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Dynamics of single and multibody earth orbiting systems Sharma, Subhash Chander 1977

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DYNAMICS OF SINGLE A%D MULTIBODY EARTH ORBITING SYSTEMS by SUBHASH CHANDER SHARMA Tech.  (Hons.), Indian I n s t i t u t e o f Technology, Kharagpur, 1970  M.S., U n i v e r s i t y o f Hawaii, Honolulu, 1972  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in  THE FACULTY OF GRADUATE STUDIES (Department o f Mechanical Engineering)  We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1977  <§)  Subhash Chander Sharma, 1977  In for  presenting  this  an advanced degree  agree  that  copying  f u l f i l m e n t of the requirements  at the University  I further  of this thesis  of British  this  that  thesis  my w r i t t e n  agree  that  f o rscholarly  permission  publication,  i n part  f o r f i n a n c i a l gain  for refer-  f o r extensive  I t i s  o r i n whole, o r the copying  s h a l l n o t be a l l o w e d  permission.  Department o f Mechanical  University  Engineering  of British  V a n c o u v e r , V 6 T 1W5,  Canada  Columbia  I  p u r p o s e s may b e g r a n t e d b y  SUBHASH CHANDER  The  Columbia,  H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s .  understood of  i n partial  t h e l i b r a r y s h a l l make i t f r e e l y a v a i l a b l e  ence and study.  the  thesis  SHARMA  without  i  . ...dzdlzatzd to H-im, ^oh. He -is the. mai>tnt ofa all tkt cfitatZon. and fitghtzou&m&A  ABSTRACT  The multibody  t h e s i s aims a t s t u d y i n g  systems w i t h  t h e dynamics o f s i n g l e and  a variety of spacecraft  cations  i n c l u d i n g c o n f i g u r a t i o n c o n t r o l f o r an  payload  deployed  Station  (SSPS),  etc.  from  a spacecraft,  a Space S h u t t l e  The p r o b l e m  the Solar  supported  i s approached  oriented  appli-  instrumentation Satellite  tethered  Power  payload,  i n an i n c r e a s i n g o r d e r  of  complexity. In tribution ration and  are considered.  variation  equations  o f motion  are analyzed  and resonance  integration.  characteristics  provided  over  a range of t h e o r b i t a l initial  onstrated  considerable  disturbances.  through Next,  moment a r e d e r i v e d  gravity  oriented  into  Poincare-  of the solutions The c l o s e d - f o r m  estab-  character  periodic soluFurthermore,  the system  eccentricity,  inertia  behaviour parameter  A p p l i c a t i o n o f t h e a n a l y s i s i s dem-  equations  Test  Satellite  of librational  motion,  f o r an a r b i t r a r i l y - s h a p e d , r i g i d  closed-form  configu-  Butenin's  o f the system.  the Gravity Gradient  general  approximate  insight  dis-  nonautonomous  using  useful i n identifying  they  and  nonlinear,  approach i n conjunction w i t h the  through numerical  the s o l u t i o n s proved  tions  and  gravity oriented, rigid  The g o v e r n i n g  o f parameter  dynamics and f o r c e  expansion method, and t h e v a l i d i t y  lished of  librational  f o ran axisymmetric,  coupled  type  the beginning  solutions obtained  systems using  the Poincare-type  (GGTS). f o r c e and  spacecraft  f o r s p i n n i n g and analysis.  The  approach y i e l d s disturbances  useful information  concerning  as a f f e c t e d by t h e system p a r a m e t e r s .  i s .applied t o several configurations: package deployed  from  Finally, multibody  system,  connecting The  Also  link  of motion  a general  i n space  applied  in a circular  results  coupled,  are linearized f o r forces  orbit,  XX, an  formulation with  instrument  fora  an e l a s t i c  nonlinear,  triaxial inter-  nonautonomous  exact  This  analytical  t h e emphasis  only  to appreciate  characteristics  of the system.  analysis  prove  range of e x i s t i n g  useful i n studying  and f u t u r e  spacecraft.  presented.  t o o r i e n t an  interest. a general  solution.  significant  The g e n e r a l  equa-  procedure i s  i s on e v o l v i n g  o f t h e problem and i t s a c c e p t a b l e are presented  solution  a n d moments r e q u i r e d  to several configurations of practical  should  The method  and t h e SSPS.  and t h e i r  are obtained.  Throughout, mulation  dynamical  to external  i n t h e f o r m o f a t e t h e r o r a beam i s d e v e l o p e d .  expressions  object  Explorer  t h e Space S h u t t l e  highly complicated  tions  response  character  for-  Numerical response of the  the dynamics o f a wide  iv  TABLE  OF  CONTENTS  Chapter  1.  2.  Page  INTRODUCTION  1  1.1  Preliminary  Remarks  1.2  L i t e r a t u r e Review  1.3  Purpose  3  and Scope o f t h e I n v e s t i g a t i o n  A T T I T U D E AND FORCE A N A L Y S E S FOR GRAVITY ORIENTED S A T E L L I T E S  RIGID  . . . .  12  AXISYMMETRIC .  15  2.1  Preliminary  2.2  Formulation of the Problem  16  2.3  Approximate  20.  2.3.1  Remarks  15  Analytical Solutions  Butenin's v a r i a t i o n of method  3.  1  parameters  . . .  20  2.3.2  Poincare-type expansion  2.3.3  Accuracy  method  of the approximate  Solutions  23  solutions  .  and S t a b i l i t y  24  2.4  Periodic  24  2.5  Resonance  29  2.6  Force  31  2.7  L i b r a t i o n a l Response and F o r c e  2.8  Concluding  Distribution  A T T I T U D E AND  Distribution . .  Remarks  35 42  FORCE A N A L Y S E S FOR  RIGID TRIAXIAL  SYSTEMS  43  3.1  Preliminary  3.2  Formulation of the Problem  46  3.3  Motion  i n the Small  49  3.3.1  S p i n n i n g systems  50  3.3.1a  Remarks  Approximate  43  analytical  solution  . . .  50  Chapter  Page 3.3.1b 3.3.2  Stability  56  G r a v i t y o r i e n t e d systems  .  58  3.3.2a  Approximate a n a l y t i c a l s o l u t i o n . .  58  3.3.2b  Stability  61  3.4  Force and Moment A n a l y s i s  3.5  L i b r a t i o n a l Response and  63 Time H i s t o r y  of  the Forces 3.6  70  S a t e l l i t e S o l a r Power S t a t i o n 3.6.1  Configuration  (SSPS)  . . . .  82  c o n t r o l f o r maximum  power generation  82  3.6.2  S t a b i l i t y of the SSPS  3.6.3  Force d i s t r i b u t i o n and c o n t r o l moments O r b i t a l p e r t u r b a t i o n s due to the  3.6.4  . . . . . . . .  s o l a r r a d i a t i o n pressure 3.7 4.  SPINNING AND  85 .  Concluding Remarks  DYNAMICS OF  84  85 90  GRAVITY ORIENTED  MULTIBODY SYSTEMS  92  4.1  Preliminary  4.2  Formulation of the Problem  93  4.3  Motion i n the Small  97  4.3.1  99  4.3.2 4.3.3  4.4 4.5  Remarks  92  S o l u t i o n f o r the t e t h e r case  S o l u t i o n f o r the beam case S o l u t i o n f o r l i b r a t i o n s of end-bodies Force and Moment A n a l y s i s R e s u l t s and D i s c u s s i o n  101 the  105 107 108  vi  Chapter  Page 4.6  5.  C o n c l u d i n g Remarks  12 6  C L O S I N G COMMENTS  BIBLIOGRAPHY APPENDIX I  .  130  - T R A N S F O R M A T I O N M A T R I C E S USED I N E Q U A T I O N S  (4.1) APPENDIX I I  12 8  AND (4.15)  - E Q U A T I O N S OF MOTION FOR THE S Y S T E M OF C A B L E OR BEAM CONNECTED S Y M M E T R I C E N D - B O D I E S . .  135 136  vii  LIST  OF  TABLES  Table 3.1 4.1  Page V a l u e s o f system parameters i n the examples of the t r i a x i a l s p a c e c r a f t under c o n s i d e r a t i o n V a l u e s of system parameters i n the example of t h e Space S h u t t l e s u p p o r t i n g a p a y l o a d (M = 2 0 0 k g ) b y a c a b l e o r a b e a m n  . .  73  109  viii  LIST OF FIGURES Figure 1-1 1-2 1-3 1-4  1-5  1- 6 2- 1 2-2  2-3 2-4 2-5  2-6 2-7  Page A s c h e m a t i c diagram o f the RAE s a t e l l i t e w i t h l o n g f l e x i b l e antennas  2  T e t h e r e d O r b i t i n g I n t e r f e r o m e t e r (TOI) e x p e r i m e n t proposed by the A p p l i e d P h y s i c s L a b o r a t o r y . . . .  4  Canada/U.S.A. Communications Technology (CTS) . Note the l a r g e s o l a r p a n e l s  5  Satellite  A s c h e m a t i c diagram o f the S a t e l l i t e S o l a r Power S t a t i o n (SSPS) i n the geosynchronous e q u a t o r i a l orbit  6  A s i m p l e model f o r m u l t i b o d y systems r e p r e s e n t i n g two r i g i d b o d i e s connected by a r i g i d or a f l e x i b l e l i n k i n the form o f a beam o r a t e t h e r . .  7  A s c h e m a t i c diagram o f the proposed p l a n o f study  14  A s c h e m a t i c diagram of the G r a v i t y G r a d i e n t T e s t S a t e l l i t e (GGTS)  17  Reference c o o r d i n a t e systems and geometry o f the s a t e l l i t e motion w i t h ty, ty and A r e p r e s e n t i n g the l i b r a t i o n a l degrees of freedom  18  A comparison between n u m e r i c a l and B u t e n i n ' s solutions  25  A t y p i c a l example comparing n u m e r i c a l , B u t e n i n ' s and l i n e a r s o l u t i o n s  26  Typical i n i t i a l conditions resulting i n periodic l i b r a t i o n a l m o t i o n . Note t h a t over a wide range of i n i t i a l c o n d i t i o n s and m o d i f i e d i n e r t i a parameter the p e r i o d i c s o l u t i o n s a r e u n s t a b l e . . .  30  Dumbbell-type s a t e l l i t e : (a) l i b r a t i o n a l a n g l e s ty, ty; (b) boom f o r c e s . . .  32  T y p i c a l time h i s t o r i e s o f l i b r a t i o n s and f o r c e components a t the mass : (a) e = 0, K = 0.75, ty = ty = 0, ty' = ty' = 0.5 i  Q  Q  Q  Q  .  36  ix  Figure  Page  (b)  increase  i n the  orbital  eccentricity  (c) r e d u c t i o n i n t h e m o d i f i e d parameter . . . (d) 2- 8  3- 1 3^2  3-3  3-4 3-5  3-6  3-7 3-8  3-9  3-10  d i s p l a c e m e n t and  . . . .  37  inertia 38  impulsive type d i s t u r b a n c e  .  39  A t y p i c a l example o f t h e G r a v i t y G r a d i e n t T e s t S a t e l l i t e (GGTS) s h o w i n g i t s l i b r a t i o n a l and force histories  41  Geometry o f t h e I n t e r n a t i o n a l S a t e l l i t e ( E x p l o r e r XX)  45  Ionospheric  R e f e r e n c e c o o r d i n a t e s y s t e m s and t h e s a t e l l i t e l i b r a t i o n a l m o t i o n w i t h a, degrees o f freedom  geometry 3 and y  A c o m p a r i s o n between t h e a p p r o x i m a t e b a t i o n and n u m e r i c a l s o l u t i o n s f o r a spinning s a t e l l i t e  perturtriaxial  S t a b i l i t y diagram f o r spinning satellites in circular orbits  of 47  55  axisymmetric 57  A c o m p a r i s o n between t h e a p p r o x i m a t e b a t i o n and n u m e r i c a l s o l u t i o n s f o r a gravity oriented s a t e l l i t e  perturtriaxial 62  S t a b i l i t y diagram f o r t r i a x i a l 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 i n a r b i t r a r y o r b i t s of small e c c e n t r i c i t i e s (e < 0.2)  64  F o r c e e q u i l i b r i u m o f a mass e l e m e n t Am maintain i t at a desired p o s i t i o n  65  T y p i c a l p l o t s showing the spinning s a t e l l i t e i n the external disturbance  to  c o n d i t i o n of a a b s e n c e o f any 71  Response o f a s p i n n i n g s a t e l l i t e t o an a r b i t r a r y disturbance. Note t h e h i g h f r e q u e n c y a m p l i t u d e m o d u l a t i o n s o f l i b r a t i o n s ( r o l l , yaw) and t r a n s v e r s e f o r c e components E f f e c t of response: (a) (b)  s y s t e m p a r a m e t e r s on  the  72  satellite  i n c r e a s e i n the o r b i t a l e c c e n t r i c i t y r e d u c t i o n i n t h e i n e r t i a asymmetry parameter K  . . . .  74 75  X  Figure  3-11  3-12  3-13 3-14  3-15  Page (c) r e d u c t i o n i n the i n e r t i a parameter I (d) i n c r e a s e i n the s p i n parameter a (e) d i f f e r e n t i n i t i a l c o n d i t i o n s .  76 77 78  Force components on an i n s t r u m e n t a t i o n package deployed from a n o n l i b r a t i n g t r i a x i a l g r a v i t y o r i e n t e d s a t e l l i t e (Space S h u t t l e )  80  E f f e c t of i n i t i a l c o n d i t i o n s and system parameters on l i b r a t i o n a l motion and f o r c e components f o r the Space S h u t t l e supported i n s t r u m e n t a t i o n package  81  C o n f i g u r a t i o n c o n t r o l of the SSPS f o r the maximum power generation  83  S t a b i l i t y diagram f o r a t r i a x i a l s p a c e c r a f t i n the geosynchronous e q u a t o r i a l o r b i t with the s o l a r panels f a c i n g the sun. Note t h a t the c o n f i g u r a t i o n (I = K = 0.15) corresponding to the SSPS l i e s i n the unstable r e g i o n  85  D i s t r i b u t i o n of f o r c e per u n i t mass on s o l a r panel (y = 100 m): . (a) z = 0, 2 km;  3-16  3- 17 4- 1 4-2 4-3  4-5  (b) x = 3 km  87  Rotations and c o n t r o l moments as f u n c t i o n s of the s o l a r aspect angle f o r maximum power generation  88  Time h i s t o r y of the c o n t r o l moments f o r s e v e r a l values of the s o l a r aspect angle  89  Reference c o o r d i n a t e s and the geometry of motion for a multibody e a r t h o r b i t i n g system  94  Force e q u i l i b r i a of the mass elements Am. to maintain them at d e s i r e d p o s i t i o n s  95  Modal r e p r e s e n t a t i o n (a) f i r s t and (b) t h i r d and  4-4  the  and  Am  f o r cable v i b r a t i o n s :  second modes f o u r t h modes  I n i t i a l displacement as represented four modes  110 Ill by the  first  Response of the system when the t e t h e r i s d i s turbed at the c e n t e r :  112  xi  Figure  Page (a) time h i s t o r y o f the t r a n s v e r s e displacement of the t e t h e r , U component (b) time h i s t o r y of the t r a n s v e r s e displacement of the t e t h e r , W component  4-6  4-7 4-8  4-9 4-10  113 114  Time h i s t o r y o f l i b r a t i o n s and f o r c e s a c t i n g at the center of mass of the o r b i t e r a f t e r the c a b l e i s d i s t u r b e d a t the c e n t e r  116  Time h i s t o r y of l i b r a t i o n s and f o r c e s f o r a set o f i m p u l s i v e i n i t i a l c o n d i t i o n s  117  Time h i s t o r y o f the s a t e l l i t e l i b r a t i o n s as a f f e c t e d by two d i f f e r e n t v a l u e s o f s p i n parameter  118  U and W v i b r a t i o n s f o r d i f f e r e n t 0 a f t e r the c a b l e encounters a m i c r o m e t e o r i t e impact  119  C o n s t i t u e n t mode shapes o f U and W v i b r a t i o n s e x c i t e d due t o a m i c r o m e t e o r i t e impact  120  4-11  Modal r e p r e s e n t a t i o n f o r beam v i b r a t i o n s  122  4-12  U and W v i b r a t i o n s f o r d i f f e r e n t 0 when the beam encounters a micrometeorite impact Time h i s t o r y o f l i b r a t i o n s and f o r c e s a t the c e n t e r o f mass of the o r b i t e r due t o the micrometeorite impact on the beam  4-13  4-14  L i b r a t i o n a l h i s t o r y of a s p h e r i c a l s a t e l l i t e due t o : (a) a micrometeorite impact on the cable; (b) a micrometeorite impact on the beam . .  123 12 4  12 6  xii  ACKNOWLEDGEMENT  The V.J.  Modi  author wishes  this The  Council  his gratitude  f o r the help given during this  acknowledgement typing  t o express  i salso  due t o Anjna  research.  (the author's  to Professor The wife) f o r  thesis. investigation  o f Canada,  Grant  was s u p p o r t e d  No. A-2181.  by t h e N a t i o n a l  Research  x i i i  LIST OF SYMBOLS  a, b  • , a, u  l  b  w  . i  c u i ' cwi a.,b.,b.  1 1 1 , 3  a..  .  1 J  a.  2  amplitudes o f v i b r a t i o n s o f beam i n e q u a t i o n (4.12) amplitudes o f v i b r a t i o n s of c a b l e i n e q u a t i o n (4.10)  .  amplitudes o f 6 , y i n equations (3.7) and (3.13)  amplitudes o f 3-, y• (i=l,2) i n equations and (4.14c)  (4.14b)  amplitudes o f and <p i n equations (2.7b) , r e s p e c t i v e l y  and  1  1  (2.7a)  c. m.,S  s a t e l l i t e c e n t e r o f mass  d, . .  elements o f square matrix [D, ] expressed i n equations (4.12e) and (4.12fT  3  d .. C 1 : J  e e* bi  elements o f square matrix [D ] expressed i n equations (4.10e)and (4.10f) c  e c c e n t r i c i t y o f the o r b i t (£, + L c o s n, . ) r, . T, • — + -oT^-lt.£n, . £k, . 1 bi bi  + £~cosh k, .) 2 bi  "^bi bi + £„ ( s i n n, . + i s i n h k,.)/£ 2 n, . bi k, . bi bi bi s  v  2 + ( s i n n, . + q, .cos n, . - q, . ) /n, . bi ^bi bi ^bi bi 2 + (s, . - s, . cosh k, . - r, . s i n h k, . )/k, . ; i = l , 2 , . bi bi bi bi bi bi sin n . e e  * c i  2 n  2  . ci  £ •+ £„cos n . i_ _ i I 21 . In  .  ci  i  =  i  2  , 1 1 , ^ , . .  xiv  £, + L c o s 1 2 £n,. bi  n. . b i  '*  c o s n, . - 1 b i 2 n, . bi  n  ;  £ s i n h k, . 2 bi_ £k,. bi  1 - cosh  0  n r  2 b i  % i  "  U  +  X  3  £ 2  3  ;  1  ;  1  1  2 ~  1  £  1  1  ._, ~ ' '' * 1  ,-1  V  .  £  forcing functions (i=l,2) used (2.9) a n d ( 3 . 5 ) , r e s p e c t i v e l y a c c e l e r a t i o n s due t o g r a v i t y (i=l,2), respectively e  . ( e , . + q, . e . + r , . e 5i l i ^ b i2 i b i 3i  c  0  function (4.10h) angular  v  o f £,, £ ,  £, n  9  momentum  0  +  4  n  stiffness  b i  k  s  )  i n equations  a t S, Am a n d  + s, • e . . ) b i 4i'  ., T , c i u n  (2.6),  Am.  ; i = l , 2, . . ' '  e i n equation  p e r u n i t mass o f t h e s a t e l l i t e  angle between t h e e q u a t o r i a l e c l i p t i c (23.5 )  [ { ( T  X  . _, ~ ' '' •  k, . bi_  .2 k, . bi 2  _  . , „ ' '• *  ._, _ ' ' ' '  1  + £„cosh k, . s i n h k, . 2 bi_ _ bi_ £k^. ,2 bi k, . bi  1  1  bi  £ s i n n, . 2 ba £n, . bi £,  s i n n. . b i 2 n, .  V  2  +  parameter  T  0  }  /  (  2  k  s  )  f o r beam,  length of the connecting a l s o u s e d as an i n d e x  link  length  link  of the connecting  plane  and t h e  '"  i = 1  (EI)^/p£  > >-2  4*2 8  (boom, c a b l e o r beam); •  i n Figures  2-6 a n d 4-1  p o i n t m a s s e s shown i n F i g u r e mass o f t h e  2-6  satellite  reduced masses o f t h e system expressed i n e q u a t i o n s (2.28b)  -  2  _  I  « ± l  (  )  (  '1 - 2 e '  1  L  +  .  3K  4 (a + 2e)  2'  l l  m  2 - I - K 1 - K I  + K - 2  2 - I (a i  ~  m  i  + 1)  ;  i H  '  i=l,2 i  =  1  /  2  a r b i t r a r y mass e l e m e n t s o f t h e s a t e l l i t e ( o r c o n n e c t i n g l i n k ) and t h e end-body, r e s p e c t i v e l y [{(Tp  + 4n^  V(Y,0) M  i*"*  1  k  ±  s  )  V  dY  b i  2  - T }/(2k ) ] Q  ; v = u,w  1  /  ;  2  s  ; V = U,W  r o o t .of t h e c h a r a c t e r i s t i c  ;  i=l,2,. i=l,2,..  equation  (4.10g)  1  r V ( Y , 0 ) M ^ dY  ; v = u,w  ; V = U,W  ;  i=l,2,  0 1 3V  r  (-T5-L-J J  0  \^  90 10=0' " b i  1  ,3V  . .) M 3  =0  dY ; v = u,w  . dY  ci  ; v = u,w  ; V = U,W  ; i=l,2,  ; V = U,W  ;  i=l,2,  xvi  (3K )  n.  1/2  i  (3K  ±  + 1)  1/2  {i(£_L_L_ + 3) a + ) - A +  3 K ,1/2  K  n 11  U l  l  - 2e  +  3 )  ( 1  2>  +  4  n 12  fT(  n 21  {4 ( I - 1 ) / ( 1 - K) }  g  + 1 ) , K _ ) ( 1 + 2") " 1  4  +  ( a + 2e) 3K  }  ,1/2  + ( a + 2e) •  n  }  4  2 e  1 / 2  ( I + K - 1) 1/2  n 22  1/2  ni l l  {I (a i  i  + 4) - 4}  n. , « il2  {I (a  i  + 1) - 1} 1/2  P  frequency (3.14a)  i  beam,  of the i 2 1/2 (1 + n ^ )  frequency  bwi  ;  i=l,2  t  (3.8a) and  mode o f U v i b r a t i o n  h  f o r the  ; i=l,2,.. mode o f W v i b r a t i o n  f o r the  th frequency o f the i mode o f U v i b r a t i o n cable, ( 1 + T n .) / ; i=l,2,.. ' 0 c i  f o r the  beam, n  cui  i=l,2  of l i b r a t i o n , equations  frequency  bui  ;  of the i*"*  1  ; i=l,2,..  f a i  2  1  2  n  t h CWl  frequency cable,  P ,P X  2  Pil'Pi2  (T  of the i Q  n \ )  1  /  2  mode o f W v i b r a t i o n ;  i=I,2,..  characteristic (3.14a)  roots  of equations  characteristic  roots  of equation  frequency  o f a motion,  f o r the  (-3K/I)  1/2  (3.8a) a n d  (4.14f)  xvii  P^rPy  tP p,Pq ]  frequency of 3,  )  y,  \p  and tt motions,  Pg  frequency o f o r b i t a l motion, w  p.  momentum conjugate to g e n e r a l i z e d  respectively  coordinate X  A q  bi  " bi " S  £ e  5i l £  ( n  bi 3i e  "  k  bi li  + ee,. 5 i (e-.S, 3i " k, b i. e 2, .iA .r ) ] :  ( e  £  k  S  )  [  k  £  • i=l,2,.  1  £  e  bi  th i r  bi  [  generalized  - bi n  - e  +  4 i  e e  5i " {  coordinate  l i  e  +  )}][k . + ee b  n  5 i  ; i=l,2,  bi 2iV e  £  +  s  ( 2i e  b i  (e . - k . e ^ / A ) ]3  1  fa  ; i 1, 2 , .  #  r.^  distance  between  and an a r b i t r a r y mass element  on the i ^ end-body t  1  ; i=l,2  r  d i s t a n c e between S and an a r b i t r a r y mass element on the s a t e l l i t e  s, . bi  [n, . s i n h k, . - k,. s i n n, . + e e . { ( l ~ bi bi bi bi 5i 2 c  + L c o s n, . ) x 1 bi y  x (n, . e-.. - k,.e,.)/£ + (e, . - n, . e~ . £, /£) s i n h k, . bi 3i bi l i ' l i b i 2 i 1' bi -  ( e . - k, . e . Jl,/Jl) s i n n, . } ] [k, . (cosh k, . 