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UBC Theses and Dissertations

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

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DYNAMICS OF SINGLE A%D MULTIBODY EARTH ORBITING SYSTEMS by SUBHASH CHANDER SHARMA Tech. (Hons.), Indian I n s t i t u t e of Technology, Kharagpur, 1970 M.S., University of Hawaii, Honolulu, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1977 <§) Subhash Chander Sharma, 1977 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 a d v a n c e d d e g r e e 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 a g r e e 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 a nd s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t 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 p u b l i c a t i o n , i n p a r t o r i n w h o l e , o r t h e c o p y i n g 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 . SUBHASH CHANDER SHARMA D e p a r t m e n t o f M e c h a n i c a l 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 C o l u m b i a V a n c o u v e r , V6T 1W5, Canada i . ...dzdlzatzd to H-im, ^oh. He -is the. mai>tnt ofa all tkt cfitatZon. and fitghtzou&m&A ABSTRACT The t h e s i s a i m s a t s t u d y i n g t h e d y n a m i c s o f s i n g l e and m u l t i b o d y s y s t e m s w i t h a v a r i e t y o f s p a c e c r a f t o r i e n t e d a p p l i -c a t i o n s i n c l u d i n g c o n f i g u r a t i o n c o n t r o l f o r an i n s t r u m e n t a t i o n p a y l o a d d e p l o y e d f r o m a s p a c e c r a f t , t h e S o l a r S a t e l l i t e Power S t a t i o n ( S S P S ) , a S p a c e S h u t t l e s u p p o r t e d t e t h e r e d p a y l o a d , e t c . The p r o b l e m i s a p p r o a c h e d i n an i n c r e a s i n g o r d e r o f c o m p l e x i t y . I n t h e b e g i n n i n g l i b r a t i o n a l d y n a m i c s a nd f o r c e d i s -t r i b u t i o n f o r a n a x i s y m m e t r i c , g r a v i t y o r i e n t e d , r i g i d c o n f i g u -r a t i o n a r e 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 , n o n a u t o n o m o u s and c o u p l e d e q u a t i o n s o f m o t i o n a r e a n a l y z e d u s i n g B u t e n i n ' s v a r i a t i o n o f p a r a m e t e r a p p r o a c h i n c o n j u n c t i o n w i t h t h e P o i n c a r e -t y p e e x p a n s i o n m e t h o d , a nd t h e v a l i d i t y o f t h e s o l u t i o n s e s t a b -l i s h e d t h r o u g h n u m e r i c a l i n t e g r a t i o n . The c l o s e d - f o r m c h a r a c t e r o f t h e s o l u t i o n s p r o v e d u s e f u l i n i d e n t i f y i n g p e r i o d i c s o l u -t i o n s a n d r e s o n a n c e c h a r a c t e r i s t i c s o f t h e s y s t e m . F u r t h e r m o r e , t h e y p r o v i d e d c o n s i d e r a b l e i n s i g h t i n t o t h e s y s t e m b e h a v i o u r o v e r a r a n g e o f t h e o r b i t a l e c c e n t r i c i t y , i n e r t i a p a r a m e t e r and i n i t i a l d i s t u r b a n c e s . A p p l i c a t i o n o f t h e a n a l y s i s i s dem-o n s t r a t e d t h r o u g h t h e G r a v i t y G r a d i e n t T e s t S a t e l l i t e (GGTS). N e x t , g e n e r a l e q u a t i o n s o f l i b r a t i o n a l m o t i o n , f o r c e a n d moment a r e d e r i v e d f o r an a r b i t r a r i l y - s h a p e d , r i g i d s p a c e c r a f t a n d a p p r o x i m a t e c l o s e d - f o r m s o l u t i o n s o b t a i n e d f o r s p i n n i n g a n d g r a v i t y o r i e n t e d s y s t e m s u s i n g t h e P o i n c a r e - t y p e a n a l y s i s . The a p p r o a c h y i e l d s u s e f u l i n f o r m a t i o n c o n c e r n i n g r e s p o n s e t o e x t e r n a l d i s t u r b a n c e s a s a f f e c t e d by t h e s y s t e m p a r a m e t e r s . The method i s . a p p l i e d t o s e v e r a l c o n f i g u r a t i o n s : E x p l o r e r XX, an i n s t r u m e n t p a c k a g e d e p l o y e d f r o m t h e S p a c e S h u t t l e and t h e SSPS. F i n a l l y , a g e n e r a l d y n a m i c a l f o r m u l a t i o n f o r a t r i a x i a l m u l t i b o d y s y s t e m , i n a c i r c u l a r o r b i t , w i t h an e l a s t i c i n t e r -c o n n e c t i n g l i n k i n t h e f o r m o f a t e t h e r o r a beam i s d e v e l o p e d . The h i g h l y c o m p l i c a t e d c o u p l e d , n o n l i n e a r , n o n a u tonomous e q u a -t i o n s o f m o t i o n a r e l i n e a r i z e d and t h e i r e x a c t s o l u t i o n p r e s e n t e d . A l s o e x p r e s s i o n s f o r f o r c e s and moments r e q u i r e d t o o r i e n t an o b j e c t i n s p a c e a r e o b t a i n e d . T h i s a n a l y t i c a l p r o c e d u r e i s a p p l i e d t o s e v e r a l c o n f i g u r a t i o n s o f p r a c t i c a l i n t e r e s t . T h r o u g h o u t , t h e e m p h a s i s i s on e v o l v i n g a g e n e r a l f o r -m u l a t i o n o f t h e p r o b l e m a nd i t s a c c e p t a b l e s o l u t i o n . N u m e r i c a l r e s u l t s a r e p r e s e n t e d o n l y t o a p p r e c i a t e s i g n i f i c a n t r e s p o n s e c h a r a c t e r i s t i c s o f t h e s y s t e m . The g e n e r a l c h a r a c t e r o f t h e a n a l y s i s s h o u l d p r o v e u s e f u l i n s t u d y i n g t h e d y n a m i c s o f a w i d e r a n g e o f e x i s t i n g and f u t u r e s p a c e c r a f t . i v TABLE OF CONTENTS C h a p t e r Page 1. INTRODUCTION 1 1.1 P r e l i m i n a r y Remarks 1 1.2 L i t e r a t u r e R e v i e w 3 1.3 P u r p o s e and S c o p e o f t h e I n v e s t i g a t i o n . . . . 12 2. ATTITUDE AND FORCE ANALYSES FOR RIGID AXISYMMETRIC GRAVITY ORIENTED S A T E L L I T E S . 15 2.1 P r e l i m i n a r y Remarks 15 2.2 F o r m u l a t i o n o f t h e P r o b l e m 16 2.3 A p p r o x i m a t e A n a l y t i c a l S o l u t i o n s 20. 2.3.1 B u t e n i n ' s v a r i a t i o n o f p a r a m e t e r s m e t hod . . . 2 0 2.3.2 P o i n c a r e - t y p e e x p a n s i o n m e thod 23 2.3.3 A c c u r a c y o f t h e a p p r o x i m a t e s o l u t i o n s . 24 2.4 P e r i o d i c S o l u t i o n s a n d S t a b i l i t y 24 2.5 R e s o n a n c e 29 2.6 F o r c e D i s t r i b u t i o n 31 2.7 L i b r a t i o n a l R e s p o n s e and F o r c e D i s t r i b u t i o n . . 35 2.8 C o n c l u d i n g Remarks 42 3. ATTITUDE AND FORCE ANALYSES FOR RIGID T R I A X I A L SYSTEMS 4 3 3.1 P r e l i m i n a r y Remarks 43 3.2 F o r m u l a t i o n o f t h e P r o b l e m 46 3.3 M o t i o n i n t h e S m a l l 49 3.3.1 S p i n n i n g s y s t e m s 50 3.3.1a A p p r o x i m a t e a n a l y t i c a l s o l u t i o n . . . 50 Chapter Page 3.3.1b S t a b i l i t y 56 3.3.2 Gravity oriented systems . 58 3.3.2a Approximate a n a l y t i c a l solution . . 58 3.3.2b S t a b i l i t y 61 3.4 Force and Moment Analysis 63 3.5 L i b r a t i o n a l Response and Time History of the Forces 70 3.6 S a t e l l i t e Solar Power Station (SSPS) . . . . 82 3.6.1 Configuration control for maximum power generation 82 3.6.2 S t a b i l i t y of the SSPS . . . . . . . . 84 3.6.3 Force d i s t r i b u t i o n and control moments 85 3.6.4 O r b i t a l perturbations due to the solar radiation pressure . 85 3.7 Concluding Remarks 90 4. DYNAMICS OF SPINNING AND GRAVITY ORIENTED MULTIBODY SYSTEMS 92 4.1 Preliminary Remarks 92 4.2 Formulation of the Problem 93 4.3 Motion i n the Small 97 4.3.1 Solution for the tether case 99 4.3.2 Solution for the beam case 101 4.3.3 Solution for l i b r a t i o n s of the end-bodies 105 4.4 Force and Moment Analysis 107 4.5 Results and Discussion 108 v i C h a p t e r Page 4.6 C o n c l u d i n g Remarks 12 6 5. CLOSING COMMENTS 12 8 BIBLIOGRAPHY . 130 APPENDIX I - TRANSFORMATION MATRICES USED I N EQUATIONS (4.1) AND (4.15) 135 APPENDIX I I - EQUATIONS OF MOTION FOR THE SYSTEM OF CABLE OR BEAM CONNECTED SYMMETRIC END-BODIES . . 136 v i i L I S T OF TABLES T a b l e Page 3.1 V a l u e s o f s y s t e m p a r a m e t e r s i n t h e e x a m p l e s o f t h e t r i a x i a l s p a c e c r a f t u n d e r c o n s i d e r a t i o n . . 73 4.1 V a l u e s o f s y s t e m p a r a m e t e r s i n t h e e x a m p l e o f t h e S p a c e S h u t t l e s u p p o r t i n g a p a y l o a d (M n = 200 kg) by a c a b l e o r a beam 109 v i i i LIST OF FIGURES Figure Page 1-1 A schematic diagram of the RAE s a t e l l i t e w i t h long f l e x i b l e antennas 2 1-2 Tethered O r b i t i n g Interferometer (TOI) experiment proposed by the Ap p l i e d Physics Laboratory . . . . 4 1-3 Canada/U.S.A. Communications Technology S a t e l l i t e (CTS) . Note the la r g e s o l a r panels 5 1-4 A schematic diagram of the S a t e l l i t e S o lar Power S t a t i o n (SSPS) i n the geosynchronous e q u a t o r i a l o r b i t 6 1-5 A simple model f o r multibody systems r e p r e s e n t i n g two r i g i d bodies connected by a r i g i d or a f l e x i b l e l i n k i n the form of a beam or a t e t h e r . . 7 1- 6 A schematic diagram of the proposed plan of study 14 2- 1 A schematic diagram of the G r a v i t y Gradient Test S a t e l l i t e (GGTS) 17 2-2 Reference coordinate systems and geometry of the s a t e l l i t e motion w i t h ty, ty and A re p r e s e n t i n g the l i b r a t i o n a l degrees of freedom 18 2-3 A comparison between numerical and Butenin's s o l u t i o n s 25 2-4 A t y p i c a l example comparing numerical, Butenin's and l i n e a r s o l u t i o n s 26 2-5 T y p i c a l i n i t i a l c o n d i t i o n s r e s u l t i n g i n p e r i o d i c l i b r a t i o n a l motion. Note that over a wide range of i n i t i a l c o n d i t i o n s and modified i n e r t i a parameter the p e r i o d i c s o l u t i o n s are unstable. . . 30 2-6 Dumbbell-type s a t e l l i t e : (a) l i b r a t i o n a l angles ty, ty; (b) boom forces . . . 32 2-7 T y p i c a l time h i s t o r i e s of l i b r a t i o n s and f o r c e components at the mass : (a) e = 0, K i = 0.75, tyQ = tyQ = 0, ty'Q = ty'Q = 0.5 . 36 i x F i g u r e Page (b) i n c r e a s e i n the o r b i t a l e c c e n t r i c i t y . . . . 37 (c) r e d u c t i o n i n the m o d i f i e d i n e r t i a parameter . . . 3 8 (d) displacement and i m p u l s i v e type d i s t u r b a n c e . 39 2- 8 A t y p i c a l example of the G r a v i t y Gradient T e s t S a t e l l i t e (GGTS) showing i t s l i b r a t i o n a l and f o r c e h i s t o r i e s 41 3- 1 Geometry of the I n t e r n a t i o n a l Ionospheric S a t e l l i t e ( E x p l o r e r XX) 45 3^2 Reference c o o r d i n a t e systems and the geometry of s a t e l l i t e l i b r a t i o n a l motion wi t h a, 3 and y degrees of freedom 47 3-3 A comparison between the approximate p e r t u r -b a t i o n and numerical s o l u t i o n s f o r a t r i a x i a l s p i n n i n g s a t e l l i t e 55 3-4 S t a b i l i t y diagram f o r s p i n n i n g axisymmetric s a t e l l i t e s i n c i r c u l a r o r b i t s 57 3-5 A comparison between the approximate p e r t u r -b a t i o n and numerical s o l u t i o n s f o r a t r i a x i a l g r a v i t y o r i e n t e d s a t e l l i t e 62 3-6 S t a b i l i t y diagram f o r t r i a x i a l g r a v i t y o r i e n t e d s a t e l l i t e s i n a r b i t r a r y o r b i t s of small e c c e n t r i c i t i e s (e < 0.2) 64 3-7 Force e q u i l i b r i u m of a mass element Am to maintain i t a t a d e s i r e d p o s i t i o n 65 3-8 T y p i c a l p l o t s showing the c o n d i t i o n of a s p i n n i n g s a t e l l i t e i n the absence of any e x t e r n a l d i s t u r b a n c e 71 3-9 Response of a s p i n n i n g s a t e l l i t e to an a r b i t r a r y d i s t u r b a n c e . Note the high frequency amplitude modulations of l i b r a t i o n s ( r o l l , yaw) and t r a n s v e r s e f o r c e components 72 3-10 E f f e c t of system parameters on the s a t e l l i t e response: (a) i n c r e a s e i n the o r b i t a l e c c e n t r i c i t y . . . . 74 (b) r e d u c t i o n i n the i n e r t i a asymmetry parameter K 75 X Figure Page (c) reduction i n the i n e r t i a parameter I 7 6 (d) increase i n the spin parameter a 77 (e) d i f f e r e n t i n i t i a l conditions . 78 3-11 Force components on an instrumentation package deployed from a nonlibrating t r i a x i a l gravity oriented s a t e l l i t e (Space Shuttle) 80 3-12 E f f e c t of i n i t i a l conditions and system parameters on l i b r a t i o n a l motion and force components for the Space Shuttle supported instrumentation package 81 3-13 Configuration control of the SSPS for the maximum power generation 8 3 3-14 S t a b i l i t y diagram for a t r i a x i a l spacecraft in the geosynchronous equatorial o r b i t with the solar panels facing the sun. Note that the configura-tion (I = K = 0.15) corresponding to the SSPS l i e s in the unstable region 85 3-15 D i s t r i b u t i o n of force per unit mass on the solar panel (y = 100 m): . (a) z = 0, 2 km; (b) x = 3 km 87 3-16 Rotations and control moments as functions of the solar aspect angle for maximum power generation 88 3- 17 Time history of the control moments for several values of the solar aspect angle 89 4- 1 Reference coordinates and the geometry of motion for a multibody earth o r b i t i n g system 94 4-2 Force e q u i l i b r i a of the mass elements Am. and Am to maintain them at desired positions 95 4-3 Modal representation for cable vibrations: (a) f i r s t and second modes 110 (b) t h i r d and fourth modes I l l 4-4 I n i t i a l displacement as represented by the f i r s t four modes 112 4-5 Response of the system when the tether i s d i s -turbed at the center: x i Figure Page (a) time history of the transverse displacement of the tether, U component 113 (b) time history of the transverse displacement of the tether, W component 114 4-6 Time history of l i b r a t i o n s and forces acting at the center of mass of the orbi t e r after the cable i s disturbed at the center 116 4-7 Time history of l i b r a t i o n s and forces for a set of impulsive i n i t i a l conditions 117 4-8 Time history of the s a t e l l i t e l i b r a t i o n s as affected by two d i f f e r e n t values of spin parameter 118 4-9 U and W vibrations for d i f f e r e n t 0 a f t e r the cable encounters a micrometeorite impact 119 4-10 Constituent mode shapes of U and W vibrations excited due to a micrometeorite impact 120 4-11 Modal representation for beam vibrations 122 4-12 U and W vibrations for d i f f e r e n t 0 when the beam encounters a micrometeorite impact 123 4-13 Time history of l i b r a t i o n s and forces at the center of mass of the or b i t e r due to the micrometeorite impact on the beam 12 4 4-14 L i b r a t i o n a l history of a spherical s a t e l l i t e due to: (a) a micrometeorite impact on the cable; (b) a micrometeorite impact on the beam . . 12 6 x i i ACKNOWLEDGEMENT The a u t h o r w i s h e s t o e x p r e s s h i s g r a t i t u d e t o P r o f e s s o r V . J . M o d i f o r t h e h e l p g i v e n d u r i n g t h i s r e s e a r c h . The a c k n o w l e d g e m e n t i s a l s o due t o A n j n a ( t h e a u t h o r ' s w i f e ) f o r t y p i n g t h i s t h e s i s . The i n v e s t i g a t i o n was s u p p o r t e d by t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a , G r a n t No. A - 2 1 8 1 . x i i i LIST OF SYMBOLS a, • , a, . amplitudes of vibrations of beam i n equation b u l b w i (4.12) c u i ' cwi amplitudes of vibrations of cable i n equation (4.10) a.,b.,b. . amplitudes of 6, y i n equations (3.7) and (3.13) 1 1 1 , 3 a.. amplitudes of 3-, y• (i=l,2) i n equations (4.14b) 1 J . and (4.14c) 1 1 a. amplitudes of and <p i n equations (2.7a) and 2 (2.7b) , respectively c. m.,S s a t e l l i t e center of mass d, . . elements of square matrix [D, ] expressed i n 3 equations (4.12e) and (4.12fT d .. elements of square matrix [D ] expressed i n C 1 : J equations (4.10e)and (4.10f) c e e c c e n t r i c i t y of the o r b i t (£, + L c o s n, . ) r, . e* T, • — + -oT^-lt.- + £~cosh k, .) b i £n, . £k, . 1 2 b i b i b i "^bi s b i + £„ ( si n n, . + i sinh k,.)/£ 2 vn, . b i k, . b i b i b i 2 + (sin n, . + q, .cos n, . - q, . ) /n, . b i ^ b i b i ^ b i b i 2 + (s, . - s, . cosh k, . - r, . sinh k, . )/k, . ; i=l,2,. b i b i b i b i b i b i sin n . £ •+ £„cos n . e* 2 i _ _ i I 21 . i = i 2 e c i 2 In . , 1 1 , ^ , . . n . c i c i x i v £, + L c o s n. . s i n n. . 1 2 b i b i . , „ £n,. 2 '* 1 X ' ' • * b i n, . b i £ nsin n, . c o s n, . - 1 2 ba b i ._, _ £n, . 2 ; 1 1 ' 1 ' ' ' b i n, . b i £, + £„cosh k, . s i n h k, . 1 2 b i _ _ b i _ . _, ~ £k^. ,2 ; 1 1 ' ' ' • b i k, . b i £ 0sinh k, . 1 - c o s h k, . 2 b i _ b i _ ._, ~ £k,. . 2 ; 1 1 ' ' ' * b i k, . b i 2 2 2 r n b i _ U X + £ 2 ~ £ 1 V ,-1 . % i " 3 3 £ f o r c i n g f u n c t i o n s ( i = l , 2 ) u s e d i n e q u a t i o n s ( 2 . 6 ) , (2.9) a n d ( 3 . 5 ) , r e s p e c t i v e l y a c c e l e r a t i o n s due t o g r a v i t y a t S, Am a n d Am. ( i = l , 2 ) , r e s p e c t i v e l y e c . (e, . + q, . e 0 . + r, . e v + s, • e. . ) ; i = l , 2 , . . 5 i l i ^ b i 2 i b i 3 i b i 4 i ' ' ' f u n c t i o n o f £,, £ 9 , £, n ., T n , e i n e q u a t i o n (4.10h) c i u a n g u l a r momentum p e r u n i t mass o f t h e s a t e l l i t e a n g l e b e t w e e n t h e e q u a t o r i a l p l a n e a n d t h e e c l i p t i c (23.5 ) [ { ( T 0 + 4 n b i k s ) V 2 + T 0 } / ( 2 k s ) '" i = 1 > 2 > - -4*2 s t i f f n e s s p a r a m e t e r f o r beam, (EI)^/p£ 8 l e n g t h o f t h e c o n n e c t i n g l i n k (boom, c a b l e o r beam); • a l s o u s e d as an i n d e x l e n g t h o f t h e c o n n e c t i n g l i n k i n F i g u r e s 2-6 and 4-1 p o i n t m a s s e s shown i n F i g u r e 2-6 mass o f t h e s a t e l l i t e r e d u c e d m a s s e s o f t h e s y s t e m e x p r e s s e d i n e q u a t i o n s (2.28b) 2 - I ( « ± l ) ( 1 + L . 3K '1 - 2e' 2' 4 (a + 2e) _ m l l 2 - I - K 1 - K I + K - 2 2 - I i ( a i + 1) ; i = l , 2 ~ m i H ' i = 1 / 2 a r b i t r a r y mass e l e m e n t s o f t h e s a t e l l i t e ( o r c o n n e c t i n g l i n k ) a n d t h e e n d - b o d y , r e s p e c t i v e l y [ { ( T p + 4 n ^ ± k s ) V 2 - T Q } / ( 2 k s ) ] 1 / 2 ; i = l , 2 , . V(Y,0) M b i dY ; v = u,w ; V = U,W ; i = l , 2 , . . i * " * 1 r o o t .of t h e c h a r a c t e r i s t i c e q u a t i o n (4.10g) 1 r V(Y,0) M ^ dY ; v = u,w ; V = U,W ; i = l , 2 , 0 1 r 3V 90 10=0' " b i 0 1 (-T5-L-J \ ^ dY ; v = u,w ; V = U,W ; i = l , 2 , J ,3V . .) M . dY ; v = u,w ; V = U,W ; i = l , 2 , 3 = 0 c i x v i n. ( 3 K i ) 1/2 ( 3 K ± + 1) 1/2 n 11 {i(£_L_L_ + 3) a + K ) - A + 3 K ,1/2 U l l - 2e + 3 ) ( 1 + 2> 4 + 4 (a + 2e) } n 12 f T (g + 1 ) n , K 3K ,1/2 _ 2 e ) (1 + 2") " 1 + 4 ( a + 2e) } • n 21 {4 ( I - 1 ) / ( 1 - K) }1 / 2 n 22 ( I + K - 1) 1/2 n i l l { I i ( a i + 4) - 4} 1/2 ; i = l , 2 n. , « i l 2 { I i ( a i + 1) - 1} 1/2 ; i = l , 2 P b u i b w i f r e q u e n c y o f l i b r a t i o n , e q u a t i o n s (3.8a) and (3.14a) f r e q u e n c y o f t h e i t h mode o f U v i b r a t i o n f o r t h e 2 1/2 beam, (1 + n ^ ) ; i = l , 2 , . . f r e q u e n c y o f t h e i * " * 1 mode o f W v i b r a t i o n f o r t h e beam, n f a i ; i = l , 2 , . . c u i C W l t h f r e q u e n c y o f t h e i mode o f U v i b r a t i o n f o r t h e c a b l e , ( 1 + T n n 2 . ) 1 / 2 ; i = l , 2 , . . ' 0 c i t h f r e q u e n c y o f t h e i mode o f W v i b r a t i o n f o r t h e c a b l e , ( T Q n \ ) 1 / 2 ; i = I , 2 , . . P X,P 2 c h a r a c t e r i s t i c r o o t s o f e q u a t i o n s (3.8a) a nd (3.14a) P i l ' P i 2 c h a r a c t e r i s t i c r o o t s o f e q u a t i o n ( 4 . 1 4 f ) f r e q u e n c y o f a m o t i o n , (-3K/I) 1/2 x v i i P^rPy tP]p,Pq) frequency of 3, y, \p and tt motions, respectively Pg frequency of o r b i t a l motion, w p. momentum conjugate to generalized coordinate X A q b i " S b i " £ e 5 i £ l ( n b i e 3 i " k b i S l i ) [ k b i £ : 5 i ( e 3 i £ " k b i e 2 i £ r r. + ee,. -.S, - , . ,.A.)] 1 • i=l,2,. t h i generalized coordinate ; i=l,2, r b i [- n b i + e e 5 i { " e l i + n b i e 2 i V £ + s b i ( e 2 i - e 4 i ) } ] [ k b . + e e 5 i ( e 3 . - kfa. e ^ / A ) ] - 1 ; i 1, 2 , . #^ distance between and an ar b i t r a r y mass element on the i t ^ 1 end-body ; i=l,2 r distance between S and an ar b i t r a r y mass element on the s a t e l l i t e s, . [n, . sinh k, . - k,. sin n, . + e e c . { ( l ~ + L c o s n, . ) x bi b i b i b i b i 5i 2 1 b i y x (n, . e-.. - k,.e,.)/£ + (e, . - n, . e~ . £, /£) sinh k, . b i 3i b i l i ' l i b i 2i 1' b i - (e 0. - k, . e 0 . Jl,/Jl) sin n, . } ] [k, . (cosh k, . 3i b i 2i i b i b i b i - cos n, . ) + ee c. { (e, . - e n • ) (k, .A- + £, sinh k, . ) /I b i 5i 4 i 2 i ' b i 2 1 b i + e_. (cosh k, . - cos n, . ) - k, . Jl, (e~.cosh k, . 3i b i b i b i 1 2i b i - e 4 i c o s n b i)/£}] 1 ; i=l,2,.. t; t^,t2 time; time at instants 1 and 2, respectively x v i i i u,w v i b r a t i o n s i n x and z d i r e c t i o n s , r e s p e c t i v e l y U s ' u x ' u y , . . u n i t v e c t o r s i n t h e d i r e c t i o n o f t h e s u n , x a x i s , y a x i s , . . . , r e s p e c t i v e l y x , y , z p r i n c i p a l b ody c o o r d i n a t e s o f t h e s a t e l l i t e w i t h o r i g i n a t S and h a v i n g y a x i s a l o n g t h e l i n e S 1 S 2 x . , y ^ , z . p r i n c i p a l b ody c o o r d i n a t e s o f t h e i end - b o d y l o c a t e d a t ; i = l , 2 1 1 , j r i l ' i l [ i n t e r m e d i a t e b ody c o o r d i n a t e s l o c a t e d a t S d u r i n g x z ( t h e m°dified E u l e r i a n r o t a t i o n s i>, cj>, A and y, 3, 1 2 , y i 2 ' i 2 j a , r e s p e c t i v e l y X Q , y Q , Z Q r o t a t i n g c o o r d i n a t e s y s t e m a t S w i t h y^ and Z Q a x e s a l o n g l o c a l 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 {A, }, {A } c o l u m n v e c t o r s c o n t a i n i n g e l e m e n t s a, ,, a, bv c v . • - b v l ' b v 2 ' a c v l ' a c v 2 ' * ' " ; r e s P e c t l v e l Y ; v = u,w A.,A. . f u n c t i o n s o f e, K, I , a u s e d i n e q u a t i o n s (3.7) , D and (3.13) A* ,A** . f u n c t i o n s o f e, K. and i n i t i a l c o n d i t i o n s u s e d m, n l , j i i n e q u a t i o n (2.17) {B* v),{B* v} c o l u m n v e c t o r s c o n t a i n i n g e l e m e n t s P k V ] _ k k v l ' P b v 2 b b v 2 ' . " - ; P c v l b c v l ' P C v 2 b c v 2 ' - - - ; r e s p e c t i v e l y ; v = u,w [ D ^ ] , [ D ] s q u a r e m a t r i c e s c o n t a i n i n g e l e m e n t s d, . . 