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Effect of solar radiations on the attitude dynamics of gravity oriented satellites Kumar, Krishna 1972

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EFFECT OF SOLAR RADIATIONS ON THE ATTITUDE DYNAMICS OF GRAVITY ORIENTED SATELLITES by KRISHNA KUMAR B.Sc, University of Allahabad, India, 1964 B.Tech. (Hons.), I.I.T. Kharagpur, India, 19 68 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF May, BRITISH COLUMBIA 1972 In presenting this thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that publication, i n part or i n whole, or the copying of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. KRISHNA KUMAR Department of Mechanical Engineering The University of B r i t i s h Columbia, Vancouver 8, Canada . Date ABSTRACT Influence of the solar radiations on the attitude dynamics of gravity gradient s a t e l l i t e s i s examined and a p o s s i b i l i t y of using the minute solar force to advantage for general attitude control i s explored. A s i m p l i f i e d preliminary analysis emphasizes the s i g n i f i c a n c e of f l e x i b i l i t y i n this class of problems. As the governing equations of motion with periodic c o e f f i c i e n t s are highly non-linear, non-autonomous and involve a large number of system variables, the subject i s explored, using approximate a n a l y t i c a l as well as numerical techniques, i n several stages representing an increasing degree of generalization. The f i r s t phase deals with the planar attitude dynamics of c y l i n d r i c a l s a t e l l i t e s i n an a r b i t r a r y e c l i p t i c o r b i t . The approximate closed form solution, obtained using the W.K.B.J, and Harmonic Balance methods, i s found to predict the l i b r a t i o n a l response with considerable accuracy for small amplitude motion. The concept of i n t e g r a l manifold has been e f f e c t i v e l y applied to obtain the s t a b i l i t y bounds of the system. Furthermore, analysis conclusively shows that the solar pressure, which, i n general, affects the s a t e l l i t e performance adversely can, in conjunction with a suitable c o n t r o l l e r , achieve l i b r a -t i o n a l damping and attitude c o n t r o l . Effectiveness of the c o n t r o l l e r i n a c i r c u l a r o r b i t i s f i r s t investigated through several approximate a n a l y t i c a l methods, which are found to predict the damping trends quite accurately. This i s followed by a comprehensive numerical analysis leading to the control system design data for s a t e l l i t e s i n an a r b i t r a r y o r b i t . In the next stage, the study i s extended to a more general s i t u a t i o n of coupled motion. Even for th i s complex problem, i t i s possible to obtain approximate a n a l y t i c a l solutions suitable for preliminary design purposes. The e f f e c t of various parameters on s a t e l l i t e response i s studied through the "system p l o t s " . The s t a b i l i t y charts have been obtained which indicate the bounds of s a t e l l i t e operation for non-tumbling motion. The attention i s then focused on the p o s s i b i l i t y of achieving s p a t i a l attitude control through the solar radiation pressure. The study leads to the development of a semi-passive c o n t r o l l e r capable of imparting a great degree of v e r s a t i l i t y to communications s a t e l l i t e s and space stations of the future. F i n a l l y , an attempt has been made to investigate the planar l i b r a t i o n a l dynamics of a s a t e l l i t e with appendages undergoing e l a s t i c deformations induced by the d i f f e r e n t i a l solar heating. The s t r u c t u r a l f l e x i -b i l i t y , i n general, adversely affects the l i b r a t i o n a l response and s t a b i l i t y . TABLE OF CONTENTS Chapter Page 1. INTRODUCTION 1 1.1 Preliminary Remarks 1 1.2 A B r i e f Review of the Literature . . . 2 1.3 Purpose and Scope of the Investigation 10 2. EFFECT OF THE SOLAR RADIATION PRESSURE ON THE PLANAR ATTITUDE DYNAMICS OF SATELLITES 14 2.1 Formulation of the Problem 15 2.2 System Response 21 2.2.1 A n a l y t i c a l and Numerical Approaches 21 2.2.2 Response Plots and Discussion of Results 27 2.2.3 Analog Simulation 37 2.3 L i b r a t i o n a l S t a b i l i t y 39 2.4 L i b r a t i o n a l Damping and Attitude Con-t r o l Using the Solar Radiation Pressure 45 2.4.1 Approximate A n a l y t i c a l Solutions for S a t e l l i t e s i n C i r c u l a r Orbits: Y I |=0, c m a x=oo . . . . 46 2.4.2 Accuracy of the Approximate A n a l y t i c a l Results 54 2.4.3 Numerical Results: Motion i n a C i r c u l a r Orbit 58 2.4.4 Numerical Results: Motion i n an Eccentric Orbit 63 2.4.5 A F e a s i b i l i t y Study 70 Chapter v Page 2.5 Concluding Remarks 81 3. COUPLED LIBRATIONAL DYNAMICS AND GENERAL ATTITUDE CONTROL OF SATELLITES IN THE PRESENCE OF THE SOLAR RADIATION PRESSURE. . . 85 3.1 Coupled L i b r a t i o n a l Dynamics 86 3.1.1 Formulation of the Problem . . . 86 3.1.2 Approximate A n a l y t i c a l and Numerical Approaches 9 6 3.1.3 System Response and Discussion of Results 104 3.1.4 L i b r a t i o n a l S t a b i l i t y . . . . . . 114 3.2 L i b r a t i o n a l Damping and Spatial Attitude Control Using the Solar Radiation Pressure 114 3.2.1 Proposed Controller Model and the Equations of Motion 116 3.2.2 Results and Discussion 121 3.3 Concluding Remarks 133 4. PLANAR LIBRATIONS OF A SATELLITE WITH FLEXIBLE APPENDAGES . . . . 135 4.1 Formulation of the Problem 136 4.2 Thermal Deflection of the Appendages . . 14 3 4.3 Results and Discussion 151 4.3.1 System Response 151 4.3.2 S t a b i l i t y 154 4.4 Concluding Remarks 155 5. CLOSING COMMENTS 158 5.1 Summary of the Conclusions 1 5 8 5.2 Recommendations for Future Work 160 BIBLIOGRAPHY 1 6 3 LIST OF FIGURES Figure Page 1.1 Schematic diagram of the proposed plan of study 13 2.1 Geometry of the planar l i b r a t i o n a l motion. . 16 2.2 The solar radiations and the s a t e l l i t e geometry 18 2.3 Vari a t i o n of the functions |F| and | j | m a x with system parameters 25 2.4 I n s t a b i l i t y induced by the solar r a d i a t i o n pressure, s a t e l l i t e i n e r t i a , and o r b i t a l e c c e n t r i c i t y 28 2.5 E f f e c t of the radiation pressure on the l i b r a t i o n a l response of a s a t e l l i t e obtained using a n a l y t i c a l and numerical methods: (a) s a t e l l i t e motion i n a c i r c u l a r o r b i t , ; e=0; 30 (b) s a t e l l i t e motion i n an eccentric o r b i t , e=0.1 31 2.6 System plots showing the maximum l i b r a t i o n a l amplitude and average period for a range of e c c e n t r i c i t y as affected by: (a) the solar parameter; 33 (b) the i n e r t i a parameter and the aspect r a t i o ; 34 (c) the solar aspect angle and i n i t i a l s a t e l l i t e p o sition 8Q at the instant of disturbance; 35 (d) i n i t i a l disturbances ipQ and 1 36 2.7 Analog simulation c i r c u i t using 9 as an independent variable . 38 2.8 A comparison of the system response obtained through analog simulation with d i g i t a l computer results 40 2.9 A t y p i c a l l i m i t i n g i n t e g r a l manifold . . . . 43 v i i Figure Page 2.10 (a) a family of l i m i t i n g manifold sections at 6=0 44 (b) a s t a b i l i t y chart showing the e f f e c t of the solar parameter and ec c e n t r i c i t y 44 2.11 A t y p i c a l v a r i a t i o n of the c h a r a c t e r i s t i c function J (x) 51 a 2.12 The solar damping of a 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 as predicted by a n a l y t i c a l and numerical methods: (a) small impulsive Q'=0.5, ^Q=0) or angular (^Q=10°, ^Q'=0) disturbances; . 56 (b) large impulsive (^^'=1.0, ^Q=0) o r angular (^Q=30O, ^Q'=0) disturbances. . 57 2.13 Typical examples of the s a t e l l i t e response i n a c i r c u l a r o r b i t showing the e f f e c t of: (a) c and i n i t i a l disturbances, max V*'- 59 (b) and i n i t i a l disturbances, Y ^ T T / 2 ; 60 • 61 ( c ) a n d v V 0 , 5 2 ' 2.14 System plots showing the damping character-i s t i c s of the 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 as affected by: (a) y^ and v^, y^O; 64 (b) c and K. , y .=0; 64 max I ' 4> (c) y^ and v^, y^=ti/2} 65 (d) and y^ denoting the allowable minimum for stable operation, 6 5 (e) y^ and v^, Y^=0.52; 66 (f) c and K. , Y ,=0.52; 66 max l ' ' (g) Y^ , and K. 67 v i i i Figure Page 2.15 E f f e c t of e and ^  on the s a t e l l i t e response i n eccentric o r b i t s 68 2.16 System plots showing damping characteris-t i c s of the s a t e l l i t e i n an eccentric o r b i t as affected by: (a) (b! (c (d (e (f (g (h (i (J (a (b (c cmax a n d V V 0 ' ' ' ' ^ a n d v v 0 ; • * • K. and u^, y^O; . . . K. and e, Y =0; . . . . c and e, Y =0, v,=0; max V ^ c and e, Y=0, v.=10; max ' ' 4> cmax a n d e ' V * 7 2 ' * . . K., and v^, Y ^ = T T / 2 ; cmax a n d V Y^=0.52; . e and y 2.17 A f e a s i b i l i t y study: geometry of the proposed c o n t r o l l e r con-fi g u r a t i o n ; schematic diagram of the c o n t r o l l e r servosystem; v a r i a t i o n of the dynamical state during control operation 71 71 72 72 73 73 74 74 75 75 77 78 80 3.1 The s a t e l l i t e undergoing general s p a t i a l motion 87 3.2 Comparison between coupled l i b r a t i o n a l response of the s a t e l l i t e as obtained using a n a l y t i c a l and numerical methods . . 10 5 3.3 System plots showing the. maximum amplitude and average period as affected by: (a) the solar parameter; (b) the i n e r t i a parameter; (c) the aspect r a t i o ; . . (d) the solar aspect angle; (e) (f) the i n i t i a l disturbance 1; the i n i t i a l disturbance 3Q1 108 108 109 109 110 110 ix Figure Page 3.4 Approximate estimate of i n i t i a l conditions for periodic (period=2Tr) l i b r a t i o n s 112 3.5 Typical examples of approximate periodic l i b r a t i o n s 113 3.6 A s t a b i l i t y p l o t showing the e f f e c t of the solar parameter on allowable impulsive disturbances . 115 3.7 Schematic diagram of the s a t e l l i t e -c o n t r o l l e r : (a) equilibrium; 117 (b) instantaneous control configuration; e=<j>, ij,=g=A=10°, ijj » = 3 '=V=0 .1 H7 3.8 Typical examples of the general attitude control using the proposed solar c o n t r o l l e r i n : (a) a c i r c u l a r o r b i t , i=0; 123 (b) a c i r c u l a r o r b i t , i=23.5°; 124 (c) an eccentric o r b i t , e=0.1, i=23.5°. . . . 125 3.9 System plots showing damping c h a r a c t e r i s t i c s of the s a t e l l i t e for attitude control along the l o c a l v e r t i c a l , Y .=0 126 1 3.10 System plots showing the attitude control c h a r a c t e r i s t i c s as affected by: (a) :v and y^; 128 (b) K. and Y 129 l j 3.11 Solar r a d i a t i o n control system for the next generation of space vehicles: (a) communications s a t e l l i t e ; 130 (b) space st a t i o n 130 3.12 Geometry of earth shadow 132 4.1 Geometry of a s a t e l l i t e with f l e x i b l e appendages 137 X Figure Page 4.2 (a) coordinate system for thermal analysis 147 (b) thermal d e f l e c t i o n of the panel 14 7 4.3 Solar radiation induced temperature d i f f e r -ence across the f l a t plate appendage-thickness 14 8 4.4 System plots showing the maximum l i b r a t i o n a l amplitude and average period as affected by: (a) the i n e r t i a parameter; 152 (b) the f l e x i b i l i t y and appendage-size parameter 153 4.5 S t a b i l i t y charts showing e f f e c t of the solar parameter, the f l e x i b i l i t y parameter and e c c e n t r i c i t y 156 ACKNOWLEDGEMENT The author wishes to express his profound sense of gratitude to Dr. V.J. Modi for guidance given through-out the preparation of the th e s i s . His help and encourage-ment have been invaluable. The inves t i g a t i o n reported i n this 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 appendage half-thickness, Figure 4.2 (a) amplitudes of in-plane and out-of-plane l i b r a t i o n s i n the complementary functions, respectively, equation(3.12) width and length of appendages, respectively, Figure 4 . 2 (b) 3 solar parameter, R SAh (l+p/3-x)/(yc'I) P P ve l o c i t y of l i g h t c(8+4g)/3TT ±c, ±g, respectively, equation (2.11) physical l i m i t on c,,c„,c^ as imposed by the co n t r o l l e r design ^ RpSA p£ 1(l+p-T)/(yc'I) RpSA p£ 2(l+p-T)/(yc ,I) RpSApm1 (l + p-x)/(yc ,I) e c c e n t r i c i t y of o r b i t non-linear functions, equation (3.11) approximations to f^, f^i respectively, equation (3.14) aspect r a t i o , (iTr/2h) • (1-p-x) / (l + p/3-x) functions of 6, equation (4.14) length and radius of a c y l i n d r i c a l s a t e l l i t e , respectively length and radius of the c y l i n d r i c a l aluminized mylar surface fFigure 2.17(a) distance of the geometric centre of the s a t e l l i t e from the mass centre angular momentum per unit mass of the s a t e l l i t e i n c l i n a t i o n of the o r b i t a l plane with respect to the e c l i p t i c unit vectors along x,y,z-axes, respectively, Figure 2 . 