 1-1. A s a t e l l i t e , when s t a b i l i z e d by one o f t h e s e moments, tends t o a t t a i n t h e p r e f e r r e d d i r e c t i o n a s s o c i a t e d w i t h t h a t moment. The r e m a i n i n g moments which a c t i n d i f f e r e n t d i r e c t i o n s c o n s t i t u t e d i s t u r b a n c e s . The chosen t e c h n i q u e , t h e r e f o r e , must have a l a r g e maximum t o r q u e compared t o t h e d i s t u r b i n g moments so t h a t the p e r t u r b e d motion I s w i t h i n t h e a l l o w a b l e l i m i t s . T a b l e I R e p r e s e n t a t i v e 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 5 S a t e l l i t e GEGS - A Moments o f i n e r t i a , P r o j e c t e d a r e a , A. I I xx r zz 615.3 s l u g f t 2 20.8 s l u g f t 2 13.1 f t 2 O f f s e t between c e n t r e o f mass and c e n t r e o f a r e a , 5.75 f t R e s i d u a l magnetic moment, fft, 302 p o l e cm 1.2 G r a v i t y - G r a d i e n t S t a b i l i z e d S a t e l l i t e s The dominance o f t h e g r a v i t y - g r a d i e n t t o r q u e over a l a r g e range o f a l t i t u d e s has l e d t o c o n s i d e r a b l e i n t e r e s t i n t h i s t e c h n i q u e o f s t a b i l i z i n g t h e a t t i t u d e o f a r t i f i c i a l s a t e l l i t e s . A s u r v e y o f t h e l i t e r a t u r e r e v e a l s t h a t t h e a n a l y s i s o f t h e problem has proceeded e s s e n t i a l l y a l o n g two p a t h s . The major b u l k o f the l i t e r a t u r e i s concerned e i t h e r w i t h the t h e o r e t i c a l a n a l y s i s o f i d e a l i z e d models under r e s t r i c t e d c o n d i t i o n s o r w i t h t h e d e t a i l e d s i m u l a t i o n o f s p e c i f i c c o n f i g u r a t i o n s o The p u r e l y t h e o r e t i c a l a n a l y s i s o f t h e problem i s l i m i t e d by t h e f a c t t h a t the g o v e r n i n g n o n - l i n e a r c o u p l e d e q u a t i o n s o f motion do not possess a c l o s e d form s o l u t i o n . Moran and Yu found t h a t some s i m p l i f i c a t i o n o f t h e prob-lem i s p o s s i b l e as t h e p e r t u r b a t i o n s o f the o r b i t due t o 6 — Gravity gradient — Aerodynamic — Solar pressure -— Magnetic — Cosmic dust 1 0 - + - 1 0 10 \ GEOS-A Geometry Approximate limit of magnetic field of earth — 1 — ^ T " " * " 10 2 5 10 2 IO1 IO-F i g u r e 1-1 Altitude miles Magnitude e f f o r c e s a c t i n g on a r e p r e s e n t a t i v e s a t e l l i t e 7 t h e l i b r a t i o n a l m o t i o n o f t h e s a t e l l i t e a r e n e g l i g i b l y -s m a l l . T h i s makes i t p o s s i b l e t o d e s c r i b e t h e o r b i t a l m o t ion u s i n g t h e s i m p l e K e p l e r i a n e q u a t i o n s . N e l s o n and L o f t s t u d i e d s m a l l a m p l i t u d e l i b r a t i o n s o f a r i g i d body i n a c i r c u l a r o r b i t u s i n g l i n e a r i z e d equa-t i o n s o f mot i o n . The a p p r o x i m a t i o n r e s u l t e d i n t h e decou-p l i n g o f t h e motions i n and normal t o the o r b i t a l p l a n e . Q Klemperer gave t h e e x a c t s o l u t i o n f o r p l a n a r l i b r a -t i o n s o f 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 . S c h e c h t e r " ^ a t t e m p t e d t o extend t h i s s o l u t i o n t o t h e case o f s m a l l o r b i t a l e c c e n t r i c i t y by t h e method o f p e r t u r b a t i o n s . The method has l i m i t e d a p p l i c a b i l i t y as the r e s u l t i n g p e r t u r -b a t i o n s grow w i t h o u t bound. Baker"'"''" found p e r i o d i c s o l u t i o n s w i t h o r b i t a l f r e -quency f o r a dumbbell 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 . He showed t h a t t h e a m p l i t u d e o f t h e motion i s a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e e c c e n t r i c i t y o f t h e o r b i t . 12 A r e c e n t paper by Z l a t o u s o v et a l i s o f c o n s i d e r -a b l e i n t e r e s t . These a u t h o r s a l s o o b t a i n e d p e r i o d i c s o l u -t i o n s o f the p l a n a r e q u a t i o n s o f moti o n . The s o l u t i o n s were f u n c t i o n s o f t h e o r b i t e c c e n t r i c i t y and a parameter w h i c h d e s c r i b e d the geometry o f t h e s a t e l l i t e . I t was found t h a t , i n a d d i t i o n t o t h e s o l u t i o n s p r e d i c t e d by.Baker, t h e r e may be two o t h e r s o l u t i o n s f o r the same v a l u e s o f t h e parameters. I n f i n i t e s i m a l p e r t u r b a t i o n s about t h e s e s o l u t i o n s were i n v e s t i g a t e d f o r s t a b i l i t y . I t was shown t h a t s t a b l e p e r i o d i c motion was p o s s i b l e f o r a l l o r b i t e c c e n t r i c i t i e s 8 by t h e p r o p e r c h o i c e o f s a t e l l i t e geometry. N o n - l i n e a r e f f e c t s i n t h e p e r t u r b a t i o n e q u a t i o n s were not c o n s i d e r e d so t h a t t h e magnitude o f a f i n i t e d i s t u r b a n c e w h i c h would y i e l d s t a b l e m otion was not determined. These a u t h o r s appear t o have been t h e f i r s t t o ana-l y z e t h e problem u s i n g t h e concept o f a s t r o b o s c o p i c phase-p l a n e . A p l o t i n t h i s p l a n e may be r e g a r d e d as t h e r e s u l t o f r e p e a t e d a p p l i c a t i o n o f a p o i n t t r a n s f o r m a t i o n . S t a b l e m o t i o n i s r e p r e s e n t e d by c l o s e d i n v a r i a n t c u r v e s and i s a s s o c i a t e d w i t h a p e r i o d i c s o l u t i o n w hich appears as a s e t of f i x e d p o i n t s . The a n a l y s i s i n v o l v i n g t h e t h r e e degrees o f freedom of a r i g i d s a t e l l i t e i s v e r y d i f f i c u l t . I f t h e o r b i t i s c i r c u l a r , t h e H a m i l t o n i a n i s c o n s t a n t which s p e c i f i e s bounds 13 on i n i t i a l c o n d i t i o n s t o g u a r a n t e e s t a b i l i t y . D e B r a ^ f o r m u l a t e d t h e problem o f t h e l i b r a t i o n o f a r i g i d a r b i t r a r i l y shaped s a t e l l i t e i n an e l l i p t i c , orbit... He c o n s i d e r e d t h e g e n e r a l case w i t h t h r e e degrees o f freedom i n t h e presence o f a s p e c i f i c form o f damping. The response of t h e s a t e l l i t e was d e t e r m i n e d f o r a l i m i t e d s e t o f i n i t i a l c o n d i t i o n s . I n s t a b i l i t y was a t t r i b u t e d t o t h e n o n - l i n e a r c o u p l i n g e x i s t i n g between t h e degrees o f freedom. The. d e t a i l e d s i m u l a t i o n t e c h n i q u e has n e a r l y always been concerned w i t h s a t e l l i t e s c o n s i s t i n g o f s e v e r a l b o d i e s which a r e h i n g e d t o g e t h e r . The j o i n t s o f such systems a r e c o n v e n t i o n a l l y equipped w i t h s p r i n g s and energy d i s s i p a t i n g mechanisms. The i n t r o d u c t i o n o f a r t i c u l a t e d b o d i e s i n c r e a s e s t h e c o m p l e x i t y o f t h e problem b u t , as p o i n t e d out by Hartbaum et a l , " ^ t h e c o n f i g u r a t i o n p o s s e s s e s c o n s i d e r a b l e m e r i t . The major advantages a r e v e r y f a s t t r a n s i e n t damping a t a l l a m p l i t u d e s o f motion and c o n s i d e r a b l e d e s i g n f l e x i b i l i t y . Z a j a c " ^ has a n a l y z e d t h e s m a l l a m p l i t u d e p l a n a r m otion of a two-body s a t e l l i t e ( F i g u r e 1-2-a) i n a c i r c u l a r o r b i t . I t was shown t h a t i n t h e presence o f v i s c o u s damping t h e c o n f i g u r a t i o n reduces t h e t i m e c o n s t a n t t o 0.137 o f t h e o r b i t a l p e r i o d . M u l t i b o d y s a t e l l i t e s have been i n v e s t i g a t e d i n d e t a i l 17 18 by s e v e r a l a u t h o r s . E t k i n ' has d e r i v e d t h e e q u a t i o n s o f m o t i o n f o r a s a t e l l i t e c o n s i s t i n g o f r i g i d b o d i e s ( F i g u r e 1-2-b). F o r an o r b i t o f low e c c e n t r i c i t y t h e e q u a t i o n s o f motion were l i n e a r i z e d by assuming s m a l l a m p l i t u d e l i b r a t i o n s o f t h e c o n s t i t u e n t b o d i e s . The r o o t s o f the r e s u l t i n g c h a r a c t e r i s t i c e q u a t i o n were e v a l u a t e d f o r a wide range o f c o n f i g u r a t i o n s . T h i s showed t h a t t h e motion c o u l d be h i g h l y damped. 19 F l e t c h e r , Rongved and Yu 7 f o r m u l a t e d t h e e q u a t i o n s o f m o t i o n f o r a two-body communications s a t e l l i t e w h i c h was 20 proposed o r i g i n a l l y by P a u l , West and Yu ( F i g u r e 1-2-c). A c o n s i d e r a b l e amount o f d e t a i l e d s i m u l a t i o n showed t h a t t h e performance o f such a d e v i c e would be s a t i s f a c t o r y . 15 21 Hartbaum et a l ' as w e l l as Hughes have a t t e m p t e d t o o p t i m i z e t h e c o n f i g u r a t i o n o f a r t i c u l a t e d s a t e l l i t e s ( F i g u r e s 1-2-d and 1-2-b r e s p e c t i v e l y ) w i t h r e s p e c t t o p o i n t i n g a c c u r a c y . S e v e r a l c o n f i g u r a t i o n s have been proposed which a r e s i m p l e r t h a n t h e m u l t i - b o d y s a t e l l i t e d i s c u s s e d above. I n 22 1963, P a u l i n v e s t i g a t e d a s a t e l l i t e i n which a mass was suspended from a " l o s s y " s p r i n g ( F i g u r e 1-2-e). The d e v i c e damped o s c i l l a t i o n s o n l y about axes p e r p e n d i c u l a r t o t h a t o f t h e s p r i n g . T h i s d i f f i c u l t y was e l i m i n a t e d i n a st u d y 23 by B u x t o n , Campbell and Losch where t h e s p r i n g was a l s o p e r m i t t e d t o execute t o r s i o n a l o s c i l l a t i o n s ( F i g u r e 1 - 2 - f ) . Systems o f t h i s t y p e a r e c h a r a c t e r i z e d by a m p l i t u d e dependent damping. S a t e l l i t e s d e s i g n e d f o r g r a v i t y - g r a d i e n t s t a b i l i z a t i o n a r e n e c e s s a r i l y v e r y l o n g . R e c e n t l y much a t t e n t i o n has been f o c u s e d on t h e e f f e c t s o f t h e r e s u l t i n g f l e x i b i l i t y . K a t u c k i and M o y e r 2 ^ have c o n s i d e r e d t h i s t o be a major f a c t o r a f f e c t -i n g t h e l i b r a t i o n a l dynamics as s o l a r h e a t i n g can produce 2 5 l a r g e changes i n t h e c o n f i g u r a t i o n . A s h l e y i n v e s t i g a t e d a n a l y t i c a l l y t h e s t r u c t u r a l dynamics o f s e v e r a l f l e x i b l e b o d i e s when e x c i t e d by t h e g r a v i t a t i o n a l g r a d i e n t f i e l d . Dow et a l have p r e s e n t e d t h e r e s u l t s o f an e x t r e m e l y e l a b o r a t e s i m u l a t i o n o f f l e x i b i l i t y e f f e c t s . The s i m u l a t i o n s t u d i e s o f t h i s n a t u r e have been s u c c e s s f u l i n p r e d i c t i n g t h e performance o f e x i s t i n g g r a v i t y -g r a d i e n t s a t e l l i t e s . The G e o d e t i c E a r t h O r b i t i n g S a t e l l i t e , GEOS-A, 5 and t h e G r a v i t y G r a d i e n t Test S a t e l l i t e , GGTS, 2 7 a r e p e r f o r m i n g as e x p e c t e d . © Hinge equipped with damper F i g u r e 1-2 Models o f m u l t i - b o d y s a t e l l i t e s 12 1.3 Purpose and Scope of I n v e s t i g a t i o n The main purpose of t h i s i n v e s t i g a t i o n i s t o o b t a i n t h e l i m i t i n g i n i t i a l c o n d i t i o n s f o r a g r a v i t y - g r a d i e n t s t a b i l i z e d s a t e l l i t e as a f u n c t i o n o f d e s i g n p arameters. The secondary purpose i s t o i n v e s t i g a t e t h e n a t u r e o f t h e m o t i o n and t o e s t a b l i s h p r o c e d u r e s based on t h i s knowledge wh i c h w i l l speed and s i m p l i f y t h e a n a l y s i s . To t h e s e ends, s e v e r a l models a r e s t u d i e d . I n each case o n l y t h o s e f o r c e s s p e c i f i c a l l y mentioned a r e i n c l u d e d i n t h e a n a l y s i s . The f i r s t model i s t h a t o f a r i g i d s a t e l l i t e e x e c u t -i n g l i b - r a t i o n s i n t h e p l a n e o f t h e o r b i t . The i n v e s t i g a t i o n assumes a n o n - d i s s i p a t i v e c o n f i g u r a t i o n ( F i g u r e 2-1). Model number two i n c l u d e s d i s s i p a t i o n by t h e a d d i t i o n 22 o f a damper o f the form proposed by P a u l . P l a n a r motion i s e s s e n t i a l f o r t h e p r o p e r f u n c t i o n i n g o f t h e model ( F i g u r e 3-1). The t h i r d model assumes t h e s a t e l l i t e t o be non-d i s s i p a t i v e but s u b j e c t t o c o n s i d e r a b l e d i s t o r t i o n due t o s o l a r h e a t i n g . The e f f e c t s o f v a r y i n g t h e s a t e l l i t e ' s p h y s i c a l p r o p e r t i e s a r e c o n s i d e r e d ( F i g u r e 4-1). The l a s t model i n v e s t i g a t e d i s an a x i - s y m m e t r i c r i g i d s a t e l l i t e ( F i g u r e 5-1). The r e s t r i c t i o n t o p l a n a r m otion i s removed a l t h o u g h t h e o r b i t i s assumed t o be c i r c u l a r . The f i r s t model i s the s i m p l e s t . I t was s p e c i f i c a l l y chosen t o p r o v i d e a b a s i c u n d e r s t a n d i n g o f t h e n a t u r e o f t h e m otion. I n g e n e r a l t h e l i b r a t i o n a l motion i s t w o - d i m e n s i o n a l and model f o u r s e r v e s as an a p p r o p r i a t e e x t e n s i o n . The i n f l u e n c e o f i n t e r n a l damping, w h i c h i s always p r e s e n t , i s i d e a l i z e d i n model two. The importance o f t h e r m a l d i s t o r -2/i- 26 t i o n has been p o i n t e d out by s e v e r a l i n v e s t i g a t o r s . The t h i r d model p r o v i d e s a c o n v e n i e n t way o f s t u d y i n g t h e s e e f f e c t s . 2. PLANAR LIBRATIONAL MOTION OF A RIGID SATELLITE 2.1 F o r m u l a t i o n o f t h e Problem The p l a n a r motion o f a r i g i d s a t e l l i t e i n a c i r c u l a r o o r b i t has been s o l v e d by Klemperer i n terms o f t h e e l l i p t i c s i n e f u n c t i o n . I n an e l l i p t i c o r b i t , t h e v a r i a t i o n s i n t h e o r b i t a l a n g u l a r v e l o c i t y and t h e l o c a l g r a v i t a t i o n a l g r a d i e n t p r o v i d e t h e s a t e l l i t e w i t h a mechanism f o r exchanging energy between th e l i b r a t i o n a l and o r b i t a l degrees o f freedom. I n g e n e r a l , t h i s l e a d s t o a r e d u c t i o n i n t h e range of i n i t i a l c o n d i t i o n s t h a t r e s u l t i n s t a b l e l i b r a t i o n a l motion as com-par e d t o t h e c o r r e s p o n d i n g range f o r a c i r c u l a r o r b i t . T h i s c h a p t e r i n v e s t i g a t e s t h e bounds t h a t must be p l a c e d on t h e i n i t i a l c o n d i t i o n s as f u n c t i o n s o f o r b i t e c c e n t r i c i t y and s a t e l l i t e geometry t o g u a r a n t e e s t a b l e m o t i o n . C o n s i d e r a r i g i d s a t e l l i t e o f a r b i t r a r y shape w i t h c e n t r e o f mass a t S e x e c u t i n g p l a n a r l i b r a t i o n a l motion w h i l e moving i n an e l l i p t i c o r b i t about t h e c e n t r e o f f o r c e 0 ( F i g u r e 2-1). The mass d i s t r i b u t i o n o f t h e c e n t r a l body i s assumed s p h e r i c a l so t h a t the o r b i t d e f i n e s a p l a n e . The p o s i t i o n o f t h e s a t e l l i t e i s ' g i v e n by t h e o r b i t a n g l e , 0, measured from t h e p e r i c e n t r e , P, i n the d i r e c t i o n o f t h e o r b i t a l m o t i o n . L e t xyz be a s e t o f o r t h o g o n a l body c o - o r d i n a t e s w i t h t h e y - a x i s normal t o t h e p l a n e o f the o r b i t . The a n g l e F i g u r e 2-1 Geometry of p l a n a r m otion o f a r i g i d s a t e l l i t e 16 between t h e l o c a l v e r t i c a l , OS, and t h e z - a x i s i n t h e sense o f t h e o r b i t a l motion d e f i n e s t h e l i b r a t i o n a n g l e , . F o r an element o f mass, dm^, o f t h e s a t e l l i t e t h e e x p r e s s i o n s f o r t h e 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 can be w r i t t e n as JJ = J*> |[re Cos ijj - rSin </> + (e * f)i\ = |V + rZQz + fd + y)V^) (2.1) - Sin - xrC*5i}>) and {dm, ~ju dmh r (2.2) I f t h e o r i g i n o f t h e xyz a x e s , S, i s a t t h e c e n t r e o f mass, J dri^ ^ ( 2 . 3 ) f*J»b = fyH = = °. {2'k) 17 Moreover, i f t h e axes a r e chosen t o be t h e p r i n c i p a l axes 2 (iyy 4 Izz ~~ Iyx ) = JL(1 +1 - J ) = 0. W i t h t h e s e r e l a t i o n s t h e e x p r e s s i o n s f o r t h e 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 become T U s i n g t h e L a g r a n g i a n f o r m u l a t i o n t h e e q u a t i o n s o f motion c o r r e s p o n d i n g t o t h e t h r e e degrees o f freedom can be w r i t t e n as rza \ rLoL (Q + a)).. - c o n s t a n t •- -ft 3M- 5 i n (p Cos <p = O (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) These e q u a t i o n s a r e e s s e n t i a l l y t h o s e o f Yu.' The terms i n v o l v i n g moments of i n e r t i a i n e q u a t i o n s (2.8) and (2.9) r e p r e s e n t p e r t u r b a t i o n s o f t h e u s u a l two-body e q u a t i o n s o f m o t i o n . T h e i r presence can be a t t r i b u t e d t o t h e f i n i t e d i m e n s i o n s of the s a t e l l i t e . Moran and Yu found a p p r o x i -mate s o l u t i o n s t o (2.10) and showed t h a t f o r t y p i c a l s a t e l l i t e s t h e c o n t r i b u t i o n o f t h e p e r t u r b a t i o n terms i s e x t r e m e l y s m a l l . N e g l e c t i n g the p e r t u r b a t i o n s i n t h e o r b i t due t o t h e l i b r a t i o n a l m o tion of t h e s a t e l l i t e , t h e s o l u t i o n o f e q u a t i o n s (2.8) and (2.9) l e a d s t o the c l a s s i c a l K e p l e r i a n r e l a t i o n s r 2 9 = * e (2 . i l ) r fx (i + e Cos 9} ' N o t i n g t h a t cit 4 J (2."12) ' rz 49 and = 9 (2.13) r4 Je* e q u a t i o n (2.10) can be r e w r i t t e n as 19 T h i s form o f t h e e q u a t i o n o f motion was p r e s e n t e d i n d e p e n -12 d e n t l y by Z l a t o u s o v et a l . I n g e n e r a l , the g o v e r n i n g n o n - l i n e a r , non-autonomous d i f f e r e n t i a l e q u a t i o n w i t h p e r i o d i c c o e f f i c i e n t s does not admit o f any c l o s e d form s o l u t i o n . The n o n - l i n e a r i t y i s s i m i l a r t o t h a t o f a " s o f t " s p r i n g t h u s r a i s i n g t h e p o s s i b i l -i t y o f a m p l i t u d e dependent i n s t a b i l i t y . 