3i bi 2i i bi bi bi 0  0  - cos n, . ) + e e . { (e, . - e • ) (k, .A- + £, s i n h k, . ) /I bi 5i 4i 2i' bi 2 1 bi c  n  + e_. (cosh k, . - cos n, . ) - k, . Jl, (e~.cosh k, . 3i bi bi b i 1 2i bi - e cos n )/£}] 4 i  t; t ^ , t 2  b i  1  ; i=l,2,..  time; time a t i n s t a n t s 1 and 2, r e s p e c t i v e l y  xviii  u,w  U  vibrations  s' x' y,.. u  x.,y^,z.  12  ,  y  1 2 S  p r i n c i p a l body located a t  il' i l [ z ( i2' i2j  j r  x  coordinates ; i=l,2  of the i  h  e  m  r o t a t i n g coordinate system a t axes a l o n g l o c a l v e r t i c a l and tal, respectively  {A, }, {A } bv cv  column vectors  A.,A.  f u n c t i o n s o f e , K, I , a u s e d a n d (3.13)  . , D  A* m, n ,A** l ,. j  a  cvl'  v  v  ]  cv2'*'"  ;  equation  bv2 bv2'."; v = u,w b  S  r  e  s  e  c  t  l  v  e  l  ZQ  bv2'  i n e q u a t i o n s (3.7)  and i n i t i a l  conditions  used  (2.17)  ;  containing  elements  Pk ]_kkvl'  P c v l c v l ' P v2 cv2'--b  b  V  respectively  ;  C  square matrices containing d ^_.'s, r e s p e c t i v e l y  elements  c  (EI)^  f l e x u r a l s t i f f n e s s o f t h e beam  F  librational  F F ) a ' b ' c'I „ „ ) ca' cb J  w i t h y^ a n d horizon-  c o n t a i n i n g e l e m e n t s a, ,, a, . • - b v l ' P Y ; v = u,w  o f e , K.i  column vectors P  [D^],[D  a  functions in  {B* ),{B* }  end-body  intermediate body c o o r d i n a t e s l o c a t e d a t S d u r i n g ° d i f i e d E u l e r i a n r o t a t i o n s i>, cj>, A a n d y , 3, a, r e s p e c t i v e l y  t  XQ,yQ,ZQ  F  axis,  p r i n c i p a l body c o o r d i n a t e s o f t h e s a t e l l i t e with o r i g i n a t S and having y a x i s along t h e l i n e S  ,  respectively  unit vectors i n the direction of the sun, x y axis,..., respectively  u  x,y,z  11  i n x and z d i r e c t i o n s ,  d, . . s a n d -' 1  1  force  s  f o r c e s a c t i n g o n m ,m^,m , e t c . i n e q u a t i o n s (2.28a) a  and expressed  c o m p o n e n t s o f F i n x , y , z , E,, ty a n d tt respectively  directions,  f o r c e n e c e s s a r y t o keep its position  a m a s s e l e m e n t Am i n  f o r c e n e c e s s a r y t o keep its position  a mass e l e m e n t  the f o r c e i n e q u a t i o n of the end-bodies inertia  (4.18)  f o r small  parameter of s a t e l l i t e , I  p r i n c i p a l moments o f i n e r t i a x, y, z a x e s , r e s p e c t i v e l y values ^i'  z  i  of principal a  x  e  s  end-body, M L X  2  a n (  ^  inertia  2  2  + ^U  I^/I  d i s t a n c e between end-bodies  the centers  distance  S and  Lagrangian,  about x., t h ^~  i=l,2  parameter, 1 -  between  about  - 3£ £ )  2  inertia  of satellite  parameter f o r the i  asymmetry parameter, 1 modified  librations  / I  moments o f i n e r t i a  respectively;  + M L  x  Am. i n  ;  1/1  o f mass o f t h e  i=l,2  T - P  moments d e f i n e d respectively  i n equations  (3.18)  and  i ^  mode o f t h e b e a m  vibrations,  sin  n, .Y + q, . c o s n, . Y + r , . s i n h k, . Y bi ^ b i b i b i b i  + s, . c o s h k,.Y + eh, . (Y bi b i b i  1  I,/I)  (4.17),  XX  M ^  i  t h  sin M  ,M  ,M  n  CI  .Y + e h . { L ( c o s C1  M*  M M /(M T'2' l  M. l  mass o f t h e i  {N, }, {N } bv cv  column v e c t o r s  n  0  +  V i i n  n  J.  -Y ox  +  Y}  directions,  M ) "2' 0  th  n  ,, n cvl' cv2' respectively  N  vibrations,  components o f M i n x, y, z respectively  ^  x  mode o f t h e c a b l e  end-body  ; i=l,2  c o n t a i n i n g e l e m e n t s n, , , n, ^ b v l ' bv2 f o r t h e beam a n d t h e c a b l e , ' ; v = u,w  i n i t i a l displacement condition e x p r e s s e d i n F i g u r e 4-4  C  for the cable  N^,N^  i n i t i a l impulse c o n d i t i o n s a p p l i e d t o the cable ' and t o t h e beam a s e x p r e s s e d i n F i g u r e s 4-9 and 4-12, r e s p e c t i v e l y  {N/ },{N* } bv cv  column v e c t o r s ncvl 1  0  nc v 2  center  P,P ,P  of  1  c o n t a i n i n g e l e m e n t s n ' ,, n ' ^ b v l bv2 r e s p e c t i v e l y ; v = u,w ' r  attraction  t o t a l , e l a s t i c and g r a v i t a t i o n a l energies, respectively  g  R R R i ' a' b ' l „ )  potential  s  n  is. K T  m' Rp  .  mi  d i s t a n c e s from 0 t o S Am., r e s p e c t i v e l y  ( o r m ) , m^, c  m^,  Am a n d  1  J radius  at  perigee t h  center T T  e  TQ  ^  as  kinetic tension  o f mass o f t h e i energy i n the connecting ^  tension parameter,  3M*L/p&  end-body  link 2  ; i=l,2  xxi  n o n d i m e n s i o n a l i z e d v a l u e s o f u, w and a s u/£, w/£ a n d y / £ , respectively inertial  frame of  modified  E u l e r i a n r o t a t i o n s of  satellite (Figure with  3-2)  and  y  link  to  t o as are  motions,  respect  and  respect  referred a  with  reference  of the  S - j ^  yaw  and  a l s o i n - p l a n e and  respectively,  (Figure  the  triaxial  , ZQ  system  end-body  line  of  the  given  0  t o XQ ,  the  pitch,  at  y  (i=l,2)  (Figure roll,  4-1)  respectively;  out-of-plane connecting  4-1)  infinitesimal  variation  of  ;  i=l,2,.. y  phase a n g l e s u s e d i n e x p r e s s i o n s f o r 3 and e q u a t i o n s (3.7) and (3.13) ; i = l , 2 phase angles (i=l,2)  and  used  i n expressions  expressed  phase angles used equations (2.7) flexibility  i n equation  i n expressions  parameter,  p£  for  and  (4.14e)  ;  +  i  characteristic  true  6.  y^ j=l,2  f o r IJJ a n d ty i n  /I*  r a t i o o f t h e f o r c e due t o s o l a r r a d i a t i o n t o t h e f o r c e o f g r a v i t a t i o n on a s a t e l l i t e n.9  in  pressure  ; i=l,2 root of  equation  (4.12g)  anomaly  r o t a t i o n about a x i s o f symmetry o f the axisymmetric gravity oriented satellite (Figure 2-2) functions equations  of s a t e l l i t e parameters (3.8b) and (3.14b)  expressed  in  xxii  th of the i end-body i n e q u a t i o n (4.14g)  functions expressed  gravitational  (i=l,2)  parameters  y  earth's  E,  reference  p  mass p e r u n i t l e n g t h o f t h e c o n n e c t i n g ( b o o m , c a b l e o r beam)  length  constant  shown i n F i g u r e  (2-6b)  t h s p i n ' p a r a m e t e r s o f t h e s a t e l l i t e and o f t h e i end-body ( i = l , 2 ) , d e f i n e d as a'(0) and a|(0) w i t h a(0)=0 and a^(0)=0, r e s p e c t i v e l y  O,CK  a*  c h a r a c t e r i s t i c roots (real, imaginary c o n j u g a t e ) o f e q u a t i o n s (2.21)  x  period  ty,  link  ty  of orbital  o r complex  motion  r o t a t i o n s a c r o s s and i n t h e o r b i t a l p l a n e , respectively, of the axisymmetric gravity oriented s a t e l l i t e (Figure 2-2)  typ • ,ty-p•  p e r i o d i c s o l u t i o n s o f ty a n d ty, r e s p e c t i v e l y , w i t h p e r i o d 2TTJ ; j = l , 2 , . .  ty , V v  small  Y  ty  perturbations  1.  n  around  respectively  tj)  Dots and primes respectively.  indicate  orbital  the variables  '0' a n d  i n the text  Y  angular  rate,  with  are nondimensional  (with  Chapters  2 a n d 4, r e s p e c t i v e l y ) ,  2TT/T  respect  'o' i n d i c a t e  and t h e f i g u r e s ,  diagrams  J  a n g l e between t h e sun and t h e  differentiation  Subscripts  ty_ . a n d d>_ . , P, i P,i ' J  solar aspect angle, l i n e o f nodes average  of  •  0,  conditions  respectively.  Force  . 2 * 2 £w a n d M £co i n a otherwise indicated.  respect unless  initial  t o t and  to  m  1  1.  1.1  Preliminary The  dimension  Remarks  l a s t two  to  spacecraft.  INTRODUCTION  decades have seen the a d d i t i o n of a  man's c a p a b i l i t y f o r space e x p l o r a t i o n , In t h i s short time, the use  l i t e s in majority  important.  of s p a c e c r a f t  the  of a r t i f i c i a l  satel-  of the telecommunications, weather s t u d i e s ,  m i l i t a r y s u r v e i l l a n c e and ceedingly  new  g e o l o g i c a l surveys has  Ahead l i e new  and  become  ex-  exciting possibilities  a p p l i c a t i o n s i n e x p l o i t i n g t e r r e s t r i a l and  t e r r e s t r i a l resources,  e.g.,  the abundant and  extra-  pollutant-free  s o l a r energy. Exacting  demands on s a t e l l i t e a t t i t u d e w i t h  to  a  f i x e d or an o r b i t i n g frame of  in  a  v a s t body of l i t e r a t u r e  reference  constituting  has  respect  resulted  the f a m i l i a r f i e l d  1 2 of l i b r a t i o n a l dynamics ' . opment of newer  technology  As and  ated w i t h the present estimate t h i s respect  usually  of the  necessarily f l e x i b l e spacecraft. of such l a r g e  Astronomy E x p l o r e r  (RAE)  s a t e l l i t e used 750 3  frequency  extraterrestrial ratory  (APL)  In  signals  radio  c o n s i s t i n g of two  ft  (Figure 1-1).  sources, the A p p l i e d  of the Johns Hopkins  spacecraft  We  University  trend  have  configurations.  g r a v i t a t i o n a l l y s t a b i l i z e d Tethered O r b i t i n g (TOI)  associ-  f u t u r e demands.  a few  low  devel-  knowledge i s u s u a l l y  witnessed  detect  the case,  a common consensus c l e a r l y suggests a strong  towards l a r g e r and already  is  antennas  Radio to  For i d e n t i f y i n g Physics has  Labo-  proposed a  Interferometer  connected by a t e t h e r  2-6  km  Antenna boom (750ft long tube,l/2in dia.)  orbit  t  To Earth  Figure  1-1  A s c h e m a t i c d i a g r a m o f t h e RAE with long f l e x i b l e antennas  satellite  3  long  (Figure  Technology  1-2).  Satellite  s o l a r p a n e l s , 1.14 However, our space the  More r e c e n t l y ,  but  of modular  x  future  stations  earth  m  (CTS)  launched  7.32  m  efforts  which  each,  i n January  be  1976  to generate  a r e aimed  cannot  h a v e t o be  t h e Canada/U.S.A.  1.2  launched  The  carries kW  in their  concept  of  of  1-3).  gigantic  entirety  through  Solar  two  (Figure  at construction  c o n s t r u c t e d i n space  subassemblies.  Communications  from  integration  Satellite  Power 5  Station  (SSPS)  (Figure  1-4)  i n the g e o s t a t i o n a r y o r b i t  and  the  futuristic  as p r o p o s e d  designs of  space  by  Glaser  colonies  at  the  6 7 earth-moon belong  to this  Shuttle which  libration  points  category.  s e r v e as  the r o l e  o f an A  of  orbiting  complex system  generated  can  thus  for specific  scientific  i s limitless,  fundamental  of configurations  level:  through The  they rigid  thesis  such  Literature  they  do  deployed  to  play  offer  flexible  aims a t l a y i n g  links  example,  of  be  package A l -  encountered  similarity  (Figure  a merely  experiments.  t h a t may  a degree  of  instrument  a l l represent multibody or  and  consist  an  precisely  future  decade  vehicle.  d e p l o y i n g an  the v a r i e t y  Space  this  expected  type v e h i c l e  though  of  so  of  onboard  i t i s also  et a l . '  i s the  t h e end  the Skylab  positioned  connected  launch  future  for testing  Furthermore,  O'Neill  i n t e r c o n n e c t e d s u b a s s e m b l i e s o r , as  represent  1.2  f a r i n the  a laboratory  instrument packages.  study  Not  f o r w a r d by  scheduled f o r o p e r a t i o n towards  will  series  as p u t  systems  at  in a  inter-  1-5).  a foundation f o r the  dynamical  systems. Review  Librational  dynamics  and  control  of gravity  oriented  as  Figure  1-2  Tethered Orbiting Interferometer (TOI) e x p e r i m e n t p r o p o s e d by t h e A p p l i e d Physics Laboratory  local vertical  I (1.14m x 7-32m)  orbit normal To Earth  Canada/U.S.A. Communications T e c h n o l o g y S a t e l l i t e Note t h e l a r g e s o l a r panels.  (CTS)  Solar c o l l e c t o r  (mass = 4 k t o n )  Receiving antenna Solar flux  Microwave beam  5000 Mw  Transmitting antenna (dia.=ikm,mass = 2.4k ton)  Synchronous orbit 0.2 km  Figure  1-4  A schematic diagram of the S a t e l l i t e (SSPS) i n t h e g e o s y n c h r o n o u s o r b i t  Solar  Power  Station  (Tl  Instrument package having specified location and orientation  Rigid or flexible link, beam or tether  --..Trajectory  Triaxial space station  Figure  1-5  A simple model f o r multibody systems r e p r e s e n t i n g two r i g i d b o d i e s c o n n e c t e d by a r i g i d o r a f l e x i b l e l i n k i n t h e f o r m o f a beam o r a t e t h e  8  w e l l as s p i n n i n g study f o r q u i t e  s a t e l l i t e s has some time.  As  been a s u b j e c t  of  considerable  i n d i c a t e d b e f o r e , the v a s t body  of l i t e r a t u r e accumulated over years has  been reviewed by  several  1 2 authors i n c l u d i n g Modi et a l . ' aspects of the  Hence  only more s i g n i f i c a n t  l i t e r a t u r e d i r e c t l y relevant  to the present study  are b r i e f l y reviewed here. The  pioneering  work on  gravity oriented  satellites  g  was c a r r i e d out by Klemperer (1960), who o b t a i n e d the exact s o l u t i o n f o r p l a n a r l i b r a t i o n s o f a dumbbell s a t e l l i t e i n a c i r 9  c u l a r o r b i t , and  by Baker  (1960) who  determined p e r i o d i c  solu-  t i o n s of the problem f o r a small o r b i t e c c e n t r i c i t y . Beletskii"'"^ (1963) focused the a t t e n t i o n on resonance e f f e c t s f o r  satellites  i n e l l i p t i c o r b i t s w h i l e Schechter " " (1964) attempted, w i t h 1  1  l i m i t e d success, to extend Klemperer's s o l u t i o n t o o r b i t a l motion by a p e r t u r b a t i o n 13 (1964) and  Brereton and  Modi  i c a l methods, i n v o l v i n g the to analyze the  method.  non-circular 12 Zlatousov et a l .  (1967) s u c c e s s f u l l y employed numeruse  of the  stroboscopic  s t a b i l i t y of p l a n a r motion i n the  large  of a r b i t r a r y e c c e n t r i c i t y . They a l s o i n v e s t i g a t e d 14 15 sponding p e r i o d i c motion cal  '  (1969) and  e c c e n t r i c i t y f o r s t a b i l i t y the only  a p e r i o d i c one.  phase plane,  the  for orbits corre-  showed t h a t a t the  available solution i s  More r e c e n t l y , S h r i v a s t a v a " ^  (1970) s t u d i e d  coupled l i b r a t i o n a l motion of axisymmetric g r a v i t y o r i e n t e d l i t e s under the forces. tricity  The and  criti-  i n f l u e n c e of i n e r t i a , e c c e n t r i c i t y and  the satel-  atmospheric  a n a l y s i s emphasized r a t h e r adverse e f f e c t s of eccenatmospheric torque on  the  system  stability.  17 Nurre  (1968) c o n s i d e r e d a more complex model of  an  9  asymmetric  gravity  investigated  the  oriented  stability  linearized  analysis.  bilize  librational  lite  the  the  satellite of  The  in a  circular orbit  i t s equilibrium  feasibility  motion of  a  of  position  adding  triaxial  a  using  rotor  gravity  and  to  a sta-  oriented  satel-  18 e s t a b l i s h e d by C r e s p o da S i l v a (1970) . 19 Thomson (1962) a n a l y z e d , t h r o u g h l i n e a r i z a t i o n ,  was  related  problem of  slowly  spinning  satellites  in circular  20 orbits.  Kane and  Barba  elliptic  orbits using  (1966) a t t e m p t e d  the  Floquet  to  study  theory while  the  motion  Wallace  in  and  21 Meirovitch analysis  (1967) e m p l o y e d , w i t h  i n conjunction  o t h e r h a n d , M o d i and  with  questionable  Liapunov's  success,  asymptotic  d i r e c t method.  On  the  i n v e s t i g a t e d r o l l dynamics of a 22 23 s p i n n i n g s a t e l l i t e u s i n g t h e W. K. B. J . (1968) and numerical (1968) m e t h o d s . T h e c o n c e p t o f i n t e g r a l m a n i f o l d s was successfully  extended  to  the  Neilson  study  of  three  degrees of  freedom motion  in c i r -  24 cular  orbits  their  variational stability A  (1969) .  The  preliminary  periodic  s o l u t i o n s were 25 established (1970).  study  of  spinning  found  unsymmetrical  and  satellites  2 6 was  c a r r i e d out  bility an  in a  but of  had a  solution  spinning little  triaxial  inertia  influence  on  to  Shippy  '  equations  of  was  the  presence of satellite  i n presence 29  Sperling  satellite to  ( 1 9 6 3 ) who 27 2 8  Cochran  satellite  success.  orthogonal The  Kane and  circular orbit.  approximate  triaxial  by  orbital the  and  to  about  external  for  a  torques,  showed t h a t the  sta-  determine  a t t i t u d e motion of  their  the  spin  principal axis  of  plane.  various  dynamics  attempted  (1972)  possible  examined  perturbing control  has  forces been  and  their  discussed  10  i n d e t a i l by s e v e r a l a u t h o r s .  Pande ^ and Van der Ha " " have 31  3  1  presented e x c e l l e n t reviews o f the work on a t t i t u d e and o r b i t a l c o n t r o l , r e s p e c t i v e l y , u s i n g the environmental f o r c e s . The study of multibody systems i n v o l v i n g f l e x i b l e has been i n progress f o r q u i t e some time. of g r a v i t y g r a d i e n t  In an e a r l i e r  s t a b i l i z e d e x t e n s i b l e dumbbell  links  treatment  satellite  32 systems  (cable mass n e g l e c t e d ) ,  Paul  (1963) showed a strong  c o u p l i n g between l i b r a t i o n a l and v i b r a t i o n a l motions.  The  feasi-  b i l i t y o f a t e t h e r connected g r a v i t a t i o n a l l y s t a b i l i z e d multibody 33 c o n f i g u r a t i o n was e s t a b l i s h e d by Robe  (1968), who  showed  that  the e l a s t i c and damping p r o p e r t i e s o f the t e t h e r have l i t t l e i n 34 f l u e n c e on the s t a b i l i t y of the system.  Singh e t a l .  (1972)  s t u d i e d the dynamics o f a heavy i n e x t e n s i b l e f l e x i b l e an e l l i p t i c o r b i t .  string in  The r e l a t i v e motion of the s t r i n g w i t h  to the s a t e l l i t e along  the l o c a l v e r t i c a l was found t o be 4  due to the i n s t a b i l i t y a t the f r e e end.  Bainum e t a l .  other  hand, S t u i v e r  two body system.  (1972) has determined i m p l i c i t  unstable  (1972)  e s t a b l i s h e d the lower l i m i t s o f t e t h e r damping and s p r i n g necessary f o r the s t a b i l i t y of a t e t h e r e d 35  respect  constants On the  expressions  f o r the r e q u i r e d c o n t r o l f o r c e s and c o r r e s p o n d i n g t r a j e c t o r i e s i n the study of a two body s a t e l l i t e system maintained along  the l o c a l  vertical. A pioneering  c o n t r i b u t i o n i n the area of r o t a t i n g two body 36  s p a c e c r a f t i s due to Chobotov cable mass and e l a s t i c i t y  (1963), who s t u d i e d the e f f e c t s o f  f o r the case o f point-mass end bodies  undergoing two d i m e n s i o n a l motion i n a c i r c u l a r o r b i t . gradient  The g r a v i t y  e f f e c t s on the s t a b i l i t y i n the small were found to be  11  negligible. entirely orbital  Hence t h e s t a b i l i t y  i n terms motions  of  of  criteria  the c a b l e ' s n a t u r a l  the  station,  and  could  be  expressed  frequencies,  v i s c o u s damping  angular  and  parameters.  37 Tai  et a l .  station under  has  the  neutral ing  (1965)  showed  a stable  influence  stability.  circular  solution  Austin  The  double  orbit,  but  et a l .  '  the  wave f o r m and  All  frequencies associated  cable would  space  oscillate  show a s t a t e  of v e l o c i t y - p r o p o r t i o n a l  (1965,  of  The  freedom  1970)  equations for a  mass s y s t e m .  cable connected  g r a d i e n t and  e l a s t i c degrees 3 8 39  bration the  spinning,  application  to the dynamical  connected,  a  of gravitational  devices to control  effective.  that  with  an  w i t h the angular v e l o c i t y . The d y n a m i c a l r e s p o n s e a n d plane of the o r b i t ) f i n i t e s i z e d  derived  found  the  showed  with  dampto  be  exact  free-rotating,  results  frequency v a r i a t i o n  was  of  flexibly  the r a d i a l v i -  the r o t a t i o n a l  H - s e c t i o n beam w e r e  speed.  found  to  increase the  s t a b i l i t y of a rotating (in space s t a t i o n connected 40  through who  a massless  showed t h e  motions) a  c a b l e were e x a m i n e d by  significance  to obtain  of  rotational  reasonable time  spinning spring-mass  Stabekis et a l . damping  constants.  (1970)  (of t h e end  In-plane motion  body of  f o r combined e x t e n s i b l e and l a t e r a l 41 v i b r a t i o n s was s t u d i e d b y C r i s t e t a l . ( 1 9 7 0 ) . The v i s c o u s a x i a l d a m p i n g was f o u n d t o b e i n e f f e c t i v e i n r e d u c i n g t h e t r a n s v e r s e o s 42 c i l l a t i o n s . Nixon (1972) i n v e s t i g a t e d t h e d y n a m i c a l e q u i l i b r i u m states an  i n three dimensions  arbitrary  analysis. cable  system  f o r a c o m p l e t e l y undamped  number o f c a b l e s ,  During the  connected  study of  system,  but the  d i d not  carry  out  system any  with  stability  three-dimensional motion of a 4 3 Bainum e t a l . (1974) d e f i n e d t h e l o w e r  12  limit that  f o r the damping c o n s t a n t o f the presence  of  a spinning  the r o t a t i o n a l  damping  system  and  showed  (of t h e end  bodies)  44 was  necessary to ensure  studied and  to  the e f f e c t  found  have  stability.  of g r a v i t y t o be  Coming t o the  librational  and  relatively  Sharma and  small  little 46  Stuiver  liptic  respectively.  of  force  was  found  The  interplay  knowledge  subtle  is  stages  discussion  by  t o be  between doubt  h i s t o r y of development of 1.3 Purpose and Scope o f From the tion  of a t r i a x i a l  systems the  has  problem  tially  received, of  be  The  attributed  linear,  nonautonomous, and  a  number o f p a r a m e t e r s  large  cise  analysis.  The  in circular  the a x i a l  reason  and e l -  component  i n the  infinite  pursuit  number o f  excellent  A  outline  such recent of  i t i s apparent  the general  as  little  that  flexibly  for this  limited  complexity of  control.  main purpose  of  has  been  effort  thesis  i s to  essen-  could,  the problem.  amenable t o any the  mo-  Likewise,  coupled equations of motion are not  the  interappended  attention.  in satellites  of  components.  and  forces  to the  the s u b j e c t 45 Saeed and S t u i v e r  i t s modest b e g i n n i n g .  as w e l l  reason  resonance.  s p a c e c r a f t dynamics Investigation  relatively,  librational  overlooked.  perhaps,  foregoing, satellite  and  p r o v i d e s an  the the  of  station  finds  the normal  T h r o u g h an  s c i e n c e evolves from 47  space  later  have r e p o r t e d p r e l i m i n a r y  satellite  greater than  (1977)  one  (1975)  the  absence  forces,  In general,  neverending.  Likins  i n the  (1974)  f o r a dumbbell-type  the  authors  attention.  investigations orbits,  The  g r a d i e n t t o r q u e on  i t s influence  received  (1974),  the  The  non-  involving  simple gain  con-  13  fundamental  understanding  tion  assess  and  to  control  analyzed  i n three  order  complexity.  of  of  a pure  and  stages  gravity  expressions  derived.  The  Gradient  Test  beginning,  the  The  coupled  general  problem an  a n a l y s i s i s applied to  the  study  Satellite  use  of  (GGTS).  the method of  equations  of motion  expressions chapter.  The  considered  Space S h u t t l e d e p l o y i n g  studies of a payload  the a  motion  satellite  Gravity  simple  model  approach. systems  and  their  gradient configurations, along  f o r f o r c e s and Case  of  such  for triaxial  gravity  is  librational  satellite is  mo-  increasing  f o r associated forces i n a dumbbell-type  with  third  moments.  gradient axisymmetric  f o r s p i n n i n g and  the  dynamics of  representing, i n general,  solution the  f o r c e s and  In the  helped i n e s t a b l i s h i n g The  concerning the  moments f o r m t h e E x p l o r e r XX,  subject  SSPS, and  emphasize usefulness  of  of  the the  results. Chapter  4 investigates librational  tribution  and  multibody  systems  to may  the  case  of  a l s o prove  involved a  orientation  in circular  a payload  1-6  i n the  plan of  for cable  orbits.  deployed  u s e f u l i n the  Figure  coherent  control  dynamics,  from  study  or  program to explore  the  connected  analysis i s applied  the  Space S h u t t l e .  of the  The  dis-  The  SSPS and  schematically illustrates study.  beam  force  approach  subject.  the  It  space c o l o n i e s . various  appears  to  aspects  represent  DYNAMICS OF SINGLE AND MULTIBODY EARTH ORBITING SYSTEMS flexible (circular  r i g i d systems ( e l l i p t i c orbits) —  .  1  —  1  :  1  [  l i b r a t i o n a l forces: .dumbbell-type systems  l i b r a t i o n a l dynamics (Lagrange's method)  l i b r a t i o n a l dynamics (Lagrange's method)  f o r c e response: .numerical  Chapter 2  F i g u r e 1-6  —1 1 — librational f o r c e s , moments  the SSPS: . stability . control system response .numerical .linear  \  gravitationally stabil i z e d end-bodies  1  .periodic solutions & their s t a b i l i t y (Floquet) , resonance system response: .numerical .Butenin .linear  ,  1  t r i a x i a l systems (spinning & g r a v i t y o r i e n t e d )  axisymmetric systems (gravity oriented) •  •—•  systems orbits)  stability: .linearized system  stability: .linearized system  system response: .numerical  Chapter 3  A schematic diagram of the proposed p l a n o f study  .. —1 dynamic analysis (Hamilton' principle)  system response: . linear  Chapter 4  15  2.  A T T I T U D E AND  FORCE A N A L Y S E S  FOR  GRAVITY ORIENTED  2.1  Preliminary  time, to  librational  has been a s u b j e c t  a c a r e f u l study  date  have c o n f i n e d  dynamics w i t h ution  of  AXISYMMETRIC  SATELLITES  Remarks  Although control  RIGID  very  forces  dynamics o f s a t e l l i t e s  of, c o n s i d e r a b l e  of the literature  study  interest  as t o t h e c h a r a c t e r  imposed on a g i v e n  system.  forces  is  satellite  important  siderations tific  i s found  not only but also  experiments.  i n literature  from design serves  The k n o w l e d g e  ployment o f an i n s t r u m e n t  package,  taining i t at a given spatial the Space S h u t t l e ) , i s indeed As possess  the formidable  any known c l o s e d  out before,  i n an  idealized  .  The i n f o r m a t i o n integrity  i n the planning  con-  f o r de-  importantly  i n main-  l o c a t i o n and o r i e n t a t i o n (e.g., vital. equations  solution,  of motion  an a p p r o x i m a t e  is  from  do n o t  analysis  u n d e r t a k e n u s i n g an e x t e n s i o n o f t h e K r y l o v 48 method ( v a r i a t i o n o f parameters) as suggested  Bogoliubov 49 by B u t e n i n wxth  certain  Poincare-type  modifications.  An a p p l i c a t i o n o f t h e  •  of scien-  of the force required  governing  form  '  and more  distrib-  46  and s t r u c t u r a l  as a b a s i s  orbital  and  As p o i n t e d  45  some  that investigations  attention primarily to attitude or  little  their  f o r quite  reveals  a preliminary analysis of the l i b r a t i o n a l  dumbbell  and  and  50 e x p a n s i o n method simplifies cal  to the linearized  the solution without  character  o f the system.  system o f equations  substantially  A response  study  affecting  further the physi-  establishes the ac-  c e p t a b i l i t y of these s o l u t i o n s during small amplitude librations. The a n a l y s i s ( B u t e n i n ' s s o l u t i o n ) i s a p p l i e d t o s t u d y t h e  16  effects and  of eccentricity  initial  disturbances  forces.  A typical  Gradient  Test  2.2  S-xyz  bital  the local  executing of  i2'  , Y^' z  s  orientation  I  m  '  a  n  ^  ^ about  by t h e x, y,  corresponding  2  + R 9 ) 2  2  *  (Figure  ty  -  a  x  i  s  2-2)  system o f the out-  to the o r -  of the s a t e l l i t e  about  by a s e t  the x^-axis  resulting  yielding  i nthe  the final  z-axes. f o rkinetic  and p o t e n t i a l  3 0 ( 1 / R ) may b e w r i t t e n a s : +  (I  sin  X + I  2  cos  2  X)ty  2  +  Z ,  T  (I  cos  2  X  X  ,A. , x„ (0 + ty) sin ty}' J  „  - L - I  2  ..(2.1a)  - I ) <j>(9 + 40 c o s ty s i n 2X X  um R  the y ^ 2  expressions  ,  Z  coordinate  the z ^ - a x i s  X  (I  2-1)  with i t s  of force  m o t i o n may b e s p e c i f i e d  2 , . , • . r, 2 2 r," s i n X) (9 + ty) c o s tj> + I {X y  z  P = -  e  body  Eulerian rotations:  S  + I  +  (R  (Figure  satellite,  The o r i e n t a t i o n  _  to the order  T = i [m A  t  rigid  and t h e x ^ - a x i s p a r a l l e l  ^ i a x e s , $ about  given  The energies  z  s  induced  so chosen as t o d i r e c t y ^ - a x i s  S-XQYQZQ  librational  modified  ^ i 2 ' i 2 Y  and  the analysis  the center  principal  momentum v e c t o r .  successive  shaped  about  vertical  arbitrary  giving x  a t S, i n a n o r b i t  angular  parameter  the configuration of the Gravity  an a r b i t r a r i l y  with the triad  along  dynamics  (GGTS) c o m p l e m e n t s  be t h e o r t h o g o n a l  satellite ward  using  inertia  o f the Problem  Consider mass c e n t e r  modified  on t h e l i b r a t i o n a l  example  Satellite  Formulation  Let  of the orbit,  _J1_  4R  (I  x  +1  y  + I ) - 3{I - (I z y z  - I ) c o s 2 X } s i n ty x 2  3  3 ( 1 - I ) s i n 2ty sinty z x  2 2 x c o s ty s i n ct - 3 ( 1  Z  s i n 2X - 3 { l + Y"  +1  X  ( I - I ) c o s 2X} z x  2 2 " - I ) c o s ty c o s ty Y  .(2.lb)  17  Figure  2-1  A schematic diagram of T e s t S a t e l l i t e (GGTS)  the  Gravity  Gradient ~~  Figure  2-2  Reference c o o r d i n a t e systems and geometry o f the s a t e l l i t e motion with ty, ty and A r e p r e s e n t i n g the l i b r a t i o n a l degrees of freedom  19  For  an a x i s y m m e t r i c s a t e l l i t e , I  = I X  A does n o t appear e x p l i c i t l y rendering  .  i n the expression  t h e c o n j u g a t e momentum p  Consequently,  z  f o r the Lagrangian  a constant of the motion,i . e . ,  A  9L p, =  = I  8A  {A. -  For  a nonspinning  The  classical  tions  (6 + ijj)sincj)} = c o n s t a n t  .  ....(2.2)  Y  satellite  (A = 0 ) , t h e c o n s t a n t m u s t b e  Lagrangian formulation  yields  zero.  the governing  equa-  of motion as:  R - R0  + 1L. = — R 2m R s  2  (I  J  R 0 + 2R R0 = ^ { s  _ j ) ( i _ 3 o s i i j cos (j)) , 2  -  Y  (ty + 0)cos^cj) +  ty + 0 - 2 (ty + Q)ty tanty  + lH. . R K  J  $ + { (ty + 6)  2  + lH. .  R Neglecting tions  (2.3b)  in<jj cosij; = 0 ,  (2.3c)  cose)) = 0 .  2  (2.3d)  1  orbital  (2.3a) and  s  (ty + Q)ty sin2ty},  1  cos ^}sinot  K  (2.3a)  2  C  x  perturbations (2.3b)  lead  due t o t h e a t t i t u d e  to the c l a s s i c a l  motion,  Keplerian  relations,  R 9 .= hg ,  (2.4a)  2  i  =  ( 1  equa-  ecos0) ,  +  (2.4b)  h„ hence,  the equations  governing  the l i b r a t i o n a l  motion  become,  3K r  ,  _  2  (  e  sine 1 + e cos0  ,  t  }  (  +  ]  _  +  J-  s  i  n  i  j  J  1 + e cos9  C  O  S  l  J  ;  r  =  r  0  ' .... ( 2 . 5 a )  and *" " 1  +  e cos8  *'  +  { (  ^'  +  1 }  +  1  +  e cos9  c  o  s  ^  }  s  i  n  *  c  o  s  *  =  0  • (2.5b)  20  2.3  Approximate A n a l y t i c a l In  coupled, motion, eters  absence of  nonlinear the  and  2.3.1  the  problem  the  any  i s analysed  After  replacing  the  degrees,  equations  (2.5)  can  of  the  librational param-  methods.  parameters  ignoring  s o l u t i o n of  Butenin's v a r i a t i o n of  method^^  trigonometric  s e r i e s e x p a n s i o n s and  higher  using  expansion  v a r i a t i o n of  1  known c l o s e d - f o r m  nonautonomous e q u a t i o n s  Poincare-type  Butenin s  their  and  Solutions  terms of be  i n IJJ a n d  functions the  written  fourth  <j> b y  and  as  2 ^"  + n  ,p =  1  2e  sine  +  f  ,  ±  ....(2.6). <J>" +  n  2 2  *  =  f  2  '  where: f  l  f  2  =  2 l^  n  e  +  2e\p'  =  2e(i)  +  n  Note  C  ,  O  S  " 1*  0  sing  ,  sine  + |  the  For compared  2  the  their  while  2  <H  '(  2<$>$\  + ^ ) 2  +  the  left  The  be  cose  .  higher f^,  term side  degrees f  of  this  in e  have  .  i n f^  found  s o l u t i o n of  2  3  functions  hand  1 -3 4> )  +  <b V  n-^ty  e  s e c o n d and  s o l u t i o n can  parameters.  + ip '  1  + i  amplitude motion each  t e r m s on  assumed i n the  +  d e r i v i n g the  approximate  of  -  terms c o n t a i n i n g  small  to  variation  • V =  c^'  3  2 2 i|J >  J  2  been n e g l e c t e d  is  cf) -  +  ( l - e c o s 9 ) ( J ) ( | <$>  2 1  that  hence  2 2 )  and  f  equations using  the  system of  2  is  small  (2.6), method  of  equations  form, 2e  a-  (9)  1  2  sin{n j  +  (e) } + — n^  sxne  2  2  -  1  ,  ....  (2 ,7a)  21  <J> =. a  (9) s i n { n 9  2  +  2  5^(6)}  ,  (2.7b)  w h e r e a-, ( 9 ) , a„ ( 9 ) , 6, ( 0 ) , 6~ (9) a r e now c o n s i d e r e d t o b e 2 2 2 ^2 s l o w l y v a r y i n g p a r a m e t e r s . S i n c e t h e s e f o u r v a r i a b l e s c a n n o t be determined  from  four  overspecified.  initial  Hence  four  conditions  alone,  the s o l u t i o n i s  constraint relations are obtained  as  follows: Equating of  the l i n e a r  a.  sin?, 2  2  (2.7) t o  system gives  two o f t h e c o n s t r a i n t  relations:  ? •'  cos?.  2  ; i  = l , 2 .  (2.6) i n c o n j u c t i o n w i t h  - a.  that  ....(2.8)  2  (2.7) y i e l d  n. c o s ? •  = 0  2  of motion  equations  a.  d e r i v a t i v e of equation  + a•  Equations in  the f i r s t  the remaining  n.6.  sin?-  =  f-  t h e assumed s o l u t i o n -  two c o n s t r a i n t ; i = 1, 2  relations, ....(2.9)  ;  where f  i  =  f  i ^  a  l  s  i  n  ^ i  '  i  a  i ^i  n  cos  i  a 2  sinc  2  , a  2  i = 1,  Solving equations  (2.8) a n d  (2.9) s i m u l t a n e o u s l y  n  cos?  2  2  );  2.  f o r a. 2  and  1  6. 1  yields, 2  I-  i  =  2  0  f.  n.  I  I  COS?.  i  i = 1,  1  JT*  f. a  For a.  small  i  n  _ 2  i  1  •  sin?. 2  (2.10)  ,  X  amplitude motion  a n d <5. are slowly 2 2 v a l u e s o v e r one p e r i o d  2  r  varying  f  are small.  parameters.  X  gives  the following:  Using  Consequently, their  average  22 ^ a  l  =  ( 1 / 8 T f  n  2  2TT 2TT 2TT  i  }  0  0  0  ^ /  b  T  O  ) ' ' ' 0 0 0  3  U  C  S  ?  2TT 2TT 2TT  fi' = -M/8TT n a i i i 6  *  i  f  T  2  J 2  i  d  *  f 1  l  ?  2  i  d  2  ?  2  . S  i  n  ? 1  d  ,  i 2  d  ? X  >  0  2  d  C  a x  0  = 1, 2.  (2 .11)  (2.11) may be e v a l u a t e d t o g i v e :  = 0 ,  2  i  1  = -  6  2  ( 1  /  4 )  i  n  %  A f t e r i n t e g r a t i n g equation for  a. ,6. i n equation 2 2 can be w r i t t e n as:  (2.12) w i t h r e s p e c t  (1 - a  2  ....(2.12)  t o 6 and s u b s t i t u t i n g  (2.7), s o l u t i o n f o r the l i b r a t i o n a l motion  2  sin{n  = 1, 2.  >  = a, sin{n,(1 - a /4)8 + 6, } + {2e/(n 2 2 20 cf> = a  d  Z  i  The r i g h t hand s i d e o f equations  ,_  N  ] 2 2 2  /4)9 + & } , ^2 ^20  2  - 1) }sin0 ,  (2.13a)  (2.13b)  2  where: a  1  &  2  =  ±  [ >  2  ±  6  - 2e/(n . - l ) } / n 2  {<JJQ  = ± [cJ) + 4 > Q / n ] 2  6,  +  2  2  1  /  2  9  n  ,  (2.13c)  (2.13d)  2  n  n  n  20  ]  = a r c t a n [4> n,/{iJ; ~ 2e/(n, - 1) } J  = a r c t a n {<J> n /<j> } •  ?  2  ,  1/2  i  20  2  f  ....(2.13e)  ....(2.13f)  23 5Q  I  2.3.2  Poincare-type expansion method In equation  (2.6),  after neglecting  the second or h i g h e r  degree terms i n ty, ty, ty', ty', the f o l l o w i n g  l i n e a r r e l a t i o n s are  obtained: ty" +  = e{2(ip' + l ) s i n 9 + n^ty cos0} ,  n ty 2  (2.14) ty" +  n^ty =  sin6 + n^ty cos0} .  e{2ty'  Since the value of e c c e n t r i c i t y i s u s u a l l y small  (e < 0.2), the  s e r i e s o f the form  ty (0) =  eV  E  i=0  (0) , (2.15)  OO  ty (0)  =  £  e ty. (9) , J  j=0  :  l e a d to the s o l u t i o n of equation ty = ip +  ' + (^ /n )sin  cos n 0  0  1  Q  (l/n ){ - 2/(n  + {n  1  ^  2  Q  2 2 n^/{1 - 4n.^) }cos n^Q  2  - l ) } s i n 0 - {n  ty (l  ±  Q  n^Q + e [{ - 3ty  1  - 1) + ^ Q ( 2 - 5 n ) / ( l - 4n ) }sin  2  1  + {2/(n  (2.14) as:  ( l - n /2)/.[l  Q  2  -  1  n ; L  n / 2 ) / ( l + 2 ^ ) }cos (1 • + n _) 0  +  1  ]  )}cos(l  - n _) 0 ]  - {ipgd + n / 2 ) / ( l + 2 n ) } s i n ( l + n^O 1  n^  2  -  1  {ty' (I  -  Q  2n )}sin(l - n )0] 1  1  ....(2.16a) i  ty = ty cos n 0 Q  2  x cos n 0 2  U  ( n 0  + ((j> /n )sin n 0 0  + {tpQ (2 - n  2  2  2  2 2 2 + e [ { j ) ( n - 4 n ) / ( l - 4n )} (  - 4n )/n (l 2  2  0  - 4n )}sin 2  2 + n / 2 ) / ( l + 2n ) }cos-(l + n ) 2  2  1  2  2  2  n0 2  24  + U (n 0  x  - n^/2)/(l  2  (1 + 2 n ) } s i n ( l 2  x s i n ( l - n )0] 2  2  + n )9  -  2  Ug  {<b' (--  - n ) 6 -  - 2n )}cos(l  Q  2  (n  2  - n^/2)/n (l 2  +  2  -  n /2)/n 2  2  2n )} 2  . (2.16b)  2.3.3  Accuracy of the At  solutions.  t h a t g i v e n by  obtained using  control  on  are  For t h i s ,  an  IBM  370  also  for large  the Runge-Kutta computer  local  e x a m i n a t i o n o f F i g u r e 2-4  tions  of  2-3,  found The  2-4).  t o be  solutions A  correspondence  careful between  However, the approximate  the coupling  error  surprisingly  approximate  showed a c l o s e r  numerical solutions.  with  For a l l the  amplitude modulations.  are not able to determine  degrees 2.4  and  were compared  method w i t h b u i l t - i n  (Figures  disturbances.  able to follow  Butenin's  solutions  the  of the exact equations of  t h e a g r e e m e n t i n r e s p o n s e was even  to assess the accuracy of  the  a numerical integration  motion,  close  solutions  t h e o u t s e t , i t was e s s e n t i a l  approximate  cases,  approximate  solu-  b e t w e e n vp a n d (J)  freedom.  Periodic  Solutions  and  Stability 51  As of  the  shown by  Brereton  , equations  (2.5)  permit  solution  form: oo  f  P ( I 1  =  Z m  =  A  1  m  n  sin(m/n) 6  ;  n = 1,  2, . . .  ;  '  (2.17)  CO  *  p F  '3  With  + =  2 A i=i  ic A  sin(i/j)0  a p r o p e r c h o i c e o f e,  ;  j = 1,  and  initial  2,...  conditions  ;  a vast  e = 0 . 2 , K 0 . 8 5 + =10°, <t> = 5°, i|/=0.3, ct>;-0.5 — numerical ---- Butenin's r  f  0  -20  ;  o  0  26  (a) numerical ,  (b) Butenin's,  e=0 , K 1  t = t  r  (c) linear =  0  '  i  =  i=°5  Orbits Figure  2-4  A t y p i c a l example linear solutions  comparing numerical,  Butenin's  and  27  majority  of the c o e f f i c i e n t s A* m,n  reducing  the above equations to the f o l l o w i n g form:  J  1  ty  P, n  r  n  , A * * . can be made to v a n i s h x,:  s i n (v/n) 0 ,  = A*  v,n  . (2 .18)  /n) 0 , P,n = A *K*, n s i n ( K' '  < >_, Y)  where the combined ty and ty motion has a p e r i o d of 2frn. displacement i n i t i a l leads to  conditions  to be zero,  Specifying  i . e . , ty^ = ty^ = 0,  = 6 = 0 i n equation (2.13). For a c i r c u l a r •^20 20 (e = 0), equation (2.13) may be r e w r i t t e n as:  oxbit  9  Z  i  2  *  ty = (ty^/n^  2  *  s i n { n ( l - ty' /4n )8} , 1  Q  1  (2.19)  ' 2 2  t  ty =  [ty /n ) Q  2  sin {n (l 2  Equation and  -  ty^/An^Q]  .  (2.19) now has the same form as equation  both ty and ty are of the form ty^  n  By a l l o w i n g d i f f e r e n t i n t e g e r values conditions  (2.18)  and ty^ , r e s p e c t i v e l y . n  f o r v, K , n, the  initial  f o r p e r i o d i c s o l u t i o n s may be d e r i v e d as:  ty = ± 2n,(1 - v/nn,)  1/2  n  ty' = ± 2 n ( l - K / n n ) Q  2  1 / 2  2  ,  (2.20)  where p, = v/n and p, = K / n . ty  c  L  ty  S t a b i l i t y of the p e r i o d i c s o l u t i o n can now be s t u d i e d by 52 using v a r i a t i o n a l a n a l y s i s i n conjunction  w i t h the F l o q u e t  theory  Substituting  r  in  T  P,n  equations  leads  '  (2.5)  and  linearizing  with  respect to  and <|)  to  \p r  v  cj)  v  r  It  v  =  F,  =  "Jt G,  1  r  1  r  + F„ 4) 2 v  f  v  + F_ 3  r  (JJ  '  ^  +  v  G  ^  +  2 v 0  Y  Y  v  <J>  + F . cb 4 v T  (2  if G_  cJ) v  3  I  +  G.  4 v Y  6  '  ;  where: * i sin6/(1 -L  ,n  ir  F„  = - 3K.  --*  1  K,n  F  3  =  2{  F  4  =  ^P,n  +  G  l  2 ( l | J  = "  G*  =  G*  = 2e  G  4  =  cos  t a n  2ty  1 ) t a n  Since  / ( l + e cos0) ,  *P,n  *P,n  +  1 ) ( J )  ^P,n  +  1  {3K./2(1  "  )  { 2 e  P,n s  i  n  s  1 1 1 0  ^ K i /  +  *P,n  2  /^  2  ^  +  + e  1  +  +  e  e  cos9) } djip  D  2  (n =  1,  + l)tan<|)  c o s 0 ) } s i n 2i|;  t  a  p  n  p  *p  /  n  ' s i n 2ch  cos0),  + ( 3 ^ / ( 1 + e cos6)  for a circular 2im  1  + e c o s 0 ) } s i n 2\p  sin0/(l t^p^n  "  '  orbit  , A ' P ,n P ,n the F l o q u e t r  period  + e cos0) ,  ,n ir / n  +  P,n (  ir  2,...),  T  Jcos^^Jcos ,  F* 1  theory  2cJ) ^ p  n  F* e t c . are of 2 is applicable.  29  Taking  solutions  i n t h e form:  i|)(0 + 2im) = a* ty{6) , (J> (6 +  2iTn) =  a*  and  solving equation  the  stability  \Rl  ,  (2.21)  f o r d i f f e r e n t values  o f a* l e a d s  to  criteria:  < 1  a?I  <)) (0)  ,  i = 1, 2,  3, 4 ; s t a b l e (2.22)  | a * |  > 1  The uated  over  tions  studied  are  that  odic  conditions  equation  i n Figure  lead  forperiodic  2-5.  (2.22).  s o l u t i o n s were  The c a s e (a) c o r r e s p o n d s  motion.  seven  The c a s e  Note  to unstable  that  t h e same  cycles  the majority  results  to the  (b) r e p r e s e n t s  o f ty, a n d n i n e  cycles  orbits.  