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 o f 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 of 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  COMMITTEE IN CHARGE Chairman:  I . McTo Cowan  J.P. Duncan G,V. P a r k i n s o n A.C. Soudack  V.Jo Modi C.R. H a z e l l E.V. Bohn  E x t e r n a l Examiner: J . Mar Head, Space Mechanics S e c t i o n Defence R e s e a r c h T e l e c o m m u n i c a t i o n s Establishment Ottawa, O n t a r i o  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 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 b e r i c a l l y and a n a l y t i c a l l y o 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 m p u t a t i o n and hence are 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 of the p l a n a r motion of a r i g i d s a t e l l i t e l e a d s t o the concept of an i n v a r i a n t s u r f a c e or 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 of the e q u a t i o n of m o t i o n i s employed t o determine the manifolds. 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 of the parameters d e s c r i b i n g the s a t e l l i t e , the 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 motion c o r responds t o the l a r g e s t i n v a r i a n t s u r f a c e w h i c h can be found. I t i s a l s o demonstrated t h a t the m a n i f o l d s are i n t i m a t e l y c o n n e c t e d w i t h p e r i o d i c s o l u t i o n s of the e q u a t i o n of motion and t h i s knowledge p e r m i t s determini n g l i m i t s on the parameters so as t o 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 c h a r t s s u i t a b l e f o r d e s i g n purposes are p r e s e n t e d .  The p l a n a r 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 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 the motion e v e n t u a l ! 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 of the undamped c a s e . 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 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 p e r m i t s d e t e r m i n a t i o n 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 e q u a t i o n of 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 the s t a b i l i t y pro-  v i d e d t h a t t h e f l e x i b l e member i s n o t t o o longThe case of 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 also consideredI 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 i n some cases 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 guaranteed i f the H a m i l t o n i a n i s l e s s than a p r e s c r i b e d value„ Values o f the H a m i l t o n i a n l a r g e r than t h i s may a l s o p e r m i t 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 of the g e n e r a l motion i s somewhat g r e a t e r than t h a t f o r the p l a n a r motion. Charts are presented g i v i n g the 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.  GRADUATE STUDIES  F i e l d o f Study:  Mechanical  Space Dynamics ( I and I I ) Mechanical V i b r a t i o n s High Speed Gas Dynamics Mechanics o f R a r i f i e d Gases N o n - l i n e a r Systems Analogue Computers  Engineering V . J . Modi CAo B r o c k l e y G.V. P a r k i n s o n G.V. P a r k i n s o n A.C. Soudack E.V. Bonn  PUBLICATIONS  Modi, V . J . , and B r e r e t o n , 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 U s i n g the WKBJ Method," J o u r n a l o f A p p l i e d Mechanics, V o l . 33, No. 3, S e p t . 1966, pp. 676-678. B r e r e t o n , R.C., a Dumbbell J o u r n a l of Dec. 1966,  and Modi, V . J . , "On the S t a b i l i t y of 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 , " the R o y a l A e r o n a u t i c a l S o c i e t y , V o l . 70, pp. 1098-1102.  B r e r e t o n , R.C., and Modi, V . J . , " S t a b i l i t y of the P l a n a r L i b r a t i o n a l M o t i o n o f a S a t e l l i t e i n an E l l i p t i c O r b i t , " P r o c e e d i n g s o f the X V I I 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, M a d r i d , 19 67. M o d i , V . J . , and B r e r e t o n , R.C., " S t a b i l i t y B o u n d a r i e s f o r P l a n a r 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 5 t h Aerospace S c i e n c e s Meeti n g , New York, New Y o r k , J a n . 23-26, 1967. Modi, V . J . , and B r e r e t o n , 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 M o t i o n o f 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 , " P r o c e e d i n g s o f the X V 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, B e l g r a d ( i n press) .  A STABILITY STUDY OF GRAVITY ORIENTED SATELLITES by ROBERT CLOUDESLEY BRERETON B. Eng., M c G i l l U n i v e r s i t y , 1959  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the Department  of Mechanical  Engineering  We accept t h i s t h e s i s as conforming required  t o the  standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1967  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e -  ments f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y able f o r r e f e r e n c e for  and study.  I f u r t h e r agree that  avail-  permission  e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes  may be granted by the Head of my Department or by h i s r e p r e sentatives.  I t i s understood that  this thesis f o r financial my w r i t t e n  gain  copying or p u b l i c a t i o n of  s h a l l not be allowed without  permission.  R.C.  Department  of Mechanical  The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada  Engineering  Columbia  Brereton  ABSTRACT The s t a b i l i t y satellites  of g r a v i t a t i o n a l gradient  i s examined  oriented  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 o u t 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  c o m p u t a t i o n and  employed  hence a r e p a r t i c u l a r l y  involve considerable  s u i t e d t o s o l u t i o n by a  digital  computer. The a n a l y s i s o f t h e p l a n a r m o t i o n o f a r i g i d leads  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  manifold. is  Numerical,integration  employed  f o r s p e c i f i e d values  or integral  of the equation  t o determine the manifolds.  of motion  I t i s shown t h a t  o f the 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  s t a b l e motion corresponds t o the l a r g e s t i n v a r i a n t w h i c h c a n be f o u n d .  satellite  with  surface  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 the  equation  o f m o t i o n and t h i s knowledge  permits  determin-  i n g l i m i t s on t h e p a r a m e t e r s s o a s t o e n s u r e s t a b l e by a s t u d y Several  of the s o l u t i o n of the v a r i a t i o n a l  charts s u i t a b l e f o r design  equation.  purposes a r e  The p l a n a r m o t i o n o f a s a t e l l i t e  motion  presented.  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 m o t i o n 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 c a s e .  The t h i r d m o d e l r e p r e s e n t s  a flexible  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 .  s a t e l l i t e free to An a n a l y s i s o f  the temperature 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 permits determination  o f t h e shape o f t h e s a t e l l i t e  of i t s p o s i t i o n . and  The r e s u l t i n g  equation  i t i s shown t h a t f l e x i b i l i t y  s o l e l y i n terms  of motion i s derived  does n o t 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 t h e f l e x i b l e member i s n o t t o o l o n g . The c a s e o f a n a x i - s y m m e t r i c orbit  i s also considered.  manifolds  Stability  a t times  apparently  case  ergodic  c a n be g u a r a n t e e d i f t h e H a m i l -  t o n i a n i s l e s s than a p r e s c r i b e d value. 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 p e r m i t this  i n a circular  I t i s shown t h a t i n t h i s  also exist although  motion can occur.  satellite  Values  of the Hamil-  s t a b l e motion and i n  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 t h e g e n e r a l m o t i o n i s somewhat  g r e a t e r than that f o r the planar motion.  Charts  are presented  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 m o t i o n .  TABLE OF  CONTENTS Page  Chapter 1  2  Introduction  .  .  .  .  .  .  .  .  .  .  .  .  1  .  .  .  .  1  1.1  P r e l i m i n a r y Remarks  1.2  Gravity-Gradient Stabilized  1.3  Purpose and Scope o f I n v e s t i g a t i o n .  .  .  .  Planar L i b r a t i o n a l Motion of a R i g i d 2.1  F o r m u l a t i o n o f t h e Problem  2.2  Simple Exact S o l u t i o n s  12  Satellite . . . .  .  .  (e = 0)  .  14  .  .  19  .  .  19  Circular Orbit  2.2.2  P e r i o d i c Solutions Using the Method o f Harmonic B a l a n c e  24  Numerical Determination of Periodic Solutions  29  Approximate S o l u t i o n s  .  14  2.2.1  2.2.3 2.3  .  5  Satellites  .  .  .  .  .  35  .  .  .  .  .  35  Balance  47  2.3.1  WKBJ M e t h o d  2.3.2  P r i n c i p l e o f Harmonic  2.3.3  Perturbation of Periodic Solutions . . . . . 0  .  .  56  2.4  Phase Space and I n v a r i a n t S u r f a c e s .  71  2.5  A c c u r a c y o f t h e Method  92  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 2.6.1 2.6.2 2.6.3  2.7  .  .  .  .  .  Solutions  97  The R e l a t i o n s h i p B e t w e e n 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  97  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 .  99  The D e g r e e o f S t a b i l i t y  C o n c l u d i n g Remarks  . . . .  105 109  V  Chapter 3  4  5  6  Page 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 3.1  F o r m u l a t i o n o f t h e Problem  3.2  Numerical  3-3  Conclusions  Results  .  .  .  .  .  118 118  .  .  .  . .  . .  123  . .  .  132 134  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 4.1  P r e l i m i n a r y Remarks .  4.2  F o r m u l a t i o n o f t h e Problem  136  4.3  T h e r m a l 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 B a l a n c e  4.5  T h e r m a l D e f l e c t i o n o f t h e Boom  4.6  Stability  4.7  C o n c l u d i n g Remarks  Two D i m e n s i o n a l Satellite .  Analysis  Motion  .  .  .  .  .  .  .  .  .  Equation  .  -134  .  .  .  . . . . .  .  .  .  151 156 166 182  o f an  Axi-Symmetric  I84  $.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 a n d Z e r o - V e l o c i t y C u r v e s .  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 R e s u l t s  5.7  Concluding Remarks  .  192  198 210  Concluding Remarks  215  6.1  General Conclusions  215  6.2  R e c o m m e n d a t i o n s f o r F u t u r e Work . . . .  Bibliography  .  .  .  .  .  .  .  .  .  216 219  L I S T OF TABLES Page  Table I II  Representative Gravity-Gradient S a t e l l i t e Characteristics o o o » « « » « . V a r i a t i o n o f R e s u l t s O b t a i n e d by I n t e g r a t i n g the Equations of Motion With S e v e r a l Step SXZ6S  III  5  »  a  o  o  o  o  o  •  o  o  •  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 Configurations . . . . . . . . .  »  •  .  .  •  93  135  L I S T OF FIGURES Figure 1-1  Page Magnitude of f o r c e s S3.t)6lllt6 o  o  •  acting o  •  e  on a o  representative •  *  .  • .  • .  • .  6  1- 2  Models o f multi-body s a t e l l i t e s  2- 1  Geometry o f p l a n a r m o t i o n o f a r i g i d satellite 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  15  s o l u t i o n when e = 0  20  2-2  .  •  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  2-4  I n i t i a l angular v e l o c i t i e s required 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 . . . . 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)  2-5 2-6 2-7 2-8  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 e c c e n t r i c i t y (K^ = 0,3, n = 1)  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 . . . .  .  .  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 numerical determination of periodic solutions . solutions with  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 The v a r i a t i o n o f F w i t h  period  22 23 31 32  Periodic  2-12  .  of  2-$  2-11  11  33 34  o f 4K  . . .  37  o r b i t angle and o r b i t  e c c e n t r i c i t y (K^ = 1) . . . . . . . . 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 m o t i o 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 a n d t h e a p p r o x i m a t e WKBJ method (K^ = 1 , e = 0 . 1 ) o . . . . . . . . .  39  44  viii Page  Figure 2-13  2-14  45  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 determined by t h e a p p r o x i m a t e WKBJ method o v e r e i g h t o r b i t s (K^ ~~ 1 j 6 0 © 3) • • • • • ° ° * •  46  Determination of the f i r s t sine series solution  2-15  2-16  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 m o t i o 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 a n d t h e a p p r o x i m a t e WKBJ m e t h o d (K. 1, e = 0 . 3 )  -i  K  i  - i i  K  i  "~* I t 0 1  0  o  o  ©  ©  0©9*  *  *  •  *  • *  0o'7°  °  °  •  •  * *  - i i i  K. l  -iv  K. - 0 . 5 l  -V  K  i  -vi  K  i  two t e r m s o f t h e  O  .  c  .  .  .  .  50 51  ©  ©  ©  ©  o  52  53  = 01  54  ™~ 0 © 1 ©  e  »  o e ©* .  .  .  .  .  55  Values of the i n i t i a l d e r i v a t i v e required t o p r o d u c e s o l u t i o n s w i t h p e r i o d o f 271 a s d e t e r m i n e d b y t h e f i r s t two t e r m s o f t h e s i n e series solution .  57  2-17  Comparison o f exact p e r i o d i c s o l u t i o n s w i t h two a n d t h r e e t e r m s i n e s e r i e s s o l u t i o n s  5S  2-18  V a l u e s o f Kj 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 o f p e r i o d 2TC .  69  2-19  V a l u e s o f K. a n d 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 47t •  70  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 ( K j = 1) f o r s o l u t i o n s o f p e r i o d 27T . .  72  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  «  1  2-22  S c h e m a t i c v i e w o f an i n v a r i a n t  surface  0  «  *  <  73 77  ix Page  Figure 2-23  A s p e c i f i c s o l u t i o n which i l l u s t r a t e 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 .  78  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^ = 0.7, e = 0.2)  80  2-25  Typical invariant surface 0«25  «  •  K. = 0.7, e = 0.2  .  .  -i  =  - i i  X«j 6  =  •  •  *>  •  •  82  2-26  Typical invariant surface with " 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 motion  2-30  K  = 1.0  86  -ii  K. = 0 . 9  87  -iii  K.  0.7  88  -iv  K  = 0.5  89  -v  K. = 0.3  90  -vi  K =  91  ±  l  ±  ±  0.1  Comparison o f t h e i n v a r i a n t s u r f a c e s generated using d i f f e r e n t i n t e g r a t i o n step s i z e s -i  Non-limiting surfaces  - i i  Limiting surfaces  .  94  . . . .  95  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 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 t h e e q u a t i o n o f m o t i o n i s i n t e g r a t e d o v e r 271 ( K^ — 1 , e — 0 )  2-31  85  - l  2-28  2-29  81  •  O  .  »  O  .  .  .  O  .  B I  100  102  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 m o t i o n i s i n t e g r a t e d o v e r 2K 103  2-32  Determination o f a complete s e t o f f i x e d of the transformation ( K ^ = l , e = 0 )  points 104  X  Page  Figure 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 m o t i o n i s i n t e g r a t e d o v e r 27t ( ~-Lj6 0ol)o O O 9 • O O • • »  106  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 m o t i o n i s i n t e g r a t e d o v e r 2K (K« G"~0»1) O o O O • o a • • •  107  2-35  Determination o f a complete s e t o f f i x e d o f t h e t r a n s f o r m a t i o n (K^ = 1, e = 0.1)  108  2-36  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 satellite  2-33  —  2-34  -i  K  -ii  i  " 3- 0  K. *"•" 0*9 l  0  s  •  *  .  .  •  *  •  *  *  .  . I l l  O  O  O  O  •  O  °  •  •  •  •  O  points  110  -iii  K  i  ~~ 0 o 7 °  *  *  °  •  •  •  *  •  •  .  .  112  -iv  K  i  ~" 0 « 5 °  °  °  •  •  °  •  •  •  •  .  .  113  -V  K  .  .  114  -vi  K  .  .  115  .  .  119  ± = 0.3 i  = 0.1  . . .  .  3-1  G e o m e t r y o f m o t i o n o f a damped s a t e l l i t e  3-2  S o l u t i o n of equation of motion i l l u s t r a t i n g the e f f e c t o f t h e damper .  125  3-3  S t r o b o s c o p i c phase plane o f t h e 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  126  3-4  T y p i c a l s t r o b o s c o p i c p h a s e p l a n e o f damped Sclt @!X.X i t 6 o o o o o o » o o » » o  127  3-5  T y p i c a l s t r o b o s c o p i c p h a s e p l a n e o f damped S cl t @X *L j. t 6 o e o o o o t o o o o  *  128  3-6  Limit cycles  (K^ = 1 . ,  e = 0.1)  .  .  .  .  129  3-7  Limit  cycles  (K^ = 1 . ,  e = 0.2)  .  .  .  .  130  3- 8  Limit  cycles  (K^ = 1 . , e = 0 . 3 )  .  .  .  .  131  4- 1  Geometry o f m o t i o n o f f l e x i b l 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  satellite boom .  137 146  XI  Figure  Page  4-3  Heat balance f o r an element o f the s a t e l l i t e  4-4  Geometry o f 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  .  .  .  .  .  149  .  4-5  Thermal d e f l e c t i o n o f the s a t e l l i t e boom .  4-6  Shape o f t h e r m a l l y d e f l e c t e d  4-7  I l l u s t r a t i o n o f the p r i n c i p a l axes o f the deflected s a t e l l i t e . . . . 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 length  163  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 angle between the sun and the x - a x i s .  164  4-8 4-9 4-10  4-12  4-13  .  157  s a t e l l i t e boom .  .  l 6 l 162  o f the . . .  (e = 0 . 2 , c< = 0 ° , • • o © © » •  T y p i c a l invariant surface l) * • ° ° o o  4-11  .  •  l6*7  V a r i a t i o n o f the 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 with o r b i t angle (e = 0 . 2 , (X = 0 , L*= 1) . . .  169  V a r i a t i o n o f the 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 angle (e = 0 . 2 , CK - 3 0 ° , L* = 1) . . . . . . . . . .  170  Range o f v a l u e s o f the d e r i v a t i v e when f = 0 = 0 f o r s t a b l e motion (L*= 1)  —1  — O  X  —iv 4-14  o  30 o  — IX — DL X  o  0^  (X  60  —  =  o  «  .  .  .  .  o  o  .  .  o  o  O  O  O  o © A  .  a  o  .  .  .  ©  o  o  .  .  .  .  .  .  .  .  90°  171 172 173  .  • •  174  Range o f v a l u e s o f the d e r i v a t i v e when ifJ = & = 0 f o r s t a b l e motion (L* = 2) -i  tX = 0 ° .  .  .  .  .  .  .  .  175  -ii  = 30° .  .  .  .  .  .  .  .  .  .  .  .  -iii  = 60° .  .  .  .  .  .  .  .  .  .  .  .  -xv  W =  .  176 .  177  xii Page  Figure 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 t e r m s t a b i l i t y —  1  °" 1 1  Xj'"" !  1  ly  2o  O  O  9  •  O  9  9  9  9  9  •  180  O  O  O  •  9  9  9  9  9  9  181  5-1  Geometry f o r t h e two d i m e n s i o n a l m o t i o n o f a  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)  195  5-3  I n v a r i a n t surface r e s u l t i n g from t h e f i r s t class of solutions . . . . . . . .  199  5-4  Cross-section of a surf ace-.similar t o that p r e s e n t e d ^ i n ' F i g u r e 5-3 when d> = 0.  201  5-5  The c r o s s - s e c t i o n <p = 0 i n p h a s e s p a c e i l l u s t r a t i n g t h e ergodic nature o f the second class of solutions . . . . . . . .  202  The c r o s s - s e c t i o n tp = 0 i n p h a s e s p a c e 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 simple " 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 " . . . . . . . . . .  203  5-7  Typical invariant surface  205  5-8  Limiting  5-6  - 1 .  invariant surface  -i  C  H  " "•0o5*  -ii  C  H  -  -iii  C  H '•  5-9  when C^ >  0  O  o  0• 5  .  °  °  °  O  o  o  o  O  °  °  °  °  186  _  9  •  206  9  9  207  9  9  208  O  0  9  9  S o l u t i o n of the equations o f motion f o r s p e c i f i c i n i t i a l conditions, i l l u s t r a t i n g the q u a s i - p e r i o d i c nature of the motion ;  5-10  209  Allowable v a r i a t i o n s i n the angular v e l o c i t i e s w h i c h may be i m p o s e d on a n 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  -ii  Ci = ± 15°  •"111  ^ ~~* ~~ Zf- 5  o  o  o  o  o  o  o  e  o  o  o  e  211 212  °  °  <•  °  *  ©  Q  «  O  •  O  213  ACKNOWLEDGEMENT The  author wishes t o express h i s thanks t o  Dr. V . J . M o d i f o r t h e g u i d a n c e g i v e n t h r o u g h o u t t h e preparation of the thesis. ment h a v e b e e n The  invaluable.  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 British  Columbia.  gratefully  H i s h e l p and encourage-  Centre o f the U n i v e r s i t y o f  The u s e o f t h e s e , f a c i l i t i e s i s  acknowledged.  :  L I S T OF SYMBOLS A  Coefficient  A  i n equation  (2.100)  Aerodynamic r e f e r e n c e area  a  Ae  Area from which r a d i a t i o n i s emitted  A,  Amplitude o f the i * ^ s a t e l l i t e boom  1  Area A  n o r m a l mode o f t h e  on w h i c h r a d i a t i o n i s i n c i d e n t  m Fourier coefficient i n the solution of e q u a t i o n (2.14) o f p e r i o d 2n n t h  n  m  '>  Projected area of the s a t e l l i t e A  Arbitrary coefficient i n the solution of the v a r i a t i o n a l e q u a t i o n (2.64)  £ 5  33 f3g  C o e f f i c i e n t i n e q u a t i o n (2.100) Arbitrary coefficient i n the solution of the v a r i a t i o n a l e q u a t i o n (2.64)  G  Coefficient  i n equation  (2.100)  Coefficient  i n equation  (4.59)  Cj,  n  Cc  Constant  of integration  CL|  Constant  value of the Hamiltonian, equation  ^•PWV" C o e f f i c i e n t  i n equation  i n equation  (2.16)  (2.23)  Cp  Aerodynamic p r e s s u r e  C r\  n Fourier c o e f f i c i e n t expressing the solar h e a t i n p u t t o t h e s a t e l l i t e boom  CyC^C^C^  Arbitrary  X)  Coefficient  S)  coefficient  constants In equation  (2.100)  Parameter determining t h e nature o f a conic £  (5.26)  Modulus o f e l a s t i c i t y  section  p  Function  &  (2.41)  defined i n equation  Dissipation function Ff  G e n e r a l i z e d f o r c e a c t i n g on t h e i * " * o f t h e s a t e l l i t e boom  1  Longitudinal force i n the s a t e l l i t e p  F  Functions Function  H~f  Hamiltonian  $4" 1  J  b  T 1  I  function strength  L a r g e moment o f i n e r t i a satellite  o f an  Inertia of the cross-section of the s a t e l l i t e boom Inertia of the i mode o f v i b r a t i o n o f t h e 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 )  ^xxJ^vy^zz. . . . P r i n c i p a l moments o f i n e r t i a Inertia variation  K  parameter  boom  of the s a t e l l i t e = I,. \ wax'  r  , E l l i p t i c integral of the f i r s t kind 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 K  axi-symmetric  . Inertia of a s a t e l l i t e with a r i g i d  T  boom  (2.40)  defined i n equation  Magnetic f i e l d  I  mode  (5.29)  defined i n equation  Q  normal  K., *' *  l_ Xj  I n e r t i a parameter Inertia variation '" e q u a t i o n (4.02)  =  (l x~ ^zz)/\y X  parameters, defined i n  Length o f the s a t e l l i t e  boom  Lagrangian function  1_  Dimensionless  9ft  R e s i d u a l m a g n e t i c moment o f t h e s a t e l l i t e  Frr\,Y\  Coefficient  length of the s a t e l l i t e  i n equation  (2.25)  boom =  xvi  Quantity o f heat Radius of curvature of t h e s a t e l l i t e Kinetic  boom  energy  Temperature o f the s a t e l l i t e  boom  The n Fourier coefficient describing 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 a t e l l i t e boom t  h  Reference temperature Vibrational Potential Elastic  kinetic  energy  energy  potential  Gravitational  energy  potential  energy  Wronskian determinant Shape f u n c t i o n o f t h e i s a t e l l i t e boom  n o r m a l mode o f t h e  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 Value of the f i r s t s o l u t i o n e q u a t i o n when Q = 27Cn Diameter of the s a t e l l i t e  of the v a r i a t i o n a l  boom  Value of the d e r i v a t i v e of the 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 © = 2 T r n Wall thickness of the s a t e l l i t e Value o f t h e second s o l u t i o n e q u a t i o n when 0 = 2%n Specific boom  boom  of the variational  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  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  . Orbit  eccentricity  -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 in rotating co-ordinates  boom a s s e e n  $  A n g u l a r momentum p e r 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  .... F r e q u e n c y e i g e n v a l u e  K  .. T h e r m a l 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 t h e s a t e l l i t e boom  kj  .Spring constant  of the s a t e l l i t e  o f t h e damper  . L e n g t h o f an element o f t h e s a t e l l i t e JLfeL ' ^Xftyi^z, /)#•  X. -  boom  boom  R e f e r e n c e l e n g t h o f an element o f t h e s a t e l l i t e boom D i r e c t i o n cosines of the l o c a l p r i n c i p a l co-ordinates  vertical  i n the  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 a e r o d y n a m i c f o r c e a n d t h e c e n t r e o f mass  faH^  Yf)  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 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 a n d t h e c e n t r e o f mass . Integer Mass o f t h e s a t e l l i t e  y/)j W  . Mass o f t h e damper 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  T^^tfj y% p  f  boom  Integers Integer 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 c o o r d i n a t e q^  XV111  p  Radiation  r  Cjate  pressure  .Momentum c o n j u g a t e  t o theX  co-ordinate  Momentum c o n j u g a t e  to the 0  co-ordinate  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  ...Rate a t w h i c h h e a t Rate o f heat unit area Generalized  °j f rt  r  area  co-ordinate i s i n c i d e n t on u n i t  area  input from i n t e r n a l sources p e r  Rate a t which heat Rate o f heat  perunit  input from e x t e r n a l sources per  Rate a t which heat Rate o f heat u n i t area  i s absorbed  i s reflected perunit  area  input from t h e sun per u n i t  area  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 e l e m e n t o f mass  S  Distance along t h e s a t e l l i t e  t  Time  V  Velocity . P r i n c i p a l body  ^ity^* XutfLrti,  ^. ^,  b  P  3  Pyrt  co-ordinates  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 t h 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 E u l e r i a n r o t a t i o n s <^> , <f> , A respectively Damper o f f - s e t , Damper  distance  displacement  , Aerodynamic P  boom  torque  Moment i n t h e s a t e l l i t e  boom  Moment due t o g r a v i t a t i o n a l Magnetic  torque  gradient  xix Radiation  torque  Function describing the error i n the numerically determined 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 solution Variable defined  i n equation  (2.38)  Complementary s o l u t i o n s o f e q u a t i o n P a r t i c u l a r i n t e g r a l o fequation Angle around s a t e l l i t e the sub-solar point Solar aspect  + 0  from  angle boom m a t e r i a l t o  o f t h e r m a l e x p a n s i o n o f boom  S p e c i f i c value y  (2.39)  boom a s m e a s u r e d  Absorptivity of satellite s o l a r energy Coefficient material  (2.39)  0(  of  - (X  Characteristic  root  Perturbation 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 Perturbation Distance  i n ip  1  between elements a l o n g  satellite  boom  . C o - o r d i n a t e o f d e f l e c t e d boom P o s i t i o n angle o f s a t e l l i t e measured from p e r i c e n t r e S p e c i f i e d value  e  - %  of S  i n i t s o r b i t as  XX  Period of a periodic  solution  Co-ordinate defining the angle o f spin o f a satellite 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) Gravitational field  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 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 b o o m . ( F i g u r e 4-4) Function defined i n equation Perturbation i n  (2.45)  ^  C o - o r d i n a t e o f d e f l e c t e d boom Atmospheric  density  D e n s i t y o f boom m a t e r i a l Stefan-Boltzmann  constant  S t r e s s i n boom m a t e r i a l Damper t i m e  constant  t h Time c o n s t a n t o f t h e n coefficient F o u r i e r a n a l y s i s o f t h e temperature 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 Orbital  i n the  period  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 angle normal t o t h e o r b i t a l A n g l e s u b t e n d e d by a n e l e m e n t boom o f l e n g t h ^  plane  of the s a t e l l i t e  b  Angle between t h e sun and t h e a x i s o f an e l e m e n t o f t h e s a t e l l i t e boom •Librational angle i n the o r b i t a l V a l u e o f (/> a t S  plane  = 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  (j)'  - 7T/2  t / / Cos 0  (p^lpi (^»*  Complementary s o l u t i o n -. P a r t i c u l a r  integral  of equation  of equation  Maximum a l l o w a b l e s i z e Solid  60j  N a t u r a l f r e q u e n c y o f t h e damper  L  angle  ,. N a t u r a l f r e q u e n c y o f t h e i s a t e l l i t e boom Function defined i n equation  Mr; ^/**^6  A n g u l a r v e l o c i t i e s about  n o r m a l mode o f t h e (2.45)  the principal  (J  Orbital  cJ*  D i m e n s i o n l e s s damper n a t u r a l  &  (2.37)  of impulse f o r s t a b i l i t y  03  co.  (2.37)  axes  f r e q u e n c y = 2 7t/t© frequency  Subscripts  "f  Final  I  Integer  ma/  Maximum  Y\ steely o  value  Value a t & Steady Initial  = 27Tn  state conditions  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 a n d ©> respectively  1. 1.1  P r e l i m i n a r y Remarks The  aspects and  INTRODUCTION  m o t i o n o f a s p a c e v e h i c l e i n v o l v e s two  of i n t e r e s t , namely, the a n a l y s e s  of i t s o r i e n t a t i o n .  as t h e  The  former,  an  of i t s t r a j e c t o r y  generally referred to  o r b i t a l motion, i s concerned w i t h the  o f t h e m o t i o n o f t h e mass c e n t r e and extension of c l a s s i c a l  dynamical  may  determination  be t h o u g h t o f  c e l e s t i a l mechanics.  On  the  o t h e r h a n d , t h e m o t i o n o f a s a t e l l i t e a b o u t i t s own o f mass i s c a l l e d  as  centre  libration.  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 it  i s desirable to maintain a s a t e l l i t e  in a fixed orienta-  t i o n r e l a t i v e t o the e a r t h . , For example, proper f u n c t i o n i n g of 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 or of weather s a t e l l i t e s attitude control. satellite,  cloud cover  requires  U n f o r t u n a t e l y , the o r i e n t a t i o n of  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  deviates i n time a n c e s , e.g.  scanning  impacts,  beginning,  magnetic f i e l d  solar radiation  interactions.  t o u n d e s i r a b l e l i b r a t i o n a l m o t i o n w h i c h must be f o r the s u c c e s s f u l o p e r a t i o n of the  This  pressure, leads  controlled  satellite.  S e v e r a l methods o f a t t i t u d e c o n t r o l a r e be  the  under the i n f l u e n c e of e x t e r n a l d i s t u r b -  micrometeorite  g r a v i t a t i o n a l and  T h e y may  antennae  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  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 the  available.  techniques.  expenditure  of  2 energy which i s a very  expensive  ment p a c k e d s p a c e c r a f t .  commodity a b o a r d an i n s t r u -  The m a i n a d v a n t a g e o f t h i s  nique i s i t s a b i l i t y t o maintain  tech-  the specified orientation  w i t h almost any d e s i r e d degree o f accuracy. Passive s t a b i l i z a t i o n techniques, can p r o v i d e  the necessary  attitude control i fthe orienta-  t i o n requirements a r e not t o o severe. obtained  w h i c h u s e no power,  Stabilization i s  by e m p l o y i n g t h e n o n - u n i f o r m i t i e s  of the environ-  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 satellite..  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 s o l a r , magnetic, and aerodynamic e f f e c t s . The  gravitational,  1 2 '  g r a v i t a t i o n a l moment a r i s e s b e c a u s e o f t h e l o c a l  v a r i a t i o n of the g r a v i t a t i o n a l acceleration within the satellite.  I t t e n d s t o make t h e " l o n g " a x i s  (the axis of  minimum moment o f i n e r t i a ) o f t h e s p a c e c r a f t p o i n t i n t h e local vertical direction.  T h e r e i s no d i s c r i m i n a t i o n  b e t w e e n " u p " a n d "down."  The maximum g r a v i t y - g r a d i e n t  torque  i s g i v e n by  (1.1) The  electromagnetic  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 a n d h e n c e when a b s o r b e d o r r e f l e c t e d a pressure.  A s a t e l l i t e with a l a r g e surface area  asymmetrically experience  with respect  t o t h e c e n t r e o f mass  exerts  placed will  a moment w h i c h may be u t i l i z e d t o e s t a b l i s h a  3 preferred orientation.  The r a d i a t i o n p r e s s u r e  i n the v i c i n i t y  o f t h e e a r t h i s g i v e n by  f> =  = 9.7 Xf0'  lb/ft*  3  r  ( 1 : 2 )  so t h a t t h e 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 The  =  A<lf  in  f o r some d i s t a n c e  i ti s r e l a t i v e l y stable  a s t h e i n v e r s e t h i r d power o f t h 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  w i n d a n d becomes q u i t e u n s t e a d y . field  extends  Within ten earth-radii  v a r i e s approximately  distance.  (1.3)  m  earth's magnetic f i e l d  i n t o space. and  frK  The e a r t h ' s m a g n e t i c  c a n 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  three d i s t i n c t  ways.  Rotary motion o f t h e conducting m a t e r i a l i n t h e spacecraft effect  induces  eddy c u r r e n t s w h i c h d i s s i p a t e e n e r g y .  The  i s t o p r o v i d e a moment w h i c h o p p o s e s t h e m o t i o n .  R o t a t i o n w i t h respect t o t h e earth's magnetic f i e l d o f ferromagnetic m a t e r i a l s present  i n the s a t e l l i t e results i n  h y s t e r e s i s l o s s e s a n d h e n c e i n a d a m p i n g moment.  The  i n t e r a c t i o n b e t w e e n t h e r e s i d u a l m a g n e t i c moment o f t h e spacecraft with the earth's f i e l d In  a l s o p r o d u c e s a moment.  c o n t r a s t t o t h e p r e v i o u s two c a s e s , t h i s  conservative. moment i s  The maximum t o r q u e  interaction i s  due t o t h e m a g n e t i c  r;  =  7 . 3 8 x , o  The p r o b l e m 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 t h e earth's the p o s i t i o n of t h e s a t e l l i t e  a f t  a  field  change w i t h  i ni t s orbit.  U n d e r c e r t a i n c o n d i t i o n s , a e r o d y n a m i c f o r c e s may provide  a n 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  the v e l o c i t y v e c t o r . satellite  Unfortunately  these  f o r c e s cause t h e  t o r e - e n t e r t h e e a r t h ' s atmosphere thus  the a p p l i c a t i o n o f t h e technique maximum a e r o d y n a m i c t o r q u e  limiting  t o a short i n t e r v a l .  The  i s g i v e n by  (1  King-Hele able  has discussed t h e determination  o f jO i n c o n s i d e r a  detail. The v a r i a t i o n o f t h e moments g i v e n b y 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  configuration^ described  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, t e n d s  to attain the preferred direction associated with moment.  The r e m a i n i n g  moments w h i c h a c t i n d i f f e r e n t  directions constitute disturbances.  The c h o s e n  t h e r e f o r e , must h a v e a l a r g e maximum t o r q u e d i s t u r b i n g moments s o t h a t t h e p e r t u r b e d the a l l o w a b l e  limits.  that  technique,  compared t o t h e  motion I s w i t h i n  5 Table Representative  Gravity-Gradient  I Satellite  Configuration  GEGS - A  Satellite Moments o f i n e r t i a ,  I I  Projected  area,  xx zz  615.3  slug f t  2  20.8  slug f t  2  13.1  A. r  f t  2  O f f s e t b e t w e e n c e n t r e o f mass a n d centre  of area,  5.75 f t  R e s i d u a l m a g n e t i c moment, 1.2  Gravity-Gradient The  fft,  Stabilized  302  pole  Satellites  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  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 h i s technique satellites.  of s t a b i l i z i n g  A survey  over a  interest  the a t t i t u d e of a r t i f i c i a l  of the l i t e r a t u r e reveals that the  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 along paths.  cm  two  The m a j o r b u l k o f t h e l i t e r a t u r e i s c o n c e r n e d e i t h e r  w i t h t h e 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 restricted specific The  conditions or with the detailed simulation of  configurationso 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 of 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 t h e g o v e r n i n g equations Moran lem  non-linear  o f m o t i o n do n o t p o s s e s s a c l o s e d f o r m  and Yu  coupled solution.  f o u n d t h a t some s i m p l i f i c a t i o n o f t h e p r o b -  i s p o s s i b l e a s t h e p e r t u r b a t i o n s o f t h e o r b i t due t o  6  — Gravity gradient — Aerodynamic — Solar pressure -— Magnetic — Cosmic dust  \ 10-+  GEOS-A Geometry  -10  10  — —^T""*" 10 2 5 10 2  Approximate limit of magnetic field of earth  1  Altitude Figure  1-1  IO  1  miles  M a g n i t u d e e f f o r c e s a c t i n g on a representative s a t e l l i t e  IO  -  7 the  l i b r a t i o n a l motion of the s a t e l l i t e are negligibly-  small.  This  motion using  makes i t p o s s i b l e  the simple Keplerian  N e l s o n and L o f t of a r i g i d  studied  the o r b i t a l  equations.  small amplitude l i b r a t i o n s  body i n a c i r c u l a r o r b i t u s i n g  t i o n s of motion. pling  to describe  l i n e a r i z e d equa-  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 d e c o u -  o f t h e m o t i o n s i n and n o r m a l t o t h e o r b i t a l  plane.  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  t i o n s of a dumbbell s a t e l l i t e  i n a circular  libra-  orbit.  S c h e c h t e r " ^ attempted t o extend t h i s s o l u t i o n t o t h e case of small  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  perturbations.  The method h a s l i m i t e d a p p l i c a b i l i t y a s t h e r e s u l t i n g p e r t u r bations  grow w i t h o u t b o u n d . Baker"'"''" f o u n d p e r i o d i c s o l u t i o n s w i t h  quency f o r a dumbbell s a t e l l i t e showed t h a t proportional  i n an e l l i p t i c  fre-  orbit.  He  the amplitude of the motion i s approximately to the e c c e n t r i c i t y of the o r b i t . 12  A recent able  orbital  interest.  p a p e r by Z l a t o u s o v e t a l  i s of  These a u t h o r s a l s o o b t a i n e d  t i o n s of the planar  equations of motion.  consider-  periodic  solu-  The s o l u t i o n s w e r e  functions  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 which  described  the geometry o f t h e s a t e l l i t e .  i n addition to the solutions predicted  I t was f o u n d  by.Baker, there  that, may  be two o t h e r s o l u t i o n s f o r t h e same v a l u e s o f t h e p a r a m e t e r s . Infinitesimal perturbations  about t h e s e s o l u t i o n s were  investigated f o r stability.  I t was shown t h a t  stable  p e r i o d i c m o t i o n was p o s s i b l e  for a l l orbit eccentricities  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 g e o m e t r y .  Non-linear  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 m a g n i t u d e yield  of a f i n i t e disturbance which  s t a b l e m o t i o n was  not  would  determined.  These a u t h o r s a p p e a r t o have been t h e f i r s t  to  l y z e the problem u s i n g the concept of a s t r o b o s c o p i c plane. of  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  phaseresult  repeated application of a point transformation.  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  ana-  Stable  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 h i c h appears as a s e t of  fixed  points.  The of  a n a l y s i s i n v o l v i n g the t h r e e degrees of  a rigid satellite  circular,  i s very d i f f i c u l t .  freedom  I f the o r b i t i s  the H a m i l t o n i a n i s constant which s p e c i f i e s  bounds  13 on i n i t i a l  conditions to guarantee  DeBra^  f o r m u l a t e d the problem of the l i b r a t i o n  rigid arbitrarily He  stability.  shaped 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 . . .  c o n s i d e r e d the g e n e r a l case w i t h t h r e e degrees of  i n t h e p r e s e n c e o f a s p e c i f i c f o r m o f damping. of  t h e s a t e l l i t e was  conditions.  The  Instability  was  response  been c o n c e r n e d w i t h s a t e l l i t e s which are hinged together.  initial  a t t r i b u t e d to the non-linear 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 h a s n e a r l y  The  always  consisting of several bodies j o i n t s of such systems  c o n v e n t i o n a l l y e q u i p p e d w i t h s p r i n g s and e n e r g y The  freedom  determined f o r a l i m i t e d set of  c o u p l i n g e x i s t i n g between t h e degrees o f  mechanisms.  of a  are  dissipating  i n t r o d u c t i o n of a r t i c u l a t e d bodies  increases  the  complexity  o f t h e p r o b l e m b u t , as p o i n t e d  o u t by H a r t b a u m  et a l , " ^ the c o n f i g u r a t i o n possesses c o n s i d e r a b l e major advantages are very  Z a j a c " ^ has a n a l y z e d  I t was  design  flexibility.  the small amplitude planar  ( F i g u r e 1-2-a) i n a c i r c u l a r  shown t h a t i n t h e p r e s e n c e o f v i s c o u s  c o n f i g u r a t i o n reduces the time constant orbital  The  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 m o t i o n and c o n s i d e r a b l e  of a two-body s a t e l l i t e  merit.  motion  orbit.  damping t h e  t o 0.137  of the  period. Multibody  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  by s e v e r a l a u t h o r s .  Etkin  motion f o r a s a t e l l i t e 1-2-b).  F o r an o r b i t  18 '  has d e r i v e d t h e e q u a t i o n s  consisting of r i g i d  bodies  of the c o n s t i t u e n t bodies. characteristic configurations.  equation This  (Figure  of low e c c e n t r i c i t y the equations  m o t i o n w e r e l i n e a r i z e d by a s s u m i n g s m a l l a m p l i t u d e  of  of  librations  The r o o t s o f t h e r e s u l t i n g  were e v a l u a t e d  f o r a wide range o f  showed t h a t t h e m o t i o n c o u l d be h i g h l y  damped. 19 F l e t c h e r , R o n g v e d a n d Yu formulated the equations o f m o t i o n f o r a t w o - b o d y c o m m u n i c a t i o n s s a t e l l i t e w h i c h was 20 7  p r o p o s e d o r i g i n a l l y by P a u l , West a n d Yu A considerable the  (Figure  amount o f d e t a i l e d s i m u l a t i o n showed  1-2-c). that  p e r f o r m a n c e o f s u c h a d e v i c e w o u l d be s a t i s f a c t o r y . 15 21 H a r t b a u m e t a l ' a s w e l l a s Hughes have a t t e m p t e d  to optimize  the c o n f i g u r a t i o n of a r t i c u l a t e d  ( F i g u r e s 1-2-d a n d 1-2-b pointing  accuracy.  satellites  respectively) with respect  to  Several simpler  configurations  have been p r o p o s e d w h i c h a r e  than the multi-body s a t e l l i t e  discussed  above.  In  22 1963,  Paul  investigated a satellite  suspended from a " l o s s y " s p r i n g damped o s c i l l a t i o n s of the s p r i n g . by  only  This  (Figure  1-2-e).  The d e v i c e  about axes p e r p e n d i c u l a r  d i f f i c u l t y was e l i m i n a t e d 23  Buxton, Campbell and Losch  permitted  i n w h i c h a mass was  t o that  i n a study  w h e r e t h e s p r i n g was a l s o  t o execute t o r s i o n a l o s c i l l a t i o n s  Systems o f t h i s type a r e c h a r a c t e r i z e d  (Figure  1-2-f).  by a m p l i t u d e  dependent  damping. S a t e l l i t e s designed f o rgravity-gradient are  necessarily very long.  stabilization  R e c e n t l y much a t t e n t i o n h a s b e e n  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 .  Katucki  and  M o y e r ^ have c o n s i d e r e d  ing  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 25  2  large  t h i s t o be a m a j o r f a c t o r a f f e c t -  changes i n t h e c o n f i g u r a t i o n .  Ashley  investigated  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  flexible  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 Dow e t a l elaborate  successful gradient GEOS-A, are  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 simulation  The  simulation  of f l e x i b i l i t y studies  effects.  o f t h i s n a t u r e have been  i n p r e d i c t i n g the performance of e x i s t i n g g r a v i t y -  satellites. 5  field.  The G e o d e t i c E a r t h  Orbiting  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 ,  performing as expected.  Satellite, GGTS,  2 7  © Hinge equipped with damper F i g u r e 1-2  Models o f multi-body  satellites  12 1.3  P u r p o s e and The  the  main purpose of t h i s  limiting  stabilized The  Scope of I n v e s t i g a t i o n i n v e s t i g a t i o n i s to  i n i t i a l conditions for a gravity-gradient  s a t e l l i t e as a f u n c t i o n of d e s i g n  parameters.  