2.2 S i m p l e E x a c t S o l u t i o n s 2.2.1 C i r c u l a r O r b i t (e = 0) When t h e o r b i t o f t h e s a t e l l i t e i s c i r c u l a r , equa-t i o n (2.14) assumes the autonomous form w h i c h has t h e f i r s t i n t e g r a l V* + 3Ki Jiff = constant = Cc . ( 2 - i 6 ) E q u a t i o n (2.16) d e f i n e s r e g i o n s o f s t a b i l i t y i n t h e ijj-phase-plane. For v a l u e s o f t h e c o n s t a n t l e s s t h a n 3K^ t h e t r a j e c t o r i e s a r e c l o s e d and t h e r e s u l t i n g motion i s p e r i o d i c . F i g u r e 2-2 i l l u s t r a t e s t h e e f f e c t o f v a r y i n g t h e c o n s t a n t of i n t e g r a t i o n . D i f f e r e n t v a l u e s o f r e s u l t i n d i f f e r e n t t r a j e c -20 21 t o r i e s w h i c h f o r C c = 3K^ a r e shown i n F i g u r e 2-3. I t i s n o t e w o r t h y t h a t t h e s e c u r v e s a r e i n v a r i a n t w i t h r e s p e c t t o G , hence the r e g i o n s o f s t a b i l i t y may be c o n s i d e r e d t o be c y l i n d e r s w i t h t h e c r o s s - s e c t i o n s d e p i c t e d h e r e . For p e r i o d i c m otion ( C c < 3 ^ ) e q u a t i o n (2.15) y i e l d s a s o l u t i o n i n terms of t h e e l l i p t i c s i n e f u n c t i o n , where = 0 and i/i = Jc^ = / 3 I 7 S i n ^ m & x a t 9 = 6 0 . The change i n t h e o r b i t a l a n g l e d u r i n g one complete c y c l e o f t h e l i b r a t i o n a l motion i s g i v e n by Ae = j= K(Si" rWx). (2-ia) Of p a r t i c u l a r i n t e r e s t a r e s o l u t i o n s where A 0 = 2 TT n/m i n d i c a t i n g m o s c i l l a t i o n s i n n o r b i t s . The i n i t i a l c o n d i t i o n s r e q u i r e d t o g e n e r a t e t h e s e s o l u t i o n s may be t a k e n t o be i>hl(o) = o ^(o) = 73K; Sir. (p^ . (2.19) The v a r i a t i o n o f d) (0) r e q u i r e d t o produce p e r i o d i c ' P;» s o l u t i o n s w i t h s p e c i f i e d v a l u e s o f m and n i s p l o t t e d i n F i g u r e 2-4 as a f u n c t i o n o f K^. -2 -90 -60 -30 0 30 60 90 lp, Degrees F i g u r e 2-3 L i m i t i n g phase p l a n e t r a j e c t o r i e s f o r e = 0 2 K| F i g u r e 2 -4 I n i t i a l a n g u l a r v e l o c i t i e s r e q u i r e d t o produce s p e c i f i e d p e r i o d i c s o l u t i o n s (e = 0) 24 2.2.2 P e r i o d i c S o l u t i o n s U s i n g t h e Method o f Harmonic Balance The n o n - l i n e a r term i n t h e g o v e r n i n g e q u a t i o n of motion (2.14) may be r e p r e s e n t e d by t h e T a y l o r ' s s e r i e s Sin f Cos 0> = Sin zy Consider' now a s o l u t i o n o f e q u a t i o n (2.14) of t h e form 00 V = X - A m n S i " ^ > (n = l > 2 ' 5 - ) w h i c h has a p e r i o d o f 2Tfn. The f i r s t two d e r i v a t i v e s may be w r i t t e n as (2.20) (2.21) 1 no=/ ; (2.22) 3 and t h e e x p a n s i o n f o r l e a d s t o an e x p r e s s i o n o f t h e form CO 00 oo V £ , & w " 25 mr-i vfi m-t f 5 K^,-%)6 + S i n fr»t4Pb-^)e n n J . •(2.23) 5 7 S i m i l a r e x p r e s s i o n s can be o b t a i n e d f o r , tyt> , ..... so t h a t t h e T a y l o r ' s s e r i e s e x p a n s i o n f o r Sin2^V i n ( 2 . 2 0 ) i n t r o d u c e s o n l y s i n e terms. R e c o g n i z i n g t h a t M = ' (2.24) <x> z e q u a t i o n ( 2 . 1 4 ) becomes 26 m 9 - 0 . ( 2 . 2 5 ) The p r i n c i p l e o f harmonic b a l a n c e r e q u i r e s t h e co-e f f i c i e n t o f each t r i g o n o m e t r i c term t o be i n d i v i d u a l l y z e r o t h u s p r o v i d i n g a s u f f i c i e n t number o f e q u a t i o n s t o s o l v e f o r f o r a s i g n i f i c a n t number o f terms i n t h e assumed s o l u t i o n , are d i f f i c u l t t o s o l v e . However, t h e i m p o r t a n t c o n c l u s i o n can be drawn, t h a t t h e assumed form o f t h e s o l u t i o n i s c o r r e c t . P e r i o d i c s o l u t i o n s o f (2.14) a r e odd f u n c t i o n s o f © . The n o n - l i n e a r i t y o f t h e e q u a t i o n s a l s o i n d i c a t e s t h a t t h e r e may be more t h a n one s o l u t i o n f o r s p e c i f i c v a l u e s o f the parameters. There a r e o t h e r f a m i l i e s o f p e r i o d i c s o l u t i o n s which a r e c l o s e l y r e l a t e d t o t h o s e a l r e a d y i n v e s t i g a t e d . L e t t h e A n ^ ( i = 1,2,....). The e q u a t i o n s a r e n o n - l i n e a r and, e Q - TC (2.26) t h e n S i n 6 — 5in Q* ( 2 . 2 7 ) ' Cos e - C o s . a* 27 J<9 de* Jz _ j o * ~" do**-and e q u a t i o n (2.14) becomes +• 3Ki 5in(]JCosy « 0-E q u a t i o n (2.»28) has e s s e n t i a l l y t h e same form as (2.14) except f o r t h e s i g n o f e, hence t h e same form o f s o l u t i o n i s v a l i d and p e r i o d i c s o l u t i o n s o f t h e form (2.29) e x i s t . These s o l u t i o n s appear as odd f u n c t i o n s about t h e p o i n t 6 = TT . I n some c a s e s , t h e s o l u t i o n s t h u s d e t e r m i n e d ar e i d e n t i c a l t o t h o s e which a r e odd about the p o i n t 0 = 0 , G t ^ o • n "_ 1 o The f a c t t h a t t h e e c c e n t r i c i t y i s n e g a t i v e i m p l i e s t h a t t h e a p o c e n t r e c o r r e s p o n d s t o 0 = 0 . I f t h e s o l u t i o n i s such t h a t a t p e r i c e n t r e ifj ^ 0, the s o l u t i o n i s d i f f e r e n t from t h a t o b t a i n e d e a r l i e r . There a l s o e x i s t s a t h i r d f a m i l y o f p e r i o d i c s o l u t i o n s . L e t (2.27) c o n t ' d (2.28) 28 (2.30) t h e n d © 4 d©* (2.3D = - S i n (|^ Cos ^ so t h a t (2.14) becomes ( if eC«e)«F # - Z e 5 m © ( ^ + i ) - J K i ^ d ; # a $ ^ = o. (2.32) T h i s i s i d e n t i c a l t o e q u a t i o n (2.14) w i t h K., r e p l a c e d by In p h y s i c a l t e r m s , t h e motion r e p r e s e n t s an o s c i l l a t i o n about t h e l o c a l h o r i z o n t a l . I t i s i n t e r e s t i n g t o note t h a t 12 Z l a t o u s o v et a l o b t a i n e d p e r i o d i c s o l u t i o n s o f t h i s t y p e n u m e r i c a l l y f o r n = 1 w i t h o u t e s t a b l i s h i n g t h e g e n e r a l form of t h e s o l u t i o n p r e s e n t e d h e r e . -K^. The e q u a t i o n may be s o l v e d u s i n g t h e t e c h n i q u e s d i s cussed above and y i e l d s the s i m i l a r s o l u t i o n (2.33) rw = i P e r i o d i c s o l u t i o n s d e t e r m i n e d w i t h n e g a t i v e v a l u e s o f e and t h u s c o r r e s p o n d t o r e a l i z a b l e s i t u a t i o n s when t h e parameters a r e r e s t r i c t e d t o t h e i n t e r v a l between z e r o and one. L a r g e r v a l u e s o f t h e s e parameters have no p h y s i c a l meaning as f o r e > 1 t h e o r b i t a l motion i s not p e r i o d i c and > 1 i s p h y s i c a l l y i m p o s s i b l e . 2.2.3 N u m e r i c a l D e t e r m i n a t i o n o f P e r i o d i c S o l u t i o n s The p r e c e d i n g s e c t i o n h a s . i n d i c a t e d a method o f d e t e r m i n i n g t h e p e r i o d i c s o l u t i o n s o f (2.14). The a c t u a l s o l u t i o n o f t h e r e s u l t i n g e q u a t i o n s i s q u i t e i n v o l v e d as t h e number o f terms r e q u i r e d f o r an a c c u r a t e s o l u t i o n i s a s t r o -n o m i c a l . F o r t u n a t e l y , the knowledge o f t h e form h e l p s c o n s i d e r a b l y i n t h e n u m e r i c a l e v a l u a t i o n o f t h e p e r i o d i c s o l u t i o n s . The n u m e r i c a l d e t e r m i n a t i o n o f the p e r i o d i c s o l u t i o n s was a c c o m p l i s h e d as f o l l o w s . A d i g i t a l computer was p r o -grammed t o s o l v e e q u a t i o n (2,14) u s i n g a n u m e r i c a l a l g o r i t h m . I n i t i a l c o n d i t i o n s were chosen c o n s i s t e n t w i t h t h e known form and e q u a t i o n (2.14) was i n t e g r a t e d u n t i l © = 27Tn. The f i n a l v a l u e s o f ^ and <^ r which were, i n g e n e r a l , d i f f e r e n t from (2„34) were n o t e d . A c o r r e c t i o n was t h e n made t o t h e v a l u e o f y/Q so as t o cause t h e f i n a l c o n d i t i o n t o become i d e n t i c a l w i t h t h e i n i t i a l c o n d i t i o n . When a p e r i o d i c s o l u t i o n e x i s t s , t h e p r o c e s s converges t o g i v e t h e r e q u i r e d v a l u e o f ijJ0 and the s o l u t i o n o f t h e i n t e r v a l . T y p i c a l p e r i o d i c s o l u t i o n s w i t h o r b i t a l f r e q u e n c y ( i . e . n = 1) a r e p r e s e n t e d i n F i g u r e s 2-5 and 2-6. The i i n i t i a l d e r i v a t i v e s , ^ ; ( 0 ), r e q u i r e d t o produce s o l u t i o n s o f t h i s t y p e a r e p l o t t e d i n F i g u r e 2-7. A s i m i l a r diagram was p r e s e n t e d i n d e p e n d e n t l y i n r e f e r e n c e 12. The diagram i n d i c a t e s t h a t f o r l e s s t h a n 1/3 t h e r e i s o n l y one p e r i o d i c s o l u t i o n w h i l e f o r l a r g e r v a l u e s o f t h e r e may be as many as t h r e e . T h i s r e s u l t i s i n a c c o r d w i t h t h e cu r v e s p r e s e n t e d i n F i g u r e 2-4. The n u m e r i c a l t e c h n i q u e can produce p e r i o d i c s o l u t i o n s o f any d e s i r e d a c c u r a c y . A r e l i a b l e e s t i m a t e o f the e r r o r may be made by computing t h e f u n c t i o n over t h e i n t e r v a l 0 ^ 0 £n7t. The exact s o l u t i o n i s odd w i t h r e s p e c t t o t h e p o i n t 8 = njt so t h a t €3 s h o u l d be i d e n t i -c a l l y z e r o . S e v e r a l t y p i c a l s i t u a t i o n s a r e shown i n F i g u r e 2-8. The maximum observed v a l u e of 6? , which o c c u r r e d a t h i g h e c c e n t r i c i t y , was .003 r a d i a n s and may be a t t r i b u t e d t o t h e n e a r l y d i s c o n t i n u o u s b e h a v i o u r o f t h e s o l u t i o n . I t i s a l s o p o s s i b l e t o s e a r c h f o r s o l u t i o n s o f l o n g e r 90 180 e, Degrees 270 360 F i g u r e 2-5 P e r i o d i c s o l u t i o n s as f u n c t i o n s o f e c c e n t r i c i t y (K^ =1, n = 1) 33 _2 I i i i i l i i i i i -1 0 1 Orbit Eccentricity F i g u r e 2 - 7 I n i t i a l d e r i v a t i v e r e q u i r e d t o produce s o l u t i o n s w i t h p e r i o d o f 231 34 Figure 2 - 8 Typical variations of the error found i n the numerical determination of periodic solutions 35 p e r i o d (n > 1 ) . The n u m e r i c a l t e c h n i q u e i s q u i t e v e r s a t i l e i n t h i s r e s p e c t . F i g u r e 2-9 i l l u s t r a t e s p e r i o d i c s o l u t i o n s w i t h a p e r i o d o f 4 R . Note t h e d e g e n e r a t i o n o f a s o l u t i o n w h i c h o s c i l l a t e s t h r e e t i m e s a t e = 0 i n t o one which o s c i l -l a t e s o n l y t w i c e a t e « 0 .35° I n i t i a l v a l u e s o f t h e d e r i v a t i v e s , y/' ^  2 ^ ' ^ o r t^ a e s o l u t l o n s a r e p r e s e n t e d i n F i g u r e 2 - 1 0 . V e r t i c a l t a n g e n t s t o t h e s e c u r v e s c o r r e s p o n d t o t h e p o i n t s where t h e s o l u t i o n s become i d e n t i c a l t o t h o s e p e r i o d i c over 2 J t . 2 . 3 Approximate S o l u t i o n s 2 . 3 . 1 WKBJ M e t h o d 2 3 F o r s m a l l a m p l i t u d e m o t i o n , e q u a t i o n ( 2 . 1 4 ) may be l i n e a r i z e d by i n t r o d u c i n g t h e a p p r o x i m a t i o n s S i n i f f ( 2 . 3 6 ) C o s f « I . The e q u a t i o n o f motion r e d u c e s t o ( i + e C o $ e ) / - ^ e S i n e ^ ' + J K t f = zts<*e ( 2 . 3 7 ) w h i c h may be t r a n s f o r m e d by means o f t h e t r a n s f o r m a t i o n tj) « ( | -r- € C o s S ) <|> ( 2 . 3 8 ) or 37 -1 1 1 1 / ° ' / 1 1 1 1 . I 1 n = 2 1 1 i i -0.4 -0.2 0 0.2 0.4 Eccentricity F i g u r e 2-10 I n i t i a l d e r i v a t i v e r e q u i r e d t o produce p e r i o d i c s o l u t i o n s w i t h p e r i o d o f 4^1 38 t o ( 2 . 3 9 ) where 3K? e Cog 9 I + e Cos e ( 2 . 4 0 ) The complementary s o l u t i o n t o (2.39) can be o b t a i n e d a p p r o x i m a t e l y u s i n g t h e WKBJ method p r o v i d e d t h e f u n c t i o n G (0) s a t i s f i e s t h e i n e q u a l i t y F = Mil « I ( 2 . 4 1 ) F i g u r e 2-11 shows t h e v a r i a t i o n o f F w i t h 6 f o r s e v e r a l v a l u e s o f e and K^ = 1. I t i s e v i d e n t t h a t , even f o r l a r g e v a l u e s o f t h e e c c e n t r i c i t y , t h e i n e q u a l i t y i n ( 2 . 4 1 ) i s r e a s o n a b l y w e l l s a t i s f i e d . The approximate s o l u t i o n t o ( 2 . 3 9 ) i s t h e n g i v e n by ( 2 . 4 2 ) where ( 2 . 4 3 ) F i g u r e 2-11 The v a r i a t i o n o f F w i t h o r b i t a n g l e and o r b i t e c c e n t r i c i t y (K. = 1) 40 and r e p r e s e n t s t h e p a r t i c u l a r i n t e g r a l o b t a i n e d u s i n g t h e method o f v a r i a t i o n o f parameters j * _ T f e Z e Sin 6 $ J@ " f t J o * * ' ~ ' I n g e n e r a l , t h e e v a l u a t i o n o f the WKBJ s o l u t i o n can be a c h i e v e d o n l y n u m e r i c a l l y and i n v o l v e s a l a r g e amount o f computation- C o n s i d e r a b l e s i m p l i f i c a t i o n i s p o s s i b l e w i t h -out s u b s t a n t i a l l y a f f e c t i n g t h e a c c u r a c y o f t h e WKBJ a p p r o x i -m a t i o n by a d o p t i n g t h e f o l l o w i n g p r o c e d u r e . N e g l e c t i n g second and h i g h e r degree terms i n e and p u t t i n g G O * = 3KF 3 K i -(2.45) V t h e r e q u i r e d f u n c t i o n s can be ap p r o x i m a t e d as Gj & 6oL (i - ye Cos e) (2.46) 41 C o s (2.46) cont'd Cos £ J L 0 Thus, within a m u l t i p l i c a t i v e constant, the solutions (2.43) become T^ 7 « Cos cJuQ +ev>(i ^ ) c o 5 ^ L + > ) 9 A * Sin + e v ( i " 5in(«JL+i)© (2.47) - 4 l ) ^ ( ^ ' ) e . (2.48) The f i r s t derivative i s missing from equation (2.39) so that the denominators i n (2.44) are equal to a constant which i s (2.49) The p a r t i c u l a r i n t e g r a l already involves the f i r s t power of 42 the e c c e n t r i c i t y so tha t 2e Cos 6 J L 0 J^ 5,» 9 Sin ^ 0 Je^±0(gf) r 9 T- /CJ . Sin 8 - Sm 0 ) •+ 0(ez); (uL± i ) m Us ing the t r a n s f o r m a t i o n (2.38) the s o l u t i o n f o r (j) i s (2.50) (2.5D where ^ = Cos CJl9 4 e ( - | - - ^ "2 )Co3(H+i)e + e(£ + ^ - i ) c o s ^ , - ' ) e - ' 0 ( e i ) 5)» fa-i)6 + 0(e*-) (2.52) F i g u r e s 2-12 and 2-13 compare the WKBJ s o l u t i o n 43 ( 2 . 4 2 - 2 . 4 4 ) and i t s a p p r o x i m a t i o n ( 2 . 5 1 , 2 . 5 2 ) w i t h t h e ex a c t n u m e r i c a l s o l u t i o n o f e q u a t i o n ( 2 . 1 4 )» I t i s apparent t h a t t h e s i m p l i f i c a t i o n t o t h e WKBJ method does not i n c r e a s e t h e e r r o r a l r e a d y p r e s e n t . The a p p r o x i m a t i o n t o t h e WKBJ s o l u t i o n f o l l o w s t h e exact s o l u t i o n i n a g e n e r a l f a s h i o n f o r a l o n g p e r i o d o f t i m e . T h i s i s i l l u s t r a t e d i n F i g u r e 2 - 1 4 , where t h e l i b r a -t i o n a l m o tion i s shown over e i g h t r e v o l u t i o n s , and d i s c u s s e d on page 4 7 . The WKBJ s o l u t i o n i s not p l o t t e d because i t i s n e a r l y c o i n c i d e n t w i t h t h e approximate r e s u l t . The a n a l y s i s does p r o v i d e u s e f u l i n f o r m a t i o n about t h e maximum v a l u e o f t h e a m p l i t u d e o f l i b r a t i o n . From ( 2 . 5 1 ) , w h i c h has a minimum when c, c ze (2*54) p r o v i d e d CJ^ > (e/fa - 3 )^) • Hence t h e a m p l i t u d e o f l i b r a t i o n i s always g r e a t e r t h a n £ c / < j k J j ? ~ ' ) + 0(gZ)° The method p r e -d i c t s t h e " p e r i o d " o f l i b r a t i o n w i t h c o n s i d e r a b l e a c c u r a c y . \ \ / / F r / t // Kj = l, e = o.l E x a c t WKBJ A p p r o x i m a t i o n to WKBJ 9 0 ° 180° 2 7 0 ° 3 6 0 ° 4 5 0 ° 5 4 0 0 F i g u r e 2-12 Comparison o f t h e e x a c t S o l u t i o n o f t h e e q u a t i o n o f motio n w i t h t h a t d e t e r m i n e d by t h e WKBJ method and t h e a p p r o x i m a t e ¥KBJ method (K^ = 1, e. = 0.1) 30" 20* 10° -10' -20° -30° -40° / \\ / - - M // It / W\ >\ \ ' \ X \ \ \ \ / 1, i / / ' -/ / f ^  —V~\ '( / // / 1/ V-V V / / / 1 1 / ' / v \ \ ' \ \ ' v> // ii Kj = l, e = 0.3 Exact WKBJ Approximation to WKBJ 90* F i g u r e 2-13 180° 270° e 360 ° 450 ° 540* Comparison o f t h e e x a c t s o l u t i o n o f t h e e q u a t i o n o f motio n w i t h t h a t d e t e r m i n e d by the WKBJ method and the a p p roximate WKBJ method (K. = 1 , e = 0.3) VJ1 f l fl f ' 1 1 in ' > 1 1 1 1 /'ll / Mi f l |i f I ' A , 1 Ai n— Ii 1 ll ii 1 1 f 'x Ml li \ i i i ii 1 f l , I' ll // i 1 1 |l 1 (I* / \ / i / V fi \ i \ II J / 11 ' 1 / 1 l /' 1 /' \ 1 L IV / i V r>/ > I j i T V 1/ / V Exact Kj = l, e - 0. li 3 Approximation to WKBJ * ) s ° *'(0) = 0 F i g u r e 2=14 2 3 4 5 6 7 8 Orbit Comparison o f t h e ex a c t s o l u t i o n o f t h e e q u a t i o n o f motion w i t h t h a t d e t e r m i n e d by t h e approximate WKBJ method o v e r e i g h t o r b i t s ( K i = 1, e = 0.3) 47 For t h e p a r t i c u l a r case o f e = 0 and K^ = 1, i t p r o v i d e s t h e w e l l known r e s u l t t h a t , f o r s m a l l a m p l i t u d e s , t h e l i b r a t i o n a l f r e q u e n c y i s y/3 t i m e s t h e o r b i t a l f r e q u e n c y . F u r t h e r , t h e presence o f a p a r t i c u l a r s o l u t i o n i n d i c a t e s t h a t a body i n an e l l i p t i c o r b i t always e x e c u t e s l i b r a t i o n a l m o t i o n . I t may be p o i n t e d out t h a t t h e f r e q u e n c y spectrum a s s o c i a t e d w i t h t h e l i b r a t i o n a l motion c o n s i s t s o f t h e o r b i t a l f r e q u e n c y , t h e f r e q u e n c y o f l i b r a t i o n i n a c i r c u l a r o r b i t , and t h e m o d u l a t i o n p r o d u c t s o f t h e two f o r e g o i n g f r e q u e n c i e s . The r e s u l t i n g m otion a c q u i r e s an a p p a r e n t l y random c h a r a c t e r due t o t h e s u p e r p o s i t i o n o f t h e v a r i o u s f r e q u e n c i e s . T h i s a l s o e x p l a i n s t h e u n u s u a l i r r e g u l a r i t i e s w h ich may be no t e d i n F i g u r e 2-14. The f r e q u e n c y o f l i b r a -t i o n i s a l s o dependent on t h e a m p l i t u d e o f t h e motio n . T h i s i n t r o d u c e s a phase s h i f t between t h e ex a c t and approximate s o l u t i o n s . The major s o u r c e o f e r r o r i s t h e n o n - l i n e a r i t y o f e q u a t i o n ( 2 . 1 4 ) . The a m p l i t u d e s o f d e f i n i t e l y i n d i c a t e t h a t t h e system i s o p e r a t i n g i n t h e n o n - l i n e a r r e g i o n . How-e v e r , i t appears t h a t t h e a p p r o x i m a t i o n t o t h e WKBJ method p r e s e n t e d here may prove adequate f o r p r e l i m i n a r y d e s i g n purposes. 2.3«2 P r i n c i p l e o f Harmonic B a l a n c e The method o u t l i n e d i n s e c t i o n 2.2.2 may be c o n s i d e r e d e x a c t . U n f o r t u n a t e l y t h e amount o f c o m p u t a t i o n a l work i n -v o l v e d i s p r o h i b i t i v e . 4a C o n s i d e r f o r example t h e t h r e e term s e r i e s s o l u t i o n (2.55) The S i n Cos \p term may be r e p r e s e n t e d by t h e f i r s t two terms o f t h e T a y l o r ' s s e r i e s (2.20) so t h a t i t r e t a i n s i t s i n h e r e n t n o n - l i n e a r c h a r a c t e r . S u b s t i t u t i n g e q u a t i o n (2.55) i n t o (2.14), c o l l e c t i n g terms and a p p l y i n g t h e p r i n c i p l e o f harmonic b a l a n c e r e s u l t s i n t h e t h r e e e q u a t i o n s A,,, [6 K, - £ - 3K.- (A J, + 2 Aj, + 2^,)] s 3e (Ahl + A3, .) / \ h f J 6 K * -» -3Kf (A3*, + ZA*,, + 2 A * ) ] = Se A^, + K J A ^ C J A J , - ^ (2.56) T h i s s e t o f e q u a t i o n s does not posses s any s i m p l e s o l u t i o n , hence an i t e r a t i v e p rocedure was adopted. For example, when e = 0.3 and = 1 t h r e e s o l u t i o n s were o b t a i n e d . The c o e f f i c i e n t s were found t o be : 49 A x 1 = 0 . 3 3 3 ; 0 . 9 7 5 ; - 1 . 2 8 6 A 2 1 = - 0 . 1 1 6 ; - 0 . 1 1 4 ; 0 . 0 9 7 ( 2 . 5 7 ) A 3 = 0 . 0 2 1 ; 0 . 0 6 5 ; - 0 . 0 1 0 . Even w i t h t h i s t h r e e term a p p r o x i m a t i o n , the amount of c o m p u t a t i o n a l e f f o r t i s comparable t o t h a t f o r t h e exact n u m e r i c a l s o l u t i o n ( s e c t i o n 2 . 2 . 3 ) . The procedure can be s i m p l i f i e d f u r t h e r by p u t t i n g A^ = 0, i« e« by c o n s i d e r i n g a two term s e r i e s s o l u t i o n . The e q u a t i o n s r e l a t i n g t h e co-e f f i c i e n t s t h e n reduce t o which can be r e w r i t t e n i n a more c o n v e n i e n t form as - e ( 2 . 5 8 ) ( 2 . 5 9 ) i d - - (2.60) 3K 5 2. 