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Dynamics of a large class of satellites with deploying flexible appendages Lips, Kenneth Wayne 1980

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DYNAMICS OF A LARGE CLASS OF SATELLITES WITH DEPLOYING FLEXIBLE APPENDAGES by KENNETH WAYNE LIPS B.A.Sc. U n i v e r s i t y of Toronto; Toronto, Canada 1967 M.A.Sc. U n i v e r s i t y of Toronto ( I n s t i t u t e f o r Aerospace S t u d i e s ) ; Toronto, Canada 1971 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of 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 August 1980 © Kenneth Wayne L i p s 1980 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t 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 f o r r e f e r e n c e and st u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of Mechanical E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Kenneth Wayne L i p s ABSTRACT A g e n e r a l f o r m u l a t i o n i s p r e s e n t e d f o r t h e l i b r a t i o n a l dynamics o f s a t e l l i t e s h a v i n g an a r b i t r a r y number, t y p e , and o r i e n t a t i o n o f f l e x i b l e appendages, each c a p a b l e o f d e p l o y i n g i n -d e p e n d e n t l y . I n p a r t i c u l a r , t h e case o f beam-type appendages de-p l o y i n g from a s a t e l l i t e i n an a r b i t r a r y o r b i t i s c o n s i d e r e d . The g o v e r n i n g n o n l i n e a r , nonautonomous, c o u p l e d system e q u a t i o n s a r e not amenable t o any c l o s e d form s o l u t i o n , hence a r e i n t e g r a t e d n u m e r i c a l l y u s i n g a d i g i t a l computer. E f f e c t o f i m p o r t a n t system parameters i s a s s e s s e d t h r o u g h i l l u s t r a t i v e c o n f i g u r a t i o n s r e p r e -s e n t i n g a l a r g e c l a s s o f g r a v i t y g r a d i e n t and s p i n n i n g s p a c e c r a f t . R a t h e r t h a n a c c u m u l a t i o n o f a l a r g e amount o f d a t a , t h e emphasis i s on e v o l u t i o n o f a g e n e r a l i z e d and o r g a n i z e d methodology f o r c o p i n g w i t h such complex d y n a m i c a l systems. The a n a l y s i s examines t h e degree o f i n t e r a c t i o n between f l e x i b i l i t y , deployment, and a t t i t u d e m o t i o n t h r o u g h s y s t e m a t i c v a r i a t i o n o f system parameters. A s t u d y o f appendage v i b r a t i o n c h a r a c t e r i s t i c s s u g g e s t t h a t an o r b i t i n g beam cannot be t r e a t e d s i m p l y as a r o t a t i n g beam because o f t h e p r e s e n c e o f t h e g r a v i t a -t i o n a l f i e l d . Rate o f r o t a t i o n p l a y s a dominant r o l e i n s t i f f e n i n g the' beam as e v i d e n c e d by t h e n o t i c e a b l e s t r a i g h t e n i n g o f t h e e i g e n -f u n c t i o n s f o r even r e l a t i v e l y low s p i n r a t e s (2 rpm). R e s u l t s a l s o show t h a t t h e d e p l o y m e n t - r e l a t e d C o r i o l i s f o r c e can p l a y a major r o l e i n c a u s i n g l a r g e i n - p l a n e d e f o r m a t i o n s . T h i s i m p l i e s t h a t , i n some c a s e s , deployment s h o u l d be c a r r i e d o u t i n s t a g e s so as t o l i m i t . t h e t i m e a v a i l a b l e t o b u i l d up l a r g e a m p l i t u d e o s c i l l a t i o n s . I n v e s t i g a t i o n o f l i b r a t i o n a l r e s p o n s e shows t h a t t h e c o u p l e d c h a r a c t e r o f t h e m o t i o n can s i g n i f i c a n t l y a f f e c t system dynamics, hence c a u t i o n s h o u l d be e x e r c i s e d i n u t i l i z i n g r e s u l t s based on s i m p l i f i e d p l a n a r a n a l y s e s . Depending on o r b i t a l p a rameters and p h y s i c a l p r o p e r t i e s o f booms, t h e r e a r e c r i t i c a l v a l u e s o f appendage l e n g t h and deployment r a t e f o r w h i c h th e s a t e l l i t e can tumble o v e r . On t h e o t h e r hand, i n g e n e r a l , appendage o f f s e t and s h i f t i n g c e n t e r o f mass were found t o have i n s i g n i f i c a n t e f f e c t on r e sponse f o r t h e c a s e s c o n s i d e r e d . T h i s may p e r m i t c o n s i d e r a b l e s i m p l i f i c a t i o n o f t h e complex h y b r i d e q u a t i o n s w i t h a s s o c i a t e d s a v i n g i n c o m p u t a t i o n a l t i m e and e f f o r t . A l s o , t h e s m a l l a m p l i t u d e o s c i l l a t i o n s e v i d e n t b o t h w i t h t h e g r a v i t y g r a d i e n t and s p i n - s t a b i l i z e d c o n f i g u r a t i o n s tends t o s u b s t a n t i a t e t h e a d o p t i o n o f a l i n e a r v i b r a t i o n a n a l y s i s . The s i m u l a t i o n o f such d i v e r s e c l a s s e s o f s a t e l l i t e s w i t h r e l a t i v e ease demonstrates th e v e r s a t i l i t y o f t h e f o r m u l a t i o n . i v TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 1.1 Preliminary Remarks 1 1.2 Literature Review 5 1.2.1 Background 5 1.2.2 Equations of motion 10 1.2.3 Appendage dynamics 14 1.2.4 S t a b i l i t y and control of f l e x i b l e spacecraft 17 1.2.5 Transient response and deployment dynamics 2 0 1.3 Purpose and Scope of the Investigation 24 2 GENERAL ATTITUDE EQUATIONS OF MOTION 28 2.1 Configuration and Reference Coordinate Systems ... 2 8 2.2 Lagrangian Formulation 33 2.2.1 Background 33 2.2.2 System k i n e t i c and poten t i a l energies .... 34 2.2.3 The Lagrange equations and an alternative momentum formulation 36 2.3 Governing Nonlinear Three-Axis Equations 39 3 APPENDAGE EQUATIONS OF MOTION 45 3.1 Background 45 3.2 Kine t i c and Potential Energy of a Deploying Beam Undergoing General Librations • • 46 3.2.1 Beam configuration and coordinates 47 3.2.2 Treatment of a x i a l foreshortening 47 3.2.3 Kinetic energy density 5 0 3.2.4 Potential energy density 52 3.2.4.1 Strain energy 52 3.2.4.2 Gravitational p o t e n t i a l 54 V Chapter Page 4 SIMPLIFIED APPENDAGE DYNAMICS 62 4.1 L i n e a r i z e d E q u a t i o n s f o r T r a n s v e r s e V i b r a t i o n s o f a D e p l o y i n g , O r b i t i n g Beam-Type Appendage 62 4.2 S o l u t i o n o f the L i n e a r i z e d V i b r a t i o n E q u a t i o n s .... 65 4.3 'Free' V i b r a t i o n C h a r a c t e r i s t i c s o f S p i n n i n g , D e p l o y i n g , O r b i t i n g Beam-Type Appendages 70 4.3.1 G o v e r n i n g e q u a t i o n s 70 4.3.2 R e s u l t s and d i s c u s s i o n 73 4.4 C o n c l u d i n g Remarks 85 5 PLANAR LIBRATIONS OF A TYPICAL GRAVITY GRADIENT CONFIGUR-ATION 87 5.1 S i m p l i f i e d S p a c e c r a f t C o n f i g u r a t i o n and System E q u a t i o n s 87 5.2 E q u a t i o n s Based on ' D i s c r e t e ' D e f o r m a t i o n C o o r d i n a t e s and ' O r b i t a l ' Time 90 5.3 R e s u l t s and D i s c u s s i o n 92 5.4 C o n c l u d i n g Remarks 99 6 GENERAL THREE-AXIS ATTITUDE MOTION 102 6.1 S p a c e c r a f t C o n f i g u r a t i o n and System E q u a t i o n s 102 6.1.1 C o m p u t a t i o n a l c o n s i d e r a t i o n s 105 6.2 R e s u l t s and D i s c u s s i o n 107 6.2.1 Two-boom g r a v i t y g r a d i e n t c o n f i g u r a t i o n ... 109 6.2.2 Four-boom s p i n - s t a b i l i z e d c o n f i g u r a t i o n ••• 118 6.2.3 CTS-type c o n f i g u r a t i o n 12 0 6.2.4 Asymmetric deployment o f appendages 123 6.3 C o n c l u d i n g Remarks 128 7 CLOSING COMMENTS 130 7.1 On F o r m u l a t i n g System E q u a t i o n s o f M o t i o n 130 7.2 C h a r a c t e r i s t i c s A s s o c i a t e d W i t h a D e p l o y i n g , O r b i t i n g , S p i n n i n g , Beam-Type Appendage 131 v i Chapter Page 7 (continued) 7.3 O v e r a l l System Response 132 7.4 Recommendations f o r Future Work 133 BIBLIOGRAPHY 135 Appendix I GENERAL EQUATIONS OF LIB-RATION BASED ON TRUE ANOMALY... 153 II SYSTEM MOMENTS OF INERTIA 160 11.1 A r b i t r a r y Appendage 160 11.2 Beam-Type Appendage 164 11.3 I n e r t i a s of S p a c e c r a f t Having A r b i t r a r y Appendages 167 11.4 I n e r t i a s o f S p a c e c r a f t With Beam-Type Appendages 170 IT.4.1 Continuous c o o r d i n a t e s 17 0 I I . 4.2 Assumed-mode format 174 •II. 5 Time Rate of Change of I n e r t i a s f o r S p a c e c r a f t With Beam-Type Appendages 178 I I I EVALUATION OF {r }, {h} AND {T} FOR BEAM-TYPE c APPENDAGES 185 I I I . l S h i f t i n g Center of Mass L o c a t i o n { r c ) and A s s o c i a t e d Time D e r i v a t i v e s 185 I I I . 2 L o c a l Angular Momentum {h} 188 I I I . 2.1 Appendages wi t h a r b i t r a r y o r i e n t a t i o n , continuous c o o r d i n a t e s 188 III.2.2 Assumed-mode format f o r appendages i n the x-y, x-z planes 18 9 I I I . 3 L o c a l Torque {T} 193 III . 3 . 1 Appendages wi t h a r b i t r a r y o r i e n t a t i o n , continuous c o o r d i n a t e s 193 v i i A p pendix Page I I I ( c o n t i n u e d ) I I I . 3 . 2 Assumed-mode format f o r appendages i n the x-y, x-z p l a n e s 195 IV A USEFUL INTEGRAL THEOREM 202 V APPLICATION OF HAMILTON'S PRINCIPLE TO A DEPLOYING CONTINUUM 204 VI MODAL INTEGRAL COEFFICIENTS 210 V I I A METHOD FOR ISOLATING SECOND DERIVATIVES OF COMPLEX COUPLED SECOND ORDER SYSTEMS 215 V I I . l A n a l y s i s 215 V I I . 2 A p p l i c a t i o n 225 v i i i L I S T OF TABLES T a b l e Page 4.1 S y s t e m e i g e n v a l u e s d e m o n s t r a t i n g i n d i v i d u a l and combined i n f l u e n c e s o f o r b i t a l m o t i o n , s p i n and d e p l o y m e n t 7 5 ix LIST OF FIGURES Figure Page 1- 1 Outline of the research program 27 2- 1 Geometry of s a t e l l i t e motion: (a) i n e r t i a l , rotating, and body-fixed coordinate systems; (b) modified Eulerian rotations H,,A,$ defining a r b i t r a r y orientation of the central r i g i d body during l i b r a t i o n s 29 2- 2 A general spacecraft configuration showing s h i f t i n g center of mass, appendage o f f s e t , deployment, and deformations 31 3- 1 Beam a x i a l foreshortening caused by transverse defor-mations v,w 48 4- 1 Model of deploying, o r b i t i n g , l i b r a t i n g , beam-type appendage experiencing f l e x u r a l o s c i l l a t i o n both i n [v(x,t)] and out [w(x,t)] of the o r b i t a l plane 71 4-2 Frequency parameter for in-plane vibrations covering a wide range of spin parameter values - no deployment. 74 4-3 E f f e c t of o r b i t a l motion and spin on frequency parameter i n absence of deployment 76 4-4 I s o l a t i o n of deployment rate and acceleration e f f e c t s on frequency parameter 78 4-5 Influence of changes i n length, deployment rate, and spin rate on (in-plane) frequency 79 4-6 Influence of spin rate on system eigenfunctions i n the absence of deployment 81 4-7 Modal changes associated with length for a spinning deploying beam 82 4-8 Planar response of a deploying, rotating, beam-type appendage to i n i t i a l t i p displacement; ¥,A = 0 84 X Figure Page 5-1 Configuration of a representative gravity gradient s a t e l l i t e , with two in-plane f l e x i b l e deploying uniform booms, undergoing planar l i b r a t i o n and defor-mation 8 9 5-2 E f f e c t of the f l e x i b l e boom length on system response for the planar case 93 5-3 Transient response of a gravity gradient s a t e l l i t e showing the e f f e c t of f l e x i b i l i t y and deployment, <F=A=0 , 95 5-4 E f f e c t of the deployment rate on pit c h and v i b r a t i o n a l response of a gravity gradient s a t e l l i t e 97 5-5 E f f e c t of i n i t i a l e l a s t i c deformations on system response for three d i f f e r e n t i n i t i a l conditions, ¥=A=0 98 5-6 Typical planar response as affected by the s h i f t i n g center of mass and appendage o f f s e t 100 6-1 Configuration representing a large class of spacecraft chosen for detailed study. Note, the arrangement shows appendages i n the x-y plane coinciding with the pi t c h plane (p) and the x-z plane perpendicular to the pi t c h plane (o) ' 103 6-2 Three-axis response of a s a t e l l i t e to an impulsive p i t c h disturbance 110 6-3 Three-axis response of a s a t e l l i t e with f u l l y deployed appendages to an impulsive out-of-plane disturbance: (a) r i g i d booms; (b) f l e x i b l e booms I l l 6-4 E f f e c t of boom deployment on the three-axis response of a s a t e l l i t e to an impulsive out-of-plane disturance: (a) r i g i d booms; (b) f l e x i b l e booms 113 6-5 E f f e c t of magnitude of an impulsive out-of-plane d i s -turbance on three-axis response 114 6-6 Planar response of the gravity gradient configuration to d i f f e r e n t i n i t i a l e l a s t i c deformations 115 6-7 Three-axis response of the gravity gradient configur-ation to d i f f e r e n t i n i t i a l e l a s t i c deformations 117 6-8 Three-axis response of a spinning spacecraft during deployment of r i g i d or f l e x i b l e appendages 118 x i Figure Page 6-9 Three-axis response of a spinning spacecraft during deployment of f l e x i b l e appendages with one boom i n i t i a l l y deformed 121 6-10 Three-axis response of a spinning spacecraft with f l e x i b l e deploying appendages when subjected to out-of-plane attitude disturbances 122 6-11 E f f e c t of stored momentum and f l e x i b i l i t y f o r : (a) r i g i d appendages, no momentum wheel; (b) r i g i d appendages with added momentum; (c) f l e x i b l e appendages with added momentum 124 6-12 Response of a CTS-type spacecraft to an i n i t i a l yaw rate disturbance during deployment of: (a) r i g i d booms; (b) f l e x i b l e booms 125 6-13 Three-axis reponse of a two-boom gravity gradient s a t e l l i t e with asymmetrically deployed appendages: (a) r i g i d booms; (b) f l e x i b l e booms 126 6-14 E f f e c t of asymmetric boom deployment on three-axis response of a two-boom gravity gradient s a t e l l i t e : (a) r i g i d booms; (b) f l e x i b l e booms 127 I I - l General displacement of a mass element i n the presence of f l e x i b i l i t y (e^), geometric o f f s e t (a^) and a s h i f t i n g center of mass (r ) 160 3 —c VII-1 Functional dependence of terms used i n the description of system equations 217 VII-2 Computational procedure for updating system derivatives 226 x i i ACKNOWLEDGEMENT The p a t i e n c e , g u i d a n c e , and c o n t i n u e d s u p p o r t p r o v i d e d by Dr. V . J . Modi have made t h i s t h e s i s p o s s i b l e . A v e r y s p e c i a l n ote o f a p p r e c i a t i o n i s a l s o extended t o Susann and our c h i l d r e n (Vanessa, H y p a t i a , U r s u l a and D a r t a n i o n ) f o r t h e i r c o o p e r a t i o n and u n d e r s t a n d i n g . F o r her e x p e r t t y p i n g o f t h e f i n a l m a n u s c r i p t a note o f tha n k s i s o f f e r e d t o Maureen Skuce. The i n v e s t i g a t i o n r e p o r t e d here was funded from t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada, Grant No. 67-2181. x i i i LIST OF SYMBOLS A(x) area of beam cross section, // dy dz A . acceleration of mass element dm. due to v i b r a t i o n and deployment, " r ' 1 + (V.-V)V. . [B.] modal i n t e g r a l c o e f f i c i e n t s as defined i n Appendix 3 VI. [B. r] , [B. ,,] modal i n t e g r a l c o e f f i c i e n t s found using admissible functions E n ( x ) , H n(x) respectively; see section 4.2 c ( ), s ( ) cosine ( ), sine ( ), respectively c., c o e f f i c i e n t i d e n t i f y i n g appendage orien t a t i o n 3 r e l a t i v e to the l o c a l v e r t i c a l , equation (3.18) cm spacecraft center of mass, Figures 2-1 and 2-2 {C.} modal i n t e g r a l c o e f f i c i e n t s as defined i n Appendix 3 VI {C. p}, {C. w} modal i n t e g r a l c o e f f i c i e n t s found using admissible functions E (x), H (x), respectively; see section 4.2 n n CPU a unit of computer time, Central Processing Unit d^ vector locating dnu for the undeformed appendage and measured with respect to 0. (d a + d1?) ; Figure 2-2 i i i d^ vector d^ p r i o r to deployment d- net change i n cL r e s u l t i n g from deployment t h i n f i n i t e s i m a l element of s p a t i a l domain of i appendage, Figure 2-2 drrK elemental mass of i appendage, Figure 2-2 ds elemental length as measured along neutral axis of the beam, Figure 3-1 d D i x i v e l e m e n t a l l e n g t h as measured a l o n g x^ d i r e c t i o n t h s p a t i a l domain o f i appendage (D = c e n t r a l r i g i d body) . 0 o r b i t a l e c c e n t r i c i t y a f u n c t i o n o f e and 0 d e f i n e d f o r co n v e n i e n c e i n e q u a t i o n (4.3c) e l a s t i c d i s p l a c e m e n t o f dm. r e l a t i v e to 0 , u c 1 C C , 1 i . + v . j . + w . k . ; e q u a t i o n (2.5) —1 c , l — 1 c , l — 1 ^ e l a s t i c d i s p l a c e m e n t o f dm^ r e l a t i v e t o O Q , U ^ _ i ^ + v i 2i + w^ k^; see F i g u r e 2-2 and e q u a t i o n (2.2a) 11*1 Young's modulus o f e l a s t i c i t y , i appendage e f f e c t i v e l o a d a c t i n g i n t r a n s v e r s e 'y* d i r e c t i o n f o r beam-type appendage u n d e r g o i n g p i t c h l i b r a t i o n s o n l y , e q u a t i o n s (4.10c) an a r b i t r a r y f u n c t i o n o f s e f f e c t i v e l o a d a c t i n g i n t h e j d i r e c t i o n f o r beam-t y p e appendage u n d e r g o i n g g e n e r a l l i b r a t i o n , e q u a t i o n s (4.5c) an e f f e c t i v e a x i a l l o a d r e s u l t i n g from the i n e r t i a l and g r a v i t a t i o n a l f o r c e f i e l d s , e q u a t i o n (3.21d) n normal mode o f a s i m p l e u n i f o r m c a n t i l e v e r beam, e q u a t i o n (4.7) u n i v e r s a l g r a v i t a t i o n c o n s t a n t l o c a l a n g u l a r momentum r e s u l t i n g from v i b r a t i o n and deployment r e l a t i v e t o O C , e q u a t i o n (2.13a) a c o n s t a n t ( s t o r e d ) c o n t r i b u t i o n t o the l o c a l a n g u l a r momentum o r b i t a l a n g u l a r momentum, e q u a t i o n (2.1a) a n g u l a r momentum a s s o c i a t e d w i t h m o t i o n o f t h e space-c r a f t r e l a t i v e t o the i n s t a n t a n e o u s c e n t e r o f mass, e q u a t i o n (2.11) h i g h e r o r d e r terms u n i t v e c t o r s a l o n g t h e x, y, z axes r e s p e c t i v e l y X V o v e r a l l system mass moments o f i n e r t i a w i t h r e s p e c t t o x, y, z axe s , t u I ] , [ g l ] + I ^ 1 ] / e q u a t i o n ( I I . 7) c o n t r i b u t i o n o f r i g i d c e n t r a l ' c o r e ' body o f t h e s p a c e c r a f t t o [I] c o n t r i b u t i o n t o [I] r e s u l t i n g s o l e l y from the g e o m e t r i c o f f s e t o f t h e undeformed appendage, e q u a t i o n ( I I . 4 a ) c o n t r i b u t i o n o f t h e undeformed appendage t o [I] w i t h a = 0, e q u a t i o n s ( I I . 3 ) and (I I . 4 b ) change i n system mass moments o f i n e r t i a due t o a s h i f t i n l o c a t i o n o f s a t e l l i t e c e n t e r o f mass, e q u a t i o n ( I I . 7 ) change i n system mass moments o f i n e r t i a a s s o c i a t e d w i t h f l e x i b i l i t y , e q u a t i o n s ( I I . 7 ) and ( I I . 9 ) t o t a l c o n t r i b u t i o n o f t h e undeformed system t o [ I ] , + l2I] + [ 3 I ] , e q u a t i o n ( I I . 7 ) t h a r e a moment of i n e r t i a f o r t h e i beam w i t h r e s p e c t t o t h e l o c a l j , k axes c o n s t a n t o f p r o p o r t i o n a l i t y r e l a t i n g t h e n e i g e n v a l u e t o t h e s p i n parameter, e q u a t i o n (4.11) s t i f f n e s m a t r i x a s s o c i a t e d w i t h t r a n s v e r s e v i b r a t i o n s o f an a r b i t i n g , d e p l o y i n g , s p i n n i n g , beam-type appendage, e q u a t i o n (4.10c) s t i f f n e s s m a t r i x a s s o c i a t e d w i t h v i b r a t i o n i n t h e l o c a l j d i r e c t i o n f o r a beam-type appendage under-d o i n g g e n e r a l l i b r a t i o n , e q u a t i o n s (4.5c) i n s t a n t a n e o u s appendage l e n g t h e f f e c t i v e appendage l e n g t h measured a l o n g t h e l o c a l d i r e c t i o n w i t h f o r e s h o r t e n i n g a c c o u n t e d f o r i n i t i a l beam l e n g t h p r i o r t o deployment n e t change i n beam l e n g t h as a r e s u l t o f deployment f u l l y d e p l o y e d l e n g t h mass o f i t h appendage X V I t o t a l mass o f t h e s a t e l l i t e , Em., where i = 0 f o r i t h e r i g i d c e n t r a l body mass o f a t t r a c t i n g body l o c a t e d a t f o c u s 0 o f t h e o r b i t 1 t o t a l number o f second o r d e r e q u a t i o n s g o v e r n i n g system dynamics i n s t a n t a n e o u s c e n t e r o f mass, F i g u r e s 2-1 and 2-2 t h p o i n t a t wh i c h t h e i appendage i s a t t a c h e d t o t h e c e n t r a l body, F i g u r e 2-2 c e n t e r o f f o r c e o f t h e o r b i t c e n t e r o f mass o f undeformed s p a c e c r a f t t h k g e n e r a l i z e d c o o r d i n a t e t h g e n e r a l i z e d f o r c e a s s o c i a t e d w i t h k degree o f freedom g e n e r a l i z e d f o r c e r e s u l t i n g from t h e g r a v i t a t i o n a l f i e l d and a s s o c i a t e d w i t h t h e k t h degree o f freedom v e c t o r l o c a t i n g t h e i n s t a n t a n e o u s c e n t e r o f mass 0 c w i t h r e s p e c t t o t h e c e n t e r o f mass OQ o f t h e unde-formed body, F i g u r e 2-2 v e c t o r l o c a t i n g dm^ r e l a t i v e t o 0 c, F i g u r e 2-2 skew symmetric m a t r i x c o n s t r u c t e d from v e c t o r - d , l z, . 0 - x , . d , i d , i - y , . x. . 0 • -M,! d , i v e c t o r l o c a t i n g diru o f t h e undeformed appendage r e l a t i v e t o O Q , see s e c t i o n 2.2.2 r e v o l u t i o n s p e r minute v e c t o r l o c a t i n g 0 c r e l a t i v e t o i n e r t i a l r e f e r e n c e X, Y, Z; F i g u r e s 2-1 and 2-2 X V I I v e c t o r l o c a t i n g ditK o f the deformed s p a c e c r a f t r e l a t i v e t o i n e r t i a l r e f e r e n c e X, Y, Z; see equa-t i o n s (2.5) and F i g u r e 2-2. l o c a t i o n o f dirK as measured a l o n g t h e n e u t r a l a x i s o f t h e beam s u b r o u t i n e used t o d e f i n e t h e f i r s t o r d e r s t a t e v e c t o r d e r i v a t i v e s as d e r i v e d from the second o r d e r system e q u a t i o n s ' r e a l ' t i m e two d i f f e r e n t i n s t a n t s i n t i m e t r a c e o f m a t r i x [ ] e l a s t i c d i s p l a c e m e n t o f t h e mass element diru measur-ed a l o n g the x^, y^, z^ d i r e c t i o n s , r e s p e c t i v e l y , F i g u r e 2-2 t h s h o r t e n i n g o f t h e i appendage a l o n g the a x i a l (x^) d i r e c t i o n due t o t r a n s v e r s e o s c i l l a t i o n s , e q u a t i o n (3.4) deployment v e l o c i t y i n t h e x^ d i r e c t i o n , F i g u r e 2-2 g e n e r a l deployment v e l o c i t y o f dnu r e l a t i v e t o CK v e l o c i t y o f dnu w i t h r e s p e c t t o 0 c due t o v i b r a t i o n and deployment, e q u a t i o n s (2.5) b o d y - f i x e d c o o r d i n a t e system w i t h o r i g i n a t 0 c , F i g u r e s 2-1 and 2-2 i n t e r m e d i a t e l o c a t i o n o f x,y,z axes u n d e r g o i n g a s e t o f m o d i f i e d E u l e r i a n r o t a t i o n s , F i g u r e 2-1(b) l o c a l appendage c o o r d i n a t e system w i t h o r i g i n a t CK , F i g u r e 2-2 b o d y - f i x e d c o o r d i n a t e system w i t h o r i g i n a t 0 Q , F i g u r e 2-2 i n e r t i a l c o o r d i n a t e system w i t h o r i g i n a t f o c u s 0^, F i g u r e s 2-1 and 2-2 x v i i i X_, Y^, Z„ o r b i t i n g reference frame with o r i g i n at 0 , X along O O v_) c \j l o c a l v e r t i c a l , Y Q tangent to the o r b i t , and Z Q p a r a l l e l to the o r b i t normal. E (x.) n assumed admissible function associated with the n 1 n 1 v^ degree of freedom, equation (4.4) F. external force applied to the beam along the j -1 d i r e c t i o n generalized force associated with the generalized € coordinate € tf_(x,) n assumed admissible function associated with the w. degree of freedom, equation (4.4) L,jC Lagrangian, T-(/» equations (2.8), (3.20) and (V.2) 0^ c o e f f i c i e n t s as defined i n equations (4.7) T,7" t o t a l system k i n e t i c energy th T^ k i n e t i c energy of i appendage only I/, V t o t a l system p o t e n t i a l energy 1/ e l a s t i c p o t e n t i a l energy, equations (2.6), (3.9), e and (3.13) V g r a v i t a t i o n a l p o t e n t i a l , equations (2.6) and (3.16) . generalized work function which can include noncon-n servative forces, equation (V.l) a dummy variable commonly used as a variable of integration cu c o e f f i c i e n t s defined by equation (3.17) 3 frequency parameter for the n ^ eigenf unction, p o o 2 £ 4 / n EJ n 3 n - r p / 3 0 p frequency parameter associated with the *v1 and 'w' ' ' beam vibrations, respectively, equation (4.12) Yj coeffdcients defined by equations (4.1c) {T} torque due to v i b r a t i o n and deployment r e l a t i v e to the center of mass, equation (2.13b) x i x v a r i a t i o n o f ( ) i d e l t a f u n c t i o n / g g dx, equ a t i o n (4.>7) Q m n t o t a l f o r e s h o r t e n i n g e f f e c t experienced by the beam, see p. 49 g e n e r a l i z e d c o o r d i n a t e r e p r e s e n t i n g continuous v a r i a b l e s , i . e . , e = u ( x , t ) , v ( x , t ) , or w(x,t); equation (2.8), (3.21) and (V.3) n r e a l e i g e n f u n c t i o n , F i g u r e 4-7 s t r a i n t e n s o r o f i appendage t h g e n e r a l i z e d c o o r d i n a t e i n the n mode f o r the 'v' t h degree o f freedom of the i appendage, equation (4.4) v a r i a b l e d e f i n e d i n equation (3.5), x + u ^ g t r u e anomaly of the o r b i t , F i g u r e 2-1(a) di m e n s i o n l e s s deployment a c c e l e r a t i o n parameter, p £ £ 4 / E J •2 4 dime n s i o n l e s s deployment r a t e parameter, pi I / E J 2 4 s p i n parameter, P<*> & / E J g r a v i t a t i o n a l c o n s t a n t , GM term appearing i n the l i n e a r p a r t i a l d i f f e r e n t i a l equations governing v i b r a t i o n along the j d i r e c t i o n , equations (4.1c) g e n e r a l i z e d c o o r d i n a t e i n n mode f o r the 'v' degree t h of freedom of the i appendage, eq u a t i o n (4.4) t h a t c o n t r i b u t i o n t o the system Lagrangian a s s o c i a t e d w i t h the undeformed c o n f i g u r a t i o n o n l y , equations (2.8) and (2.9) net o f f s e t o f 0. from 0 , (o_. - r ) , F i g u r e 2-2 3_ C 1 c t h l i n e a r mass d e n s i t y f o r the i appendage X X a ( i ) (0) Z summation a* g e n e r a l i z e d s t r e s s t e n s o r g e o m e t r i c o f f s e t o f 0^ and O Q , F i g u r e 2-2 T t h a t c o n t r i b u t i o n t o t h e system L a g r a n g i a n a s s o c -i a t e d w i t h system f l e x i b i l i t y , e q u a t i o n s (2.8) and (2.9) {X } m a t r i x o f d i r e c t i o n c o s i n e s between R and x , y , z ; a — c e q u a t i o n (2.7) {x^} m a t r i x o f d i r e c t i o n c o s i n e s between x,y,z and X^'Y^' z^; e q u a t i o n (2.2c) i ^ i m o d i f i e d E u l e r i a n r o t a t i o n s from x,y,z t o x ^ , y ^ , z ^ e s t a b l i s h i n g appendage o r i e n t a t i o n ¥,A,$ m o d i f i e d E u l e r i a n r o t a t i o n s d e f i n i n g l i b r a t i o n a l m o t i o n o f t h e b o d y - f i x e d x,y,z axes r e l a t i v e t o t h e o r b i t i n g r e f e r e n c e X 0 , Y Q , Z 0 ; F i g u r e 2-1(b) a) i n e r t i a l a n g u l a r v e l o c i t y o f x,y,z axes; e q u a t i o n s (2.5) to s e f f e c t i v e s p i n r a t e as g i v e n i n e q u a t i o n s (4.10c) t h tt^ n a t u r a l f r e q u e n c y o f t h e n e i g e n f u n c t i o n ^ n d i m e n s i o n l e s s f r e q u e n c y , tt^/Q , e q u a t i o n s (4.10c) o s c a l a r p r o d u c t , ( ) 1 d( ) / d t , d( )/d9 8( )/8a ( ) e x p r e s s e d i n terms o f l o c a l c o o r d i n a t e s , X^'Y^> z^; see p. 32 ( ) e x p r e s s e d i n terms o f c e n t r a l c o o r d i n a t e s x , y , z ; see p. 32 ( ) n o n d i m e n s i o n a l i z e d such t h a t ( ) = ( )/£, e x c e p t f o r tt = tt /Q a: n n' TT 1 = ( T T / £ ) ' , e t c .  0 nd S,' = V/l, £" = I"/I; whereas ' ' xx i ( ) ( ) d e n s i t y , e.g. L = L a g r a n g i a n d e n s i t y { } column m a t r i x [ ] square m a t r i x [ ] skew symmetric m a t r i x , see [ r , .] f o r example m a t r i x t r a n s p o s e d Note t h a t a l l symbols remain as d e s c r i b e d here u n l e s s s p e c i f -i c a l l y d e f i n e d o t h e r w i s e i n t h e t e x t . i , j , k a r e dummy i n d i c e s w i t h i b e i n g used e x c l u s i v e l y t o denote t h e i appendage and j , k g e n e r a l l y i d e n t i f y v e c t o r / t e n s o r components o r r e f e r e n c e axes (as t h t h. r e f e r r e d t o b e l o w ) . A l s o m,n i d e n t i f y t h e m , n assumed-mode. Fo r c l a r i t y x,y,z a r e commonly r e p l a c e d by s u b s c r i p t s 1,2,3 r e s p e c t i v e l y . F i n a l l y , a s i d e from 'rpm*, MKS u n i t s and symbols a r e used t h r o u g h o u t i n t h e p r e s e n t a t i o n o f r e s u l t s . 1 1. INTRODUCTION 1.1 P r e l i m i n a r y Remarks The s t u d y o f s a t e l l i t e m o t i o n assumed p r a c t i c a l i m p o r t a n c e w i t h t h e l a u n c h i n g o f t h e S o v i e t Union's S p u t n i k I i n October o f 1957. A key component i n t h i s s t u d y i s t h e a n g u l a r r o t a t i o n e x p e r -i e n c e d w i t h r e s p e c t t o an o r b i t i n g c e n t e r o f mass. Of c o u r s e , i t i s t h e p r e d i c t i o n and c o n t r o l o f a s p a c e c r a f t ' s o r i e n t a t i o n w h i c h u l t i m a t e l y d e t e r m i n e s i t s u s e f u l n e s s . As a r e s u l t o f t h e r e l a t i v e l y s i m p l e geometry o f t h e e a r l y s a t e l l i t e s , p r e l i m i n a r y i n v e s t i g a t i o n s o f a t t i t u d e b e h a v i o u r were c a r r i e d out u s i n g a r i g i d , s i n g l e - b o d y r e p r e s e n t a t i o n . I n many i n s t a n c e s t h i s t u r n e d out t o be a s u f f i c i e n t l y a c c u r a t e model. However, some o f t h e s a t e l l i t e s were n e i t h e r e n t i r e l y r i g i d nor c o u l d t h e y be r e p r e s e n t e d as a s i n g l e body; f o r example, E x p l o r e r I had f o u r f l e x i b l e antennae r a d i a t i n g out from t h e main body. As t h e number and c o m p l e x i t y o f t a s k s grew, so t o o d i d t h e c o m p l e x i t y o f a s p a c e c r a f t ' s c o n f i g u r a t i o n . A g e n e r a l d e f i n i t i o n o f a contemporary s a t e l l i t e would be: "any c o l l e c t i o n o f o r b i t i n g , a r b i t r a r i l y - s h a p e d , i n t e r c o n n e c t e d b o d i e s . " Embedded i n t h i s c o n c e p t a r e a number o f c h a r a c t e r i s t i c s r e n d e r i n g t h e p r o c e s s o f m a t h e m a t i c a l m o d e l l i n g q u i t e d i f f i c u l t . O v e r a l l , s a t e l l i t e c o n f i g u r a t i o n s can assume q u i t e i r r e g u l a r forms s i n c e t h e y a r e n o t c o n s t r a i n e d by t h e aerodynamics t o t h e same degree as an a i r c r a f t o r a m i s s i l e . Each s p a c e c r a f t m i s s i o n and 2 hence each c o n f i g u r a t i o n t e n d s t o be unique making i t d i f f i c u l t t o e v o l v e a g e n e r a l a n a l y s i s . The e x t e n t o f f l e x i b i l i t y may v a r y t h u s making t h e system a h y b r i d c o n s t r u c t i o n o f t h e c l a s s i c a l f u l l y - r i g i d and f u l l y - e l a s t i c p a r t s . C o n s i d e r a b l e r e l a t i v e m o t i o n between components can o c c u r as a r e s u l t o f a r t i c u l a t i o n , d e p l o y -ment o r v i b r a t i o n . C o n s e q u e n t l y , time-dependent i n e r t i a s as w e l l as a s h i f t i n g c e n t e r o f mass a r e i n t r o d u c e d , t h u s compounding t h e problem c o n s i d e r a b l y . F l e x i b i l i t y i s a d e s i g n c h o i c e d i c t a t e d i n p a r t by a d i c h o t -omy o f extremes i n t h e f o r c e environment: v e r y h i g h a c c e l e r a t i o n s d u r i n g d e l i v e r y t o o r b i t f o l l o w e d by v e r y low a c c e l e r a t i o n s d u r i n g most, i f not a l l , o f t h e o p e r a t i o n a l l i f e . O f t e n s t r u c t u r e s h a v i n g l a r g e d i m e n s i o n s a r e r e q u i r e d t o c a r r y o u t e x p e r i m e n t s , p r o v i d e s t a b i l i z a t i o n , and g e n e r a t e power. However, c o n f i g u r a t i o n s i z e and w e i g h t can be s e v e r e l y c o n s t r a i n e d as a r e s u l t o f l a u n c h v e h i c l e l i m i t a t i o n s o r s t r u c t u r a l s t r e n g t h o f t h e s a t e l l i t e com-ponents. As a s o l u t i o n s p a c e c r a f t a r e i n i t i a l l y packaged as com-p a c t r i g i d b o d i e s . Once i n o r b i t v a r i o u s elements d e p l o y t o e s t a b l i s h t h e d e s i r e d c o n f i g u r a t i o n . Deployment, i n many i n s t a n c e s , accompanies t h e a t t i t u d e a c q u i s i t i o n phase d u r i n g w h i c h l a r g e a n g l e manoeuvres t a k e p l a c e . An a d d i t i o n a l c o m p l i c a t i o n i s c o n t r i b u t e d by t h e p r e s e n c e o f such e n v i r o n m e n t a l f o r c e s as s o l a r r a d i a t i o n p r e s s u r e , t h e e a r t h ' s magnetic f i e l d , f r e e m o l e c u l a r f o r c e s , e t c . , c a p a b l e o f e x c i t i n g e l a s t i c degrees o f freedom. The f a c t t h a t f l e x i b i l i t y i s an i m p o r t a n t f a c t o r not t o be o v e r l o o k e d was dem o n s t r a t e d q u i t e c o n v i n c i n g l y by two i n c i d e n t s . E x p l o r e r I (1958) was i n i t i a l l y s p i n n i n g a b o u t t h e a x i s o f minimum 3 moment of i n e r t i a . Four antennae located normal to the spin axis allowed for energy d i s s i p a t i o n v i a dynamic coupling between pre-cessional and v i b r a t i o n a l degrees of freedom. The end r e s u l t was a tumbling about the axis of maximum moment of i n e r t i a i n a state of minimum k i n e t i c energy - a r e s u l t not previously recognized by c l a s s i c a l r i g i d body s t a b i l i t y theory. Canada's f i r s t s a t e l l i t e , A l l ouette I (1962), a s p i n - s t a b i l i z e d system with four booms up to about 23 m i n length l y i n g i n the spin plane, experienced a premature decay i n spin rate. Subsequent analysis suggested that a solar-thermal induced asymmetry i n boom shape resulted i n a net opposing moment from the r a d i a t i o n pressure. Thus, what was i n i t i a l l y considered to be anomalous behaviour was attributed, by p o s t - f l i g h t analysis, d i r e c t l y to f l e x i b i l i t y e f f e c t s . In t h i s context i t i s relevant to mention the Orbiting Geophysical Obser-vatory (0G0 I I I ) , one of the more elaborate c o l l e c t i o n s of r i g i d and e l a s t i c bodies. Launched i n 1966, i t demonstrated that i n t e r -action between control torques and e l a s t i c deformations could r e s u l t i n attitude i n s t a b i l i t y . With an increasing use of f l e x i b l e appendages, the problem grows more c r i t i c a l as stationkeeping and pointing requirements become stringent. For example, the j o i n t Canada/U.S.A. Communi-cations Technology S a t e l l i t e (CTS/Hermes) launched i n 1976 c a r r i e s two solar panels measuring 1.1 m by 7.3 m each to generate 1.2 kW. The 'Galileo', scheduled f o r launch i n 1982, has a r t i c u l a t e d members with both spinning and nonspinning sections making up the main body. Attached to the spinning part are f l e x i b l e booms up to 11 m long. In addition, wide variations are expected to occur 4 i n the i n e r t i a p r o p e r t i e s over the l i f e o f the m i s s i o n due t o a r e l a t i v e l y l a r g e r a t i o of p r o p e l l a n t - t o - s p a c e c r a f t mass. E l a s t i c members used i n the c o n s t r u c t i o n o f any f u t u r e S o l a r Power S a t e l l i t e s (SPS) would have dimensions measurable i n k i l o m e t e r s . With the advent o f the Space S h u t t l e , t e t h e r - s u p p o r t e d s a t e l l i t e systems extending t o 100 km are a n t i c i p a t e d . As has been i m p l i e d , p r e d i c t i n g s a t e l l i t e a t t i t u d e motion i s by no means a simple p r o p o s i t i o n , even i f the system i s r i g i d . The time-bound c h a r a c t e r of most p r o j e c t s r e s t r i c t s a t t e n t i o n to a g i v e n c o n f i g u r a t i o n w i t h dynamic s i m u l a t i o n c o n f i n e d to phases c o n s i d e r e d most important. I t i s t h e r e f o r e understandable why the m a j o r i t y of p u b l i s h e d papers d i s c u s s o n l y steady s t a t e motion i n the s m a l l . T r a n s i e n t behaviour a s s o c i a t e d w i t h the c r i t i c a l phase of a t t i t u d e a c q u i s i t i o n and/or deployment r e l a t e d manoeuvres has been l a r g e l y i g n o r e d . I t should be mentioned t h a t deployment e f f e c t s , a lthough o f a t r a n s i e n t nature, may be f e l t over a l o n g p e r i o d o f time as a r e s u l t o f r e l a t i v e l y s m a l l e x t e n s i o n r a t e s which can be a s s o c i a t e d w i t h l o n g appendages.