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On coupled librational dynamics of gravity oriented axi-symmetric satellites Shrivastava, Shashi Kant 1970

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( ON COUPLED LIBRATIONAL DYNAMICS OF GRAVITY ORIENTED AXI-SYMMETRIC SATELLITES by SHASHI KANT SHRIVASTAVA B. Tech. (Hons.), Indian Institute of Technology, Kharagpur, 19 6 7 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of MECHANICAL ENGINEERING We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 19 70 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 and 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 by t h e Head o f my De 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 . SHASHI KANT SHRIVASTAVA De 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 8, Canada Dedicated to my -parents S r i Bhaiya L a i Shrivastava (1901-1970) and Srimati Siyarani Shrivastava (1912-1968) who passed away witnessing my involve-ment in the process of education which they valued most ABSTRACT 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 on the attitude dynamics of gravity oriented, non-spinning, axi-symmetric s a t e l l i t e s , executing general l i b r a -t i o n a l motion i s investigated using a n a l y t i c a l , numerical and analog techniques. The problem i s studied i n the i n -creasing order of complexity. For the case of a c i r c u l a r o r b i t , the autonomous, con-servative system represented by constant Hamiltonian yi e l d s zero-velocity curves and motion envelopes which i d e n t i f y regions of i n s t a b i l i t y from conditional and guaranteed stable motion. The non-linear, coupled equations of motion are solved using approximate a n a l y t i c a l techniques: Butenin 1s v a r i a t i o n of parameter method and invariant i n t e g r a l approach. A comparison with the numerical response, establishes th e i r s u i t a b i l i t y i n studies involving motion i n the small. The invariant i n t e g r a l method maintains reasonable accuracy even for larger, predominantly planar, disturbances. However, for a general motion i n the large, the a n a l y t i c a l solutions provide only q u a l i t a t i v e information and one i s forced to resort to numerical, analogic or hybrid procedures. The analysis suggests strong dependence of system response on the in-plane disturbances and s a t e l l i t e i n e r t i a . The l i b r a t i o n a l and o r b i t a l frequencies are of the same order of magnitude. It also shows that the stable solution, when represented i n a three dimensional phase space may lead to 'regular', 'ergodic' or 'island' type regions. The l i m i t i n g i n t e g r a l manifolds, given here for a few represen-tative values of Hamiltonian, provide a l l possible combin-ations of i n i t i a l conditions, which a s a t e l l i t e can withstand without tumbling. The r e s u l t s , for a range of s a t e l l i t e i n e r t i a , are condensed i n the form of design p l o t s , i n d i c a t i n g allowable disturbances for stable motion. In general, the slender s a t e l l i t e s exhibit better s t a b i l i t y c h a r a c t e r i s t i c s . The presence of aerodynamic torque destroys the symmetry properties of the i n t e g r a l manifolds. The s t a b i l i t y of the equilibrium configuration, which now deviates from the l o c a l v e r t i c a l , i s established through Routh's as well as Liapunov's c r i t e r i a . As the system i s s t i l l autonomous and conservative, the Hamiltonian remains constant leading to the bounds of l i b r a t i o n . Numerical analysis of the system response i n -dicates increased s e n s i t i v i t y to planar disturbances. The d i s t o r t i o n and contraction of the regular, ergodic and island type s t a b i l i t y regions show the adverse e f f e c t s of aerodynamic torque. The design plots suggest that the shorter s a t e l l i t e s , normally not preferred from gravity-gradient considerations, could exhibit better s t a b i l i t y characteris-t i c s i n the presence of large aerodynamic torque. An alternate, economical approach to the dynamical analysis of the s a t e l l i t e s i s attempted using an analog computer. A comparison with the d i g i t a l data establishes the s u i t a b i l i t y of the method for design purposes and r e a l time simulation. As the regular surface represents the only usable s t a b i l i t y region from design considerations, a detailed study to e s t a b l i s h the bound between regular and ergodic type s t a b i l i t y was undertaken. The periodic solutions, obtained numerically using variable secant i t e r a t i o n show t h e i r spinal character with the body of s t a b i l i t y region b u i l t around them. Of p a r t i c u l a r significance i s the fundamental periodic solution ( t w o planar o s c i l l a t i o n s i n one out-of-plane cycle) associated with the regular region, suitable for p r a c t i c a l operation of a s a t e l l i t e . The remaining periodic solutions represent degeneration of the i s l a n d - l i k e areas surrounding the mainland. The results lead to a set of fundamental periodic solutions over a wide range of system parameters. Floquet's v a r i a t i o n a l analysis i s used to estab-l i s h the c r i t i c a l disturbance (C„ - 0.8), beyond which no cr stable motion can be expected. The periodic solutions to-gether with the regular s t a b i l i t y region are presented here as functions of Hamiltonian, s a t e l l i t e i n e r t i a and aerodynamic torque. The case study of GEOS-A s a t e l l i t e i s also included. In e l l i p t i c o r b i t , the Butenin's analysis of coupled forced systems i s found to give an approximate solution of good accuracy. However for t h i s non-autonomous s i t u a t i o n , V where Hamiltonian i s no longer a constant of the motion, the concept of i n t e g r a l manifold breaks down. Fortunately, the design plots can s t i l l be generated by d i r e c t u t i l i z a t i o n of the response c h a r a c t e r i s t i c s . In general the s t a b i l i t y region diminishes with increasing e c c e n t r i c i t y and disappears completely for e > 0.35. The presence of atmosphere adds to the complex behaviour of t h i s non-autonomous system, where even the equilibrium configuration now becomes periodic i n character. The s t a b i l i t y regions are further reduced with i n s t a b i l i t i e s normally i n i t i a t i n g i n the planar degree of freedom. F i n a l l y , a p o s s i b i l i t y of using the atmospheric forces i n attitude control i s explored. The use of a set of horizontal flaps i n conjunction with a semi-passive, v e l o c i t y -sensitive c o n t r o l l e r appears to be promising. With a suitable choice of system parameters even a large disturbance can be damped i n approximately two o r b i t s . TABLE OF CONTENTS CHAPTER PAGE 1 INTRODUCTION 1 1.1 Preliminary Remarks 1 1.2 Literature Review 2 1.3 Purpose and Scope of the Investigation . . 6 2 LIBRATIONAL RESPONSE AND STABILITY IN CIRCULAR ORBIT 10 2.1 Preliminary Remarks 10 2.2 Formulation of the Problem 11 2.3 Hamiltonian and Bounds of L i b r a t i o n . . . . 15 2.4 Approximate Solutions and System Response 20 2.4.1 Variation of Parameter Method (Butenin) . . . . . 20 2.4.2 Invariant Integral Method 32 2.5 Nature of Stable Solutions 38 2.6 S t a b i l i t y Plots 46 2.7 Concluding Remarks . 50 3 EFFECT OF AERODYNAMIC TORQUE ON SYSTEM RESPONSE AND STABILITY 53 3.1 Preliminary Remarks 53 3.2 Formulation of the Problem 54 3.2.1 Aerodynamic Torque 5 4 3.2.2 Lagrangian and Equation of Motion 5 8 3.2.3 Hamiltonian 60 v i i CHAPTER PAGE 3.3 Dynamic E q u i l i b r i a and S t a b i l i t y i n the Small 61 3.3.1 Equilibrium Configurations 61 3.3.2 In f i n i t e s i m a l Technique (Routh) . . 62 3.3.3 Liapunov's Direct Method 64 3.4 Bounds of Li b r a t i o n . . . . 67 3.5 Numerical Solution .72 3.6 Discussion of Results 79 3.7 Analog Simulation of the System .• 85 3.7.1 Accuracy of the Simulation 88 3.8 Concluding Remarks 9 3 4 REGULAR STABILITY AND PERIODIC SOLUTIONS . . . . 95 4.1 Preliminary Remarks 9 5 4.2 Analysis 97 4.2.1 Limiting S t a b i l i t y and Periodic Solutions 100 4.2.2 V a r i a t i o n a l S t a b i l i t y of Periodic Solutions 101 4.3 Discussion of Results 105 4.4 Concluding Remarks 114 5 LIBRATIONAL RESPONSE AND STABILITY IN ELLIPTIC ORBITS 117 5.1 Preliminary Remarks 117 5.2 Formulation of the Problem 118 5.3 Approximate Solution and System Response 119 5.3.1 Variation of Parameter Method (Butenin) 119 v i i i CHAPTER PAGE 5.3.2 Accuracy of the Solution 12 2 5.4 S t a b i l i t y Bound 128 5.4.1 A n a l y t i c a l Approach 128 5.4.2 Numerical Approach 129 5.5 E f f e c t of Aerodynamic Torque on System Response and S t a b i l i t y 134 5.5.1 Equations of Motion 134 5.5.2 Equilibrium Positions 136 5.5.3 L i b r a t i o n a l Response . 136 5.5.4 S t a b i l i t y Plots 143 5.6 Concluding Remarks 145 6 AERODYNAMIC DAMPING 147 6.1 Preliminary Remarks 147 6.2 F e a s i b i l i t y of the Concept 148 6.3 Response Analysis 153 6.4 Concluding Remarks . 159 7 CLOSING COMMENTS 160 7.1 Summary 160 7.2 Recommendations for Future Work 162 BIBLIOGRAPHY 165 L I S T OF TABLES TABLE PAGE 4-1 C r i t i c a l conditions as affected by s a t e l l i t e i n e r t i a and aerodynamic torque . 110 LIST OF FIGURES FIGURE PAGE 1- 1 Schematic diagram of the proposed plan of study 9 2- 1 Geometry of s a t e l l i t e motion 12 2-2 E f f e c t of s a t e l l i t e i n e r t i a and Hamiltonian on zero-velocity curves 16 2-3 E f f e c t of s a t e l l i t e i n e r t i a and Hamiltonian on motion envelope (<(>' = 0 ) : (a) l i m i t i n g region for guaranteed bounded l i b r a t i o n (C =-1.0) 18 n (b) l i b r a t i o n s bounded i n 4> only (C H=-0.5) 19 2-4 E f f e c t of s a t e l l i t e i n e r t i a on l i b r a t i o n a l response obtained using numerical and v a r i a t i o n of parameter methods: (a) impulsive disturbance; ijj =<|> =0, ij> • =<|>' =0 . 5 °. ? 25 o o (b) disturbance in the o r b i t a l plane; i|> =15° ,^'=0.25, <J> =<j>'=0 . 26 r o o o o (c) disturbance across the o r b i t a l plane; 4) =^ '=0,<j> =15° ,<j>'=0.25 27 o ro o o 2-5 Stroboscopic phase plane (0=0), obtained using numerical and v a r i a t i o n of parameter methods: (a) impulsive disturbance 30 (b) angular disturbance 31 2-6 Accuracy of approximate solutions: (a) response to large disturbance along one of the degrees of freedom 34 (b) response to predominantly planar disturb-ance 35 2-7 Numerically generated response to a large ar b i t r a r y disturbance showing ef f e c t s of s a t e l l i t e i n e r t i a 37 x i FIGURE PAGE 2-8 The cross-section 4>=0 i n phase space i n d i c a t i n g types of stable solutions generated by d i f f e r -ent i n i t i a l conditions for a given Hamiltonian: (a) regular and ergodic .40 (b) islands and t h e i r breakdown into ergodic 40 2-9 E f f e c t of s a t e l l i t e i n e r t i a on nature of stable solutions: <J>=0 , C =-1.0 41 ti 2-10 E f f e c t of i n i t i a l conditions and Hamiltonian on the l i m i t i n g i n t e g r a l manifolds (K^=0.5): (a) regular; C =-1.0 44 (b) i s l a n d type; C =-1.0 44 n (c) regular; C =0.4 45 n 2-11 Design plots showing allowable impulsive disturbance for stable motion (a) 0=0 47 (b) |<j>|=30° 48 2- 12 E f f e c t of s a t e l l i t e i n e r t i a on allowable impulsive disturbance for stable motion . . . . . 49 3- 1 Geometry of s a t e l l i t e motion i n presence of atmosphere 55 3-2 Vari a t i o n of stable equilibrium position due to aerodynamic torque and s a t e l l i t e i n e r t i a 66 3-3 E f f e c t of s a t e l l i t e i n e r t i a , Hamiltonian and aerodynamic torque on zero-velocity curves . 68 3-4 Regions of bounded motion 70 3-5 Influence of aerodynamic torque on motion envelope 71 3-6 Ef f e c t s of aerodynamic torque and i n e r t i a parameter on the response of s a t e l l i t e s to an impulsive disturbance 73 x i i FIGURE PAGE 3-7 Response of a s a t e l l i t e to a disturbance i n one degree of freedom i n presence of aerodynamic torque: (a) ^ 0=15°, 1/^=0.25; 4>o=<f>;=0 74 (b) ^O=^Q=0; <f>0=15°, <j>^ =0.25 74 3-8 Representative cross-sections of motion envelopes and l i m i t i n g i n t e g r a l manifolds i n d i c a t i n g influence of aerodynamic torque . . . . 76 3-9 The cross-section cf>=0 i n phase space in d i c a t i n g types of stable solutions generated by d i f f e r -ent i n i t i a l conditions for given Hamiltonian . . . 77 3-10 E f f e c t of i n e r t i a parameter on motion en-velope and l i m i t i n g i n t e g r a l manifolds for given aerodynamic moment and Hamiltonian 78 3-11 E f f e c t of aerodynamic moment on l i m i t i n g i n t e g r a l manifold; K.=1.0, C =-1.5 80 1 ri 3-12 Limiting i n t e g r a l manifold for large Hamiltonian 81 3-13 Design plots i n d i c a t i n g allowable impulsive disturbances at equilibrium positions for Stable motion: (a) K. = 1.0, 0.5 82 l (b) K. = 0.75, 0.25 83 l 3-14 Analog simulation c i r c u i t using 6 as the independent variable 8 7 3-15 Representative response plots obtained using analog simulation; =(}> =0, =^0 ."5 s ° ° r o T o (a) i n absence of aerodynamic torque 89 (b) in presence of aerodynamic torque 90 3-16 Allowable impulsive disturbances at equilibrium positions for stable motion - a comparison between numerical and analog results 9 2 4-1 Stroboscopic phase plane at o)=0 in d i c a t i n g types of stable solutions generated by various i n i t i a l conditions at given Hamiltonian: x i i i FIGURE PAGE (a) i n absence of aerodynamic torque 9 8 (b) i n presence of aerodynamic torque 99 4-2 Periodic response: (a) i n absence of aerodynamic torque . 102 (b) i n presence of aerodynamic torque 10 3 4-3 E f f e c t of i n e r t i a and atmosphere on the impulsive disturbances for stable periodic motion; \p =\b , d> -0 10 7 r o re' ro 4-4 E f f e c t of system parameters on the region of regular s t a b i l i t y and fundamental periodic solution P 2 1 109 4-5 Variations of the period and the trace of f i n a l condition matrix with Hamiltonian for the stable periodic solution 4-6 Reduction of the allowable impulsive disturb-ance for stable motion due to non-regular solution and atmosphere 113 4- 7 Allowable impulsive disturbance at equilibrium position, for regular motion and corresponding periodic solutions: (a) e f f e c t of i n e r t i a 115 (b) e f f e c t of atmosphere 115 5- 1 Representative comparison of the responses, generated using Butenin's approach and numerical method, showing effects of: (a) s a t e l l i t e i n e r t i a . . . 123 (b) o r b i t e c c e n t r i c i t y . . . . . 124 (c) i n i t i a l conditions 125 5-2 Numerical results i n d i c a t i n g the e f f e c t of o r b i t e c c e n t r i c i t y on the s a t e l l i t e response: (a) no disturbance . 127 (b) large disturbance 127 5-3 Stroboscopic phase plane at 4>=0 showing breakdown of the i n t e g r a l manifold concept for non-autonomous system 131 x i v FIGURE PAGE 5-4 E f f e c t of s a t e l l i t e i n e r t i a and o r b i t e c c e n t r i c i t y on the allowable impulsive disturbances for stable motion; 9 =\b = < b =0 : o r o o (a) K. = 1.0, 0.5 132 l i (b) K i = 0.75, 0.25 133 5-5 Varia t i o n of aerodynamic c o e f f i c i e n t and stable equilibrium configuration with 6 and e . 137 5-6 Typical system responses showing the e f f e c t of o r b i t e c c e n t r i c i t y and (a) atmospheric torque 138 (b) s a t e l l i t e i n e r t i a 139 (c) i n i t i a l conditions 140 5-7 I n s t a b i l i t y excited by change of system parameters 142 5- 8 E f f e c t of aerodynamic disturbance for stable motion, K.=1.0, 6 =<J> =0, =ty 144 ' l o o ' ro re 6- 1 Aerodynamic damping and s t a b i l i z a t i o n : (a) s a t e l l i t e configuration . 149 (b) possible arrangements of s t a b i l i z e r s . . . . 150 6-2 Aerodynamically damped response ( x ^ =2.0) showing the effects of: m a x (a) proportionality constants 155 (b) s a t e l l i t e i n e r t i a and aerodynamic torque 156 (c) i n i t i a l conditions 157 ACKNOWLEDGEMENT The author wishes to express his gratitude to Dr. V.J. Modi for guidance given throughout the preparation of the thesis. His help and encouragement have been invaluable. The suggestions by Dr. A.C. Soudack, Dr. J.E. Neilson and Mr. C. Tschann are appreciated. The investigation reported i n the thesis was supported (in part) by the National Research Council, Grant No. A-2181, and the Defence Research Board of Canada, Grant No. 9551-18. LIST OF SYMBOLS amplitude of l i b r a t i o n i n o r b i t a l plane constants, equation (2.22) amplitude of l i b r a t i o n across o r b i t a l plane e c c e n t r i c i t y of o r b i t nonlinear functions, equation (2.10) approximations to f and g, respectively, due to constraints, equation (2.15) nonlinear functions, equation (5.3) approximations to f^ and g^, respectively, due to constraints, equation (5.9) acceleration due to gravity angular momentum per unit mass of s a t e l l i t e moduli of Jacobian e l l i p t i c functions, equation (2.24) distance of a mass element from s a t e l l i t e ' s mass center moment arm, equation (6.4) mass of s a t e l l i t e frequency of l i b r a t i o n i n o r b i t a l plane, (3K.) V 2 frequency of l i b r a t i o n across o r b i t a l plane, (3K.+1)1/2 l momentum conjugate to generalized coordinate distance between center of attr a c t i o n and s a t e l l i t e ' s center of mass X V I I time unit vector along x~-axis, equation (3.2b) o r b i t a l v e l o c i t y p r i n c i p a l body coordinate with z along the axis of symmetry intermediate body coordinates with o r i g i n at center of mass during modified Eulerian rotations respectively periodic c o e f f i c i e n t s i n v a r i a t i o n a l equation (4.1) flap area surface area of s a t e l l i t e c o e f f i c i e n t s i n c h a r a c t e r i s t i c equation (3.15) aerodynamic c o e f f i c i e n t i n c i r c u l a r o r b i t , e l l i p t i c o r b i t and at perigee, respectively r a t i o of transverse to a x i a l cross-sectional areas of s a t e l l i t e , TTD /4L ' o / o drag and l i f t c o e f f i c i e n t s , respectively non-dimensionalized Hamiltonian center of pressure c y l i n d r i c a l s a t e l l i t e ' s diameter and length, respectively Hamiltonian I =1 >I xx yy zz moments of i n e r t i a about x,y,z axes, respectively amplitude scaling factors appearing i n analog simulation i n e r t i a parameter, (I-I )/I 2 Z Lagrangian aerodynamic moment center of a t t r a c t i o n normal pressure due to atmosphere components of P along x^, y^, axis, respectively periodic solution, i planar o s c i l l a t i o n s (ijj) corresponding to j out of plane cycles (o>) universal gas constant radius of the earth centre of mass of the s a t e l l i t e r a t i o of s a t e l l i t e v e l o c i t y to average molecular v e l o c i t y , V //( 2gRT , 1 / 2 S s i n £ o k i n e t i c energy ambient temperature dimensionless time, l i b r a t i o n a l period/ o r b i t a l period trace of the condition matrix 0 at time t wall temperature of the s a t e l l i t e aerodynamic potential g r a v i t a t i o n a l p o t e n t i a l Liapunov function 2 ( , r ° - 5 f V ( S o » 2 d S ' J o o XIX a s c a l i n g f a c t o r a p p e a r i n g i n a n a l o g s i m u l a t i o n 8 d e p e n d e n t v a r i a b l e , e q u a t i o n (2.22) 6 - ^ , 6 2 p h a s e a n g l e s , e q u a t i o n s ( 2 . 1 2 ) , (5.6) 5q^ i n f i n i t e s i m a l i n c r e m e n t o f q^ £ d i s t a n c e between c e n t e r o f mass and c e n t e r o f p r e s s u r e o f s a t e l l i t e C n 1 8 + 3 1 n . n 2 9 + 3 2 0 a n g u l a r p o s i t i o n o f t h e s a t e l l i t e , m e a s u r e d f r o m p e r i c e n t e r * * * 0, 01,6^ ph a s e a n g l e s , e q u a t i o n s ( 2 . 2 2 ) , (2.24) A r o t a t i o n a b o u t a x i s o f symmetry e i g e n v a l u e s y g r a v i t a t i o n a l c o n s t a n t y. p r o p o r t i o n a l i t y c o n s t a n t i n c o n t r o l l e r c h a r a c t e r -i s t i c r e l a t i o n £ a n g l e between s a t e l l i t e s u r f a c e and a i r v e l o c i t y p a t m o s p h e r i c d e n s i t y o s u r f a c e r e f l e c t i o n c o e f f i c i e n t f o r t a n g e n t i a l momentum t r a n s f e r a' s u r f a c e r e f l e c t i o n c o e f f i c i e n t f o r n o r m a l momentum t r a n s f e r x s h e a r s t r e s s due t o at m o s p h e r e T ^ , T 2 n o n d i m e n s i o n a l i z e d maximum s t a b i l i z i n g t o r q u e max max i n IJJ and <J> d e g r e e s o f f r e e d o m , r e s p e c t i v e l y <j> r o t a t i o n a c r o s s t h e o r b i t a l p l a n e ^ r o t a t i o n i n t h e o r b i t a l p l a n e XX Subscripts cr c r i t i c a l value of parameter for s t a b i l i t y e value of parameter at stable equilibrium configuration o i n i t i a l condition p value of parameter at pericenter v v a r i a t i o n a l P periodic R l i m i t i n g value of parameter for guaranteed regular s t a b i l i t y Dots and primes indicate d i f f e r e n t i a t i o n with respect to t and 9, respectively; ^ i n diagrams represent c r i t i c a l value for s t a b i l i t y as given by the Floquet theory. 