3Ki A J ^ I F o r s p e c i f i e d v a l u e s o f e and e q u a t i o n s (2.59) and (2.60) d e f i n e c u r v e s i n an A^ , Ag -^-plane ( F i g u r e s 2 - 1 5 - i t o 2-15-vi) where t h e p o i n t s o f i n t e r s e c t i o n g i v e t h e r e q u i r e d v a l u e s o f A-^  -^  and Ag ^° S i n c e 50 Equation (2.59) Equation (2-60) Solution F i g u r e 2 - 1 5 - i D e t e r m i n a t i o n o f t h e f i r s t two terms o f t h e s i n e s e r i e s s o l u t i o n (K. = 1„0) 51 Equation (2-59) Equation (2-60) Solution F i g u r e 2 - 1 5 - i i D e t e r m i n a t i o n o f t h e f i r s t two terms of the s i n e s e r i e s s o l u t i o n ( K i = 0.9) ^1,1 Equation (2.59) Equation (2.60) Solution F i g u r e 2 - 1 5 - i i i D e t e r m i n a t i o n o f t h e f i r s t two terms of t h e s i n e s e r i e s s o l u t i o n (K. = 0.7) 53 .4 •2 -•4 1—I—1 • ' ' i t 1 ' 1 ' 1 ' 1 1 t • 1 1 1 1 , , , 1 , 1 1 ' 1 1 1 1 , 1 1 1 , 1 1 1 , 1 1 1 1 ' ' . 1 , 1 1 -' 1 1 1 1 ' ' ' ; •' •' •' ; / / / • i i i \ • • 1 — \ i • ; : \ ; i i i j i i • — ' ' \ 1 1 ' 1 1 \ • 1 —*-Ar-l _ _ \ X ' ' 1 . » ^^ "-^  X. \ i i i \ i x . \ ^ ' i i Tv *^s***slfc7s^^ L • 1 1 1 X ^^mmm^ mmmm^ ^ ^ x ^ ^ S . 1 1' i - ^ S ^ 1 1 Kj = 0.5 L i i • i I * I • i i i • i i • I I I ! • 1 1 I 1 I 1 1 • 1 • 1 1 | 1 1 1 I 1 1 > 1 • • 1 | 1 t • 1 1 1 1 f\ " 1 • 1 1 r r 1 • 1 i I i ' i i i i '. • 1 ' . i 1 1 .1 • i i • i > i • » i » i i i • i i 1 i 1 i ' i 1 i • -3 i i - U 1 ^ — . .i — ^ A-_e -1.0 -5 0 5 1.0 Equation (2.59) Equation (2-60) Solution F i g u r e 2 - 1 5 - i v D e t e r m i n a t i o n o f t h e f i r s t two terms o f t h e s i n e s e r i e s s o l u t i o n {K. = 0„5) 54 -.41 -1.0 0 .5 Equation (2-59) Equation (2-60) Solution 1.0 F i g u r e 2-15-v D e t e r m i n a t i o n o f the f i r s t two terms o f t h e s i n e s e r i e s s o l u t i o n (K. = 0.3) , \ e X i -2 -1 Kj = 0.1 .2 e \ j -1.0 -.5 0 A n 1.0 Equation 2.59 Equation 2-60 Solution F i g u r e 2 - 1 5 - v i D e t e r m i n a t i o n o f t h e f i r s t two terms of t h e s i n e s e r i e s s o l u t i o n (K^ = 0.1) t h e i n i t i a l 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 can a l s o be e s t i m a t e d . F i g u r e 2-16 shows the v a l u e o f \l> , (0) ' P >—• o b t a i n e d i n t h i s manner. A comparison w i t h F i g u r e 2-7 i n d i c a t e s t h a t t h e g e n e r a l c h a r a c t e r i s t i c s o f t h e diagram a r e q u i t e a c c u r a t e . I n p a r t i c u l a r , t h e approximate scheme p r e d i c t s the maximum v a l u e o f the e c c e n t r i c i t y and t h e minimum v a l u e o f f o r wh i c h t h r e e p e r i o d i c s o l u t i o n s e x i s t w i t h c o n s i d e r a b l e p r e c i s i o n . F i g u r e 2-17 compares the r e s u l t s o f t h e s i m p l e two and t h r e e term a n a l y s i s w i t h t h e exact n u m e r i c a l s o l u t i o n o f t h e e q u a t i o n o f motion f o r t h e t h r e e cases l i s t e d i n e q u a t i o n (2.57). The a c c u r a c y o f t h e two term s o l u t i o n i s r a t h e r poor (maximum e r r o r ^  20%). The a d d i t i o n o f t h e t h i r d term improves t h e a c c u r a c y o n l y o f t h e s o l u t i o n o f s m a l l e s t a m p l i t u d e . To a c h i e v e g r e a t e r a c c u r a c y r e q u i r e s t h a t more terms i n t h e T a y l o r ' s e x p a n s i o n o f t h e n o n - l i n e a r term be r e t a i n e d . 2.3»3 P e r t u r b a t i o n o f P e r i o d i c S o l u t i o n s C o n s i d e r t h e p e r i o d i c s o l u t i o n (p^ n d e v e l o p e d i n 2.2.2 or 2.2.3» L e t 6 r e p r e s e n t a p e r t u r b a t i o n so t h a t t h e a c t u a l l i b r a t i o n a l a n g l e i s g i v e n by 1.5 -15 0 1 -2 .3 Eccentricity F i g u r e 2-16 V a l u e s o f t h e i n i t i a l d e r i v a t i v e r e q u i r e d t o produce s o l u t i o n s w i t h p e r i o d o f 2tt as deter m i n e d by the f i r s t two terms o f t h e s i n e s e r i e s s o l u t i o n F i g u r e 2-17 Comparison o f two and t h r e e e xact p e r i o d i c s o l u t i o n s w i t h t e r m s i n e s e r i e s s o l u t i o n s 59 Y = tyP,n + S <2.62) S u b s t i t u t i n g (2.62) i n t o (2.14) r e s u l t s i n ( l + e C o s 9 ) ^ n - Z e 5 ^ Q ( ^ n + ») + (| + e C o 5 6 ) S " - 2 e S m © S' (2.63) w h i c h f o r s m a l l $ reduces t o t h e v a r i a t i o n a l e q u a t i o n (l+eCo96)$"- 2e-5r*e S* + 3 K; Coc 2ipp^n S = 04 (2.64) T h i s i s a 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 w i t h p e r i o d i c c o e f f i c i e n t s w h i c h p o s s e s s e s two l i n e a r l y independent s o l u t i o n s , ^ ^ ( 0 ) and £ 2 ^ ' d e f i n e d by t n e i n i t i a l c o n d i t i o n s i,io) = / S,'(o) - o ( 2 . 6 5 ) Sz(o) = 0 &» » / .. The s o l u t i o n t o (2.64) s u b j e c t t o the i n i t i a l c o n d i t i o n s 60 Ho) - K (2.66) S'(o) = J o i s g i v e n by S(e) = * e f y & ) * yjx(&) . (2.67) Now, f o r 0 = 2Jt , e q u a t i o n (2.67) g i v e s (2.68) $'(2X) = + So ^(XX) and s i n c e e q u a t i o n (2.64) i s i n v a r i a n t when 0 i s r e p l a c e d by 0 + 2TL , t h e s o l u t i o n i n (2.67) can be extended over t h e i n t e r v a l 2% ^ 0 ^ 4 ^ by c o n s i d e r i n g (2.68) t o be new i n i t i a l c o n d i t i o n s and w r i t i n g t h e analogous r e l a t i o n £(&) - [Ks\(*••*•) + X Si(*K}}S,(B) , ( 2 . 6 9 ) 4 {&/(**•) + IS . T h i s p r o c e s s may be c o n t i n u e d i n d e f i n i t e l y . Thus t h e complete s o l u t i o n f o r 3 may be w r i t t e n i n terms o f the s o l u t i o n o v er the i n t e r v a l 0 ^ 0 ^ 27T. Because e q u a t i o n (2.64) remains i n v a r i a n t when S i s r e p l a c e d by - 6 and 0 by - 0 ? t h e <Sg s o l u t i o n must be an odd f u n c t i o n o f 0. i . e . , 61 ix[-6) = -Sg(6) ( 2 . 7 0 ) (2.7D The i n i t i a l c o n d i t i o n s i(o) = - Sjzn) S'(o) = ^(zx) d e f i n e a s o l u t i o n Sfe) = - Sz(2K)il(e) + Sz(zx)S2(&) ( 2 . 7 2 ) i n t h e i n t e r v a l 0 ^ 0 ^ 271 which matches t h e £ 2 s o l u t i o n i n t h e i n t e r v a l -2TT ^ 0 £ 0 . The f i n a l c o n d i t i o n s a t 0 = 27t must, t h e n be (2.73) Hence, from (2,72) ( 2 . 7 4 ) and ff^r; = -&z(vc),^(2K) + $i(m)Si(zit)- 1 (2.75) 62 1 o 6 o = S'(ZJt) (2.76) and &,far)£/vr) -j/ftjr^fcjr) = / - Wfzrt). (2.77) The l e f t hand s i d e o f (2 . 7 7 ) i s t h e Wronskian o f t h e d i f f e r e n t i a l e q u a t i o n and can be det e r m i n e d from t h e co-29 e f f i c i e n t s . 7 For e q u a t i o n (2.64) t h e r e l a t i o n i s JW_ _ Ze Sine J6 (2t78) W l -t e Cos & or and t h e r e f o r e W(o) =• K/^JT^) = /. (2.80) C o n s i d e r t h e s o l u t i o n (2.67) when 9 = 2nTC and l e t S(zKn) = (2.81) s'(zrcn) -t h e n , by (2.68) 63 (2.82) where b - (2.83) <y = 81 (m) and (2.82) r e p r e s e n t s a p a i r o f l i n e a r d i f f e r e n c e e q u a t i o n s . T a k i n g a s o l u t i o n o f t h e form •S 0 (2.84) Be V 10 and s u b s t i t u t i n g i n' ( 2 . 8 2 ) r e s u l t s i n A s J s (2.85) B, - A,c*- h + 6sdV" o r 64 A -n f s I \ ( 2 . 8 6 ) - Ag c + E>5 C « - = O . c o n t ' d F o r a n o n - t r i v i a l s o l u t i o n , i t i s r e q u i r e d t h a t - ( a 4- c l )V (aol - be) = O. ( 2 . 8 7 ) 30 I t may be p o i n t e d out t h a t t h e a n a l y s i s due t o F l o q u e t ^ y i e l d s an e q u a t i o n f o r t h e c h a r a c t e r i s t i c r o o t s t h a t , i s : i d e n t i c a l t o e q u a t i o n ( 2 . 8 7 ) w h i c h s i m p l i f i e s , u s i n g ( 2 . 7 6 ) and ( 2 . 7 7 ) , t o - 2cxY + J = O ( 2 . 8 8 ) g i v i n g YL = a + V a * - 1 ; (£ ( 2 . 8 9 ) and e q u a t i o n ( 2 . 8 9 ) when s u b s t i t u t e d i n t o ( 2 . 8 6 ) g i v e s ^ = fa-ft) Aft = + \/E_A ; . j (i-1,2). (2.90) The s o l u t i o n ( 2 . 8 4 ) may be w r i t t e n as t h e sum o f t h e two s o l u t i o n s *n = Af/ , " + A ^ " (2.9D . J&LLL_/\S jr."_ j/EZL/t, K* ( 2 , 9 2 ) 65 hence T a k i n g t h e sum and d i f f e r e n c e o f ( 2 . 9 1 ) and (2.93) g i v e s f^7 ( 2 . 9 4 ) whose p r o d u c t y i e l d s t h e r e l a t i o n K - £ r £ = 4\Ahh*J=4As,% . ( 2 . 9 5 , T h i s r e s u l t shows t h a t , i f the s o l u t i o n o f e q u a t i o n ( 2 . 6 4 ) i s i n s p e c t e d each time the independent v a r i a b l e e q u a l s 2 r c n , the v a l u e s o f t h e f u n c t i o n and i t s d e r i v a t i v e , when p l o t t e d i n t h e £,^ J - p l a n e , l i e on a c e r t a i n c u r v e . For ja|> 1 t h e curve i s a h y p e r b o l a so t h a t 6 e v e n t u a l l y becomes v e r y l a r g e . On t h e o t h e r hand i f |a| < 1 t h e cur v e i s an e l l i p s e and £ i s bounded.. I t may be co n c l u d e d t h a t i n t h e l a t t e r c a s e , s m a l l p e r t u r b a t i o n s about t h e p e r i o d i c s o l u t i o n s a r e s t a b l e and f o r 0 = 2J[ n may be found on a s i n g l e curve i n t h e ,y/ - p l a n e which s u r r o u n d s t h e p o i n t ( ^ p ( 0 ) , ^ p ( 0 ) ) . F o r v a l u e s o f 0 o t h e r t h a n z e r o , a p o i n t on the curve 66 t r a n s f o r m s i n t o (2.96) S'(e) = *&',(9) * where ^ and l i e on t h e curve d e f i n e d by ( 2 . 9 5 ) . Hence * - &{&)S*'f9) - S'MSJe) ,2.97, j = ^ f e ) - ( 2 . 9 8 ) J W(6) w h i c h when s u b s t i t u t e d i n t o ( 2 . 9 5 ) determine the shape o f th e c u r v e a t the s p e c i f i e d v a l u e o f © - zS(e)S'(o) fa)s/(e) - sJQSfapj (2,99) t h a t i s , (2.100) The n a t u r e o f t h e curve a t t h e s p e c i f i e d v a l u e o f 0 i s d e t e r m i n e d by t h e s i g n o f the parameter such t h a t f o r % > 0 t h e curve i s an e l l i p s e and f o r < a h y p e r b o l a . From e q u a t i o n s (2.99) and (2.100) t h e r e i s o b t a i n e d = ~ - 0 T - *<•) t h e r e f o r e , £) < 0 i f |a| > 1 and § ) > 0 i f | a | * 1. Thus, depending upon t h e v a l u e o f |a| t h e curve d e f i n e d by s u c c e s s i v e passages o f t h e s o l u t i o n a t f i x e d 6 i s e i t h e r an e l l i p s e o r a h y p e r b o l a . I n t h e f i r s t case a t u b u l a r s u r f a c e i s d e f i n e d . L i n e a r p e r t u r b a t i o n a n a l y s i s p r e d i c t s t h a t i n i t i a l c o n d i t i o n s which do not l e a d t o e x a c t l y p e r i o d i c motion may s t i l l p e r m i t t h e m o t i o n t o remain i n t h e neighbourhood o f the p e r i o d i c s o l u t i o n . There i s a l s o t h e p o s s i b i l i t y t h a t t h e m otion may d r i f t away from t h e g e n e r a t i n g s o l u t i o n . The c r i t e r i o n d e t e r m i n i n g t h e k i n d o f motion i s t h e magni-tude o f the &2 - t y p e o f s o l u t i o n o f the v a r i a t i o n a l e q u a t i o n a t 9 = 23T o For | c S ^ (2JT ) j > 1, t h e v a r i a t i o n i n c r e a s e s w i t h t i m e , but i f l ^ g ^ ^ J I ) ! < 1 t h e d i f f e r e n c e between t h e a c t u a l and t h e p e r i o d i c s o l u t i o n s remains bounded. I n f a c t , i n the l a t t e r c a s e , the s o l u t i o n s l i e on a s u r f a c e i n <f> , </>', 6 -space which always s u r r o u n d s t h e p e r i o d i c s o l u t i o n and has 67 (2.101) 0, an e l l i p t i c c r o s s s e c t i o n . F i g u r e 2-18 p r e s e n t s t h e r e g i o n s i n an e,K^ parameter space f o r which t h e p e r t u r b a t i o n s about t h e v a r i o u s p e r i o d i c s o l u t i o n s o f o r b i t a l f r e q u e n c y a r e s t a b l e . T h i s diagram was d e t e r m i n e d n u m e r i c a l l y and r e p r e s e n t s an e x t e n s i o n i n t h e 12 r e g i o n near e = 1 o f t h e work by Z l a t o u s o v et a l . I n g e n e r a l t h e r e a r e t h r e e p e r i o d i c s o l u t i o n s t o c o n s i d e r . One, w i t h l a r g e p o s i t i v e i n i t i a l d e r i v a t i v e , always l e a d s t o un-s t a b l e p e r t u r b a t i o n s . The o t h e r two s o l u t i o n s , h a v i n g e i t h e r a s m a l l p o s i t i v e o r l a r g e n e g a t i v e i n i t i a l d e r i v a t i v e may y i e l d s t a b l e p e r t u r b a t i o n s . F i g u r e 2-19 p r e s e n t s s i m i l a r d a t a f o r p e r i o d i c s o l u -t i o n s which complete t h r e e o s c i l l a t i o n s i n two o r b i t s . I t may be emphasized t h a t t h e r e i s o n l y one c l a s s o f s o l u t i o n i n t h i s case (n = 2 ) because s o l u t i o n s w i t h t h e p o s i t i v e s l o p e a t 6 = 0 r e q u i r e d i n F i g u r e 2.10 have a t 6 = 2TT t h e n e g a t i v e s l o p e a l s o found i n t h i s F i g u r e (see a l s o F i g u r e 2-9). When the parameter a i n e q u a t i o n (2.88) l i e s between +1 and - 1 , t h e p e r t u r b a t i o n s a r e c o n f i n e d t o a s u r f a c e w i t h e l l i p t i c a l c r o s s - s e c t i o n . I n g e n e r a l , as t i m e i n c r e a s e s t h e p o i n t o f i n t e r s e c t i o n o f t h e t r a j e c t o r y w i t h t h e 0 = 0 p l a n e moves around t h e e l l i p s e . I n f a c t , s i n c e / — — 5 - UQ> Y = a t i V 1 " A " E ( 2 . 1 0 3 ) where 6 9 F i g u r e 2-18 V a l u e s o f K. and e w h i c h l e a d t o v a r i a t i o n a l l y s t a b l e p e r i p d i c s o l u t i o n s o f p e r i o d 27T 70 F i g u r e 2-19 V a l u e s o f K. and e w h i c h l e a d t o v a r i a t i o n a l l y s t a b l e p e r i o d i c s o l u t i o n s o f p e r i o d 4TC 71 © = Cos a., ( 2 . 1 0 4 ) i t i s e v i d e n t f rom e q u a t i o n s (2.82) t h a t t h e p o i n t r e p r e -s e n t i n g t h e p e r t u r b a t i o n s o l u t i o n i n t h e \\) , </> - p l a n e r o t a t e s around t h e e l l i p s e a t t h e average r a t e o f Q r a d i a n s per o r b i t . I n p a r t i c u l a r , i f © i s a r a t i o n a l s u b - m u l t i p l e o f 27C t h e p o i n t s d e f i n e d by (2.82) appear s t a t i o n a r y . Thus th e p e r t u r b a t i o n s o l u t i o n has i t s e l f become p e r i o d i c . T h i s o c c u r r e n c e i s t h e m a n i f e s t a t i o n o f t h e c o a l e s c e n c e w i t h the o r i g i n a l p e r i o d i c s o l u t i o n o f a n o t h e r s o l u t i o n c h a r a c t e r i z e d by a p e r i o d e q u a l t o t h a t o f t h e p e r t u r b a t i o n . F o r example, r e f e r r i n g t o F i g u r e s 2 - 7 , 2 - 9 , and 2 - 1 0 i t i s seen t h a t one t y p e o f s o l u t i o n o f p e r i o d 4TT becomes i d e n t i c a l w i t h t h a t o f p e r i o d 27t a t e « 0 . 4 3 . F i g u r e s 2 - 2 0 and 2 - 2 1 i n d i c a t e t h a t f o r both p e r i o d i c s o l u t i o n s |a| = 1 a t t h i s p o i n t , hence © = 0 , 1 8 0 ° . 2 . 4 Phase Space and I n v a r i a n t S u r f a c e s The e q u a t i o n o f motion (2 . 1 4 ) may be w r i t t e n as a p a i r o f f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n s I ( 2 . 1 0 5 ) dj^' _ Z e S*A 9 (VV /) -3K,5tn<j)Cos ip d 0 / y- e Cos B T h i s p a r a l l e l s t h e u s u a l H a m i l t o n i a n f o r m u l a t i o n where t h e a 0 -2 -3 n = 1 / - ^ 1 (0 )>0 / (Small) - Hip1iO>>0/(Large) -(|/P/1lo)<0 \ • - 5 -.4 -.3 F i g u r e 2-20 -2 1 .1 2 •3 Orbit Eccentricity V a r i a t i o n " o f parameter a w i t h o r b i t e c c e n t r i c i t y ( K i = 1) f o r s o l u t i o n s o f p e r i o d 27T d i s t i n c t i o n between c o - o r d i n a t e s and momenta n e a r l y d i s a p p e a r s . The s o l u t i o n o f ( 2 . 1 0 5 ) depends on t h e s t a r t i n g v a l u e s o f both <// and ip/, as w e l l as t h e v a l u e o f t h e independent v a r i a b l e , a t t h e i n i t i a l p o i n t (8 = 0 Q ) . T h i s s u g g e s t s t h a t t h e s t a t e o f t h e system can be r e p r e s e n t e d by a p o i n t i n a phase space formed by t h e t h r e e o r t h o g o n a l c o - o r d i n a t e s ijf, <f', and 0. As shown i n s e c t i o n 2.2.1 f o r e = 0 , t h e e x i s t e n c e o f a f i r s t i n t e g r a l (2.16) i n d i c a t e s t h a t t h e s t a b l e s t a t e o f t h e system i s r e p r e s e n t e d by a p o i n t w h i c h l i e s on t h e c l o s e d c u r v e , l|/ Z +• 3 K ; = Cc £ 3Kf . (2.106) The c u r v e can be thought o f as d e f i n i n g a c y l i n d r i c a l s u r f a c e w i t h o v a l c r o s s - s e c t i o n ( F i g u r e 2 - 3 ) . The g o v e r n i n g n o n - l i n e a r , non-autonomous d i f f e r e n t i a l e q u a t i o n does not admit o f a s i m p l e s o l u t i o n f o r e ^ 0 . How-e v e r , i t seems l o g i c a l t o expect t h a t c l o s e d c u r v e s analogous t o t h o s e o f F i g u r e 2-3 s h o u l d c o n t i n u e t o e x i s t f o r non-zero e c c e n t r i c i t y . These c u r v e s would t h e n be f u n c t i o n s o f 6 and t h u s d e f i n e a s u r f a c e i n t h e t h r e e d i m e n s i o n a l ty, (p\ 0 -space. I t i s apparent t h a t e q u a t i o n (2»14) i s p e r i o d i c i n 0 w i t h p e r i o d 2JT. , hence t h e s u r f a c e need o n l y be d e t e r m i n e d over t h a t i n t e r v a l . * 31 The s u r f a c e may be g e n e r a t e d by what Henon and H e i l e s ^ r e f e r t o as a " n u m e r i c a l e x p e r i m e n t . " An i n i t i a l p o i n t , (}J = (p Q i ip' = iff'Q, 0 = 0 i s chosen and e q u a t i o n s (2 . 1 0 5 ) are i n t e g r a t e d u n t i l 6 e q u a l s 271. T h i s produces a "consequent" p o i n t = <P Q, = ^ Q, 8 = 2?T which may be c o n s i d e r e d as a new i n i t i a l p o i n t w i t h 0 = 0 . The p r o c e s s may be thought o f as a t r a n s f o r m a t i o n , d e f i n e d by e q u a t i o n s (2.105), o f th e i n i t i a l p o i n t . The new s t a r t i n g p o i n t may i t s e l f be t r a n s f o r m e d r e p e a t e d l y t h u s l e a d i n g t o a s e r i e s o f p o i n t s i n t h e , ty' - p l a n e a t 8 = 0. I f any o f t h e t r a n s f o r m e d p o i n t s l i e s o u t -s i d e t h e r e g i o n |^|£7C/2, t h e n a l l t h e p o i n t s d e t e r m i n e d by t h e p r o c e s s l e a d t o t u m b l i n g m otion and may be p l o t t e d i n t h e u n s t a b l e r e g i o n . A l t e r n a t i v e l y , t h e p o i n t s may l i e i n s i d e t h e r e g i o n i n d i c a t i n g s t a b l e o p e r a t i o n and, when p l o t t e d , appear t o d e f i n e a c u r v e . T h i s i s an i n v a r i a n t c u r v e o f t h e t r a n s f o r m a t i o n . That i s , t h e t r a n s f o r m a t i o n o f t h e curve l y i n g i n t h e 0 = 0 p l a n e r e s u l t s i n t h e same curve b e i n g g e n e r a t e d a t 0 = 2X . The two cu r v e s a r e con-n e c t e d by an i n f i n i t y o f t r a j e c t o r i e s t h e r e b y d e f i n i n g a s u r f a c e which may be c a l l e d an " i n v a r i a n t s u r f a c e " o r J'iht.egral ;>manif b i d . "Tl The - e x i s t encef of u suchi s u r f acese f o r l i b r a t i o n a l m o tion i n a c i r c u l a r o r b i t i s e v i d e n t from 32 e a r l i e r d i s c u s s i o n , and, as p o i n t e d out by Moser, t h e i r e x i s t e n c e can be proven f o r e ^ 0 as f o l l o w s . The concept o f a t r a n s f o r m a t i o n w h i c h c o n v e r t s a p o i n t {tyQi ty'Q) i n the 0 = 0 p l a n e i n t o a n o t h e r (i^c» ^' c) i n t h e 0 = 27T p l a n e i s v e r y h e l p f u l i n t h i s r e g a r d . P e r i o d ! s o l u t i o n s appear as s e t s o f f i x e d p o i n t s i n t h e two p l a n e s and hence a r e c h a r a c t e r i z e d by an i d e n t i t y t r a n s f o r m a t i o n . The p e r i o d o f t h e s o l u t i o n s may be t a k e n as l o n g as d e s i r e d by making n -*•<». Hence t h e r e w i l l be a c o u n t a b l y i n f i n i t e number o f p o i n t s w h i c h s a t i s f y an i d e n t i t y t r a n s -f o r m a t i o n . S i n c e t h e d e n s i t y o f t h e s e p o i n t s may be made as h i g h as d e s i r e d , t h e mapping d e f i n e d i n t h i s manner must be a r e a p r e s e r v i n g . Now, t h e p e r i o d o f o s c i l l a t i o n o f t h e system f o r e = 0 i s g i v e n by (2.18) as a m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n o f dj m„v° The p e r i o d i s bounded by 2TC < A © < <*>) (O ^ | ^ ( 2 . 