* In a d d i t i o n , deployment a f f e c t s the f o r c e f i e l d a c t i n g on the f l e x i b l e members, thus i n f l u e n c i n g e l a s t i c response, s t r u c t u r a l i n t e g r i t y , and the l i b r a t i o n i t s e l f . As can be expected, the dynamics i s s t r o n g l y c o n f i g u r a t i o n dependent. Few i n v e s t i g a t i o n s have been r e p o r t e d which apply t o more than one type of s p a c e c r a f t . As a r u l e the more g e n e r a l the f o r m u l a t i o n i n terms o f c o n f i g u r a t i o n the l e s s developed i s the * For example, e x t e n s i o n of a 200 m boom t y p i c a l l y r e q u i r e s 2000 s. 5 analysis. Ultimately a s p e c i f i c case i s chosen with i t s attendant s i m p l i f i c a t i o n s . An attempt i s made i n t h i s thesis to demonstrate the use of a systematic methodology for dealing with complex o r b i t i n g dynamical systems. This i s achieved by formulating and solving the equations of motion applicable to a large class of spacecraft configurations. 1.2 Li t e r a t u r e Review 1.2.1 Background Over the years the accumulated l i t e r a t u r e pertaining to the various aspects of s a t e l l i t e system response, s t a b i l i t y , and con-t r o l has indeed become enormous. Hence to review t h i s l i t e r a t u r e i n depth would present one with a formidable task not to mention the space required. Furthermore, several excellent review papers have been written covering s p e c i f i c areas of the subject [Noll et a l . (1969), 1 Shrivastava et a l . (1969), 2 L i k i n s (1970, 1974, 1976, 1977), 3' 4' 5' 6 L i k i n s and Bouvier (1971), 7 Modi (1974), 8 Williams (1976), 9 Garg et a l . ( 1 9 7 8 ) / 1 0 Stuhlinger (1979), 1 1 Roberson 12 (1979) ]. Therefore, the objective here i s to trace the general evolution of the subject, problems faced i n modelling and analysis, achievements and shortcomings, and more importantly, to indicate the r o l e of the present contribution within the o v e r a l l scheme of progress to date. Only the more important contributions d i r e c t l y relevant to the thesis subject i n hand are recorded here. 6 G e n e r a l s p a c e c r a f t m o t i o n c o n s i s t s o f : ( i ) t r a n s l a t i o n o f t h e c e n t e r o f mass ( o r b i t a l ) , ( i i ) r o t a t i o n w i t h r e s p e c t t o t h e c e n t e r o f mass ( l i b r a t i o n a l ) , and ( i i i ) r e l a t i v e m o t ions between th e c o n s t i t u e n t p a r t s (e.g. v i b r a t i o n , deployment, f u e l movement, e t c . ) . To f i r s t o r d e r , t h e e f f e c t o f l i b r a t i o n and e l a s t i c d e f o r m a t i o n on 13 14 o r b i t a l m o t i o n i s n e g l i g i b l e [Moran (1961), Yu (1964), M i s r a and 15 Modi (1977) ] . C o n s e q u e n t l y , an a p r i o r i s o l u t i o n e x i s t s f o r t h e t r a j e c t o r y as r e p r e s e n t e d by t h e c l a s s i c a l K e p l e r r e l a t i o n s . 12 A c c o r d i n g t o Roberson (1979), t h e f i r s t p u b l i s h e d paper de-v o t e d t o a t t i t u d e m otion o f an a r t i f i c i a l e a r t h s a t e l l i t e i s by K l e m p e r e r and Baker (1956). I t d i s c u s s e d t h e p l a n a r m o t i o n o f a r i g i d dumbbell c o n f i g u r a t i o n moving i n a c i r c u l a r o r b i t . F o l l o w i n g t h i s , t h e model o f a s i n g l e r i g i d body t r a v e l l i n g a l o n g a c i r c u l a r p a t h was adopted f o r much o f t h e e a r l y work [e.g. De B r a and Delp 17 (1961) ] . P r i m a r i l y , c o n f i g u r a t i o n s were t a k e n t o be e i t h e r sym-m e t r i c (1963, 1964, 1962, 1 9 6 6 ) , 1 8 - 2 1 a x i s y m m e t r i c ( 1 9 6 6 ) , 2 2 or a-23 24 symmetric (1965, 1963). ' The l i b r a t i o n i t s e l f was assumed t o be 17 23 25 c o u p l e d , l i n e a r (1961, 1 9 6 5); ' p l a n a r , n o n l i n e a r (1969); e t c . The v a r i o u s e n v i r o n m e n t a l i n f l u e n c e s c o n s i d e r e d ( s o l a r , m a g n e t i c , aerodynamic, g r a v i t a t i o n a l , e t c . ) a r e d i s c u s s e d i n r e p o r t s such as t h o s e o f Evans ( 1 9 6 4 ) , 2 6 S i n g e r ( 1 9 6 4 ) , 2 7 and Roberson ( 1 9 6 4 ) . 2 8 I n g e n e r a l , t h e e f f e c t o f t h e e a r t h ' s o b l a t e n e s s on a t t i t u d e dynamics was found t o be n e g l i g i b l e . The l i t e r a t u r e s u g g e s t s t h a t even f o r t h e s i m p l e s t o f s p a c e c r a f t , as r e p r e s e n t e d by a s i n g l e r i g i d body, t h e number o f parameters i n v o l v e d i n t h e problem i s s u f f i c i e n t l y l a r g e t h a t a n a l y s t s d e a l w i t h a s p e c i f i c s e t o f s a t e l l i t e parameters o n l y . 29 A welcome e x c e p t i o n was t h e work o f B e l e t s k i i (1965) i n w h i c h an a x i s y m m e t r i c body i n an e c c e n t r i c o r b i t i s s u b j e c t e d t o a v a r i e t y o f e x t e r n a l t o r q u e s . 7 The importance of configuration as a variable i n the analysis increases with the use of multibody systems. One of the simpler concepts involves a r e l a t i v e motion within a main r i g i d body creat-ing an 'ef f e c t i v e ' i n t e r n a l s t i f f n e s s and/or damping. A planar case was examined by Brereton (1967) 3^ and Tschann (1970). 3"*" Considered too were dampers with: (a) one degree of freedom para-l l e l to the spin axis, i . e . a nutation damper [Kane and Levinson 32 (1976) ]; (b) two degrees of freedom normal to the spin axis, i . e . 33 34 a precession damper [Cloutier (1976) ]. Cochran et a l . (1980) compared the performance of nutation versus precession.dampers for both axisymmetric and asymmetric s a t e l l i t e s . Dampers have been 35 applied to the dual spin concept as well by Vigneron (1971). For control purposes two single-degree-of-freedom gyros were oper-25 ated i n a 'ro l l - v e e ' mode by Morrison (1969); on the other hand, a single two-degree-of-freedom gyro was studied by Kane and Athel (1972). 3 6 Frequently encountered i n the l i t e r a t u r e are spacecraft made 37 up of connected r i g i d masses. Both planar [Paul (1963) ] and 3 8 coupled [Pringle (1968) ] l i b r a t i o n a l behaviour has been studied 39 for an o r b i t i n g dumbbell. C r i s t and E i s l e y (1969) have presented l i n e a r as well as some nonlinear analysis for two spinning systems: (i) a f l e x i b l e dumbbell; ( i i ) a small mass spring-connected to a 40 large mass. Connell (1969) sought to optimize pointing accuracy and attitude s t a b i l i t y using a hinged two-body system. Also, con-siderable attention has been directed towards the analysis of 41 cable-connected two-body systems. For example, Chobotov (1963) 42 and Bainum and Evans (1976) have examined pote n t i a l e x c i t a t i o n of 43 such a system by the gravity gradient. Tai and Loh (1965) and 8 Stabekis and Bainum (1970)" dealt with planar response while 45 Bainum and Evans (1975) evaluated the use of a multiple cable 47 design. The model of Modi and Sharma (1977) allowed for both string-type as well as beam-type connection. F i n a l l y , Austin and 48 Zetkov (1974) have discussed simulations for a class of f l e x i b l e two-body s a t e l l i t e s . A more general configuration i s the a r t i c u l a t e d type, i . e . to the main body(ies) are appended secondary smaller bodies. Etkin 49 50 (1962) and Maeda (1963) presented the l i n e a r i z e d uncoupled equations together with some preliminary response data for a gravity-s t a b i l i z e d configuration i n which a number of rods are symmetric-a l l y pinned to a central body with both s t i f f n e s s and damping 51 present at the j o i n t s . Hughes (1966) added t i p masses to the rods of t h i s configuration and optimized both transient and steady state pointing performance with respect to system parameters. At synchronous a l t i t u d e maximum pointing errors of the order of two degrees and damping times less than one o r b i t are achieved. Lips 52 (1967) derived the l i n e a r i z e d planar equations for a similar system but augmented the gravity gradient e f f e c t by attaching the t i p masses to the rods by means of long 'massless' strings. Later, the effectiveness of a number of d i f f e r e n t rod-string-tip-mass com-binations i n providing gravity gradient s t a b i l i z a t i o n was invest-53~ igated by Garg (1969). Another series of complex configurations i s t y p i f i e d by the g r a v i t y - s t a b i l i z e d Radio Astronomy Explorer (RAE) 54 s a t e l l i t e studied by Dow et a l . (1966). In t h i s case four long (228 m), f l e x i b l e , deployable booms l i e i n the o r b i t a l plane and a set of r i g i d damper booms are skewed with respect to the o r b i t a l plane. The s p i n - s t a b i l i z e d Alouette I and II had four (11 m to 36 m) booms 9 attached to a central r i g i d body. Described by Charyk (1977) J U i s a series of communications s a t e l l i t e s which have as appendages: antennae booms, antennae dishes, and/or solar panels. Janssens 57 (197 6) discusses the dynamics of a spinning r i g i d body to which are attached appendages which act as spherical pendulums. F i n a l l y , 58 i t should be mentioned that the SPS of Glaser (197 6) could bring into question the r o l e of the appendage as a secondary body. In addition to r i g i d or hybrid r i g i d / e l a s t i c forms which a s a t e l l i t e might assume, there i s also the prospect of a f u l l y e l a s t i c system. Ashley (1967),^ 9 Modi and Brereton (1968),^ 61 62 Bainum et a l . (1978, 1980) ' have studied the dynamics of a 6 3 beam-type s a t e l l i t e . Also Breakwell and Andeen (1977) dealt with a chain of beads to be used for communications, while Chobotov 64 (1977) proposed a chain of solar arrays aligned along the l o c a l v e r t i c a l as a method of c o l l e c t i n g solar energy. The introduction of f l e x i b i l i t y into the design has proven to be a major source of complication. With respect to l i b r a t i o n i t acts as a d i s s i p a t i v e feedback mechanism int e r a c t i n g with the control system and environmental forces. Provided s t r u c t u r a l i n -t e g r i t y i s maintained, large amplitude (or even unstable) v i b r a -tions are of concern only to the degree that they a f f e c t the l i b r a -t i o n a l behaviour.* Consequently, f l e x i b i l i t y need be included i n the analysis only i f i t i s expected to i n t e r f e r e with the attitude motion. However, i t i s important to emphasize that a p r i o r i know-lege of f l e x i b i l i t y e f f e c t s i s , i n general, not ava i l a b l e . * Unless the f l e x i b l e member must also s a t i s f y some other design c r i t e r i a , e.g., support a magnetometer. 10 N e g l e c t o f f l e x i b i l i t y has produced a h o s t o f s u r p r i s i n g d y n a m i c a l b e h a v i o u r , some o f w h i c h was i d e n t i f i e d e a r l i e r . Among th e f i r s t t o r e v i e w f l e x i b i l i t y e f f e c t s based on p o s t - f l i g h t a n a l -y s i s were N o l l e t a l . ( 1 9 6 9 ) . 1 I n t e r a c t i o n problems a r e shown t o m a n i f e s t t h e m s e l v e s as t r a n s i e n t phenomena, l i m i t c y c l e o s c i l l a -t i o n s , o r i n s t a b i l i t i e s . Such u n f o r e s e e n o c c u r e n c e s a r e a t t r i b u t e d t o d e f i c i e n c i e s i n the s t r u c t u r a l dynamics a n a l y s i s and/or know-l e d g e o f t h e f l i g h t environment. S i m i l a r f i n d i n g s a r e e x p r e s s e d 65 7 by a NASA document (1969) and by L i k i n s and B o u v i e r (1971). 5 6 The s u r v e y was updated by L i k i n s (1976, 1977) ' who r e p o r t e d some s u c c e s s e s as w e l l as f a i l u r e s i n t r e a t i n g f l e x i b l e systems. Modi (1974) o f f e r s a comprehensive s t a t e o f t h e a r t assessment o f e f f o r t s made i n d e a l i n g w i t h b o t h t h e l i b r a t i o n a l r e s p o n s e and t h e appendage dynamics i t s e l f . P a r t i c u l a r l y h e l p f u l i s t h e attempt t o u n i f y t h e problem by s i m u l t a n e o u s l y b r i n g i n g t o g e t h e r d i f f e r e n t a s p e c t s o f c o n t r o l system, s t r u c t u r a l , and l i b r a t i o n a l dynamics. 1.2.2 E q u a t i o n s o f m o t i o n I n g e n e r a l , t i m e and e f f o r t i n v o l v e d i n t h e d e r i v a t i o n o f gov-e r n i n g e q u a t i o n s o f m o t i o n can be enormous hence, one s e r i o u s l y i n -q u i r e s about t h e most e f f i c i e n t p r o c e d u r e a v a i l a b l e . The fundamen-t a l c h o i c e s range from D'Alembert's P r i n c i p l e and t h e Newton-Euler 6 6 v e c t o r approach t o t h e use o f t h e Gibb's f u n c t i o n and t h e 67 H a m i l t o n - L a g r a n g e v a r i a t i o n a l model. The i s s u e w i l l be r e v i e w e d o n l y b r i e f l y here s i n c e i t has been debate d e x t e n s i v e l y i n t h e l i t e r a t u r e by R u s s e l l (1969, 1 9 7 6 ) , 6 8 , 6 9 Hooker ( 1 9 7 0 ) , 7 0 S t i c h i n e t a l . (1970, 1 9 7 5 ) , 7 4 , 7 5 Anand and Whisnant ( 1 9 7 1 ) , 7 3 Ho (1974, 1 9 7 7 ) , 7 4 , 7 5 L i k i n s (1974, 1 9 7 5 ) , 4 , 7 6 W i l l i a m s ( 1 9 7 6 ) , 9 McDonough 11 77 78 (1976), Jerkovsky (1978), and others. The most common objec-t i o n raised regarding the Newton-Euler ap p l i c a t i o n i s the need to include constraint forces even when they are not s p e c i f i c a l l y sought. This d i f f i c u l t y i s eliminated with the Lagrangian pro-cedure but here the k i n e t i c energy expression can become unwieldy to derive and even more so to d i f f e r e n t i a t e , as indicated by 69 7 9 Russell (1976) and Lips (1978). Overall, no one approach 7 6 appears to be superior i n a l l respects. L i k i n s (1975), for example, points out that any apparent advantage i n structure of the Lagrange quasicoordinate equations over the Euler equations disappears when the issue i s c a r e f u l l y examined. If any trend i s discernable at a l l i t i s toward acceptance of a r e s u l t s i m i l a r to that of Kane et a l . (1965, 1 9 6 8 ) 8 0 , 8 1 and Jerkovsky (1978) 7 8 i n which the v a r i a t i o n a l a p p l i c a t i o n of D'Alembert's p r i n c i p l e y i e l d s equations s i m i l a r i n forms to those of Newton and Euler. That i s , for n r i g i d bodies having k degrees of freedom the governing equa-tions of motion can be represented by [Likins (1977)^]: n 9R. 3(£. Z [(F.-m.R.) o _ - l + (M.-H ) o —1] = 0; i,...,k (1.1) j=l ~3 dq± ~3 ~3 dq± where, t h F_. = force acting on j body, M. = moment acting about center of mass of j*"* 1 body, t h H. = angular momentum for the j body, t h m_. = mass of j body, Rj = i n e r t i a l p o s i t i o n vector, q^ = i generalized coordinate, -D t h = angular v e l o c i t y vector for the j body. Constraints and redundant equations vanish as a r e s u l t of the summed 12 8 2 dot m u l t i p l i c a t i o n s . A recent paper by Kane and Levinson (1980) suggests that such an approach involves the least e f f o r t . Levinson 8 3 (1977) has developed a computer program for constructing equations of motion for r i g i d systems v i a symbolic manipulation, based on equation (1.1). The d i v e r s i t y with which the problem i s approached i s empha-84 — 89 sized by the following c o l l e c t i o n of papers. Meirovitch et a l . have consistently adopted the v a r i a t i o n a l approach to investigate s t a b i l i t y of spacecraft having e l a s t i c appendages. A unique set of adiabatic invariants i s provided by M i t c h e l l and L i n g e r f e l t 90 (1970) when studying the e f f e c t s of 'slow' changes i n material 91 volume and e l a s t i c i t y . Keat (1970) was among the f i r s t to out-l i n e a systematic method for deriving nonlinear equations, combin-ing an Euler approach for attitude equations with a Lagrange approach for appendage dynamics. A l t e r n a t i v e l y , i n applying the 92 Lagrange equations, Samin and Willems (197 5) used quasicoordmates when dealing with the attitude dynamics while generalized coordin-ates describe the vibrations. Russell (1976) 6 9 preferred the use of f i r s t order momentum equations. A D'Alembert derivation i s im-93 plemented by Bodley and Park (1972) who employed projections of momentum as the dynamic variables, a choice f i r s t advanced by 71 Vance and S t i c h i n (1970). Exact or 'global' equations of motion 77 for an e l a s t i c body are offered by McDonough (1976) as a check against approximate analyses. A number of studies have been carried out which consider the motion of a f l e x i b l e system to be simply a 94 perturbed form of the r i g i d body solution [Grote et l . (1971), Huang and Das (1973), 9 5 Morton et a l . (1973), 9 6 Kraige and Junkins (1976) 9 7]. Pringle (1972) 9 8 has implemented a perturbation formal-13 ism v i a c a n o n i c a l t r a n s f o r m a t i o n s . Of c o u r s e , one cannot over-emphasize t h e i m p o r t a n c e o f c h o i c e o f r e f e r e n c e c o o r d i n a t e s i n any f o r m u l a t i o n p r o c e d u r e , as i t can have a p r o f o u n d e f f e c t i n e i t h e r s i m p l i f y i n g o r c o m p l i c a t i n g t h e 9 r e s u l t a n t g o v e r n i n g e q u a t i o n s . W i l l i a m s (1976) has recommended a s a t e l l i t e a t t i t u d e r e f e r e n c e frame f i x e d t o t h e c e n t e r b o d y . N a t u r a l l y , t h i s may not be u s e f u l f o r s p a c e c r a f t w h i c h a r e n o t a l l -go f l e x i b l e , as p o i n t e d out by Veubeke (197 6) and C a n a v i n and L i k i n s ( 1 9 7 7 ) . 1 0 0 A l s o , an a r r a y o f v a r i a b l e s such as t h e E u l e r a n g l e s , d i r e c t i o n c o s i n e s , E u l e r p a r a m e t e r s , q u a t e r n i o n s , C a y l e y -K l e i n and E u l e r - R o d r i g u e z p arameters can be used t o i d e n t i f y v e h i c l e a t t i t u d e . R e c e n t l y , many a u t h o r s such as Davenport ( 1 9 7 3 ) , 1 0 1 Ohkami ( 1 9 7 6 ) , 1 0 2 and N a z a r o f f ( 1 9 7 9 ) 1 0 3 have reexamined t h e g e n e r a l p r o blem o f t r a n s f o r m a t i o n between r e f e r e n c e frames i n 104 a t h r e e d i m e n s i o n a l space. W i l k e s (197 9) d e r i v e s an e x p r e s s i o n i n w h i c h t h e elements o f t h e t r a n s f o r m a t i o n a r e e x p r e s s e d e x p l i c i t -l y as f u n c t i o n s o f any t h r e e r o t a t i o n a n g l e s and t h e c o r r e s p o n d i n g r o t a t i o n axes. What has become common f o r s p a c e c r a f t n a v i g a t i o n a p p l i c a t i o n s i s t h e use o f t h e H a m i l t o n q u a t e r n i o n s because o f t h e i r r e l a t i v e compactness - f o u r e lements v e r s u s n i n e f o r d i r e c t i o n c o s i n e s [Garg ( 1 9 7 8 ) , 1 0 Mayo ( 1 9 7 9 ) 1 0 5 ] . I c k e s ( 1 9 7 0 ) 1 0 6 i n t e -g r a t e d t h e c o n c e p t o f q u a t e r n i o n s i n t o a d i g i t a l a t t i t u d e c o n t r o l 107 system. A l s o h e l p f u l i s t h e work o f Klumpp (1976), S p u r r i e r ( 1 9 7 8 ) , 1 0 8 Sheppard ( 1 9 7 8 ) , 1 0 9 and G r u b i n ( 1 9 7 9 ) 1 1 0 i n e n a b l i n g one t o e f f i c i e n t l y e x t r a c t a q u a t e r n i o n from a d i r e c t i o n c o s i n e m a t r i x . 14 1.2.3 Appendage dynamics A c r i t i c a l step i n completing the development of the govern-ing equations for a s a t e l l i t e i s the dynamical modelling of any f l e x i b l e elements present. Much of the background to the problem i s presented by Modi (1974). Because of i t s importance, some of the main features and conclusions are discussed here along with a review of the more recent l i t e r a t u r e . Pioneering contributions to 111 3 the f i e l d may be attributed to Vigneron (1968), L i k i n s (1970), Hughes and Garg (1973), 1 1 2 and M e i r o v i t c h . 8 4 8 9 B a s i c a l l y , e l a s t i c members are treated either as a series of interconnected r i g i d bodies whose dynamics i s governed by a set of ordinary d i f f e r e n t i a l equations, or as a continuum generating a system of p a r t i a l d i f f e r -e n t i a l equations. The continuum set can be ' d i s c r e t i z e d 1 i n a mathematical sense by means of an assumed-mode solution, thus pro-viding a set of 'distributed' or 'modal' coordinates* governed by 3 ordinary d i f f e r e n t i a l equations. Li k i n s (1970) encouraged the v adoption of a hybrid system combining the attitude variables describ-ing l i b r a t i o n with the d i s t r i b u t e d coordinates defining f l e x i b l e behaviour. Depending on the accuracy required, one can truncate the modal series representation appropriately, thus s i g n f i c a n t l y reducing the number of degrees of freedom while at the same time avoiding troublesome high frequency interactions i n the simulation. 112 Truncation i s r a t i o n a l i z e d by Hughes and Garg (1973). More 69 quantitative c r i t e r i a are prescribed by Russell (1976), and Likins 113 et a l . (1976). No truncation i s needed i f one works m the * Coordinates representing the shape or motion of a continuum at p a r t i c u l a r instant. 15 114 frequency domain as suggested by Kulla (1972), Larsen and L i k i n s ( 1976), 1 1 5 and Poelart (1977); 1 1 6 however, t h i s i s not possible for a nonlinear system. Determination of spacecraft v i b r a t i o n c h a r a c t e r i s t i c s can con-sume a large proportion of the analysis e f f o r t . I d e a l l y the assumed-modes chosen i n solving for d i s t r i b u t e d coordinates would be the 3 exact eigenfunctions. The approach taken by L i k i n s (197 0) i s to derive modal c h a r a c t e r i s t i c s based on the assumption of a constant 112 average angular motion. On the other hand, Hughes and Garg (1973) carr i e d out an ambitious and d e t a i l e d study for f l e x i b l e solar array c h a r a c t e r i s t i c s , both 'constrained' (no l i b r a t i o n ) and 'unconstrain-ed' ( l i b r a t i o n f u l l y integrated into the v i b r a t i o n equations). 117 118 Nguyen and Hughes (1976) and Gupta (1976) applied the f i n i t e element method to the same end. Structural dynamics associated with 119 the CTS i s also studied by Vigneron (1975) who examined 'free' v i b r a t i o n c h a r a c t e r i s t i c s at one-g (ground level) and zero-g (in-120 orbit) states, and by Sincarsin (1977) who includes the gravity 121 gradient e f f e c t . Hughes and Sharpe (1975) generalized the model somewhat by including a source of i n t e r n a l angular momentum when deriving appendage c h a r a c t e r i s t i c s . Meirovitch (1974, 1975, 122 123 124 1976) ' ' has presented a general eigenvalue approach for a r r i v i n g at o v e r a l l spacecraft modes. In e f f e c t , t h i s represents an extension of the normal coordinate method to any hybrid repre-sentation which can be modelled as a l i n e a r gyroscopic system. The second order system i s replaced by a set of f i r s t order equa-tions expressed i n terms of the state variables. As pointed out 125 by Nelson and Glasgow (1979) , one could work with second order eigenrelations. S t i l l another a l t e r n a t i v e i s to construct vehicle 16 modes by s y n t h e s i s o f t h e modes o f t h e c o n s t i t u e n t elements as sug-g e s t e d by Rubin ( 1 9 7 5 ) 1 2 6 and H i n t z ( 1 9 7 5 ) . 1 2 7 One o f t h e more common s p a c e c r a f t appendages i s t h e boom used i n communications and i n p r o v i d i n g g r a v i t a t i o n a l s t a b i l i t y . Be-cause o f i t s l o n g s l e n d e r n a t u r e i t can, i n g e n e r a l , be m o d e l l e d as a beam. F o r t h i s r e a s o n t h e problem o f r o t a t i n g beams i s o f 128—135 p a r t i c u l a r i n t e r e s t and has been s t u d i e d e x t e n s i v e l y . I n 136 p a r t i c u l a r , Kaza and K v a t e r n i k (1977) have s u r v e y e d s e v e r a l methods f o r l i n e a r as w e l l as second degree n o n l i n e a r r e p r e s e n t a -t i o n s o f t h e problem. Independent o f t h i s r e s u l t , L i p s and Modi 137 138 (1977, 1978) ' worked out second degree e q u a t i o n s f o r t h e more 139 g e n e r a l c a s e o f a d e p l o y i n g r o t a t i n g beam. Nguyen (1978) de-r i v e d and at t e m p t e d t o s o l v e (not alw a y s s u c c e s s f u l l y ) t h e f u l l y n o n l i n e a r e q u a t i o n s . I t i s u s e f u l t o r e c o g n i z e t h a t i t i s not always n e c e s s a r y t o d e a l w i t h n o n l i n e a r e q u a t i o n s . I f , f o r example, t h e n o n l i n e a r i t y i s a r e s u l t o f l a r g e s t e a d y s t a t e d e f o r m a t i o n s (as might be t h e c a s e w i t h h i g h s p i n r a t e s ) , t h e n o n t r i v i a l e q u i l i b -r i u m shape can be e s t a b l i s h e d t o g e t h e r w i t h l i n e a r e q u a t i o n s de-14 0 s c r i b i n g o s c i l l a t i o n s w i t h r e s p e c t t o i t [ F l a t l e y (1966), M e i r o v i t c h e t a l . ( 1 9 7 6 ) , 1 4 1 K i s s e l b a c h ( 1 9 7 6 ) , 1 4 2 H a b l a n i e t a l . 143 (1978) J-^ J] . A number o f a u t h o r s have f o c u s e d a t t e n t i o n on i n v e s t i g a t i n g 144-146 t h e dynamics o f ' l a r g e ' f l e x i b l e s p a c e c r a f t . C a n a v i n and M e i r o v i t c h ( 1 9 7 9 ) 1 4 7 have p o i n t e d out t h a t w i t h an i n c r e a s e i n s i g n i f i c a n c e o f f l e x i b i l i t y , i t s e f f e c t s can no l o n g e r always be c o n s i d e r e d as s i m p l y a p e r t u r b a t i o n on t h e r i g i d body r e s p o n s e . 99 Veubeke (1976) has p r e s e n t e d a method f o r d e a l i n g w i t h t h e non-l i n e a r dynamics o f c o m p l e t e l y f l e x i b l e s p a c e c r a f t . 17 As a f i n a l p o i n t , one s h o u l d bear i n mind t h a t f l e x i b l e e lements i n space a r e s u s c e p t i b l e t o e n v i r o n m e n t a l e x c i t a t i o n . Modi (1974) has r e v i e w e d most o f t h e s i g n i f i c a n t i n v e s t i g a t i o n s r e p o r t e d i n t h e l i t e r a t u r e . I n a d d i t i o n , t h e r m a l e f f e c t s on appen-144 dage dynamics have been s t u d i e d by Lorenz (1975), T s u c h i y a ( 1 9 7 7 ) , 1 4 8 F a r r e l l ( 1 9 7 7 ) , 1 4 9 and F r i s c h ( 1 9 8 0 ) . 1 5 0 A l s o more 151 r e c e n t l y , Kumar (1976), making use o f q u a s i - s t e a d y a s s u m p t i o n s about e l a s t i c r e s p o n s e , has a s s e s s e d t h e r m a l and r a d i a t i o n p r e s s u r e 152 e f f e c t s on l i b r a t i o n a l s t a b i l i t y . J o s h i and Kumar (1980) make d i r e c t use o f s o l a r r a d i a t i o n p r e s s u r e t o o f f s e t t h e e f f e c t o f o r b i t a l e c c e n t r i c i t y on a t t i t u d e performance. 1.2.4 S t a b i l i t y and c o n t r o l o f f l e x i b l e s p a c e c r a f t Perhaps a good i n d i c a t o r as t o j u s t how i m p o r t a n t f l e x i b i l i t y e f f e c t s have become i s t h e l a r g e body o f l i t e r a t u r e d e v o t e d t o s t a b i l i t y a n a l y s i s a l o n e f o r n o n - r i g i d s a t e l l i t e s . C l a s s i c a l r e -s u l t s based on r i g i d body i n e r t i a s p r o v e d i n a d e q u a t e . Many s t u d i e s d e a l s p e c i f i c a l l y w i t h systems h a v i n g l o n g , f l e x i b l e , beam-type 153-155 appendages/ f o r example: V i g n e r o n e t a l . (1976, 197 0) , ^ i . x. i /•-.m-ix 156 ^ , ,->nnA\ 157 ^ 142,158,159 Hughes e t a l . (1971), Dong e t a l . (1974), e t c . T y p i c a l l y i t i s found t h a t : ( i ) t h e ' m a j o r - a x i s ' theorem i s a n e c e s s a r y but not a s u f f i c i e n t c o n d i t i o n f o r s t a b i l i t y o f a f l e x -i b l e system; ( i i ) n e c e s s a r y c o n d i t i o n s f o r s t a b i l i t y can a l s o be d e r i v e d u s i n g such approaches as t h a t o f Lyapunov; ( i i i ) assuming t h e s p a c e c r a f t t o be r i g i d c o u l d l e a d d i r e c t l y t o e r r o n e o u s r e s u l t s ; f o r example, t h e e f f e c t o f e l a s t i c i t y i s t o d e s t a b i l i z e 1 c r o s s e d - d i p o l e 1 c o n f i g u r a t i o n s w h i c h a r e s t a b l e when c o n s i d e r e d 154 t o be r i g i d b o d i e s . How f a r booms can be extended w i t h o u t 18 160 causing attitude i n s t a b i l i t y was discussed by Meirovitch (1974) for spinning systems. Barbera and L i k i n s ( 1 9 7 3 ) 1 6 1 and Meirovitch 84 — 8 9 et a l . have developed s t a b i l i t y c r i t e r i a for a large class of continuum systems using the Lyapunov method. Very general c r i t e r i a 162 are developed also by L i k i n s (1972) and are applicable to any l i n e a r dynamical system including that represented by spacecraft. The Lyapunov technique was put i n a state variable format by Smith 163 and G i l l (1974). Nonzero products of i n e r t i a associated with f l e x i b i l i t y were found to a s s i s t i n s t a b i l i z i n g the dual-spin con-164 f i g u r a t i o n studied by Tseng and P h i l l i p s (1976). During the s t a b i l i t y study of a gyrostat having f l e x i b l e appendages, Calico ( 1 9 7 6 ) 1 6 5 found i t s u f f i c i e n t to represent each e l a s t i c displace-16 6 ment by a single d i s c r e t e coordinate. Boland et a l . (1974) dealt quite generally with s t a b i l i t y of a system of interconnected deformable bodies. Although b r i e f , such an outline should convince the reader of the need to include f l e x i b i l i t y when carrying out any s t a b i l i t y studies. Even when f l e x i b i l i t y does not upset the i n t r i n s i c s t a b i l i t y of a configuration i t can, nonetheless, interefere with the a b i l i t y to control attitude to the desired degree. Several al t e r n a t i v e s e x i s t i f one has to deal with unwanted v i b r a t i o n s . The control system can be modified. Also, the e l a s t i c behaviour i t s e l f may be controlled. S p e c i f i c examples of i n t e r a c t i o n between a spacecraft's control system and i t s s t r u c t u r a l degrees of freedom have been 167 168 documented by L i k i n s et a l . (1970), M i l l a r (1970), Malich (1975), 1 6 9 Loesch et a l . (1976), 1 7 0 and others. More recently, 171 Yocum and Sla f e r (1978), during t h e i r study of 'severe* i n t e r -actions , observed the p o s s i b i l i t y of a 'beating* phenomenon i n a 19 172 multi-boom s a t e l l i t e system. Hughes (1976) has discussed the use of a passive damper to reduce such inter a c t i o n s . With respect to c o ntrol, Hughes has also considered the implications of f l e x -173 l b i l i t y for the p i t c h attitude control of the CTS and for f l e x -174 i b l e spacecraft i n general. B a s i c a l l y a continuum mechanics representation i s adopted followed by a truncated modal series approximation which r e s u l t s i n additional degrees of freedom form-ing a feedback loop r e l a t i v e to the attitude response. Zach 175 (1970) attempted optimal control of a d i s t r i b u t e d gravity-s t a b i l i z e d system. Also, a design approach based on pole a l l o c a -17 6 ti o n has been advanced by Tseng and Mahn (1978). M i l l a r et a l . 