secondary purpose i s t o i n v e s t i g a t e the nature  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 b a s e d on t h i s  which w i l l  s p e e d and  s e v e r a l models a r e  s i m p l i f y the a n a l y s i s .  studied.  s p e c i f i c a l l y mentioned The ing  obtain  first  lib-rations  are  of  knowledge  To  these  I n each case o n l y t h o s e i n c l u d e d i n the a n a l y s i s .  plane of the  orbit.  execut-  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 M o d e l number two  ends, forces  model i s t h a t of a r i g i d s a t e l l i t e  i n the  the  2-1).  i n c l u d e s d i s s i p a t i o n by t h e  addition  22 o f a damper o f t h e is  f o r m p r o p o s e d by P a u l .  e s s e n t i a l f o r the proper f u n c t i o n i n g of the  (Figure  t h i r d m o d e l assumes t h e  d i s s i p a t i v e but  subject  solar heating.  The  model  The satellite  to considerable  distortion  considered  (Figure  nondue  removed a l t h o u g h first  chosen t o p r o v i d e  the  The  restriction  4-1).  simplest.  a basic understanding  In general  to planar  o r b i t i s assumed t o be  model i s the  to  satellite's  l a s t m o d e l 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 ( F i g u r e 5-1).  The  s a t e l l i t e t o be  e f f e c t s of v a r y i n g the  p h y s i c a l p r o p e r t i e s are  motion.  motion  3-1). The  is  Planar  I t was of the  rigid  motion  circular. specifically nature  of  the  the l i b r a t i o n a l motion i s two-dimensional  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 idealized  i n model two.  The i m p o r t a n c e o f t h e r m a l  distor-  2/i- 26 t i o n h a s b e e n p o i n t e d o u t by s e v e r a l The t h i r d m o d e l p r o v i d e s a c o n v e n i e n t effects.  investigators. way  of studying  these  2. 2.1  PLANAR LIBRATIONAL MOTION OF  Formulation The  o r b i t has  of the  A RIGID S A T E L L I T E  Problem  planar motion of a r i g i d s a t e l l i t e o b e e n s o l v e d by K l e m p e r e r  sine function.  I n an  elliptic  o r b i t a l angular  v e l o c i t y and  in a  circular  i n terms of the  elliptic  o r b i t , the v a r i a t i o n s i n the  the l o c a l g r a v i t a t i o n a l  gradient  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 e x c h a n g i n g b e t w e e n t h e l i b r a t i o n a l and  energy  o r b i t a l degrees of freedom.  g e n e r a l , t h i s leads t o a r e d u c t i o n i n the range of  initial  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 m o t i o n as pared  t o the  corresponding  range f o r a c i r c u l a r  chapter  i n v e s t i g a t e s t h e b o u n d s t h a t must be  initial  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  In  com-  orbit.  This  p l a c e d on  eccentricity  the and  s a t e l l i t e geometry t o guarantee s t a b l e motion. Consider  a rigid  satellite  of 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 m o t i o n w h i l e m o v i n g i n an  elliptic  ( F i g u r e 2-1).  The  o r b i t about the  centre of f o r c e  mass d i s t r i b u t i o n o f t h e  0  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 t h e o r b i t d e f i n e s a p l a n e . p o s i t i o n of the 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 ,  m e a s u r e d f r o m t h e p e r i c e n t r e , P, orbital  The  i n the d i r e c t i o n of  0,  the  motion. L e t x y z be a s e t o f o r t h o g o n a l  the y - a x i s normal to the  plane  body c o - o r d i n a t e s  of the o r b i t .  The  angle  with  F i g u r e 2-1  Geometry of p l a n a r m o t i o n o f a satellite  rigid  16 b e t w e e n t h e l o c a l v e r t i c a l , OS, a n d t h e z - a x i s i n t h e s e n s e of the o r b i t a l motion defines the l i b r a t i o n angle, an e l e m e n t o f mass, dm^, for  the kinetic  JJ  of the s a t e l l i t e the expressions  Cos ijj - rSin </> + (e * f)i\  r  =  For  a n d p o t e n t i a l e n e r g i e s c a n be w r i t t e n a s  J*> |[e  =  .  |V  + rQ Z  z  -  +  Sin  (2.1)  fd + y)V^)  - xrC*5i}>)  and  {dm,  ~ju dm  h  r  I f the o r i g i n  (2.2)  o f t h e x y z a x e s , S, i s a t t h e c e n t r e o f m a s s ,  J dri^ f* »b J  =  fyH  (2.3)  ^  =  =  °.  '  {2  k)  17 M o r e o v e r , i f t h e a x e s a r e c h o s e n t o be t h e p r i n c i p a l  2 (iyy I 4  zz  =  =  With these r e l a t i o n s potential  JL(1  ~~ I  +1 - J  axes  )  yx  (2.5)  )  0.  the expressions f o r the k i n e t i c  and  e n e r g i e s become  T  (2.6)  (2.7)  Using the Lagrangian  f o r m u l a t i o n the equations of motion  c o r r e s p o n d i n g t o t h e t h r e e d e g r e e s o f f r e e d o m c a n be written  as (2.8)  ra z  \  rLoL (Q + a)).. 3M-  constant  (2.9)  •- -ft  5 i n (p Cos <p =  O  (2.10)  These e q u a t i o n s  are  e s s e n t i a l l y t h o s e o f Yu.'  i n v o l v i n g moments o f i n e r t i a represent motion.  perturbations  satellite.  mate s o l u t i o n s t o  ( 2 . 1 0 ) and  due  Moran  (2.8)  and  (2.9)  Yu  (2.9)  the  approxi-  typical  p e r t u r b a t i o n terms i s  perturbations  leads  of  finite  found  showed t h a t f o r  c o n t r i b u t i o n of the Neglecting  and  to the l i b r a t i o n a l motion of the  of equations  and  be a t t r i b u t e d t o t h e  dimensions of the  extremely small.  (2.8)  terms  o f t h e u s u a l two-body e q u a t i o n s  T h e i r p r e s e n c e can  s a t e l l i t e s the  i n equations  The  i n the  orbit  s a t e l l i t e , the s o l u t i o n  to the  classical  Keplerian relations  r 9  =  2  *  r Noting  e  (2 . i l )  fx ( i + e Cos 9}  '  that  4  cit  r  z  J 49  (2."12) '  and  =  9 (2.13)  r  equation  (2.10) can  4  be r e w r i t t e n  Je*  as  19  T h i s form  of the equation of motion  was  presented  indepen-  12 d e n t l y by Z l a t o u s o v e t a l . I n g e n e r a l , t h e g o v e r n i n g n o n - l i n e a r , non-autonomous differential  equation with periodic  coefficients  a d m i t o f any  c l o s e d form  The  solution.  does not  non-linearity  is  s i m i l a r to t h a t of a " s o f t " s p r i n g thus r a i s i n g the p o s s i b i l ity  of amplitude  2.2  Simple  2.2.1  dependent  Exact S o l u t i o n s  Circular  O r b i t (e =  When t h e o r b i t tion  instability.  0)  of the s a t e l l i t e i s c i r c u l a r , equa-  ( 2 . 1 4 ) a s s u m e s t h e autonomous  which  has  the f i r s t  V* Equation  form  integral  +  3Ki Jiff  = constant = C  c  (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  phase-plane.  .  ( -i6) 2  ijj-  F o r 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^  the  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 m o t i o n  is  F i g u r e 2-2  constant  of  illustrates  the e f f e c t  of v a r y i n g the  periodic.  integration. Different  values of  result  i n different  trajec-  20  21 t o r i e s which f o r C  = 3K^  c  a r e shown i n F i g u r e 2-3.  noteworthy that these  curves  G , hence the r e g i o n s  o f s t a b i l i t y may  c y l i n d e r s w i t h the  (C  a s o l u t i o n i n terms of the  = 0 The  i n v a r i a n t with respect be  considered  cross-sections depicted  For p e r i o d i c motion  where  are  i/i  and  =  <  3^)  elliptic  =  Jc^  change i n t h e  c  It is  to  (2.15) y i e l d s  sine function,  /3I7  Sin  ^  m  &  at  x  d u r i n g one  c y c l e of the l i b r a t i o n a l motion i s given  9 =  The  initial  s o l u t i o n s may  -  (2  s o l u t i o n s where  A0  = 2 TT  to  i> (o)  ia)  n/m  orbits.  conditions required to generate  be t a k e n  0  by  Si  in n  6.  complete  Ae = j= K( " rWx). indicating m oscillations  be  here.  equation  o r b i t a l angle  Of p a r t i c u l a r i n t e r e s t a r e  to  these  be  = o  hl  (2.19)  = 73K;  ^(o) The  v a r i a t i o n o f d)  Sir.  (p^.  (0) r e q u i r e d t o p r o d u c e p e r i o d i c  ' P;»  solutions with s p e c i f i e d values F i g u r e 2-4  as a f u n c t i o n o f  K^.  o f m and  n i s plotted in  -2 -90  -60  -30  0  30  60  lp, Degrees  F i g u r e 2-3  L i m i t i n g phase p l a n e for e = 0  trajectories  90  2  K| F i g u r e 2-4  I n i t i a l angular v e l o c i t i e s required to 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  motion  n o n - l i n e a r term  (2.14) may  i n the governing  be r e p r e s e n t e d by t h e T a y l o r ' s  Sin f Cos 0> =  C o n s i d e r ' now  equation  of  series  zy  Sin  a s o l u t i o n of equation  (2.20)  (2.14) o f t h e  form  00  = X- mn  V which  A  has  S i  a p e r i o d of 2Tfn.  be w r i t t e n  "  ^ The  > ( first  two  n  =  l  > ' -) 2  no=/  and t h e e x p a n s i o n  derivatives  £,  may  for  ;  (2.22)  3  l e a d s t o an e x p r e s s i o n o f  form  V  5  as  1  CO  (2.21)  00  &  oo  w  "  the  25  m -i r  f  vfi  m-t  K^,-%)6  5  +  S  i  n  n  n  5  Similar expressions  c a n be o b t a i n e d  for  only s i n e terms.  M =  '  <x>  z equation  (2.14)  becomes  Recognizing  J.  7  , tyt> , ..... s o  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 introduces  •(2.23)  fr»t4Pb-^)e  i n (2.20)  that  (2.24)  26  m 9  -  0.  (2.25)  The p r i n c i p l e o f h a r m o n i c b a l a n c e r e q u i r e s t h e c o efficient  o f each t r i g o n o m e t r i c  t e r m t o be i n d i v i d u a l l y  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 the  A  n  ^ ( i = 1,2,....).  for a significant  The e q u a t i o n s  zero  to solve f o r  are non-linear  and,  number o f t e r m s i n t h e assumed s o l u t i o n ,  are  difficult  to solve.  However, t h e i m p o r t a n t  conclusion  can  be d r a w n , t h a t t h e assumed f o r m o f t h e s o l u t i o n i s  correct. 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  also indicates that  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  ©.  there  of the  parameters. There a r e other 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 are  c l o s e l y r e l a t e d t o those already  Q  e  -  investigated.  Let (2.26)  TC  then Sin  6  —  Cos  e  -  5in  Cos.  Q*  a*  (2.27) '  27  de*  J<9  J jo*  (2.27) cont'd  _ ~" do**-  z  a n d e q u a t i o n (2.14)  becomes  (2.28)  +• 3Ki 5in(]JCosy Equation  «  0-  (2.»28) h a s e s s e n t i a l l y t h e same f o r m a s  (2.14) e x c e p t f o r t h e s i g n o f e, h e n c e t h e same f o r m 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)  exist. point  These s o l u t i o n s appear a s odd f u n c t i o n s 6 = TT .  about t h e  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  are  i d e n t i c a l t o t h o s e w h i c h a r e odd a b o u t t h e p o i n t  Gt^o  • n " 1o  0=0,  _  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 that  the apocentre corresponds t o 0 = 0 .  i s such that a t p e r i c e n t r e from that  I f the solution  ifj ^ 0, t h e s o l u t i o n i s d i f f e r e n t  obtained 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 Let  solutions.  28  (2.30)  then  (2.3D  d©  d©*  4  =  so t h a t  -  S i n (|^  ( 2 . 1 4 ) becomes  (if e C « e ) « F  #  - Z e 5 m © ( ^ + i ) - J K i ^ d ; a $ ^ = o. #  This i s i d e n t i c a l t o equation -K^.  Cos ^  (2.32)  (2.14) w i t h K., r e p l a c e d by  The e q u a t i o n may b e 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 t h e s i m i l a r  solution  (2.33) rw = i In p h y s i c a l terms, 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  that  12  Zlatousov et a l  obtained periodic solutions of t h i s  numerically f o rn = 1 without establishing the general of the s o l u t i o n presented  here.  type form  P e r i o d i c s o l u t i o n s determined of e and  thus  the parameters and  one.  correspond  values  t o r e a l i z a b l e s i t u a t i o n s when  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  Larger v a l u e s o f these parameters  meaning as f o r e > >  with negative  1 t h e o r b i t a l motion  h a v e no p h y s i c a l  i s n o t p e r i o d i c and  1 i s physically impossible.  2.2.3  Numerical The  Determination of Periodic Solutions  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  determining t h e p e r i o d i c s o l u t i o n s of (2.14). solution of the resulting  The a c t u a l  equations i s q u i t e i n v o l v e d as t h e  number o f t e r m s r e q u i r e d f o r a n a c c u r a t e s o l u t i o n i s a s t r o nomical.  F o r t u n a t e l y , t h e knowledge o f t h e form  helps  considerably i n the numerical evaluation of the periodic solutions. The  numerical determination of the periodic  was a c c o m p l i s h e d  as f o l l o w s .  grammed t o s o l v e e q u a t i o n Initial  A digital  solutions  c o m p u t e r was p r o -  (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 .  c o n d i t i o n s w e r e c h o s e n c o n s i s t e n t w i t h t h e known  form  and  equation  (2.14) was i n t e g r a t e d u n t i l  f i n a l values of ^ from  a n d <^r w h i c h  (2„34) w e r e n o t e d .  © = 27Tn.  The  were, i n g e n e r a l , d i f f e r e n t  A c o r r e c t i o n was t h e n made t o t h e  value  o f y/ s o a s t o c a u s e t h e f i n a l  c o n d i t i o n t o become  Q  identical with the i n i t i a l  condition.  s o l u t i o n e x i s t s , the process value  of  ijJ  0  When a p e r i o d i c  converges t o g i v e t h e r e q u i r e d  and t h e s o l u t i o n o f t h e i n t e r v a l .  Typical periodic solutions with orbital ( i . e . n = 1)  i n F i g u r e s 2-5  are presented  frequency  and 2-6.  The  i  initial  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. was p r e s e n t e d  i n r e f e r e n c e 12.  independently  indicates that f o r  A similar  1/3  l e s s than  curves  presented  t h e r e may  This r e s u l t i s i n accord with the 2-4.  i n Figure  The n u m e r i c a l  The d i a g r a m  t h e r e i s o n l y one  periodic solution while f o r l a r g e r values of be a s many a s t h r e e .  diagram  technique  o f any d e s i r e d a c c u r a c y .  can produce p e r i o d i c s o l u t i o n s  A reliable  estimate  of the error  may be made by c o m p u t i n g t h e f u n c t i o n  o v e r t h e i n t e r v a l 0 ^ 0 £n7t.  The e x a c t  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 = n j t s o t h a t €3 s h o u l d be i d e n t i cally  zero.  2-8.  The maximum o b s e r v e d  high  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 value  o f 6? , w h i c h o c c u r r e d a t  e c c e n t r i c i t y , was .003 r a d i a n s a n d may be a t t r i b u t e d  to the nearly discontinuous It  behaviour  of the solution.  i s a l s o possible t o search f o rs o l u t i o n s of longer  90  180  270  360  e, Degrees  Figure  2-5  P e r i o d i c s o l u t i o n s as e c c e n t r i c i t y (K^ = 1 ,  functions n = 1)  of  33  _2 I -1  i  i  i  i  l 0  i  i  i  O r b i t Eccentricity Figure 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  i  i 1  34  Figure 2 - 8  T y p i c a l v a r i a t i o n s of the e r r o r found i n the numerical determination of periodic solutions  35 period  (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  versatile  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  solutions  in this respect.  with a period of 4R . which  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  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 w h i c h 0.35°  lates only twice at e «  I n i t i a l values of the  derivatives,  y/' ^ 2 ^ '  Figure 2-10.  V e r t i c a l tangents t o these curves  ^  o r  t  ^  a e  oscil-  s  o  l  u  t  l  o  n  s  a  r  e  presented i n correspond  t o t h e p o i n t s w h e r e 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  For small amplitude motion,  equation  ( 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  Sin  f  if  (2.36)  Cos The  f  equation o f motion  «  I .  reduces t o  (i + e C o $ e ) / - ^ e S i n e ^ ' + J K  t  f  = zts<*e  (2.37)  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 ) <|>  (2.38)  or  37  1  1  1  1  1  1  1  i  i  °'/  /  -1  n =2  .  -0.4  I  -0.2  1  1  0  1  0.2  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  0.4  38 to  (2.39) where  3K  ?  I  +  e Cog 9  (2.40)  e Cos e  The c o m p l e m e n t a r y s o l u t i o n t o (2.39) c a n 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)  satisfies  the inequality  Mil  F = F i g u r e 2-11  « I  (2.41)  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 a n d K^ = 1.  I t i s evident that,  even f o r l a r g e  values o f t h e e c c e n t r i c i t y , the i n e q u a l i t y i n (2.41) i s reasonably w e l l  satisfied.  The a p p r o x i m a t e 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  (2.42)  where  (2.43)  Figure  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 o r b i t e c c e n t r i c i t y (K. = 1)  and  40 and  represents  the p a r t i c u l a r i n t e g r a l obtained  using  t h e method o f v a r i a t i o n o f p a r a m e t e r s  j * _  T  f  Z e Sin 6  e  $  J@  " f t J o * * ' In g e n e r a l ,  the evaluation  be a c h i e v e d o n l y n u m e r i c a l l y computationout  Considerable  ~  o f t h e WKBJ s o l u t i o n c a n  a n d i n v o l v e s a l a r g e amount o f  simplification  i s possible  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  m a t i o n by a d o p t i n g Neglecting  '  withapproxi-  the f o l l o w i n g procedure. second and h i g h e r  degree terms i n e and  putting G O *  3K  =  F  (2.45) 3 K i -  V the  required functions  Gj  &  c a n be a p p r o x i m a t e d a s  6o (i L  ye  Cos  e) (2.46)  41  Cos (2.46) cont'd Cos  £J 0 L  Thus, w i t h i n a m u l t i p l i c a t i v e constant, the s o l u t i o n s  (2.43)  become  T^7  Cos cJ Q  «  u  +ev>(i  ^)co5^ +>)9 L  (2.47)  A  *  Sin  + ev(i"  5in(«JL+i)© (2.48)  -  The f i r s t  4  l ) ^ ( ^ ' ) e  .  d e r i v a t i v e i s m i s s i n g from equation  (2.39)  so t h a t the denominators i n (2.44) are equal t o 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 a l r e a d y i n v o l v e s the f i r s t  power o f  42 the  eccentricity  so  that  6 J 0 J^5,»  Cos  9 Sin ^ 0  L  Je^±0(gf) (2.50)  r  2e 9  T- /CJ. Sin 8 - Sm  •+ 0(e ); z  Using the  transformation  (u ±  i)  L  (2.38) t h e  0 )  m  solution for  (j) i s  (2.5D  where  ^ = Cos CJ 9 l  4 e(-|- - ^"2)Co3(H+i)e  + e(£ + ^ - i ) c ^ , - ' ) e - ' 0 ( e ) i  o s  (2.52)  5)»  Figures  2-12  and 2-13  fa-i)6 + 0(e*-)  compare t h e WKBJ  solution  43 (2.42  - 2.44) and i t s a p p r o x i m a t i o n  exact  numerical  s o l u t i o n of equation  that the simplification the  (2.51,  error already  2.52) w i t h t h e  (2.14)»  I t i s apparent  t o t h e WKBJ method d o e s n o t i n c r e a s e  present.  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 general  time.  This i s i l l u s t r a t e d  tional  m o t i o n i s shown o v e r e i g h t r e v o l u t i o n s , a n d d i s c u s s e d  on page 4 7 .  fashion f o r a long period of i n F i g u r e 2-14,  where t h e l i b r a -  The WKBJ s o l u t i o n i s n o t p l o t t e d b e c a u s e i t i s  n e a r l y c o i n c i d e n t w i t h the approximate The a n a l y s i s d o e s p r o v i d e t h e maximum v a l u e  result.  u s e f u l i n f o r m a t i o n about  o f the amplitude  of libration.  From  (2.51),  w h i c h h a s a minimum when  c,  ze  c  provided  CJ^ > (e/fa  - 3^))  i s always g r e a t e r than  •  (2  Hence t h e a m p l i t u d e  £c/<jkJj?~')  +  dicts the "period" of l i b r a t i o n with  0(g )° Z  *  of libration  The m e t h o d  considerable  pre-  accuracy.  54)  \  \  /  /  Fr  //  /t  Kj =  l,  e=  o.l  Exact  WKBJ Approximation  to 90°  180°  270°  360°  WKBJ  450°  0 F i g u r e 2-12  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 m o t i o n w i t h t h a t d e t e r m i n e d b y t h e WKBJ m e t h o d a n d t h e a p p r o x i m a t e ¥KBJ m e t h o d (K^ = 1 , e. = 0.1)  540  30"  / \\  20*  /  /  --  M  W\ >\  10°  '  //  \  X  \  // ^/ ' f 1, —V~\ i/  It  /  \\  '(  -10'  V-  V  /  /  1 1  / // / 1/ /  V /' 'ii/ \ \ ' v>  -20°  v  -30°  //  \ \  Kj = l, e = 0.3 Exact  WKBJ Approximation  to -40°  \\  90* F i g u r e 2-13  180°  270°  e  360°  WKBJ 450°  540*  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 m o t i o n w i t h t h a t d e t e r m i n e d b y t h e WKBJ m e t h o d a n d t h e a p p r o x i m a t e WKBJ method (K. = 1 , e = 0.3)  VJ1  fl  fl1  in '>  f'  f '  1  /'ll  11  1  i  li  \11 i  i i ii  //  f l  1  ,1  i  Ai  ' A  I' ll  |l  1  1  J  /  '1/ L r>/ > /' 1 /'  Ij i  V  1/  V  V  \ 1  IV /  T  / Kj = l, e - 0.  Approximation to WKBJ 3  4  5  li V  Exact  F i g u r e 2=14  ll  II ii  i \  / i  l/  1  2  Ii  fi \  (I*/ \  1  n—  , 1  fI  / Mi  f l |i  x  Ml 1  1  1  6  *  )  3 s  °  *'(0) = 0 7  8  Orbit 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 m o t i o n w i t h t h a t d e t e r m i n e d by t h e a p p r o x i m a t e WKBJ m e t h o d o v e r e i g h t o r b i t s ( K = 1, e = 0.3) i  47 F o r t h e p a r t i c u l a r c a s e o f e = 0 a n d 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 frequency presence  i s y/3 t i m e s t h e o r b i t a l  frequency.  Further, the  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  executes l i b r a t i o n a l  motion.  I t may be p o i n t e d o u t t h a t t h e f r e q u e n c y 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 orbital orbit,  consists ofthe  frequency, the frequency of l i b r a t i o n and t h e modulation  frequencies.  spectrum  ina  circular  p r o d u c t s o f t h e two f o r e g o i n g  The r e s u l t i n g m o t i o n  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 frequencies. which tion  This also explains the unusual  may be n o t e d i n F i g u r e 2-14.  irregularities  The f r e q u e n c y o f l i b r a -  i s a l s o d e p e n d e n t on t h e a m p l i t u d e o f t h e m o t i o n .  This  i n t r o d u c e s a phase s h i f t between t h e e x a c t and approximate solutions. The equation  major source 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  (2.14).  t h a t t h e system ever, i t appears  The a m p l i t u d e s  of  definitely  indicate  i s operating i n the non-linear region.  How-  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 h e r e may p r o v e a d e q u a t e 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 The  exact.  Balance  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  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 w o r k i n -  volved i s prohibitive.  