1 0 7 ) Moser's theorem t h e n a s s e r t s t h a t t h e r e e x i s t c u r v e s i n t h e neighbourhood o f t h e curv e s (2 . 1 0 6 ) which remain i n v a r i a n t under t h e mapping f o r s m a l l e. F i g u r e 2-22 r e p r e -s e n t s such an i n v a r i a n t s u r f a c e s c h e m a t i c a l l y . The s u r f a c e s g e n e r a t e d by e q u a t i o n ( 2 , 1 4 ) have c e r t a i n symmetry p r o p e r t i e s . T h i s d i f f e r e n t i a l e q u a t i o n remains un-changed i f both 0 and dJ change s i g n . Thus, t h e s o l u t i o n and i t s d e r i v a t i v e d e f i n e d by t h e c o n d i t i o n s (0) = 0 , <//(0) = </^ , a r e odd and even f u n c t i o n s r e s p e c t i v e l y . Conse-q u e n t l y , the p o i n t s d e f i n e d i n t h e 0 = 2TCm (m = 1,2,...) i p l a n e s a r e r e f l e c t i o n s about the - a x i s o f t h e p o i n t s d e f i n e d i n t h e 0 = 27X m (m = - 1 , - 2 , ...) p l a n e s . Hence the c r o s s - s e c t i o n s o f t h e s u r f a c e a t 0 = 0 and 27T a r e sym-m e t r i c a l about the <p" - a x i s . T h i s i s i l l u s t r a t e d i n F i g u r e 2-23. 77 F i g u r e 2-22 Schematic view o f an i n v a r i a n t s u r f a c e F i g u r e 2-23 A s p e c i f i c s o l u t i o n w hich i l l u s t r a t e s t h e symmetry p r o p e r t i e s o f t h e i n v a r i a n t s u r f a c e F u r t h e r , t h e p o i n t s d e f i n e d i n t h e 0 = 0 p l a n e w i l l be t h e m i r r o r image about the (//-axis o f t h e p o i n t s d e f i n e d i n t h e 0 =-8 p l a n e , or e q u i v a l e n t l y i n t h e 0 = 2K - 0 p l a n e . Thus the c r o s s - s e c t i o n o f t h e i n v a r i a n t s u r f a c e a t 0 = •§ i s . a m i r r o r image about th e t// - a x i s o f the c r o s s -s e c t i o n a t 0 = 2 X - 0. Hence, the c r o s s - s e c t i o n a t 0 = K i s a l s o symmetric about t h e - a x i s . S e v e r a l c r o s s - s e c t i o n s o f an i n v a r i a n t s u r f a c e t a k e n a t v a r i o u s o r b i t a n g l e s a r e p r e s e n t e d i n F i g u r e 2-24 f o r s p e c i f i c v a l u e s o f e and K^. An i n i t i a l p o i n t t a k e n w i t h i n a g i v e n m a n i f o l d gener-a t e s a d i f f e r e n t t r a j e c t o r y and hence a new s u r f a c e . The p r o p e r t y o f u niqueness g u a r a n t e e s t h a t the new t r a j e c t o r y does not i n t e r s e c t t h e o l d one. The new i n v a r i a n t s u r f a c e must t h e r e f o r e l i e c o m p l e t e l y w i t h i n t h e o r i g i n a l . On t h e o t h e r hand, an e x t e r n a l i n i t i a l c o n d i t i o n g e n e r a t e s an e x t e r n a l s u r f a c e p r o v i d e d t h a t t h e motion con-t i n u e s t o be stable.. The d e s i r e d r e g i o n o f s t a b i l i t y may be r e p r e s e n t e d as t h e l a r g e s t c l o s e d i n v a r i a n t s u r f a c e t h a t can be c o n s t r u c t e d . T y p i c a l i n v a r i a n t s u r f a c e s a r e shown i n F i g u r e s 2 - 2 5 - i and 2-25-ii° The symmetry p r o p e r t i e s a r e r e a d i l y o b s e r v e d . T h i s concept o f a l i m i t i n g s u r f a c e i n t h e phase space i s v e r y i m p o r t a n t . For g i v e n v a l u e s o f t h e parameters i t p r o v i d e s a l l p o s s i b l e c o m b i n a t i o n s o f i n i t i a l a n g l e s and v e l o c i t i e s t o w h i c h a s a t e l l i t e may be s u b j e c t e d a t any p o i n t i n i t s o r b i t w i t h o u t c a u s i n g i t t o become u n s t a b l e . 80 1.0 F i g u r e 2-24 C r o s s - s e c t i o n s o f an i n v a r i a n t s u r f a c e a t v a r i o u s o r b i t a n g l e s ( K i = 0.7, e = 0.2) 330' 300° icr JO*-iff 6(f /A ' V ' 1 90' 120" ,50" ^Trajectory, Kj = 1., e = 0.25 F i g u r e 2 - 2 5 - i T y p i c a l i n v a r i a n t s u r f a c e {K± = 1, e = 0.25) 83 I n t h i s r e s p e c t i t i s an improvement o f t h e work o f Z l a t o u s o v et a l . 1 2 The e x i s t e n c e o f an i n t e g r a l m a n i f o l d r a i s e s doubts as t o t h e a p p l i c a b i l i t y o f Lyapunov's d i r e c t method i n t h e d e t e r m i n a t i o n o f s t a b i l i t y . I n t h i s method a Lyapunov V-f u n c t i o n , which may be t a k e n t o be p o s i t i v e d e f i n i t e , i s sought. A c c o r d i n g as i t s time d e r i v a t i v e i s n e g a t i v e de-f i n i t e , z e r o , o r p o s i t i v e d e f i n i t e t h e motion i s e i t h e r a s y m p t o t i c a l l y s t a b l e , n e u t r a l l y s t a b l e , o r u n s t a b l e . When an i n t e g r a l m a n i f o l d e x i s t s , t h e p o i n t r e p r e -s e n t i n g t h e s t a t e o f motion always l i e s on t h e same s u r f a c e . T h e r e f o r e t h e motion must be n e u t r a l l y s t a b l e . Hence t h e d e r i v a t i v e o f the V - f u n c t i o n i s i d e n t i c a l l y z e r o and t h e V-f u n c t i o n must be a c o n s t a n t o f t h e motion. That i s , t h e i n t e g r a l m a n i f o l d i s a s u r f a c e on wh i c h t h e V - f u n c t i o n i s c o n s t a n t . Near such a s u r f a c e an approximate V - f u n c t i o n w i l l p o s s e s s a time d e r i v a t i v e o f v a r i a b l e s i g n so t h a t no i n f o r -m a t i o n r e g a r d i n g s t a b i l i t y can be o b t a i n e d by an approximate a n a l y s i s . On t h e o t h e r hand, t h e d e t e r m i n a t i o n o f t h e V-f u n c t i o n w i t h dV/dt = 0 i s e x a c t l y e q u i v a l e n t t o t h e d e t e r -m i n a t i o n o f t h e i n t e g r a l m a n i f o l d so t h a t no s a v i n g i n e f f o r t i s t o be ex p e c t e d . U s i n g an approximate V - f u n c t i o n and r e q u i r i n g t h a t t h e s t a t e v e c t o r be l a r g e r than some s p e c i f i e d v a l u e , i t i s p o s s i b l e t o e s t i m a t e bounds ; on t h e c o n d i t i o n s w h i c h l e a d t o i n s t a b i l i t y . These bounds a r e o n l y approximate and o n l y t h e more e x a c t n u m e r i c a l l y d e t e r m i n e d bounds a r e c o n s i d e r e d . I t i s p o s s i b l e t h a t a p a r t i c u l a r s e t o f parameters (e, K^) may be a s s o c i a t e d w i t h more t h a n one r e g i o n o f s t a b i l i t y . T h i s i s i l l u s t r a t e d i n F i g u r e 2 - 2 6 . There i s a s i n g l e c e n t r a l " m a i n l a n d " accompanied by one o r more " i s l a n d s . " The s m a l l e r i n v a r i a n t s u r f a c e s wrap themselves around t h e main s u r f a c e i n a h e l i c a l manner. The r e g i o n between the s u r f a c e s r e p r e s e n t s u n s t a b l e i n i t i a l c o n d i t i o n s . The symmetry p r o p e r t i e s o f t h e i n v a r i a n t s u r f a c e p r o -v i d e a means o f condensing c o n s i d e r a b l e i n f o r m a t i o n i n t o a s i n g l e diagram. The l i n e i n phase space d e f i n e d by t h e i n t e r s e c t i o n o f t h e p l a n e s = 0 and 6 = 0 forms an a x i s o f sym-metry o f the s u r f a c e . The i n t e r c e p t s made by t h e l i m i t i n g m a n i f o l d on t h i s l i n e r e p r e s e n t bounds t h a t must be p l a c e d on t h e d e r i v a t i v e f o r t h e g i v e n c o n f i g u r a t i o n ( t|> = 6 = 0) t o ensure s t a b l e motion. For a s p e c i f i e d v a l u e o f t h e p o i n t s o f i n t e r s e c t i o n can be p l o t t e d as a f u n c t i o n o f e c c e n t r i c i t y ( F i g u r e s 2 - 2 7 - i t o 2 - 2 7 - v i ) . Q u a l i t a t i v e l y such a diagram measures t h e s i z e o f t h e r e g i o n o f s t a b i l i t y . The s p i k e s i n t h e diagrams i n d i c a t e t h e p r e s ence o f t h e secondary i s l a n d s o f s t a b i l i t y d i s c u s s e d e a r l i e r . The . i r r e g u l a r edges o f t h e s t a b l e r e g i o n s a r e caused by t h e p r e s e nce of many a d d i t i o n a l s m a l l s p i k e s . The F i g u r e s a l s o i n d i c a t e t h a t the r e g i o n o f s t a b i l i t y s h r i n k s r a p i d l y w i t h i n c r e a s i n g e c c e n t r i c i t y . At some upper l i m i t 5 e , t h e s t a b i l i t y r e g i o n s h r i n k s t o a p o i n t ; or i n the phase space 4 t F i g u r e 2-26 T y p i c a l i n v a r i a n t s u r f a c e w i t h " i s l a n d s " ca \J1 0 -1 2 .3 .4 .5 e Stable ( periodic solutions with Unstable I period = 2?tn. I Transition points as determined by perturbation analysis. F i g u r e 2-27-i Range of v a l u e s o f the d e r i v a t i v e when y = 8 = 0 f o r s t a b l e m otion (K. = 1.0) 87 T 1 1 r I I I I l 1 0 .1 2 -3 .4 5 e S tab le ( per iodic solutions with Unstable \ per iod = 2 *n . X Transition points as determined by perturbat ion analysis. F i g u r e 2 - 2 7 - i i Range o f v a l u e s o f t h e d e r i v a t i v e when <f = 0 = 0 f o r s t a b l e m otion ( K i = 0 .9 ) 89 Stable j periodic solutions with Unstable ( period = 27rn. X Transition points as determined by perturbation analysis. F i g u r e 2 - 2 7 - i v Range o f v a l u e s o f t h e d e r i v a t i v e when = 0 f o r s t a b l e m otion {K± = 0 . 5 ) 90 1.5 .].5 I i ' ' • 0 .1 -2 .3 4 e Stable ( periodic solutions with Unstable ( period = 2fln. X Transition points as determined by perturbation analysis. F i g u r e 2-27-v Range o f v a l u e s o f the d e r i v a t i v e when y = 6 = 0 f o r s t a b l e m otion ( K i = 0.3) -1-5 " 1 1 1 1  0 -1 2 3 4 -5 e Stable ( periodic solutions with Unstable ( period = 2fln. I Transition points as determined by perturbation analysis. F i g u r e 2 - 2 7 - v i Range o f v a l u e s o f t h e d e r i v a t i v e when f = 6 = 0 f o r s t a b l e m otion ( K i = 0.1) r e p r e s e n t a t i o n t h e i n v a r i a n t s u r f a c e d egenerates t o a s i n g l e t r a j e c t o r y . Thus, beyond a c e r t a i n c r i t i c a l v a l u e of eccen-t r i c i t y , s t a b l e m otion i s not p o s s i b l e . At s t i l l h i g h e r e c c e n t r i c i t i e s , s t a b i l i t y may r e t u r n but the s i z e o f t h e r e g i o n i s so s m a l l as t o be o f no p r a c t i c a l i m p o r t a n c e . 2.5 A c c u r a c y o f t h e Method There a r e s e v e r a l s o u r c e s o f e r r o r i n t h e method ou t -l i n e d i n s e c t i o n 2.4° They a r e a l l due t o t h e f i n i t e n a t u r e o f t h e n u m e r i c a l p r o c e s s . A l a r g e d i s c r e p a n c y may a r i s e i n t h e s o l u t i o n due t o t h e t r u n c a t i o n e r r o r o f t h e n u m e r i c a l i n t e g r a t i o n p r o c e s s and " r o u n d o f f " g e n e r a t e d i n the computer. The t r u n c a t i o n e r r o r v a r i e s as f o r t h e Adams-33 B a s h f o r t h and Runge-Kutta t e c h n i q u e s employed. J The round-o f f e r r o r , on t h e o t h e r hand, tends t o i n c r e a s e as 1/h. Thus t h e r e i s an optimum s t e p s i z e which m i n i m i z e s t h e t o t a l e r r o r . At t i m e s a s i t u a t i o n may a r i s e where the c r i t i c a l s t e p s i z e i s t o o s m a l l f o r r a p i d c o m p u t a t i o n and t h e r e s u l t -i n g p r e c i s i o n may not be e s s e n t i a l f o r t h e a n a l y s i s . To i l l u s t r a t e t h i s p o i n t , s e v e r a l s t e p s i z e s were chosen f o r t h e i n t e g r a t i o n o f e q u a t i o n (2.14) over 30 o r b i t s . The r e s u l t i n g v a l u e s o f the f i n a l c o n d i t i o n s (Table I I ) i n -d i c a t e t h a t t h e e r r o r s can be reduced t o .0001 r a d i a n s by employing a s t e p s i z e o f 3°° F i g u r e 2 - 2 8 - i p r e s e n t s th e i n v a r i a n t c u r v e i n t h e 6 = 0 p l a n e as g i v e n by t h e most a c c u r a t e s o l u t i o n (h = 1.5°) TABLE I I The E f f e c t o f V a r y i n g I n t e g r a t i o n Step S i z e I n e r t i a parameter, = 1.0 O r b i t e c c e n t r i c i t y , e = 0.1 I n i t i a l d i s p l a c e m e n t , <Po = 0 I n i t i a l v e l o c i t y , ty* - 1,18 h <//(607T) \p' (607T) Degrees Radians 30 U n s t a b l e a f t e r 4.5 o r b i t s 15 U n s t a b l e a f t e r Id.3 o r b i t s 7*5 -0.18556 1.12262' 3 -0.18910 1,11949 1.5 -0.18911 1.11948 A l s o p l o t t e d i n t h e F i g u r e a r e s e l e c t e d p o i n t s d e t e r m i n e d from t h e n u m e r i c a l s o l u t i o n o b t a i n e d w i t h h = 7.5°. I t i s i n t e r e s t i n g t o note t h a t t h e d i f f e r e n c e between t h e two c r o s s - s e c t i o n s i s much s m a l l e r t h a n t h a t between t h e two s o l u t i o n s i n d i c a t e d i n T a b l e I I , M i t r o p o l s k i y ^ i n d i c a t e s t h a t t h i s i s f r e q u e n t l y observed i n systems which possess i n t e g r a l m a n i f o l d s . There does not appear t o be a t h e o r e t i -c a l e x p l a n a t i o n f o r t h i s b e h a v i o u r , but i t does p e r m i t t h e use o f l a r g e s t e p s i z e s . A l t h o u g h t h e r e s u l t i n g s o l u t i o n s have l a r g e e r r o r s t h e i n v a r i a n t s u r f a c e s a r e u s u a l l y o f s u f f i c i e n t a c c u r a c y . The e r r o r s appear t o cause d i s p l a c e -ments around t h e m a n i f o l d r a t h e r t h a n normal t o i t . 94 y 6 ^ 3 24 / 4 K-1, e =.1 •28 e = 27tn / • h = 1.5° f6 o h = 75° Invariant curve + 1.0 + •5 • 30 u -t 1 1 1-• 8 - 5 30 \ \ , 0 \ 23 11 ^ 2 2 V •5 2 0 \ 9 X 20' 12 s is J--1.0 \ \ n = 7\ 16< 27 H 1 1 H (Rod.) 25 7 3/ 5; / Figure 2 - 2 8 - i Comparison of the invariant surfaces generated using diff e r e n t integration step sizes (Non-limiting surface) F i g u r e 2-26*-ii Comparison o f t h e 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 u s i n g d i f f e r e n t i n t e g r a t i o n s t e p s i z e s . ( L i m i t i n g s u r f a c e s ) The d e t e r m i n a t i o n o f t h e l i m i t i n g i n v a r i a n t s u r f a c e i n v o l v e s a d d i t i o n a l d i f f i c u l t i e s . Here, even a s m a l l d i s -c repancy can l e a d t o erroneous r e s u l t s r e g a r d i n g s t a b i l i t y . As i n d i c a t e d by T a b l e I I t h e e r r o r s b r i n g about e a r l y i n -s t a b i l i t y and hence cause t h e s i z e o f t h e l i m i t i n g i n v a r i a n t s u r f a c e t o be u n d e r e s t i m a t e d . (F'3UgJ^per^-r£6^iii.)j.; The r e s u l t s suggest t h a t any e r r o r which d i s p l a c e s t h e r e p r e s e n t a t i v e p o i n t i n t o the u n s t a b l e r e g i o n l e a d s t o f u r t h e r growth o f t h i s e r r o r . Any subsequent e r r o r which a c t s towards th e s t a b l e r e g i o n w i l l have t o be somewhat l a r g e r t h a n t h e o r i g i n a l e r r o r t o r e g a i n s t a b i l i t y . On t h e o t h e r hand, th e e f f e c t o f an e r r o r which t a k e s t h e r e p r e s e n -t a t i v e p o i n t i n t o t h e s t a b l e r e g i o n may be i m m e d i a t e l y c a n c e l l e d by an e r r o r o f t h e same s i z e but i n t h e o p p o s i t e sense. Thus, assuming t h a t b o t h k i n d s o f e r r o r a r e e q u a l l y p r o b a b l e , t h e r e w i l l be a d r i f t towards i n s t a b i l i t y from a t h i n " s k i n " w h i c h l i e s i n s i d e t h e l i m i t i n g i n v a r i a n t s u r f a c e I n t h e m a j o r i t y o f cases s t u d i e d t h i s tendency towards i n s t a b i l i t y was n o t e d . A second e r r o r i n t h e d e t e r m i n a t i o n o f t h e l i m i t i n g m a n i f o l d i s due t o t h e t e r m i n a t i o n o f the n u m e r i c a l i n t e g r a -t i o n a f t e r a f i n i t e t i m e . T h i s may r e s u l t i n an u n s t a b l e s o l u t i o n a p p e a r i n g s t a b l e . C a r e f u l p l o t t i n g o f t h e r e s u l t s can u s u a l l y d e t e c t any tendency of t h i s t y p e . T h i s e r r o r a c t s t o i n c r e a s e t h e s i z e o f t h e l i m i t i n g i n v a r i a n t s u r f a c e . T h i r d l y , t h e p r o c e s s o f n u m e r i c a l e x p e r i m e n t a t i o n i s n e c e s s a r i l y d i s c r e t e . That i s , bounds can be p l a c e d on i n i t i a l c o n d i t i o n s w h i c h s e p a r a t e s t a b l e and u n s t a b l e s o l u -t i o n s . These bounds may be made as f i n e as d e s i r e d p r o v i d -i n g s u f f i c i e n t computing t i m e i s used. There i s , t h e r e f o r e , a p r a c t i c a l l i m i t t o the p r e c i s i o n w i t h which t h e l i m i t i n g m a n i f o l d can be d e t e r m i n e d . T h i s r e s u l t s i n t h e n u m e r i c a l l y e v a l u a t e d l i m i t i n g m a n i f o l d l y i n g i n s i d e t h e t r u e one. The m a j o r i t y o f t h e n u m e r i c a l i n t e g r a t i o n s were p e r -formed u s i n g a s t e p s i z e o f 7 . 5 ° . The r e s u l t i n g l i m i t i n g s u r f a c e s were compared w i t h more ex a c t r e s u l t s i n s e v e r a l cases u s i n g the d a t a a t 0 = = 0 as a s t a n d a r d . The r e s u l t s w i t h h = 7 . 5 ° l a y w i t h i n - . 0 7 and +.05 u n i t s i n o f t h e more p r e c i s e r e s u l t s . I n t h e m a j o r i t y o f cases the e r r o r was l e s s t h a n . 0 3 u n i t s . 2.6 The S i g n i f i c a n c e o f P e r i o d i c S o l u t i o n s 2.6.1 The R e l a t i o n s h i p Between M a n i f o l d s and P e r i o d i c S o l u t i o n s I n s e c t i o n 2.4 i t was shown t h a t an i n i t i a l c o n d i t i o n chosen i n s i d e a s p e c i f i c m a n i f o l d r e s u l t s i n t h e g e n e r a t i o n o f a new m a n i f o l d which l i e s c o m p l e t e l y w i t h i n t h e f i r s t . A s u c c e s s i o n o f i n i t i a l c o n d i t i o n s may t h u s be chosen which determine p r o g r e s s i v e l y s m a l l e r m a n i f o l d s , f i n a l l y r e s u l t i n g i n a s u r f a c e which has zero c r o s s - s e c t i o n . Because o f t h e p e r i o d i c i t y e x h i b i t e d by t h e i n v a r i a n t s u r f a c e , t h i s mani-f o l d must t h e n r e p r e s e n t a p e r i o d i c s o l u t i o n . Hence p e r i o d i c s o l u t i o n s a c t as s p i n e s upon which the i n v a r i a n t s u r f a c e s a re 9 8 b u i l t . The g e n e r a l m o t i o n can t h e n be thought o f as a bounded p e r t u r b a t i o n about t h e a p p r o p r i a t e p e r i o d i c s o l u t i o n . The n u m e r i c a l r e s u l t s p r e s e n t e d i n s e c t i o n 2.4 I n d i c a t e t h a t w i t h i n c r e a s i n g e c c e n t r i c i t y t h e s i z e o f t h e 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 h r i n k s . U l t i m a t e l y , the c r o s s - s e c t i o n becomes a s e t o f p o i n t s so t h a t the m a n i f o l d degenerates i n t o a p e r i o d i c s o l u t i o n . The importance o f t h e p e r i o d i c s o l u t i o n s i s now e v i d e n t ; as a t t h e l a r g e s t o r b i t e c c e n t r i c i t y f o r s t a b l e m otion t h e o n l y a v a i l a b l e s o l u t i o n i s a p e r i o d i c one. Because a t t h i s c r i t i c a l v a l u e o f e c c e n t r i c i t y t h e i n v a r i a n t s u r f a c e i s i n f i n i t e s i m a l i n s i z e , t h e l i n e a r p e r t u r b a t i o n a n a l y s i s s h o u l d c o r r e c t l y p r e d i c t a change from s t a b l e t o u n s t a b l e p e r t u r b a t i o n s . The maximum e c c e n t r i c i t y f o r s t a b l e m o t i o n can t h u s be de t e r m i n e d w i t h g r e a t p r e c i s i o n by t h e v a r i a t i o n a l a n a l y s i s o f t h e a p p r o p r i a t e p e r i o d i c s o l u t i o n . The d e t a i l s o f the a n a l y s i s were p r e s e n t e d i n s e c t i o n 2,3-3 and t h e l i m i t i n g v a l u e s o f e c c e n t r i c i t y were p l o t t e d i n F i g u r e s 2-18 and 2-19° They are a l s o i n d i c a t e d i n F i g u r e s 2-27-i t o 2-27-vi t o compare t h e a c c u r a c y o f t h e s t r i c t l y n u m e r i c a l s e a r c h f o r t h e l i m i t i n g m a n i f o l d w i t h t h e more t h e o r e t i c a l d e t e r m i n a t i o n o f t h e c h a r a c t e r i s t i c exponents. The agreement i s q u i t e good, but tends t o become p o o r e r as d e c r e a s e s . T h i s i s due t o a 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 a p p e a r i n g and d i s t u r b i n g t h e n u m e r i c a l s e a r c h near t h e maximum e c c e n t r i c i t y . Because t h e n u m e r i c a l i n t e g r a t i o n s i n v o l v e d a r e performed o n l y over f i n i t e i n t e r v a l s and a r e not 99 "open ended" as i n t h e case of t h e n u m e r i c a l s e a r c h , t h e a c c u -r a c y may be improved t o any d e s i r e d degree. These r e s u l t s a l s o e x p l a i n t h e a p p a r e n t l y anomolous b e h a v i o u r o f t h e r e g i o n o f s t a b i l i t y between = 0.5 and 0.3. F i g u r e s 2-7 and 2-18 i n d i c a t e t h a t t h e upper s p i k e d i s a p p e a r s w i t h d e c r e a s i n g ( i t can be shown t h a t t h e s p i k e v a n i s h e s f o r = 1/3) w h i l e t h e l o w e r one c o n t i n u e s t o grow u n t i l ~ 0.25» O c c a s i o n a l l y 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 appear i n s i d e t h e m a n i f o l d c r o s s - s e c t i o n s . F i g u r e 2-29 p r e s e n t s s e v e r a l 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 s e v a l u a t e d f o r = 1, e = .2. The c l o s e d c u r v e s s u r r o u n d s t a b l e p e r i o d i c s o l u t i o n s and t h e p o i n t e d i n v a r i a n t p l o t s a r e a s s o c i a t e d w i t h u n s t a b l e s o l u t i o n s . T h i s i s i n agreement w i t h t h e a n a l y s i s ( s e c t i o n 2.3.3) where i t was shown t h a t t h e p e r t u r b a t i o n s l i e a l o n g e l l i p s e s o r h y p e r b o l a e i n t h e v i c i n i t y o f t h e p e r i o d i c s o l u t i o n s . I n t h e s t r o b o s c o p i c phase p l a n e s t a b l e p e r i o d i c s o l u t i o n s appear l i k e c e n t r e s w h i l e t h e 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 th e appearance of s a d d l e p o i n t s . Thus t h e i n s p e c t i o n o f t h e p e r i o d i c s o l u t i o n s p r o v i d e s q u a l i t a t i v e i n f o r m a t i o n c o n c e r n i n g t h e n a t u r e o f t h e s t r o b o s c o p i c phase p l a n e and hence o f the m o t i o n . 