1 . INTRODUCTION 1 . 1 Preliminary Remarks The advent of the space age has brought promise of a new world to mankind. Some of i t s innovations are already here, others are yet to come. Among the numerous facets of t h i s e xciting new era, communication, earth-resources, navigation and m i l i t a r y (implying public safety) are the aspects l i k e l y to involve and af f e c t major portion of humanity. In thi s respect the remarks of Arthur C. Clarke*, directed p a r t i c u l a r l y to the communication s a t e l l i t e , are pertinent: . . . . What we are building now i s the nervous system of mankind, which w i l l l i n k together the whole human race, for better or worse, i n a unity which no e a r l i e r age could have imagined . . . . Accompanying th i s new world i s a "restructuring of p o l i t i c a l , s c i e n t i f i c and business thinking""'" leading to an open global society. However, s c i e n t i f i c success demands s c i e n t i f i c pre-c i s i o n . A l l of the above mentioned missions normally require s a t e l l i t e s to maintain preferred orientations r e l a t i v e to the earth. Among the numerous methods proposed for station keeping, gravity-gradient s t a b i l i z a t i o n has gained much attention primarily due to the passive nature of the system. The earth's natural s a t e l l i t e , the moon, provides an excellent example of such attitude control. The lunar globe i s a t r i a x i a l e l l i p s o i d with i t s longer axis captured by the earth's g r a v i t a t i o n a l f i e l d . The key to thi s s t a b i l i z a t i o n p r i n c i p l e i s the fact that the gr a v i t a t i o n a l f i e l d varies over a s a t e l l i t e r e s u l t -ing in a restoring moment tending to align i t s long axis (axis of minimum moment of in e r t i a ) with the l o c a l v e r t i c a l . Unfortunately, a gravity gradient s t a b i l i z e d s a t e l l i t e , even though positioned correctly i n the beginning, deviates with time from t h i s desired orientation due to perturbing environ-mental forces such as aerodynamic and radiation pressures, g r a v i t a t i o n a l and magnetic f i e l d interactions, micrometeorite impacts, etc. Design of a' s a t e l l i t e capable of proper functioning in such a "ho s t i l e " environment demands thorough understanding of i t s dynamical behaviour. Such a study, with a p a r t i c u l a r reference to an axi-symmetric s a t e l l i t e , forms the subject of t h i s thesis. 1.2 Literature Review A survey of the pertinent l i t e r a t u r e reveals a vast body of information i n thi s area. The bulk of the i n v e s t i -gation, however, i s devoted to the r e s t r i c t e d problem of l i b r a t i o n s i n the plane of the o r b i t . The dynamic analysis of a general motion has gained r e l a t i v e l y l i t t l e attention, 3 probably due to the non-linear, coupled character of govern-ing equations. The pioneering work on pure gravity oriented s a t e l l i t e s 2 was carried 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 3 c i r c u l a r o r b i t , and by Baker (1960) who found periodic solutions of the problem for small o r b i t e c c e n t r i c i t y . Beletskiy 5(1963) focused the attention on resonance effects for s a t e l l i t e s 5 i n e l l i p t i c .orbits while Schechter (1964) attempted, with limited success, to extend Klemperer's solution to non-circular o r b i t a l motion by perturbation methods. Zlatousov et a l . (1964) and more recently, Brereton and Modi (1967) success-f u l l y employed numerical methods, involving the use of the stroboscopic phase plane, to analyze the s t a b i l i t y of planar motion i n the large for orbits of arb i t r a r y e c c e n t r i c i t y . 8 9 They also investigated the corresponding periodic motion ' (19 69) 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. Brereton"1"0 (1967) has presented an excellent review of the work on planar l i b r a t i o n s . Thomson'1"''" (19 62) analyzed, through l i n e a r i z a t i o n , the related problem of slowly spinning s a t e l l i t e s i n c i r c u l a r 12 o r b i t s . Kane and Barba (1966) attempted to study the motion in e l l i p t i c o r b i t s using Floquet theory while Wallace and 13 Meirovitch (1967) used, with questionable success, asymptotic analysis i n conjunction with Liapunov's d i r e c t method. Modi and Neilson investigated r o l l dynamics of a spinning s a t e l l i t e 1 4 1 5 using the W.K.B.J. ( 1 9 6 8 ) and numerical ( 1 9 6 8 ) methods. The concept of i n t e g r a l manifolds was successfully extended to the study of three degrees of freedom motion i n c i r c u l a r o r b i t " ^ ( 1 9 6 9 ) . The periodic solutions were found and t h e i r 17 v a r i a t i o n a l s t a b i l i t y was established ( 1 9 7 0 ) . The l i t e r a t u r e on slowly spinning s a t e l l i t e s has been summarized, quite 18 e f f e c t i v e l y , by Neilson ( 1 9 6 8 ) . The presence of various perturbing forces and t h e i r influence on s a t e l l i t e dynamics has been discussed i n some ; 19 20 d e t a i l by Roberson ( 1 9 5 8 ) , Wiggins ( 1 9 6 4 ) , Moyer and K a t u c k i 2 1 ( 1 9 6 6 ) , Anand et a l . 2 2 ( 1 9 6 9 ) and Flanagan and M o d i 2 3 ( 1 9 7 0 ) . At higher a l t i t u d e s , normally used for communication s a t e l l i t e s , the solar radiation pressure becomes quite 2 4 s i g n i f i c a n t . Sohn ( 1 9 5 9 ) indicated the use of solar radia^ tion to orient the s a t e l l i t e with respect to the sun while 2 5 Ule ( 1 9 6 3) . considered i t s application to spin an array of mirrors to achieve attitude s t a b i l i t y . A more complete analysis accounting for solar as well as the earth and earth 2 6 r e f l e c t e d radiations was attempted by Clancy and M i t c h e l l ( 1 9 6 4 ) . In addition to the inherent l i m i t a t i o n s of the approach, the r e s u l t i n g force expression,given i n an i n t e g r a l form, required numerical evaluation. This rendered t h e i r application to any comprehensive study of s a t e l l i t e attitude 27 2 8 29 dynamics impractical. Modi and Flanagan ' ' evaluated these forces, quite accurately, using the concept of cutting-5 plane distance r a t i o s and used them to analyze the environ-mental e f f e c t on the s a t e l l i t e planar dynamics. A c r i t i c a l review of the developments i n t h i s f i e l d i s presented by , Flanagan 3 0(1969). 31 Schrello (1961) pointed out the predominance of aerodynamic torque for s a t e l l i t e s at a l t i t u d e below 350 miles with variations i n equilibrium configurations d i s -32 cussed by Debra (1959). The e f f e c t of small aerodynamic 33 and g r a v i t a t i o n a l torques were treated by Beletskiy (1960) as independent perturbations to the motion of rapidly 34 spinning s a t e l l i t e s . Evans (1962) presented the aerodynamic and radiation disturbances i n the fundamental form of pres-35 sure and shear stress. Using i n f i n i t e s i m a l analysis, Garber (1963) treated the e f f e c t of constant disturbing torques on the l i b r a t i o n a l motion of a r i g i d , gravity oriented system 3 6 in a c i r c u l a r o r b i t . More d i r e c t l y , Meirovitch and Wallace (1966) established the regions of guaranteed s t a b i l i t y for a slowly spinning, axi-symmetric s a t e l l i t e i n a c i r c u l a r o r b i t with constant aerodynamic force. For two s a t e l l i t e configurations, equilibrium positions were tested for s t a b i l i t y i n the small through Liapunov's d i r e c t method. No attempt was made to obtain response of the system to an arbi t r a r y disturbance or the l i m i t i n g bounds for s t a b i l i t y . 37 Nurre (1968) considered a more complex model of an asymmetric 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 investigated the s t a b i l i t y 6 of i t s equilibrium position using l i n e a r i z e d analysis. The results of the study were substantiated for an i n f i n i -tesimal disturbance, by numerical solution of the exact equations of motion. Exploitation of g r a v i t a t i o n a l , magnetic and solar radiation forces for damping the l i b r a t i o n s and c o n t r o l l i n g the attitude has been suggested by many authors. The f e a s i b i l i t y of such a concept i n terms of solar s a i l i n g was 3 8 investigated by Garwin as early as 1958. Several configur-ations of mechanical dampers have been evolved and analyzed 39 40 41 by Debra (1962) , Kamm (1962), Paul (1963), Modi and Brereton 4 2(1969), Tschann and Modi 4 3' 4 4(1970), etc. Paul 45 et a l . (1963) suggested the use of magnetic hysteresis damper, while Mallach 4 6(1966), Modi et a l 4 7 , 4 8 ' 4 9 ( 1 9 7 0 ) proposed c o n t r o l l e r s using solar radiation pressure. A s t a b i l i t y theorem, derived by P r i n g l e ^ (1963) for a damped autonomous system, involving the Hamiltonian as a Liapunov function, i s of considerable s i g n i f i c a n c e . A c r i t i c a l analysis of the literature on the subject, as presented by Tschann^ (1970) , forms a useful contribution to the f i e l d . 1.3 Purpose and Scope of the Investigation From the foregoing, i t i s apparent that the general motion of a gravity gradient s t a b i l i z e d s a t e l l i t e and the effects of environmental forces on i t have received, r e l a t i v e l y , l i t t l e attention. The reason for t h i s limited e f f o r t could, 7 perhaps, be attributed to the complexity of the problem. The non-linear non-autonomous, coupled equations of motion involving large number of parameters are not amenable to any simple concise analysis. Such an investigation, how-5 2 ever, i s important because, as pointed out by Kane (1966) strong coupling exists between the planar and transverse motions. The main purpose of t h i s investigation, therefore, i s to gain a fundamental understanding of the dynamics of the general motion and to obtain the l i m i t i n g i n i t i a l conditions for stable motion, i n a r b i t r a r y o r b i t s , as a function of design parameters. The e f f e c t of aerodynamic forces, which become s i g n i f i c a n t for near-earth s a t e l l i t e s , on the l i b r a t i o n a l response and s t a b i l i t y i s also investigated. The p o s s i b i l i t y of aerodynamic damping and attitude control i s examined. From cost considerations, the a p p l i c a b i l i t y of analog simulation as well as a n a l y t i c a l methods i s explored. The problem i s analyzed i n f i v e stages, representing, i n general, an increasing order of complexity. In the beginning, coupled l i b r a t i o n a l motion of a pure gravity-gradient, axi-symmetric s a t e l l i t e i s considered for the auton-omous case of a c i r c u l a r o r b i t . The work i s e s s e n t i a l l y an 5 3 extension of the study i n i t i a t e d by Modi and Brereton (1968). It helps establish methods of approach for subsequent research. The influence of constant aerodynamic torque on equilibrium configurations, l i b r a t i o n a l response, nature of 8 solutions and s t a b i l i t y bounds of near-earth s a t e l l i t e s i s investigated i n the t h i r d chapter. A determination of the periodic solutions, which form the spines of s t a b i l i t y bounds, i s the main objective in the next stage. The peculiar ergodic behaviour of the t r a j e c -t o r i e s , not reported i n the case of planar l i b r a t i o n s , i s analyzed. The c r i t i c a l conditions and p r a c t i c a l l y usable bounds of s t a b i l i t y are also obtained. A case study of 54 Geodetic Earth Orbiting S a t e l l i t e (GEOS-A) emphasize the usefulness of the r e s u l t s . In Chapter 5, the analysis i s extended to the case of e l l i p t i c o r b i t s . The non-autonomous character of the system increases the complexity of the problem, e s p e c i a l l y i n presence of atmosphere. The parametric study of the response and s t a b i l i t y i n the large has p a r t i c u l a r relevance to the geophysical, earth resources, and m i l i t a r y s a t e l l i t e s . F i n a l l y , the f e a s i b i l i t y of using aerodynamic forces in the l i b r a t i o n a l damping through a semi-passive c o n t r o l l e r i s explored. . The response analysis over a large range of system parameters establishes i t s effectiveness. Figure 1-1 schematically i l l u s t r a t e s the various stages involved i n the proposed plan of study. It i s f e l t that the approach represents a coherent program to explore the subject. COUPLED LIBRATIONS OF GRAVITY-ORIENTED AX I-SYMMETRIC SATELLITES equ i l i b r i u m configura-tions & t h e i r s t a b i l i t y : •Routh •Liapunov autonomous conservative system ( c i r c u l a r orbits) g r a v i t y -gradient f i e l d g r avity-gradient & aerodynamic torques motion envelopes: •Hamilton system response: •Butenin • invariant-i n t e g r a l • numerical •analog character of stable solutions: •numerical periodic solutions & t h e i r s t a b i l i t y : •numerical •Floquet l i b r a t i o n a l s t a b i l i t y : •numerical •analog non-autonomous conservative system ( e l l i p t i c o r b i t s ) g r a v i t y -gradient f i e l d autonomous non-conservative system ( c i r c u l a r orbit) system response: •Butenin •numerical g r a v i t y -gradient & aerodynamic torques l i b r a t i o n a l s t a b i l i t y : •numerical aerodynamic damping with semi-passive c o n t r o l l e r response a n a l y s i s : •numerical Figure 1-1 Schematic diagram of the proposed plan of study 2 . LIBRATIONAL RESPONSE AND STABILITY IN CIRCULAR ORBIT 2 . 1 Preliminary Remarks Gravity-gradient s t a b i l i z a t i o n of axi-symmetric non-spinning s a t e l l i t e s , moving i n c i r c u l a r orbits and free to l i b r a t e both i n and across the o r b i t a l plane i s analyzed 53 here. The study, i n i t i a t e d by Modi and Brereton , emphasizes the e f f e c t of s a t e l l i t e i n e r t i a on the bounds of possible motion, response and s t a b i l i t y c h a r a c t e r i s t i c s . The invariant Hamiltonian, representing the f i r s t i n t e g r a l , y i e l d s regions 55 of possible motion through zero v e l o c i t y curves and establishes conditions for s t a b i l i t y . As the second order, non-linear, coupled equations of motion do not possess any known closed form solution, an approximate analysis i s undertaken using an extension of the Krylov and Bogoliubov method^ (variation of parameter) 57 . . . . as suggested by Butenin with certain modifications. A response study establishes the a c c e p t a b i l i t y of the solution for small amplitude l i b r a t i o n s . The Hamiltonian i s also used to reduce the order of the system and to obtain another approximate solution i n terms of Jacobian e l l i p t i c functions. For a motion i n the large, the equations are solved numerically using Adams Bashforth predictor corrector i n t e -5 8 gration with a Runge-Kutta s t a r t e r . The concept of i n t e g r a l manifolds ' ' ' ' i n a three dimensional phase-space i s adopted for concise presentation of the solution. Three classes of stable t r a j e c t o r i e s e x i s t : 'regular 1, 'ergodic' and 'island' type. The l i m i t i n g manifolds es t a b l i s h the s t a b i l -i t y bounds. The e f f e c t of the i n e r t i a parameter, Hamiltonian and i n i t i a l conditions are studied by t h e i r systematic variations. The massive information generated i s condensed in the form of design p l o t s , which give allowable disturb-ances for non-tumbling motion. 2.2 Formulation of the Problem Consider an a r b i t r a r i l y shaped, r i g i d s a t e l l i t e with mass center at S i n an o r b i t about center of force 0 (Figure 2-1). Let S-xyz be the p r i n c i p a l body axes of the s a t e l l i t e with the t r i a d S-x^y^Zg so chosen as to d i r e c t Zg-axis outward along the l o c a l v e r t i c a l and the y^-axis p a r a l l e l to the o r b i t a l angular momentum vector. The orientation of the s a t e l l i t e may be s p e c i f i e d by a set of successive rotations: about y^-axis giving x^y^z^-axes; <j> about x^-axis r e s u l t i n g i n the yL^y^z^ t r i a d ; and X about Z2 -axis y i e l d i n g xyz. The expression for potential and k i n e t i c energies 3 10 to 0 ( l / r ) can be written as: - 3/A [ sin * if/ { I z z - (Ixx - lyy ) CcosZ\ - emz A ) -f- A sin cos y sinA C05A sin<t>C Ix -1^) +- c o s * Y s m H U z z : - C r x 3 c - ^ ) ( 5 i n ^ -cos^ A ) ] + C05^ cos^cj) ( I ^ + I ^ - I 2 Z ) J / 4 r 3 . . . . ( 2.la) 4- 4> ( 9 4- y ) cos <t> sin A cos A ( I x x - 1^ ) + (6 + ip f cos* cb ( I x x sin* A + 1^ cos*A)/* + [ x - (0 + q O sincj>J lzz . . . . ( 2 . i b ) As for an axi-symmetric s a t e l l i t e , I =1 =1, A does not 1 xx yy appear e x p l i c i t l y i n the expression for Lagrangian thus rendering the conjugate momentum p^ a constant of the motion, i . e . , P X - dL/dX = I Z 2 L [ A - (0 + 40 Sin 4>J = constant . . . . ( 2 . 1 c ) For a non-spinning s a t e l l i t e the constant must be zero, therefore equations ( 2 . 1 a ) and ( 2 . 1 b ) assume the simpler forms: = -/JLm/r +/x( l - I Z 2 ) ( i -3C05^1|/Cos^) . . . . ( 2 . 2 a ) T = m ( r + r * 6 * ) A + l [ > * + (e + ^ C 0 5 H j / 2 . . . . ( 2 . 2 b ) Using the Lagrangian formulation, the governing equations of motion i n r,0,^ and <p degrees of freedom can be written as: . . . . ( 2 . 3 a ) rH -f £r P 0 + i [(e 4- ij)) c o s H - 2Ce + ^ )*3inctcos^J/m =0 . . . . ( 2 . 3 b ) ip+9 -2,<i>(0 + i l O t Q i r < f > + 3MKj smiy cosq^/r^ = 0 . . . . ( 2 . 3 c ) $ + [ (e + q^f + 3yuK] coszty/r5]m<\> cose}) =o . . . . ( 2 . 3 d ) Neglecting perturbations of the o r b i t a l motion due to l i b r a -t i o n s , ^ ' ^ 3 e t a*" the equations ( 2 . 3 a ) and ( 2 . 3 b ) lead to the c l a s s i c a l Keplerian r e l a t i o n s : rZB = h Q . . . . ( 2 . 4 ) .j P " ^ 0 + G C O S 9 ) . . . . ( 2 . 5 ) For a c i r c u l a r o r b i t (e=0 ) , with 6 as an independent va r i a b l e , 5 3 the equations ( 2 . 3 c ) and ( 2 . 3 d ) transform to: ty" - fc^'C^+i) tanc|> 4- 3Kj sin l|/ cos - o ( 2 . 6 a ) <\>" + + 3Kj cos* iy} sin <f> coscj) =0 ( 2 . 6 b ) The governing equation i n the X degree of freedom i s repre-sented by ( 2 . l c ) . These second order, coupled, and highly non-linear equations do not appear to possess any known closed-form solution. However, before proceeding to seek an approximate solution i t would be worthwhile to gain some insight into the problem by examining the Hamiltonian of the system. 15 2.3 Bounds of L i b r a t i o n For c i r c u l a r o r b i t s , the Lagrangian function associated with the l i b r a t i o n a l motion does not involve time e x p l i c i t l y . Hence, the corresponding Hamiltonian of t h i s conservative system i s a constant of the motion: H = P q , V + P^4> + PA A = l [ c f ) ^ + ( ^ - 0 i ) c o 5 H J A - 3 A d - i Z z ) = constant i . e . = constant . . . . (2.7) Setting UJ=cf>== 0 results i n a set of zero-velocity 55 curves , symmetrical with respect to and <$> axes (Figure 2-2), enclosing regions of possible motion. Consistent with the s t a b i l i t y c r i t e r i o n of nontumbling motion, the zero-v e l o c i t y curves are presented only over the range if;, C J > = ± T T/2. It may be observed that f o r : C < -(1+3K.) , no motion i s possible; H— 1 -(1+3K.)< C „ < -1 , the motion i s bounded; 1 — rt — -1 < C „ < 0 / the motion i s bounded i n <f> only; ri — 0 < C , unbounded motion i s possible i n both d i r e c t i o n s . . (2.8) Figure 2-2 E f f e c t of s a t e l l i t e i n e r t i a and Hamiltonian on zero-v e l o c i t y curves Furthermore, with constant Hamiltonian any one of the four state parameters (\|> 1 , <j> ,<{>' ) can be eliminated thus permitting a three-dimensional state-space represen-tation of the motion. Within the bounds imposed by the zero-velocity curves, relations defining surfaces i n i|j,if'/<l> or <\>,$,,ty space can be obtained by equating the eliminated v e l o c i t y to zero. For example, i n 1 , cj> - space motion i s bounded by C o s H ( 4 ^ - i - 3 Kj c o s ^ ) - C H =0 . . . ( 2 . 9 ) Figure 2 - 3 i l l u s t r a t e s these surfaces. They represent envel-opes of possible motion i n the state space for a given value of Hamiltonian. Note that the zero-velocity curves are merely cross-sections of these surfaces at 4 > ' = 0 . A growth of the l i m i t i n g closed envelope (C = - 1 . 0 ) with increasing K. ri 1 (Figure 2 - 3(a))suggests a more stable performance. The envel-opes are open i n ^ d i r e c t i o n for C „ > - 1 . 