2 distance of the geometric centre of the f i r s t set of c o n t r o l l e r plates from s a t e l l i t e mass centre along -z.^ and y 2~axes, respectively, Figure 3.7 distance of the geometric centre of the second set of c o n t r o l l e r plates G^ from s a t e l l i t e mass centre along Z 2 ~ a x i s , Figure 3.7 position of a representative element of the i t n appendage as measured along the length of i t s deflected configuration mass per unit length of the i t h appendage mass of the appendages non-dimensional mass of the appendages, nip/ms t o t a l mass of the s a t e l l i t e OK^1/2, (3K±+1 ) 1 / / 2 , respectively ( l - p - x ) / { 2 ( 1 + p - T ) } mass centre of the s a t e l l i t e body (without appendages) the instantaneous mass centre of the s a t e l l i t e with f l e x i b l e appendages time unit vectors i n the x-y plane normal and par a l l e to the representative element, Figure 2 . 2 unit vector along the z-axis, Figure 2 . 2 p r i n c i p a l coordinates of the s a t e l l i t e body with o r i g i n at the mass centre xiv x0' y0' Z0 x l ' y l ' Z l '2'y2'*2 J intermediate body coordinates with o r i g i n at the centre of mass during the Eulerian rotations ip,B,X, respectively, Figure 3.1 rectangular coordinate system with x normal to the set of plates , Figure 3.7 ^ x ,y , z s a s ' s rectangular coordinate axes with y s_ z s plane d i v i d i n g the c y l i n d r i c a l surface into dark and bright halves A. . 1D A diametral cross sectional area of the s a t e l l i t e , 2rh amplitude of the j"*"*1 normal mode of the i t l a appendage t o t a l area of each set of the c o n t r o l l e r plates 11 I m . l i mass moment of i n e r t i a of a r i g i d c y l i n d r i c a l s a t e l l i t e about i - a x i s , I =1 =1 xx yy mass moment of i n e r t i a of the s a t e l l i t e body about i - a x i s , I., when the s a t e l l i t e i s e n t i r e l y r i g i d K appendage-size parameter, m^£ /{I +(l/3)nip£ } thermal conductivity of the appendage material K. l i n e r t i a parameter, (1-1 /I) z z n1[l-{(a2/4)+(1/2)cf2/(n2-l)2}/{l-a2/8)}] n 2 [ l - { ( a 2 / 4 ) + ( 1 / 4 ) c f 2 / ( n 2 ( n 2 - l ) ) } ] MlfM2,M3 components of the t o t a l r a d i a t i o n moment, equation (3.7) M ,M_ ,M_ l c ' 2c' 3c components of the moment due to the radiatio n force on the cylinder surface, equation (3.5) M, ,M- components of the moment due to the radiatio n lp 3p force on the cylinder faces, equation (3.6) X V generalized forces, i=ip,3,A generalized force for j t n normal mode of the i ^ appendage f l e x i b i l i t y parameter, (1/2)(£ a a S/K) cl t distance from the centre of force to the s a t e l l i t e centre of mass earth radius perigee distance length of the earth shadow i n the o r b i t a l plane solar constant S |sin (9-Hf>-<j>) I temperature at any point of the appendage maximum and minimum temperature of the appendage, respectively average l i b r a t i o n a l period as a f r a c t i o n of the o r b i t a l period; T a V / ^ j and Tav,3 r o r p i t c h and r o l l degrees of freedom,respectively, i n the case of coupled l i b r a t i o n s time-index, the time to damp to ±0.5° k i n e t i c energy potential energy complementary functions i n the W.K.B.J, solution (1+e Cos Q)i>, Y +Y Y ' c p complementary and p a r t i c u l a r solutions i n the W.K.B.J, analysis, respectively angle between the incident radiations and the x-axis, Figure 2.2 absorptivity of the appendage material i n c l i n a t i o n of the i^-h appendage with respect to the axis of symmetry, Figure 4.1 x v i c o e f f i c i e n t of thermal expansion of the appendage material emissivity of the appendage material Sin c() (1-Cos i) , -Sin i Sin <J>, respectively, p o s i t i o n control angles, j=i^,3,X angle representing d e f l e c t i o n of the appendage, Figure 4.2 (b) phase angles i n the Butenin solution, equation (3.12) n.6+6., j=l,2 3 3 thermal d e f l e c t i o n of a representative element of the i t h appendage position of a representative element of the i t h appendage as measured from the mass centre s s a t e l l i t e p o s i t i o n angle as measured from the perigee c o n t r o l l e r gains, j=iJj,g,A rectangular coordinate system used i n the thermal analysis, Figure 4.2(a) r e f l e c t i v i t y and transmissivity of the s a t e l l i t e material, respectively radius of curvature of the appendage, Figure 4.2(b) posi t i o n angle of a representative element of the s a t e l l i t e ' s curved surface, Figures 2.2, 3.1 (I T / 2 J+Tan" 1!-Sin (6+iJj-<J))/{Sin 3 Cos (9+i|>-<|>) } ] Stefan-Boltzmann constant rotation i n the o r b i t a l plane, across the o r b i t a l plane and about the axis of symmetry, respectively, Figure 3.1 natural frequency of the j^h normal mode of the i t h appendage x v i i solar aspect angle function describing the j appendage th mode of the i i th 2 (AT)n,(AT)„ approximate values of the temperature difference AT, equations (4.11) and (4.12), respectively Dots and primes represent d i f f e r e n t i a t i o n with respect to t and 6, respectively, unless otherwise defined; subscript 0 refers to i n i t i a l conditions. 1. INTRODUCTION 1.1 Preliminary Remarks The e x c i t i n g new era of space exploration has brought to the forefront many problems i n science and technology. Demands of high speed and energy, accurate guidance and control, operation i n the h o s t i l e environment, r e l i a b i l i t y and a host of other considerations present a formidable i challenge. Although the s c i e n t i f i c e f f o r t s have had spec-tacular successes, our quest for unlocking the mysteries of nature has only been i n i t i a t e d . Among the numerous facets of the space age, communi-cations, exploration of earth-resources, navigation, weather-forecasting, surveillance of p o l l u t i o n and defence are l i k e l y to involve and a f f e c t a major portion of humanity. The success of these space-missions often demands a preferred orientation for a s a t e l l i t e r e l a t i v e to the earth. Several methods are available for attitude control which may be broadly c l a s s i f i e d as passive or active. Any active s t a b i l i z a t i o n procedure involves expenditure of energy, an expensive commodity aboard a spacecraft, leading to an increase i n weight and space requirements with a reduced s a t e l l i t e l i f e - s p a n . The main advantage of the technique i s i t s a b i l i t y to achieve a s p e c i f i e d o r i e n t a t i o n 2 with almost any desired degree of accuracy. Passive techniques, on the other hand, do not consume any power and depend for t h e i r operation on the nat u r a l l y available forces. Among these methods, the one re l y i n g on the gravity gradient represents a conceptually simple and at t r a c t i v e p o s s i b i l i t y . The key to t h i s s t a b i l i z a t i o n p r i n c i p l e l i e s i n the v a r i a t i o n of the g r a v i t a t i o n a l f i e l d over physical dimensions of a s a t e l l i t e r e s u l t i n g i n a restoring moment tending to al i g n i t s 'long axis' (the 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 . The accuracy of thi s manner of control, however, i s li m i t e d and further deteriorates due to the perturbing influence of the environmental forces i n the form of micrometeorite impacts, radiation pressure, earth's magnetic f i e l d intern-actions, aerodynamic forces, etc. Design of a s a t e l l i t e , capable of accomplishing i t s space-mission, demands a thorough understanding of the influence of the environmental forces on i t s dynamical behaviour. This thesis attempts to analyze the e f f e c t of solar radiations on the l i b r a t i o n a l performance of gravity oriented spacecrafts. 1.2 A Bri e f Review of the Lite r a t u r e Any attempt at a comprehensive review of the vast body of the ex i s t i n g l i t e r a t u r e i s considered unwarranted 3 and beyond the scope of the present t h e s i s . Fortunately, several excellent papers are available which survey s p e c i f i c aspects of s a t e l l i t e dynamics quite thoroughly.''' ^ The attention i s , therefore, focused here on the more s i g n i f i -cant contributions having d i r e c t relevance to the investigation i n hand. A l i t e r a t u r e search reveals considerable i n t e r e s t i n the attitude dynamics of the gravity oriented s a t e l l i t e s , p a r t i c u l a r l y during the past decade. The pioneering work on the gravity gradient s t a b i l i z a t i o n was c a r r i e d out by Klemperer 7 (1960), Baker 8 (1960), Beletsky 9 (1963) and Schechter^ (1964) who undertook t h e o r e t i c a l planar analyses of the s a t e l l i t e l i b r a t i o n s . A s i g n i f i c a n t study by Zlatousov et a l . " ^ (1964) investigated the problem of s t a b i l i t y using the concept of stroboscopic phase plane. 12 Modi and Brereton (1966) used the W.K.B.J, method to predict the amplitude and frequency of l i b r a t i o n s with a 13 14 considerable accuracy. Furthermore, the authors ' (1966-67) successfully employed numerical methods involv-ing use of the stroboscopic phase plane,to analyze the motion in the large for orbits of a r b i t r a r y e c c e n t r i c i t y . They 15 16 also investigated the corresponding periodic solutions ' (1969) and showed that at 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. Anand et 17 a l . (19 71) e s s e n t i a l l y followed these tracks to examine the s a t e l l i t e response i n an eccentric o r b i t using a 18 perturbation approach. Clark (19 70) investigated the response of a two-body gravity gradient system i n a near-c i r c u l a r o r b i t . An approximate a n a l y t i c a l technique was 19 used by Puri and Bainum (19 71) to study the system behaviour during the deployment of gravity gradient booms. 20 Kane (1965) pointed out that a strong coupling exists between the planar and the transverse degrees, thus emphasizing a need for the detailed i n v e s t i g a t i o n of the general s p a t i a l motion. These findings were further 21 substantiated by Breakwell and Pringle (1966). Modi and 22 Brereton (1968) studied the coupled l i b r a t i o n a l dynamics of a dumbbell 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 was extended by Modi and Shrivastava (1969-72) for 23 — 26 a r b i t r a r i l y shaped axisymmetric s a t e l l i t e s i n c i r c u l a r 2 7 as well as eccentric o r b i t s . Another important procedure for s t a b i l i z a t i o n i s through spinning, thus turning the entire s a t e l l i t e into a gyroscope. In view of the complexity of the equations of motion governing the dynamics of slowly spinning, r i g i d 2 axisymmetric s a t e l l i t e s i n the g r a v i t a t i o n a l f i e l d , Thomson (1962) presented a s t a b i l i t y c r i t e r i o n using the l i n e a r i z e d 29 analysis. Pringle (1964) investigated the s t a b i l i t y of equilibrium positions of a spinning axisymmetric body, i n a c i r c u l a r o r b i t , and established the bounds of possible 30 motion through zero v e l o c i t y curves. Kane and Barba 5 (1966) attempted to study the motion i n an e l l i p t i c o r b i t 31 using the Floquet theory while Wallace and Meirovxtch (1967) analyzed the same problem, with a questionable success, using an asymptotic analysis i n conjunction with Lyapunov's d i r e c t method. Modi and Neilson (1968) studied the r o l l dynamics of a spinning s a t e l l i t e using the 32 33 W.K.B.J. and numerical methods. The concept of i n t e g r a l manifold was successfully applied to gain an insight into 34 35 the problem of s t a b i l i t y (1969) and periodic solutions (1972). According to the c l a s s i c a l mechanics, the stable r o t a t i o n a l motion of a r i g i d body i n the absence of external forces i s possible only i f the axis of rotation i s a p r i n c i p a l axis of l e a s t or greatest i n e r t i a . If the body i s not r i g i d and energy i s dissipated through c y c l i c forces during nutation, only the motion about the axis of maximum i n e r t i a i s stable. The constraint of the "major axis spin r u l e " was subsequently removed by the introduction of the dual-spin concept which allows two sections to nominally rotate about a common axis at d i f f e r e n t rates r e l a t i v e to the i n e r t i a l space. The f i r s t fundamental contribution to the f e a s i b i l i t y of thi s approach was by 3 6 Landon and Stewart (1964). The concept was developed further by I o r i l l o 3 7 (1965), L i k i n s 3 8 (1967), M i n g o r i 3 9 (1969), P r i n g l e 4 0 (1969) and Bainum et a l . 