177 17 8 (1979) and Hughes et a l . (1979) have suggested various types of control system l o g i c one might apply. Methods for providing control torque include the use of a f l e x i b l e boom actuator [Gatlin 179 et a l . (1969) ], a double-gimbaled momentum wheel [ H i l l a r d 18 0 (1976) ], and the use of environmental forces [Pande et a l . (1974, 1976),181,182,183 e t c > ] > 184 As a means of reducing f l e x i b i l i t y e f f e c t s , Hughes (1975) 185 and Gething et a l . (1975) suggest increased damping of s t r u c t u r a l modes. More d i r e c t i s the v i b r a t i o n i s o l a t i o n adopted by Cretcher 186 and Mingori (1971) and the v i b r a t i o n suppression of Balas 1 8 7 18 8 (1979). Smith and G i l l (1977) introduced the concept of state parameter control i n which e l a s t i c degrees of freedom are i n -cluded i n the state vector to be controlled. Somewhat more sophis-t i c a t e d versions of t h i s procedure are described as appendage modal 18 9 190 control by Beysens (1976), Jonckhere (1976), Sellappan et a l . (1978), 6 1 and Meirovitch et a l . (1978, 1 9 7 9 ) . 1 9 1 - 1 9 3 Martin and 194 Bryson (1980) discuss a possible low order c o n t r o l l e r which 20 p r o v i d e s near optimal a t t i t u d e r e g u l a t i o n but i s not as s e n s i t i v e t o m o d e l l i n g e r r o r i n the appendage dynamics as i s the f u l l - o r d e r c o n t r o l l e r . I n d i c a t i o n s as t o tren d s i n the c o n t r o l o f prese n t and f u t u r e s p a c e c r a f t are p r o v i d e d by the d i s c r e t e - t i m e approach of F o l g a t e 195 196 (1976) and Kuo (1976) along w i t h the appearance o f d i g i t a l onboard computers [Kuo e t a l . ( 1 9 7 4 ) , 1 9 7 Kawato e t a l . ( 1 9 7 6 ) , 1 9 8 Van Landingham et a l . ( 1 9 7 8 ) 1 9 9 ] . De Bra ( 1 9 7 9 ) 2 0 0 suggests t h a t the added s o p h i s t i c a t i o n o f m o d e l l i n g e r r o r compensation v i a the c o n t r o l system i t s e l f [Skelton and L i k i n s (1978) 2 "^*"] may become a p r e r e q u i s i t e f o r coping w i t h f u t u r e c h a l l e n g e s . 1.2.5 T r a n s i e n t response and deployment dynamics By now we have gained some a p p r e c i a t i o n as to the c o m p l e x i t i e s a s s o c i a t e d w i t h c o n f i g u r a t i o n geometry and v e h i c l e e l a s t i c i t y . A n a l y s i s becomes even more i n v o l v e d d u r i n g e x t e n s i o n or r e t r a c t i o n o f f l e x i b l e appendages. The deployment i n t r o d u c e s a v a r i a b l e mass d i s t r i b u t i o n (and hence v a r i a b l e moments of i n e r t i a ) t o g e t h e r w i t h r e l a t i v e v e l o c i t i e s and a c c e l e r a t i o n s . Perhaps because of i t s i n -herent c o m p l e x i t y the problem has r e c e i v e d r e l a t i v e l y l i t t l e a t t e n -t i o n . In g e n e r a l , a v a i l a b l e i n v e s t i g a t i o n s tend t o be more l i m i t e d i n scope than those d e a l i n g with nondeploying f l e x i b l e s t r u c t u r e s . 202 Lang and Honeycutt (1967) approximated s p i n n i n g f l e x i b l e d e p l o y i n g rods t o p o i n t masses l o c a t e d a t the r a d i u s o f g y r a t i o n . 203 C l o u t i e r (1968) has c o n s i d e r e d s p i n n i n g systems w i t h t i p masses extended by means of w e i g h t l e s s r i g i d r o d s . On the other hand, 204 Bowers and W i l l i a m s (1970) employed r i g i d booms and synchronized t h e i r deployment i n t e r v a l s w i t h p i t c h a t t i t u d e so as to ensure 21 capture by the gravity gradient. The assumption of r i g i d booms 205 permitted Hughes (1972) to obtain an approximate closed form solution for attitude behaviour of a s p i n - s t a b i l i z e d s a t e l l i t e , during appendage deployment. A series of deployment-related s i t u -ations ranging from spacecraft detumbling to a study of s t a b i l i t y 2 0 6 — 2 1 1 during asymmetric deployment were examined by Bainum et a l . for some s p e c i f i c configurations. The elements deployed i n these studies were either r i g i d rods having a uniformly d i s t r i b u t e d mass or point masses - no f l e x i b i l i t y was taken into account. Dow et a l . 54 (1966), however, d i d consider f l e x i b l e boom deployment as i t applied to the RAE. As described by the authors the model develop-ed i s quite extensive i n that i t includes such factors as s t r u c t u r a l damping, a r i g i d damper-boom model, solar r a d i a t i o n pressures, g r a v i t a t i o n a l e f f e c t s , and 3-axis attitude motion with both analog and d i g i t a l simulations. Although i n t e r e s t i n g r e s u l t s were present-ed i n the form of pitch/yaw displacement, maximum l i b r a t i o n angles attained under d i f f e r e n t deployment and p i t c h i n i t i a l conditions, and energy v a r i a t i o n i n o r b i t ; no governing equations are given. 212 Cherchas (1971) investigated maximum nutation, precession, bending moments, and defle c t i o n s occuring for a s p e c i f i c spinning configuration with booms extending normal to the nominal spin axis. However, when evaluating v i b r a t i o n c h a r a c t e r i s t i c s deployment rate e f f e c t s were ignored and a constant spin rate was assumed. A similar approach i s adopted for the case of solar array deployment 213 perpendicular to the major spin axis [Cherchas and Gossain (1974) ]. In both the previous presentations, the so-called 'quasi-modal' approach i s used i n which eigenfunctions associated with the i n -stantaneous appendage configuration are used when forming the 22 series representation for the e l a s t i c deformations. The d i f f i c u l t issue of d i f f e r e n t i a t i n g the quasi-modes i s not addressed. A somewhat d i f f e r e n t class of deploying systems i s represent-214 ed by the cable configuration of Ebner (197 0) and Stuvier et a l . 215 (1973). In t h i s case, attitude motion i s determined without analyzing cable o s c i l l a t i o n s . Such e f f e c t s are f u l l y accounted f o r , however, by Modi et a l . (1978, 1979). ± 0 ' z ± / Kane and Levinson 218 (1977) have presented an i n i t i a l analysis of the dynamics related to a payload undergoing f r e e - f l i g h t deployment from an o r b i t i n g spacecraft at the end of a massless cable. Simulations indicate the p o s s i b i l i t y of r e t r i e v i n g a payload from 100 m i n less than 78 min-utes for a spacecraft moving i n a 90 minute o r b i t . S i g n i f i c a n t progress has been made toward generating analyses applicable to deploying systems but s t i l l r e taining some degree of generality. In the form of a progress report on the CTS, Hughes 219 (1976) deals with the e f f e c t of deployment when formulating equa-tions for a nonrotating r i g i d body with an a r b i t r a r y number of deploy-able appendages capable of small deformations. A set of general equa-tions are developed for the case i n which deployment occurs at a con-stant rate along the r e c t i l i n e a r d i r e c t i o n s of the appendage. Also, 212 213 the view i s expressed that the use of a modal analysis ' can be expected to y i e l d good r e s u l t s i f rates of deployment are gradual 138 enough, i . e . £ < °,^ Jl. Lips and Modi (1978) presented a more general formulation allowing for three dimensional nonlinear attitude motion, g r a v i t a t i o n a l e f f e c t s , an a r b i t r a r y number/type/orientation of appendages, s h i f t i n g center of mass and independent deployment rates and accelerations of a r b i t r a r y magnitude and d i r e c t i o n for each appendage. In addition, second degree nonlinear equations were de-23 rived which govern the v i b r a t i o n of deploying beam-type appendages of a r b i t r a r y cross-section. Preliminary r e s u l t s suggested the ex-istence of c r i t i c a l combinations of f l e x i b i l i t y , deployment, and spin which can give r i s e to large amplitude, and even unstable, vibrations and hence l i b r a t i o n s . A l i n e a r form of the Newton-Euler equations governing t r a n s l a t i o n and ro t a t i o n are formulated 22 0 (but not applied) by Jankovic (1980), for a single deformable body having a time variable (deploying) configuration. An important aspect, and a major area of study i n i t s own r i g h t , i s the e f f e c t of deployment on appendage v i b r a t i o n character-221 222 i s t i c s . Leech (1970), Tabarrok et a l . (1974), and Jankovic 223 (1976) have studied the behaviour of a uniform beam extending along the a x i a l d i r e c t i o n . For example, an a n a l y t i c solution exists for deployment at a constant rate since the equations can then be 222 expressed as a Bessel equation. A solution for the case of a r b i -t r a r y deployment history was obtained using a series of admissible functions. As opposed to t h i s case, the second degree v i b r a t i o n 138 equations of Lips and Modi (1978), account for the e f f e c t of de-ployment acceleration, spin rate and spin acceleration. 'Free' v i b r a t i o n c h a r a c t e r i s t i c s obtained with a l i n e a r i z e d form of these equations have been used to assess the influence of spin, deployment rate, deployment acceleration, and C o r i o l i s e f f e c t s a r i s i n g from 224 coupling of deployment and spin v e l o c i t i e s [Lips and Modi (1978) ]. 220 Jankovic (198 0) worked out v i b r a t i o n equations for the CTS solar panels modelled as a nonspinning boom, r i g i d pressure plate, and membrane. No gravity gradient or deployment acceleration e f f e c t s were considered. Governing equations were solved using both the quasi-modal approach and Galerkin polynomials. Good agreement was 24 o b t a i n e d between measured and s i m u l a t e d a c c e l e r a t i o n a t t h e a r r a y t i p f o r t h e CTS. However, t h e tremendous e f f o r t i n v o l v e d i n gener-a t i n g r e s p o n s e v i a t h e q u a s i - m o d a l approach s u g g e s t s t h e p r o g r e s s can be made o n l y by means o f some o t h e r a p p r o x i m a t e p r o c e d u r e s , such as t h r o u g h t h e use o f a d m i s s i b l e f u n c t i o n s . 1.3 Purpose and Scope o f t h e ' I n v e s t i g a t i o n The l i t e r a t u r e r e v i e w i l l u s t r a t e s t h e l a r g e number o f f a c t o r s w i t h w h i c h t h e a n a l y s t i s f a c e d when m o d e l l i n g t h e d y n a m i c a l be-h a v i o u r o f a modern day s p a c e c r a f t . E q u a t i o n s o f m o t i o n can be f o r m u l a t e d by means o f t h e E u l e r i a n , D'Alembert, o r L a g r a n g i a n p r i n c i p l e u s i n g g e n e r a l i z e d c o o r d i n a t e s o r q u a s i c o o r d i n a t e s . E l a s t i c members a r e t r e a t e d as a s e r i e s o f i n t e r c o n n e c t e d r i g i d b o d i e s as i n t h e f i n i t e element p r o c e d u r e , o r as a continuum. M a t h e m a t i c a l l y , t h e g o v e r n i n g e q u a t i o n s range from a f i n i t e s e t o f p a r t i a l d i f f e r -e n t i a l e q u a t i o n s t o an i n f i n i t e s e t o f o r d i n a r y d i f f e r e n t i a l equa-t i o n s . I n g e n e r a l , i n v e s t i g a t o r s have d e a l t e x t e n s i v e l y w i t h t h e l i n e a r s t e a d y s t a t e r e s p o n s e f o r a w e l l - d e f i n e d c o n f i g u r a t i o n . V e r y l i t t l e a t t e n t i o n has been d i r e c t e d toward t h e t r a n s i e n t phase w h i c h i s o f c o n s i d e r a b l e i n t e r e s t and i m p o r t a n c e because o f t h e p o s s i b i l i t y o f l a r g e a n g l e m o t i o n s d u r i n g deployment o f appendages. There a r e o n l y s c a t t e r e d a t t e m p t s w h i c h c o n s i d e r f l e x i b i l i t y and deployment f o r a g e n e r a l c o n f i g u r a t i o n . T h i s t h e s i s p r e s e n t s a g e n e r a l n o n l i n e a r f o r m u l a t i o n f o r l i b r a t i o n a l dynamics (Chapter 2) o f a s p a c e c r a f t i n an e c c e n t r i c o r b i t w i t h an a r b i t r a r y number, t y p e , and o r i e n t a t i o n o f d e p l o y i n g f l e x i b l e appendages. B o t h s h i f t s i n t h e c e n t e r o f mass l o c a t i o n and g e o m e t r i c o f f s e t o f t h e p o i n t o f attachment o f t h e appendage 25 from the center of mass are provided for. The beam-type appendage which i s chosen here i s t y p i c a l of what one actually finds i n use. Note that even solar panels of the CTS were supported by a dominant central boom. Second degree non-li n e a r v i b r a t i o n equations for such f l e x i b l e appendages, derived in Chapter 3, are based on a v a r i a t i o n a l approach. Taken into account are such features as variable sectional properties and a x i a l 'foreshortening'. Of course, the general equations account for coupling with the three attitude degrees of freedom. The conjugate character of the system with the l i b r a t i o n a l motion a f f e c t i n g the vibratory displacement and vice-versa, renders the problem quite challenging. Evaluation of exact appendage modal c h a r a c t e r i s t i c s can be expensive and time consuming. Improvements i n l i b r a t i o n a l predic-tions may not always j u s t i f y the e f f o r t . The s i t u a t i o n i s further obscured during deployment since the system i s then nonautonomous. Hence i t i s h e l p f u l to examine the v i b r a t i o n c h a r a c t e r i s t i c s for models more accurate than the simple Euler-Bernoulli beam and yet not as involved as the nonlinear case. For t h i s reason a l i n e a r i z e d analysis of the vibratory motion i s car r i e d out over a range of spin and deployment parameters, assuming only p i t c h attitude motion, i . e . l i b r a t i o n a l motion, i n the o r b i t a l plane (Chapter 4). However, from design considerations, the main aspect of i n t e r e s t would be the transient l i b r a t i o n a l response over a range of system parameters and i n i t i a l conditions. Thus, having formula-ted general l i b r a t i o n a l equations, one seeks t h e i r solution. Results of d i r e c t numerical integration on a d i g i t a l computer are given i n Chapter 5 for the case of planar motions and i n Chapter 6 26 f o r t h e c a s e of 3 - a x i s a t t i t u d e r e s p o n s e . The system i s so program-med as t o i s o l a t e t h e e f f e c t o f f l e x i b i l i t y , s p i n , s h i f t i n g c e n t e r o f mass, e t c . , and t o a s s e s s t h e i r s i g n i f i c a n c e i n a g i v e n s i t u a t i o n . To summarize, t h e o b j e c t i v e here i s t o e v o l v e a f o r m u l a t i o n o f a g e n e r a l c h a r a c t e r a p p l i c a b l e t o a l a r g e c l a s s o f s p a c e c r a f t . Rather t h a n t h e a c c u m u l a t i o n o f a l a r g e amount o f d a t a , t h e i n t e n t has been t o e s t a b l i s h a s y s t e m a t i c methodology f o r c o p i n g w i t h complex d y n a m i c a l systems. F i g u r e 1.1 p r o v i d e s an o v e r v i e w o f t h e r e s e a r c h e f f o r t . DYNAMICS OF A LARGE CLASS OF S A T E L L I T E S WITH DEPLOYING F L E X I B L E APPENDAGES BEAM-TYPE APPENDAGE DYNAMICS* Second Degree Nonlinear Equations Linear Equations NONLINEAR LIBRATION DYNAMICS: T r i a x i a l S a t e l l i t e with A r b i t r a r y Number and Orientation of F l e x i b l e Appendages Deploying Independently** 'Free' V i b r a t i o n C h a r a c t e r i s t i c s Gravity Gradient, Spin-Stabilized Configur-ations, Solved by simultaneous in t e g r a t i o n on d i g i t a l computer * Accounts f o r : i ) a x i a l 'foreshortening', i i ) 3-axis a t t i t u d e motion, i i i ) general deployment, iv) g r a v i t a t i o n a l e f f e c t s , v) v a r i a b l e cross section, v i ) transverse and a x i a l o s c i l l a t i o n s , v i i ) o f f s e t from center of mass. ** Accounts f o r : i ) o r b i t a l e c c e n t r i c i t y , i i ) g r a v i t y gradient, i i i ) s h i f t i n g center of mass, iv) geometric o f f s e t of appendages. Planar Motion 3-Axis Motion Rigid F l e x i b l e NondeployingJ | Deploying! Nondeployingj Deploying 1 Figure 1-1 Outline of the research program 28 2. GENERAL ATTITUDE EQUATIONS OF MOTION R e c o g n i z i n g t h a t o r b i t a l p e r t u r b a t i o n s due t o e i t h e r space-15 c r a f t l i b r a t i o n o r appendage v i b r a t i o n a r e i n g e n e r a l n e g l i g i b l e , one can d e s c r i b e t h e m o t i o n o f t h e c e n t e r o f mass a c c o r d i n g t o t h e f o l l o w i n g K e p l e r i a n r e l a t i o n s : h Q = R- 9 = c o n s t a n t ; (2.1a) o c R c = h Q 2 / y ( l + e c 6 ) . (2.1b) Of c o u r s e , i t has been t a c i t l y assumed t h a t deployment w i l l not a l t e r t h i s f i n d i n g . Thus, i n al m o s t a l l m i s s i o n s o f p r a c t i c a l i m p o r t a n c e , l i b r a t i o n a l dynamics can be s t u d i e d i n d e p e n d e n t l y o f t h e o r b i t a l m o t i o n . T h i s c h a p t e r d e r i v e s , u s i n g t h e L a g r a n g i a n p r o c e d u r e , g o v e r n i n g n o n l i n e a r l i b r a t i o n a l e q u a t i o n s v a l i d f o r a l a r g e c l a s s o f f l e x i b l e s a t e l l i t e systems. 2.1 C o n f i g u r a t i o n and R e f e r e n c e C o o r d i n a t e Systems C o n s i d e r a s p a c e c r a f t w i t h i t s i n s t a n t a n e o u s c e n t e r o f mass 0 c n e g o t i a t i n g an a r b i t r a r y t r a j e c t o r y w i t h r e s p e c t t o t h e c e n t e r o f f o r c e a t 0 [ F i g u r e 2 - 1 ( a ) J . P o s i t i o n v e c t o r R and t r u e anomaly i — c 6 d e f i n e t h e l o c a t i o n o f 0 c w i t h r e s p e c t t o t h e i n e r t i a l r e f e r e n c e X, Y, Z c e n t e r e d a t 0^. X Q, Y Q, Z Q r e p r e s e n t an o r t h o g o n a l o r b i t -i n g r e f e r e n c e frame w i t h i t s o r i g i n f i x e d a t 0 where X^ and Y^ a r e 3 c 0 0 th e l o c a l outward v e r t i c a l and h o r i z o n t a l r e s p e c t i v e l y , and Z Q i s a l i g n e d w i t h t h e o r b i t normal. 29 Orbit F i g u r e 2 - 1 Geometry o f s a t e l l i t e m o t i o n : (a) i n e r t i a l , r o t a t i n g , and b o d y - f i x e d c o o r d i n a t e systems; (b) m o d i f i e d E u l e r i a n r o t a t i o n s ¥,A f* d e f i n i n g a r b i t r a r y o r i e n t a t i o n o f t h e c e n t r a l r i g i d body d u r i n g l i b r a t i o n s . 30 The l i b r a t i o n a l r e s p o n s e i s d e f i n e d by t h e m o d i f i e d E u l e r i a n r o t a t i o n s A, $ o f t h e b o d y - f i x e d axes x, y, z w i t h r e s p e c t t o t h e o r b i t i n g r e f e r e n c e X Q , Y Q, Z Q [ F i g u r e 2 - 1 (b)] . Note t h a t t h i s sequence i s synonomous w i t h t h e B r y a n t and Kardan a n g l e s r e f e r r e d 2 2 5 t o by W i t t e n b u r g . Axes x Q , y Q , Z Q a r e p a r a l l e l t o t h e x, y, z c o o r d i n a t e s a t a l l t i m e s , but have a d i f f e r e n t o r i g i n , O Q , r e p -r e s e n t i n g c e n t e r o f mass l o c a t i o n i n t h e absence o f any f l e x i b i l i t y e f f e c t s ( F i g u r e 2 - 2 ) . The l i t e r a t u r e s u r v e y s u g g e s t s t h a t t h e c l a s s o f c o n f i g u r -a t i o n s r e p r e s e n t e d by an a r b i t r a r i l y - s h a p e d c e n t r a l r i g i d body t o w h i c h a r e a t t a c h e d an a r b i t r a r y number o f d e p l o y i n g appendages has a wide range o f a p p l i c a t i o n . The appendages can be r i g i d o r f l e x i b l e and a r e t o be c a n t i l e v e r e d t o t h e main body f o r m i n g a 8 5 s i m p l i f i e d t o p o l o g i c a l t r e e * [ M e i r o v i t c h ( 1 9 7 2 ) ] . Note b o t h t h e o r i e n t a t i o n and t h e shape o f t h e appendages v a r y . The p r e s e n c e o f a r i g i d main body a l l o w s one t o d e s c r i b e t h e r o t a t i o n a l dynamics r e l a t i v e t o a s e t o f axes x, y, z f i x e d t o t h i s main body. Such a r e f e r e n c e i s s t i l l a ' f l o a t i n g ' system by v i r t u e o f t h e o r b i t a l and t h e a t t i t u d e m o t i o n s o f t h e c e n t r a l body. As w i t h t h e o r b i t i n g axes, t h e o r i g i n i n t h i s c a s e i s t a k e n t o be c o i n c i d e n t w i t h t h e i n s t a n t a n e o u s c e n t e r o f mass. Con-s e q u e n t l y , i n such a c a s e t h e l i n e a r momentum i s z e r o a t a l l t i m e s r e s u l t i n g i n t h e s i m p l i f i c a t i o n o f k i n e t i c energy, g r a v i t a t i o n a l 100 p o t e n t i a l , and a n g u l a r momentum e x p r e s s i o n s . * i . e . , no c l o s e d l o o p s and no secondary branches o c c u r i n t h e t o p o l o g y o f t h e s t r u c t u r e . 31 F i g u r e 2-2 A g e n e r a l s p a c e c r a f t c o n f i g u r a t i o n s h o w i n g s h i f t i n g c e n t e r o f m a s s , a p p e n d a g e o f f s e t , d e p l o y m e n t , a n d d e f o r m a t i o n s . 32 The p r a c t i c a l s i g n i f i c a n c e o f t h e above c h o i c e f o r a r e f e r -ence frame becomes apparent when one r e c o g n i z e s t h a t a t t i t u d e s e n s o r s t o o a r e f i x e d t o t h e main body. T h i s f a c t p r o v i d e s some gu i d a n c e a l s o when one i s s e l e c t i n g c o o r d i n a t e s t o be used i n m o n i t o r i n g s p a c e c r a f t a t t i t u d e . I d e a l l y , t h e same c o o r d i n a t e s s h o u l d be c a p a b l e o f r e p r e s e n t i n g b o t h t h e g r a v i t y g r a d i e n t and s p i n - s t a b i l i z e d c o n f i g u r a t i o n s . Such c o n s i d e r a t i o n s l e a d one t o choose t h e E u l e r a n g l e s as d e f i n e d i n F i g u r e 2-1(b). t h E l a s t i c d e f l e c t i o n s u^, v^, w^ o f t h e i appendage a r e measured w i t h r e s p e c t t o t h e i r undeformed c o n f i g u r a t i o n as s p e c i f i e d by t h e x^, y^, z^ system o f c o o r d i n a t e axes w h i c h , i n t u r n , a r e o b t a i n e d t h r o u g h t h e m o d i f i e d E u l e r r o t a t i o n s I|K , A . ^ , <|K r e l a t i v e t o t h e x^, y Q , ZQ c o o r d i n a t e s . These r o t a t i o n s a r e used t o c o n s t r u c t t h e t r a n s f o r m a t i o n m a t r i x [ X j J > a l l o w i n g c o n v e r s i o n between l o c a l appendage c o o r d i n a t e s e f 1 ^ and c e n t r a l c o o r d i n a t e s (2.2a) e ! 1 — 1 (0). (2.2b) CX.S|J. (2.2c) 33 where: s ( ) = s i n e ( ) ; c ( ) = c o s i n e ( ). The o r i g i n o f t h e l o c a l appendage c o o r d i n a t e s y s t e m , CK, h a s a n e t o f f s e t TT . f r o m t h e i n s t a n t a n e o u s mass c e n t e r as a r e s u l t o f f l e x i -— l b i l i t y and d e p l o y m e n t , as g i v e n by r , and due t o g e o m e t r i c o f f s e t a_^. E a c h appendage i s shown d e p l o y i n g i n d e p e n d e n t l y w i t h v e l o c i t y IK a l o n g t h e x^ a x i s ( F i g u r e 2 - 2 ) . N o t e t h a t t h e c e n t e r o f mass s h i f t i s m e a s u r e d f r o m 0 Q t o 0 c and a l l o w s f o r a s y m m e t r i c d e p l o y -ment. T h a t i s r = — Z f ( d a + d1? + a. + e. ) dm. , (2.3a) — c m . J —I — l — l — l I s i m . l w here: d. v e c t o r l o c a t i n g dm. f o r t h e u n d e f o r m e d appendage and i s ~^ ci ID measured w i t h r e s p e c t t o 0., d. + d,; c l — i — l d a v e c t o r d ^ p r i o r t o d e p l o y m e n t ; d ^ n e t change i n d as a r e s u l t o f d e p l o y m e n t ; B u t , Z f (d a + c. )dm. = 0, (2.3b) hence, i n general, r = — Z f (d b + e.)dm.. (2.3c) ~° m s i \ - 1 _ 1 1 2.2 Lagrangian Formulation 2.2.1 Background P r i n c i p l e methods of formulating equations of motion were outlined i n Chapter 1. Use of the Lagrangian equations were pre-34 f e r r e d here s i n c e t h e o b j e c t i v e i s t o s t u d y o v e r a l l v e h i c l e m o t i o n s , not f o r c e s o f i n t e r a c t i o n . The degrees o f freedom a s s o c i a t e d w i t h such a holonomic system can be r e p r e s e n t e d by t h e q u a t e r n i o n s i n c o n j u n c t i o n w i t h q u a s i c o o r d i n a t e s o r momentum v a r i a b l e s t o a r r i v e a t r e l a t i v e l y s i m p l e r s e t s o f r e l a t i o n s . However, t h e q u a n t i t i e s w h i c h p h y s i c a l l y d e s c r i b e t h e mot i o n become more d e e p l y b u r i e d i n an e s c a l a t i n g number o f e q u a t i o n s . They may p e r m i t more r a p i d n u m e r i c a l a n a l y s i s , however, a t t h e c o s t o f l o s t p h y s i c a l under-s t a n d i n g o f t h e t r u e n a t u r e o f t h e mo t i o n . 2.2.2 System k i n e t i c and p o t e n t i a l e n e r g i e s F i g u r e 2-2 i d e n t i f i e s t h e undeformed ( r ^ = d^+£^) and de-formed ( r , .= r . - r +e.) s p a c e c r a f t c o n f i g u r a t i o n . Such a system —d, I —I —c —I u n d e r g o i n g g e n e r a l l i b r a t i o n t o g e t h e r w i t h appendage v i b r a t i o n and deployment a t v e l o c i t y has a k i n e t i c energy w h i c h can be e x p r e s -sed, r e l a t i v e t o t h e i n e r t i a l r e f e r e n c e , i n t h e m a t r i x form: C dR, . _ dR, . 2 ms. {-5F } * { dtT } + \ E {a)} T [ X i ] T Ul1') [ X ± ] <w> i + {.fl f [ r d f i ] { V Q j i > dm. 1 m. l + l E f { v o , i > T { v o , i » a i V ( 2 - 4 ) i ./m. 35 where: - d , 1 R + r , . = R + r . + e —c —d, 1 —c — l — c , l = R + r . + e . - r ; — C —1 — x — c V i r e l a t i v e v e l o c i t y of mass element diru, due to vi b r a t i o n and deployment only, as measured with respect to the x^, 3e y . , z . axes; + (V, • V)(r. + e C j i ) ; t h instantaneous moment of i n e r t i a matrix for the i appen-dage, expressed i n ' l o c a l ' coordinates; {to} CO -CO, (0 . Ac$ + 6 (cysAs<I> - s¥c<I>) + ¥cAs<£> -As$ + 9 (c^sAc$ + s¥s<I>) + Vc$c$\ h + ec^cA - VsA (2. 5) It i s easy to recognize d i s t i n c t contributions from: (i) o r b i t a l motion associated with a t r a n s l a t i n g mass center; ( i i ) r o t a t i o n a l motion due to o v e r a l l attitude l i b r a t i o n s ; ( i i i ) r e l a t i v e motion due to appendage o s c i l l a t i o n and deployment. Total p o t e n t i a l energy (f) consists of a g r a v i t a t i o n a l (1/ ) y and an e l a s t i c (1/ ) contribution. Neglecting terms of t h i r d and higher order i n the variable (r^/R c), one can write [Etkin (1962), England (1969), 2 2 6 Meirovitch (1972), 8 5 e t c . ] : 49 1/ = 1/ + 1/ g e umc R~ - (-L) 2 tr ( [ X i ] T [ I : i ( i ) ] [XiD 2R J i 1 c 36 + + R x c 3u c, 1 a 2 3 u e dx. c , i 32u_ c , i 3 2 ' 3x.3y. ' x . 1 J 1 l a 2 3 w ' 2 l ^ r 1 ) ; (2.6) where: y m <xa> -g r a v i t a t i o n a l c o n s t a n t , GM; t o t a l combined mass o f t h e s a t e l l i t e ; v e c t o r o f d i r e c t i o n c o s i n e s between and x, y, z; c¥c$ + sysAs<l> •c1Ps$ + s'i'sAc*} = s^cA X l a , l a, 2 X a,3 (2.7) 2.2.3 The Lagrange e q u a t i o n s and an a l t e r n a t i v e momentum f o r m u l a t i o n The L a g r a n g i a n f o r t h e h y b r i d c o n f i g u r a t i o n o f F i g u r e 2-2 can be s e p a r a t e d i n t o two p a r t s : one a s s o c i a t e d w i t h t h e undeform-ed c o n f i g u r a t i o n (~) and one w i t h t h e deformed appendages ( E / T d D , ) : 1 5 6 1 1 L = T - V = E (q, q, t ) + Z / T (q,q,e,e,£ x,•..,t) d D ^ 1 (2.8) A p p l i c a t i o n o f H a m i l t o n ' s p r i n c i p l e t o such a holonomic system (Appendix V) y i e l d s t h e f o l l o w i n g form o f t h e Lagrange e q u a t i o n s f o r t h e a t t i t u d e d egrees o f freedom q k = A, $: A ( U L ) _ iL . = Q (2. 9a) 37 or, 1 f K ' ^ - ' - r - + E / [ l t <r^> - ~ ] a D i " % < 2 - 9 b > where, = generalized force associated with q^ .. Note, the equations are amenable to a control system study since prescribed control forces can be introduced d i r e c t l y as general-ized forces i n conjunction with a suitable control strategy. Applying equation (2.9) to the multibody configuration under consideration here, can be extremely involved. In fact, at t h i s point i t i s not p a r t i c u l a r l y advantageous to evaluate the Lagrangian i n d e t a i l . Rather, one can take advantage of the following r e l a -tions for the k i n e t i c energy: 3T 3T 3 ^ — o 3q, 3w 3q ' ~ * 30O r,' ' 8q k - 9q k d ,3T . 3T Q d . . d ,3T\ n 3 ^ dt ( 7 ^ } = 3^ ° dt ^ + dt W ° 7-" 9q k - 3q k - 9q} where [Samin and Willems (1975), 9 2 L i k i n s (1975) 7 6] 3d) - V ' / co — — / (2.10) Substituting the above r e l a t i o n s into equation (2.9a) provides an alte r n a t i v e form for the governing equations i n a momentum format 38 79 w h i c h c a n be e x p r e s s e d i n m a t r i x n o t a t i o n a s T T d t d t r. • o q . d q . k 9 q k 9 q k ^k ^k where: {H} = a n g u l a r momentum v e c t o r a s s o c i a t e d w i t h m o t i o n r e l a t i v e t o t h e s y s t e m mass c e n t e r , [I] {w} + Z J" m [ r ^ ^ ] {V ^} diru; i i ' ' 1 [I] = o v e r a l l s y s t e m moments o f i n e r t i a w i t h r e s p e c t t o x, y, z, a x e s , p r e s e n t e d i n s u c h a way as t o i s o l a t e f l e x i b i l i t y e f f e c t s ( A p p e n d i x I I ) ; and/ n o t e f o r example, 91/ Q , = - ^ = g e n e r a l i z e d f o r c e a s s o c i a t e d w i t h t h e g r a v i t a t i o n a l f i e l d , 3<-^> { ^ £ } [ I ] { x a K R 3 9 q k c The f o r m u l a t i o n p r e s e n t s s e v e r a l a d v a n t a g e s , p a r t i c u l a r l y when one i s f a c e d w i t h complex r o t a t i n g s y s t e m s . I t r e s u l t s i n c o n s i d e r -a b l e s a v i n g i n t h e amount o f a l g e b r a i n v o l v e d . As a r u l e , t h e more complex t h e s y s t e m t h e g r e a t e r t h e s a v i n g . T h i s i s b e c a u s e r a t h e r t h a n d i f f e r e n t i a t i n g t h e k i n e t i c e n e r g y t w i c e , one c a r r i e s o u t a s i n g l e d i f f e r e n t i a t i o n o f t h e s i m p l e r momentum f u n c t i o n t o g e t h e r w i t h w i t h some r e l a t i v e l y s t r a i g h t f o r w a r d d i f f e r e n t i a t i o n s o f t h e angu-l a r v e l o c i t y v e c t o r w. I n a d d i t i o n t o r e d u c i n g t h e a l g e b r a i c e f f o r t s u c h a method c a n be a p p l i e d s y s t e m a t i c a l l y ; a s a r e s u l t i t s h o u l d p r o v e l e s s e r r o r p r o n e . S y s t e m k i n e m a t i c s , c o n f i g u r a t i o n , and f l e x i b i l i t y e f f e c t s a r e n o t i n t e r t w i n e d t o t h e d e g r e e t h a t t h e y a r e w i t h t h e L a g r a n g e g e n e r a l i z e d c o o r d i n a t e e q u a t i o n s . C o n s e q u e n t l y , d i g i t a l s i m u l a t i o n s c a n be s e t up t o more e a s i l y accommodate c h a n g e s i n e i t h e r c o n f i g u r a t i o n o r c h o i c e o f c o o r d i n a t e s . 39 2.3 Governing Nonlinear Three-Axis Equations In such a matrix form the equations (2.11) appear rather compact. However, i n actual a p p l i c a t i o n they can be expanded r e s u l t i n g i n governing second order equations for r o l l (¥), yaw (A), and p i t c h (spin, $) as follows: ¥, R o l l degree of freedom [c 2A ( s 2 $ I i ; L + c 2 $ l 2 2 - s 2 $ I 1 2 ) + s 2 A l 3 3 + s2A (s$I 1 3+c$I 2 3) ]^ + c 2A ( s 2 $ i 1 1 + c 2 $ i 2 2 - s 2 $ i l 2 ) + s 2 A l 3 3 + s2A ( s $ I 1 3 + c $ i 2 3 ) l 1 ! 