0 ( F i g u r e 2 - 3(b)) in d i c a t ing p o s s i b i l i t y of unbounded planar l i b r a t i o n s . The actual motion of the system, however, i s dependent upon the i n i t i a l conditions as well as the Hamiltonian. Hence to est a b l i s h the character of the motion, such as amplitude and frequency, i t i s es s e n t i a l to solve the governing equations In the absence of any known closed form solution, approximate a n a l y t i c a l methods and numerical techniques have to be resorted to. Figure 2-3 E f f e c t of s a t e l l i t e i n e r t i a and Hamiltonian motion-envelope (4>'=0): (a) l i m i t i n g region for guaranteed bounded l i b r a t i o n s (C =-1.0) Figure"2-3 E f f e c t of- s a t e l l i t e i n e r t i a (b) l i b r a t i o n s bounded i n <j> and Hamiltonian on motion envelope (<j>'=0): only (C =-0.5) 2.4 Approximate Solutions and System Response 57 2.4.1 Variations of Parameter Method (Butenin ) Representing the trigonometric functions by t h e i r series, neglecting f i f t h and higher degree terms i n ,<p and thei r derivatives, and c o l l e c t i n g the nonlinear terms on the r i g h t hand side the equations of motion take the form iy" + 3Kj = 2 4)' ly'd) +z$<b + W<P/z> +2K[ t | / 3 + a + 3K , ) ( j ) = -q/^d) - J i ^ c j ) + 3Kj - M ^ V A 4-X(if3Kj)cJ)y3 . . . . (2.10) or Hi" + n [ ( V - ^ < q / ' -cj)" + n£cf> = gCH/. iy '^ .d) ' ) For small amplitude motion each term i n f and g i s small as compared to the terms on the L.H.S. of the equation (2.10), hence t h e i r approximate solution can be found using the method of v a r i a t i o n of parameters. The complementary solution of t h i s system of equa-tions i s given by a set of harmonic functions l|J = a s i n ^ 0 4- £±) . . . . (2.12) 4) = bsin(n^9 I t i s i n t e n d e d h e r e t o o b t a i n t h e s o l u t i o n e s s e n t i a l l y i n t h e s a m e f o r m a s t h e c o m p l e m e n t a r y s o l u t i o n , b u t n o w p e r m i t -t i n g t h e a m p l i t u d e s a n d p h a s e a n g l e s t o b e f u n c t i o n s o f G, i . e . , iy=a(0)sin [ n £ e + ^ ( 9 ) ] . . . . ( 2 . 1 3 ) 4> = b(e) sin [n^B - f ^ ( e ) ] N o t e t h a t n ^ a n d r e p r e s e n t t h e p r i n c i p a l f r e q u e n c i e s g i v e n b y t h e s o l u t i o n o f t h e h o m o g e n e o u s e q u a t i o n s a n d a , b , 3 j , $ 2 a r e u n k n o w n s t o b e d e t e r m i n e d . E a c h o f t h e f u n c t i o n s a ( 9 ) , b ( 8 ) , 3]_ ( 8 ) , 3 2 ( 9 ) m a y k e e x p r e s s e d a s a f u n c t i o n o f 8 p l u s a c o n s t a n t . T h u s t h e s o l u t i o n i n t h i s f o r m i n v o l v e s e i g h t u n k n o w n s a n d h e n c e i s o v e r s p e c i f i e d . I t w i l l b e , t h e r e f o r e , n e c e s s a r y t o o b t a i n f o u r m o r e r e l a t i o n s i n a d d i t i o n t o t h o s e g i v e n b y t h e i n i t i a l c o n d i t i o n s . T h i s i s a c h i e v e d b y i n t r o d u c i n g l o g i c a l c o n s t r a i n t 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 . 1 3 ) t o t h a t o f t h e h o m o g e n e o u s s o l u t i o n ( 2 . 1 2 ) g i v e s t h e t w o c o n -s t r a i n t r e l a t i o n s : a'sin^S + a § i cos s =0 • • . ( 2 . 1 4 a ) b' sin r[ 4- b ^ c o s Y[ =0 . . . ( 2 . i 4 b ) M a t h e m a t i c a l l y t h i s i m p l i e s t h a t t h e n o n l i n e a r i t i e s a r e s m a l l . P h y s i c a l l y i t m e a n s t h a t t h e s a t e l l i t e i s e x e c u t i n g s m a l l amplitude motion. Normally t h i s condition i s s a t i s f i e d by most communication, weather, or earth-resources s a t e l l i t e s . The other two re l a t i o n s are obtained by d i f f e r e n -t i a t i n g once again with respect to 0 and substituting i n the equations of motion giving Q' r\i cos^ - a n1 sin ^ = f* . . (2.i4c) b> cosr[ - b nz ^ sin = . . (2.i4d) where •f = I ( a sin *s .ariiCos-s , bsin rj_, bn^cosnj . . .(2.15) g* = g ( a sin ^ , a r\± cos ^ , b sin , b n^  cos r[) Solving the four algebraic equations i n (2.14) for the un-knowns a ' , b ' , 8 ^ , 8 2 gives b* - i/r\K g* cos K[ . . . .(2.16) a §i = - i/nj_ f* sin - i / n ^ 3* sin ^ Here f* and g* are known nonlinear functions i n a , b , 8 ^ , 8 2 / and 6. For small amplitude motion (say 10°), f* and g* are quite small (5-10%) compared to the remaining terms in the equations of motion. Hence a , b , P i ; L are slowly varying 23 parameters. Using t h e i r average values over a period i n ijj and 4> degrees of freedom yie l d s 'ZTT rlW 4. d a / d e = (i/Airzr\i) j f cos-c , d b / d G = (1/4 TT* n^) J J cos r|_ d^S drj_ &7T /£7T o 0 . . . . (2.17) Evaluating the integrals and using the conditions of ortho-gonality gives a', b' = 0 a^n, . . . . (2.18) ,2 b n„ i . e . , solution represented by equation (2.18) becomes + t Q n - M ( 3 K i ^ q / 0 / y j : ] ] 24 <() = {4f + <KY(i + 3Ki>f* S l n [ { i - ( t f + <KVCi+3K|) )/4} (1 +3Kj)V*e i t o n 1 [(l+3Kj^/<}>']] . . . . (2.19) To e s t a b l i s h the v a l i d i t y of the a n a l y t i c a l solution the equations of motion (2.6) were rewritten as a set of four f i r s t order r e l a t i o n s : d<b'/d0 = - ( ( i f 4- 3K,- C05z Hi} sin <|> cos(() . . . . (2.20) and were integrated numerically using Adams-Bashforth pre-5 8 dictor-corrector procedure with a Runge-Kutta s t a r t e r . The step size of 3° gave results of s u f f i c i e n t accuracy without 64 involving excessive computation time . The use of symmetry properties, as exhibited by the invariant nature of the equations under transformation (Q ,\\> to (6 ,-<(>) , (-6 ,<$>) or (-9 , - I J J ,-4>) s u b s t a n t i a l l y reduced the e f f o r t . The l i b r a t i o n a l response of a wide range of s a t e l l i t e s to a broad spectrum of disturbances was obtained, over f i f t y o r b i t s , by systematically varying the i n e r t i a parameter and i n i t i a l conditions, and i s compared with that given by the approximate closed form solution i n Figure 2-4. For conciseness only i n i t i a l and terminal regions are shown. 3 0 r numerical 0 analyt ical •30 30 K;=1.0 4>° 0 •30 30 -30 30 4>r 0 -30' K;=0.5 3 47 48 49 50 orbits Figure 2-4 Ef f e c t of s a t e l l i t e i n e r t i a on l i b r a t i o n a l response obtained using numerical and v a r i a t i o n of parameter methods: (a) impulsive disturbance 1^=15°^ = 0 . 2 5 , ^ 0 = ^ = 0 n u m e r i c a l a n a l y t i c a l 30 r-0 -30 30 . 0 Kj=1.0 - 3 0 30 i -J \L - 3 0 30 ~ 4>* 0 - 3 0 K--0.5 1 3 47 48 49 50 Figure 2-4 o r b i t s E f f e c t of s a t e l l i t e i n e r t i a on l i b r a t i o n a l response obtained using numerical and v a r i a t i o n of parameter methods: (b) disturbance i n the o r b i t a l plane 27 numerical -analyt ical 30 r 0 •30 30 <t> o 0 Kj= 1-0 •30 30 i -so-so -•30 K; = 0.5 1 47 48 49 50 Figure 2-4 orbits E f f e c t of s a t e l l i t e i n e r t i a on l i b r a t i o n a l response obtained using numerical and v a r i a t i o n of parameter methods: (c) disturbance across the o r b i t a l plane 28 Figure 2-4(a) indicates that the a n a l y t i c a l method determines the amplitude and frequency of the motion with con-siderable accuracy. For a disturbance of appreciable magnitude (ij^ =cf>^ =0.5) , the main discrepancy i s i n the phase which appears to be cumulative. From the p r a c t i c a l application point of view th i s may not be a serious l i m i t a t i o n as amplitude and frequency of motion provide s u f f i c i e n t i n -formation needed i n preliminary s t r u c t u r a l design of a s a t e l l i t e . The agreement, i n general, i s better for slender s a t e l l i t e s (K^-1.0) which are normally preferred for gravity-gradient s t a b i l i z a t i o n . The l i b r a t i o n a l frequency, which depends on the disturbances encountered as well as the i n e r t i a parameter, i s of the order of o r b i t a l frequency. For an i d e n t i c a l disturbance i n the two degrees of freedom, a larger amplitude and smaller frequency motion i s excited across the o r b i t a l plane (<j>). Both the a n a l y t i c a l and numerical solutions indicate decrease i n frequency and increase i n amplitude, p a r t i c u l a r l y for planar motion , with decreasing . The accuracy of the a n a l y t i c a l solutions improves when the disturbance i s r e s t r i c t e d to one degree of freedom only. As apparent from the equations (2.6), the planar disturbances (d> = <t>l=0) do not excite a transverse motion o o (Figure 2-4 (b)). However, l i b r a t i o n s i n 4> d i r e c t i o n lead to small ripples i n the degree of freedom, which increase with decreasing i n e r t i a (Figure 2-4 (c)). The a n a l y t i c a l 29 method f a i l s to predict t h i s behaviour as well as small amplitude modulations, perhaps due to the assumed form of the solution. The periodic nature of the independent variable provides yet another standard for comparison between the solutions. On the stroboscopic phase plane (Figure 2-5), the points shown represent the state of the system at 8=2TTn (n=0 ,1 ...,50). A few of the points are l a b e l l e d . Here again the c o r r e l a t i o n between the two methods appears to be quite s a t i s f a c t o r y . Any error i n the phase results only i n circumferential rotation (as against the r a d i a l departure)of the point of i n t e r s e c t i o n of the trajectory with the plane. The agreement suggests a p o s s i b i l i t y of using the a n a l y t i c a l solu-tion for s t a b i l i t y analysis by the i n t e g r a l manifold technique. It may be pointed out that the two solutions are compared here under adverse si t u a t i o n s . In actual practice, the communication s a t e l l i t e s demand extreme pointing accuracy. So i n that case the predictions made by the approximate theory are l i k e l y to be more accurate. The simple analysis presented here provides considerable insight into the physical nature of the coupled motion and appears to be adequate for preliminary design purposes. 0 . 2 2 5 0 . 1 5 0 0 .075 0 . 0 0 0 - 0 . 0 7 5 - 0 . 1 5 0 - 0 . 2 2 5 - 1 5 - 7 . 5 7 . 5 15 - 1 5 - 7 . 5 7.5 15 Figure 2-5 Stroboscopic phase-plane (6=0), obtained using numerical and v a r i a t i o n of parameter methods: (a) impulsive disturbance o 0.30 0.20 010 0.00 - 0.10 0.20 - 0.30 c -A J J ^ 9 ^ V — * ^ P i 5 ~ o — 7 SZ o9 r50 Q\ a iHS 7 • 1 4 &A t 1 1 I 11 V v i i V ; ' / \3-*\ 1  \ ^ \ H v/ / / b / / J ° o . t o : 1 ° 4» 0 -°° d>:10° 0 = 0.0 o o CP -10 -5 0 5 10 -10 - 5 0 5 10 Figure 2-5 Stroboscopic phase-plane (6=0) , obtained using numerical and v a r i a t i o n of parameter method: (b) angular disturbance 32 2.4.2 Invariant Integral Method The Hamiltonian, a constant of the motion, which gave the bounds of l i b r a t i o n , can be used to reduce the order of the system leading to yet another approximate a n a l y t i c a l solution. Multiplying equations of motion (2.6a) and (2.6b) by 2 2^' cos <j> and 2<j>', respectively, adding and integrating once y i e l d the normalized Hamiltonian (equation 2.7), which can be rearranged as, iy / J l - [ ( C H - C ^ / C O S ^ + i + 3Kj ] - 3 K ( s i n * l J / . . . . (2.21a) 4 ^ = [ c H + i + 3 K , c o s ^ - l | T * J - Q l + 3K ( cos^ij/ - I i i ' * ] S i n ^ (j) . . . . (2.21b) As i s well-known, the solution of an equation of the form, where a^ and a^ are constants, i s a Jacobian e l l i p t i c a l function: p = s i n 1 [ ( Q i / a J * Sn ( ( q ^ ( e - 6 ' ) , a ^ j ] • • • • (2*22) where 3=0 at 6=9*. 3 3 Thus equation (2.21) can be solved approximately i n the form (2.22) provided, d/d9 {(C H - < p ' * ) / c o s H } * 0 d/de { 3Kj cos^ -qj - v'1} - 0 Note that these conditions are equivalent and corres-pond to (from equation 2.6a): 4 (j) I]/ (ili'+i) t a n 4> - 0 . . . . ( 2 . 2 3 ) thus implying that the coupling terms may be ignored. Hence for systems s a t i s f y i n g t h i s condition, the solution can be approximated by: f = s i n 1 O i S n { ( 5K0A ( 0 - 9 * ) , k f J ] <t> = s i n 1 [ S n { ( i + 3 K, C O S * ( j / 0 - ) , k * j where, . . . . (2.24) k£ = 1 + [ i + (cH -^Vcos^j/SK, = i + C H / ( i + 3K,oos*i|i -l|//*) if(et): - 4>(eJ) =o The solution becomes exact i n the absence of disturbances across the o r b i t a l plane (Figure 2-6(a)i), however, the planar motion excited by a transverse disturbance i s not exhibited by the a n a l y t i c a l solution (Figure 2 - 6 ( a ) i i ) . K; =1.0 45 -45 45 ° 0 -45 301-0 -30 30 - 3 0 34 n u m e r i c a I 0 Figure 2-6 B u t e n i n . jnv. i n t e g r a l m e t h o d +0=30^0=05 , 4=^=0 ±-A± I A J L 3 47 48 49 50 orbits Ul kJI I o Accuracy of approximate solution: (a) response to large disturbance along one of the degrees of freedom Figure 2-6 Accuracy of approximate solution: (b) response to predominantly planar disturbance The method, i n general, represents a better approximation compared to Butenin 1s approach, p a r t i c u l a r l y when the d i s -turbances across the o r b i t are r e l a t i v e l y small (Figure 2 - 6 (b)). The c o r r e l a t i o n improves with increasing i n e r t i a , e s p e c i a l l y i n the i|) d i r e c t i o n . Thus, while the widely used l i n e a r i z a t i o n techniques may be acceptable for small disturbances ( 5 - 6 deg. amplitude of l i b r a t i o n ) , the v a r i a t i o n of parameter approach gives a good approximation for r e l a t i v e l y large disturbances ( 2 0 - 2 5 deg.). The invariant i n t e g r a l method appears to extend t h i s range considerably ( 3 0 - 4 0 deg.), p a r t i c u l a r l y i f cross-motion i s small. However; for very large disturbances, when the amplitude modulations due to coupling become substantial, a numerical integration i s unavoidable for response and s t a b i l i t y analysis. Figure 2 - 7 shows over six o r b i t s , the e f f e c t of s a t e l l i t e i n e r t i a on the exact response to a large a r b i t r a r y disturbance. Decrease i n tends to make a sat-e l l i t e more sensitive to a given disturbance with a marked reduction i n the l i b r a t i o n a l frequency. The e f f e c t appears to be greater i n the d i r e c t i o n compared to that i n the. <j> d i r e c t i o n . The modulation of the amplitude due to a coupling between the two degrees of freedom, more pronounced i n the (j) degree of freedom /increases with decreasing . The i n f l u -ence of harmonics may be expected to increase with disturb-ances . Figure 2-7 Numerically generated response to a large arb i t r a r y disturbance showing ef f e c t s of s a t e l l i t e i n e r t i a 2.5 Nature of the Stable Solutions: The application of the concept of i n t e g r a l manifolds or invariant surfaces i n a three-dimensional phase-space for studying l i b r a t i o n a l s t a b i l i t y of a gravity-oriented system has been discussed i n some d e t a i l by Brereton and 7 10 53 Modi ' ' . The method provides a c l e a r picture as to the entire spectrum of disturbances to which a s a t e l l i t e can be subjected at any point i n i t s o r b i t without causing i t to be unstable. Although the system under consideration i n -volves four state elements (i> ,i>1 , 4>,<f>1 ) , the method i s s t i l l applicable due to a constant value of the Hamiltonian.However, elimination of an element using equation (2.7) leads to an ambiguity concerning i t s sign, as pointed out by Henon and H e i l e s ^ 0 . It i s , therefore, necessary to delineate between the two p o s s i b i l i t i e s to u t i l i z e the invariant surface concept Hence, two spaces must be used to describe the state of the system, one for pos i t i v e values of the eliminated state , element and the other for i t s negative values. Here the two spaces used for presenting the solution are ip,^',<}> with $' ^  0 To obtain cross-sections of the invariant surface in (J)1 space i t was necessary to develop an interpolation scheme so that the state of the system could be ascertained for any given value of the stretching coordinate. This was achieved by Adams-Bashforth predictor-corrector method in conjunction with a polynomial f i t to the past history of the state coordinates and t h e i r derivatives. Having f i t t e d the polynomials i n 9 to the numerical solution the state of the system i s r e a d i l y determined using Newton-Raphson .. 16 i t e r a t i o n Figures 2-8 and 2-9 show the cross-sections of the invariant surfaces, at <j)=0, obtained using t h i s procedure. The cross-sections of motion envelopes,represented by (j)'=0 , are also included to f a s c i l i t a t e comparison with the region of possible motion. Due to the coupling e f f e c t s and r e l a t i v e frequency of motion i n the two degrees of free-dom the system exhibits three d i s t i n c t types of stable solutions. For a given Hamiltonian a systematic v a r i a t i o n i n the i n i t i a l conditions leads to 'regular', 'ergodic' and 'island' (Figure 2-8) type t r a j e c t o r i e s . For c e r t a i n i n i t i a l condition, integration of the equations of motion over a large number of o r b i t s leads to the phase space trajectory i n t e r s e c t i n g <J>=0 plane at a series of points defining a smooth curve (0) as shown in Figure 2-8. The selection of any i n i t i a l condition within the enclosed region leads to a nested surface and hence, a cross-section l y i n g completely within the former. In the l i m i t , t h e i n t e g r a l manifold degenerates to a l i n e , i . e . , cross-section in the ii-ty' plane reduces to a point, thus representing a periodic solution. On the other hand, i t may be emphasized that although the s t a b i l i t y of the motion i s assured for C„<-1.0, the Figure 2-8 The c r o s s - s e c t i o n <f> = 0 i n phase space i n d i c a t i n g types of s t a b l e s o l u t i o n generated by d i f f e r e n t i n i t i a l c o n d i t i o n s f o r a given Hamiltonian: (a) r e g u l a r and ergodic; (b) i s l a n d s and t h e i r breakdown i n t o ergodic - 0 . 5 h Figure 2-9 E f f e c t of s a t e l l i t e i n e r t i a on nature of stable solution; cf>=0 , C =-1.0 42 existance of a well defined i n t e g r a l manifold i s not guaran-teed as indicated by the figure. It i s possible that an ergodic trajectory may exhibit p e r i o d i c i t y over a large number of or b i t s and hence can be thought of as generating small tubular s t a b i l i t y surfaces. However, any e f f o r t at determining these surfaces would involve an enormous amount of numerical computation, which can hardly be j u s t i f i e d due to the academic nature of t h e i r usefulness. Further change i n the i n i t i a l condition leads to a regrouping of the erogodic intersections i n a well defined chain of islands (Figure 2-8 (b)) , which, i n t h i s case, rep-resent two cycles of the planar l i b r a t i o n for every three o s c i l l a t i o n s across the o r b i t a l plane. Although the s t a b i l i t y i s assured here, proximity of the islands to the s t a b i l i t y bound makes operation of a s a t e l l i t e i n t h i s region undesir-able. Thus, for a l l p r a c t i c a l purposes, the regular 'main-land' represents the only stable region for safe operation of a s a t e l l i t e . Subjecting the system to any further v a r i a t i o n i n the external disturbances results i n the breakdown of the islands into the ergodic behaviour. The process of regener-ation of islands and t h e i r degeneration into ergodic behaviour appears to progress i n d e f i n i t e l y approaching the boundary of possible motion, where the margin of s t a b i l i t y vanishes. 43 A comment concerning the e f f e c t of the i n e r t i a parameter on the character of the solution would be appropriate here. For a given Hamiltonian, the systematic reduction of K^, from 1.0 (Figures 2-8) to 0.5 and 0.25 diminishes the p o s s i b i l -i t y of ergodicity (Figure 2-9). Thus, decreasing the slender-ness of a s a t e l l i t e appears to confine the region of ergodic-i t y closer to the bound of possible motion. For a given C H several i n t e g r a l manifolds, represent-ing the region of stable motion, are possible depending on the i n i t i a l conditions. The largest of these may be called, the l i m i t i n g i n t e g r a l manifold. Figures 2-10(a) and 2-10(b) indicate the l i m i t i n g regular manifold and that corresponding to the islan d type solution, respectively. In general, the l a t t e r winds around the regular manifold and i s associated with a d i f f e r e n t periodic solution. The increase i n Hamilton-ian causes a marked reduction i n the s t a b i l i t y region (Figure 2-10c) suggesting the p o s s i b i l i t y of a c r i t i c a l value, C„ , beyond which the manifolds cease to e x i s t . Thus, for cr a value of the Hamiltonian greater than the c r i t i c a l (a0.8), gravity-gradient s t a b i l i z a t i o n of a s a t e l l i t e i s not possible. It should be noted that during the integration of the equations no attempt i s made to reduce the order of the system using the Hamiltonian. Rather, the Hamiltonian, a constant of the motion, i s computed along with the cross-section data and i s used as a check on the o v e r a l l accuracy of the method. 