2.6.2 D e t e r m i n a t i o n o f a Complete Set o f P e r i o d i c S o l u t i o n s S e c t i o n 2.2.2 i n d i c a t e d t h e e x i s t e n c e o f p e r i o d i c s o l u -t i o n s t h a t c o u l d be r e p r e s e n t e d as a s i n e s e r i e s . An e x t e n -s i v e n u m e r i c a l s e a r c h f o r t h e s e s o l u t i o n s r e s u l t e d i n t h e F i g u r e 2-29 I n v a r i a n t s u r f a c e i l l u s t r a t i n g t h e appearance of s t a b l e and 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 i n the s t r o b o s c o p i c phase p l a n e d a t a p r e s e n t e d i n s e c t i o n 2 . 2 . 3 « No attempt was made, how-e v e r , t o determine i f t h e s o l u t i o n s o b t a i n e d c o n s t i t u t e d a complete s e t . The importance of p e r i o d i c s o l u t i o n s i n d e t e r m i n i n g t h e l i m i t i n g v a l u e s o f e c c e n t r i c i t y and t h e g e n e r a l shape o f t h e i n v a r i a n t s u r f a c e s makes t h e knowledge of a complete s e t d e s i r a b l e . E a r l i e r t h e p r o c e s s o f i n t e g r a t i o n o f e q u a t i o n ( 2 . 1 4 ) from 6 = 0 t o -G = 2TTn was d e s c r i b e d as a t r a n s f o r m a t i o n between th e two p l a n e s . I n terms o f such a t r a n s f o r m a t i o n , a p e r i o d i c s o l u t i o n appears as a f i x e d p o i n t . Thus enumera-t i o n o f a complete s e t o f p e r i o d i c s o l u t i o n s r e q u i r e s t h e d e t e r m i n a t i o n o f a l l t h e f i x e d p o i n t s o f t h e t r a n s f o r m a t i o n . A l t h o u g h t h i s i s a s i m p l e concept i t i n v o l v e s an enormous amount of work because o f t h e l a r g e number o f t r a j e c t o r i e s w h i c h must be computed. F i g u r e 2 -30 p r e s e n t s f o r = 1 , e = 0 , c o n t o u r s i n t h e 0 = 2TT p l a n e which c o r r e s p o n d t o l i n e s o f c o n s t a n t ^ i n t h e 0 = 0 p l a n e . A curve may be drawn which passes t h r o u g h t h o s e p o i n t s such t h a t i^ i i s i n v a r i a n t under th e t r a n s f o r m a t i o n . The c o r r e s p o n d i n g p l o t s f o r c o n s t a n t d/ a r e shown i n F i g u r e 2-31« The p o i n t s o f i n t e r s e c t i o n o f t h e yp - i n v a r i a n t and t h e ^ ' - i n v a r i a n t c u r v e s c o n s t i t u t e a complete s e t o f f i x e d p o i n t s f o r t h e g i v e n v a l u e s o f the parameters ( F i g u r e 2 - 3 2 ) . Note t h a t because t h e e c c e n t r i c i t y i s z e r o , t h e r e a r e an i n f i n i t e number of i n v a r i a n t p o i n t s which d e f i n e a c l o s e d 102 -1.5 -1.0 -.5 0 .5 1.0 1.5 WW (Rod) F i g u r e 2-30 The t r a n s f o r m a t i o n o f l i n e s o f c o n s t a n t */> when t h e e q u a t i o n o f motion i s i n t e g r a t e d over 27T (K.. = 1, e = 0) 103 Figure 2-31 The transformation of l i n e s of constant ^' when the equation of motion i s integrated over 27T (K i = 1, e = 0) 104 2 lp'(27f) 0 -l 1 1 1 - / / / / / f 1 ' i \ ' 1 ' / 1 ll I . / i ! . _ D l / 1 / 1 i \ 1 / \-K| = 1. e = 0 i 1 i - ff • i 1 i -1-5 -1.0 -.5 0 .5 1-0 1.5 4i(27t), Rod lp Invariant lp' Invariant ip and ip1 Invariant Figure 2-32 Determination of a complete set of fix e d points of the transformation (K^ = 1, e = 0) c u r v e . T h i s i s c o n s i s t e n t w i t h t h e ex a c t s o l u t i o n a r r i v e d a t i n s e c t i o n 2.2.1. F i g u r e s 2 - 3 3 , 2 - 3 4 , and 2 - 3 5 p r e s e n t the c o r r e s p o n d -i n g s e t o f c u r v e s f o r = 1, e = 0.";.1. I n t h i s case t h e t h r e e s o l u t i o n s on t h e (//-axis a r e i m m e d i a t e l y e v i d e n t . Because t h e t r a n s f o r m a t i o n s t u d i e d i s f o r n = 1, t h e s e t h r e e s o l u t i o n s a l s o l i e on t h e (//-axis a t 6 = Tf and hence a r e a l s o members o f t h e second f a m i l y o f p e r i o d i c s o l u t i o n s . There a r e a l s o s o l u t i o n s o f t h e t h i r d f a m i l y a t f = ±1t/2 as i n d i c a t e d by 12 Z l a t o u s o v et a l . No o t h e r p e r i o d i c s o l u t i o n s e x i s t so t h a t t h o s e d e t e r -mined i n s e c t i o n 2.2.2 form a complete s e t f o r n = 1. 2 . 6 . 3 The Degree o f S t a b i l i t y As a l r e a d y mentioned, any s t a t e o f mo t i o n w i t h i n t h e r e g i o n o f s t a b i l i t y w i l l g i v e r i s e t o a s u r f a c e which l i e s w i t h i n t h e l i m i t i n g s u r f a c e a t a l l t i m e s . S i n c e t h e major d i s t u r b a n c e s a r e e s s e n t i a l l y s t o c h a s t i c i n n a t u r e , t h e d i s -t a n c e between t h e s u r f a c e c o r r e s p o n d i n g t o t h e a c t u a l motion o f t h e s a t e l l i t e and t h e l i m i t i n g s u r f a c e would be a measure o f t h e l o n g term s t a b i l i t y . F u r t h e r , s i n c e t h e phase space r e p r e s e n t a t i o n shows t h a t t h e v a r i o u s s u r f a c e s a r e n e s t e d , t h i s d i s t a n c e becomes a maximum when t h e s u r f a c e becomes as s m a l l as p o s s i b l e , i . e . a s i n g l e t r a j e c t o r y . T h i s s t a t e c o r r e s p o n d s t o a p e r i o d i c m otion o f t h e s a t e l l i t e w i t h p e r i o d 2Tf . The n e c e s s a r y momentum change a t 0 = 0 can be o b t a i n e d 106 F i g u r e 2-33 The t r a n s f o r m a t i o n o f l i n e s o f c o n s t a n t $ when t h e e q u a t i o n o f motion i s i n t e g r a t e d over 27T (K. - 1, e = 0 . 1 ) 107 F i g u r e 2-34 The t r a n s f o r m a t i o n o f l i n e s o f c o n s t a n t when the e q u a t i o n o f motion i s i n t e g r a t e d over 27C (K± = 1, e = 0.1) 108 -1.5 -1.0 -5 0 -5 10 15 4/(271), R o d ijj Invariant _— \y' invariant o ip and ijV Invariant F i g u r e 2-35 D e t e r m i n a t i o n o f a complete s e t o f f i x e d p o i n t s o f t h e t r a n s f o r m a t i o n (K. = 1 , e = 0.1) 109 from F i g u r e s 2 - 2 7 - i t o 2 - 2 7 - v i . The p l o t s i n F i g u r e s 2 - 3 6 - i t h r o u g h 2 - 3 6 - v i p r e s e n t t h e s i z e o f t h e d e s t a b i l i z i n g i m p u l s e as a f u n c t i o n o f o r b i t a n g l e f o r s e v e r a l v a l u e s o f e c c e n t r i c -i t y and i n e r t i a parameter. Only th e l a r g e s t i n v a r i a n t s u r -f a c e s have been used t o p r e p a r e t h e s e diagrams. Moreover, t h e presence o f s p i k e s has been i g n o r e d so t h a t t h e margin o f s t a b i l i t y shown i n t h e F i g u r e s r e p r e s e n t s a c o n s e r v a t i v e e s t i m a t e . I t i s apparent t h a t o r b i t a n g l e does not a f f e c t t h e degree o f s t a b i l i t y g r e a t l y . . E c c e n t r i c i t y o f t h e o r b i t , on t h e o t h e r hand, i s a p o w e r f u l d e s t a b i l i z i n g f a c t o r . V a l u e s o f e g r e a t e r t h a n 0 . 15 l e a d t o a s u b s t a n t i a l l o s s o f s t a b i l i t y . 2 . 7 C o n c l u d i n g Remarks The e s s e n t i a l f e a t u r e s o f the a n a l y s i s p r e s e n t e d i n t h i s c h a p t e r and t h e c o n c l u s i o n s based on them may be sum-m a r i z e d as f o l l o w s : ( i ) The a n a l y s i s demonstrated t h e e x i s t e n c e o f v a r i o u s p e r i o d i c s o l u t i o n s which may be d etermined by a n a l y t i c a l o r n u m e r i c a l means. ( i i ) The concept o f a t h r e e d i m e n s i o n a l phase space has been i n t r o d u c e d . T h i s has t h e v a l u a b l e p r o p e r t y t h a t t h e t r a j e c t o r i e s d e s c r i b e d i n t h e phase space by t h e r e p r e -s e n t a t i v e p o i n t s a r e unique and n o n - i n t e r s e c t i n g . ( i i i ) The p e r i o d i c s o l u t i o n s appear as h e l i c a l t r a j e c t o r i e s i n the phase space. The g e n e r a l c h a r a c t e r o f t h e motion i s d i s p l a y e d by t h e I n t e g r a l m a n i f o l d s which have been d e t e r m i n e d n u m e r i c a l l y . The m a n i f o l d s a r e a l s o non-Am 110 e = 0 0 . 0 5 ^ - ' 0 . 1 0 , ^ ' 0 . 1 5 ^ _ _ 0 . 2 5 _ .030 ' 0 3 5 5 K: = 1.0 0 . 3 0 - . 0 - 2 5 _ _ . £ , 2 0 " 0.15~~ 0 . 0 5 9 0 ° 1 8 0 c e 2 7 0 3 6 0 F i g u r e 2 - 3 6 - i Maximum momentum change r e q u i r e d t o d e s t a b i l i z e a s a t e l l i t e (K^ = 1.0) -1 - 2 1 1 1 e = 0 \ -~ \ ^ . 0 J 0 _ " \ V \ \ \ \ 0 - 3 2 ^ / 1 1 4 / y * . -i / ' » _ / " v . 0 - 3 2 ^ ^ 0 . 9 o . i o — . ^ " 0 i i i 111 0° 9 0 ' 1 8 0 ° e 2 7 0 c 3 6 0 F i g u r e 2 - 3 6 - i i Maximum.momentum change r e q u i r e d t o d e s t a b i l i z e a s a t e l l i t e (K. = 0 .9) 112 e 360 F i g u r e 2 - 3 6 - i i i Maximum momentum change r e q u i r e d t o d e s t a b i l i z e a s a t e l l i t e (K^ = 0.7) F i g u r e 2-36-iv Maximum momentum change r e q u i r e d t o d e s t a b i l i z e a s a t e l l i t e ( K i = 0 . 5 ) Ai j ) -2 114 Kj = 0-3 e = 0 0.05, 0.10 -QilO 0-05 0" 90 180 e 270 360 F i g u r e 2-36-v Maximum momentum change r e q u i r e d t o d e s t a b i l i z e a s a t e l l i t e (K^ = 0.3) 115 Kj = 0.1 e = 0 ML e 0.05 0.05" 0° 90° 180 e 270 360 F i g u r e 2-36-vi Maximum momentum change r e q u i r e d t o d e s t a b i l i z e a s a t e l l i t e (K. = 0.1) 116 i n t e r s e c t i n g and e x h i b i t a c l o s e r e l a t i o n s h i p w i t h t h e p e r i o d i c s o l u t i o n s . ( i v ) The r e g i o n o f s t a b i l i t y i s r e p r e s e n t e d by t h e l a r g e s t i n t e g r a l m a n i f o l d which can be c o n s t r u c t e d . The i m p o r t a n c e o f such a s u r f a c e cannot be over emphasized as f o r g i v e n v a l u e s o f the parameters i t p r o v i d e s a l l p o s s i b l e c o m b i n a t i o n s o f d i s t u r b a n c e s t o which a s a t e l l i t e may be s u b j e c t e d a t any g i v e n p o i n t i n i t s o r b i t w i t h o u t c a u s i n g i t t o tumble. (v) For a c i r c u l a r o r b i t t h e i n v a r i a n t s u r f a c e s i n t h e phase space a r e c y l i n d e r s w i t h s i m p l e c r o s s - s e c t i o n s . F o r f i n i t e e c c e n t r i c i t y t h e s u r f a c e s a r e h e l i c a l and e x h i b i t s u b s t a n t i a l v a r i a t i o n i n e r o s s - s e c t d o n e w i t h l o r b l t a l l a n g l e : . ( v i ) As e c c e n t r i c i t y i n c r e a s e s , the s i z e o f t h e l i m i t -i n g s u r f a c e d e c r e a s e s and f o r e = e i t c o l l a p s e s t o a l i n e , ° max ' i . e . t o a p e r i o d i c s o l u t i o n . There i s a l i m i t t o the v a l u e o f o r b i t e c c e n t r i c i t y f o r s t a b l e l i b r a t i o n a l m otion. T h i s c r i t i c a l v a l u e depends on t h e geometry of t h e s a t e l l i t e . The n u m e r i c a l l y o b t a i n e d v a l u e was checked by l i n e a r p e r t u r b a t i o n a n a l y s i s o f t h e p e r i o d i c s o l u t i o n s . ( v i i ) The a n a l y s i s s u g g e s t s t h a t a s m a l l v a l u e o f e c c e n t r i c i t y and a l a r g e v a l u e o f i n e r t i a parameter would h e l p t o ensure s t a b i l i t y . F o r e l a r g e r t h a n about O.38 p r a c t i c a l g r a v i t a t i o n a l g r a d i e n t s t a b i l i z a t i o n o f a s a t e l l i t e i s not p o s s i b l e . I f t h e s i z e o f t h e l i m i t i n g i n v a r i a n t s u r -f a c e i s i n t e r p r e t e d as a measure of t h e d i s t u r b a n c e s which a s a t e l l i t e w i l l t o l e r a t e and s t i l l c o n t i n u e t o execute l i b r a t i o n a l m o t i o n , i t has been shown t h a t even q u i t e moderate v a l u e s o f e c c e n t r i c i t y would s e r i o u s l y reduce t h e a b i l i t y o f t h e s a t e l l i t e t o w i t h s t a n d e x t e r n a l d i s t u r b a n c e s 3. PLANAR LIBRATIONS OF A DAMPED SATELLITE 3.1 F o r m u l a t i o n o f t h e Problem The a n a l y s i s o f t h e p l a n a r l i b r a t i o n s o f a r i g i d s a t e l -l i t e was p r e s e n t e d i n t h e p r e c e d i n g c h a p t e r . By c o n s t r u c t i n g t h e s a t e l l i t e so t h a t r e l a t i v e m o tion can o c c u r between v a r i o u s members and i n s e r t i n g energy d i s s i p a t i n g mechanisms which oppose t h i s r e l a t i v e m o t i o n i t i s p o s s i b l e t o h a s t e n t h e cap-t u r e o f t h e s a t e l l i t e by t h e g r a v i t y - g r a d i e n t f i e l d and t o reduce t h e e f f e c t s o f e x t e r n a l d i s t u r b a n c e s on i t s o r i e n t a t i o n . S e v e r a l d e s i g n s have been proposed i n t h e l i t e r a t u r e ( s e c t i o n 1.2). Some of t h e s e a r e q u i t e complex as t h e y attempt t o s t a b i l i z e t h e s a t e l l i t e about a l l t h r e e body axes. I f o n l y p l a n a r l i b r a t i o n s a r e c o n s i d e r e d , t h e damper proposed by 22 P a u l i s adequate. T h i s d e v i c e c o n s i s t s o f two p o i n t masses c o n s t r a i n e d t o move a l o n g t h e a x i s o f t h e s a t e l l i t e and con-n e c t e d by a l i n e a r s p r i n g - d a s h p o t arrangement. The c o n f i g u r a t i o n s t u d i e d by P a u l i s u n d u l y r e s t r i c t i v e and a more g e n e r a l c o n f i g u r a t i o n has been s e l e c t e d f o r s t u d y ( F i g u r e 3-1). Note t h a t t h e mean p o s i t i o n o f t h e damper mass i s o f f s e t f r om t h e c e n t r e o f mass of t h e main body. 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 t h e system may be w r i t t e n as 119 F i g u r e 3-1 Geometry o f motion of a damped s a t e l l i t e 120 \Z  ( 3 ' 1 ) 1 (3-2) W r i t i n g t h e d i s s i p a t i o n f u n c t i o n as and u s i n g Lagrange's f o r m u l a t i o n g i v e s t h e e q u a t i o n s o f m o t i o n i n t h e <^> and z-^  degrees o f freedom (Iyy + "\i(f. + l/X© + r) + 2 M a (£ +J. )|, (6 4-(3.4) a (3-5) 121 P u t t i n g K; = r"V"In-J . v v J.yy 1 = Jilt* (3.6) ^ = t u f a t h e g o v e r n i n g e q u a t i o n s may be r e w r i t t e n as The use of the awkward r e l a t i o n s gove rn ing r, 6 , and • & as f u n c t i o n s of t ime can be avo ided by changing the i n d e -pendent v a r i a b l e to 0 u s i n g the r e l a t i o n s (2 .11 )- (2 .13 ) and 122 Fo r an e l l i p t i c a l o r b i t i t i s p o s s i b l e t o w r i t e a r e l a t i o n -s h i p f o r 0 i n terms o f t h e o r b i t a l p e r i o d A 1» _ m. h+ec*,ef_ w (n-eCosef (3.10) so t h a t t h e parameters d e s c r i b i n g t h e damper may be w r i t t e n / _ 9 0d = e (3.11) U s i n g t h e s e r e l a t i o n s , t h e system e q u a t i o n s (3.8) and (3«9) may be w r i t t e n as .3 (ft + Kd (l + *)Z) stn (PCos(l> = o i + eCosQ '  r (3.12) and (I - ezP 2e Sin 6 123 4 l(i+eCose)+ I - 3C o5 I + e Cos a (3.13) •/V+.f.- ± z J C o l ± where (3.H) The system i n v o l v e s a l a r g e number o f v a r i a b l e s which comp l i c a t e s the a n a l y s i s . To b e t t e r unders tand the b a s i c c h a r a c t e r o f the system i t i s conven ient to c o n s i d e r a p a r t i c u l a r s i t u a t i o n where the mass, m^, i s c r i t i c a l l y damped, i . e . ' = (3.15) T Fu r the rmore , the parameter w i l l be taken to be u n i t y thus r e p r e s e n t i n g a s l e n d e r , dumbbel l type s a t e l l i t e . 3 . 2 . Numer ica l R e s u l t s The damping present i n the model causes the phase space t r a j e c t o r i e s to g r a d u a l l y approach a l i m i t c y c l e . The d i s -s i p a t i o n o f energy i n the damping mechanism p re c l udes employ-i n g t h e p r o c e s s w h i c h made p o s s i b l e t h e g e n e r a t i o n o f an i n v a r i a n t s u r f a c e . For a damped system t h e r e i s no c l o s e d i n v a r i a n t s u r f a c e , however, t h e r e p r e s e n t a t i o n o f t h e be-h a v i o u r o f t h e system i n t h e s t r o b o s c o p i c phase p l a n e i s s t i l l v e r y h e l p f u l . F o r a g i v e n v a l u e o f 6 , a p o i n t i n t h e s t r o b o s c o p i c j (^/-plane d e t e r m i n e s t h e s t a t e o f t h e s a t e l l i t e w h i l e a c o r r e s p o n d i n g p o i n t i n a Z, Z T - p l a n e s p e c i f i e s t h e s t a t e o f the damper. For s i m p l i c i t y o n l y t h e phase p l a n e r e p r e -s e n t i n g t h e s t a t e o f t h e s a t e l l i t e i s s t u d i e d h e r e . A t y p i c a l s h o r t h i s t o r y o f t h e s a t e l l i t e m o tion i s p r e s e n t e d i n F i g u r e 3-2. The c o r r e s p o n d i n g s t r o b o s c o p i c phase p l a n e i s shown i n F i g u r e 3-3. The diagrams i n d i c a t e t h a t t h e damper i s c a u s i n g t h e a m p l i t u d e o f t h e m otion t o decay. S i m i l a r r e s u l t s f o r s e v e r a l cases a r e p r e s e n t e d i n F i g u r e s 3-4 and 3-5. Note t h a t t h e apparent c u r v e s i n t h e s e diagrams do not i n d i c a t e t h e a c t u a l m otion o f t h e s a t e l l i t e , but s e r v e o n l y t o i n d i c a t e t h e i n w a r d t r e n d o f s u c c e s s i v e p o i n t s . The number o f apparent c u r v e s i s o f no s p e c i a l s i g n i f i c a n c e . • I t may be p o i n t e d out t h a t a f t e r a c o n s i d e r a b l e p e r i o d o f t i m e s u c c e s s i v e p o i n t s i n t h e s t r o b o s c o p i c phase p l a n e f a l l p r o g r e s s i v e l y c l o s e r t o g e t h e r . The system t h u s approaches a p e r i o d i c s o l u t i o n or l i m i t c y c l e . F i g u r e s 3 - 6 t o 3 - 8 compare the l i m i t c y c l e s g i v e n by d i f f e r e n t dampers w i t h t h e p e r i o d i c s o l u t i o n s o b t a i n e d i n s e c t i o n 2 . 