1 s and d c^_.'s, r e s p e c t i v e l y 1-' ( E I ) ^ f l e x u r a l s t i f f n e s s o f t h e beam F l i b r a t i o n a l f o r c e F F F s) a' b' c'I f o r c e s a c t i n g o n ma,m^,m , e t c . a n d e x p r e s s e d „ „ ) i n e q u a t i o n s (2.28a) c a ' c b J c o m p o n e n t s o f F i n x, y, z, E,, ty and tt d i r e c t i o n s , r e s p e c t i v e l y f o r c e n e c e s s a r y t o k e e p a mass e l e m e n t Am i n i t s p o s i t i o n f o r c e n e c e s s a r y t o k e e p a mass e l e m e n t Am. i n i t s p o s i t i o n t h e f o r c e i n e q u a t i o n (4.18) f o r s m a l l l i b r a t i o n s o f t h e e n d - b o d i e s i n e r t i a p a r a m e t e r o f s a t e l l i t e , I x / I p r i n c i p a l moments o f i n e r t i a o f s a t e l l i t e a b o u t x, y, z a x e s , r e s p e c t i v e l y v a l u e s o f p r i n c i p a l moments o f i n e r t i a a b o u t x., t h ~^ ^ i ' z i a x e s a n ( ^ i n e r t i a p a r a m e t e r f o r t h e i e n d - b o d y , r e s p e c t i v e l y ; i = l , 2 X L 2 + M 2 L 2 + ^ U 2 - 3£ £ ) M as y m m e t r y p a r a m e t e r , 1 - I^/I m o d i f i e d i n e r t i a p a r a m e t e r , 1 - 1/1 d i s t a n c e b e t w e e n t h e c e n t e r s o f mass o f t h e e n d - b o d i e s d i s t a n c e b e t w e e n S a n d ; i = l , 2 L a g r a n g i a n , T - P moments d e f i n e d i n e q u a t i o n s (3.18) a n d ( 4 . 1 7 ) , r e s p e c t i v e l y i ^ mode o f t h e beam v i b r a t i o n s , s i n n, .Y + q, . c o s n, . Y + r, . s i n h k, . Y b i ^ b i b i b i b i + s, . c o s h k,.Y + eh, . (Y - I,/I) b i b i b i 1 XX t h M ^ i mode o f t h e c a b l e v i b r a t i o n s , s i n n .Y + eh . { L ( c o s n -Y - + Y} C I C 1 J . ox M ,M ,M c o m p o n e n t s o f M i n x, y, z d i r e c t i o n s , x ^ r e s p e c t i v e l y M* M nM 0/(M n + M 0) M. l T ' 2 ' V i i l "2' t h mass o f t h e i e n d - b o d y ; i = l , 2 {N, }, {N } c o l u m n v e c t o r s c o n t a i n i n g e l e m e n t s n, , , n, bv c v ^ b v l ' bv2 n , , n f o r t h e beam and t h e c a b l e , c v l ' c v 2 ' ' r e s p e c t i v e l y ; v = u,w N C i n i t i a l d i s p l a c e m e n t c o n d i t i o n f o r t h e c a b l e as e x p r e s s e d i n F i g u r e 4-4 N ^ , N ^ i n i t i a l i m p u l s e c o n d i t i o n s a p p l i e d t o t h e c a b l e ' and t o t h e beam a s e x p r e s s e d i n F i g u r e s 4 - 9 and 4-12, r e s p e c t i v e l y {N/ },{N* } c o l u m n v e c t o r s c o n t a i n i n g e l e m e n t s n' ,, n' b v c v ^ b v l b v 2 n 1 n r e s p e c t i v e l y ; v = u,w c v l c v 2 1 ' r 0 c e n t e r o f a t t r a c t i o n P , P g , P t o t a l , e l a s t i c and g r a v i t a t i o n a l p o t e n t i a l e n e r g i e s , r e s p e c t i v e l y R R R si ' a' b ' l d i s t a n c e s f r o m 0 t o S ( o r m c ) , m^, m^, Am and „ n ) Am., r e s p e c t i v e l y is. T K . 1 m' mi J Rp r a d i u s a t p e r i g e e t h c e n t e r o f mass o f t h e i en d - b o d y ; i = l , 2 T k i n e t i c e n e r g y T t e n s i o n i n t h e c o n n e c t i n g l i n k e ^ 2 TQ t e n s i o n p a r a m e t e r , 3M*L/p& ^ x x i n o n d i m e n s i o n a l i z e d v a l u e s o f u, w a n d y g i v e n as u/£, w/£ and y/£, r e s p e c t i v e l y i n e r t i a l f r a m e o f r e f e r e n c e a t 0 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 o f t h e t r i a x i a l s a t e l l i t e w i t h r e s p e c t t o X Q , , ZQ s y s t e m ( F i g u r e 3-2) and o f t h e e n d - b o d y ( i = l , 2 ) w i t h r e s p e c t t o t h e l i n e S - j ^ ( F i g u r e 4-1) r e f e r r e d t o a s p i t c h , yaw and r o l l , r e s p e c t i v e l y ; a and y a r e a l s o i n - p l a n e and o u t - o f - p l a n e m o t i o n s , r e s p e c t i v e l y , o f t h e c o n n e c t i n g l i n k ( F i g u r e 4-1) i n f i n i t e s i m a l v a r i a t i o n o f ; i = l , 2 , . . p h a s e a n g l e s u s e d i n e x p r e s s i o n s f o r 3 a n d y i n e q u a t i o n s (3.7) and (3.13) ; i = l , 2 p h a s e a n g l e s u s e d i n e x p r e s s i o n s f o r a n d y^ ( i = l , 2 ) and e x p r e s s e d i n e q u a t i o n (4.14e) ; j = l , 2 p h a s e a n g l e s u s e d i n e x p r e s s i o n s f o r IJJ and ty i n e q u a t i o n s (2.7) f l e x i b i l i t y p a r a m e t e r , p £ /I* r a t i o o f t h e f o r c e due t o s o l a r r a d i a t i o n p r e s s u r e t o t h e f o r c e o f g r a v i t a t i o n on a s a t e l l i t e n.9 + 6. ; i = l , 2 i c h a r a c t e r i s t i c r o o t o f e q u a t i o n (4.12g) t r u e a n o m a l y r o t a t i o n a b o u t a x i s o f symmetry o f t h e a x i s y m m e t r i c g r a v i t y o r i e n t e d s a t e l l i t e ( F i g u r e 2-2) f u n c t i o n s o f s a t e l l i t e p a r a m e t e r s e x p r e s s e d i n e q u a t i o n s (3.8b) and (3.14b) x x i i t h f u n c t i o n s o f t h e i e n d - b o d y ( i = l , 2 ) p a r a m e t e r s e x p r e s s e d i n e q u a t i o n (4.14g) y e a r t h ' s g r a v i t a t i o n a l c o n s t a n t E, r e f e r e n c e l e n g t h shown i n F i g u r e (2-6b) p mass p e r u n i t l e n g t h o f t h e c o n n e c t i n g l i n k (boom, c a b l e o r beam) t h O,CK s p i n ' p a r a m e t e r s o f t h e s a t e l l i t e a n d o f t h e i e n d - b o d y ( i = l , 2 ) , d e f i n e d as a ' ( 0 ) a n d a | ( 0 ) w i t h a ( 0 ) = 0 and a ^ ( 0 ) = 0 , r e s p e c t i v e l y a* c h a r a c t e r i s t i c r o o t s ( r e a l , i m a g i n a r y o r c o m p l e x c o n j u g a t e ) o f e q u a t i o n s (2.21) x p e r i o d o f o r b i t a l m o t i o n ty, ty r o t a t i o n s a c r o s s a nd i n t h e o r b i t a l p l a n e , r e s p e c t i v e l y , o f t h e a x i s y m m e t r i c g r a v i t y o r i e n t e d s a t e l l i t e ( F i g u r e 2-2) typ • , ty-p • p e r i o d i c s o l u t i o n s o f ty and ty, r e s p e c t i v e l y , w i t h p e r i o d 2TTJ ; j = l , 2 , . . ty , ty s m a l l p e r t u r b a t i o n s a r o u n d ty_ . a n d d>_ . , V Y v 1. • n P, i Y P , i r e s p e c t i v e l y ' J J tj) s o l a r a s p e c t a n g l e , a n g l e b e t w e e n t h e s u n and t h e l i n e o f n o d e s a v e r a g e o r b i t a l a n g u l a r r a t e , 2 T T / T D o t s and p r i m e s i n d i c a t e d i f f e r e n t i a t i o n w i t h r e s p e c t t o t a n d 0, r e s p e c t i v e l y . S u b s c r i p t s '0' and 'o' i n d i c a t e i n i t i a l c o n d i t i o n s o f t h e v a r i a b l e s i n t h e t e x t a n d t h e f i g u r e s , r e s p e c t i v e l y . F o r c e . 2 * 2 d i a g r a m s a r e n o n d i m e n s i o n a l ( w i t h r e s p e c t t o m £w and M £co i n a C h a p t e r s 2 and 4, r e s p e c t i v e l y ) , u n l e s s o t h e r w i s e i n d i c a t e d . 1 1. INTRODUCTION 1.1 Preliminary Remarks The l a s t two decades have seen the addition of a new dimension to man's c a p a b i l i t y for space exploration, the spacecraft. In t h i s short time, the use of a r t i f i c i a l s a t e l -l i t e s in majority of the telecommunications, weather studies, m i l i t a r y surveillance and geological surveys has become ex-ceedingly important. Ahead l i e new and e x c i t i n g p o s s i b i l i t i e s of spacecraft applications i n e x p l o i t i n g t e r r e s t r i a l and extra-t e r r e s t r i a l resources, e.g., the abundant and pollutant-free solar energy. Exacting demands on s a t e l l i t e attitude with respect to a fixed or an o r b i t i n g frame of reference has resulted in a vast body of l i t e r a t u r e constituting the f a m i l i a r f i e l d 1 2 of l i b r a t i o n a l dynamics ' . As i s usually the case, devel-opment of newer technology and knowledge i s usually associ-ated with the present estimate of the future demands. In t h i s respect a common consensus c l e a r l y suggests a strong trend towards larger and necessarily f l e x i b l e spacecraft. We have already witnessed a few of such large configurations. Radio Astronomy Explorer (RAE) s a t e l l i t e used 750 f t antennas to 3 detect low frequency signals (Figure 1-1). For i d e n t i f y i n g e x t r a t e r r e s t r i a l radio sources, the Applied Physics Labo-ratory (APL) of the Johns Hopkins University has proposed a g r a v i t a t i o n a l l y s t a b i l i z e d Tethered Orbiting Interferometer (TOI) consisting of two spacecraft connected by a tether 2-6 km Antenna boom (750ft long tube,l/2in dia.) orbit t To Earth F i g u r e 1-1 A s c h e m a t i c d i a g r a m o f t h e RAE s a t e l l i t e w i t h l o n g f l e x i b l e a n t e n n a s 3 l o n g ( F i g u r e 1 - 2 ) . More r e c e n t l y , t h e Canada/U.S.A. C o m m u n i c a t i o n s T e c h n o l o g y S a t e l l i t e (CTS) l a u n c h e d i n J a n u a r y 1976 c a r r i e s two s o l a r p a n e l s , 1.14 m x 7.32 m e a c h , t o g e n e r a t e 1.2 kW ( F i g u r e 1 - 3 ) . However, o u r f u t u r e e f f o r t s a r e a i m e d a t c o n s t r u c t i o n o f g i g a n t i c s p a c e s t a t i o n s w h i c h c a n n o t be l a u n c h e d i n t h e i r e n t i r e t y f r o m t h e e a r t h b u t h a v e t o be c o n s t r u c t e d i n s p a c e t h r o u g h i n t e g r a t i o n o f m o d u l a r s u b a s s e m b l i e s . The c o n c e p t o f S o l a r S a t e l l i t e Power 5 S t a t i o n (SSPS) i n t h e g e o s t a t i o n a r y o r b i t as p r o p o s e d by G l a s e r ( F i g u r e 1-4) and t h e f u t u r i s t i c d e s i g n s o f s p a c e c o l o n i e s a t t h e 6 7 e a r t h - m o o n l i b r a t i o n p o i n t s as p u t f o r w a r d by O ' N e i l l e t a l . ' b e l o n g t o t h i s c a t e g o r y . N o t f a r i n t h e f u t u r e i s t h e S p a c e S h u t t l e s c h e d u l e d f o r o p e r a t i o n t o w a r d s t h e end o f t h i s d e c a d e w h i c h w i l l s e r v e a s a l a b o r a t o r y f o r t e s t i n g o n b o a r d and d e p l o y e d i n s t r u m e n t p a c k a g e s . F u r t h e r m o r e , i t i s a l s o e x p e c t e d t o p l a y t h e r o l e o f an o r b i t i n g l a u n c h v e h i c l e . A c o m p l e x s y s t e m s o g e n e r a t e d c a n t h u s c o n s i s t o f a s e r i e s o f i n t e r c o n n e c t e d s u b a s s e m b l i e s o r , as an e x a m p l e , m e r e l y r e p r e s e n t t h e S k y l a b t y p e v e h i c l e d e p l o y i n g an i n s t r u m e n t p a c k a g e p o s i t i o n e d p r e c i s e l y f o r s p e c i f i c s c i e n t i f i c e x p e r i m e n t s . A l -t h o u g h t h e v a r i e t y o f c o n f i g u r a t i o n s t h a t may be e n c o u n t e r e d i n f u t u r e i s l i m i t l e s s , t h e y do o f f e r a d e g r e e o f s i m i l a r i t y a t a f u n d a m e n t a l l e v e l : t h e y a l l r e p r e s e n t m u l t i b o d y s y s t e m s i n t e r -c o n n e c t e d t h r o u g h r i g i d o r f l e x i b l e l i n k s ( F i g u r e 1 - 5 ) . The t h e s i s a i m s a t l a y i n g a f o u n d a t i o n f o r t h e d y n a m i c a l s t u d y o f s u c h s y s t e m s . 1.2 L i t e r a t u r e R e v i e w L i b r a t i o n a l d y n a m i c s a n d c o n t r o l o f g r a v i t y o r i e n t e d a s F i g u r e 1-2 T e t h e r e d O r b i t i n g I n t e r f e r o m e t e r (TOI) e x p e r i m e n t p r o p o s e d by t h e A p p l i e d P h y s i c s L a b o r a t o r y local vert ical I (1.14m x 7-32m) orbit normal To Earth Canada/U.S.A. C o m m u n i c a t i o n s T e c h n o l o g y S a t e l l i t e (CTS) N o t e t h e l a r g e s o l a r p a n e l s . Solar flux Solar col lector (mass = 4 k t o n ) Receiving antenna Microwave beam 5000 Mw Transmitting antenna (dia.=ikm,mass = 2.4k ton) Synchronous orbit 0.2 km F i g u r e 1-4 A s c h e m a t i c d i a g r a m o f t h e S a t e l l i t e S o l a r Power S t a t i o n (SSPS) i n t h e g e o s y n c h r o n o u s o r b i t ( T l Instrument package having specified location and orientation Rigid or flexible link, beam or tether --..Trajectory Triaxial space station F i g u r e 1-5 A s i m p l e m o d e l f o r m u l t i b o d y s y s t e m s r e p r e s e n t i n g two r i g i d b o d i e s c o n n e c t e d by a r i g i d o r a f l e x i b l e l i n k i n t h e f o r m o f a beam o r a t e t h e 8 well as spinning s a t e l l i t e s has been a subject of considerable study for quite some time. As indicated before, the vast body of l i t e r a t u r e accumulated over years has been reviewed by several 1 2 authors including Modi et a l . ' Hence only more s i g n i f i c a n t aspects of the l i t e r a t u r e d i r e c t l y relevant to the present study are b r i e f l y reviewed here. The pioneering work on gravity oriented s a t e l l i t e s g was c a r r i e d out by Klemperer (1960), who obtained the exact solution for planar l i b r a t i o n s of a dumbbell s a t e l l i t e i n a c i r -9 cular o r b i t , and by Baker (1960) who determined periodic solu-tions of the problem for a small o r b i t e c c e n t r i c i t y . Beletskii"'"^ (1963) focused the attention on resonance effects for s a t e l l i t e s i n e l l i p t i c o r b i t s while Schechter1"1" (1964) attempted, with lim i t e d success, to extend Klemperer's solution to non-circular 12 o r b i t a l motion by a perturbation method. Zlatousov et a l . 13 (1964) and Brereton and Modi (1967) successfully employed numer-i c a l methods, involving the use of the stroboscopic phase plane, to analyze the s t a b i l i t y of planar motion in the large for orbits of a r b i t r a r y e c c e n t r i c i t y . They also investigated the corre-14 15 sponding periodic motion ' (1969) and showed that at the c r i t i -c a l e c c e n t r i c i t y for s t a b i l i t y the only available solution i s a periodic one. More recently, Shrivastava"^ (1970) studied the coupled l i b r a t i o n a l motion of axisymmetric gravity oriented s a t e l -l i t e s under the influence of i n e r t i a , e c c e n t r i c i t y and atmospheric forces. The analysis emphasized rather adverse e f f e c t s of eccen-t r i c i t y and atmospheric torque on the system s t a b i l i t y . 17 Nurre (1968) considered a more complex model of an 9 a s y m m e t r i c g r a v i t y o r i e n t e d s a t e l l i t e i n a c i r c u l a r o r b i t and i n v e s t i g a t e d t h e s t a b i l i t y o f i t s e q u i l i b r i u m p o s i t i o n u s i n g a l i n e a r i z e d a n a l y s i s . The f e a s i b i l i t y o f a d d i n g a r o t o r t o s t a -b i l i z e t h e l i b r a t i o n a l m o t i o n o f a t r i a x i a l g r a v i t y o r i e n t e d s a t e l -18 l i t e was e s t a b l i s h e d by C r e s p o d a S i l v a (1970) . 19 Thomson (1962) a n a l y z e d , t h r o u g h l i n e a r i z a t i o n , t h e r e l a t e d p r o b l e m o f s l o w l y s p i n n i n g s a t e l l i t e s i n c i r c u l a r 20 o r b i t s . Kane a n d B a r b a (1966) a t t e m p t e d t o s t u d y t h e m o t i o n i n e l l i p t i c o r b i t s u s i n g t h e F l o q u e t t h e o r y w h i l e W a l l a c e a n d 21 M e i r o v i t c h (1967) e m p l o y e d , w i t h q u e s t i o n a b l e s u c c e s s , a s y m p t o t i c a n a l y s i s i n c o n j u n c t i o n w i t h L i a p u n o v ' s d i r e c t m e t h o d . On t h e o t h e r h a n d , M o d i and N e i l s o n i n v e s t i g a t e d r o l l d y n a m i c s o f a 22 23 s p i n n i n g s a t e l l i t e u s i n g t h e W. K. B. J . (1968) a n d n u m e r i c a l (1968) m e t h o d s . The c o n c e p t o f i n t e g r a l m a n i f o l d s was s u c c e s s f u l l y e x t e n d e d t o t h e s t u d y o f t h r e e d e g r e e s o f f r e e d o m m o t i o n i n c i r -24 c u l a r o r b i t s (1969) . The p e r i o d i c s o l u t i o n s w e r e f o u n d a n d 25 t h e i r v a r i a t i o n a l s t a b i l i t y e s t a b l i s h e d ( 1 9 7 0 ) . A p r e l i m i n a r y s t u d y o f s p i n n i n g u n s y m m e t r i c a l s a t e l l i t e s 2 6 was c a r r i e d o u t by Kane and S h i p p y (1963) who e x a m i n e d t h e i r s t a -2 7 2 8 b i l i t y i n a c i r c u l a r o r b i t . C o c h r a n ' a t t e m p t e d t o d e t e r m i n e an a p p r o x i m a t e s o l u t i o n t o e q u a t i o n s o f a t t i t u d e m o t i o n f o r a t r i a x i a l s p i n n i n g s a t e l l i t e i n p r e s e n c e o f e x t e r n a l t o r q u e s , 29 b u t had l i t t l e s u c c e s s . S p e r l i n g (1972) showed t h a t t h e s p i n o f a t r i a x i a l s a t e l l i t e was p o s s i b l e a b o u t t h e p r i n c i p a l a x i s o f i n e r t i a o r t h o g o n a l t o t h e o r b i t a l p l a n e . The p r e s e n c e o f t h e v a r i o u s p e r t u r b i n g f o r c e s and t h e i r i n f l u e n c e on s a t e l l i t e d y n a m i c s a n d c o n t r o l h a s b e e n d i s c u s s e d 10 i n d e t a i l by several authors. Pande31^ and Van der Ha3"1" have presented excellent reviews of the work on attitude and o r b i t a l control, respectively, using the environmental forces. The study of multibody systems involving f l e x i b l e l i n k s has been i n progress for quite some time. In an e a r l i e r treatment of gravity gradient s t a b i l i z e d extensible dumbbell s a t e l l i t e 32 systems (cable mass neglected), Paul (1963) showed a strong coupling between l i b r a t i o n a l and v i b r a t i o n a l motions. The f e a s i -b i l i t y of a tether connected g r a v i t a t i o n a l l y s t a b i l i z e d multibody 33 configuration was established by Robe (1968), who showed that the e l a s t i c and damping properties of the tether have l i t t l e i n -34 fluence on the s t a b i l i t y of the system. Singh et a l . (1972) studied the dynamics of a heavy inextensible f l e x i b l e s t r i n g in an e l l i p t i c o r b i t . The r e l a t i v e motion of the s t r i n g with respect to the s a t e l l i t e along the l o c a l v e r t i c a l was found to be unstable 4 due to the i n s t a b i l i t y at the free end. Bainum et a l . (1972) established the lower l i m i t s of tether damping and spring constants necessary for the s t a b i l i t y of a tethered two body system. On the 35 other hand, Stuiver (1972) has determined i m p l i c i t expressions for the required control forces and corresponding t r a j e c t o r i e s i n the study of a two body s a t e l l i t e system maintained along the l o c a l v e r t i c a l . A pioneering contribution i n the area of rotating two body 3 6 spacecraft i s due to Chobotov (1963), who studied the ef f e c t s of cable mass and e l a s t i c i t y for the case of point-mass end bodies undergoing two dimensional motion i n a c i r c u l a r o r b i t . The gravity gradient e f f e c t s on the s t a b i l i t y i n the small were found to be 11 n e g l i g i b l e . Hence t h e s t a b i l i t y c r i t e r i a c o u l d be e x p r e s s e d e n t i r e l y i n t e r m s o f t h e c a b l e ' s n a t u r a l f r e q u e n c i e s , a n g u l a r and o r b i t a l m o t i o n s o f t h e s t a t i o n , a nd v i s c o u s d a m p i n g p a r a m e t e r s . 