44 Figure 2-10 E f f e c t of i n i t i a l conditions and Hamitonian on the l i m i t i n g i n t e g r a l manifolds (Ki=0.5): (a) regular, C =-1.0 / (b) islan d type, C = -1.0 2.6 S t a b i l i t y Plots The r e s u l t s displayed i n Figure 2-10 may be presented in a more informative manner, p a r t i c u l a r l y , for design pur-poses. If ip and cf> are held fix e d , a constant value of G„ H describes, i n ijj'-tj)1 plane, an e l l i p s e , which degenerates into a c i r c l e for 0=0. For CH<_-1, i . e . , when the s t a b i l i t y i s guaranteed, the solution i s stable over the entire e l l i p s e . However, beyond t h i s , a bounded motion i s possible only over varying arcs of constant e l l i p s e s , corresponding to the l i m i t i n g invariant surfaces. Figures 2-11 and 2-12 show the allowable impulsive disturbances for non-tumbling motion for several values of i n e r t i a parameters and angular disturbances. The computational e f f o r t involved i n obtaining these plots i s enormous. Fortunately the symmetry properties discussed, e a r l i e r , which also make the plots of the system with 0 Q X 4>o=0 symmetrical about 4>'=0 axis, keep the analysis manageable. The figures, which also include for comparison the 53 results for dumbbell s a t e l l i t e s obtained by Brereton , show better s t a b i l i t y c h a r a c t e r i s t i c s for slender s a t e l l i t e s . I t i s observed that most s a t e l l i t e s can stand larger disturbance across the o r b i t a l plane compared to that i n the plane of the o r b i t . Furthermore, the a b i l i t y of a s a t e l l i t e to with-stand larger negative ' disturbance for given ty0A0t$Q i s of i n t e r e s t . In general, the s t a b i l i t y bound diminishes with increasing angular disturbances. The peculiar shape Figure 2-11 Design plots showing allowable impulsive disturbance for stable motion: (a) 0=0 Figure 2-12 E f f e c t of s a t e l l i t e i n e r t i a on allowable impulsive disturbance for stable motion of these curves may be attributed to the coupled non-linear nature of governing equations which also give r i s e to a few exceptions to the findings mentioned above. 2.7 Concluding Remarks The s i g n i f i c a n t aspects of the analysis may be summar-ized as follows: (i) I n e r t i a parameter plays an important role i n the response and s t a b i l i t y c h a r a c t e r i s t i c s of a s a t e l -l i t e . Slender s a t e l l i t e s (large K^) are l i k e l y to exhibit better s t a b i l i t y , ( i i ) Zero-velocity curves and motion envelopes can be u t i l i z e d p r o f i t a b l y to i d e n t i f y regions of possible motion. They also provide information concerning conditional and guaranteed s t a b i l i t y . For i n i t i a l conditions leading to the Hamiltonian s a t i s f y i n g the inequality -(l+3Ki)<_ C R <_ -1 ; the s t a b i l i t y of r e s u l t i n g l i b r a t i o n a l motion i s assured, ( i i i ) The analysis suggests conditional s t a b i l i t y for -1< C < C„ =0.8. The system i s l i k e l y to show cr better performance i n <J> degree of freedom. The actual character of the motion i s governed by the i n i t i a l conditions, (iv) The approximate closed form solution through the Butenin's approach can determine the l i b r a t i o n a l 51 frequency and amplitude quite accurately, e s p e c i a l l y for slender s a t e l l i t e s , even with disturbances of appreciable magnitude. A small phase discrepancy, cumulative i n time, causes only a circumferential s h i f t i n the stroboscopic phase-plane. The method can provide considerable insight into the system behaviour and gives results suitable for preliminary design purposes, (v) The constant Hamiltonian can be used to reduce the order of the system and leads to yet another a n a l y t i c a l solution, which, i n general, gives better approximation, p a r t i c u l a r l y when the motion across the o r b i t i s small, (vi) Both the solutions f a i l to predict the coupling e f f e c t s , which increase with increasing disturbances and decreasing (v i i ) The l i b r a t i o n a l and o r b i t a l frequencies are of the same order of magnitude. An i d e n t i c a l disturbance in and <j> excites higher frequency, smaller ampli-tude motion i n the o r b i t a l plane than that across f t . ( v i i i ) The system exhibits three d i s t i n c t l y d i f f e r e n t solutions: regular, island type and ergodic. However, from p r a c t i c a l considerations only the regular solution provides usable bounds for stable motion. The concept of in t e g r a l manifold used 52 here for s t a b i l i t y study gives, for given i n e r t i a parameter and Hamiltonian, a l l possible combin-ations of external disturbances, to which a s a t e l l i t e can be subjected, without causing i t to tumble, (ix) For a given Hamiltonian the s t a b i l i t y bound i s represented by the l i m i t i n g i n t e g r a l manifold. On the other hand, degeneration of the invariant sur-face to a l i n e corresponds to a periodic solution of the problem. Thus periodic solutions may be thought of as spines around which i n t e g r a l manifolds are b u i l t , (x) The plots of allowable impulsive disturbances should prove useful during s a t e l l i t e design. The symmetry properties considerably extend t h e i r range of application. 3. EFFECT OF AERODYNAMIC TORQUE ON SYSTEM RESPONSE AND STABILITY 3.1 Preliminary Remarks Presence of various perturbing forces necessarily complicates the problem under study. More s i g n i f i c a n t of 20 these, for close-earth s a t e l l i t e s , are the aerodynamic forces , which may be e f f e c t i v e even at four to f i v e hundred miles a l t i t u d e . This chapter investigates the e f f e c t of aerodynamic moment on the coupled l i b r a t i o n a l motion of the c y l i n d r i c a l s a t e l l i t e negotiating a c i r c u l a r t r a j e c t o r y . In the beginning the stable equilibrium configurations are established through Routh's c r i t e r i a as well as Liapunov's d i r e c t method. The regions of guaranteed and conditionally stable motion are given as functions of i n e r t i a parameter, Hamiltonian and aerodynamic torque. The numerically determined response to a, variety of disturbances helps i n establishing the influence of system parameters. The concept of i n t e g r a l manifolds again proves useful i n analyzing the character of stable t r a j e c -t o r i e s and to obtain the s t a b i l i t y bounds. The design p l o t s , in d i c a t i n g allowable impulsive disturbances for s t a b i l i t y , reveal the adverse e f f e c t of atmosphere. In view of the computational cost involved, an alternate economical approach using analog simulation, 54 normally used i n real-time studies, i s attempted. A compar-ison of the response and s t a b i l i t y data with the numerical results establishes the s u i t a b i l i t y of the method. 3.2 Formulation of the Problem 3.2.1 Aerodynamic Torque Consider a r i g i d , axi-symmetric s a t e l l i t e , with mass-centre at S, executing coupled l i b r a t i o n a l motion while moving i n a c i r c u l a r o r b i t about the centre of a t t r a c t i o n 0 (Figure 3-1). As before, x, y, z are p r i n c i p a l body axes of the s a t e l l i t e , whose orientation i s s p e c i f i e d by modified Eulerian rotations: i n the o r b i t a l plane; (j) across the o r b i t a l plane; and X about the axis of symmetry. The s a t e l -l i t e i s subjected to g r a v i t a t i o n a l and aerodynamic torques, which are evaluated using the following simplifying but r e a l i s t i c assumptions: (i) g r a v i t a t i o n a l potential can very c l o s e l y be approximated by a truncated se r i e s ; ( i i ) a i r density p and gravity f i e l d are functions of height only. Variations of p over s a t e l l i t e dimensions are ignored; ( i i i ) r e l a t i v e v e l o c i t y of the s a t e l l i t e with respect to the surrounding atmosphere i s taken to be the same as s a t e l l i t e ' s o r b i t a l v e l o c i t y , i . e . , i t s v a r i a t i o n due to the s a t e l l i t e ' s l i b r a t i o n a l motion and atmospheric rotation are neglected; (iv) ambient condition i s represented by free molecular flow; (v) C"D, based on projected area and usually a very complex function, i s taken to be a constant; (vi) centre of pressure i s assumed to be coincident with the geometrical centre of the s a t e l l i t e and small changes i n i t s position due to l i b r a t i o n a l motion are ignored. At altitudes of about 100 miles and over, the mean free path of molecular motion i s large i n r e l a t i o n to t y p i c a l s a t e l l i t e dimensions, and the flow regime i s c l a s s i f i e d as free molecular flow. Surface forces i n t h i s regime are due to molecular impingement on i t s surface and th e i r subsequent re-emission. For convex bodies, the aerodynamic force at a point on the surface of a s a t e l l i t e at s p e c i f i e d conditions i s a function only of the angle between the impinging stream of molecules and the surface. Consequently, i t i s possible to integrate the surface force over the fr o n t a l area of the s a t e l l i t e , y i e l d i n g the net l i f t , drag and moment on the ; s a t e l l i t e i n terms of attitude angles I|J,4>,A. For l i b r a t i o n a l motion only the aerodynamic moment i s of i n t e r e s t , slow decay of the o r b i t because of drag being neglected. • 6 6 Using the methods of Schaaf and Chambre the normal pressure and shear stress on a s a t e l l i t e surface exposed to, free molecular flow can be expressed as P = Cs v ^ i s f ) L"{$-«-') St/ft 4- r ' ( T w / T Q J V 4 e O ( Vx. + * ) + r ' s ; ( J T T W / T 0) V V ^ 5 ( i 4- erf (s; ) j | . . . . (3.1) T - - ( S v ^ c r c o s ^ S 0 ^ ) [ e ^ V # S 0 / ( l - r e r f ( 5 0 / ) ) ] • • • • (3*2} The moment on the body i s given by V\Q• - ( I X ( P + f ) d A . . . . (3.3a) A which, using symmetry properties of the s a t e l l i t e and i n t e r -mediate body coordinates x 2 y 2 z 2 , can be written as Ho = U * J . L\ {\ + \ ) - ^ ( \ ) J d A (3.3b) Here u represents a unit vector along x„ axis. The evalu-ation of the i n t e g r a l , which, i n general, can be achieved only numerically, involves substantial computational e f f o r t s . However, for c y l i n d r i c a l s a t e l l i t e s , i t can be approximated, 36 6 7 quite accurately, i n a closed form by the expression ' M q = - 0 - 5 ? v * c D € cosl|/[D eL 0|cosi|i| - f T r ^ s i n l | / / 4 j . . . .(3.3c) The absolute sign ensures a p p l i c a b i l i t y 'of the expression for a l l values of ^. 32,34,67,68,et a l Variations i n p,v,CD,e being usually small, they are assumed to remain constant for a given 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 . The analysis, however, 58 can be extended to accommodate t h e i r variations without too much d i f f i c u l t y . Thus aerodynamic moment i s a function of l i b r a t i o n a l angle ip and not of angular v e l o c i t i e s . Consequently, a potential function can be derived that i s equal to the negative of the work required to move the s a t e l l i t e from a reference orientation (0,0,0) to the orientation given by For a symmetric s a t e l l i t e i t i s = ? Cj> 6 AL 0 D|/ + Sin HI (cos IJi +(jrDa/4Lo)S\"V}] = (IQZ^/Z)i^ + sin q i ( c o s q J - f C i S i n i U ) ] ( f o r mm) . . . . (3.4) where the dimensionless aerodynamic c o e f f i c i e n t , B^, i s repre-sented by B F = c D ^ DO L 0 z/zie* • • • • <3-5) and q - . W D . / 4 L , = 7r{(i-K ()/3(l+K0} i /7^ • • ( 3 . 6 ) 3.2.2 Lagrangian and Equations of Motion As the expressions of k i n e t i c and poten t i a l energies due to gravity-gradient (equations 2.2) remain unchanged, the Lagrangian becomes L - T - ( Ug + U Q ) = m(r*+r*e*)A + I 0 * 4 - (O+fyf cos* <t)]/Z - M ( i - I « ) ( i - 3cos8-Oi c o s H ) A r 3 + f nQ d<|/ • • • • »-7> 0 As before, ignoring the o r b i t a l perturbations due to l i b r a t i o n a l motion and using 6 as the independent va r i a b l e , the equations of motion become: iy• - Z <t>' (q/' + i) tan (f> -4-3K] sin cosi^ • + B f (I cos l|i | + Ci sm lp) cos-ip/cos* cb - 0 4>"+- '{(lU'+if 4- 3Kj cos* cp | sin (f) cos4> =0 • • • . (3*8) i t should be noted that the equations governing the <j) as well as X (relation 2.1c) degrees of freedom are unaffected by the presence of aerodynamic moment. Even for t h i s s i m p l i f i e d s i t u a t i o n the second order, non-linear, coupled equations of motion have no known closed form solution. A numerical technique under t h i s condition can again be used to advantage. However, considerable use-f u l information concerning equilibrium positions, regions of possible motion, zones of conditional and guaranteed 6 0 s t a b i l i t y , e t c . , c a n b e o b t a i n e d w i t h o u t e v e n s o l v i n g t h e e q u a t i o n s . T h i s p r o v i d e s s o m e i n s i g h t i n t o t h e b e h a v i o u r o f t h e s y s t e m w i t h o u t i n v o l v i n g a p p r e c i a b l 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 . 3 . 2 . 3 H a m i l t o n i a n S i n c e t h e l i b r a t i o n a l m o t i o n i s s t i l l c o n s e r v a t i v e a n d a s t h e L a g r a n g i a n o f t h e s y s t e m d o e s n o t i n v o l v e t i m e e x p l i c i t l y t h e H a m i l t o n i a n r e p r e s e n t s a c o n s t a n t o f t h e s y s t e m , i . e . , - c o n s t a n t ( 3 . 9 ) o r , i n a n o n - d i m e n s i o n a l f o r m : C H = 2 , H / i e * c o n s t a n t (-for | l | i |<JTA) ( 3 . 1 0 ) 3.3 Dynamic E q u i l i b r i a and S t a b i l i t y i n the Small 3.3.1 Equilibrium Positions At equilibrium p o s i t i o n , the potential i s an extremum, hence = 0 e eu/dcL. COS l)J[3 Kj cos* £ s\r\\\) +-ty\costy\ f %Cj.Sinl|iJ = 0 • S i n Z<\> [ i 4- 3K( Cos* l | /J = 0 . . . . (3.11) In i n f i n i t e number of solutions are possible. However, i f the s t a b i l i t y of l i b r a t i o n a l motion i s defined such that the l i b r a t i o n a l angles remain within ±TT/2 from the l o c a l v e r t i c a l (non-tumbling motion), only the following nine equilibrium positions are of i n t e r e s t : = ± TT/Z , o ; = ±ir/z = t a n ^ - i / C i ) , 4 = = t a n i ( - B f / ( 3 K l + B f (3.12) S t a b i l i t y of these equilibrium positions can be tested by i n f i n i t e s i m a l technique or Liapunov's d i r e c t method without having to solve for the perturbed motion about an equilibrium p o s i t i o n . 62 3.3.2 In f i n i t e s i m a l Technique (Routh) Linearized perturbation equations for motion about an equilibrium configuration can be written as o~ljj" - d V (2<Ktan(t>J + 5l|J [3K,cosm-{an2| •C. cos bf/cos^e]-8^{ltm^H)yS^4),o (3.13a) + Scf) [ COS %,§>e (C + 4- 3K| C03 l^jje)} 4- [sin 4-1) 4- 3 s«n^l|4 s in ^ ( t u f t U 0 (3.13b) For the solution of the form 8 = § % e A l t the equations give - 3 K | sin 2, iye sin = 6 > Q e X | t [-^>>i((|4+i)tQn(j)e K + cos2,<t)e{(itJe/+ L*4>. 0 0 (3.14) 63 leading to the c h a r a c t e r i s t i c equation A A* 4-B A3i + CX\ + DXi +-E =0 • • • • (3-is) where A =1 E> = - 2 tan (j)e C = 3 Kj cos 2 iye - B f [sin 2iJJe - qcos 2(|4}/cos^ e + CO5 Z<(>e[C+ i f 4-3K ( CO5*H)e) 4-4(lU^+ifsin^ D = - ^ t Q n ( ( ) e cos^ 4 ) e { ( ^ 4 - l f 4- 3K,- cos*(|4] - 3 K , sin Ztye Sin£(f)etan ( ^ ( l ^ + i ) E = C O S £<t> e[(li> e' f i f + 3 K ' l COS^i ) 4]{3K f C O S ^ l p e - B f (Sin 2lp e - C d cos ^^ e)/co5*c() ej € * e s , n ^ e S ' , n The application of the s t a b i l i t y conditions A , E> , C, B , E > 0 (BC - AD) D - B5 1 E > 0 64 showed a l l the equilibrium positions to be unstable except the last.one, i . e . , ljJ£ .= tan 1 [- B f/(3Kj Cj.)j , (j>e = 0 (3.16) 3.3.3 Liapunov's Direct Method A simpler and more useful approach to the s t a b i l i t y study of an equilibrium position i s by Liapunov's second or di r e c t method. Using Hamiltonian as a Liapunov's function \ - C H - C : $ + cos*4) ((P'*" - 1 - 3K( cos*-tJJ) -H3 f(iy + sin iKcos ip+qs\n i W - f C o s ^ f i +3K-, e o s ^ e ) ] - B f [iye+sin ipe(cos^L4- q s w (jpj . . . . (3.17) Therefore, from equation (3.8) . . . (3.18) Thus the system i s p l a i n stable i f V T > 0. It can be shown that for V L >^  0, > 0 6 5 i.e d \ / c > q j d ( f ) " ^ 0 . . . (3.19) This implies a minimum value for pote n t i a l energy at the equilibrium position. For the matrix to,be p o s i t i v e d e f i n i t e 2 c o s I 1)4 ( 3K- , C o s * (|>e + B f q ) - Z B^ s\n 2,l|Je £ 0 2 C o s £c|)e ( 1 + 3 K - , c o s ^ 4 J e ) ^ 0 - 3K-, Sin 2,cj) Sin t VP >0 (3.20) Testing the equilibrium positions l i s t e d e a r l i e r (equation 3.12), confirms the conclusions of\the i n f i n i t e s -imal analysis. The 'plain s t a b l e 1 equilibrium position i s a function of i n e r t i a parameter as well as aerodynamic c o e f f i c i e n t . Figure 3-2 shows the stable equilibrium configuration as a function of aerodynamic c o e f f i c i e n t for four^representative s a t e l l i t e s . It i s apparent that the equilibrium position rapidly changes with aerodynamic coefficient,;, p a r t i c u l a r l y i n the range of small B f and K^. However, beyond Bf= 4.75 the trend reverses and most s a t e l l i t e s tend to a t t a i n a uniform attitude around 50 - 60 deg. \ gure 3-2 Variation of stable equilibrium position due to aerodynamic torque and s a t e l l i t e i n e r t i a 67 3 .4 Bounds of Librations Hamiltonian, being a constant of the motion, can be used quite e f f e c t i v e l y to study the general behaviour of the system through zero v e l o c i t y curves defined by CH =-cos*4)(i+3K]COS*(JJ)+ B f ( v + s i n t p C c o s l / y f C i S i n . . . . (3.21) Figure 3-3 presents these plots for various values of Hamiltonian, i n e r t i a parameter and aerodynamic c o e f f i c i e n t s . The curves, enclosing the regions of r e a l v e l o c i t i e s , repre-sent the bounds ...of l i b r a t i o n a l motion. It may be observed that f o r : W C H e = - i - 3 K j H - ^ t a n 1 t - B f / ( 3 K i + - B f C 1 ^ C H t no motion i s possible; I. C H E < C H 4 - i - B f (rr/z - CO , the motion i s bounded; ( i i i ) - i - B f (TT/Z - q ) < c H < -^(n/z - c i ) > the motion i s bounded i n 0 only; unbounded motion i s possible i n both d i r e c t i o n s . . . . . (3.22) -90 0 90-90 0 90 Figure 3-3 E f f e c t of s a t e l l i t e i n e r t i a , Hamiltonian and aero-dynamic torque on zero-velocity curves The presence of aerodynamic moment destroys the symmetry of the curves about cj) axis by s h i f t i n g them towards -ijj d i r e c t i o n . A study of the largest region enclosed by the bounded curves suggests that although the slender s a t e l l i t e s , i n general, exhibit better s t a b i l i t y character-i s t i c s , t h e i r performance degenerates sub s t a n t i a l l y when subjected to appreciable aerodynamic torque. The plots o also indicate the s a t e l l i t e ' s increased s u s c e p t i b i l i t y to planar disturbances. An increase i n aerodynamic c o e f f i c i e n t reduces the value of the l i m i t i n g Hamiltonian for guaranteed s t a b i l i t y . Figure 3 - 4 shows these regions as functions of Hamiltonian and aerodynamic c o e f f i c i e n t s for several values of i n e r t i a parameter. It i s apparent that slender s a t e l l i t e s are l i k e l y to exhibit better s t a b i l i t y c h a r a c t e r i s t i c s for small aerodynamic moment, however, larger i s expected to reverse t h i s trend. The motion envelope i n ty,ty,<j> -space i s defined by CH -cosz<i>(i\)'Z- i - 3 K , cos* IP) + Sin(jy(cos^ -T-Cj_sin \\) )] =0 . . . . ( 3 . 2 3 ) In Figure 3 - 5 a t y p i c a l motion envelope shows that the aero-dynamic torque causes a breakdown of the symmetry about ijj=0 plane and increases the p o s s i b i l i t y of i n s t a b i l i t y . Having established the equilibrium positions, t h e i r s t a b i l i t y , and the bounds of l i b r a t i o n s , the next l o g i c a l Figure 3-4 Regions of bounded motion Figure 3-5 Influence of aerodynamic torque on motion envelope step would be to investigate the system response and s t a b i l i t y . 3.5 Numerical Solution The equations of motion (equation 3.8) can be written as a set of four f i r s t order r e l a t i o n s : dv/de = V ' ; dd)/de = & dV'/dQ = Z<bXy'+i)tQr\$ - 3K-, Sini|Jcosl|J - B f (cos V + q sin qJ) c o s i j j / c o s ^ d(f)Vd0 = - {((jZ-fl)*- + 3K,- cos^q;} cos<|>sin<f) . . . .(3.24) Adams-Bashforth predictor-corrector integration pro-cedure was used for numerical solution of these equations. Note that the system exhibits an invariant character only under the transformation (Q ,ip to (B ,ip , - $ ) . Figures 3-6 and 3-7 show the response charts thus obtained for a set of s a t e l l i t e s subjected to systematically varying aerodynamic c o e f f i c i e n t and i n i t i a l disturbances. A step size of 3° was chosen for integration. The solution involving four dependent variables i>,i>1, 0 and cj)1 defines a trajectory i n a four-dimensional Figure 3 - 6 Effects of aerodynamic torque and i n e r t i a parameter on the response of s a t e l l i t e to an impulsive disturbance K; =1.0 30°F • 0 - 3 0 0 Bf = 0 0 ; 0.2 ; 1.0 . ' A N ' \ _ - ^  ^  ,\ !