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  solution  (2.55)  The  Sin  Cos \p  t e r m may be r e p r e s e n t e d by t h e f i r s t  terms o f t h e T a y l o r ' s s e r i e s  (2.20) s o t h a t i t r e t a i n s i t s  inherent non-linear character. into  two  Substituting  equation  (2.55)  (2.14), c o l l e c t i n g t e r m s a n d a p p l y i n g t h e p r i n c i p l e o f  harmonic balance r e s u l t s i n t h e three  A,,, [6 K,  equations  - £ - 3K.- (A J, + 2 Aj, + 2^,)]  (2.56)  3e (A  s  hl  /\  h f  J 6 K * -»  -3K  f  (A *, 3  + A3,.) + ZA*,,  + 2  A*)]  = Se A^, + K J A ^ C J A J , - ^ T h i s s e t o f e q u a t i o n s does n o t possess s o l u t i o n , hence an i t e r a t i v e procedure e x a m p l e , when e = 0.3 a n d obtained.  any s i m p l e  was a d o p t e d .  = 1 three solutions  The c o e f f i c i e n t s w e r e f o u n d t o be :  For  were  49 A  x  1  =  0.333;  0.975;  -1.286  A  2  1  = -0.116;  -0.114;  0.097  A  3  =  0.021;  0.065;  (2.57)  -0.010.  E v e n w i t h t h i s t h r e e t e r m a p p r o x i m a t i o n , t h e amount of computational e f f o r t numerical solution  i s comparable t o t h a t f o r t h e exact  (section 2.2.3).  s i m p l i f i e d f u r t h e r b y p u t t i n g A^ a two t e r m s e r i e s s o l u t i o n .  The p r o c e d u r e 0, i« «  =  e  c a n be  by c o n s i d e r i n g  The e q u a t i o n s r e l a t i n g t h e c o -  e f f i c i e n t s then reduce t o  (2.58)  which  c a n 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 (2.59)  3K  id-  2.  5  For s p e c i f i e d v a l u e s o f e and and  (2.60) d e f i n e c u r v e s i n a n A^  -  3Ki  (2.60) AJ^I  equations  (2.59)  , Ag -^-plane ( F i g u r e s  2 - 1 5 - i t o 2 - 1 5 - v i ) 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-^ -^ a n d Ag ^°  Since  50  Equation (2.59) Equation (2-60) Solution Figure 2-15-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 Figure 2-15-ii  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 = 0.9) i  ^1,1  Equation (2.59) Equation (2.60) Solution  Figure  2-15-iii  D e t e r m i n a t i o n o f t h e f i r s t two t e r m s o f t h e s i n e s e r i e s s o l u t i o n (K. = 0.7)  53  .4  1—I— '  1 ' 1 • 1 1 , 1 , ' 1 1 , 1 1 , 1 1 , 1 1  1  .  1  ,  1  '  ' 1 1 , 1 1 1 1 1  t  1 , 1 1  ' -  1  1  1  1  '  '  '  •' •' •'  ;  /  /  •  i  i  / i  • •  \i  \;  i i  j  '  i i  ' \  i •  1  1  \•  1  '  _ _ \ X  1  —*-Ar-l '  '  1  . »  i  i  i  \ i  ' i • •• • I • 1 •  i I  i 1i i  I  1  1 > 1  1 | I 1 |  1  1  1  •  1i i  !  1  X. \ x .  \ ^  ^****s 7^^ s  s  lfc  I  ^^ ^m ^  1i X • 1'  mmm  mmm  i  I  L  L  ^^x^^S. -^S^  ^  —  .  i  1 1 1 •  1  "  f\  1  rr  1  i i  ^  ' i  ' . i.1  1  i • » i i  •  i  A_e  i i i i  1 1  '  1  .i —  t  1  i i i i i  i  1  1  '. • • > » i •  *  1  I i  i i  I  1  •  1  ^^"-^  Tv  1 1 • 1 1 • 1  I  1  •  Kj = 0.5  —  1  • ; :  1  1  1  1  1  \  —  '  1  '  ;  •2  •  1  ' i t  • -3  -•4  i i  U  1 -1.0  -5  0  5  1.0  Equation (2.59) Equation (2-60) Solution Figure  2-15-iv  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  1.0  Equation (2-59) Equation (2-60) Solution Figure  2-15-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 t e r m s 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)  ,  \  Kj = 0.1  .2  e  X i  -2  -  1  e \  -1.0  -.5  1.0  0  A  n  Equation  2.59  Equation 2-60 Solution Figure 2-15-vi  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^ = 0.1)  j  the i n i t i a l  d e r i v a t i v e o f t h e 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 t h e 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 c o m p a r i s o n 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 are q u i t e accurate.  I n p a r t i c u l a r , t h e a p p r o x i m a t e scheme  p r e d i c t s t h e maximum v a l u e o f t h e e c c e n t r i c i t y a n d t h e minimum v a l u e o f with considerable  f o r which three p e r i o d i c s o l u t i o n s e x i s t precision.  F i g u r e 2-17 c o m p a r e s t h e 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 t e r m a n a l y s i s w i t h t h e e x a c t n u m e r i c a l of t h e equation o f motion f o r t h e t h r e e cases equation  (2.57).  r a t h e r poor third  term  The a c c u r a c y  (maximum e r r o r ^  o f t h e two t e r m 20%).  improves t h e accuracy  smallest amplitude.  2.3»3  listed i n solution i s  The a d d i t i o n o f t h e  only of the s o l u t i o n of  To a c h i e v e g r e a t e r a c c u r a c y  t h a t more t e r m s i n t h e T a y l o r ' s e x p a n s i o n term  solution  requires  of the non-linear  be r e t a i n e d . Perturbation of Periodic Solutions Consider  2.2.2 o r 2.2.3»  t h e p e r i o d i c s o l u t i o n (p^ Let 6  a c t u a l l i b r a t i o n a l angle  n  developed i n  represent a p e r t u r b a t i o n so t h a t t h 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  Values of the i n i t i a l derivative required t o p r o d u c e s o l u t i o n s w i t h p e r i o d o f 2tt a s d e t e r m i n e d by t h e f i r s t t w o t e r m s o f t h e s i n e series solution  Figure  2-17  Comparison of exact 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 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 =  +  e C o s 9 ) ^  (| + e C o  5  This  n  -  +  < .62) 2  S  n  (2.14) r e s u l t s i n  Z e 5 ^ Q ( ^  6 ) S " -  which f o r small $  (l+eCo96)$"-  P  (2.62) i n t o  Substituting (l+  ty ,  2eSm©  + »)  n  S'  (2.63)  reduces t o the v a r i a t i o n a l  2e-5r*e  S* + 3 K; Coc 2ip ^ p  i s a linear differential  equation  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 solutions,  equation  ^ ^ ( 0 ) a n d £ 2 ^ ' d e f i n e d by  t  n  S= 0  4  (2.64)  with periodic  independent n  e  initial  conditions  i,io)  =/  S,'(o) - o (2.65)  Sz(o)  =  &»  »  The s o l u t i o n t o (2.64) s u b j e c t  0  / .. to the i n i t i a l  conditions  60  Ho)  -  K  S'(o)  =  Jo  (2.66)  i s g i v e n by  S(e)  =  * fy&)  *  e  Now, f o r 0 = 2Jt , e q u a t i o n  .  yj (&) x  (2.67)  (2.67) g i v e s  (2.68) $'(2X) and s i n c e e q u a t i o n  =  +  So  ^(XX)  (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) c a n be e x t e n d e d o v e r t h e i n t e r v a l 2% initial  ^  0 ^  4^  by c o n s i d e r i n g  (2.68) t o be new  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  £(&)  -  [K \(*••*•) + X  Si(*K}}S,(B)  s  ,  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 . the complete s o l u t i o n f o r 3  i s r e p l a c e d by - 6  27T.  (2.64) r e m a i n s i n v a r i a n t when  a n d 0 by - 0  odd f u n c t i o n o f 0. i . e . ,  Thus  may be w r i t t e n i n t e r m s o f  the s o l u t i o n over the i n t e r v a l 0 ^ 0 ^ Because e q u a t i o n  (2.69)  ?  t h e <Sg  S  s o l u t i o n must be an  61  i [-6) =  -S (6)  x  g  (2.70)  The i n i t i a l  conditions  i(o)  = - Sjzn)  S'(o) define  a  (2.7D  ^(zx)  =  solution  Sfe)  =  -  Sz(2K)i (e) l  + Sz(zx)S2(&)  in  t h e i n t e r v a l 0 ^ 0 ^ 271 w h i c h m a t c h e s t h e £ 2  in  t h e i n t e r v a l -2TT ^  0 £ 0.  The f i n a l  (2.72)  solution  conditions at  0 = 27t must, t h e n be  (2.73)  Hence, from  (2,72)  (2.74) and  ff^r; =  -& (vc),^(2K) z  + $i(m)Si(zit)-  1  (2.75)  62 1 o6o  =  (2.76)  S'(ZJt)  and  &,far)£/vr)  -j/ftjr^fcjr)  = / - Wfzrt).  l e f t hand s i d e o f (2.77)  The  differential  (2.77)  i s t h e Wronskian of t h e  e q u a t i o n a n d c a n be d e t e r m i n e d f r o m t h e c o -  29 efficients.  For equation  7  JW_ W  (2.64) t h e r e l a t i o n i s  Ze Sine l -t e Cos &  _  J6  (2t78)  or  and  therefore  W(o)  =• K/^JT^)  Consider the s o l u t i o n  =  /.  ( 2 . 6 7 ) when 9 = 2nTC  (2.80)  and l e t  S(zKn) = (2.81) s'(zrcn)  t h e n , by  (2.68)  -  63  (2.82)  where  and  b  -  <y  =  (2.83)  81 (m)  (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  equations.  Taking a s o l u t i o n o f the form •S  Be and  substituting  A  V  10  results i n  J  (2.85)  s  s  B, or  in'(2.82)  (2.84)  0  -  A,c*-  h  + 6 dV" s  64 -n  A  - Ag For  +  c  f s  I\  (2.86)  E>5 C « - =  O .  cont'd  a n o n - t r i v i a l solution, i ti s required  -  (a  4- c l ) V  (aol  -  =  be)  that  O.  (2.87) 30  I t may be p o i n t e d  out that  t h e a n a l y s i s due t o  Floquet^  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 to equation and  (2.87) which s i m p l i f i e s , using  (2.76)  (2.77), t o -  2cxY  +  J  =  (2.88)  O  giving  Y and  ^  equation  =  =  a  +  V *a  ;  1  (  ( 2 . 8 9 ) when s u b s t i t u t e d i n t o  fa-ft) Aft + \ / E _ A . j ;  =  The two  L  (2.89)  £  (2.86)  gives  (i-1,2).  (2.90)  s o l u t i o n ( 2 . 8 4 ) may be w r i t t e n a s t h e sum o f t h e  solutions  *n = A f / , " + A ^ "  .  J&LLL_/\  S  jr."_ j/EZL/t,  (2.9D  K*  ( 2 , 9 2 )  65 hence  T a k i n g t h e sum a n d d i f f e r e n c e o f ( 2 . 9 1 ) a n d (2.93) g i v e s  f^7  whose p r o d u c t  (2.94)  yields the r e l a t i o n  K - £ r £  =  4\A h*J=4As,%  .  h  (2.95,  T h i s r e s u l t shows t h a t , i f t h e s o l u t i o n o f e q u a t i o n (2.64)  i s i n s p e c t e d each time t h e independent  variable  e q u a l s 2 r c n , t h e 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 ja|> 1 t h e curve i s a hyperbola  For  becomes v e r y l a r g e .  curve.  so t h a t 6 e v e n t u a l l y  On t h e o t h e r h a n d i f |a| < 1 t h e c u r v e  i s a n e l l i p s e a n d £ i s bounded.. I t may be c o n c l u d e d  that  i n t h e l a t t e r case, 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 a n d f o r 0 = 2J[ n may be f o u n d single  curve i n t h e  ,y/ - p l a n e w h i c h  surrounds  on a  the point  (^ (0),^p(0)). p  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 t h e c u r v e  66 transforms  into  (2.96)  S'(e)  where ^  =  *&',(9) *  l i e on t h e c u r v e d e f i n e d by ( 2 . 9 5 ) .  and  * j  -  =  &{&)S*'f9)  ^fe)  J  w h i c h when s u b s t i t u t e d  into  -  Hence  S'MSJe)  ,2.97,  -  ( 2  W(6) (2.95)  determine  .  9 8 )  t h e shape o f  the curve at the s p e c i f i e d value o f ©  - zS(e)S'(o)fa)s/(e)-  sJQSfapj  (2,99)  that i s ,  (2.100) The  nature o f t h e curve a t t h e s p e c i f i e d value o f 0  67 is  d e t e r m i n e d by t h e s i g n o f t h e  parameter  (2.101) such that f o r % > a hyperbola.  0 t h e c u r v e i s an e l l i p s e a n d f o r  From e q u a t i o n s (2.99) and  <  0,  (2.100) t h e r e i s  obtained  ~ -0T-  = t h e r e f o r e , £) <  0 i f |a| >  1 and § ) >  T h u s , d e p e n d i n g upon t h e v a l u e defined is  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 .  Linear  0 i f | a | * 1. o f |a| t h e c u r v e  passages of the s o l u t i o n at f i x e d 6  by s u c c e s s i v e  tubular surface  *<•)  In the f i r s t  case a  i s defined.  perturbation analysis predicts that  initial  c o n d i t i o n s w h i c h do n o t l e a d t o e x a c t l y p e r i o d i c m o t i o n still the  permit the motion t o remain i n the neighbourhood of  periodic solution.  t h e m o t i o n may The  drift  There i s a l s o the p o s s i b i l i t y  away f r o m t h e g e n e r a t i n g  c r i t e r i o n determining  t u d e o f t h e &2 a t 9 = 23T o time,  that  solution.  the k i n d of motion i s the  magni-  -type of s o l u t i o n of the v a r i a t i o n a l equation  F o r | c S ^ (2JT ) j >  but i f l ^ g ^ ^ J I ) !  <  1, t h e v a r i a t i o n i n c r e a s e s  with  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 r e m a i n s b o u n d e d . the  may  In f a c t , i n  l a t t e r c a s e , t h e s o l u t i o n s l i e on a s u r f a c e  i n <f> , </>',  - s p a c e w h i c h a l w a y s 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  6  an e l l i p t i c  cross section.  Figure  2-18  presents  the regions  i n a n 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 solutions of o r b i t a l frequency are stable. 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  This diagram  an e x t e n s i o n  was  i n the  12 r e g i o n n e a r e = 1 o f t h e w o r k by Z l a t o u s o v general  stable perturbations.  The  o t h e r two s o l u t i o n s , h a v i n g initial  either  d e r i v a t i v e may  stable perturbations. F i g u r e 2-19  presents  t i o n s which complete three  in  One,  d e r i v a t i v e , always l e a d s t o un-  a small p o s i t i v e or large negative  may  In  there are three p e r i o d i c solutions to consider.  with large positive i n i t i a l  yield  et a l .  s i m i l a r data  for periodic solu-  o s c i l l a t i o n s i n two o r b i t s .  be e m p h a s i z e d t h a t t h e r e i s o n l y one t h i s case  (n = 2 )  c l a s s of s o l u t i o n  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 negative  It  h a v e a t 6 = 2TT  slope a l s o found i n t h i s Figure  (see a l s o  the  Figure  2-9). When t h e p a r a m e t e r a i n e q u a t i o n  (2.88) l i e s  between  +1 a n d - 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 elliptical  cross-section.  I n g e n e r a l , as time i n c r e a s e s  point of i n t e r s e c t i o n of the t r a j e c t o r y with the p l a n e moves a r o u n d t h e e l l i p s e .  In f a c t ,  where  =  a t  i  V1"A  the  0=0  since  UQ>  /——5Y  with  "  E  (2.103)  69  Figure  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  ©  it  i s evident  =  Cos  (2.104)  a.,  from equations  (2.82) t h a t t h e p o i n t  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 \\) , </>  repre-  -plane  r o t a t e s around the e l l i p s e a t the average r a t e o f Q per  orbit.  o f 27C the  In particular, i f © i s a rational  the points defined  radians  sub-multiple  by (2.82) a p p e a r s t a t i o n a r y .  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  occurrence i s the manifestation  become p e r i o d i c .  Thus This  of the coalescence with 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 of another s o l u t i o n characterized by a p e r i o d e q u a l For it  to that of the perturbation.  example, r e f e r r i n g t o F i g u r e s  2-7,  2-9,  and  i s s e e n 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  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 « and 2-21  becomes  Figures  i n d i c a t e t h a t f o r b o t h p e r i o d i c s o l u t i o n s |a|  a t t h i s p o i n t , hence © = 2.4  0.43.  2-10  0,  pair of f i r s t  of motion  order  =1  180°.  Phase Space and I n v a r i a n t The e q u a t i o n  2-20  Surfaces ( 2 . 1 4 ) may be w r i t t e n a s a  differential  equations  I dj^'  d0 This  _  Z e S*A 9 (VV  /) -3K,5tn<j)Cos ip  / y- e Cos  p a r a l l e l s the usual Hamiltonian  (2.105) B  f o r m u l a t i o n where t h e  a  0  -  -2  1  /  - Hip iO>>0 (Large)  /  n = 1  ^ (0)>0 (Small) 1  /  -(|/ lo)<0 P/1  -3  -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 = 1 ) f o r s o l u t i o n s o f p e r i o d 27T i  distinction The  ip , a s w e l l a s t h e v a l u e o f t h e i n d e p e n d e n t point  (8 = 0 ) .  This suggests  Q  both  variable,  /  the i n i t i a l  disappears.  ( 2 . 1 0 5 ) d e p e n d s on t h e s t a r t i n g v a l u e s o f  s o l u t i o n of  <// and at  b e t w e e n c o - o r d i n a t e s and momenta n e a r l y  that the s t a t e  o f t h e s y s t e m c a n be r e p r e s e n t e d by a p o i n t i n a p h a s e s p a c e f o r m e d by t h e t h r e e o r t h o g o n a l  co-ordinates f o r e = 0,  As shown i n s e c t i o n 2.2.1 a first  integral  The  curve  +•  Z  by a p o i n t w h i c h l i e s  3K;  =  c a n be t h o u g h t  the e x i s t e n c e of  governing  C  £  c  3Kf  on t h e  closed  surface  2-3). differential  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.  to  o f F i g u r e 2-3  those  eccentricity.  that closed curves  i s apparent  How-  analogous  should continue t o e x i s t f o r non-zero  T h e s e c u r v e s w o u l d t h e n be f u n c t i o n s o f 6  thus d e f i n e a surface i n the three dimensional It  curve,  (2.106)  n o n - l i n e a r , non-autonomous  e v e r , i t seems l o g i c a l t o e x p e c t  the  .  o f as d e f i n i n g a c y l i n d r i c a l  with oval cross-section (Figure The  0.  (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  system i s represented  l|/  ijf, <f', and  that equation  ty,  (p\  and  0  -space.  (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. , h e n c e t h e s u r f a c e n e e d o n l y be d e t e r m i n e d  over  that  interval. * The  s u r f a c e may  r e f e r t o as a " n u m e r i c a l (}J = (p  Q i  ip' =  be g e n e r a t e d  by what Henon and H e i l e s ^  experiment."  iff' , 0 = 0 i s c h o s e n and Q  i n t e g r a t e d u n t i l 6 e q u a l s 271.  31  An i n i t i a l  point,  equations  (2.105)  T h i s produces a  are  "consequent"  = <P ,  point  = ^ ,  Q  Q  a new i n i t i a l  8  =  2?T w h i c h may be c o n s i d e r e d  point with 0 = 0 .  The p r o c e s s  may be t h o u g h t  o f a s 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 the  initial The  repeatedly -plane  (2.105), o f  point. new s t a r t i n g p o i n t may i t s e l f be  transformed  a t 8 = 0.  by t h e p r o c e s s  I f any o f t h e t r a n s f o r m e d  lead t o tumbling  i n the unstable  region.  points l i e s  of the transformation.  nected  lying  s t a b l e o p e r a t i o n a n d , when T h i s i s an i n v a r i a n t  That i s , t h e t r a n s f o r m a t i o n  i n the 0 = 0  plane  b e i n g g e n e r a t e d a t 0 = 2X . by a n i n f i n i t y  determined  m o t i o n a n d may be p l o t t e d  p l o t t e d , appear t o d e f i n e a curve.  of t h e curve  out-  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  inside the region indicating  curve  , ty'  thus l e a d i n g t o a s e r i e s of points i n the  side the region |^|£7C/2, then a l l the points  curve  as  r e s u l t s i n t h e same  The two c u r v e s  of t r a j e c t o r i e s thereby  a r e con-  defining a  s u r f a c e w h i c h may be c a l l e d a n " i n v a r i a n t s u r f a c e " o r J ' i h t . e g r a l ;>manifb i d . "Tl T h e - e x i s t e n c e f o f u s u c h i s u r f a c e s e f o r l i b r a t i o n a l motion i n a c i r c u l a r  orbit  i s evident  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 existence The point  {ty  Qi  o u t by M o s e r ,  their  c a n be p r o v e n f o r e ^ 0 a s f o l l o w s . concept o f a t r a n s f o r m a t i o n which converts ty' ) Q  i n the 0 = 0  i n t h e 0 = 27T p l a n e  plane  i n t o another  (i^ »  i s very h e l p f u l i n t h i s regard.  s o l u t i o n s a p p e a r a s s e t s o f f i x e d p o i n t s i n t h e two and  h e n c e a r e c h a r a c t e r i z e d by a n i d e n t i t y  a  c  ^' ) c  Period! planes  transformation.  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 a s l o n g a s  d e s i r e d b y m a k i n g n -*•<». infinite  Hence t h e r e w i l l  countably  number o f p o i n t s w h i c h s a t i s f y a n i d e n t i t y  formation.  Since  trans-  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  be  preserving.  area  be a  i n t h i s manner must  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 s y s t e m f o r e = 0 i s given  by (2.18) a s a m o n o t o n i c a l l y  f u n c t i o n o f dj „ ° m  2TC  increasing  The p e r i o d i s bounded by  v  <  A©  <  <*>) (O ^  |  ^  Moser's theorem then a s s e r t s t h a t there i n t h e neighbourhood o f t h e curves  such an i n v a r i a n t s u r f a c e The  surfaces  symmetry p r o p e r t i e s .  Figure  g e n e r a t e d by e q u a t i o n This d i f f e r e n t i a l  i t s derivative defined  remain 2-22 r e p r e -  (2,14) have c e r t a i n  equation  remains un-  Thus, t h e s o l u t i o n  by t h e c o n d i t i o n s  (0) = 0 ,  <//(0) =  </^, a r e o d d a n d e v e n f u n c t i o n s r e s p e c t i v e l y .  quently,  the points defined  the  i n t h e 0 = 27X m (m = - 1 ,  2-23.  -axis of the points - 2 , ...) p l a n e s .  cross-sections of the surface  m e t r i c a l a b o u t t h e <p" - a x i s .  Conse-  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 t h e defined  curves  schematically.  c h a n g e d i f b o t h 0 a n d dJ change s i g n . and  exist  (2.106) which  i n v a r i a n t u n d e r t h e m a p p i n g f o r s m a l l e. sents  (2.107)  This  Hence  a t 0 = 0 a n d 27T a r e symi sillustrated  i n Figure  77  F i g u r e 2-22  S c h e m a t i c v i e w o f an i n v a r i a n t  surface  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 which i l l u s t r a t e s 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 surface  the  Further, the p o i n t s defined i n the 0 = 0 be t h e m i r r o r image a b o u t t h e in  t h e 0 =-8  plane.  plane,  Thus t h e  o r e q u i v a l e n t l y i n t h e 0 = 2K  - 0.  -axis.  o f an i n v a r i a n t s u r f a c e t a k e n i n F i g u r e 2-24 initial  t// - a x i s o f t h e  Several cross-sections are  f o r s p e c i f i c v a l u e s o f e and  K^.  point taken w i t h i n a given manifold h e n c e a new  i n t e r s e c t t h e o l d one.  must t h e r e f o r e l i e c o m p l e t e l y  generates  an  stable..  The  invariant surface original. condition  d e s i r e d r e g i o n of s t a b i l i t y  constructed.  T y p i c a l i n v a r i a n t surfaces are  and  2-25-ii°  The  conmay that  shown  symmetry p r o p e r t i e s a r e  observed. This concept of a l i m i t i n g  is  trajectory  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  Figures 2-25-i  readily  w i t h i n the  The  e x t e r n a l surface provided t h a t the motion  be r e p r e s e n t e d  in  new  t h e o t h e r h a n d , an e x t e r n a l i n i t i a l  t i n u e s t o be  c a n be  The  gener-  surface.  p r o p e r t y o f u n i q u e n e s s g u a r a n t e e s t h a t t h e new  On  cross-  at v a r i o u s o r b i t angles  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  does not  at  Hence, the c r o s s - s e c t i o n a t 0 = K  a l s o symmetric about t h e  An  - 0  c r o s s - s e c t i o n of the i n v a r i a n t s u r f a c e  section at 0 = 2 X  presented  will  (//-axis o f t h e p o i n t s d e f i n e d  0 = •§ i s . a m i r r o r image a b o u t t h e  is  plane  very important.  s u r f a c e i n the phase space  For g i v e n values of the parameters i t  provides a l l p o s s i b l e combinations 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 point i n i t s orbit without  causing  of i n i t i a l  angles  be s u b j e c t e d a t  and any  i t t o become u n s t a b l e .  