4 1 (1970). The presence of perturbing environmental forces 42 has been discussed i n some d e t a i l by Roberson (1958), Wiggins 4 3 (1964), Clancy and M i t c h e l l 4 4 (1964), et a l . At higher a l t i t u d e s , the solar radiation pressure becomes quite s i g n i f i c a n t . A r e l a t i v e l y recent study by Flanagan 45 and Modi (1970) established that the only predominant force above 6,0 00 miles i s due to the solar r a d i a t i o n pressure. The perturbing e f f e c t of t h i s force on s a t e l l i t e 46 47 orb i t s was.investigated by Parkinson (1960), Musen (1960), B r y a n t 4 8 (1961), K o z a i 4 9 (1961), et a l . Modi and Flanagan i n i t i a t e d a simple study of the planar l i b r a t i o n a l dynamics of a f l a t plate s a t e l l i t e under the influence of 50 the solar radiation force (1971). The environmental e f f e c t was found to be, i n general, detrimental to the s a t e l l i t e performance. However, i t was suggested that an appropriate control of the solar torque can be u t i l i z e d to maintain a s a t e l l i t e i n a desired att i t u d e . 51 Garwin (1958) was probably the f i r s t one to explore the p o s s i b i l i t y of employing the minute force to advantage 52 through "solar s a i l i n g " for interplanetary t r a v e l . Sohn (1959) suggested a s p e c i f i c configuration using the plates of large surface area to orient a s a t e l l i t e with respect 53 to the sun while Ule (1963) considered i t s application to spin an array of mirrors to achieve attitude s t a b i l i t y . 54 Galitskaya and Kiselev (1965) studied the p r i n c i p l e of 7 l i b r a t i o n a l control of space probes about three axes. The qu a l i t a t i v e study provided useful information about design and control of vanes although no attempt was made to solve the precise equations of motion. Almost simultaneously, 55 Mallach (1966) proposed a system for solar damping of a gravity oriented s a t e l l i t e . His phase plane analysis i s oversimplified through l i n e a r i z a t i o n and the use of average torques. Modi and Flanagan"^ (1971) examined the planar attitude control of a gravity oriented s a t e l l i t e i n an e c l i p t i c o r b i t using the solar torque to achieve damping through a c o n t r o l l e r of the form c=c(ip'). A generalization of t h i s c o n t r o l l e r concept (c=c(^, Y^) ) w a s l a t e r 57 proposed by Modi and Tschann (1971). Reference should be made to an important class of problems involving a multibody system. Apart from i t s proposed use as, an o r b i t i n g interferometer, the concept has gained considerable si g n i f i c a n c e due to i t s obvious relevance to future space stations and non-polluting power generation devices i n space. Of p a r t i c u l a r i n t e r e s t i s the s i g n i f i c a n t contribution by Stuiver et a l . " ' 8 ^ who have analyzed deployment dynamics and small amplitude l i b r a t i o n s of a tethered system, force analysis of a dumb-b e l l s a t e l l i t e and configuration control of a dual body. It i s int e r e s t i n g to recognize that the orient a t i o n control of a multibody system and the solar c o n t r o l l e r 8 concept studied here have a degree of s i m i l a r i t y . Another area of i n t e r e s t has been the l i b r a t i o n a l dynamics of gravity oriented s a t e l l i t e s accounting for the i r e l a s t i c character. A need for such an inv e s t i g a t i o n was f i r s t f e l t when the anomalous behaviour of E x p l o r e r - I 6 2 (1958), Explorer-XX 6 3 (1963) and A l o u t e t t e - I 6 4 (1967) was attributed to t h e i r f l e x i b l e appendages. Ejver since the influence of s t r u c t u r a l f l e x i b i l i t y has 65 received considerable attention. Dow et a l . (1966) presented results of the extremely elaborate analog and d i g i t a l simulations of the f l e x i b l e booms c a r r i e d by the radio astronomy s a t e l l i t e (RAE). Attempt was also made to simulate the system response and check these re s u l t s 6 6 against the f l i g h t data (1968). The National Aeronautics and Space Administration's space vehicle design c r i t e r i a 5 document (1969) presents an excellent review of the work in t h i s f i e l d and puts the current and anticipated future 6 7 problems in proper perspective. The paper by N o l l et a l . (1969) i s e s s e n t i a l l y a summary of the NASA document. A variety of approaches have been proposed for formulation of the governing equations of motion accounting for 6 8 - 7 2 str u c t u r a l f l e x i b i l i t y , although no comprehensive attempt has been made to solve them, perhaps, due to t h e i r complexity. Ef f e c t s of thermal d i s t o r t i o n on the dynamics of the f l e x i b l e members of the spacecrafts was investigated, 9 73 74 in some d e t a i l s , by Beshara (1966), F r i s c h (1967), Y u 7 5 (1969), Vigneron 7 6 (1970), Graham 7 7 (1970), Ahmed 7 8 (1971), et a l . Its influence on the dynamic behaviour of spinning s a t e l l i t e s was studied by Et k i n , Hughes and 7 9 — 81 Cherchas (1965-69). In general, the spin decay due to a loss of energy through e l a s t i c i t y led to undesirable nutation of the spin axis. E f f e c t of thermal bending of booms on the l i b r a t i o n a l performance of gravity gradient 8 2 systems was analyzed by Modi and Brereton (1968), Kanning (1969) and England (1969). The analysis showed a s i g n i f i c a n t influence of the thermal deformations on the s a t e l l i t e ' s response to external disturbances. The long range e f f e c t was a substantial reduction i n the s t a b i l i t y region and the corresponding c r i t i c a l e c c e n t r i c i t y 85 for non-tumbling motion. A recent paper by Hughes (19 72) studies c h a r a c t e r i s t i c modes of twist/pitch o s c i l l a t i o n s associated with a large, f l e x i b l e solar panel of the proposed Communications Technology S a t e l l i t e (CTS) . The author has also indicated a procedure for including the f l e x i b i l i t y e f f e c t s through a modified control system simulation. These investigations, however, mark only a beginning. With a constant increase i n the power require-ment for better communication and the associated increase i n size of the solar panels, the problem of f l e x i b i l i t y i s bound to demand more attention of dynamicists i n the coming decade. 1.3 Purpose and Scope of the Investigation From the review of the relevant l i t e r a t u r e , i t i s apparent that the dynamics of gravity oriented s a t e l l i t e s accounting for the environmental forces has considerable ground to cover. Among the natural forces, the solar radiation pressure appears to be predominant at high 50 56 57 a l t i t u d e s . Modi, Flanagan and Tschann ' ' attempted to assess i t s influence on the l i b r a t i o n a l dynamics and control of gravity oriented s a t e l l i t e s i n the e c l i p t i c plane. The results of t h i s preliminary study, confined to a highly s i m p l i f i e d plate model for s a t e l l i t e s i n planar l i b r a t i o n s , did emphasize a need for the generalized treatment of the problem. Such a study with reference to a more r e a l i s t i c c y l i n d r i c a l model for s a t e l l i t e s i n high a l t i t u d e o r b i t s forms the main object of t h i s i n v e s t i g a t i o n . It i s intended here to analyze the problem i n several stages representing, i n general, an increasing order of complexity. The study i s i n i t i a t e d with an inv e s t i g a t i o n of planar l i b r a t i o n a l dynamics of s a t e l l i t e s i n an arbitr a r y e c l i p t i c o r b i t . The system response i s analyzed using approximate a n a l y t i c a l , numerical and analog techniques. The concept of i n t e g r a l manifold has been e f f e c t i v e l y applied to study s t a b i l i t y of the system. Furthermore, analysis conclusively shows that t h i s minute force can be e f f e c t i v e l y used to achieve the planar 1 1 l i b r a t i o n a l damping and attitude c o n t r o l . The concept offers a great degree of v e r s a t i l i t y i n positioning the s a t e l l i t e at any i n c l i n a t i o n to the l o c a l v e r t i c a l through a judicious choice of system parameters. An example sub-stantiates the f e a s i b i l i t y of the concept. In the next phase, the r e s t r i c t i o n of a planar motion i s removed thus increasing, s u b s t a n t i a l l y , the complexity of the problem. The vast amount of design information i s condensed i n the form of "system p l o t s " showing the e f f e c t of various parameters on the maximum amplitude and average period of the motion. The s t a b i l i t y plots have been obtained which indicate the allowable impulsive disturbances tyQ1 and 3 Q' for non-tumbling motion. The attention i s then focused on a p o s s i b i l i t y of using the solar r a d i a t i o n pressure to achieve three dimensional l i b r a t i o n a l damping and attitude c o n t r o l . The analysis i s kept quite general to include s a t e l l i t e s i n arb i t r a r y e l l i p t i c , n o n - e c l i p t i c o r b i t s . The study, lead-ing to the development of a semi-passive solar c o n t r o l l e r for the s p a t i a l attitude control, i s p a r t i c u l a r l y relevant to the next generation of communications s a t e l l i t e s and future space stations. F i n a l l y , an attempt has been made to investigate the planar l i b r a t i o n a l dynamics of a s a t e l l i t e with appendages (e.g., solar panels) undergoing thermoelastic deformations induced by the d i f f e r e n t i a l solar heating. The analysis suggests t h e i r strong d e s t a b i l i z i n g influence and emphasizes the fact that i n the dynamic study of future s a t e l l i t e s , f l e x i b i l i t y i s bound to play an important r o l e . 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. I t i s f e l t that the approach represents a coherent program to explore the subject. EFFECT OF SOLAR RADIATIONS ON THE ATTITUDE DYNAMICS OF GRAVITY ORIENTED SATELLITES Planar l i b r a t i o n a l dynamics Planar l i b r a t i o n a l damping and attitude control Coupled l i b r a t i o n a l dynamics General three dimensional l i b r a t i o n a l damping and attitude control L i b r a t i o n a l dynamics accounting for thermoelastic ef f e c t s System response: •W.K.B.J.+ Harmonic Balance •Analog •Numerical L i b r a -t i o n a l s t a b i l i t y •Numerical System response, e=0: Linear A i r y Numerical System response, e^O: Numerical System response, e=0: Linear Butenin •Numerical System response and l i b r a -t i o n a l s t a b i l i t y , e^O: •Numerical System response: •Numerical System response and l i b r a -t i o n a l s t a b i l i t y : •Numerical Figure 1.1 Schematic diagram of the proposed plan of study to 2. EFFECT OF THE SOLAR RADIATION PRESSURE ON THE PLANAR ATTITUDE DYNAMICS OF SATELLITES This chapter deals with the planar l i b r a t i o n a l motion of gravity oriented c y l i n d r i c a l s a t e l l i t e s i n an arb i t r a r y e c l i p t i c o r b i t under the influence of the solar radiation pressure. I t emphasizes the importance of the solar parameter, aspect r a t i o and i n e r t i a parameter on 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 s a t e l l i t e s . As the governing non-linear, non-autonomous d i f f e r e n t i a l equation does not possess a known closed form solution, an approximate study i s undertaken using the 12 86 W.K.B.J. ' and Harmonic Balance methods. The v a l i d i t y of the approximate response data i s assessed through the numerical integration of the exact equation. The concept 13 14 89 of invariant surface ' ' has been successfully applied to develop the s t a b i l i t y charts which provide a considerable insight into the character of the l i b r a t i o n a l motion. S u i t a b i l i t y of using the analog technique, normally preferred for economic, real-time simulation, i s also explored. Next, a p o s s i b i l i t y of employing the minute solar force to achieve the l i b r a t i o n a l damping and attitude control of the s a t e l l i t e i s investigated. The analysis describes the planar motion of the s a t e l l i t e s equipped with the proposed solar c o n t r o l l e r . Apart from a few attempts to obtain the approximate closed form solutions i n c i r c u l a r o r b i t s , the study i s primarily c a r r i e d out using numerical methods. Toward the end the p r a c t i c a l f e a s i b i l i t y of the concept has been investigated and system plots presented which should prove useful i n s e l e c t i n g the appropriate c o n t r o l l e r parameters. 