having  eval-  of the solu-  Some o f t h e t y p i c a l  t o ty a n d <j> l i b r a t i o n s  leading  initial  period the peri-  o f ty a r e  of the  initial  solutions.  Resonance The  mation  about  study  of equations  the dynamical  a satellite  (i.e. 0 <  resonance  (2.16)  behaviour  i n an e l l i p t i c  = 1 / 3 , 1/12,  Since the  through  i n five  unstable  o f v, K and n and t h e s t a b i l i t y  a range  s o l u t i o n where  2.5  conditions  of the orbital  completed  For  i = 1, 2, 3, 4 ;  initial  presented  conditions as  ,  < 1, t h e c o n d i t i o n i n ty m o t i o n  n  2  an i m p o r t a n t  of the linearized  orbit,  respectively)  gives  the values  lead = 1/2  infor-  system.  n ^ = 1,  1/2  t o resonance  i n ty m o t i o n .  (K  leading  i s not possible.  i  = - 1/4)  to  30  e-0-,  [aj § - 5 - i . [ b ] ^-^5.5-%; stable  —, unstable  =<4>=o '0  T  o  —  —,  —  3  4  0 075 Figure  2-5  '\  I  I  I  0-8  0-85  0-9  0-95  Typical i n i t i a l conditions r e s u l t i n g i n periodic l i b r a t i o n a l motion. Note t h a t over a wide range of i n i t i a l c o n d i t i o n s and modified i n e r t i a parameter the p e r i o d i c s o l u t i o n s are u n s t a b l e .  31  2.6  Force  Distribution  A vast majority of the axisymmetric lites  may  be r e p r e s e n t e d  m a s s e s m^ (Figure  , m^  2-6).  , m  by t h e d u m b b e l l  connected  c  Here m  i s taken  c  mass o f t h e s a t e l l i t e . distances  from  b y a boom  inertial  coordinate  system X  2  +  where  }  =  {  g  }  {AF} r e p r e s e n t s  Am i n i t s p o s i t i o n . equation  2 {R„} m X L  dt  ( 2 . 2 3 ) may  where  ,Y  J  F o r a m a s s e l e m e n t Am  i  , Y  reference  to the  i  , Z  c a n be w r i t t e n as  ..(2.23)  the force necessary  t o keep  The a c c e l e r a t i o n o n t h e l e f t  d0  %r(  d0  of  at a  AF  be e x p r e s s e d  length  corresponding  ^ >/Am ,  0 —[0  ,Z  p per unit  to the  of motion with  i  m  o f mass  satelpoint  t o be l o c a t e d a t t h e c e n t e r  0, r e s p e c t i v e l y .  E, f r o m S, t h e e q u a t i o n  { R  configuration with  , R^ a n d R r e f e r  distance  dt  gravity oriented  t h e mass hand  element  side of  as  [01 {R  } )], m x ,y ,z ' ' J  i  Q  Q  Q  ...(2.24a)  the transformation matrix i s , 1 [6]  0  0  0  cos0  -sinf  0  sine  cosf  (2.24b)  and r  {R„> m  The  W  z  o  =  £ sine  . < E, cosijj coscj) + R | E, s i n i j j cos<j)  s u b s c r i p t s here  a vector  (2.24c)  represent  i s expressed.  t h e frame  The e x p r e s s i o n  of  reference  i n which  f o r the gravitational  F i g u r e 2-6  Dumbbell-type s a t e l l i t e :  (a) l i b r a t i o n a l angles ty, ty; (b) boom f o r c e s  33  acceleration  {g}  = -  may b e w r i t t e n a s  (y/R^)  [6]{R } m  m  where  1/R  ,  x  X  0' 0'  (2.25a)  0  Y  may b e c a l c u l a t e d f r o m e q u a t i o n  1  R~ m  m  (1  R  3E  - r — costy  (2.25b)  coscb)  2  3  E  Note  t h a t t h e terms  containing  (2.4a)  (2.25b).  £  , i^. , t c . are neglected i n R e  R equation  (2.24c),  For small values  of eccentricity  a n d ( 2 . 4 b ) may b e u s e d t o d e r i v e  equations  the following relations:  y = co R p ( l + e ) , 2  3  . (2.26) o)(l + e cos0)  After  substituting  (2.23)  {R  }  and u s i n g  equations  , 0  , y  0'  0  equations  R m  (2.24a) (2.24c),  , y a n d 6,  a n d (2.25a) (2.25b),  an e x p r e s s i o n  into  (2.26)  equation  to replace  f o r t h e f o r c e may b e  r e w r i t t e n as { AF}  (1 + e c o s G )  2  d_ de  (1 + e c o s 0 )  2 d_  de  [6]  r E, s i m j i { E, costy  E sinty r  COScJ) +  R  +  (1 + e cos6)  [0 ]  coscj)  E, sine})  ^-2^  cosf  E, sirup  cost}) + cos<p  R^  Am  (2.27)  By a p p r o p r i a t e l y  s e l e c t i n g £ and Am, the f o r c e a t any p o i n t  on the s a t e l l i t e can be determined and hence, by i n t e g r a t i o n , the r e s u l t a n t f o r c e on a s e c t i o n o f the boom.  Of course, f o r e q u i -  librium, {.F } a  { F  +  }  B  { F  +  } +  C  { F  ic  O  { F} = m £ t o a a  hence:  b  { F  { F  C  { F  ca  { F  cb  }  " b  =  }  m  -  }  =  {  = " c  }  F  }  { F  +  C  B  } =  0,  >  }  {F}  2  ca  m  A  {F}  {  £ c o  m*  C  F  ,  >  }  m  {  F  }  b  • (2.28a)  where : m  * = a  (m /m ){m a  s  + ixa^l^/l)  b  + (1/2) p U ,  m. = ( /m ){m + m U / £ ) + (1/2)pi} , m  b  b  *c  m  m  =  ( m  *  c  / m  s  s  )  a  ^  (  m b  ca  p[(£/m ){m  m*  - p[U/m ){m  b  m  s  s  s  =  m  a  +  m  b  +  m  c  +  p  l  c  £  /  £  2  " a  )  m  2  ( A  i  / J l )  + (1/2) p U  - l)} ,  2  ±  b  + m (£ /£) + (1/2) p U - l /2 ] ,  a  + m U /Jl)  c  c  1  2  ±  + (1/2) p U - £ / 2 J , 2  '  (2,28b) with the components of { F } i n the  E,, \p  r  tj) and X , Y ' , Z ' frames 1  35  of  reference  related  through  sincj>  h,i>,<p  costy  0  {F  the following, cost})  sinty  COSI(J  - sinijj - COSIJJ sine})  cost})  cost})  - sinty  [ E  ^  T  { F }  X',Y',Z'  sintf) (2.29)  Simplification  of equation  (2.29) y i e l d s  the force  components  as:  Fg. =  (1 + e c o s 9 )  x cos cb 2  F^  F^  [ 1 - 3 c o s i ( ) cos ty -  3  2  (1 + e c o s 0 ) { ( i j /  2  + 1)  2  + Cb' }] , 2  =  (1 + e c o s e )  -  2e ( '  =  (1 + e c o s e )  -  2 e ty s i n G  [ ( 1 + e cose){ty"  3  + l)sin0  cost}) + 3 s i n ^  costy  + 3 c o s ty sinty  1)sinty]  cosij; cost})] ,  [ ( 1 + e cos6){<}>" +  3  - 2ty' (ty' +  costy]  (ty' + 1)  2  sin<|>  costy}  . (2.30)  2.7  Librational  Response  Important of  and Force  Distribution  system parameters were v a r i e d over  possible interest  t o assess  their  amount o f i n f o r m a t i o n so g e n e r a t e d  for  conciseness,  establish  trends  Figure  1  planes  results  however,  sufficient to  here.  2-7 s h o w s t i m e h i s t o r i e s o f t h e l i b r a t i o n s a n d f o r c e  components f o r g i v e n a n d ty-ty  i s r a t h e r enormous,  a few o f t h e t y p i c a l  are presented  range  i n f l u e n c e on t h e response.  The  only  a wide  e,  yield  and i n i t i a l c o n d i t i o n s .  closed trajectories  P h a s e p l o t s i nty-ty'  corresponding  to periodic  e= 0,K 0.75 r  1  t = <t>o=°'  ;  2  3 Orbits  1=^=0.5  4  5  6  37  T y p i c a l time h i s t o r i e s of l i b r a t i o n s and (b) i n c r e a s e components at the mass m. orbital eccentricity  force i n the  38  39  0.8  T y p i c a l time h i s t o r i e s o f l i b r a t i o n s and f o r c e components a t the mass m : (d) d i s p l a c e m e n t and i m p u l s i v e type d i s t u r b a n c e  40  solutions motion the  as  given  i s at  axial  2-7a). Planar  effect  may  be  2-7(a).  Figure  2-7(d)  bance.  Note  gradient =  16  get  cal tain  the  with  as  forces  at  planar  preferred  an  the  plane Also  (Figure 2-7(b).  amount o f  ampli-  t h e ty m o t i o n The  value  slender type  repeats  repeats  effect  along  impulsive  to  represented was  the  with as  of  Figure  well  of  as  .  system of  ex-  distur-  itself  after  librational  satellite, the  considered  an  a  typical  GGTS (m  (Figures  dynamics  =  gravity  0.2 8  2-1,  kg,  2-8).  The  the f e a s i b i l i t y of the g r a v i t y g r a d i e n t  at  Here the  local  line  the  by  the  the  force  vertical.  corresponds  It i s interesting  resultant force  'control force'  instantaneous  I t i s a measure of  orientation.  component of  a rather  existing  altitudes.  solid  F^  librations  i n the  motion here  f o r an  t i p mass a l o n g the  or  2-7 ( c )  the  of  oscillations.  i n 1966  high  orientations.  by  of  w e l l as  a difference i n forces  represented  of  a decrease  response  out  motion.  oscillations.  Figure  periods  o b j e c t i v e was t o s t u d y  the  inplane  show c o n s i d e r a b l e  some a p p r e c i a t i o n a s  launched  stabilization sents  and  increase  that  F^  studying  c o n f i g u r a t i o n as  m)  mission  by  shows t h e  associated  the  the  i s shown i n F i g u r e  l a r g e number o f  l a r g e number o f To  y  a  a displacement  extremely  and  to  than  Note  phase p l o t s i n d i c a t e t h a t  amplitude  tend  to  w e l l as  appreciated The  method.  increasing e  The  after  forces  posed  frequency  of as  tude modulations. only  Butenin  c o m p o n e n t i s much l a r g e r t h a n  librations  itself  the  a higher  force  The  by  to  and  repre-  local  required  to  vertimain-  The  'resultant force'  the  force  to note  that  i s maximum n e a r t h e  in  this  the  perigee  axial and  41  e = 0.1, K.-1;  T = 2 2 h r s . 10mts.,uj = d> = o IO  1  'O  =0-3,4>' = 0.2 'o  o  o ,o  -151Control force  Resultant  2.5  X10 ),N O A  -2.5^ 1  x10 ),N 0 8  x10 ),No 8  ,  ,. -  *  Sr:  \  -1.5 0 Figure 2-i  I  I  I 3 Orbits  I  4  I  A t y p i c a l example o f t h e G r a v i t y G r a d i e n t Test S a t e l l i t e (GGTS) s h o w i n g i t s l i b r a t i o n a l a n d force histories  6  42  apogee, whereas, near  the  minor  2.8  Concluding  the  transverse  a x i s of  the  on (i)  t h e m may A  be  general  The  i t s maximum  orbit.  the  is  librational  i n v e s t i g a t i o n and  as  formulation  coupled,  ation  of  summarized  satellites (ii)  reaches  Remarks  Important aspects based  component  conclusions  follows:  for axisymmetric  gravity oriented  presented, nonlinear,  motion  are  o f p a r a m e t e r s and  nonautonomous e q u a t i o n s analysed the  using  of  Butenin's  Poincare-type  vari-  expansion  methods. (iii)  Stability Floquet  of  orbits  Expressions satellite ning  (v)  of  The the  packages deployed  the  conditions  are  derived.  forces  They  in a  should  are  applied  satellite  in  be  dumbbell-type useful in  plan-  presented  to  the  flight  tested  (GGTS) .  here  has  o r i e n t a t i o n c o n t r o l of  the  from  for resonance  the  experiments.  procedures  gradient  using  established,  for librational  information  p o s i t i o n and  are  scientific  Analytical gravity  periodic solutions i s studied  a n a l y s i s and  elliptic (iv)  the  space v e h i c l e s .  direct  relevance  instrumentation  to  43  3.  A T T I T U D E AND  FORCE A N A L Y S E S  RIGID TRIAXIAL 3.1  Preliminary The  SYSTEMS  Remarks  study  of  librational  dynamics of 17  for  a  few  major Of  preliminary  limitation  course,  this  -  axisymmetric  simplification •.  There has  are  an  satellite  of  communications (ii)  The  governing  Even w i t h solution ential  the of  is  of  Solar  of  of  the  from  one  configuration.  several  consid-  the  satellite the  indeed  first  Canadian  Intelsat series  of  etc. motion  of  are  extremely  a x i s y m m e t r y , one  nonlinear,  a vast  Skylab,  belong  Another  complicated.  i s faced  nonautonomous s e t  majority  a v a r i e t y of nonexistent. proposed 5  to  this  with  of  the  differ-  orbital  (tensile  to  present  configurations The  1-4),  keep  have c o n f i n e d  dynamics w i t h or  the  Canadian  space  day  missions  where an  ax-  Communications  of the colonies  Satellite and  6  category.  important point date  of  configurations  (SSPS , Figure  investigations to  character  Sputnik,  several  assumption  i s often  Anik,  or  of  s i t u a t i o n s where a  equations  coupled,  Power S t a t i o n  attitude  , suffers  equations.  symmetry  many o t h e r s  the  1,  space v e h i c l e s of  satellite  '  i s d i c t a t e d by  satellites,  Nevertheless, employ  '  character  symmetry, e.g.,  Alouette  except  •  a number o f  axis  satellites,  26 2 8  investigations  erations : (i)  FOR  compressive,  very  i n mind  i s the  fact  that  a t t e n t i o n p r i m a r i l y to little  p e r i o d i c or  i n t e r e s t as aperiodic,  to  high  the the or  44  low  frequency,  tribution hand, of  of the forces  such  Force  would  the  this  chapter,  procedure.  spinning  this  solar  panels  power and  the sun f o r the  the important  general  For  station.  torque maximum  critical  rigid  system  equations  insight  using  the i n e r t i a  initial  of  into  aspects  of  motion.  through  i n space.  numerical  launched  over  provide  t h e Space  and  analysis, general which  (Figure  should  location  Application of the analysis i s  from  values  eccentricity,  at a given  XX  consid-  a range o f  the Newtonian  an o b j e c t  Validity  integration of  the o r b i t a l  configurations of the Explorer  ment package  method.  The s o l u t i o n s  using  using the  s o l u t i o n s f o r both  a n d moments a r e o b t a i n e d  prove u s e f u l i n maintaining orientation  a perturbation  and s p i n parameters,  f o r forces  are derived  motion  c o n f i g u r a t i o n s are sought f o r  through  Next,  of librational  closed-form  the system behaviour  disturbances.  expressions  spacecraft  oriented  the s o l u t i o n s i s checked exact  equations  Approximate  and g r a v i t y  formidable  erable of  among  an a r b i t r a r i l y - s h a p e d ,  Lagrangian  the  On t h e o t h e r  design. In  of  dis-  and a s s o c i a t e d c o n t r o l system.  them towards  rank  or  be a p r e r e q u i s i t e t o t h e p l a n n i n g  on t h e g i g a n t i c s o l a r  to position  etc.)  system.  the case o f the o r b i t i n g  distribution  efficiency its  imposed on a g i v e n  experiment  consider  required  of fatigue failure,  i n f o r m a t i o n would  a scientific  example,  of  possibility  and  illustrated  3 - 1 ) , an i n s t r u -  S h u t t l e , and t h e SSPS.  Telemetry  antenna  mass=1  Total mass = 49 kg  Geometry o f t h e I n t e r n a t i o n a l  Ionosheric  Satellite  (Explorer  XX)  46  3.2  Formulation  of the Problem  Consider  an a r b i t r a r i l y - s h a p e d s a t e l l i t e  o f mass a t S m o v i n g force is  0  (Figure  completely  Eulerian pitch  3-2).  specified  principal  reference  coordinate  inertial  three  coordinate  about t h e center o f  successive  modified  2  system X  1  , Y  1  1  yaw a n d  the attitude ofthe  respect  , Z  coordinates,  to the orbital before.  An  i s l o c a t e d a t 0.  the expressions  With  for kinetic  c a n be w r i t t e n a s :  + R 0 ) + I X (6 c o s y 2  define  s y s t e m X Q , y^ , Z Q d e f i n e d  to the principal  [mS ( R  They  i t s center  orientation of the satellite  axes x, y, z w i t h  p o t e n t i a l energies  T = i z  through  respectively.  satellite  and  Any s p a t i a l  trajectory  r o t a t i o n s y , 3, a n d a , r e f e r r e d t o a s r o l l ,  (spin),  respect  i n an a r b i t r a r y  with  2  *  *  cos3 - Y s i n 3 + a )  2  •  + I (-8 s i n y cosa + 3 cosa + 0 cosy s i n g s i n a + y cos3 s i n a ) ^ + I (8 s i n y s i n a - 6 s i n a Z  2T + 0 cosy  sinB c o s a  ym P =  +  -^T 4R  IX  + y cos3 c o s a ) {-  2  J  ,  ....(3.1a)  + 3(1 + c o s 2 B ) s i n y } + I„ {1 Y  2 + 3 s i n 2a s i n 2y sin3 + 3 c o s 2a c o s 2y + 3 s i n y 2  x c o s 3 ( c o s 2a - 1 ) } + I -  {1 - 3 s i n 2a s i n 2y sin3  3 c o s 2a c o s 2y - 3 s i n y c o s 3  Using  2  the Lagrangian  2  ( c o s 2a + 1 ) } ]  .  ....(3.1b)  procedure and r e c o g n i z i n g t h a t t h e o r b i t a l  motion remains e s s e n t i a l l y  unaffected  by s a t e l l i t e  librations,  Figure  3-2  Reference c o o r d i n a t e systems and the geometry of s a t e l l i t e l i b r a t i o n a l motion w i t h a , 3 and y degrees of freedom  48  the'governing  equations of a t t i t u d e motion i n the y, $, and a  l i b r a t i o n a l degrees of freedom can be w r i t t e n as: Y  (roll)  •  2 2 sxn $ + cos 8 -  (I  2 2 " 2 2 cos a cos 8)Y - 2 ( 1 s i n 8 + cos 8  K  - K c o s a cos 8) , 1 + e cos0 2  Y' +  2  ;  S  I  N  A  6  K  s  i  n  2  a  c  o  s  8"  B  2 2  -  {I  cosy cos 2 8 - 2  2 cos 8 + K.(l - 2 cos a s i n 8)  COSY  2  - K sin8 s i n 2a sinY + . — f * - — 3 - s i n 2a cos8} i + e cost) s :  T • n " , I smS a + + K cos8  n 9  COSY  8'  2el sin8 . - , _ „ T— 5 — sm8 + I sinY cos8 1 + e cosQ  i  s i n 2a sin8 - sinY cos 2a)  (COSY  K s i n 2a sin3 ^  „"2 , , 8 + (I T  a , ,, 2 , ' „ i . „ „• 1 + K cos a)Y 3 s i n 28 T  -  + K Y ' 3 ' s i n 2a c o s 3 - ( I - K cos 2a)a'8' cos8 2  " 1 + e. cos6  +  i  , s i n 2Y  2 COS  K s i n 2Y 2 x  (l v  +  {  (  "  1  8  r  "  1  K  c  ,  T  ( I  -  1  o  s  a  )  c o s  !  s i n 28 - K sinY cos8 s i n 2a}  3  ,  +  Y  +  E  C  O  ,  S  E  (I  cos 2a s i•n 28 cos 2a ,+ 3 1 + e cos! n  r \ 1 + e cose'  u  1 n  -  +  K  2 > cos^a) } i  +  K  s i " 2a cos 2Y sin8  n • ' • • • • (3 • 2 s.)  8  (yaw)  n v c ^ . ! (a" - 2e3' sine ^ (1 - K s m a) (8 - i + cos0 2  )  e  - 2  COSY  2 cos 8  +  K(COSY  K Y " cos8 s i n 2a , 2  2 cos 23 cos a +  {  COSY  / T  Y  c  o  s  2  _  n  2 s i n a - sinY sin8 s i n  49  e. s i n 9 cosg cose  s  +  . i  n  e  + cosy  sing  )  J  -  Y  1  COSY  +  r x {I cosy  • n , / • • sin3 + K ( s i n y s i n  2"  C  O  Y  2  .  s i n  4  cos3  T  (  1  +  e  c  o  s  9  _  n  )  ~ 23 0  K s i n 2 Y cos3 s i n 2a ., +  2a  . „ 2e sin0 r,, • .2 s i n 23 - T — ; 7r{K(sinY sin a 1 + e cosG ' 2 . -, 3 ( I - 1 + K c o s a) . 2 . sin } s i n Y s i n 23  Y  S  ->  (I + K cos 2a)Y'a'  12 Y  s i n 3 s i n 2a) 2 2 . I - 2 + K cos a .2 ,  +  L  c o s 2a)} a' - K 3'a' s i n 2a +  2 - 1 + K c o s a) T 2  (I  ' +  *i  n  2 a  3  vJ- + 1 +  cosS  e  _ „  v  "  ;  U  '" (3.2b)  a  (pitch) "  T  la  -  -  \  -  {I  , K3 + —  +  K  2e I a sine i + e cos6 ~ s i n 23 c o s  COSY  ~ 2  H ^  2a)  s i n 2a  -  . - i ismy  0  2  F T  +  0  2  Y'  ,, . „ - K ( s m Y cos3  „ c o s 2a ,  T  -  K  Y'  2  cos 3 2  s i n  2a  s i n 3 c o s 2a)}3*  COSY  , _ ,, 1 t ( I + K c o s 2a) Y 3  / T  2  N  cos3  \ \ ^^^^  s i n 2a  „ cos3  2e I sin6 - i—; n1 + e cosa  ,  COSY  D  COSB  2  { s i n Y s i n 2a - c o s y s i n 3 s i n 2a + s i n 2 y s i n 3 c o s 2a  2  3 + -=.—• ( s i n Y cos3 1 + e cose ' = 0. . equations  1  ....(3.2c)  (3.2) r e p r e s e n t  dynamics o f a t r i a x i a l  2  c o s 2a - c o s 2 y s i n 2a - s i n y c o s 3 s i n 2a)}  K  The  . sinp -  sin3 + K ( s i n Y  COSY  •2  T  system  a complete negotiating  s e t governing  attitude  an a r b i t r a r y e l l i p t i c  trajectory. 3.3  Motion The  too  i n the Small coupled,  complicated  nonlinear,  t o admit  nonautonomous e q u a t i o n s  any known c l o s e d - f o r m  solution.  (3.2) a r e Although  50  a numerical solution obtained,  i t often  importance a  through a d i g i t a l  fails  advantage  sometimes  provided  can  always  t o i m p a r t u n d e r s t a n d i n g as t o  of the p h y s i c a l  situation,  computer  parameters  a linearized  i t retains  involved.  analysis  essential  Under  may  be  features  the such  used of  be  to  the  system.  3.3.1  Spinning  3.3.1a  systems  Approximate Replacing  equations  (3.2)  and  degree  higher  by  analytical  the t r i g o n o m e t r i c their terms  of  the a-equation as,  a"  -  {2e  sin6/(l  solution  series i n y,  functions  i n y and  3 in  e x p a n s i o n s and  ignoring  second  3,  y',  + e c o s 9 ) } ( a ' + 1)  -  3*  results  i n decoupling  {3K/2I(1 + e cos0)}sin  2a =  0 .  • ••• ( 3 * 3 ) Since  values of the o r b i t  parameter  are usually  eccentricity  small  (e,|K|<  and  the  asymmetry  0.2), a P o i n c a r e - t y p e  expansion , 0  a (6) =  0  E s=0  CO  I q=0  e  q  leads  to the s o l u t i o n  terms  i n e and  a = a8  +  K  S  a  (6) , q  ,  (3.4a)  S  o f e q u a t i o n (3.3)  up  to the f i r s t  degree  K,  2e(a + 1)(6  - sine) + ( 3 K / 4 a I ) { 6 - ( l / 2 a ) s i n 2 a 6 } , (3.4b)  where  the spin  parameter  a i s defined  as  a = a'(0)  with  a(0)  =  0.  51  Substituting  equation  (3.4b)  into  the governing  equations f o r  the  roll  a n d yaw  degrees  o f freedom and l i n e a r i z i n g  set  of equations  of the  form:  y"  + n  2  l  Y  + m  B*  L 1  =  f  gives  a  ;  1 ±  (3.5)  3" + n  2 2  3 +  Y  where c o m p l i c a t e d  f  l l  ' = f  forcing  1  .  2  functions are represented  = e [ { ( 2 l o + 51 - 3 ) Y -  2 l ( o+ l)B'}cos9  + 2(B  + K [ {4 + a + 3 / 4 ( a + 2 e ) - 21 - I a / 2 } y c o s  -  I - 3/4(a  + a (Y'  f  1  2  =  x c o s 2a9 -  =  (Y -  -  {(B"  2a6 +  {l/2(a  ]  +  1)  + B)/2  { Y " / 2 + Y ( 2 + a)  y and B by  00  oo  E s=0  E q=0  e  q  K  S Y  B')sin9}  + Y') + 1 ( 1 + a)(B  + 2e) - 1 - a ) ( B  Replacing  Y(6)  + Y*)cos9  + a)(B  {(3/4(a  c o s 2a6 -  + y')sin0  2a9 J ;  +B)}sin  2e{I(l  + K[  + 2e) - a ) B '  by :  -  aB'}sin  2o0 ] .  