2 . 3 f o r t h e same v a l u e s o f e and K.. Of t h e t h r e e •8 4> Rad o i r T r 0/27T F i g u r e 3-2 S o l u t i o n o f e q u a t i o n o f m o tion i l l u s t r a t i n g t h e e f f e c t o f t h e damper 126 -.3 e = 2Tin, n=o,i,2' e = .2 K j = 1. K = . 0 5 o-= 3 . r*= 6 . o +10 f-5 10 -.2 -.1 8 .1 + -5 2 H Rad ° n Point representing the state of the system at 6=2Ttn„ • Limit cyclei F i g u r e 3-3 S t r o b o s c o p i c phase p l a n e o f the s o l u t i o n i l l u s t r a t e d i n F i g u r e 3-2 127 e = 0.2 0< • 1.0 K. = i 1.0 °2 0.01 «*= 3. 6. -.4 6 7 + •5 \10 13 19 16 22 25 ..o o -3 2 0 ^ 29p Q 36°-2 o \ 3|3 23 \ > 1 5 J8 P21 °«34 •' c§7 / 2 4 - - o " ° 2 7 30 a o •<>.... 20 17 14 + - 5 •+- Rad ^11 Point representing the state of the system at 9=2Ttn. Limit cycle. Apparent curves defined by groups of points. F i g u r e 3-4 T y p i c a l s t r o b o s c o p i c phase pl a n e o f a damped s a t e l l i t e 128 v n Point representing the state of the system at 0 = 2 ? ™ . • Limit cycle Apparent curves defined by groups of points. F i g u r e 3 - 5 T y p i c a l s t r o b o s c o p i c phase p l a n e o f a damped s a t e l l i t e G.Deg F i g u r e 3-7 L i m i t c y c l e s {K± = 1.0, e = 0.2) 132 p e r i o d i c s o l u t i o n s a v a i l a b l e f o r a g i v e n s e t o f parameters (e, K^) o n l y t h e one w i t h t h e s m a l l e s t a m p l i t u d e was s u i t a b l e f o r comparison. For s m a l l dampers (K^«l) and s u f f i c i e n t l y 2 2 h i g h n a t u r a l f r e q u e n c i e s » 3 ^ Q ) t h e two t y p e s o f s o l u t i o n a r e v i r t u a l l y i d e n t i c a l . The i n t r o d u c t i o n o f damping causes t h e l i m i t c y c l e t o l a g b e h i n d t h e p e r i o d i c s o l u t i o n o b t a i n e d f o r = 0. The v a r y i n g g r a v i t a t i o n a l g r a d i e n t and o r b i t a l a n g u l a r v e l o c i t y i n t e r a c t w i t h t h i s l a g and can o f t e n l e a d t o an i n c r e a s e i n t h e a m p l i t u d e o f t h e motion a l t h o u g h t h e n o n l i n e a r i t y o f t h e system p r e c l u d e s any d e f i n i t e p r e d i c t i o n o f t h i s n a t u r e . Note t h a t a v e r y low n a t u r a l f r e q u e n c y o f t h e damper can c r e a t e a problem because t h e c o e f f i c i e n t o f e q u a t i o n ( 3 . 1 3 ) w h i c h r e p r e s e n t s t h e s p r i n g " c o n s t a n t " becomes n e g a t i v e . . The motion o f t h e damper mass t h e n becomes un-s t a b l e and t h e e q u a t i o n s cease t o d e s c r i b e t h e system. 3 . 3 C o n c l u s i o n s The a d d i t i o n o f a damper t o a r i g i d s a t e l l i t e r e s u l t s i n t h e d i s a p p e a r a n c e o f t h e i n v a r i a n t m a n i f o l d s d i s c u s s e d i n C h a p t e r 2. The accompanying l i m i t c y c l e s a r e independent o f t h e i n i t i a l c o n d i t i o n s and, f o r s m a l l dampers, a r e n e a r l y i d e n t i c a l w i t h one o f t h e p e r i o d i c s o l u t i o n s o b t a i n e d f o r t h e undamped case. The e q u a t i o n s o f motion i n t h e two cases a r e s i m i l a r and t h e presence o f damping me r e l y causes t h e t r a j e c t o r y i n phase space t o s p i r a l i n w a r d s . The l i m i t s o f s t a b i l i t y d e t ermined i n C h a p t e r 2 may be r e g a r d e d as a p p r o x i m a t e l y v a l i d f o r t h e l i g h t l y damped s a t e l l i t e . The presence o f damping causes t h e s a t e l l i t e t o become s t a b l e e v e n t u a l l y , i r r e s p e c t i v e o f t h e i n i t i a l c o n d i -t i o n . However, i f t h e s t a t e o f t h e s a t e l l i t e can be ex-p r e s s e d by a p o i n t i n s i d e t h e a p p r o p r i a t e l i m i t i n g m a n i f o l d o f Chapter 2, and t h e damping i s s m a l l , t h e subsequent motion s h o u l d remain s t a b l e and approach t h e l i m i t c y c l e . I n d o i n g so t h e r e p r e s e n t a t i v e p o i n t d r i f t s i nwards a c r o s s the i n t e r m e d i a t e i n v a r i a n t s u r f a c e s which were determined f o r t h e undamped case. T h e r e f o r e , i t may be c o n c l u d e d t h a t t h e s m a l l a m p l i t u d e p e r i o d i c s o l u t i o n s o b t a i n e d f o r t h e undamped s a t e l l i t e r e p r e s e n t a l i m i t i n g case approached by a l l r e a l s a t e l l i t e s . 4. PLANAR LIBRATIONS OF A LONG FLEXIBLE SATELLITE 4.1 P r e l i m i n a r y Remarks G r a v i t y - g r a d i e n t s t a b i l i z a t i o n o f s a t e l l i t e s r e q u i r e s t h a t t h e i n e r t i a o f t h e s a t e l l i t e be l a r g e so t h a t t h e a v a i l -a b l e t o r q u e i s s u f f i c i e n t l y g r e a t t o overcome t h e e f f e c t s o f e x t e r n a l d i s t u r b a n c e s . To a c e r t a i n e x t e n t , t h i s problem has been overcome by t h e use o f t h e de H a v i l l a n d STEM ( S e l f -s t o r i n g T u b u l a r E x t e n s i b l e M o d u l e ) 2 ^ w h i c h i s ca p a b l e o f e x t e n d i n g a t u b u l a r boom up t o s e v e r a l hundred f e e t l o n g (Table I I I ) . The STEM boom i s t y p i c a l l y about one i n c h i n d i a m e t e r , or l e s s , w i t h a w a l l t h i c k n e s s o f about .002 i n c h e s . T h i s t h i n , s l e n d e r , t u b u l a r member i s q u i t e f l e x i b l e and hence s u s c e p t i b l e t o e x t e r n a l d i s t u r b a n c e s . Thus a l o n g boom whic h p r o v i d e s a l a r g e s t a b i l i z i n g e f f e c t can a l s o i n t r o d u c e s u b s t a n t i a l e x t e r n a l l y i n d u c e d f o r c e s . P r e l i m i n a r y a n a l y s i s 2 ^ i n d i c a t e s t h a t among t h e v a r i o u s forms o f d i s t u r b -ance ( s e c t i o n 1.1) t h e t h e r m a l d e f o r m a t i o n o f t h e boom i s l i k e l y t o have t h e most s i g n i f i c a n t e f f e c t on t h e performance. The problem I n g e n e r a l i s e x t r e m e l y complex as i t i n -v o l v e s t h e s o l u t i o n o f a s e t o f s i m u l t a n e o u s d i f f e r e n t i a l e q u a t i o n s w i t h a l a r g e number o f parameters. To i n i t i a t e t h e s t u d y o f such a d i f f i c u l t problem, a s i m p l i f i e d model i s c o n s i d e r e d . 135 TABLE I I I CHARACTERISTICS OF REPRESENTATIVE STEM CONFIGURATIONS 3 5 , 3 6 M a t e r i a l R a d i u s , a^ W a l l t h i c k n e s s , D e n s i t y , Thermal c o n d u c t i v i t y , k^ S p e c i f i c h e a t , c^ C o e f f i c i e n t o f t h e r m a l e x p a n s i o n , <*t A b s o r p t i v i t y ( s o l a r ) , <x s E m i s s i v i t y , . € b Bending s t i f f n e s s , 6*1 M a s s / l e n g t h , "m A l o u e t t e I S t e e l 0.475 0.006 0.286 26 0 .11 6 .5x10° 0 . 9 0 . 8 351 0 . 0 6 8 A l o u e t t e I I U n i t s B e r y l l i u m copper 0 . 2 5 0 . 0 0 2 0 . 3 2 -6 50 0.092 10x10 0.45 0.25 15.5 0.0142 i n c h e s i n c h e s 3 l b / i n -BTU/hr f t °F BTU/lb °F o F - l l b f t ' l b / f t T h i s c h a p t e r s t u d i e s t h e p l a n a r l i b r a t i o n s o f a s l e n d e r , f l e x i b l e s a t e l l i t e o f c o n s t a n t c r o s s - s e c t i o n under t h e i n f l u -ence o f s o l a r h e a t i n g . The s o l a r f l u x i s t a k e n t o be d i r e c t and c o n t i n u o u s , and t h e r e s u l t i n g bending i s assumed t o t a k e p l a c e i n t h e p l a n e o f t h e o r b i t . .The concept o f t h e phase space r e p r e s e n t a t i o n o f t h e motion i s extended t o a deform-a b l e system and t h e c o r r e s p o n d i n g 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 a r e o b t a i n e d . C h a r t s a r e p r e s e n t e d which i n d i c a t e t h e e f f e c t o f o r b i t e c c e n t r i c i t y , s o l a r a s p e c t a n g l e , and t h e s a t e l l i t e ' s 136 p h y s i c a l p r o p e r t i e s on t h e a l l o w a b l e d i s t u r b a n c e s f o r s t a b l e o p e r a t i o n . 4.2 F o r m u l a t i o n o f t h e Problem C o n s i d e r a s l e n d e r f l e x i b l e s a t e l l i t e w i t h c e n t r e o f mass a t S, deformed due t o s o l a r h e a t i n g , and e x e c u t i n g p l a n a r l i b r a t i o n a l motion w h i l e moving i n an e l l i p t i c o r b i t about t h e c e n t r e o f f o r c e 0 ( F i g u r e 4-1). The k i n e t i c energy f o r an element o f t h e s a t e l l i t e l o c a t e d a t % * ( 5 , t ) (4.1) can be w r i t t e n as (4.2) F i g u r e 4-1 Geometry o f motion of f l e x i b l e s a t e l l i t e 138 Because S i s t h e c e n t r e o f mass, t h e r e l a t i o n s ' fldmy = f i d * = f } \ =fi\ = 0 p e r m i t w r i t i n g t h e t o t a l k i n e t i c energy as (4.3) (4.4) where The q u a n t i t y (4.6) r e p r e s e n t s t h e a n g u l a r momentum of t h e s a t e l l i t e w i t h r e s p e c t t o t h e r o t a t i n g x, z-axes. The c h o i c e ^ = o <^> i n d i c a t e s t h a t the axes r o t a t e i n a manner e q u i v a l e n t t o t h a t o f axes w h i c h a r e f i x e d i n a r i g i d body. T h i s d e v i c e s e p a r a t e s t h e l i b r a t i o n a l degree o f freedom, d e s c r i b e d by the c o - o r d i n a t e uV , and t h e v i b r a t i o n a l degrees o f freedom w h i c h employ t h e x , z - c o - o r d i n a t e s . I f t h e x,z-axes a r e t a k e n t o be t h e p r i n c i p a l axes o f t h e deformed body, t h e p o t e n t i a l energy due t o t h e 139 g r a v i t a t i o n a l f i e l d i s ( e q u a t i o n (2.7)) There i s a l s o t h e e l a s t i c p o t e n t i a l energy , J ._ / / ds (4.9) Us i n g Lagrange's e q u a t i o n s t h e mot i o n o f t h e c e n t r e o f mass can be d e s c r i b e d by t h e e q u a t i o n s 9 L (4.10) +ra(i-35ih>Ji (4.11) w h i l e t h e l i b r a t i o n a l m o tion i s governed by which r e q u i r e s t h a t t h e d e f l e c t i o n o f t h e beam be known. T h i s i s a v e r y complex problem. As a f i r s t a p p r o x i -m a t i o n l e t 140 To t h i s degree o f a p p r o x i m a t i o n f « and hence t h e c o - o r d i n a t e s o f an element o f t h e boom a r e g i v e n by (4.14) (4.15) The d e f l e c t i o n o f an i s o l a t e d beam w i t h no e x t e r n a l f o r c e s p r e s e n t i s g i v e n by t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n ' 3 z r 2s- £1 IK (4.16) w h i c h has t h e s o l u t i o n oo where t h e mode shapes, X ^ ( s ) , and t h e a m p l i t u d e s , A ^ ( t ) s a t i s f y t h e d i f f e r e n t i a l e q u a t i o n s Ai + ^ Al = ° Clh d s * J (4.17) (4.18) (4.19) 141 s u b j e c t t o t h e a p p r o p r i a t e i n i t i a l and boundary c o n d i t i o n s . The X^ may be thought o f as g e n e r a l i z e d c o - o r d i n a t e s and t h e A^ as t h e a m p l i t u d e s . The X^ a r e o r t h o g o n a l ; t h a t i s (4.20) also., f o r a f r e e - f r e e beam, so t h a t t h e x, z-axes remain t h e p r i n c i p a l axes o f t h e deformed body. The k i n e t i c energy o f v i b r a t i o n may be w r i t t e n i n terms o f t h e A. (4.22) fit £ M « < H The e x p r e s s i o n f o r t h e e l a s t i c p o t e n t i a l energy 142 (4.23) may be i n t e g r a t e d by p a r t s t w i c e t o g i v e L i L ' 4 . 2 (4.24) J S i n c e t h e s a t e l l i t e i s c o m p l e t e l y f r e e , t h e bending moment, CIh& and t h e sh e a r f o r c e , ^ ( ^ r i , | ^ ) > a r e z e r o a t t h e e n d s ^35 and hence u, (4.25) CO j <4\ h The i n e r t i a s o f t h e beam a r e g i v e n a p p r o x i m a t e l y by * Z I I = - L A i T c = * fs*\ = I, (4.26) 143 CO * * T (4.26) c o n t ' d yy xxx' -"-zz ' 4 — The e x p r e s s i o n s f o r t h e 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 may-th e n be w r i t t e n as 1=1 + {>! c=/ and t h e r e f o r e t h e e q u a t i o n s o f motion g o v e r n i n g t h e a m p l i -t u d e s o f t h e v i b r a t i o n a l modes a r e (4.27) (4.28) Here t h e F^ r e p r e s e n t t h e g e n e r a l i z e d f o r c e s due t o t h e r m a l bending. The dependence o f t h e s e f o r c e s on t h e 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 l e a d s t o c o u p l i n g between e q u a t i o n s (4.12) and ( 4 . 2 8 ) . T h i s a n a l y s i s p a r a l l e l s t h a t o f E t k i n and Hughes-^ who t o o k 2 » e 2 , JJJT^ . The n a t u r a l f r e q u e n c i e s o f a u n i f o r m , f r e e - f r e e beam 144 37 can be dete r m i n e d f r o n r CO; = (kL) (4.29) where Cos(kl\ Co5^(l(L). = I. ( 4 . 3 0 ) The v a l u e s o f ( k L ) ^ a r e ( k L ) . (4.3D 1 4.73004 2 7.85320 3 10.99561 4 14.13717 5 17.27876 i £ 6 ( 2 i + 1)^. For t h e r e p r e s e n t a t i v e STEM c o n f i g u r a t i o n s (Table I I I ) the l o w e s t n a t u r a l f r e q u e n c y i s . * _ I7S-X 10 ( 4 . 3 2 ) (L i n f e e t ) w h i c h f o r l e n g t h s t y p i c a l o f p r e s e n t p r a c t i c e (50-500 f e e t ) l e a d s t o v a l u e s f o r t h e fundamental p e r i o d o f from 3.75 t o 375 seconds. I t may be no t e d t h a t the p e r i o d o f t h e v i b r a t i o n i s much l e s s t h a n t h a t o f t h e o r b i t a l m o t i o n . The r e s u l t s o f Chapter 2 showed t h a t t h e v a l u e o f ip 145 i s o f t h e same o r d e r as 6 . A l s o fA/r i s a p p r o x i m a t e l y e q u a l * 2 t o 8 so t h a t i n e q u a t i o n (4.28) a l l terms due t o t h e g r a v i t a t i o n a l f i e l d can be c o m p l e t e l y i g n o r e d compared t o th e r e l a t i v e l y h i g h n a t u r a l f r e q u e n c y o f t h e boom. The d e f l e c t i o n o f t h e boom can be approx i m a t e d by t h e normal modes t h u s r e d u c i n g the e q u a t i o n s o f motion c o r r e s p o n d i n g t o t h e e l a s t i c degrees o f freedom t o (4.33) 4.3 Thermal A n a l y s i s o f t h e Boom At t h i s s t a g e i n f o r m a t i o n c o n c e r n i n g t h e d e f o r m a t i o n o f t h e s a t e l l i t e under t h e i n f l u e n c e o f s o l a r h e a t i n g i s e s s e n t i a l t o proceed f u r t h e r . F i g u r e 4-2 i l l u s t r a t e s t h e assumed c r o s s - s e c t i o n o f t h e f l e x i b l e s a t e l l i t e w h i c h may be thought o f as a p p r o x i m a t i n g t h e STEM. C o n s i d e r an element o f t h e beam as shown i n F i g u r e 4-3. T a k i n g t h e t h i c k n e s s o f t h e w a l l t o be much l e s s t h a n i t s r a d i u s , i g n o r i n g l o n g i t u d i n a l c o n d u c t i o n and p e r f o r m i n g a heat b a l a n c e f o r the element g i v e s t h e e q u a t i o n < 5n z T h i s e q u a t i o n r e p r e s e n t s a s i g n i f i c a n t improvement on t h e 3 5 work o f E t k i n and Hughes^' i n t h a t t h e e f f e c t o f t h e r m a l c o n d u c t i v i t y i s c o n s i d e r e d . The t h e r m a l i n p u t t o the boom from t h e sun can be 146 F i g u r e 4-2 Assumed c r o s s - s e c t i o n o f s a t e l l i t e boom Sun F i g u r e 4-3 Heat b a l a n c e f o r an element o f t h e s a t e l l i t e boom 148 w r i t t e n as 1 O ( 4 . 3 5 ) elsewhere where 0 * i s t h e a n g l e between t h e l o n g i t u d i n a l a x i s o f t h e boom and t h e sun. T h i s may be w r i t t e n as a F o u r i e r s e r i e s co ci = Q <x Sin 6 J Cs h Cos n n where ft / 2 n = 0 n = 1 n even, ^ 0 n odd, ^  1. ( 4 . 3 6 ) ( 4 . 3 7 ) To determine t h e amount o f r a d i a t i o n i n c i d e n t on t h e i n t e r i o r o f t h e boom c o n s i d e r t h e r a d i a t i o n e m i t t e d by an element o f a r e a (dA ) on t h e i n s i d e o f t h e tube ( F i g u r e 4 - 4 ) 6 J Q X (A+n)]. ( 4 . 3 8 ) Assuming t h a t t h e r a d i a t i o n obeys Lambert's l a w , t h e r a d i a -t i o n w h i c h f a l l s on t h e second element o f a r e a (dA. ) i s F i g u r e 4-4 Geometry of r a d i a n t heat t r a n s f e r i n t h e i n t e r i o r o f the s a t e l l i t e boom 150 cJAe d A i n - 4 6, <T ,4 (4.39) where d&j i s the s o l i d ang le subtended by d A i n as seen from dA. d c o = i n (4.40) From F i g u r e 4-4 i t i s ev iden t t ha t X*. = X r — A i r (4.41) (4-42) and (4.43) so t ha t c**K = 0 - fr) (4.44) The r a d i a t i o n i n c i d e n t on dA. f rom dA i s then g i v e n by (4.45) and s i n c e 151 <JA e - G ^ J ^ O / A r ( 4 . 4 6 ) (4.47) /-c ©5 4«4 S o l u t i o n o f t h e Heat Bal a n c e E q u a t i o n R e p r e s e n t i n g t h e te m p e r a t u r e d i s t r i b u t i o n i n t h e boom by a s e r i e s o f the form 00 where i t may be assumed t h a t l e a d s t o t h e s e r i e s f T f c ^ + a ) ] * T i o (t) HTji)[_Tk/„£)Cosnfa+siX (4.50 Hence t h e r a d i a t i o n i n c i d e n t upon t h e i n t e r i o r i m m e d i a t e l y f o l l o w i n g e m i s s i o n i s 152 2TT T 4 4 .4 T 3 r~ T^ Co3ha 1 V + * v Z _ I- J (4.5D A f r a c t i o n , € b , o f t h i s r a d i a t i o n i s abs o r b e d , r V f r 4 A T 3 V T ^ ^ " ^ l and a q u a n t i t y (1 - 6.^) i s r e f l e c t e d , n f l z (4.52) (4.53) I f t h e r e f l e c t i o n i s assumed d i f f u s e , t h e r e f l e c t e d r a d i a t i o n i s a g a i n i n c i d e n t upon t h e i n t e r i o r o f t h e boom i n a manner analogous t o t h a t employed i n t h e d e r i v a t i o n o f e q u a t i o n (4.51). The t o t a l i n c i d e n t heat f l u x on the i n -t e r i o r o f t h e boom w a l l i s t h e n g i v e n by I n' ffiso Y)«l v (4.54) Now, 153 I (4.56) v4n* so t h a t <x> ^ «-T x 4 ^ V T C o 3 n n (4.57) The f i n a l form o f t h e heat b a l a n c e e q u a t i o n can now be w r i t t e n ' b (4.58). 3 r 2-^ " ^ ~o 5 Kin - 3€br\A 7"T^ C w h i c h i s of t h e form 154 a> ) C C o s n a = O where the n depend on t i m e , but are independent o f JTL . T h i s l e a d s t o the s e r i e s of equat ions where r z n = 1 n even; ^ 0 n odd , ^ 1. (4.59) (4.60) (4.61) (4.62) The f i r s t o f these equat ions i s n o n - l i n e a r . The rema in ing e q u a t i o n s , a l though l i n e a r , c o n t a i n t ime v a r y i n g c o e f f i c i e n t s which depend on the s o l u t i o n o f ( 4 .60 ) . F o r t u n a t e l y i t i s not necessa ry to s o l v e the equat ions i n d e t a i l . C o n s i d e r a b l e i n f o r m a t i o n can be ob ta ined by de te rm in ing the approximate t ime constant and the s teady s t a t e s o l u t i o n . 155 The t i m e c o n s t a n t o f t h e n t h term o f t h e s e r i e s ( 4 . 4 8 ) i s g i v e n by r . -and t h e s t e a d y s t a t e s o l u t i o n f o r t h e same term i s 3 5 The a n a l y s i s o f E t k i n and Hughes, r e f e r r e d t o e a r l i e r , i g n o r e s t h e term i n v o l v i n g c o n d u c t i v i t y i n t h e denominator o f e q u a t i o n ( 4 . 6 4 ) . T h i s i s u s u a l l y t h e dominant term and hence t h e i r a n a l y s i s o v e r e s t i m a t e s t h e e f f e c t o f s o l a r h e a t i n g . F o r t h e r e p r e s e n t a t i v e c o n f i g u r a t i o n s (Table I I I ) Y * -L±— ( 4 . 6 5 ) and t h e r e f o r e Xt I s v e r y s h o r t , o f t h e o r d e r o f 12 seconds. The s h o r t t i m e c o n s t a n t i m p l i e s t h a t t h e F^ depend d i r e c t l y on t h e s o l a r a s p e c t a n g l e as t h e t h e r m a l l a g i s n e g l i g i b l e compared t o t h e l i b r a t i o n a l and o r b i t a l p e r i o d s . The presence o f i n t e r n a l damping w i l l cause the complemen-t a r y s o l u t i o n s o f e q u a t i o n ( 4 . 