1 - s^sAcA (2s 2$I i : L-s2$I 1 2+2s 2As$I 1 3) 6^ + [s2A ( I 3 3 - s 2 $ I i ; L - c 2 $ I 2 2 + s2$I 1 2) + 2c2A (s$I 1 3+c*I 2 3)JAY + { c 2 A [ s 2 $ { ^ 1 1 ~ 1 2 2 ) + ( l - c 2 * ) I 1 2 l + s2A(c$I 1 3-s$I 2 3)l>W + (3y/R3) { [ (s 2As 2$-c 2$) I 1 1 + (s 2Ac 2$-s 2$) I 2 + c 2 A l 3 3 - (l+s 2A) s 2 $ I 1 2 - s2A (s$I 1 3+c$I 2 3) ] s^c^F + c2'FsA [s*c$ ( I 1 1 _ I 2 2 ) " c 2 < i > 112 ] " c 2 ^ c A ( c $ I i 3 " s $ I 2 3 ^ + cA ( s $ r 1 + c $ r 2 ) - sA - r 3 + {[cA(s*c* ( I I ; L - I 2 2 ) - c 2 * i 1 2 i + sA ( c $ I 1 3 - s $ I 2 3 ) A + { s A [ s * c * ( I 2 2 - I 1 1 ) + c 2 * I 1 2 l + cA ( c * I 1 3 - s $ I 2 3 ) } A 2 + {cA ( s * c * ( i 1 1 ~ i 2 2 ) " c 2 $ * 1 2 ~ h 3 ] + sA[-s$ ( i 2 3 + h l ) + c$(il3-h2)]}k + {c^ (c2As2$+c2<I>) l n + [cY (s 2$+c2Ac 2$) - sVsAs2*] I 2 2 - c¥c2Al 3 3 + [c^s2$(l-c2A) 40 2svl'sAc2$] I 1 2 + 2cA[ (2c¥sAs* - s¥c*)I 1 3 + (2cvFsAc* s¥c$)I 2 3] }6A + cA[c2$ ( I 1 ; L - I 2 2 ) + s 2 $ I 1 2 - I 3 3 ] $ A - [cAfsfcl^ c $ I 2 3 ) - s A l 3 3 ] $ + cA ( s * I 2 3 - c $ l ' 1 3 ) h2 - {cA [s$ ( i 1 3 + h 2 ) c $ ( I 2 3 - h 1 ) ] + s A l 3 3 } $ + { cA[cYsAs2* - sYc2$) d - ^ - I ^ ) s ^ I ^ ] - 2cA(sys2<£> + cvFsAc2<l') I 1 2 + 2 (s^sAsS - cYc 2Ac$) l " l 3 2c ,Fc 2As$I 2 3}0$ + [cH'sAcA(s 2*I 1 1 + c2®122~133) + s ^ c A s ^ c ^ ( I 22 11±) + cA(sVc2$ - c¥sAs2$)I 1 2 - (c^c2As$ + s xFsAc$)I 1 3 (s^sAs* - c¥c2Ac*)I 2 3]9 + {sVcV[ (s 2Ac 2$ - s 2 $ ) I 2 2 (s2As2<£> + c 2 * ) I 1 1 + c 2 A l 3 3 ] + c2vi'sAs$c* ( I 1 i ~ I 2 2 ) [sA (c2<Fc21> - s 2 >F) + sYcys2$]I 1 2 - c2VcAc$I 1 3 - cA (s2¥sAc* c24's$)I 2 3 } e 2 + {c^sAcA ( s 2 * I i ; L + c 2 $ i 2 2 " I 3 3 ) s^cAs^c^ ( i 2 2 ~ 1 i i ) " cA(c1'sAs2* - sH!c2*) I^-cY [s* ( c 2 A l 1 3 h 2) + c $ ( c 2 A l 2 3 - h 1 ) ] - s¥sA[c$(i 1 3-h 2) - s * ( I 2 3 - h 1 ) ] sVcAh^e = Q^; (2.12a) 41 A, Yaw degree of freedom (c2<l>I11 + s 2 $ I 2 2 + s2$I 1 2)A + ( c 2 $ i i : L + s2§I22 + s 2 $ f 1 2 ) A + [ s 2 $ ( I 2 2 - I n ) + 2c2$I 1 2]$A + (3y/R3) {s 2^sAcA ( s 2 $ I i ; L + c2<l>I22 - I 3 3 - s2$I 1 2) - s¥c2A(s$I 1 3 + c $ I 2 3 ) + sVcH'cA [ s * c * ( I 1 ; L - I 2 2 ) - c2$I 1 2] + s^cYsA (c*T- 1 3 - S $ I 2 3 ) } + ( c * ^ - s$T 2) - ( c * I 1 3 - s $ I 2 3 ) $ + (s«I 1 3 + c $ I 2 3 ) $ - [ c * ( I 1 3 + h 2) - s $ ( i 2 3 - h ^ ) ] * + [ (c>FsAc2$ + sYs2$) (I - I 2) + c V s A l ^ - 2(sVc2* - cVsAs2*)I 1 2 + IcVciX ( s * I 1 3 + c $ I 2 3 ) ] 6 $ + { c A [ c 2 * ( I 1 1 - I 2 2 ) + I 3 3 + 2s2$I 1 2] - 2sA(sOI 1 3 + c * I 2 3 ) } H + { c A [ s * c * ( I 1 1 - I 2 2 ) - c 2 * I 1 2 ] + s A ( c $ I 1 3 - s*I 2 3)}$ + | s A c A ( s 2 $ I 1 1 + c 2 $ I 2 2 - I 3 3 - s2$I 2) - c 2 A ( s $ I 1 3 + c $ I 2 3 ) ] $ 2 + { c A [ s * c * ( I 1 1 - I 2 2 ) - c 2 $ i 1 2 + h 3] + s A [ s $ ( - i 2 3 + h±) + c * ( I 1 3 + h2)}y + {cY[c2A(I 3 3- - 3 2 * I i ; L - c 2 $ I 2 2 ) - ( c 2 $ I i ; L + s 2 $ I 2 2 ) ] - [cVs2$(l + s 2A) - 2sH'sAc2$]I l 2 + sYsAs2*(I 2 2 - - 2c¥s2A(s$I 1 3 + c * I 2 3 ) + 2s¥cA(c$I 1 3 - s $ I 2 3 ) } 8 $ + {cYsA[s*c$ d i ; L - I 2 2 ) - c 2 $ I 1 2 - sH ,(c 2$I 1 1 42 + s ^ $ I 2 2 + s 2 * I 1 2 ) - c^cA ( c $ I 1 3 - s$I 2 3)}6 +'{sVcVcAIs^c* - I 2 2 ) - c 2 * I 1 2 ] - c 2¥sAcA(s 2$I i : L + c 2 * I + I - s2$I 1 2) + c 2¥c2A(s$I 1 3 + c$T_23) + sfc'FsA ( c $ I 1 3 - s $ I 2 3 ) }6 2 + {c¥sA[s*c$ ( i i ; L - i 2 2 ) - c2<s>i12 + h 3] - s y ( c 2 $ i i : L + s 2 $ i 2 2 + s 2 $ i 1 2 - c¥cA[c$(i 1 3 + h 2) - s $ ( i 2 3 - h±)}Q = QA; (2.12b) $, P i t c h (spin) degree of freedom I 3 3 * + I 3 3 $ + (3y/R3) {c2$ [sYcYsA ( 1 ^ - I 2 2 ) + (c2V - s 2 , F s 2 A ) I 1 2 + • s$c$ [ (e 2f - s 2¥s 2A) ( I 2 2 - + 2s2 , l'sAl 1 2] - sWcAcS ( s Y s A l 1 3 - cVl23) + sVcAsfc (c¥I 1 3 + s * s A l 2 3 ) } + T 3 - [ s A l 3 3 + cA(s<2>I13 + c*I 2 3)]$.+ {c 2A [s$c$(I 2 - I ) + c2*I 2] - sAcA(c$I 1 3 - s $ i 2 3 ) ] y 2 - { c A [ s $ d 1 3 - h 2) + c $ ( i 2 3 + h x) + s A l 3 3 }V + {cA[(c^sAs2$ - s¥c2*) (I - I u ) - s¥l 3 3] + (c¥s2Ac2$ + 2s ,FcAs2$)I l 2 + 2cvFc2A ( c $ I 1 3 - s*I 2 3)}6Y + {cA[c2$(I 2 2 - 1^) - I - 2s2$I l 2] + 2sA(s$I 1 3 + c*I 2 3)AY - ( c * I 1 3 - s $ I 2 3 ) A - [ s $ c $ ( I 2 2 - I x) + c 2 $ I 1 2 ] A 2 + [ s $ ( I 2 3 - h±) - c * ( I 1 3 - h 2)]A 43 + { c f s A [ c 2 * ( I 2 2 - - I 3 3 - 2(c¥sAs2$ - s¥c2<I>) T_12 - 2cVcA ( s $ I 1 3 + c $ I 2 3 ) } + {c¥[cAI 3 3 - s A ( s $ I 1 3 + c * I 2 3 ) ] + s¥(c$i 1 3 - s*i 2 3 ) } e + {-(s2¥ - c2vs2h) [s$c$ d 2 2 - I u ) + c 2 * I 1 2 ] - sWcVsA ( c 2 * d 2 2 - - 2 s 2 $ I 1 2 + cA [As ( c * I 1 3 - S < J > I 2 3 ) + sVcV ( s * I 1 3 + c * I 2 3 ) ] } 9 2 + {-(c^sAs-D - sVc*) ( I 1 3 - h 2) - (cTsAc$ + s^s$) ( I 2 3 + h±) + c l f c A i 3 3 } e = Q $ ; (2.12c) A s i d e from i n s t a n t a n e o u s moments o f i n e r t i a and t h e v a l u e o f r ^ , t h e i n f l u e n c e o f f l e x i b i l i t y and deployment exte n d s t o t h e terms: h. 1 {h} = {h 2} = I / [r„ ,] {V„ ,} dm,; (2.13a)  I f [ r d , i ] { v o , i } d mi< i _ i ro-l l . , J l = a l o c a l a n g u l a r momentum r e s u l t i n g from t h e r e l a t i v e v e l o c i t i e s a s s o c i a t e d w i t h v i b r a t i o n and deployment; F 3 ' [ r , .] (A_ .} dm.; (2.13b) d , l 0 , 1 l i n . l = a l o c a l l y a p p l i e d i n e r t i a t o r q u e r e s u l t i n g from t h e r e l a t i v e a c c e l e r a t i o n s a s s o c i a t e d w i t h v i b r a t i o n and deployment. The e x p r e s s i o n s a r e worked o u t i n d e t a i l i n Appendix I I I . W i t h t h e e q u a t i o n s i n t h i s form, one a p p r e c i a t e s t h e complex n o n l i n e a r , nonautonomous, and c o u p l e d c h a r a c t e r o f t h e system. Even t h e s i m p l e s t o f t h e e q u a t i o n s (2.12c) c o n t a i n s more t h a n 44 s e v e n t y terms! The problem i s f u r t h e r a g g r a v a t e d by t h e f a c t t h a t d e f o r m a t i o n s u^, v^, w^ a p p e a r i n g i n t h e above e x p r e s s i o n s a r e t h e m s e l v e s f u n c t i o n s o f l i b r a t i o n a l m o t i o n . O b v i o u s l y , even f o r r e l a t i v e l y s i m p l e s i t u a t i o n s , one can hope t o s e a r c h f o r a g e n e r a l s o l u t i o n o n l y t h r o u g h n u m e r i c a l methods. Even t h a t i s a f o r m i d a b l e t a s k ! F i n a l l y , t h e range o f a p p l i c a t i o n can be i n d i c a t e d by summar-i z i n g e s s e n t i a l f e a t u r e s o f t h e f o r m u l a t i o n : • a r b i t r a r y s a t e l l i t e geometry; • a r b i t r a r y n u m b e r / t y p e / o r i e n t a t i o n o f f l e x i b l e appendages; • i n d e p e n d e n t l y d e p l o y i n g appendages; • appendage o f f s e t w i t h r e s p e c t t o t h e c e n t e r o f mass; • system c e n t e r o f mass moves a l o n g an a r b i t r a r y t r a j e c t o r y ; • t h r e e degrees^ o f freedom a r e a s s o c i a t e d w i t h b o t h a t t i t u d e b e h a v i o u r and appendage d e f o r m a t i o n ; • n o n l i n e a r a t t i t u d e m o t i o n ; • b o t h s p i n - s t a b i l i z e d and g r a v i t y - s t a b i l i z e d o r i e n t a t i o n s a r e d e s c r i b e d by t h e m o d i f i e d E u l e r i a n r o t a t i o n s chosen; • g e n e r a l i z e d f o r c e terms a r e r e t a i n e d . 45 3 . NONLINEAR APPENDAGE DYNAMICS 3 . 1 Background I n t h e d e r i v a t i o n so f a r , t h e p h y s i c a l c h a r a c t e r i s t i c s o f t h e appendages have been l e f t u n s p e c i f i e d ; f o r example, one c o u l d be d e a l i n g w i t h a s t r i n g , beam, membrane, s h e l l , e t c . As found i n C h apter 2 , t h e a t t i t u d e e q u a t i o n s r e q u i r e a d e s c r i p t i o n o f t h e e l a s t i c d i s p l a c e m e n t f i e l d [e^ (£.3/ ^ ) ] o n l y and a r e not d i r e c t l y dependent on t h e t y p e o f appendages i n v o l v e d . U l t i m a t e l y , however, t o o b t a i n a s o l u t i o n o f t h e l i b r a t i o n a l dynamics one must s p e c i f y t h e t y p e o f appendage and s o l v e t h e a s s o c i a t e d e q u a t i o n s g o v e r n i n g f l e x i b i l i t y . T h i s can i n v o l v e even g r e a t e r e f f o r t t h a n t h e a t t i -t u de e q u a t i o n s t h e m s e l v e s . In t h e f o l l o w i n g development t h e s p a c e c r a f t i s assumed t o have beam-type appendages. They a r e r e p r e s e n t a t i v e o f antennae, s t a b i l i z i n g booms, and t h e s u p p o r t i n g b a r s a s s o c i a t e d w i t h e x p e r i -m e n t a l packages and s o l a r a r r a y s . F u r t h e r m o r e , one would e x p e c t l o n g t r u s s - l i k e s t r u c t u r e s t o d i s p l a y an o v e r a l l beam-type be-h a v i o u r . T a k i n g t h e beam t o be o f t h e E u l e r - B e r n o u l l i t y p e makes i t e s s e n t i a l l y one d i m e n s i o n a l . Hence, i t s c h a r a c t e r i s t i c s a r e s p e c i f i e d by o n l y one s p a t i a l v a r i a b l e ( x ^ ) . Such a ' s l e n d e r ' system i s assumed t o e x p e r i e n c e s i m p l e f l e x u r e o n l y , i . e . , e f f e c t s o f r o t a r y i n e r t i a and shear d e f o r m a t i o n a r e c o n s i d e r e d n e g l i g i b l e . 2 26 F u r t h e r m o r e , t o r s i o n i s n o t d e a l t w i t h h e r e . England ( 1 9 6 9 ) s t a t e s t h a t t h e fundamental f r e q u e n c y o f boom i n t o r s i o n i s s e p a r a t e d from t h e bend i n g f r e q u e n c y and, as suc h , t h e s e o s c i l l a -46 a t i o n s may be c o n s i d e r e d u n c o u p l e d . The s i g n f i c a n c e o f t w i s t i s f u r t h e r r e d u c e d f o r contemporary space booms w h i c h t e n d t o be z i p -13 9 p e r e d t h u s e n s u r i n g a h i g h t o r s i o n a l s t i f f n e s s [Nguyen (1978) ]. An i m p o r t a n t a s p e c t o f t h e beam-type appendages i s t h e i r r e l a t i v e l y low f l e x u r a l r i g i d i t y w h i c h makes them q u i t e s u s c e p t i b l e t o l a r g e a m p l i t u d e o s c i l l a t i o n s when exposed t o e n v i r o n m e n t a l d i s -t u r b a n c e s and c o n t r o l manoeuvres. I d e a l l y , one would l i k e t o a n a l y z e t h e appendage n o n l i n e a r dynamics w i t h t h e utmost a c c u r a c y ; however, i n g e n e r a l , t h i s would be q u i t e a c h a l l e n g i n g t a s k even w i t h t h e 139 227 h e l p o f a computer [Nguyen (1978), A l m r o t h e t a l . (1978), H e l l i w e l l ( 1 9 7 8 ) , 2 2 8 J a n k o v i c (1980) 2 2°]. A l s o i t can be expen-s i v e and o f t e n may n o t be q u i t e n e c e s s a r y . I n most s i t u a t i o n s one can o b t a i n r e s u l t s o f adequate a c c u r a c y by i n c l u d i n g o n l y t h e more i m p o r t a n t n o n l i n e a r c o n t r i b u t i o n s . W i t h t h i s i n mind, g e n e r a l v i b r a t i o n e q u a t i o n s a r e d e r i v e d w h i c h r e t a i n terms o n l y up t o 2 n d degree. T h i s i s c o n s i s t e n t w i t h t h e use o f 2 degree v i b r a t i o n -r e l a t e d terms as found i n t h e i n e r t i a , momentum, and t o r q u e c a l c u -l a t i o n s f o r t h e a t t i t u d e e q u a t i o n s . 3.2 K i n e t i c and P o t e n t i a l Energy o f a D e p l o y i n g Beam Undergoing G e n e r a l L i b r a t i o n s I n d e r i v a t i o n o f e q u a t i o n s o f m o t i o n f o r a c o n t i n u a t h e methods o f a n a l y t i c a l dynamics have an i n t u i t i v e a p p e a l i n t h a t t h e y a p p l y t o any 'system' - r i g i d , f l e x i b l e , o r a h y b r i d c o l l e c -t i o n o f such b o d i e s . F o r example, t h e a p p l i c a t i o n o f H a m i l t o n ' s P r i n c i p l e t o t h e beam-type appendage under c o n s i d e r a t i o n h e r e , can y i e l d a complete s e t o f boundary c o n d i t i o n s i n a d d i t i o n t o t h e g o v e r n i n g e q u a t i o n s . 47 3.2.1 Beam c o n f i g u r a t i o n and c o o r d i n a t e s t h Consider the i appendage to be a beam d e p l o y i n g w i t h l o c a l v e l o c i t y IL along the x^ d i r e c t i o n , where the x^ a x i s c o i n -c i d e s w i t h the undeformed n e u t r a l a x i s of the appendage (Figur„e 2-2). Note the appendage attachment p o s i t i o n i s o f f s e t from the o v e r a l l system c e n t e r of mass. The l i n e a r d e n s i t y p^, s t i f f n e s s E^, and c r o s s s e c t i o n a l i n e r t i a J..^. ^  are allowed to vary along the l e n g t h of the boom. A l s o , the appendage i s p e r m i t t e d to have any a r b i t r a r y i n i t i a l o r i e n t a t i o n i n space and i s f r e e t o undergo t r a n s -134 v e r s e as w e l l as a x i a l deformations. Fang (1975) p o i n t s out the need f o r c o n s i d e r i n g a x i a l and t r a n s v e r s e degrees of freedom s i m u l -t a n e o u s l y when d e a l i n g w i t h l a r g e amplitude problems. N e g l e c t i n g appendage t h i c k n e s s i n the y^, z^ d i r e c t i o n s , the g e n e r a l d i s p l a c e -ment of elemental mass dm. l o c a t e d a t x. w i t h r e s p e c t to the i n -I l ^ stantaneous mass ce n t e r of the o v e r a l l s p a c e c r a f t i s : r , . = [x, + a, , - x + u . ( x . , t ) - u,^ , ( x , , t ) ] i , — d , i l l , i c i i f s , i i — l + [ t f i • - y + v. (x.,t)] j . + [a, . - z + w. (x.,t)] k. . (3.1) where a x i a l f o r e s h o r t e n i n g (-u^ •) due to t r a n s v e r s e displacements 1 S / 1 has been i n c l u d e d e x p l i c i t l y . 3.2.2 Treatment of a x i a l f o r e s h o r t e n i n g Assessment of the f o r e s h o r t e n i n g e f f e c t has presented s e v e r a l 133 problems i n the p a s t [Vigneron (1975), Kaza and K v a t e r n i k 13 6 (1977) ]. The approach adopted i n t h i s t h e s i s u n i f i e s some of 48 t h e e a r l i e r p r o c e d u r e s . C o n s i d e r a beam o f undeformed l e n g t h £ as measured a l o n g t h e x a x i s ( F i g u r e 3-1). I n t h e p r e s e n c e o f t r a n s v e r s e d i s p l a c e m e n t v, w t h e e f f e c t i v e l e n g t h a t any i n s t a n t becomes . That < I can be e s t a b l i s h e d by n o t i n g t h a t any deformed beam element o f l e n g t h ds has a p r o j e c t e d l e n g t h dx and: (3.2) z F i g u r e 3-1 Beam a x i a l f o r e s h o r t e n i n g caused by t r a n s v e r s e d e f o r m a t i o n s v, w. As i n d i c a t e d , a number o f d i f f e r e n t methods have been d e v i s e d t o d e a l w i t h t h i s d i f f e r e n c e . F o r t h e problem o f r o t a t i n g beam v i b -229 230 r a t i o n H u r t y e t a l . (1964) and M e i r o v i t c h (1967) c o n s i d e r t h e e f f e c t as a w o r k i n g a x i a l d i s p l a c e m e n t (ds - dx) a c t i n g on t h e c e n t r i f u g a l l o a d i n g . I t i s i n t r o d u c e d t h r o u g h t h e l i m i t s o f i n t e -g r a t i o n by Hughes and Fung ( 1 9 7 1 ) a s : f ( s ) ds = / f ( x ) [1 + \{v 2 + w 2 ) ] dx + H.O.T.; (3.3) o "'o where: f ( s ) = any a r b i t r a r y f u n c t i o n o f l o c a t i o n (s) a l o n g t h e n e u t r a l a x i s a t any i n s t a n t ; I = i n s t a n t a n e o u s beam l e n g t h as measured a l o n g t h e x a x i s , A I - A; 2 7 (v 2 + w 2 ) dx- v = 9v w = 9w. x x ) dx, x 8 x , x 3 x ,0 H.0.T.= h i g h e r o r d e r terms. 133 V i g n e r o n (1975), on t h e o t h e r hand, i n c l u d e s i t d i r e c t l y as a c o n t r i b u t i o n t o t h e assumed a x i a l d i s p l a c e m e n t f i e l d e q u a l t o : x u f s ~~ k f ( v a 2 + w a 2 > d a - ( 3 ' 4 ) 0 The. d i f f e r i n g approaches suggested by e q u a t i o n s (3.3) and (3.4) can be shown t o y i e l d i d e n t i c a l r e s u l t s . D e f i n i n g n = x + u f s , (3.5) i t f o l l o w s t h a t a t x = 0; u f s = 0 , n = 0; and a t x = £ - A; u ^ = A , n = T h e r e f o r e : 1 2 2 dn = dx + d u f s = [1 + j ( v x + w x )] dx. r i d S u b s t i t u t i n g i n (3.3) g i v e s t h e f o l l o w i n g r e l a t i o n v a l i d t o 2 degree i n v, w, 50 / f ( s ) d s = / f ( X) [ l + f ( v v 2 + w v 2 ) ] d x = / f (n-u.c)dn -j Q j 0 z x x ^ Q r s (3.6) C l e a r l y the f o r e s h o r t e n i n g can be d e a l t w i t h by c o n s i d e r i n g i t to be an a d d i t i o n a l displacement -u,. , thus j u s t i f y i n g the assumption 133 136 of Vigneron. Kaza and K v a t e r n i k (1977) p o i n t out t h a t t h i s e f f e c t can be d e a l t w i t h i m p l i c i t l y by working w i t h n o n l i n e a r s t r a i n -displacement r e l a t i o n s but a l i n e a r displacement f i e l d . T h i s l a t t e r approach r e q u i r e s t h a t one r e t a i n terms through f o u r t h degree i n the energy e x p r e s s i o n s i n order to o b t a i n the 2 n d degree equations. For the purpose of t h i s t h e s i s the use of a m o d i f i e d displacement i s r cl 136 p r e f e r r e d as terms o n l y up to 3 degree are necessary. How-ever, i n order t o apply t h i s method to a beam of v a r i a b l e c r o s s -s e c t i o n , the theorem of Appendix (IV) i s r e q u i r e d as w e l l . I t i s worth n o t i n g t h a t to o b t a i n even the l i n e a r v i b r a t i o n equations f o r a r o t a t i n g beam, one must take i n t o account geometric 136 n o n l i n e a r i t i e s thus again emphasizing the s i g n i f i c a n c e of f o r e -s h o r t e n i n g terms. The i m p l i c i t approach of u s i n g n o n l i n e a r 130 s t r a i n - d i s p l a c e m e n t r e l a t i o n s was demonstrated by L i k i n s (197 3) 92 and Samin and Willems (1975). 3.2.3 K i n e t i c energy d e n s i t y K i n e t i c energy terms a s s o c i a t e d w i t h appendage o s c i l l a t i o n s are a l r e a d y i m p l i c i t i n equation (2.4). Here they are expanded f o r a beam-type appendage. As the f o r m u l a t i o n i s w i t h r e s p e c t t o the o v e r a l l system c e n t e r of mass the v i b r a t i o n s are u n a f f e c t e d by the o r b i t a l motion. 51 C o n s e q u e n t l y , k i n e t i c energy a s s o c i a t e d w i t h t h e f l e x i b l e b e h a v i o u r o f t h e i appendage can be e x p r e s s e d as T. = {u)} T / [r ~ . ] (V_ .} dm 1 Jm. d ' x 0 , 1 1 l / ( v o , i } T { v o , i } d m i - ( 3 - 7 ) "m. l S u b s t i t u t i n g f o r t h e a n g u l a r v e l o c i t y , deployment v e l o c i t y , p o s i -t i o n v e c t o r , e t c . i n terms o f l o c a l c o o r d i n a t e s , e x p a n d i n g , and o m i t t i n g t h e s u b s c r i p t i w h i c h denotes t h e i appendage; t h e k i n e t i c r d energy t o 3 degree can be w r i t t e n as T = k / * P { ( T T - , 4. + u. + Uu ) 2 2 J 1, t t x 0 2 + ( l T _ . + V^ + UV ) + (TT _ , + W + UW ) 2,t t x 3, t t x + U [ 2 ( T T 1 ^ t + u t ) + U ( l + 2 u x ) ] - 2 [ T T l , t + U t + U < 1 + V ] ( u f s , t + ^ f s ^ + [ ( T T 2 + v ) 2 + (TT 3 + w ) 2 ] u ^ 2 2 2 2 + [ ( T T 1 + X + U) + (TT 3 + W) - 2 (IT 1 + x + u) u f g ] w 2 2 2 2 + [ ( T T ^ + x + u) + (TT 2 + v) - 2 ( T V 1 + x + u) u f g ] CO3 2 [ ( T T 1 + X + U ~ u f s ) (TT 2 +V) a)1(jo2 + (^3+w) ^2^3 + ( T T 1 + x + u - u f s ) (^3+w) w2_ w3] 52 + 2 [ i r l f t + ( u t + Uu x) - ( u f S f t + U u f S / X ) + U][Tr 3 + w) o>2 - (TT2 + v) w 3] + 2(TT 2 t + v f c + Uv x) [(TT 1 + x + u - u f g ) o ) 3 - (TT3 + w)a) 1] + 2 (TT3 t + wfc + Uwx) [ (TT2 + V)(JO1 - (TT1 + x + u - u f g ) u 2 ] } d x = f T (x, e , e x , e x x , e t , e ^ , t ) dm, (3.8) ^m where e = g e n e r a l i z e d c o o r d i n a t e o f t h e continuum, u, v, o r w i n t h i s c a s e . 3.2.4 P o t e n t i a l energy d e n s i t y The appendage b e i n g s t u d i e d i s under t h e i n f l u e n c e o f two c o n s e r v a t i v e f o r c e f i e l d s : one t h e r e s u l t o f a v a r i a t i o n i n s t r e n g t h o f t h e g r a v i t a t i o n a l f o r c e d i s t r i b u t e d o v er t h e body, t h e o t h e r a consequence o f t h e e l a s t i c r e s t o r i n g moments p r e s e n t d u r i n g b e nding. 3.2.4.1 S t r a i n energy A g e n e r a l e x p r e s s i o n f o r t h e e l a s t i c s t r a i n energy f o r a homo-geneous i s o t r o p i c continuum w i t h no d i s s i p a t i v e elements but ex-p e r i e n c i n g l a r g e s t r a i n s i s : u e = ifff a * T £ s d x d y d z ' ( 3 - 9 ) where: T a* = t r a n s p o s e o f g e n e r a l i z e d s t r e s s t e n s o r ; e g = s t r a i n t e n s o r . 53 S t r a i n can be t a k e n s m a l l w h i l e s t i l l a l l o w i n g f o r l a r g e r e l a t i v e d e f l e c t i o n and r o t a t i o n w i t h i n t h e system. F o r t h e case o f s m a l l s t r a i n s , Hooke's Law can be i n t r o d u c e d i n t o e q u a t i o n (3.9) g i v i n g [ L i k i n s e t a l . (1973) 1 3 0 ] : V e = IX/7 E S T E £ S D X D ^ D Z ; ( 3 - I O ) where: E = Young's modulus f o r t h e m a t e r i a l . Green's s t r a i n t e n s o r i s d e v e l o p e d f o r a s l e n d e r o n e - d i m e n s i o n a l 136 E u l e r - B e r n o u l l i beam by Kaza and K v a t e r n i k (1977) , g i v i n g s t r a i n as a f u n c t i o n o f d i s p l a c e m e n t . Only t h e a x i a l s t r a i n i s r e l e v a n t here. F o r e s h o r t e n i n g i s i n t r o d u c e d e x p l i c i t l y as i n t h e e v a l u a t i o n o f k i n e t i c energy so t h a t , u s i n g t h e beam c o o r d i n a t e s o f F i g u r e 2-2: 1 2 2 2 e , n = (u-u.c ) + [ (u-u, ) + v + w ] ; s , l l f s x 2 f s x x x 1 2 = u - yv - zw + (u - yv - zw ) ; (3.11) x J xx xx 2 x J xx xx ' and E s , 2 2 E s , 3 3 £ s , 1 2 £ s , 1 3 £ s , 2 3 °" S q u a r i n g ( 3 . 1 1 ) , 2 2 2 2 2 e n i = u + y v + z w - 2 u (yv + zw ) s , l l x J xx xx x w xx xx 3 2 + 2yzv w + u - 3u (yv + zw ) J xx xx x x J xx xx 54 2 2 2 2 + 3u (y v + z w + 2yzv w ) x J xx xx J xx xx -3-3 o 2 2 2 - y v - z w° - 3yz v W y v - 3y zv w •* X X X X J X X X X X X X X + H.O.T. (3.12) S u b s t i t u t i n g t h i s r e s u l t i n t o ( 3 .10), i n t e g r a t i n g , and t a k i n g advantage o f t h e o n e - d i m e n s i o n a l c h a r a c t e r o f t h e system J*J"ydydz = J"J*zdydz = J"J*yzdydz = 0, g i v e s t h e p o t e n t i a l due t o f l e x u r e t o th e 3 degree E [ A ( u 2 + u 3 ) + (1 + 3u x) ( J 2 2 W x x + J 3 3 V x x ) ] d X ' 0 (3.1.3) where: E = E (x) ; A(x) = beam c r o s s s e c t i o n a l a r e a , ff dydz; J 2 2 = ffz2dYdz' J-,-, = ff y^dydz; ) i n e r t i a ! 2 2 _ „ > c r o s s s e c t i o n a l a r e a moments o f 33 3.2.4.2 G r a v i t a t i o n a l p o t e n t i a l To t h e 2 degree i n (r./R ), t h e g r a v i t a t i o n a l p o t e n t i a l o f any body w i t h f i n i t e d i m e n s i o n s i s g i v e n by e g u a t i o n (2.6) a s : ym^ g R ^ c + 2R M 3 { * * ) T t X i ] T I I 1 1 > 1 t X i ] 55 - t r ( [ X i ] T [ I ^ l [ X ± ] ) } . (3-14) In o r d e r t o d e r i v e g o v e r n i n g appendage e q u a t i o n s o f m o t i o n t h i s e x p r e s s i o n i s expanded u s i n g l o c a l appendage c o o r d i n a t e s . A x i a l f o r e s h o r t e n i n g i s i n t r o d u c e d e x p l i c i t l y i n t h e d i s p l a c e m e n t f i e l d as d i s c u s s e d i n t h e p r e v i o u s s e c t i o n s . A l s o , use i s made o f t h e i n t e g r a l theorem from Appendix ( I V ) . The p o t e n t i a l r e l a t e d t o moti o n o f t h e c e n t e r o f mass i s u n c o u p l e d from f l e x i b i l i t y and th u s i s i g n o r e d . E x t e n d i n g t h e energy d e n s i t y c o n c e p t t o moments o f i n e r t i a , one can w r i t e , f o r a g i v e n appendage: I ., = / I .. d D i . (3.15) D k J D k I U s i n g t h i s d e f i n i t i o n one i s p r o v i d e d w i t h a c o n c i s e d e s c r i p t i o n o f t h e v i b r a t i o n - r e l a t e d g r a v i t a t i o n a l p o t e n t i a l , V * = ( I r T 3 ~ ) ^ H ^ l + °22l22 + C 3 3 5 3 3 + c 1 2 5 1 2 c + C 1 3 i 1 3 + C 2 3 i 2 3 ) ' ( 3 ' 1 6 ) t h where, f o r c l a r i t y t h e s u b s c r i p t i i s o m i t t e d f o r t h e i append-age, and j , k a r e dummy i n d i c e s r e f e r r i n g t o t h e l o c a l x^, y^, z^ axes o f any g i v e n appendage as 1, 2, 3, r e s p e c t i v e l y . A l s o c., = c o e f f i c i e n t i d e n t i f y i n g appendage o r i e n t a t i o n r e l a t i v e 3 t o t h e l o c a l v e r t i c a l R . — c For c o n v e n i e n c e l e t : a l = { 3 x a , l 2 - 1 ) ; a 2 = ( 3 X a / 2 2 - 1 ) ; ( 3 x a , 3 " D i X a , l Xa,2 ; x a , l xa,3' Xa,2 Xa,3* a l X j l 2 + a 2 X j 2 2 + a 3 X j 3 2 + 6 ( a 4 X j l X j 2 + a 5 X j l X j 3 + a 6 X j 2 x j 3 } ' j = 1, 2, 3; - 2 ( a l X l l X 2 1 + a 2 X 1 2 X 2 2 + ^ 3 X 1 3 X 2 3 ) - 6 [ a 4 ( X l l X 2 2 + X 1 2 X 2 1 ) + ^ 5 ( X 1 1 X 2 3 +X 1 3X 2 1) + « 6 ( x 1 2 x 2 3 + x 1 3 x 2 2 ) l ; - 2 ( a l X l l X 3 1 + ^ 2 X 1 2 X 3 2 + a 3 X 1 3 X 3 3 ) - 6 [ a 4 ( X l l X 3 2 + X 1 2 X 3 1 ) + « 5 ( X 1 1 X 3 3 + X 1 3 X 3 1 ) + a g ( x 1 2 x 3 3 + x 1 3 x 3 2 ) ; - 2 ( a l X 2 3 X 3 1 + a2 X22 x32 + 0 13 X23X33 ) - 6 [ a 4 ( x 2 l X 3 2 + X 2 2 X 3 1 ) + a 5 ( X 2 l X 3 3 + X 2 3X 3 1> 57 + a 6 ( x 2 2 X 3 3 + X 2 3 X 3 2 } - { 3 - 1 8 b ) S u b s t i t u t i n g t h e i n e r t i a d e n s i t i e s from Appendix I I , e q u a t i o n ( I I . 6 ) , i n t o (3.16) y i e l d s t h e f o l l o w i n g e x p r e s s i o n f o r g r a v i t a t i o n a l p o t e n t i a l a p p l i c a b l e t o a beam-type appendage h a v i n g an a r b i t r a r y o r i e n t a t i o n i n space: % = { ( C 2 2 + C 3 3 ) ( X + U ) 2 c + [2 ( C 2 2 + C 3 3 ) 7 T 1 + C 1 2 ( 7 T 2 + V ) + °13 ( 7 T 3 + w ) ] ( x + u ) + [ C12^1 + < c n + c 2 3 ) ( 2 7 T 2 + V ) + °23'n3]v + [C13*1 + C 2 3 ( U 2 + V ) + ( C 1 1 + C 2 2 ) ( 2 T r 3 + w ) ] w } X P [ 2 ( c 2 2 + C 3 3 ) ( T r 1 + a + u ) + c^^+v) v 2 +w 2 + C I 3 ( T T 3 + W ) ] da} {— —) . (3.19) By i n s p e c t i o n one a p p r e c i a t e s t h e c o n s i d e r a b l e s i m p l i f i c a t i o n s p o s s i b l e i f g e n e r a l o r i e n t a t i o n and o f f s e t terms a r e n o t p r e s e n t . Of c o u r s e , t h e c o m p l e x i t y i s f u r t h e r reduced i f t h e e q u a t i o n i s c a r r i e d j u s t t o t h e 2 degree. 3.3 N o n l i n e a r E q u a t i o n s G o v e r n i n g T r a n s v e r s e and A x i a l V i b r a t i o n s To s t a r t w i t h , a p p l y i n g H a m i l t o n ' s P r i n c i p l e (Appendix V) l e a d s t o an e q u i v a l e n t s e t o f Lagrange e q u a t i o n s a p p r o p r i a t e f o r t h i s system. Making use o f t h e energy e x p r e s s i o n s d e v e l o p e d f o r 58 such a beam-type appendage, t o g e t h e r w i t h t h e theorem o f Appendix IV, one can e s t a b l i s h t h e L a g r a n g i a n d e n s i t y o f t h e beam. That i s , i n t h i s c a s e : 1 = t h e d i f f e r e n c e between t h e k i n e t i c and p o t e n t i a l energy p e r u n i t l e n g t h ; = T - V = I (x, e, e x , e x x , e ^ , V t ) (3.20) S u b s t i t u t i n g (3.20) i n t o t h e Lagrange e q u a t i o n s d i s c u s s e d above y i e l d s t h e 2 degree e q u a t i o n s f o r v i b r a t i o n : a x i a l o s c i l l a t i o n (e = u) P { u t t " U f s , t t + 2 U ( u x t - U f s , x t > + " ( u x - u f s f x ) 2 + U (U - U_ ) + 13 + TT v xx f s , x x ' l , t t - [co2 + co2 - 2 ( c 2 2 + c 3 3 ) ( y / 2 R c 3 ) ] ( T ^ + x + u - u f g ) + [-w3 t + u 1w 2 + c 1 2 ( y / 2 R c 3 ) ] ( T T 2 + v) + [w + w,u>3 + C l 3 ( y / 2 R C 3 ) ] (^3 + W ) " 2 3 ( 7 T 2 , t + V t + Uv x) + 2a) 2 ( T T 3 f t + w t + Uw x)} + P x U [ u t - U f s , t + U ( u x - u f s f x ) + U + " l . t - ^ - ^ ^ 2 + v) + ( O 2 ( T T 3 + w) ] - \ { ( E A ) x ( 2 + 3 u x ) u x + 3 [ ( E J 3 3 ) x V 2 x + <EJ 2 2> ^ 1 } 59 - E [ A ( 1 + 3u )u + 3 ( J _ 0 v v + J 0 0 w w )] = F, ; 1 v x' xx 33 xx xxx 22 xx xxx 1 (3.21a) t r a n s v e r s e o s c i l l a t i o n (e = v) p{v.. + 2Uv . + Uv + U 2v + TT_ K t t x t x xx 2 , t t + [o3 3 t + 0 ) ^ 2 + c 1 2 ( y / 2 R c 3 ) ] ( T ^ + x + u - u f g ) - [ t o 1 2 + w 3 2 - 2 ( C i ; l + c 3 3 ) ( y / 2 R c 3 ) ] (TT2 + v) + t + w 2oa 3 + c 2 3 ( y / 2 R c 3 ) ] (TT3 + w) + 2a) 3 l*1>t + u t - u f 8 f t + U ( u x - U f g f X ) + U] - 2 U l ( T r 3 f t + w t + Uw x)} + p x l l [ v t + U v x + 7 T2 t + w 3 ^ 1 + X + U ~ U f s ^ ~ ^ 1 ^ 3 + w ^ ( F A v ) + (EJ_,V ) (1 + 3u ) A x x 33 xx xx x + 3[2 ( E J 0 , v ) u + E J _ , v u V= F„; (3.21b) 33 xx x xx 3 3 xx xxx 2 t r a n s v e r s e o s c i l l a t i o n (e = w) p { w t t + 2Uw x t + uw x + U 2 w x x + * 3 f t t 3 ' + [-u>2 t + w1(*>3 + c 1 3 ( y / 2 R c ) ] (T^ + x + u - u f g ) + [ a ) l f t + u) 2a) 3 + c 2 3 ( y / 2 R c 3 ) ] ( u 2 + v) - [co 1 2 + w 2 2 - 2 ( c i ; L + c 2 2 ) ( y / 2 R c 3 ) ] (TV3 + w) 60 " 2 " 2 [ T r l , t + u t " u f s , t + U ( u x " u f s f x ) + U ] + 2 a , 1 ( 1 r 2 f t + v t + U v x ) } + p xU|> 3 t + wfc + Uw x - ^ 2 ^ 1 + x + u ~ u f s ^ + ^ 1 ^ 2 + V ^ (F.W ) + (EJ„„w ) (1 + 3u ) A x x 2 2 xx xx x + 3[2(EJ_„w ) u + EJ_„w u ] = F_; (3.21c) 22 xx x xx 22 xx xxx 3 where: F A = an e f f e c t i v e a x i a l l o a d r e s u l t i n g from t h e i n e r t i a l and g r a v i t a t i o n a l f o r c e f i e l d t t " u t t - 2 U u a t " U U a J { P { ~ * 1 , - U 2 U - U + [ u 2 2 + w 3 2 - 2 ( c 2 2 + c 3 3 ) ( p/2R c 3)] (TT 1 + a + u) + [ a ) 3 , t " C 01 W2 ~ C 1 2 ( y ^ 2 R c 3 ) ] ( 1 T2 + V ) 3 t u 2 , t + ^1^3 + c 1 3 ^ y / , 2 R c ^ ^ 3 + w^ + 2 0 ) 3 ( T T 2 F T + V T + U V A ) - 2 0 ) 2 ( T T 3 ; t + W T + U W A ) } > aU [ T T 1 t + u f c + U u a + U - 0 3 3 ( T T 2 + v) + u>2(it3 + w ) ] j d a . - P (3.21d) Here (*) = t o t a l t i m e r a t e o f change as measured i n l o c a l c o o r d i n -8 0 . I T 3 0 a t e s , TTV- + U T 3t 9x 61 A l l components a r e d e r i v e d w i t h r e s p e c t t o t h e l o c a l x^, y^, axes as i n d i c a t e d by s u b s c r i p t s 1, 2, 3, r e s p e c t i v e l y . As i n t h e case o f g e n e r a l l i b r a t i o n a l m o t i o n , t h e e q u a t i o n s g o v e r n i n g t r a n s l a t i o n a l o s c i l l a t i o n s o f t h e e l a s t i c appendages a r e a l s o seen t o be n o n l i n e a r , nonautonomous, and c o u p l e d . Together, t h e l i b r a t i o n a l and v i b r a t i o n a l d e g r e e s o f freedom form a c o n j u g a t e system hence, t h e y must be s o l v e d s i m u l t a n e o u s l y . As can be e x p e c t e d , t h e o v e r a l l system i s t o o complex t o be amenable t o any c l o s e d - f o r m s o l u t i o n . ^ The e s s e n t i a l f e a t u r e s i n c l u d e d when m o d e l l i n g t h e o s c i l -l a t i o n s o f a beam-type appendage a r e summarized below: • a r b i t r a r y t r a j e c t o r y ; • g r a v i t a t i o n a l e f f e c t s ; • 3 - a x i s l i b r a t i o n s ; • s h i f t i n g c e n t e r o f mass; r • g e o m e t r i c o f f s e t o f appendage p o i n t o f attachment from t h e mass c e n t e r ; • t r a n s v e r s e as w e l l as a x i a l o s c i l l a t i o n s ; ncL • n o n l i n e a r (2 degree) e f f e c t s ; • v a r i a b l e mass d e n s i t y , f l e x u r a l r i g i d i t y , and a r e a o f t h e beam c r o s s - s e c t i o n ; • a r b i t r a r y deployment v e l o c i t y and deployment a c c e l e r a t i o n ; • a r b i t r a r y appendage o r i e n t a t i o n . 62 SIMPLIFIED APPENDAGE DYNAMICS 4.1 L i n e a r i z e d Equations f o r Transverse V i b r a t i o n s o f a Deploy-i n g , O r b i t i n g Beam-Type.Appendage The second degree v i b r a t i o n equations of Chapter 3 are ex-tremely i n v o l v e d making even a numerical s o l u t i o n e l u s i v e i n the 227 228 g e n e r a l case. ' Through l i n e a r i z a t i o n and c o n s i d e r i n g the appendage t o be uniform the problem becomes somewhat more t r a c t -a b l e . T h i s , t o g e t h e r w i t h the r e a l i s t i c assumption of a x i a l r i g i d -i t y as w e l l as c o n t i n u i t y c o n s i d e r a t i o n s , r e s u l t i n the deployment v e l o c i t y being uniform along the l e n g t h of the appendage [Tabarrok et a l . ( 1 9 7 4 ) 2 2 2 ] , i . e . U(x,t) = U(t) = £(t) and d U ^ , t } = U(t) = £(t). A p p l i c a t i o n of such c o n s i d e r a t i o n s to equation (3.21) l e a d s t o the governing f i r s t degree equations f o r the i * " * 1 appendage i n the v and w degrees of freedom p{v. . + 2£v . + £V + I 2 v + TT _ . . K t t x t x xx 2 , t t + Y (x + T T 1 ) - Y 2 ( v + 7 r 2 ) + Y 3 ( w + ^3* + 2w3 ( T T 1 T + I) - 2 W l ( T T 3 t + w t + £w x)} - ( F A v ) + EJ__V = F „ ; (4.1a) A x x 33 xxxx 2 • 2 p{w*.*. + 2£w . + £w + £ w + T T 0 H t t x t x xx 3 , t t + Y 4 (X + T T 1 ) + Y 5 (V + TT2) - Yg (W + T T 3 ) 63 20) 2 ( 7 T 1 / t + I) + 2 U l ( ^ ^ + V t + £ V x ) } (F_w ) + E J . 0 w = F_. (4.1b) A x x 22 x x x x 3 Here 0 K / '"c Y„ = y/2R 3 ; Y ] _ = o)3. + 0)^2 + C 1 2 Y 0 ; Y 2 = 0 ) x 2 + w 3 2 - 2 ( c i ; L + C 3 3 ) Y 0 ; Y 3 = - U l + o) 2w 3 + C 2 3 Y 0 ; Y 4 = -o) 2 + 0)^3 + C 1 3 Y Q ; Y 5 = W L + 0) 20) 3 + C 2 3 Y 0 ; Y g = o ) x 2 + u ) 2 2 - 2 ( c n + c 2 2 ) Y Q ; Y 7 = U ) 2 2 + W 3 2 " 2 ( C 2 2 + C 3 3 ) Y 0 ; Yg = a>3 - U l a ) 2 - C 1 2 Y 0 ; Y Q = a) 1 + 0)^3 + C 1 3 Y Q ; Y10 = _ 7 r l , t t " * + ^7*1 + ^ 2 + Y 9 U 3 + 2 u 3 i r 2 , t " ^ 2 ^ 3 , t ; Y l l = Y10 + £ ; F A = p[y10U - x) + ( | ) Y ? U 2 " x 2 ) ] . (4.1c) 64 The number of parameters can be reduced by use of a nondimen-sional form of the equations. Defining: x = x/Hj t = §t; v = v/&; w = w/£; 6 = h Q/R c 2; 6 = -26 2es6/(l + ec9); (4.2) and substituting into (4.1) y i e l d s : v e e + 2(£' - e i ) v 0 + 2£'v x e + (£" - 2e 1£' ~ Y 2 ^ + (V' - 2 e 1 J ' + 2A , 2.