the expansions  „  ,  of the  form:  (0) ;  c q  + Y'/2)/(1  S  (3.6) 00  CO 8 ( 6 )  and  =  E s=0  retaining  considerable  E q=0  e  the f i r s t  q  K  S  B  (9) ; q  degree  ,  S  terms  amount o f a l g e b r a i c  i n e and K g i v e s , a f t e r  manipulations,  the solution  a as  - 2e) }  52  s i n p. 0 i  y)  2  cos p.( l  a. cos6. + e b. , cos6 •  1  1  -L  1  X^X  f  1  I  i=l ^cos  -X sin p f  p 6 i  + K b.  cos5.  9  0  }  1,2  + K A ^  sin(  2  +  K  A  2,j +4  c o s  P  4  and  and p  1  2 - (n  + n  l x  A^, A  2  2 1 2  (P  b. = ± i,n  (P£  6  - (-D e :  + 6) &  + 6 )J  - (-D 2a0  e  j  £  Z  I = 2 f o r j = 2, 4.  2  12  (3.7)  frequencies obtained  2 2 + n . n-, , = 0 , 11 "12  from ....(3.8a)  1  1 2 Pj 2 2 i " l2 P  (3.8b)  ; i = l ,2  n  (3.7) a. , b. ,5. ,6. are d e f i n e d as l ' i,n I ' i,n +  '0 3  I-  m  l l Pi  In equations ^  a. = ±  c o s  £  m )p  1;L  i,j+4  £  0 - (-l) 2ae + 6 ) "|  are c h a r a c t e r i s t i c - m  A  j  £  are g i v e n by  2  n 11 " P i m  e  sin(p 0 - (-l) 9 + 6 )  j  P j i  1 = 1 f o r j = 1, 3 ;  Here p  e  4 + Z j=l  + K b. „ s i n 6 . ^ 1,2  . a. sin6• + e b. , s i n 6 . , \ i i i , l i , l  i  i  ^ P 2  2  VkPkf - \pj  , Ao k A  lp A 2  ' 4n " k P k 4 n - f - A p +. A J Z  A  Z  3  2  2  l P l  1  +  QPk - A  1/2  B  P l  2 /  / 4n- P " y A p - A Z  2  x  2  Z  k  P l  2  An-l\^  1/2  53  f 0 - arctan T 3  V k  +  +  6.  i  \ x p 2  \  k  P  x  -  2  P2 1 "  Pl 2  A  *  X  l P l  \ , o.  ^ , 4n " k k 4n-3 „ = arctan — r " 2 2 l P l Z  X  n  X  x -  ;  A  p  +  X  Z^ , Z  4n-3  =  "  (  '4n-2  A  4n-l  = ~  n , l  = "  " n  +  A  n,5  +  n , 2  A  +  C n,3 P2  +  [>n,5 Pl  +  ^ n,7  +  A  {  {  {  A  { p  )  n , 6  [ n,l Pl  A  4n  2  p  "  2  X  -  2  1  P  \  -±  -—  4n-l kJ  Z  A  ;  , e t c . given by:  2  ( A  l  4n- Pk  Z  = 1, 2 ; i = 1, 2 ; k = 2 / i  with  Z  Z  T  1  :  p  2  s  )  i  c  n  6  °  s  l  6  (2a)  (  2  a  )  "  l  ( A  "  ( A  n-l }  }  +  ^a)"" }  +  1  (  2  a  )  n1  }  +  i  n,4  A  A  +  n,7  + \  1 1 _ i  n1  n,3  A N  2  { p  , 6  n,8  { p  A  +  n,4  A  )  s  i  n  n,8  )  c  °  { p  2  "  P l  {  2  -  ±  "  (  2  s  6  '  2  '  Uo)^ }^  cos6  1  a  )  n-ln  R  c  o  s  2  a  )  R±  }  ]  s  i  n  6  1  2  6  sin  ^^'^1  "  (  2  6  2  6 l  '"  = 1, 2,  and  ( A .  .  \  =  (  [  n  l l "  (p  m  "  M L ) (2a)  1  (-1) ( 2 a )  1  3  V ]  [ n ^ -  -1 + m  1;L  rn  1 2  (p  m  -  j  V  ( p  m  - (-1)  3  ( 2 a )  2  ]  54  ri?„ 12  (p„ m  m 1 2 ( pm v t  X.  (-1) ( 2 a ) j  -  m  .  (-1) (2a) j  1  V  1  1  m  )  n  l l  l l  (  "  p  m  (p  m  "  j (  "  j  2  (  a  )  1  2  a  )  X )  1  1  )  2  i  V i,J+4j X  where X, .' = a { I ( a + 1) (1 + X P ) + 3 ( 1 1, j m mm X  l,j 4 +  *2,j  =  = m <° a  {l  +  1 ) ( X  (-l) X (l  m  P > " ("D^ .* W  +  m  X  2,  j +  4  = -  +  3 ( A  m  P >/  +  +  +  2  e  (-l) a(X  (-l) a(l  + p X )  +  ( K a  1)/(1 -  2e)  P  m  +  m  m  + p )  j  "  )  m  j  + p )} .  ;  m  m  + 4 + a + 3/4(a +  2e)  + 1 ) ( I - 1/2) - 3 / 4 ( a + 2 e ) } ;  m  4 ( a  j  }  1  - 21 - l a / 2 - X p { ( a m  (-1) (X  m  - P^/2)/2  j  - l)/2 -  a p  m  " 2)  +  (a + l ) X ( l / ( l m  -  2}/2  - 2 e ) - 1}  ;  i =. 1 , 2 ; j = 1, 2, 3 , 4 - ; m = 1 f o r j = 1, 2 ; m = 2 f o r j = 3, 4.  To solution,  assess  of  the approximate  i t was c o m p a r e d w i t h t h e r e s u l t s  integration  of the exact  R u n g e - K u t t a method computer  accuracy  equations  with built-in  (Figure 3-3).  of the numerical  of motion,  error  The agreement  analytical  control  o b t a i n e d by t h e o n a n I B M 370  i n frequency  and  amplitude  55  e = K = 0.1 , 1= 2 ; a=2 ,  7 = 10°,  numerical,  'o  y= (3 =0,  [3=0-5  o  o  'o  analytical  20r-  -20  a  Orbits Figure  3-3  A comparison between the approximate p e r t u r b a t i o n and numerical s o l u t i o n s f o r a t r i a x i a l s p i n n i n g s a t e l l i t e  56  are  predicted  3.3.1b  an a c c e p t a b l e  accuracy.  Stability For  lite,  with  y and 6 m o t i o n s o f a s p i n n i n g  the stable  the c h a r a c t e r i s t i c  frequencies  p^ and p  2  should  satelbe  real,  i.e., p  2  =  ±[n  -  4n  + n  2 ±1  2 x  n  2 2  Furthermore,  p  j- 1/4,  2  and  p  2  >  -  2 2  1 / 2  m i l  ]  m  1 2  ± {( n ^ + n  2  2  for noncircular  , 9a  2  -  m^m^)  2  (3.9a)  1,4,9,  , 4a  2  > 0.  orbital  motion  (e ^ 0 ) ,  ,  (3.9b)  f o r s t a b i l i t y o f asymmetric  / a  2  , 16a  2  systems  (K ^ 0 ) ,  ,  ....(3.9c) 53  The above c o n d i t i o n s a  c a r e f u l study  reveals here, are  follow  from the Mathieu equation  .  of the s t a b i l i t y diagram f o r the Mathieu  that with small  values  o f e and K  t h e s u p e r h a r m o n i c s p = 2, 3, 4,  ( e , K < 0.2)  However, equation  considered  , 3a, 4a, 5a,  not excited. Figure  3-4  presents  and  a f o r an a x i s y m m e t r i c  the  amount o f s p i n  rather  rapidly with  a s t a b i l i t y chart  i n terms o f I  satellite in a circular  required  f o r the stable motion  an i n c r e a s e  i n I.  orbit.  Note,  diminishes  57  e = K=0  100i  75  50h  stable 25  unstable 0. 0  0-25  _L_  0.5  " T T T -  075  I  F i g u r e 3-4  S t a b i l i t y diagram f o r s p i n n i n g satellites in circular orbits  axisymmetric  58  3.3.2  Gravity oriented  3.3.2a  systems  Approximate a n a l y t i c a l s o l u t i o n Replacing  the t r i g o n o m e t r i c  t h e i r s e r i e s expansions i n equations second and the  a" -  2 2  = e[  + ™ 3' 2 1  leads  ignoring  the  y' r e s u l t s i n  2{y'  +  (1 - I - K)3}sin9 - K) ;  (3.10a)  3 + m y ' = 2e ( B ' - y ) s i n 9 ;  (3.10b)  22  (3K/I)a = e{-  For the  and  y hy  equations:  + 3y(I - I ) c o s 0 ] / ( 1 3" + n  (3.2)  i n a , 3, and  h i g h e r degree terms i n a , 3, Y, a ' , 8*,  following  y" + n ^ y  functions  a " cos9 + 2 ( a ' + l ) s i n 0 } .  stable gravity gradient  ....(3.10c)  c o n f i g u r a t i o n a i s bounded, which  t o , f o r I > 0, - °° <: K < 0 .  An  expansion  a (0)  leads a - a  +  =  Z q=0  e  4  0  cos  p 0 +  *a  n  2  + (2/(p  2  equation  (3.10c) as  1 2 2 ( a / p ) s i n p 0 + ef {3a„p /(4p - 1)}cos p 0 0 ' a 0 a a a  (l/p ){2/(l - p ) a  (3.11a)  q  to the s o l u t i o n of  n  (0)  a  L  + a'(5p  - l)}sin9 - { a  Q  2  - 2)/(4p  L  2  L  - l)}sin  p ( l + p /2)/(2p a  a  a  p 0 a  + 1) }cos ( p  a  +  1)9  59  + {a p (Q  1 + P /2)/(2p  a  { a ^ l  a  a  - l)}cos(p  + p /2)/(2p„ + l ) } s i n ( a  a  + 1)  P n i  a  + ( a ' ( - 1 + p /2)/(2p - l)}sin(p a  where  p  a  = (- 3 K / I )  a  - 1)  a  - 1)9] ,  a  (3.11b)  1/2  S i m i l a r l y , assuming  Y(6) =  E e q=0  4  (Q) ,  y q  (3.12) 3(6)  yields  the  =  E e q=0  q  solution  (y \  3(6), q  of equations  s i n p, 2 E i=l  cos  A.cos p.8 1  *  1  -  £  E j=l V  =  for  1  where p^ and p p and  4  2 - (n  2 1  L  j  +  4  2  + n  *  Vi a  1  (-D^e  cos(p 8 -  ( - 1 )  £  j = 1, 3 ;  ^  I =  2  2 2 2  - m  2  21  m )p 2 2  2  S i n 6  i  +  e  b  i  s i n 6  i,lJ  + 6) ^ £  3  +  Q  6) £  J  f o r j = 2, 4 ;  are c h a r a c t e r i s t i c  A^ are given by, 2  a^ cosS^ + e b^ cos6^  p.  - A . s i n p.(  1  (A^ s i n ( p e  1  (3.10b) as  > .  + e  (3.10a) and  + n  (3.13)  f r e q u e n c i e s o b t a i n e d from 2 2 1  n  2 2 2  = 0 ,  ....(3.14a)  60  In  equation  (3.13) a ^ , b  , 5  ±  a. = ± 1  A p 2  - X  2  1/2  +  p  +  P A  l P l  2  -  1  A  P l  2  4 - W l \ /"2^ k " - V k^ -A p .+ A A l2 ~ i ; $ Y, k k x 2 l " = a r c tan ^ \ A p - A p o k 0 k, , ,4 ~k k l H 2 - 2 l\ ,• = a r c tan x —-— 2  2  b.l = ±  Z  +  2  l P l  2  i  6. i  X  +  0  P  0  u  V 2  P  p  p  /  p  K  1  A  6• 1,1 i  —^  \ -A p 2  , Z  1  —  2  , z  2  3  , Z  Z  1  = - (A + A )sin6-  Z  2  = - (A^ + A ) c o s 6  Z  3  = -  1  2  6  A  ' "  A  A  "  3 (p  A  A  +  1  2  S  ?  X  p  A  P  p  T  —  Z p  l P l  2  - Z A 3  R  k /  g i v e n by:  4  - (A + A ) s i n 6 ;  L  3  ( P  A  6  ( P  A  ,  , A  D  2  ;  1 " 1) cos6^  + 1) + 8 (p  2  2  7  A  + 1) +  4  - (A + Ag)cos6  1  + 1) + 4 (p  2  P l  7 (P  and A.^ , A (A.  (  P  1  Z  + A  + 1) + A,  ±  5  /  ; k= 2/i ;  = 1, 2  with Z  =  P  x  A  1  Y  Z  4  -j^ are d e f i n e d as  ±  ' o V k k Y f-*o\ W * ' e +  Z  , 6  ±  2  - 1) c o s 6 ; 2  1" 2  1) sin6 ^  - •1) s i n 6  2  ;  d e f i n e d as  \  2 [n 21  (P  " ( - D ) ] [n j  m  2  22 " <m " p  (" ) ) J 1  J  2  61  + ni  2  n  1  m  22  -m  (p  2 2  -  2 2  (Pm "  (p  -  m  -1  Jx2 - (-!)-»)  m  (  "  1  )  j  )  m  2  (-l)-i)  n  2 1  (p  m  -  2 21  -  (-1)3)  X  ( p - (-1) m  m  V  l J + 4j X  where: X  j  =  "  (  -  1  X. . - = 3+4  )  J  {  "  1  (-1) a J  " (1 + m  K ) } a  m\n  p ) nrm  A  "  +  1  - ^ 5  1  "  ^ V '  -  1  K)  ;  j = 1,. 2, 3, 4 ; m = 1 f o r j = 1, 2 ; m = 2 f o r j = 3, 4 . As  before, f o r assessing the accuracy  mate a n a l y t i c a l s o l u t i o n , (3.11b) and numerical 3-5).  g i v e n by  (3.13) were compared w i t h t h o s e  integration  of the exact  i s surprisingly  3.3.2b  equations  equations  obtained  through  of motion  a  (Figure solu-  close.  Stability A study  (i)  results  The a g r e e m e n t b e t w e e n t h e e x a c t a n d t h e a n a l y t i c a l  tions  given  response  of this approxi-  of equations  (3.10)  leads to the conclusions  below: F o r bounded condition  eccentric  addition p^  ft 1 / 4 ,  i n a c i r c u l a r orbit, the  following  must be s a t i s f i e d , -co  In  a-motion  <  K  <  0.  ...(3.15a)  orbits, the s t a b i l i t y  t o t h e above 1, 4,  9,  of a i s ensured  i f ,i n  condition, (3.15b)  ;  62  e = 0-02,  —  a  K—0-5,1-2,  numerical  (Y,|3,a)=5° , (T,p,a)=0 — analytical 0  o  Oh  -10h 0 Figure  3-5  0-5  1 Orbits  1-5  A comparison between the approximate p e r t u r b a t i o n and numerical s o l u t i o n s f o r a t r i a x i a l gravityoriented s a t e l l i t e  63  (ii)  For  the  stability  Pp  m u s t be  p 2  =  5 21 [ n  -  4n  real.  +  n  y, 6 m o t i o n s  Thus,  22  "  n2 }  2 1  of  1 / 2  2  m  21  ]  m  In e c c e n t r i c o r b i t s , p  ji 1/4,  2  in  1,  The r e g i o n o f  extends  21  from  the  the  represents  the  gravity  P  a  , p^  , p^  (e < 0.2)  by  abscissa to  > 1 are  considered  the  axisymmetric  the  results  3.4  Force  of and  Let  effect = p^  not  systems  I and  K  1) . of  3 and  1/2,  1.  The  effect  of  the a motion y  note =  the  eccentricity for frequencies  t h a t the  1/12,  unstable 1/3,  small,  the  i s experienced  Note  cir-  considered  representing  line  a  in  the  superharmonics values  conditions  which  agree  of  with  Moment A n a l y s i s AF  be  the  force necessary  in i t s position  tion  Am  i n t e r m s o f AF  be  of  6,  2.5.  Am  can  values  For  the  on  Also  for  element of  =  for a,  diagram  excited for small eccentricity  here.  section  2  i s comparatively  (i.e. I + K =  p^  )  y motions  =  expressed  22  satisfied.  stability  P  region  m  oriented satellites.  s t a b l e 3 and  similar  21  ( 3 . 1 5 c ) , m u s t be  i s to promote resonance A  m  (3.15d)  the  1.  "  ,  of  1/2,  22  condition  systems  orbit  n  (3.15c)  axisymmetric  a  +  ot i s s t a b l e f o r a l l t h e  orbit,  here. and  3-6  { ( n  and  (3.14a) t h a t  .  9,  motions of t r i a x i a l  cular  1  addition to equation Figure  and y  4,  circular orbit,  i t f o l l o w s from equation  22  > 0  in a  expressed  as  as  to maintain  shown i n F i g u r e  ( i n the  x,  y,  3-7.  z system)  a certain The  and  mass  acceleragravity,  g  64  a  unstable (e*0)  |3,T  unstable (e^O)  -co  4  -3  a stable (3,Y unstable  >/  *y  gravity oriented axi symmetric systems  S j>  /  /  /  .^-PP -V2 r  7  -co  Figure  3-6  S t a b i l i t y diagram f o r t r i a x i a l satellites i n arbitrary orbits (e < 0.2)  gravity of small  oriented eccentricities  F i g u r e 3-7  Force e q u i l i b r i u m of a mass element Am maintain i t at a d e s i r e d p o s i t i o n  to  66  ^2<[C]  ( R } ) = [ C ] {g } m  m  +  I C  ] if!>  ....(3.16)  f  where:  R  m  = p o s i t i o n vector =  to mass element Am  (R s i n y cos3 + x)u + ( R cosy cosa + R s i n y sinS  + y ) u + (-R cosy s i n a + R s i n y sin3 cosa + z ) u y  g y  = -  m  (y/R ) 3  R  m m  l l  =  c o s  3  ,  ,  [C] = t r a n s f o r m a t i o n  C  sina  matrix w i t h elements C^_. ,  cosy /  - s i n a sin3 cosy - cosa siny , C  13  C  21  =  C  ~  O  S  a  c o s | 3  s i n  3  s i n  cosy + s i n a s i n y ,  Y  cos0 + sin3 sine ,  = cos6(sina 23  =  c  ^31  =  c o s  C^2  =  C  C^^  o  s  ^ ( 3  sin3 siny + cosa cosy) - sine cos3 s i n a , sin3 s i n y - s i n a cosy) - sin6  c o s a  siny sine - sin3 cos6 ,  sinG(sina  sin3 siny + cosa cosy) + cosG cos3 s i n a ,  = sin6(cosa  Using equations 1  1  cos3 cosa ,  sinB  s i n y - s i n a cosy) + cos6 cosa cosB .  (2.26) and the approximation  3  — 5 - - —-- [1 - — {x s i n y cos3 + y ( s i n a s i n y s i n 3 + cosy cosa) R R m J  J  R  + z(cosa the  siny sin3 - s i n a cosy)}] ,  f o r c e e x p r e s s i o n can be r e w r i t t e n  AF = u ) ( l + e c o s 6 ) ( A F u  + AF u  xx  yy  2  3  Here AF^ , position  AF^. , AF  z  are r a t h e r  c o o r d i n a t e s x, y,  ments, v e l o c i t i e s and  (3' -  ....(3.17)  lengthy e x p r e s s i o n s  z of Am  and  librational  involving displace-  ' 3 -  (1 + e cos8){( y  2 cos3 + cosy sin3)  siny) }] 2  + y [ - 3 siny cos3(siny - 2e s i n 6 ( - y '  sin3 s i n a + cosy cosa)  cos3 cosa + 3' s i n a - siny  - cosy sin3 cosa) + + 2y'  .  zz  2  .= x [ l - 3 s i n y cos +  + AF u ) Am  accelerations:  2 AF  as,  sina  (1 + e cos0){-y" cos3 cosa  sin3 (- cosy sin3 s i n a + s i n y cosa)  + 2y a  cos3 s i n a - y  + 23'a'  cosa + 2a  sm3  cos3 s i n a + 3  (cosy sin3 s i n a - siny  + cosy cos3(cosy sin3 s i n a - s i n y  sina  cosa)  cosa)}]  + z[3 s i n y cos3(cosy s i n a - s i n y sin3 cosa) - 2e s i n 0 ( y '  cos3 s i n a + 3* cosa - siny cosa  + cosy sin3 sina) + - 2y'  (1 + e cos9){y" cos3 s i n a  sin8 (cosy sin3 cosa + s i n y  sina)  12  i i  n  + 2y a  cos3 cosa - y  sin3 cos3 cosa + 3  - 23'a'  s i n a + 2a' (cosy sin3 cosa + s i n y  cosa  sina)  + cosy c o s 3 ( c o s y sinB cosa + siny s i n a ) } ]  ;  x [ - 3 s i n y cos3(cosy cosa + s i n y s i n g sina) -  2e s i n 0 ( y  cosB cosa - 3  2  2  sina  (1 + e cos0){y" cos3 cosa  + cosy s i n 3 cosa) + + 2y'  sina + siny  2  cosy cos 3 s i n  I  I  - 2 '3'  a  Y  s i n 3 cosa •  s i n 3 cos3 s i n a - 3 " s i n a + 2 3 ' cosy cos3  - y'  + cosy cos3(cosy s i n 3 s i n a - s i n y  cosa  cosa)}] 2  + y [ l - 3(cosy cosa + s i n y s i n 3 sina) +  (1 + e c o s 0 ) { 2 y '  cos3 s i n a ( c o s y s i n 3 s i n a  - s i n y cosa) + y'p' cos3 s i n 2 a + 2 y ' a ' s i n 3 2  1  2  2  (1 - cos 3 s i n a) + 2 3  - y'  \  „ i 2 . 2  -j s i n 2y  - sin y 2  +  i2  2  3  z[-  .  cosy cos3 - a  2  - cos 3 s i n a +  s i n a ( c o s y s i n 3 cosa „ '  s i n a - 2a  + s i n y sina) - 3 2  '  sin^3)  (cosy cosa + s i n y s i n 3 sina) }] sin3  -  3 2 2 a + -j s i n 2 a (cos y  cos  2e s i n 0 ( y ' s i n 3 - cosy cos3 -  (1 + e cos0){y" s i n 3 + 2 y ' cos8 s i n a ( s i n y 1  + 23 '  1  2  1  + cosy s i n 3 cosa) '+ 2y .3  Y  cos3 cos a +  c o s a ( s i n y s i n a + cosy s i n 3 cosa) 2  s i n 2 a - a" 2  1  2  s i n 2y s i n 3 cos 2 a 2  + j s i n 2 a ( c o s y s i n 3 - s i n y)}] ;  1  2  —  a')  sina 2  cos 3 s i n  x[3 siny cos3(cosy s i n a - siny sinp - 2e s i n e ( ~ y  cos3 s i n a - 3  cosa)  cosa + siny  cosa  - cosy s i n 3 sina) + (1 + e cose){-y" s i n 3 s i n a + 2y' cosy cos 3 cosa + 2y'3' s i n 3 s i n a ,2 y  ti  - -2~ s i n 23 cosa - 3 J  i  cosa - 23  ,  cosy cos3 s i n a  + cosy cos3 (cosy s i n 3 cosa + siny s i n a ) } ] +  Yl  -  3  s  i  2 2 2 2a (cos y - s i n y s i n 3)  n  2  s i n 2y s i n 3 cos 2a + 2e sine (y ' s i n 3 - a it  - cosy cos3) + (1 + e cos0){-y  sin3  + 2y ' cos3 cosa (cosy s i n 3 s i n a - siny •  2 y' 2 cos3 s i n a + -~2~ cos 3 s m  1  cosa)  2  - 2y 3  2a  + 23' s i n a (siny cosa - cosy s i n 3 sina) 12  3  •  -  sin  , s i n 2a + 2"  ,  o  2a  ,  (~  "  + a S  I  .2  N  Y  s i n 2y -  2—  . „ S  I  N  P  0  cos  2a  , 2 . 2. , , + cos y s i n 3) / J n  2 + z [ l - 3(cosy s i n a - siny s i n 3 cosa) +  (1 + e c o s 0 ) { 2 y ' a ' s i n 3 - y'ft' s i n 2a cos3  + 2y ' cos3 cosa (siny s i n a + cosy cosB cosa) - y ' ( l - c o s 3 c o s a ) + 23' cosa (siny cosa 2 2 • - cosy s i n 3 sina) - 3' cos a - 2a cosy cos3 12 2 - a - (siny s i n 3 cosa - cosy sina) - cos 3 cos a}]. 2  2  2  2  2  70  The of  the  moment r e q u i r e d  satellite  can  M  to  t h u s be  obtained  r  =  attain  x AF  the  desired orientation  from  .  ....(3.18)  m  m s 3.5  Librational A  and  their  vast  v a r i e t y of  systematic  information. typical  R e s p o n s e and  are  presented  nents  at  a given  system parameters in  Figures  correspond  3-8  symmetry but of  fuel  hold  develops  (Figure  the  sence of  any  XX  3-8  transverse  forces  of F  the  roll  Similarly,  and the  the  force  3.1),  satellite  w h i c h has  a  to  correspond  those  to  compo-  affected are  p h y s i c a l parameters  the  satellite  and  various a'  yaw axial  There  pitch F  y  quency  spinning  a s y m m e t r y due  represents  the  V a r i a t i o n of  of  by  given  closely  geometric  the  expenditure required  to  in position. .  periodic variation  3-9.  few  p l o t s as  external disturbance.  and  a  phase plane  (Table  Forces  the  disturbance  only  problem  extensive  motion,  The  inertia  3-1).  i o n probe Figure  and  3-10.  to Explorer  the  librational  for a triaxial  through  to  here.  f o r the  location,  Forces  resulted in rather  for conciseness,  Response data  the  variables inherent  variation  However,  results  Time H i s t o r y o f  i s no  rate.  vanish.  c o n d i t i o n i n the  Also,  roll as  Influence  or can  of  yaw,  1  system parameters  librations force  on  expected, arbitrary -  z  remains  just  be  an  ab-  essentially set the  in with  i s recorded  in  unchanged but significant  i o n probe remains  Figure high  fre-  modulations.  almost  constant  71  1-2,  CT  = 220,  Vli-£|5'-0  350  a  Y = (3=0  275  200  3h  F -F -0 y  z  e = 0.1,K = o,0.1  6 F (X10),2 X  N  1.5  0.5  0  Orbits Figure  3-8  T y p i c a l p l o t s showing t h e c o n d i t i o n o f a s a t e l l i t e i n t h e absence o f any e x t e r n a l  spinning disturbance  72  e = K = 0.1 , l = 2a =220, ;  To=  5,  B-V-0, (3 = 0.5  °,-> ' ^ 0  TP,a  0  R(x10),  Figure  3-9  Response of a s p i n n i n g s a t e l l i t e to an a r b i t r a r y d i s t u r b a n c e . Note the high frequency amplitude modulations of l i b r a t i o n s ( r o l l , yaw) and t r a n s v e r s e f o r c e components  73  but  relatively  The  phase p l o t s  over of  a large  the  the  3-10a) . effect  A  quency of  tial tions  an  influence  roll  increase  expected,  3.1  and  a decrease  and  increase  in a  of  a change  t o be  The  summarizes  librations  and  (b)  3-10  International Ionospheric S a t e l l i t e (Figure 3-1)  a corresponding e f f e c t of  rise  3-10(c).  amplitude  of  both  A  As  can  affects  the  0  0.1  0.1  2  220  0.15  2  0.15  the be  I  .15  in  libra-  affects  3-10d).  -0  The  of  T  100 mts.  90 mts.  -1  frethe  substan-  the  K  0  the  i n the  e  The S a t e l l i t e S o l a r P o w e r Station (Figure 1-4)  (Figure  a decrease  i n Figure  disturbance  0  as  a s y m m e t r y p a r a m e t e r K.  (Figure  The S p a c e S h u t t l e ( 1 7 5 , 0 0 0 k g ) m a i n t a i n i n g an instrument p a c k a g e (200 kg) a t t h e end o f a r i g i d m a s s l e s s 400 m l o n g boom a l o n g t h e local vertical  in  indicates  Values of system parameters i n the examples the t r i a x i a l s p a c e c r a f t under c o n s i d e r a t i o n configuration  effect  well  eccentricity  spin parameter  initial  as  the  orbital  The  and  manner  i n the  the  3-9  I is illustrated  similar  only  substantial increase  librations.  i s noticed.  i s p e r i o d i c but  Note a  i n the  frequency  appear.  3-10  inertia  appears  frequency  Figure  Figures  i n the  at high  motion  frequency  yaw  i n the  forces  performance  Table  to  parameter  and  the  orbits.  comparison of  significant  forces  system v a r i a b l e s .  