3 3 ) t o damp out l e a v i n g o n l y t h e f o r c e d m otion o f t h e boom. As t h e F^ depend e s s e n t i a l l y 156 on t h e p o s i t i o n o f the s a t e l l i t e , t h e r a t e o f change o f 0 • w h i c h i s o f t h e o r d e r o f t h e o r b i t a l p e r i o d , t h e term i n (4.33) may a l s o be n e g l e c t e d so t h a t A; - ^ (4.66) Thus t h e i n s t a n t a n e o u s c o n f i g u r a t i o n o f t h e s a t e l l i t e i s r e p r e s e n t e d by i t s s t e a d y s t a t e d e f l e c t i o n c o r r e s p o n d i n g t o t h e o r i e n t a t i o n . T h i s i n e f f e c t e l i m i n a t e s e q u a t i o n s (4.33) l e a v i n g o n l y (4.12), c o n t a i n i n g v a r i a b l e i n e r t i a s , t o be s o l v e d . 4»5 Thermal D e f l e c t i o n o f t h e Boom C o n s i d e r an element o f the boom o f w i d t h a^dO. ( F i g -u r e 4-5). The l e n g t h o f t h e s t r i p i s J J n ) = (Rc + a t C c S f i ) ^ . (4.67) I f & r e f denotes t h e o r i g i n a l l e n g t h o f t h e s t r i p a t a r e f e r -ence t e m p e r a t u r e , r^ef> t h i s l e n g t h can be o b t a i n e d by a l t e r -i n g t h e te m p e r a t u r e and s t r e s s i n g t h e m a t e r i a l 4(a)-Jr(/[i+*t(T(n)-Tn/)+ M»] (4.68) hence (4.69) 157 F i g u r e 4-5 Thermal d e f l e c t i o n of the s a t e l l i t e boom 158 The l o n g i t u d i n a l f o r c e on t h e element i s [T J <r(s±)ablob da (4.70) where = jPfff/z) = mean l e n g t h o f t h e element. D u r i n g o r b i t a l motion t h e l o n g i t u d i n a l f o r c e on t h e s a t e l -l i t e i s n e g l i g i b l y s m a l l so t h a t <<ref w h i c h r e p r e s e n t s t h e l o n g i t u d i n a l t h e r m a l e x p a n s i o n . The moment produced by t h e s t r e s s e s on t h e s e c t i o n (4.72) I f the frequency of the thermal driving force i s much lower than the natural frequency of the beam, the beam w i l l be i n equilibrium and the moment w i l l be zero. That i s 159 <£>h a / = ft Jref a* ( 4 . 7 3 ) L e t t h e £ ^ , Y\ ^ - c o - o r d i n a t e system be d e f i n e d so t h a t t h e fc^, V |^-plane l i e s i n the pl a n e o f mot i o n w i t h t h e sun on t h e n e g a t i v e V ^ - a x i s ( F i g u r e 4 - 6 ) . From elementary c a l c u l u s L_ = ' •*/< m I (4.74) o r where 0(. T a k i n g f= [J^t + 4ekirJ30 (A=£k^ (4.76) g i v e s (4.75) yo) = sjfi°) * ° (4-77) ( 4 . 7 8 ) 1 6 0 T h i s s o l u t i o n i s i l l u s t r a t e d i n F i g u r e 4-6. I f s i s t h e a r c l e n g t h o f t h e curve Mw  + T) (4.79) and t h e s o l u t i o n can be w r i t t e n i n t h e p a r a m e t r i c form s. = I 5m" ~T«nh (*/**)] % = Jt* Cosh (S/Jt*) t (4.80) (4.81) A s a t e l l i t e o f g i v e n l e n g t h can be e a s i l y f i t t e d a l o n g an a r c o f t h e cur v e i n F i g u r e 4-6, hence i t s c e n t r e of,mass, o r i e n t a t i o n o f p r i n c i p a l a x e s , and t h e c o r r e s p o n d -i n g moments o f i n e r t i a can be de t e r m i n e d as i n d i c a t e d i n F i g u r e 4-7. The r e s u l t s o f t h i s a n a l y s i s a r e p l o t t e d i n F i g u r e s 4-8 and 4-9. The v a r i a t i o n i s e x p r e s s e d by t h e f a c t o r s L - I tain I -tt (4.82) where I r e p r e s e n t s t h e i n e r t i a o f t h e r i g i d s a t e l l i t e about the x- and y-axes. I t i s i n t e r e s t i n g t o note t h a t I - I v v min i s n e a r l y t h r e e t i m e s I zz and max 1 6 2 F i g u r e 4-7 I l l u s t r a t i o n of the p r i n c i p a l axes o f t h e d e f l e c t e d s a t e l l i t e F i g u r e 4-8 Maximum i n e r t i a v a r i a t i o n s as f u n c t i o n s o f boom l e n g t h 164 F i g u r e 4-9 R e l a t i v e i n e r t i a v a r i a t i o n as a f u n c t i o n o f t h e a n g l e between the sun and t h e x - a x i s 165 x 5-so t ha t 1zz. z zMa/ (4.84) 2 Not ing aga in tha t J J dt de J 2 _ J* and r e p r e s e n t i n g r y Je d$ lyy * rr(i-2I5^,s) equation (4.12) becomes 4 e5fn©0"2TS,'n*/5)} (4.85) (4.86) (4.87) 166 + 3 (I ~ 41 Sth*j8)5rncpCo« f » O . (4.87) V ' I ' c o n t ' d 4.6 S t a b i l i t y A n a l y s i s The gove rn ing n o n - l i n e a r , non-autonomous d i f f e r e n t i a l equa t i on (4.87) w i th p e r i o d i c c o e f f i c i e n t s i s o f the same form as (2.14). Hence the s o l u t i o n of the equa t ion when r e p r e s e n t e d i n the th ree d imens iona l phase space genera tes i n t e g r a l man i f o lds w i th p r o p e r t i e s s i m i l a r t o those d i s c u s s e d i n Chapter 2. The s o l u t i o n of the equa t ion was ob ta ined f o r a wide range o f i n i t i a l c o n d i t i o n s u s i n g a d i g i t a l computer. The s a t e l l i t e was taken to be s t a b l e i f i t d i d not tumble w i t h i n f i f t y o r b i t s . The t r a j e c t o r y s t a r t i n g f rom s t a b l e i n i t i a l c o n d i t i o n s genera tes a s u r f a c e i n phase space o f the form shown i n F i g u r e 4-10. I t i s s t i l l s u f f i c i e n t to extend the phase space co-o rd i na t e 6 on l y up t o 271 because when a t r a j e c t o r y a r r i v e s at 6 = 2K w i th c e r t a i n v a lues o f p and ip' , i t may be extended by c o n s i d e r i n g another t r a j e c t o r y s t a r t i n g from 9 = 0 w i th the same va lues o f lp and ^ ' . T h i s i s i d e n t i c a l w i th the procedure adopted i n Chapter 2. The sea r ch f o r the l i m i t i n g i n i t i a l c o n d i t i o n s which r e s u l t i n s t a b l e l i b r a t i o n a l mot ion i s thus a sea rch f o r 168 the largest such surface. Any state of motion which l i e s with-i n t h i s surface generates a tra j e c t o r y , and hence a new sur-face, which remains inside the l i m i t i n g manifold. Therefore, the region enclosed by the l i m i t i n g surface i s the region of s t a b i l i t y . The usefulness of such a phase space was explained i n section 2.4. For given e c c e n t r i c i t y , s a t e l l i t e character-i s t i c s , and solar aspect angle i t provides a l l possible com-binations of disturbances to which the s a t e l l i t e may be subjected without causing i t to tumble. The effect of solar aspect angle on the s t a b i l i t y region i s i l l u s t r a t e d i n Figures 4-11 and 4-12 where cross-sections of the s t a b i l i t y region at several values of the orbit angle are presented. A convenient condensation of data may be effected by pl o t t i n g the intercepts of the (f/ -axis with the phase space cross-sections at 0 = 0 as shown i n Figures 4-13 and 4-14. It i s apparent that with increasing e c c e n t r i c i t y the s t a b i l -i t y region decreases i n size and beyond a c r i t i c a l value ceases to exist. The accuracy with which the l i m i t i n g manifolds were determined i s approximately the same as discussed i n Chap-ter 2. The calculations were carried out with the same precision and therefore the error i n the region of s t a b i l i t y i s approximately ± , 0 3 units i n (p'. Several symmetry properties exhibited by equation (4.^7) simplify the presentation of the numerical r e s u l t s . F i g u r e 4-11 V a r i a t i o n o f t h e c r o s s - s e c t i o n o f a l i m i t i n g i n v a r i a n t s u r f a c e w i t h o r b i t a n g l e (e = 0.2, OC = 0°, L* = 1) 170 di de -90 -60° •30° 30 60° 90° F i g u r e 4-12 V a r i a t i o n o f t h e c r o s s - s e c t i o n o f a l i m i t i n g i n v a r i a n t s u r f a c e w i t h o r b i t a n g l e (e = 0 . 2 , = 30° , L* = 1) 172 173 175 179 Note tha t s u b s t i t u t i n g 0( + IX f o r oC r e s u l t s i n no change i n the t r a j e c t o r i e s as the equa t ion i s i n v a r i a n t under t h i s s u b s t i t u t i o n . For nega t i v e (X , say o< = - c * e > t n e s o l u t i o n f o r (p ob t a ined by i n t e g r a t i n g i n the d i r e c t i o n o f i n c r e a s -i n g 9 f rom the i n i t i a l c o n d i t i o n s ij> = 0, = ^ , i s e x a c t l y o p p o s i t e i n s i g n to tha t ob ta ined by i n t e g r a t i n g backwards f rom the same i n i t i a l c o n d i t i o n s but w i th o( = ^ . e Hence the t r a j e c t o r i e s formed i n the phase space f o r -1>< become m i r r o r images o f those ob ta ined f o r + <X. In p a r t i c -u l a r , the c r o s s - s e c t i o n s f o r 0( - - #_ at 6 = 0,71 are m i r r o r images about the i//-axis o f the c r o s s - s e c t i o n s ob ta ined f o r <tf = + 0/ . Thus the i n t e r c e p t s o f the tP*'-axis w i th the e l i m i t i n g s u r f a c e a t 9 = 0 do not change when <X changes s i g n or i n c r e a s e s by 1 8 0 ° . Hence the s t a b i l i t y r e g i o n v a r i e s p e r i o d i c a l l y w i th ex. As the s o l a r aspec t ang le would va ry due t o o r b i t p r e c e s -s i o n and the mot ion o f the p l ane t about the s u n , on l y those r e g i o n s o f the s t a b i l i t y cha r t s which a c t u a l l y o ve r l ap guarantee l ong term s t a b i l i t y ( F igure 4-15) . The s p i k e s appear ing i n F i g u r e s 4-13 and 4-14 r ep resen t the secondary s t a b i l i t y r e g i o n s . They are a s s o c i a t e d w i th d i f f e r e n t p e r i o d i c s o l u t i o n s to equa t ion ( 4 . 8 7 ) , and appear as s m a l l h e l i c a l r e g i o n s su r round ing the main s t a b l e r e g i o n of phase space . As the narrowness o f these secondary r e g i o n s makes them u n s u i t a b l e f o r any p r a c t i c a l o p e r a t i o n , the appea r -ance o f sp i ke s reduces the p r a c t i c a l upper l i m i t on e c c e n t r i c i t y f o r s t a b l e o p e r a t i o n . 180 181 _2 i 1 1 1 i I 0 1 2 -3 4 -5 e Figure 4-15-ii Range of value of the derivative when = 9 = 0 f o r long term s t a b i l i t y (L* = 2) 182 4 . 7 Concluding Remarks Based on the analysis the following observations may be made; (i) The method employed i n t h i s chapter i s r e s t r i c t e d to "short" s a t e l l i t e s f o r two reasons. The fundamental frequency of the beam must be much higher than the o r b i t a l frequency i n order that the approximation employed i n equation ( 4 . 3 3 ) remain v a l i d . Also, the relations ( 4 . 8 4 ) which represent the variable i n e r t i a s introduce considerable error f o r long s a t e l l i t e s . ( i i ) The s t a b i l i t y l i m i t s for a slender f l e x i b l e s a t e l l i t e , free to deform under the action of solar heating, have been obtained using the concepts of phase space and i n t e g r a l manifolds. This determines the c r i t i c a l values of i n i t i a l disturbances to which a s a t e l l i t e may be subjected without causing i t to tumble. ( i i i ) In general a small value of e c c e n t r i c i t y would help to ensure s t a b i l i t y . The c r i t i c a l value of e c c e n t r i c i t y i s affected by the dimensionless length L* , as well as the solar aspect angle, 0( . When L* i s s p e c i f i e d , the c r i t i c a l e c c e n t r i c i t y varies p e r i o d i c a l l y with o< i n such a manner that i t increases with, increasing (X* for 0 < o ( < 9 0 0 . For the cases considered, i t appears that g r a v i t a t i o n a l gradient s t a b i l i z a t i o n of an undamped s a t e l l i t e i s not possible under any circumstances for e > 0 . 4 2 5 (Figure 4 - 1 2 ) . (iv) The f l e x i b l e nature of the s a t e l l i t e causes a reduction i n the size of the s t a b i l i t y region for almost a l l 183 va l ues o f <X . T h i s r e d u c t i o n i s not s e ve r e . I t i s conc luded tha t s a t e l l i t e f l e x i b i l i t y o f t h i s na ture does not have a s t r o n g d e s t a b i l i z i n g i n f l u e n c e . (v) The c r i t i c a l e c c e n t r i c i t y based on the l ong term s t a b i l i t y a n a l y s i s would be c o n s i d e r a b l y lower than the one s p e c i f i e d above i n ( i i i ) . T h i s i s because o f the p e r i o d i c v a r i a t i o n of the s t a b i l i t y r e g i o n w i t h (X.. ( v i ) The presence o f s p i k e s f u r t h e r reduces the p r a c t i c a l upper l i m i t on e c c e n t r i c i t y . I f the depth o f the s t a b i l i t y diagrams ( F igures 4 - H , 4 - 1 2 , 4 -13 ) 1st i n t e r p r e t e d as a measure o f the d i s t u r b a n c e which the s a t e l l i t e would t o l e r a t e w i thout becoming u n s t a b l e , i t i s apparent t ha t even q u i t e moderate va lues o f e c c e n t r i c i t y s e r i o u s l y reduce the a b i l i t y o f the s a t e l l i t e to w i ths t and e x t e r n a l d i s t u r b a n c e s . The e f f e c t o f i n c r e a s i n g L i s s i m i l a r but not as s e ve r e . ( v i i ) I t must be p o i n t e d out tha t e c l i p s e s o f the sun by the p l ane t were not cons ide r ed i n t h i s a n a l y s i s and may c o n t r i b u t e to a f u r t h e r l o s s of s t a b i l i t y . These occu r rences would e x c i t e the v i b r a t i o n a l modes o f the boom and a more d e t a i l e d a n a l y s i s o f the decay of the v i b r a t i o n s would be r e q u i r e d . 5. . TWO .DIMENSIONAL MOTION OF AN AXI-SYMMETRIC SATELLITE IN A CIRCULAR ORBIT 5.1 Introductory Remarks The review of the l i t e r a t u r e (Section 1.2) suggests that the planar motion of a r i g i d s a t e l l i t e i n a gravity-gradient f i e l d has been the subject of considerable i n v e s t i -gation. In contrast, the dynamical study of a s a t e l l i t e executing l i b r a t i o n a l motion out of the o r b i t a l plane has received comparatively l i t t l e attention. Such an i n v e s t i -ng gation i s important because, as pointed out by Kane, for large amplitudes the transverse motion i s strongly coupled with that i n the plane. The lack of information may be partly attributed to the fact that the governing equations of motion are non-l i n e a r , non-autonomous, and coupled. They also involve a large number of parameters and hence are not amenable to any simple concise analysis. Some s i m p l i f i c a t i o n of the problem i s achieved by r e s t r i c t i n g the s a t e l l i t e to move i n a c i r -13 cular o r b i t . For t h i s case, as indicated by Auelmann, ^ closed zero-velocity curves exist under certain conditions which l i m i t the amplitude of motion. In t h i s chapter, the s t a b i l i t y bounds f o r coupled l i b r a t i o n a l motion of an axi-symmetric s a t e l l i t e i n a c i r c u -l a r orbit are obtained numerically. The zero-velocity curves suggest possible regions of s t a b i l i t y and i n s t a b i l i t y . Regu lar and e rgod i c t ypes o f s t a b l e mot ion are d i s c u s s e d and the behav iour o f the system i n the t r a n s i t i o n r e g i o n i s e x p l a i n e d . Us ing the concept o f an i n v a r i a n t s u r f a c e , i t i s shown tha t s t a b l e mot ion can r e s u l t even when the zero v e l o c -i t y curves are open.. ^ L im i t i ng i n v a r i a n t s u r f a c e s are p r e -sen ted and p rov ide a comprehensive summary o f the i n i t i a l d i s t u r b a n c e s to which a s a t e l l i t e may be sub j e c t ed .w i t hou t caus ing i t t o become u n s t a b l e . 5.2. Fo rmu la t i on o f the Problem Cons ide r a r i g i d s a t e l l i t e w i th mass cen t re at S i n an o r b i t about the cen t re o f f o r c e 0 ( F igure 5-1). Let S-xyz be the p r i n c i p a l body axes o f the s a t e l l i t e and the t r i a d S-x „ y z„ be chosen so tha t the z -ax i s i s d i r e c t e d o J o o o outward a long the l o c a l v e r t i c a l and the y Q - a x i s i s p a r a l l e l to the o r b i t a l angu l a r momentum v e c t o r . The, p o s i t i o n o f the mass cen t r e i s g i v e n by the d i s t a n c e r between 0 and S and the o r b i t ang le 6. The o r i e n t a t i o n o f the s a t e l l i t e may be s p e c i f i e d by a set o f r o t a t i o n s taken i n the f o l l o w i n g o r d e r : a r o t a t i o n , , about the y Q - a x i s , g i v i n g the x^y^z^-axes; a r o t a t i o n , 0 , about the x-^-axis r e s u l t i n g i n the X2^2Z2 t r^-a<^5 a n c * a r o t a t i o n , X , about the Zg-ax i s which y i e l d s the p r i n c i p a l body axes x y z . Us ing the p r i n c i p a l axes the k i n e t i c energy of the s a t e l l i t e can be w r i t t e n as 18? N o t i n g t h a t u) = <jf> Cos X + (& + j>) C o ^ ?^ so t h a t ( 5 . 1 ) = L-0J/>»A + (e-h^)coj^ COJ \ ( 5 . 2 ) + :-L4> (Co^z\ Txx -f J/**A Iyy) + <jt>(& + f)C<>S<fr5fo\ C*2\(lX/-Iyy) ( 5 . 3 ) To determine the p o t e n t i a l energy , c o n s i d e r an element o f mass, dm, w i th co-o rd ina t e s x , y , z , . The d i s t a n c e between the mass element and the cen t re o f f o r c e can be w r i t t e n as ./t = [(* * rJ„t + (y <- rjf , , rj^f] » , 5.4) 188 where th e d i r e c t i o n c o s i n e s J t v , , Z„ between t h e outward x y z l o c a l v e r t i c a l and the x y z - a x e s a r e ix = -SintpCosX + CosipSinjiSmX Jhj = 5fntfj5ln\ + C<&<f>S?/10C*3 A K = Cos p Cos ft (5.5) The p o t e n t i a l energy o f t h e s a t e l l i t e i s 1* U = -A -v2 (5.6) As S i s t h e c e n t r e o f mass - O (5.7) and s i n c e xyz a r e p r i n c i p a l axes 189 With t h i s the expression for the potential energy s i m p l i f i e s U = -W*" * # r \lAi-tf>p-qh#'-3fi*i  l5'9> Now, (5.10) so that there results U = - r ^ + & (Txx + T y y + I „ ) + 4 Si« if1 Cos (J; 5m 0 5 in A Cos A (ixx " Tyy) 190 (5.11) z / O cont'd + Cc/ip Co* f ( l x / + l y y " I * For a slender axi-symmetric s a t e l l i t e I x x = Iyy = T > I Z z (5.12) hence equations (5.3) and (5.11) assume the much simpler forms T = -j t\ (r 2 + + \ I (t* + (®+ 4»/C»> ) N ? (5.13) U = - 7 ^ r (5.14) The generalized momenta can be expressed as (5.15) »v - -If - ^ 191 Because = 0 i n t h i s c a s e , the momentum conjugate to the co-o rd ina t e A i s c ons t an t . T h i s momentum can be i d e n t i f i e d w i th the s p i n o f the s a t e l l i t e about the z - a x i s , ^ = J^^X - (© + ^ )5^ <6J= constant. (5.16) For a non-sp inn ing s a t e l l i t e the constant must be z e r o , t h e r e f o r e j = x ^ ( r V re*) +11 (j>\ is-") Us ing the Lagrang ian f o r m u l a t i o n the gove rn ing equa -t i o n s o f mot ion f o r the ijj and 0 degrees of freedom can be w r i t t e n as (5.18) 192 N o t i n g t h a t til • 2 ii " p = <f>'e + and f o r a c i r c u l a r o r b i t © - O hence t h e e q u a t i o n s o f motion may be w r i t t e n as // (5 .21 ) (5 .22 ) + 3ft 5fh Cosip = o <f>" -h fo'+'f* 3KiG**pjSf>lfiG>3<f> = 0 ^ ( 5 . 2 3 ) 5.3 The H a m i l t o n i a n and Z e r o - V e l o c i t y Curves I g n o r i n g t h e r , 8, and ^  c o - o r d i n a t e s t h e L a g r a n g i a n f o r t h e system can be w r i t t e n as 193 ( 5 . 2 4 ) Thus f o r a c i r c u l a r o r b i t , the Lagrang ian f u n c t i o n c o r r e -sponding to the l i b r a t i o n a l mot ion does not i n v o l v e t ime e x p l i c i t l y hence the co r r e spond ing Hami l t on i an i s a constant o f mot ion and i s g i v e n by H = Hfrii - t and u s i n g ( 5 . 2 1 ) e q u a t i o n ( 5 . 2 5 ) may be r e w r i t t e n as 13 Auelmann's paper ' c on t a i n s an e r r o r i n the statement o f t h i s e q u a t i o n . D e f i n i n g a new v a r i a b l e y)' =. (jJ Cc5 <jt> ( 5 . 2 7 ) equa t ion ( 5 . 2 6 ) becomes 0'Z+ $ = C*3*$(H 3KfCos*<p)+ CHt ( 5 . 2 8 ) 194 Setting 0 ' = = 0 gives the zero-velocity curves for the motion { \ty \ , j0|<f7T/2) which are presented f o r various values of i n Figure 5 - 2 . Since the right hand side of (5.28) i s a maximum at 0 = ip = 0, the sum of the squares of the v e l o c i t i e s i s positive only inside the zero-velocity curves. Therefore the zero-velocity curves represent bounds for the motion. It i s thus possible to conclude that f o r : Cp| < - ( 1 + 3 K ^ ) , no motion i s possible, - ( 1 + 3K^)^C^ ^ - 1 , the motion i s bounded, -1 < ^ 0, i n s t a b i l i t y can arise only i n the yV - d i r e c t i o n , 0 < C^, unbounded motion i s possible i n both directions, 5 . 