37 T a i e t a l . (1965) showed t h a t a s p i n n i n g , c a b l e c o n n e c t e d s p a c e s t a t i o n h a s a s t a b l e c i r c u l a r o r b i t , b u t t h e c a b l e w o u l d o s c i l l a t e u n d e r t h e i n f l u e n c e o f g r a v i t a t i o n a l g r a d i e n t and show a s t a t e o f n e u t r a l s t a b i l i t y . The a p p l i c a t i o n o f v e l o c i t y - p r o p o r t i o n a l damp-i n g d e v i c e s t o c o n t r o l e l a s t i c d e g r e e s o f f r e e d o m was f o u n d t o be 3 8 39 e f f e c t i v e . A u s t i n e t a l . ' ( 1 9 6 5 , 1970) d e r i v e d t h e e x a c t s o l u t i o n t o t h e d y n a m i c a l e q u a t i o n s f o r a f r e e - r o t a t i n g , f l e x i b l y c o n n e c t e d , d o u b l e mass s y s t e m . The r e s u l t s showed t h e r a d i a l v i -b r a t i o n wave f o r m a n d f r e q u e n c y v a r i a t i o n w i t h t h e r o t a t i o n a l s p e e d . A l l t h e f r e q u e n c i e s a s s o c i a t e d w i t h an H - s e c t i o n beam w e r e f o u n d t o i n c r e a s e w i t h t h e a n g u l a r v e l o c i t y . The d y n a m i c a l r e s p o n s e and s t a b i l i t y o f a r o t a t i n g ( i n t h e p l a n e o f t h e o r b i t ) f i n i t e s i z e d s p a c e s t a t i o n c o n n e c t e d 40 t h r o u g h a m a s s l e s s c a b l e w e r e e x a m i n e d by S t a b e k i s e t a l . (1970) who showed t h e s i g n i f i c a n c e o f r o t a t i o n a l d a m p i n g ( o f t h e end b o d y m o t i o n s ) t o o b t a i n r e a s o n a b l e t i m e c o n s t a n t s . I n - p l a n e m o t i o n o f a s p i n n i n g s p r i n g - m a s s s y s t e m f o r c o m b i n e d e x t e n s i b l e a nd l a t e r a l 41 v i b r a t i o n s was s t u d i e d b y C r i s t e t a l . (1970). The v i s c o u s a x i a l d a m p i n g was f o u n d t o be i n e f f e c t i v e i n r e d u c i n g t h e t r a n s v e r s e o s -42 c i l l a t i o n s . N i x o n (1972) i n v e s t i g a t e d t h e d y n a m i c a l e q u i l i b r i u m s t a t e s i n t h r e e d i m e n s i o n s f o r a c o m p l e t e l y undamped s y s t e m w i t h an a r b i t r a r y number o f c a b l e s , b u t d i d n o t c a r r y o u t any s t a b i l i t y a n a l y s i s . D u r i n g t h e s t u d y o f t h e t h r e e - d i m e n s i o n a l m o t i o n o f a 4 3 c a b l e c o n n e c t e d s y s t e m , B a i n u m e t a l . (1974) d e f i n e d t h e l o w e r 12 l i m i t f o r t h e d a m p i n g c o n s t a n t o f a s p i n n i n g s y s t e m and showed t h a t t h e p r e s e n c e o f t h e r o t a t i o n a l d a m p i n g ( o f t h e e n d b o d i e s ) 44 was n e c e s s a r y t o e n s u r e t h e s t a b i l i t y . The a u t h o r s (1975) l a t e r s t u d i e d t h e e f f e c t o f g r a v i t y g r a d i e n t t o r q u e on t h e s p a c e s t a t i o n a n d f o u n d i t s i n f l u e n c e t o be s m a l l i n t h e a b s e n c e o f r e s o n a n c e . C oming t o t h e l i b r a t i o n a l f o r c e s , one f i n d s t h e s u b j e c t 45 t o h a v e 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 . S a e e d a n d S t u i v e r 46 ( 1 9 7 4 ) , and Sharma and S t u i v e r (1974) h a v e r e p o r t e d 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 f o r a d u m b b e l l - t y p e s a t e l l i t e i n c i r c u l a r a n d e l -l i p t i c o r b i t s , r e s p e c t i v e l y . I n g e n e r a l , t h e a x i a l c o m p o nent o f t h e f o r c e was f o u n d t o be g r e a t e r t h a n t h e n o r m a l c o m p o n e n t s . The i n t e r p l a y b e t w e e n d o u b t and r e a s o n i n t h e p u r s u i t o f k n o w l e d g e i s n e v e r e n d i n g . T h r o u g h an i n f i n i t e number o f s u c h s u b t l e s t a g e s s c i e n c e e v o l v e s f r o m i t s m o d e s t b e g i n n i n g . A r e c e n t 47 d i s c u s s i o n by L i k i n s (1977) p r o v i d e s an e x c e l l e n t o u t l i n e o f t h e h i s t o r y o f d e v e l o p m e n t o f t h e s p a c e c r a f t d y n a m i c s and c o n t r o l . 1.3 P u r p o s e a n d S c o p e o f t h e I n v e s t i g a t i o n F rom t h e f o r e g o i n g , i t i s a p p a r e n t t h a t t h e g e n e r a l mo-t i o n o f a t r i a x i a l s a t e l l i t e as w e l l a s f l e x i b l y i n t e r a p p e n d e d s y s t e m s h a s 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 . L i k e w i s e , t h e p r o b l e m o f l i b r a t i o n a l f o r c e s i n s a t e l l i t e s h a s b e e n e s s e n -t i a l l y o v e r l o o k e d . The r e a s o n f o r t h i s l i m i t e d e f f o r t c o u l d , p e r h a p s , be a t t r i b u t e d t o t h e c o m p l e x i t y o f t h e p r o b l e m . The n o n -l i n e a r , n o n a u t o n o m o u s , and c o u p l e d e q u a t i o n s o f m o t i o n i n v o l v i n g a l a r g e number o f p a r a m e t e r s a r e n o t a m e n a b l e t o any s i m p l e c o n -c i s e a n a l y s i s . The m a i n p u r p o s e o f t h e t h e s i s i s t o g a i n 13 f u n d a m e n t a l u n d e r s t a n d i n g c o n c e r n i n g t h e d y n a m i c s o f t h e g e n e r a l mo-t i o n and t o a s s e s s c o n t r o l f o r c e s and moments. The p r o b l e m i s a n a l y z e d i n t h r e e s t a g e s r e p r e s e n t i n g , i n g e n e r a l , an i n c r e a s i n g o r d e r o f c o m p l e x i t y . I n t h e b e g i n n i n g , c o u p l e d l i b r a t i o n a l m o t i o n o f a p u r e g r a v i t y g r a d i e n t a x i s y m m e t r i c s a t e l l i t e i s c o n s i d e r e d and e x p r e s s i o n s f o r a s s o c i a t e d f o r c e s i n a d u m b b e l l - t y p e s a t e l l i t e d e r i v e d . The a n a l y s i s i s a p p l i e d t o t h e s t u d y o f t h e G r a v i t y G r a d i e n t T e s t S a t e l l i t e (GGTS). The u s e o f s u c h a s i m p l e m o d e l h e l p e d i n e s t a b l i s h i n g t h e m e thod o f a p p r o a c h . The e q u a t i o n s o f m o t i o n f o r t r i a x i a l s y s t e m s a n d t h e i r s o l u t i o n f o r s p i n n i n g and 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 s , a l o n g w i t h t h e e x p r e s s i o n s f o r f o r c e s a n d moments f o r m t h e s u b j e c t o f t h e t h i r d c h a p t e r . C a s e s t u d i e s o f E x p l o r e r XX, S S PS, and t h e S p a c e S h u t t l e d e p l o y i n g a p a y l o a d e m p h a s i z e u s e f u l n e s s o f t h e r e s u l t s . C h a p t e r 4 i n v e s t i g a t e s l i b r a t i o n a l d y n a m i c s , f o r c e d i s -t r i b u t i o n a nd o r i e n t a t i o n c o n t r o l f o r c a b l e o r beam c o n n e c t e d m u l t i b o d y s y s t e m s i n c i r c u l a r o r b i t s . The a n a l y s i s i s a p p l i e d t o t h e c a s e o f a p a y l o a d d e p l o y e d f r o m t h e S p a c e S h u t t l e . I t may a l s o p r o v e u s e f u l i n t h e s t u d y o f t h e SSPS and s p a c e c o l o n i e s . F i g u r e 1-6 s c h e m a t i c a l l y i l l u s t r a t e s t h e v a r i o u s a s p e c t s i n v o l v e d i n t h e p l a n o f s t u d y . The a p p r o a c h a p p e a r s t o r e p r e s e n t a c o h e r e n t p r o g r a m t o e x p l o r e t h e s u b j e c t . DYNAMICS OF SINGLE AND MULTIBODY EARTH ORBITING SYSTEMS r i g i d systems ( e l l i p t i c orbits) . — — •—• f l e x i b l e systems (ci r c u l a r orbits) 1 axisymmetric systems (gravity oriented) 1 t r i a x i a l systems (spinning & gravity oriented) , \ g r a v i t a t i o n a l l y s t a b i -l i z e d end-bodies • 1 : [ 1 1 — 1 1 — .. —1 l i b r a t i o n a l dynamics (Lagrange's method) l i b r a t i o n a l forces: .dumbbell-type systems .periodic solutions & t h e i r s t a b i l i t y (Floquet) , resonance l i b r a t i o n a l dynamics (Lagrange's method) l i b r a t i o n a l forces, moments the SSPS: . s t a b i l i t y . control dynamic analysis (Hamilton' pri n c i p l e ) s t a b i l i t y : .linearized system system response: .numerical .Butenin .linear force response: .numerical Chapter 2 system response .numerical .linear s t a b i l i t y : .linearized system Chapter 3 system response: .numerical system response: . l i n e a r Chapter 4 Figure 1-6 A schematic diagram of the proposed plan of study 15 2. ATTITUDE AND FORCE ANALYSES FOR RIGID AXISYMMETRIC GRAVITY ORIENTED S A T E L L I T E S 2.1 P r e l i m i n a r y Remarks A l t h o u g h l i b r a t i o n a l d y n a m i c s o f s a t e l l i t e s a n d t h e i r c o n t r o l h a s b e e n a s u b j e c t o f , c o n s i d e r a b l e s t u d y f o r q u i t e some t i m e , a c a r e f u l s t u d y o f t h e l i t e r a t u r e r e v e a l s t h a t i n v e s t i g a t i o n s t o d a t e h a v e c o n f i n e d a t t e n t i o n p r i m a r i l y t o a t t i t u d e o r o r b i t a l d y n a m i c s w i t h v e r y l i t t l e i n t e r e s t as t o t h e c h a r a c t e r a n d d i s t r i b -u t i o n o f f o r c e s i m p o s e d on a g i v e n s y s t e m . As p o i n t e d o u t b e f o r e , a p r e l i m i n a r y a n a l y s i s o f t h e l i b r a t i o n a l f o r c e s i n a n i d e a l i z e d 45 46 d u m b b e l l s a t e l l i t e i s f o u n d i n l i t e r a t u r e ' . The i n f o r m a t i o n • i s i m p o r t a n t n o t o n l y f r o m d e s i g n a n d s t r u c t u r a l i n t e g r i t y c o n -s i d e r a t i o n s b u t a l s o s e r v e s as a b a s i s i n t h e p l a n n i n g o f s c i e n -t i f i c e x p e r i m e n t s . The k n o w l e d g e o f t h e f o r c e r e q u i r e d f o r d e -p l o y m e n t o f an i n s t r u m e n t p a c k a g e , and more i m p o r t a n t l y i n m a i n -t a i n i n g i t a t a g i v e n s p a t i a l l o c a t i o n and o r i e n t a t i o n ( e . g . , f r o m t h e S p a c e S h u t t l e ) , i s i n d e e d v i t a l . A s t h e f o r m i d a b l 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 do n o t p o s s e s s any known c l o s e d f o r m s o l u t i o n , an a p p r o x i m a t e a n a l y s i s i s u n d e r t a k e n u s i n g an e x t e n s i o n o f t h e K r y l o v a n d B o g o l i u b o v 48 49 method ( v a r i a t i o n o f p a r a m e t e r s ) as s u g g e s t e d by B u t e n i n w x t h c e r t a i n m o d i f i c a t i o n s . An a p p l i c a t i o n o f t h e P o i n c a r e - t y p e 50 e x p a n s i o n m e t h o d t o t h e l i n e a r i z e d s y s t e m o f e q u a t i o n s f u r t h e r s i m p l i f i e s t h e s o l u t i o n w i t h o u t s u b s t a n t i a l l y a f f e c t i n g t h e p h y s i -c a l c h a r a c t e r o f t h e s y s t e m . A r e s p o n s e s t u d y e s t a b l i s h e s t h e a c -c e p t a b i l i t y o f t h e s e s o l u t i o n s d u r i n g s m a l l a m p l i t u d e l i b r a t i o n s . The a n a l y s i s ( B u t e n i n ' s s o l u t i o n ) i s a p p l i e d t o s t u d y t h e 16 e f f e c t s o f e c c e n t r i c i t y o f t h e o r b i t , m o d i f i e d i n e r t i a p a r a m e t e r and i n i t i a l d i s t u r b a n c e s on t h e l i b r a t i o n a l d y n a m i c s a nd i n d u c e d f o r c e s . A t y p i c a l e x a m p l e u s i n g t h e c o n f i g u r a t i o n o f t h e G r a v i t y G r a d i e n t T e s t S a t e l l i t e (GGTS) c o m p l e m e n t s t h e a n a l y s i s ( F i g u r e 2-1) 2.2 F o r m u l a t i o n o f t h e P r o b l e m C o n s i d e r an a r b i t r a r i l y s h a p e d r i g i d s a t e l l i t e , w i t h i t s mass c e n t e r a t S, i n an o r b i t a b o u t t h e c e n t e r o f f o r c e ( F i g u r e 2-2) L e t S - x y z be t h e o r t h o g o n a l p r i n c i p a l b ody c o o r d i n a t e s y s t e m o f t h e s a t e l l i t e w i t h t h e t r i a d S - X Q Y Q Z Q so c h o s e n a s t o d i r e c t y ^ - a x i s o u t -w a r d a l o n g t h e l o c a l v e r t i c a l a n d t h e x ^ - a x i s p a r a l l e l t o t h e o r -b i t a l a n g u l a r momentum v e c t o r . The o r i e n t a t i o n o f t h e s a t e l l i t e e x e c u t i n g a r b i t r a r y l i b r a t i o n a l m o t i o n may be s p e c i f i e d by a s e t o f s u c c e s s i v 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 : ty a b o u t t h e x ^ - a x i s g i v i n g , Y^' z ^ i _ a x e s , $ a b o u t t h e z ^ - a x i s r e s u l t i n g i n t h e x i 2 ' ^ i 2 ' z i 2 s Y s t e m ' a n ^ ^ a b o u t t h e y ^ 2 - a x i s y i e l d i n g t h e f i n a l o r i e n t a t i o n g i v e n b y t h e x, y, z - a x e s . The c o r r e s p o n d i n g e x p r e s s i o n s f o r k i n e t i c a n d p o t e n t i a l 3 e n e r g i e s t o t h e o r d e r 0 ( 1 / R ) may be w r i t t e n a s : T = i [m ( R 2 + R 2 9 2 ) + ( I s i n 2 X + I c o s 2 X)ty2 + ( I c o s 2 X A I S X Z X 2 , . , • . r, 2 2 , , T r," ,A. , x „ J „ - L - I 2 + I s i n X) (9 + ty) c o s tj> + I {X - (0 + ty) sin ty}' z y + ( I - I ) <j>(9 + 40 c o s ty s i n 2X Z X . . ( 2 . 1 a ) um P = * _J1_ R 4 R 3 ( I + 1 + I ) - 3 { I - ( I - I ) c o s 2 X } s i n 2 ty x y z y z x - 3 ( 1 - I ) s i n 2ty sinty s i n 2X - 3 { l + ( I - I ) c o s 2X} z x Y" z x 2 2 2 2 " x c o s ty s i n ct - 3 ( 1 + 1 - I ) c o s ty c o s ty Z X Y . ( 2 . l b ) 17 F i g u r e 2-1 A s c h e m a t i c d i a g r a m o f t h e G r a v i t y G r a d i e n t T e s t S a t e l l i t e (GGTS) ~~ Figure 2-2 Reference coordinate systems and geometry of the s a t e l l i t e motion with ty, ty and A representing the l i b r a t i o n a l degrees of freedom 19 F o r an a x i s y m m e t r i c s a t e l l i t e , I = I . C o n s e q u e n t l y , X z A d o e s n o t a p p e a r e x p l i c i t l y i n t h e e x p r e s s i o n f o r t h e L a g r a n g i a n r e n d e r i n g t h e c o n j u g a t e momentum p a c o n s t a n t o f t h e m o t i o n , i . e . , A 9L p , = = I {A. - (6 + ijj)sincj)} = c o n s t a n t . . . . . ( 2 . 2 ) 8A Y F o r a n o n s p i n n i n g s a t e l l i t e (A = 0 ) , t h e c o n s t a n t must be z e r o . The c l a s s i c a l L a g r a n g i a n f o r m u l a t i o n y i e l d s 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 a s : R - R 0 2 + 1L. = — ( I _ j ) ( i _ 3 C o s 2 i i j cos 2(j)) , (2.3a) RJ 2m R x Y s R 0 + 2R R0 = ^ { - (ty + 0)cos^cj) + (ty + Q)ty sin2ty}, (2.3b) s ty + 0 - 2 (ty + Q)ty tanty + lH. K . sin<jj cosij; = 0 , (2.3c) R J 1 $ + { (ty + 6) 2 + lH. K . c o s 2 ^ } s i n o t cose)) = 0 . (2.3d) R 1 N e g l e c t i n g o r b i t a l p e r t u r b a t i o n s due t o t h e a t t i t u d e m o t i o n , e q u a -t i o n s (2.3a) a nd (2.3b) l e a d t o t h e c l a s s i c a l K e p l e r i a n r e l a t i o n s , R 29 .= hg , (2.4a) i = ( 1 + e c o s 0 ) , (2.4b) h„ h e n c e , t h e e q u a t i o n s g o v e r n i n g t h e l i b r a t i o n a l m o t i o n become, 3K r , _ 2 ( e s i n e , t } ( + ]_ + J- s i n i j J C O S l J ; = 0 1 + e c o s 0 1 + e c o s 9 r r ' .... (2.5a) a n d *" " 1 + e c o s 8 *' + { ( ^ ' + 1 } + 1 + e c o s 9 c o s ^ } s i n * c o s * = 0 • (2.5b) 20 2.3 A p p r o x i m a t e A n a l y t i c a l S o l u t i o n s I n t h e a b s e n c e o f any known c l o s e d - f o r m s o l u t i o n o f t h e c o u p l e d , n o n l i n e a r and n o n a u tonomous e q u a t i o n s o f l i b r a t i o n a l m o t i o n , t h e p r o b l e m i s a n a l y s e d u s i n g B u t e n i n ' s v a r i a t i o n o f p a r a m -e t e r s and t h e P o i n c a r e - t y p e e x p a n s i o n m e t h o d s . 2.3.1 B u t e n i n 1 s v a r i a t i o n o f p a r a m e t e r s m e t h o d ^ ^ A f t e r r e p l a c i n g t h e t r i g o n o m e t r i c f u n c t i o n s i n IJJ and <j> by t h e i r s e r i e s e x p a n s i o n s and i g n o r i n g t e r m s o f t h e f o u r t h and h i g h e r d e g r e e s , e q u a t i o n s (2.5) c a n be w r i t t e n a s 2 ^" + n 1 ,p = 2e s i n e + f± , . . . . ( 2 . 6 ) . <J>" + n 2 2 * = f 2 ' w h e r e: 2 2 2 2 2 1 2 fl = n l ^ e C O S 0 " 1* ) + J i|J > + 2<H ' ( 1 + ip ' + -3 4> ) + 2e\p' s i n g , f 2 = 2e(i) , s i n e + | cf)3 - c ^ ' 2 - 2<$>$\ + i <b3V + n 1 2 ( l - e c o s 9 ) ( J ) ( | <$>2 + ^ 2 ) + e n-^ty c o s e . N o t e t h a t t h e t e r m s c o n t a i n i n g s e c o n d and h i g h e r d e g r e e s i n e h a v e b e e n n e g l e c t e d w h i l e d e r i v i n g t h e f u n c t i o n s f ^ , f . F o r s m a l l a m p l i t u d e m o t i o n e a c h t e r m i n f ^ a n d f 2 i s s m a l l c o m p a r e d t o t h e t e r m s on t h e l e f t hand s i d e o f e q u a t i o n s ( 2 . 6 ) , h e n c e t h e i r a p p r o x i m a t e s o l u t i o n c a n be f o u n d u s i n g t h e m e thod o f v a r i a t i o n o f p a r a m e t e r s . The s o l u t i o n o f t h i s s y s t e m o f e q u a t i o n s i s assumed i n t h e f o r m , • 2e V = a1- (9) s i n { n j + (e) } + — 2 s x n e , .... (2 ,7a) 2 2 n ^ - 1 21 <J> =. a 2 (9) s i n { n 2 9 + 5 ^ ( 6 ) } , (2.7b) w h e r e a-, ( 9 ) , a„ ( 9 ) , 6, ( 0 ) , 6~ (9) a r e now c o n s i d e r e d t o be 2 2 2 ^2 s l o w l y v a r y i n g p a r a m e t e r s . S i n c e t h e s e f o u r v a r i a b l e s c a n n o t be d e t e r m i n e d f r o m f o u r i n i t i a l c o n d i t i o n s a l o n e , t h e s o l u t i o n i s o v e r s p e c i f i e d . Hence f o u r c o n s t r a i n t r e l a t i o n s a r e o b t a i n e d as f o l l o w s : E q u a t i n g t h e f i r s t d e r i v a t i v e o f e q u a t i o n (2.7) t o t h a t o f t h e l i n e a r s y s t e m g i v e s two o f t h e c o n s t r a i n t r e l a t i o n s : a. s i n ? , + a • ? •' c o s ? . = 0 ; i = l , 2 . . . . . ( 2 . 8 ) 2 2 2 2 2 E q u a t i o n s o f m o t i o n (2.6) i n c o n j u c t i o n w i t h t h e assumed s o l u t i o n -i n e q u a t i o n s (2.7) y i e l d t h e r e m a i n i n g two c o n s t r a i n t r e l a t i o n s , a. n. c o s ? • - a. n.6. s i n ? - = f- ; i = 1, 2 ; . . . . ( 2 . 9 ) w h e r e f i = f i ^ a l s i n ^ i ' a i n i cos^i i a 2 sinc 2 , a 2 n 2 c o s ? 2 ) ; i = 1, 2. S o l v i n g e q u a t i o n s (2.8) a n d (2.9) s i m u l t a n e o u s l y f o r a. and 1 2 6. y i e l d s , 1 2 I- = f . C O S ? . i 0 n. I i _ 2 I 2 1 JT* • r f . s i n ? . , a i n i 1 X 2 i = 1, 2 (2.10) F o r s m a l l a m p l i t u d e m o t i o n f a r e s m a l l . C o n s e q u e n t l y , a. a nd <5. a r e s l o w l y v a r y i n g p a r a m e t e r s . U s i n g t h e i r a v e r a g e 2 X2 v a l u e s o v e r one p e r i o d g i v e s t h e f o l l o w i n g : 22 ^ 2TT 2TT 2TT * a l 2 = ( 1 / 8 T f n i } 0 0 0 f i C O S ? i 2 d ? l 2 d ? 2 2 d 0 > ^ 2TT 2TT 2TT * . , N ,_ fi' = -M/8TT 3n a ) ' ' ' f i S i n ? i d ? ] d C 2 d 0  6 i 2 U / b T T i i 2 J 0 0 0 1 12 X 2 Z2 i = 1, 2. (2 .11) The r i g h t hand side of equations (2.11) may be evaluated to give: a = 0 , x2 612 = - ( 1 / 4 ) n i % > i = 1, 2. ....(2.12) After integrating equation (2.12) with respect to 6 and substituting for a. ,6. in equation (2.7), solution for the l i b r a t i o n a l motion 2 2 can be written as: = a, sin{n,(1 - a 2 /4)8 + 6, } + {2e/(n 2 - 1) }sin0 , (2.13a) 2 2 20 cf> = a sin{n (1 - a 2 /4)9 + & } , (2.13b) 2 2^ ^20 where: a1 = ± [ > 2 + {<JJQ - 2e/(n 2. - l ) } 2 / n 2 ] 1 / 2 , (2.13c) & 2 = ± [cJ)2 + 4>Q 2/n 2] 1 / 2 , (2.13d) i 2 6, = arc tan [4>nn,/{iJ;n ~ 2e/(n, - 1) } J f ....(2.13e) ±20 6 ? = arc tan {<J>n 9/<j>n} • ....(2.13f) 20 23 I 5 Q 2.3.2 Poincare-type expansion method In equation (2.6), after neglecting the second or higher degree terms i n ty, ty, ty', ty', the following l i n e a r relations are obtained: ty" + n2ty = e{2(ip' + l)s i n 9 + n^ty cos0} , (2.14) ty" + n^ty = e{2ty' sin6 + n^ty cos0} . Since the value of e c c e n t r i c i t y i s usually small (e < 0.2), the series of the form ty (0) = E e V (0) , i=0 (2.15) OO ty (0) = £ eJty. (9) , j = 0 : lead to the solution of equation (2.14) as: ' 2 2 ty = ip 0 cos n 10 + (^ Q/n 1)sin n^Q + e [{ - 3ty Q n^/{1 - 4n.^ ) }cos n^Q + (l/n 1){ - 2/(n 2 - 1) + ^Q(2 - 5 n 2 ) / ( l - 4n 2) }sin n ^ + {2/(n 2 - l)}sin0 - {n± tyQ(l + n 1 / 2 ) / ( l + 2^) }cos (1 • + n]_) 0 + {n 1 ^ Q ( l - n1/2)/.[l - 2 n ; L ) } c o s ( l - n]_) 0 - {ipgd + n 1 / 2 ) / ( l + 2 n 1 ) } s i n ( l + n^O - {ty'Q(I - 2 n 1 ) } s i n ( l - n 1)0] ....(2.16a) i 2 2 2 ty = tyQ cos n 20 + ((j> 0/n 2)sin n 20 + e[{ (j) 0(n 1 - 4 n 2 ) / ( l - 4n 2)} x cos n 20 + {tpQ (2 - n 2 - 4 n 2 ) / n 2 ( l - 4n 2)}sin n 20 U 0 ( n 2 + n 2 / 2 ) / ( l + 2n 2) }cos-(l + n 2) 24 + U 0 ( n 2 - n ^ / 2 ) / ( l - 2 n 2 ) } c o s ( l - n 2 ) 6 - {<b'Q(--2 + n 2 / 2 ) / n 2 x (1 + 2 n 2 ) } s i n ( l + n 2 ) 9 - Ug ( n 2 - n ^ / 2 ) / n 2 ( l - 2 n 2 ) } x s i n ( l - n 2 ) 0 ] . (2.16b) 2.3.3 A c c u r a c y o f t h e a p p r o x i m a t e s o l u t i o n s A t t h e o u t s e t , i t was e s s e n t i a l t o a s s e s s t h e a c c u r a c y o f t h e a p p r o x i m a t e s o l u t i o n s . F o r t h i s , t h e s o l u t i o n s w e r e c o m p a r e d w i t h t h a t g i v e n by a n u m e r i c a l i n t e g r a t i o n o f t h e e x a c t e q u a t i o n s o f m o t i o n , o b t a i n e d u s i n g t h e R u n g e - K u t t a method w i t h b u i l t - i n e r r o r c o n t r o l o n an IBM 370 c o m p u t e r ( F i g u r e s 2-3, 2 - 4 ) . F o r a l l t h e c a s e s , t h e a g r e e m e n t i n r e s p o n s e was f o u n d t o be s u r p r i s i n g l y c l o s e e v e n f o r l a r g e d i s t u r b a n c e s . The a p p r o x i m a t e s o l u t i o n s a r e a l s o a b l e t o f o l l o w l o c a l a m p l i t u d e m o d u l a t i o n s . A c a r e f u l e x a m i n a t i o n o f F i g u r e 2-4 showed a c l o s e r c o r r e s p o n d e n c e b e t w e e n B u t e n i n ' s a n d n u m e r i c a l s o l u t i o n s . H o wever, t h e a p p r o x i m a t e s o l u -t i o n s a r e n o t a b l e t o d e t e r m i n e t h e c o u p l i n g b e t w e e n vp a n d (J) d e g r e e s o f f r e e d o m . 2.4 P e r i o d i c S o l u t i o n s a nd S t a b i l i t y 51 As shown by B r e r e t o n , e q u a t i o n s (2.5) p e r m i t s o l u t i o n o f t h e f o r m : oo f P ( I 1 = Z A m n s i n ( m / n ) 6 ; n = 1, 2, . . . ; m = 1 ' (2.17) CO ic A * p + = 2 A s i n ( i / j ) 0 ; j = 1, 2,... ; F ' 3 i = i W i t h a p r o p e r c h o i c e o f e, and i n i t i a l c o n d i t i o n s a v a s t e = 0 . 2,K r 0 . 8 5 ; +o=10°, <t>0= 5°, — numerical i|/=0.3, ct>;-0.5 ---- Butenin's f 0 - 2 0 26 (a) numerical , (b) Butenin's, (c) linear e=0 , K r1 t = t = 0 ' i= i=°5 Orbits F i g u r e 2-4 A t y p i c a l e x a m p l e c o m p a r i n g n u m e r i c a l , B u t e n i n ' s and l i n e a r s o l u t i o n s 27 majority of the c o e f f i c i e n t s A* , A**. can be made to vanish J 1 m,n x,: reducing the above equations to the following form: tyn = A* sin (v/n) 0 , rP, n v,n <)>_, = A** sin (K/n) 0 , YP,n K,n ' ' . (2 .18) where the combined ty and ty motion has a period of 2frn. Specifying displacement i n i t i a l conditions to be zero, i . e . , ty^ = ty^ = 0, leads to = 6 9 =0 i n equation (2.13). For a c i r c u l a r oxbit •^ 20 Z20 (e = 0), equation (2.13) may be rewritten as: i * 2 2 * ty = (ty^/n^ s i n { n 1 ( l - ty'Q /4n 1)8} , (2.19) t ' 2 2 ty = [tyQ/n2) s i n { n 2 ( l - ty^/An^Q] . Equation (2.19) now has the same form as equation (2.18) and both ty and ty are of the form ty^ n and ty^ n , respectively. By allowing d i f f e r e n t integer values for v, K, n, the i n i t i a l conditions for periodic solutions may be derived as: 1/2 tyn = ± 2n,(1 - v/nn,) ty'Q = ± 2 n 2 ( l - K / n n 2 ) 1 / 2 , (2.20) where p, = v/n and p, = K/n. cty L ty S t a b i l i t y of the periodic solution can now be studied by 5 2 using v a r i a t i o n a l analysis i n conjunction with the Floquet theory S u b s t i t u t i n g r TP,n r v ' i n equations (2.5) and l i n e a r i z i n g with r e s p e c t to and <|) leads to \p = F, f + F„ 4) + F_ <J> + F . cb r v 1 r v 2 r v 3 Y v 4 T v It "Jt ' ^ if I cj) = G, (JJ + G 0 ^ + G_ cJ) + G . 