2.1 Formulation of the Problem: Consider a c y l i n d r i c a l s a t e l l i t e o r b i t i n g i n the e c l i p t i c plane about the centre of force 0 and executing planar l i b r a t i o n a l motion (Figure 2.1). Let x,y,z represent the p r i n c i p a l axes of the s a t e l l i t e such that the y-axis i s normal to the o r b i t a l plane. Orientation of the z-axis with respect to the l o c a l v e r t i c a l defines the l i b r a t i o n a l angle ty» Neglecting o r b i t a l perturbations due to the l i b r a t i o n a l motion and the solar pressure (for s a t e l l i t e s having large mass/area ra t i o ) / t h e equation of motion i n the ifi- degree of freedom can be written a s ^ ' ^ ' ^ 7 . ( i+ e Cos 0) \ p - - 2 e SLYI G (i+v|/';+ 3 K, Sunty Cos^ = R b ^ / a i (2.1) For response and s t a b i l i t y analysis, i t i s e s s e n t i a l to evaluate the generalized force Q,. Considering the Figure 2.1 Geometry of the planar l i b r a t i o n a l motion 17 r e f l e c t i o n to be specular, the expression for solar force dF c on a representative element of the curved surface (Figure 2.2) and the corresponding moment dMc about the centre of mass can be written as: dFc = - < 5 r k / c ' ) lCos*| Coso-[Coso<Cos<r(u-P-'r) 5u r -Cos* SLncrC l-p - t jUo -- S t v\<X ( l - P-t)£ "]cUr d H t « - 7 (Sr -k/c' ) Cos<* Coscr [ - f^ lCoscxl CosV(i+pjt) I Cos*| Si/n o- 0- f-tr) + r SLY\CX COS O-(I- f^tr)]do-• ••• (2 • 3) Hence the moment M becomes c = _j ( 5 A / C ) [ ( 2 / 3 ) ^ 1 Cos :oc|{(l+p^) + ( I - f - t ) / ^ 4-Or/4) r Stnoc ( i - p - t ) ] C o s < * Figure 2.2 The solar radiations and the s a t e l l i t e geometry 19 Moment M p due to radiations on the f l a t ends can be shown to be Hja =• T (SY*b/c') SiY\oC Cose* ( l - f - t ; .... (2.5) where b i s the distance of the c y l i n d r i c a l face from the mass centre. From the p r i n c i p l e of v i r t u a l work, i s found to be equal to the t o t a l moment leading to = f a ^ A ^ / 3 c ' ) { ( i + f - ^ ; + ( » -p -* ) /2 } Cos <X [ ( C o s « [ - G Sc™<x] .... (2.6) where Q = l3TTr/fl{^) ( 2 b / ^ - 1 ) . (2.7) [ ) / {o + p-to + a - p - r )/z} ] Noting that: o< = 0 + V - 4 > - TT/Z r(^/2J + ft.,, , ( 2 - f e - l /2 )7r<(e4\|r -4>)^ (2*+ ' l/2 ) TT b = Utyz)-•*>.,, > <2& + i / z ) i r < f f r + ^ - * ) s ( 2 l + 3/4)TT «»•• (2.8) where k i s an integer, the expression for takes the form = ( S A ^ / c ' M n - f / 3 - t ) . [ | S v n ( 9 + ^ - 4 > ; | + 9 (Cos ( 0 + ^ - 6 ) 1 ] Here, g represents the aspect r a t i o (nr/2h) ( 1 - p - x ) / (l+p/3-T) . With t h i s , equation (2.1) takes the form (1+ -e Cos 6) V - 2 e Stv\ 0 ( i + t y ' H 3 Ki Sen Cos ^ = { c ( i + e ) 3 / 0 + e C o s £ ) 3 j S t n ( 0 + ^ - 4 0 . [ I S t * ( e + ^ - 4 > ) l + 3 I Cos <e+Hr-4>)|] where c = { ( R * 5 A ^ ; / C i u c ' i ) ] . ( i + e/s-v 2 1 2 . 2 System Response 2 . 2 . 1 A n a l y t i c a l and Numerical Approaches The non-linear, non-autonomous d i f f e r e n t i a l equation ( 2 . 1 0 ) does not possess a known closed form s o l u t i o n . It i s therefore proposed to obtain an approximate a n a l y t i c a l solution without s u b s t a n t i a l l y a f f e c t i n g the physical character of the system. One of the p o s s i b i l i t i e s of 8 6 attaining t h i s objective i s through the W.K.B.J. analysis. Any attempt at obtaining an a n a l y t i c a l solution would require presenting the equation as (1+ e Cos 0) V - 2 e S i / n G U + V j + 3 K j Siy\ "ty Cos if/ - { c m C i + e ) 3 / n + e Cos©) 5 ] . Sl-n (Q^- <P). ( 2 . 1 1 ) [ Stri(0+\|/-<t>; + g m Cos(e+^-<t>) ] where 2 i T r <(e+Hr-4>;$ {zk + 1/2)* (2 ft + # T < (9 + <- C 2 *. ^  i ) (2£+l )7T< (^+^-4))<^ ( 2 ^ + 3/2)7T (2&+3/^Tr<(e+*M>)^ C2^+2)TT 22 with the matching of the response at ( 0 + ^ r - O » = 2&TT+ - riTT/z k f n being integers. The W.K.B.J, method i s applicable to a l i n e a r system and requires removal of the f i r s t d erivative term. Intro-ducing the transformation Y=i|;(l+e Cos 6), l i n e a r i z i n g the re s u l t i n g equation and neglecting the second and higher order terms i n e, c m , the equation of motion reduces to V " + G 2 ( 0 ) = ? ( 0 ) •••• (2.12) where G 2(0) = 3Kj [ i - eCosQ ( l - 1 / 3 ^ - ( c > n / 3 K , ) ( l + 3 e - 4 e C o s 0 ) . {3>iy\ 2(9-<t>) + 9 ™ Cos Z(Q-<p)} ] P(0) = 2 -e Scr> 0+[cCw,/2) ( l + 3 € - 3 e C o s 0 ; . { 1- Cos Z(Q-<t>) +• 9n, S e n ZC0-4))] ] For the W.K.B.J, method to be successful, i t i s necessary that the inequality 23 F = GZ\(G?/zs)-(SM.(<k%f\ « 1 •••• ( 2 * 1 3 ) be s a t i s f i e d . The complementary solution can then be written as Y c = A i X i ( G ) + A 2 X 2 ( 0 ) .... (2.14) where X , (0) = G"''(01 COS { f0 G(Q)de] Xa (0) = G , / 2 ( 0 ) Sc«n t i q 8 G ( e ) i o } Further s i m p l i f i c a t i o n i n the expressions for and X2 can be achieved through the Binomial expansions of G(9), -1/2 G ' (9) and the consistent assumption concerning the second and higher order terms thus giving: l/2 G ( 9 ) = ( 3 K i ) [ i - f i C o s O C l - l / 3 K ( ) / 2 - ( C r « / 6 K , - ) . (1 + 3 € - 4 e Cos 0j {Sir, 2(0 - 4>) + 3mCbs 2(0-<t)j}J 24 d%) = ( 3 K 0 [1+ eCosG ( 1 - l/3Ki)/4 + <£m/12 fc) . ( i+3 e - Cos e). {s ty» z c e - <D;+8 m Cos 2 (e - d>;J J { G(0)dl0 = ( 3 K i ) [ 0 - e S c n O C i - l/3ki)/2 - ( c m / i 2 k , ) ( i 4 3 e - A e C o s e ; . { C o s 2 ^ + §MSL» 2<t>-Cos2(9-<0) +9wxSln provided that the following inequality i s s a t i s f i e d , J = | e C o s 0 ( i - 1/3K.0 +(c v v 1 /3Ki ) ( l + 3 € - 4 e C o s 0 ) . { S t n 2 ( 0 - * ; + 9 V M C o s 2 ( 0 - 4 » } | < < 1 (2.15) Figure 2.3 shows the maximum values of F and J over a wide range of e c c e n t r i c i t y and solar parameter. For a gravity oriented s a t e l l i t e i n a c i r c u l a r or near- c i r c u l a r o r b i t with a representative value of c-o.l the i n e q u a l i t i e s (2.13) and (2.15) are s a t i s f i e d reasonably w e l l , thus suggesting a p o s s i b i l i t y of a s a t i s f a c t o r y s o l u t i o n . 25 Figure 2.3 0.1 e V a r i a t i o n of the functions with system parameters 0.2 F a n d max 'max 26 The method of Harmonic Balance was applied to obtain a p a r t i c u l a r solution of the form = A 4- 6 S(.y\ 0 + C Cos 0 + D Cos29 -»-E SiY) ZQ (2.16) with the constants given by A , 0 C Z ZCZ -A, c 4 -2C*. C 2 0 ^ 2 C 3 0 C 2 Here, Ci = 3 K | c 2 ^ e ( t - 3 k i ) / 2 . c 3 Cw,(i+3e) {SIY\ 2<f -9^Cc7s2<J)3/z c 7 = C A =. Cyvx(l+3e) {Cos24>+9mSiv^2^3/2. (2.17) Or = c , - c 3 - i A r b i t r a r y constants A.^  and A 2 as determined from the i n i t i a l conditions are: AJL - [lY u/X,(0.)}-{Y^^)/X l(0 aj]-{Yo7x ,V) } + A, = [ {Y 0 /x , (e j j - { v^e . )/x . (e . ) } - Az { x^eu/x .c©, )^ To check the v a l i d i t y of the approximate closed form solution, the equation of motion (2.10) was integrated numerically using an IBM 360-67 d i g i t a l computer. The Adams-Bashforth predictor-corrector procedure with a fourth order Runge-Kutta s t a r t e r was used i n conjunction with a suitable stepsize of 3°. 2.2.2 Response plots and Discussion of Results The d e s t a b i l i z i n g influence of the solar r a d i a t i o n pressure and other system parameters i s v i v i d l y i l l u s t r a t e d i n Figure 2.4. I n s t a b i l i t y induced by an increase i n the value of the solar parameter and e c c e n t r i c i t y or a reduction i n the gravity gradient torque characterized by a smaller emphasizes the importance of these parameters. 28 g = o.5 <t>=o • o = 0 1 c = 0 K i = 1 e = o C =1 K i = 1 e = o A, C = 0 Kj - 0.2 e = o C = 0 Kj = 1 e = o.3 / » • 1 0 1 2 3 4 5 6 0 -O rb i t s Figure 2.4 I n s t a b i l i t y induced by the solar r a d i a t i o n pressure, s a t e l l i t e i n e r t i a , and o r b i t a l e c c e n t r i c i t y 29 Figure 2.5 compares, for several t y p i c a l s i t u a t i o n s , the l i b r a t i o n a l response of a s a t e l l i t e as determined by numerical and a n a l y t i c a l methods. System behaviour i n absence of the radiatio n pressure (c=0), obtained numeric-a l l y , i s also included to help e s t a b l i s h trends. I t i s apparent that for r e l a t i v e l y small disturbances, the two responses agree quite w e l l . As can be expected, the influence of n o n - l i n e a r i t i e s becomes s i g n i f i c a n t at larger amplitudes leading to discrepancies between the two solutions. The r e s u l t s of an extensive analysis involving the e f f e c t of various system parameters and i n i t i a l conditions on the response c h a r a c t e r i s t i c s have been condensed i n the form of "system p l o t s " . Figure 2.6(a) shows the e f f e c t of solar parameter on the maximum amplitude and average period of the motion as given by the two methods. It establishes the area d i s t r i b u t i o n , which af f e c t s c, to be as important a parameter as the i n e r t i a d i s t r i b u t i o n factor K^, normally a c o n t r o l l i n g element i n the design of a gravity oriented system. The W.K.B.J, method predicts | ^ l m a x and T & v quite • accurately for s a t e l l i t e s i n c i r c u l a r o r b i t s . However, the c o r r e l a t i o n deteriorates with an increase i n e c c e n t r i c i t y and for negative c. It i s in t e r e s t i n g to note that i n the case of e l l i p t i c o r b i t s , the minimum l i b r a t i o n a l amplitude does not correspond to c=0. Thus, a suitable choice of c could minimize the l i b r a t i o n a l amplitude. Furthermore, 4.0 2.0 -2.0 ° - -4 .0 4j0 e = 0 C = 0.1 Kj= 1.0 T <)> = 0 l|£ = 0.05 Numer ica I Analytical C = 0 9 = 0.5 g=o 2.0 0 -2 .0 -4J0 0 - Orbits Figure 2.5 E f f e c t of the rad i a t i o n pressure on the l i b r a t i o n a l response of a s a t e l l i t e obtained using a n a l y t i c a l and numerical methods: (a) s a t e l l i t e motion i n a c i r c u l a r o r b i t , e=0 00 o 15.0 75 0 - 7 5 -15.0 15.0 75 0 -7.5 -15.0 e = o.i c = o.i Kj = 1.0 4> = o Numerical Analy t ica I C = 0 T g =0.5 g=o 0 1 2 3 4 5 6 ^ 0 -Orb i ts Figure 2.5 E f f e c t of the radiatio n pressure on the l i b r a t i o n a l response of a s a t e l l i t e obtained using a n a l y t i c a l and numerical methods: (b) s a t e l l i t e motion i n an eccentric o r b i t , e=0.1 32 negative c, i n general, leads to high frequency l i b r a t i o n s . Influence of the i n e r t i a parameter and aspect r a t i o i s indicated i n Figure 2.6(b). The gravity gradient moment continues to be e f f e c t i v e i n l i m i t i n g the l i b r a t i o n a l amplitude even i n the presence of the solar radiations. Its ef f e c t on the l i b r a t i o n a l period i s less pronounced. The aspect r a t i o does not appear to have any s i g n i f i c a n t influence on the frequency although, the amplitude, i n general, increases with an increase i n g. For predicting a long range s a t e l l i t e l i b r a t i o n a l dynamics, i t i s important to study influence of the solar aspect angle, i . e . , p o s i t i o n of the sun i n the e c l i p t i c plane, on system behaviour. The analysis shows that the solar aspect angle does not s i g n i f i c a n t l y a f f e c t the maximum response for a 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 , which i s usually the case of i n t e r e s t . However, <j> can a f f e c t I ti> I considerably for s a t e l l i t e s i n e l l i p t i c t r a j e c t o r i e s ' 1 max (Figure 2.6(c)). S i m i l a r l y , the point of application (6Q) of a disturbance i s of p a r t i c u l a r importance only for e l l i p t i c o r b i t s . The period does not show any regular trend with changes i n <j> or 0Q. Figure 2.6(d) points out the l i m i t a t i o n s of the gravity gradient s t a b i l i z a t i o n i n the presence of severe disturbances. It may be noted that the minimum l i b r a t i o n a l response does not necessarily correspond to zero i n i t i a l e = e g = o Ki = 1.0 $ = o iy„= o • Numerica I Analytical e=e Kj = 1.0 +o=0 g = o.5 <)) = o ip0= o •Numerical Analytical Figure 2.6 System plots showing the maximum l i b r a t i o n a l amplitude and average period fo r a range of e c c e n t r i c i t y as affected by: (a) the solar parameter e = e Kj = 1.0 ^o=o c = 0.1 (j> =o i|^ = o Numerical Analytical 1.0 0.5 0.1 e = 0.2 0.1 e = o.2 -0.1 0 1 1 _ J — 0.2 0.3 g 0.4 0.5 Figure 2.6 System plots showing the maximum l i b r a t i o n a l amplitude and average period for a range of e c c e n t r i c i t y as affected by: (b) the i n e r t i a parameter and the aspect r a t i o <t>° 9° Figure 2.6 System plots showing the maximum l i b r a t i o n a l amplitude and average period for a range of e c c e n t r i c i t y as affected by: (c) the solar aspect angle and i n i t i a l s a t e l l i t e position 6 n at the instant of disturbance Figure 2.6 System plots showing the maximum l i b r a t i o n a l amplitude and average period for a range of e c c e n t r i c i t y as affected by: (d) i n i t i a l disturbances 37 conditions. Furthermore,the period of l i b r a t i o n s does not seem to follow any d e f i n i t e trend with changes i n the external disturbance. 