80  1.0  Figure  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 = 0.7, e = 0.2) i  330' 300°  ,50" 120" 90'  6(f iff  icr /A  ' V '  1  JO*-  ^Trajectory, Kj = 1., e = 0.25  Figure 2-25-i  Typical  invariant  surface  {K  ±  = 1,  e = 0.25)  83 In  t h i s respect  et a l .  1  i t i s an improvement o f t h e work o f Z l a t o u s o v  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 determination  of s t a b i l i t y .  d i r e c t method i n t h e  I n t h i s method a L y a p u n o v V-  f u n c t i o n , w h i c h 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.  According  finite,  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-  zero, 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  asymptotically stable, neutrally stable, or unstable. When a n i n t e g r a l m a n i f o l d  exists, the point  repre-  s e n t i n g t h e s t a t e o f m o t i o n a l w a y s l i e s on t h e same s u r f a c e . Therefore  t h e m o t i o n must be n e u t r a l l y s t a b l e .  Hence t h e  derivative of the V-function i s i d e n t i c a l l y  z e r o a n d t h e V-  f u n c t i o n must be a c o n s t a n t  That  o f the motion.  i s , the  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 w h i c h t h e V - f u n c t i o n i s constant. Near such a s u r f a c e an a p p r o x i m a t e V - f u n c t i o n w i l l p o s s e s s a t i m e d e r i v a t i v e o f v a r i a b l e s i g n s o t h a t no i n f o r mation regarding s t a b i l i t y analysis.  effort  by a n a p p r o x i m a t e  On t h e o t h e r h a n d , t h e d e t e r m i n a t i o n  f u n c t i o n w i t h dV/dt = 0 mination  c a n be o b t a i n e d  o f t h e V-  i s exactly equivalent t o the deter-  o f t h e i n t e g r a l m a n i f o l d s o t h a t no s a v i n g i n  i s t o be e x p e c t e d . Using  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 t h a n some s p e c i f i e d v a l u e , i t i s possible t o estimate  b o u n d s 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 ;  instability.  T h e s e bounds a r e o n l y a p p r o x i m a t e and  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 It ( e , K^)  be  stability.  a s s o c i a t e d w i t h more t h a n one  This  is illustrated  "islands."  The  region  unstable  symmetry p r o p e r t i e s o f t h e  The  line  s e c t i o n of the planes metry of the s u r f a c e .  = 0 and The  conditions. pro-  information into  to ensure s t a b l e motion.  6=0  be  placed  c o n f i g u r a t i o n ( t|> = 6  =  of  0)  the  p l o t t e d as a f u n c t i o n o f  (Figures 2-27-i to 2-27-vi).  such a diagram measures the  inter  limiting  bounds t h a t must be  For a s p e c i f i e d value  p o i n t s o f i n t e r s e c t i o n can  a  f o r m s an a x i s o f sym-  i n t e r c e p t s made by t h e  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  The  initial  region  i n p h a s e s p a c e d e f i n e d by t h e  on t h i s l i n e r e p r e s e n t  eccentricity  The  invariant surface  v i d e a means o f c o n d e n s i n g c o n s i d e r a b l e  manifold  o r more  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 t h e m s e l v e s  between the s u r f a c e s r e p r e s e n t s  s i n g l e diagram.  of  There i s  one  a r o u n d t h e m a i n s u r f a c e i n a h e l i c a l manner.  The  parameters  i n F i g u r e 2-26.  a s i n g l e c e n t r a l " m a i n l a n d " a c c o m p a n i e d by  the  considered.  i s p o s s i b l e that a p a r t i c u l a r set of  may  only  Qualitatively  s i z e of the r e g i o n of  stability.  s p i k e s i n the diagrams i n d i c a t e the presence  the secondary i s l a n d s of s t a b i l i t y  discussed  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 p r e s e n c e o f many a d d i t i o n a l s m a l l s p i k e s .  earlier.  c a u s e d by The  of The .  the  Figures  also  i n d i c a t e t h a t the r e g i o n of 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 increasing  eccentricity.  At some u p p e r l i m i t  5  e  , the  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  4t  F i g u r e 2-26  Typical invariant surface with  "islands" ca \J1  0  I  F i g u r e 2-27-i  -1  2  Stable  ( periodic solutions with  Unstable I  Transition  e  .3  .4  .5  period = 2?tn.  points as determined  by perturbation  analysis.  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 y = 8 = 0 f o r s t a b l e m o t i o n (K. = 1.0)  87  T  1  1  r  I  I  I  I  l  1  0  .1  2  -3  .4  5  e Stable Unstable X  Transition  ( periodic \  with  period = 2*n. points  perturbation Figure 2-27-ii  solutions  as  determined  by  analysis.  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 o t i o n ( K = 0 . 9 ) i  89  X  Stable  j  Unstable  (  solutions  with  p e r i o d = 27rn.  Transition points as perturbation  Figure 2-27-iv  periodic  determined  by  analysis.  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 o t i o n {K = 0 . 5 ) ±  90  1.5  .].5 I 0  i  '  '  •  .1  -2  .3  4  e Stable X F i g u r e 2-27-v  ( periodic solutions with  Unstable ( period = 2fln. Transition points as determined by perturbation analysis. 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 y = 6 = 0 f o r s t a b l e m o t i o n ( K = 0.3) i  -1-5 " 0  1  1  1  -1  2  1  3  4  -5  e Stable  ( periodic solutions with  Unstable ( I  Figure 2-27-vi  period = 2fln.  Transition points as determined by perturbation analysis. 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 o t i o n ( K = 0.1) i  representation trajectory. tricity,  the  invariant surface  Thus, beyond a c e r t a i n c r i t i c a l  s t a b l e m o t i o n i s not  eccentricities, stability region 2.5  i s so  There are  several  numerical  possible.  At s t i l l  r e t u r n but  the  o f no  eccen-  higher  s i z e of  the  p r a c t i c a l importance.  s o u r c e s o f e r r o r i n t h e method o u t -  They a r e  a l l due  to the  finite  nature  process.  A l a r g e d i s c r e p a n c y may e r r o r of the  a r i s e i n the  the  truncation  and  " r o u n d o f f " generated i n the The  value of  single  Method  i n s e c t i o n 2.4°  of the  may  s m a l l as t o be  Accuracy of the  lined  degenerates to a  truncation  s o l u t i o n due  to  numerical i n t e g r a t i o n process computer.  e r r o r v a r i e s as  f o r the  Adams-  33 Bashforth off  error.  ing  Runge-Kutta t e c h n i q u e s employed.  e r r o r , on t h e  Thus t h e r e  step  and  i s an  p r e c i s i o n may To  Figure  a r i s e where the  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 not  be  e s s e n t i a l f o r the  e r r o r s can s i z e of  final be  by  total  critical the r e s u l t -  s i z e s were  (2.14) o v e r 30 o r b i t s .  conditions  reduced to  .0001  (Table I I ) i n radians  by  3°°  2-28-i presents the  p l a n e as g i v e n  1/h.  analysis.  t h i s point, several step  r e s u l t i n g v a l u e s of the  employing a step  as  round-  s i z e which minimizes the  i n t e g r a t i o n of equation  d i c a t e t h a t the  6=0  optimum s t e p  illustrate  chosen f o r the The  o t h e r hand, tends t o i n c r e a s e  At t i m e s a s i t u a t i o n may  s i z e i s too  The  J  the  i n v a r i a n t curve i n  the  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  I n e r t i a parameter,  =  1.0  Orbit e c c e n t r i c i t y , e  =  0.1  Initial  <Po = 0  displacement,  Initial velocity, h  ty*  -  <//(607T)  Degrees  Size  1,18 \p' (607T)  Radians  30  Unstable a f t e r  4.5  orbits  15  U n s t a b l e a f t e r Id.3  orbits  7*5  -0.18556  1.12262'  3  -0.18910  1,11949  1.5  -0.18911  1.11948  Also p l o t t e d i n the Figure from t h e numerical  are selected points  s o l u t i o n obtained  determined  w i t h h = 7.5°.  It is  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 cross-sections  i s much s m a l l e r t h a n t h a t b e t w e e n t h e two  s o l u t i o n s i n d i c a t e d i n Table I I ,  Mitropolskiy^  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 i n t e g r a l manifolds. cal  explanation  use  of large step  indicates possess  T h e r e does n o t a p p e a r t o be a t h e o r e t i -  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 sizes.  Although the 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 s u f f i c i e n t accuracy.  are usually of  The e r r o r s a p p e a r t o c a u s e d i s p l a c e -  ments a r o u n d t h e m a n i f o l d  r a t h e r than normal t o i t .  94  y  6  24  11  ^3  ^ 22V  + 1.0  20\ 9  X  20'  \ /  •28 /  \  K-1, e =.1  4  e = 27tn  n =7  + •5  \  • h = 1.5°  f  6  o h = 75°  Invariant curve  • 30  u  1  -t  1  16< H  1-  1  1  27  H  -5  •8  (Rod.)  30  3/  25  ;  5  7  \ •5  /  \ , 0  \ 12  is  23  s  J--1.0  Figure 2 - 2 8 - i  Comparison of the i n v a r i a n t surfaces generated using d i f f e r e n t i n t e g r a t i o n step s i z e s (Non-limiting surface)  F i g u r e 2-26*-ii  Comparison of the i n v a r i a n t s u r f a c e s generated using d i f f e r e n t i n t e g r a t i o n step s i z e s . (Limiting surfaces)  The  determination  of the l i m i t i n g  involves additional d i f f i c u l t i e s . crepancy  invariant  Here, even a s m a l l  can l e a d t o e r r o n e o u s r e s u l t s r e g a r d i n g  As i n d i c a t e d by T a b l e stability  and  dis-  stability.  I I the e r r o r s b r i n g about e a r l y i n -  hence cause t h e s i z e o f the l i m i t i n g  s u r f a c e t o be u n d e r e s t i m a t e d . The  surface  r e s u l t s suggest  invariant  (F'3UgJ^per^-r£6^iii.)j.;  t h a t any  e r r o r which d i s p l a c e s  the r e p r e s e n t a t i v e p o i n t i n t o the unstable r e g i o n l e a d s f u r t h e r growth of t h i s  error.  Any  subsequent e r r o r which  a c t s t o w a r d s t h e s t a b l e r e g i o n w i l l h a v e t o be l a r g e r than the o r i g i n a l o t h e r hand, the  sense.  On  e r r o r which takes the  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 c a n c e l l e d by an  somewhat  error to regain s t a b i l i t y .  e f f e c t o f an  be  e r r o r o f t h e same s i z e b u t  immediately i n the  opposite  p r o b a b l e , t h e r e w i l l be a d r i f t t o w a r d s i n s t a b i l i t y  In  i n s i d e the l i m i t i n g  the m a j o r i t y of cases  instability  was  i s due  equally from a  invariant  surface  s t u d i e d t h i s tendency towards  noted.  A second e r r o r i n the d e t e r m i n a t i o n manifold  the  represen-  Thus, assuming t h a t both k i n d s of e r r o r are  t h i n " s k i n " which l i e s  to  of the  limiting  t o the t e r m i n a t i o n of 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 time. s o l u t i o n appearing  stable.  c a n u s u a l l y d e t e c t any  T h i s may  result  i n an  Careful plotting  tendency of t h i s type.  acts t o i n c r e a s e the s i z e of the l i m i t i n g T h i r d l y , the process  of numerical  unstable  of the  results  This error  invariant  surface.  experimentation  is  necessarily discrete.  T h a t i s , bounds c a n be p l a c e d  on  initial  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  tions.  T h e s e bounds may be made a s f i n e a s d e s i r e d  ing  sufficient  computing time i s used.  soluprovid-  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 with which the l i m i t i n g manifold  c a n be d e t e r m i n e d .  evaluated  l i m i t i n g manifold  The m a j o r i t y formed u s i n g surfaces  a step  This r e s u l t s i n the lying  of the numerical size of 7.5°.  the data at 0 =  r e s u l t s with h = 7.5°  i n t e g r a t i o n s were p e r -  The r e s u l t i n g l i m i t i n g  = 0 as a s t a n d a r d .  l a y within -.07  o f t h e more p r e c i s e r e s u l t s . e r r o r was l e s s t h a n  a n d +.05  In the majority  units i n of cases the  Solutions  The R e l a t i o n s h i p B e t w e e n M a n i f o l d s Solutions  and P e r i o d i c  I n s e c t i o n 2.4 i t was shown t h a t a n i n i t i a l chosen i n s i d e a s p e c i f i c m a n i f o l d o f a new m a n i f o l d succession  which l i e s  of i n i t i a l  r e s u l t s i n the  completely  a surface  periodicity  condition generation  within the f i r s t .  A  c o n d i t i o n s may t h u s be c h o s e n w h i c h  determine p r o g r e s s i v e l y smaller manifolds, in  The  .03 u n i t s .  The S i g n i f i c a n c e o f P e r i o d i c  2.6.1  one.  w e r e c o m p a r e d w i t h more e x a c t r e s u l t s i n s e v e r a l  cases using  2.6  inside the true  numerically  which has zero  cross-section.  finally resulting Because o f t h e  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  f o l d must t h e n r e p r e s e n t s o l u t i o n s a c t as s p i n e s  a periodic solution.  mani-  Hence p e r i o d i c  upon w h i c h t h e i n v a r i a n t s u r f a c e s  are  98  built.  The g e n e r a l m o t i o n c a n t h e n be t h o u g h t o f a s a bounded  perturbation The  about t h e a p p r o p r i a t e  solution.  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  that with  increasing  invariant  surface  shrinks.  solution.  i s now e v i d e n t ;  Indicate  eccentricity the size of the limiting  becomes a s e t o f p o i n t s a periodic  periodic  Ultimately, the cross-section  so t h a t  the manifold  degenerates  The i m p o r t a n c e o f t h e p e r i o d i c  into  solutions  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 motion the only  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 surface  i s infinitesimal  analysis  i n size, the l i n e a r  perturbation  s h o u l d c o r r e c t l y p r e d i c t a change f r o m s t a b l e t o  unstable perturbations.  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 c a n t h u s be d e 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 analysis of the appropriate The  periodic  solution.  d e t a i l s o f t h e 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 a n d 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 w e r e i n Figures  2-18 a n d 2-19°  They a r e a l s o  plotted  indicated i n Figures  2 - 2 7 - i t o 2 - 2 7 - v i 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 numerical search f o r the l i m i t i n g manifold  with  t h e more  t h e o r e t i c a l determination o f t h e c h a r a c t e r i s t i c exponents. The  a g r e e m e n t i s q u i t e g o o d , b u t t e n d s t o become p o o r e r a s decreases.  T h i s i s due t o a m u l t i p l i c i t y  of periodic  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 the  maximum e c c e n t r i c i t y .  involved  a r e performed only  Because t h e n u m e r i c a l over f i n i t e  near  integrations  i n t e r v a l s and a r e n o t  99 "open ended" as i n t h e case o f t h e n u m e r i c a l s e a r c h , t h e r a c y may  be i m p r o v e d t o any  d e s i r e d degree.  These  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 of s t a b i l i t y  between  = 0.5  and  0.3.  i n d i c a t e t h a t the upper s p i k e disappears w i t h (it  c a n be shown t h a t t h e s p i k e v a n i s h e s  the lower  one  region  and  2-18  decreasing  for ~  c o n t i n u e s t o grow u n t i l  results  of the  F i g u r e s 2-7  = 1/3)  while  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 the manifold c r o s s - s e c t i o n s . c r o s s - s e c t i o n s of i n v a r i a n t The  surfaces evaluated f o r  .2.  and  the pointed i n v a r i a n t p l o t s are a s s o c i a t e d w i t h  1,  stable periodic solutions unstable  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 or hyperbolae  i n the v i c i n i t y  of the p e r i o d i c  I n the 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  centres w h i l e the unstable p e r i o d i c  s o l u t i o n s have the appearance of s a d d l e p o i n t s . i n s p e c t i o n of the p e r i o d i c s o l u t i o n s provides information concerning  the nature  p l a n e and  motion.  2.6.2  =  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) w h e r e i t was  solutions.  surround  inside  presents several  e =  solutions.  closed curves  F i g u r e 2-29  accu-  hence of the  Thus t h e  qualitative  of the s t r o b o s c o p i c phase  Determination Solutions  of a Complete Set of P e r i o d i c  S e c t i o n 2.2.2  i n d i c a t e d the existence of 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  exten-  s i v e numerical search f o r these s o l u t i o n s r e s u l t e d i n the  Figure  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 the 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 the s t r o b o s c o p i c phase p l a n e  i n s e c t i o n 2.2.3«  data presented  No  a t t e m p t was  ever, to determine i f the s o l u t i o n s obtained complete s e t . mining  The  made, how-  constituted a  importance of p e r i o d i c s o l u t i o n s i n deter  the l i m i t i n g values  o f e c c e n t r i c i t y and  the  general  s h a p e 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 k n o w l e d g e o f a complete set d e s i r a b l e . E a r l i e r the process f r o m 6 = 0 t o -G = 2TTn was b e t w e e n t h e two  planes.  (2.14)  of i n t e g r a t i o n of equation d e s c r i b e d as a  transformation  I n terms of 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 a p p e a r s as a f i x e d p o i n t .  T h u s enumera-  t i o n of a complete set of 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 determination Although  of a l l the f i x e d p o i n t s of the  t h i s i s a simple  transformation.  c o n c e p t i t i n v o l v e s an  amount o f w o r k b e c a u s e o f t h e l a r g e number o f w h i c h must be  through  presents  plane.  those  A c u r v e may  p o i n t s such t h a t  transformation.  The  shown i n F i g u r e  2-31«  The  = 1,  for  plane which correspond  i n the 0 = 0  enormous  trajectories  computed.  F i g u r e 2-30 t h e 0 = 2TT  the  e = 0,  contours  in  to l i n e s of constant  ^  be d r a w n w h i c h i^i  corresponding  passes  i s i n v a r i a n t under p l o t s f o r constant  the d/  are  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  the ^ ' - i n v a r i a n t curves  c o n s t i t u t e a complete set of  points f o r the given values  of the parameters  Note t h a t because t h e e c c e n t r i c i t y  fixed  (Figure 2-32).  i s zero, there are  an  i n f i n i t e number o f i n v a r i a n t p o i n t s w h i c h d e f i n e a c l o s e d  102  -1.5  -1.0  -.5  0  .5  1.0  WW  (Rod)  Figure  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 m o t i o n i s i n t e g r a t e d o v e r 27T (K.. = 1, e = 0)  1.5  103  Figure  2-31  The t r a n s f o r m a t i o n of l i n e s of constant ^' when the equation of motion i s i n t e g r a t e d over 27T (K = 1, e = 0) i  104 2  lp'(27f) 0  1  1  - /// / / /f  Dl  1  1  \  -l  '  1  /  /'  1  i  i\  I  /  1  1  '  ll  1  - ff .  /  i  !  ._  •  -1-5  -1.0  \-  i  -.5  K| = 1. e = 0 i  F i g u r e 2-32  0  4i(27t), Rod lp  1  i  .5  1  1-0  i  1.5  Invariant  lp' Invariant ip and ip Invariant 1  Determination of a complete s e t o f f i x e d p o i n t s o f the t r a n s f o r m a t i o n (K^ = 1, e = 0)  curve.  This i s c o n s i s t e n t w i t h t h e exact  solution arrived  a t i n s e c t i o n 2.2.1. Figures  2 - 3 3 , 2 - 3 4 , and 2-35 p r e s e n t  ing s e t o f curves  for  the  = 1, e = 0.";.1. I n t h i s  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  case t h e t h r e e  evident.  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 a l s o l i e on t h e (//-axis a t 6 = Tf  correspond-  Because  three solutions  a n d h e n c e a r e a l s o members  of 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 . solutions of the t h i r d family at f  There a r e a l s o  = ±1t/2 a s i n d i c a t e d by  12 Zlatousov  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  deter-  m i n e d i n s e c t i o n 2.2.2 f o r m a c o m p l e t e s e t f o r n = 1. 2.6.3  The D e g r e e o f S t a b i l i t y As a l r e a d y m e n t i o n e d , a n y s t a t e o f m o t i o n w i t h i n t h e  r e g i o n of s t a b i l i t y w i l l give r i s e t o a surface which within the l i m i t i n g disturbances tance  surface at a l l times.  Since  lies  the major  are e s s e n t i a l l y stochastic i n nature, the d i s -  between t h e s u r f a c e c o r r e s p o n d i n g  of t h e s a t e l l i t e and t h e l i m i t i n g of t h e long term s t a b i l i t y .  t o the a c t u a l motion  s u r f a c e w o u l d be a m e a s u r e  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 a s 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 .  This  corresponds t o a p e r i o d i c motion of the s a t e l l i t e  state with  p e r i o d 2Tf . The  necessary  momentum c h a n g e a t 0 = 0 c a n be  obtained  106  Figure  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 m o t i o n i s i n t e g r a t e d o v e r 27T (K. - 1, e = 0.1)  107  Figure  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 t h e e q u a t i o n o f m o t i o n i s i n t e g r a t e d o v e r 27C (K = 1, e = 0.1) ±  108  -1.5  -1.0  -5  0  -5  10  15  4/(271), R o d  ijj _— o F i g u r e 2-35  Invariant  \y' invariant  ip and ijV Invariant  Determination o f a complete set 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 Figures 2-27-i to 2-27-vi. through  2-36-vi present  i n e r t i a parameter.  f o r several values  t h e p r e s e n c e o f s p i k e s has  estimate.  these  the degree of s t a b i l i t y g r e a t l y . . on t h e o t h e r h a n d , i s a p o w e r f u l  2.7  Concluding The  this  conservative  does n o t  affect  E c c e n t r i c i t y of the  orbit,  destabilizing factor.  l e a d to a s u b s t a n t i a l l o s s of  Values  stability.  Remarks  and  t h e c o n c l u s i o n s b a s e d on them may  be  in  sum-  as f o l l o w s : (i)  The  a n a l y s i s demonstrated the existence of  p e r i o d i c s o l u t i o n s w h i c h may numerical  be  various  d e t e r m i n e d by a n a l y t i c a l  or  means.  (ii) has  margin  e s s e n t i a l f e a t u r e s of the a n a l y s i s presented  chapter  marized  Moreover,  b e e n i g n o r e d so t h a t t h e  I t i s apparent that o r b i t angle  0.15  of e c c e n t r i c -  diagrams.  shown i n t h e F i g u r e s r e p r e s e n t s a  of e greater than  impulse  Only the 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  of s t a b i l i t y  2-36-i  plots i n Figures  the s i z e of the d e s t a b i l i z i n g  as a f u n c t i o n o f o r b i t a n g l e i t y and  The  The  concept of a t h r e e dimensional  been i n t r o d u c e d .  