6 ; v 1 r v 2 Yv 3 v 4 Y v ' where: * i s i n 6 / ( 1 + e cos0) , -L i r , n i r , n F„ = - 3K. cos 2ty / ( l + e cos0) , --* 1 ir / n F 3 =  2{K,n + 1 ) t a n * P , n ' F4 = ^P,n t a n * P , n " { 2 e s 1 1 1 0 / ^ 1 + e cos9) } djip + l)tan<|)p  + 2 ( l | JP,n + 1 ) ( J )P,n + ^ K i / 2 ^ 1 + e c o s 0 ) } s i n 2i|; p t a n * p / n G l = " (^P,n + 1 ) s i n 2*P,n ' G * = {3K./2(1 + e c o s 0 ) } s i n 2\p s i n 2ch G * = 2e s i n 0 / ( l + e cos0), 4 = " t ^ p ^ n + D 2 + ( 3 ^ / ( 1 + e cos6) Jcos^^Jcos 2cJ) p ^ n (2 G Since f o r a c i r c u l a r o r b i t , A ' , F* F* e t c . are of rP ,n TP ,n 1 2 p e r i o d 2im (n = 1, 2,...), the F l o q u e t theory i s a p p l i c a b l e . 29 T a k i n g s o l u t i o n s i n t h e f o r m : i|)(0 + 2im) = a* ty{6) , (J> (6 + 2iTn) = a* <)) (0) , and s o l v i n g e q u a t i o n (2.21) f o r d i f f e r e n t v a l u e s o f a* l e a d s t o t h e s t a b i l i t y c r i t e r i a : \Rl a?I < 1 , i = 1, 2, 3, 4 ; s t a b l e (2.22) | a * | > 1 , i = 1, 2, 3, 4 ; u n s t a b l e The i n i t i a l c o n d i t i o n s f o r p e r i o d i c s o l u t i o n s w e r e e v a l -u a t e d o v e r a r a n g e o f v , K and n and t h e s t a b i l i t y o f t h e s o l u -t i o n s s t u d i e d t h r o u g h e q u a t i o n ( 2 . 2 2 ) . Some o f t h e t y p i c a l r e s u l t s a r e p r e s e n t e d i n F i g u r e 2-5. The c a s e (a) c o r r e s p o n d s t o t h e i n i t i a l c o n d i t i o n s l e a d i n g t o ty and <j> l i b r a t i o n s h a v i n g t h e same p e r i o d as t h a t o f t h e o r b i t a l m o t i o n . The c a s e (b) r e p r e s e n t s t h e p e r i -o d i c s o l u t i o n w h e r e s e v e n c y c l e s o f ty, and n i n e c y c l e s o f ty a r e c o m p l e t e d i n f i v e o r b i t s . N o t e t h a t t h e m a j o r i t y o f t h e i n i t i a l c o n d i t i o n s l e a d t o u n s t a b l e s o l u t i o n s . 2.5 R e s o n a n c e The s t u d y o f e q u a t i o n s (2.16) g i v e s an i m p o r t a n t i n f o r -m a t i o n a b o u t t h e d y n a m i c a l b e h a v i o u r o f t h e l i n e a r i z e d s y s t e m . F o r a s a t e l l i t e i n an e l l i p t i c o r b i t , t h e v a l u e s n^ = 1, 1/2 ( i . e . = 1/3, 1/12, r e s p e c t i v e l y ) l e a d t o r e s o n a n c e i n ty m o t i o n . S i n c e 0 < < 1, t h e c o n d i t i o n n 2 = 1/2 ( K i = - 1/4) l e a d i n g t o t h e r e s o n a n c e i n ty m o t i o n i s n o t p o s s i b l e . 30 3 4 e-0-, [aj § -5 - i . [b ] ^-^5.5-%; stable —, unstable — =<4>=o ' 0 To —, — ' \ I I I 0 075 0-8 0-85 0-9 0-95 Figure 2-5 Typical i n i t i a l conditions r e s u l t i n g in periodic l i b r a t i o n a l motion. Note that over a wide range of i n i t i a l conditions and modified i n e r t i a parameter the periodic solutions are unstable. 31 2.6 F o r c e D i s t r i b u t i o n A v a s t m a j o r i t y o f t h e a x i s y m m e t r i c g r a v i t y o r i e n t e d s a t e l -l i t e s may be r e p r e s e n t e d by t h e d u m b b e l l c o n f i g u r a t i o n w i t h p o i n t m asses m^ , m^ , m c c o n n e c t e d b y a boom o f mass p p e r u n i t l e n g t h ( F i g u r e 2 - 6 ) . H e r e m c i s t a k e n t o be l o c a t e d a t t h e c e n t e r o f mass o f t h e s a t e l l i t e . , R^ and R r e f e r t o t h e c o r r e s p o n d i n g d i s t a n c e s f r o m 0, r e s p e c t i v e l y . F o r a mass e l e m e n t Am a t a d i s t a n c e E, f r o m S, t h e e q u a t i o n o f m o t i o n w i t h r e f e r e n c e t o t h e i i i i n e r t i a l c o o r d i n a t e s y s t e m X , Y , Z c a n be w r i t t e n a s d t 2 { R m } = { g } + ^ A F>/Am , ..(2.23) w h e r e {AF} r e p r e s e n t s t h e f o r c e n e c e s s a r y t o k e e p t h e mass e l e m e n t Am i n i t s p o s i t i o n . The a c c e l e r a t i o n on t h e l e f t h a n d s i d e o f e q u a t i o n (2.23) may be e x p r e s s e d as {R„} d t 2 L m JX ,Y , Z 0 — [ 0 %r( [01 {R } ) ] , d0 d0 m J x Q , y Q , z Q ' i ' w h e r e t h e t r a n s f o r m a t i o n m a t r i x i s , . . . ( 2 . 2 4 a ) [6] and {R„> 1 0 0 0 c o s 0 - s i n f 0 s i n e c o s f r £ s i n e m W z o = . < E, cosijj coscj) + R | E, s i n i j j cos<j) (2.24b) (2.2 4 c ) The s u b s c r i p t s h e r e r e p r e s e n t t h e f r a m e o f r e f e r e n c e i n w h i c h a v e c t o r i s e x p r e s s e d . The e x p r e s s i o n f o r t h e g r a v i t a t i o n a l Figure 2-6 Dumbbell-type s a t e l l i t e : (a) l i b r a t i o n a l angles ty, ty; (b) boom forces 33 a c c e l e r a t i o n may be w r i t t e n a s {g} = - (y/R^) [ 6 ] { R m } x , m m X 0 ' Y 0 ' 0 (2.25a) w h e r e 1/R may be c a l c u l a t e d f r o m e q u a t i o n ( 2 . 2 4 c ) , 1 3E ( 1 - r — costy coscb) R~ (2.25b) m R 2 3 E £ N o t e t h a t t h e t e r m s c o n t a i n i n g , i ^ . , e t c . a r e n e g l e c t e d i n R R e q u a t i o n ( 2 . 2 5 b ) . F o r s m a l l v a l u e s o f e c c e n t r i c i t y e q u a t i o n s (2.4a) a n d (2.4b) may be u s e d t o d e r i v e t h e f o l l o w i n g r e l a t i o n s : y = co2 R p ( l + e ) 3 , . (2.26) o ) ( l + e c o s 0 ) A f t e r s u b s t i t u t i n g e q u a t i o n s (2.24a) a n d (2.25a) i n t o e q u a t i o n (2.23) and u s i n g e q u a t i o n s ( 2 . 2 4 c ) , ( 2 . 2 5 b ) , (2.26) t o r e p l a c e {R } , , y and 6, an e x p r e s s i o n f o r t h e f o r c e may be 0 , y 0 ' 0 R m r e w r i t t e n as { AF} (1 + e cosG) 2 d_ de (1 + e c o s 0 ) 2 d _ de [6] r E, s i m j i { E, costy COScJ) + R E sinty coscj) + (1 + e c o s 6 ) [0 ] r E, sine}) ^ - 2 ^ c o s f cost}) + R^ E, sirup cos<p Am (2.27) By appropriately selecting £ and Am, the force at any point on the s a t e l l i t e can be determined and hence, by integration, the resultant force on a section of the boom. Of course, for equi-librium, {.Fa} + { F B } + { F C } + { F C A } + { F C B } = 0, ic O hence: { F } = m £ t o { F } a a { F b } = " m b { F } > { F C } - m* £ c o 2 { F } , { F c a } = m c a { F } > { F c b } = " m c b { F } • where : * m = a m. (ma/ms){mb + i x a ^ l ^ / l ) + (1/2) pU , b = (mb/ms){ma + m c U 2 / £ ) + (1/2)pi} , m*c = ( m c / m s ) ^ m b ( £ 2 / £ ) " m a ( A i / J l ) + (1/2) p U 2 - l±) } , p[(£/ms){mb + m c(£ 1/£) + (1/2) pU - l±/2 ] , * m ca (2.28a) m*b - p[U/m s){m a + m cU 2 /J l ) + (1/2) pU - £ 2/2 J , m s = m a + m b + m c + p l ' (2,28b) with the components of { F } i n the E,, \p r tj) and X 1 , Y ' , Z' frames 35 o f r e f e r e n c e r e l a t e d t h r o u g h t h e f o l l o w i n g , {Fh,i>,<p [ E ^ T { F } X ' , Y ' , Z ' sincj> costy cost}) sinty cost}) 0 - s i n i j j COSI(J cost}) - COSIJ  sine}) - sinty sintf) (2.29) S i m p l i f i c a t i o n o f e q u a t i o n (2.29) y i e l d s t h e f o r c e c o m p o n e n t s a s : Fg. = (1 + e c o s 9 ) 3 [ 1 - 3 cos 2i() cos2 ty - (1 + e c o s 0 ) { ( i j / + 1) 2 x cos 2cb + Cb' 2}] , F^ = (1 + e c o s e ) 3 [ ( 1 + e cose){ty" costy - 2ty' (ty' + 1)sinty] - 2e ( ' + l ) s i n 0 cost}) + 3 s i n ^ cosij; cost})] , F^ = (1 + e c o s e ) 3 [ ( 1 + e cos6){<}>" + (ty' + 1) 2 sin<|> costy} - 2e ty s i n G + 3 c o s ty sinty costy] . (2.30) 2.7 L i b r a t i o n a l R e s p o n s e a nd F o r c e D i s t r i b u t i o n I m p o r t a n t s y s t e m p a r a m e t e r s w e r e v a r i e d o v e r a w i d e r a n g e o f p o s s i b l e i n t e r e s t t o a s s e s s t h e i r i n f l u e n c e on t h e r e s p o n s e . The amount o f i n f o r m a t i o n so g e n e r a t e d i s r a t h e r e n o r m o u s , h o w e v e r , f o r c o n c i s e n e s s , o n l y a few o f t h e t y p i c a l r e s u l t s s u f f i c i e n t t o e s t a b l i s h t r e n d s a r e p r e s e n t e d h e r e . F i g u r e 2-7 shows t i m e h i s t o r i e s o f t h e l i b r a t i o n s and f o r c e c o m p o n e n t s f o r g i v e n e , and i n i t i a l c o n d i t i o n s . P h a s e p l o t s i n ty-ty ' a n d ty-ty1 p l a n e s y i e l d c l o s e d t r a j e c t o r i e s c o r r e s p o n d i n g t o p e r i o d i c e = 0 , K r 0 . 7 5 ; t = <t>o=°' 1=^=0.5 1 2 3 4 5 6 Orbits 37 Typical time h i s t o r i e s of l i b r a t i o n s and force components at the mass m. o r b i t a l e c c e n t r i c i t y (b) increase i n the 3 8 39 0.8 T y p i c a l time h i s t o r i e s of l i b r a t i o n s and force components at the mass m : impulsive type disturbance (d) displacement and 40 s o l u t i o n s a s g i v e n by t h e B u t e n i n m e t h o d . N o t e t h e o u t o f p l a n e m o t i o n i s a t a h i g h e r f r e q u e n c y t h a n t h e i n p l a n e m o t i o n . A l s o t h e a x i a l f o r c e c omponent i s much l a r g e r t h a n o r F^ ( F i g u r e 2 - 7 a ) . The e f f e c t o f i n c r e a s i n g e i s shown i n F i g u r e 2 - 7 ( b ) . P l a n a r l i b r a t i o n s a s w e l l a s F^ show c o n s i d e r a b l e amount o f a m p l i -t u d e m o d u l a t i o n s . The p h a s e p l o t s i n d i c a t e t h a t t h e ty m o t i o n r e p e a t s i t s e l f o n l y a f t e r a l a r g e number o f o s c i l l a t i o n s . The e f f e c t o f may be a p p r e c i a t e d by s t u d y i n g F i g u r e 2-7 (c) a l o n g w i t h F i g u r e 2 - 7 ( a ) . The a m p l i t u d e and p e r i o d s o f t h e l i b r a t i o n s as w e l l a s f o r c e s t e n d t o i n c r e a s e w i t h a d e c r e a s e i n t h e v a l u e o f . F i g u r e 2-7(d) shows t h e r e s p o n s e o f a r a t h e r s l e n d e r s y s t e m e x -p o s e d t o a d i s p l a c e m e n t a s w e l l a s an i m p u l s i v e t y p e o f d i s t u r -b a n c e . N o t e t h a t t h e p l a n a r m o t i o n h e r e r e p e a t s i t s e l f a f t e r an e x t r e m e l y l a r g e number o f o s c i l l a t i o n s . To g e t some a p p r e c i a t i o n a s t o t h e l i b r a t i o n a l d y n a m i c s and a s s o c i a t e d f o r c e s f o r a n e x i s t i n g s a t e l l i t e , a t y p i c a l 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 s r e p r e s e n t e d by t h e GGTS (m = 0.2 8 k g , y = 16 m) l a u n c h e d i n 1966 was c o n s i d e r e d ( F i g u r e s 2-1, 2 - 8 ) . The m i s s i o n o b j e c t i v e was t o s t u d y t h e f e a s i b i l i t y o f t h e g r a v i t y g r a d i e n t s t a b i l i z a t i o n a t h i g h a l t i t u d e s . H e r e t h e ' c o n t r o l f o r c e ' r e p r e -s e n t s a d i f f e r e n c e i n f o r c e s a t t h e i n s t a n t a n e o u s a n d l o c a l v e r t i -c a l o r i e n t a t i o n s . I t i s a m e a s u r e o f t h e f o r c e r e q u i r e d t o m a i n -t a i n t h e t i p mass a l o n g t h e l o c a l v e r t i c a l . The ' r e s u l t a n t f o r c e ' r e p r e s e n t e d by t h e s o l i d l i n e c o r r e s p o n d s t o t h e f o r c e i n t h i s p r e f e r r e d o r i e n t a t i o n . I t i s i n t e r e s t i n g t o n o t e t h a t t h e a x i a l c o m ponent o f t h e r e s u l t a n t f o r c e i s maximum n e a r t h e p e r i g e e and 41 e = 0.1, K.-1; T=22hrs. 10mts.,uj = d> = o =0-3,4>' = 0.2 1 IO 'O 'o o o ,o -151-2.5 X10 ),N OA - 2 . 5 ^ 1 x108),N 0 x108),No -1.5 0 F i g u r e 2 - i Control force Resultant 3 Orbits 4 A t y p i c a l e x a m p l e o f t h e G r a v i t y G r a d i e n t T e s t S a t e l l i t e (GGTS) s h o w i n g i t s l i b r a t i o n a l a n d f o r c e h i s t o r i e s I I \ I I , ,. - * Sr: I 6 42 a p o g e e , w h e r e a s , t h e t r a n s v e r s e component r e a c h e s i t s maximum n e a r t h e m i n o r a x i s o f t h e o r b i t . 2.8 C o n c l u d i n g Remarks I m p o r t a n t a s p e c t s o f t h e i n v e s t i g a t i o n a n d c o n c l u s i o n s b a s e d on them may be s u m m a r i z e d as f o l l o w s : ( i ) A g e n e r a l f o r m u l a t i o n f o r a x i s y m m e t r i c g r a v i t y o r i e n t e d s a t e l l i t e s i s p r e s e n t e d , ( i i ) The c o u p l e d , n o n l i n e a r , n o n a u t o n o m o u s e q u a t i o n s o f l i b r a t i o n a l m o t i o n a r e a n a l y s e d u s i n g B u t e n i n ' s v a r i -a t i o n o f p a r a m e t e r s and t h e P o i n c a r e - t y p e e x p a n s i o n m e t h o d s . ( i i i ) S t a b i l i t y o f t h e p e r i o d i c s o l u t i o n s i s s t u d i e d u s i n g t h e F l o q u e t a n a l y s i s and t h e c o n d i t i o n s f o r r e s o n a n c e i n e l l i p t i c o r b i t s a r e e s t a b l i s h e d , ( i v ) E x p r e s s i o n s f o r l i b r a t i o n a l f o r c e s i n a d u m b b e l l - t y p e s a t e l l i t e a r e d e r i v e d . They s h o u l d be u s e f u l i n p l a n -n i n g o f s c i e n t i f i c e x p e r i m e n t s . (v) A n a l y t i c a l p r o c e d u r e s a r e a p p l i e d t o t h e f l i g h t t e s t e d g r a v i t y g r a d i e n t s a t e l l i t e (GGTS) . The i n f o r m a t i o n p r e s e n t e d h e r e has d i r e c t r e l e v a n c e t o t h e p o s i t i o n and o r i e n t a t i o n c o n t r o l o f t h e i n s t r u m e n t a t i o n p a c k a g e s d e p l o y e d f r o m s p a c e v e h i c l e s . 43 3. ATTITUDE AND FORCE ANALYSES FOR RIGID T R I A X I A L SYSTEMS 3.1 P r e l i m i n a r y Remarks The s t u d y o f l i b r a t i o n a l d y n a m i c s o f s a t e l l i t e s , e x c e p t 17 2 6 2 8 f o r a few 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 ' ' , s u f f e r s f r o m one m a j o r l i m i t a t i o n - a x i s y m m e t r i c c h a r a c t e r o f t h e c o n f i g u r a t i o n . Of c o u r s e , t h i s s i m p l i f i c a t i o n i s d i c t a t e d by s e v e r a l c o n s i d -e r a t i o n s : •. • ( i ) T h e r e a r e a number o f s i t u a t i o n s w h e r e a s a t e l l i t e i n d e e d has an a x i s o f s y m m e t ry, e . g . , S p u t n i k , t h e f i r s t C a n a d i a n s a t e l l i t e A l o u e t t e 1, s e v e r a l o f t h e I n t e l s a t s e r i e s o f c o m m u n i c a t i o n s s a t e l l i t e s , e t c . ( i i ) The 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 a r e e x t r e m e l y c o m p l i c a t e d . E v e n w i t h t h e a s s u m p t i o n o f a x i s y m m e t r y , one i s f a c e d w i t h t h e s o l u t i o n o f c o u p l e d , n o n l i n e a r , n o n a u tonomous s e t o f d i f f e r -e n t i a l e q u a t i o n s . N e v e r t h e l e s s , a v a s t m a j o r i t y o f t h e p r e s e n t day m i s s i o n s e m p l o y s p a c e v e h i c l e s o f a v a r i e t y o f c o n f i g u r a t i o n s w h e r e a n a x -i s o f symmetry i s o f t e n n o n e x i s t e n t . The C a n a d i a n C o m m u n i c a t i o n s s a t e l l i t e A n i k , S k y l a b , p r o p o s e d c o n f i g u r a t i o n s o f t h e S a t e l l i t e S o l a r Power S t a t i o n ( S S P S 5 , F i g u r e 1 - 4 ) , s p a c e c o l o n i e s 6 and many o t h e r s b e l o n g t o t h i s c a t e g o r y . A n o t h e r i m p o r t a n t p o i n t t o k e e p i n m i n d i s t h e f a c t t h a t t h e i n v e s t i g a t i o n s t o d a t e h a v e c o n f i n e d a t t e n t i o n p r i m a r i l y t o t h e a t t i t u d e o r o r b i t a l d y n a m i c s w i t h v e r y l i t t l e i n t e r e s t a s t o t h e c h a r a c t e r ( t e n s i l e o r c o m p r e s s i v e , p e r i o d i c o r a p e r i o d i c , h i g h o r 44 l o w f r e q u e n c y , p o s s i b i l i t y o f f a t i g u e f a i l u r e , e t c . ) o r d i s -t r i b u t i o n o f t h e f o r c e s i m p o s e d on a g i v e n s y s t e m . On t h e o t h e r h a n d , s u c h i n f o r m a t i o n w o u l d be a p r e r e q u i s i t e t o t h e p l a n n i n g o f a s c i e n t i f i c e x p e r i m e n t and a s s o c i a t e d c o n t r o l s y s t e m . F o r e x a m p l e , c o n s i d e r t h e c a s e o f t h e o r b i t i n g s o l a r power s t a t i o n . F o r c e d i s t r i b u t i o n on t h e g i g a n t i c s o l a r p a n e l s and t o r q u e r e q u i r e d t o p o s i t i o n them t o w a r d s t h e s u n f o r t h e maximum e f f i c i e n c y w o u l d r a n k among t h e i m p o r t a n t c r i t i c a l a s p e c t s o f i t s d e s i g n . I n t h i s c h a p t e r , g e n e r a l e q u a 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 f an a r b i t r a r i l y - s h a p e d , r i g i d s p a c e c r a f t a r e d e r i v e d 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 . A p p r o x i m a t e c l o s e d - f o r m s o l u t i o n s f o r b o t h t h e s p i n n i n g and g r a v i t y o r i e n t e d c o n f i g u r a t i o n s a r e s o u g h t f o r t h i s f o r m i d a b l e s y s t e m u s i n g a p e r t u r b a t i o n m e t h o d . V a l i d i t y o f t h e s o l u t i o n s i s c h e c k e d t h r o u g h n u m e r i c a l i n t e g r a t i o n o f t h e e x a c t e q u a t i o n s o f m o t i o n . The s o l u t i o n s p r o v i d e c o n s i d -e r a b l e i n s i g h t i n t o t h e s y s t e m b e h a v i o u r o v e r a r a n g e o f v a l u e s o f t h e i n e r t i a a n d s p i n p a r a m e t e r s , t h e o r b i t a l e c c e n t r i c i t y , and i n i t i a l d i s t u r b a n c e s . N e x t , u s i n g t h e N e w t o n i a n a n a l y s i s , g e n e r a l e x p r e s s i o n s f o r f o r c e s and moments a r e o b t a i n e d w h i c h s h o u l d p r o v e u s e f u l i n m a i n t a i n i n g an o b j e c t a t a g i v e n l o c a t i o n and o r i e n t a t i o n i n s p a c e . A p p l i c a t i o n o f t h e a n a l y s i s i s i l l u s t r a t e d t h r o u g h c o n f i g u r a t i o n s o f t h e E x p l o r e r XX ( F i g u r e 3 - 1 ) , an i n s t r u -ment p a c k a g e l a u n c h e d f r o m t h e S p a c e S h u t t l e , and t h e SSPS. Telemetry antenna mass=1 Total mass = 49 kg G e o m e t r y o f t h e I n t e r n a t i o n a l I o n o s h e r i c S a t e l l i t e ( E x p l o r e r XX) 46 3.2 F o r m u l a t i o n o f t h e P r o b l e m C o n s i d e r an a r b i t r a r i l y - s h a p e d s a t e l l i t e w i t h i t s c e n t e r o f mass a t S m o v i n g i n an a r b i t r a r y t r a j e c t o r y a b o u t t h e c e n t e r o f f o r c e 0 ( F i g u r e 3-2). Any s p a t i a l o r i e n t a t i o n o f t h e s a t e l l i t e i s c o m p l e t e l y s p e c i f i e d t h r o u g h t h r e e s u c c e s s i v 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 y, 3, and a, r e f e r r e d t o a s r o l l , yaw a n d p i t c h ( s p i n ) , r e s p e c t i v e l y . They d e f i n e t h e a t t i t u d e o f t h e s a t e l l i t e p r i n c i p a l a x e s x , y, z w i t h r e s p e c t t o t h e o r b i t a l r e f e r e n c e c o o r d i n a t e s y s t e m X Q , y^ , Z Q d e f i n e d b e f o r e . An i n e r t i a l c o o r d i n a t e s y s t e m X 1 , Y 1 , Z 1 i s l o c a t e d a t 0. W i t h r e s p e c t t o t h e p r i n c i p a l c o o r d i n a t e s , t h e e x p r e s s i o n s f o r k i n e t i c a n d p o t e n t i a l e n e r g i e s c a n be w r i t t e n a s : T = i [m ( R 2 + R 20 2) + I (6 c o s y cos3 - Y sin3 + a ) 2 z S X * * • + I (-8 siny cosa + 3 cosa + 0 cosy sing s i n a + y cos3 s i n a ) ^ + I (8 siny s i n a - 6 s i n a Z 2T + 0 c o s y sinB c o s a + y cos3 c o s a ) J , ....(3.1a) ym P = + -^T I X {-2 + 3(1 + c o s 2 B ) s i n y}+ I„ {1 4R Y 2 + 3 s i n 2a s i n 2y sin3 + 3 c o s 2a c o s 2y + 3 s i n y 2 x c o s 3 ( c o s 2a - 1 ) } + I {1 - 3 s i n 2a s i n 2y sin3 - 3 c o s 2a c o s 2y - 3 s i n 2 y c o s 2 3 ( c o s 2a +1)}] . ....(3.1b) 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 a n d r e c o g n i z i n g t h a t t h e o r b i t a l m o t i o n r e m a i n s e s s e n t i a l l y u n a f f e c t e d by s a t e l l i t e l i b r a t i o n s , Figure 3-2 Reference coordinate systems and the geometry of s a t e l l i t e l i b r a t i o n a l motion with a , 3 and y degrees of freedom 4 8 the'governing equations of attitude motion in the y, $, and a l i b r a t i o n a l degrees of freedom can be written as: Y (r o l l ) • 2 2 2 2 " 2 2 ( I sxn $ + cos 8 - K cos a cos 8)Y - 2 ( 1 sin 8 + cos 8 - K cos 2a cos 28) , ; S I N 6 A Y' + K s i n 2 a c o s B 8" 1 + e cos0 2 2 2 2 - { I cosy cos 2 8 - 2 C O S Y cos 8 + K.(l - 2 cos a sin 8) C O S Y - K sin8 sin 2a sinY + . — f s : * - n 9 — 3 - sin 2a cos8} 8' i + e cost) T • n " , 2el sin8 . - , _ „ I smS a + T— 5— sm8 + I sinY cos8 1 + e cosQ + K cos8 ( C O S Y sin 2a sin8 - sinY cos 2a) i a K sin 2a sin3 „"2 , , T , , T, 2 , ' „ i . „ „• ^ 8 + ( I - 1 + K cos a)Y 3 sin 28 + K Y'3' sin 2a cos 23 - ( I - K cos 2a)a'8' cos8 " 1 + e. cos6 { ( 1 " 1 " K c o s a ) c o s Y sin 28 - K sinY cos8 sin 2a} 2 , s i n 2Y C O S 8 r T , , 3 , n 2 > i + i ( I - 1 + ! + E C O S E ( I - 1 + K cos^a) } K s i n 2Y 2 • 2n 2 , 3 cos 2a sin 8 cos a + 1 + e cos! + K s i " 2a cos 2Y sin8 x ( l + r \ - n • v 1 + e cose' u ' • • • • (3 • 2 s.) 8 (yaw) n v c ^ 2 . ! (a" - 2e3' sine ^ K Y " cos8 sin 2a , / T _ n (1 - K sm a) (8 - i + e cos0 ) 2 { Y c o s 2 2 2 2 - 2 C O S Y cos 8 + K ( C O S Y cos 23 cos a + C O S Y sin a - sinY sin8 sin 49 e. s i n 9 cosg . n * i ' L r x • n , / • • -> - + e c o s e s i n 2 a ) J Y + { I c o s y sin3 + K ( s i n y s i n 2a + c o s y s i n g c o s 2a)} a' - K 3'a' s i n 2a + ( I + K c o s 2 a ) Y ' a ' cos3 2 (I - 1 + K c o s a) 12 . „ 2e sin0 rT, , • .2 - T Y s i n 23 - T — ; 7 r { K ( s i n Y sin a 2 1 1 + e cosG ' 2 , C O S Y sin3 s i n 2a) . -, 3 ( I - 1 + K c o s a) . 2 . _ n + 2 sin Y} - 2 ( 1 + e c o s 9 ) sin Y s i n 23 2 . I - 2 + K c o s a .2 . ~ 0 + 2" C O S Y s i n 23 K s i n 2Y cos3 s i n 2a ., 3 v _ „ + 4 vJ- + 1 + e cosS ; " U '" (3.2b) a ( p i t c h ) T " 2e I a sine T H . 