due  of  inertia  suggest that  amplitude  forces  transverse  number o f  important  both  large  24 hrs.  74  e-0.2, K = 0.1, 1 = 2 . a = 220, v = 5°, |3-Y'-0,  F i g u r e 3-10  E f f e c t of system parameters on the s a t e l l i t e r e sponse: (a) i n c r e a s e i n the o r b i t a l e c c e n t r i c i t y  p'-a5  75  e-0-1, K = 0, l=2 a = 220, Y=5°, P=-/=0, [£=0-5 ;  Figure  3-10  o  o  E f f e c t o f system p a r a m e t e r s on t h e s a t e l l i t e r e sponse: (b) r e d u c t i o n i n t h e i n e r t i a a s y m m e t r y parameter K  76  F (x10), N 0 v  Y  ^  3  £(x10), N 0  F i g u r e 3-10  E f f e c t of system parameters on the s a t e l l i t e r e sponse: (c) r e d u c t i o n i n the i n e r t i a parameter I  77  e=K=0.1,1=2; cr = 300, Y= 5° p =Y^0, ^=0.5 Q  F i g u r e 3-10  o  E f f e c t of system parameters on the s a t e l l i t e response: (d) i n c r e a s e i n the s p i n parameter a  78 e=K-0.1, 1=2, a =220, Y=R=0, Y'=0.5, |3=-5° o  O  F i g u r e 3-10  'o  o  E f f e c t of system parameters on the s a t e l l i t e response: (e) d i f f e r e n t i n i t i a l c o n d i t i o n s  'o  79  resulting the  r e s p o n s e ,but  same  only  (Figure  3-10e).  Typical  response  data  s y s t e m a r e shown i n F i g u r e s eters  closely  instrument (Table  correspond  package  fora triaxial  3-11 a n d 3-12.  correspond  negative  3-11 r e p r e s e n t s  rather of  i n F^ a n d  leading  force  tional  an e x t e n s i v e  motion  initial  rigid  required  boom  to hold the  However,  and f o r c e s  significant  Interestingly,  i n t h e boom.  Figure  concerning  form.  t o the system  F^ i s a l w a y s  3-12  and asymmetry parameters i n a compact  leads  the forces are  fore / 0  amount o f i n f o r m a t i o n  inertia,  disturbance  components on the probe i n t h e As e x p e c t e d ,  are noticed.  to tension  eccentricity,  massless  a 200 k g  i n position.  i n a circular orbit.  variations  oriented  supporting  t o those  absence o f any e x t e r n a l d i s t u r b a n c e . constant  gravity  character  The p h y s i c a l p a r a m -  a t t h e e n d o f a 400m l o n g  package  Figure  the overall  t o t h e Space S h u t t l e  3 . 1 ) . The f o r c e s  instrument  locally,leaving  summarizes the effect  on t h e l i b r a -  The p r e s e n c e  librations with  o f an  p  < p„  6  a < p^ cies In of  (case  a).  Also  (relatively,  note  the fluctuating  see Figure  forces  a t higher  frequen-  3-11) a c t i n g o n t h e i n s t r u m e n t  package.  g e n e r a l , t h e f r e q u e n c y o f F was f o u n d t o be g r e a t e r F o r F . The e f f e c t o f an i n c r e a s e i n e c c e n t r i c i t y y z x  significantly  increase  substantially  affecting  asymmetry  (case  c  inertia  of F  x  B, Y a n d F  of a, F (case  and  b).  without  An i n c r e a s e i n  c) i s t o r e d u c e p ^ and p ^ and t o i n c r e a s e  complex modulation o f F frequency  the amplitudes  than that i sto  y  and F  z  i s also noticed.  parameter  i s just  i n addition t o a decrease The e f f e c t  the opposite  (case  p^.  i nthe  o f an i n c r e a s e d).  Note  A  i nthe  that the  80  Librational Forces (I =2) (b)e=0-1,K=-0.15 (2) 45 to x,y,z axes  (a)e=0, K =-0-15,-0.25 (1) along local vertical •, ;  0.1 F ,N X  a,2  -  a,1 b,1 ;  0  0  a,2  Fy ,N -0-2  a," b,1  -04  a,1  0 F ,N 7  -0-03  a,2 0  Figure  0-25 3-11  0-5 Orbits  075  Force components on an i n s t r u m e n t a t i o n package deployed from a n o n l i b r a t i n g t r i a x i a l g r a v i t y o r i e n t e d s a t e l l i t e (Space Shuttle)  81  (a) e-0.K--0.15, I-2-. (b)e-0.02,K—0.15,l-2,-(c)e-0,K—0.5,l-2(d)e-0,K—Q15.I-3  a[3,a) = 5 ,  (Y,P,a)-0;  0  ;  10i—  -0-3 0-04rL  F,,N  0  -0-04  0  Figure  0-5 3-12  1  Orbits  1.5  E f f e c t o f i n i t i a l c o n d i t i o n s and s y s t e m p a r a m e t e r s on l i b r a t i o n a l m o t i o n and f o r c e c o m p o n e n t s f o r t h e Space S h u t t l e s u p p o r t e d i n s t r u m e n t a t i o n package  82  a x i a l component o f the f o r c e constant 3.6  (F ) i s negative  and e s s e n t i a l l y  f o r a l l the cases i n v e s t i g a t e d .  S a t e l l i t e S o l a r Power S t a t i o n (SSPS) The  S a t e l l i t e S o l a r Power S t a t i o n as proposed by G l a s e r ^  i s one o f the more e l e g a n t and c h a l l e n g i n g concepts o f r e c e n t times.  A schematic diagram o f the SSPS i n the geosynchronous equa-  t o r i a l o r b i t generating was shown e a r l i e r  8,000 MW  (Figure 1-4).  through p h o t o v o l t a i c  conversion  The energy i s t r a n s m i t t e d by a  microwave beam t o give 5,000 MW d.c. power on the e a r t h . T h i s s e c t i o n aims a t s t u d y i n g the dynamics, and  stability  c o n t r o l o f the SSPS using the a n a l y s i s developed e a r l i e r i n  t h i s chapter.  I t a l s o summarizes the a v a i l a b l e i n f o r m a t i o n on  the e f f e c t of s o l a r r a d i a t i o n pressure  on the o r b i t a l  perturba-  t i o n s o f the space power s t a t i o n . 3.6.1  C o n f i g u r a t i o n c o n t r o l f o r maximum power  generation  For the maximum e f f i c i e n c y of conversion i n t o e l e c t r i c a l energy, the panels Obviously, panels  from the s o l a r  of the SSPS must face the sun.  t h i s would r e q u i r e d i r e c t i o n a l c o r r e c t i o n s of the  to account f o r the motion of the SSPS i n the geosynchronous  o r b i t and of the e a r t h i n the e c l i p t i c .  This i s e a s i l y  achieved  through the r o l l r o t a t i o n y and p i t c h r o t a t i o n a about the l o c a l h o r i z o n t a l and the x - a x i s , r e s p e c t i v e l y . F i g u r e 3-13 shows the e q u a t o r i a l and e c l i p t i c planes . P o s i t i o n o f the sun with represented  r e s p e c t t o the l i n e  by <p^ , the s o l a r aspect a n g l e .  o f nodes i s Let u  g  and u  be  Equatorial plane  R Geostationary  orbit  Ecliptic Line of nodes  Sun  Figure  3-13  Configuration  control  o f t h e SSPS  f o r t h e maximum p o w e r  generation  CO  84  the  unit  vectors  respectively.  i n t h e d i r e c t i o n o f t h e sun and  F o r u ^ t o be u  s  i n the d i r e c t i o n of  . u  = 1  y  the u  y-axis,  g  ,  ....(3.19a)  i.e., -  siny  s i n i sincj) + cose)) c o s y c o s (a + s s -  sincj>  8)  cos i cosy s i n ( a +  s  0)  =  1  (3.19b)  This y i e l d s : Y = - arc sin(sin a = Thus, can  [0  i sin<f> ) ;  + arc tan(tan$  the r o l l  s  cos i ) ] .  and p i t c h r a t e s  be w r i t t e n  necessary  ^s ., . 2. . 2, ,1/2 s (1 - s i n l s i n <J> ) '  . '  [1 +  ;  l  n  1  C  O  S  9  ,i  where  = -  ° 2 *g] 1 - s i n i s i n <J> s C  revolutions  Stability The  imately  taken  and h a v i n g  to control  the  SSPS  value  a = -1  the region  (3.21b)  orbital  rate  around the  sun  SSPS  of spin  as - 1 .  ....(3.21a)  per day).  of the  3.3.1, a s t a b i l i t y  that  the earth's  s  3.6.2  1  2  <$>' r e p r e s e n t s  (1/365.25  ....(3.20b)  as:  S  a'  ....(3.20a)  p a r a m e t e r o f o r t h e S S P S may  Using  the analysis  presented  diagram f o r the spacecraft  roll  a n d yaw  approx-  section  in a circular  c a n be d r a w n a s shown i n F i g u r e  of stable  in  be  3-14.  motion i s very  orbit  Note small.  85  e = 0,  Figure  3-14  O = -1;  unstable--  S t a b i l i t y diagram f o r a t r i a x i a l spacecraft i n the geosynchronous e q u a t o r i a l o r b i t w i t h the s o l a r panels f a c i n g t h e sun.- N o t e t h a t t h e c o n f i g u r a t i o n (I=K=0.15) c o r r e s p o n d i n g t o t h e SSPS l i e s i n t h e u n s t a b l e r e g i o n .  86  In  f a c t the  (Table  configuration of  3.1)  w o u l d be  i s indeed  required  with  respect  3.6.3  Force  to  distribution defined  by  obtain  of  (3.17)  forces  equations  and  and  ( 3 . 1 8 ) may  moments t o  (3.20a)  distribution  as  of  3-16 the  they  Note  that  Newton-meters.  control  moments f o r f i v e  Note,  the  transverse  axial  component.  the  x =  3 km,  In  Figure  A  an  preliminary  40  i t s capacity  3.6.4  per  Orbital perturbations The  orbital  cent of  o r i e n t the  i n t e r e s t i n the  perturbations  of  the  due  to  corresponding i s shown i n  y  a n <  ^  a  '  the  forces  the  order  the  small,  of  sun  also  several of  are  shown.  of magnitude higher  SSPS t o  the  f o r max-  histories  analysis suggests  of  Figure  required  are  order  time  d i f f e r e n t p o s i t i o n s of  moments a r e  d i r e c t i o n at  c o n t r o l moments a r e  librational  3-17,  as  shows  to o r i e n t the panels  corresponding the  determine  3-15(a)  the  v a r i a t i o n s of  to  (0.5  i t s panels  orientations  axial  0,2km, w h i l e  maximum p o w e r r e q u i r e d MW  c o n t r o l moment  to  Figure  r e s u l t i n enormous c o n t r o l moments, o f  million  the  (3.20b).  station,  although  used  achieve  solar aspect angle The  be  mass a l o n g  shows t h e  imum p o w e r g e n e r a t i o n . included.  kg  a transverse  Figure  functions  per  and  longitudinal stations, z =  (b) .  Glaser  o r i e n t a t i o n of  c o n t r o l moments  two  3-15  desired  and  force  at  by  Thus a c o n t i n u o u s  the  the  distribution  proposed  sun.  distribution  Equations  SSPS as  unstable.  to  the  the  power  face  the  that sun  the is  about  generation).  solar radiation  solar r a d i a t i o n pressure SSPS d a t e s b a c k  than  to  the  pressure  induced  days of  i t s  (b)  x = 3km  Force,N 5  (X10)  ^(x10 )  0  1  ^  r  -2 Figure  3-15  ^  ^  \  1  1  1  -1  0  1  2  z(km) D i s t r i b u t i o n of f o r c e per u n i t mass on the s o l a r panel (y = 100 m): (a) z = 0, 2 km; (b) x = 3 km  88  Figure  3-16  R o t a t i o n s a n d c o n t r o l moments a s f u n c t i o n s o f t h e s o l a r a s p e c t a n g l e f o r maximum p o w e r g e n e r a t i o n  89  o  = 0  -5' 0 F i g u r e 3-17  , -45  ! 0.25  o  , 45—-  I 0.5  0(orbits)  o  , ±90  I 075  Time h i s t o r y o f the c o n t r o l moments f o r s e v e r a l v a l u e s o f the s o l a r a s p e c t a n g l e  I  1  90  conception.  However,  a detailed  mathematical  analysis  of the  31 topic e*  was i n i t i a t e d  only  = 0.0001 f o r t h e SSPS  with  an area/mass  summarized (i)  as  b y V a n d e r Ha  .  (assumed  as a p e r f e c t l y  black  - 5 ) , the long  SSPS v a r i e s  term o r b i t a l e f f e c t s  maximum v a l u e  The m a j o r  axis  (iii)  The a n g l e  between  increase  of the i n i t i a l l y  periodically  (ii)  The a n g l e the of  with  a period  3.7  essentially  constant,  t h e l i n e o f nodes  and p e r i g e e  shows an  182.5 days  f o r <}>  one y e a r  followed  by  i s zero  the orbit  periodically  = T T / 2 , 3TT/2  a  again, o f t h e SSPS a n d  with  a n d 365 d a y s  i n the value  sudden  a  period  for  = 0,  o f i i s about  = T T / 2 , 3TT/2 . cycle  i s completed  the i n i t i a l  The as  varies  The maximum v a r i a t i o n  Concluding  of the  remains  of the orbit  o f i n c l i n a t i o n between  The r e g r e s s i o n  of  be  o f e i s 0.1.  plane  node  may  365 d a y s .  o f 180° o v e r  0.25° f o r Q (v)  body  o f about  equatorial  n.  taking  circular orbit  j u m p o f 180° when t h e e c c e n t r i c i t y (iv)  By  follows:  The e c c e n t r i c i t y  The  recently  of the longitude  i n about  325 y e a r s  eccentricity  and s o l a r  o f the ascending and i s independent aspect  angle.  Remarks  significant  aspects  of the analysis  may  be  summarized  follows: (i)  A general going  formulation  f o r the t r i a x i a l  l i b r a t i o n a l motion  presented.  satellite  i n an a r b i t r a r y  orbit i s  under-  91  (ii)  Highly  complicated  equations yields ably  are  coupled,  solved using  approximate the  numerical  solutions  not  only  also  give  nonautonomous  a perturbation procedure which  solutions that  with  but  nonlinear,  analysis.  compare q u i t e The  closed-form  reduce computational  better  i n s i g h t i n t o the  favor-  time  and  physics  of  effort the  problem. (iii)  Expressions and  for forces  o r i e n t an  should  prove  from o r b i t i n g (iv)  object useful  of  Shuttle the  are  i n planning  obtained.  scientific  to p o s i t i o n They experiments  space s t a t i o n s ,  practical launching  SSPS.  moments r e q u i r e d  i n space  A n a l y t i c a l procedures tions  and  are  applied  importance: an  instrument  to  three  Explorer package  XX,  configurathe Space  i n space  and  92  4.  DYNAMICS OF SPINNING AND  GRAVITY  ORIENTED MULTIBODY SYSTEMS 4.1  P r e l i m i n a r y Remarks As d i s c u s s e d e a r l i e r , ever i n c r e a s i n g demand on a d d i -  t i o n a l power has l e d to l a r g e r and hence n e c e s s a r i l y f l e x i b l e spacecraft. the RAE,  TOI,  We  have a l r e a d y looked at a few of them i n c l u d i n g  CTS  and SSPS.  Although  the c o n f i g u r a t i o n s are  v a s t l y d i f f e r e n t , they a l l belong to the f a m i l i a r c l a s s of multibody  systems i n t e r c o n n e t e d through  r i g i d or  flexible  links. This chapter s t u d i e s the l i b r a t i o n a l dynamics, f o r c e d i s - ' t r i b u t i o n and o r i e n t a t i o n c o n t r o l of a system of two bodies, i n t e r c o n n e c t e d through  a t e t h e r or a beam, and nego-  tiating a circular trajectory. motion f o r the complicated  finite  The governing equations  of  system are o b t a i n e d accounting f o r  g y r o s c o p i c and g r a v i t a t i o n a l torques.  An approximate c l o s e d  form s o l u t i o n of the h i g h l y n o n l i n e a r , nonautonomous and coupled equations i s o b t a i n e d through v a l i d i t y assessed through  l i n e a r i z a t i o n and i t s  the numerical a n a l y s i s of the exact  equations of motion f o r a p a r t i c u l a r case.  I t provides consid-  e r a b l e i n s i g h t i n t o the dynamical  and  the system.  behaviour  s t a b i l i t y of  Next, g e n e r a l e x p r e s s i o n s f o r f o r c e s and moments  a c t i n g a t the end bodies are d e r i v e d . i s a p p l i e d to a c o n f i g u r a t i o n of  Finally,  practical  the a n a l y s i s  interest:  the  Space S h u t t l e s u p p o r t i n g a s u b s a t e l l i t e or an i n s t r u m e n t a t i o n  93  package. 4.2  Formulation  o f the Problem  Consider  a s y s t e m o f two a r b i t r a r i l y  mass  , M2 w i t h  through  an e l a s t i c  the  c e n t e r s o f mass a t S-^ and S t ( l e n g t h JI) , i n a c i r c u l a r  (Figure 4-1).  with  the o r i e n t a t i o n  of the l i n e  d e f i n e d by t h e E u l e r i a n r o t a t i o n s y orbit)  and a  ( i n the plane  deformations w(y)  while  librational a  i  of the  '  3  i'  Y  t h e end b o d i e s motion i^  ~  link  2  T  u  s  t o undergo  general  specified  t h e system has e i g h t  Using inertial  o f the connecting  1  to specify  link.  F i g u r e s 4-1, 4-2 and r e c o g n i z i n g t h a t  frame o f r e f e r e n c e X ,  by  degrees  f r e e d o m i n a d d i t i o n t o t h e number o f modes u s e d deformation  of the  a r e d e n o t e d by u ( y ) and  are permitted  n  of centers (S^S^  The t r a n s v e r s e  (with r e s p e c t t o the l i n k ) ' ^ *  around  The s y s t e m i s f r e e  (out o f t h e plane  o f the o r b i t ) .  o f the connecting  orbit  Here S r e p r e s e n t s t h e  c e n t e r o f mass o f t h e c o n f i g u r a t i o n .  librate  connected  2  link  center of force 0  overall to  their  shaped b o d i e s o f  ( i n the  Y , Z'), 1  (4.la) z. 1  94  Figure  4-1  Reference c o o r d i n a t e s and the geometry of motion f o r a multibody e a r t h o r b i t i n g system  95  Figure  4-2  F o r c e e q u i l i b r i a o f t h e m a s s e l e m e n t s Am. t o m a i n t a i n them a t d e s i r e d p o s i t i o n s  and  Am  96  and  o  u +  R :  [6]  (4.1b)  [y] [ a ]  0  where [cu]  transformation are defined  gravitational  1  [ 0 ] , [y],  matrices  i n Appendix  potential  I, the expressions  energies  R  .  .  mi  R  9  -  R  M.  yp R  mi  M.  The  m  -JL  E i=l  cable  for kinetic  and  be w r i t t e n a s :  . dm. + f l mi i 2  2 =  may  [B^] a n d  '2 ^  * i=l  P  [ a ] , [y^],  -JL  expressions  a n d beam,  m  dy  (4.2a)  ,  J  (4.2b)  dy  m  f o r the e l a s t i c  f o r small  R  u a n d w,  potential  have  energy  of the  the form:  JL  T  . 8u> 2  r  ec  2  {  3 ^  (  . 3w> 2 -, ,  +  (  ay  }  }  d  (4.2c)  '  y  -JL  [  eb  (  E  I  )  f  2  (  i!u,2  +  8y  .  +  T  e  {  (  | S , 2  +  (  |2,2  } ] d y  ,.  3y (4.2d)  where  T  e  tension  = - F  Note,  i n the connecting  = y  here  y=L  link  (T ) i s g i v e n  (4.2e)  F  :  the tension  by,  y=-L  n  i s assumed t o be  constant.  97  Substituting  from equation  (3.17) i n equation  (4.2e) g i v e s ,  for a c i r c u l a r trajectory,  T  * •2 *2 2 • * = M L{(y - 9 )cos a + y 0 s i n y  g  + a  2  + 2a 0 cosy + 40  With the energy may  2  cos y  s i n 2a cos a}  2  2  .  ....(4.2f)  e x p r e s s i o n s known, Hamilton's  principle  be used by a l l o w i n g the v a r i a t i o n i n each of the degrees  of  freedom to be zero, i . e . , t  2  t  r 2  L  a  dt =  5L  dt = 0 .  ....(4.3)  a  T h i s leads to the equations of l i b r a t i o n s (y, a; y^ and v i b r a t i o n s of I ^ =  ( u  '  4.3  w )  (u, w).  are l i s t e d  cable  =  Motion  ( u  '  w  '  The  equations f o r a p a r t i c u l a r  i n Appendix I I .  9 7 ' 37 beam " }  0  derivatives,  y" + 4y +  Here, f o r y = - l ^ , & ; 2  *  a, y.  , 6^ , U, W and  the equations i n Appendix II may  (1/1*)  - } ( y + y . ) - {(1 + a.)I . - 2 1 .}&!] yi ' ' i ' l xi yi l v  their  be l i n e a r i z e d :  2 Z [I .(y" + y") + 4{(1 + a . ) / 4 ) I . 1=1 J  - I  case  i n the Small  Assuming small v a l u e s of y,  y:  , 3^ , a^)  98  {Y - (£ /£) } (U" + 4U) dY = 0 ;  - e  a:  a" + 3a + ( l / l * )  ±  E I„,(a + a.) i=l  {Y - (£ /£)}(W" + 3W)  + e  Y :  i  :  (Y- + Y " ) I  V  1  l  &  - 2l  a. :  y i  + 4(Y  y i  ±  +  { ( 1  + Y){(1 + < J . / 4 ) I  }(  Y  V ^ i . - V  +  0 ;  ^  (4.4b)  " I >  x i  ±  0; '  i  (4.5a)  +{ ( 1  +  a  i  xi  ) z  + y!) = 0 ;  r  (4.5b)  a. + a" = 0 x  The of  dY  1  - $ . { ( l + a . ) I . - 21 .} = i l x i y i  5  (4.4a)  1  (4.5c)  corresponding l i n e a r i z e d  equations  f o rthe vibrations  t h e c a b l e may be w r i t t e n a s : U:  9  2  U 2 + U - (Y - £ / £ ) ( y " + 4y) - T  = 0 ;  x  90  2 9 W + 30  (Y  (a" + 3a)  -  -  TQ  2  3 9Y  Similarly,  0  W  9Y  . . (4.6a)  2  2 = 0  (4.6b)  t h e e q u a t i o n s f o r t h e v i b r a t i o n s o f t h e beam  (of u n i f o r m c r o s s s e c t i o n ) r e d u c e t o : 4 U:  k  9 S  9Y  U - T. 4  0  9  2  U  9Y  2  2 . 9 U + U - (Y + 90'  ( y " + 4y) = 0 ; (4.7a)  99  3W  i^W _ _  4.3.1  (4.4a)  expressions  y  y:  2  Solution f o r the the tether  Substitution tions  3W  ,  2  +  and  of equations  (4.4b),  case  (4.5a) and  respectively,  leads  (4.5c) i n t o  equa-  to the simplified  f o r y and a motion as  4y  -  e  (Y -  ( U " + 4U) d Y = 0 ,  ....(4.8a)  0 1 a:  a  + 3a + e  (Y - £ / £ ) ( w " + 3W) 1  which ly,  i n conjunction with equations  yield  linear,  These v i b r a t i o n separation  dY = 0 ,  (4.6a) and ( 4 . 6 b ) , r e s p e c t i v e -  homogeneous a n d u n c o u p l e d e q u a t i o n s  equations  ....(4.8b)  may b e s o l v e d u s i n g  i n u a n d W.  t h e method o f  of variables'* , i . e . , 4  U ( Y , 6 )  =  V (Y) 1  W(Y,0) = W ( Y ) 1  U  2  ( 0 )  ,  (4.9a)  W (6).  (4.9b)  2  The  resulting  s o l u t i o n s f o r U a n d W may b e w r i t t e n a s  (a„  • cos p , . 8 + b  oo U =  I  CU1  r  C U l  CU1  . s i np *CU1  .0)M. ,  ....(4.10a)  C l  oo W =  Z (a i=l c  w  l  . cos p . 0 + b . s i np .0)M . • cwi cwi ^cwi c i  ....(4.10b)  100  Here, the c o e f f i c i e n t s a . ,a . , e t c . are determined from ' cui cwi following  relations  {  A  cv  }  [  c  D  =  v  The  (v = u o r w) :  =  {B* }  ]  _  1  {  [D ]-  cv  N  }  ....(4.10c)  ;  {N^}  1  c  .  elements of the square matrix  d  . . Cl]  _  i^j  tD ]  cos Z h  e  n  cos(n 2(n  + n  cp  + n  cp  e  2  . h  2 £,  {  _L_  ) -  cm  2 2£ £,  hi  cm  m =  £  C  ci  J  .  cp  ) -  cm cm  .  n ££  sin n  .  £1' n."cj  sin(n . + c i n . + n cj c  n .) c i . i }  .  c i . ; J  •, n  9±  2 n . c i  1  )  ]  '  ; p = ij/m ;  ....(4.10e)  £,  cj  n  - n  1 - cos n .  +  .  £1 +  '  n „c,i  1  l+  2 n . cj i ,j  - n  cp  (1 -  2  .  9.1 + 2e h  .  2  . - n .) J + n . - n . cj c i C  c  s i n 2n  n  cos(n 2(n  1  i  C  sin n  1 - cos n  £ *  cos  1  )  . [h c] 3  sin(n  O 0  -  n  T  h  2  ci  CJJ  by:  £, 1 - c o s i{ £ n P  -  n  o +  , .. = h 2  given  e  +  cp  d  r  sin n  2£  [ cm  m  +  a  . - n .) s i n ( n . + n .) c i cj c i c] n . - n . " n . + n . cx CJ Cl CJ  2  +  ....