T h i s has  phase space  the valuable property  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 p h a s e s p a c e by t h e  that repre-  s e n t a t i v e p o i n t s a r e u n i q u e and n o n - i n t e r s e c t i n g . (iii)  The  p e r i o d i c s o l u t i o n s a p p e a r as  t r a j e c t o r i e s i n the phase space.  The  helical  general character  t h e m o t i o n 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 been d e t e r m i n e d n u m e r i c a l l y .  The  manifolds  of  w h i c h have  are a l s o  non-  110 e=  0  0.05^-' 0.10,^'  0.15  ^  __0.25_  .030 ' Am  0 K:  =  355  1.0 0.30 -.0-25 _ _ . £ , 2 0 "  0.15~~  0.05  90°  180  c  270  e Figure 2-36-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)  360  1  1  111  1  e = 0  \ ~  -  \  ^.  0 J 0 _  \  " \  V  \ \  0-32^ \  1  i  /  . -  /  4  /'  »  _  1  / " v . 0-32^  ^  y *  0.9 -1  o . i o — .  ^"  0  i  -2  0° Figure  2-36-ii  90'  i 180°  e  i 270  c  360  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 Figure 2-36-iii  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 =0.5) i  114  Kj = 0-3  e=0  0.05, 0.10 Aij)  -QilO 0-05  -2  0"  F i g u r e 2-36-v  90  180  e  270  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)  360  115  Kj = 0.1  e =0  0.05  ML  e  0.05"  0°  F i g u r e 2-36-vi  90°  180  e  270  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)  360  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 the p e r i o d i c  solutions. (iv)  The  region of s t a b i l i t y  i s represented  by  l a r g e s t i n t e g r a l m a n i f o l d w h i c h c a n be  constructed.  i m p o r t a n c e o f s u c h a s u r f a c e c a n n o t be  over  the The  emphasized  as  f o r g i v e n values of the parameters i t provides a l l p o s s i b l e combinations  o f d i s t u r b a n c e s t o w h i c h a s a t e l l i t e may  s u b j e c t e d a t any it  given point i n i t s o r b i t without  be  causing  t o tumble. (v)  For a c i r c u l a r o r b i t the i n v a r i a n t surfaces i n  the phase space are 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 . For 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  exhibit  substantial variation i n eross-sectdonewithlorbltallangle:. (vi)  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 of the  ing surface decreases ° i.e.  and  for e = e  i t collapses to a max  to a periodic solution.  of o r b i t  There i s a l i m i t  t o the  line, '  value  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 motion.  c r i t i c a l value  limit-  This  depends on t h e g e o m e t r y o f t h e s a t e l l i t e .  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  The  c h e c k e d 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 of the p e r i o d i c s o l u t i o n s . (vii)  The  e c c e n t r i c i t y and  a n a l y s i s suggests a large value  h e l p to ensure s t a b i l i t y .  that a small value  of i n e r t i a parameter would  For e l a r g e r than about  p r a c t i c a l 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 a i s not  possible.  of  I f the s i z e of the l i m i t i n g  O.38 satellite  invariant sur-  f a c e i s i n t e r p r e t e d as a measure o f 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 l i b r a t i o n a l motion,  i t has  still  continue to  execute  b e e n shown t h a t e v e n q u i t e  moderate v a l u e s of e c c e n t r i c i t y would s e r i o u s l y reduce ability  the  of the s a t e l l i t e to withstand e x t e r n a l disturbances  3. 3.1  PLANAR LIBRATIONS OF  Formulation of the The  l i t e was  Problem  a n a l y s i s of the planar l i b r a t i o n s of a r i g i d  presented  the s a t e l l i t e members and  A DAMPED S A T E L L I T E  i n the preceding  chapter.  satel-  By c o n s t r u c t i n g  so t h a t r e l a t i v e m o t i o n can o c c u r between v a r i o u s  inserting  energy d i s s i p a t i n g mechanisms w h i c h  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 ture of the s a t e l l i t e  the  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  capto  r e d u c e 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 p r o p o s e d i n t h e ( s e c t i o n 1.2).  literature  Some o f t h e s e a r e q u i t e c o m p l e x a s t h e y  t o s t a b i l i z e t h e s a t e l l i t e a b o u t a l l t h r e e body a x e s . 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 p r o p o s e d  attempt I f only  by  22 Paul  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 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 The and  is  arrangement.  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  a more g e n e r a l c o n f i g u r a t i o n h a s  ( F i g u r e 3-1).  con-  restrictive  been s e l e c t e d f o r s t u d y  N o t e t h a t t h e mean p o s i t i o n o f t h e damper mass  o f f s e t f r o m t h e c e n t r e o f mass o f t h e m a i n b o d y . The  k i n e t i c and  be w r i t t e n a s  p o t e n t i a l energies of the system  may  119  F i g u r e 3-1  G e o m e t r y o f m o t i o n o f a damped  satellite  120  \Z  ( 3  '  1 )  (3-2)  1  W r i t i n g the d i s s i p a t i o n  function  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-^ d e g r e e s o f f r e e d o m  (Iyy  + "\i(f. l/X© +  +  r) + 2 M (£ +J. )|, (6 4a  (3.4)  a  (3-5)  121 Putting  K; = "V" r  In  J .  v  v  J.yy  1  =  ^  the governing  The u s e  =  e q u a t i o n s may  of the  (3.6)  Jilt*  t u f a  be r e w r i t t e n  awkward r e l a t i o n s  as  g o v e r n i n g r,  and  6,  •  & as f u n c t i o n s pendent  of time  variable  to  can be a v o i d e d by c h a n g i n g t h e  0 u s i n g the  relations  inde-  (2.11)-(2.13)  and  122 For  an e l l i p t i c a l  orbit  i ti s possible to write a relation-  s h i p f o r 0 i n terms o f t h e o r b i t a l  A  1» _ m.  so t h a t  /  h+ec*,ef_  w  the parameters d e s c r i b i n g  _  period  (n-eCosef  .  (3 10)  t h e damper may be w r i t t e n  e  9  (3.11)  0 = d  Using these r e l a t i o n s , and  t h e system e q u a t i o n s (3.8)  ( 3 « 9 ) may be w r i t t e n a s  (3.12)  .3 (ft + Kd (l + *) ) stn (PCos(l> = o Z  i + eCosQ  '  r  and (I  - ezP  2e Sin 6  123  4  I I+  l(i+eCose)+  •/V+.f.-  3Co5  e Cos a  (3.13)  ± z J C o l ±  where  (3.H)  The s y s t e m i n v o l v e s complicates character  of the  particular damped,  the  analysis. system i t  number o f v a r i a b l e s  To b e t t e r is  understand the  convenient  which  basic  to  consider a  s i t u a t i o n where t h e m a s s , m^, i s  critically  i.e.  '  T Furthermore,  the  Numerical  =  parameter  representing a slender, 3.2.  a large  will  be t a k e n t o be u n i t y  dumbbell type  thus  satellite.  Results  The damping p r e s e n t trajectories  (3.15)  to g r a d u a l l y  s i p a t i o n of energy  i n the  i n t h e model causes t h e approach a l i m i t  cycle.  phase The  damping mechanism p r e c l u d e s  space  disemploy-  ing  the  process  invariant  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  surface.  F o r a damped s y s t e m t h e r e  of  i s no  an  closed  i n v a r i a n t s u r f a c e , however, the r e p r e s e n t a t i o n of the  be-  haviour  is  still  of the  very  system i n the  s t r o b o s c o p i c phase p l a n e  helpful.  For a given value  of 6,  j (^/-plane d e t e r m i n e s t h e corresponding  s t a t e of the  p o i n t i n a Z,  o f t h e damper.  a point i n the  Z  -plane  T  stroboscopic  s a t e l l i t e while  s p e c i f i e s the  For s i m p l i c i t y o n l y the phase p l a n e  s e n t i n g the s t a t e of the s a t e l l i t e  i s studied  a  state repre-  here.  A t y p i c a l short h i s t o r y of the s a t e l l i t e motion i s presented  i n F i g u r e 3-2.  phase p l a n e  i s shown  The  the amplitude  stroboscopic  diagrams i n d i c a t e of the motion t o  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 are presented  F i g u r e s 3-4  and  d i a g r a m s do n o t but  corresponding  i n F i g u r e 3-3.  t h a t t h e damper i s c a u s i n g decay.  The  serve  points.  3-5.  Note t h a t the apparent curves  i n d i c a t e the a c t u a l motion of the  only to i n d i c a t e the The  inward  trend of  number o f a p p a r e n t c u r v e s  in  in these  satellite,  successive  i s o f no s p e c i a l  significance. • 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  of time successive fall  p o i n t s i n the stroboscopic  progressively closer together.  a p e r i o d i c s o l u t i o n or l i m i t F i g u r e s 3-6 different  to 3-8  The  phase  system thus  plane approaches  cycle.  compare t h e l i m i t  cycles given  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  section 2.2.3  f o r the  period  same v a l u e s  o f e and  K..  Of t h e  by  in three  •8  i  r  r  T  4> Rad o  0/27T F i g u r e 3-2  S o l u t i o n of  e q u a t i o n of  motion i l l u s t r a t i n g  the  effect  of  the  damper  126  +10  o  e = 2Tin, n=o,i,2'  e = .2 Kj=  1.  o-=  3.  K=.05  r*=  f-5  6.  8  10  -.2  -.3  -.1  .1  H  2  + -5 °  n  •  Point representing the state of the 6=2Ttn„ Limit  Figure  3-3  system at  cyclei  S t r o b o s c o p i c phase p l a n e o f t h e 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  Rad  127  e  0<• 1.0  = 0.2  °2  K.i = 1.0 0.01  «*=  3. 6.  6  + •5 7  \>  \10  13  16  19  1 5  J8 22 25 ..o oP21 32 ^ °«34 •' 29p c§7 / 2 4 Q 36°- - - o " ° 2 7 2 o \ 3|3 30 •+23 a o •<>.... 20 14 17 ^11 0  -.4  Rad  + -5  Point  representing  Limit  cycle.  Apparent  F i g u r e 3-4  curves  the  state  defined  of  the  by groups  Typical stroboscopic satellite  system  of  at 9=2Ttn.  points.  p h a s e p l a n e o f a damped  128  v  n  •  Figure 3 - 5  Point representing the state of the system at  0=2?™.  Limit cycle Apparent curves defined by groups of points.  T y p i c a l s t r o b o s c o p i c p h a s e p l a n e o f a damped satellite  G.Deg F i g u r e 3-7  Limit  cycles  {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 given set of ( e , K^)  parameters  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  f o r comparison.  F o r s m a l l dampers 2  high natural frequencies s o l u t i o n are v i r t u a l l y The  (K^«l) and  suitable  sufficiently  2 »3 ^  Q  )  t h e two t y p e s o f  identical.  i n t r o d u c t i o n o f damping causes t h e l i m i t  cycle = 0.  to l a g behind the p e r i o d i c s o l u t i o n obtained f o r  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 the amplitude of the motion although the n o n l i n e a r i t y of the s y s t e m 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 . N o t e t h a t a v e r y l o w 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  of equation  ( 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 negative..  The m o t i o n o f t h e damper mass t h e n becomes u n -  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 3.3  system.  Conclusions The  a d d i t i o n o f a damper t o a r i g i d  satellite  results  i n the disappearance of the i n v a r i a n t manifolds discussed i n C h a p t e r 2.  The  of the 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  accompanying  limit  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 t h e undamped c a s e .  The  cycles are  independent  solutions obtained f o r  e q u a t i o n s o f m o t i o n i n t h e two  cases  a r e s i m i l a r and t h e p r e s e n c e o f d a m p i n g m e r e l y c a u s e s 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 The  limits  of s t a b i l i t y  inwards.  determined i n Chapter 2  may  be r e g a r d e d a s a p p r o x i m a t e l y v a l i d satellite.  f o rthe l i g h t l y  The p r e s e n c e o f d a m p i n g c a u s e s 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 tion.  of the i n i t i a l  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  manifold  C h a p t e r 2, a n d t h e d a m p i n g i s s m a l l , t h e s u b s e q u e n t  m o t i o n s h o u l d remain s t a b l e and approach t h e l i m i t In  condi-  c a n be e x -  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 of  damped  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  cycle.  inwards across  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 w h i c h were d e t e r m i n e d  for  t h e undamped c a s e .  the  small amplitude periodic solutions obtained f o r the  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  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 all  real  satellites.  case approached by  4. 4.1  PLANAR LIBRATIONS OF A LONG F L E X I B L E S A T E L L I T E  Preliminary  Remarks  Gravity-gradient that the i n e r t i a able  requires  o f t h e s a t e l l i t e be l a r g e s o t h a t t h e a v a i l -  torque i s s u f f i c i e n t l y great  external disturbances. has  s t a b i l i z a t i o n of s a t e l l i t e s  t o overcome t h e e f f e c t s o f  To a c e r t a i n e x t e n t ,  this  problem  b e e n overcome by t h e u s e o f t h e de H a v i l l a n d STEM  (Self-  s t o r i n g Tubular E x t e n s i b l e Module) ^ which i s capable o f 2  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 h u n d r e d f e e t  long  (Table I I I ) . The  STEM boom i s t y p i c a l l y a b o u t one i n c h i n d i a m e t e r ,  or l e s s , w i t h thin,  slender,  a wall thickness  o f a b o u t .002 i n c h e s .  t u b u l a r member i s q u i t e f l e x i b l e a n d h e n c e  susceptible to external disturbances. which provides substantial  a large s t a b i l i z i n g  Thus a l o n g  e f f e c t can a l s o  e x t e r n a l l y induced forces.  boom introduce  Preliminary  a n a l y s i s ^ i n d i c a t e s t h a t among t h e v a r i o u s 2  ance  This  forms o f d i s t u r b -  ( 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  likely  t o h a v e t h e most s i g n i f i c a n t The  volves  problem I n general  e f f e c t on t h e p e r f o r m a n c e .  i s e x t r e m e l y complex as i t i n -  the s o l u t i o n of a s e t of simultaneous  equations with  a l a r g e number o f p a r a m e t e r s .  the  study o f such a d i f f i c u l t  is  considered.  differential To  initiate  problem, a s i m p l i f i e d  model  135 TABLE I I I CHARACTERISTICS OF REPRESENTATIVE STEM  CONFIGURATIONS  Alouette I I  Alouette I  Beryllium  Units  copper  Material  Steel  Radius, a^  0.475  0.25  inches  Wall thickness,  0.006  0.002  inches  Density,  0.286  0.32  l b / i n -3  Thermal conductivity,  26  k^  50  S p e c i f i c heat, c^  0.11  0.092  Coefficient of thermal expansion, <*  6.5x10°  10x10  Absorptivity ( s o l a r ) , <x  0.9  0.45  0.8  0.25  t  s  Emissivity,.  €  Bending stiffness,  6*1  Mass/length,  "m  b  351  BTU/hr f t °F B T U / l b °F  -6  o -l F  lb f t '  15.5  0.068  35,36  0.0142  lb/ft  This chapter studies the planar l i b r a t i o n s of a slender, flexible  satellite  of constant  ence o f s o l a r h e a t i n g . and  c r o s s - s e c t i o n under t h e i n f l u -  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  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 c o n c e p t space r e p r e s e n t a t i o n o f t h e motion a b l e system  i s extended  o f t h e phase t o a deform-  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  are obtained.  Charts a r e presented which  surfaces  indicate the effect  of o r b i t e c c e n t r i c i t y , s o l a r aspect 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 operation. 4.2  Formulation  of the Problem  Consider a slender f l e x i b l e mass a t S, d e f o r m e d planar l i b r a t i o n a l  s a t e l l i t e with centre  due t o s o l a r h e a t i n g , and  for  executing  m o t i o n w h i l e moving i n an e l l i p t i c  about the c e n t r e of f o r c e 0 ( F i g u r e 4-1). an element o f t h e s a t e l l i t e  %  *(5,t  of  The k i n e t i c  orbit energy  located at  )  (4.1)  c a n be w r i t t e n a s  (4.2)  F i g u r e 4-1  Geometry of  motion of  flexible  satellite  138 B e c a u s e S i s t h e c e n t r e o f mass, t h e  ' fldm  =  y  fid*  permit w r i t i n g  = f } \  relations  =fi\  =  (4.3)  0  the t o t a l k i n e t i c energy as  (4.4)  where  The  quantity  (4.6)  r e p r e s e n t s t h e a n g u l a r momentum o f t h e s a t e l l i t e w i t h to the r o t a t i n g  x, z-axes.  ^ indicates  The  choice  = o  <^>  t h a t t h e a x e s 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. separates  respect  This  device  the l i b r a t i o n a l degree of freedom, d e s c r i b e d  the co-ordinate w h i c h employ  by  uV , and t h e v i b r a t i o n a l d e g r e e s o f f r e e d o m  the x,z-co-ordinates.  I f t h e x , z - a x e s a r e t a k e n t o be t h e p r i n c i p a l a x e s o f t h e deformed body, t h e p o t e n t i a l  e n e r g y due t o t h e  139 gravitational field  i s (equation  There i s a l s o t h e e l a s t i c  , J  Using  ._  /  Lagrange's  (2.7))  potential  energy  ds  /  equations  (4.9)  t h e motion o f t h e centre  o f mass c a n be d e s c r i b e d b y t h e e q u a t i o n s  9  (4.10)  L  +r (i-35ih>Ji a  (4.11)  w h i l e t h e l i b r a t i o n a l m o t i o n i s g o v e r n e d by  w h i c h 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. This i sa very mation l e t  complex problem.  As a f i r s t  approxi-  140 To t h i s  degree o f a p p r o x i m a t i o n (4.14) f  «  a n d h e n c e t h e c o - o r d i n a t e s o f a n e l e m e n t o f t h e boom a r e g i v e n by (4.15)  The d e f l e c t i o n o f a n 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  ' 3z r£1  2s-  differential  equation  IK  (4.16)  which has t h e s o l u t i o n oo  (4.17)  w h e r e t h e mode s h a p e s , X ^ ( s ) , a n d t h e a m p l i t u d e s , A ^ ( t ) satisfy  the d i f f e r e n t i a l  equations  Ai + ^ Al = ° h  Cl  (4.18)  (4.19)  ds*  J  141 subject t o the appropriate i n i t i a l The X^ may  and b o u n d a r y c o n d i t i o n s .  be t h o u g h t o f a s g e n e r a l i z e d  a n d t h e A^ a s t h e a m p l i t u d e s .  co-ordinates  The X^ a r e o r t h o g o n a l ;  that  is (4.20)  a l s o . , f o r a f r e e - f r e e beam,  so t h a t t h e x , z - a x e s r e m a i n t h e p r i n c i p a l a x e s o f t h e deformed  body.  The k i n e t i c  e n e r g y o f v i b r a t i o n may  be w r i t t e n i n  t e r m s o f t h e A.  (4.22)  fit The  £M«<H  expression f o r the e l a s t i c  potential  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  (4.24)  '4.2  J Since the s a t e l l i t e  i s completely  ^35 CI & and t h e shear and hence  force, ^(^ i,|^)>  u,  h  f r e e , t h e bending  r  a  r  e  z  e  r  o  a  t  t  h  moment, e  e  n  d  s  (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  I I  = =  -L *  * fs*\  by  Z  Ai  T  c  = I,  (4.26)  143 *  CO  yy The  x  '  x x ' -"-zz  *T  (4.26) cont'd  4 —  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-  t h e n be w r i t t e n a s  1=1  + (4.27)  {>!  c=/  and t h e r e f o r e t h e e q u a t i o n s  of motion governing  the ampli-  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  (4.28) H e r e 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 d e p e n d e n c e  of the s a t e l l i t e and  (4.28).  of these  f o r c e s on t h e o r i e n t a t i o n  l e a d s t o c o u p l i n g between  equations  (4.12)  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  H u g h e s - ^ who t o o k  2  » e , JJJT^ 2  .  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 can  37 be d e t e r m i n e d f r o n r CO;  =  (4.29)  (kL)  where  Cos(kl\ The v a l u e s  Co ^(l(L). = I.  (4.30)  5  o f (kL)^ are (kL). 1  4.73004  2  7.85320  3  10.99561  4  14.13717  5  17.27876  i£  6  ( 2 i + 1)^.  F o r 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 the lowest  (4.3D  (Table I I I )  n a t u r a l frequency i s .  *  _  I7S-X 10  (4.32)  (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 present  (50-500 f e e t ) l e a d s  f o r t h e fundamental period o f  t o values  f r o m 3 . 7 5 t o 375 s e c o n d s .  practice  I t may be n o t e d t h a t t h e 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 motion. The r e s u l t s o f C h a p t e r 2 showed t h a t t h e v a l u e  o f ip  145 A l s o fA/r  o f t h e same o r d e r a s 6 .  is  i s approximately  equal  *2  to 8  so t h a t i n e q u a t i o n  gravitational field  (4.28) a l l t e r m s due t o t h e  c a n be c o m p l e t e l y i g n o r e d c o m p a r e d t o  the r e l a t i v e l y high n a t u r a l frequency  o f t h e boom.  d e f l e c t i o n o f t h e boom c a n be a p p r o x i m a t e d modes t h u s r e d u c i n g t h e e q u a t i o n s o f m o t i o n to the e l a s t i c  The  by t h e n o r m a l corresponding  degrees o f freedom t o  (4.33)  4.3  T h e r m a l 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  of 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  further.  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 be t h o u g h t  o f as approximating  s a t e l l i t e which  may  t h e STEM.  C o n s i d e r a n e l e m e n t o f t h e beam a s 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  than  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 performing a heat  <  balance  5n  f o r t h e element g i v e s t h e e q u a t i o n  z  This equation represents a s i g n i f i c a n t  i m p r o v e m e n t on t h e  35 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 conductivity i s considered. The  t h e r m a l i n p u t t o t h e boom f r o m t h e s u n c a n be  146  F i g u r e 4-2  Assumed c r o s s - s e c t i o n  of 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  (4.35)  1  elsewhere  O  where 0 * i s t h e a n g l e between boom a n d t h e s u n .  the longitudinal axis  T h i s may be w r i t t e n  as a F o u r i e r  of the series  co  ci  =  Q  <x Sin  6 JC  s  h  Cos  (4.36)  nn  where  ft /  2  n = 0 n = 1 (4.37) n even, ^ 0 n odd, ^  1.  To d e t e r m i n e 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 a n e l e m e n t o f a r e a (dA ) on t h e i n s i d e o f t h e t u b e ( F i g u r e 6  JQ  Assuming t h a t  4-4)  X (A )]. +n  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 which f a l l s  on t h e s e c o n d e l e m e n t o f a r e a (dA. ) i s  (4.38)  Figure  4-4  Geometry of r a d i a n t heat t r a n s f e r i n i n t e r i o r o f t h e s a t e l l i t e boom  the  150  ,4 cJAe  (4.39)  4 6, <T  dAin  where  d&j i s  the  s o l i d angle  s u b t e n d e d by d A  i n  as s e e n  from  dA.  dco  From F i g u r e  in  =  4-4 i t  X*.  