0 F T . N T, , . „ „ l a - i + e cos6 ~ ^ sinp - i i s m y cos3 - K ( s m Y cos3 c o s 2a , - \ C O S Y s i n 23 c o s 2a) - 2\ \ ^^^^ Y ' - K Y ' 2 c o s 2 3 s i n 2a - { I C O S Y sin3 + K ( s i n Y s i n 2a + C O S Y sin3 c o s 2a)}3* • 2 , K3 s i n 2a / T , _ , , 1 0 t „ 2e I sin6 , D + — ~ - ( I + K c o s 2a) Y 3 cos3 - i — ; n- C O S Y C O S B 2 1 + e cosa K 2 2 2 + { s i n Y s i n 2a - c o s y s i n 3 s i n 2a + s i n 2y sin3 c o s 2a 3 2 2 + -=.—• K ( s i n Y cos3 c o s 2a - c o s 2y s i n 2a - s i n y c o s 3 s i n 2a)} 1 + e cose ' 1 = 0. . . . . . ( 3 . 2 c ) The e q u a t i o n s (3.2) r e p r e s e n t a c o m p l e t e s e t g o v e r n i n g a t t i t u d e d y n a m i c s o f a t r i a x i a l s y s t e m n e g o t i a t i n g an a r b i t r a r y e l l i p t i c t r a j e c t o r y . 3.3 M o t i o n i n t h e S m a l l The c o u p l e d , n o n l i n e a r , n onautonomous e q u a t i o n s (3.2) a r e t o o c o m p l i c a t e d t o a d m i t any known c l o s e d - f o r m s o l u t i o n . A l t h o u g h 50 a n u m e r i c a l s o l u t i o n t h r o u g h a d i g i t a l c o m p u t e r c a n a l w a y s be o b t a i n e d , i t o f t e n f a i l s t o i m p a r t u n d e r s t a n d i n g as t o t h e i m p o r t a n c e o f t h e p h y s i c a l p a r a m e t e r s i n v o l v e d . U n d e r s u c h a s i t u a t i o n , s o m e t i m e s a l i n e a r i z e d a n a l y s i s may be u s e d t o a d v a n t a g e p r o v i d e d i t r e t a i n s e s s e n t i a l f e a t u r e s o f t h e s y s t e m . 3.3.1 S p i n n i n g s y s t e m s 3.3.1a A p p r o x i m a t e a n a l y t i c a l s o l u t i o n R e p l a c i n g t h e t r i g o n o m e t r i c f u n c t i o n s i n y and 3 i n e q u a t i o n s (3.2) by t h e i r s e r i e s e x p a n s i o n s and i g n o r i n g s e c o n d and h i g h e r d e g r e e t e r m s i n y, 3, y', 3* r e s u l t s i n d e c o u p l i n g o f t h e a - e q u a t i o n a s , a" - {2e s i n 6 / ( l + e c o s 9 ) } ( a ' + 1) - { 3 K / 2 I ( 1 + e c o s 0 ) } s i n 2a = 0 . • ••• (3*3) S i n c e v a l u e s o f t h e o r b i t e c c e n t r i c i t y a nd t h e a s y m m e t r y p a r a m e t e r a r e u s u a l l y s m a l l (e,|K|< 0 . 2 ) , a P o i n c a r e - t y p e e x p a n s i o n , 0 0 CO a (6) = E I e q K S a (6) , (3.4a) s=0 q=0 q , S l e a d s t o t h e s o l u t i o n o f e q u a t i o n (3.3) up t o t h e f i r s t d e g r e e t e r m s i n e and K, a = a8 + 2 e ( a + 1 ) ( 6 - sine) + ( 3 K / 4 a I ) { 6 - ( l / 2 a ) s i n 2a6} , (3.4b) w here t h e s p i n p a r a m e t e r a i s d e f i n e d a s a = a ' ( 0 ) w i t h a ( 0 ) = 0. 51 S u b s t i t u t i n g e q u a t i o n (3.4b) i n t o t h e g o v e r n i n g e q u a t i o n s f o r t h e r o l l and yaw d e g r e e s o f f r e e d o m and l i n e a r i z i n g g i v e s a s e t o f e q u a t i o n s o f t h e f o r m : y" + n 2 l Y + m L 1 B* = f 1 ± ; (3.5) 3" + n 2 2 3 + Y ' = f 1 2 . w h e r e c o m p l i c a t e d f o r c i n g f u n c t i o n s a r e r e p r e s e n t e d b y : f l l = e [ { ( 2 l o + 51 - 3 ) Y - 2 l ( o + l ) B ' } c o s 9 + 2 ( B + y ' ) s i n 0 ] + K [ {4 + a + 3 / 4 ( a + 2e) - 21 - I a / 2 } y c o s 2a6 + { l / 2 ( a + 1) - I - 3/4 ( a + 2e) - a ) B ' c o s 2a6 - { ( B " + B)/2 + a ( Y ' + B ) } s i n 2a9 J ; f 1 2 = 2 e { I ( l + a ) ( B + Y * ) c o s 9 - (Y - B ' ) s i n 9 } + K[ { ( 3 / 4 ( a + 2e) - 1 - a ) ( B + Y') + 1 ( 1 + a ) ( B + Y ' / 2 ) / ( 1 - 2e) } x c o s 2a9 - { Y"/2 + Y ( 2 + a) - a B ' } s i n 2o0 ] . R e p l a c i n g y and B by t h e e x p a n s i o n s o f t h e f o r m : 00 oo Y ( 6 ) = E E e q K S Y „ c (0) ; s=0 q=0 q , S (3.6) CO 00 8 ( 6 ) = E E e q K S B (9) ; s=0 q=0 q , S and r e t a i n i n g t h e f i r s t d e g r e e t e r m s i n e and K g i v e s , a f t e r a c o n s i d e r a b l e amount o f a l g e b r a i c m a n i p u l a t i o n s , t h e s o l u t i o n as 52 y) 2 I i = l s i n p. 0 cos p.( i 1 l ^ c o s p i6 - X i s i n p i f a. cos6. + e b. , cos6 • - L 1 X ^  X 1 f 1 . a. sin6• + e b. , sin6. , \ i i i , l i , l + K b. 9 cos5. 0} + K b. „ sin6. ^ 1,2 1,2 4 + Z j = l e sin(p £0 - (-l) j9 + 6 £) e Ai,j+4 c o s ( P £ 6 - (-D :e + 6 &) + K A 2 ^ s i n ( P j i 0 - (-l) j2ae + 6 £) "| + K A 2 , j + 4 c o s ( P £ e - (-D j2a0 + 6Z)J 1 = 1 for j = 1, 3 ; I = 2 for j = 2, 4. (3.7) Here p 1 and p 2 are c h a r a c t e r i s t i c frequencies obtained from 4 2 2 2 2 2 P - ( n l x + n 1 2 - m1;L m 1 2)p + n 1 . n-, , = 0 , ....(3.8a) 11 "12 and A^, A 2 are given by n 11 " Pi m12 Pj m l l Pi 2 2 P i " n l 2 ; i = l , 2 (3.8b) In equations (3.7) a. , b. , 5 . ,6. are defined as ^ l ' i , n I ' i,n a. = ± I-'30 + V k P k f , Ao Ak + BQPk ^ 2 P 2 - \ p j l p 2 A 1 - P l A 2 / 1/2 b. = ± i, n ' Z4n " A k P k Z 4 n - 3 f / Z4n- 2P k " ZAn-l\^ - A 2p 2 +. A l P l J y A xp 2 - P l A 2 1/2 53 + f 3 0 + V k P k P 2 A 1 " P l X 2 6 . - a r c t a n T * i \ x 2 p 2 - x l P l \ , ^ , Z 4 n " X k p k Z 4 n - 3 X l p 2 " X 2 P 1 \ o. „ = a r c t a n T ; — r x - -± - — 1 " A 2 p 2 + X l P l Z 4 n - 2 P k - Z 4 n - l A k J \ n = 1, 2 ; i = 1, 2 ; k = 2 / i ; w i t h Z^ , Z 2 , e t c . g i v e n by: :4n-3 = " ( A n , l + A n , 2 ) s i n 6 l " ( A n , 3 + A n , 4 ) s i n 6 2 ' '4n-2 ( A n , 5 + A n , 6 ) c ° s 6 l " ( A n , 7 + A n , 8 ) c ° s 6 2 ' n - l 4 n - l = ~ [ A n , l { P l + ( 2 a ) 1 1 _ i } + \ i 2 { p ± - Uo)^1}^ c o s 6 1 C An,3 {P2 + ( 2 a ) n 1 } + A n , 4 { p 2 " ( 2 a ) R c o s 6 2 n - l n Z 4 n = " [ > n , 5 { P l + ^ a ) " " 1 } + A N , 6 { P l " ^ ^ ' ^ 1 s i n 6 l " ^ A n , 7 { p 2 + ( 2 a ) n 1 } + A n , 8 { p 2 " ( 2 a ) R ± } ] s i n 6 2 '" n = 1, 2, and ( A . . \ = ( [ n l l " ( pm " M L ) 3 ( 2 a ) 1 V ] [ n ^ - ( p m - ( - 1 ) 3 ( 2 a ) 2 ] + m1;L r n 1 2 ( p m - (-1) j ( 2 a ) 1 V -1 54 ri?„ - (p„ - ( - 1 ) j ( 2 a ) 1 V 12 m m (p m - ( - 1 ) j ( 2 a ) 1 1 ) 1 2 v t m m l l ( p m " j ( 2 a ) 1 X ) n l l " ( pm " j ( 2 a ) 1 1 ) 2 X. . i V X i , J + 4 j w h e r e X, .' = a { I (a + 1) (1 + X P ) + 1, j m m m 3 ( 1 - l ) / 2 - (-1) j (X + p ) } . X l , j + 4 = a m { l < ° + 1 ) ( X m + P m> " ( " D ^ 1 . * W } ; * 2 , j = ( - l ) j X m ( l - P ^ / 2 ) / 2 + ( - l ) j a ( X m + p m ) + 4 + a + 3 / 4 ( a + 2e) - 21 - l a / 2 - X m p m { ( a + 1 ) ( I - 1/2) - 3 / 4 ( a + 2 e ) } ; X 2 , j + 4 = 3 ( A m + P m > / 4 ( a + 2 e ) " a p m + " 2) - ( - l ) j a ( l + p m X m ) + (a + l ) X m ( l / ( l - 2e) - 1} + P m ( K a + 1 ) / ( 1 - 2e) - 2}/2 ; i =. 1, 2 ; j = 1, 2, 3 , 4 - ; m = 1 f o r j = 1, 2 ; m = 2 f o r j = 3, 4. To a s s e s s a c c u r a c y o f t h e a p p r o x i m a t e a n a l y t i c a l s o l u t i o n , i t was c o m p a r e d w i t h t h e r e s u l t s o f t h e n u m e r i c a l i n t e g r a t i o n o f t h e e x a c t e q u a t i o n s o f m o t i o n , o b t a i n e d by t h e R u n g e - K u t t a method w i t h b u i l t - i n e r r o r c o n t r o l on an IBM 370 c o m p u t e r ( F i g u r e 3 - 3 ) . The a g r e e m e n t i n f r e q u e n c y a n d a m p l i t u d e 5 5 e = K = 0.1 , 1= 2 ; a=2 , 7 = 10°, y= (3 =0, [3=0-5 'o o 'o o numerical, analytical 20r-- 2 0 a Figure 3-3 Orbits A comparison between the approximate perturbation and numerical solutions for a t r i a x i a l spinning s a t e l l i t e 56 a r e p r e d i c t e d w i t h an a c c e p t a b l e a c c u r a c y . 3.3.1b S t a b i l i t y F o r t h e s t a b l e y and 6 m o t i o n s o f a s p i n n i n g s a t e l -l i t e , t h e c h a r a c t e r i s t i c f r e q u e n c i e s p ^ and p 2 s h o u l d be r e a l , i . e . , p 2 = ±[n2±1 + n 2 2 - m i l m 1 2 ± { ( n ^ + n 2 2 - m ^ m ^ ) 2 - 4 n 2 x n 2 2 > 1 / 2 ] > 0. (3.9a) F u r t h e r m o r e , f o r n o n c i r c u l a r o r b i t a l m o t i o n (e ^ 0 ) , p 2 j- 1/4, 1 , 4 , 9 , , (3.9b) and f o r s t a b i l i t y o f a s y m m e t r i c s y s t e m s (K ^ 0 ) , 2 2 2 2 2 p / a , 4a , 9a , 16a , . . . . ( 3 . 9 c ) 53 The a b o v e c o n d i t i o n s f o l l o w f r o m t h e M a t h i e u e q u a t i o n . However, a c a r e f u l s t u d y o f t h e s t a b i l i t y d i a g r a m f o r t h e M a t h i e u e q u a t i o n r e v e a l s t h a t w i t h s m a l l v a l u e s o f e and K ( e , K < 0.2) c o n s i d e r e d h e r e , t h e s u p e r h a r m o n i c s p = 2, 3, 4, , 3 a , 4 a , 5 a , a r e n o t e x c i t e d . F i g u r e 3-4 p r e s e n t s a s t a b i l i t y c h a r t i n t e r m s o f I and a f o r a n a x i s y m m e t r i c s a t e l l i t e i n a c i r c u l a r o r b i t . N o t e , t h e amount o f s p i n r e q u i r e d f o r t h e s t a b l e m o t i o n d i m i n i s h e s r a t h e r r a p i d l y w i t h an i n c r e a s e i n I . 5 7 100i e = K=0 75 5 0 h 25 unstable stable 0. 0 0-25 _L_ 0.5 I " T T T -075 Figure 3-4 S t a b i l i t y diagram for spinning axisymmetric s a t e l l i t e s i n c i r c u l a r o r b i t s 58 3.3.2 Gravity oriented systems 3.3.2a Approximate a n a l y t i c a l solution Replacing the trigonometric functions i n a, 3, and y hy t h e i r series expansions i n equations (3.2) and ignoring the second and higher degree terms i n a, 3, Y, a', 8*, y' results i n the following equations: y" + n ^ y + ™ 2 1 3 ' = e[ 2{y' + (1 - I - K)3}sin9 + 3y(I - I)cos0]/(1 - K) ; (3.10a) 3" + n 2 2 3 + m 2 2y' = 2e ( B ' - y ) s i n 9 ; (3.10b) a" - (3K/I)a = e{- a" cos9 + 2(a' + l)sin0} . ....(3.10c) For the stable gravity gradient configuration a i s bounded, which leads to, for I > 0, - °° <: K < 0 . An expansion a (0 ) = Z e 4 a ( 0 ) (3.11a) q=0 q leads to the solution of equation (3.10c) as 1 2 2 a - an cos p 0 + ( a n / p ) s i n p 0 + ef {3a„p /(4p - 1)}cos p 0 0 * a 0 ' a L 0L a a L a + ( l / p a ) { 2 / ( l - p 2) + a'(5p 2 - 2)/(4p 2 - l ) } s i n p a 0 + (2/(p 2 - l)}sin9 - { a Q p a ( l + p a/2)/(2p a + 1) }cos (p a + 1)9 5 9 + { a Q p a ( - 1 + P a/2)/(2p a - l ) } c o s ( p a - 1) { a ^ l + p a/2)/(2p„ + l ) } s i n ( P n i + 1) a a + ( a ' ( - 1 + p a/2)/(2p a - l ) } s i n ( p a - 1)9] , 1/2 where p a = (- 3K/I) S i m i l a r l y , assuming (3.11b) Y(6) = E e 4 y (Q) , q=0 q 3(6) = E e q 3 ( 6 ) , q=0 q y i e l d s the solution of equations (3.10a) and (3.10b) as (3.12) ( y \ 2 E i=l s i n p, cos p. A.cos p.8 -A.sin p.( . 1 * 1 1 * 1 a^ cosS^ + e b^ cos6^ ^ > V ai S i n 6 i + e b i s i n 6 i , l J (A^ sin(p £ e - (-D^e + 6 £) ^ + e E j=l L j + 4 cos(p £8 - ( - 1 ) 3 Q + 6 £) V J 1 = 1 for j = 1, 3 ; I = 2 for j = 2, 4 ; (3.13) where p^ and p 2 are c h a r a c t e r i s t i c frequencies obtained from 4 2 2 2 2 2 p - ( n 2 1 + n 2 2 - m 2 1 m 2 2)p + n 2 1 n 2 2 = 0 , ....(3.14a) and A^ are given by, 2 2 6 0 In equation (3.13) a^ , b ± , 5 ± , 6 ± -j^  are defined as a. = ± 1 b. = ± l 'eo + Vk pkY + f-*o\ + W*' A 2 p 2 - X l P l 2 4 - W l \ 2 +/ Z^ " ^ V 2 -A 2p 2.+ A l P l P 2 A 1 - P l A 2 "2Pk - V k 1/2 A / l p2 ~ p i x ; / i $0 + Y 0 Xk Pk p 2 A l " P 1 X ? 6 . = arc tan ^ u, K x 1 1 1 2 i \ A p - A p YoAk + S0 pk, , , Z4 ~ Ak Pk Zl H P2 - A 2 p l \ 6 • ,• = arc tan — ^ — — T x —-— 1,1 \ - A 2 p 2 + A l P l Z 2p R - Z 3 A k / i = 1, 2 ; k = 2/i ; with Z 1 , Z 2 , z3 , Z 4 given by: Z 1 = - (A 1 + A 2)sin6- L - (A 3 + A 4 ) s i n 6 2 ; Z 2 = - (A^ + A 6 ) c o s 6 1 - (A 7 + Ag)cos6 2 ; Z 3 = - A± + 1) + A, ( P1 " 1) cos6^ ' A3 (p 2 + 1) + A4 (p 2 - 1) cos6 2 ; Z4 = " A5 ( P l + 1) + A6 ( P 1 " 1) sin6 ^  " A7 (P 2 + 1) + A8 (p 2 - •1) s i n 6 2 ; and A.^  , A , , A D defined as (A. \ [n 2 21 (P m " (-D j) 2] [n 22 " <pm " ( " 1 ) J ) 2 J 61 Jx2 -1 + n i 2 1 m 2 2 ( p m - (-!)-») n 2 2 - (Pm " ( " 1 ) j ) 2 m 2 1 ( p m - (-1)3) n221 - ( p m - (-1) V m X l X J + 4j - m 2 2 ( p m - ( - l ) - i ) w h e r e : X j = " ( - 1 ) J { 1 " " K ) } am\n " + 1 - 5 ^ 1 " ^ V ' 1 - K ) ; X. . - = - (-1) J a (1 + A p ) ; 3 + 4 m nrm j = 1,. 2, 3, 4 ; m = 1 f o r j = 1, 2 ; m = 2 f o r j = 3, 4 . As b e f o r e , f o r a s s e s s i n g t h e a c c u r a c y o f t h i s a p p r o x i -mate a n a l y t i c a l s o l u t i o n , r e s p o n s e r e s u l t s g i v e n by e q u a t i o n s (3.11b) a nd (3.13) w ere c o m p a r e d w i t h t h o s e o b t a i n e d t h r o u g h a n u m e r i c a l i n t e g r a t i o n o f t h e e x a c t e q u a t i o n s o f m o t i o n ( F i g u r e 3 - 5 ) . The a g r e e m e n t b e t w e e n t h e e x a c t a n d t h e a n a l y t i c a l s o l u -t i o n s i s s u r p r i s i n g l y c l o s e . 3.3.2b S t a b i l i t y A s t u d y o f e q u a t i o n s (3.10) l e a d s t o t h e c o n c l u s i o n s g i v e n b e l o w : ( i ) F o r b o u n d e d a - m o t i o n i n a c i r c u l a r o r b i t , t h e f o l l o w i n g c o n d i t i o n m ust be s a t i s f i e d , -co < K < 0. . . . ( 3 . 1 5 a ) I n e c c e n t r i c o r b i t s , t h e s t a b i l i t y o f a i s e n s u r e d i f , i n a d d i t i o n t o t h e ab o v e c o n d i t i o n , p ^ ft 1/4, 1, 4, 9, (3.15b) 62 e = 0-02, K—0 - 5 , 1 - 2 , — numerical (Y,|3,a)0=5° , (T,p,a)o=0 — analytical a Oh -10h 0 Figure 3 -5 0-5 1 Orbits 1-5 A comparison between the approximate perturbation and numerical solutions for a t r i a x i a l gravity-oriented s a t e l l i t e 63 ( i i ) F o r t h e s t a b i l i t y o f y, 6 m o t i o n s i n a c i r c u l a r o r b i t , a nd Pp must be r e a l . T h u s , i t f o l l o w s f r o m e q u a t i o n (3.14a) t h a t p 2 = 5 [ n 2 1 + n 2 2 " m 2 1 m 2 2 1 { ( n 2 1 + n 2 2 " m 2 1 m 2 2 ) 2 - 4 n 2 1 n 2 2 } 1 / 2 ] > 0 . ( 3 . 1 5 c ) I n e c c e n t r i c o r b i t s , t h e c o n d i t i o n p 2 ji 1/4, 1, 4, 9, , (3.15d) i n a d d i t i o n t o e q u a t i o n ( 3 . 1 5 c ) , must be s a t i s f i e d . F i g u r e 3-6 r e p r e s e n t s t h e s t a b i l i t y d i a g r a m f o r a , 6, and y m o t i o n s o f t r i a x i a l g r a v i t y o r i e n t e d s a t e l l i t e s . F o r a c i r -c u l a r o r b i t , ot i s s t a b l e f o r a l l t h e v a l u e s o f I and K c o n s i d e r e d h e r e . The r e g i o n o f s t a b l e 3 and y m o t i o n s i s c o m p a r a t i v e l y s m a l l , and e x t e n d s f r o m t h e a b s c i s s a t o t h e l i n e r e p r e s e n t i n g t h e a x i s y m m e t r i c s y s t e m s ( i . e . I + K = 1) . The e f f e c t o f e c c e n t r i c i t y o f t h e o r b i t i s t o p r o m o t e r e s o n a n c e o f t h e a m o t i o n f o r f r e q u e n c i e s P a = 1/2, 1. A s i m i l a r e f f e c t on 3 and y i s e x p e r i e n c e d i n t h e r e g i o n e x p r e s s e d by p ^ = p^ = 1/2, 1. N o t e t h a t t h e s u p e r h a r m o n i c s P a , p^ , p^ > 1 a r e n o t e x c i t e d f o r s m a l l e c c e n t r i c i t y v a l u e s (e < 0.2) c o n s i d e r e d h e r e . A l s o n o t e t h e u n s t a b l e c o n d i t i o n s o f t h e a x i s y m m e t r i c s y s t e m s f o r = 1/12, 1/3, w h i c h a g r e e w i t h t h e r e s u l t s o f s e c t i o n 2.5. 3.4 F o r c e a n d Moment A n a l y s i s L e t AF be t h e f o r c e n e c e s s a r y t o m a i n t a i n a c e r t a i n mass e l e m e n t Am i n i t s p o s i t i o n as shown i n F i g u r e 3-7. The a c c e l e r a -t i o n o f Am i n t e r m s o f AF ( i n t h e x, y, z s y s t e m ) a n d g r a v i t y , g c a n be e x p r e s s e d a s 64 a unstable (e*0) -co 4 - 3 |3,T unstable (e^O) a stable (3,Y unstable >/ *y / / gravity oriented S axi symmetric j> systems / .^ -PrP7-V2 -co F i g u r e 3-6 S t a b i l i t y d i a g r a m f o r t r i a x i a l g r a v i t y o r i e n t e d s a t e l l i t e s i n a r b i t r a r y o r b i t s o f s m a l l e c c e n t r i c i t i e s (e < 0.2) Figure 3-7 Force equilibrium of a mass element Am to maintain i t at a desired position 66 ^ 2 < [ C ] ( R m } ) = [C] {gm} + I C ] if! > f ....(3.16) where: R = position vector to mass element Am m = (R siny cos3 + x)u + (R cosy cosa + R siny sinS sina + y ) u y + (-R cosy sina + R siny sin3 cosa + z)u , g = - ( y / R 3 ) R , ym m m [C] = transformation matrix with elements C^ _. , C l l = c o s 3 cosy / - sina sin3 cosy - cosa siny , C13 = C O S a s i n 3 cosy + sina siny , C21 ~ c o s | 3 s i n Y cos0 + sin3 sine , = cos6(sina sin3 siny + cosa cosy) - sine cos3 sina , C23 = c o s ^ ( c o s a sin3 siny - sina cosy) - sin6 cos3 cosa , ^31 = c o s 3 siny sine - sin3 cos6 , C^2 = sinG(sina sin3 siny + cosa cosy) + cosG cos3 sina , C^^ = sin6(cosa sinB siny - sina cosy) + cos6 cosa cosB . Using equations (2.26) and the approximation 1 1 3 — 5 - - —-- [1 - — {x siny cos3 + y ( s i n a siny sin3 + cosy cosa) R J R J R m + z(cosa siny sin3 - sina cosy)}] , the force expression can be rewritten as, AF = u) 2(l + e cos6) 3(AF u + AF u + AF u ) Am . ....(3.17) x x y y z z Here AF^ , AF^. , AF z are rather lengthy expressions involving position coordinates x, y, z of Am and l i b r a t i o n a l displace-ments, v e l o c i t i e s and accelerations: 2 2 ' 2 AF .= x [ l - 3 s i n y cos 3 - (1 + e cos8){( y cos3 + cosy sin3) + (3' - siny) 2}] + y [ - 3 siny cos3(siny sin3 sina + cosy cosa) - 2e sin6(-y' cos3 cosa + 3' sina - siny sina - cosy sin3 cosa) + (1 + e cos0){-y" cos3 cosa + 2y' sin3 (- cosy sin3 sina + siny cosa) + 2y a cos3 sina - y sm3 cos3 sina + 3 sina + 23'a' cosa + 2a (cosy sin3 sina - siny cosa) + cosy cos3(cosy sin3 sina - siny cosa)}] + z[3 siny cos3(cosy sina - siny sin3 cosa) - 2e sin0(y' cos3 sina + 3* cosa - siny cosa + cosy sin3 sina) + (1 + e cos9){y" cos3 sina - 2y' sin8 (cosy sin3 cosa + siny sina) i i 12 n + 2y a cos3 cosa - y sin3 cos3 cosa + 3 cosa - 23'a' sina + 2a' (cosy sin3 cosa + siny sina) + cosy c o s 3(cosy sinB cosa + siny sina)}] ; x[- 3 siny cos3(cosy cosa + siny sing sina) - 2e sin0(y cosB cosa - 3 sina + siny sina + cosy sin3 cosa) + (1 + e cos0){y" cos3 cosa 2 2 I I + 2 y' cosy cos 3 s i n a - 2 Y'3' sin3 cosa 2 • - y' sin3 cos3 sina - 3 " sina + 2 3 ' cosy cos3 cosa + cosy cos3(cosy sin3 sina - siny cosa)}] 2 + y [ l - 3(cosy cosa + siny sin3 sina) + (1 + e cos0 ) { 2 y ' cos3 sina(cosy sin3 sina - siny cosa) + y'p' cos3 s i n 2 a + 2 y'a' sin3 1 2 2 2 ' - y' (1 - cos 3 s i n a) + 23 sina(cosy sin3 cosa \ „ i 2 . 2 „ ' . i2 + siny sina) - 3 s i n a - 2 a cosy cos3 - a 2 2 2 - cos 3 s i n a - (cosy cosa + siny sin3 sina) }] 3 3 2 + z[- -j sin 2y s i n 3 cos 2 a + -j sin 2 a (cos y - s i n 2 y sin^3) - 2e sin0(y' sin3 - cosy cos3 - a') + (1 + e cos0){y" sin3 + 2 y' cos8 sina(siny sina 1 1 2 Y 1 2 2 + cosy sin3 cosa) '+ 2y .3 cos3 cos a + — cos 3 sin + 23 cosa(siny sina + cosy sin3 cosa) ' 2 1 - s i n 2 a - a" - s i n 2y sin3 cos 2 a 1 2 2 2 + j sin 2a(cos y sin 3 - sin y)}] ; x[3 siny cos3(cosy sina - siny sinp cosa) - 2e sine(~y cos3 sina - 3 cosa + siny cosa - cosy sin3 sina) + (1 + e cose){-y" sin3 sina + 2y' cosy cos 3 cosa + 2y'3' sin3 sina ,2 y ti i , - J-2~ s i n 23 cosa - 3 cosa - 23 cosy cos3 sina + cosy cos3 (cosy sin3 cosa + siny sina)}] 3 2 2 2 + Y l 2 s i n 2a (cos y - s i n y s i n 3) - s i n 2y sin3 cos 2a + 2e sine (y ' sin3 - a it - cosy cos3) + (1 + e cos0){-y sin3 + 2y ' cos3 cosa (cosy sin3 sina - siny cosa) • 1 2 y ' 2 2 - 2y 3 cos3 sin a + -~2~ cos 3 s m 2a + 23' sina (siny cosa - cosy sin3 sina) 1 2 3 • o , " s i n 2y . „ 0 - s i n 2a + a - 2— S I N P cos 2a , sin 2a , .2 , 2 . 2n. , , + 2" (~ S I N Y + cos y s i n 3) / J 2 + z [ l - 3(cosy sina - siny sin3 cosa) + (1 + e cos0){2y'a' sin3 - y'ft' sin 2a cos3 + 2y ' cos3 cosa (siny sina + cosy cosB cosa) - y ' 2 ( l - cos 23 cos 2a) + 23' cosa (siny cosa 2 2 • - cosy sin3 sina) - 3' cos a - 2a cosy cos3 12 2 - a - (siny sin3 cosa - cosy sina) - cos 23 cos 2a}]. 70 The moment r e q u i r e d t o a t t a i n t h e d e s i r e d o r i e n t a t i o n o f t h e s a t e l l i t e c a n t h u s be o b t a i n e d f r o m M = r x AF . . . . . ( 3 . 1 8 ) m m s 3.5 L i b r a t i o n a l R e s p o n s e and Time H i s t o r y o f t h e F o r c e s A v a s t v a r i e t y o f v a r i a b l e s i n h e r e n t t o t h e p r o b l e m and t h e i r s y s t e m a t i c v a r i a t i o n r e s u l t e d i n r a t h e r e x t e n s i v e 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 a few o f t h e t y p i c a l r e s u l t s a r e p r e s e n t e d h e r e . R e s p o n s e d a t a f o r t h e l i b r a t i o n a l m o t i o n , f o r c e compo-n e n t s a t a g i v e n l o c a t i o n , and p h a s e p l a n e p l o t s as a f f e c t e d by s y s t e m p a r a m e t e r s f o r a t r i a x i a l s p i n n i n g s a t e l l i t e a r e g i v e n i n F i g u r e s 3-8 t h r o u g h 3-10. The p h y s i c a l p a r a m e t e r s c l o s e l y c o r r e s p o n d t o E x p l o r e r XX ( T a b l e 3 . 