2.2.3 Analog Simulation The re s u l t s presented so far are confined only to a few t y p i c a l s i t u a t i o n s . The numerical approach aided by a d i g i t a l computer, though informative, tends to be expensive. Hence i t was decided to explore the p o s s i b i l i t y of using the analog technique, normally preferred for economic r e a l -time simulation. The governing equation of motion (2.10)can be re-written as -{3 Ki/ 2(1+e Cos &)} Sin 2 ^ (2..16 {"| SLY^ ( e 4--UV--4)) | + % \ Cos C 9 V U / - 0) | ] This equation was programmed with 0 as the independent variable of the analog computer. The functions depending on the variable if> were obtained through the generalized 8 7 integration technique. The absolute values were generated by using diodes. Figure 2.7 shows the simulation c i r c u i t where [ioc(i+e)3{ isinO^ip-^) l+9ICos(0++-<t>)l}/Ci*ecose)4] Figure 2.7 Analog simulation c i r c u i t using 9 as an independent variable 39 3 0.3 K i Cos 2 ^ o 2.5" c ^ = 0-5 e ^ 7 = 2c ( i4 -e ) ^ = 0.2. ja+e)^ The analog computer PACE 231-R5 was used for the simulation. Amplitude scal i n g , i n conjunction with a reference voltage of ±100V, was c a r r i e d out to minimize the errors. Figure 2.8 shows a close agreement between the response r e s u l t s as obtained using the analog and numerical techniques. Analog simulation i s p a r t i c u l a r l y h e l p f u l for an extensive parametric study of the system i n r e a l time. Its usefulness i n s a t e l l i t e design could be enhanced considerably through hybridization with a d i g i t a l computer to take advantage of the l a t t e r ' s memory and l o g i c 2.3 L i b r a t i o n a l S t a b i l i t y Analysis of motion and s t a b i l i t y in the large provides valuable design c r i t e r i a . I t gives the l i m i t and margin of s t a b i l i t y for a s a t e l l i t e and hence 0 2 4 6 0 2 4 Q - O r b i ts Figure 2.8 A comparison of the system response obtained through analog simulation with d i g i t a l computer res u l t s provides the region of i t s capture. Since the governing 88 equation with periodic c o e f f i c i e n t s s a t i s f i e s the L i p s c h i t z conditions for uniqueness, i t i s possible to undertake the 89 "numerical experiment" to generate an i n t e g r a l manifold. Trajectories emanating from 6 = 6g+2nTr (n=0 , 1 , 2) with given i n i t i a l conditions i n and ' being the same, the time history of a representative point i n the three dimensional phase space 6,^,^' results i n a succession of t r a j e c t o r i e s o r i g i n a t i n g from 6=0 and terminating at 0=2TT where, l o g i c a l l y , 4> ( 0 ) and ^ ' ( 0 ) take the values of i> -. (2TT) and 1 , ( 2 T T ) . T n n r n - l r n - l Such a series of t r a j e c t o r i e s defines an invariant surface or i n t e g r a l manifold. It should be noted that any one point on the surface i s s u f f i c i e n t to generate and thus define the en t i r e surface. L o g i c a l l y , the concept of i n t e g r a l manifold breaks down for unstable tumbling motion. Thus, the l i m i t of s t a b i l i t y i s defined by the largest or the l i m i t i n g invariant surface. The importance of the i n t e g r a l manifold concept cannot be over-emphasized as i t provides a l l possible combinations of external disturbances to which the s a t e l l i t e can be subjected without causing i t to tumble. In general, i t consists of a "mainland" together with "islands" which, i n the l i m i t , shrink to a l i n e or a set of l i n e s , respectively, representing periodic solutions. However, from p r a c t i c a l design considerations, only the major envelope of motion 42 (mainland), which i s invariably associated with the fundamen-t a l periodic solution, represents the region of i n t e r e s t . Figure 2.9 shows a t y p i c a l l i m i t i n g surface which gives the bounds of s t a b i l i t y for the given set of system parameters. For a parametric analysis, c h a r a c t e r i s t i c s of the system can be better represented by a cross section of the invariant surface. Figure 2.10 (a) shows these sections at 6=0 f o r a range of e c c e n t r i c i t y . This concise represen-t a t i o n i s useful i n providing an in s i g h t into the influence of design parameters on the s t a b i l i t y bounds of the system. The concept of invariant surface has been further extended to develop s t a b i l i t y charts. These charts are constructed using some s p e c i f i c intercept of the l i m i t i n g surface (e.g., at ip=0=O) , plotted against e c c e n t r i c i t y , as a measure of s t a b i l i t y . I t may be noted that an increase in the value of e c c e n t r i c i t y or the solar parameter normally results i n a reduction of the s t a b i l i t y bound (Figure 2.10 (b)). The "spikes" i n the s t a b i l i t y p l o t s , associated with emergence of the periodic solutions, provide information concerning the c r i t i c a l e c c e n t r i c i t y , beyond which the gravity gradient s t a b i l i z a t i o n i s not possible. 10 Figure 2.9 A t y p i c a l l i m i t i n g integral manifold Figure 2.10 (a) a family of l i m i t i n g manifold sections at 6=0 (b) a s t a b i l i t y showing the e f f e c t of the solar parameter and ec c e n t r i c i t y 2.4 L i b r a t i o n a l Damping and Attitude Control Using the Solar Radiation Pressure The system 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 , investigated over a wide range of system parameters,clearly showed a substantial adverse influence of the solar r a d i a t i o n pressure. However, the r e s u l t s of t h i s study, as also the 56 57 findings of e a r l i e r preliminary investigations ' , suggest that the r a d i a t i o n force can provide an e f f e c t i v e damping torque to maintain a s a t e l l i t e i n a desired a t t i t u d e . This can be accomplished by c o n t r o l l i n g c, the solar para-meter, according to the r e l a t i o n s : c = (2.19) with c < c , k being an integer. Here c represents 1 1 max' ^ ^ max * a physical l i m i t as imposed by the s a t e l l i t e design, and the position control parameter c a n take, i n general, any desired value ( 0<y^<2i\) . The analysis describes the planar motion of a c y l i n d r i c a l s a t e l l i t e , with the proposed c o n t r o l l e r i n an e c l i p t i c o r b i t . Apart from a few attempts to obtain approximate closed form solutions i n c i r c u l a r o r b i t , the study i s primarily numerical i n character. 46 2.4.1 Approximate A n a l y t i c a l Solutions for S a t e l l i t e s i n C i r c u l a r Orbits: Y,=0, c =°° ' i\> max The equation of planar l i b r a t i o n s with the solar damping has the form ( 1 + e Cos 0 ) V - 2 e S C r > G C l + V ) + 3 kvSc<nHf Cos^V & - { ^ V + ty-fy)} { c i + i + C o s e ) 3 } # where For the p a r t i c u l a r case of s a t e l l i t e s i n c i r c u l a r orbits,the equation s i m p l i f i e s to + (9/2)|Sir» 2 ( 9 + ^ - 4 0 1 ] •••• ( 2 ' 2 0 ) Using the transformation 0-(j>=x and li n e a r i z i n g , t h e equation can be rewritten as V + 2 f i l ( X ) V + S ( X ) ^ = 0 (2.21) 47 where dLCx) = [(AIM,/2).{SCI?X + (9/2 )|Scnzxin S(x) = [ 3 K ; + z-ty{ScnX4- C3/a)|SCy\2X\j] and primes denote d i f f e r e n t i a t i o n with respect to x. Two d i f f e r e n t approaches were adopted to solve t h i s l i n e a r d i f f e r e n t i a l equation with variable c o e f f i c i e n t s : (a) Piecewise l i n e a r i z a t i o n with constant c o e f f i c i e n t s In equation (2.21), the functions d(x) and s(x) having a period IT represent the variable damping and spring s t i f f -ness, respectively. The approach here replaces these terms by t h e i r average values over f i n i t e i n t e r v a l s of x leading to V + 2 V + Se 4* = 0 (2.22) where X (My/4) { 1+ (2/TT) ( 9 - i ) } ] ; ( * - «WTT< X £ U U </4JTT I ( A ^ ) { i + - ( 2 / T T ) (9 + 1)}] ; {k + l/<jTT<X£(fe+3fo)7T 48 f [ 3 K i + (Z'a|r/2){l 4-(2/TT)(9-t)}]i (k- i/Ajrr<x<C^+ '/^ TT : [ 3 Ki + ( V y l 2 ) { i + (2/ n ) .+1) 5]j ^ + W 7 1 " ^ x ^ ( k +• 3 ^ ) 7 T In the solution of l i n e a r d i f f e r e n t i a l equation (2.22), i t i s easy to i d e n t i f y three d i s t i n c t cases: 2 (i) d g <s e: The roots of the c h a r a c t e r i s t i c equation are complex conjugate leading to underdamped o s c i l l a t i o n s given by A i r e e x b C - d € x ) . [ A t C o s ( C S e - d e ) x J T r (2.23) where , B^ as determined by the i n i t i a l conditions are: 1/2 A. = ^ b U e * ) . [ t f c C o s{(S e-<*£; X0} - {C d e ^ ) / ( s e - d i / / 2 j Sen {(s e - d e a f x . } J + { O0'+ d e ^ 0 ) / ( s c - d ^ , / 2 - } Cos{ LSerde)%}] 2 ( i i ) d g = s e : The roots of the c h a r a c t e r i s t i c equation are negative and equal leading to c r i t i c a l l y damped o s c i l l a t i o n s -\|/ = ( A z x + 62.) ( - d e * } (2.24) 49 with K^' B2 < j i v e n t*Y: 2 ( i i i ) d g >s e: The roots of the c h a r a c t e r i s t i c equation are r e a l and negative leading to overdamped o s c i l l a t i o n s (2.25) where constants A^, have the values: (b) Airy's Method This approach requires introducing the c l a s s i c a l change i n variable Z = l | r . ex\> ( J * a U ) cbd) •••• (2.26) 50 transforming the equation (2.21) into (2.27) where 2. -(Hv/z) (Stv\2X + 9 Cos 2*.)] ; I T T < x <, ( t + ' / 2 ) n -(/U^/2) (Scy> 2.x- 9. C 0 s 2 x ) ] ; (^4-I/2.;TT<X$ (!>+t)rr and [(MV'IM { X-(1/2)5^2% + x (9/2.) ( i - Cos 2 x ) ] J j ^ T T < x ^ ( ^ + V 2 - ) r r {d(x)cU= [CM.vW { x - c i / z ) S U i 2 . ) c -(9/2) ( i - C 0 s 2 x ) ( t + V 2 K < x ^ * . + 0 T r It may be noted that J (x) i s a periodic function 3. with period TT (Figure 2.11). Fortunately, i t can be approximated by a pair of stra i g h t l i n e s (over each period): 52 J,W=. 'U, X +- \>, ; 0 < X $. 7T/2. _U2.)C+ \>2_ ; rr/2 < x ^ 7T Here, the constants u^, v^, and are determined using the following c r i t e r i a : JO 4CX}-(U,X+Uj d x = J JaCx)-CU 2X+-l> 2)ax=:0 ,n/2 1 { J A(u,x.+v,)V( n/2 -x) d x C (2.28) The f i r s t c r i t e r i o n assures that the approximation does not a l t e r the area between the curve J (x) and the a x-axis while the second condition ascertains a constant area moment about X = T T / 2 . The solution of the algebraic equations represented by (2.28) leads to: 1», = [ ( C / T T a ; { ( A i y / ^ ) - ^ - / A ^ ^ + ^ K\ lXw(3+f)/3z} + ( 1/TT) { z ^ J - ( g - y U ^ } +C3/TTJ ( ( U ^ i 6 ) g. J 53 Ua. = [( + ( S / T T ^ C / U ^ ) ^ ] + { ( M v / 4 ) 9 + M^5-C9/TT) I t r e a d i l y follows that: = U ,^ , tf^fc^ s fell U , ; ^ T T < X < ( U l / 2 ) T T U 2 W = U2-fc + 2 = Vz+ (fc + l/2)TTU2>(fe+l/z)TV < X^(^ +l)TT F i n a l l y , introducing the substitution 2./3 U\ x vtf-,-= -u i or ) transforms equation (2.27) into the desired form (d - UT2 - o (2.29) with the solution i n terms of Airy's functions of the f i r s t and second k i n d ^ as 54 2 = A A . Ai tuj) + . B i (w-) Here the constants of integration A^, B^, evaluated using the i n i t i a l conditions, are: A 4 = [ { z . 