\ / , / \ / \ I \ i \ Figure 3-7 Response of a s a t e l l i t e to a disturbance i n one degree of freedom i n presence of aerodynamic torque: (a) IJJ =15°, ^'=0.25, <(> =<j>'=0; (b) ^o=^;=0, <J>0=15°, *;=0.25 o o —i phase-space. The invariant Hamiltonian (equation 3-10) permits determination of any one of the variables i n terms of the other three. Hence i t i s possible to present the solution as a trajectory traced by the representative point in a three dimensional phase space. For example the solution i s completely defined i n i(>,ijj',(j> - space for cf>'^  0. The procedure for generating an i n t e g r a l manifold was discussed in Chapter 2. Figure 3-8 shows the cross-section of such a l i m i t i n g surface as affected by the aerodynamic c o e f f i c i e n t . The sections of motion envelopes are also included for comparison. A comment concerning the influence of i n i t i a l con-diti o n s on the nature of the solution and hence on the associated invariant surface would be appropriate here. As before, for C^ = -1.5, Figure 3-8 shows regular invariant surface cross-section. However, for the same value of C„ n but d i f f e r e n t i n i t i a l conditions, Figure 3-9 shows formation of six islands surrounding the main regular region. Further variations i n the i n i t i a l conditions r e s u l t i n complete breakdown of the invariant surface, as shown by ergodic solution, followed by reformation of the second set of islands. Note that,throughout,the l i b r a t i o n a l response of the s a t e l l i t e i s bounded and hence stable i n accordance with the stated c r i t e r i o n . Figure 3-10 shows representative sections of l i m i t -ing invariant surfaces and motion envelopes for a wide Figure 3-8 Representative cross-sections of motion envelopes and l i m i t i n g i n t e g r a l manifolds i n d i c a t i n g i n f l u -ence of aerodynamic torque Figure 3-9 The cross-section 0=0 i n phase space i n d i c a t i n g types of stable solution generated by d i f f e r e n t i n i t i a l conditions for given Hamiltonian Figure 3-10 E f f e c t of i n e r t i a parameter on motion envelope and l i m i t i n g i n t e g r a l manifolds for given aero-dynamic moment and Hamiltonian 79 range of s a t e l l i t e i n e r t i a parameter. The eff e c t s of aero-dynamic moment and Hamiltonian on l i m i t i n g i n t e g r a l manifolds themselves are shown i n Figures 3 - 1 1 and 3 - 1 2 , respectively. It i s apparent that increase i n and C H a f f e c t the region of s t a b i l i t y adversely. As explained i n Section 2 . 6 , for a given Hamiltonian, the plot of vs. i))1 i s an e l l i p s e reducing to a c i r c l e for 4 > E = 0 . In the region where the s t a b i l i t y i s assured the solution i s bounded over the entire e l l i p s e . However, for C R > - 1 — B ^ (TT/2-C^) stable motion occurs only over a portion of the constant C „ e l l i p s e s . Figure 3 - 1 3 shows the influence n of i n e r t i a parameter and aerodynamic c o e f f i c i e n t on the allowable impulse for s t a b i l i t y . The symmetry of the plots about cj>'=0 axis i s retained which helps i n reducing rather extensive computations. 3 . 6 Discussion of Results In general, response of a s a t e l l i t e depends on i t s physical properties, aerodynamic moment and disturbances encountered. For the given i n e r t i a parameter and disturbance, the amplitude of motion i n the o r b i t a l plane and sharpness of the peaks i n the di r e c t i o n increases with increasing B ^ , however, motion across the o r b i t a l plane i s r e l a t i v e l y unaffected. So far as the e f f e c t of i n e r t i a i s concerned, 80 8 4 decrease i n tends to make a s a t e l l i t e more sensitive to a given disturbance and the frequency of response shows marked reduction (Figure 3 - 6 ) . Figure 3 - 7(a) shows motion across the o r b i t a l plane to be r e l a t i v e l y less affected by a transverse disturbance even i n presence of an aerodynamic moment. However, a disturbance across the o r b i t a l plane excites appreciable in-plane motion, which grows with increasing B f (Figure 3 - 7 (b)). A larger amplitude, smaller frequency motion i s observed i n the o r b i t a l plane than that across i t for an i d e n t i c a l disturbance i n the two degrees of freedom. The frequencies are of the order of o r b i t a l frequency. As before, the stable solutions of the system may lead to three d i s t i n c t classes of t r a j e c t o r i e s i n the phase space (Figures 3 - 8 , 3 - 9 ) referred to as 'regular 1, 'island type', or 'ergodic'. However, the aerodynamic moment destroys the symmetry of manifold cross-sections. Although tthe solution in each case represents stable motion, island and ergodic type of behaviour, being of l i t t l e use from p r a c t i c a l design considerations, lead to substantial reduction i n s t a b i l i t y . It i s in t e r e s t i n g to note that the p o s s i b i l i t y of ergodicity diminishes with decreasing , as region between in t e g r a l manifold and motion envelope i s reduced (Figure 3 - 1 0 ) . The i n t e g r a l manifolds as well as motion envelopes shrink i n size with increasing aerodynamic moment and Hamil-tonian. Thus there i s a l i m i t i n g value of C , dependent upon 85 B^, beyond which stable motion i s not possible. A convenient condensation of the r e s u l t s for design purposes (Figure 3-13) shows bounds of impulsive disturbances at equilibrium position for several d i f f e r e n t s a t e l l i t e s under a set of aerodynamic moments. I t i s apparent that the presence of B^, i n general, decreases the bound for stable motion, reduction being more pronounced for slender s a t e l l i t e s . The s a t e l l i t e s with large show better s t a b i l i t y character-i s t i c s when no or small aerodynamic moment i s present. How-ever, for large B^, shorter s a t e l l i t e s (small K^) are l i k e l y to have better performance. The results presented here are confined only to a few s i t u a t i o n s . The numerical approach, though informative, tends to be quite expensive. Hence the p o s s i b i l i t y of using the analog technique, normally preferred for economic,real-time simulation, i s explored. 3.7 Analog Simulation , Using trignometric i d e n t i t i e s the equations of motion (for |^|_<Tr/2) can be rewritten as: l)j" = [ztf (V'±L)s\nZ$) - B f (1+ cos Z V 4- q sinW}/(i 4- COS 2<$) ) - 3K| sin $ = -{(ip'+if 4 - 3 K j ( l - r - c o s £ W / 2 } s i n . . . . (3.25) 86 With 8, as the independent variable of the analog 69 computer the equations were programmed by the general method . The trignometric functions of dependent variables were generated e x p l i c i t l y using generalized integration technique, i . e . , . . . . (3.26) Figure 3-14 shows a schematic of the simulation c i r c u i t used i n conjunction with the analog computer PACE 231-R5. The computer has a reference voltage of ±100 V. Characteris-t i c s of the m u l t i p l i e r s and the divider suggested the need for suitable amplitude sc a l i n g . The r e l a t i v e l y large values of cj> often made the output of the divider grow rapidl y . However, i t can be shown that, i n general, for | <f> | > 75° the s a t e l l i t e s become unstable i n i\>. Using known maximum values of the variables, the scaling factors K, and a were adjusted (e.g., K = = 25, a = 0.25) to arrive at a balanced, well-scaled c i r c u i t . The simulation was about 1000 times faster than the actual system (1 second =. 1 radian) . Further improvement i n the speed can be accomplished by suitably adjusting the . integrators' gains. However, any attempt at speeding-up the process beyond a factor of 5,000 showed res u l t s to be unstable through accumulated error. 87 Figure 3-14 Analog simulation c i r c u i t using 6 as the independent variable 8 8 Any combination of disturbances can be provided by setting the i n i t i a l conditions of the integrators, as i n -dicated. The i n e r t i a parameter and aerodynamic c o e f f i c -ient can be varied by changing the corresponding potentiometer s e t t i n g . In absence of an atmosphere the same c i r c u i t may be used by disconnecting the dotted block i n Figure 3-14. ; 3.7.1 Accuracy of Simulation The l i b r a t i o n a l response of a wide range of s a t e l l i t e s under a variety of atmospheric conditions and disturbances was examined by systematically varying the i n e r t i a parameter, aerodynamic c o e f f i c i e n t and i n i t i a l conditions. The planar as well as the out of plane l i b r a t i o n s were recorded as a function of the s a t e l l i t e ' s o r b i t a l p o sition on a x-t p l o t t e r . In a l l cases studied, the r e s u l t s agreed well with numerical solution. Figure 3-15 shows a t y p i c a l plot of s a t e l l i t e ' s response to impulsive disturbance as a function of i n e r t i a parameter and aerodynamic c o e f f i c i e n t . The analysis confirms the e a r l i e r findings and suggests the s u i t a b i l i t y of the simulation for quantitative investigation. To es t a b l i s h the accuracy of the method for s t a b i l i t y studies the analog solutions were compared with the d i g i t a l r e s u l t s . Plots of allowable impulsive disturbances, at equilibrium p o s i t i o n , for non-tumbling motion were found 89 0 I n l i l i i i i i i i i t i i i l i l i l l 20 - 2 0 i — i — i — i — h 20? - 2 0 - i 4 i i iiiiiiiiiiiiiiiiiiiiiiiiiimiiiii 8 O R B I T S 0 8 f I aft i <i =1.0 0 4 8 iMftOhttiiVihiiiliMii'mi'iffriiHiimiiti'iMii'iM 201 - 2 0 ? O 4> - 2 0 -20? 1111111111111 20 - 2 0 l l l l l l l l l l l i l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 1 i H H — I 1 1 — I — h -20S K j=0 .75 0 4 8 ,l tyn^ tT) 1111 HI n 111111 ill 11 n i ] i n 111 ] 11 • i, i ri t I'I 11 K:=0.25 Figure 3-15 Representative response plots obtained using analog simpulation; = $ q = 0 , ^=^=0.5: (a) i n absence of aerodynamic torque 90 |iiii|hiVjYif<ji>ii|iw^'i'iijirti|fmpirr[)rnl 8 ORBITS 9 - 2 0 --20r - 4 0 ^ 40-o l - 4 0 -Kj =0.5 : B f =0.2 80^ - 8 0 -K; =0.5 ; Bf =1.0 Figure 3-15 Representative response plots obtained using analog simulation; ^ 0 = ()> 0= 0' ^ = 4 ^ = 0 . 5: (b) i n presence of aerodynamic torque 91 for several combinations of and . Due to absence of any inherent l o g i c and memory i n the analog computer, point by point checking was necessary for generating these plots. Their symmetry about axis, however, helped i n reducing the e f f o r t considerably. The impulsive disturbance at equilibrium position was varied systematically and the s a t e l l i t e response was observed over 50 o r b i t s . The computer has a safety c h a r a c t e r i s t i c of going into hold mode i n case of overload. This feature may be u t i l i z e d to serve as a log i c unit. The outputs \p and <p were scaled i n such a manner as to overload the computer as soon as tumbling occurred. The feature also proved useful i n l i m i t i n g the integration to 50 o r b i t s . Figure 3-16 compares, for several representative situations, the d i g i t a l and analog simulation r e s u l t s . In general, the agreement appears to be acceptable except for minor discrepancies i n the v i c i n i t y of "spikes" and "islands." Fortunately, t h i s does not appear to be c r i t i c a l as t h e i r proximity to the s t a b i l i t y bound would, i n any case, be con-sidered unsuitable for s a t e l l i t e operation. In spite of the inherent l i m i t a t i o n s of an analog computer, the simulation presented here i s economical and s u f f i c i e n t l y accurate for quantitative analysis. Its usefulness i n , s a t e l l i t e design and r e a l time studies could be enhanced considerably through hybridization with the d i g i t a l computer to take advantage of l a t t e r ' s memory and l o g i c . Figure 3-16 Allowable impulsive disturbances at equilibrium positions for stable motion - a comparison between numerical and analog results 93 3.8 Concluding Remarks The important aspects of the investigation and relevant conclusions may be summarized as follows: (i) The analysis presented here, involving several simplifying but r e a l i s t i c assumptions, can be applied readily to actual systems with s u f f i c i e n t accuracy. It gives a complete picture concerning response and s t a b i l i t y of s a t e l l i t e s under the influence of aerodynamic moment, which cannot be ignored for near earth operation, ( i i ) The l o c a l v e r t i c a l i s no longer the equilibrium p o s i t i o n i n the presence of an aerodynamic moment. The stable equilibrium orientations, found using i n f i n i t e s i m a l technique as well as Liapunov's d i r e c t method, and bounds of l i b r a t i o n s obtained through the Hamiltonian of the system, are strongly affected by i n e r t i a c h a r a c t e r i s t i c s and aerodynamic moments. ( i i i ) The amplitude of motion, esp e c i a l l y i n the o r b i t a l plane, increases considerably with increasing B^. The e f f e c t of disturbance to transverse motion i s more pronounced for the generalized co-ordinate i n the plane of the o r b i t , (iv) The system exhibits three, d i s t i n c t l y d i f f e r e n t , stable t r a j e c t o r i e s : regular, islands, and ergodic. However, from p r a c t i c a l considerations, 94 only the regular solution provides bounds for stable motion, (v) The reduction i n size of i n t e g r a l manifolds with increasing and C H suggests a c r i t i c a l value of Hamiltonian for stable motion, (vi) Plots of allowable impulsive disturbances, which a s a t e l l i t e at equilibrium can sustain without tumbling, show s a t e l l i t e s with large i n e r t i a to be r e l a t i v e l y more stable at higher a l t i t u d e s (small B^). However, shorter s a t e l l i t e s exhibit better s t a b i l i t y c h a r a c t e r i s t i c s i n the presence of a large aerodynamic moment, (vii ) The analog real-time simulation of the problem gives res u l t s of s u f f i c i e n t accuracy. The discrepancies with d i g i t a l results are confined to the regions which are of l i t t l e importance from design con-siderations. The analysis and results presented here should prove useful i n s t a b i l i t y and design considerations of near-earth s a t e l l i t e s . 4. REGULAR STABILITY AND PERIODIC SOLUTIONS 4.1 Preliminary Remarks For the axi-symmetric s a t e l l i t e s executing coupled l i b r a t i o n s i n c i r c u l a r o r b i t , i t was found that the stable conditions may not, i n a l l cases, lead to a well defined 'regular' surface i n phase space. In the region of guaranteed s t a b i l i t y , as indicated by closed zero-velocity curves, i t was also possible to obtain, a chain of 'islands' or 'ergodic' solutions i n the t r a n s i t i o n region. The same behaviour per-s i s t e d i n the presence of aerodynamic torque. The study of the l i b r a t i o n a l motion, governed by the nonlinear, coupled, autonomous equation (3.8) suggested that the largest regular surface represents the only usable stable region from design considerations. 8 9 Modi and Brereton ' studied periodic solutions associated with the planar gravity oriented systems. They • emphasized the importance of the solutions by pointing out the fact that, at the largest e c c e n t r i c i t y for s t a b i l i t y , the only possible motion i s a periodic one. A s i g n i f i c a n t r e l a t i o n s h i p between i n t e g r a l manifold and periodic solution becomes apparent. A succession of i n i t i a l conditions may be chosen to determine progressively smaller manifolds, which degenerate, i n the l i m i t , to a l i n e . Because of the periodic nature of the invariant surface, t h i s l i n e must, then, rep-96 resent a periodic solution. Hence the periodic t r a j e c t o r i e s must act as spines around which the manifolds are b u i l t . 17 Modi and Neilson extended the concept to an a x i -symmetric spinning s a t e l l i t e l i b r a t i n g i n presence of gravity gradient torques. I n i t i a l conditions for periodic solutions were presented over a range of system parameters for motion in c i r c u l a r and e l l i p t i c o r b i t s . V a r i a t i o n a l s t a b i l i t y of periodic solution was examined using extension of Floquet's 70 c r i t e r i o n to the fourth order system. The coupled l i b r a t i o n a l motion of an axi-symmetric s a t e l -l i t e i n the presence of aerodynamic torque i s investigated here with p a r t i c u l a r emphasis on the bound between regular and ergodic type of s t a b i l i t y . Transition of the periodic solution ?21' a s s ° c i a t e d with the regular s t a b i l i t y region, to P^ |- and P23' c o r r e s P o n d i n g to the chains of islands, i s studied through cross sections of the i n t e g r a l manifolds with a systematic v a r i a t i o n of disturbances. I n i t i a l con-dit i o n s for regular, stable, periodic motion are obtained over a range of i n e r t i a and aerodynamic parameters and the l i m i t i n g s t a b i l i t y conditions are established, p r e c i s e l y , using the Floquet analysis. Representative response data are also included to show the v a r i a t i o n of the associated period. F i n a l l y , a set of design p l o t s , i n d i c a t i n g region of regular s t a b i l i t y as a function of system parameters, are presented. 97 4.2 Analysis As pointed out before (Sections 2.5 and 3.5), the con-stant Hamiltonian (equations 2.7 and 3.10) makes i t possible to represent the stable motion concisely by an i n t e g r a l manifold i n ^ , ^ ' , 4 ) - space. Consider, for example, i t s cross-section at <j>=0 revealing three d i s t i n c t classes of stable solutions (Figure 4-1). It i s apparent that the most pre-dominant of these i s the well defined, nested, regular solution. Its degeneration to a point, achieved through appropriate choice of i n i t i a l conditions, would represent a periodic solution ^21' w n i c n acts as a spine of the manifold. On the other hand, an alternate set of disturbance, though l y i n g within the region of guaranteed s t a b i l i t y as suggested by the closed motion envelopes, may give r i s e to a chain of f i v e 'islands' surrounding the l i m i t i n g regular region. In the three dimensional phase-space they would appear as a h e l i c a l tubular surface around the largest regular manifold. Its degeneration to a h e l i x obviously represents another periodic solution The t h i r d type of stable solution, represented by apparently 'ergodic' character of the trajectory f i l l i n g the; t r a n s i t i o n zone between the regular s t a b i l i t y region and islands, may also involve p e r i o d i c i t y over a large number of o r b i t s . Subjecting the system to further v a r i a t i o n i n the external disturbances may r e s u l t i n the formation of new islands, corresponding to a d i f f e r e n t periodic solution 98 + Figure 4-1 Stroboscopic phase plane at $=Q i n d i c a t i n g types of stable solutions generated by various i n i t i a l conditions at given Hamiltonian: (a) i n absence of aerodynamic torque 9 9 1.5 1.0 0-5 0 f - 0 - 5 - 1 . 0 -1 .5 m o t i o n e n v e l o p e 4>'= o r e g u l a r l i m i t i n g reg j i s l a n d , i s l and2 e r g o d i c p e r i o d i c - 9 0 Figure 4-1 90 Stroboscopic phase plane at cj)=0 in d i c a t i n g types of stable solutions generated by various i n i t i a l conditions at given Hamiltonian: (b) i n presence of aerodynamic torque 100 (e.g., P23^* This process of formation of islands and t h e i r degeneration i s limited only by the approach of the motion envelope. Although stable, proximity of islands to the motion envelope and the i r r e g u l a r nature of the ergodic solutions render them of questionable value. Thus, the l i m i t i n g regular region remains the only p r a c t i c a l l y usable s t a b i l i t y bound. Rest of the discussion i s , therefore, confined to th i s region. 4.2.1 Limiting S t a b i l i t y and Periodic Solutions The determination of regular i n t e g r a l manifold was accomplished numerically. In general, the equation (3.8) was integrated over 40-50 orb i t s for a few representative disturbances within the motion envelope. For the case of regular s t a b i l i t y t h i s leads to a well defined cross-section i n the stroboscopic phase plane at 4>=0. The l i m i t i n g region of s t a b i l i t y was established by choosing a condition corres-ponding to the mid-point of the smallest intercept on ^=^e between the regular and other t r a j e c t o r i e s . The process was repeated, u n t i l the intercept approached zero. Usually 4-5 i t e r a t i o n s were found to be s u f f i c i e n t . So far as the periodic solution associated with the regular region i s concerned, i t was necessary to e s t a b l i s h i t s degeneration to a point. This was accomplished by s e l e c t i n g , successively, i n i t i a l conditions corresponding to 101 the mid-point of the intercept by the regular region on ty=tyQ. This variable secant i t e r a t i o n process converged quite r a p i d l y , leading to the desired periodic solution i n 2-3 cycles. The same technique can be applied to determine the l i m i t i n g s t a b i l i t y for island type t r a j e c t o r i e s and associated periodic solution. The period of the solutions was established through response analysis (Figure 4-2). 4.2.2 V a r i a t i o n a l S t a b i l i t y of Periodic Solutions As periodic solutions play an important role i n the. l i b r a t i o n a l dynamics of a s a t e l l i t e , i t was thought approp-r i a t e to explore the conditions for t h e i r s t a b i l i t y . However, i t should be emphasized that, although the cross-sectioning, concept i s r e l a t i v e l y simple and y i e l d s considerable insight into the nature of the motion i n the large, the numerical character of th i s approach involves a substantial amount of computer time. This i s p a r t i c u l a r l y true for a precise determination of the c r i t i c a l disturbance beyond which even the periodic - solutions show i n s t a b i l i t y ( i . e . , the i n t e g r a l manifolds cease to e x i s t ) . A need for v a r i a t i o n a l s t a b i l i t y analysis of periodic solutions i s , therefore, quite apparent. Substituting ijj=ij)p+^v, 4>=<|>p+<j>v i n equation (3.8) and l i n e a r i z i n g with respect to I|J and <$>^  y i e l d s : = + %ty f B 3 ^ 4 - B 4 ^ . . . .