+ Y ?x + Y 1 0 ) v x + [V2 - Y 1 0 d - x) - | Y 7 d - * 2 ) 3 v x x + ( E J 3 3 / p e 2 £ 4 ) v x x x x r /\ .A. ./\ - 2 0 ) ^ - (2£'u)1 - Y 3)w - 2£'u)1wx + Y l x + y 2 = F 2 / p 0 2 £ ; < 4- 3 a> + (£" - 2 e 1 i ' - Y.6)w + (£" - 2e1l' + 2£' 2 + Y ?x + Y l 0 ) w x + [V2 - Y , 0 ( l - x) - \ Y 7 d - x 2 ) ] + ( E J 2 2 / P e V ) w x x x x • y a ^ •v /\ /\ 65 + 2to.v Q + (2oo,£' + Y r ) v + 2io.,£'v~ 1 8 1 5 1 x + Y 4 X + U 3 = ( F 3 / p 6 2 £ ) ; (4.3b) where y ^ , OJ_. (j = 0, ... , (11)) e x p r e s s e d u s i n g '9' as a measure o f time a r e based on e q u a t i o n s 2.5, 4.1c, 4.2; and: (*) = d( ) / d t ; ( ) • = d( )/d8; ( ) 0 = 3( )/3x; C) = ( )/£, ex c e p t note £' = £'/£, = I"/I; e 1 = e s 9 / ( 1 + e c 8 ) ; y 2 ( 9 ) = T T 2 " + 2(£' - e-^Tr^' - 2e 1£ ITr 2 + 2® {v^ + V - ii + £') - 2a) 3(TT 3' + £ ' T T 3 ) + YJTT-L + (£" ~ Y 2 ) ^ 2 + Y3^3'" y 3 ( 9 ) = T T 3 " + 2(£' - e 1 ) T r 3 " - 2 e 1 & , T T 3 - 2u2(v1l + 2 ' ^ + £ ' ) + 2 C O 1 ( T T 2 I + £ ' T T 2 ) + Y 4 ^ - L + Y 5 ^ 2 " Ye^3' (4.3c) 4.2 S o l u t i o n o f t h e L i n e a r i z e d V i b r a t i o n E q u a t i o n s The assumed-mode approach i s adopted t o s o l v e e q u a t i o n s (4.3) 230 as e x p l a i n e d by M e i r o v i t c h (1967). A c a r e f u l t r u n c a t i o n o f t h e number o f modes used can e f f e c t a c o n s i d e r a b l e r e d u c t i o n i n t h e o r d e r o f t h e system w i t h o u t s a c r i f i c i n g e s s e n t i a l d y n a m i c a l c h a r -a c t e r i s t i c s . E l a s t i c d i s p l a c e m e n t s a r e r e p r e s e n t e d here by a l i n e a r c o m b i n a t i o n o f known f u n c t i o n s o f t h e s p a t i a l v a r i a b l e x m u l t i p l i e d by time-dependent g e n e r a l i z e d c o o r d i n a t e s as f o l l o w s : 66 v(x,0) = E E n ( x ) ? n ( 9 ) n w(x,0) = E H n ( x ) c n ( e ) n (4.4) Care must be exercised when evaluating such derivatives as 9v _ 9v "5"0 90 9v 90 x fixed + dv 9x x fixed 9x 96 since for a deploying system 9x 96 = -SL'x ? 0 ; in general. Sp a t i a l dependence of the c o e f f i c i e n t s i n the equations can be replaced by constant c o e f f i c i e n t s dependent only on the selected t h E , Hn- The procedure involves multiplying throughout by the m assumed mode shape and integrating the equation for the v^, w^  degrees of freedom over the domain x = 0 - l . The r e s u l t i n g equations can be expressed i n the following matrix form; {£»} + 2(rB . E]£' - [B E ] e 1 ) { ? ' } + [K2]{0 - 2 u ) 1 [ B 1 ^ H ] U ' } + ( Y 3 [ B 1 H] - 2co1£' [ B 2 Q H.] ) iO = ( f 2 ) ; (4.5a) U" } + 2 ( I B 2 0 / H ] ^ ' " [ B l , H ] e l ) { ? ' } + t K 3 ] { ? } + 2 u l [ B l , H ] { ^ } + ( Y 5 [ B 1 , E ] + 2 < V ' [ B 2 0 , E ] ) { g } = { f 3 > ; ( 4 ' 5 b ) where: [K 2] = 2 " [ B 3 1 f E ] + £ ' 2 [ B 2 9 / E ] + 2 e 1 £ ' [ B 3 2 / E ] + T ^ t B ^ ^ ] 1 E J 3 3 + ^ 7 [ B 1 1 , E ] " V B 1 , F J + < ^ 4 > £ B 0 , E ] ; [K 3] = £'TB 3 1 /„] + ^ 2 I B 2 9 / H ] + 2 e i V [ B 3 2 / f / ] + Y l l l B 1 0 f „ ] 1 E J22 + 2 V B H , H ] - V B I , H ] + { ^ A ) [ B O t H ] t {f 2> = - V c 4 ^ E } + ( F 2 / p 0 2 £ - y 2 ) { c i , E } ; { f 3 } = ~ Y4 { C4,H } + ( F 3 / P 9 2 * " ^ 3 ) { C i , H } ; (4.5c) with y • and y _ , y , given i n equations (4.1c) and (4.3c), respective-3 2 3 l y . Modal c o e f f i c i e n t s [B^], {C\} are as defined i n Appendix VI. The analysis i s s i m p l i f i e d by taking the beam cross-section to be symmetric, i . e . , J„„ . = J.,., .. Since the appendage i s uniform as well, i t i s reasonable to assume sim i l a r shape functions i n both the y. and z. d i r e c t i o n s . Ideally one would use the exact J l l J eigenfunctions for each boom i n a manner sim i l a r to that employed , . 126,127 in the component-mode synthesis technique. However, as indicated i n the introduction, the evaluation of such character-i s t i c s can be quite d i f f i c u l t and expensive for complex systems even i f the eigenvalue problem can be c l e a r l y defined. The problem i s further complicated by the fact that during deployment, system 193 c h a r a c t e r i s t i c s vary with time. Meirovitch et a l . (1979) have concluded that for any l i n e a r gyroscopic system i t i s s u f f i c i e n t to use a set of admissible functions provided they are complete. For a function to be considered admissible i t must s a t i s f y the geometric boundary conditions and be d i f f e r e n t i a b l e to order p for 68 a system of order 2p. In the p a r t i c u l a r case studied here a con-venient set of modes are provided by the eigenfunctions of a •v 231 232 simple uniform cantilevered beam 9 n ( x ) / ' which s a t i s f y the following equation and boundary conditions. 4 g ~~~~ - 3 a =0; 3n,xxxx n ^ n 4 2 4 3 n = p J T J T / E J ; g (0) = g -(0) = g --(1) = g ---(1) = 0. (4.6) ^n ^n,x ^n,xx an,xxx The solution of (4.6) and some of i t s properties are: g (x) = coshg x - cosg x - 0 ( s i n h 3 x - sing x); 3 n n n TI n n 0 = ( c o s h 3 „ + c o s 3 ) / ( s i n h 3 „ + sing J ; T I n n n n 1 + cosh3 cos3 = 0; n n r1 I g mg„dx = [ B ] = 0/ m f n; J Q m n 1 =1, m = n. (4.7) Note, E ( x ) = H (x) = g n ( x ) ; n n n so that, [ B j , E ] = [ B j , H ] = [ B j ] ' {C, F} = {C. I I ) = {C.}. (4-8) 69 The n e t r e s u l t o f t h i s a p p r o a c h i s t o e l i m i n a t e t h e s p a t i a l d e p e n d e n c e t r a n s f o r m i n g e a c h p a r t i a l d i f f e r e n t i a l e q u a t i o n i n t o a t i m e - d e p e n d e n t s e t o f c o u p l e d o r d i n a r y d i f f e r e n t i a l e q u a t i o n s f o r t h e d i s c r e t e g e n e r a l i z e d c o o r d i n a t e s . A l s o , t h e f o l l o w i n g r e l a t i o n s a r e u s e f u l when e v a l u a t i n g { r c } / {h}, {T}, [ i ] : v = Z g E, - x£g -5 ; t v^n^n y n , x n n ' v ~ = E [g ~k - £ (g " + x g ] ; x t ^n,x^n 3 n , x 3 n , x x n n ' = £{g 1 - 2x£g ^\ + [ (2£ 2 - £)xg - + x 2 £ 2 g } ; t t ^n n ^n,x^n ^n,x ^n,xx n n v t = £(v + i v ) = £ M g J n + £ ( g n - x g n x ) 5 n l ; n v x t = ht+ lh = I ( g n , x ^ n " *K,A); V t t = ( v t t + 2 £ v t + £ v ) ' Z£{g 5 + 2 £ ( g n - x g X K + U ( g n - x g x > + x 2 A 2 g x x K n > ; n v . . = v ~ . + 2 £ v ~ + £ v ~ ; x t t x t t x t x £{g - 2x£g - ^ l + [2? (x g + 2xg ~~) ^ y n,x^n y n , x x s n v y n , x x x ^n,xx x£g — ]£ }. . (4.9) 3 n , x x ^ n 70 4.3 'Free' Vibration C h a r a c t e r i s t i c s of Spinning, Deploying, Orbiting, Beam-Type Appendages 4.3.1 Governing equations As discussed, both i n section 4.2 and by Hughes and Garg 112 (1973), the use of 'exact' appendage modal c h a r a c t e r i s t i c s may not be necessary when assessing p o t e n t i a l i n t e r a c t i o n e f f e c t s such as resonance, between the st r u c t u r a l dynamics and the o r b i t a l , a t t i -121 tude, or control dynamics [Hughes and Sharpe (1975) ]. The high degree of coupling makes even a parametric numerical study d i f f i c u l t for equations (4.5). In order to obtain some appre-c i a t i o n as to the fundamental character of the vibrations, the simpler system of Figure 4-1 i s examined. Here planar (pitch only) attitude motion i s allowed. Neglecting o f f s e t and considering the vibrations to be free (F 2 = = 0), equations (4.5) can be expressed, for boom orientations 0 ) ^ = 0 , TT as: U " ) + 2(£'[B 2 Q] - e 1[B 13){?'} + ([K] -w s 2[B 1 ]){C} ='{f}; (4.10a) U " } + 2(£'IB 2 0] - e 1 [ B 1 ] ) { c ' } + [K]U> = {0}; (4.10b) where: [K] = £"[B 3 1] + £ , 2 [ B 2 9 ] - 2e 1£'[B 3 2] + |[(1 + $') 2 + (| + | c2$)][B ± 1] + ( 6 n 2 - 1) [ B 1 ] ; U s2 = ( i + $-) 2 + (| - | c2$); 71 I (focus) F i g u r e 4-1 Model o f d e p l o y i n g , o r b i t i n g , l i b r a t i n g , beam-t y p e appendage e x p e r i e n c i n g f l e x u r a l o s c i l l a t i o n b o t h i n t v ( x , t ) ] and out [ w ( x , t ) ] o f t h e o r b i t a l p l a n e . 72 = (nn/e) 2 {f} Equations (4.10a, 4.10b), being coupled and non-autonomous are, i n general, not amenable to any simple closed form solution. It i s int e r e s t i n g to note that the equations are e s s e n t i a l l y s i m i l a r i n form. E f f e c t i v e spin (u)g) r e s u l t i n g from o r b i t a l motion, pi t c h , and the gravity gradient serves to reduce s t i f f n e s s for the in-plane coordinate £ r e l a t i v e to.out-of-plane motion £. Also, the in-plane degree of freedom experiences an additional loading r e s u l t i n g from the C o r i o l i s force associated with deployment and rotation, forces due to spin acceleration, and those of the gr a v i -t a t i o n a l f i e l d . Deployment a l t e r s s t i f f n e s s while introducing an e f f e c t i v e negative damping into the system. Type of tr a j e c t o r y i s speci f i e d through e^. A truncated set of the equations (4.10a, 4.10b) i s treated as a discrete eigenvalue problem at any given instant i n time. The ri g h t hand side i s taken to be zero when solving for the 'free' eigenvalues and eigenfunctions. The analysis i s c a r r i e d out over a large range of parameter values. Of course, the c h a r a c t e r i s t i c s found i n t h i s way are v a l i d only over that period of time for which 224 the c o e f f i c i e n t s can be considered constant [Lips and Modi (1978) ]. This technique also forms the basis for the 'quasi-modal' approach 212 of Cherchas (1971). A si m i l a r concept was presented by Worden 233 (1980) during a study of ship motions. The approach adopted here makes use of the eigenvalue analysis, only to assess fundamental v i b r a t i o n c h a r a c t e r i s t i c s and t h e i r parametric v a r i a t i o n . Response i s then based on a d i r e c t numerical integration. 73 4.3.2 Results and discussion From the time of o r b i t a l i n j e c t i o n u n t i l steady state attitude equilibrium i s achieved, a s a t e l l i t e can experience high rates of spin r e s u l t i n g i n a very s i g n i f i c a n t influence on f l e x i b l e appendage c h a r a c t e r i s t i c s . Figure 4-2 demonstrates t h i s e f f e c t , for in-plane v i b r a t i o n , i n an e f f i c i e n t and compact manner by p l o t t i n g frequency 4 2 4 parameter 3 n = P&n& /EJ over a large range of spin parameter values 4 2 4 A = pco £ /EJ. Note that with newer generation spacecraft increas-s s ingly employing longer members, spin parameter values w i l l also tend to be larger than i n the past. Equations 4.10 are truncated so as to include only the f i r s t three modes. The re l a t i o n s h i p between the eigenvalue and the spin parameter i s e s s e n t i a l l y l i n e a r , i . e . , B n = V s . (4.11) There i s no need to plot the out-of-plane r e s u l t since, comparing equations for the case of 'free' v i b r a t i o n and spin only: 3 4 n n = 8 4 T „ + A 4 ; (4.12) n,OP n,IP s as indicated i n Table 4.1. The r e s u l t s allow one to assess the ef f e c t of variatio n s i n the natural v i b r a t i o n frequency (^ n), phys-i c a l c h a r a c t e r i s t i c s of the beam (p, EJ, £), and spin rate. Note the dramatic changes which could occur i n c h a r a c t e r i s t i c v i b r a t i o n frequencies and spin-up/spin-down. For booms aligned along the l o c a l radius vector, R^ ,, v a r i a -tions of the in-plane frequency parameter are not l i n e a r for small values 0 f the spin parameter (0 < As < 6), as indicated i n Figure 4-3. Also, when the spin parameter r e s u l t s s o l e l y from 74 F i g u r e 4-2 Frequency parameter f o r i n - p l a n e v i b r a t i o n s c o v e r i n g a wide range o f s p i n parameter v a l u e s - no deployment. 75 T a b l e 4.1 System E i g e n v a l u e s D e m o n s t r a t i n g I n d i v i d u a l and Combined I n f l u e n c e s o f O r b i t a l M o t i o n , S p i n and Deployment.t B 00 B! W rt D * i < w z o "2 &4 H O < E e m H > to co «^ VD 00 01 Ol 01 rH CN ID lO in *r cn *r •sr ^" •» in in p-p- io co rH 00 CO o 00 Ol rH rH rH rH 1 o o rH rH X X o ID O o O O O in in p- p-o o A c o 0 o H rH X X 01 rH O N O O T o P- rH o o in in o o o o o o A A A A 4J 4J +> J-> 0 0 0 0 m CQ CQ ca en c •rH o rH a 2 p- m p-Ol (Ti oi oi ID i£ 10 10 ^ i ^ ^ i ^ * ^ p- p- O co i o co 00 03 CO 03 • C D ' C D K X i * 0 o \ o \ CD cu c c CD CD ft) Id C C H H IB Id 0< CU rH rH I I CM CU UH <+H 1 I 0 O C C I I H H -U 4J 3 3 O O c .2 Xi 3 -en io co in rH in Ql CO Ul r) ^ ffl n oi in ^ N ^ rO CO rH CD rH rH C N T m ro rH in o o o o o o o o o o o o o o o o o o O O O O O O CN O O IN O O O CN O O CN O O O O O O O CD CD c c id fl rH rH CU CU I I c c CD CD OJ c c c CD id id id C rH rH rH id Cu cu cu H i l l CU MH UH UH I o o o G i l l M JJ JJ 4-> 3 3 3 O 6 O tn c •rl — I - P o § 3 „ c u — o«<x> o o rH rH X X CO t- CTi vo cn d o in in O o o o A A s s CT1 e -rl o rH CU CD Q in r- 10 p*- Ol rH CO o 10 rH CN cn CTi rH m rH rn cn •qi CTi 10 in (TV *r (Ti in T *r <* •<* rH (Tl CO lO CN o ID m o r- (N rH CO CTi CO o CO (Tl CO cn o rH rH rH rH rH rH rH m m m (M m 1 1 O O 1 O 'o 1 O o o rH rH rH rH H rH rH X X X X X X X CN CN o o m CN 10 ID CTi o O O O ID oi o *r T in in T T in f P- r- r» P- p- p* d d o o O o o o o o o o rH H rH rH rH X X X X X 01 p- CO o CN 01 o o o t» cn rH *r o ID cn o r» rH p- p- rH o d o o O in in in m o o o o o o o o o o •CD <D KD *CD "CD 'CD 'CD O O O O O O O CD CD CD CD CD CD id C c C c C e rH id id id id id id Cu rH rH rH rH rH rH 1 Cu cu Cu CU CU CU UH 1 c c c c c c ? H t-H M M M H 3 o tn c •rl o + rH ft CD Q = 2 •J U -•H " O n U u o a a o c - H IO 10 tO (0 10 ID CO in p- O f o if v in p* p- r- P- P-^ if If f o m m n CM I I I I I 0 o o o o rH rH rH rH rH X X X X X CO r - O l r - p 01 01 01 Ol Ol ^ * ^ p- p- p- r- p* o o o o o i i I i i 0 o o o o r-\ r-\ r-t X X X X X CO CO 01 P~ CO 01 cn cn cn cn ^ i ^ i *^ *^ *^ p- p- p^  r^ - p-o d o d o in in o o o o o o o in o in in o o in o o m o o o o o o o o o o CN CN IN IN CM O O O O O oj aj CD CD id id c C rH id id cu rH rH rH rH I Cu CU CU CU UH I I I I o C C C C I H M H M 4-> 3 O tn tn C tn •rH C S + l r j S •H 0 -H a c ja ii tjrj—' M O ' C D a CD a o II w Q . • e A 4JO OO H II 0 OJ ~ CO id 3 O C O CD -C C N CO rH S P I N N I N G O N L Y 0 O R B I T I N G n 0 O R B I T I N G II TT/2 S P I N N I N G n 0 O R B I T I N G n 0 O U T O F P L A N E R E S O N A N C E 0 F i g u r e 4-3 Frequency parameter d u r i n g o r b i t a l m o t i o n o n l y o r s p i n n i n g o n l y , a t s m a l l v a l u e s o f s p i n parameter - no deployment. 77 o r b i t a l m o t i o n , boom f r e q u e n c i e s a r e s i g n i f i c a n t l y h i g h e r t h a n t h o s e f o r t h e case o f pure s p i n i n t h e r e g i o n X > 1. The r e a s o n f o r t h i s i s t h e a d d i t i o n a l s t i f f e n i n g p r o v i d e d by t h e g r a v i t a t i o n a l f i e l d ( E q u a t i o n 4.10). A s i m i l a r phenomenon e x i s t s f o r o u t - o f - p l a n e o s c i l l a t i o n s . Note, resonance i s i n d i c a t e d when s p i n r a t e e q u a l s t h e n a t u r a l f r e q u e n c y f o r t h e case D f i n - p l a n e v i b r a t i o n s . T h i s 139 f i n d i n g was a l s o d i s c u s s e d by Nguyen (1978). The i n h e r e n t l y un-s t a b l e n a t u r e o f a beam p o i n t i n g a l o n g a t a n g e n t t o t h e o r b i t i s a l s o d i s p l a y e d i n F i g u r e 4-3. I s o l a t e d i n F i g u r e 4-4 i s t h e i n f l u e n c e o f deployment r a t e and deployment a c c e l e r a t i o n . R e g a r d l e s s o f whether t h e beam i s e x t e n d i n g o r r e t r a c t i n g , a d e c r e a s e o c c u r s i n t h e v i b r a t i o n f r e q u e n c y w i t h an i n c r e a s e i n t h e magnitude o f £. F o r a g i v e n deployment v e l o c i t y t h e tendency toward n o n - o s c i l l a t o r y b e h a v i o u r i n c r e a s e s f o r l o n g e r booms. The t r e n d i s s i m i l a r f o r booms a c c e l e r a t i n g o u t from t h e s p a c e c r a f t . However, t h e system becomes s t i f f e r d u r i n g d e c e l e r a t i o n . Note t h a t a l t h o u g h deployment i s i t s e l f c a p a b l e o f a s i g n i f i c a n t i n f l u e n c e , i n p r a c t i c a l a p p l i c a t i o n , i t s e f f e c t can be n u l l i f i e d by o r b i t a l e f f e c t s a l o n e . Much o f t h e i n f o r m a t i o n c o n t a i n e d i n F i g u r e 4-5 i s i m p l i c i t i n e a r l i e r r e s u l t s , n e v e r t h e l e s s , i t w i l l s e r v e t o emphasize some o f t h e major f a c t o r s a f f e c t i n g f r e q u e n c y as t h e beam ex t e n d s . F o r 2 4 t h e n o n s p i n n i n g c a s e , Qn a 1/Z hence l a r g e v a r i a t i o n s o c c u r up t o about 200 meters ( F i g u r e 4-5). However, s p i n s t i f f e n s t h e system c o n s i d e r a b l y such t h a t t h e s e l a r g e v a r i a t i o n s i n f r e q u e n c y l a s t o n l y up t o 100 meters a t 2.0 rpm. D u r i n g deployment, b u t i n t h e absence o f s p i n and/or o r b i t a l e f f e c t s beam b e h a v i o u r becomes non-o s c i l l a t o r y . The g r e a t e r t h e r a t e o f deployment t h e s h o r t e r t h e 78 F i g u r e 4-4 I s o l a t i o n of deployment r a t e and deployment a c c e l e r a t i o n e f f e c t s on frequency parameter. 79 0 I i i 1 i J 0 5 0 100 150 2 0 0 2 5 0 i , m 4-5 I n f l u e n c e o f changes i n l e n g t h , deployment r a t e , o r s p i n r a t e on ( i n - p l a n e ) f r e q u e n c y . 80 length at which t h i s occurs. The natural frequency of v i b r a t i o n , associated with the imagin-ary part of the system eigenvalue, i s of prime importance. However, Table 4.1 demonstrates that the r e a l part can also prove to be of inter e s t during deployment. In t h i s case i t i s n o n t r i v i a l and, i n f a c t , can be greater than zero implying i n s t a b i l i t y or, by analogy, a negative damping thus helping to explain some of the previous r e s u l t s (Figure 4-5). Increasing the deployment v e l o c i t y can de-crease frequency (or eliminate i t altogether) while increasing the magnitude of the r e a l part. On the other hand, spin improves s t a b i l -i t y as implied, for example, by a beam deploying at 2 rpm. In t h i s case a l l r e a l parts of the eigenvalue become negative. Additional r e s u l t s contained i n the table allow one to judge combined e f f e c t s of deployment v e l o c i t y , deployment acceleration, and spin. Figure 4-6 c l e a r l y demonstrates that the s t i f f e n i n g e f f e c t of spin rate on eigenvalues also extends to the eigenfunctions. Even at 2 rpm the e f f e c t i s substantial with a l i m i t i n g shape being reached by 10 rpm. The s t i f f e n i n g behaviour of the eigenfunctions i n Figure 4-7 i s c l e a r l y due to an increase i n length only and i s not affected by deployment rate. However, eigenfunctions have been altered by higher deployment rates (e.g., I = 1.0 m/s). Another e f f e c t of deployment i s to produce complex sets of eigenfunctions. When evaluating the r e s u l t s contained i n Figures 4-6 and 4-7 one should bear i n mind that for a simple nondeploying, nonrotating beam, the shape of the eigenfunctions remains invariant with length. F i g u r e 4-6 S p i n r a t e and i t s i n f l u e n c e on system e i g e n f u n c t i o n s i n t h e absence o f deployment. 82 2 r- • 77 (same as pure flexure case) 2L 3 L [36576 m] W e a l 0 -1 - 2 - 4 P/EJ = 00335 s - m 2 rpm / = 003048 m/s = 0 -2 I— l— — J — 1 ' • 0 0.2 04 0-6 0.8 10 A X F i g u r e 4-7 Mode changes a s s o c i a t e d w i t h l e n g t h of a s p i n n i n g d e p l o y i n g beam. 83 Boom response to an i n i t i a l t i p displacement equal to 5% of the length i s displayed i n Figure 4-8 by a plot of the generalized coordinate associated with the f i r s t admissible function ( i . e . , n = 1). The contribution of the second assumed mode was found to be <<5% i n a l l situations. This i s i n part expected because of the nature of the i n i t i a l condition, but i t does support the conclusion 233 stated by Jankovic (1976), that modal coupling i s n e g l i g i b l e between the f i r s t two modes for deployment v e l o c i t i e s of t h i s order. S t i f f e n i n g caused by spin i s r e f l e c t e d by an increase i n the o s c i l l a t i o n frequency. No amplitude change occurs for such a con-servative system. The deploying beam operates at a small amplitude because of the smaller i n i t i a l length and hence smaller i n i t i a l con-d i t i o n . I t i s deployment rate i t s e l f which s h i f t s the frequency of the response. Despite the smaller i n i t i a l condition the deploying, o r b i t i n g beam s t i l l experiences an increase i n amplitude. This i s a conseqence of the C o r i o l i s force contained i n the {f} matrix (Equation 4.10c). In fact, t h i s amplitude increase i s a prelude to the case of the deploying beam rotating at 0.2 rpm and experiencing nonlinear displacement. An order of magnitude check reveals that the 1 '-related term i s capable of severely loading the boom. Although not shown here, t h i s e f f e c t could be either augmented or reduced by 217 spin accelerations. Misra and Modi (1979) also describe the possible build-up of deflections due to C o r i o l i s e f f e c t on an or b i t i n g f l e x i b l e tether system. Out-of-plane vibrations occur at a higher frequency, but do not experience the C o r i o l i s e f f e c t which caused the excessive in-plane displacements. F i n a l l y , i t should be emphasized that the presence of t h i s additional load during deployment makes the duration time associated i (0)m /m/s a)s, rpm FLEXURE ONLY SPINNING • 1 0 r 05 «i 0 -.05 -.10 - DEPLOYING II IN ORBIT " + SPIN L L 140 140 140 0 0 0-15 0-15 0-15 0 0-2 0 0-004 0-2 DEPLOYMENT TIME , s 0 0 200 200 100 \ A- Vy v u,Q) = -05 JL J R„ =12,378 km P/EJ =-00335 s2 -m" 4 IN-PLANE 0 2000 4000 t s 6000 8000 F i g u r e 4 - 8 P l a n a r r e s p o n s e o f a d e p l o y i n g , r o t a t i n g beam-type appendage t o i n i t i a l t i p d i s p l a c e m e n t , ¥=A= 0 . oo 85 w i t h t h e deployment p r o c e s s an i m p o r t a n t parameter. I n a p r a c t i c a l s i t u a t i o n , s h o u l d v i b r a t i o n s become e x c e s s i v e , t h e deployment p r o c e s s c o u l d be t e r m i n a t e d u n t i l a m p l i t u d e s r e t u r n t o a c c e p t a b l e l e v e l s . 4.4 C o n c l u d i n g Remarks I n summary, p r e s e n t e d i s a method o f s o l u t i o n f o r a d e p l o y i n g , o r b i t i n g , l i b r a t i n g , beam-type s p a c e c r a f t appendage c a p a b l e o f t r a n s v e r s e o s c i l l a t i o n b o t h i n and o u t o f t h e p l a n e o f r o t a t i o n . The o b j e c t i s t o p r o v i d e some a p p r e c i a t i o n as t o t h e i n f l u e n c e de-ployment and r o t a t i o n p a r a m e t e r s have e i t h e r s e p a r a t e l y , o r when combined. The more s a l i e n t o b s e r v a t i o n s a r e : ( i ) An o r b i t i n g beam cannot be t r e a t e d s i m p l y as a r o t a t i n g beam because o f t h e p r e s e n c e o f t h e g r a v i t a t i o n a l f i e l d w h i c h can c o n t r i b u t e t o h i g h e r f r e q u e n c i e s , depending on th e r e l a t i v e magnitude o f t h e s p i n parameter. ( i i ) The ' f r e e ' v i b r a t i o n c h a r a c t e r i s t i c s o f o u t - o f - p l a n e m o t i o n d u r i n g s p i n a r e i d e n t i c a l w i t h i n - p l a n e m o t i o n e x c e p t t h a t i t o c c u r s a t a h i g h e r f r e q u e n c y ( E q u a t i o n 4.10). ( i i i ) Resonance can o c c u r between i n - p l a n e appendage v i b r a t i o n s and t h e s p i n degree o f freedom. ( i v ) I n t h e absence o f r o t a t i o n , deployment r a t e i n t r o d u c e s i n -s t a b i l i t y r e g a r d l e s s o f t h e d i r e c t i o n o f e x t e n s i o n . On t h e o t h e r hand, a c c e l e r a t i o n e f f e c t s a r e dependent on d i r e c t i o n ( e x t e n s i o n o r r e t r a c t i o n ) . 86 (v) The change i n l e n g t h i t s e l f , as opposed t o deployment r a t e o r deployment a c c e l e r a t i o n , remains one o f t h e s t r o n g e s t f a c t o r s i n f l u e n c i n g f r e q u e n c y v a r i a t i o n s ( s p i n o r n o - s p i n ) . ( v i ) S p i n a c c e l e r a t i o n s do not a f f e c t system e i g e n v a l u e s o r e i g e n f u n c t i o n b u t , r a t h e r , c o n t r i b u t e an a d d i t i o n a l t r a n s -v e r s e l o a d i n g t o t h e beam. ( v i i ) Rate o f r o t a t i o n p l a y s a dominant r o l e i n s t i f f e n i n g t h e system as e v i d e n c e d by t h e s t r a i g h t e n i n g o f t h e e i g e n -f u n c t i o n s (Table 4.1, F i g u r e s 4-5 t o 4-8). ( v i i i ) Deployment r e l a t e d C o r i o l i s f o r c e s can p l a y a major r o l e i n c a u s i n g l a r g e i n - p l a n e d e f o r m a t i o n s . T h i s i m p l i e s i n some c a s e s t h a t deployment s h o u l d be c a r r i e d out i n s t a g e s so as t o l i m i t t h e t i m e a v a i l a b l e t o b u i l d up a l a r g e a m p l i t u d e r e s p o n s e . Once t h e deployment has been t u r n e d o f f t h e o s c i l l a t i o n s can be damped o u t . Note, r e s u l t s g i v e n here a p p l y t o b o t h s p i n n i n g and g r a v i t y -g r a d i e n t s p a c e c r a f t d u r i n g and a f t e r a t t i t u d e a c q u i s i t i o n . The f r e q u e n c y parameter d a t a s h o u l d be p a r t i c u l a r l y u s e f u l i n d e a l i n g w i t h problems o f i n t e r a c t i o n between t h e s t r u c t u r a l , c o n t r o l system, and v e h i c l e dynamics. 87 5. PLANAR LIBRATIONS OF A TYPICAL GRAVITY GRADIENT CONFIGURATION Having obtained rather general equations for both 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 next l o g i c a l step would be to apply them to a class of representative systems. However, the problem in i t s utmost generality i s so complex that the physical character of the system i s l i k e l y to get l o s t i n the immense amount of algebra involved. As a f i r s t step i n assessing the s i g n i f i c a n c e of system parameters and i n order to e s t a b l i s h some of the basic c h a r a c t e r i s t i c s of the motion a simple yet r e a l i s t i c configuration consisting of a r i g i d central body having two long f l e x i b l e booms free to l i b r a t e and deform i n the o r b i t a l plane was considered. Although t h i s r e s u l t s i n some s i m p l i f i c a t i o n of the governing equations, they s t i l l remain nonlinear, nonautonomous, and coupled and hence quite challenging. The main objective i s to get some appreciation as to the i n t e r a c t i o n between f l e x i b i l i t y and l i b r a -t i o n a l motion during the steady state as well as transient phases as represented by deploying appendages. 5.1 Simplified Spacecraft Configuration and System Equations Figure 5-1 i l l u s t r a t e s the s p e c i f i c spacecraft studied here which i s a s i m p l i f i e d form of the general configuration presented in Figure 2-2. Cantilevered to a central r i g i d body are two diametrically opposed uniform f l e x i b l e beam-type appendages, which can be deployed independently. In the nominal equilibrium con-88 d i t i o n the undeformed appendages are aligned along the l o c a l v e r t i c a l (F^) with boom number 1 pointing i n the outward d i r e c t i o n (cj>^ = 0) r e l a t i v e to the center of force 0 . If attitude motion i s r e s t r i c t e d such that r o l l and yaw remain unexcited then p i t c h motion ($) takes place i n the o r b i t a l plane only, the condition referred to as 'planar' motion. Furthermore, the boom vibrations are also assumed to be confined to the o r b i t a l plane. This type of configuration i s t y p i c a l of the many g r a v i t y - s t a b i l i z e d concepts. Although t h i s represents a degree of approximation i t provides a st a r t i n g point i n the analysis of such a complex system. Governing equations are arrived at by applying the general r e s u l t s of Chapter 2 and 4 to the s p e c i f i c configuration represent-ed i n Figure 5-1. Note the use of a hybrid system of coordinates with nonlinear attitude equations (so important during the transient attitude a c q u i s i t i o n stage) together with l i n e a r appendage equa-tions. The s i m p l i f i e d expressions here omit the e f f e c t of o f f s e t ( T T) of the appendage attachment point. Considering ¥ = A = u = w = 0 , evaluating appendage coordinate transformation matrices [x-^] and [ x 2 l > expanding out [I], [I], {h}, { r } as i n Appendices II and III and substituting into equation (2.12c) y i e l d the following equation for p i t c h attitude motion: I 3 3 (6 + «) + i 3 3 (6 + *) + I ( " J i 3 ) [ ( I 2 2 - I u ) s 2 * + 2I 1 2c2$] + I P i / 0 1 { x i ( V i , t t + 2 V i , x t - L v . [ 1 + 4 - x.)c«J> .v ]} dx. =0; i = 1,2 (5.1a) i i 2 i i l l i F i g u r e 5-1 C o n f i g u r a t i o n o f a r e p r e s e n t a t i v e g r a v i t y g r a d i e n t s a t e l l i t e , w i t h two i n - p l a n e f l e x i b l e d e p l o y i n g u n i f o r m booms, undergoing p l a n a r d e f o r m a t i o n . oo 90 where: 1 -I. J 3 3 = 1*33 + I { p i [ I £ i 3 + / o V 2 " 2 x i u f s f i > d X i ] } ;  ( I 2 2 - I 1 1 ) = <1 I22-1 I11 ) + I {?i[¥i3 ~ 7 0 1 ( Vi 2 + ^ f s . i * d X i ] > r £ i I-, 0 = -,I-,<j + X [p. / (x. - u r .) v . d x . ] . 12 1 12 ^ I y n v l f s , i I I A l s o w i t h a) 1 = u 2 = 0, e q u a t i o n s (4.1a) f o r l i n e a r i n - p l a n e v i b r a t i o n s become: p. [v. + 21, [(6 + $) + v. ] + 1(9 + h 2 + (-3) (| K i i , t t x i , x t R j z c - § C 2 $ ) ] V ; L + [ (6 + $ ) 2 + (| + § c 2 $ ) } x i v i / X R c + [ (6 + 'i) + |(- i i 3-) s 2 $ ] X i + {I 2 + - x ± ) R c c + E . J _ , .v. = 0. (5.1b) 1 3 3 , i i , x x x x 5.2 E q u a t i o n s Based on ' D i s c r e t e ' D e f o r m a t i o n C o o r d i n a t e s and ' O r b i t a l ' Time The s o l u t i o n o f e q u a t i o n s (5.1) i s o b t a i n e d u s i n g t h e assumed-mode p r o c e d u r e f o r r e p r e s e n t i n g e l a s t i c d e f o r m a t i o n s , as e s t a b l i s h e d i n C h a pter 4. I n terms o f o r b i t a l t i m e and u s i n g t h e n o n d i m e n s i o n a l form o f t h e v i b r a t i o n e q u a t i o n s ( 4 . 3 a ) , t h e e q u a t i o n g o v e r n i n g p i t c h can be e x p r e s s e d a s : 91 I 3 3 $ " + ( I 3 3 - 2e1I33) (1 + *•) + | ( 1 + ec6) [ ( I 2 2 - I I : L ) S 2 * + 2 l 1 2 c 2 $ ] + T 3 - 2 e x h 3 = 0; (5.2) where: I , , = . i „ + Z {p,.*.3 [ i + Z E (B , - ^B T J C 1 ? 1 ] } ; 33 1 33 . " i l l3 mn,l 2 mn,ll m n 1 mn ( I 2 2 - I 1 1 ) = (1 X22 " I 1 ! ! * + 1 { P i £ i 3 I i " I Z ( Bmn,l x mn + \ B m n ) }; Z mn,ll mm I , = N I + E [p . £ . 3 ( E C . 5 1 ) ] ; 12 1 1 2 . K x x m, 4 m x m I ' = E { p . £ . 2 E E (B , - 75- B ..) (2£. S^'si + 3 A ' ? 1 ^ 1 ) } ; 33 . x x mn,l 2 mn.ll 1 m n m^^ n x mn h 0 = z [ p . £ . 2 z (c .l.E,1' + c ,„£'. 3 . ^x x m,4 x^ m m,12 x m^ x m r_ = z P . £ . { z z (4 B , nc<j). £ . I " K X Z , X ) + Z [ C A-2KX" 3 . Kx x 2 mn,10 1 x x m n m,4 x m^ x mn ' m ' + 2C ,,£.£'. + (C m,ll x x^ m m,12 x x + C .„£.' 2) E 1 ] } ; m, 14 x m^ J i = 1 , 2; (JK = 0, TT. 92 B o t h appendages a r e governed by e q u a t i o n (4.10a). Consequent-l y p i t c h e x c i t a t i o n can g e n e r a t e asymmetric o s c i l l a t i o n s o n l y . T h i s i s n o t t h e c a s e , however, f o r d i f f e r i n g boom i n i t i a l c o n d i t i o n s . A l s o d i f f e r e n c e s i n boom p h y s i c a l p r o p e r t i e s o r deployment w i l l a l t e r t h e r e s p o n s e . System e q u a t i o n s g o v e r n i n g t h e p l a n a r dynamics (4.10a and 5.2) were s o l v e d s i m u l t a n e o u s l y w i t h t h e h e l p o f an AMDAHL 470/V6-I I d i g i t a l computer. The n u m e r i c a l i n t e g r a t i o n r o u t i n e was based on t h e i m p l i c i t Adam's method w i t h b u i l t - i n e r r o r c o n t r o l . V i g n e r o n ^ ^ has p o i n t e d o u t s e v e r a l d i f f i c u l t i e s e n c o u n t e r e d i n t h e n u m e r i c a l t r e a t m e n t o f t h i s c l a s s o f problems. F l a n a g a n 234 (1969) a l s o r e f e r s t o d e f i c i e n c i e s r e l a t e d t o d i g i t a l compu-t a t i o n . F o r t u n a t e l y , w i t h advances i n computer t e c h n o l o g y and b e t t e r i n t e g r a t i o n r o u t i n e s a v a i l a b l e t o d a y , no such d i f f i c u l t i e s were e n c o u n t e r e d . 