8i(uj- 0) -E o ' ft ;Cur 0)}/{ A i ( ^ ) . D i W 0 ) - A U u M B \ ( o > „ ) } ] It should be emphasized that i n the above approximate techniques: (i) matching of the i n i t i a l conditions would be required at the commencement of each i n t e r v a l (kTr/2< x < (k+l)iT/2). Fortunately, the e f f i c i e n t character of the solar damping r e s u l t s i n a short transient, ( i i ) accuracy of the solution can be su b s t a n t i a l l y improved by choosing smaller i n t e r v a l s . However, thi s would involve a penalty i n terms of excessive computation time. 2.4.2 Accuracy of the Approximate A n a l y t i c a l Results The v a l i d i t y of the approximate approaches was assessed by comparison with the "exact"numerical solution of governing equation (2.20). It was observed that the solar c o n t r o l l e r can e f f e c t -i v e l y damp the l i b r a t i o n a l motion to an adequate l e v e l , i n approximately one o r b i t . Figure 2.12 presents a few representative response plots corresponding to angular and impulsive disturbances. At the outset one recognizes a successful estimate of the damping trend. Discrepancies i n the l o c a l d e t a i l s are l i k e l y to be of l i m i t e d concern from design considerations as the general character of the response i s predicted with f a i r accuracy. Of the two approximate techniques adopted here, piecewise l i n e a r i z a t i o n with constant c o e f f i c i e n t s appears to fare better. I t i s in t e r e s t i n g to note that larger values of i n r e l a t i o n to y^ improve the accuracy of the a n a l y t i c a l solutions. The approximate procedures discussed here aid i n : better understanding of the c o n t r o l l e r ' s damping mechanism. The gain y^ i s associated with variable damping while and govern the e f f e c t i v e system s t i f f n e s s . The l i n e a r -ized analysis suggests diminished importance of the i n e r t i a parameter for a s a t e l l i t e with the solar c o n t r o l l e r , as an increase i n can compensate for i t s low value. Further guided by the l i n e a r system's behaviour, one would anticipate better c o n t r o l l e r performance for gains y^ and so chosen as to approximate the conditions of c r i t i c a l damping. This would correspond to e = 0 g = 0.5 |I = 10 y = 0 Numerical , - A i r y Kj=0.8 9 = 0 C m a x = co - -L inear A r\\ v5 t=° ^°'5 10 5 0 10 5 0 + o = 1 0 ° + o = 0 * * * * M v^=io (|4'=0-5 \ \ V 1 0 + o = 1 0 ° +0=° It %» i t \ t u t \v \ V I 1 0 1 2 0 1 2 0 -Orbi ts Figure 2.12 The solar damping of a 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 as predicted by an a l y t i c a l and numerical methods: (a) small -impulsive (ip *=0.5, iK=0) or angular =10°, TJ>0'=0) disturbances 0 0 30 15 e = o Kj= 0.8 g = o.5 (j)= 0 Numerica I Airy max = CO Linear 30 15 V 10 •>• t= 0 •> 1 30 - 15 -l|j = 30° l|f=0 - 30 - 15 l|io= 30 \\>= 0 ) 1 2 0 1 0 - O r b i t s Figure 2.12 The solar damping of a 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 as predicted by a n a l y t i c a l and numerical methods: (b) large impulsive (ip '=1.0, i>n=0) or angular (^=30°, i|>0'=0) disturbances 0 0 58 Even though a n a l y t i c a l results are able to depict certain features of the system, they f a i l to give indications as to the e f f e c t of a l i m i t e d c . Moreover, although max ^ implementation of the approximate methods would s t i l l be possible for y^=T\/2, i t s application appears out of question for intermediate values of t h i s parameter, due to the absence of s t a t i c equilibrium p o s i t i o n about which l i n e a r i z a t i o n could be performed. Hence for a comprehensive analysis of i the c o n t r o l l e r performance, numerical integration represents the only a l t e r n a t i v e . 2.4.3 Numerical Results: Motion i n a C i r c u l a r Orbit Figure 2.13, showing the l i b r a t i o n a l response i n a few t y p i c a l s i t u a t i o n s , serves to i l l u s t r a t e the effectiveness of the solar radiation pressure i n c o n t r o l l i n g the attitude of s a t e l l i t e s moving i n c i r c u l a r o r b i t s . A b i l i t y of the c o n t r o l l e r to precisely a l i g n a s a t e l l i t e along the l o c a l v e r t i c a l as well as the l o c a l horizontal, normally an unstable equilibrium configuration i n the gravity gradient sense, i s remarkable (Figures 2.13 (a), (b)). In the absence of any s t a t i c equilibrium configuration, a l i m i t cycle behaviour for intermediate positions (0<y^<n/2) i s expected (Figure 2.13 (c) ) . Figure 2.14 (a) shows the influence of y^ and on the time-index T^, i . e . , the time required to a l i g n the e = o 90 60 l|i°30 0 - 3 0 90 60 f 3 0 0 - 3 0 0 9 = 0 . 5 <t> = 0 1 ^max ~ 0.5 <l»o = 6 0 ° . - \ ...  i . 0 - O r b i t s ; . 2 90 60 30 0 -30 Kj =1.0 rV 5 V 5 Cmqx = 1 +o = 0 0 -o -30 n 2 0 6 Cmax= 2 = 0 A = 2 C ill max - 2 = 60° J = 0 e Figure 2.13 Typical examples of the s a t e l l i t e response i n a c i r c u l a r o r b i t showing the e f f e c t of: (a) c and i n i t i a l disturbances, Y=0 max ' 'ty e = o 120 -90 -+°60 30 120 90 h +°60 30h 0 g =0.5 <))=o V 2 \ = 3 0 +0 = 60 C 120 90 60 30 2 ° 0 120 V 4 0 + 0 = 60° 0 -Orb i ts Ki =1.0 [1^=30 C m a x = 2 +0=0 1 V 3 0 +o=0 V 4 0 0 2 0 120 V 5 0 + 0=0 V 6 0 + o = ° 0 Figure 2.13 Typical examples of the s a t e l l i t e response i n a c i r c u l a r o r b i t showing the e f f e c t of: (b) and i n i t i a l disturbances, Y^=tt/2 o e=o 90 f 4 S g=as (j>=o KI=I.O Y ^ 0 - 5 2 c m a x = 2 <|>0=60 «p0=o • 1 • 90 i - • -r\,= 10 \ V^= 2 45 • n 1 0 -Orb i ts 90 45 1 — -\ V^=10 • e e e f f e c t of: (c) and y^=0.52 Figure 2.13 Typical examples of the s a t e l l i t e response in a c i r c u l a r o r b i t showing the 62 s a t e l l i t e along the l o c a l v e r t i c a l within ±0.5°. A lack of any regular trend i s apparent. This makes the choice of optimum gains somewhat d i f f i c u l t . Increase i n c af f e c t s c ^ max T^ only up to a certain value beyond which the dependence becomes rather weak (Figure 2.14(b)). Of course, t h i s optimum value of c i s also controlled by system parameters c max J J c K. and g. Figure 2.14(c) indicates the influence of y^ and on the time-index for the case J^=T\/2 . Again, as before, suitable set of y^ and can be chosen that would permit quick s t a b i l i z a t i o n . The importance of proper s e l e c t i o n of these parameters i s further emphasized i n Figure 2.14(d) which provides the minimum value of required to insure capture i n the horizontal equilibrium p o s i t i o n . The system behaviour for the intermediate value of i s indicated i n Figures 2.14(e), (f) for a range of system parameters. I t may be noted that amplitude of the o s c i l l a -tions and average position of a s a t e l l i t e vary considerably depending on the system parameters. This i s i n sharp con-t r a s t to the case where Y^ =0 o r I T / 2 . It i s i n t e r e s t i n g to recognize that an average position s i g n i f i c a n t l y d i f f e r e n t from the l o c a l v e r t i c a l can only be obtained for high values of and at the cost of larger amplitudes. However, a reduction i n K. and an increase i n c have a ' I max favourable influence i n attaining t h i s configuration. The next l o g i c a l step would be to vary the pos i t i o n control parameter systematically and ascertain i t s e f f e c t on the average po s i t i o n and the amplitude of l i m i t cycle around i t (Figure 2.14 (g)). The r e s u l t s suggest that a s a t e l l i t e with as small as 0.2, which has poor s t a b i l i t y i n the gravity gradient f i e l d , can be positioned almost anywhere between the l o c a l horizontal and the l o c a l v e r t i c a l with the maximum l i b r a t i o n a l amplitude of =2°. However, with larger values of , increasing y^ tends to create more d i f f i c u l t y for the system to compromise between the gravity gradient f i e l d and the solar c o n t r o l l e r . As a r e s u l t the amplitude of l i m i t cycle increases. Of course, the l i m i t cycle disappears for YljJ=7r/2,when l o c a l horizontal represents the s t a t i c equilibrium configuration. 2.4.4 Motion i n an Eccentric Orbit E c c e n t r i c i t y of the o r b i t introduces a forcing function on the system generally leading to a worsening of the l i b r a t i o n a l response. With the l i b r a t i o n a l motion always accompanying the o r b i t a l motion i n an eccentric o r b i t , one i s concerned about the steady state amplitude. Hence i t i s taken as a c r i t e r i o n to test the influence of system parameters. Figure 2.15 i l l u s t r a t e s the e f f e c t i v e -ness of the solar c o n t r o l l e r i n achieving a desired s a t e l l i t e o r i e n t a t i o n . It i s p a r t i c u l a r l y successful i n orienting the s a t e l l i t e along the l o c a l v e r t i c a l or at small i n c l i n a t i o n s to i t with r e l a t i v e l y small l i b r a t i o n s . System plots showing the damping ch a r a c t e r i s t i c s of the 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 as affected by: (a) u. and v., y.=0; (b) c m a x and K i, 20 10 •amp 0 30 20 o -| lav 10 0 e = o g = o.5 (j)= 0 1 K; -max = 1 = 1.5 = 0.52 20 1.5 Figure 2.14 40 60 80 0 0.5 1.0 \i^r Cmax System plots showing the damping char a c t e r i s t i c s of the 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 as affected by: K., y,=0.52 (e) y^ and v v ° - 5 2 ; (f) c and max 67 Figure 2.14 System plots showing the damping c h a r a c t e r i s t i c s of the 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 as affected by: (g) y and K ± g=0.5 <}>=0 Kj = l Cmax = 1.5 Ll^ = 80 V^=30 lft=0 90 e =o.i 60 ^ = 30 -30 V 0 1 60 o 3 0 0 9 0 60 3 0 0 — J .. - . • • e = o.i -Y*= 0.26 -• o = 0 " 1 0 4 9 - O r b i t s 0 8 6 0 30 0 - e = 0.2 90 e = o.2 90 -Y * = 0 n +0 = 30° - 6 0 0.26 • o = 0 6 0 3 0 30 — i 0 • 0 e = 0.1 Y*=TT/2 t = 6 0 ° e = 0.2 Y^=n/2 * = 60° 6 Figure 2.15 E f f e c t of e and y^ on the s a t e l l i t e response in eccentric orbits o 00 To better understand the dynamical behaviour of the s a t e l l i t e , numerous response plots were obtained for a range of system parameters and the r e s u l t i n g information was condensed i n the form of system plots as before. Figure 2.16(a) presents the v a r i a t i o n of amplitude as a function of the gain y^ and the damping design con-s t r a i n t c , i n absence of the p o s i t i o n control (Y,=0). For a given c , increasing y, improves the c o n t r o l l e r performance only up to a l i m i t e d extent. However, with increasing (Figure 2.16(b))and (Figure 2.16(c)) the performance i s generally improved, e s p e c i a l l y at lower values of y^. On the other hand, for large values of y^ r v, and K. have but a small e f f e c t on ^ 1 ' T'max Turning to the e f f e c t of e c c e n t r i c i t y on the performance of the system, i t i s observed that the steady state amplitude i s quite s e n s i t i v e to e, p a r t i c u l a r l y for c <0.5 (Figures 2.16(d)-(f)). max 3 Figures 2.16(g), (h) show the influence of system . parameters i n c o n t r o l l i n g the attitude along the l o c a l h o r i z o n t a l . It may be observed that for c <1.5, the J max amplitude of motion i s considerably affected by e c c e n t r i c i t y . However, for large values of c m a x / amplitude exhibits only a weak dependence although large e continues to a f f e c t the c o n t r o l l e r performance adversely. I t may be noted that i n thi s case the predominant parameter i s since i t governs the s a t e l l i t e ' s a b i l i t y to a t t a i n l o c a l horizontal con-fig u r a t i o n , p a r t i c u l a r l y for small (Figure 2.16 (h)). For high values of y^, should be kept low to achieve better performance i f the i n e r t i a parameter i s small. On the other hand, high i s a warrant for the s a t e l l i t e to a t t a i n the mean steady state along the l o c a l .horizontal when i s large. F i n a l l y , the attention was focused on the influence of system parameters on the s a t e l l i t e ' s a b i l i t y to a t t a i n any general orientation (Figures 2.16 ( i ) , ( j ) ) . In t h i s case, even i n c i r c u l a r o r b i t , the system cannot reach any asymptotic equilibrium and the e c c e n t r i c i t y only further increases the amplitude of steady state l i b r a t i o n s . As pointed out i n the case of a c i r c u l a r o r b i t , high values of c are desirable for smaller amplitude l i m i t cycles and max ^ •* a s i g n i f i c a n t deviation from the l o c a l v e r t i c a l . However, increasing v^, while increasing ^ a v » tends to increase the l i b r a t i o n a l amplitude as well, y^ continues to be an . important parameter i n a f f e c t i n g any desired change i n the s a t e l l i t e ' s preferred o r i e n t a t i o n . 2.4.5 A F e a s i b i l i t y Study The results c l e a r l y point out a rather outstanding performance of such a radiatio n c o n t r o l l e r i n e f f e c t i v e l y damping the l i b r a t i o n a l motion. V e r s a t i l i t y of the con-t r o l l e r i s further enhanced by i t s a b i l i t y to s t a b i l i z e 0 20 40 60 80 0 10 20 30 Figure 2.16 System plots showing damping c h a r a c t e r i s t i c s of the s a t e l l i t e i n an eccentric o r b i t as affected by. (a) c and ^ ' Y,.