is  =  evident  X  r  —  (4.40)  that (4.41)  A i r  (4-42) and (4.43)  so  that  c**K = The r a d i a t i o n given  incident  0on dA.  fr) f r o m dA  (4.44)  is  then  by  (4.45)  and  since  151  <JA  -  e  (4.46)  G^J^O/Ar  (4.47)  /-c  4«4  Solution  ©5  o f t h e Heat Balance  Equation  Representing the temperature by a s e r i e s  d i s t r i b u t i o n i n t h e boom  o f t h e form 00  w h e r e i t may be a s s u m e d  that  leads t o the series  fTfc^+a)]  *  T ( t ) HTji)[_T „£)Cos fa+siX io  Hence t h e r a d i a t i o n i n c i d e n t following  emission i s  k/  n  upon t h e i n t e r i o r  immediately  (4.50  152 2TT  T V  4  A fraction,€ ,  and a q u a n t i t y  +  1  I-  J  o f t h i s r a d i a t i o n i s absorbed,  b  r  r~^  T Co3ha  4.4T * v Z _ 3  (4.5D  V f r  AT V ^ ^ " ^ l  4  3  T  (4.52)  (1 - 6.^) i s r e f l e c t e d , nfl z  (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 radiation i s again incident  u p o n t h e i n t e r i o r o f t h e boom  i n a manner a n a l o g o u s t o t h a t equation terior  (4.51).  The t o t a l  o f t h e boom w a l l  employed i n t h e d e r i v a t i o n o f  incident  h e a t f l u x on t h e i n -  i s t h e n g i v e n by  (4.54) In  '  Now,  ffiso  Y)«l  v  153  (4.56)  v * I  4n  so  that  ^ «-T  The be  <x>  x 4^VT  C o 3n  n  (4.57)  f i n a l f o r m o f t h e h e a t b a l a n c e e q u a t i o n c a n now  written  ' 3 r 2  - 3€ r\ b  A  7"T^ ^"^  w h i c h i s of t h e form  C ~o  5  Kin  b  (4.58).  154 a>  ) C where  the  This leads  d e p e n d on t i m e ,  n  to  the  series  = O  n a  C o s  of  but a r e  (4.59)  independent  ofJTL.  equations (4.60)  (4.61)  where  r  z  n = 1 n even;  (4.62)  ^ 0  n o d d , ^ 1.  The f i r s t  of these  remaining equations,  equations  although l i n e a r ,  c o e f f i c i e n t s w h i c h depend on t h e Fortunately in detail.  is  non-linear. c o n t a i n time  s o l u t i o n of  not n e c e s s a r y  The  to  varying  (4.60).  solve  the  equations  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 o b t a i n e d by  determining the state  it  is  solution.  approximate time  constant  and t h e  steady  155 The t i m e c o n s t a n t o f t h e n t h t e r m o f t h e s e r i e s (4.48)  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 t e r m i s  3 5  The a n a l y s i s  o f E t k i n and Hughes,  referred to earlier,  ignores 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 of equation  T h i s i s u s u a l l y t h e dominant term and  (4.64).  hence t h e i r a n a l y s i s heating.  overestimates the effect of solar  For the representative Y  and  therefore  X  t  The s h o r t  *  I  i n t h e denominator  s  configurations  (Table I I I )  -L±—  very short,  (4.65)  o f t h e o r d e r o f 12 s e c o n d s .  time constant implies  that  t h e F^ d e p e n d  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 negligible  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  The p r e s e n c e o f i n t e r n a l damping w i l l tary solutions the  forced  of equation  (4.33)  m o t i o n o f t h e boom.  periods.  c a u s e t h e complemen-  t o damp o u t l e a v i n g  only  As t h e F^ d e p e n d 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 t h e s a t e l l i t e , t h e r a t e o f change o f 0 •  which i s o f t h e order  of the orbital  (4.33) may a l s o be n e g l e c t e d  A; Thus t h e i n s t a n t a n e o u s represented the  period, the  term i n  so t h a t  ^  (4.66)  configuration of thes a t e l l i t e i s  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  orientation.  leaving only  This  i n e f f e c t eliminates equations  (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 ,  to  (4.33)  t o be  solved. 4»5  T h e r m a l D e f l e c t i o n o f t h e Boom C o n s i d e r a n e l e m e n t o f t h e boom o f w i d t h  ure  4-5).  The l e n g t h  &  r  e  f  = (R  c  denotes t h e o r i g i n a l  e n c e t e m p e r a t u r e , ^ f>  this  re  ing  (Fig-  of the s t r i p i s  J J n ) If  a^dO.  + a  t  C c  length length  S  f i ) ^ .  (4.67)  of the strip at a referc a n be o b t a i n e d  by a l t e r -  t h e temperature and s t r e s s i n g t h e m a t e r i a l  4(a)-J [i+* (T(n)-T ) r(/  t  n/  +  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 e l e m e n t i s  J  [T  <r(s±)a lo b  b  da (4.70)  where  = jPfff/z) During o r b i t a l motion lite  = mean l e n g t h  o f t h e element.  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 -  i s n e g l i g i b l y s m a l l so t h a t  <<ref  which  represents the l o n g i t u d i n a l thermal The  moment p r o d u c e d  expansion.  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 d r i v i n g force i s much lower than the n a t u r a l frequency of the beam, the beam w i l l be i n e q u i l i b r i u m and the moment w i l l be zero.  That i s  159  <£>h  /  a  ft  =  (4.73)  a*  Jref  L e t t h e £ ^ , Y\ ^ - c o - o r d i n a t e t h e fc^, V | ^ - p l a n e l i e s  i n t h e plane  on t h e n e g a t i v e V ^ - a x i s  s y s t e m be d e f i n e d s o t h a t o f motion w i t h t h e sun  (Figure 4 - 6 ) .  From e l e m e n t a r y  calculus  L_  ' •*/<  =  m  (4.74)  I  or  (4.75)  where  f=  [J^t  0(.  + 4e irJ (A=£k^ 3  k  0  (4.76)  Taking  yo) = jfi°) * ° s  -  (4  77)  gives (4.78)  160  This solution i s i l l u s t r a t e d  i n Figure  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 and  t h e s o l u t i o n c a n be w r i t t e n  s. =  I  +  T)  i n the parametric  Cosh  ( /Jt*) S  A s a t e l l i t e o f g i v e n l e n g t h c a n be e a s i l y a l o n g an a r c o f t h e curve  F i g u r e 4-7.  (4.81)  t  fitted  and t h e correspond-  c a n be d e t e r m i n e d a s i n d i c a t e d i n  The r e s u l t s o f t h i s  F i g u r e s 4-8 a n d 4-9.  (4.80)  i n F i g u r e 4-6, h e n c e i t s c e n t r e  of p r i n c i p a l axes,  i n g moments o f i n e r t i a  form  5m" ~T«nh (*/**)]  % = Jt*  of,mass, o r i e n t a t i o n  (4.79)  analysis  are plotted i n  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  factors  L  -  I tain  (4.82)  I -tt where I  represents the i n e r t i a  the x- and y-axes.  of the r i g i d s a t e l l i t e  about  I t i s i n t e r e s t i n g t o note that I - I  i s nearly three times  I  zz max  and  v  v  min  162  Figure  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 deflected s a t e l l i t e  the  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 a s f u n c t i o n s o f boom l e n g t h  164  Figure  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 b e t w e e n t h e s u n and t h e x - a x i s  165  5-  x  so  that  zz.  1  zz  Ma/ (4.84) 2  Noting  again  that  J de  J dt  J and  2  (4.85)  J*  _  r  y  Je  d$  representing  l  yy  * r (i-2I5^,s) r  (4.86)  equation (4.12) becomes  (4.87)  4 e5fn©0" ' */ )} 2TS, n  5  166  + 3 (I ~ 41 Sth*j8)5rncpCo« f V  '  Stability  4.6  I  (4.87)  with periodic  (2.14).  form as  non-autonomous  coefficients  differential  i s o f t h e same  d i m e n s i o n a l phase s p a c e  manifolds with properties  i n Chapter  cont'd  Hence t h e s o l u t i o n o f t h e e q u a t i o n when  represented i n the three integral  (4.87)  O .  Analysis  The g o v e r n i n g n o n - l i n e a r , equation  »  '  similar  generates  to those  discussed  2.  The s o l u t i o n o f t h e e q u a t i o n was o b t a i n e d f o r a w i d e range  of i n i t i a l  satellite fifty  conditions using a d i g i t a l  was t a k e n t o be s t a b l e  orbits.  The t r a j e c t o r y  conditions generates shown i n F i g u r e phase space trajectory  4-10.  starting  a surface It  if it  computer.  d i d not tumble from s t a b l e  This  within  initial  i n phase s p a c e o f t h e f o r m  is still  sufficient  t o extend the  c o - o r d i n a t e 6 o n l y up t o 271 b e c a u s e when a arrives  a t 6 = 2K  with certain  values  ip' , i t may be e x t e n d e d by c o n s i d e r i n g a n o t h e r starting  The  f r o m 9 = 0 w i t h t h e same v a l u e s  of p  and  trajectory  o f lp a n d ^ ' .  i s i d e n t i c a l w i t h t h e p r o c e d u r e a d o p t e d i n C h a p t e r 2. The s e a r c h f o r t h e l i m i t i n g  result  i n stable  librational  initial  conditions  which  motion i s thus a search f o r  168 the l a r g e s t such s u r f a c e . i n t h i s s u r f a c e generates  Any s t a t e of motion which l i e s w i t h a t r a j e c t o r y , and hence a new s u r -  f a c e , which remains i n s i d e the l i m i t i n g m a n i f o l d .  Therefore,  the r e g i o n enclosed by the l i m i t i n g s u r f a c e i s the r e g i o n of stability. The u s e f u l n e s s of such a phase space was e x p l a i n e d i n s e c t i o n 2.4.  For g i v e n 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 s o l a r aspect angle i t provides a l l p o s s i b l e comb i n a t i o n s of d i s t u r b a n c e s t o which the s a t e l l i t e may be s u b j e c t e d without  causing i t t o tumble.  aspect angle on the s t a b i l i t y  The e f f e c t of s o l a r  region i s i l l u s t r a t e d  4-11 and 4-12 where c r o s s - s e c t i o n s of the s t a b i l i t y at  i n Figures region  s e v e r a l v a l u e s of the o r b i t angle are presented. A convenient  condensation  of data may be e f f e c t e d by  p l o t t i n g the i n t e r c e p t s of the (f/ - a x i s w i t h the phase space c r o s s - s e c t i o n s at 0 = 0 as shown i n F i g u r e s 4-13 and 4-14. I t i s apparent  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 the s t a b i l -  i t y r e g i o n decreases  i n s i z e and beyond a c r i t i c a l  value  ceases t o e x i s t . The determined ter  2.  accuracy with which the l i m i t i n g manifolds were i s approximately  the same as d i s c u s s e d i n Chap-  The c a l c u l a t i o n s were c a r r i e d out with the same  p r e c i s i o n and t h e r e f o r e the e r r o r i n the r e g i o n of s t a b i l i t y i s approximately  ± , 0 3 u n i t s i n (p'.  Several symmetry properties e x h i b i t e d by equation (4.^7) s i m p l i f y 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 of the c r o s s - s e c t i o n of 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  F i g u r e 4-12  -60°  •30°  30  60°  V a r i a t i o n of the c r o s s - s e c t i o n of 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)  90°  172  173  175  179 Note t h a t i n the  trajectories  substitution. for  (p  0( + IX  substituting  as t h e  equation i s  For negative  (X , s a y  o b t a i n e d by i n t e g r a t i n g  ing 9 from the exactly  initial  from the  i n no  invariant  o< =  - c  * >  t  e  that  same i n i t i a l  ij> = 0 ,  n  this  solution  e  = ^  o b t a i n e d by  change  under  i n the d i r e c t i o n of  conditions  opposite i n sign to  backwards  oC r e s u l t s  for  increas,  is  integrating  c o n d i t i o n s but w i t h  o( =  ^  . e  Hence t h e t r a j e c t o r i e s become m i r r o r ular,  the  formed i n the  phase s p a c e f o r -1><  images o f t h o s e o b t a i n e d f o r + <X.  cross-sections for  0( - - # _  at  In  6 = 0,71  particare  m i r r o r images a b o u t t h e i / / - a x i s o f t h e c r o s s - s e c t i o n s o b t a i n e d f o r <tf = + 0/ . Thus t h e i n t e r c e p t s o f t h e tP*'-axis w i t h t h e e limiting  surface  or i n c r e a s e s  As t h e  9 = 0 do not  change when  stability  s o l a r aspect  region varies  regions of the  stability  long term s t a b i l i t y  planet  about t h e  periodic  regions.  s o l u t i o n s to  equation  regions  of  As t h e n a r r o w n e s s  makes them u n s u i t a b l e  for  stable  operation.  precesthose  overlap  and 4 - 1 4  guarantee  represent  associated  (4.87),  of these  any p r a c t i c a l  with  and a p p e a r  s u r r o u n d i n g t h e main s t a b l e  ance o f s p i k e s r e d u c e s the p r a c t i c a l for  4-13  They a r e  as s m a l l h e l i c a l phase s p a c e .  orbit  4-15).  The s p i k e s a p p e a r i n g i n F i g u r e s  different  sign  with  sun, only  c h a r t s which a c t u a l l y  (Figure  secondary s t a b i l i t y  periodically  a n g l e w o u l d v a r y due t o  s i o n and t h e m o t i o n o f t h e  the  <X changes  by 1 8 0 ° .  Hence t h e ex.  at  region  secondary  o p e r a t i o n , the  upper l i m i t  on  regions appear-  eccentricity  180  181  _2 i  0  Figure 4-15-ii  1  1  1  1  2  -3  i  I  4  -5  e Range of value of the d e r i v a t i v e 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 a n a l y s i s the f o l l o w i n g o b s e r v a t i o n s  may  be made; (i) to  The method employed i n t h i s chapter i s r e s t r i c t e d  " s h o r t " s a t e l l i t e s f o r two  reasons.  The  fundamental  frequency  of the beam must be much h i g h e r than the  frequency  i n order t h a t the approximation  equation  (4.33)  remain v a l i d .  orbital  employed i n (4.84)  A l s o , the r e l a t i o n s  which r e p r e s e n t the v a r i a b l e i n e r t i a s i n t r o d u c e c o n s i d e r a b l e e r r o r f o r long (ii)  satellites.  The  s t a b i l i t y l i m i t s f o r a slender 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 the a c t i o n of s o l a r h e a t i n g , have been obtained u s i n g the concepts i n t e g r a l manifolds.  T h i s determines  of phase space and the c r i t i c a l  initial  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  without  causing i t t o tumble. (iii)  v a l u e s of  be s u b j e c t e d  In g e n e r a l a s m a l l value of e c c e n t r i c i t y would  h e l p to ensure  stability.  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 a f f e c t e d by the dimensionless s o l a r aspect angle, 0( .  l e n g t h L* , as w e l l as the  When L* i s s p e c i f i e d , the  e c c e n t r i c i t y v a r i e s p e r i o d i c a l l y with  critical  o< i n such a manner  t h a t i t i n c r e a s e s with, i n c r e a s i n g (X* f o r 0 < o ( < 9 0 . 0  For the  cases c o n s i d e r e d , i t appears t h a t 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 of an undamped s a t e l l i t e i s not p o s s i b l e under any  circumstances (iv)  for e > 0.425  (Figure 4 - 1 2 ) .  The f l e x i b l e nature of the s a t e l l i t e  causes  a  r e d u c t i o n i n the s i z e of the s t a b i l i t y r e g i o n f o r almost a l l  183 values that  of  <X .  This reduction is  satellite  flexibility  not s e v e r e .  of t h i s  nature  It  is  concluded  does n o t have a  strong d e s t a b i l i z i n g influence. (v) stability  The c r i t i c a l  variation  of the  (vi)  (iii).  stability  region with  as a measure o f t h e without  The e f f e c t  it  the depth of  4-13)  is  the  would  apparent that  s e r i o u s l y reduce  satellite  to withstand  external  is  similar  p l a n e t were n o t to a f u r t h e r  and a more d e t a i l e d a n a l y s i s w o u l d be r e q u i r e d .  analysis  l o s s of s t a b i l i t y .  o c c u r r e n c e s would e x c i t e the v i b r a t i o n a l of the  decay  the  severe.  e c l i p s e s of  considered in this  even  disturbances.  but n o t as  must be p o i n t e d out t h a t  the  interpreted  1st  eccentricity  It  may c o n t r i b u t e  reduces  of  of i n c r e a s i n g L  (vii)  If  4-12,  becoming u n s t a b l e ,  one  periodic  d i s t u r b a n c e which the s a t e l l i t e  q u i t e moderate v a l u e s of the  4-H,  term  (X..  on e c c e n t r i c i t y .  (Figures  than the  because o f the  The p r e s e n c e o f s p i k e s f u r t h e r  diagrams  s u n by t h e  This is  stability  p r a c t i c a l upper l i m i t  ability  b a s e d on t h e l o n g  a n a l y s i s w o u l d be c o n s i d e r a b l y l o w e r  s p e c i f i e d above i n  tolerate  eccentricity  the and  These  modes o f t h e boom of the  vibrations  5. . TWO .DIMENSIONAL MOTION OF AN  AXI-SYMMETRIC  SATELLITE IN A CIRCULAR ORBIT 5.1  I n t r o d u c t o r y Remarks The review of the l i t e r a t u r e  ( S e c t i o n 1.2)  suggests  t h a t the p l a n a r motion of a r i g i d s a t e l l i t e i n a g r a v i t y g r a d i e n t f i e l d has been the s u b j e c t of c o n s i d e r a b l e i n v e s t i gation.  In c o n t r a s t , 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 r e c e i v e d comparatively l i t t l e a t t e n t i o n . g a t i o n i s important l a r g e amplitudes  Such an  has  investi-  ng  because, as p o i n t e d out by Kane,  for  the t r a n s v e r s e motion i s s t r o n g l y coupled  with t h a t i n the plane. The l a c k of i n f o r m a t i o n may  be p a r t l y a t t r i b u t e d t o  the f a c t t h a t the governing equations of motion are nonl i n e a r , non-autonomous, and  coupled.  They a l s o i n v o l v e a  l a r g e number of parameters and hence are not amenable t o any simple c o n c i s e a n a l y s i s .  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 t o move i n a c i r 13 cular orbit.  For t h i s case, as i n d i c a t e d by Auelmann, ^  c l o s e d z e r o - v e l o c i t y curves e x i s t under c e r t a i n c o n d i t i o n s 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 l a r o r b i t are obtained n u m e r i c a l l y . suggest  satellite in a circu-  The  zero-velocity  p o s s i b l e r e g i o n s of s t a b i l i t y and i n s t a b i l i t y .  curves  Regular the  and e r g o d i c t y p e s  behaviour  explained.  of the  system i n the  Using the  shown t h a t  stable  ity  are  curves  of stable  concept  motion are transition  region  o f an i n v a r i a n t  m o t i o n can r e s u l t  open.. ^ L i m i t i n g  d i s c u s s e d and  surface,  even when t h e  invariant  causing i t  to which  to  become  Formulation  5.2.  Consider an o r b i t  about  a satellite  surfaces  are  a rigid  satellite  centre  triad  S - x „ y z„ be c h o s e n so t h a t o o o  the  principal  mass c e n t r e the  orbit  is  local vertical  ,  given  angle  0,  about  by t h e  Q  rotation, X  (Figure  of the the  satellite  z -axis o  is  in  and  the  directed  Q  r  S  Let  5-1).  and t h e y - a x i s  distance  of the  taken  the y - a x i s , the  0  at  is  parallel  The, p o s i t i o n o f between  0 and S and  i n the  may be s p e c i f i e d  following  order:  g i v i n g the x^y^z^-axes; a  x-^-axis r e s u l t i n g , about  satellite  the  i n t h e 2^2 2 X  Z  Z g - a x i s which y i e l d s  tr  a  rotation,  ^- ^5 a<  the  Using the  principal  c a n be w r i t t e n  as  axes the  kinetic  by  rotation,  a n c  *  a  principal  body a x e s x y z .  satellite  the  6.  of r o t a t i o n s  about  subjected.without  a n g u l a r momentum v e c t o r .  The o r i e n t a t i o n a set  body a x e s  J  orbital  pre-  initial  w i t h mass c e n t r e  of force  be t h e  to  veloc-  Problem  S-xyz  outward a l o n g the  is  unstable.  of the  the  may be  it  zero  s e n t e d and p r o v i d e a c o m p r e h e n s i v e summary o f t h e disturbances  is  energy  of  the  18?  (5.1)  Noting  that u)  =  <jf> Cos X  =L-0J/>»A  so  +  +  (& + j>)  C  o  ^  (e-h^)coj^  ?^  (5.2)  COJ \  that  + :-L4> (Co^ \ T z  xx  -f J/**A  I ) yy  + <jt>(& + f)C<>S<fr5fo\ C*2\(l -Iyy)  (5.3)  X/  To d e t e r m i n e t h e p o t e n t i a l o f m a s s , dm, w i t h t h e mass element  energy,  co-ordinates x , y , z , . and t h e c e n t r e  of force  c o n s i d e r an element  The d i s t a n c e  c a n be w r i t t e n  ./t = [(* * rJ„t (y <- rjf , +  between as  , rj^f] »  ,. 5  188 where t h e d i r e c t i o n c o s i n e s J t ,  , Z„ y and t h e x y z - a x e s a r e v  x  local vertical  i  x  = -SintpCosX +  Jhj = K  The  5fntfj5ln\  =  potential  U = -A  Cos  between t h e  outward  z  CosipSinjiSmX  + C<&<f>S?/10C*3 A  (5.5)  p Cos ft  energy of the s a t e l l i t e i s  1*  -v  2  (5.6)  As S i s t h e c e n t r e o f mass  -  and s i n c e x y z a r e p r i n c i p a l axes  O  (5.7)  189  With t h i s the e x p r e s s i o n f o r the p o t e n t i a l 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 t h e r e  U  results  = - r ^  +  &  + 4 Si« if  1  (Txx + T  Cos (J; 5 m  y y  + I„)  0 5 in A Cos A (ixx " Tyy)  190  z  /  + Cc/ip Co* f ( l  x  /  + lyy "I*  For a s l e n d e r axi-symmetric  I  =  x x  hence equations  Iyy  =  T  >  (5.11)  O  cont'd  satellite  I z  (5.12)  Z  (5.3) and (5.11) assume the much s i m p l e r  forms  T = -j t\ (r + 2  + \ I (t* + (® 4»/C»> ) +  N?  U  =  -  (5.13)  7 ^ (5.14)  r  The g e n e r a l i z e d momenta can be expressed as  »v - -If  -^  (5.15)  191  Because the  = 0 in this  co-ordinate  A  i d e n t i f i e d w i t h the  is  case,  constant.  s p i n of the  t h e momentum c o n j u g a t e  to  T h i s momentum c a n be satellite  about the  z-axis,  ^ = J^^X - (© + ^)5^<6J= constant.  For a non-spinning s a t e l l i t e  the  constant  must be  (5.16)  zero,  therefore  j=  x ^(rV Using the  tions  Lagrangian formulation the governing  of motion f o r the  written  is-")  re*) + 1 1 (j>\  ijj and 0 d e g r e e s  equa-  o f f r e e d o m can be  as  (5.18)  192 Noting  that  til  p and  =  for a circular  •  2  ii "  <f>'e +  orbit  (5.21)  © hence t h e e q u a t i o n s  -  O  o f m o t i o n may be w r i t t e n a s  // (5.22)  + 3ft 5fh <f>"  5.3  -h  fo'+'f*  Cosip  =  o  3KiG**pjSf>lfiG>3<f> = 0 ^  The H a m i l t o n i a n a n d Z e r o - V e l o c i t y C u r v e s I g n o r i n g t h e r , 8, a n d ^  f o r t h e s y s t e m c a n be w r i t t e n a s  co-ordinates t h e Lagrangian  (5.23)  193  (5.24)  Thus f o r  a circular  sponding to the explicitly  orbit,  librational  hence the  o f m o t i o n and i s  the  Lagrangian f u n c t i o n  m o t i o n does n o t  involve  corresponding Hamiltonian is  given  corre-  a  time constant  by  H = Hfrii - t  and u s i n g  (5.21)  equation  (5.25)  may  be r e w r i t t e n a s  13 Auelmann's paper ' this  contains  an e r r o r  i n the  statement  of  equation. D e f i n i n g a new y)'  equation  (5.26)  0' + Z  variable =.  (jJ Cc5  (5.27)  <jt>  becomes  $  =  C*3*$(H  3KfCos*<p)+  C  Ht  (5.28)  194 Setting 0 ' = { \ty \  motion  v a l u e s of  = 0 gives  the z e r o - v e l o c i t y curves f o r the  , j0|<f7T/2) which a r e presented f o r i n Figure  5-2.  various  Since the r i g h t hand s i d e o f  (5.28) i s a maximum at 0 = ip = 0, the sum o f the squares of the v e l o c i t i e s i s p o s i t i v e only curves.  T h e r e f o r e the z e r o - v e l o c i t y curves represent bounds  f o r t h e motion.  