5.3 R e s u l t s and D i s c u s s i o n C a l c u l a t i o n s were c a r r i e d o u t f o r t h e two-boom g r a v i t y g r a d -i e n t c o n f i g u r a t i o n a t an o r b i t a l a l t i t u d e o f 6000 km. The p h y s i c a l c h a r a c t e r i s t i c s o f t h e appendages c o i n c i d e w i t h t h o s e o f t h e RAE 2 antennas (p=0.023024 kg/m, E J ^ = 7.85 Nm ). P r i n c i p a l i n e r t i a s 2 and mass f o r t h e c e n t r a l r i g i d body a r e 18 Nm and 150 kg r e s p e c t -i v e l y . Through a s y s t e m a t i c v a r i a t i o n o f t h e l a r g e number o f v a r i a b l e s i n h e r e n t t o t h e system one can g e n e r a t e e x t e n s i v e amounts o f i n f o r m a t i o n . ' However, f o r c o n c i s e n e s s , o n l y t y p i c a l r e s u l t s s u g g e s t i n g t r e n d s a r e r e c o r d e d h e r e . R e c o g n i z i n g t h a t t h e f l e x i b i l i t y e f f e c t s a r e l i k e l y t o i n -c r e a s e w i t h l e n g t h o f t h e appendages, F i g u r e 5-2 s t u d i e s p i t c h and v i b r a t i o n a l r e s ponse o f a s a t e l l i t e w i t h two g r a v i t y g r a d i e n t booms 93 | R c = 12,378 km , e = 0 , $(0) = 9 • EJ 3 3 = 7.85 Nm 2 • 1(0) = 0 • 9=0.023024 kg /m • i - 0.2 m/s-L= 50 m = 200 m = 250 m 6 0 0 0-1 0.2 0-3 04 0-5 O R B I T S are 5-2 E f f e c t of the f l e x i b l e boom l e n g t h on system response f o r the p l a n a r case. 94 e x t e n d i n g t o 50, 200 and 250 m. The s a t e l l i t e i s i n a c i r c u l a r o r b i t w i t h t h e booms e x t e n d i n g a t a r a t e o f 0.2 m/s. I t i s s u b j e c t -ed t o an i m p u l s i v e d i s t u r b a n c e o f $(0) = 9. I n a d d i t i o n t o t h e v e r y l a r g e a m p l i t u d e e x p e r i e n c e d by t h e p i t c h r e s p o n s e t h e f i g u r e shows t h r e e p o i n t s o f i n t e r e s t : ( i ) There i s a q u i c k r e v e r s a l i n t h e d i r e c t i o n o f p i t c h m o t i o n due t o i n c r e a s e i n t h e moment o f i n e r t i a about t h e p i t c h a x i s ( c o n s e r v a t i o n o f momentum) broug h t about by deployment. ( i i ) A m p l i t u d e o f p i t c h o s c i l l a t i o n s t e n d s t o i n c r e a s e w i t h an i n -c r e a s e i n t h e f i n a l deployment l e n g t h , L. T h i s appears t o be a d i r e c t consequence o f t h e v i b r a t o r y m o t i o n w h i c h s e t s i n a t l a r g e r l e n g t h s . ( i i i ) F l e x i b l e appendages undergo s m a l l a m p l i t u d e a n t i s y m m e t r i c m o t i o n . F i g u r e 5-3 compares t h e re s p o n s e o f r i g i d and f l e x i b l e s a t e l -l i t e s d u r i n g appendage deployment from 180 t o 200 m. C o r r e s p o n d i n g performance w i t h t h e appendage l e n g t h f i x e d a t 200 m i s a l s o i n -c l u d e d . I t i s o f i n t e r e s t t o r e c o g n i z e t h a t f o r t h e n o n d e p l o y i n g c o n d i t i o n , f l e x i b l e appendages remain v i r t u a l l y u n e x c i t e d r e s u l t i n g i n a p i t c h r e s p o n s e t h a t i s i d e n t i c a l t o t h e r i g i d c a s e . However, t h e e f f e c t o f deployment o f r i g i d appendages i s t o reduce t h e maximum a m p l i t u d e from 35° t o 30°. The i n f l u e n c e o f f l e x i b i l i t y i s t o f u r t h e r a c c e n t u a t e t h i s t r e n d w i t h t h e a m p l i t u d e reduced t o around 20°. Note a l s o t h e h i g h f r e q u e n c y m o d u l a t i o n o f t h e l i b r a -t i o n a l r e s p o n s e due t o v i b r a t o r y m o t i o n o f t h e f l e x i b l e appendages. T h i s i s a d i r e c t r e s u l t o f t h e f l e x i b i l i t y i n t e r a c t i o n , t h a t i s , 95 R c = 12,378 km , e = 0 , $(0) = e E J 3 3 = 7.85 Nm2 , 9=0-023024 kg/m i(0),m i,m/s L, m 0.05 r 1,2 - 0.05 L 40° r 180 180 0-2 0.2 200 RIGID 200 FLEXIBLE 200 RIGID 200 FLEXIBLE 0.1 Am 0.2 0.3 ORBITS F i g u r e 5 -3 T r a n s i e n t r e s p o n s e o f a g r a v i t y g r a d i e n t s a t e l l i t e s h o w i n g t h e e f f e c t o f f l e x i b i l i t y a n d d e p l o y m e n t , ¥=A=0. 96 the booms serve as a feedback mechanism for the attitude motion. E f f e c t of the deployment rate on the system dynamics i s i n -dicated i n Figure 5-4 where the appendages are deployed from the i n i t i a l length of 200 m to a f u l l y deployed value of 250 m. In-creasing the deployment rate from 0.1 m/s to 0.2 m/s does not seem to a f f e c t the p i t c h response s u b s t a n t i a l l y except for a d i s t i n c t s h i f t i n phase, however, the v i b r a t i o n a l response becomes quite sensitive to the deployment rate and at a c r i t i c a l value of around 0.22 m/s the v i b r a t i o n a l motion becomes unbounded leading to i n -s t a b i l i t y of the p i t c h v i b r a t i o n s . Boom response to an i n i t i a l t i p displacement equal to 5% of the length i s displayed i n Figure 5-5 by a plot of the generalized coordinate associated with the f i r s t admissible function of boom number 1 and the corresponding p i t c h l i b r a t i o n . Symmetric i n i t i a l displacements of the booms produce no pitching while antisymmetric i n i t i a l conditions (tip displacement = 5% of boom length) r e s u l t i n a pi t c h - 9°. Disturbing only one boom i n i t i a l l y y i e l d s l i b r a t i o n s less than 5°. A considerable difference e x i s t s i n the frequency of vi b r a t i o n a l response for the symmetric case as opposed to the other two. The high frequency behaviour i s eliminated along with the . p i t c h response for the symmetric case. This i s because the high frequency modulation i s a d i r e c t r e s u l t of coupling with the p i t c h motion, which i s not excited during the symmetric case. During the formulation of the governing equations i t was recognized that i n c l u s i o n of the s h i f t i n g center of mass and/or geometric o f f s e t of the appendages considerably adds to the complex-i t y of the problem. Hence i t was considered desirable to assess t h e i r e f f e c t s on the general response. This i s examined i n Figure 97 R c= 12,378 km , e = 0 EJ33 = 7.85 Nm 2 • 9 =0.023024 kg/m • I =0.10 m/s = 0.20 m/s = 0.22 m/s $(0) 1(0) L 9 • 200 m 250 m 0.10 r 0 0.2 0.3 ORBITS 0.4 0.5 F i g u r e 5-4 E f f e c t o f t h e d e p l o y m e n t r a t e o n p i t c h a n d v i b r a t i o n a l r e s p o n s e o f a g r a v i t y g r a d i e n t s a t e l l i t e . 98 R c = 12,378 km , e=0 , L = 200 m EJ 3 3=7,85 Nm 2 , 9 = 0.023024 kg/m BOOM INITIAL CONDITIONS: BOOM 1 ONLY SYMMETRIC ANT I SYMMETRIC 0.05 r 0 0.1 0.2 03 ORBITS 0.4 F i g u r e 5-5 E f f e c t o f i n i t i a l e l a s t i c d e f o r m a t i o n s on system response f o r t h r e 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 , ¥ = A = 0 . 99 5-6 w h i c h shows t h e e f f e c t t o be m i n i m a l f o r t h e case c o n s i d e r e d . T h i s i s q u i t e i m p o r t a n t as c o n s i d e r a b l e s i m p l i f i c a t i o n o f t h e e q u a t i o n s and subsequent s a v i n g i n t h e c o m p u t a t i o n a l e f f o r t can be a c h i e v e d w i t h o u t s a c r i f i c i n g a c c u r a c y . 5.4 C o n c l u d i n g Remarks The p l a n a r dynamics r e v e a l s some i m p o r t a n t f e a t u r e s a s s o c i a t e d w i t h t h e m o t i o n o f g r a v i t y g r a d i e n t systems h a v i n g f l e x i b l e d e p l o y -i n g appendages: ( i ) D i g i t a l c o m p u t a t i o n now p e r m i t s s o l u t i o n o f a t t i t u d e dynamics problems p r e v i o u s l y i n t r a c t a b l e w i t h such an approach. ( i i ) P i t c h m o t i o n e x c i t e s a n t i s y m m e t r i c boom d e f o r m a t i o n s o n l y . ( i i i ) The r e s u l t s i n d i c a t e t h a t even l a r g e a m p l i t u d e l i b r a t i o n a l m o t i o n f a i l s t o e x c i t e s u b s t a n t i a l appendage d e f o r m a t i o n s . On t h e o t h e r hand, appendage d e f o r m a t i o n s , caused by i n i t i a l c o n d i t i o n s o r o t h e r w i s e , can have s u b s t a n t i a l e f f e c t s on response as e v i d e n c e d by t h e c o n s i d e r a b l e m o d u l a t i o n o f p i t c h a m p l i t u d e t o g e t h e r w i t h a s i g n i f i c a n t i n c r e a s e i n f r e q u e n c y o f appendage o s c i l l a t i o n when compared w i t h t h e u n c o u p l e d c a s e . ( i v ) Deployment can r e s u l t i n a s i g n i f i c a n t i n c r e a s e i n v i b r a t i o n a m p l i t u d e . I n f a c t , depending on t h e o r b i t a l p a r a m e t e r s and p h y s i c a l p r o p e r t i e s o f t h e booms, t h e r e e x i s t s c r i t i c a l com-b i n a t i o n s o f boom l e n g t h and deployment r a t e f o r w h i c h t h e s a t e l l i t e can tumble o v e r . 100 R c = 12,378 km , e = 0 , L = 200 m• j E J 3 3 = 7.85 Nm 2 , 9= 0.023024 kg/m-ANTISYMMETRIC BOOM INITIAL CONDITIONS cm. fixed cm. shifting c.m. shifting + offset = 2 m °n = 1 m 10 r 5 h $ 0 I - 5 h - 1 0 1 0.2 0.3 ORBITS 0.5 F i g u r e 5-6 T y p i c a l p l a n a r r e s p o n s e as a f f e c t e d by t h e s h i f t i n g c e n t e r o f mass and appendage o f f s e t . 101 Deployment i n c r e a s e s t h e d e g r e e o f c o u p l i n g between t h e a t t i t u d e and v i b r a t i o n a l d e g r e e s o f f r e e d o m . I n c e r t a i n c a s e s t h e e f f e c t o f s h i f t s i n c e n t e r o f mass and o f f s e t c a n be n e g l i g i b l e . 102 6. GENERAL THREE-AXIS ATTITUDE MOTION The p l a n a r d y n a m i c a l s t u d y though u s e f u l s h o u l d be c o n s i d e r e d a p p r o x imate and p r o v i d e s m e r e l y p r e l i m i n a r y i n f o r m a t i o n c o n c e r n i n g t h e system b e h a v i o u r . To an e x t e n t , i t g i v e s a mechanism f o r check-i n g t h e enormous amount o f a l g e b r a and a l s o i n v o l v e s a s i m p l i f i e d v e r s i o n o f t h e g e n e r a l computer program. W i t h t h i s as background, the p r e s e n t c h a p t e r a p p l i e s t h e a n a l y s i s t o t h e case o f g e n e r a l t h r e e - a x i s a t t i t u d e m o t i o n . 6.1 S p a c e c r a f t C o n f i g u r a t i o n and System E q u a t i o n s The two-boom g r a v i t y g r a d i e n t c o n f i g u r a t i o n o f C h a p t e r 5 i s b u t one p a r t i c u l a r c a s e o f t h a t c l a s s o f s p a c e c r a f t d e p i c t e d i n F i g u r e 6-1. The e q u i l i b r i u m a t t i t u d e i s t a k e n t o be such t h a t t h e x-y p l a n e c o i n c i d e s w i t h t h e o r b i t a l p l a n e . A l s o , any p i t c h o r s p i n m o t i o n o c c u r s i n t h e x-y p l a n e hence i t i s r e f e r r e d t o as t h e ' s p i n ' p l a n e and c o n t a i n s booms numbered 1 t h r o u g h 4 h a v i n g t h e a r b i t r a r y o r i e n t a t i o n (|> as i n d i c a t e d ( F i g u r e 6-1). Booms 5 and 6 l i e i n t h e x-z p l a n e . A l t h o u g h a maximum o f s i x appendages a r e i l l u s t r a t e d , t h e g o v e r n i n g e q u a t i o n s c o n s i d e r e d i n t h i s s e c t i o n a p p l y t o a c o n f i g u r a t i o n h a v i n g an a r b i t r a r y number o f booms i n each o f t h e s e p l a n e s . C o o r d i n a t e t r a n s f o r m a t i o n m a t r i c e s r e q u i r e d t o r e l a t e l o c a l appendage c o o r d i n a t e s t o t h e c e n t r a l axes i n c l u d e : [ XP ] c<b 1 -scf> s* I ( f o r a p l a n a r boom); 1 0 3 spin plane ^ p = X p= 0 , <Pp arbitrary * > 0 = X 0 = 0 , tf0 arbitrary F i g u r e 6-1 C o n f i g u r a t i o n r e p r e s e n t i n g a l a r g e c l a s s o f space-c r a f t chosen f o r d e t a i l e d study. Note the arrangement shows appendages i n the x-y plane c o i n c i d i n g w i t h the p i t c h plane (p) and i n the x-z plane p e r p e n d i c u l a r t o the p i t c h plane (o). 104 t x 0 l = 0 sip. 0 1 0 0 (for a boom i n the x-z plane). (6.1) Possible gravity gradient configurations represented by Figure 6-1 include that of two diametrically opposed appendages i n the spin plane (Figure 5-1), an RAE-type arrangement involving four booms symmetrically placed i n the spin plane, or perhaps a set of six mutually perpendicular booms. S i m i l a r l y , one could i d e n t i f y configurations applicable to s p i n - s t a b i l i z e d spacecraft such as the Alouette series having two sets of antennae of d i f f e r e n t lengths l y i n g i n the spin plane. On the other hand, a CTS-type configur-ation could be modelled using just two booms (numbers 5 and 6) perpendicular to the spin plane. Of course, r i g i d unsymmetrical s a t e l l i t e s with no appendages can also be included within t h i s c l a s s . Obviously the representation of Figure 6-1 has a wide range of a p p l i c a b i l i t y . However, i t i s but one p a r t i c u l a r case of the more general configuration o r i g i n a l l y presented i n Figure 2-2, thus emphasizing the v e r s a t i l i t y offered by that formulation. Appropriate equations governing l i b r a t i o n a l and v i b r a t i o n a l behaviour can be arrived at by applying the more general r e s u l t s of Chapters 2 through 4 to the configuration under study here. The main assumption made i s that an a r b i t r a r y number of f l e x i b l e , de-ploying, uniform booms l i e i n the x-y and the x-z planes. As i n Chapters 4 and 5, appendage motion i s governed by a set of p a r t i a l d i f f e r e n t i a l equations and advantage i s taken of the assumed-mode procedure i n a r r i v i n g at a f i n a l solution. Again, a x i a l o s c i l -105 la t i o n s are neglected. With a description available for the appen-dage motion one can proceed as i n Chapter 5 to evaluate those terms in the l i b r a t i o n equations dependent on f l e x i b i l i t y and deployment. Consequently, the system equations of motion consist of equation (2.12) for the attitude dynamics and the set of equations (4.5) for each boom. A study of these equations reveals how an already complex system becomes more complicated when one takes into account such factors as s h i f t i n g center of mass, appendage o f f s e t , a x i a l fore-shortening and deployment. 6.1.1 Computational considerations As before the solution was sought by d i r e c t numerical in t e -gration on a d i g i t a l computer. It i s perhaps of int e r e s t to out-l i n e the basic programming approach adopted for dealing with such an involved system. The motion i s governed by 3 l i b r a t i o n a l and N-3 v i b r a t i o n a l second order ordinary d i f f e r e n t i a l equations which are transformed to 2N f i r s t order equations by constructing a state vector of the N zeroth order (displacement) and the N f i r s t order (velocity) terms. In t h i s form, a l l degrees of freedom are solved for simultaneously as a f i r s t order i n i t i a l value problem. Used i s an integration routine provided by the UBC Computing Services which i s based on an i m p l i c i t Adam's method. For the procedure to succeed one must use the l a t e s t data available when updating derivatives. Consequently, the second order derivative associated with each degree of freedom must be expressed as a function of the lower order derivatives only 235 [Conte (1965) ]. This presents one with a considerable challenge 106 f o r t h e complex h i g h l y c o u p l e d system a t hand. The amount o f a l g e b r a i n v o l v e d i s m i n i m i z e d by t a k i n g advantage o f t h e n u m e r i c a l t e c h n i q u e o u t l i n e d i n Appendix V I I . O r g a n i z a t i o n o f t h e computer program c e n t e r e d on a s e r i e s o f s u b r o u t i n e s . A main program d i r e c t e d t h e i n t e g r a t i o n p r o c e s s c a l l i n g t h e s y s t e m - s u p p l i e d r o u t i n e s as d e s i r e d and p r o v i d e d t h e needed i n p u t / o u t p u t s e r v i c e s . The i n t e g r a t i o n package r e q u i r e d a r o u t i n e (SYSTM) t o d e f i n e t h e system dynamics i n terms of e x p l i c i t e x p r e s s i o n s f o r t h e f i r s t o r d e r d e r i v a t i v e o f t h e s t a t e v e c t o r . The g o v e r n i n g e q u a t i o n s a r e employed d i r e c t l y i n SYSTM i n two d i s t i n c t s t a g e s t o d e a l w i t h t h e l i b r a t i o n a l and v i b r a t i o n a l c o n t r i b u t i o n s . I n a d d i t i o n , a s e p a r a t e s u b r o u t i n e was w r i t t e n f o r each o f t h e q u a n t i t i e s { r c } , {h}, { r } , [ I ] , [ I 1 ] r e q u i r e d by SYSTM. I n each case f i r s t d egree, second degree, and f o r e s h o r t e n i n g e f f e c t s due t o f l e x i b i l i t y were grouped i n s e p a r a t e b l o c k s . O v e r a l l t h e modu-l a r approach adopted was i n t e n d e d t o p e r m i t easy e x t e n s i o n o f t h e program t o i n c l u d e a d d i t i o n a l appendage e q u a t i o n s and t o a l l o w f o r i s o l a t i o n o f t h e e f f e c t s i n d i c a t e d above. The 3 - a x i s program was s e t up t o accommodate an a r b i t r a r y number o f assumed modes and s i x booms, f o u r i n t h e x-y p l a n e and two i n t h e x-z p l a n e . Assuming a two mode r e p r e s e n t a t i o n r e s u l t s i n a system o f f i f t y - f o u r f i r s t o r d e r e q u a t i o n s . Modal i n t e g r a t i o n c o e f f i c i e n t s were d e t e r m i n e d i n d e p e n d e n t l y by n u m e r i c a l q u a d r a t u r e . Where p o s s i b l e , t h e s e i n t e g r a l s were e v a l u a t e d a n a l y t i c a l l y as w e l l . Accompanied by a l i b e r a l use o f comment c a r d s t h e program exceeded t h r e e thousand f i v e hundred l i n e s . However, no s t o r a g e l i m i t a t i o n s were e n c o u n t e r e d ; a l t h o u g h e x e c u t i o n t i m e s c o u l d n o t be i g n o r e d as CPU v a l u e s o f 50-100 were not uncommon. P a r t i c u l a r l y t i m e consum-107 ing are integrations involving small appendage lengths. A similar 236 finding was pointed out by Misra and Modi (1979). To cope with the r e l a t i v e l y small step size demanded by the high frequency o s c i l -l a t i o n s a two-stage integration procedure i s established thus allow-ing for a complete change i n such parameters once during the course of the integration. The program was coded i n FORTRAN using double pre c i s i o n variables throughout. 6.2 Results and Discussion The endeavour here has aimed at developing a model which tests the transient and steady state e f f e c t s of f l e x i b i l i t y and deployment in a r e l a t i v e l y general manner, the ultimate objective being to assess t h e i r i n t e r a c t i o n with the attitude dynamics. P r a c t i c a l d i f f i c u l t i e s a r i s e i f one wishes to simulate the behaviour of an actual spacecraft. F i r s t l y , only limited response data i s available i n the open l i t e r a t u r e and what there i s r a r e l y applies to nonlinear deploying sit u a t i o n s . Even for the ex i s t i n g data one tends to f i n d but an incomplete i d e n t i f i c a t i o n of those parameters needed for carrying out a meaningful comparative simulation. Secondly, addi-t i o n a l refinements to the model developed here may have to be made in order to include c h a r a c t e r i s t i c s unique only to the system under consideration. In many cases the e f f o r t required to take into account new features may not be great since dynamic simulation of the f l e x i -b i l i t y e f f e c t s has already been carried out. For example, introduction of momentum biasing as used on the CTS would simply mean the adding of a constant term to the h^ compo-nent of the l o c a l momentum vector. S i m i l a r l y , i t i s r e l a t i v e l y 108 s t r a i g h t f o r w a r d t o i n t r o d u c e damping i n t o the s t u d y . However, t h i s i n j e c t s a degree o f u n c e r t a i n t y i n t o t h e s i m u l a t i o n s i n c e no s i n g l e t h e o r y has emerged w h i c h a d e q u a t e l y d e s c r i b e s t h e damping c h a r a c t e r -i s t i c s ; a p o i n t emphasized by t h e r e c e n t f i n d i n g s o f Garg e t a l . 237 (1979). T h i s i s one r e a s o n i t has not been d e a l t w i t h i n t h e c u r r e n t i n v e s t i g a t i o n . A l s o i t was c o n s i d e r e d u n n e c e s s a r y t o f u r t h e r c o m p l i c a t e an a l r e a d y c o m p l i c a t e d t a s k when one c o u l d be r e a s o n a b l y c e r t a i n as t o t h e e f f e c t o f t h e phenomenon - energy d i s s i p a t i o n a l o n g w i t h a t t e n u a t i o n o f t h e a m p l i t u d e o f v i b r a t i o n . A l t h o u g h comparisons o f s i m u l a t i o n s w i t h a c t u a l f l i g h t con-d i t i o n s i s t h e i d e a l , t h e r e e x i s t , n o n e t h e l e s s , a l t e r n a t i v e measures one can t a k e t o e s t a b l i s h c o n f i d e n c e i n t h e w o r k i n g o f t h e program. A check on t h e a l g e b r a e x i s t s , t o some e x t e n t a t l e a s t , d u r i n g a p p l i c a t i o n o f t h e assumed-mode s o l u t i o n t o such terms as {h}, { r } , e t c . A s i d e from t h e u s u a l symmetry one e x p e c t s o f t h e e x p r e s s i o n s f o r t h e g e n e r a l t h r e e d i m e n s i o n a l c a s e , one a l s o f i n d s t h e g r o u p i n g o f c o e f f i c i e n t s f a l l i n g i n t o a f a m i l i a r p a t t e r n so t h a t any d e v i a t i o n l e a d s one t o a g a i n r e v i e w t h e d e r i v a t i o n . A c t u a l response can be checked by p u r s u i n g such t r i v i a l c a s e s as r e s p o n s e t o z e r o i n i t i a l c o n d i t i o n s . The program can be r u n i n t h e v i b r a t i o n o r p i t c h modes o n l y so t h a t a m p l i t u d e s and p e r i o d s can be p r e c i s e l y checked. P l a n a r n o n l i n e a r r e s p o n s e o f a r i g i d c o n f i g u r a t i o n i n an e c c e n t r i c o r b i t was compared w i t h t h a t by B r e r e t o n (1978). 3 0 Comparison o f s i m u l t a n e o u s response o f b o t h l i b r a t i o n a l and v i b r a t i o n a l m o t ions i s p o s s i b l e u s i n g a p l a n a r program d e r i v e d i n d e p e n d e n t l y o f t h e g e n e r a l program. A l s o , peak p i t c h d i s p l a c e m e n t and p i t c h r e v e r s a l p r e d i c t e d f o r the RAE by Dow e t a l . (1966)54 and Bowers e t a l . (1970) 2 0 4 a r e s i m i l a r t o t h e response g e n e r a t e d h e r e . 109 6.2.1 Two boom g r a v i t y g r a d i e n t c o n f i g u r a t i o n Having g a i n e d some f a m i l i a r i t y w i t h t h e p l a n a r r e sponse o f a two-boom g r a v i t y g r a d i e n t system, a l o g i c a l s t e p was t o examine i t s f u l l y c o u p l e d a t t i t u d e b e h a v i o u r . R e p r e s e n t a t i v e r e s u l t s p r e -s e n t e d here assume t h e same o r b i t , c o n f i g u r a t i o n , and boom c h a r a c t e r -i s t i c s as d e s c r i b e d i n s e c t i o n 5.3. F i g u r e 6-2 compares t h e r e s p o n s e o f r i g i d and f l e x i b l e s a t e l l i t e s d u r i n g appendage deployment from 0-100 m. C o r r e s p o n d i n g r e s p o n s e w i t h the appendage l e n g t h f i x e d a t 100 m i s a l s o i n c l u d e d . In a l l t h e c a s e s , t h e system i s s u b j e c t e d t o t h e i n i t i a l i m p u l s i v e p i t c h ( p l a n a r ) e x c i t a t i o n o f $(0) = 6 w i t h r o l l and yaw degrees o f freedom l e f t u n d i s t u r b e d . As can be e x p e c t e d , l a r g e a m p l i t u d e p i t c h m o t i o n r e s u l t s , however, i t i s o f i n t e r e s t t o r e c o g n i z e t h a t t h e r e a r e v i r t u a l l y no c o u p l i n g e f f e c t s as r o l l and yaw m o tions a r e e s s e n t i a l l y a b s e n t , so i s the v i b r a t o r y response o f t h e f l e x i b l e appendages. Note t h a t f o r t h e n o n d e p l o y i n g c o n d i t i o n , near absence o f t h e f l e x i b l e appendage v i b r a t i o n r e s u l t s i n p i t c h r e s p o n s e t h a t i s i d e n t i c a l t o t h e r i g i d c a s e . However, d u r i n g deployment, s l i g h t v i b r a t i o n o f t h e f l e x i b l e members i n t h e e a r l y s t a g e does b r i n g about a n o t i c e a b l e d i f f e r e n c e i n t h e r e s u l t i n g p i t c h r e s p o n s e . T h i s i s analogous t o t h e b e h a v i o u r o b s e r v e d e a r l i e r d u r i n g t h e p l a n a r m o t i o n ( F i g u r e s 5-2, 5-3). J u s t how s t r o n g c o u p l i n g e f f e c t s can be i s demonstrated by a p p l y i n g an 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 o f H'(O) = 6 t o t h e r o l l degree o f freedom o n l y , F i g u r e ( 6 - 3 ) . Large a m p l i t u d e s r e s u l t i n b o t h yaw and p i t c h as w e l l as f o r v i b r a t i o n s . I n f a c t , t h e o v e r a l l m o t i o n becomes u n s t a b l e w i t h i n h a l f an o r b i t . T h i s i s i n marked 110 R C = 12,378 km , e--0 , | ( 0 ) = e E J 3 3 = 7 .85Nm 2 , 9=0-023024 kg/m -I 1 1 = .,133 = 1535kgm 2 , ^I22 = 1 8 k g m 2 . A JKOI.m 100 100 0 0 0.05 r (,m/s L , m * 100 RIGID 100 FLEXIBLE * 0-2 100 RIGID 0.2 100 FLEXIBLE •— F i g u r e 6-2 T h r e e - a x i s response o f a s a t e l l i t e t o an i m p u l s i v e p i t c h d i s t u r b a n c e . R c= 12,378 km . e = 0 , L = 100m • E J 3 3 = 7.85 Nm 2 . 9 = 0-023024 kg/m • \jr ( 0) = e • ^ - 5i $ 1*11 = i l 3 3 = 1 5 3 5 k g m 2. 1I 2 2 = I8kgm' 111 0.10 -I 0.05 0 I] A-0.05 -0-10 ORBITS F i g u r e 6 -3 T h r e e - a x i s r e s p o n s e o f a s a t e l l i t e w i t h f u l l y d e p l o y e d appendages t o an i m p u l s i v e o u t - o f - p l a n e d i s t u r b a n c e : (a) r i g i d booms; (b) f l e x i b l e booms. 112 contrast to the stable response•associated with the planar i n i t i a l condition of the same magnitude. Note also the s i g n i f i c a n t e f f e c t of deployment on the nature of the coupled response (Figure 6-4). Large displacements are also experienced i n t h i s case within less than 0.5 o r b i t . F l e x i b i l i t y , however, has minimal e f f e c t except near the point of i n s t a b i l i t y where i t a l t e r s p i t c h response quite dramatically. Having considered two very d i f f e r e n t types of attitude d i s t u r -bances, the next l o g i c a l step was to assess system s e n s i t i v i t y to a given disturbance. To t h i s end the system was subjected to a set of three impulsive r o l l v e l o c i t i e s of increasing magnitude (Figure 6-5). Note the strong coupling e f f e c t s continue to p e r s i s t even i n the presence of a small disturbance. The larger the r o l l rate, the e a r l i e r the i n s t a b i l i t y sets i n . The results also suggest that large displacements i n l i b r a t i o n a l and v i b r a t i o n a l degrees of free-dom are cl o s e l y related. Boom response to an i n i t i a l t i p displacement equal to 5% of the length i s displayed i n Figure 6-6 by a plot of the generalized coordinate associated with the f i r s t admissible function and the corresponding p i t c h l i b r a t i o n . Symmetric i n i t i a l displacements of the booms produce no pitching while antisymmetric i n i t i a l conditions r e s u l t i n a peak p i t c h « 8°. Disturbing only one boom i n i t i a l l y y i e l d s l i b r a t i o n s less than 5°. Note a considerable difference i n frequency between the v i b r a t i o n a l response for the symmetric case as opposed to the other two situations. Such high frequeney behaviour i s eliminated during symmetric o s c i l l a t i o n since p i t c h i t s e l f i s not excited. ^ R c = 12,378 km . e = 0 . L = 100m • E J 3 3 » 7.85 N m 2 . 9= 0023024 kg/m • 4^ (0) = G • ^ 1 A $ 1^ 11 = i l 3 3 = 1 5 3 5 k g m 2, 1I 2 2 =18 kg m ' 4(0) = 0- JL = 0.2 m/s 113 -20 -40 Q2 0 3 O R B I T S 0.10 A 0.05 0 I] 4-0.05 -010 0.5 F i g u r e 6-4 E f f e c t o f boom deployment on t h e t h r e e - a x i s r e s p o n s e o f a s a t e l l i t e t o an i m p u l s i v e o u t - o f - p l a n e d i s t u r b a n c e : (a) r i g i d booms; (b) f l e x i b l e booms. 114 R c = 12,378 km, e = 0, L = 100m , 9/EJ= 0 00293 s 2 m " 4 1*11 = i l 3 3 = 1535kgm2 ,^22 = 18kgm2 *(0) = O) 0.019. (b) 0.10 9, (c) 9 • A -20 Y -40 -0.10 0 0.1 0.2 0.3 0.4 0.5 O R B I T S F i g u r e 6-5 E f f e c t o f magnitude o f an i m p u l s i v e o u t - o f - p l a n e d i s t u r b a n c e on t h r e e - a x i s r e s p o n s e . R c = 12,378 km , e=0 , L = 100m • 1*11 = l l 3 3 = 1 5 3 5 k g m 2 , , I 2 2 = 18kgm 2 , 0 / E J = 0-00293 s 2 m - 4 -F i g u r e 6-6 P l a n a r r e s p o n s e o f t h e g r a v i t y g r a d i e n t c o n f i g u r a t i o n t o d i f f e r e n t i n i t i a l e l a s t i c d e f o r m a t i o n s . 116 Figure 6-7 describes three-dimensional l i b r a t i o n a l response when t i p of the appendage i s displaced (in the o r b i t a l plane) by an amount equal to one per cent of i t s length. As expected, the case of symmetric appendage disturbance c l o s e l y resembles the planar response data given e a r l i e r i n Figure 6-6(c). There i s , however, approximately one degree of yaw apparent aft e r half an o r b i t . This i s i n contrast to the antisymmetric case where both r o l l and yaw remain unexcited [Figure 6-7(b)]. Pitch responds i n a manner analo-gous to that i n the planar case except that now the peak amplitude i s only around 1.5°. However, a s t r i k i n g l y dramatic e f f e c t of coupling i s revealed when the system i s subjected to a disturbance i n the form of t i p displacement of one of the booms [Figure 6-7(a)]. I n i t i a l l y , up to around a quarter of an o r b i t , only a small amplitude p i t c h l i b r a t i o n a l motion i s unexcited. However, subsequently both yaw and r o l l appear, grow i n magnitude monotonically and i n turn cause large amplitude v i b r a t i o n d r i v i n g the system unstable within half an Orbit! This i s i n marked contrast to the apparently stable behaviour i n the planar case, even with more severe i n i t i a l conditions, as given i n Figure 6-6(a). This emphasizes significance of coupling e f f e c t s i n a study of the class of spacecraft with f l e x i b l e appendages. Although not shown here, r e s u l t s were also obtained to assess ef f e c t s of several other parameters on dynamics of the two boom gravity gradient configuration free to undergo three-axis l i b r a t i o n s . The use of higher modes to represent appendage v i b r a t i o n showed only minor difference i n amplitude without a f f e c t i n g general character of the response. S i m i l a r l y , the e f f e c t of s h i f t i n g center of mass, o f f - s e t of the appendage attachment, and the appendage foreshortening R c = 12,378 k m . e = 0 . L = 1 0 0 m A $ 1*11 z i l 3 3 = 1 5 3 5 k g m 2 . ^l22 = 1 8 k g m , o / E J = 0 . 0 0 2 9 3 s 2 m" 4 . 0 (a) A 0 0 (C) / / / ) k 1 1 1 1 0 0.1 0.2 0.3 ORBITS 0.4 0.5 -0.05 0 0.2 0.3 ORBITS 0.5 F i g u r e 6-7 T h r e e - a x i s response of the g r a v i t y g r a d i e n t c o n f i g u r a t i o n to d i f f e r e n t i n i t i a l e l a s t i c deformations. 118 during transverse o s c i l l a t i o n s was found to be n e g l i g i b l e on l i b r a -t i o n a l response (amplitude change less than 5%). More noticeable was the s h i f t i n the phase which was also present during deployment of the appendage. Also, as found i n Chapter 5, deployment can a f f e c t the system response sub s t a n t i a l l y and under c e r t a i n c r i t i c a l combin-ation of parameters can drive i t unstable. 6.2.2 Four-boom s p i n - s t a b i l i z e d configuration In addition to the g r a v i t y - s t a b i l i z e d concept, another e q u i l -ibrium orientation involves a s a t e l l i t e spinning at a rate much greater than the o r b i t a l rate, with the axis of spin normal to the o r b i t a l plane. Using coordinates as defined i n Figure 2-2, the x, y body-fixed axes l i e i n the spin plane ( o r b i t a l plane). This section studies dynamics of a system having four mutually orthogonal, f l e x i b l e , deploying, uniform beam-type appendages numbered 1 through 4 l y i n g i n the spin plane (Figure 6-1). O r b i t a l c h a r a c t e r i s t i c s together with boom properties p, EJ^^ are the same as i n the gravity-s t a b i l i z e d case. The length of each pair of diametrically opposed booms i s sim i l a r to that of the Allouette II s a t e l l i t e . Presented i n Figure 6-8 i s the three-axis attitude response of the system ( i n i t i a l l y spinning at 0.1 rad s ^) during deployment of appendages at 0.10 ms Although a l l booms have the same sta r t i n g length and deploy at the same rate, booms numbered 2 and 4 stop deploying at 10 m whereas 1 and 3 extend to 35 m. Results for r i g i d appendages are also included for comparison. Despin of the pi t c h degree of freedom i s according to the conservation of angular momentum. The configuration i s highly stable with the p i t c h rate attaining a constant value following deployment, and there i s no -1 $(0) =0.10 r a d s 9/EJ = 0-00293 S 2 m " 4 . i l j j = 1 8 k g ™ 2 ; 1 = 1.2.3. i 4>j = 0 , | , i r f | r ; j=l,--,4- 1 1.(0)= 5 i , j,-, =010 ms"1-L 1 ,L 3= 35 m t L 2,L 4=10 m rigid j <|>J - f lexible' <j> 1 1 9 20 -20" -^ , A r a d s 0.05 lot. -0.0 5 0.02 0.03 0.04 0.05 $ R B I J S ^ F i g u r e 6-8 T h r e e - a x i s response of a s p i n n i n g s p a c e c r a f t d u r i n g deployment of r i g i d or f l e x i b l e appendages. 120 out-of-plane l i b r a t i o n s . Note the e f f e c t of f l e x i b i l i t y i s essen-t i a l l y n e g l i g i b l e . This i s consistent with the low l e v e l of vib r a -tions. In fact, the appendage v i b r a t i o n i s v i r t u a l l y absent u n t i l the f i r s t set of booms stop deploying. Even afte r 0.05 o r b i t (685 s) o s c i l l a t i o n s at the t i p stay much less than 1% of the boom length. Displacing boom 1 [4>^  = 0,£^(0) = 5m] by 0.25 m at the t i p i n the spin plane at the s t a r t of deployment s t i l l f a i l s to excite any roll/yaw motion (Figure 6-9). However, considerable i n t e r a c t i o n between the p i t c h and the f l e x i b l e appendages leads to high frequency modulation of the pit c h rate, a r e s u l t similar to that observed i n the gravity gradient case. Figure 6-10 presents response of the system to an impulsive roll/yaw disturbance equal to 10% of the nominal i n i t i a l spin rate. Large amplitude displacements r e s u l t leading to tumbling motion i n less than eleven minutes. Furthermore, not only the p i t c h rate but also the yaw rate decreases s i g n i f i c a n t l y . On the other hand, the r o l l rate appears to grow. Note that the strong r o l l coupling effects experienced i n the gravity gradient case are not dominant here. Also the appendage o s c i l l a t i o n s are minimal. 