(4.10d)  sin(n  n  ±r  the  .{- . °  sin n n  . cj  cj  3  .  5-^} 2  (3 - 4 c o s n  n  „ +  2 e  „ , h .{^ c i 3  2£  n  2  J  . + cos °  1 2  ,  sin n -  2n  .) °  3  n . c i J  .  °1  3  101  1 - cos n  +  In  22) +  2  ^  I 4<| + o 2  s i n 2n  £2  2  cj  . - 8 s i nn .  Sl) }  4n C  t h e above e x p r e s s i o n s , n ^ i s t h e i * " *  tic  '  J  (4.10f)  root of the  1  characteris-  equation s i n n  . + e h  .{£  (cos n . - l ) / £  + l } = 0,  . . . . ( 4 . lOg)  where , h  ci  =  (  sin n . , c i —~2 n . ci  n  r  In-.  2  }  c i  £, (I. - £„)  , 1 1  +  J> J> 2 ^ 1 2 4-  s i nn n  solving  substituting  f o r y and a leads  T  2  0 n  ~  .  +  r  ~  e  {  n . - 1 c i £. 1 -  C l .  . C  Finally,  ~  [  7z  I  2JT  and  2 n . c i  £, + X,_ c o s n . T 1 2 c i , 0  5—(cos  In .  1  C  n  2£_ - J> 2 1 61 "  . C  (4.10h) •  1  f o r U, u " , e t c . i n t o to the following  1) ) ] •  1  e q u a t i o n s (4.8)  expressions:  •  Y =  YA  Yr, 1  c o s 26 + -~- s i n 20 + e 0 2  CO  I e*.{a . (cos p .0 - c o s 20) c i cui cui r  P + b  .( s i n p .0 cui cui r  a' a = a . c o s /38 + — 0  sin  2  s i n 20)} , '  /I8  - e  .... ( 4 . 1 1 a )  I e*.{a .(cos p .0 - c o s / 3 8 ) c i cwi cwi r  P + b  .(sin p CWI  4.3.2  Solution  r  .0  CWl  s i n /30) } .  f o r t h e beam  Substituting  (4 . l i b )  ^  from  case  equations  (4.8a) and  (4.8b)  into  equations  102  (4.7a) and  (4.7b), r e s p e c t i v e l y , leads to a s e t of l i n e a r , homo-  geneous and uncoupled  equations i n U and W f o r the beam case.  An  a p p l i c a t i o n of the method of s e p a r a t i o n of v a r i a b l e s as before to the v i b r a t i o n equations r e s u l t s i n the s o l u t i o n s which may  be writ-  ten as :  = ^ / ^ u i  U  W  =  i^^bwi  C  C  O  O  Pbui  S  S  bwi  P  9  6  +  +  b  b  bui  bwi  S  S  i  i  Pbui  n  9 ) M  b i '•  P b w i  n  9  )  ....(4.12a)  M  b i  (  4  -  Here the c o e f f i c i e n t s a, . , a, . , e t c . are determined bui bwi ' f o l l o w i n g r e l a t i o n s (v = u or w),  { A  bv  }  =  [ D  b  ] _ 1  { N  bv  }  { B  bv  }  =  [ D  b  ] _ 1  { N  bv  }  '  1  b  +  ( q  bi  + -TT-, 2 (n +  (q  L  b  b i  +  b i  q  bj  )  - n  {  1  K4  c  o  s  (  n  b  + b  i  n b j  i  b j  )>]  T-[ (q, . q, . + l ) s i n ( n , . - n. .) . ) b i bj bi by' H  " q ){cos(n . b j  ....(4.12c)  - D s i n ( n , , + n. .) -'»-"i" - «  K  L V M  b ;  )  [D, ] o  [ (q . q ^ ^ b i bj  -  b  from the  , e t c . of the square matrix  n  2 ( n . + n .)  2  ....(4.12d)  w i t h the c o e f f i c i e n t s d, , , d, bll oiz given as:  *bij|i^j  •  1  b  - n  b j  )  -  1}]  103  + £ —= ^—{ (q n + k bm bp m  +  2  b  n, m  b  s,  m  b  + r.  p  b  p  b  k, ) c o s h k, P  p  sinn  b  (qi_ k, s, - r, n, ) s i n h k, ^bm bp bp bp bm bp  b  m  c o s n, bm  + ( bqm^ k, bp r, bp - n, bm s, bp ) ( c o s h k, bp c o s n, bm - 1) M  +  (^bm q^  + —=  b p + k,bp s,bp) s i n h k, bp  s i n n, bm }  _ { ( , . r, . k, . - s, . s, . k, .) c o s h k, . s i n h k, . v b i bj bi bx bj bj bx bj " bj r  ,2  2  bi  k  ubm  n  k  + (r, • k, . s, . - r, . s, . k, .) ( c o s h k, . c o s h k . - 1) bx bx bj bj bx bj bx bj v  +  +  +  (s, . s, . k, . - k, . r, . s, . ) s i n h k, . c o s h k, . bx b j bx b ] bj bx bx bj (  b i  s  b i bj " bj bj b i  k  r  I bm  £  h  S  [ ( s i n  n  R  bp  +  q  r  bp  C  O  )  s  i  n  h  k  "bp "  S  q  b i  S  bp  i  n  ) / n  b j  h k  }  bp 2  +  bp " bp  ( s  r  - h +  ( 1 / n  ( r  + e  2  b  h  bp  S  ±  n  " bp r  cosh k  p  b  i  h  %  h  f a  " bp S  bp  / k  + s  b p  j(1/3  b  ) / £  p  +  l  C  °  h  k  2 {{  sinh +  -  S  C  r  b p  At  —  bj  bp  "bp  S  +  q  bp  S  i  n  2  ; m = i , j  2  I  N  H  2  K  K' h 3  bp  ) / n  bp  +  ~ 2 n  bj  ; p = ij/m; (4.12e)  _  q S  n  b p  i /l )  . + s . 4 k  °  ) / k  k )/k }/£]  2  +  bp  cos  2n. b  ^  .)  104  + 2 k  bj  — (cosh 2k, ^  - 1) + -= ? { ( r , . k. . n . •+ k . ^ 3 bi b: r  2  2  b  b  + <3t, • s, • n, .) cosh k, . s i n n, . + (q, . s, . k, . - r, . n, .)x ^bj b j b j ' bj bj bj bj bj bj bj' KH  . k, . + q, . r, , n, .) s i n h k, . s i n n x cos n,b]. s i n h k,bj. + (s, bj b] ^bj bj bj bj bj +  ( q  b j b j b j " b j bj r  + 2e h + s +  b j  k  b j  S  ]  [ £ { ( - cos n 2  (  c  b j  b  s  n  + q  b j  bj  c  o  s  h  bj "  k  sin n  b j  b j  cos n  b j  )  }  + ( r  b j  b j  - q  b j  cosh k ) / k . . ] + e h 2  bj  )/n  1  b  cosh  j  k  b j  - r /k .)/£  b j  2  b j  o  sinh k ) / k } / £ -  (sin n . + q  - s  n  2 j  b j  )/n  2 j  b  + (s  (l/3 -  b j  - r  b  j  sinh  k  b j  + £ /£ ) 2  2  (4.12f) The f r e q u e n c i e s of v i b r a t i o n of the beam may be c a l c u l a t e d u s i n g the values of n ^  , the i * " *  b  r o o t of the c h a r a c t e r i s t i c  1  equation 5i s, . - [n, . (cosh k, . - cos n, . ) + z-. bi bi bi bi' k ^ e  b  {k, . (e . - n, . e~ . £, /£) bi l i b i 2 i 1' n  x cosh k, . - n, . ( e . - e„. k, . £,/£)cos n, . + (n, . e bi b i 3i 2i b i 1 bi b i 3x 0  bi^l - k, . e, . ) (1 bi l i £  v  n  s i n n, . ) } ] [n, . s i n n, . + k, . s i n h k, . bi bi bi bi bi  s  5i + e i {n, . (e_ . - e. . £, k, ./£)sin n, . + k, • (e. . - e„.) k, . b i 3 i 4i 1 b i bi b i 4i 2i bi e  x (1 - cosh k ) b i  +  k b  i  (e  3  i  - k  b i  e  2 i  £- /£)sinh k > ] L  b i  1  x  = 0. (4.12g)  Using the s o l u t i o n s f o r U and W i n equations  (4.12), y  x  105  and  a f o r the beam may be found as YQ  y = Y  cos 20 +  0  + b  b u i  (sin  s i n 20 + e  p  b u i  _  a = a  0  Ji^  a b u i  (  c o s  Pbui  6  "  c  o  s  2  0  )  (4.13a)  a' y--  - cos /3 9) + b  4.3.3  e  i=l p, . - - ^ i s i n 20)} ,  0  cos /30 + —  n  E  Solution  b w ±  s i n /30 - e  (sin  P  b w ±  E e*.{a, . (cos p, .0 b i bwi ^bwi p, . = ^ s i n /30) } /3~  0  (4.13b)  f o r the l i b r a t i o n s of the end-bodies  Using equation  (4.5c), a s o l u t i o n  f o r the p i t c h motion  of the end bodies may be w r i t t e n as, (CK + «Q)0 + aQ - a ; i = 1, 2.  =  S i m i l a r l y u s i n g equations and  ....(4.14a)  (4.5a) and (4.5b), the s o l u t i o n  3^ ( i = 1, 2) i s expressed as 2 y, = E a.. s i n ( p * 0 + 6 j_l J J i J  f o r y^  follows: ) - y  ;  (4.14b)  E a.. A . , cos (p* .0 + 6. .) j=l i j !J !D --3  ;  (4.14c)  x  X  2  3- = where a. l j  ( f o r k = 2/j) =  g M  (  "  i 0  +  P  (  Y  0  +  Y  iQ ik  Pik 2  U  3  i2 i2 - Pil i l A  A  +  {  -^Q  +  ^Q^ik Pi2 i l A  +  B  i0  " il P  A  i2 . (4.14d)  106  i0 6 . . = a r c t a n (J^ 3  +  P._  J  0 - i O —i k  ( Y  +  Y  X . _  *i2  ii £ —ki i i x  U  -  p.-  i 2  P  P  X . ,  * i l i  -  i 2 i± i l i " ii l A  Y  l  P  +  n  X  ii 2 i — i i  Y-AJX..  '0  +  g.  p..  n  'lO' l k  )  lO " l k  (4.14e) and  p*-^ , * p. * i  a  r  e  ^  It governing  2 i l l  *2  m  P  i s of interest  librations  P  t o recognize  (section  3.3.2),  accuracy  i n the present  i ti s reasonable  the small.  useful  t o expect  stability  thermore,  i n terms o f bounded  stability  the librations  a l s o bounded.  system  (U = W = y  =  OL  i n f o r m . The  been e s t a b l i s h e d  similar  order o f  motion  i n the connecting  link,  trigonometric func-  o f the vibrational  The s t a b i l i t y  o f motion  o f the  o f the connecting  y anda, which  i s appro-  equations  stability  Even i n t h e absence o f damping  t o ensure  link  o f the motion  o f the linearized  Thus t h e m a t e r i a l s t i f f n e s s  sufficient  II) fora  (3.2) h a v i n g  i n f o r m a t i o n about  U and W can be e x p r e s s e d tions.  (Appendix  case.  The s o l u t i o n  yields  that the equations  connecting  o f equations  A comment c o n c e r n i n g  normally  (4.14g)  (3.2) w i t h e = K = 0 a r e i d e n t i c a l  o f the solution  are  «  o f t h e endbodies  validity  here.  (4.14f)  * m • -i il2 i j . *2 2 n.*", „ i j " il2  non-vibrating, non-librating  0) a n d e q u a t i o n s  i  P  * ~" i j  ™ i l l  1 D  in  from  n  X . .  priate  frequencies obtained  n  n  =  characteristic  e  4 , 2 ^ 2 > *2 ^ 2 2 - ( n . , , + n . , „ - m... m.,~)P+ n. n.,„ = 0 , i l l i l 2 i l li l 2 ^ i i l l i l 2  with  with  n  link i s  motion.  Fur-  a r e i n f l u e n c e d b y U a n d W,  o f the motion  y + Y^  a  n  d 3j_ i n  107  t e r m s o f 1 ^ a n d o\ by  using  Figure  , i = 1, 2, f o r t h e e n d - b o d i e s may  3-4.  Note  relatively  higher  4.4  a n d Moment  Force Let  AF  values  a large region  of inertia  element  Am o n t h e c o n n e c t i n g  of  be e x p r e s s e d  and AF^  ( i nx^  to maintain  link  a n d Anu  as shown i n F i g u r e  f o r c e s may  of stable motion f o r  and s p i n parameter.  ( i n x, y, z system)  be t h e f o r c e s n e c e s s a r y  the  studied  Analysis  system)  respectively,  be  4-2.  , y^  , z^  an a r b i t r a r y  on t h e i  Through  mass  end-body,  Newtonian a n a l y s i s ,  i n terms of t h e a c c e l e r a t i o n  components  t h e mass e l e m e n t s a s :  {AF}  {AF..} =  [a ] ±  =  T  [a]  T  [y]  [B.] [y.] T  T  T  [ 6 ]  [a]  T  T  2 [-^ dt [ ] Y  {R } -  T  [ 6 ]  T  {g }]Am ;  2 ° [-^{R dt  } "  (4.15a)  } ] Anu. (4.15b)  Here  the transformation matrices  Appendix ^ given may  I.  earlier  The v e c t o r s (equations  be e x p r e s s e d  g  Equations  R  m  and R  4.1)  [cu] , [B^]-, e t c . a r e g i v e n i n . (ini n e r t i a l mi  reference)  were  and t h e a c c e l e r a t i o n due t o g r a v i t y  as:  g ^m  = - 4r R 3 m m  . mi  =  ;  ....(4,16a)  R  ~ R . . 3 mi mi  (4.16b)  ]  ( 4 . 1 5 ) c a n now b e u s e d  for calculating  the force  acting  108  at any p o i n t o f the connecting l i n k and the end-body. ate the f o r c e s a c t i n g a t the c e n t e r s o f mass ( S end-bodies, or L  2  equation  1  , S ) of the  (3.17), w i t h e = S = x = z = 0  , may be used as the s o l u t i o n o f e q u a t i o n  To e v a l u -  2  and y = -L^  (4.15a) .  The moment r e q u i r e d to maintain the end-body i n a d e s i r e d o r i e n t a t i o n can be o b t a i n e d  M.  =  r  l  i  X  A  from  i  F  •  ....(4.17)  M.  X  Furthermore, tions  f o r s m a l l y, a, y  i  , B  i  and t h e i r d e r i v a t i v e s , equa-  (4.5) r e p r e s e n t l i b r a t i o n s o f the end-bodies.  small amplitude  motion, e q u a t i o n  _ M.  i  =  •'_  Thus f o r the  (4.17) may be r e w r i t t e n as,  _  r . x AF* J i i  (4 .18)  M. x  where AF? i s o b t a i n e d from equation  (3.17) w i t h e = 0 and by  r e p l a c i n g x, y, z, y, 3, a, y', 3'/ a', y", 3", a" with x z  ±  3V 4.5  , y + y  ±  ,  B  i  i  , y  i  ,  , a + cu , y' + y' , 3]_ , a' + a | , y" + yV , ±  , a" + aV , r e s p e c t i v e l y . R e s u l t s and D i s c u s s i o n To demonstrate the u s e f u l n e s s o f the a n a l y s i s , i t was ap-  p l i e d t o a system o f c o n s i d e r a b l e i n t e r e s t , the Space S h u t t l e . The example s t u d i e d here c o n s i d e r s a 175,000 kg o r b i t e r i n a 90 minute o r b i t which supports a s u b s a t e l l i t e o r an instrument package by a  109  cable  o r a beam.  Table  4.1.  with  Note  Figures  first  cable  4-3 t h r o u g h  system.  parameter  reduces  increase appears  trary  Effect  w i t h e = 0.  t o be l e s s 4-4  pronounced a n d 4-5  t h r e e modes  on t h e  frequency  line  forces  Table  Beam  i s reflected  .  i n an  flexibility  response  and t h e i r  was o b s e r v e d  total  from  presents  In a l l the cases,  The l i b r a t i o n a l  as w e l l  for arbi-  t o be s l i g h t l y  as t h e time  a t t h e c e n t e r o f mass o f t h e o r b i t e r  4.1  of the  The c o n t r i b u t i o n  to the exact response.  f o r t h e W component.  connecting  Cable  the stiffness  show t h e v i b r a t i o n  represented by N  (S^S^)  the  a t h i g h e r modes.  of U vibrations  of centers  Here,  arbitrary  Influence of  i s significant  approximation  that  e  of the cable-  parameter  As e x p e c t e d ,  o f t h e mode.  i n i t i a l conditions  close  the  of the f l e x i b i l i t y  t o t h e mode s h a p e d u e t o a n  i n amplitude  first  than  parameter  4-10 s h o w t h e r e s p o n s e  w i t h a n i n c r e a s e i n e, w h i c h  Figures  the  are listed i n  f o u r modes o f t h e c a b l e i s shown i n F i g u r e 4-3.  tension  a  investigated  the increase i n the f l e x i b i l i t y  s i n e wave c o r r e s p o n d s  the  cases  I.  p and  connected  Different  a,  motion history  f o r these  higher y of  of the initial  Values o f system parameters i n t h e example o f t h e S p a c e S h u t t l e s u p p o r t i n g a p a y l o a d (M^ = 2 0 0 k g ) by a c a b l e o r a beam link  case  I ,m  p,kg/m  (EI) ,N b  a b c  400 400 1000  0 .06 0 .12 0 .12  -  d  400  0 .30  1460  m  2  k  s  -  —  0.15  T 0 n  e  0 1 25 12 .5 0 2 5 0 5 5  0 .5  110  [a] T =25, £=0-1[b] T = 12.5, £ = 02 [c] T = 5, £=0-5 G  ;  0  ;  First mode  a ----- -  Figure  4-3  Modal r e p r e s e n t a t i o n (a) f i r s t a n d s e c o n d  f o r cable modes  vibrations:  G  Ill  F i g u r e 4-3  Modal r e p r e s e n t a t i o n f o r c a b l e v i b r a t i o n s : (b) t h i r d and f o u r t h modes  112  N=[U(Y,0)=U W(Y,0)-W, |U(Y,0) c  O)  o  Q  First mode  0=0  3  U,W(x10)  U,W(x10)0  U,W(X10)0  U,W(x10)0  3 U,W(X10)  Figure  4-4  I n i t i a l displacement f i r s t f o u r modes  as represented  by t h e  113  U(Y,0) = U  o  &u(Y,e>  0=0  Y=0-5  0;  U(x10) 0 3  -5  Second mode  . 1  U(x10) oK\-  U(x10) 0 4  -5 5 3  Total  l'\ V' /A  a I'A X  0  ,'  /  A  » \  v ' V \  V  \l  •  •V' \\/  0.125 Figure  4-5  0-25 Orbits  0-375  R e s p o n s e o f t h e s y s t e m when t h e t e t h e r i s d i s t u r b e d at the center: (a) t i m e h i s t o r y o f t h e t r a n s v e r s e d i s p l a c e m e n t o f t h e t e t h e r , U component  114  W(Y,0) = W, o  ^W(Y,9) =0 9=0  ;  Y=0.5  First mode 3_ W(x10) 0  Second mode W(xiO) 0  W(x10) 0  W(x10) 0 0-125 Figure  4-5  0.25 Orbits  0-375  0-5  Response of the system when the t e t h e r i s d i s t u r b e d a t the c e n t e r : (b) time h i s t o r y of the t r a n s v e r s e displacement of the t e t h e r , W component  115  conditions of  about  The  are p l o t t e d i n Figure  4-6.  180° b e t w e e n t h e i n d u c e d  axial  component o f t h e f o r c e i s almost  noted  that the forces presented  to  those  acting at the center  Figures  motion  a n d 4-8  disturbance  induced  parameters. 1^  4-7  Here,  information  control  are equal  show t h e r e s p o n s e  of the s a t e l l i t e  3-4)  should  t o ensure  prove  i s noticed.  constant.  o f mass o f t h e  the spin parameter  = 0.1, s e e F i g u r e  The  here  applied to the cable  librations  a phase d i f f e r e n c e  y and a motions  be  impulsive  Here,  I t should  and  opposite  satellite. t o an  line.  initial Note  the cable  f o r two d i f f e r e n t  spin  cu i s s o c h o s e n ( f o r  s t a b l e £L  and y^  useful i n designing  an  motions. appropriate  system. Figures  vibrational  4-9  a n d 4-10  response  the  tether center.  the  size  show t h e c o r r e s p o n d i n g  to the impulsive  disturbance  transverse applied at  The m a g n i t u d e o f t h e d i s t u r b a n c e  (mass = 1 y g , s p e c i f i c  gravity =2)  km/sec) o f t h e b i g g e s t m i c r o m e t e o r i t e  i s based  and v e l o c i t y  expected  on  (45  to h i tthe cable  55 in  an e i g h t hour p e r i o d  .  ing  v i b r a t i o n s i s extremely  are  a l m o s t t h e same. Figures  ysis using  Note small  4-11 t h r o u g h  that the magnitude o f the r e s u l t and t h e f r e q u e n c i e s  4-13 p r e s e n t  the results  f o r t h e case o f the Space S h u t t l e s u p p o r t i n g a beam.  satisfy  a  The p h y s i c a l p a r a m e t e r s o f t h e s y s t e m  o f U and W  of the analsubsatellite (Table  4.1)  the condition 42(EI)  b  >>  T  I  2  e  ,  (4 .19)  116  Figure  4-6  Time h i s t o r y of l i b r a t i o n s and f o r c e s a c t i n g at the center of mass of the o r b i t e r a f t e r the cable i s d i s t u r b e d at the c e n t e r  117  To o=0 =a  Figure  4-7  Time h i s t o r y of impulsive  Y„«<4-0.05  f  (N) , (c)  o f l i b r a t i o n s and f o r c e s i n i t i a l conditions  c  fora set  118  e = K -0, l 0.1; ^ = a=0-05, (Y, a,Y (3,^,^=0,(N ) ,(c) 0^50aplOO: r  lf  c  •(X2)  a, 50  49.91 0 Figure  o  0-25 4-8  0-5 Orbits  075  Time h i s t o r y of the s a t e l l i t e l i b r a t i o n s as a f f e c t e d by two d i f f e r e n t v a l u e s of the s p i n parameter  119  [C];N=;{u,w, | u C  4  0  U  Y^O-5'  a0  1=0, { d.U Y£0-5'o  0-30°  U0  Y=0-5  , i WY=0-5>o d0  W  Z  U,W[x10]0 _  4  L  4r-  0 = 60  7.  U,W[x10]0 -4 4  0 = 90  U,W[X10] 0 7  -4 4  0 = 120  UW[x10 J 0 — 7  -44r-  0 = 150  U,W[x10j 0 7  F i g u r e 4-9  U and W v i b r a t i o n s f o r d i f f e r e n t 0 a f t e r the cable encounters a micrometeorite impact  J  120  [N ], c  [c], 9 = 90°,  U  ,  W  First mode  0 F i g u r e 4-10  0-25  0-5 Y  075  C o n s t i t u e n t mode shapes of U and W v i b r a t i o n s e x c i t e d due to a micrometeorite impact  1  121  derived  by u s i n g  potential  (EI), — 2 —  energy,  2  f2 l  b  t h e beam modes i n t h e e x p r e s s i o n equation  i n the following  .2 T r2 .9 w^2, -j e (—j) } d y » -2%  0  0  ,.9 u 2  {(—j)  (4.2d),  +  N  *i  f o rt h e e l a s t i c  ;  . . ,3u,2  U^)  form,  • . _ . ,9w>2-,  + (g^)  ) dy .  i  (4.20) Here,  t h e system has t h e v i b r a t i o n  fixed  beam  than  o f a cable  noted of  (i.e.,  t h e beam  case(d)  spond  TQ  first  of Figure  Here,  (Figure  using  r  a  ^  n  e  r  I t i s t o be  the f i r s t curve)  mode  i n  ( a ) , (b) a n d ( c ) c o r r e -  T Q = .0, c o r r e s p o n d s  the least  4-12) a s w e l l  In  effective  a s shown i n c a s e ( a ) . response f o r  o f t h e beam.  mass o f t h e m i c r o m e t e o r i t e  the remaining  Both  i n k ,  mode s h a p e s d i f f e r e n t l y .  (N^) a t t h e c e n t e r  case,  maintained  t h e changes  4-12 a n d 4-13 s h o w t h e v i b r a t i o n a l  the previous  indicated.  t o t h e beam  Interestingly,  T Q = 0, e x h i b i t s  disturbance  as b e f o r e .  cases  beam a r e shown  i n t h e p a r a m e t e r s o f t h e beam a s  (maximum a m p l i t u d e )  be 6 yg here w i t h  values  i s derived  modes o f t h e v i b r a t i n g  4-11.  t h e beam w i t h  impulsive  against  ^2^'  by i t s s t a t i c d e f l e c t i o n  and e i n f l u e n c e t h e d i f f e r e n t  Figures  to  (4.19)  the local horizontal.  stiffness  an  three  that the case(a),  general,  = 0 f o ry =  fixed-  (4.20).  t o t h e changes  along  of a  u = w = 0 f o r y = -l-^, i ^ ) •  (approximated  The  Note  (i.e.,  = ^  that the expression  expression  in  u = w =  characteristics  t h e time  parameters  history  having  As  i s taken t h e same  o f t h e beam c o n f i g u r a t i o n  as l i b r a t i o n a l motion  and f o r c e s  (Figure  122  [a]k-0.15, T =£ = 0 [b]k = 0-15, T=5 , £ = 0 [c]k = 0-25, T = 5, £ = 0 [d]k = 0.15, T= 5, £ = 0-5 s  o  s  ;  G  s  ;  Q  s  0  First mode  0-25 Figure  4-11  0-5 Y  075  Modal r e p r e s e n t a t i o n f o r beam v i b r a t i o n s  U,W[x10] 0  U,W[X10 ] 0  UW[x10] 0  U,W[X10] 0  F i g u r e 4-12  u and W v i b r a t i o n s f o r d i f f e r e n t 9 when the beam encounters a micrometeorite impact  124  T°[X10 ] 0 4  o° [X10 ] 0 4  f^[xlO] 0  y  F  F [X10J 7  0 0-25  Figure  4-13  0-5 Orbits  075  Time h i s t o r y o f l i b r a t i o n s and f o r c e s a t t h e c e n t e r o f mass o f t h e o r b i t e r due t o t h e m i c r o m e t e o r i t e i m p a c t o n t h e beam  4-13)  are  recorded The  (1^ =  here.  librational  1), supported  disturbances  by  -1 i s due  to the  the  resulting  roll  can  scanning  4.6  A  as  elastic  equations  reduces  the  and  should from A  is  the  change  spin  system.  satellite  parameter  Note  in pitch  performance of  aspects  coupled,  are  of  the  rate, a  that though  satellite  investigation  may  The  f o r the  orient  prove  design of  an  useful space  criterion  is  system  presented.  the  their  solution  time  f o r c e s and  and  exact not  effort  physics of  only  but the  solu-  also problem,  moments r e q u i r e d t o  o b j e c t i n space are i n planning  obtained.  scientific  posiThey  experiments  stations. i n terms of  a beam t o e n s u r e  presented.  