4 Phase Space and Trajectories The equation of motion ( 5 . 1 2 ) and ( 5 . 2 3 ) may be written as a set of four f i r s t order d i f f e r e n t i a l equations rj0 CoS# d i L = Tars $ + 2j>5ih$ JO ( 5 . 2 9 ) - F, >$)</>') F i g u r e 5-2 Z e r o - v e l o c i t y c u r v e s f o r an a x i - s y m m e t r i c 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 (K^ = 1) 196 do d<j>' = - { ( ^ + ¥ + 3 * ^ } ****** (5.29) c o n t ' d These may he r ea r r anged i n the .form 7Q - C°» * W = & - ^ - d * ' ( 5 . 3 0 ) F, " <t>' Fz which d e f i n e s a t r a j e c t o r y i n a f o u r - d i m e n s i o n a l phase space . The Hami l t on i an (5.28) permi ts de te rmin ing any one phase space co-o rd ina t e i n terms o f the o ther t h r e e . S o l v -i n g f o r Cos 0 g i v e s , o r , + c • - ' P- / j ^ i L l i k _ / ( 5 . 3 1 ) The ambigu i t y as to s i g n i n d i c a t e s tha t the i n f o r m a t i o n d e r i v e d from the Hami l t on i an cannot d i f f e r e n t i a t e between 197 +09 -0i 7T + 0 , and TT - 0 . T h i s i n f o r m a t i o n can u s u a l l y be ob t a i ned from a c o n s i d e r a t i o n o f the c o n t i n u i t y of the s o l u t i o n . I f 0 i s taken to be p o s i t i v e and l e s s than 7 T / 2 , equa t i on ( 5 . 3 1 ) i s unambiguous and equa t ion ( 5 . 3 0 ) d e f i n e s a unique t r a j e c t o r y ( 5 . 3 2 ) i n a t h r e e - d i m e n s i o n a l {p, , 0 ' - s p a c e . A s i m i l a r equa t ion ho lds f o r 0 ^ 0 and d e f i n e s e q u a l l y unique t r a j e c t o r i e s i n another phase space . C e r t a i n symmetry p r o p e r t i e s , however, permit the e l i m i n a t i o n o f one o f the spaces . 5o5 Symmetry P r o p e r t i e s On s u b s t i t u t i n g e = - © </J = - </> ( 5 . 3 3 ) i n equa t ions ( 5 . 2 2 - 5 .23 ) i t i s observed tha t y/ = , 0*=,.r_0£i==•;>•_ !0* '=. .^' 'and the equa t ions o f mot ion are unchanged. Thus a t r a j e c t o r y d e f i n e d by ( 5 . 2 2 - 5 . 2 3 ) or the equ i v a l en t ( 5 . 3 2 ) , which passes through the po in t ( (Jpf, 0 ' ) possesses a m i r r o r t r a j e c t o r y which passes 198 through the po in t (-^, (j) 1 , - 0 ' ) . The r e fo r e the t r a j e c t o r y which passes through the po in t ( 0 , 0) e x h i b i t s symmetry about the (p ' - a x i s . When the s u b s t i t u t i o n s a = - e yj ^ - y ( 5 . 3 4 ) are made i n equat ions ( 5 . 2 2 - 5*23 ) j p ' = c p ' , 0 T = 0 ' and the form o f the equat ions i s i n v a r i a n t . T h e r e f o r e the t r a j e c t o r y pa s s i ng th rough the po in t ( cp, <p ], 0 ' ) w i th 0 ' > 0 possesses a m i r r o r t r a j e c t o r y which passes through the p o i n t (-<p, 0 T ) where 0 < 0 . Hence the t r a j e c t o r y d e f i n e d by the equat ions o f mot ion i n the phase space i s , f o r 0 > 0 , a m i r r o r image about the (p ' , 0 f - p l a n e o f the t r a j e c t o r y d e f i n e d f o r 0 < 0 . 5 . 6 Numer i ca l r e s u l t s I t was p o i n t e d out i n s e c t i o n ( 5 . 3 ) tha t i n i t i a l c o n d i t i o n s co r r e spond ing to - ( 1 + 3 K . ) ^ ^ -1 a lways l e a d to s t a b l e mot ion . However, t he re appear t o be two t ypes o f t r a j e c t o r i e s which i n d i c a t e s the e x i s t e n c e o f two c l a s s e s o f s o l u t i o n s . The f i r s t c l a s s o f s o l u t i o n s i s i l l u s t r a t e d i n F i g u r e 5 - 3 . Here an i n v a r i a n t s u r f a c e i s d e f i n e d i n the t h r ee-d imens iona l phase space . That i s an " i s o l a t i n g " i n t e g r a l 199 C H = - 1 - 5 (J) ^ 0 Figure 5-3 Invariant surface re s u l t i n g from the f i r s t class of solutions 200 has been determined numerically. Figure 5-4 indicates the cross-section of a s i m i l a r surface i n the plane i/ ' = 0. It should be noted that the points of intersection of the trajectory with the plane = 0 define a smooth boundary. The second type of behaviour i s i l l u s t r a t e d i n Figure 5-5« As before i t represents the state of motion of the system i n the plane ijs = 0 with the i d e n t i c a l co-ordinates. However, i n t h i s case the points appear to be scattered randomly over regions i n the plane indicating the "ergodic" nature of the motion. 38 Kane has indicated that the motion normal to the o r b i t a l plane may exhibit a type of beat phenomenon with a very long period, t y p i c a l l y 35 to 45 o r b i t s . This type of behaviour would lead to a plot of the type presented i n Figure 5-5• A large number of points would have to be determined before p e r i o d i c i t y becomes evident. Such motion could best be described as "quasi-ergodic." The behaviour of the solution i n the t r a n s i t i o n region between the large simple surfaces of the f i r s t type and the ergodic behaviour of the second type lends support to the concept of quasi-ergodicity. In the t r a n s i t i o n region, "chains of islands" appear which become smaller and more 31 numerous as the ergodic region i s approached (Figure 5-6). For > -1 there i s a p o s s i b i l i t y that the motion may be unstable. The numerical results indicate that a stable i n i t i a l condition r e s u l t s i n a solution of the f i r s t type and hence i n the generation of an invariant surface 201 F i g u r e 5-4 C r o s s - s e c t i o n o f a s u r f a c e s i m i l a r t o t h a t p r e s e n t e d i n F i g u r e 5-3 when = 0 202 F i g u r e 5-5 The c r o s s - s e c t i o n = 0 i n phase space i l l u s t r a t i n g t h e e r g o d i c n a t u r e of t h e second c l a s s o f s o l u t i o n s 203 F i g u r e 5-6 The- c r o s s - s e c t i o n i/> = 0 i n phase space i l l u s t r a t i n g the t r a n s i t i o n from a l a r g e s i m p l e " mainland" t o an e r g o d i c t r a j e c t o r y v i a a number of " I s l a n d s " ( F i g u r e 5-7). C e r t a i n l y , i f t h e r e i s such a s u r f a c e , t h e t r a j e c t o r y can n e v e r l e a v e i t and s t a b i l i t y i s g u a r a n t e e d . F u r t h e r , t h e n u m e r i c a l a n a l y s i s s u g g e s t s t h a t t h e e r g o d i c t y p e o f t r a j e c t o r i e s a r e not c o n s i s t e n t w i t h s t a b i l i t y f o r C H > " l o F o r a g i v e n v a l u e o f C^, s u i t a b l e i n i t i a l c o n d i t i o n s can be chosen t o g e n e r a t e d i f f e r e n t s u r f a c e s . The l a r g e s t such s u r f a c e i s r e f e r r e d t o as a l i m i t i n g s u r f a c e . Thus t h e i n t e r i o r o f a l i m i t i n g s u r f a c e r e p r e s e n t s a l l p o s s i b l e s t a t e s o f t h e s y s t e m , c o n s i s t e n t w i t h t h e f i x e d v a l u e o f t h e H a m i l -t o n i a n , c o r r e s p o n d i n g t o s t a b l e m o t i o n . The n u m e r i c a l work i n d i c a t e s t h e e x i s t e n c e o f i n v a r i a n t s u r f a c e s up t o a t l e a s t C^ = 0„5. F o r C^ ^ 0.6 i n v a r i a n t s u r f a c e s do not appear t o e x i s t . F i g u r e s 5-8-i t o 5-8 - i i i show s e v e r a l l i m i t i n g s u r f a c e s f o r r e p r e s e n t a t i v e v a l u e s o f C H (0 ^ 0)° ^he symmetry p r o p e r t i e s o f s e c t i o n 5.5 ( i l l u s -t r a t e d i n F i g u r e 5 - S - i i i ) may be Used t o d e t e r m i n e t h e s u r f a c e s f o r 0 < 0. The l i m i t i n g s u r f a c e s p r o v i d e u s e f u l i n f o r m a t i o n c o n -c e r n i n g t h e n a t u r e o f t h e m o t i o n . At h i g h v a l u e s o f C^ t h e l i m i t i n g s u r f a c e s f o r b o t h 0 ^ 0 and 0 ^ 0, when drawn i n t h e same d i a g r a m ( F i g u r e 5 - 8 - i i i ) , a p p e a r as a t w i s t e d f i g u r e e i g h t . I t i s i n t e r e s t i n g t o n o t e t h a t b o t h m o t i o n s a r e q u a s i - p e r i o d i c w i t h t h e p e r i o d o f t h e o u t - o f - p l a n e m o t i o n a p p r o x i m a t e l y d o u b l e t h a t o f the i n - p l a n e l i b r a t i o n s . F i g u r e 5-9 i l l u s t r a t e s t h i s b e h a v i o u r f o r a s p e c i f i c s e t o f 205 F i g u r e 5-7 T y p i c a l i n v a r i a n t s u r f a c e when C H > -1 206 207 F i g u r e 5 - 8 - i i L i m i t i n g i n v a r i a n t s u r f a c e ( C R 208 209 F i g u r e 5-9 S o l u t i o n o f t h e e q u a t i o n s o f motion f o r s p e c i f i c i n i t i a l c o n d i t i o n s , i l l u s t r a t i n g t h e q u a s i - p e r i o d i c n a t u r e o f t h e motion 210 i n i t i a l conditions. The r e s u l t s displayed i n Figures 5 - 8-i to 5 - 8 - i i i may be presented i n a more informative manner. If </* and 0 are f i x e d , a constant value of describes a c i r c l e i n a 0 ?-plane. A point i n t h i s plane s p e c i f i e s value of the angular v e l o c i t i e s of the body and hence gives a complete set of i n i t i a l conditions. I f the v e l o c i t i e s are zero, the point l i e s at the o r i g i n and the Hamiltonian has i t s minimum value. For values of Lp and 0 such that = =1 defines a r e a l radius, there exists a c i r c l e inside which s t a b i l i t y i s guaranteed. Larger values of C^ result i n s t a b i l i t y for varying arc lengths of the constant C^ c i r c l e s . Figures 5-10-1 to 5 - 1 0 - i i i show the stable regions, i n $ ' , 0 '-planes f o r various combinations of (p and 0 .. It i s possible to make an observation concerning the r e l a t i v e s e n s i t i v i t y of the s a t e l l i t e i n the position of stable equilibrium to disturbances i n the ip a n d 0 directions. It i s evident from Figure 5 -10-i that for <p = 0 = 0, the coupled motion can remain stable even when subjected to the angular v e l o c i t i e s ip = - 1 , 15 , 0 f . = ±1.75.. The resultant v e l o c i t y of approximately 2,1 i s considerably above the value of v/J which holds f o r the.planar case. Thus motion r e s t r i c t e d to the o r b i t a l plane appears to be less stable than the more general two dimensional motion. 5»7 Concluding Remarks The re s u l t s show that there are two d i s t i n c t types 2 1 1 -2 A 0 1 2 IP F i g u r e 5 - 1 0 - i A l l o w a b l e v a r i a t i o n s , i n t h e a n g u l a r v e l o c i t i e s w hich may be imposed on an a x i - s y m m e t r i c s a t e l l i t e when i n a s p e c i f i e d o r i e n t a t i o n (0. = 0 ) 212 F i g u r e 5-10-i i A l l owab le v a r i a t i o n s i n the angu la r v e l o c i t i e s which may be imposed on an ax i-symmetr ic s a t e l l i t e when i n -a—spec i f i ed o r i e n t a t i o n (0 = ±15 ) 213 214 of s t a b i l i t y a s s o c i a t e d w i th the coup led l i b r a t i o n a l motion of a g r a v i t y - g r a d i e n t o r i e n t e d s a t e l l i t e . For va lues of l e s s than -1 u n c o n d i t i o n a l s t a b i l i t y i s guaranteed by the e x i s t e n c e o f c l o s e d zero v e l o c i t y curves about the e q u i l i b r i u m p o s i t i o n . On the o the r hand , i f -1 < C„ < C„ ^ 0 .55 , the max system possesses c o n d i t i o n a l s t a b i l i t y which depends upon the i n i t i a l c o n d i t i o n s imposed and cor responds to the ex i s t ence o f an i n v a r i a n t s u r f a c e i n the phase space . Thus s t a b l e mot ion can r e s u l t even when the zero v e l o c i t y curves are not c l o s e d . From the 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 p resen ted here i t can be conc luded tha t w i th i n c r e a s i n g the r e g i o n of s t a b i l i t y d im in i shes r a p i d l y i n s i z e . For ~ 0.55 i t ceases to e x i s t a l t o g e t h e r . T h i s imposes upper bounds on the d i s t u r b a n c e s which a s a t e l l i t e can t o l e r a t e w i thout becoming u n s t a b l e . The l i b r a t i o n a l mo t ions , both i n and normal to. the o r b i t a l p l a n e , are q u a s i - p e r i o d i c . T h i s i s p a r t i c u l a r l y n o t i c e a b l e at h i gh v a l ues o f where the mot ion i n the o r b i t a l p lane occurs at a f r equency approx imate l y double t ha t o f the mot ion normal to the o r b i t a l p l a n e . The a n a l y s i s suggests t ha t the coup led l i b r a t i o n a l mot ion i s more s e n s i t i v e to the i n-p l ane d i s t u r b a n c e s com-pared t o those normal to the o r b i t a l p l a n e . T h i s i n d i c a t e s tha t des ign ana l y ses performed u s i n g p l ana r mot ion are c o n -s e r v a t i v e and tha t the a c t u a l mot ion i s at l e a s t as s t a b l e as i n d i c a t e d by the s i m p l i f i e d s tudy . 6. CONCLUDING REMARKS 6,1 G e n e r a l C o n c l u s i o n s The r e s u l t s p r e s e n t e d have shown t h e u s e f u l n e s s o f a d o p t i n g a phase space w h i c h p o s s e s s e s a s u f f i c i e n t number o f d i m e n s i o n s so t h a t t h e s t a t e o f t h e system under s t u d y i s u n i q u e l y r e p r e s e n t e d by t h e c o - o r d i n a t e s o f a p o i n t . . In t h e g r e a t m a j o r i t y o f t h e c a s e s s t u d i e d , t h e n u m e r i c a l i n t e -g r a t i o n o f the e q u a t i o n s o f m o t i o n l e d t o t h e g e n e r a t i o n o f an i n v a r i a n t s u r f a c e . That i s , an " i s o l a t i n g " i n t e g r a l m a n i -f o l d c o u l d be o b t a i n e d w h i c h forms a s u r f a c e i n t h e phase space and r e p r e s e n t s a r e d u c t i o n i n t h e number o f c o - o r d i n a t e s by u n i t y . In t h o s e c a s e s where no i n t e g r a l m a n i f o l d c o u l d be f o u n d , t h e m o t i o n was e i t h e r u n s t a b l e o r a t y p e o f l o n g p e r i o d beat phenomenon was o b s e r v e d . . I n t h e l a t t e r c a s e , t h e r e i s e v i d e n c e t h a t t h e i n v a r i a n t s u r f a c e b r e a k s up i n t o a t o r t u o u s s t r u c t u r e w h i c h g i v e s t h e m o t i o n a random a p p e a r a n c e . The i n v a r i a n t s u r f a c e s a r e n o n - i n t e r s e c t i n g . T h i s i s t h e i r most i m p o r t a n t p r o p e r t y as i t i n c r e a s e s t h e i r u s e f u l -n e s s i n the s t u d y o f t h e g e n e r a l m o t i o n and t h e c o n d i t i o n s w h i c h y i e l d s t a b i l i t y . T h e r e i s an i n h e r e n t l i m i t a t i o n t o t h e p r a c t i c a l a p p l i c a t i o n o f t h e concept i n t h a t i t i s d i f f i c u l t t o c o n -c e i v e o f a space p o s s e s s i n g more t h a n t h r e e d i m e n s i o n s . 216 Hence systems r e q u i r i n g more than th ree co-o rd ina t e s f o r t h e i r d e s c r i p t i o n are too complex f o r t h i s approach to y i e l d r e a l l y u s e f u l i n f o r m a t i o n . Where the system i n v o l v e s t h r ee s t a t e v a r i a b l e s , or l e s s , and p a r t i c u l a r l y i n the case o f the g e n e r a l second o rde r d i f f e r e n t i a l e q u a t i o n , the method i s ex t reme ly v a l u a b l e . 6.2 Recommendations f o r Future Work There a re many p o s s i b i l i t i e s f o r ex t ens i on of the work p resen ted h e r e . . I t wou ld , f o r example, be u s e f u l to o b t a i n an approximate a n a l y s i s o f the p l ana r mot ion o f a r i g i d s a t e l l i t e which i s capable of p r e d i c t i n g the s i z e and shape o f the l i m i t i n g i n v a r i a n t s u r f a c e . One p o s s i b l e way o f making t h i s s tudy would be to i n v e s t i g a t e the t r a n s f o r m a t i o n concept i n t r o d u c e d i n S e c t i o n 2.4 and t o determine approx imat ions . to the mapping which y i e l d the r e s u l t s w i th a c o n s i d e r a b l e r e d u c t i o n i n computa t iona l e f f o r t . The a n a l y s i s o f the damped s a t e l l i t e cou ld be extended c o n s i d e r a b l y . The damper employed i n Chapter 3 i s complex and i n e f f i c i e n t as compared to o the r d e v i c e s . The c o n f i g u r a -t i o n s t u d i e d by Zajac"'"^ may prove t o be u s e f u l i n approx imat -i ng p r a c t i c a l s a t e l l i t e s . There i s a l a r g e amount of l i t e r a -t u r e conce rn ing the sma l l ampl i tude mot ions of these d e v i c e s , but ve r y l i t t l e e f f o r t has been made i n c o n s i d e r i n g the i nhe ren t n o n - l i n e a r i t y o f the equa t ions or the problem o f capture o f the s a t e l l i t e by the g r a v i t a t i o n a l g r a d i e n t f i e l d . The problem of the " l o n g " e l a s t i c s a t e l l i t e a l s o 217 remains to be treated. There i s at least one s c i e n t i f i c experiment that has been proposed which would require very 39 long antennae. The analysis becomes much more d i f f i c u l t as the parametric excitation of the boom becomes s i g n i f i c a n t as a resu l t of the low natural frequency and interacts with the non-linearity of the large thermal deflections. A study of the general motion of a non-spinning s a t e l l i t e would involve a massive amount of work. I f the orbit i s e l l i p t i c a l , the' Hamiltonian varies with time and hence the state of the system (equations 5.18 and 5«19) depends on the f i v e variables; & , d; , <p', <f> , ^>'as well as the parameters and e. By f i x i n g the values of Q , K^ ., and e and assuming that an i n t e g r a l manifold e x i s t s , i t should be possible to determine 0 , for example, as a function of (ft , Ly', and cf>'' £'or a s p e c i f i e d set of i n i t i a l conditions. The determination of such a function would be extremely i n t e r e s t -ing, but i t would involve the integration of the equations of motion f o r approximately one thousand orbits i f the r e s u l t -ing accuracy i s to be equivalent to that i n Chapter 5° The determination of the l i m i t i n g manifold w i l l thus be a very time consuming process. A comprehensive study of the effects of variations i n the parameters and e thus appears to be u n r e a l i s t i c . A contribution could be made by attempting an approxi-mate solution of the equations of motion and by comparing the re s u l t s with the numerical work. This i s a more elaborate problem than that proposed f o r the approximate solution of 218 t h e p l a n a r motion and i t i s assumed t h a t t h e p l a n a r work would be completed b e f o r e t h e more c o m p l i c a t e d a n a l y s i s i s attempted. I t . w o u l d a l s o be u s e f u l t o pe r f o r m a d e t a i l e d s i m u l a t i o n of an a c t u a l s a t e l l i t e i n o r d e r t o a s s e s s more a c c u r a t e l y t h e magnitude o f t h e e f f e c t s o f v a r i o u s d i s t u r b a n c e s . The d e s i g n o f a s a t e l l i t e might w e l l form t h e u l t i m a t e g o a l o f t h i s work and such a s i m u l a t i o n would be r e q u i r e d i n any e n g i n e e r i n g s t u d y and would be e s s e n t i a l i n d e t e r m i n i n g the e f f e c t s o f d e s i g n changes on t h e performance. BIBLIOGRAPHY 1 Jensen, J., Townsend, G., Kork, J. , and Kraft, D., Design  Guide to Orb i t a l F l i g h t , McGraw-Hill, New York, 1962, pp. 752-753. 2 Wiggins, Lyle E., "Relative Magnitudes of the Space Environment Torques oh a S a t e l l i t e , " AIAA Journal, Vol. 2, No. 4, A p r i l 1964, pp. 770-771. 3 Glasstone, Samuel, Sourcebook on the Space Sciences, Van Nostrand, Princeton, N.J., 1965, Chap. 8. 4 King-Hele, Desmond, S a t e l l i t e s and S c i e n t i f i c Research, revised ed., Routledge and Kegan, London, 1962, p. 114. 5 Piscane, Vincent L., Pardoe, Peter P., and Hook, P. 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