6.2.3 CTS-type configuration A completely d i f f e r e n t class of s a t e l l i t e s i s represented by the CTS-type configuration b r i e f l y referred to i n Chapter 1. It i s characterized by two f l e x i b l e appendages (numbered 5 and 6, Figure 6-1) and a momentum wheel perpendicular to the o r b i t a l plane. The general formulation of Chapter 2 i s re a d i l y adapted to t h i s configur-ation as well by simply adding the momentum wheel e f f e c t to the {h} vector. $10) = 0.10 rad s"1-9/EJ= 0 00293 s 2 m ' 4 . ^ j j = 18 kg m 2 . j = 1,2,3-4>i =0 ,T | , i r , | r . i = i , - , 4 -§1(0) = 0.05 1 i(0) = 5 m l JL = 010 m s " 1 . L 1 T L 3 = 35 m f L 2,L 4=10 m 121 20( -20* /• _ _ 0.05 -0.0 5 0.02 0.03 0.04 0.05 O R B I T S F i g u r e 6-9 T h r e e - a x i s response o f a s p i n n i n g s p a c e c r a f t d u r i n g deployment o f f l e x i b l e appendages w i t h one boom i n i t i a l l y deformed. $(0) =0.10 rad s -1 9 -4 9/EJ= 0 00293 s^m • 18 kg m 2 . j = 1,2,3 "$"(0)= A(0) = 0.01 rad s r1 11]] = •i = 1|(0) = L1.L3-0 , H , T r , 3 7 T . i = 1, • *f4 2 2 ' 5 m , J^. =0-10 ms 35 m , L 2,L 4=10 m. -1 A 5 122 20 OP -20c -40 0.10 \ % 1 i \ , \ \Jy X x x. ' x X. % x x X \ X x > x X X V X *X 0.05 rad s 0 1 -0.05 0.01 0.02 0.03 ORBITS 0.04 V 1 \ » _ \ -\ — 1 1 0.05 0 §. -0.05 0.05 F i g u r e 6-10 Three - a x i s response of a s p i n n i n g s p a c e c r a f t w i t h f l e x i b l e d e p l o y i n g appendages when su b j e c t e d t o out-of-plane a t t i t u d e d i s t u r b a n c e s . 123 Some representative response data i s given by Figures 6-11, 6-12. The e f f e c t of including a stored momentum and of f l e x i b i l i t y , i n the absence of deployment, i s emphasized i n Figure 6-11. As expected, the additional momentum has a s t a b i l i z i n g e f f e c t [compare Figures 6-11 (a), (b)]. On the other hand, f l e x i b i l i t y tends to make the system unstable [Figure 6-11(c)]. Deployment ef f e c t s are i l l u s t r a t e d by Figure 6-12. Note a marked difference i n the p i t c h behaviour due to in t e r a c t i o n with the boom vibrations. Furthermore, the i n s t a b i l i t y appears to set in somewhat e a r l i e r compared to the r i g i d case. 6.2.4 Asymmetric deployment of appendages Also of i n t e r e s t i s the case of asymmetry introduced by the appendages. This could occur i n the event of f a i l u r e of a boom to deploy f u l l y or i f the fully-deployed configuration of the o v e r a l l spacecraft i s i t s e l f asymmetric (e.g. Pioneer IV). An equivalent e f f e c t would be present during modular construction of very large space structures such as the SPS. Also, asymmetric deployment has 17 9 2 08 been proposed as a useful means of attitude control. ' Figures 6-13 and 6-14 compare response of r i g i d and f l e x i b l e asymmetric configurations to planar e x c i t a t i o n for the two-boom gravity gradient s a t e l l i t e studied i n sections 5.3 and 6.2.1. Figure 6-14 involves appendage deployment from 0-100 m for the boom aligned along the outward-pointing v e r t i c a l , and from 0-50 m for the second boom located 18 0 degrees with respect to the f i r s t . Corresponding performance with the appendage length fixed at 100 m and 50 m respectively, i s also included, (Figure 6-13). It i s CTS-TYPE CONFIGURATION , A(0) = 0 ^ R c= 12,378 km , e = 0 , L = 100m . TJT A $ 2 A •jl-l-l = 120 , -,122 = 110 > -1I33 = 85 kgm , g/EJ = 0 00293 s2nrT*. (a) RIGID, h Q = 0 ; (b) RIGID, hQ = 20 ; (c) FLEXIBLE, h = 20 N ms-o , S o , S 0 , 0 4 0 ° i 7 (a) / 0 /-~\ / 2 0 " \ / / 0 ^ — y \ - 2 0 ° • \ 0 \ " 4 0 0 0.1 0 . 2 0 . 3 0 . 4 0 0.1 0 . 2 0 . 3 0 . 4 0 . 5 0 0.1 0 . 2 0 . 3 0 . 4 0 . 5 O R B I T S O R B I T S O R B I T S F i g u r e 6-11 E f f e c t o f s t o r e d momentum and f l e x i b i l i t y f o r : (a) r i g i d appendages, no momentum wheel; (b) r i g i d appendages w i t h added momentum; (c) f l e x i b l e appendages w i t h added momentum. 125 CTS-TYPE CONFIGURATION , A(0) = S • R c= 12,378 k m . e = 0 , L = 100m . ^ 9/EJ= 0 00293 S 2 m ' 4 . ^ A <£ = 120 , -,I22 = 110 , - , I 3 3 = 85 kgm2. 1(0) - 50 m , I - 0-10 ms'1 , =20 N ms-F i g u r e 6-12 Response o f a CTS-type s p a c e c r a f t t o an i n i t i a l yaw r a t e d i s t u r b a n c e d u r i n g deployment o f : (a) r i g i d booms; and (b) f l e x i b l e booms. 126 R c = 12,378 km , e = o , j ( 0 ) = e E J 3 3 = 7.85 Nm 2 , 9=0023024 kg/m • -jIn = 1I33 = 1535kgm 2 , ^I22 = 18kgm 2 = 0 , v 2 = x . L 1 = 100 m , L 2 = 50 m A • — $ $1 ^ . A i 0.10 A 0.05 -0.05 J-0.10 0.2 03 O R B I T S 04 0-5 F i g u r e 6-13 T h r e e - a x i s r e sponse o f a two-boom g r a v i t y g r a d i e n t s a t e l l i t e w i t h a s y m m e t r i c a l l y d e p l o y e d appendages: (a) r i g i d booms; (b) f l e x i b l e booms. R c = 12,378 km , e = 0 , § { 0 ) Z Q E J 3 3 = 7.85 Nm 2 , 9=0023024 kg /m 1*11 = 1I33 = 1535kgm 2 , ^I22 = 18kgm 2 ^ = 0 . <P2 = x • h = 1 0 0 m, |_2= 50 m ^(0)= f2(0) = 1 m , ^ = f 2 = 0-20 ms"1-A $ 127 0.10 -I 0.05 - 2 0 ° h -40 ' 0 § 4 0.05 0-10 0.2 0.3 O R B I T S F i g u r e 6-14 E f f e c t o f asymmetric boom deployment on t h r e e - a x i s r e s ponse o f a two-boom g r a d i e n t s a t e l l i t e : (a) r i g i d booms; (b) f l e x i b l e booms. 128 demonstrated that f l e x i b i l i t y can r e s u l t i n a p i t c h response of up to 10 degrees larger for a nondeploying asymmetric configuration as opposed to the symmetric case (Figure 6-13). Also, a weak coupling of the roll/yaw motions becomes apparent a f t e r 0.5 o r b i t s . Com-parisons based on f l e x i b l e deploying booms are even more dramatic. The asymmetric condition (Figure 6-14) produces s i g n i f i c a n t vibra-tions which eventually become unstable as a r e s u l t of the large amplitude roll/yaw behaviour induced by coupling e f f e c t s . 6.3 Concluding Remarks Examination of the governing system equations together with some t y p i c a l three dimensional simulations presented i n t h i s chapter leads to the following conclusions: (i) The ease with which such diverse classes of s a t e l l i t e con-figurations have been simulated demonstrates the v e r s a t i l i t y of the general formulation. ( i i ) Coupled character of the motion s i g n i f i c a n t l y a f f e c t s the system dynamics, hence caution should be exercised i n u t i l i z -ing r e s u l t s based on the planar analysis. ( i i i ) S i g n i f i c a n t s i m p l i f i c a t i o n i n the equations can occur with appendages having a s p e c i f i c o rientation or, i f one can ignore such factors as appendage o f f s e t , foreshortening, s h i f t s i n the center of mass location, f l e x i b i l i t y , deployment, or higher modes used i n the assumed v i b r a t i o n solution. Elimin-ation of even one of these parameters such as TT e f f e c t s con-siderable savings i n algebra with associated reduction i n com-putational time and e f f o r t . 1 2 9 ( i v ) P i t c h and a s s o c i a t e d i n - p l a n e v i b r a t i o n s do not e x c i t e r o l l / yaw degrees o f freedom. On t h e o t h e r hand, a r o l l d i s t u r b a n c e can e x c i t e t h e y a w / p i t c h m o t i o n . (v) S t a b l e l i b r a t i o n s do not e x c i t e s i g n i f i c a n t appendage m o t i o n whereas i n i t i a l boom d i s p l a c e m e n t s can r e s u l t i n v e r y n o t i c e -a b l e changes i n a t t i t u d e . ( v i ) I n t e r a c t i o n between f l e x i b i l i t y and l i b r a t i o n l e a d s t o an i n c r e a s e i n t h e f r e q u e n c y o f appendage o s c i l l a t i o n t o g e t h e r w i t h a h i g h f r e q u e n c y m o d u l a t i o n o f t h e a t t i t u d e r e s p o n s e . ( v i i ) The s m a l l a m p l i t u d e o s c i l l a t i o n s e v i d e n t b o t h w i t h t h e g r a v i t y g r a d i e n t and s p i n - s t a b i l i z e d r e s p o n s e j u s t i f y a l i n e a r v i b r a t i o n a n a l y s i s . ( v i i i ) There a r e c o m b i n a t i o n s o f f l e x i b i l i t y , deployment, and i n i t i a l c o n d i t i o n s f o r w h i c h a s a t e l l i t e can tumble o v e r . ( i x ) F l e x i b i l i t y c o n s i d e r a t i o n s can be p a r t i c u l a r l y s i g n i f i c a n t i n t h e s t u d y o f asymmetric deployment as i t can g r e a t l y i n c r e a s e t h e magnitude o f a t t i t u d e r e s p o n s e and t h e degree o f c o u p l i n g t o t h e p o i n t o f c a u s i n g t u m b l i n g . 130 7. CLOSING COMMENTS O v e r a l l , t h e t h e s i s p r e s e n t s a u n i f i e d p r o c e d u r e , based on t h e methods o f a n a l y t i c a l dynamics, f o r d e r i v i n g and s o l v i n g system e q u a t i o n s g o v e r n i n g g e n e r a l s p a c e c r a f t l i b r a t i o n a l m o t i o n , w h i c h i n c l u d e s t h e e f f e c t s o f f l e x i b i l i t y and deployment. A p p l i c a t i o n o f t h e method i s i l l u s t r a t e d t h r o u g h a g e n e r a l i z e d c o n f i g u r a t i o n r e p r e s e n t a t i v e o f an i m p o r t a n t c l a s s o f problems. T h i s i s p a r t i c u l -a r l y h e l p f u l t o a d e s i g n e n g i n e e r as t h e r e i s no need t o c o n t i n u a l l y r e d e r i v e a complete s e t o f e q u a t i o n s f o r each new s p a c e c r a f t . R a t h e r t h a n t h e a c c u m u l a t i o n o f a l a r g e amount o f d a t a , t h e emphasis i s on e v o l u t i o n o f a g e n e r a l i z e d and o r g a n i z e d methodology f o r cop-i n g w i t h such complex n o n l i n e a r , nonautonomous, and c o u p l e d d y n a m i c a l systems. E f f e c t i v e n e s s o f t h e approach i s i l l u s t r a t e d t h r o u g h an e x t e n s i v e r e s ponse e v a l u a t i o n o f g r a v i t y g r a d i e n t , s p i n - s t a b i l i z e d and CTS-type c o n f i g u r a t i o n s . I m p o r t a n t f e a t u r e s o f t h e f o r m u l a t i o n p r o c e d u r e and c o n c l u s i o n s based on t h e r e s ponse r e s u l t s a r e p r e -s e n t e d i n the f o l l o w i n g s e c t i o n s . 7.1 On F o r m u l a t i n g System E q u a t i o n s o f M o t i o n A l t h o u g h s e v e r a l s t u d i e s have been c a r r i e d out on t h e formu-l a t i o n a l o n e f o r f l e x i b l e s a t e l l i t e a t t i t u d e dynamics, none has a t t a c k e d t h e problem t o t h i s degree of g e n e r a l i t y . I t s h o u l d be emphasized t h a t t h e momentum f o r m u l a t i o n o f C h a p t e r 2 need not be r e s t r i c t e d t o t h e s t u d y o f s a t e l l i t e dynamics a l o n e , b u t i s a g e n e r a l r e s u l t o f p a r t i c u l a r v a l u e i n a n a l y z i n g any complex r o t a t i n g system. Note a l s o t h a t t h e a t t i t u d e e q u a t i o n s u l t i m a t e l y i n v o l v e o n l y t h e 131 generalized coordinates associated with the appendage degrees of freedom. Consequently, for any s a t e l l i t e configuration, governing l i b r a t i o n a l equations remain the same. However, the vi b r a t i o n equations change to r e f l e c t the character of the appendage. Develop-ing appendage equations i n terms of ' l o c a l ' coordinates means that they can be analyzed d i r e c t l y , the point emphasized also by 135 118 Laurenson (1976) and Gupta (1978). Overall, the formulation related to appendage vibrations presented i n t h i s thesis represents a s i g n i f i c a n t extension to the Euler-Bernoulli beam theory. In the analysis of such involved systems the key element which makes a solution f e a s i b l e i s adoption of a continuum representation i n conjunction with an assumed-mode solution. Note, i n general, any approximate shape functions s a t i s f y i n g geometric boundary con-d i t i o n s can be used, including ones found by means of a f i n i t e element method. 7.2 Cha r a c t e r i s t i c s Associated with a Deploying, Orbiting, Spinning Beam-Type Appendage The dominant influences on system eigenvalues for most p r a c t i c a l applications are the spin parameter and appendage length. Changes in spin rate i t s e l f however do not a f f e c t the natural v i b r a t i o n c h a r a c t e r i s t i c s but act as an additional external boom loading. Also of importance i s the fact that resonance can occur between the spin degree of freedom and in-plane o s c i l l a t i o n s . Deployment rate and changes i n deployment rate a l t e r e f f e c t i v e s t i f f n e s s of the boom. In addition, deployment rate introduces a term into the equations which can be viewed as a negative damping. 132 S p e c i f i c a l l y , d e p l o y m e n t - r e l a t e d C o r i o l i s f o r c e s can r e s u l t i n l a r g e a m p l i t u d e d e f o r m a t i o n s i n t h e s p i n p l a n e and hence must be c o n s i d -e r e d when a r r i v i n g a t any deployment s t r a t e g y . O u t - o f - p l a n e e i g e n f r e q u e n c i e s become s h i f t e d by an amount p r o p o r t i o n a l t o t h e s p i n parameter. T h i s r e s u l t i s m o d i f i e d f o r an o r b i t i n g beam s i n c e , i n g e n e r a l , t h e c h a r a c t e r i s t i c s o f an o r b i t -i n g beam can v a r y s i g n i f i c a n t l y from t h o s e f o r a beam r o t a t i n g i n t h e absence o f t h e g r a v i t y g r a d i e n t i n f l u e n c e . 7 . 3 O v e r a l l System Response R e s u l t s suggest t h a t l a r g e a m p l i t u d e a t t i t u d e b e h a v i o u r can o c c u r s i m u l t a n e o u s l y w i t h s m a l l a m p l i t u d e o s c i l l a t i o n s t h u s j u s t i -f y i n g a l i n e a r v i b r a t i o n a n a l y s i s i n such c a s e s . The p l a n a r s t u d y c a r r i e d out i n C h a p t e r 5 d e m o n s t r a t e s t h a t l a r g e a m p l i t u d e p i t c h m o t i o n does not n e c e s s a r i l y e x c i t e t h e appen-dage v i b r a t i o n s ; t h e r e v e r s e , however, i s n o t t r u e . I n f a c t , once t h e appendages a r e e x c i t e d , s i g n i f i c a n t c o u p l i n g between v i b r a t i o n a l and l i b r a t i o n a l m o t i o n r e s u l t s i n a h i g h f r e q u e n c y m o d u l a t i o n of the p i t c h r e s ponse t o g e t h e r w i t h a s h i f t i n t h e e x p e c t e d f r e q u e n c y o f t h e appendage v i b r a t i o n . I n g e n e r a l , p l a n a r m o t ions do n o t cause o u t - o f - p l a n e r o t a t i o n s whereas, f o r example, a r o l l d i s t u r b a n c e r e s u l t s i n a t h r e e - a x i s a t t i t u d e r e s p o n s e . Hence c a r e must be t a k e n when i n t e r p r e t i n g d a t a based on a p l a n a r a n a l y s i s o n l y s i n c e t h e c o u p l i n g e f f e c t s can become q u i t e s i g n i f i c a n t , t o t h e p o i n t o f domin-a t i n g t h e r e s p o n s e . T h r e e - a x i s a n a l y s i s s u g g e s t s t h a t , i n g e n e r a l , parameters such as s h i f t i n g c e n t e r o f mass, appendage o f f s e t , and t h e use o f a l a r g e r 133 number o f modes ( a l l o f w h i c h s u b s t a n t i a l l y c o m p l i c a t e the formu-l a t i o n ) have l i t t l e e f f e c t on t h e magnitude o f t h e r e s p o n s e . Hence one can n e g l e c t them, a t l e a s t d u r i n g t h e p r e l i m i n a r y d e s i g n s t a g e , w i t h c o n s i d e r a b l e s a v i n g i n c o m p u t a t i o n a l t ime and e f f o r t . A l s o , i t was found t h a t l i b r a t i o n d u r i n g asymmetric deployment o f appen-dages can become u n s t a b l e when f l e x i b i l i t y i s t a k e n i n t o a c c o u n t , even though no such i n s t a b i l i t y may e x i s t f o r the r i g i d c o n f i g u r a -t i o n . Depending on o r b i t a l p a rameters and p h y s i c a l p r o p e r t i e s o f booms, r e s u l t s show t h a t t h e r e e x i s t c r i t i c a l v a l u e s o f appendage l e n g t h and deployment r a t e f o r w h i c h a s a t e l l i t e can tumble o v e r . A p p l i c a t i o n o f t h e a n a l y s i s t o t h r e e d i s t i n c t l y d i f f e r e n t c l a s s e s o f s a t e l l i t e s ( g r a v i t y g r a d i e n t , s p i n - s t a b i l i z e d , CTS-type) demonstrates th e v e r s a t i l i t y o f t h e g e n e r a l f o r m u l a t i o n . I t i s o f i n t e r e s t t o note t h a t t h e s i z e , speed, and a c c u r a c y o f t h e modern day d i g i t a l computer p e r m i t s d y n a m i c a l s i m u l a t i o n s w h i c h would not have been p o s s i b l e a decade e a r l i e r . 7 . 4 Recommendations f o r F u t u r e Work ( i ) A comprehensive n u m e r i c a l i n v e s t i g a t i o n c o u l d be c a r r i e d out v a r y i n g i n i t i a l c o n d i t i o n s and major s p a c e c r a f t c h a r a c t e r i s -t i c s i n a s y s t e m a t i c manner. U s i n g such a m a t r i x o f con-d i t i o n s one c o u l d g e n e r a t e a parameter map i d e n t i f y i n g e s s e n t i a l r e s ponse f e a t u r e s f o r a f a m i l y o f c o n f i g u r a t i o n s , e.g., g r a v i t y g r a d i e n t , s p i n n i n g , CTS and RAE c l a s s e s o f s a t e l l i t e s , e t c . 134 ( i i ) The f o r c e d r e s ponse o f t h e system can be r e a d i l y examined by i n t r o d u c i n g t h e a p p r o p r i a t e g e n e r a l i z e d f o r c e s . Of p r i m a r y c o n c e r n would be t h e e f f e c t o f e n v i r o n m e n t a l f o r c e s such as t h o s e r e s u l t i n g from s o l a r r a d i a t i o n p r e s s u r e , aerodynamic and, p e r h a p s , c o n t r o l f o r c e s . ( i i i ) G o v e r n i n g e q u a t i o n s can be f o r m u l a t e d f o r a d d i t i o n a l t y p e s of appendages. ( i v ) A s i g n i f i c a n t e x t e n s i o n t o t h e m o d e l l i n g o f appendages would be t o r e l a x c o n s t r a i n t s a t t h e r o o t so as t o a l l o w f o r c a n t i -l e v e r e d and/or h i n g e d c o n n e c t i o n s w i t h / w i t h o u t s t i f f n e s s and d i s c r e t e damping. S t r u c t u r a l damping c o u l d a l s o be a l l o w e d f o r . I n a d d i t i o n , t h e a n a l y s i s c o u l d be extended t o i n c l u d e appendages u n d e r g o i n g a c o n t r o l l e d v a r i a t i o n i n o r i e n t a t i o n w i t h t i m e - a f e a t u r e o f p o t e n t i a l m i l i t a r y i m p o r t a n c e . (v) An i n t e r e s t i n g s t u d y would be t o c r e a t e a computer-generated v i s u a l i z a t i o n o f t h e s p a c e c r a f t r e s p o n s e . F o r i n s t a n c e , one might w i s h t o l o o k a t a s i m u l a t e d d y n a m i c a l h i s t o r y s t a r t i n g w i t h t h e i n i t i a l u ndeployed h i g h - s p i n s t a t e a t o r b i t a l i n -j e c t i o n , t h r o u g h th e a t t i t u d e a c q u i s i t i o n phase i n v o l v i n g deployment o f appendages, t o t h e s t e a d y s t a t e e q u i l i b r i u m c o n d i t i o n . A l s o , i t c o u l d prove r e w a r d i n g t o v i s u a l i z e t h e e f f e c t o f a t t i t u d e c o n t r o l manoeuvres made u s i n g e x t e n s i o n and r e t r a c t i o n o f the appendages. ( v i ) Having such a g e n e r a l f o r m u l a t i o n a t hand, t h e n e x t l o g i c a l s t e p t o f u l l y e x p l o i t i t s p o t e n t i a l would be t o d e v i s e an a p p r o p r i a t e c o n t r o l s t r a t e g y and seek t h e o p t i m a l d e s i g n . 135 BIBLIOGRAPHY 1. N o l l , R.B. et a l . 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Guidance and Control. 1 (2), 109-116. 172. Hughes, P.C. (1976) Passive damper analysis for reducing attitude c o n t r o l / f l e x i b i l i t y i n t e r a c t i o n . J. Spacecraft and  Rockets, 13(5), 271-274. 173. Hughes, P.C. (1972) F l e x i b i l i t y considerations for the p i t c h attitude control of the Communications Technology S a t e l l i t e . Trans. Canadian Aeronautics and Space I n s t i t u t e , 5(1). 174. Hughes, P.C. (1973) Dynamics of f l e x i b l e space vehicles with active attitude control. C e l e s t i a l Mech. 9, 21-39. 175. Zach., C. (1970) Time-optimal control of gravity-gradient s a t e l l i t e s with disturbances. J. Spacecraft and Rockets, 7 (12), 1434-1440. 176. Tseng, G.T. and Mahn, J r . , R.H. (1978) F l e x i b l e spacecraft control design using pole a l l o c a t i o n technique. AIAA J.  Guidance and Control, 1(4), 279-281. 177. M i l l a r , R.A. and Vigneron, F.R. (1979) Attitude s t a b i l i t y of a pseudorate j e t - c o n t r o l l e d f l e x i b l e spacecraft. AIAA J.  Guidance and Control, 2(2), 111-118. 178. Hughes, P.C. and Abdel-Rahman, T.M. (1979) S t a b i l i t y of pro-port i o n a l - p l u s - d e r i v a t i v e - p l u s - i n t e g r a l control of f l e x i b l e spacecraft. AIAA J. Guidance and Control, 2(6), 499-503. 148 179. Gat l i n , J.A. et a l . (1969) S a t e l l i t e attitude control using a torqued, 2-axis-gimbaled boom as the actuator. J. Space- c r a f t and Rockets, 6(9), 1013-1018. 180. % H i l l a r d , S.E. (1976) Attitude control of synchronous s a t e l -l i t e s possessing f l e x i b l e solar arrays using a double-gimbaled momentum wheel. Proc. ESA Symp. on Dynamics and Control of  Non-Rigid Spacecraft, ESA SP 117, 113-124. 181. Pande, K.C. et a l . (1974) Time-optimal p i t c h control of s a t e l l i t e s using solar r a d i a t i o n pressure. J. Spacecraft and  Rockets, 11(8), 601-603. 182. 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(1976) Accuracy improvement of a 3-axis s t a b i l i z a t i o n by the use of an onboard computer and the modern control theory. XXVIIth Congress International Astr- nautical Federation, Anaheim, C a l i f . Paper No. IAF-76-017. 199. Van Landingham, H.F. and Meirovitch, L. (1978) D i g i t a l con-t r o l of spinning f l e x i b l e spacecraft. AIAA J. Guidance and  Control, 1(5), 347-351. 200. De Bra, D.B. (1979) Control technology challenges for g r a v i t a t i o n a l physics experiments i n space. AIAA J. Guidance  and Control, 2(2), 147-151. 201. Skelton, R.E. and L i k i n s , P.W. (1978) Orthogonal f i l t e r s for model error compensation i n the control of nonrigid space-c r a f t . AIAA J. Guidance and Control, 1(1), 41-49. 202. Lang, W. and Honeycutt, G.H. (1967) Simulation of deploy-ment dynamics of spinning spacecraft. NASA-TN-D-4074. 203. Cl o u t i e r , G.J. (1968) Dynamics of deployment of extendible booms from spinning space vehicles. J. 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(1977) E f f e c t of gravity-gradient toruqes on the dynamics of a spinning spacecraft with t e l e -scoping appendages. To be presented at the AIAA Symp. on  Dynamics and Control of Large F l e x i b l e Spacecraft, Blacksburg, V i r g i n i a . 210. Bainum, P.M. and Sellappan, R. (1977) The use of a movable telescoping end mass system for the time-optimal control of spinning spacecraft. International Astronautical Federation  XXVIIIth Congress, Prague, Paper No. 77-227. 211. Sellappan, R. and Bainum, P.M. (1976) Dynamics of spin-s t a b i l i z e d spacecraft during deployment of telescopic appen-dages. J. Spacecraft and Rockets, 13(10), 605-610. 212. Cherchas, D.B. (1971) Dynamics of s p i n - s t a b i l i z e d s a t e l l i t e s during extension of long f l e x i b l e booms. J. Spacecraft and  Rockets, 8(7), 802-804. 213. Cherchas, D.B. and Gossain, D.M. (1974) Dynamics of a f l e x i b l e solar array during deployment from a spinning space-c r a f t . Trans. Canadian Aeronautics and Space In s t i t u t e , 7(1), 10-18. 214. Ebner, S.G. (1970) Deployment dynamics of rotating cable-connected space stations. J. Spacecraft and Rockets, 7(10), 1274-1275. 215. Stuiver, W. and Bainum, P.M. (1973) A study of planar de-ployment control and l i b r a t i o n damping of a Tethered Orbiting Interferometer s a t e l l i t e . J. Astronautical Sciences. XX(6), 321-346. 216. Modi, V.J. and Misra, A.K. (1978) Deployment dynamics and control of tethered s a t e l l i t e systems. AIAA/AAS Astro- dynamics Conf. Palo Alto, C a l i f . Paper No. 78-1398. 217. Misra, A.K. and Modi, V.J. (1979) Dynamics of a tether connected payload deploying from the space shuttle. 2nd VPI  & SU/AIAA Symp. On Dynamics and Control of Large F l e x i b l e  Spacecraft, Blacksburg, V i r g i n i a . June 21-23, 591-609. 151 218. Kane, T.R. and Levinson, D.A. (1977) Deployment of a cable supported payload from an o r b i t i n g spacecraft. J. Spacecraft  and Rockets, 14(7), 409-413. 219. Hughes, P.C. (1976) Deployment dynamics of the Communications Technology S a t e l l i t e - a progress report. Proc. ESA Symp. on  Dynamics and Control of Non-Rigid Spacecraft, ESA SP 117, 335-340. 220. Jankovic, M.S. (1980) Deployment dynamics of f l e x i b l e space-c r a f t . Ph.D. d i s s e r t a t i o n , University of Toronto I n s t i t u t e  for Aerospace Studies. 221. Leech, CM. (1970) The dynamics of beam under the influence of convecting i n e r t i a l forces. Ph.D. d i s s e r t a t i o n , University  of Toronto. 222. Tabarrok, B. et a l . (1974) On the dynamics of an a x i a l l y moving beam. J. Franklin I n s t i t u t e , 297(3), 201-220. 223. Jankovic, M.S. (1976) La t e r a l vibrations of an extending rod. University of Toronto I n s t i t u t e for Aerospace Studies, Tech. Note No. 202. 224. Lips, K.W. and Modi, V.J. (1978) Dynamical c h a r a c t e r i s t i c s associated with deploying, o r b i t i n g , beam-type appendages. AIAA/AAS Astrodynamics Conf. Palo Alto, C a l i f . Paper No. 78-1399. 225. Wittenburg, J. (1977) Dynamics of systems of r i g i d bodies. B.G. Teubner, Stuttgart, W. Germany. 226. England, F.E. (1969) A normal mode analysis of a s a t e l l i t e employing long f l e x i b l e booms. Ph.D. d i s s e r t a t i o n , University  of Maryland. 227. Almroth, B.D., et a l . (1978) Automatic choice of global shape functions i n s t r u c t u r a l analysis. AIAA J. 16(5) 525-528. 228. H e l l i w e l l , W.S. (1978) A fa s t i m p l i c i t i t e r a t i v e numerical method for solving multi-dimensional p a r t i a l d i f f e r e n t i a l equations. AIAA J. 16(7), 663-666. 229. Hurty, W.C and Rubinstein, M.F. (1964) Dynamics of structures. Prentice-Hall. 230. Meirovitch, L. (1967) A n a l y t i c a l methods i n vibrations. MacMillan Co. 231. Young, D. and Felgar, J r . , R.P. (1949) Tables of character-i s t i c functions representing normal modes of vi b r a t i o n of a beam. Bureau of Engineering Research, University of Texas, Publication No. 4913. 152 232. Felgar, R.P., J r . (1950) Formulas for inte g r a l s containing c h a r a c t e r i s t i c functions of a v i b r a t i n g beam. Bureau of  Engineering Research, University of Texas, Austin, Texas, C i r c u l a r No. 14. 233. Worden, D. (1979) Slamming motions of a rectangular-section barge model i n harmonic waves. M.A.Sc. d i s s e r t a t i o n , University of B r i t i s h Columbia. 234. Flanagan, R.C. (1969) E f f e c t of environmental forces on the attitude dynamics of gravity oriented systems. Ph.D. d i s s e r t a t i o n , University of B r i t i s h Columbia, p. 8. 235. Conte, S.D. (1965) Elementary numerical analysis. McGraw- H i l l . 236. Modi, V.J. and Misra, A.K. (1979) On the deployment dynamics of tether connected two-body systems. Acta Astronautica, 6(9), 1183-1197. 237. Garg, S.C, et a l . (1979) F l i g h t r e s u l t s on s t r u c t u r a l dynamics from Hermes. J. Spacecraft and Rockets, 16(2), 81-87. 153 APPENDIX I GENERAL EQUATIONS OF LIBRATION ON TRUE ANOMALY The e q u a t i o n s o f Chapter 2 can be e x p r e s s e d i n a somewhat more e l e g a n t manner i f one t a k e s advantage o f t h e K e p l e r r e l a t i o n s t o d e s c r i b e t h e o r b i t a l v a r i a b l e s w h i l e a l s o t a k i n g t h e o r b i t a l anomaly as a measure o f ti m e as i n Chapter 4. ^ D e g r e e o f Freedom ( R o l l ) CclA (s*i l„ * c'il„ - sz$Jlt * s*J I33 + **A OII,i t c$Ix3)JY"' * /c*A £s*2 {lg - ze, I„) * cl$ (In - Ze, Itt)J * sxA (In ~ z */ I33 ) * stA [s* ( I,j -2e,Il3) * c?(7*'$ - ze,Iis)J - sFsAcA <2sl2I„ -si}Itl * Zs'As?I/3) J ¥' + fstA (I33 - sl21„ - c*2fi2 * stilt) * ZctJL Csl 1,3 * ci Iti)JA' f + { clA Csti (1„-I»> t a- ctf)InJ * stA <rfl9 - s2 7^)} I'f * [ 3/'</•>• ecd)J/ f (s'As22- c'f)I„ t (sti czI - SZIJ In * C*AI„ - 0+sU)s2?Ia ~ 154 (siIa * dl^JsYcY * czYsALs}c$(I„ -T») -czIInJ - czYcA CcII/3 - siIZ3)J + cA[s$ (ft - Ze,h,) * cl (a - le,h,)J - sA(Q- Ze,h3) t {cA [aid U„-ln) - ci2l,z] + sA (cl 1,3 - si Iz3) ) A" * /sA Cslcl (Jtt -I„) + c*2 I,zl * cA (cill3 - sIJZ3)J(Af + /cA sic I [(1/-Ze,I„) - (Izi -2e,Itt) - cA czl (lft - Ze,Ilz) - cA h3 * sA icI(Tl3 - Ze,Jn - hz) - si (Iz3 - 2e,Iz3 * h,)J * cY(ciAs'I +clI)I„ + [cY(s*I * czA clI) - jYjAszIJJzz - cYctA I33 + [cYszi (1-ciA) - ZsYsAc'IJIn + ZcACdcY sAj! - sYei)Jls * (ZcYsAd + sVcDIzslJ A' t cAC ct$ (I„ -I21) + szi Ilt - J33 J 1'A' - [ sAJ3i - cA lslll3 + cl I13)J I" t cA (si J 2 3 - c§In)(2')Z 1- { -sA d/3 - 2e, I33) - cA [si <I,3 - Z'lln +ht) * c$(Il3 -ie, Izi-h,)] 155 •>• cA [sYIn + (cVsAszi - sYcz$) (I„-Iiz)] - ZcA (sfsii +c¥sAct})I,i * 2 (sYsAsS - c¥ cUcf)I,a * 2cYczAs$ Ii3J I' + { c¥sAcA fs*I(I,, - 2 e,I„) / cz$ (Izz - 2e,Izi) - C i > ? - Z '/ Its)J + sYcAsZd [ (Izz ' 2*Izz) - df, -2.e,I„)J ~ cA (c VsAsiZ -sYcti)(Ifz -ze,I,t) - (c¥eiJLs$ + s¥sAci)(I,3 ~ 2e,I,3) - (cYczA cl - sYsA si) (I{i - 2 e, li3) + (cYd -sY*A si) h, - (cYsi-s¥sAc€)hz + s¥cAh3 + sYcY[(s2Acli -Witt ~ CsM*1! * c'*)It t c*Al33 f etY SJLsIc* (Irlzi) - CsA(c*rci$ - ) + sYcYjiIJ 1,2 - czYcAd 1,2 - cA (sifsAd - CzYsDItlJ = &y (6) > ( I . l a ) 156 -A- Degree of Freedom (Yaw) (cli In * j*fltt * ati In) A" * /'c'l (In " 2 4 I„) * s*I(l£~Ze,Iu) + Ml (In - Zeil/i)] A' + [-sz$ (1,,-Izt) + 2ci§ In] 2'A! + C 3/(1 * ece)J {s'YsAcA * c'f I2Z-T33 - sz$ In ) - sYczA (sfI/3 *• c$Iz3) + WcYcA [jfc I (I,, -lit) -czl I,zJ * sVc VsA (cl I)3- siIzi)J f c$(f-2e,Jb) - s§ (Pz - ze, hz) - (ci Il3 - S$ IZ3 ) §" + C**I,3 * cfZtt)(2')*- + C - c$ (1,3 - 2. e, Il3 +hz ) * (lis - ze,Iz3 - h, ) + sActt * sYsz$)(I„ -!„) t cYsA 1st ~ 2 (sYczI - cfjJL*t?)In + ZcfcA (s$I,3 + cSIa)J £ y * { iALo-2 (J„ -Iu) +133 *szl IltJ - ZsA (tll,3 * c* Its ) J F'l' * { cA [ si d (I„ -Itt) - ai I/z7 * sA (c$I,}-sIIti)}Y" * fsAcA (slf I„ f c1? iu - i n ) - czA (siil3 + dizjjcr'f 157 + { cA sid [ (I/,- Ze, J„) - (Ttt - Ze, I2t)J - CA czl (Iii - 2e,In) / sA fcf - Ze, I,3) - jj (Zz'3 -Ze,J23)J t sA(s!h, + clhz) t cAh3 + cY fciA (I3} - s'lI„-clIJZi) - Cell,, * s*IIZz) - CiY sti (ltszA) - 2 s YjJ. ciIJ J,z - sYsJL si I (1,,-Jtd - 2 cYszA (siItt t cilt$) * 2sYcA (cl J „ - sil^jY' + I cYsA sicl f (If, - 2e, l„) - (Iii - ze, Izz) - cYsActl (lit - ze,I,z) - s¥[clI (1,1 - Ze, I„) + szI (Jit - Ze, In) + jzl ( Z,i - ze,l,t)J - cYcA C d (l,3 -ze,i,i)- si (lis - ^iIu)J * cYcA (sih,-clhz) t cYsA h3 + cFsYcA Cslcf (Iff-Izt) - cil In! ~ czYsA cA ( s*II„ * czI I2i +I33 - sz$ J/z)tclfczA (sila + cIJzs) + c¥s¥sA (dl,* -sli2i)j - Px (6) ; ( i . i b ) 158 «i? Degree of Freedom ( P i t c h , Spin) I33 3" * <I» - 2clJ33)I' t C3/(/+ cce)J { etl CsYc YsA (I„-In) + (c'Y-s'Ys'A) lit] + ** ci [U*Y- ftYs*A)(Itt ~T„) + 2szYsA I,t] * sY cA Csl UYT» * sYsAIi3) ' (s?sA I/3- c¥ Izi)]} * /} -2e,h3 ~ [ sAI3i f cA CsIIl3 * ci Its)] Y" * i c *A fsld CItz - In) -f- etS lit] ~ sAcA Cdla - s$Iz3)J(Y')Z + {-*A Hi's - 2*/ls») - cA Csl CI,i - 2e,In) * cl (1^ - ze/lu)] - cA Ccf h, -jiht) * cAUcYsA si$ - sYczDCIu-1„) - SYI33] + (cYstAcii + ZsYcAsz$)In t 2cYclA (cil,3 -siIxt)}Y' + { cA Cai (In-T„) - T33 - Zsz$ lit] + isA (si Il3 + cl IZ3)}A'r' - (dl,3 -sIIz3)A" t [sici (I„ -Izz) - Clil^ J * {-C* (I 13 -Ze,Il3) t si (l/3 - Ze/Iti) 4 si h, + ci hz 159 t cYsA [cti (Itx-Iu) - I33J ~ 2 (cYsAszI - sYcz$)I,2 - ZcYcA Csl Il3 + cl IM)J A' + { cYcAUn - Ze,J,3) ~ (cYjAsI -sYel) (I/3 - ze,I,3 -ht) - CcYsAd * sYs!) ( J « - *4 its - (s*r-c'rsVDCsid(izz-i,,) + czi In ~ sYcYsA [cil (Izz-I,,) - 2sz$ I,zJ * cA Cc'YsA (cfln -sfltt) * cYsY (sf 1,3 + cll2})]} = fia (e) . d . i c ) 160 APPENDIX I I SYSTEM MOMENTS OF INERTIA I I . 1 A r b i t r a r y Appendage Moments o f i n e r t i a a r e c a l c u l a t e d w i t h r e s p e c t t o t h e i n -s t a n t a n e o u s c e n t e r o f mass o f t h e s a t e l l i t e (° c) but a r e e x p r e s s e d i n terms o f l o c a l appendage c o o r d i n a t e s . P e r t i n e n t system geometry i s g i v e n i n F i g u r e I I - l . F i g u r e I I - l G e n e r a l d i s p l a c e m e n t o f a mass element i n t h e p r e s e n c e o f f l e x i b i l i t y (e^), g e o m e t r i c o f f s e t ( a ^ ) and a s h i f t -i n g c e n t e r o f mass (r ). 161 N e g l e c t i n g s u b s c r i p t i , t h e l o c a t i o n o f a n y m a s s e l e m e n t dm o f t h e i t h a p p e n d a g e c a n b e e x p r e s s e d : Kit di! 1) w h e r e : tt_. = j t n c o m p o n e n t o f n e t o f f s e t v e c t o r a l o n g t h e l o c a l a x e s . B a s e d o n t h i s d e s c r i p t i o n o f t h e d i s p l a c e m e n t f i e l d w i t h r e s p e c t t o t h e i n s t a n t a n e o u s c e n t e r o f m a s s , o n e c a n e v a l u a t e a p p e n d a g e c o n -t r i b u t i o n s t o t h e i n e r t i a t e n s o r o f t h e o v e r a l l s y s t e m : In - / fa? + rf>) d m > f J(—)dm = Jm (-)dm 162 * h3*)Jdm + + - Z f [ ur)dm + J[v2+ urz + z (y * 2>2 )v + Z(f Izz = J <nlt + tfsldm ; xc + (2 + l3)% Jdm - Z f(xc u + $c ur) dm + J[u2 + urz + Z (x+ 2>/) it + z (y + i$)wjdm ; 1*3 = J (>~J + r-J)dm ; = m (if + li) +j[xz+ if* + z(l,x + It y)Jdm + rn (xi f yl) - zf[(x+gt) Xc + (f + It)jf>c Jdm - ZJ(xcol + fcir) dm + JCu*z + ir* + 2(x + 2,) a + 2(y 163 + 2>z ) rJdm ; In = / rdi fad*; = J ( xy + x hz + y 2>/ + 11 lz ) dm + JW ~ J(% u ^ xcr) dm +/C (tf+&z) + + hi) V r u. irjdm ; S i m i l a r l y ; = J (x%> + x%3 + flt + 2>ih)dm + m Xcfr - JC(? + 2,3) *c + (x- +2>i)fc Jdm - j(%U + xcur)dm i f C (?+2>3)U> + + li) ur + aurjdm • Iz3 - J rdl rd3 dtn ; - S (ft + 'fri* + hh)dm + m 164 + &2)ur + vur ]dm * ( i i . 2 ) I t i s s e e n t h a t t h e r e s u l t i n g e x p r e s s i o n s a r e made up o f t e r m s a s s o c i a t e d w i t h t h e u n d e f o r m e d c o n f i g u r a t i o n , f l e x i b i l i t y , and n e t o f f s e t . I I . 