1 ) , w h i c h h a s a g e o m e t r i c s ymmetry b u t d e v e l o p s i n e r t i a a s y m m e t r y due t o t h e e x p e n d i t u r e o f f u e l ( F i g u r e 3 - 1 ) . F o r c e s c o r r e s p o n d t o t h o s e r e q u i r e d t o h o l d t h e i o n p r o b e i n p o s i t i o n . . F i g u r e 3-8 r e p r e s e n t s t h e s a t e l l i t e c o n d i t i o n i n t h e ab-s e n c e o f any e x t e r n a l d i s t u r b a n c e . T h e r e i s no r o l l o r yaw, j u s t t h e p e r i o d i c v a r i a t i o n o f t h e p i t c h r a t e . A l s o , as c a n be e x p e c t e d , t h e t r a n s v e r s e f o r c e s F and F v a n i s h . I n f l u e n c e o f an a r b i t r a r y y z -1 d i s t u r b a n c e a n d v a r i o u s s y s t e m p a r a m e t e r s i s r e c o r d e d i n F i g u r e 3-9. V a r i a t i o n o f a' r e m a i n s e s s e n t i a l l y u n c h a n g e d b u t h i g h f r e -q u e n c y r o l l a n d yaw l i b r a t i o n s s e t i n w i t h s i g n i f i c a n t m o d u l a t i o n s . S i m i l a r l y , t h e a x i a l f o r c e on t h e i o n p r o b e r e m a i n s a l m o s t c o n s t a n t 71 1 - 2 , CT = 220, V l i - £ | 5 ' - 0 350 a 275 200 Y = (3=0 3h F y - F z - 0 6 FX(X10),2 N e = 0.1,K = o,0.1 0 0.5 Orbits 1.5 F i g u r e 3-8 T y p i c a l p l o t s s h o w i n g t h e c o n d i t i o n o f a s p i n n i n g s a t e l l i t e i n t h e a b s e n c e o f any e x t e r n a l d i s t u r b a n c e 7 2 e = K = 0.1 , l = 2 ;a =220, T o =5, B-V-0, (3 = 0.5 °,->0 ' ^ T P ,a 0 R(x10), Figure 3-9 Response of a spinning s a t e l l i t e to an ar b i t r a r y disturbance. Note the high frequency amplitude modulations of l i b r a t i o n s ( r o l l , yaw) and transverse force components 73 b u t r e l a t i v e l y l a r g e t r a n s v e r s e f o r c e s a t h i g h f r e q u e n c y a p p e a r . The p h a s e p l o t s s u g g e s t t h a t t h e m o t i o n i s p e r i o d i c b u t o n l y o v e r a l a r g e number o f o r b i t s . F i g u r e 3-10 s u m m a r i z e s t h e e f f e c t o f i m p o r t a n t s y s t e m v a r i a b l e s . N o t e a s u b s t a n t i a l i n c r e a s e i n b o t h t h e a m p l i t u d e and f r e q u e n c y o f t h e l i b r a t i o n s as w e l l as t h e f o r c e s due t o an i n c r e a s e i n t h e o r b i t a l e c c e n t r i c i t y ( F i g u r e 3-10a) . A c o m p a r i s o n o f F i g u r e s 3-9 and 3-10 (b) i n d i c a t e s t h e e f f e c t o f a d e c r e a s e i n t h e i n e r t i a a s y m m e t r y p a r a m e t e r K. The s i g n i f i c a n t i n f l u e n c e a p p e a r s t o be a c o r r e s p o n d i n g r i s e i n t h e f r e -q u e n c y o f r o l l a n d yaw l i b r a t i o n s . The e f f e c t o f a d e c r e a s e i n t h e i n e r t i a p a r a m e t e r I i s i l l u s t r a t e d i n F i g u r e 3 - 1 0 ( c ) . A s u b s t a n -t i a l i n c r e a s e i n t h e f r e q u e n c y a n d a m p l i t u d e o f b o t h t h e l i b r a -t i o n s and f o r c e s i s n o t i c e d . The s p i n p a r a m e t e r a f f e c t s t h e p e r f o r m a n c e i n a s i m i l a r manner ( F i g u r e 3 - 1 0 d ) . As c a n be e x p e c t e d , a c h a n g e i n t h e i n i t i a l d i s t u r b a n c e a f f e c t s t h e T a b l e 3.1 V a l u e s o f s y s t e m p a r a m e t e r s i n t h e e x a m p l e s o f t h e t r i a x i a l s p a c e c r a f t u n d e r c o n s i d e r a t i o n c o n f i g u r a t i o n e K I 0 T I n t e r n a t i o n a l I o n o s p h e r i c S a t e l l i t e ( F i g u r e 3-1) 0.1 0.1 2 220 100 m t s . The S p a c e S h u t t l e (175,000 kg) m a i n t a i n i n g an i n s t r u m e n t p a c k a g e (200 kg) a t t h e end o f a r i g i d m a s s l e s s 400 m 0 -0 .15 2 90 l o n g boom a l o n g t h e l o c a l m t s . v e r t i c a l The S a t e l l i t e S o l a r Power S t a t i o n ( F i g u r e 1-4) 0 0.15 0.15 -1 24 h r s . 74 e -0.2, K = 0.1, 1 = 2 . a = 220, v = 5°, |3-Y'-0, p'-a5 Figure 3-10 E f f e c t of system parameters on the s a t e l l i t e re-sponse: (a) increase i n the o r b i t a l e c c e n t r i c i t y 75 e-0-1, K = 0, l=2 ; a = 220, Yo=5°, Po=-/=0, [£=0-5 F i g u r e 3-10 E f f e c t o f s y s t e m p a r a m e t e r s on t h e s a t e l l i t e r e -s p o n s e : (b) r e d u c t i o n i n t h e i n e r t i a a s y m m e t r y p a r a m e t e r K 7 6 Fv(x10), Y N 0 ^ 3 £(x10), N 0 Figure 3-10 E f f e c t of system parameters on the s a t e l l i t e re-sponse: (c) reduction i n the i n e r t i a parameter I 77 e=K=0.1,1=2; cr = 300, YQ= 5° po=Y^0, ^=0.5 Figure 3-10 E f f e c t of system parameters on the s a t e l l i t e response: (d) increase i n the spin parameter a 7 8 e=K-0.1, 1=2, a =220, Y=R=0, Yo'=0.5, |3=-5° O 'o o ' o Figure 3-10 E f f e c t of system parameters on the s a t e l l i t e response: (e) d i f f e r e n t i n i t i a l conditions 79 r e s u l t i n g r e s p o n s e ,but o n l y l o c a l l y , l e a v i n g t h e o v e r a l l c h a r a c t e r t h e same ( F i g u r e 3 - 1 0 e ) . T y p i c a l r e s p o n s e d a t a f o r a t r i a x i a l g r a v i t y o r i e n t e d s y s t e m a r e shown i n F i g u r e s 3-11 and 3-12. The p h y s i c a l p aram-e t e r s c l o s e l y c o r r e s p o n d t o t h e S p a c e S h u t t l e s u p p o r t i n g a 200 k g i n s t r u m e n t p a c k a g e a t t h e end o f a 400m l o n g m a s s l e s s r i g i d boom ( T a b l e 3 . 1 ) . The f o r c e s c o r r e s p o n d t o t h o s e r e q u i r e d t o h o l d t h e i n s t r u m e n t p a c k a g e i n p o s i t i o n . F i g u r e 3-11 r e p r e s e n t s f o r c e c o m p o n e n t s on t h e p r o b e i n t h e a b s e n c e o f any e x t e r n a l d i s t u r b a n c e . As e x p e c t e d , t h e f o r c e s a r e c o n s t a n t i n a c i r c u l a r o r b i t . H o wever, f o r e / 0 s i g n i f i c a n t v a r i a t i o n s i n F^ and a r e n o t i c e d . I n t e r e s t i n g l y , F^ i s a l w a y s n e g a t i v e l e a d i n g t o t e n s i o n i n t h e boom. F i g u r e 3-12 s u m m a r i z e s r a t h e r an e x t e n s i v e amount o f i n f o r m a t i o n c o n c e r n i n g t h e e f f e c t o f e c c e n t r i c i t y , i n e r t i a , and a s y m m e t r y p a r a m e t e r s o n t h e l i b r a -t i o n a l m o t i o n a nd f o r c e s i n a c o m p a c t f o r m . The p r e s e n c e o f an i n i t i a l d i s t u r b a n c e l e a d s t o t h e s y s t e m l i b r a t i o n s w i t h p < p„ a 6 < p ^ ( c a s e a ) . A l s o n o t e t h e f l u c t u a t i n g f o r c e s a t h i g h e r f r e q u e n -c i e s ( r e l a t i v e l y , s e e F i g u r e 3-11) a c t i n g on t h e i n s t r u m e n t p a c k a g e . I n g e n e r a l , t h e f r e q u e n c y o f F x was f o u n d t o be g r e a t e r t h a n t h a t o f F o r F . The e f f e c t o f an i n c r e a s e i n e c c e n t r i c i t y i s t o y z s i g n i f i c a n t l y i n c r e a s e t h e a m p l i t u d e s o f a , F and w i t h o u t s u b s t a n t i a l l y a f f e c t i n g B, Y and F ( c a s e b ) . An i n c r e a s e i n a s y m m e t r y ( c a s e c) i s t o r e d u c e p ^ and p ^ and t o i n c r e a s e p ^ . A c o m p l e x m o d u l a t i o n o f F and F i n a d d i t i o n t o a d e c r e a s e i n t h e c y z f r e q u e n c y o f F x i s a l s o n o t i c e d . The e f f e c t o f an i n c r e a s e i n t h e i n e r t i a p a r a m e t e r i s j u s t t h e o p p o s i t e ( c a s e d ) . N o t e t h a t t h e 8 0 Librational Forces (I =2) (a)e=0, K (1) along =-0-15,-0.25 ; local vertical •, (b)e=0-1,K=-0.15 (2) 45 to x,y,z axes 0.1 F X ,N -a,2 -0 a,1 ; b,1 0 Fy ,N -0-2 - 04 a,2 a," b,1 0 F7 ,N -0-03 0 Figure 3-11 a,1 a,2 0-25 0-5 Orbits 075 Force components on an instrumentation package deployed from a nonlibrating t r i a x i a l gravity oriented s a t e l l i t e (Space Shuttle) 81 a[3,a) 0 = 5 , (Y,P,a)-0; (a) e-0.K--0.15, I-2-. (b)e-0.02,K—0.15,l-2,-(c)e-0,K—0.5,l-2;(d)e-0,K—Q15.I-3 10i— -0-3 L 0-04r-F,,N 0 -0-04 0 F i g u r e 3-12 0-5 1 Orbits 1.5 E f f e c t o f i n i t i a l c o n d i t i o n s and s y s t e m p a r a m e t e r s on l i b r a t i o n a l m o t i o n and f o r c e c o m p o n e n t s f o r t h e S p a c e S h u t t l e s u p p o r t e d i n s t r u m e n t a t i o n p a c k a g e 82 a x i a l component of the force (F ) i s negative and e s s e n t i a l l y constant for a l l the cases investigated. 3.6 S a t e l l i t e Solar Power Station (SSPS) The S a t e l l i t e Solar Power Station as proposed by Glaser^ i s one of the more elegant and challenging concepts of recent times. A schematic diagram of the SSPS i n the geosynchronous equa-t o r i a l o r b i t generating 8,000 MW through photovoltaic conversion was shown e a r l i e r (Figure 1-4). The energy i s transmitted by a microwave beam to give 5,000 MW d.c. power on the earth. This section aims at studying the dynamics, s t a b i l i t y and control of the SSPS using the analysis developed e a r l i e r in th i s chapter. It also summarizes the available information on the e f f e c t of solar radiation pressure on the o r b i t a l perturba-tions of the space power sta t i o n . 3.6.1 Configuration control for maximum power generation For the maximum e f f i c i e n c y of conversion from the solar into e l e c t r i c a l energy, the panels of the SSPS must face the sun. Obviously, this would require d i r e c t i o n a l corrections of the panels to account for the motion of the SSPS in the geosynchronous o r b i t and of the earth i n the e c l i p t i c . This i s e a s i l y achieved through the r o l l rotation y and p i t c h rotation a about the l o c a l horizontal and the x-axis, respectively. Figure 3-13 shows the equatorial and e c l i p t i c planes . Position of the sun with respect to the l i n e of nodes i s represented by <p^ , the solar aspect angle. Let u g and u be Equatorial plane R Sun Geostationary orbit Ecl ipt ic Line of nodes F i g u r e 3-13 C o n f i g u r a t i o n c o n t r o l o f t h e SSPS f o r t h e maximum power g e n e r a t i o n CO 84 t h e u n i t v e c t o r s i n t h e d i r e c t i o n o f t h e s u n a n d t h e y - a x i s , r e s p e c t i v e l y . F o r u ^ t o be i n t h e d i r e c t i o n o f u g u . u = 1 , . . . . ( 3 . 1 9 a ) s y i . e . , - s i n y s i n i sincj) + cose)) c o s y c o s (a + 8) s s - sincj> s c o s i c o s y s i n ( a + 0) = 1 (3.19b) This yields: Y = - a r c s i n ( s i n i sin<f> ) ; . . . . ( 3 . 2 0 a ) a = - [0 + a r c t a n ( t a n $ s c o s i ) ] . . . . . ( 3 . 2 0 b ) T h u s , t h e r o l l a n d p i t c h r a t e s n e c e s s a r y t o c o n t r o l t h e SSPS c a n be w r i t t e n a s : S l n 1 C O S ^ s ., . . . . . ( 3 . 2 1 a ) , i . 2. . 2, ,1/2 9 s ' (1 - s i n l s i n <J> ) ' a' = - [1 + C ° 2 1 2 *g] ; (3.21b) 1 - s i n i s i n <J> s w h e r e <$>'s r e p r e s e n t s t h e e a r t h ' s o r b i t a l r a t e a r o u n d t h e s u n (1/365.25 r e v o l u t i o n s p e r d a y ) . 3.6.2 S t a b i l i t y o f t h e SSPS The v a l u e o f s p i n p a r a m e t e r o f o r t h e SSPS may be a p p r o x -i m a t e l y t a k e n as - 1 . U s i n g t h e a n a l y s i s p r e s e n t e d i n s e c t i o n 3.3.1, a s t a b i l i t y d i a g r a m f o r t h e s p a c e c r a f t i n a c i r c u l a r o r b i t a n d h a v i n g a = -1 c a n be d r a w n a s shown i n F i g u r e 3-14. N o t e t h a t t h e r e g i o n o f s t a b l e r o l l a n d yaw m o t i o n i s v e r y s m a l l . 85 e = 0, O = -1; unstable--F i g u r e 3-14 S t a b i l i t y d i a g r a m f o r a t r i a x i a l s p a c e c r a f t i n t h e g e o s y n c h r o n o u s e q u a t o r i a l o r b i t w i t h t h e s o l a r p a n e l s f a c i n g t h e sun.- N o t e t h a t t h e c o n f i g u r a t i o n (I=K=0.15) c o r r e s p o n d i n g t o t h e SSPS l i e s i n t h e u n s t a b l e r e g i o n . 86 I n f a c t t h e c o n f i g u r a t i o n o f t h e SSPS as p r o p o s e d by G l a s e r ( T a b l e 3.1) i s i n d e e d u n s t a b l e . Thus a c o n t i n u o u s c o n t r o l moment w o u l d be r e q u i r e d t o o b t a i n t h e d e s i r e d o r i e n t a t i o n o f i t s p a n e l s w i t h r e s p e c t t o t h e s u n . 3.6.3 F o r c e d i s t r i b u t i o n a nd c o n t r o l moments E q u a t i o n s (3.17) and (3.18) may be u s e d t o d e t e r m i n e d i s t r i b u t i o n o f f o r c e s and moments t o a c h i e v e o r i e n t a t i o n s as d e f i n e d by e q u a t i o n s (3.20a) and ( 3 . 2 0 b ) . F i g u r e 3 -15(a) shows t h e f o r c e d i s t r i b u t i o n p e r k g mass a l o n g t h e a x i a l d i r e c t i o n a t two l o n g i t u d i n a l s t a t i o n s , z = 0 , 2 k m , w h i l e t h e c o r r e s p o n d i n g d i s t r i b u t i o n a t a t r a n s v e r s e s t a t i o n , x = 3 km, i s shown i n F i g u r e 3-15 (b) . F i g u r e 3-16 shows t h e v a r i a t i o n s o f y a n < ^ a ' r e q u i r e d a s f u n c t i o n s o f t h e s o l a r a s p e c t a n g l e t o o r i e n t t h e p a n e l s f o r max-imum power g e n e r a t i o n . The c o r r e s p o n d i n g c o n t r o l moments a r e a l s o i n c l u d e d . N o t e t h a t a l t h o u g h t h e l i b r a t i o n a l f o r c e s a r e s m a l l , t h e y r e s u l t i n enormous c o n t r o l moments, o f t h e o r d e r o f s e v e r a l m i l l i o n N e w t o n - m e t e r s . I n F i g u r e 3-17, t h e t i m e h i s t o r i e s o f c o n t r o l moments f o r f i v e d i f f e r e n t p o s i t i o n s o f t h e s u n a r e shown. N o t e , t h e t r a n s v e r s e moments a r e an o r d e r o f m a g n i t u d e h i g h e r t h a n t h e a x i a l c o m p o n e n t . A p r e l i m i n a r y a n a l y s i s s u g g e s t s t h a t t h e maximum power r e q u i r e d t o o r i e n t t h e SSPS t o f a c e t h e s u n i s a b o u t 40 MW (0.5 p e r c e n t o f i t s c a p a c i t y o f power g e n e r a t i o n ) . 3.6.4 O r b i t a l p e r t u r b a t i o n s due t o t h e s o l a r r a d i a t i o n p r e s s u r e The i n t e r e s t i n t h e s o l a r r a d i a t i o n p r e s s u r e i n d u c e d o r b i t a l p e r t u r b a t i o n s o f t h e SSPS d a t e s b a c k t o t h e d a y s o f i t s Force,N 5 0 (X10) Figure 3-15 (b) x = 3 k m ^(x101) ^ r ^ ^ \ 1 1 1 -2 -1 0 1 2 z(km) D i s t r i b u t i o n of force per unit mass on the solar panel (y = 100 m): (a) z = 0, 2 km; (b) x = 3 km 88 F i g u r e 3-16 R o t a t i o n s a n d c o n t r o l moments a s f u n c t i o n s o f t h e s o l a r a s p e c t a n g l e f o r maximum power g e n e r a t i o n 89 o o o = 0 , -45 , 45—- , ±90 -5' ! I I I 0 0.25 0.5 075 1 0(orbits) Figure 3-17 Time h i s t o r y of the c o n t r o l moments f o r s e v e r a l values of the s o l a r aspect angle 90 c o n c e p t i o n . However, a d e t a i l e d m a t h e m a t i c a l a n a l y s i s o f t h e 31 t o p i c was i n i t i a t e d o n l y r e c e n t l y b y Van d e r Ha . By t a k i n g e* = 0.0001 f o r t h e SSPS (assumed as a p e r f e c t l y b l a c k b o d y w i t h an a r e a / m a s s - 5 ) , t h e l o n g t e r m o r b i t a l e f f e c t s may be s u m m a r i z e d a s f o l l o w s : ( i ) The e c c e n t r i c i t y o f t h e i n i t i a l l y c i r c u l a r o r b i t o f t h e SSPS v a r i e s p e r i o d i c a l l y w i t h a p e r i o d o f a b o u t 365 d a y s . The maximum v a l u e o f e i s 0.1. ( i i ) The m a j o r a x i s o f t h e o r b i t r e m a i n s e s s e n t i a l l y c o n s t a n t , ( i i i ) The a n g l e b e t w e e n t h e l i n e o f n o d e s a nd p e r i g e e shows an i n c r e a s e o f 180° o v e r one y e a r f o l l o w e d by a s u d d e n jump o f 180° when t h e e c c e n t r i c i t y i s z e r o a g a i n , ( i v ) The a n g l e o f i n c l i n a t i o n b e t w e e n t h e o r b i t o f t h e SSPS and t h e e q u a t o r i a l p l a n e v a r i e s p e r i o d i c a l l y w i t h a p e r i o d o f 182.5 d a y s f o r <}> = T T / 2 , 3TT/2 a n d 365 d a y s for = 0, n. The maximum v a r i a t i o n i n t h e v a l u e o f i i s a b o u t 0.25° f o r Q = T T / 2 , 3TT/2 . (v) The r e g r e s s i o n c y c l e o f t h e l o n g i t u d e o f t h e a s c e n d i n g node i s c o m p l e t e d i n a b o u t 325 y e a r s a n d i s i n d e p e n d e n t o f t h e i n i t i a l e c c e n t r i c i t y a n d s o l a r a s p e c t a n g l e . 3.7 C o n c l u d i n g Remarks The s i g n i f i c a n t a s p e c t s o f t h e a n a l y s i s may be s u m m a r i z e d as f o l l o w s : ( i ) A g e n e r a l f o r m u l a t i o n f o r t h e t r i a x i a l s a t e l l i t e u n d e r -g o i n g l i b r a t i o n a l m o t i o n i n an a r b i t r a r y o r b i t i s p r e s e n t e d . 91 ( i i ) H i g h l y c o m p l i c a t e d c o u p l e d , n o n l i n e a r , n o n a u tonomous e q u a t i o n s a r e s o l v e d u s i n g a p e r t u r b a t i o n p r o c e d u r e w h i c h y i e l d s a p p r o x i m a t e s o l u t i o n s t h a t c o m p a r e q u i t e f a v o r -a b l y w i t h t h e n u m e r i c a l a n a l y s i s . The c l o s e d - f o r m s o l u t i o n s n o t o n l y r e d u c e c o m p u t a t i o n a l t i m e and e f f o r t b u t a l s o g i v e b e t t e r i n s i g h t i n t o t h e p h y s i c s o f t h e p r o b l e m . ( i i i ) E x p r e s s i o n s f o r f o r c e s a n d moments r e q u i r e d t o p o s i t i o n and o r i e n t a n o b j e c t i n s p a c e a r e o b t a i n e d . They s h o u l d p r o v e u s e f u l i n p l a n n i n g s c i e n t i f i c e x p e r i m e n t s f r o m o r b i t i n g s p a c e s t a t i o n s , ( i v ) A n a l y t i c a l p r o c e d u r e s a r e a p p l i e d t o t h r e e c o n f i g u r a -t i o n s o f p r a c t i c a l i m p o r t a n c e : E x p l o r e r XX, t h e S p a c e S h u t t l e l a u n c h i n g an i n s t r u m e n t p a c k a g e i n s p a c e a n d t h e SSPS. 92 4. DYNAMICS OF SPINNING AND GRAVITY ORIENTED MULTIBODY SYSTEMS 4.1 Preliminary Remarks As discussed e a r l i e r , ever increasing demand on addi-t i o n a l power has led to larger and hence necessarily f l e x i b l e spacecraft. We have already looked at a few of them including the RAE, TOI, CTS and SSPS. Although the configurations are vastly d i f f e r e n t , they a l l belong to the f a m i l i a r class of multibody systems interconneted through r i g i d or f l e x i b l e l i n k s . This chapter studies the l i b r a t i o n a l dynamics, force d i s - ' t r i b u t i o n and orientation control of a system of two f i n i t e bodies, interconnected through a tether or a beam, and nego-t i a t i n g a c i r c u l a r t r a j e c t o r y . The governing equations of motion for the complicated system are obtained accounting for gyroscopic and g r a v i t a t i o n a l torques. An approximate closed form solution of the highly nonlinear, nonautonomous and coupled equations i s obtained through l i n e a r i z a t i o n and i t s v a l i d i t y assessed through the numerical analysis of the exact equations of motion for a p a r t i c u l a r case. It provides consid-erable insight into the dynamical behaviour and s t a b i l i t y of the system. Next, general expressions for forces and moments acting at the end bodies are derived. F i n a l l y , the analysis i s applied to a configuration of p r a c t i c a l i n t e r e s t : the Space Shuttle supporting a s u b s a t e l l i t e or an instrumentation 93 package. 4.2 Formulation of the Problem Consider a system of two a r b i t r a r i l y shaped bodies of mass , M2 wit h t h e i r c e n t e r s of mass a t S-^ and S2t connected through an e l a s t i c l i n k (length JI) , i n a c i r c u l a r o r b i t around the c e n t e r o f f o r c e 0 (Figure 4-1). Here S r e p r e s e n t s the o v e r a l l c e n t e r o f mass of the c o n f i g u r a t i o n . The system i s f r e e to l i b r a t e w i t h the o r i e n t a t i o n of the l i n e o f c e n t e r s ( S ^ S ^ d e f i n e d by the E u l e r i a n r o t a t i o n s y (out of the plane of the o r b i t ) and a ( i n the plane of the o r b i t ) . The t r a n s v e r s e deformations o f the connecting l i n k are denoted by u(y) and w(y) while the end bodies are p e r m i t t e d to undergo ge n e r a l l i b r a t i o n a l motion (with r e s p e c t to the l i n k ) s p e c i f i e d by a i ' 3 i ' Y i ^  ~ ' 2^ * T n u s the system has e i g h t degrees of freedom i n a d d i t i o n to the number of modes used to s p e c i f y the deformation o f the connecting l i n k . Using F i g u r e s 4-1, 4-2 and r e c o g n i z i n g t h a t ( i n the i n e r t i a l frame of r e f e r e n c e X 1, Y 1, Z'), (4.la) z . 1 94 Figure 4-1 Reference coordinates and the geometry of motion for a multibody earth o r b i t i n g system 95 F i g u r e 4-2 F o r c e e q u i l i b r i a o f t h e mass e l e m e n t s Am. and Am t o m a i n t a i n them a t d e s i r e d p o s i t i o n s 96 and o R : 0 + [6] [y] [a] u (4.1b) where t r a n s f o r m a t i o n m a t r i c e s [ 0 ] , [y], [ a ] , [y^], [B^] and [cu] a r e d e f i n e d i n A p p e n d i x I , t h e e x p r e s s i o n s f o r k i n e t i c and g r a v i t a t i o n a l p o t e n t i a l e n e r g i e s may be w r i t t e n a s : 1 * i = l R . . R . dm. + f l mi mi i 2 '2 ^ R m R dy , m J (4.2a) M. - J L 2 P = - E 9 i = l M. m i - J L yp R m dy (4.2b) The e x p r e s s i o n s f o r t h e e l a s t i c p o t e n t i a l e n e r g y o f t h e c a b l e and beam, f o r s m a l l u and w, have t h e f o r m : JL T e c 2 - J L r . 8u> 2 . 3w> 2 -, , { ( 3 ^ + ( a y } } d y ' (4.2c) eb 2 [ ( E I ) f ( i ! u , 2 + + T e { ( | S , 2 + ( |2,2 } ] d y,. 8y . 3y (4.2d) where t e n s i o n i n t h e c o n n e c t i n g l i n k (T ) i s g i v e n by, T = - F e y = F y=L : (4.2e) y=-L n N o t e , h e r e t h e t e n s i o n i s assumed t o be c o n s t a n t . 97 Substituting from equation (3.17) i n equation (4.2e) gives, for a c i r c u l a r trajectory, * • 2 * 2 2 • * T g = M L{(y - 9 )cos a + y 0 siny sin 2a + a 2 + 2a 0 cosy + 40 2 cos 2y cos 2a} . ....(4.2f) With the energy expressions known, Hamilton's p r i n c i p l e may be used by allowing the v a r i a t i o n i n each of the degrees of freedom to be zero, i . e . , t t 2 r 2 L dt = a 5L dt = 0 . ....(4.3) a This leads to the equations of l i b r a t i o n s (y, a; y^ , 3^ , a^) and vibrations (u, w). The equations for a p a r t i c u l a r case of I ^ = are l i s t e d in Appendix I I . Here, for y = - l ^ , &2; ( u ' w ) c a b l e = ( u ' w ' 9 7 ' 37 }beam " 0 * 4.3 Motion i n the Small Assuming small values of y, a, y. , 6^ , U, W and t h e i r derivatives, the equations in Appendix II may be l i n e a r i z e d : 2 y: y" + 4y + (1/1*) Z [I .(y" + y") + 4{(1 + a.)/4)I . 1=1 J - I -}(y+y.) - {(1 + a.)I . -21 .}&!] y i v' ' i ' l x i y i l 9 8 - e {Y - (£ 1/£) } (U" + 4U) dY = 0 ; (4.4a) a: a" + 3a + ( l / l * ) E I„,(a + a.) i=l + e {Y - (£1/£)}(W" + 3W) dY 0 ; (4.4b) Y ±: (Y- + Y " ) I y i + 4(Y ± + Y){(1 + < J . / 4 ) I x i " I ±> - $ . { ( l + a . ) I . - 21 .