-<4.i) Figure 4-2 Periodic response: (a) i n absence of aerodynamic torque Figure 4-2 Periodic response: (b) i n presence of aerodynamic torque 104 where A i = Z<$L tan (J) = fc(lp;+i)tan<t>p A 4 = [ ^ ( H i p + 1 ) ~ B f tan (|)p(i4- Cos2(|jp 4- q s i n i ^ ] / o ) ^ p B i = - ( q j ; + O s i n S l ( | ) p 0 B3= 3K| £({J p^in &(|)p/£ B4= "EC + i f + 3 Ki Cos* q; pJ c o s This fourth order l i n e a r system has p e r i o d i c a l l y varying c o e f f i c i e n t s of a common period, say T p . Thus, Floquet theory i s applicable. The s t a b i l i t y c r i t e r i o n can be expressed as: ^ ]_ i = 1,2,3,4; stable ' . . .(4.2) y i = 1,2,3,4; unstable The system being autonomous, one of the c h a r a c t e r i s t i c multi-p l i e r s i s unity as the derivative of the periodic solution s a t i s f i e s the v a r i a t i o n a l equation. Existence of constant Hamiltonian, a f i r s t i n t e g r a l of motion, makes another multi-p l i e r unity. For the system having these properties along : with the invariant nature of phase space representation, i t can be shown that TTX^ = 1. If X^ = = 1, then A^A^ = 1. 1 0 5 Hence the two free exponents X^ and X^, which determine the s t a b i l i t y of the solution, must l i e on the unit c i r c l e or the r e a l axis i n the complex plane. Thus the s t a b i l i t y c r i t e r i o n becomes 4 i=i unstable <4 or >0 ; stable . • . . ( 4 . 3 ) A f i n a l condition matrix 0(T p) i s computed from 0(0) equal 4 to i d e n t i t y matrix. As T r [ 0 ( T p ) ] = E X . , the s t a b i l i t y of i = l 1 the periodic solution i s determined by T r [ 6 ( T p ) ] - 2 > 2, ] unstable < z ] stable . . . . ( 4 . 4 ) 4 . 3 Discussion of Results The closed motion envelope i n Figure 4 - 1(a) guarantees a non-tumbling motion leading to three d i s t i n c t l y d i f f e r e n t characters of phase-space representations regular, ergodic and i s l a n d type each associated with a periodic solution. The presence of aerodynamic moment destroys the symmetry of the motion envelope as well as the i n t e g r a l manifold cross-section at <£ = 0 (Figure 4 - 1 (b) ) . Both the regular and is l a n d type s t a b i l i t y regions are distorted even for small B^. This makes the determination of s t a b i l i t y bound as well as the periodic solutions somewhat d i f f i c u l t . Figure 4 - 2 shows the periodic response plotted over six o r b i t s during which at least two cycles are completed. 106 In absence of aerodynamic torque the fundamental periodic solution associated with the regular region executes two planar o s c i l l a t i o n s for one transverse cycle. On the other hand, island s t a b i l i t y regions are associated with the periodic solutions P^^ and P 2 3 (Figure 4-2 (a)). Although the presence of atmosphere s h i f t s the response towards the -IJJ d i r e c t i o n the basic c h a r a c t e r i s t i c s , indicated above, remain e s s e n t i a l l y unaffected (Figure 4-2 (b)). In a l l the cases investigated, the smaller frequency response was associated with larger amplitudes. The determination of a complete set of periodic , solutions would involve a c a r e f u l scanning of the region of possible motion. However, the invariant nature of the i n t e g r a l manifold and predominance of the fundamental periodic solution render i t s u f f i c i e n t , from the point of view of usefulness, to give only a set of i n i t i a l conditions generating E ^ i * Figure 4-3 shows the effects of s a t e l l i t e i n e r t i a and aerodynamic torque on the impulsive disturbances to excite a stable, fundamental periodic motion. The l i m i t s of t h e i r s t a b i l i t y , as obtained from the Floquet theory, are also indicated. Besides being symmetrical about <f>'=0, the plots suggest r e l a t i v e l y larger demand on transverse disturbance. In a l l cases P 2 1 requires a negative planar impulse - T J J ^ , the magnitude of which reduces, i n general, with increasing B^ and K.. For comparison, generating conditions corresponding 107 141 -1.5 -1.0 - 0 . 5 0 0.5 1 0 1.5 1 Ki =1-0 B f = 0 . 2 \ \\ V \\ \\ 1 1 fc^Bf=0.2 T K; =0.5 1.0 r21 1. i J -1.0 -0 .5 0 - 1 0 - 0 . 5 0 -1.0 -0 .5 0 to' Figure 4-3 E f f e c t of i n e r t i a and atmosphere on the impulsive disturbances for stable periodic motion; =ip , 4> =0 ° E e to some of the other periodic solutions are also included for the p a r t i c u l a r case of a dumbbell s a t e l l i t e operating i n absence of atmosphere. Relatively (compared to that for P2^) large p o s i t i v e planar impulses excite P 2 3 while the large negative ones lead to £ 4 5 * The system, being dependent upon a number of variables, would involve an enormous amount of computation for any compre-hensive analysis. Furthermore, massive information so generated has to be presented i n a concise form for ease of application. One way would be to represent an i n t e g r a l manifold by i t s intercept, with a convenient axis, as a measure of s t a b i l i t y . The v a r i a t i o n of ip' intercept with the l i m i t i n g regular, manifold as a function of Hamiltonian i s shown i n Figure 4 - 4 . The fundamental periodic solutions and the c r i t i c a l conditions for t h e i r s t a b i l i t y are also indicated. The plots c l e a r l y emphasize the influence of s a t e l l i t e i n e r t i a and aerodynamic moment. It may be observed that appreciable reduction i n s t a b i l i t y would r e s u l t for s a t e l l i t e s with smaller , p a r t i c u -l a r l y i n the presence of an atmosphere. The spinal character of the periodic solutions i s quite apparent. The plot indicates that, at c r i t i c a l Hamiltonian ( C „ ) for stable "cr motion, the only available solution i s a periodic one. In conjunction with the e a r l i e r r esults concerning the l i m i t i n g motion envelope, equation (3.22), the plots provide better insight into the nature of the solutions as affected by the Hamiltonian. 109 Figure 4-4 E f f e c t of system parameters on the region of regular s t a b i l i t y and fundamental periodic solution P~, For small values of C R the system behaviour i s regular. This continues u n t i l C , representing the bound beyond which the R island type and ergodic solutions also appear, i s attained. As the condition for guaranteed bounded motion ( 3 . 2 2 ) i s approached, the t r a j e c t o r i e s show increasing tendency towards non-regular behaviour, p a r t i c u l a r l y for p o s i t i v e planar impulses. For larger values of Hamiltonian, however, the stable solutions appear, only as regular t r a j e c t o r i e s . At C„ the manifolds cease to e x i s t , leaving the fundamental cr periodic solution to be the only stable condition. The following table shows the impulsive disturbances at e q u i l i b -rium configuration required to excite at C„ , determined 2 1 H c r using Floquet's v a r i a t i o n a l analysis. The corresponding l i b r a t i o n a l period T p i s shown as a f r a c t i o n of o r b i t a l cr period. From the consideration of p r a c t i c a l application, the values C„ are also included, which emphasize the reduced R s t a b i l i t y condition. Table 4 - 1 C r i t i c a l conditions as affected by s a t e l l i t e i n e r t i a and aerodynamic torque K i B f C H p CH ^O T P R cr cr cr 1 . 0 0 . 0 - 2 . 5 6 0 . 8 2 7 - 1 . 3 3 7 0 . 7 1 5 7 1 . 0 0 . 2 - 2 . 8 0 0 . 7 6 6 - 1 . 1 5 5 0 . 7 3 6 5 1 . 0 1 . 0 - 3 . 3 2 - 1 . 0 5 5 - 0 . 2 5 2 0 . 6 3 6 2 0 . 7 5 0 . 0 . - 2 . 0 4 0 . 8 1 2 - 1 . 2 3 5 0 . 7 6 1 1 0 . 5 0 . 0 - 1 . 9 2 0 . 7 9 6 - 1 . 1 2 6 0 . 8 1 6 1 0 . 5 0 . 2 - 2 . 0 5 0 . 8 0 2 - 0 . 9 5 7 0 . 8 1 3 3 0 . 5 1 . 0 - 2 . 3 6 - 1 . 2 8 8 - 0 . 1 8 1 0 . 7 5 6 0 0 . 2 5 0 . 0 - 1 . 7 1 0 . 7 8 7 - 0 . 9 9 5 0 . 8 8 2 3 For stable, fundamental, periodic motion T p increases with C„ and B... Increase i n K. , however, causes i t s reduction n r 1 (Figure 4 - 5 ) . The trace of the f i n a l condition matrix 0(Tp), appearing as Floquet's s t a b i l i t y c r i t e r i o n , also varies s u b s t a n t i a l l y with the system parameters. For preliminary design of a s a t e l l i t e , i t would be more pertinent to have, from s t a b i l i t y considerations, i n f o r -mation about the s a t e l l i t e ' s a b i l i t y to withstand impulsive disturbances for regular behaviour. This could be accomplished quite r e a d i l y recognizing the fact that, at equilibrium p o s i t i o n , the p l o t 1 vs. <j)1 , for a given Hamiltonian, i s a c i r c l e . When C„ i s small a l l conditions on t h i s c i r c l e lead n. to regular t r a j e c t o r i e s . However, for C„ < C„< C„ , i . e . , . l i _ H ri R cr when i s l a n d , ergodic type or unstable solutions are possible, regular behaviour can occur only over the arc corresponding to a l i m i t i n g intercept (Figures 4 - 1 , 4-4). Figure 4 - 6 compares, for a dumbbell s a t e l l i t e , the allowable impulsive disturbance for regular behaviour with , c r i t i c a l conditions for s t a b i l i t y as obtained by B r e r e t o n . ^ The plots are symmetrical about I J J ^ -axis. The reduction { - 2 7 . 5 % i n area) i n the bound due to the ergodic and i s l a n d type behaviour i s p a r t i c u l a r l y s i g n i f i c a n t i n \p degree of freedom. The presence of atmosphere further deteriorates the s i t u a t i o n . The physical parameters indicated i n the diagram approximately correspond to the gravity-gradient s t a b i l i z e d s a t e l l i t e GEOS-A, launched on November 6 , 1 9 6 5 Figure 4-5 Variation of the period and the trace of f i n a l condition matrix with Hamiltonian for the stable periodic solution P„, 113 Figure 4-6 Reduction of the allowable impulsive disturbance for stable motion due to non-regular solution and atmosphere 114 (data: semi-major axis - 5 , 0 0 0 miles; e = 0 . 0 7 ; I = 6 1 5 . 3 slug, f t 2 ; I = 7 1 6 . 0 slug, f t 2 ; I = 2 0 . 8 slug, f t 2 ; D L = yy z z o o 2 5 4 1 3 . 1 f t ; e= 5 . 7 5 ft) . It i s apparent that the s a t e l l i t e would have i t s regular s t a b i l i t y reduced by = 4 5 % at 2 5 0 miles a l t i t u d e , where = 1 . The e f f e c t of i n e r t i a and atmosphere on the regular s t a b i l i t y region i s shown i n Figure 4 - 7 . Although the re-duction i n available s t a b i l i t y bound due to ergodic and islan d type solutions i s s i g n i f i c a n t for s a t e l l i t e s with larger , they s t i l l show better s t a b i l i t y c h a r a c t e r i s t i c s p a r t i c u l a r l y when B^ i s small. 4 . 4 Concluding Remarks The important conclusions based on the analysis may be summarized as follows: (i) The; investigation emphasizes the usefulness of the concept of i n t e g r a l manifolds by pointing out the fact that, beside providing the bound of s t a b i l i t y , t h e i r degeneration leads to periodic solutions. Thus the periodic solutions act as spines around which the s t a b i l i t y regions are b u i l t , ( i i ) The system exhibits three d i s t i n c t l y d i f f e r e n t solutions even when the bounded motion i s guaran-teed. The regular behaviour corresponds to the periodic solution P 2 1 ' w n i l e the i s l a n d type rep-resentation i n the phase plane i s associated with Figure 4-7 Allowable impulsive disturbance at equilibrium position for regular motion and corresponding periodic solutions: (a) e f f e c t of i n e r t i a ; (b) e f f e c t M of atmosphere m v. 116 other periodic solutions, e.g., ^ 2 3 ' P 4 5 ' e t c • The ergodic behaviour i n the t r a n s i t i o n region i s probably i n d i c a t i v e of long period l i b r a t i o n s . ( i i i ) Proximity of the islands to the motion envelope and the unpredictable nature of the ergodic solutions render the l i m i t i n g regular region to be the only useful s t a b i l i t y bound for s a t e l l i t e design. (iv) The fundamental period corresponding to the solution P^^ i s close to the o r b i t a l rate. It increases with decreasing K. and increasing C„ or B_. 1 n r (v) In general, the c r i t i c a l Hamiltonian for stable motion decreases with decreasing . The reduction i s enhanced during the presence of aerodynamic torque. As the only available solution i s a periodic one for C„ , i t can be determined quite cr readily and accurately using extension of Floquet's s t a b i l i t y c r i t e r i o n to the fourth order system. 5. LIBRATIONAL RESPONSE AND STABILITY IN ELLIPTIC ORBITS 5.1 Preliminary Remarks The l i b r a t i o n a l analysis, so f a r , was r e s t r i c t e d to the s a t e l l i t e s negotiating c i r c u l a r t r a j e c t o r i e s . The s i m p l i f i c a t i o n was necessary because of the complex character of the governing equations which then became autonomous. The next l o g i c a l step would be the consideration of a more general s i t u a t i o n involving motion i n an e l l i p t i c t r ajectory. It may be pointed out that the analysis of the r e s u l t i n g non-autonomous, gravity oriented system, even i n absence of environmental forces,remains unexplored. On the other hand, the importance of such a study becomes apparent when one recognizes the fact that meteorological, earth resources, m i l i t a r y reconnaissance, etc., s a t e l l i t e s using close earth or b i t s for better resolutions, can have t h e i r l i f e span increased through use of e l l i p t i c t r a j e c t o r i e s . This chapter investigates coupled l i b r a t i o n a l dynamics of such non-autonomous systems. In the beginning an approx-imate closed form a n a l y t i c a l solution i s obtained for the system i n absence of aerodynamic moment using modification of 57 Butenin 1s approach. This i s followed by numerical response and s t a b i l i t y analysis i n the large over a wide range of i n e r t i a parameter. Next, the e f f e c t of aerodynamic moment on the equilibrium configuration, system response, and s t a b i l i t y o 118 are s t u d i e d i n d e t a i l . As the concept of i n t e g r a l m a n i f o l d breaks down due to the non-autonomous c h a r a c t e r of the system, the amount of computational e f f o r t i n v o l v e d i s enormous. A convenient condensation of the response d a t a , e f f e c t e d through p l o t s showing a l l o w a b l e i m p u l s i v e d i s t u r b a n c e s over a s e t of e c c e n t r i c i t i e s , r e p r e s e n t s an attempt at p r o v i d i n g i n f o r -mation of p a r t i c u l a r use d u r i n g p r e l i m i n a r y design of a s a t e l l i t e . 5.2 Formulation of the Problem In absence of atmosphere (Figure 2-1) , equations (2.1c) and (2.3) represented the g e n e r a l motion of the system. Using the K e p l e r i a n r e l a t i o n s and n o t i n g t h a t d/dfc = 0 d / d e = ( h e / r * ) d / d e d y d - ezayde" + 8 d / d e = ( h e Y r 4 ) d V d e * - 2(^e/r5)(dr/de)4/de . . . . (5.1) the equations i n the l i b r a t i o n a l degrees of freedom t r a n s -form t o : l |T ( i 4- e c o s G ) - 2esin0(l|/'+i) - *4>'(l|/flXl-f e c o s O ) t a n c f > 4- 3Kj sin l\) cos lp -0 . . . . . (5.2a) 0>"(1 4-e cos 9) -^e^s in 6 # + [ ( 1 +• V'f ( 1 + e c o s 0 ) 4- 3 Kj cos* (JJ J -sin cf) cos $ - 0 . . . . (5.2b) These second order, coupled, non-linear, non-autonomous equations of motion remain invariant under the transformation ( 6 , i p , 4 > ) to (9 , i { i , -< |>) , ( - 6 , - ^ , 0 ) or ( - 6 , - ^ , - 0 ) . 5.3 Approximate Solution and System Response 5.3.1 Variation of Parameter Method (Butenin) In absence of known, exact, closed form solution, i t was decided to analyze the problem approximately using modification of Butenin's v a r i a t i o n of parameter technique. The method was described e a r l i e r i n Section 2.4.1. Replacing the trignometric functions of the depen-dent variables by t h e i r s e r i e s , ignoring f i f t h and higher degree terms i n \p, <f>, and t h e i r derivatives, and c o l l e c t i n g non-linear terms and forcing function on the r i g h t side, equation (5.2) takes the form: + 3 K i 1(1 ^ Xes in 6 -+ Z [esin 9 y'/(L+ecos6) + &fy(W±t) +ZV f/3 4- 3Kj (|V 3 j . . (5.3a) <t>" +(i4-3K'0<t) ^ O e s i n 8 (J>'/(i+ecosf l ) ± + 3Kj/( i4 - ecos 6 ) 3 / 3 - Z V ' ( p - f % -4ip'cf)3/3 + 3 K j i p 1 ( { ) / ( l +ecose)J or ty" + (|J - zes\n 6 + j) ' , 9) <t)"4- n ^ = ^ Y ^ . f ,9) (5.3b) (5.4) 120 The solution of the corresponding l i n e a r system ( i . e . , f^=g^=0) i s given as: l|j = as in (n i 9+^ i ) -f-2esine/(n^ - i ) • • • • (5.5a) (j) = bsin (n^g + . . . . ( 5 . 5 b ) where a,b,$^ and ^ are constants which can be determined from i n i t i a l conditions. A solution of the sim i l a r form i s sought for the case under consideration, allowing, however, the amplitude and phase angles to be functions of 6, i . e . , I)) = a(S) sin (n i 9 + ^(6)) + 2es in e/(nf- i) ( 5.6a) <l>= b(9)s\n(n^ e+§^ (0)) .... (5.6b) a,b,B^ and c a n ^ e expressed as functions of 0 plus a constant. Thus the solution i n the present form involves eight unknowns, four of which can be determined by the i n i t i a l conditions while the remaining have to be found through the imposition of constraints. Keeping the f i r s t derivative of equations ( 5 . 6 a ) and ( 5 . 6 b ) to be similar to the homogeneous solution gives two of the constraint r e l a t i o n s : a'sin'S + Q ^ - c o s = ° . . . . ( 5 . 7 a ) b' sin rj_ 4- b ^ c o s = o . . . . ( 5 . 7 b ) Other two relations are obtained by substituting ( 5 . 6 ) i n equations of motion ( 5 . 4 ) giving 121 a'^cos^ - Q ^ i ^ i s i n ^ = h - - •• • (5-7c) bn^cosi^ - bn z^ Sin = cj* . . . . ( 5.7d) where il = fi[QS\W^ + ZesinO/(n£-1) , an icos "S + 2 e cos 0 / ( ^ - 1 ) , bsin ^ , bn^cos r[ , 9 j g* =g^[QSin^ + ^ es'm6/<fn£-l), a q c o s ^ - f -2 e c o s e / ( n f - l \ b sin r^ , b n ^ c o s ^ , 0J . . . . ( 5 . 8 ) solving the equations ( 5 . 7 ) y i e l d s Q' = f \ cos - ^ / n i b' = g* cos n^/n^ Qi = - ^  sin ^ / a n A ^ = - 9 * s i r i . . . . ( 5 . 9 ) * * f l ' g l keing small for small disturbances, a,b,8 1 and B 2 are slowly varying parameters. Using t h e i r average values over a period gives i / i / r^ n ( %ir * da/de = (l/air 3 n i ) [ [ { f / c o s ^ d - ^ dn_de d b/d 9 = c i/8 ff3 ) f' i f * f' j J cos n_ d ^  d T. d e d^ /de=-( 1/8TT 3n La)l l ' J 1 ' s , n ^ d^ d a de d^/de=-(i/8/r3n;ib)j[MjM fV^n ^ d-> d^ de (5.10) or, 122 = a sin [tsKO^ & 4- @A] 6 / ( 3 K , -1) ( s . i i a ) cf) =. bsin [ ( 3 K j + i ) 0+£>*] . . . . (s.iib) where a = [Hi* + [<lT - 2 e / ( 3 K ( - D f / s K i f * ^ - - [ b* { i W K i / C l - e ^ J - 3 a * K f ( i - 1/(1-. + A eV(3K| -l)j0/(4 ( 1 + 3K| fx] + tan1 [(H-3K0V*<fe/£I . . . . (5.12) 5.3.2 Accuracy of the Solution To es t a b l i s h the accuracy of t h i s a n a l y t i c a l technique the equations of motion (5.2) were integrated numerically. The l i b r a t i o n a l response as affected by s a t e l l i t e i n e r t i a , o r b i t a l e c c e n t r i c i t y and external disturbance was obtained over f i f t y o rbits using a step-size of 3°. However, for con-ciseness, the comparison between the two methods i s limited to i n i t i a l and terminal regions i n Figure 5-1. As i n the case of a c i r c u l a r o r b i t , the solution appears to agree well with the numerical r e s u l t s , p a r t i c u l a r l y for the motion across the o r b i t a l plane, even for a disturbance of appreciable magnitude (^=0^=0.5). The e f f e c t of eccen-t r i c i t y i s r e f l e c t e d through motion modulations. Both methods show that a larger amplitude, smaller frequency motion, with G e =0.025 123 numerical ; analyt ical 3 0 r -Figure 5-1 Representative comparison of the responses generated using Butenin 1s approach and numerical method, showing e f f e c t s of: (a) s a t e l l i t e i n e r t i a K; = 1.0 •numerical • analyt ical 124 30 r •30 30 0 e =0.1 4>' Figure 5-1 Representative comparison of the responses, gener-ated using Butenin's approach and numerical method, showing ef f e c t s of: (b) o r b i t e c c e n t r i c i t y K; = 1.0 e = 0 1 *,= *>'= 125 numerical *, — analytical 301-0 - 3 0 3 0 4>= <b'=o 'o T o <t>° - 3 0 3 0 T -o-- 3 0 3 0 • «15;<|g0.25 0 - 3 0 0 Figure 5-1 J---L-L 3 4 7 orbits 48 49 50 Representative comparison of the responses, generated using Butenin's approach and numerical method, showing ef f e c t s of: (c) i n i t i a l conditions 1 2 6 a period of the order same as that of the o r b i t , i s excited i n the o r b i t a l plane when the s a t e l l i t e i s subjected to i d e n t i c a l disturbances i n the two degrees of freedom. The phase d i s -crepancy between the solutions appears to grow with time. The l i b r a t i o n a l amplitude predicted by the approach i s , i n general, smaller than the actual. Thus the r e s u l t i n g ana-l y t i c a l s t a b i l i t y bound'is l i k e l y to be larger. Although, the agreement deteriorates with decreasing slenderness of the s a t e l l i t e (Figure 5 - 1(a)) and increasing e c c e n t r i c i t y (Figure 5 - 1 (b)), the analysis continues to predict the general behaviour, at least q u a l i t a t i v e l y . Reduction of or increase i n e -enhances the amplitude modulation, e s p e c i a l l y for planar degree of freedom. Both the solutions indicate that i n absence of any i n i t i a l d i s -turbance, appreciable o s c i l l a t i o n s i n the o r b i t a l plane are excited due to e c c e n t r i c i t y of the o r b i t (Figure 5 - 1 (c)) . A presence of any cross motion appears to induce small perturbations i n the planar l i b r a t i o n s , however, the a n a l y t i c a l approach f a i l s to predict t h i s phenomenon. As i n actual practice, the gravity gradient s a t e l l i t e s possess large , normally move in c i r c u l a r or almost c i r c u l a r o r b i t s , and exhibit moderate pointing accuracy, the a n a l y t i c a l solution can be applied with confidence, at least for preliminary design purposes. The e f f e c t of e c c e n t r i c i t y i s indicated i n Figure 5 - 2 , which compares the numerically generated response of a Figure 5-2 Numerical results indicating the e f f e c t of o r b i t e c c e n t r i c i t y on the s a t e l l i t e response: (a) no disturbance; (b) large disturbance 128 dumbbell s a t e l l i t e . Although, i n a c i r c u l a r o r b i t no motion i s excited i n absence of disturbances, appreciable planar motion, which grows with e c c e n t r i c i t y , i s noticed i n e l l i p t i c t r a j e c t o r i e s . Considerable amplitude modulations at higher e c c e n t r i c i t i e s , p a r t i c u l a r l y of the motion i n the o r b i t a l plane, suggest an increased tendency towards i n s t a b i l i t y for the s a t e l l i t e subjected to an ar b i t r a r y disturbance. 