2 Beam-Type Appendage C o n s i d e r i n g t h e beam t o be s l e n d e r , y — z ^ 0 . F o r a u n i f o r m c r o s s s e c t i o n dm = pdx t h u s a l l o w i n g some o f t h e i n t e g r a l s o f e q u a-t i o n ( I I . 2 ) t o be e v a l u a t e d e x p l i c i t l y . The n e t c o n t r i b u t i o n (ex-c l u d i n g r_ c) o f a g i v e n appendage o f t h i s t y p e t o t h e o v e r a l l i n e r t i a t e n s o r i s , u s i n g l o c a l c o o r d i n a t e s ; n (it + 2S) +fJ[*Crz2(*t "M* j maf + t})+fi1 (ii + h) + f{*C(u> m(if+iz2) t fJ'dJ+h) +Sj/1 + (x + 2>i)T + (U--*fs)V J doc ; -ml,h ~ifX%-fjJrh In ' in ~ J-31 ' -In = 'In = 165 f (x + Z,) if t (u-U/s) c*rjc/x ; 1 ( I I . 3 ) When d i f f e r e n t i a t i n g s u c h e x p r e s s i o n s , n o t e t h a t & = £(t) b u t _a - c o n s t a n t . S e p a r a t i n g o u t t h e e f f e c t s a s s o c i a t e d w i t h g e o m e t r i c o f f s e t o n l y : ZIZZ « m(i***2) j 2J/3 = ; ( I I . 4 a ) and t h e 'undeformed' beam: 9I„ = 0 ; ,In « if? I* 3Izt - fXl(iJL+M y ,1a ' if*'** ,I3i = fI'd** 2d ) 3J" = 0 ( I I . 4 b ) I n t h e d e r i v a t i o n o f v i b r a t i o n e q u a t i o n s t h e i n e r t i a d e n s i t y ( I . , ) i s r e q u i r e d a s an e x p l i c i t f u n c t i o n o f e l a s t i c d e f l e c t i o n s j k u s i n g t h e l o c a l c o o r d i n a t e s . E q u a t i o n s (II. 2 ) c a n be u s e d i n t h e 166 d e r i v a t i o n . Those terms which are not e x p l i c i t f u n c t i o n s of u, v, w are not included i n the f o l l o w i n g . G enerally *J* = X JJA d m ' (II-5) For the beam-type appendage adopted here, which i n c l u d e s a x i a l f o r e s h o r t e n i n g and p = p ( x ) : eIn - if C ir, (*>+<0 + *s<rj + J> + t fur2 - [ 2f <irf + * +u) c/<k] [ { (nz + «rx*)J . + ftf* - [ £ 2f(irt + a. ¥ u) d<x J [ + urx2)J ) eI/z - /> [ cn + v)(xttt) +.ir,irj - CJxf (irt + t) doc ][{ (irxz + urf)J . 167 eI/3 - f [(n3+ ur)(x + il) + 7r,urJ - [ J* j> (ir3 ( I I . 6 ) The theorem o f Appendix IV has been a p p l i e d when d e a l i n g w i t h . Fo r example: ~ " ' * /*{ "' -fJ[XZfWi+ d« J C j s i n c e , I I . 3 I n e r t i a s o f S p a c e c r a f t Having A r b i t r a r y Appendages Adding up t h e r e s u l t s o f e q u a t i o n ( I I . 2 ) , o v e r a l l s a t e l l i t e moments o f i n e r t i a can be a r r i v e d a t wh i c h s e p a r a t e out t h e e f f e c t s o f f l e x i b i l i t y . The e x p r e s s i o n can be s i m p l i f i e d by e l i m i n a t i n g t h e appearance o f r_ c components i n l o c a l c o o r d i n a t e s . T h i s i s p o s s i b l e s i n c e , a s i d e from deployment e f f e c t s , v a r i a t i o n s i n c e n t e r of mass l o c a t i o n a r e n o t independent but u l t i m a t e l y a r e f u n c t i o n s o f l o c a l appendage d e f l e c t i o n s . T o t a l s a t e l l i t e mass moment o f i n e r t i a , w h i c h i n c l u d e s t h e e f f e c t s o f f l e x i b i l i t y , g e o m e t r i c o f f -168 s e t o f appendage r o o t , c e n t e r o f mass s h i f t , and d e p l o y m e n t , c a n be p u t i n t h e f o l l o w i n g u s e f u l f o r m w i t h r e s p e c t t o t h e c e n t r a l r e f e r e n c e s y s t e m x, y, z: [I]' [,U * ftJl * fsIJ * feTJ * fen, I], t = fuJJ * felJ * fcmU i -Jit * % X'?c Y M -ult3+">s&tc M T ul33 -l»3<*!*& L 169 / C ore + v* t (xt *},,<)n : jt ul < it.-1 n: s i Y Xi M M £ T R I C ( I I . 7 ) 170 Note t h a t a x i a l f o r e s h o r t e n i n g terms can be accommodated by r e p l a c i n g u- by (u. - u , .) i n t h i s e x p r e s s i o n . A l s o t h e [x•] m a t r i c e s a r e x J x f s , x ^ x c o n s t a n t w i t h r e s p e c t t o t i m e i n t h i s a n a l y s i s . I I . 4 I n e r t i a s o f S p a c e c r a f t W i t h Beam-Type Appendages I I . 4 . 1 C o n t i n u o u s c o o r d i n a t e s The e f f e c t on s p a c e c r a f t moments of i n e r t i a o f appendage f l e x -i b i l i t y i s worked out i n d e t a i l here f o r an a r b i t r a r y number of beam-type appendages. G e n e r a l r e l a t i o n s c o n v e r t i n g components from l o c a l c o o r d i n a t e s ( e q u a t i o n s I I . 2 and I I . 3 ) t o c e n t r a l c o o r d i n a t e s "are p r e s e n t e d . Then, based on t h e c o o r d i n a t e t r a n s f o r m a t i o n s d e f i n e d i n C h a p t e r 6, t h e terms I . , a r e expanded o u t f o r t h e case o f appen-e 3 K dages r e s t r i c t e d t o t h e x-y and x-z p l a n e s as i d e n t i f i e d by s u b s c r i p t s p ( p l a n a r ) and o ( o u t - o f - p l a n e ) r e s p e c t i v e l y . C o n s i d e r i n g t h e i appendage t o have an a r b i t r a r y , but f i x e d c ~ 2 (X/2tL Xzitl I/2,i * %l*>i Xs2,lJ/3,i 171 - 2 ( X/3,1 Xz3,i iJft + X,3,i X33,i I/3,'i + Xz3ti XS3t£ ) J ; ^.(Xn,i Xit,i In,i * Xz,,i Xtt,i 222, i +Xj/fi Xj2tC ~ (X„,i Xiz,i * Xn,c Xif, i)I/i,i -(XthC X3zt£ + X/t,£ to'.*') - ( Xn,i ^ [ X,hi X/3,i I/f'i + Xv,i Xl3,i l21,i + X3l,c ' (X//,i Xj3,i + Xi$,i X$lti) ~(Xz,,i Xt*,L + Xz3,iXsi,i) Iu/<J J 2- [ X,z,i X/3,i I/hi + X%t,i Xistt 2ttlc + XnS Xn,i Is$li ~ (*n,£Xf.i +X,$,iX2iti)Infi - ( X,i, i Xss,£ + Xa, c Xst, i) 1/hc ~ ( Xx*, c #33, i + Xi*,; X3Z/c)Jz3,iJ ( I I . 8 ) 17 2 S u b s t i t u t i n g t h e [ X p ] , [ X Q ] m a t r i c e s o f t h e c o n f i g u r a t i o n s t u d i e d i n C h a p t e r 6 i n t o e q u a t i o n ( I I . 8 ) , g i v e s t h e f o l l o w i n g f l e x i b i l i t y c o n t r i b u t i o n s : + <? tty i- sift (*p y 2,f/>) <Vp - [ si J 4j>J* + s'ft (J/-*/) + (sift 2z,f> Xp 2>?,o (So - (x0 +2>hc) n% - T-ST.%/ira<U xc elzz = £ Cfp f*s*# fy1 + iff + Z 32fF liff ntf c 173 + 2 2,,, (JfXrjJ J*, 1 * fCf* { + s2?, * S * + 2lzfo V + 2sH Jiff (JU -*.)J JJx.J . IZ £Cff, J/' {7 s*n> rf * £ - C2& kf 4)j>M t (stfy 2,,, f city It,,) (Jr-xp)J t sft it,0 ure + si>, -utr, - ey, [£%dot * lt,c CJ.-x.)J ("«*l"»*.)J</x,J } J ~€ I/s = fCffJ'f S(ff> Hp ty f [5f/> ' c<ff> 174 £ C / {cfy h,p * f«P/> **tr Ocp + 2,,p)J or? + cfy ~ sfy L^foj><k + £ C P 1 { c% h3,o - s% (X0 t Z>lto)J% ( ) j J*.D . ( I I .9) 1 7 5 I I . 4 . 2 Assumed-mode format The s o l u t i o n f o r e l a s t i c d i s p l a c e m e n t assumed i n C h a p t e r 4 i s s u b s t i t u t e d i n t o e q u a t i o n s ( I I . 9 ) . Combining modal c o e f f i c i e n t s , s i m p l i f y i n g , and r e a r r a n g i n g y i e l d s t h e f o l l o w i n g w o r k i n g form f o r the i n e r t i a f l e x i b i l i t y e f f e c t s : •In 35 C | | { 0>-»> s,J/ *f»f»P) + Bm;i fiJliCfmK + C** fm ) 2>iff> ) Jp 8mn, io ] (£m ?n + fm fn) ~ ifo [S2% Bmn,,, jf - Cszr, Z3fo * 2s2K l>,o) Bmnftc A?J (tmfc + fm X) } + £ {[Cm,, (2c2<ff> 2>i,P + sify I,,?) + Cmt4 m 176 * # c g t (smn,, Cs,j; (s>?, $: $: * * f. ?mt:j -if,Cc ft 8mn,n Jf t C-szfy 22,/> + 2cx<fp 2>,,?) 8mn, io JlpJ (£m (t, + ?mfn) ~ z f0 ( 8mn,n if + * 2>,o 8m»,,o jf )(?m $' + in)} + m ( Sr*? I Cl*'** ' st*r 2'><>) C»>< p p j ~ SZfp Cm,+ Ipjjfm + 2 &$p Cmtl fnt J ($m$n + ?m.f*)J ~ z1 fp ( 8mn, ,i lp + 2 i/fp Sm^o JpZ')(&'&'+ fm?S) 'If, if £c*% 8m*,n £ + <s% 2>3,c + I CX% lt,o)B*',»J tfmfi,' * fit:)) + Jo SiK Cm,4d } ) J 177 £ ^ C £ £ {#»>», < f { frtf *2<Pr + f,J? fm tn ] + j f, Ap [7stft 3/iW,// Jf + (SZfy 2>l,P + C2<Ppil,p) 8mn,loJ Ctm C + tiX) J + g (fid, C«*t 2>x,p - cxfy 2>i,p) Cm,i - czfy Cm,+ AfJjfa Ac] Pm + 5% 2>t,o Cm,, ?m J )J ; £ ± C & % { 8„n,, Sff & tn -jfjo* P P S*% tmfn ) + {fp A/ h,P 'ft 8mi,,,o (?m £ + fmfn) ' jfi Ac [7 Si% 8mn,„ Ac + (St% 2>,,o ' cz(& l*>») £m»,io](tmfn + fm tn) J + £ (/A/{ *4>f Cm,, fa + C C*<Pp it* - c<fp 2/,p) Cm,i ~ eft C»,4 Jp] fm J Cn,,+ AoJ Z£ ) J 178 C?m tn * ' J foJo* S¥>c 2,Zso ($m tn * tA X)J} + £ (f/t Jf {cfr Z$tP Cm,i tm * C<c<f/> + *<Pp ll,p)Cn*,i + s<fpC„)4 J,]fmJ+f0J,*{c%Z2.o Cm,, fa + C (c<& 23,0 - SKI,,,) Cm,, -S% Cm,+ £oJ?m J )J . (11.10) I I . 5 Time Rate of Change of I n e r t i a s f o r S p a c e c r a f t w i t h Beam-Type Appendages Not a l l elements c o m p r i s i n g t h e s p a c e c r a f t i n e r t i a t e n s o r v a r y w i t h t i m e . [^1] and ^ I ] , f o r example, a r e f i x e d . [ ^ I ] , however, can v a r y d u r i n g deployment by d i f f e r e n t i a t i n g t h e l o c a l elements as g i v e n by e q u a t i o n ( I I . 6 ) , t r a n s f o r m i n g t o c e n t r a l c o o r d i n a t e s , and summing over a l l appendages. I n terms of l o c a l c o o r d i n a t e s t h e s e d e r i v a t i v e s a r e : 179 3I33 « j>jxcjL+*ii) . s i 2 } = o (11.11) C o n t r i b u t i o n s a s s o c i a t e d w i t h t h e ch a n g i n g p o s i t i o n o f t h e c e n t e r o f mass ( O T n I ^ u ) can be found d i r e c t l y from e q u a t i o n ( I I . 7 ) : 0 - Z M s (% % + h i > 0 / 0 cm^zt ~ + cm %31 ~ - zms C *c*c J • cn 2/z " ™s C*cfc 0 cm 2/3 J * cm 2Z$ 0 The most i n v o l v e d e x p r e s s i o n s a r e a s s o c i a t e d w i t h f l e x i b i l i t y . They can be d e t e r m i n e d most e f f i c i e n t l y by d i f f e r e n t i a t i n g t h e assumed-mode form o f I . as g i v e n by e q u a t i o n (11.10). That i s , t h e t o t a l e f f e c t from a l l appendages i s : 1 8 0 ±*C£±( Bm»,,{ 2fpjp* (C2(ff # * ' t faff) t& * & fS) +fjll (tit: < gSC)]} - z ff-lf ( Apt* <st(Pp 2>*i<> 4 2s*<Pr 8™*, ,o + 3 s*fe J!FJ ((m'tf + f£ ff) + 2jtp [sty 3mn,it' Jp + (sz<fp 2z,p +25*$ 8r»»,t*J (&$n + fmf*')J -jf0lo{Jof-2 ** Smn)fc + 3s*% 8»n,»JeJ(fcti + [sz% 8mn,n Jl0 ' CS%% l3)0 f 2 sx% 2,,c)8m»,l*] (tiC + &x)j) + gfajt, {c (2c*<Pp h,r tszfy h,P )Cm91 t *2<fp C„f4Jp](zlp f£ + Jp fc) f SZfp Cm/4 JfJp fm * * h? C»»< (2h fm tip 6?)} * + Z 2>l,c Cm,, (zlefc + h?m )} ) D ; 181 + SttfJ, fa till -jf,J, { C*'V/> 8»">,>' + fmfn) + 2 £c*0, 6mn,u + (Sift Ztjf * Zc*ff I/,,) 8m»,iojtf(fmf£fcJL C (3 8m»,// 1" 2 Z/,o 8mn,/o) JcM (fn ffm fi) + Z C8mn,„ + 2l/,o jftflfS* kfn)j) + £(ff,Jtp { [zs*fr - stfpl,,,) Cm,t - svff Cm,4 JpJ(2Xp fm t JLpfc) * Z Z9,p Cm,/(2 Jp tm +Jf>?m.'>) " S2.<pPCn,+ Jlpi, & } + 2J>. 4, l3,c Cm,, (Zl + io fm ) )D ; ei33 « cz*(Bwh/pj; +z /j! (fm t: + s*% X) + s [j>j;j, ft? X + Ja J?J. + fm f/)lj -jfijp [ (4 Z,,p Sn*,,* + S 8mn,,i JpKtjf&'tjlftS) 182 * P P 4 z(Z l/,p 8mn,io + S/T>»,//Jp)Jp (9m. $n 4 ?/ tn)] - {fi l{ £ Z(sz% *t,o + iSfo 2>,.o) Jto 8»w,/o + 3 C2% 8»«,» jLjCfcti + tiii) + i£(st& h,o +2c*fo 2,,*) + C2<f>o 3»*,t, JkJ L (k fit ' tn)}) * ™ ( z fp^p Cn,)/ & +Jpfc) difc + * *** Jul, ?£})J} ** (ti tn' * * 3 (ijjfjlf Sf,fj!&' * j 5 £ I s f i & &)1 * iff 4 /11 Cst<f>> 2/,P + CZ(fp Zt,p) Bmtijio + 2 sfl> Bnw,// Jp] 0 li,P + cz<fp lt,P) Bmfi,ioJJtp (fmff + ?m tn)} ~fol* c% S»n>>c ^ I <t»K * fm ft) 183 0 (Slfp lz,f> ~Ct(pp l,,p)Cm,, - CZfp Cm,+ Ap] (zip $m + J/S*) - czfi Cm,* Jpif + J0J,(C ( C¥e It, a * SVo l3,o)Cm,, * C% Cm^AcJ (*L $£ + A*$m\) t Sto Cm,, lz,o (Z& + M f*) t C% Cm* lolo fm})j -J* - itc£%t 8m*>, Cj,JJs*<teti*£to - A if St9i fa fi + 3(fp£plip '<Pf> tmtf [ Ap (fm' ti< fmfnP) t Ap (titn + tit/)] - / fcAc { CUszft Z,,c ' czn 2>3,o) 8mn,,0 + }sz% 8mn,n A.J L C (i ti * ii ti) + Z [ Cst%Z,l0 - ci%li,o ) Smnj/0 + / St Bmrij// AeJ Ao (fm + £( j>pAP { Sfp h>P Cm,, (ZApti + Ap-ii) f € <*fy2z,p - ctph,t)Cfi,,, - cfy Cm, 4 ApJ 184 (tjp fm +4/> tm) ~ eft Cm,4 JfJf $i } + f0Jc { C' (cz% + si?* *9,o)Cm,i + a% Cm,* LJ (*<lo tm. + tm) + C*<fi Cn>>4 Jt,J, p o + fj/c% (?i ti ffmt*) 4 S(jJ}Jp <f,Z»SnP * J9 JLiL C % (i tn ) - J> Jp ShP S<fp 8mn,,0 [ Jf i C t i + fitf) + Jf (i>£?* + titn)] f f j , lzt* *% Sm»tio [ L * tit;) f 1 (fat; * i i (2lf tm t Jp tm) * *<fP Cm}4 Jp ^P fm J+fJ, £H23,c~S% h,o)Cm,t ~ *%Cm*JU](ti0ti + JU $i) + c% Cm,' (&M fi +Jo tm) ~ S%Cm,+ JeJc $i J) J . (n.13) 185 APPENDIX I I I EVALUATION OF { r }, {h}, AND {T} FOR BEAM-TYPE APPENDAGES c The g e n e r a l n o n l i n e a r a t t i t u d e e q u a t i o n s o f C h a p t e r 2 remain q u i t e l e n g t h y d e s p i t e t h e compact form i n w h i c h f l e x i b i l i t y and de-ployment e f f e c t s a r e r e p r e s e n t e d . E v a l u a t i o n o f t h e s e e f f e c t s i n f a c t r e q u i r e s a major e f f o r t when d e a l i n g w i t h t h e t h r e e d i m e n s i o n a l case c o n t a i n i n g o f f s e t and a r b i t r a r y appendage o r i e n t a t i o n . Upon a p p l y i n g t h e d e f i n i t i o n , f o r {h} and {T} and i n p a r t i c u l a r , t o t h e c o n f i g u r a t i o n o f F i g u r e 2 -2 one q u i c k l y l o s e s any n a i v e t e ! I I I . l S h i f t i n g C e n t e r o f Mass L o c a t i o n { r } and A s s o c i a t e d Time D e r i v a t i v e s The e x p r e s s i o n s d e v e l o p e d here a r e based on t h e d e f i n i t i o n o f r ^ g i v e n i n Chapter 2 w h i c h a l l o w s f o r a change i n c e n t e r o f mass l o c a t i o n due t o e l a s t i c m o t ions o r deployment. An a r b i t r a r y number of appendages l i e i n t h e x-y, x-z p l a n e s . D e f o r m a t i o n s a r e s o l v e d f o r as i n C h a p t e r 4 . t SH Jf g (Cm.t fm)J} > 186 $m * # jf ±(Cm,,$m>] } ; <</>**) C 2fj>„ j; £ (Cm,, ft)] + £{ C-i'r. a.')* * cv.jfg (cn,,, . ( I I I . l ) <'/>**) { £j^prcfP iPbj, ~ spP ± c„3,i, ( Ap(m * 2Jp{m)] * £ f0[ct>o lcb L f Sfc £ Cm,, Ac (Ac ?n! + * Ac, fm)J'j ; m (i/ms){£[ff, J/Jf + c<pP£ Cm,,lf (J, (m * Zi„ fm)]} . (//ms) {£ j>pl£ c„„ J? (J, if * *i, # f ! 7 * £. a[-stiJfj, t cj/i do £ Cm,, o J0 m lJL,f* tti.fl)]) . ( I I I . 2 ) 187 xc = O/ws) C f jp I cf? (j/ - Jtp Jf ) - sty + Jf)t*J} * £j { c ¥, (J/ * JfX)+S% £ Cm,iC 1? fm +4-JcJ„k + 2( J*l + JL*) tmJ}D } I = <'/m,)C f ff>{ sfr (j,* t Jf Jf ) + 'tr £ Cm,, CJ> fm * 4 JfJf + 2 (4>J/> f ll) (m J) * * *L { * Cm,, CM* & i 4jeJof£ y 2 U, L y Jo) 5m JJJ ; | = <//m,) C * fp{£ Cm,, [J/ fa t 4ip Jf y z (Jf Jf i Jf1) t £ j, / -sfe (J.* •> JobJo) + cto £ Cm, i t rn %JJJ > 188 I I I . 2 L o c a l A n g u l a r Momentum {h} A component m a t r i x d e f i n i t i o n i s g i v e n i n C h a p t e r 2 f o r a n g u l a r momentum a p p l i e d about t h e system i n s t a n t a n e o u s c e n t e r o f mass s o l e l y as a r e s u l t o f m o t i o n r e l a t i v e t o t h e b o d y - f i x e d axes x, y, z. I n t h i s case t h e r e l a t i v e m o t i o n r e f e r r e d t o i s caused by appendage v i b r a t i o n o r deployment and does n o t i n c l u d e l i b r a t i o n a l e f f e c t s ; hence t h e term ' l o c a l ' . I I I . 2 . 1 Appendages w i t h a r b i t r a r y o r i e n t a t i o n , c o n t i n u o u s c o o r d i n a t e s C o n s i d e r e d , i s a beam-type appendage h a v i n g an a r b i t r a r y o r i e n t a t i o n , u n d e r g o i n g t r a n s v e r s e o s c i l l a t i o n s o n l y , and a l s o i n c l u d e d i s t h e e f f e c t o f a x i a l f o r e s h o r t e n i n g . Components o f momenta a r e e v a l u a t e d a l o n g t h e c e n t r a l c o o r d i n a t e axes. However, a p p e n d a g e - r e l a t e d v a r i a b l e s a r e based on c o o r d i n a t e s x^, y^, z^; f o r c o n v e n i e n c e t h e s u b s c r i p t i dropped. The c o n t r i b u t i o n o f t h e i t h appendage t o {h} i s demonstrated f o r t h e component a l o n g t h e x a x i s . The r e s u l t a n t h 0 , h^ terms a r e analogous. - Xiz **) + <*** X,z ' XzzX,z)x + (XnXst - Xzz Xm) "rj Vt + C (X33 ~ Xn V3) + (X*$ Xl2 - X3Z X,*)x + (X33XZZ- XszZs) 1 8 9 Xn ' Xzi Xn) nr * (X32 Xn ' Xx Xn) * + C (Xz3 **> ' Xtt + (Xn Xx* -x* Xu)* f (XszXzs ' X33 Xxx) ur J Vx + £ (X33 *k - X31 **) + (x,z X33 - Xf3 X3l) * + (Xtt X33 ~ XZ3 X32) Vj urx } 4 (Xn V9 " Xl3 > * + ( Xtt ^3 - Xz3 "* ) * 4 ( Xjz i$ - X33**) <f - { (Xn 7T3 - Xa irz) C i c*£ + *£>J + ( X/3 "1 ~ Xn %) £ ^x Vct + «TX urxt + (/ (V* tfx* * ^x"rxx)J } (X-x)J dx . ( I I I . 4 ) I I I . 2 . 2 Assumed-mode format f o r appendages i n t h e x-y, x-z p l a n e s . The more g e n e r a l r e s u l t s o f t h e p r e v i o u s s e c t i o n a r e a p p l i e d t o t h e c a s e o f booms l y i n g i n t h e x-y o r x-z p l a n e s o n l y , u s i n g t h e c o o r d i n a t e t r a n s f o r m a t i o n s g i v e n i n Chapter 6. Appendage d e f o r m a t i o n s a r e r e p l a c e d by t h e assumed s e r i e s s o l u t i o n s o f Chapte 4 . S u b s t i t u t i n g i n t o t h e e x p r e s s i o n s d e v e l o p e d i n t h e p r e v i o u s s e c t i o n and s i m p l i f y i n g : 190 ( id? n,t -**,fJr)J t f0A,[Ft,o ' *3,o *zto + (j J, frttC - v2,0ie)3 4 £ ifiJlp ['Cm,, cfy ir9/f Jtp fi{ i-JLpCCm,, Hx,t + *<fp Cms J/>) /m + Cfp CCm,i %S dp ' Cm,,o *3tp Jp)fm + (- Cm, 1 dp + Cm,,o *z,f dp + Cm,,x *<Pp JpJp ) fm ] + fcd0 [ Jo (~Cm,, ntt6 + Cms SKJC}61 4 (Cm,/ *is Ic - Cm,,o fl3,o Jo ' Sft Cm,nJ* JJC * *i,o Cm,,lo tm t *9L ( - Cm,, ^2,0 do + Cm,,o *t,o Jo ) ?m J j t' mt ( fpJP tSmn>'° 5<fp + + S?P ('{ Bm«,,o 7TSf, Jp t Bmn,„ 1 9 1 (fa fa * f»k ) + *%(-{ Smntlo *zto io + Bmn,n 7*1,0 Jo (fm, f/\ + fm f*) -r cfyj, ( 8mnti Jc ft* i° '(m tn ) ( i **,p1p - n,p Jp)J * j>eJ, rn,c rM - irh0 - i J* Cs% rrh0 t eft fr3j0 ) + ie Cs% t c?a t £ {fpJp [ Cm,, *<f> ip im + Jp (Cm,, +Cm,+ C<fpip) fm + 3<P/> C-Cm,i fy,? Jp 1- Cm,,c *3,'jp) fa + (' Cpm,/ VhP Jp * Cm,,o %,fi)p + Cm,,, cfy Jpj'p)?* J t J Jo [Jo (Cm,, S% nr$t0 - C^, c% frl/0 -cMf4Jo) " Cm, tt Jte + ( 'ft ' SK Cm,, Jo ) tnJ J 192 * m^T i fp^ ^ Smn,io Vhp Jp (fm fn + t£ $n) + (~i 8mn,io cfyit^t Jp t 8mn,n Hrlp)(&fa<tifnP) - 8mn,v S<fp JpIF fm tn J t /<, Jo C~ Bmn,,o (S % #1,0 + C% irho)lo($mti * ?m tn )+ <{ Bm*,* (S% 7f/>0 + eft i $ ) 0 ) L - Bmn,n tsK ± £ < fpJp C %,r*-,r -*z,r*<,f- + < S(Pf *,,r ~ dp *ttt)Jp] + fcL Cvh* «x,o" *i,o Vho + eft Ci *zto M - l*t,o Jo )J + Z {j>pSp [ Sp (CmtiCfy*hp +C«,,sfy*tl, •h Crrijf JLp )fm + (- Cm,/ (cfp *,,/> iSfy fft,p) * Cm, 10 (cfy vhp + sty ir2tP ) J, ¥ Cm,,z JpJf)?mFJ * j>ol0 [Jc (Cm,, %,o + C% Cm* A)fm* + (Cm/fo irltCi0 * Cm,,* '% Jo I'Cm,i%,dc)fm\ 193 " Cm,, f fo %,o io ?m * St* (Cm,/ Kx,c J, ' Cm,to m n * Jr r <S4P%,, - efy ir,,,) A, ( £fc * M i ' ) + ({ Bmn,io (s<pF - cty TT,,, )J/> - emn> l 7 Cjfy %lf - cfy irliP)jf )(ggt %T) ] * fj, * C/e (-{ B inn, 10 IT?,* J, + 8mn,n J„) * %t tif) * MM ( Bm*,, L (fa ?: in) - 8 m * , u J L } ) • ( I I I . 5 ) I I I . 3 L o c a l Torque {D T h i s s e c t i o n i s analogous t o s e c t i o n I I I . 2 . I n t h i s c a s e , however, t o r q u e s a c t i n g about t h e system c e n t e r o f mass as a r e s u l t o f f l e x i b i l i t y and deployment a r e worked o ut i n d e t a i l as opposed t o momenta. I I I . 3 . 1 Appendages w i t h a r b i t r a r y o r i e n t a t i o n , c o n t i n u o u s c o o r d i n a t e s t h As i n p a r t I I I . 2 . 1 , t h e c o n t r i b u t i o n o f t h e i appendage t o {D i s demonstrated f o r t h e component a l o n g the x a x i s f o r t h e case o f a r b i t r a r y boom o r i e n t a t i o n . R e s u l t a n t T^r terms a re analogous. 194 - Xzz*j) + ( xn jcz3 - %3 Xtz) * + (Xn XZ3 -Xn XzzJurJ (%*+ IV <v»* + V2 tfX* -r C/t Vx) + [ (X33 Xi' XiZ rs) + (XnXti-XuXst)* + (XiiXn-XuXtJ + Ut [ (Xn *t - Xn *,) + (ytx X,3'X2^X/X) V + (XtxXn'XnXiz) + (X,z*3 - Xn Kz)* + CXzt #3 'X<* ) ^ + (Xsz £3 ~ X$s Wi) & - CXn ~ Xtz TTs) [ V1 (Vx Vxx + < <S*x + Z 1/ (fx **t+ **t)J - (I-*) { Cirs -t Ut (Xn-Xn)lU (4? ****)] + (Xi9 ( I I I . 6 ) 1 9 5 I I I . 3 . 2 Assumed mode format f o r appendages i n t h e x-y, x-z p l a n e s P r o c e e d i n g as i n s e c t i o n I I I . 2 . 2 , t h e g e n e r a l r e l a t i v e m o t i o n t o r q u e e x p r e s s i o n s can be a p p l i e d t o t h e case of appen-dages l y i n g i n t h e x-y and x-z p l a n e s . f _ m (i *$,*JP -*t,rX?)] t Ah C**,o «s - *s,c + sVo Ci n,0 Jto - X> ) J {fpJLp [-Cm,, c<pp V9lP JLp + JLp (Cm,t ir2lP + Cms sft$p) ' zc">"> c<fp^s,p lp U f 2-&P (Cm,,***,? +Cm,lt sfp Jp ) tm f cfP ( Cms *9,f ip -1*3,? CCm,u>jp- G*s* ti/JLp )) tm ' C-Cm,, (n,P + *<Pf JP Up + ^i/p ( Cm,/o J-f ~ Cm, ,$ Jp /JLp ) + s<f/> (Cmi„Jplp + Cm, ,4 Jp ) ) J + fo^o [ (Cms S*JL* - Cm,i irhc)l0 fr + 2 (- Cm,,o irh0 + Cm,,, fm + C Cm,, 0t o >m 196 ^3,0 So + (~ Cm,i3 7*3,0 + s(Po Cm, 14. 3<?* C„,,x jto ) Jtc)/m* t C% CCm,, n,oJo tm° ftCm,,* %,o Jo tm * C Cm,,3 W2t0 Jo /M 4 Cm, to **.*Jo ' Cm,, n,o Jo ) tm J } *0 ^9,t Jp ( B/tjn,z3 C tut $n 4 tm tn) ~ 8nm,n ($m $n + tmf/) ) + ( { 8mn,,o ( KP 00 * 0 00 - fl~3,i> * *<Pf If )!'/> + Sty fr3,e CSm/i,/s 4p t 8mn,xi jf/Jp )) -fmif) ~ Bmx,, c<P/> J* (£' f/* &t ?/) -2 cfr J, M C &mn,n tn ' ~ 8mn,z° ^ » ) 4 c<Pr C 8mn,z, Jpj'p + 8mn,zz Jp ) £»' f/ J * J-o ^ & mn, IP 197 Jo (f/n fa + fm tn) ~ &nm,,% s<& ^,0 io (fm fa + fm tn) + (" z Bm/1,/0 (^3,0 ~ Ft,* ~ + Sti, Vz,o CSmntz<,lX/io + Bnm,,tZ )) (fa f>l + fm K) + 8J»»,,'* Jo (fa tn -faff) ' 2 8mn,« cfoJoJ, fa fn* + 2 Bm*,%0 cfoJoJo fm + C<rl (8mri,izJZ + 8m»,z, i i ) fa f * J } > J £ ^ < fpJp E %,f #,,p - n,f r3tfi - cfr < { **,rJP ' *3,PJP )J tfiJL E *3,o £3,0 - j *s,o + Sfo )J* + (eU + 3% )Jo] + £ { frjp E' Cm,i S<(F ir,,r JP fa ~ JP (Cm,, irhr + Cm* * * p - tCm,io 1*3,? Ip fm ~ Z ( Cm,10 7Tf/p + Cm, 11 C<fpIp)Jp ?mp - s<fp (Cm,, %P Jp + W3jP (Cm,,ojp +Cm,ttJp))fa + (Cm,, ( KP 198 + c<fp JP )JP - C ir/tP + cfyJF) C£%,*4> Jf/Jf )) tm] + J.1* CCm,i 09i n,* - C% fT,,,) - Cm,+ JL*)1* tm + (2 Cm," CsVe r3,o - c&<irho) - Cm.uM )Jo jm *4 + *3,o - cV**i,o) (Cm,,* JUM + Cm,* J /JLo ) - Cm,Jo Jo ~ Cm,i+Jo + Cm,i Ceti *„o-*¥. r,,.)J. ) fa J } C<fp 1*1,* Jp fn + & ?t t k i : + titsr - i*z,p j[, (( Bmn,ix * 2 Sf,m/Ii )($m n + tm fif ) lf Crf,,f - IT,,, + cfrjp) + cfy f%, ( Bmn,it * 8m,ts J?/Jf ) ) (ti X +f*f»') - 8mn,i S<fpj;W?S < t£l')-l*4fJp ip (6m*,* $m 1f -Sm,t0 sfr ( 8mn,2i J-pJp 1" Jp1 ) tin trf J 1 9 9 ^ fcJ° t~ 8mn,io * * o " (C ft %,o + 5<eo1fhC) lo (8m»tZ3 (4^ & " 8rjtnttz ( $m %rx + ?m fn.) ) - (I BM„t,e (ifho - irhc + S*lo + + (e%V3to + SW> Tr,,o)(8rymttt, JOVM * 8m*,n lo )) (tmfn * fmU)Jj}; ±±. < plp L%P ft* - n*,, ' i Of, \ e - c<fP Us)JL? + (*fr *>,r - ^ol)J t £{fpJPCCm,i Ccff whf + s<?pn,t) + Cm^Jtf)Jp$i + tJlp (Cm, 10 (cfy * s<pF frtff.) + Cm,,,!?)?/ + (~ Cm,, Sp (eft fr\p + s<fp fTZ/P t Jp)t fa,,, Iplp * Cm,* t (cfp fr,tp t sfp jrtt,) 200 (Cm ,/odp + Cm, /? JL? /J, ))tm*3 * file r (Cm,, irh0 t eft Cm,* Jo) Jo fa t Z (Cmft0 + Cm,,, Jo) M fa + (- Cm,, #,,o ic f (Cm,,o K,,o + Cm,,t cVoJc)Io + (Cm,,? K,o + Cm,>* 'VoJo ) (Jo*/Jo )&i -SK ( Cm,, frz,o Jo Zi * Z Cm,,o ^Z,oJc fa + (C>">» *L,O Jo + Cm,,3 JoVA - Cm, i fie A)) tJj} + ££ {j>p Jp C- Bmn,,o (Sty 1Th, -cfy^^jf (fa fa + fa x > impfa + irt:)-%>P - cfy ntr)lf><8n,n,x%<fm*fa+imtHP) " &nm,ii- ( (m $tf +fJHih)) +({ 8mn,io ( $zt* - "\r * sfyJp -c<Pf>Jf>)Jp ' (s<fp ' cfp irXff> ( Bmn,lf Jp + 8mn,x<. Jp/Jp )) (fafa + tmf£)J + f0lo/:Bmn,,o C9i %,oJ> (fe K * fm K+fctS * fcti ) 201 + c<fo 1*1,0 io ( Bmn,i3 ( tm\ $1 * fm ttt ) - Bnm,* (U fa * ikk))* B»">«> (n,o - ho - m L ) ic + c<t>c *x,0 ( 6m»,u, j*/!* + B*nf,i Io )) (fc & + fm K) + Smn,! A" Ck ft ~ fa £°) + ZSV* Jo A (Bmn, ft) Jo Jo + Bm»,xx Jo* ) t^]})' ( I I I . 7 ) 202 APPENDIX IV A USEFUL INTEGRAL THEOREM THEOREM: C & * 0 X (IV.1) P r o o f : F o r t h e p u r p o s e s o f t h i s d e r i v a t i o n c o n s i d e r u, v as a r b i t r a r y f u n c t i o n s . . I n t e g r a t i n g by p a r t s Jb udv - uv/* - J**/fdu ( i v . 2 ) C o n s i d e r : UCx) « / y V * V * —— dlL = du Jx = f(x)dx y If Cx*> = - I^fC^ J* —^ dv m $ cx> dx Then: 0 " * c x x*a 203 + JJlJ:(x)[JxJl$C°l)d«J<lx . (IV.3) Q.E.D. D i s c u s s i o n When f o r e s h o r t e n i n g i s i n t r o d u c e d e x p l i c i t l y a s an a d d i t i o n a l a x i a l d i s p l a c e m e n t i t r e m a i n s as an i n t e g r a l f u n c t i o n i n t h e s y s t e m k i n e t i c e n e r g y f o r example. T h i s t h e o r e m p r o v i d e s a method f o r t r a n s f o r m i n g t h e s e t e r m s i n t o an e f f e c t i v e a x i a l l o a d i n t h e f i n a l e q u a t i o n s o f m o t i o n f o r beams w i t h v a r y i n g s e c t i o n a l p r o p e r t i e s . 204 APPENDIX V APPLICATION OF HAMILTON'S PRINCIPLE TO A DEPLOYING CONTINUUM The g e n e r a l form of Hamilton's P r i n c i p l e can be w r i t t e n 67 ( M e i r o v i t c h , 1970) 4 where: £ = Lagrangian, which i n c l u d e s the p o t e n t i a l energy of con-s e r v a t i v e f o r c e s ; = g e n e r a l i z e d work f u n c t i o n which can i n c l u d e n onconservative f o r c e c o n t r i b u t i o n s . Note, f o r a giv e n appendage spanning a r e g i o n D, X * T-V * J(V-V)JD - /Zc/D-(V.2) For the case of a d e p l o y i n g beam-type appendage i n the presence of a g r a v i t a t i o n a l f i e l d , the Lagrange d e n s i t y f u n c t i o n r d assumes the form of equation (3.20) when taken t o 3 degree: 205 C a r r y i n g out t h e v a r i a t i o n on t h i s f u n c t i o n a l as i m p l i e d by e q u a t i o n (V.1): (V.4) where: Sir = %e Se ; £ D = 0 f o r a continuum s i n c e O •= D C jTt , ) o n l y (Thus a g r e e i n g w i t h t h e c o n c l u s i o n o f T a b a r r o k e t a l . 222 1974) The system i s t a k e n t o be d e p l o y i n g a l o n g t h e l o c a l x d i r e c t i o n o n l y , w i t h v e l o c i t y [/('Xj~t) , so t h a t : d^ ^ = t o t a l t i m e d e r i v a t i v e r e l a t i v e t o t h e l o c a l appendage tft. c o o r d i n a t e s ; — + V *_ . (V.5) 2% W i t h t h i s i n mind t h e many terms of e q u a t i o n (V.4) can be e v a l u a t e d by f r e e l y making use o f i n t e g r a t i o n by p a r t s . 206 t/ 1 *e* i. 2x (V. 6a) • / 'V J J a ¥ Sexx JX )dt tJ / L 9£*x dx \ ?exx / * J0 (V. 6b) 207 (V. 6c) But: T-i T h e r e f o r e : (V.7a) 208 S i m i l a r l y , i t can be shown t h a t : - V oT<f / + /*[ (<Lf.)- £_ ii )lfeJxte. (V . 7 b ) S u b s t i t u t i n g e q u a t i o n s (V.6) and (V.7) i n t o ( V . 4), i t i s seen t h a t a s t a t i o n a r y v a l u e o f t h e H a m i l t o n f u n c t i o n a l demands t h a t : (V.8a) 209 and, \SL - (±£ )- [r l£ - 2- (ii- ) + l/x iL h - o, a t . X. — & j '. (V.8b) (V.8a) r e p r e s e n t t h e g o v e r n i n g e q u a t i o n s o f m o t i o n d e f i n e d i n terms of t h e Lagrange d e n s i t y ; (V.8b) r e p r e s e n t a complete s e t o f boundary c o n d i t i o n s , g e o m e t r i c and dynamic. 210 APPENDIX V I MODAL INTEGRAL COEFFICIENTS D e f i n e d here a r e a s e r i e s o f i n t e g r a l s a s s o c i a t e d w i t h a chosen shape f u n c t i o n g n ( x ) . These i n t e g r a l s appear as co-e f f i c i e n t s i n t h e s o l u t i o n o f t h e g o v e r n i n g e q u a t i o n s o f l i b r a -t i o n and v i b r a t i o n . I n g e n e r a l , g n can be any assumed . 2 2 0 f u n c t i o n , such as a G a l e r k m p o l y n o m i a l [ J a n k o v i c (1980) J . Modal I n t e g r a l C o e f f i c i e n t s C n j j _ s Cflt3 = ^ > Cn,s ~ *f *3h,£ J* j Cn,i = / ' * 0», 2* d* ; Cn,/o ~ Cn,i + Cn,* ~ C»,s Cn, II — Cn, 4 * C„f $ ~ Cn, 6 Chfll — Cn,+ + Crt,s ~ Cn,6 ~t Cn,tZ 3 5 Cn>3 - 2 Cn,7 + C»,i Cn, /+ ~ Cn,i ~ 2C*,S + Cn,? Modal I n t e g r a l C o e f f i c i e n t s B . ^ mn i J Bmn,i ~ £ d*" ^x 213 P, — Bmn.z. ~ 8nm,z 3mh,/c = Bnm,c - Bmn, & mn Bmn, /? — Bmn, I + 8ft m,z. ~ 8nm, (, 8mn,zo = Bmn, i + Bmn,z, ~ Bmn, & $ mn,i — Bnm,i) ~ (8mn,<) ~ Bnm,i) / Bmn, 23 ~ 2(8mn,3 ~ Bnm,iz ~~ Bmntz ) BtrW,z4- ~ %(-8mn>3 + 8/tm,iz) + Bmn, it. ; Bnin,z? — 2 Bnm,iz + 8nm,/3 + 8mn,/+ ; B. mn,Z(, — Bnm,f ~ 2 Bryin,g + 2 BnmfZ = Bnm,5 " z &mn>& " z Bnm}/Z ~ 2 Bmn,/3 ~. Bmn, t4 Bmn, 18 — Bmn,z ~ Bmn, Bmn, zf 2 C Bmn, z Bmn, 6 ~~ Bmn,/o ) i~ B mn, Brnn, 30 — Bmn, 1 ~ B nm, c B mn, 31 — 8 rnn,zc ~ Bmn, Bmn,3z ~ £mn%3e ~ Brnn, 10 (VI.2) APPENDIX VII A METHOD FOR ISOLATING SECOND DERIVATIVES OF COMPLEX COUPLED SECOND ORDER SYSTEMS A common approach adopted when integrating a system of second order equations numerically i s to transform them to a system of f i r s t order equations. An esse n t i a l element i n the strategy i n -volves expressing the second order deri v a t i v e of each independent variable i n terms of lower order derivatives only. Such a require-ment i s e a s i l y met for uncoupled systems. For more complex f u l l y coupled systems, as represented by the l i b r a t i o n a l and v i b r a t i o n a l motions dealt with i n t h i s t h esis, one i s faced with a considerable amount of algebraic manipulation. The e f f o r t needed i s a l l e v i a t e d somewhat by taking advantage of numerical techniques where appropriate. V I I . l Analysis A l l second order derivatives appearing i n the l i b r a t i o n a l equations are to be expressed e x p l i c i t l y i n terms of attitude degrees of freedom. To begin with the general r e s u l t of Chapter 2 can bp written: U ( V I I . l ) 0 216 where: i n *-h<= a n v e r n i n a e a u a t i o n f o r Q l o w e r t h a n second <\ * = terms n t e g o g q0 o r d e r : ~ ;Of** = c o n t r i b u t i o n t o t h e e q u a t i o n s f - , due t o t h e combined e f f e c t o f t h e r e l a t i v e m o t i o n t o r q u e s ; <V = c o n t r i b u t i o n o f t h e second o r d e r terms o f t h e y degree ^Jf o f freedom t o t h e e q u a t i o n ; b u t not i n c l u d i n g t h e 2* e f f e c t . r As i n d i c a t e d i n F i g u r e V I I - 1 , t h e r e l a t i v e m o t i o n t o r q u e e f f e c t i s u l t i m a t e l y a f u n c t i o n o f t h e Q . The v a r i o u s c o e f f i c i e n t s n e c e s s a r y t o d e s c r i b e t h i s r e l a t i o n s h i p q u a n t i t a t i v e l y a r e d e v e l o p -ed as f o l l o w s : p ^ p L V 1 * n ' C n ) J r,o r,is °x (VII.2) where: c o n t a i n s a l l terms h a v i n g d e r i v a t i v e s l e s s t h a n second o r d e r ; <V = c o n t r i b u t i o n s o f second o r d e r terms o f t h e QM degree of 9 l l freedom t o t h e OC - ' e f f e c t . F o r t h e a t t i t u d e e q u a t i o n s c o n s i d e r e d h e r e : ATTITUDE EQUATIONS COMBINED TORQUE EFFECT «l* (fj) RELATIVE MOTION TORQUE _ 0, , *• , APPENDAGE GENERALIZED COORDINATES ANGULAR ACCELERATION, LOCAL COMPONENTS ¥ <&) 0 — € J t '•>>U F i g u r e V I I - 1 F u n c t i o n a l d e p e n d e n c e o f t e r m s u s e d i n t h e d e s c r i p t i o n o f s y s t e m e q u a t i o n s . 218 ^ - CA ( s i r / > r + d r 2 > y ) - S A r s , r ; uc**rh9 * ci r , , 9 ) - SAT,,?; as — P rSt9. (VII.3) Thus t h e * V c o e f f i c i e n t s a r e i n t u r n f u n c t i o n s o f t h e P c o e f f i c i e n t s where i t can be shown: r, ' r, C , ? „ l ( * > / ) ] • • J J -= r. A + r> a 9 • ( V I I . 4 ) where: P /» c o n t a i n s a l l terms h a v i n g d e r i v a t i v e s l e s s t h a n second o r d e r ; 219 P = c o n t r i b u t i o n s o f second o r d e r terms o f t h e o degree 'M ' o f freedom t o t h e fl e f f e c t . J C o n s i d e r t h e case o f appendages i n t h e x-y p l a n e o n l y : ?M F * ->p r m + if ?«) ~ cf'4f> Bn,n,i( **,m ?n p * ( r / Cm,, * *fr -if Cm,* ) f/ ]} < ^ = f I ~ " [ cff *' Smn"a r***'m *' 220 + sfr ifz ) Cmfi 4 Ip Cn,4.) K,mJ} P ( V I I . 5 ) D e r i v a t i v e s f o r t h e g e n e r a l i z e d c o o r d i n a t e s o f t h e appendage v i b r a -t i o n s c a n be p u t i n t h e f o r m : VI * til * m - KI - Kih- (VII. 6 ) where: erms h a v i n g d e r i v a t i v e s l e s s t h a n s e c o n d / Y C I lr1Cl c o n t a i n a l l t VO I'UOJ o r d e r ; IV C I iyj 1 1 c o n t r i b u t i o n s o f s e c o n d o r d e r t e r m s o f t h e q k d e g r e e 2k ' * M) o f f r e e d o m t o the jf, j?' d e r i v a t i v e s f o r t h e i t h appendage. I n p a r t i c u l a r : 221 (VII.7) c Equation (VII.7) makes use of the c o e f f i c i e n t s £f . _ appear-in g i n e x p r e s s i o n s f o r components of angular a c c e l e r a t i o n s taken along l o c a l appendage axes. That i s : aJ = *L * eJ>lM it1 <WII-8) where: £ , c o n t a i n s terms having d e r i v a t i v e s l e s s than second JfO order; * £ t — c o n t r i b u t i o n s of second,order terms of the DJZ Jj^jl degree of freedom to & , determined by numerical methods. S u b s t i t u t i n g back from equation (VII.8) through to ( V I I . l ) y i e l d s : 9 6 + *4 9* '• ( V I I . 9) 222 where: C J * i n c l u d e s a l l c o n t r i b u t i o n s o f t h e O degree o f J,JL freedom t o t h e Q e q u a t i o n . •* T h i s system o f t h r e e e q u a t i o n s i n t h r e e unknowns {¥, A , & ) can be s o l v e d u s i n g r o u t i n e methods o f e l i m i n a t i o n and s u b s t i t u t i o n from t h e t h e o r y o f e q u a t i o n s : w i t h : • * (VII.10) 0, = /*/fczA Cs'f I„ + c2} Izt - szl I,t) + $2A I j 3 f s z ^ (si 1,3 * C $ Izs)J-et = cA [sic? (I„ -Izz) - czi IIZJ + sA ('? 1,3 - Si 7z3> 223 sA I 3 3 - cA (siI,3 + c$Iz3); •/(c2II/, t SZIIZZ + Szl I/z); s i Tz3 ~ cl I/3 ; cA [sicI (I„ -fzt) - cz$ - sAe. '-/Is*-' I33 ~ CA (si I/3 + C l Iz3)> IzS - C l J / 3 ; ' ej[cA(s$r/)lf - cl fa) - sArdjr]; - e,[cA(s$r/fJi + c$r2)A)-sAr3jX + 0ZJ - 0,£cA(sIfo + ci fa)-sAfa* 0jJ; - e4( d rjjY - sl%y + e6 h 07 ( r 3 t yr + 0g ) ; 0? C PS/JL + *i )* Syr + Syr ( I " £JL ) > 225 V I I . 2 A p p l i c a t i o n The a c c u r a c y , and hence t h e u l t i m a t e s u c c e s s , o f a n u m e r i c a l i n t e g r a t i o n depends on use o f t h e most ' c u r r e n t ' i n f o r m a t i o n a v a i l a b l e when-computing d e r i v a t i v e s . F o r t h a t c l a s s o f problems a n a l y z e d i n t h e p r e c e d i n g s e c t i o n where t h e s t a t e v e c t o r i s a h y b r i d c o n s t r u c t i o n o f two main groups o f c o o r d i n a t e s - l i b r a t i o n a l and f l e x i b l e - t h e c a l c u l a t i o n s a r e o r g a n i z e d as shown i n F i g u r e V I I - 2 . 226 SET f = i ' = M = 0 COMPUTE { $ I * i t J COMPUTE fj j f ) COMPUTE COMPUTE COEFFICIENTS COMPUTE Yj A t E q u a t i o n (VII.11) COMPLETE EVALUATION OF F i g u r e V I I - 2 C o m p u t a t i o n a l p r o c e d u r e f o r u p d a t i n g s y s t e m d e r i v a t i v e s . 

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