} = 0; i l x i y i ' (4.5a) 5 i : V & l + { ( 1 + V ^ i . - V ^ i + { ( 1 + a i ) z x i - 2 l y i } ( Y r + y!) = 0 ; (4.5b) a. : a. + a" = 0 1 x (4.5c) The c o r r e s p o n d i n g l i n e a r i z e d e q u a t i o n s f o r the v i b r a t i o n s o f the c a b l e may be w r i t t e n a s : U: 2 9 U 2 90 2 9 W 3 0 2 + U - (Y - £ x/£)(y" + 4y) - T 0 9Y 2 = 0 ; + ( Y - (a" + 3a) - T Q 3 W 2 9Y = 0 . . (4.6a) (4.6b) S i m i l a r l y , t he e q u a t i o n s f o r the v i b r a t i o n s o f the beam (of u n i f o r m c r o s s s e c t i o n ) reduce t o : U: k 4 9 U - T. 2 2 9 U . 9 U S 9Y 4 0 9Y 2 + 9 0 ' + U - (Y - (y" + 4y) = 0 ; (4.7a) 99 i^W _ _ 3 2W , 3 2W 4.3.1 S o l u t i o n f o r t h e t h e t e t h e r c a s e S u b s t i t u t i o n o f e q u a t i o n s (4.5a) a nd (4.5 c ) i n t o e q u a -t i o n s (4.4a) and ( 4 . 4 b ) , r e s p e c t i v e l y , l e a d s t o t h e s i m p l i f i e d e x p r e s s i o n s f o r y and a m o t i o n a s y: y + 4y - e a: a + 3a + e 0 1 (Y - (U" + 4U) dY = 0 , . . . . ( 4 . 8 a ) (Y - £ 1/£)(w" + 3W) dY = 0 , . . . . ( 4 . 8 b ) w h i c h i n c o n j u n c t i o n w i t h e q u a t i o n s (4.6a) a nd ( 4 . 6 b ) , r e s p e c t i v e -l y , y i e l d l i n e a r , homogeneous a n d u n c o u p l e d e q u a t i o n s i n u a n d W. T h e s e v i b r a t i o n e q u a t i o n s may be s o l v e d u s i n g t h e m e t h o d o f s e p a r a t i o n o f v a r i a b l e s ' * 4 , i . e . , U ( Y , 6 ) = V1(Y) U 2 ( 0 ) , (4.9a) W(Y,0) = W 1(Y) W 2 ( 6 ) . (4.9b) The r e s u l t i n g s o l u t i o n s f o r U a n d W may be w r i t t e n a s oo U = I (a„ • c o s p , . 8 + b . s i n p . 0 ) M . , . . . . ( 4 . 1 0 a ) CU1 r C U l CU1 *CU1 C l oo W = Z (a . c o s p . 0 + b . s i n p . 0 ) M . • . . . . ( 4 . 1 0 b ) i = l c w l c w i c w i ^ c w i c i 100 H e r e , t h e c o e f f i c i e n t s a . , a . , e t c . a r e d e t e r m i n e d f r o m t h e ' c u i c w i f o l l o w i n g r e l a t i o n s (v = u o r w) : { A c v } = [ D c ] _ 1 { N c v } ; . . . . ( 4 . 1 0 c ) { B * v } = [ D c ] - 1 { N ^ } . . . . . ( 4 . 1 0 d ) The e l e m e n t s o f t h e s q u a r e m a t r i x tD ] a r e g i v e n b y : d . . C l ] n s i n ( n . - n .) s i n ( n . + n .) _ ±r c i cj c i c ] i i ^ j 2 n . - n . " n . + n . ' c x CJ C l CJ c o s n s i n n £, 1 - c o s n + e Z h [ 2£ + - i { ££ m cm n 2 £ n c p n C P c o s ( n + n ) - 1 c o s ( n - n ) - 1 cp cm cp cm •, 2 ( n + n ) 2 ( n - n ) n c p cm cp cm o T s i n n . s i n n . + e 2 h . h . [h - 1 2 ( 1 - 9± £1' c i c ] 3 £ 2 n„, n. c i "cj 2 £, s i n ( n . - n .) s i n ( n . + n .) + _ L _ { C J C 1 + c i c i } O 0 2 n . - n . n . + n . 2£ c j c i c j c i £, 1 - c o s n . 1 - c o s n . hi c l + c i . ] . £ * 2 + 2 ; J ' n . n . c j c i m = i , j ; p = i j / m ; . . . . ( 4 . 1 0 e ) , s i n 2n . £, d . . = h 9.1 + 2e h . { - . n (3 - 4 c o s n . + c o s 2n .) CJJ 2 c j ° 3 c j ° 3 ° 3 c o s n . s i n n . „ „ , 2£ n , s i n n . - £1 + 5-^} + e 2 h 2 . { ^ - - °1 n . 2 c i 3 1 2 n . c i n . J c i J c j J 101 1 - c o s n I s i n 2n . - 8 s i n n . + 2 22) + 4<| + £2 Sl) } ^ o2 2 4n ' cj C J ( 4 . 1 0 f ) I n t h e a b o v e e x p r e s s i o n s , n ^ i s t h e i * " * 1 r o o t o f t h e c h a r a c t e r i s -t i c e q u a t i o n s i n n . + e h .{£ ( c o s n . - l ) / £ + l } = 0 , . . . . ( 4 . l O g ) w h e r e 2 s i n n . £, + X,_ c o s n . T n n . 2£_ - J> , , c i 1 2 c i , r 0 c i r 2 1 h c i = ( — ~ 2 In-. } [~ 2 ~ ~ e { 61 " n . c i T n n . - 1 c i 0 c i £, (I. - £„) J> J> s i n n . £. , 1 1 2 ^ 1 2 C l . 1 -+ 2 4- 7z + 5 — ( c o s n . - 1) ) ] • 2JT I n . In . C 1 C 1 C 1 (4.10h) • F i n a l l y , s u b s t i t u t i n g f o r U, u", e t c . i n t o e q u a t i o n s (4.8) an d s o l v i n g f o r y a n d a l e a d s t o t h e f o l l o w i n g e x p r e s s i o n s : • Y A CO Y = Yr, c o s 26 + -~- s i n 20 + e I e * . { a . ( c o s p .0 - c o s 20) 10 2 c i c u i r c u i P + b . ( s i n p .0 s i n 2 0 ) } , .... (4.11a) c u i r c u i 2 ' a' a = a . c o s /38 + — s i n /I8 - e I e * . { a . ( c o s p .0 - c o s /38) 0 c i c w i r c w i P + b . ( s i n p .0 s i n /30) } . (4 . l i b ) CWI r C W l ^ 4.3.2 S o l u t i o n f o r t h e beam c a s e S u b s t i t u t i n g f r o m e q u a t i o n s (4.8a) and (4.8b) i n t o e q u a t i o n s 102 (4.7a) and (4.7b), respectively, leads to a set of l i n e a r , homo-geneous and uncoupled equations i n U and W for the beam case. An application of the method of separation of variables as before to the v i b r a t i o n equations results in the solutions which may be writ-ten as : U = ^ / ^ u i C O S Pbui 6 + b b u i S i n P b u i 9 ) M b i '• ....(4.12a) W = i ^ ^ b w i C O S P b w i 9 + bbwi S i n P b w i 9 ) M b i ( 4 - 1 2 b ) Here the c o e f f i c i e n t s a, . , a, . , e t c . are determined from the bui bwi ' following relations (v = u or w), { A b v } = [ D b ] _ 1 { N b v } ' • ....(4.12c) { B b v } = [ D b ] _ 1 { N b v } ....(4.12d) with the c o e f f i c i e n t s d, , n , d, , etc. of the square matrix [D, ] b l l oiz o given as: 1 [ (q K. q K^ - Dsin(n, , + n. .) * b i j | i ^ j 2(n b. + n b.) L ^ b i 4 b j - ' » - " i " b i - « b j + ( q b i + q b j ) { 1 - c o s ( n b i + n b j ) > ] + -TT-, T-[ (q, . q, . + l)sin(n, . - n. .) 2 ( n b i - n b ; . ) L V M b i Hbj b i by' + ( q b i " q b j ) { c o s ( n b . - n b j ) - 1}] 103 + £ —= ^—{ (q n, s, + r. k, ) cosh k, s i n n m n + k 2 b m b m b p b p b p b P b m bm bp + (qi_ k, s, - r, n, ) s i n h k, cos n, ^bm bp bp bp bm bp bm + (q^ k, r, - n, s, ) (cosh k, cos n, - 1) Mbm bp bp bm bp bp bm + (q^ n u + k, s, ) s i n h k, s i n n, } ^bm bm bp bp bp bp bm + —= _ { ( r , . r, . k, . - s, . s, . k, .) cosh k, . s i n h k, . , 2 v 2 b i bj b i bx bj bj bx bj k b i " k b j + (r, • k, . s, . - r, . s, . k, .) (cosh k, . cosh k . - 1) v bx bx bj bj bx bj bx bj + (s, . s, . k, . - k, . r, . s, . ) s i n h k, . cosh k, . bx b j bx b] bj bx bx b j + ( s b i k b i r b j " S b j R b j r b i ) s i n h k b i S i n h k b j } + £ I h b m [ ( s i n n b p + q b p C O S "bp " q b p ) / n b p 2 + ( s b p " r b p S ± n h % " S b p C ° S h k b p ) / k b p - h ( 1 / n b p " r b p / k b p ) / £ + l2{{- C ° S "bp + q b p S i n n b p ) / n b p + ( r b p cosh k b p + s b p s i n h k b p)/k b p}/£] + e 2 h b i h f a j ( 1 / 3 - + i2/l2) ; m = i , j ; p = i j / m ; 2 _ r . + s . q + At — S I N H 2 K K ' + ~ cos 2n. .) 4 k b j h 3 2 n b j b ^ (4.12e) 104 + — (cosh 2k, - 1) + -= ? r { ( r , . k. . 2 k b j ^ n 2. •+ k 2. b^ b3 b i b: + <3t, • s, • n, .) cosh k, . s i n n, . + (q, . s, . k, . - r, . n, .) ^bj b j b j ' bj bj KHbj bj bj bj bj' x x cos n, . sinh k, . + (s, . k, . + q, . r, , n, .) sinh k, . sin n b] bj bj b] ^bj bj bj bj bj + ( q b j r b j k b j " S b j nbj] ( c o s n b j c o s h k b j " 1 ) } + 2e h b j [ £ 2 { ( - cos n b j + q b j sin n b j ) / n b j + ( r b j cosh k b j + s b j sinh k b j ) / k b j } / £ - - r b j/k b.)/£ + (sin n b. + q b j cos n b j - q b j ) / n 2 j + ( s b j - r b j sinh k b j - s b j cosh k b j)/k 2..] + e 2 h 2 j ( l / 3 - + £ 2/£ 2) (4.12f) The frequencies of vibrati o n of the beam may be calculated using the values of n b^ , the i * " * 1 root of the c h a r a c t e r i s t i c equation e 5 i s, . - [n, . (cosh k, . - cos n, . ) + z-. {k, . (e n . - n, . e~ . £, /£) x b i b i b i b i ' k b^ b i l i b i 2i 1' x cosh k, . - n, . (e 0. - e„. k, . £,/£)cos n, . + (n, . e v b i b i 3i 2i b i 1 b i b i 3x n b i ^ l - k, . e, . ) (1 - s s i n n, . ) } ] [n, . s i n n, . + k, . sinh k, . b i l i £ b i b i b i b i b i e 5 i + e i {n, . (e_ . - e. . £, k, ./£)sin n, . + k, • (e. . - e„.) x k, . b i 3i 4i 1 b i b i b i 4i 2i b i x (1 - cosh k b i ) + k b i ( e 3 i - k b i e 2 i £-L/£)sinh k b i>] 1 = 0. (4.12g) Using the solutions for U and W i n equations (4.12), y 105 and a for the beam may be found as YQ y = Y 0 cos 20 + sin 20 + e E e J i ^ a b u i ( c o s Pbui 6 " c o s 2 0 ) i = l p, . + b b u i ( s i n p b u i 0 - - ^ i s i n 20)} , (4.13a) _ a ' a = a n cos /30 + — s i n /30 - e E e*.{a, . (cos p, .0 0 y-- b i bwi ^bwi p, . - cos /3 9) + b b w ± ( s i n P b w ± 0 = ^ sin /30) } (4.13b) /3~ 4.3.3 Solution for the l i b r a t i o n s of the end-bodies Using equation (4.5c), a solution for the pit c h motion of the end bodies may be written as, = (CK + «Q)0 + aQ - a ; i = 1, 2. ....(4.14a) Sim i l a r l y using equations (4.5a) and (4.5b), the solution for y^ and 3^ (i = 1, 2) i s expressed as follows: 2 y, = E a.. sin(p* 0 + 6 ) - y ; (4.14b) j _ l x J X J i J 2 3- = E a.. A . , cos (p* .0 + 6. .) ; (4.14c) j = l i j !J !D --3 where (for k = 2/j) a. = M ( g i 0 + ( Y 0 + Y i Q U i k Pik 32 + {-^Q + ^ Q ^ i k + B i 0  l j " P i 2 A i 2 - P i l A i l Pi2 A i l " P i l A i 2 . (4.14d) 106 3 i 0 + ( Y 0 + Y i O U i k P i k P i 2 A i l " P i l X i 2 6 . . = a r c t a n (J^ - — i£—iii x i± i i i i — i i ) J P . _ X . _ - p . - X . , - Y n + Y - A J X . . + g . n p.. * i 2 i 2 * i l i l '0 ' l O ' l k l O " l k (4.14e) and p*-^ , a r e ^ n e c h a r a c t e r i s t i c f r e q u e n c i e s o b t a i n e d f r o m * 4 , 2 ^ 2 > *2 ^  2 2 n p . - ( n . , , + n. , „ - m... m.,~)P- + n . n i n.,„ = 0 , * i i l l i l 2 i l l i l 2 ^ i i l l i l 2 w i t h 2 *2 * n i l l m • -i « i l 2 P i j . * *2 2 m i l l P i j P i j " n. , „ i l 2 (4 .14f) X . . 1 D ™ ~ " *" (4.14g) 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 e q u a t i o n s g o v e r n i n g l i b r a t i o n s o f t h e e nd b o d i e s ( A p p e n d i x I I ) f o r a s y s t e m w i t h n o n - v i b r a t i n g , n o n - l i b r a t i n g c o n n e c t i n g l i n k (U = W = y = OL = 0) and e q u a t i o n s (3.2) w i t h e = K = 0 a r e i d e n t i c a l i n f o r m . The v a l i d i t y o f t h e s o l u t i o n o f e q u a t i o n s (3.2) h a v i n g b e e n e s t a b l i s h e d ( s e c t i o n 3 . 3 . 2 ) , i t i s r e a s o n a b l e t o e x p e c t s i m i l a r o r d e r o f a c c u r a c y i n t h e p r e s e n t c a s e . A comment c o n c e r n i n g s t a b i l i t y o f t h e m o t i o n i s a p p r o -p r i a t e h e r e . The s o l u t i o n o f t h e l i n e a r i z e d e q u a t i o n s o f m o t i o n n o r m a l l y y i e l d s u s e f u l i n f o r m a t i o n a b o u t s t a b i l i t y o f t h e m o t i o n i n t h e s m a l l . E v e n i n t h e a b s e n c e o f da m p i n g i n t h e c o n n e c t i n g l i n k , U and W c a n be e x p r e s s e d i n t e r m s o f bo u n d e d t r i g o n o m e t r i c f u n c -t i o n s . Thus t h e m a t e r i a l s t i f f n e s s o f t h e c o n n e c t i n g l i n k i s s u f f i c i e n t t o e n s u r e s t a b i l i t y o f t h e v i b r a t i o n a l m o t i o n . F u r -t h e r m o r e , t h e l i b r a t i o n s y a n d a, w h i c h a r e i n f l u e n c e d b y U and W, a r e a l s o b o u n d e d . The s t a b i l i t y o f t h e m o t i o n y + Y^ a n d 3j_ i n 107 t e r m s o f 1^ and o\ , i = 1, 2, f o r t h e e n d - b o d i e s may be s t u d i e d by u s i n g F i g u r e 3-4. N o t e a l a r g e r e g i o n o f s t a b l e m o t i o n f o r r e l a t i v e l y h i g h e r v a l u e s o f i n e r t i a a n d s p i n p a r a m e t e r . 4.4 F o r c e a n d Moment A n a l y s i s L e t AF ( i n x, y, z s y s t e m ) and AF^ ( i n x ^ , y^ , z^ s y s t e m ) be t h e f o r c e s n e c e s s a r y t o m a i n t a i n an a r b i t r a r y mass e l e m e n t Am on t h e c o n n e c t i n g l i n k and Anu on t h e i e n d - b o d y , r e s p e c t i v e l y , as shown i n F i g u r e 4-2. T h r o u g h N e w t o n i a n a n a l y s i s , t h e f o r c e s may be e x p r e s s e d i n t e r m s o f t h e a c c e l e r a t i o n c o m p o n e n t s o f t h e mass e l e m e n t s a s : 2 {AF} = [ a ] T [ y ] T [ 6 ] T [ - ^ {R } - {g }]Am ; (4.15a) d t 2 ° {AF..} = [ a ± ] T [ B . ] T [ y . ] T [ a ] T [ Y ] T [ 6 ] T [ - ^ { R } " } ] Anu. d t (4.15b) H e r e t h e t r a n s f o r m a t i o n m a t r i c e s [cu] , [B^]-, e t c . a r e g i v e n i n A p p e n d i x I . The v e c t o r s R and R . ( i n i n e r t i a l r e f e r e n c e ) w e re ^ m m i g i v e n e a r l i e r ( e q u a t i o n s 4.1) and t h e a c c e l e r a t i o n due t o g r a v i t y may be e x p r e s s e d a s : g = - 4r R ; . . . . ( 4 , 1 6 a ) ^m R 3 m m g . = ]~ R . . (4.16b) m i 3 mi mi E q u a t i o n s (4.15) c a n now be u s e d f o r c a l c u l a t i n g t h e f o r c e a c t i n g 108 at any point of the connecting l i n k and the end-body. To evalu-ate the forces acting at the centers of mass (S 1 , S 2) of the end-bodies, equation (3.17), with e = S = x = z = 0 and y = -L^ or L 2 , may be used as the solution of equation (4.15a) . The moment required to maintain the end-body i n a desired or i e n t a t i o n can be obtained from M. = l r i X A F i • ....(4.17) M. X Furthermore, for small y, a, y i , B i and t h e i r derivatives, equa-tions (4.5) represent l i b r a t i o n s of the end-bodies. Thus for the small amplitude motion, equation (4.17) may be rewritten as, _ • ' _ _ M. = r. x AF* i J i i (4 .18) M. x where AF? i s obtained from equation (3.17) with e = 0 and by replacing x, y, z, y, 3, a, y', 3'/ a', y", 3", a" with x i , y i , z ± , y + y± , B i , a + cu , y' + y'± , 3]_ , a' + a| , y" + yV , 3V , a" + aV , respectively. 4.5 Results and Discussion To demonstrate the usefulness of the analysis, i t was ap-pl i e d to a system of considerable i n t e r e s t , the Space Shuttle. The example studied here considers a 175,000 kg o r b i t e r i n a 90 minute o r b i t which supports a s u b s a t e l l i t e or an instrument package by a 109 c a b l e o r a beam. D i f f e r e n t c a s e s i n v e s t i g a t e d a r e l i s t e d i n T a b l e 4.1. N o t e t h e i n c r e a s e i n t h e f l e x i b i l i t y p a r a m e t e r e w i t h p and I. F i g u r e s 4-3 t h r o u g h 4-10 show t h e r e s p o n s e o f t h e c a b l e -c o n n e c t e d s y s t e m . E f f e c t o f t h e f l e x i b i l i t y p a r a m e t e r on t h e f i r s t f o u r modes o f t h e c a b l e i s shown i n F i g u r e 4-3. H e r e , t h e s i n e wave c o r r e s p o n d s t o t h e mode s h a p e due t o an a r b i t r a r y t e n s i o n p a r a m e t e r w i t h e = 0. As e x p e c t e d , t h e s t i f f n e s s o f t h e c a b l e r e d u c e s w i t h an i n c r e a s e i n e, w h i c h i s r e f l e c t e d i n an i n c r e a s e i n a m p l i t u d e o f t h e mode. I n f l u e n c e o f f l e x i b i l i t y a p p e a r s t o be l e s s p r o n o u n c e d a t h i g h e r modes. F i g u r e s 4-4 and 4-5 show t h e v i b r a t i o n r e s p o n s e f o r a r b i -t r a r y i n i t i a l c o n d i t i o n s r e p r e s e n t e d b y N . The c o n t r i b u t i o n f r o m t h e f i r s t t h r e e modes i s s i g n i f i c a n t and t h e i r t o t a l p r e s e n t s a c l o s e a p p r o x i m a t i o n t o t h e e x a c t r e s p o n s e . I n a l l t h e c a s e s , t h e f r e q u e n c y o f U v i b r a t i o n s was o b s e r v e d t o be s l i g h t l y h i g h e r t h a n t h a t f o r t h e W c o m p o n e n t . The l i b r a t i o n a l m o t i o n a , y o f t h e l i n e o f c e n t e r s (S^S^) as w e l l as t h e t i m e h i s t o r y o f t h e f o r c e s a t t h e c e n t e r o f mass o f t h e o r b i t e r f o r t h e s e i n i t i a l T a b l e 4.1 V a l u e s o f s y s t e m p a r a m e t e r s i n t h e e x a m p l e o f t h e S p a c e S h u t t l e s u p p o r t i n g a p a y l o a d (M^ = 200 kg) by a c a b l e o r a beam c o n n e c t i n g l i n k c a s e I ,m p,kg/m ( E I ) b , N m 2 k T n s 0 e a 400 0 .06 - 25 0 1 C a b l e b 400 0 .12 - - 12 .5 0 2 c 1000 0 .12 — 5 0 5 Beam d 400 0 .30 1460 0.15 5 0 .5 110 [a] TG=25, £=0-1;[b] T0 = 12.5, £ = 02; [c] TG = 5, £=0-5 First mode a ----- -F i g u r e 4-3 M o d a l r e p r e s e n t a t i o n f o r c a b l e v i b r a t i o n s : (a) f i r s t a n d s e c o n d modes I l l Figure 4-3 Modal representation for cable vibrations: (b) t h i r d and fourth modes 112 Nc=[U(Y,0)=UO) W(Y,0)-Wo, |QU(Y,0) 3 U,W(x10) U,W(x10)0 U,W(X10)0 U,W(x10)0 3 U,W(X10) First mode 0=0 F i g u r e 4-4 I n i t i a l d i s p l a c e m e n t a s r e p r e s e n t e d by t h e f i r s t f o u r modes 113 U(Y,0) = Uo &u(Y,e> 0=0 0; Y=0-5 U(x103) 0 -5 . 1 U(x10) oK\-Second mode U(x104) 0 -5 3 5 a l'\ V / I'A / ,' X A ' A 0 » \ \l • v ' V • ' \ / \ V \ V Total 0.125 0-25 Orbits 0-375 F i g u r e 4-5 R e s p o n s e o f t h e s y s t e m when t h e t e t h e r i s d i s t u r b e d a t t h e c e n t e r : (a) t i m e h i s t o r y o f t h e t r a n s v e r s e d i s p l a c e m e n t o f t h e t e t h e r , U component 114 W(Y,0) = Wo, W^(Y,9) 9=0 =0; Y=0.5 First mode 3 _ W(x10) 0 Second mode W(xiO) 0 W(x10) 0 W(x10) 0 0-125 0.25 Orbits 0-375 0-5 Figure 4-5 Response of the system when the tether i s disturbed at the center: (b) time history of the transverse displacement of the tether, W component 115 c o n d i t i o n s a r e p l o t t e d i n F i g u r e 4-6. H e r e , a p h a s e d i f f e r e n c e o f a b o u t 180° b e t w e e n t h e i n d u c e d y and a m o t i o n s i s n o t i c e d . The a x i a l c omponent o f t h e f o r c e i s a l m o s t c o n s t a n t . I t s h o u l d be n o t e d t h a t t h e f o r c e s p r e s e n t e d h e r e a r e e q u a l a n d o p p o s i t e t o t h o s e a c t i n g a t t h e c e n t e r o f mass o f t h e s a t e l l i t e . F i g u r e s 4-7 and 4-8 show t h e r e s p o n s e t o an i n i t i a l i m p u l s i v e d i s t u r b a n c e a p p l i e d t o t h e c a b l e l i n e . N o t e t h e c a b l e m o t i o n i n d u c e d l i b r a t i o n s o f t h e s a t e l l i t e f o r two d i f f e r e n t s p i n p a r a m e t e r s . H e r e , t h e s p i n p a r a m e t e r cu i s s o c h o s e n ( f o r 1^ = 0.1, s e e F i g u r e 3-4) t o e n s u r e s t a b l e £L and y^ m o t i o n s . The i n f o r m a t i o n s h o u l d p r o v e u s e f u l i n d e s i g n i n g an a p p r o p r i a t e c o n t r o l s y s t e m . F i g u r e s 4-9 and 4-10 show t h e c o r r e s p o n d i n g t r a n s v e r s e v i b r a t i o n a l r e s p o n s e t o t h e i m p u l s i v e d i s t u r b a n c e a p p l i e d a t t h e t e t h e r c e n t e r . The m a g n i t u d e o f t h e d i s t u r b a n c e i s b a s e d on t h e s i z e (mass = 1 y g , s p e c i f i c g r a v i t y = 2 ) and v e l o c i t y (45 km/sec) o f t h e b i g g e s t m i c r o m e t e o r i t e e x p e c t e d t o h i t t h e c a b l e 55 i n an e i g h t h o u r p e r i o d . N o t e t h a t t h e m a g n i t u d e o f t h e r e s u l t -i n g v i b r a t i o n s i s e x t r e m e l y s m a l l a n d t h e f r e q u e n c i e s o f U and W a r e a l m o s t t h e same. F i g u r e s 4-11 t h r o u g h 4-13 p r e s e n t t h e r e s u l t s o f t h e a n a l -y s i s f o r t h e c a s e o f t h e S p a c e S h u t t l e s u p p o r t i n g a s u b s a t e l l i t e u s i n g a beam. The p h y s i c a l p a r a m e t e r s o f t h e s y s t e m ( T a b l e 4.1) s a t i s f y t h e c o n d i t i o n 4 2 ( E I ) b >> T e I2 , (4 .19) 116 Figure 4-6 Time history of l i b r a t i o n s and forces acting at the center of mass of the orbiter after the cable i s disturbed at the center To = a o=0 Y„«<4-0.05f 117 (Nc) , (c) F i g u r e 4-7 Time h i s t o r y o f l i b r a t i o n s and f o r c e s f o r a s e t o f i m p u l s i v e i n i t i a l c o n d i t i o n s 118 e = K -0, lr0.1; ^ = ao=0-05, (Y, a,Ylf(3,^,^=0,(Nc) ,(c) 0^50- aplOO: a, 50 •(X2) 49.91 0 0-25 0-5 Orbits 075 Figure 4-8 Time history of the s a t e l l i t e l i b r a t i o n s as affected by two d i f f e r e n t values of the spin parameter 119 [C];NC= ;{u,w, | 0 u Y^O-5 ' a0 1=0, { d.U Y £ 0 - 5 ' o U0 , iW Y=0-5 d0 Y=0-5>o J 4 Z U,W[x10]0 _ 4 L 4 r -7. U,W[x10]0 -4 4 U,W[X107] 0 -4 4 UW[x107J 0 — - 4 -4r-U,W[x107j 0 U W 0-30° 0 = 60 0 = 90 0 = 120 0 = 150 Figure 4-9 U and W vibrations for d i f f e r e n t 0 afte r the cable encounters a micrometeorite impact 120 [Nc], [c], 9 = 90°, U , W First mode 0 0-25 0-5 075 1 Y Figure 4-10 Constituent mode shapes of U and W vibrations excited due to a micrometeorite impact 121 d e r i v e d by u s i n g t h e beam modes i n t h e e x p r e s s i o n f o r t h e e l a s t i c p o t e n t i a l e n e r g y , e q u a t i o n ( 4 . 2 d ) , i n t h e f o l l o w i n g f o r m , ( E I ) , fl2 2 0 .2 0 T r%2 . . • . _ b ,.9 u N 2 .9 w^2, -j e ; ,3u,2 . ,9w>2-, — 2 — { (—j) + (—j) } dy » -2- U ^ ) + (g^) ) dy . * i i (4.20) H e r e , t h e s y s t e m h a s t h 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 a f i x e d -f i x e d beam ( i . e . , u = w = = ^ = 0 f o r y = ^2^' r a ^ n e r t h a n o f a c a b l e ( i . e . , u = w = 0 f o r y = -l-^, i ^ ) • I t i s t o be n o t e d t h a t t h e e x p r e s s i o n (4.19) i s d e r i v e d u s i n g t h e f i r s t mode o f t h e beam ( a p p r o x i m a t e d b y i t s s t a t i c d e f l e c t i o n c u r v e ) i n e x p r e s s i o n ( 4 . 