=0; (b) y. and v., Y„=0 1.2 O 0.8 •max 0.4 0.2 1 1 1 1— (c) Cmax= 1.5 e = 0.1 9 = 0.5 <t>=° H f i o "—~-^—~-^L — i 1 i i 1.2 0.8 0.4 g =0.5 (f) =0 * v = 8 0 v ° e=o.3 — 0.2 0.1 • 0.05-•max = 1.5 (d) -L 0.8 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 Kj Kj Figure 2.16 System plots showing damping ch a r a c t e r i s t i c s of the s a t e l l i t e i n an eccentric 1.0 o r b i t as affected by: (c) K ± and , Y^=0; (d) K ± and e, y =0 'amp • o av 0 0.5 1.0 1.5 0 0.2 0.4 0.6 Cmax Kj Figure 2.16 System plots showing damping ch a r a c t e r i s t i c s of the s a t e l l i t e i n an eccentric o r b i t as affected by: (g) c m a x and e, y =n/2; (h) K^, y^ and , 0.5 1.0 Cmax © Figure 2.16 System plots showing damping char a c t e r i s t i c s of the s a t e l l i t e i n an eccentric o r b i t as affected by: (i) c and v., Y,=0.52; (j) e and Y, J max ij;' ' ijj • u ' 1 the system along any general p o s i t i o n including the l o c a l h orizontal. However, a question arises as to the p r a c t i c a l f e a s i b i l i t y of such a c o n t r o l l i n g device. This section b r i e f l y describes a simple arrangement for r e a l i z i n g the desired value of the solar parameter through s p e c i f i e d v a r i -ation of the centre of pressure from the mass centre. A schematic diagram of the proposed solar radiation c o n t r o l l e r i s shown i n Figure 2.17 (a). It consists of two i d e n t i c a l cylinders of highly r e f l e c t i v e , l i g h t but r i g i d material (e.g. , aluminized mylar membrane) mounted at the ends of a s l i d i n g rod with t h e i r common axis coinciding with the axis of the main s a t e l l i t e body. The arrangement enables one to vary the geometric centre by moving the s l i d i n g rod a x i a l l y . This movement would normally cause a n e g l i g i b l e s h i f t i n the mass centre with a small change i n . By providing a suitable balancing mass ,these variations can be further reduced. Such a model e f f e c t i v e l y represents the system described i n equation (2.10) where c i s governed by r e l a t i o n s (2.19). The a x i a l movement of the rod may be achieved through a servosystem, schematically shown i n Figure 2.17(b). Position and v e l o c i t y sensors aboard the s a t e l l i t e would generate signals which i n conjunction with m u l t i p l i e r s , l i m i t e r and l o g i c c i r c u i t simulate the control function commanding the movement of the s l i d i n g rod. The function 77 Geometric centre Cylindrical aluminized mylar surface Mass centre Cylindrical satellite Sliding rod 1 _ 1 Figure 2.17 A f e a s i b i l i t y study: (a) geometry of the pro-posed c o n t r o l l e r configuration Figure 2.17 A f e a s i b i l i t y study: (b) schematic diagram of the co n t r o l l e r servosystem 00 of the l i m i t i n g c i r c u i t i s to es t a b l i s h correspondence between the maximum current and the maximum movement of the s l i d i n g rod. I t should be noted that control of the solar parameter involves occasional sudden changes from ± c m a x t o *°max* Fortunately t h i s does not pose any problem as, i n t h i s region, damping torque i t s e l f i s small. Recognizing that a degree of o r b i t a l movement at the synchronous al t i t u d e corresppnds to four minutes, the c o n t r o l l e r has ample time to a t t a i n the new configuration. This i s c l e a r l y shown i n Figure 2.17(c) where, for the case considered,the change i n sign occurs approximately over 15°. A condition on the values of gains involved, i . e . K^/K^/K^, i s r e a d i l y deduced: K, K* = fly • {1/(4*')} I > e ' l / { 0 + A s a t e l l i t e of given mass and i n e r t i a parameter w i l l have, depending on a mission, demands on i t s pointing accuracy. This w i l l e s t a b l i s h the required c . Now the u max system plots presented e a r l i e r , can be used to se l e c t the appropriate c o n t r o l l e r parameters to meet thi s requirement. 30 15 0 2.0 C 0 -2.0 1.5 F 0 -1.5 1.5 Torque -1.5 e = 0 g = 0.5 <J> = 0 Kj = 0.6 C m a x=2 ^ = 2 0 V^=0 V=0 +0= 30° 4 j f ° i [ F= Sin(9++-<j)) [ i S i n O + ^ - ^ l * 9lCos(Q + i[j-([))|] Figure 2.17 A f e a s i b i l i t y study: operation 1 • 0 -Orbits (c) v a r i a t i o n of the dynamical state during control CO o 81 Regulation of the tr a n s a l a t i o n a l motion being r e l a t i v e l y simpler, the c o n t r o l l e r has apparent advantage over the deployment control of an unfurlable material 57 suggested by Modx and Tschann. Inherent s i m p l i c i t y of the concept assures r e l i a b i l i t y , so v i t a l for space operations. Furthermore, the proposed model presents a . p o s s i b i l i t y of positioning the servosystem within the s a t e l l i t e rather than leaving i t exposed to the space environment. The main objective of the study has been to i l l u s -trate the immense potential of the solar ra d i a t i o n pressure for l i b r a t i o n a l damping and attitude control and i t s f e a s i b i l i t y from p r a c t i c a l considerations. Details of the hardware design and control c i r c u i t r y although somewhat routine, may constitute an in t e r e s t i n g project i n i t s e l f . However, th i s i s outside the scope of the present investigation, 2.5 Concluding Remarks The s a l i e n t features of the analysis and the con-clusions based on them may be summarized as follows: (i) For a small amplitude motion,which i s usually the case of in t e r e s t , the W.K'.B.J. analysis can predict the l i b r a t i o n a l response of a s a t e l l i t e , subjected to the gravity gradient and solar radiation forces, with a considerable accuracy. The 82 discrepancies, which become s i g n i f i c a n t during severe disturbances, primarily a f f e c t the d e t a i l s of the response without s u b s t a n t i a l l y a l t e r i n g the maximum amplitude, ( i i ) The solar r a d i a t i o n pressure has, i n general, a substantial detrimental influence on the l i b r a t i o n a l dynamics of a s a t e l l i t e . For non-zero c, s a t e l l i t e always executes l i b r a t i o n a l motion. Hence neglect-ing the radiation forces when predicting the attitude behaviour of the gravity gradient s a t e l l i t e s could r e s u l t i n gross errors, ( i i i ) Analog procedures may be preferred over the numer-i c a l techniques for r e a l time simulation or where a large amount of computation has to be performed for a s p e c i f i c configuration. The hyb r i d i z a t i o n of analog and d i g i t a l units may help resolve several inherent l i m i t a t i o n s i n l o g i c and memory. (iv) The concept of i n t e g r a l manifolds i n a three dimensional phase space can be e f f e c t i v e l y applied to study the s t a b i l i t y of the system. The l i m i t -ing invariant surface provides a l l possible com-binations of disturbances which the s a t e l l i t e can withstand without tumbling. Furthermore, the Concept leads to the periodic solutions, which act as spines around which the s t a b i l i t y regions are b u i l t , giving information concerning c r i t i c a l 83 e c c e n t r i c i t y beyond which stable operation of the s a t e l l i t e i s not possible, (v) The solar parameter c s u b s t a n t i a l l y a f f e c t s the s t a b i l i t y of the motion and hence merits equal consideration with the e c c e n t r i c i t y and i n e r t i a parameter. The s t a b i l i t y charts presented here should prove useful during the preliminary design of a s a t e l l i t e . (vi) Through the proposed solar c o n t r o l l e r , i t i s possible to a l i g n the s a t e l l i t e i n a c i r c u l a r o r b i t p r e c i s e l y along the l o c a l v e r t i c a l or the l o c a l h o rizontal, even i n the presence of severe disturbances. So far as the attitude control of a s a t e l l i t e i n an ar b i t r a r y p o s i t i o n away from the l o c a l v e r t i c a l i s concerned, a lower value of i n e r t i a parameter leads to better c o n t r o l l e r per-formance. Through a suitable choice of control parameters, the l i b r a t i o n a l motion can be damped within a f r a c t i o n of an o r b i t , ( v i i ) The solar c o n t r o l l e r continues to be e f f e c t i v e for s a t e l l i t e s i n eccentric o r b i t s , ( v i i i ) The po s i t i o n control angle y^ represents a rather important parameter. Through i t s suitable choice, i t i s possible to change the ori e n t a t i o n of a s a t e l l i t e i n o r b i t thus enabling i t to undertake diverse missions. (ix) The approximate methods used to study the con-t r o l l e r effectiveness are able to predict the general character of the response. The a n a l y t i c a l treatment of the problem c l e a r l y establishes that gravity gradient booms are no longer necessary for a s a t e l l i t e with the solar c o n t r o l l e r . A suitable choice of v, can compensate for low K.. 3. COUPLED LIBRATIONAL DYNAMICS AND GENERAL ATTITUDE CONTROL OF SATELLITES IN THE PRESENCE OF THE SOLAR RADIATION PRESSURE The l a s t chapter investigated the influence of the solar r a d i a t i o n pressure on the planar attitude dynamics of a s a t e l l i t e . The study not only established importance of the solar parameter i n determining the l i b r a t i o n a l response, but also demonstrated how the minute force could be used to advantage i n attitude c o n t r o l . The object, here, i s to extend t h i s analysis to a more r e a l i s t i c s i t u a t i o n of general three dimensional motion. Such an inv e s t i g a t i o n i s important 20 because, as pointed out by Kane , for large amplitudes, the transverse motion i s strongly coupled with that i n the plane. The investigation i s complicated by the highly non-l i n e a r , non-autonomous, coupled character of the governing equations of motion, which are not amenable to known a n a l y t i c a l procedures. For a general analysis, one i s forced to resort to numerical techniques. The p a r t i c u l a r cases involving motion i n c i r c u l a r , e c l i p t i c o r b i t s do o f f e r some s i m p l i f i c a t i o n and have been studied to assess the pot e n t i a l of several approximate a n a l y t i c a l approaches. The attention i s then directed towards the p o s s i b i l -i t y of using the solar force to achieve three dimensional l i b r a t i o n a l damping and attitude control. The analysis i s 86 kept quite general to include the s a t e l l i t e s i n e l l i p t i c o r b i t s i n an a r b i t r a r y plane. Due to a strong coupling between the three degrees of freedom, numerous p o s s i b i l i t i e s had to be t r i e d before f i n a l l y a r r i v i n g at a successful configuration. 3.1 Coupled L i b r a t i o n a l Dynamics 3.1.1 Formulation of the Problem Consider a c y l i n d r i c a l s a t e l l i t e o r b i t i n g i n the e c l i p t i c plane about the centre of force 0 and executing coupled l i b r a t i o n a l motion (Figure 3.1). Let x,y,z represent 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 X0'^0' Z0 S O c ^ o s e n a s t o d i r e c t the z-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. Any orie n t a t i o n 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: >J> about the yg-axis giving x^,y^,z^; 3 about the y^-axis r e s u l t i n g i n *2'Y2'Z2' a n < ^ ^ about the Z 2~axis y i e l d i n g the f i n a l body coordinates x,y,z. The expressions of k i n e t i c and potential energies 3 to 0(1/R ) for an axisymmetric s a t e l l i t e ( 1 ^ = 1 = 1) can be shown to be: Figure 3.1 The s a t e l l i t e undergoing general s p a t i a l motion 88 Using the Lagrangian formulation, the governing equations of l i b r a t i o n a l motion can be written as: - r%%W/olt)[{X-C6+iir;5Lyxp}SCinp] « ^ (32) + I^CG + ^ H X - C O ^ j S c * ^ Cos (3 -(d[.ctt) { X_cd + \V)SLrvp5 = Q X where Q^(i=^,£,A) are the generalized forces due to the solar r a d i a t i o n pressure. 89 Using the Keplerian o r b i t a l r e lations and recogniz-ing that: { 0 + e CosQ)ty -2 eScvvO ( i + V ) } - 2 ( | + e C o s ^ ( l + ^ p V a A i P + 3K{Siv\tyCosv|/ - ( i-Ki ) [ ( i+e^ . ( l ^Qp/ jU l ) (d[dt) = 0 (olid©; = C^ 9/R z)(d|d&) = ( * e / K4) ( a z / a ' 0 2 ; - 2. ( ^ e / ^ ( a R / d 0 ) . (d(d0; the equations of motion take the form: (3.3) 90 The next step that naturally follows i s the evalu-ation of the generalized forces Q^. This i s accomplished through the v i r t u a l work analysis of the torque produced by specular r e f l e c t i o n of the radiations. For convenience, a new set of x g,y ,y g-axes i s defined such that the y -z plane divides the c y l i n d r i c a l surface into the dark and bright halves. Considering f i r s t a represen-t a t i v e element of the curved surface (Figure 3.1), the expression for the force dF c can be written as ol F c = - ( £ r L j c ' ) |{CoS(cr+-<re) Sen (0 +-l|/-(J)) + 5iY\(cr+cr.)Swfi • Cos(a+lp-<|)j}|. [ {Cos L<r±<r,)Siv\(&+^r-<t>) + SCv\Cc-+<^;. SCv> ji Cos C 0 +ty-<t>)} C 1+ P - t ) +-{ 5C-«Ccr+o^)5iVi (0+ -^<J)J -Cos(<r-+<r 0/Stn|J Cos ( &+1]/ - <fc) } ( I- P ~ t ) U r (3.4) The components of the moment due to th i s force along s x s , s y s , s z s are integrated over the s u n - l i t region of the c y l i n d r i c a l surface obtaining: 91 [ S A n f } / 2 C / ] . [ ( t + P - r ) { ^ ( e + ^ - 4 ) ) ( l + 0/3)Co2cy7) + SL>fyCo5(e + ty-40 ( » -0/3) Cos 2xr0)+ (1/3) SlY) p Sin 2 (G +A|T- <f>)} +- (»/2) (I - f-r; { (U (3>) S L vf ( 0+Hr- <J»). Cos 2cr„ + (4/3)SUK (J SiAn2(0+ty-<fcjSin 2<rj -( ^ / a j s u ^ Cosc&+ty-<t>) <^s 2o- 0 ] ] Coscr 0 + S i n |& CcnC&+ty~<b) SiY\cr0} [S A ^  / 2 c'] P-^){-ty3)SiT\ 2o-0SCv?