I t i s thus p o s s i b l e t o conclude that  Cp| < - ( 1 + 3 K ^ ) , -(1  i n s i d e the z e r o - v e l o c i t y  for:  no motion i s p o s s i b l e ,  + 3 K ^ ) ^ C ^ ^ - 1 , the motion i s bounded, -1  <  ^  0, i n s t a b i l i t y can a r i s e only i n the yV - d i r e c t i o n ,  0  <  C^, unbounded motion i s p o s s i b l e  i n both  directions,  5.4  Phase Space and T r a j e c t o r i e s The  written  ( 5 . 1 2 ) and ( 5 . 2 3 ) may be  equation of motion  as a s e t of f o u r f i r s t  rj0 d i L  order d i f f e r e n t i a l  equations  CoS# Tars $  =  +  2j>5ih$ (5.29)  JO  -  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  (5.29) cont'd  d<j>'  = -{(^  +¥ + 3 * ^ }  ******  T h e s e may he r e a r r a n g e d i n t h e .form  7Q  -  C°» *  W  -  &  =  F, which space.  defines  a trajectory  The H a m i l t o n i a n  phase s p a c e  co-ordinate  ^  "  -  <t>'  d  * '  (5.30)  Fz  i n a f o u r - d i m e n s i o n a l phase  (5.28)  permits  i n terms  d e t e r m i n i n g any one  of the other  three.  Solv-  i n g f o r Cos 0 g i v e s ,  or  ,  + c  • - ' P- / j ^ i L l i k _  The a m b i g u i t y  as t o s i g n i n d i c a t e s  d e r i v e d from t h e H a m i l t o n i a n cannot  /  that  (5.31)  the information  differentiate  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  be o b t a i n e d f r o m a c o n s i d e r a t i o n o f t h e  usually  continuity  of  the  solution. If equation  0 is  t a k e n t o be p o s i t i v e  (5.31)  a unique  is  and l e s s  unambiguous and e q u a t i o n  than  7T/2, defines  (5.30)  trajectory  (5.32)  in a three-dimensional holds f o r another permit 5o5  0 ^ 0 and d e f i n e s  phase s p a c e . the  , 0'-space.  {p,  equation  e q u a l l y unique t r a j e c t o r i e s  C e r t a i n symmetry p r o p e r t i e s ,  e l i m i n a t i o n o f one o f t h e  Symmetry  A similar  in  however,  spaces.  Properties  On s u b s t i t u t i n g  e  = - ©  </J =  in  equations  (5.22  0*=,.r_0£i==•;>•_ unchanged. the (  5.23 ) i t  (Jp , 0 ' )  is  !0*'=..^''and the  Thus a t r a j e c t o r y  equivalent f  -  - </>  (5.32),  (5.33)  observed that  y/  =  equations of motion  d e f i n e d by  (5.22  which passes through the  possesses a mirror t r a j e c t o r y  -  5.23)  , are or  point  which  passes  198 through the which  point  (-^,  1  passes through the  about the  point  =  yj ^  made i n  equations  the form of the  the  -  y  equations  point  d e f i n e d by t h e  equations  0 > 0,  trajectory 5.6  -  is  5*23 ) j p ' = c p ' , invariant.  a mirror  conditions  was  of  the with  0')  passes  through  Hence t h e  trajectory  phase s p a c e  (p ' , 0 - p l a n e  of  f  is, the  results p o i n t e d out  motion.  of t r a j e c t o r i e s  classes  and  =0'  T  0.  in  corresponding to  lead to s t a b l e types  0 <  <p ],  which  of motion i n the  image a b o u t t h e  defined for  0  Therefore  ( cp,  point  where 0 < 0 .  T  Numerical It  symmetry  (5.34)  p a s s i n g through the  0 )  section  -(1 + 3K.)  However, which  (5.3) ^  that  initial  ^  -1 a l w a y s  t h e r e appear t o  indicates  the  be  existence  two of  two  solutions.  The f i r s t 5-3.  e  (5.22  (-<p,  for  -  0 possesses a mirror t r a j e c t o r y  0' >  exhibits  0)  substitutions  a  trajectory  (0,  trajectory  (p ' - a x i s .  When t h e  are  T h e r e f o r e the  (j) , - 0 ' ) .  class  of solutions  H e r e an i n v a r i a n t  d i m e n s i o n a l phase s p a c e .  surface That  is is  is  illustrated  d e f i n e d i n the an " i s o l a t i n g "  in  Figure  threeintegral  199  C  H  =  (J)  F i g u r e 5-3  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 the c l a s s of s o l u t i o n s  -1-5  ^  first  0  200 has been determined n u m e r i c a l l y .  F i g u r e 5-4  i n d i c a t e s the  c r o s s - s e c t i o n of a s i m i l a r s u r f a c e i n the plane  i/' = 0.  It  should be noted t h a t the p o i n t s of i n t e r s e c t i o n of the t r a j e c t o r y with the plane The 5-5«  = 0 d e f i n e a smooth boundary.  second type of behaviour i s i l l u s t r a t e d i n F i g u r e  As before i t r e p r e s e n t s the s t a t e 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 p o i n t s appear to be s c a t t e r e d randomly over r e g i o n s i n the plane nature  i n d i c a t i n g the  "ergodic"  of the motion. 38 Kane  has  o r b i t a l plane may  i n d i c a t e d t h a t the motion normal to the e x h i b i t a type  of beat phenomenon with a  very long p e r i o d , t y p i c a l l y 35 to 45 o r b i t s .  T h i s type  behaviour would l e a d to a p l o t of the type presented F i g u r e 5-5•  A l a r g e number of p o i n t s would have to  determined before p e r i o d i c i t y becomes e v i d e n t . could best be d e s c r i b e d as The  in be  Such motion  "quasi-ergodic."  behaviour of the s o l u t i o n i n the t r a n s i t i o n  between the l a r g e simple  of  s u r f a c e s of the f i r s t  region  type and  ergodic behaviour of the second type lends support  the  to the  concept of q u a s i - e r g o d i c i t y . In the t r a n s i t i o n r e g i o n , "chains of i s l a n d s " appear which become s m a l l e r and more 31 numerous as the ergodic r e g i o n i s approached For may  >  be u n s t a b l e .  (Figure  5-6).  -1 there i s a p o s s i b i l i t y that the motion The  numerical  r e s u l t s i n d i c a t e that a  s t a b l e i n i t i a l c o n d i t i o n r e s u l t s i n a s o l u t i o n of the type and hence i n the g e n e r a t i o n  first  of an i n v a r i a n t s u r f a c e  201  Figure  5-4  Cross-section of a surface s i m i l a r to p r e s e n t e d i n F i g u r e 5-3 when = 0  that  202  Figure  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 the ergodic nature of the second c l a s s of s o l u t i o n s  203  Figure  5-6  The- c r o s s - s e c t i o n i/> = 0 i n p h a s e s p a c e 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 " 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 "  5-7).  (Figure  trajectory Further, type C  H  >  Certainly,  can never l e a v e  l  i s such a s u r f a c e ,  i t and s t a b i l i t y  the numerical analysis  of trajectories "  i f there  suggests that  are not c o n s i s t e n t  such s u r f a c e interior  tonian,  different  of a l i m i t i n g surface consistent  C^ = 0„5.  up t o a t l e a s t  surfaces  do n o t a p p e a r t o e x i s t .  show s e v e r a l l i m i t i n g s u r f a c e s  for 0 <  are  surfaces  eight.  It  5-8-iii  to  values of 5.5  (illus-  0. provide useful  and 0  ^  this  o f C^ t h e  0, when drawn i n  a p p e a r as a  i s i n t e r e s t i n g t o note  illustrates  information con-  At h i g h v a l u e s  f o r both 0 ^ 0  motion approximately double that 5-9  of i n v a r i a n t invariant  for representative  q u a s i - p e r i o d i c with the period of the  Figure  0.6  5-8-i  Figures  same d i a g r a m ( F i g u r e 5 - 8 - i i i ) ,  figure  of the Hamil-  the existence F o r C^ ^  the nature of the motion.  limiting  states  may be U s e d t o d e t e r m i n e t h e  The l i m i t i n g s u r f a c e s cerning  Thus t h e  a l l possible  ^he symmetry p r o p e r t i e s o f s e c t i o n  i n Figure 5-S-iii)  surfaces  conditions  motion.  surfaces  the  for  The l a r g e s t  with the fixed value  The n u m e r i c a l work i n d i c a t e s  0)°  surfaces.  represents  corresponding to stable  (0 ^  initial  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 .  of the system,  trated  the ergodic  with s t a b i l i t y  o f C^, s u i t a b l e  c a n be c h o s e n t o g e n e r a t e  H  guaranteed.  o  For a given value  C  is  the  that  twisted both motions  out-of-plane  of the i n - p l a n e l i b r a t i o n s .  behaviour f o r a s p e c i f i c  set of  205  Figure  5-7  Typical  invariant  s u r f a c e when C  H  >  -1  206  207  Figure  5-8-ii  Limiting  invariant  surface  (C  R  208  209  Figure  5-9  S o l u t i o n of the equations of motion f o r specific i n i t i a l conditions, i l l u s t r a t i n g the q u a s i - p e r i o d i c nature of the motion  210 i n i t i a l conditions. The r e s u l t s d i s p l a y e d i n F i g u r e s 5 - 8 - i to 5 - 8 - i i i be presented i n a more i n f o r m a t i v e manner. f i x e d , a constant value of 0 -plane.  may  I f </* and 0 are  describes a c i r c l e i n a  A p o i n t 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 g i v e s 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  p o i n t l i e s at the o r i g i n and the Hamiltonian has i t s minimum For values of Lp and 0 such that  value.  = =1  defines a  r e a l r a d i u s , t h e r e e x i s t s a c i r c l e i n s i d e which s t a b i l i t y i s guaranteed.  Larger values of C^ r e s u l t i n s t a b i l i t y f o r  v a r y i n g arc l e n g t h s of the constant C^  circles.  F i g u r e s 5-10-1 to 5 - 1 0 - i i i show the s t a b l e regions, i n $ ' ,  0 ' - p l a n e s f o r v a r i o u s combinations  of (p and 0 . .  It i s  p o s s i b l e to make an o b s e r v a t i o n 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 p o s i t i o n of s t a b l e e q u i l i b r i u m to d i s t u r b a n c e s i n the  ip  a n  d  0 directions.  i s evident from F i g u r e 5 - 1 0 - i that f o r <p = 0 = 0,  It  the  coupled motion can remain s t a b l e even when s u b j e c t e d to the angular v e l o c i t i e s  ip = - 1 , 1 5 , 0 . = ±1.75..  v e l o c i t y of approximately value of  v/J  f  2,1  The  resultant  i s c o n s i d e r a b l y above the  which holds f o r t h e . p l a n a r 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 l e s s s t a b l e than the more g e n e r a l two 5»7  dimensional motion.  Concluding Remarks The r e s u l t s show t h a t there are two  distinct  types  211  A  -2  Figure 5-10-i  0  1  IP  Allowable v a r i a t i o n s ,in the angular v e l o c i t i e s w h i c h may be i m p o s e d on a n 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 )  2  212  Figure  5-10-ii  Allowable v a r i a t i o n s i n the angular v e l o c i t i e s w h i c h may be i m p o s e d 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 = ±15 )  213  214 of  stability  a s s o c i a t e d with the  of a g r a v i t y - g r a d i e n t  coupled l i b r a t i o n a l  oriented s a t e l l i t e .  l e s s t h a n -1 u n c o n d i t i o n a l s t a b i l i t y existence position.  of  closed zero v e l o c i t y  On t h e  other hand, i f  is  motion  For values  of  g u a r a n t e e d by  curves  about the  -1 < C „ < C „  the  equilibrium  ^  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 initial  w h i c h depends upon t h e  c o n d i t i o n s imposed and c o r r e s p o n d s t o the  o f an i n v a r i a n t  surface  m o t i o n can r e s u l t  i n the  phase s p a c e .  even when t h e  Thus  zero v e l o c i t y  existence stable  curves  are  not  closed. From t h e  limiting  invariant  surfaces  c a n be c o n c l u d e d t h a t w i t h i n c r e a s i n g stability  diminishes rapidly  in size.  ceases  exist  This  to  altogether.  d i s t u r b a n c e s which a s a t e l l i t e  presented here  the r e g i o n For  ~  of  0.55  it  i m p o s e s u p p e r bounds on t h e  can t o l e r a t e  without  becoming  unstable. The l i b r a t i o n a l orbital  plane,  noticeable orbital that  at  plane  are  m o t i o n s , b o t h i n and n o r m a l to. t h e  quasi-periodic.  high values occurs at  of  The a n a l y s i s motion i s  servative  more s e n s i t i v e  as  and t h a t  i n d i c a t e d by t h e  orbital  suggests that  design analyses  is  particularly  a frequency approximately  to the  pared t o those normal to the that  This  where t h e m o t i o n i n  o f the motion normal t o the  the  orbital  double  plane.  coupled  in-plane  the  librational  disturbances  plane.  This  actual  simplified  motion i s study.  at  least  com-  indicates  performed using planar motion are  the  it  as  con-  stable  6. 6,1  General  Conclusions  The r e s u l t s adopting  a phase  of  dimensions  is  uniquely  the  great  gration  by  the  the  of  state  by t h e  the  equations  surface.  have  shown t h e  which possesses  represented  c o u l d be  space  space  so t h a t  an i n v a r i a n t fold  presented  majority  of  CONCLUDING REMARKS  of the  number  system  study  studied,  numerical  the  a r e d u c t i o n i n the  In inte-  generation  an " i s o l a t i n g "  o b t a i n e d which forms a s u r f a c e  and r e p r e s e n t s  under  of a p o i n t . .  the  of motion l e d to is,  of  a sufficient  co-ordinates  cases  That  usefulness  integral  i n the  number o f  of mani-  phase co-ordinates  unity. In those  found, beat  the  c a s e s where  m o t i o n was  phenomenon was  evidence  that  structure  the  invariant  important  i n the  which y i e l d There application ceive  of  study  unstable  observed. . In the  The i n v a r i a n t  ness  either  which g i v e s  t h e i r most  no i n t e g r a l  the  surface  or a type latter  breaks  m o t i o n a random  surfaces  are  p r o p e r t y as  o f the  manifold could  general  of  case,  up i n t o  long  period  there a  is  tortuous  appearance.  non-intersecting. it  be  increases  their  m o t i o n and t h e  This  useful-  conditions  stability. is  an i n h e r e n t  of the  a space  concept  possessing  l i m i t a t i o n to i n that  it  is  the  practical  difficult  more t h a n t h r e e  is  to  con-  dimensions.  216 Hence s y s t e m s  r e q u i r i n g more t h a n t h r e e  d e s c r i p t i o n are useful  t o o complex f o r  information.  variables, general  or l e s s ,  Where t h e  and p a r t i c u l a r l y  i n the  presented here..  It  which  limiting  is  of the  capable  invariant  s t u d y w o u l d be t o  introduced in Section to  reduction in  considerably.  of p r e d i c t i n g the  2.4 and t o  of the  the  as  s i z e and shape  One p o s s i b l e way o f the t r a n s f o r m a t i o n  concerning the  but v e r y l i t t l e  results  with a  capture  non-linearity of the  considerable  damped s a t e l l i t e  c o u l d be  other  devices.  There  is  a large  has  been made i n  of the  satellite  The p r o b l e m o f t h e  equations  The c o n f i g u r a in  approximat-  amount o f  considering or the  by t h e g r a v i t a t i o n a l "long"  elastic  extended  complex  small amplitude motions of these  effort  concept  effort.  compared t o  satellites.  making  determine approximations .  s t u d i e d by Zajac"'"^ may p r o v e t o be u s e f u l  ing p r a c t i c a l  obtain  rigid  The damper employed i n C h a p t e r 3 i s  and i n e f f i c i e n t  inherent  is  e x t e n s i o n o f t h e work  planar motion of a  investigate  computational  The a n a l y s i s  ture  the  e x a m p l e , be u s e f u l t o  surface.  t h e mapping w h i c h y i e l d  tion  state  F u t u r e Work  would, f o r  an a p p r o x i m a t e a n a l y s i s  this  case o f  really  e q u a t i o n , t h e method  T h e r e a r e many p o s s i b i l i t i e s f o r  of the  three  their  valuable.  Recommendations f o r  satellite  for  approach to y i e l d  system i n v o l v e s  second order d i f f e r e n t i a l  extremely 6.2  this  co-ordinates  literadevices,  the  problem of gradient  satellite  also  field.  217 remains  t o be t r e a t e d .  experiment  There  i s a t l e a s t one s c i e n t i f i c  that has been proposed which would r e q u i r e very 39  long antennae.  The a n a l y s i s becomes much more d i f f i c u l t  as the parametric e x c i t a t i o n of the boom becomes s i g n i f i c a n t as a r e s u l t o f the low n a t u r a l frequency and i n t e r a c t s with the n o n - l i n e a r i t y o f the l a r g e thermal A study o f the g e n e r a l motion s a t e l l i t e would i n v o l v e a massive orbit  deflections.  of a non-spinning  amount of work.  I f the  i s e l l i p t i c a l , the' Hamiltonian v a r i e s with time and  hence the s t a t e of the system  (equations 5.18 and 5«19)  depends on the f i v e v a r i a b l e s ; the parameters  and e.  & , d; , <p', <f> , ^>'as w e l l as  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 m a n i f o l d e x i s t s , i t should be p o s s i b l e t o determine  0 , f o r example, as a f u n c t i o n of (ft ,  Ly', and cf>'' £'or a s p e c i f i e d s e t o f i n i t i a l c o n d i t i o n s . d e t e r m i n a t i o n o f such a f u n c t i o n would be extremely ing,  interest-  but i t would i n v o l v e the i n t e g r a t i o n o f the equations  of motion ing  The  f o r approximately one thousand  o r b i t s i f the r e s u l t -  accuracy i s t o be e q u i v a l e n t t o that i n Chapter  5°  The  d e t e r m i n a t i o n of the l i m i t i n g m a n i f o l d w i l l thus be a very time consuming p r o c e s s .  A comprehensive study o f the e f f e c t s  of v a r i a t i o n s i n the parameters  and e thus appears t o be  unrealistic. A c o n t r i b u t i o n could be made by attempting an a p p r o x i mate s o l u t i o n o f the equations of motion and by comparing the r e s u l t s with the numerical work. problem than that proposed  T h i s i s a more e l a b o r a t e  f o r the approximate  s o l u t i o n of  218 t h e p l a n a r m o t i o n a n d i t i s assumed t h a t t h e p l a n a r work w o u l d be c o m p l e t e d 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 a t t e m p t e d . I t . w o u l d a l s o be u s e f u l t o p e r f o r m of an a c t u a l s a t e l l i t e  a detailed simulation  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 . of a s a t e l l i t e and  such  The  design  might w e l l form the u l t i m a t e g o a l of t h i s  work  a s i m u l a t i o n w o u l d be r e q u i r e d i n a n y e n g i n e e r i n g  s t u d y a n d w o u l d be e s s e n t i a l i n d e t e r m i n i n g d e s i g n c h a n g e s on t h e p e r f o r m a n c e .  the e f f e c t s of  BIBLIOGRAPHY 1  Jensen, J . , Townsend, G., Kork, J . , and K r a f t , D., Design Guide t o O r b i t a l F l i g h t , McGraw-Hill, New York, 1962, pp.  752-753.  2  Wiggins, L y l e E., " R e l a t i v e Magnitudes of the Space Environment Torques oh a S a t e l l i t e , " AIAA J o u r n a l , V o l . 2, No. 4, A p r i l 1964, pp. 770-771.  3  G l a s s t o n e , Samuel, Sourcebook on the Space S c i e n c e s , Van Nostrand, P r i n c e t o n , 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, r e v i s e d ed., Routledge and Kegan, London, 1962, p. 114.  5  P i s c a n e , Vincent L., Pardoe, Peter P., and Hook, P. 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I 9 6 0 , pp. 124-126. :  12  Z l a t o u s o v , V..A., Okhotsimsky, D.E., Sarghev, V.A., and Torzhevsky, A.P., " I n v e s t i g a t i o n of a S a t e l l i t e O s c i l l a t i o n s i n the Plane of an E l l i p t i c O r b i t , " Proceedings of the E l e v e n t h I n t e r n a t i o n a l Congress of A p p l i e d Mechanics, G o r t l e r , Henry, ed. , S p r i n g e r - V e r l a g , B e r l i n , 1 9 6 4 , pp.. 4 3 6 - 4 3 9 .  220 13  Auelmann, R i c h a r d R., "Regions o f . L i b r a t i o n f o r a Symmetric a l S a t e l l i t e , " AIAA J o u r n a l , V o l . 1 , No. 6 , June 1 9 6 3 , pp. 1 4 4 5 - 1 4 4 6 .  14  DeBra, D., "The Large A t t i t u d e Motions and S t a b i l i t y , Due t o G r a v i t y , of, a S a t e l l i t e With Passive Damping i n an O r b i t o f A r b i t r a r y E c c e n t r i c i t y About an Oblate Body," Ph.D. d i s s e r t a t i o n , S t a n f o r d U n i v e r s i t y , June 1 9 6 2 .  15  Hartbaum, H.., Hooker, W. , L e l i a k o v , I . , and M a r g u l i e s , G. , " C o n f i g u r a t i o n S e l e c t i o n f o r Passive 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 , " Paper presented at the Symposium on Passive G r a v i t y Gradient S t a b i l i z a t i o n , Ames Research Center, M o f f e t t F i e l d , C a l i f . , May 1 0 - 1 1 , 1 9 6 5 .  16  Zajac, E.E., "Damping o f a G r a v i t a t i o n a l l y Oriented TwoBody S a t e l l i t e , " ARS J o u r n a l , V o l . 3 2 , No. 1 2 , Dec. 1 9 6 2 , pp. 1 8 7 1 - 1 8 7 5 .  17  E t k i n , B., " A t t i t u d e S t a b i l i t y of A r t i c u l a t e d G r a v i t y O r i e n t e d S a t e l l i t e s . Part I - General Theory and Motion i n O r b i t a l Plane," Report No. 8 9 , U n i v e r s i t y o f Toronto, I n s t i t u t e o f Aerophysics, Nov. 1 9 6 2 ,  18  E t k i n , B,, "Dynamics of G r a v i t y - O r i e n t e d O r b i t i n g Systems With A p p l i c a t i o n t o P a s s i v e S t a b i l i z a t i o n , " AIAA J o u r n a l V o l . 2 , No. 6 , June 1 9 6 4 , pp. 1 0 0 8 - 1 0 1 4 .  19  F l e t c h e r , H.J., Rongved, L., and Yu, E.Y., "Dynamics A n a l y s i s o f a Two-Body G r a v i t a t i o n a l l y Oriented S a t e l l i t e , " B e l l System T e c h n i c a l J o u r n a l , V o l . 4 2 , 1 9 6 3 , pp. 2 2 3 9 2266. :  20  P a u l , B,, West, J.W,, and Yu, E.Y., "A Passive G r a v i t a t i o n a l A t t i t u d e C o n t r o l System f o r S a t e l l i t e s , " B e l l System T e c h n i c a l J o u r n a l , V o l . 4 2 , 1 9 6 3 , pp. 2195-2238"  21  Hughes, P.C., "Optimized Performance of an A r t i c u l a t e d G r a v i t y Gradient S a t e l l i t e at Synchronous A l t i t u d e , " Report No. 1 1 8 , U n i v e r s i t y o f Toronto, I n s t i t u t e o f Aerospace S t u d i e s , Nov, 1 9 6 6 .  22  P a u l , B,, "Planar L i b r a t i o n s of an E x t e n s i b l e Dumbbell S a t e l l i t e , " AIAA J o u r n a l , V o l . 1 , No. 2 , Feb. 1 9 6 3 , pp. 411-418.  23  Buxton, A.C., Campbell, D.E., and Losch, K., "Rice/Wilberf o r c e G r a v i t y - G r a d i e n t Damping System," Paper presented at the Symposium on Passive G r a v i t y Gradient S t a b i l i z a t i o n , Ames Research Center, M o f f e t t F i e l d , C a l i f . , May 1 0 - 1 1 , 1965 ; 1  221 24  K a t u c k i , R.J., and Mover, R.J., "Systems A n a l y s i s and Design f o r a C l a s s of G r a v i t y Gradient S a t e l l i t e s U t i l i z i n g V i s c o u s Coupling Between the Earth's Magnetic and G r a v i t y F i e l d s , " Paper presented at the Symposium on Passive G r a v i t y Gradient S t a b i l i z a t i o n , Ames Research Center, M o f f e t t F i e l d , C a l i f . , May 1 0 - 1 1 , 1 9 6 5 .  25  Ashley, H o l t , "Observations on the Dynamic Behaviour of Large F l e x i b l e Bodies i n O r b i t , " AIAA J o u r n a l , V o l . 5 , No. 3 , Mar. 1 9 6 7 , pp. 4 6 O - 4 6 9 .  26  Dow, P . 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