2 0 ) . The f i r s t t h r e e modes o f t h e v i b r a t i n g beam a r e shown i n c a s e ( d ) o f F i g u r e 4-11. H e r e , c a s e s ( a ) , (b) a n d (c) c o r r e -s p o n d t o t h e c h a n g e s i n t h e p a r a m e t e r s o f t h e beam a s i n d i c a t e d . N o t e t h a t t h e c a s e ( a ) , T Q = .0, c o r r e s p o n d s t o t h e beam m a i n t a i n e d a l o n g t h e l o c a l h o r i z o n t a l . I n t e r e s t i n g l y , t h e c h a n g e s i n k , T Q a n d e i n f l u e n c e t h e d i f f e r e n t mode s h a p e s d i f f e r e n t l y . I n g e n e r a l , t h e beam w i t h T Q = 0, e x h i b i t s t h e l e a s t e f f e c t i v e s t i f f n e s s (maximum a m p l i t u d e ) a s shown i n c a s e ( a ) . F i g u r e s 4-12 and 4-13 show t h e v i b r a t i o n a l r e s p o n s e f o r an i m p u l s i v e d i s t u r b a n c e (N^) a t t h e c e n t e r o f t h e beam. As a g a i n s t t h e p r e v i o u s c a s e , mass o f t h e m i c r o m e t e o r i t e i s t a k e n t o be 6 y g h e r e w i t h t h e r e m a i n i n g p a r a m e t e r s h a v i n g t h e same v a l u e s as b e f o r e . B o t h t h e t i m e h i s t o r y o f t h e beam c o n f i g u r a t i o n ( F i g u r e 4-12) a s w e l l a s l i b r a t i o n a l m o t i o n a n d f o r c e s ( F i g u r e 122 [a]ks-0.15, To=£ = 0 ; [b]ks = 0-15, TQ=5 , £ = 0 [c]ks = 0-25, TG = 5, £ = 0 ; [d]ks = 0.15, T0= 5, £ = 0-5 First mode 0-25 0-5 075 Y Figure 4-11 Modal representation for beam vibrations U,W[x10] 0 U,W[X10 ] 0 UW[x10] 0 U,W[X10] 0 Figure 4-12 u and W vibrations for d i f f e r e n t 9 when the beam encounters a micrometeorite impact 124 T°[X104] 0 o° [X104] 0 f^ [xlO] 0 Fy F7 [X10J 0 0-25 0-5 Orbits 075 F i g u r e 4-13 Time h i s t o r y o f l i b r a t i o n s a n d f o r c e s a t t h e c e n t e r o f mass o f t h e o r b i t e r due t o t h e m i c r o m e t e o r i t e i m p a c t on t h e beam 4-13) a r e r e c o r d e d h e r e . The l i b r a t i o n a l r e s p o n s e o f a 200 k g s p h e r i c a l s a t e l l i t e ( 1 ^ = 1 ) , s u p p o r t e d by a c a b l e o r a beam, t o t h e m i c r o m e t e o r i t i c d i s t u r b a n c e s i s shown i n F i g u r e 4-14. H e r e , t h e s p i n p a r a m e t e r a-^= -1 i s due t o t h e o r b i t a l m o t i o n o f t h e s y s t e m . N o t e t h a t t h e r e s u l t i n g r o l l m o t i o n and t h e c h a n g e i n p i t c h r a t e , t h o u g h s m a l l , c a n s t r o n g l y i n f l u e n c e t h e p e r f o r m a n c e o f a s a t e l l i t e s c a n n i n g a d i s t a n t g a l a x y . 4.6 C o n c l u d i n g Remarks The more s i g n i f i c a n t a s p e c t s o f t h e i n v e s t i g a t i o n may be s u m m a r i z e d as f o l l o w s : ( i ) A g e n e r a l f o r m u l a t i o n f o r a t r i a x i a l m u l t i b o d y s y s t e m w i t h an e l a s t i c i n t e r c o n n e c t i n g l i n k i s p r e s e n t e d . ( i i ) H i g h l y c o m p l i c a t e d c o u p l e d , n o n l i n e a r , n o n a u t o n o m o u s , h y b r i d e q u a t i o n s a r e l i n e a r i z e d a nd t h e i r e x a c t s o l u -t i o n o b t a i n e d . The c l o s e d f o r m s o l u t i o n n o t o n l y r e d u c e s t h e c o m p u t a t i o n a l t i m e a n d e f f o r t b u t a l s o g i v e s b e t t e r i n s i g h t i n t o t h e p h y s i c s o f t h e p r o b l e m , ( i i i ) E x p r e s s i o n s f o r t h e f o r c e s and moments r e q u i r e d t o p o s i -t i o n a nd o r i e n t a n o b j e c t i n s p a c e a r e o b t a i n e d . They s h o u l d p r o v e u s e f u l i n p l a n n i n g s c i e n t i f i c e x p e r i m e n t s f r o m o r b i t i n g s p a c e s t a t i o n s . ( i v ) A d e s i g n c r i t e r i o n i n t e r m s o f s t i f f n e s s , t e n s i o n , a n d l e n g t h o f a beam t o e n s u r e a p a r t i c u l a r mode o f v i b r a t i o n i s p r e s e n t e d . 126 / ' / rS ' . [Y,a,T1Ipra1,T,a)Tr(3ra1]o = o [a] TG = 5, £ = 0-5, Nc , [b] ks = 0.15,T0 = 5, £ = 0-5, Nt Y°[X1C)5]0 [x10]0 Y°[x104J 0 -1 5 -a -a 6 [X10]0 -5 0 Figure 4-14 L i b r a t i o n a l history of a spherical s a t e l l i t e due to: (a) a micrometeorite impact on the cable; (b) a micrometeorite impact on the beam 127 (v) A n a l y t i c a l procedures are applied to several configu-rations of the Space Shuttle supporting a s a t e l l i t e i n the o r b i t . The analysis presented here should also prove useful i n studying several other e x i s t i n g and future configurations. 128 5. CLOSING COMMENTS B e f o r e c l o s i n g , a few s u g g e s t i o n s r e g a r d i n g t h e d i r e c t i o n f o r f u t u r e i n v e s t i g a t i o n s a r e a p p r o p r i a t e h e r e . A f e a s i b i l i t y s t u d y o f t h e a t t i t u d e and o r b i t a l c o n t r o l o f t h e SSPS u s i n g t h e s o l a r r a d i a t i o n p r e s s u r e s h o u l d be o f i n t e r e s t . N o t e t h a t t h e c o n t r o l r e q u i r e m e n t s a r e known f r o m s e c t i o n 3.6. I n t h e a n a l y s i s o f a m u l t i b o d y s y s t e m h a v i n g c o n n e c t i n g members o f l a r g e c r o s s - s e c t i o n a l a r e a s , t h e yaw (3) m o t i o n o f t h e s e members s h o u l d be c o n s i d e r e d i n c o n j u n c t i o n w i t h t h e i r r o l l (y) and p i t c h (a) m o t i o n s . I n t h e c a s e o f an e x t e n s i b l e c o n n e c t i n g l i n k , i t s l o n g i t u d i n a l v i b r a t i o n s a r e l i k e l y t o i n -f l u e n c e t h e s y s t e m b e h a v i o u r s i g n i f i c a n t l y . F u r t h e r m o r e , t h e e n d - b o d i e s a r e n o t n e c e s s a r i l y s y m m e t r i c a nd r i g i d . Thus a g e n e r a l f o r m u l a t i o n o f t h e p r o b l e m s h o u l d i n c l u d e t h e l o n g i t u -d i n a l v i b r a t i o n s o f t h e c o n n e c t i n g l i n k , a s w e l l a s , f l e x i b i l i t y , a r b i t r a r y mass and g e o m e t r y d i s t r i b u t i o n f o r t h e e n d - b o d i e s . W i t h t h e S p a c e S h u t t l e s c h e d u l e d t o go i n t o t h e o r b i t i n n o t t o o d i s t a n t f u t u r e , a new e r a o f s p a c e age h a s commenced. I t p r o m i s e s t h e f e a s i b i l i t y o f o b t a i n i n g t h e m o s t d e s i r e d c o n d i -t i o n s o f z e r o - g r a v i t y and p e r f e c t vacuum d u r i n g t h e s t u d i e s o f m e t a l f o r m i n g , b i o l o g i c a l e x p e r i m e n t s , e t c . A number o f i n v e s -t i g a t i o n s w i l l i n v o l v e t h e u s e o f t e t h e r e d p a y l o a d s l a u n c h e d o r s u p p o r t e d f r o m t h e S p a c e S h u t t l e . The l i b r a t i o n a l d y n a m i c s a nd t h e i n d u c e d f o r c e s a c t i n g o n t h e s e p a y l o a d s d u r i n g t h e i r d e p l o y -ment and r e t r i e v a l f r o m t h e o r b i t e r s h o u l d be a n a l y z e d i n d e t a i l . I n n e a r - e a r t h o r b i t s , t h e s u c c e s s o f a s p a c e m i s s i o n w i l l 129 be influenced by the atmosphere. 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C r i s t , S.A., and E i s l e y , J.G., "Cable Motion of a Spinning Spring-Mass System i n Orbit," Journal of Spacecraft and  Rockets, Vol. 7, No. 11, November 1970, pp. 1352-1357. Nixon, D.D., "Dynamics of a Spinning Space Station with a Counterweight Connected by Multiple Cables," Journal of  Spacecraft and Rockets, Vol. 9, No. 12, December 1972, pp. 896-902. Bainum, P.M., and Evans, K.S., "Three-Dimensional Motion and S t a b i l i t y of Two Rotating Cable-Connected Bodies," Journal  of Spacecraft and Rockets, Vol. 12, No. 4, A p r i l 1975, pp. pp. 242-250. Bainum, P.M., and Evans, K.S., "The E f f e c t of Gravity-Gradi-ent Torques on the Three Dimensional Motion of a Rotating Space Station-Cable-Counterweight System," AIAA 13th Aero-space Sciences Meeting, Pasadena, C a l i f . , January 20-22, 1975. Saeed, I., and Stuiver, W., "Periodic Boom Forces i n Dumbbell-Type S a t e l l i t e s Moving i n a C i r c u l a r Orbit," AIAA Journal, Vol. 12, No. 4, A p r i l 1974, pp. 423-424. Sharma, S.C., and Stuiver, W., "Boom Forces i n Lib r a t i n g Dumbbell-Type S a t e l l i t e s , " AIAA Journal, Vol. 12, No. 4, A p r i l 1974, pp. 425- 426. Lik i n s , P.W., "The Influence on Dynamics and Control Theory of the Spacecraft Attitude Control Problem," Proceedings  of the 6th Canadian Congress of Applied Mechanics, Vancouver, May 29-June 3, 1977, pp. 321-335. 134 48. K r y l o v , N.M., and B o g o l i u b o v , N.M., I n t r o d u c t i o n t o N o n l i n e a r  M e c h a n i c s , P r i n c e t o n U n i v e r s i t y P r e s s , 1 9 4 3 . 49. B u t e n i n , N.V., E l e m e n t s o f N o n l i n e a r O s c i l l a t i o n s , B l a i s d e l l , 1 9 6 5 , pp. 1 0 2 - 1 3 7 , 2 0 1 - 2 1 7 . 50. N a y f e h , A.H., P e r t u r b a t i o n M e t h o d s , J o h n W i l e y a n d S o n s , I n c . , New Y o r k , 1 9 7 3 , pp. 23-27. 5 1 . B r e r e t o n , R . C , "A S t a b i l i t y S t u d y o f G r a v i t y O r i e n t e d S a t -e l l i t e s , " Ph.D. T h e s i s , 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 , November 1967, pp. 2 4 - 3 5 . 52. C u n n i n g h a m , W.J., I n t r o d u c t i o n t o N o n l i n e a r A n a l y s i s , McGraw-H i l l Book Company, New Y o r k , 1958, p p . 2 6 1 - 2 6 5 . 53.. M e i r o v i t c h , L., M e t h o d s o f A n a l y t i c a l D y n a m i c s , M c G r a w - H i l l Book Company, New Y o r k , 1970 , pp. 2 8 2 - 2 8 7 . 54. P i s k u n o v , N., D i f f e r e n t i a l a nd I n t e g r a l C a l c u l u s , P e a c e P u b l i s h e r s , Moscow, p p . 4 7 9 - 4 8 7 . 55. Ruppe, H.O., I n t r o d u c t i o n t o A s t r o n a u t i c s , V o l . 2, A c a d e m i c P r e s s , New Y o r k , 1967, pp. 4 3 3 - 4 4 6 . 135 APPENDIX I TRANSFORMATION MATRICES USED IN EQUATIONS (4.1) and (4.15) [6] = 1 0 0 cos0 0 sine 0 •sinG cose [ Y ] cosy siny 0 -siny 0 cosy 0 0 1 [a] 1 0 0 cosa 0 sina 0 -sina cosa . [Y ±] cosy. siny. •siny. 0 cosy^ 0 0 1 [ B ± ] = COS3: -sing. 0 1 sinB, 0 cos3. , [ o u ] 0 cosa sina -sina. I I cosa 136 APPENDIX I I EQUATIONS OF MOTION FOR THE SYSTEM OF CABLE OR BEAM CONNECTED SYMMETRIC END-BODIES E q u a t i o n o f y m o t i o n : 2 2 2 2 2 2 2 Z [y{ ( I . - I .) ( ( c o s a s i n y. + c o s y . ) c o s 3- + s i n a s i n 3-x i y i l l x i 1 2 2 * + Tr s i n 2a s i n y . s i n 23.) - I . - M. L. c o s a} + y { ( I . 2 x x xx x x 1 x x • 2 2 2 - I .) ( a ( c o s 2a s i n y . s i n 23. + s i n 2 a ( s i n 3. - s i n y. c o s 3. yx 'x x x ' x x 1 2 2 + y . ( ^ s i n 2a c o s y , s i n 23. - s i n a s i n 2y. c o s 3.) I 2 1 x x ' x x * • 2 2 2 + 3 . ( s i n 2a s i n y . c o s 23. + s i n 2 3 . ( s i n a s i n y. - c o s a ) ) x x x x ' x 2 • + M. L. a s i n 2a} + a ( I . - I . ) ( c o s a sinB• x x x x yx x - s i n a s i n y . c o s 3 . ) c o s y . cos3- + a{ ( I . - I . ) (- ^ - ( s i n a c o s y . x l 1 x x x x y x 2 ' l 2 • 2 x s x n 23- + c o s a s i n 2y. c o s 3-) - Y- s i n a c o s 2y. c o s 3. x x x ' x ' 1 x 1 + 3 • ( T- s i n a s i n 2y. s i n 23 + c o s a c o s y , c o s 23-) l 2 x ' l x 2 2 2 + 6 ( s i n y ( c o s 2a c o s 23^ - 2 c o s a c o s y^ c o s 3^ 2 + s i n 2a s i n y . s i n 23-) - c o s y . ( c o s a s i n 2y. c o s 3-x x x x x 2 + s i n a c o s y , s i n 23.))) - I . ( y . s i n a s i n 3. + a . ( s i n a s i n 3 . x x xx 1 x x x x 2 + c o s a s i n y . cos3.)) + 1 . ( y . s i n a c o s 3. - 3. c o s a c o s y . x l y x x x x x 2 • 2 •• 2 - 0 s i n y ) - 2M. L. 0 s i n y c o s a} + y . { ( I . - I . ) ( c o s a c o s 3-x x ' 'x xx y x 1 137 1 2 • Y i + T sina siny. sin 3.) - I . cosa} + y. {-z- sina COSY, s i n 23. + 3 i ( s i n a s i n y i cos 23^ - cosa sin 23 i) + 9 (siny (-j s i n 2a x 2 2 2 x s i n 2y^ cos 3^ + sin a cosy i s i n 23^) + cosy s i n a ( s i n 3^ 2 - cos 2y. cos 3-)) - I • a. sina cosy. + 1 .(3- sina siny. ' l l x i l ' l y i l 11 • • • + 9 cosy sina)} - 1 . 3 - sina cosy. + 3-{(I . - I . ) 9 x ' y i l ' l l x i y i 1 • 2 x (— sin 23^ (- siny s i n 2a s i n Y^'~ siny sin 2a + cosy sina s i n 2y^) + cos 23^(- siny s i n 2a siny^ + cosy cosa cosy.)) + I . a.(cosa cos3- - sina siny. sin3.) 1 X I 1 1 ' l 1 + I . 9(siny siny. - cosy cosa cosy.)) + I . a.(cosa s i n 3 . y i I ' i x i I l - sina siny^ cos3^) - I x ^ 9 a^(siny cosy^ cos3^ + cosy(sina s i n 3 ^ • 2 2 2 + cosa siny. c o s 3 . ) ) + 9 {(I . - I .) ( s i n 2y(- 2 cos y. cos 3. l I x i y i ' ' l i 2 2 2 2 2 + s i n 2a siny. s i n 3 . + 2 cos a sin y. cos 3. + 2 sin a sin 3.) l l ' i l l 2 - 2 cos 2y(sma cosy^ s i n 23^ + cosa sin 2y^ cos 3^)) 2 2 - 2M. L. s i n 2y cos a}] i i 1 + £ J 0 1 I I [p£ 2{y(- U 2 - (Y - - i ) 2 c o s 2 a + (Y - —^)W s i n 2a £ £ - W2 sin 2a) + y(- 2U U + (W(Y - —^ - 2W) + a((Y - 2 - W 2))x £1 • •• 1^ x s i n 2a + 2 (Y —)W a cos 2a) + U a ( (Y - — ) sina + W cosa) 138 £ 1 + a ( 2 ( Y - — ) W 8 s i n y s i n 2a + (- 2W U 0 c o s y - U W a li 2 - h - U(Y - — ) - 2W 6 s i n y s i n a ) s i n a + (U(Y ± ) (28 c o s y + a) £ ^ o . £. - U W + 2 U W - 2 ( Y - j ) 2 e s i n y c o s a ) c o s a ) + ij ( (Y - -^ ) c o s a - W s i n a ) + U W s i n a + u 0 c o s y ( ( Y p ) s i n a + 2W c o s a ) £ £ . 2 + W 8 s i n y ( W s i n 2a - 2 ( Y ^ ) c o s 2 a ) + 2 9 2 ( s i n 2 y ( U - (Y ^ ) 2 c o s 2 a + (Y j^)W s i n 2a - W 2 s i n 2 a ) £ + 2U c o s 2 y ( ( Y - y ) c o s a - W s i n a ) ) } + M * L { ( | | ) 2 + (|| ) 2 } x 2 • • 2 *2 2 • x {y c o s a - y a s i n 2a + 2a 0 s i n y c o s a + 20 s i n 2y c o s a } ] d Y = 0 E q u a t i o n o f a m o t i o n : 2 I [- a { ( I . - I . ) c o s 2 y . c o s 2 B . + 1 • - M. L 2 } i = l X 1 Y 1 1 1 y i i i • • 2 * 2 + a { ( I . - I . ) ( y . s i n 2y. c o s 3 . + 3 . c o s y. s i n 2 3 . ) x i y i '1 ' i * i " i ' i 1' 1 • Y + T I . 0 s i n y c o s a c o s y , s i n 2 3 . } + - M i . - I .) x 2 x i '1 1 2 x i y i x ( c o s a c o s y , s i n 2 3 . - s i n a s i n 2y. c o s 1 . ) + y{ ( I 1 3 . 1 1 x i - I y i ) c o s 2a s i n y ^ ^ s i n 2 3 i + s i n 2 a ( s i n 2 y i c o s 2 3 i - s i n 2 3 i ) ) 2 - y ^ ( s i n a c o s 2y^ c o s 3^ + c o s a s i n y ^ s i n 2 ^ - s i n a c o s 3 ^ ) 1 + 3- (TT s i n a s i n 2y. s i n 2 3 . + c o s a c o s y , c o s 2 3 . ) 1 2 1 1 '1 1 139 2 2 1 2 + 0 ( s i n y ( c o s y^ c o s 3^ - -j s i n 2a s i n y ^ s i n 23^ - 2 s i n a 1 2 2 2 - c o s 2 a c o s 3^+2 s i n y^ c o s 3^)) + c o s y ( s i n a c o s y ^ s i n 23^ 2 + c o s a s i n 2y. c o s 3.)) + 1 . ' ( - y. s i n a + a. ( s i n a sin3-1 1 x i 11 1 1 0 2 + c o s a s i n y ^ cos3^) + ^ s i n y ( c o s 2a s i n y^ + 2)) 2 y * 2 •, + 1 . 3 - c o s a c o s y . + M. L. (- 4- s i n 2a + 20 s i n y c o s a) } yx x 1 1 1 2 Y • . Y • + - s f - d • - I • ) c o s y , s i n 23. + y . { ( I . - I . ) ( ± s i n y . s i n 23. 2 x i y i '1 K x '1 x i yx 2 '1 1 2 + 3^ c o s y ^ c o s 23^ + 0 ( s i n y ( c o s a c o s 2y^ c o s 3^ 2 2 - s i n a s i n y ^ s m 23^ + c o s a s i n y^) + c o s y s i n 2y_^  c o s 8^) . + I . a. s i n y . cos3- + I -(3- c o s y . + 0 s i n y c o s a ) } x i 1 '1 1 yx x 11 ' 2 + 3. I . s i n y . + 3 - { ( I . - I . ) 0 ( c o s y c o s y. s i n 23. 1 y x '1 x x x y x ' 'x x + s i n y (- - i - c o s a s i n 2y^ s i n 23^ + s i n a c o s y ^ c o s 23^)) + I . a. c o s y . s i n 3 . + 1 . 0 s i n y s i n a c o s y . } x i 1 x x y x x • * - a- I • c o s y . cos3- + I • a. 0 s i n y ( s i n a s i n y . cos3. X X X X X X X X X X • 2 2 - c o s a sin3.) +0 { ( I . - I . ) ( s i n 2 y ( s i n a s i n 2y. c o s 8. x xx y x ' x x 2 2 2 2 - c o s a c o s y ^ s i n 23_^ ) + 2 c o s y ( s i n 2 a ( s i n y^ c o s 3^ - s i n 3^) 1 2 2 2 • - c o s 2a s i n y . s i n 23.) + -z- s i n 2 a ( s i n 3- - s i n y. c o s 3.) 'x x 2 x 'x 1 1 1 2 2 + c o s 2a s i n y . s i n 23.) + -^ M. L. (1 - 4 c o s y ) s i n 2a}] 2 i x 2 1 1 140 + £ 1 2 * [p£ {- ( (Y ^ ) 2 + W 2 ) a - Wa(2W + U0 s i n y c o s a ) 0 + U y ( ( Y j-)sina + W cosa) + y(sin 2a(j(W - (Y j p ' Y £1 ^1 * - 2 (Y - — )W 0 siny) + s i n a ( ( Y - -j-) U + 2U W 6 cosy) 1 2 2 2 2 + 20 siny ( (Y - -j-) cos a + W sin a) + cosa (- W U + U W • & -| » & , + U W 6 siny - 2 (Y ~^ ) U 9 cosy)) - (Y j±) W £ + 9 ( ( U W + U W)siny sina - U(Y - - ^ ) s i n y cosa + 2W W cosy) *2 A l ^1 2 + 9 (- 2 U ( ( Y ^ ) s i n a + W cosa) sin 2y + (Y j±) ( 1 - 4 cos y)x £ x w cos 2a + i ( ( Y 1~)2 ~ W 2) (1 - 4 cos 2y) s i n 2 a ) } + M * L { ( ( | ^ ) 2 + ( ^ ^ H a + i ( y 2 - 0 2 ) s i n 2a - 2y 8 siny cos 2a * 2 2 + 20 cos y s i n 2 a } ] d Y = 0 Equation of y^ motion: •• 2 y • {(I • - I . ) sin 6. + I .} + y. { ( I . - I . ) (- a siny. s i n 2g. 'x x i yx l yxJ 'x x i yx T i px 2 + 8• s i n 2g.) +1 . 0 siny sina s i n 2y. s i n 8-} x i xx ' 'x 1 1 2 + y{- ( I . - I . ) ( ^ sina siny. s i n 28- + cosa cos 8.) xx yx 2 'x x x • Y 1 + I . cosa} + y [ ( I . - I .){4r{— s i n 2a cosy, sin 28. XX X I y'1 £ 1 1 2 2 • 2 2 - s i n a s i n 2y. cos + a(2 sina s i n y. cos 8-'x x 'x x - cosa siny^ s i n 28^) + 8^  (cosa s i n 28^ - sina siny^ cos 28^) 141 1 2 2 + 0 ( s i n y ( — s i n 2a s i n 2y^ COS 3^ - c o s a c o s y ^ s i n 23^) 2 2 • • + COSY s i n a ( s i n 3- - c o s 2y. COS 3-))} - I - ( a + a. COSY- COS3.)X i ' i l x i 1 ' 1 * i ' * a x s i n a + I . (3- s i n y . + 0 COSY) s i n a ] - -^'(I • - 1 - ) c o s y . s i n 23. y i I l 2 x i y i ' l l + a[ ( I . - I . ) { T T s i n 2Y . c o s 3. - 3. c o s y , c o s 23-x i y i 2 ' i l i ' i l 2 + 0 ( s i n y ( - s i n a s i n y ^ s i n 23^ + c o s a s i n 3^ 2 2 + c o s a c o s 2y^ c o s 3^) + c o s y s i n 2y^ c o s 3^)} • • • + I . a. s i n y . cos3- + I •(3- c o s y . + 0 s i n y c o s a ) ] x i l ' l * i y i I ' I ' ' • • + 3 . [ ( I - - I -)0{- c o s y c o s y , c o s 23- + s i n y ( s i n a s i n 23. l x i y i i i ' i + c o s a s i n y . c o s 23-)} - I - a. cos3- + I - 0 ( c o s y c o s y . l l x i l l y i ' ' i - s i n y c o s a s i n y . ) ] - a. I . s i n g . + 1 . . a . 0 ( c o s y s i n y . cos3. + s i n y c o s a c o s y - cos3-) + 0 [ ( I - - I - ) { - s i n 2 3 - ( s i n 2y x l I x i y i l 1 2 2 x s i n a s m y ^ + s i n 2a c o s y ^ {-^  - c o s y) ) + c o s 3^(2 s i n 2y x 1 2 2 2 2 x c o s a c o s 2y^ + s i n 2y^ (-^  s i n a + 2 c o s y c o s a - 2 s i n y ) ) } ] = 0 Equation of 3^ motion: I y i & i + I y i ^ s i n a c o s Y i + Y [ d x i - I y i H j ( s i n 2 3 i ( - c o s a 2 2 • 1 + s i n a s i n y.) + s i n 2a s i n y . c o s 23.) + a{— s i n a s i n 2y. s i n 23. i l l 2 ' l l 142 + cosa cosy- cos 23-) + Y-( _ cosa sin 23. + sina siny. cos 23.) i i i I ' I M i ' + 0(cos 23^(COSY cosa COSY^ - siny cos 2a siny^) 1 2 + -~ s i n 23. (COSY sina s i n 2y. + siny s i n 2 a ( l + s i n Y-)))} + I . a.(cosa cosS• + sina siny. sin3•) + 1 .{a cosa cosy. - y^ sina siny^ + G (siny siny^ - cosy cosa cosy^) } ] • r a 2 • - I . a siny. + a [ ( I . - I . ) cos Y. s i n 23- + Y- COSY • cos 23-y i 1 x i y i 2 ' i l ' i ' i l 2 1 - 0 (- COSY cos Y^ s i n 23^ + s i n y ^ c o s a s^-n 2 Y i s ^ n 2 ^ i + sina COSY- cos 2 3 - ) ) } + I . a. COSY- s i n 3 . 1 1 X I 1 1 1 r Y i + I . cosy • (- Y • + 9 siny sina) ] + Y • [ ( I • - I • ) ( - - ? r - sin 23. y i l l l x i y i 2 l - 0(siny sina s in 23^ + cos 23^(siny cosa sinY^ - COSY COSY^ ) ) } + T- X ^ a^ cos3^ + I ^ 0 (siny cosa siny^ - COSY COSY^) ] + 1 ^ a^ 0{COSY COSY^ s i n 3 ^ + siny(sina cos3^ - cosa siny- sin3-)} + 0 ( I - - I . ) [ - sin 23-(sin 2y cosa x I i x i y i l 2 2 2 2 2 2 2 x s i n 2Y^ + 2 s i n y cos Y^ + 2 cos y cos a s i n Y ^ ~ 2 cos y s i n a 1 2 2 2 + ^ ( s i n a sin y^ ~~ cos a)} + cos 23^(2 sin 2y sina cosy^ 2 1 + 2 cos Y sin 2a siny^ - ^ s^- n 2 a siny^) ] = 0 143 E q u a t i o n o f m o t i o n : • • • • • • + y ( s i n a s i n y ^ cos3^ - c o s a sin3^) + y { a ( s i n a sin3^ + c o s a s i n y . cos3.) + y. s i n a c o s y . cos3- - 3- ( c o s a cos3-' 1 I ' I ' I I I I + s i n a s i n y ^ sin3^) - 8 ( s i n y c o s y ^ cos3^ + c o s y s i n a sin3^ • • • - • + c o s y c o s a s i n y . cos3-)} + a c o s y . cos3- + a { - y. s i n y . cos3-1 I l l I i i l - 3^ c o s y ^ sin3^ + 0 ( s i n y s i n a s i n y ^ cos3^ - s i n y c o s a sin3^)} - y. sin3. - y.{3. cos3. + 0 ( c o s y s i n y . cos3-1 1 I 1 I I I I I • • + s i n y c o s a c o s y ^ cos3^)} + 3^ 0 ( s i n y c o s a s i n y ^ sin3^ - c o s y c o s y ^ sin3^).= 0 C a b l e E q u a t i o n f o r U v i b r a t i o n : • 2 * 2 "2 2 * • U + U ( - y + 0 - 40 s i n y) + W(y s i n a + 0 s i n a - 0 s i n y c o s a ) • • • • • • . . + W(y s i n a + 2y a c o s a rt- y 0 c o s y c o s a + a 0 s i n y s i n a + 2 0 2 s i n 2y s i n a ) + (Y - (- y c o s a + 2y a s i n a • . • • • 2 - y a s i n y c o s a + y 0 c o s y s i n a - 20 s i n 2y c o s a ) 1^1 ^  IJ ? * 2 2 2 * 2 2 2 * * - — j { ( y - 0 ) c o s a + a + 4 0 c o s y c o s a + y 0 s i n y s i n 2a p£ + 2a 0 c o s y } — y = 0 9Y^ 144 E q u a t i o n f o r W v i b r a t i o n : r * 2 • 2 2 • * • 2 • • W + W{(0 - y ) s i n a + y 0 s i n y s i n 2a - a + 2a 0 c o s y * 2 2 2 • • • - 40 c o s y s i n a} + U ( - 2y s i n a + 0 s i n y c o s a ) + U ( - y s i n £ + 2y 0 c o s y c o s a + 2 0 2 s i n 2y s i n a ) + (Y - -j-) i ^ i y 2 - 0 2 ) s i n 2a . • 2 •• * 2 2 M*L «2 *2 2 - 2y 0 s i n y c o s a + a + 20 c o s y s i n 2a} - — y { ( y - 0 ) c o s a p£ • • • 2 • ' * 2 2 2 P ) W + y 0 s i n y s i n 2a + a + 2a 0 c o s y + 40 c o s y c o s a } — • j = 0 8Y Beam E q u a t i o n f o r U v i b r a t i o n : • 2 * 2 ' 2 2 • • U + U ( - y + 0 - 40 s i n y) + W(y s i n a + 0 s i n a - 0 s i n y c o s a ) • • • • • » • + W(y s i n a + 2y a c o s a + y 0 c o s y c o s a + a 0 s i n y s i n a •2 £ 1 •• + 20 s i n 2y s i n a ) + (Y - — ) ( - y c o s a + 2y a s i n a • • • 2 - y a s i n y c o s a + y 0 c o s y s i n a - 20 s i n 2y c o s a ) ^ ( E I ) b 9 4U M*L r,-2 *2 X 2 • ' • • „ „• I + j — — j - — ( y -• 8 ) c o s a + y 8 s i n y s i n 2a + 2a 0 c o s y p£ 3Y pl ^ ' 2 ^ A a 2 2 2 \3 2U + a + 4 8 c o s y c o s a } — = 0 3Y E q u a t i o n f o r W v i b r a t i o n : r * 2 ' 2 2 • • '2 •• W + W{(0 - y ) s i n a + y 0 s i n y s i n 2a - a + 2a 0 c o s y • 2 2 2 - 40 c o s y s i n a} + U ( - 2y s i n a + 0 s i n y c o s a ) 145 + U ( - y s i n a + 2y 8 c o s y c o s a + 28 s i n 2y s i n a ) ^1 r l -2 *2 • * 2 + (Y - ~ H ^ ( Y ~ 9 ) s i n 2a - 2y 8 s i n y c o s a + a , oQ2 2 • i ^ ( E I ) b 3 4W M * L / # - 2 '2. 2 + 28 c o s y s i n 2a} + j— —j - —=•{ (y - 8 ) c o s a p i 9 Y 4 pl 2 • * • 2 • • • 2 2 2 9 W + y 8 s i n y s i n 2a + a + 2a 8 c o s y + 48 c o s y c o s a} » = 

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