and  c l o s e d form  into  multibody  n o n l i n e a r , nonautonomous,  linearized  computational  orbiting  length  of  the  interconnecting link  better insight  Expressions tion  Here,  formulation for a t r i a x i a l  obtained.  gives  spherical  follows:  an  hybrid  (iv)  the  kg  galaxy.  Highly complicated  tion  200  Remarks  general  with  (iii)  and  a  a beam, t o t h e m i c r o m e t e o r i t i c  motion  more s i g n i f i c a n t  summarized  (ii)  motion  a distant  Concluding  (i)  orbital  of  4-14.  s t r o n g l y i n f l u e n c e the  The be  a cable or  i s shown i n F i g u r e  a-^=  small,  response  stiffness,  a particular  tension,  mode o f  and  vibration  126 /  ' / rS  '.  [Y,a,T p a ,T,a T (3 a ] = o [a] T = 5, £ = 0-5, N , [b] k = 0.15,T = 5, £ = 0-5, N 1I  G  c  s  r  1  )  r  r  1  o  0  Y°[X1C)]0 5  [x10]0  Y°[x10J 0 4  -1 5 -a-a 6 [X10]0  -5 0 Figure  4-14  L i b r a t i o n a l h i s t o r y of a s p h e r i c a l s a t e l l i t e due to: (a) a micrometeorite impact on the cable; (b) a micrometeorite impact on the beam  t  127  (v)  A n a l y t i c a l procedures are a p p l i e d to s e v e r a l r a t i o n s of the Space S h u t t l e i n the The  studying  supporting  a  configu-  satellite  orbit.  a n a l y s i s presented here should a l s o prove u s e f u l i n  s e v e r a l other e x i s t i n g and  future  configurations.  128  5. Before for  closing,  C L O S I N G COMMENTS a few s u g g e s t i o n s r e g a r d i n g t h e d i r e c t i o n  future investigations  study  of the attitude  solar  radiation  control  In  roll  from  Note  that the  having  t h e yaw  (3)  connecting motion  members s h o u l d be c o n s i d e r e d i n c o n j u n c t i o n w i t h  connecting fluence  link,  general  (a) m o t i o n s .  I n the case  i t s longitudinal  t h e system  end-bodies  behaviour  symmetric  mass  With  of the connecting  t h e Space S h u t t l e  not too distant  It  promises  future,  of zero-gravity  metal  forming, b i o l o g i c a l  scheduled  to i n -  and r i g i d .  Thus a  include the as w e l l  as,  f o r the  the  longituflexibility,  end-bodies.  t o go i n t o  the orbit  age has  commenced.  o f o b t a i n i n g t h e most d e s i r e d c o n d i -  and p e r f e c t vacuum d u r i n g t h e s t u d i e s o f experiments,  tigations  will  involve  supported  from  t h e Space S h u t t l e .  from  e t c . A number o f i n v e s -  the use o f t e t h e r e d payloads  forces acting  ment and r e t r i e v a l  extensible  Furthermore,  a new e r a o f s p a c e  the f e a s i b i l i t y  tions  In  link,  and geometry d i s t r i b u t i o n  in  induced  should  of  their  are likely  significantly.  are not necessarily  vibrations  o f an  vibrations  f o r m u l a t i o n of the problem  arbitrary  the  feasibility  3.6.  system  areas,  A  o f t h e SSPS u s i n g t h e  section  of a multibody  cross-sectional  (y) and p i t c h  dinal  control  s h o u l d be o f i n t e r e s t .  a r e known  the analysis  members o f l a r g e these  and o r b i t a l  pressure  requirements  are appropriate here.  on these  the orbiter  near-earth orbits,  The l i b r a t i o n a l payloads  launched  dynamics and  during their  s h o u l d be a n a l y z e d  the success  or  o f a space  deploy-  i n detail.  mission  will  129  be i n f l u e n c e d by the atmosphere. multibody  T h e r e f o r e , the a n a l y s i s of the  systems presented i n Chapter  4 should be extended  account f o r the atmospheric drag f o r c e s . the problem  to  A s i m i l a r e x t e n s i o n of  d u r i n g deployment of a payload must be g i v e n a very  important c o n s i d e r a t i o n .  130  BIBLIOGRAPHY 1.  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Chobotov, V., " G r a v i t y - G r a d i e n t E x c i t a t i o n of a R o t a t i n g Cable-Counterweight Space S t a t i o n i n O r b i t , " Transactions of the ASME, S e r i e s E: J o u r n a l of A p p l i e d Mechanics, December 1963, pp. 547-554.  133  T a i , C.L., and Loh, M.M.H., "Planar Motion o f a R o t a t i n g Cable-Connected Space S t a t i o n i n O r b i t , " J o u r n a l o f Spacec r a f t and Rockets, V o l . 2, No. 6, November-December 19 65, pp. 889-894. A u s t i n , F., "Nonlinear Dynamics o f a F r e e - R o t a t i n g F l e x i b l y Connected Double-Mass Space S t a t i o n , " Journal of Spacecraft and Rockets, V o l . 2, No. 6, November-December 1965, pp. 901A u s t i n , F., and Pan, H.H., "Planar Dynamics o f Free R o t a t i n g F l e x i b l e Beams w i t h T i p Masses," AIAA J o u r n a l , V o l . 8, No. 4, A p r i l 1970, pp. 726-733. S t a b e k i s , P., and Bainum, P.M., "Motion and S t a b i l i t y o f a R o t a t i n g Space Station-Cable-Counterweight C o n f i g u r a t i o n , " J o u r n a l o f S p a c e c r a f t and Rockets, V o l . 7, No. 8, August 1970, pp. 912-918. C r i s t , S.A., and E i s l e y , J.G., "Cable Motion o f a S p i n n i n g Spring-Mass System i n O r b i t , " J o u r n a l o f S p a c e c r a f t and Rockets, V o l . 7, No. 11, November 1970, pp. 1352-1357. Nixon, D.D., "Dynamics o f a Spinning Space S t a t i o n with a Counterweight Connected by M u l t i p l e Cables," J o u r n a l o f S p a c e c r a f t and Rockets, V o l . 9, No. 12, December 1972, pp. 896-902. Bainum, P.M., and Evans, K.S., "Three-Dimensional Motion and S t a b i l i t y o f Two R o t a t i n g Cable-Connected Bodies," J o u r n a l of S p a c e c r a f t and Rockets, V o l . 12, No. 4, A p r i l 1975, pp. pp. 242-250. Bainum, P.M., and Evans, K.S., "The E f f e c t o f G r a v i t y - G r a d i ent Torques on the Three Dimensional Motion o f a R o t a t i n g Space Station-Cable-Counterweight System," AIAA 13th Aerospace Sciences Meeting, Pasadena, C a l i f . , January 20-22, 1975. Saeed, I . , and S t u i v e r , W., " P e r i o d i c Boom Forces i n DumbbellType S a t e l l i t e s Moving i n a C i r c u l a r O r b i t , " AIAA J o u r n a l , V o l . 12, No. 4, A p r i l 1974, pp. 423-424. Sharma, S.C., and S t u i v e r , W., "Boom Forces i n L i b r a t i n g Dumbbell-Type S a t e l l i t e s , " AIAA J o u r n a l , V o l . 12, No. 4, A p r i l 1974, pp. 425- 426. L i k i n s , P.W., "The I n f l u e n c e on Dynamics and C o n t r o l Theory of the S p a c e c r a f t A t t i t u d e C o n t r o l Problem," Proceedings of the 6th Canadian Congress o f A p p l i e d Mechanics, Vancouver, May 29-June 3, 1977, pp. 321-335.  134  48.  K r y l o v , N.M., a n d B o g o l i u b o v , N.M., Introduction Mechanics, Princeton U n i v e r s i t y Press, 1943.  to  49.  B u t e n i n , N.V., Elements of Nonlinear 1965, pp. 102-137, 201-217.  50.  N a y f e h , A.H., P e r t u r b a t i o n Methods, John Wiley New Y o r k , 1 9 7 3 , pp. 23-27.  51.  Brereton, R.C, "A S t a b i l i t y S t u d y o f G r a v i t y O r i e n t e d Satellites," Ph.D. T h e s i s , the U n i v e r s i t y of B r i t i s h Columbia, November 1967, pp. 24-35.  52.  C u n n i n g h a m , W.J., Introduction to Nonlinear A n a l y s i s , H i l l B o o k C o m p a n y , New Y o r k , 1 9 5 8 , p p . 261-265.  53..  M e i r o v i t c h , L., M e t h o d s o f A n a l y t i c a l D y n a m i c s , B o o k C o m p a n y , New Y o r k , 1 9 7 0 , pp. 282-287.  54.  P i s k u n o v , N., D i f f e r e n t i a l a n d I n t e g r a l C a l c u l u s , P u b l i s h e r s , Moscow, pp. 479-487.  55.  R u p p e , H.O., Introduction to Astronautics, P r e s s , New Y o r k , 1 9 6 7 , pp. 433-446.  Oscillations,  Vol.  and  Nonlinear  Blaisdell, Sons,  Inc.,  McGraw-  McGraw-Hill  2,  Peace Academic  135  APPENDIX I T R A N S F O R M A T I O N M A T R I C E S USED I N E Q U A T I O N S 1  [6]  =  0  cos0  •sinG  0  sine  cose  1  [a]  0  0  0  0  0  cosa  -sina  0  sina  cosa  COS3:  0 1  [B ] = ±  -sing.  [Y ]  .  [Y ] ±  cos3.  (4.15)  cosy  -siny  0  siny  cosy  0  0  0  1  cosy.  •siny.  0  siny.  cosy^  0  0  sinB, 0  (4.1) and  1  0 ,  [ o u ]  cosa sina  I  -sina.  I  cosa  136  APPENDIX I I E Q U A T I O N S OF MOTION FOR THE S Y S T E M OF C A B L E OR CONNECTED Equation of y  SYMMETRIC END-BODIES  motion:  2 Z  BEAM  2  2  2  [y{ ( I . - I .) ( ( c o s a s i n y. + c o s y . ) c o s xi y i l l  1 + Tr s i n 2a s i n y . s i n 23.) 2 x x  - I  2  2  1  •  1  *  . - M. L. c o s a } + y { ( I . xx x x xx  - I .) ( a ( c o s 2a s i n y . s i n 23. + s i n 2 a ( s i n yx 'x x 1 + y . ( ^ s i n 2a c o s y , I 2 x  2 3- + s i n2a s i n2 3x i  2  2  2  3. - s i n y. c o s 3. x ' x x  2  2 3.) x  s i n 23. - s i n a s i n 2y. c o s x 'x  *•  + 3 . ( s i n 2a s i n y . c o s 23. + s i n 2 3 . ( s i n x x x x  2a s i n2y. - c o s2a ) ) 'x  2 •  + M. L. a s i n 2a} + a ( I . - I . ) ( c o s a sinB• x x xx yx x - sina  s i n y . cos3.)cosy. x l x 1  x s x n 23- + c o s a x +  1 3 • ( T- s i n a l 2  cos3- + a { ( I . - I . ) (- ^ - ( s i n a c o s y . x xx yx 2 ' l 2 • 2  s i n 2y. c o s 3-) x x  - Y- s i n a 'x  s i n 2y. s i n 23 + c o s a x  cosy, 'l  2  c o s 2y. c o s 3. ' 1 x  c o s 23-) x  2  2  + 6 ( s i n y ( c o s 2a c o s 23^ - 2 c o s a c o s y ^ c o s 3^ + s i n 2a s i n y . s i n 23-) x x  2  - c o s y . ( c o s a s i n 2y. c o s 3x x x  + sina  c o s y , s i n 23.))) x x  + cosa  s i n y . cos3.)) + 1 .(y.sina x l yx x  -  2 •  - I  .(y.sina xx x 1  2  2  s i n 3. + a . ( s i n a x x  2  c o s 3. - 3. c o s a x x  ••  sin3. x  cosy. x  2  0 s i n y ) - 2M. L. 0 s i n y c o s a } + y . { ( I . - I . ) ( c o s a c o s 3x x ' 'x xx yx 1  137  2  1 +  Y  s i n a s i n y . s i n 3.)  T  + 3 (sina siny i  • i - I . cosa} + y. {-z- s i n a COSY, s i n 23.  cos 23^ - cosa s i n 23 )  i  i  2  2  x s i n 2y^ cos 3^ + s i n a c o s y  + 9 (siny (-j s i n 2a x  2  i  s i n 23^) + cosy s i n a ( s i n 3^  2 - cos 2y. cos 3-)) - I • a. s i n a cosy. + 1 'l l xi l 'l •  .(3- s i n a yi l  •  siny. 1  1  •  + 9 cosy sina)} - 1 . 3 - s i n a cosy. + 3-{(I . - I . ) 9 x ' yi l 'l l xi yi x  1 • 2 (— s i n 23^ (- s i n y s i n 2a s i n Y^'~  s i n y s i n 2a  + cosy s i n a s i n 2y^) + cos 2 3 ^ ( - siny s i n 2a s i n y ^ + cosy cosa cosy.)) + I . a.(cosa cos3- - s i n a s i n y . s i n 3 . ) 1 XI 1 1 ' l 1 + I . 9 ( s i n y s i n y . - cosy cosa cosy.)) + I . a.(cosa yi I ' i xi I  sin3. l  - s i n a s i n y ^ cos3^) - I ^ 9 a ^ ( s i n y cosy^ c o s 3 ^ + c o s y ( s i n a x  •2  + cosa s i n y . c o s 3 . ) ) + l I  2  sin3^  2  9 {(I . - I . ) ( s i n 2 y ( - 2 cos y. cos 3. xi yi ' ' l i  + s i n 2a s i n y . s i n 3 . + 2 cos 2 a s i n 2y. cos 23. + 2 l l ' i l  s i n 2a s i n2 3.) l  2 - 2 cos 2 y ( s m a cosy^ s i n 23^ + cosa s i n 2y^ cos  2  - 2M. L. s i n 2y cos i i 1  2 a}]  1  + £  - W  2  [p£ {y(- U 2  J 0  -  2  I (Y - - i ) c o s a + 2  s i n a ) + y ( - 2U U + 2  £  x s i n 2a + 2 (Y  3^))  1 —)W  (W(Y  2  I (Y - —^)W  £ - — ^ - 2W)  s i n 2a  + a((Y -  £ 2  - W ))x 2  • •• ^1 a cos 2a) + U a ( (Y - — ) s i n a + W cosa)  138 £  1 + a(2(Y - — ) W l  8 siny  (- 2W U 0 c o s y  -  2  i  - U(Y - — )  s i n 2a +  - 2W  - U W a  h  6 siny  sina)sina  +  (U(Y  £± ) ( 2 8 c o s y  o.  ^  £.  - U W + 2 U W - 2 ( Y -  j) e  - W sina) + U W sina  + u 0 cosy((Y  siny  2  + ij ( (Y - -^)  cosa) cosa)  p)s i n a  + 2W  £  -  (Y  s i n 2a - 2(Y  ^) cos a + 2  2  (Y  ^)cos a) 2  j^)W s i n 2 a -  £ + 2U c o s 2 y ( ( Y - y ) c o s a  - W sina))}  2 • • {y c o s a - y a s i n 2a + 2a 0 s i n y  x  cosa  cosa)  £.  + W 8 siny(W  + a)  + 2 9 (sin 2y(U 2  2  W sin a) 2  2  + *L{(||)  +  2  M  (||) } 2  x  2 *2 2 • c o s a + 20 s i n 2y c o s a } ] d Y  = 0  Equation 2 I [i=l  of a  motion:  a{(I . - I .)cos y. cos B. Y 2  X  1  1  + 1 • - M. L } y i i i 2  2  1  1  • • 2 * 2 + a { ( I . - I . ) ( y . s i n 2 y . c o s 3 . + 3 . c o s y. s i n 2 3 . ) xi y i '1 ' i * i " i ' i 1' 1 • Y + I . 0 s i n y c o s a c o s y , s i n 2 3 . } + - M i . - I .) x 2 x i '1 1 2 x i y i T  x  (cosa cosy, 1  -  I  y  i  s i n 2y. c o s 1 . ) 1  .  2 c o s 2 y ^ c o s 3^ + c o s a  1 + 3- (TT s i n a 1 2  3  c o s 2a siny^^ s i n 2 3  )  - y^(sina  s i n 23. - sina  1  x i  + sin 2a(sin y  cos 3  2  i  siny^  s i n 2y. s i n 2 3 . + c o s a 1  + y{ (I  1  sin 2 ^  cosy, '1  2  i  - sina  cos 23.) 1  i  - sin 3 ) )  cos3^)  2  i  139  +  2  2  1  0 ( s i n y ( c o s y ^ c o s 3^ - -j s i n 2a s i n y ^  1  2  2  - cos 2 a c o s  3^+2  2  s i n 2y. c o s 3.)) + 1  + cosa  siny^  1  1  2  2a s i n y ^ +  1  Y•  .  + - s f - d • - I • ) c o s y , s i n 23. 2 x i y i '1 x  1  2))  2  1  2  •,  c o s a) }  Y•  + y . { (I . - I .)( ± '1 x i yx 2  K  s i n 23^  sin3-  1  2 y * L . (- 4- s i n 2a + 20 s i n y  + M.  1  cosy^  + a. ( s i n a  1  0  cosy.  + cosy(sina  . ' ( - y. s i n a x i 1  cos3^) + ^ s i n y ( c o s  + 1.3cosa yx x  - 2 sin a  2  s i n y ^ c o s 3^))  + cosa  2  s i n 23^  siny.  s i n 23.  '1  1  2 +  3^ c o s y ^ c o s 23^  + 0 (siny(cosa  c o s 2y^  cos  3^  2 - sina  siny^  sm  .  +  I  . a. s i n y . xi 1 '1  +  3.  1  I  . siny. yx '1  I  + cosa  X  - cosa  • cosy. X X  cosa)}  s i n 2y^  s i n 23^ siny  + sina sina  cosy^  cos  X  X  •2  XX  *  cos3.  X  X  {(I . - I .)(sin xx yx  s i n 23_^)  + 2 cos y ( s i n  1  23^))  cosy.} x  cos3- + I • a. 0 s i n y ( s i n a s i n y .  sin3.) +0 x  cosy^  8^)  2  2y(sina '  2 - cosa  s i n 2y_^ c o s  + 3 - { ( I . - I . ) 0 ( c o s y c o s y. s i n 23. x xx yx ' 'x x  •  I  + cosy  1  . a. c o s y . sin3. + 1 . 0 xi 1 x x yx  - a-  2  s i n y^)  cos3-1 + Iy x-(3cosy. + 0 siny x 1 '  + s i n y (- - i - c o s a +  23^  s i n 2y.  2  X  cos x  2  8. x  2  2  2 a ( s i n y ^ c o s 3^ - s i n 3^)  2  2  2•  - c o s 2a s i n y . s i n 23.) + -z- s i n 2 a ( s i n 3- - s i n y. c o s 3.) 'x x 2 x 'x 1 1 1 2 2 + + ^- M. L. (1 - 4 c o s y ) s i n 2a}] 2 c o s 2a s i n yi. s i n 23.) x 2 1 1  140  1  + £  *  2  [p£ {-  ( (Y  ^ )  2  + W ) a - Wa(2W + U0 s i n y  cosa)  2  0 + Uy((Y  j-)sina + W cosa) + y ( s i n 2 a ( j ( W  -  (Y  jp 'Y  1 ^1 * - 2 (Y - — )W 0 siny) + s i n a ( ( Y - -j-) U + 2U W 6 cosy) £  1 2 2 2 2 + 20 siny ( (Y - -j-) cos a + W s i n a) + cosa (- W U + U W •  & -|  + U W 6 s i n y - 2 (Y  »  &,  ~^) U 9 cosy)) -  (Y  j±) W  £ + 9 ( ( U W + U W)siny s i n a - U ( Y - - ^ ) s i n y cosa + 2W W *2 + 9 (- 2 U ( ( Y  A  l ^ ) s i n a + W cosa) s i n 2y + £ 1~)  x w cos 2 a + i ( ( Y + M*L{((|^)  2  +  (^^Ha  2  (Y  cosy)  ^1 j±) ( 1 - 4  2 cos y)x  ~ W ) (1 - 4 c o s y ) s i n 2 a ) } 2  2  + i ( y - 0 ) s i n 2 a - 2y 8 s i n y 2  2  cos a 2  *2 2 + 20 cos y s i n 2 a } ] d Y = 0  Equation of y^ motion: •• 2 y • { ( I • - I . ) s i n 6. + I .} + y. { ( I . - I . ) (- a s i n y . 'x xi yx l yx 'x xi yx i J  + 8• s i n 2g.) x i + y{-  +1  T  p  2 . 0 s i n y s i n a s i n 2y. s i n 8-} xx ' 'x 1  1 2 ( I . - I . ) ( ^ s i n a s i n y . s i n 28- + cosa cos 8.) xx yx 2 'x x x •  Y  + I . cosa} + y [ ( I . - I .){4r{— XX  s i n 2g. x  XI  2 2 - s i n a s i n 2y. cos 'x x - cosa s i n y ^ s i n 28^)  y'1  1  £  s i n 2a cosy,  1  s i n 28.  1  • 2 2 + a(2 s i n a s i n y. cos 8'x x + 8^ (cosa s i n 28^ - s i n a s i n y ^ cos  28^)  141  1 s i n 2a s i n 2y^ COS 23^ - c o s2a c o s y ^ s i n 23^)  + 0(siny(—  +  COSY  sina(sin  2  3-  COS  - c o s 2y.  i  'i  2  3-))}  * + I . (3- s i n y . + 0 COSY) s i n a ] yi I l  x sina  -  l  I  •  •  - ( a + a. COSY- COS3.)X 1 '1* i '  x i  a - -^'(I • 2 x i  -  - ) c o s y . s i n 23. y i ' l l  1  + a [ ( I . - I . ) { T T s i n 2Y . c o s 3. - 3. c o s y , c o s 23xi y i 2 ' i l i' i l  2 + 0(siny(-  sina  siny^  s i n 23^ + c o s a  2  + cosa  2  c o s 2y^ c o s 3^) + c o s y •  s i n 3^  s i n 2y^ c o s 3^)}  •  •  + I . a . s i n y . cos3- + I •(3- c o s y . + 0 s i n y c o s a ) ] xi l 'l * i y i I 'I ' ' •  •  + 3 . [ ( I - - I -)0{- c o s y l x i y i  c o s y , c o s 23- + s i n y ( s i n a i i '  s i n 23. i  c o s 23-)} - I - a . cos3- + I - 0 ( c o s y c o s y . l x i l l y i ' ' i  + cosa  siny. l  - siny  cosa  siny.)]  + siny  cosa  cosyl  x sina  s m y ^ + s i n 2a c o s y ^ {-^ - c o s y ) ) + c o s 3^(2 s i n 2y x  x cosa  c o s 2y^ + s i n 2y^ (-^ s i n a + 2 c o s y c o s a - 2 s i n y ) ) } ]  +1  - a. I . s i n g .  cos3-) + 0 I  ..a. 0(cosy s i n y .  [ ( I - - I -){- s i n23-(sin x i y i l  1  1  2  cos3. 2y x  2  2  2  2  2  = 0  E q u a t i o n of 3^ motion:  I  yi  &i  +  I  y i ^  s  i  n  a  c o s  Y  i  + Y [ d  x  i  -  I y  i  H j ( s i n  23 (- cos a i  2 2 •1 + s i n a s i n y . ) + s i n 2a s i n y . c o s 23.) + a{— s i n a i l l 2  s i n 2y. s i n 2 3 . ' l l  142  + cosa cosyi  cos 23-) + Y - ( cosa s i n 23. + s i n a s i n y . cos 23.) i i I ' I i ' _  M  + 0(cos 23^(COSY cosa COSY^ - s i n y cos 2a s i n y ^ ) 1 2 + -~ s i n 23. (COSY s i n a s i n 2y. + s i n y s i n 2 a ( l + s i n Y-)))} + I . a.(cosa cosS• + s i n a s i n y . sin3•) + 1  .{a cosa cosy.  - y^ s i n a s i n y ^ + G (siny s i n y ^ - cosy cosa cosy^) } ] • ra 2 • . a siny. + a [ ( I . ) cos Y. s i n 23- + Y- COSY • cos 23yi 1 xi yi 2 ' i l ' i ' i l 2 1 - 0 (- COSY cos Y^ s i n 23^ + s i n y ^ ^i ^ ^ i -  I  .  -  I  c  + s i n a COSY- cos 2 3 - ) ) } 1  +  I  1  o  s  a  . a. C O S Y 1  XI  1  s  n  2  Y  -  0 ( s i n y s i n a s i n 23^ + cos 2 3 ^ ( s i n y  +  T- ^  +  I  ^  2  1  I y i. cosy l• (- Yl• + 9 s i n y sina) ] + Yl• [ (x Ii •  a^ c o s 3 ^  n  sin3.  +  X  s  r i yI i • ) ( - 2- ? r - s i n l23. Y  -  cosa s i n Y ^ - COSY  COSY^))}  0 (siny cosa s i n y ^ - COSY C O S Y ^ ) ]  + 1 ^ a^ 0 { C O S Y C O S Y ^ s i n 3 ^ + s i n y ( s i n a c o s 3 ^ - cosa sinyI  sin3-)} + 0 i  ( I - - I . ) [ - s i n 2 3 - ( s i n 2y cosa x xi yi l  2 2 2 2 2 2 2 x s i n 2Y^ + 2 s i n y cos Y ^ + 2 cos y cos a s i n ^ ~ 2 cos y s i n a Y  1 2 2 2 + ^ ( s i n a s i n y^ ~~ cos a)} + cos 23^(2 s i n 2y s i n a cosy^ 2 1 + 2 cos Y s i n 2a s i n y ^ - ^ ^s  n 2  a  siny^) ] = 0  143  Equation ••  of  motion:  ••  •  + y(sina  siny^  cos3^ - c o s a  + cosa  siny.  cos3.) + y. s i n a I 'I  + sina  siny^  sin3^) - 8 ( s i n y  + cosy  cosa  1  -  ' 1  3^ c o s y ^  •  sin3^) + y { a ( s i n a  sin3^  c o s y . cos33- ( c o s a ' I I I  cos3-  -  cosy^  cos3^ + c o s y  ••  sina  • -  sin3^  •  s i n y . cos3-)} + a c o s y . cos3- + a { - y. s i n y . I l l I i sin3^ + 0 ( s i n y  sina  siny^  - y. s i n 3 . - y.{3. cos3. + 0 ( c o s y 1 I I I I 1  1  •  + siny  cosa  cosy^  - cosy  cosy^  cos3^ - s i n y  siny. I  I  cosa  cos3i l  sin3^)}  cos3I  •  cos3^)} + 3^ 0 ( s i n y  cosa  siny^  sin3^  sin3^).= 0  Cable Equation  f o rU vibration:  •2 * 2 U + U(- y +0 + W(y s i n a + 20  2  "2 2 *• - 40 s i n y ) + W(y s i n a •  •  + 2y a c o s a  s i n 2y s i n a )  • . - y a siny  cosa  +  •  •  rt- y 0 c o s y  (Y -  • • + y 0 cosy  cosa  (- y c o s a  sina  + 0 sina •  •  - 0 siny  .  + a 0 siny + 2y a  .  sina  sina  •2 - 20 s i n 2y c o s a )  1^1 ^ IJ ? *2 2 2 *2 2 2 * * - — j { ( y - 0 )cosa + a +40 cos y cos a + y 0 siny  p£ + 2a 0 c o s y } — y 9Y^  = 0  cosa)  s i n 2a  144  Equation  forW  vibration:  r *2 •2 2 •* W + W{(0 - y ) s i n a + y 0 siny  •2 • • s i n 2a - a + 2a 0 c o s y  -  *2 2 2 • • • 40 c o s y s i n a} + U ( - 2y s i n a  +  2y 0 c o s y  -  . • 2y 0 s i n y •  •  + y 0 siny  cosa  + 20  + 0 siny  s i n 2y s i n a )  2  +  cosa)  + U ( - ys i n  £ (Y - -j-) i ^ i y  2  -  0 )sin 2  2a  2 •• *2 2 M*L «2 *2 2 c o s a + a + 20 c o s y s i n 2a} - — y { ( y - 0 ) c o s a p£ •2 • ' * 2 2 2 P ) W s i n 2a + a + 2 a 0 c o s y + 40 cos y cos a}—•j = 0 8Y  Beam Equation  forU  vibration:  • 2 * 2 U + U(- y +0  ' - 40 •  + W(y s i n a  •  •  + 2y a c o s a  •2 + 20 s i n 2y s i n a )  - y a siny ^ + ^  (  E  '  + a  b 9 U j —— j p£ 3Y I  2  )  4  ^  A a  +48  Equation  2  cosa  2 2 • • s i n y) + W(y s i n a •  +  sina  cosa)  •  + a 0 siny  + 2y a  •2 - 20 s i n 2y  *2 2 • ' • -• 8 ) c o s a + y 8 s i n y X  \3 U cos y cos a } — = 3Y 2  sina  sina  cosa)  • „ „• I s i n 2a + 2a 0 cosy  2  0  vibration:  * 2 ' 2 2 •• W + W{(0 - y ) s i n a + y 0 siny r  -  »  cosa  1 •• (Y - — ) ( - y c o s a  • • + y 0 cosy r  forW  •  + y 0 cosy  - 0 siny  £  M*L ,-2 - — ( y pl 2  + 0 sina  •2 2 2 40 c o s y s i n a} + U ( - 2y s i n a  '2 s i n 2a - a  + 0 siny  •• + 2a 0 c o s y  cosa)  145  +  U(-  y  sina  +  (Y - ~ H ^ ( Y  ^1  , oQ2 + 28  cos  y  2y  l -2  r  2  +  8 cosy  +  +  *2 ~  9  28  s i n 2y  • * ) s  • i ^ s i n 2a} +  i  n  ( E I )  pi • * y 8 siny  cosa  •2 s i n 2a + a +  2a  -  2y  b j—  3W —j 9Y 4  4  2  8 siny M*L - —=•{ pl  • • 2a 8 c o s y  sina)  cos / #  -2 (y  a +  '2. 2 - 8 )cos a  •2 +  48  a  cos  2 y cos  2 a}  2 9  W »  =  

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