5.4 S t a b i l i t y Bound 5.4.1 A n a l y t i c a l Approach With obvious li m i t a t i o n s of the Butenin method one can hardly expect i t to be suitable for any s t a b i l i t y study i n the large. However, i t was g r a t i f y i n g to observe that, i n spite of rather d r a s t i c s i m p l i f i c a t i o n s , the method success-f u l l y establishes trend for the influence of system parameters. The form of the solution as given by the v a r i a t i o n of parameter method (equation 5.11) suggests that the s t a b i l i t y of the system i s governed by the amplitude of the harmonic terms. The s t a b i l i t y c r i t e r i a , therefore, become |Q| 4- 2 e / ( 3 K , - 1} 4 TT/Z |b| ^ TT/Z . . . . (5.13) or, i n terms of impulsive disturbance at equilibrium con-fi g u r a t i o n , 129 | i | / o ' - 2 e / ( 3 K j 4 ( 3 K j ) V * [TT/Z - ze/(5Kri)} | <}/ | f 7T ( i - r - 3 K i ) V V ^ ( 5* 1 4 ) In the ^ Q plane, these correspond to rectangular regions, symmetrical about if^-axis. It i s apparent that a decrease i n slenderness of the s a t e l l i t e or an increase i n o r b i t e c c e n t r i c i t y would, i n general, a f f e c t the s t a b i l i t y adversely, p a r t i c u l a r l y i n the i|> degree of freedom. It i s inter e s t i n g to observe that most s a t e l l i t e s should be able to withstand r e l a t i v e l y larger +ve planar impulses. At K^=l/3, the s t a b i l i t y bound i s not defined and below th i s c r i t i c a l value, most of the trends mentioned above are reversed. The Butenin's approach thus y i e l d s some q u a l i t a t i v e insight into the system s t a b i l i t y i n the large. However, for quantitative results one has to adopt numerical methods. 5.4.2 Numerical Approach For autonomous systems, use of the concept of the i n t e g r a l manifold i n conjunction with the constant Hamiltonian f a s c i l -i t a t e d the s t a b i l i t y analysis appreciably. Unfortunately, i n presence of e c c e n t r i c i t y , the concept loses i t s importance due to the obvious d i f f i c u l t y i n representing and in t e r p r e t -ing the hyper-surfaces i n phase space. Intersection by a phase plane (say, — ' ) no longer represents a cross-section of the hyper-invariant-manifold, and only leads to the scattered 1 3 0 encounter with the t r a j e c t o r i e s (Figure 5 - 3 ) . Hence an alternate approach i s necessary to get meaningful information about the system s t a b i l i t y . Here the s t a b i l i t y bounds are established by analyz-ing the l i b r a t i o n a l response, over 1 5 - 2 0 o r b i t s , to systemat-i c a l l y varied i n i t i a l conditions, s a t e l l i t e i n e r t i a , and o r b i t e c c e n t r i c i t y . The vast amount of information, thus gathered, i s condensed i n the form of design plots (Figure 5 - 4 ) , which indicate allowable impulsive disturbances ( ^ 0 = < J > 0 = 0 ) at perigee for non-tumbling motion, over a range of and e. For comparison, e a r l i e r results with c i r c u l a r orbits are also included. The e f f e c t of even s l i g h t increase i n e c c e n t r i c i t y i s to rapidly reduce the s t a b i l i t y region, p a r t i c u l a r l y for s a t e l l i t e s with smaller . The reduction, i n general, i s more severe i n the plane of the o r b i t , where the s a t e l l i t e i s able to withstand, r e l a t i v e l y , large pos i t i v e disturbance. The plots remain symmetrical about cf>' =0 as i n the case of autonomous system. The peculiar shape of a s t a b i l i t y region with numerous spikes may be attributed to the predominance of various periodic solutions. Of course, at the highest e c c e n t r i c i t y for stable motion, the only available solution 9 i s a periodic one as indicated by dots i n Figure 5 - 4 . The crossing of s t a b i l i t y bounds suggest that i n some si t u a t i o n s , increase i n e c c e n t r i c i t y may be favourable, l o c a l l y , i n system s t a b i l i z a t i o n . 131 Figure 5-3 Stroboscopic phase plane at 0=0 showing breakdown of the i n t e g r a l manifold concept for non-autonomous system Figure 5-4 E f f e c t of s a t e l l i t e i n e r t i a and o r b i t e c c e n t r i c i t y on the allowable impulsive disturbances for stable motion; 0 =Ui =d> =0 : (a) K. =1. 0 , 0 . 5 M O O O 1 w 134 Reduction of to 0.25, i . e . , a value less than the c r i t i c a l 1/3, reverses some of the trends established above. This i s apparent from i t s better s t a b i l i t y c h a r a c t e r i s t i c s compared to K^=0.5 i n eccentric o r b i t s . A 'shorter' s a t e l l i t e also exhibits an a b i l i t y to withstand larger negative impulse, Although the plots presented here are for disturbances received at perigee, averaging over a large number of or b i t s suggests th e i r a p p l i c a b i l i t y , at least approximately, to any 6 i n small e c c e n t r i c i t y o r b i t s . In p r i n c i p l e the system 7 behaviour i s si m i l a r to planar l i b r a t i o n s i n e l l i p t i c o r b i t and coupled l i b r a t i o n s i n c i r c u l a r o r b i t . It i s important to recognize that presence of cross-plane motion improves the s a t e l l i t e ' s a b i l i t y to withstand impulsive disturbances. 5.5 E f f e c t of Aerodynamic Torque on System Response and S t a b i l i t y 5.5.1 Equations of Motion / 66 Using Schaaf and Chambre''s approach for a s a t e l l i t e surface i n free molecular flow the modified pot e n t i a l function for a c y l i n d r i c a l 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 was given by equation (3.4 ). In an e l l i p t i c o r b i t , the change i n density and o r b i t a l v e l o c i t y from point to point can be expressed as: ^ = { ( r - R e ) / ( r p - R e ) f ' . . . . < 5 . i 5 a ) v* = >* (v/vp)* - v* { ( 2 rp - p + r e;/(r + r e)} . . . . (5.15b) 1 3 5 The value of exponent n varies between - 5 to - 7 i n the a l t i t u d e range of 1 0 0 - 5 0 0 miles. The aerodynamic p o t e n t i a l , thus becomes: U Q = %l{(r/rp-^/rp)/(L~Re/r^f(zrp/r - i + e ) / ( i - f e ) J { ( j J -f s i n ( p ( c o s i p f q s i n (for [(JJ| < TT/Z) • • • • (5-16> where B ^ = S C D ^ B o L ° V p / H e J . . . . ( 5 . 1 7 a ) q = TTD0/4L0 = 7T [ a - K ] ) / n a + K \ ) } i / l ( 5 - 1 7 b ) It i s apparent that, consistent with the assumptions (Section 3.2.1), the governing equation of motion i n the <J) degree of freedom (5.2b) remains unchanged and that i n the degree modifies to: ip"(i-4-e cosG) - £ e s i n e ( t y'4 - 1 ) - £ < ^ V l ) (4> ecos0)tan(b 4-3K\sin (p cos\\) -f Bf (1+ e cose)(|cos • i - q s i n l p ) c o s ^ / c o s ^ =0 . . . . ( 5 . 1 8 ) where \ = \ [ { ^ f e ) / ( ^ e c o ^ ) - ^ A P | / a -- R e / r p ) ] n ( l 4 - ^ e c o s 6 4-e^)Ci4-e)V(l+eco5ef . . . . (5.19) Note that the system retains invariant character only under 136 the transformation (0,i|;,<J>) to (6/i|> ,-<j>) . Increased complexity renders the a n a l y t i c a l techniques of questionable value, p a r t i c u l a r l y for motion i n the large. Numerical methods, therefore, have to be resorted to. 5.5.2 Equilibrium Configuration The stable equilibrium position i s given as: l(J = t a n 4 { - B f /(3K- ,+B f c p j cb = 0 • • • <5-20> e £ E J j c As B ' varies with 6 ,ip changes continuously (Figure E e 5-5). The symmetry of the plots about 0=0 i s of i n t e r e s t . The presence of e c c e n t r i c i t y tends to confine the effects of aerodynamic perturbations to the region near perigee. Even for small e c c e n t r i c i t y of orbits (e < 0.1), the aerodynamic torque becomes n e g l i g i b l e for |0| > 60°. The rate of reduction becomes steeper with increasing e and B^ , and P decreasing . 5.5.3 System Response A few representative response p l o t s , obtained numer-i c a l l y , for systematically varied i n e r t i a , o r b i t e c c e n t r i c i t y , aerodynamic c o e f f i c i e n t , and i n i t i a l conditions are shown i n Figure 5-6. As against the l i b r a t i o n a l motion about a con-stant equilibrium position i n c i r c u l a r o r b i t , presence of a forcing function along with the periodic v a r i a t i o n of 137 140 160 6 Figure 5-5 Variation of aerodynamic c o e f f i c i e n t and stable equilibrium configuration with 6 and e 20 0 - 2 0 2 0 4? - 2 0 rp=200 miles e = 0.0 ;0 .05 ,0.1 ; 0 . 2 - 1 " " " A A ^ A .A / V , ' 1 J | / 1 K i = 1 0 W W \j \) V — (unstable for e=0.1). 1 1 1 i 1 i i Figure 5-6 orbits Typical system responses showing the e f f e c t of o r b i t e c c e n t r i c i t y and (b) s a t e l l i t e i n e r t i a orbits Figure 5-6 Typical system responses showing the e f f e c t of o r b i t e c c e n t r i c i t y and (c) i n i t i a l conditions 141 aerodynamic torque and equilibrium configuration makes the r e s u l t i n g response quite complex. The modulations, which are more predominant i n the planar degree of freedom,grow rapidly with B f (Figure 5-6(a)). Even for an i d e n t i c a l disturbance P in the two degrees of freedom, the planar component appears to be more susceptible to i n s t a b i l i t y . This, i n a sense, j u s t i -f i e s the e a r l i e r s i m p l i f i e d model of planar l i b r a t i o n s used , i 2-9 et a l . by several authors. As can be expected, the forcing function a r i s i n g from o r b i t e c c e n t r i c i t y induces planar l i b r a t i o n a l motion. Due to aerodynamic influence, planar o s c i l l a t i o n s were noticed i n absence of any external disturbance, even i n c i r c u l a r o r b i t . The combined e f f e c t of e and B f results i n a considerably E larger planar motion (Figure 5-6(b)), p a r t i c u l a r l y for short s a t e l l i t e s . Irrespective of K., e or B, a s a t e l l i t e i n i t i a l l y positioned c o r r e c t l y along the l o c a l v e r t i c a l executes appreciable l i b r a t i o n s i n the o r b i t a l plane. The presence of a cross motion does not af f e c t i t noticeably (Figure 5-6(c) ). The character of the response suggests possible reduction i n the s t a b i l i t y region due to aerodynamic torque. A l l the response data presented so f a r , correspond to stable operation of the s a t e l l i t e . Its c r i t i c a l dependence on s a t e l l i t e i n e r t i a and o r b i t e c c e n t r i c i t y was shown through s t a b i l i t y p l o t s . In Figure 5-7 are shown several examples of i n s t a b i l i t y as functions of K^,B^ and e. Note that a slender s a t e l l i t e (K.=1.0), moving i n a c i r c u l a r o r b i t through Figure 5-7 I n s t a b i l i t y excited by change of system parameter 143 a pure gravity gradient f i e l d executes large amplitude stable l i b r a t i o n s when subjected to a unit impulse ($^=$^=1.0 ,1)1^ ^ = 0 ) . However, changes i n system parameters beyond the c r i t i c a l values lead to tumbling motion. For instance, reduction of to 0.5 or increase of e c c e n t r i c i t y to 0.15 lead to i n s t a b i l i t y within a short time. Increase i n B f to a unit value i n i t i a t e s 'clockwise' tumbling i n c i r c u l a r o r b i t i t s e l f . I t may be pointed out that i n a l l these cases, the motion across the or b i t remains bounded. Importance of parametric study of the system, from design considerations, i s thus apparent. 5.5.4 S t a b i l i t y Plots As no known closed form solution i s available and the int e g r a l manifold technique does not appear to be applicable, the s t a b i l i t y of the system i s established, as before, through numerically generated response. Design plots again prove useful i n condensing an enormous amount of information. The plots (Figure 5-8), symmetrical about <j>^ =0, show a l l possible combinations of allowable impulsive disturbances for s t a b i l i t y . . The corresponding r e s u l t s for a c i r c u l a r o r b i t are also included for comparison. It i s apparent that even a small e c c e n t r i c i t y of the o r b i t makes the s t a b i l i t y region shrink s u b s t a n t i a l l y . The presence of aerodynamic torque further enhances th i s trend. As i n the case of e c c e n t r i c i t y , the reduction i n the s t a b i l i t y margin i s predominantly i n the degree of freedom. The system shows, i n general, better a b i l i t y to withstand posi t i v e it* 145 planar impulses. The peculiar shapes of the boundary with spikes may be attributed, as before, to coupling e f f e c t s and the existence of a variety of periodic solutions. The aerodynamic torque represents a periodic disturb-ance. Although active only over a r e l a t i v e l y small portion of the s a t e l l i t e ' s eccentric o r b i t , i t has considerable adverse influence on the s t a b i l i t y . The extensive amount of computation involved limited the investigation to only a few representative s i t u a t i o n s . 5.6 Concluding Remarks The important aspects of the analysis and more s i g n i f -icant conclusions may be summarized as follows: (i) A simple closed form solution as given by Butenin's v a r i a t i o n of parameter method can be used e f f e c -t i v e l y during preliminary design of a s a t e l l i t e , ( i i ) In the absence of any disturbance, e c c e n t r i c i t y excites a pure planar motion having a period of the same order as the o r b i t a l rate. Coupling effects i n this case are r e l a t i v e l y less s i g n i f i c a n t . ( i i i ) In the case of the non-autonomous system, the concept of i n t e g r a l manifold i n the phase space loses i t s importance due to obvious l i m i t a t i o n of v i s u a l i z -ation and interpretation of hyper-surfaces. (iv) The s t a b i l i t y region diminishes rapidly with increase i n e c c e n t r i c i t y . The shrinkage i s more s i g n i f i c a n t 146 i n the planar degree of freedom and for shorter s a t e l l i t e s (smaller K.). 1 (v) The c r i t i c a l e c c e n t r i c i t y for stable motion decreases subst a n t i a l l y with reduction i n . Even i n absence of aerodynamic moment, the gravity gradient f a i l s to s t a b i l i z e the most stable configuration of a dumbbell s a t e l l i t e beyond e c r=0.35. The presence of atmosphere affects the s i t u a t i o n adversely. The q u a l i t a t i v e analysis suggests that at K^=l/3 no s t a b i l i t y can be expected. Any reduction of i n e r t i a parameter below t h i s c r i t i c a l value reverses the normal trends. (vi) The presence of aerodynamic torque affects the stable equilibrium configuration which changes p e r i o d i c a l l y with the position of the s a t e l l i t e i n eccentric o r b i t . The torque leads to rapid degeneration of s t a b i l i t y region. 147 6. AERODYNAMIC DAMPING 6.1 Preliminary Remarks Having gained some understanding of the behaviour of gravity-oriented systems, the next l o g i c a l step would be to explore the p o s s i b i l i t y of c o n t r o l l i n g the undesirable l i b r a t i o n s to achieve high pointing accuracy. To t h i s end several damping mechanisms have been evolved and analyzed. 39 Debra considered use of a sphere moving i n a 40 viscous media to damp the general l i b r a t i o n s . Kamm suggested the V e r t i s t a t : a set of two f l e x i b l e booms put ho r i z o n t a l l y at r i g h t angle to each other to control the motion both i n and across the o r b i t a l plane. For planar l i b r a t i o n a l control 41 Paul proposed the use of a 'lossy' spring supporting a 42 small mass. Modi and Brereton improved t h i s model through 43 44 a parametric study. Tschann and Modi ' undertook an optimization of the same model using a n a l y t i c a l methods and presented a rigorous performance comparison with the conven-t i o n a l boom dampers. Use of environmental forces i n l i b r a t i o n a l damping 45 and attitude control i s not new. Paul et a l . showed the f e a s i b i l i t y of magnetic f i e l d . The application of solar radiation pressure for a space vehicle propulsion during inter-planetary f l i g h t s has been proposed by several authors 3 8 including Garwin, who described i t as "solar s a i l i n g . " Sohn et a l . investigated s p e c i f i c configurations for s a t e l -l i t e s t a b i l i z a t i o n with respect to the sun. More d i r e c t l y , 46 Mallach suggested the use of solar ra d i a t i o n as a damping force for gravity oriented s a t e l l i t e s . Recently, Modi et a l . 47-49 established the f e a s i b i l i t y of using solar radiation pressure for an e f f i c i e n t planar damping and attitude control by adjusting the exposed areas of solar pads as a function of l i b r a t i o n a l v e l o c i t y and angle. This chapter explores the p o s s i b i l i t y of u t i l i z i n g the normally d e s t a b i l i z i n g aerodynamic moment to advantage. A semi-passive, v e l o c i t y - s e n s i t i v e c o n t r o l l e r provides restoring moment of appropriate magnitude and sense through judicious adjustment of flaps exposed to the free molecular flow. This concept of l i b r a t i o n a l damping through d i f f e r e n t i a l l i f t i s e s s e n t i a l l y an extension of the a i r c r a f t s t a b i l i z -ation techniques. 6.2 F e a s i b i l i t y of the Concept Introduction of the aerodynamic force to a g r a v i t y -gradient system presents a p o s s i b i l i t y of center of pressure not coinciding with the center of mass. This leads to the aerodynamic torque which, i f controlled e f f i c i e n t l y , can provide not only the l i b r a t i o n a l damping but also the attitude control of the s a t e l l i t e . Extending the concept of a i r c r a f t attitude control to spacecraft moving i n the r a r e f i e d atoms-phere, consider a s a t e l l i t e , with two i d e n t i c a l s t a b i l i z i n g "flaps', as shown i n Figure 6-1 (a). The f l a p s , located i n gure 6-1 Aerodynamic damping and s t a b i l i z a t i o n : (b) possible arrangements of s t a b i l i z e r s the l o c a l horizontal plane passing through the center of mass of the s a t e l l i t e and controlled independently, are free to rotate about the axes perpendicular to the l i n e of symmetry of the s a t e l l i t e . An equal and opposite rotation of the f l a p s , leads to moment about the center of mass which has s t a b i l i z i n g components i n both and $ degrees of freedom. Thus with l i b r a t i n g s a t e l l i t e s , f l a p orientation can be adjusted continuously to provide suitable correcting torque. The moment due to the forces being balanced, no rotation about the z axis (yaw) i s induced. As the s a t e l l i t e , under the action of various disturbances, s t a r t s to l i b r a t e , the flaps are i n c l i n e d appropriately with respect to the impinging stream to provide a s t a b i l i z i n g torque. The torque, i f controlled as a function of s a t e l l i t e ' s l i b r a t i o n a l v e l o c i t y , should be able to damp the motion. Figure 6-1(b) shows some of the alternate schemes for f l a p arrangement. While the scheme discussed above i s l i k e l y to be the simplest to construct i t has obvious l i m i t a t i o n s , e.g., lack of control i n an i n d i v i d u a l degree of-freedom. The triangular setting (Figure 6-1(b)i), i n which the front f l a p damps the planar motion while the rear two by th e i r opposite movement control the cross-plane l i b r a t i o n s , provides a way of governing the i n d i v i d u a l degree of freedom. For maintaining the axi-symmetric character, the rear flaps are appropriately o f f - s e t from the center l i n e of the s a t e l l i t e . A further improvement, i n terms of symmetry :and 152 magnitude of the restoring moment, i s represented by the con-fi g u r a t i o n shown i n Figure 6 - l ( b ) i i . Here the o f f - s e t i s eliminated without a f f e c t i n g independent control of the ind i v i d u a l degree. Introduction of a set of s p l i t - f l a p s (Figure 6-1 ( b ) i i i ) represents another p o s s i b i l i t y . Here the center sections of each assembly are actuated i n d i v i d u a l l y to provide torque for planar control, while the other ones damp the motion i n <J) degree of freedom. Numerous other variations can be thought of by combining these basic arrangements. The concept during actual design may be faced with several optimization problems: (i) The atmospheric density as well as the l i f t co-e f f i c i e n t reduce rapidly with increase i n a l t i t u d e . On the other hand, the l i f e time of the s a t e l l i t e s diminishes with t h e i r closeness to the earth. Thus a compromise i s indicated, ( i i ) The flaps should be l i g h t yet s u f f i c i e n t l y r i g i d and large to generate enough l i f t . Furthermore, the drag should be small to minimize o r b i t a l perturbations. ( i i i ) The flaps should be so located as to avoid i n t e r -ference with the operation of antennae, cameras and solar c e l l s , (iv) The arms supporting the flaps should be long enough for adequate moment without s a c r i f i c i n g lightness 153 and r i g i d i t y . Obviously extensive ground tests would be required. Angular movement of the well arranged flaps i s l i k e l y to have l i t t l e e f f e c t on t o t a l i n e r t i a , axi-symmetric character of the system or the position of the center of mass. 0f ; course, sensing the disturbance and operation of the flaps may involve time delay. This, however, would be of l i t t l e s i g n i f i c a n c e due to long period (order of o r b i t a l period) of the l i b r a t i o n s . 6.3 Response Analysis With l i n e a r l y proportional, v e l o c i t y - s e n s i t i v e c o n t r o l , the governing equations of motion i n a c i r c u l a r o r b i t modify to: iy" - 2<t> ( q / + i ) t a n <[) 4- 3Kj sin I jJcos^ + B f (lcost[/| 4-CjS'W l)J) C O S ^ / C O S ^ c j ) 4- yU| l ) / = 0 . . . . (6.1a) $ + [C^J'f if 4- 3.K-, cos*- qj] sinct> cosc^ 4- jd^ = 0 . . . . (6.1b) where and are positive proportionality constants and i s the constant aerodynamic c o e f f i c i e n t for the s a t e l l i t e without flaps. Due to axi-symmetric arrangement, the s t a b i l -i z e rs do not induce rotations about the z axis. The fore-going ignores any variations i n damping torque due to small X o s c i l l a t i o n s caused by coupling e f f e c t s (equation (2.1c)). 154 The physical size and location of the flaps would also impose a l i m i t on the s t a b i l i z i n g torque, i . e . , ! M i iJJ'l ^ T t max AH 4> I ^ % . . . . (6.2) where, (6.3) 2 For example, a s a t e l l i t e , with 1=600 slug, f t , i n a c i r c u l a r o r b i t at 200 miles a l t i t u d e (p = 3.0 x 10~ 1 3 s l u g / f t 3 - ARDC 6 8 1959) and provided with two 3' x 3' flaps with moment arm of 5" each, has the maximum c o e f f i c i e n t of l i f t equal to 6 6 about 0.2 and the associated x. becomes 2.0. The con-max d i t i o n (6.2) implies that the flaps would maintain t h e i r orientation for torque requirement beyond T. max Figure 6-2 shows, over 3 o r b i t s , the e f f e c t of con-t r o l l e r proportionality constants ( y ^ , u 2 ) and system parameters on l i b r a t i o n a l response. A slender s a t e l l i t e with a small aerodynamic c o e f f i c i e n t B^, undergoes subs t a n t i a l l y large motion i n absence of damping. However, a small s t a b i l i z i n g torque i n either or cf> d i r e c t i o n causes a quick reduction in amplitude (Figure 6-2 (a) ) . Increase i n \x^, considerably improves the damping e f f i c i e n c y . The time index may be as small as the o r b i t a l period. 155 3 0 r Figure 6-2 Aerodynamically damped response (i-; =2.0) showing the effects of: m a x (a) proportionality constants Jli=fX2=0.5 Kj =1 .0 , B f =0 .2 6 0 r 30 0 -|60H 4o = 6 0 °4 = 4 5 +o=<tr>0 90 0 -451 J L 3 0 orbits 1 4o'=<to=0 I Figure 6 - 2 Aerodynamically damped response ( x { = 2 . 0 ) showina the e f f e c t s of: max ^ (c) i n i t i a l conditions 158 The effectiveness of t h i s aerodynamic damping concept with reference to s a t e l l i t e s of d i f f e r e n t and B f i s suggested by Figure 6-2(b). It appears that i r r e s p e c t i v e of the transient response, which strongly depends on the system parameters, the time to damp remains r e l a t i v e l y un-affected. F i n a l configuration attained i n each case i s the stable equilibrium p o s i t i o n , which depends on and B f only. As shown i n Figure 6-2(c) the aerodynamic c o n t r o l l e r i s able to damp large disturbances without changing the proportionality constants. Of course the time index, i . e . , time to damp to the prescribed f r a c t i o n of the i n i t i a l amplitude, increases with the magnitude of a disturbance, yet even i n the worse si t u a t i o n considered i t i s limited to three o r b i t s . Bounded response to the normally d e s t a b i l i z i n g disturbance of ^o =^o = 2'^ ( F : " - 9 u r e 6 _2 (c) ) suggests improved s t a b i l i t y . The mechanism appears to be quite e f f e c t i v e i n l i b r a -t i o n a l damping of near-earth s a t e l l i t e s . Its e f f i c i e n c y i n co n t r o l l i n g general motion appears to be, at l e a s t , equal to 44 that of conventional viscous dampers and solar pressure 47-49 s t a b i l i z a t i o n i n planar motion. It i s int e r e s t i n g to note that i f B f, which involves several variable parameters, were adjusted appropriately or the s t a b i l i z i n g torque were controlled not only as a function of l i b r a t i o n a l v e l o c i t y but also of i t s angular displacement, the mechanism could s t a b i l i z e the s a t e l l i t e at any desired orientation. This would represent a simple yet powerful method 159 of attitude control. For eccentric o r b i t s , steady state performance, i . e . , time to damp as well as the l i m i t cycle amplitudes would be of i n t e r e s t . Here also the general effectiveness of the con t r o l l e r appears to be promising. 6.4 Concluding Remarks Aerodynamic damping of l i b r a t i o n a l motion appears to be quite promising. A v e l o c i t y - s e n s i t i v e , semi-passive c o n t r o l l e r can damp even large amplitude motion, both i n and across the o r b i t a l plane, i n less than two o r b i t s . The f i n a l configur-ation attained i s the stable equilibrium one. The time index depends mainly on the proportionality constants and disturb-ances encountered. The s t a b i l i t y bound i s l i k e l y to be enlarged sub s t a n t i a l l y . Several optimization problems a r i s i n g from mechanics, aerodynamics, control, and structual strength considerations do e x i s t , however, they appear to be within the reach of the present l e v e l of technology. 7. CLOSING COMMENTS 7.1 Summary As .indicated at the outset, the main objective of t h i s study has been to gain some insight into the l i b r a t i o n a l response and s t a b i l i t y of the gravity oriented s a t e l l i t e s as affected by system parameters and aerodynamic forces. Emphasis, throughout, has been on generating information suitable for design purposes. The thesis establishes several useful approaches to investigate the problems of autonomous and non-autonomous character with p a r t i c u l a r reference to the l i b r a t i o n a l dynamics. Among the few a n a l y t i c a l techniques available for the study of non-linear coupled problems, Butenin's v a r i a t i o n of parameter method appears to fare well, at least for motion in the small. The success of the method i n predicting amplitude and frequency with acceptable accuracy for both autonomous and non-autonomous systems makes i t i d e a l l y suited to the planning of the s a t e l l i t e control system. A precise,real time simulation using an analog computer i s of considerable importance where cost i s the over-riding consideration. It may prove to be of p a r t i c u l a r significance to the countries l i k e B r a z i l and India which are involved i n the design of communication s a t e l l i t e s to be used for s o c i a l reforms. 1 6 1 The study emphasizes the usefulness of zero v e l o c i t y curves a r i s i n g from constant Hamiltonian associated with the autonomous system. It provides bounds of l i b r a t i o n s , leads to approximate closed form solution for system response and constraints for guaranteed and conditional s t a b i l i t y . The concept of i n t e g r a l manifold used so successfully 7 — 9 2 7-29 by Modi et a l . ' i n the s t a b i l i t y study of the planar system can also be u t i l i z e d i n analyzing an autonomous, coupled system. It provides a l l possible combinations of external disturbances to which a s a t e l l i t e can be subjected, at any point i n i t s o r b i t , without causing i t to tumble. The fact that the degeneration of the invariant surface leads to a periodic solution further enhances the importance of the method. Application of the Floquet theory to the fourth order system helps to establish s t a b i l i t y of the periodic motion as well as the c r i t i c a l disturbance leading to a tumbling motion. Of considerable interest are the three d i s t i n c t l y d i f f e r e n t solutions - regular, islan d type, and ergodic - i n the guaranteed s t a b i l i t y domain. The regular s t a b i l i t y region being the only one usable from p r a c t i c a l considerations, i t s detailed study and the re s u l t i n g design charts represent innovations of far reaching implication. With the study of the non-autonomous case of e l l i p t i c orbits we take a modest step forward i n a f i e l d that has remained, so fa r , unexplored. The addition of atmospheric 162 effects c l e a r l y emphasize the penalty, i n terms of reduced s t a b i l i t y , one must pay to achieve longer l i f e . F i n a l l y the u t i l i z a t i o n of aerodynamic forces to advan-tage through a semi-passive controller represents the f i r s t recorded attempt. The concept presents exciting p o s s i b i l i t i e s of e f f i c i e n t l i b r a t i o n a l damping and attitude control. It i s believed that the information presented here adds to our understanding of s a t e l l i t e attitude dynamics and should prove useful during spacecrafts' design. 7.2 Recommendations for Future Work The investigation reported here brings to l i g h t numer-ous p o s s i b i l i t i e s for extension and innovations. Some of the more important problems are l i s t e d below: (i) The dynamical study of an a r b i t r a r i l y shaped, non-r i g i d s a t e l l i t e would make the analysis more r e a l i s t i c and complete. The magnitude of the d i f f i c u l t i e s , however, increases substantially as the i n e r t i a parameter becomes a time dependent function and X, the rotation about the long axis, no longer remains a c y c l i c coordinate, ( i i ) Slow o r b i t a l decay and l o c a l variations i n atmospheric conditions make the aerodynamic c o e f f i c i e n t a func-tion of time, even i n near-circular o r b i t s . This together with the influence of other environmental disturbances, such as solar and earth radiations, 163 on the general motion of s a t e l l i t e s merit investigation. ( i i i ) In spite of the complex character of the problem, the approximate closed form solutions, i n general, have proved to be of considerable p r a c t i c a l use. Hence a l l e f f o r t s should be made to improve t h e i r accuracy. The use of modified generating functions, better representation of coupling e f f e c t s , series solution, extension of W.K.B.J. Method to fourth order systems, etc., a l l appear promising, (iv) As periodic solutions represent degeneration of int e g r a l manifolds, the p o s s i b i l i t y of reconstruc-ting them from known periodic solutions should be explored. The s t a b i l i t y analysis of periodic 8 9 solution as given Modi and Brereton ' may form a start i n g point for any such attempt. Recognizing the important role played by periodic solutions i n s t a b i l i t y study, t h e i r determination for the non-autonomous case represents l o g i c a l , extension to the study i n Chapter 4. The corres-ponding solution for motion i n a c i r c u l a r o r b i t being known, perturbation analysis may prove e f f e c t i v e , at least for small e c c e n t r i c i t y o r b i t s , (v) For non-autonomous, coupled, conservative system, the concept of the i n t e g r a l manifold f a i l e d to provide useful information, primarily because of the d i f f i -164 culty i n v i s u a l i z i n g a surface i n more than a three dimensional domain. This, indeed, presents a challenging problem. Probably, the method of adiabatic invariants using slowly varying Hamil-tonian may prove to be of some use. (vi) The concept of l i b r a t i o n a l damping using aero-dynamic forces should be extended: (a) to cover non-autonomous s i t u a t i o n (b) to explore i t s effectiveness i n attitude con-t r o l by adding displacement sensitive terms. This presents an inte r e s t i n g p o s s i b i l i t y of chang-ing s a t e l l i t e orientation i n the o r b i t . BIBLIOGRAPHY 1. 'Communication S a t e l l i t e s - a New World Now' s p e c i a l issue Astronautics and Aeronautics, AIAA, Vol. 6, No. 4, A p r i l 1968, p. 32, 71. 2. Klemperer, W.B. " S a t e l l i t e Librations of Large Ampli-tude," ARS Journal, Vol. 30, No. 1, January 1960, pp. 123- 124. 3. Baker, R.M., J r . , "Librations on a S l i g h t l y Eccentric Orbit," ARS Journal, Vol. 30, No. 1, January 1960, pp. 124- 126. 4. Beletskiy, V.V., "The L i b r a t i o n of a S a t e l l i t e on an E l l i p t i c Orbit," Dynamics of S a t e l l i t e s , edited by M. Roy, Academic Press, New York, 1963, pp. 219-230. 5. Schechter, H.B., "Dumbbell Librations i n E l l i p t i c Orbits," AIAA Journal, Vol. 2, No. 6, June 1964, pp. 1000-1003. 6. Zlatousov, V.A., Okhotsimsky, D.E., Sarghev, V.A., and Torzhevsky, A.P., "Investigation of S a t e l l i t e O s c i l l a -tions i n the Plane of an E l l i p t i c Orbit," Proceedings XI International Congress of Applied Mechanics, edited by H. Gortler, Springer-Verlog, B e r l i n , 1964, pp. 436-439. 7. Brereton, R.C., and Modi, V.J., " S t a b i l i t y of Planar L i b r a t i o n a l Motion of a S a t e l l i t e i n E l l i p t i c Orbit," Proceedings XVII International Astronautical Federation  Congress, Gorden and Breach Science Publishers, Inc., New York, 1967, pp. 179-192. 8. Modi, V.J., and Brereton, R.C., "Periodic Solutions Associated with the Gravity-Gradient Oriented System; Part I: A n a l y t i c a l and Numerical Determination," AIAA  Journal, Vol. 7, No. 7, July 1969, pp. 1217-1225. 9. Modi, V.J., and Brereton, R.C., "Periodic Solutions Associated with the Gravity Gradient Oriented System; Part I I : S t a b i l i t y Analysis," AIAA Journal, Vol. 7, No. 8, August 1969, pp. 1465-1468. 10. Brereton, R.C., "A S t a b i l i t y Study of Gravity Oriented S a t e l l i t e s , " Ph.D. Thesis, University of B r i t i s h Columbia, November 1967. 1 6 6 1 1 . 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Flanagan, R.C., and Modi, V.J., Radiation Forces on a F l a t Plate i n E c l i p t i c Near-Earth Orbits," Transactions  of the Canadian Aeronautics and Space I n s t i t u t e , Vol. 3 , September 1 9 7 0 , pp. 2 4 . Sohn, R.L., "Attitude S t a b i l i z a t i o n by Means of Solar Radiation Pressure," ARS Journal, Vol. 2 9 , No. 5 , May 1 9 5 9 , pp. 3 7 1 - 3 7 3 . 2 5 . Ule, L.A., "Orientation of Spinning S a t e l l i t e by Radiation Pressure," AIAA Journal, Vol. 1 , No. 7 , July 1 9 6 3 , pp. 1 5 7 5 - 1 5 7 8 . 2 6 . Clancy, T.F., and M i t c h e l l , T.P., "Effects of Radiation Forces on the Attitude of an A r t i f i c a l Earth S a t e l l i t e s , " AIAA Journal, Vol. 2 , No. 3 , March 1 9 6 4 , pp. 5 1 7 - 5 2 4 . 2 7 . Modi, V.J., and Flanagan, R.C., "Effect of Environmental Forces on the Attitude Dynamics of Gravity Oriented S a t e l l i t e s ; Part I: High A l t i t u d e Orbits," Aeronautical  Journal, the Royal Aeronautical Society, (in press). 2 8 . Modi, V.J., and Flanagan, R.C., "Effect of Environmental Forces on the Attitude Dynamics of Gravity Oriented S a t e l l i t e s ; Part I I : Intermediate Orbits Accounting for Earth Radiations," Aeronautical Journal, the Royal Aeronautical Society"]! (in press) . 2 9 . Flanagan, R.C., and Modi, V.J., "Effect of Environmen-t a l Forces on the Attitude Dynamics of Gravity Oriented S a t e l l i t e s ; Part I I I : Close-Earth Orbits Accounting for Aerodynamic Forces," Aeronautical Journal, the Royal Aeronautical Society, (in press). 3 0 . Flanagan, R.C., "Effect of Environmental Forces on the Attitude Dynamics of Gravity Oriented S a t e l l i t e s , " Ph.D. Thesis, University of B r i t i s h Columbia, July 1 9 6 9 . 1 6 8 3 1 . Schrello, D.M., "Aerodynamic Influence on S a t e l l i t e L ibrations," ARS Journal, Vol. 3 1 , No. 3 , March 1 9 6 1 , pp. 4 4 2 - 4 4 4 . 3 2 . Debra, D.B. "The E f f e c t of Aerodynamic Forces on S a t e l l i t e Attitude," The Journal of Astronautical  Sciences, Vol. 6 , 1 9 5 9 , pp. 4 0 - 4 5 . 3 3 . Beletskiy, V.V., "Motion of an A r t i f i c a l Earth S a t e l l i t e About i t s Center of Mass," A r t i f i c i a l Earth S a t e l l i t e s , Vol. 1 , edited by L.V. Kurnosova, Plenum Press, New York, 1 9 6 0 , pp. 3 0 - 5 9 . 3 4 . Evans, W.J., "Aerodynamic and Radiation Disturbance . Torques on S a t e l l i t e s Having Complex Geometry," Journal  of Astronautical Sciences, Vol. 9 , 1 9 6 2 , pp. 9 3 - 9 9 . 3 5 . Garbar, T.B., "Influence of Constant Disturbance Torques on the Motion of Gravity Gradient S t a b i l i z e d S a t e l l i t e s , " AIAA Journal, Vol. 1 , No. 4 , 1 9 6 3 , pp. 9 6 8 - 9 6 9 . 3 6 . Meirovitch, L., and Wallace, F.B., J r . , "On the E f f e c t of Aerodynamic and Gravitational Torques on the Attitude S t a b i l i t y of S a t e l l i t e s , " AIAA Journal, Vol. 4 , No. 1 2 , December 1 9 6 6 , pp. 2 1 9 6 - 2 2 0 2 . 3 7 . Nurre, G.S., "Effect of Aerodynamic Torque on an Asymmetric Gravity S t a b i l i z e d S a t e l l i t e , " AIAA Journal, Vol. 6 , No. 9 , September 1 9 6 8 , pp. 1 0 4 6 - 1 0 5 0 . 3 8 . Garwin, R.L., "Solar S a i l i n g - A P r a c t i c a l Method of Propulsion Within the Solar System," Jet Propulsion, Vol. 2 8 , No. 3 , March 1 9 5 8 , pp. 1 8 8 - 1 9 0 . 3 9 . Debra, D., "The Large Attitude Motions and S t a b i l i t y due to Gravity of a S a t e l l i t e with Passive Damping i n an Orbit about an Oblate Body," Ph.D. Thesis, Stanford University, June 1 9 6 2 . 4 0 . Kamm, L.J., " V e r t i s t a t : an Improved S a t e l l i t e Orient-ing Device," ARS Journal, Vol. 3 2 , No. 6 , June 1 9 6 2 , pp. 9 1 1 - 9 1 3 . 4 1 . Paul, B., "Planar Librations of an Extensible Dumbbell S a t e l l i t e , " AIAA Journal, Vol. 1 , No. 2 , February 1 9 6 3 , pp. 4 1 1 - 4 1 8 . 4 2 . Modi, V.J., and Brereton, R.C., "The Planar Motion of a Damped Gravity Gradient S t a b i l i z e d S a t e l l i t e , " Transactions of the Canadian Aeronautics and Space  In s t i t u t e , Vol. 2 , No. 1 , March 1 9 6 9 , pp. 4 4 - 4 8 . 169 43. Tschann, C , and Modi, V.J. , "Linearized Analysis of a Damped Gravity Oriented S a t e l l i t e , " Aeronautical  Journal, the Royal Aeronautical Society"^ (in press) . 44. Tschann, C , and Modi, V.J., "A Comparative Study of two C l a s s i c a l Damping Mechanisms for Gravity Oriented S a t e l l i t e s , " Transactions of the Canadian Aeronautics  and Space I n s t i t u t e , (in press). 45. Paul, B., West, J.W., and Yu, E.Y., "A Passive Gravi-t a t i o n a l Attitude Control System for S a t e l l i t e s , " The  B e l l System Technical Journal, Vol. 42, September 1963, pp. 2195-2238. 46. Mallach, E.G., "Solar Pressure Damping of Librations of Gravity Gradient Oriented S a t e l l i t e , " AIAA Student  Journal, Vol. 4, No;. 4, December 1966, pp. 143-147. 47. Modi, V.J., and Flanagan, R.C., " L i b r a t i o n a l Damping of Gravity Oriented System Using Solar Radiation Pressure," Aeronautical Journal., the Royal Aeronautical Society, (in press). 48. Modi, V.J., and Tschann, C , "On the Attitude and L i b r a t i o n a l Control of a S a t e l l i t e Using Solar Radiation Pressure," to be presented at XXI International Astro-nautical Federation Congress, Constance, German Federal Republic, October 1970. 49. Tschann, C , and Modi, V.J., "Solar Radiation Damping of a Gravity Oriented S a t e l l i t e Using W.K.B. Method," (presented for publication). 50. 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