(04l^-43)-H(2/3)Scn 2cr0SCn £ . Co$2'C&4-l|r-(t))+ U / 3 ) C o s 2 o ^ S v M p > S u v » U&+i\r-(p) } + { u - P-t)/^} { S e n 2cr0 S ^ C 4?)- (2/3) SCv, 2cr; . 5cn 2js Cos C 0 + ^ - 4 > ; - C 2 / 3 K o s 2cr-0 SCw,p>Suv,2.^0+"Hr-4))5] + { TT S r A (i - p _ r ; /Ac'} Cos p Cos C e +1^-4)). -f-S C & +1|/--4>;. Sv* cr0 + Cos cr; S uw, f Crt £ & - f U 7 " <t>) } 92 Recognizing that the distance of the f l a t end, receiving radiations, from the mass centre i s given by b where b = the corresponding expressions for moment contributed by the plane ends are given by: Mij> = { TVS r A 0 - ?-t)\lkd)) Cos p» dos(G+^-4)j {Scn(Q+iU-4)) Cos o-0 Cos p> |Cos(&+ty-4>)| { Sc^(94-l^-^)CoiC-+Siy><r0ScriPCos(e+^3 M2}> = {ITS rA ( i - R ) f r O ] Cos P Cos {si* C0 Sc^cr0 — Cos CTJ SGv> Co <> f> | Co s ( 0 +ty -4>J | J S CY\ C G - 4)) Sin rr, - Co s <r„ Co s t& f^ .-(fe) SIM .. (3.6) 93 Combining the moment-components due to the curved surface and the plane ends y i e l d s : Mi = f V + Mif = {SA £|,/C2c'j} [(l + f-Tr) {SC7fce+^-4))(l + 0/3.) Cos20^+SCyf^. Cos^ C a + V-<W (» - 0/3) Cos 2<r0) + 0/3) Sen (5 SW> ZCe+ty-tySin 2xr0} 4{o-f-xY2j.{u/3)Sc>?"co+^-(D;cos 2<r-0-(^ /3;5c^Cos"(e4-i^<t>; Cos2er0 +(A |3) Sin [1 Sin 2(0+ty-<|>) SCn 2c-e }] + { T T 5 r A/(<.c')} C i-F-t) (2 -W Cosj*]Cos(9+*-<fr)|. {SCn ( e + \|r-(p) Cos o-„ + Stv\o-0 Cos (0+iV-<b?StM fi] M i = Mac Mijj 5cn Z a - 0 S i n f i Cos~( 6+ty-4>) - Cos 2.0-, Su, (J Sw» 2{e+TV-$j} -HI/3J ('~p-V){ SCw 2o-e 5c>f(e +llr-<t>)- Sin 2<ra Scr?£. Co62"Ce+1|>-^ ; - Cos 2<r0SU>p Sir\ 2(0 +ty-cD) ] + {TTSrA/Uc')} U f y M O - p - * ) Cos f> \Cos(Q + ty-<t)) 94 [SiY\CQ+ty-<$>)SiY\cr0- Co<,<r0 S i r \ (i Cos C6 + ^-<P)} (3.7) Using the p r i n c i p l e of v i r t u a l work leads to the following expressions for generalized forces: Qty = (Mi.Coso-.-t-M^StvNcr:) Cos (i 6p> - ( M, Svnou - Mz. Cos o^) (3.8) = 0 being zero, A'-(l+^') Sin 3 becomes a constant of the system. For a non-spinning s a t e l l i t e , t h i s constant must vanish leading to a considerable s i m p l i f i c a t i o n of the equations of motion: (i'+eCos 9) V - 2 e S u n Q U + V ) - 2(n-€Cos9)Ci+"U>)pta^|2..'-. + 3 Ki S i * ^ Cos ^ = R|, • (i+ef/ci+eCosG) 3 C \ + e Cos 0) f - 2 e Sen G f>' + {(n- e Cos 9 ) 0 + V A 3K; Cof^}. SW»fk Cos p = • (\+ef[(i + € Cos Of ... (3.9) where (c/Cosp) j£os<r./( \^)}.{C3/4)Stn(G4Ur-(f)(l+ 0/S)Cos2xr.) + (3/4)Scn"P Cos"(9+ -^<t)(i-0/a) Cos2a;) +0/<0 Sen (3 . SCY\2(e+iU-4>)Scrx 2cre+ }o(StvfC&+llJ-^)Cos 2a~0 -•SCvx^ p Cos (0+"^-$) Cos 2cr„ + St>v» |S SIM 2(0 - ^ - ^ S c n 2<^ )} + 3 Cos cr. Cos p I Cos (0 +^ -<t>)|{ Son (0 Cos cr; + Slncr; Cos(&+ty--q>) P } + Scnoi {(t/2)5ivA207SCv?"(9+'^ -(t)) -(I/2)5CM2C^ ScnZ|2>Cosi(6+llr-(t?)-(>/2.)Cos2c- Si*, p . Sin 2(0+1^-4?)+ 9 Cos p |Cos (0+^~4)|.(Scn(e+^-4>,) • Servo* - Cos cr0 Cos (0 + l(>-4>) Sen f> ) Pip t<v K J 4 , | > , &,4>, ty,p) c [{Si/n^/(i+K>} {(3/4)Sen(0+^)0 40^)Cos20^) + (3/4) SCrf p Cos (&+l|>-U/*)C03 2cr0)+ (l fa)Sm p. Sin 2(0 + ^ -<i>) Sen 2cr0 + 1F> (Serf (6-+"^-45) Cos 2cr0 96 9Scv^o'Cos|MCos(e+xM>)| { Sir,t& + ty-&) Cosa-b + $LY\<rQ CosC0+^-^S(.^ f> J - CosCTo { 0/2-; 5 2JC JO . Scw(0+ty-4>)- (1/2) Scyx 2.cr0 Sinf> Co/" (9+^-4) - 0/2.) Cos 2o-0 St>n (3 Sin 2.(0 + ^ -4?) + % C-o-S f> . 1 CosO+^-4>) (. ( SCr>(e+^-4>)SCncr 0 -Cos <r0 Cos (Q+ 1lr-6) S iw |2> ) ^  3.1.2 Approximate A n a l y t i c a l and Numerical Approaches For c i r c u l a r o r b i t s , the equations simplify to: 2(1+^') p'tav) p + 3 K\ Cos^ = F-^  p>"+{C^^)2-r3KvCos^^}Scr1(2>Cos ^ = Fp> 97 (a) Linearized Analysis Through d i r e c t l i n e a r i z a t i o n the equations of l i b r a -t i o n a l motion degenerate to: where and F^^ represent the f i r s t order terms i n the expressions for F^ and F^ given by: - 0 The function n a v i n 9 a period of 2TT, can be e f f e c t i v e l y described by i t s f i r s t harmonic term i n the Fourier expansion leading to a considerable s i m p l i f i c a t i o n : Solution of the above set of uncoupled equations i s rather t r i v i a l and has the closed form of: 98 \]r = T>, Cos ^ . 0 + ^ 2 . S C n n , e 4- { q Sin (0 -<*>)] V (3.io) ji = D 3 Cos r>20 4 - S e n v\29 where 0^,02,0^ and are obtained using i n i t i a l conditions: 3 > l = [ a|/ 0 - {C f/ (* f - i ) } S i n C o s *n,e 0 - C » / ^ i ) [ ^ 0 / - { ^ / C ^ - O j Cos (e.-4>)] S m - n , 0 o + I ^ - fc^(nf-i)J CosCSo-*)] Cos ^,0„ T>3 = f>„ Cos n 2 « i - C P 07^2-) ^ 3>4 - P 6 Sin r\^Q0 + ( (SoV^z) tte Y)XQ0 23 91 (b) Butenin Method ' Representing the trigonometric functions by t h e i r series, neglecting f i f t h and higher order terms i n ip, 3 and t h e i r d e r i v a t i v e s , and c o l l e c t i n g the non-linear terms on the r i g h t hand side, the equations of motion can be written as: 99 > (3.11) where i, ( ^ ^  ij;': p, p'; = 2 p p'+ 2 p p V - 0/3) p3p>V(2/3^f^ + c i / z ; - n l 1 - ^ + 0/20 2-, 1.3 For small amplitude motion, each term i n and 1^ i s comparatively small, hence the approximate solution to equations (3.11) can be found using the method of v a r i a t i o n of parameters. Accordingly, consider a solution i n the form: = UQ) Simile + £.(e)} + ^/ ( < H ; }5c*(e -< i> ; > (3.12) Note that n^ and represent the p r i n c i p a l frequencies given by the solution of the homogeneous equations and 100 a^,a2»6^/62 a r e unknown functions of 6 to be determined by introducing l o g i c a l constraints. D i f f e r e n t i a t i n g the equations (3.12) with respect to 6 gives: \|/=[a,YHCos Cn,e+£i)+{q/fc?--»;5 Cos(Q-<w] +• a,' Sen cvi,9 + sx) +• a,£/cos (n.e + S.) P> s f a ^ C o s C n t G + ^ J + a i £c^ ( ^ 9 + ^ + C o s CnzQ+Sz) Here, the terms appearing within the square brackets i n each equation correspond to the case when a^,a2/<$^ and 6^ are treated as constants. Permitting these parameters to be varying slowly, rest of the terms on the right hand side can be equated to zero giving: a,'Sen i , +- a , o V c o s i . i = o 1 y (3.13) &l SCY\ i t + a z £ 2 / C o s £-2.= 0 J Mathematically t h i s implies that the n o n - l i n e a r i t i e s are small. Physically, i t means that the s a t e l l i t e i s executing small amplitude motion. Normally t h i s condition i s s a t i s f i e d by most communications, weather or earth-resources : s a t e l l i t e s . The other two equations are obtained by d i f f e r e n -101 t i a t i n g once again with respect to 0 and subs t i t u t i n g i n equations of motion (3.11) leading to: tx/ri. Cos l\ - (k,y\, 5, SCv> £, = f ^ W2 Cos 2-2 ~ A* ^ 2.^2 SCY\ iz- {2 (3.14) where ^ ( a . S c w i , , a c c o s t , , a ^ i n i i , a ^ z C o s £ z ) Solving the four algebraic equations (3.13) and (3.14) for the four unknowns a i ' a 2 , ( - > i a n <^ ^2 y i e ^ s : 0>l- (t/r>() j - * Cos £.| For small amplitude motion, f^* and f 2 * are quite small compared to the remaining terms i n the equations of motion, Hence, using the average values of the slowly varying parameters a^,a2,6-^ and 62 leads to: a , ' = {i/cAT^r,,)] I f Cos e, ae, diz o 'a , 2TT r2.IT ^valuation of the integrals u t i l i z i n g the conditions of orthogonality gives: a,' = a z ' - 0 = --h,[{(af-/^ + i » / i K ^ / C ^ - 0 z 5 / ( i - a^/s)] Substituting the integrated expressions of 6^' and 6^' i n and leads to: 103 t l ! = Y\,e+s«= ^«[>-{^^)^Ci/2)^y(m12--,)2-]/{i-(ai-/8)]]0 The solution can now be written more conveniently i n the form: \\r D,Cos(K|0) + T>2. Sc*(k, 0 H {cj/(Y^i^Sen ( e-<t>)' p = Cos C + -D^  S i n ( K 2 Q ) where (3.15) and D^,D2/D.j and are obtained using i n i t i a l conditions: D , = [ ^ 0 - { q / c - n , x - o 5 S c n ( e 0 ~ c t > ) ] Cos ( K , Go) 104 ]\= C"1^- c f / ^ ' - | ) } 5 m ( V < t ' ) ] S in CK,0oj + CI(KI) [> . ' - { q / ( < - u } Co5(e0~4>;] Co5(K teoj £ 4 = p o Sen (Kx0o)+ (l/Ka.) p i Cos 3.1.3 System Response and Discussion of Results Figure 3.2 shows the l i b r a t i o n a l response of the s a t e l l i t e s i n two t y p i c a l s i t u a t i o n s . A comparison of the response of the s a t e l l i t e s for two values of solar parameter c c l e a r l y points out a substantial detrimental e f f e c t of the solar radiations. However, i t s e f f e c t on the cross-plane motion i s r e l a t i v e l y less s i g n i f i c a n t . Furthermore, the approximate a n a l y t i c a l solutions appear to predict the response c h a r a c t e r i s t i c s with a f a i r degree of accuracy. As a matter of fact, both the l i n e a r as well as the non-linear Butenin solutions describe the s a t e l l i t e performance (amplitude and frequency) quite well,even for r e l a t i v e l y large disturbances (-20°). The discrepancy i n phase,which i s cumulative i n nature, i s not l i k e l y to a f f e c t the pre-liminary design process where the methods are most l i k e l y to f i n d a p p l i c a t i o n . 105 20 10 l)J 0 - 1 0 - 2 0 10 o P 0 -10 20 10 l(J 0 -10 - 2 0 10 P e = o Kj= l 0 •> 0 Numerical g = 0.5 p = o.i Po= o.i p> o Butenin <(>= 0 0 •Linear C = 0.1 - l O h 3 v 4 7 0 -Orbits Figure 3.2 Comparison between coupled l i b r a t i o n a l response of the s a t e l l i t e as obtained using a n a l y t i c a l and numerical methods 106 To better understand the dynamical behaviour of the s a t e l l i t e over a wide range of system parameters and i n i t i a l conditions, numerous response plots were obtained varying the values of c, , g, <j>, and i n i t i a l disturbances. The r e s u l t i n g information was condensed i n the form of system pl o t s . Only a few of the representative p l o t s , s u f f i c i e n t to e s t a b l i s h trends, are presented here. Figure 3.3(a) exhibits e f f e c t s of the solar parameter on the maximum amplitude and the average period of the motion. P a r t i c u l a r l y s i g n i f i c a n t i s i t s influence on lip I 1 ^ 1 r 1 max Period of the pitching motion appears to be generally smaller when c i s negative. In t h i s s i t u a t i o n , the approximate methods show considerable error i n predicting the frequency of the planar motion. However, i t may be observed that the period of the in-plane l i b r a t i o n s i s , i n general, larger than that for the out-of-plane o s c i l l a t i o n s , with both having values between 0.5 and 1.0, a trend well established by the approximate methods. Influence of the i n e r t i a parameter i s indicated i n Figure 3.3(b). It may be noted that the gravity gradient torque l i m i t s the l i b r a t i o n a l amplitudes i n ip as well as 8 degrees of freedom quite e f f e c t i v e l y . On the other hand period, p a r t i c u l a r l y of the in-plane l i b r a t i o n s , remains v i r t u a l l y unaffected. 107 The aspect r a t i o appears to have r e l a t i v e l y l i t t l e e f f e c t on the l i b r a t i o n a l amplitude or period (Figure 3.3(c)). The same i s true for the e f f e c t of the solar aspect angle (Figure 3.3(d)). In general, the system plots showing the e f f e c t of <f> would be useful i n predicting the long range performance of the s a t e l l i t e . Figures 3.3(e), (f) indicate the reduced e f f e c t i v e -ness of gravity gradient s t a b i l i z a t i o n when the s a t e l l i t e i s subjected to severe disturbances. As the equations of motion are strongly coupled, a large disturbance across the o r b i t a l plane s u b s t a n t i a l l y a f f e c t s the in-plane motion, however, the converse i s not true. This i s apparent from the fact that when 3Q=6Q'=0, F^=0 and equilibrium of the r o l l degree of freedom remains unaffected by the pitching motion. It i s i n t e r e s t i n g to note that the large amplitude l i b r a t i o n s occur when the i n i t i a l impulsive disturbance i s e n t i r e l y i n the o r b i t a l plane. This i s p a r t i c u l a r l y important as i t suggests the possible application of the planar study i n the early stages of s a t e l l i t e design. Several features of the system behaviour suggested by the approximate solutions are of i n t e r e s t . Out-of-plane l i b r a t i o n s executed by the s a t e l l i t e are always p e r i o d i c . The period of the r o l l motion, e x h i b i t i n g only a'weak amplitude dependence (according to the Butenin s o l u t i o n ) , i s = 1/^2 (o r b i t a l period). On the other hand, pitching motion would be periodic only under s p e c i f i c s i t u a t i o n s : 1.0 'av,/3 0.2 1.0 0.2 60 IPI max 30 0 60 e = 0 K, = 1 g =0.5 p = o.i <t> = o P o = 0 £ = a 5 Numerical • hi ^  |p • • ii • Lf-e-rr^ • • • • Linear Butenin (a) . 1.0 0.2 b 1.0 0.2 60 30 0 60 - (b) - 30 e = o c = o.i v|/o=o i|f = o —-Numerical g = o.5 p = o.i (j> = o B = o p' = o.5 -Linear Butenin 11' inn i m MHn11H t*,nr» **n\ mmrTMai I I 0.5 0.6 0.7 „ 0.8 0.9 1.0 K i Figure 3.3 System plots showing the maximum amplitude and average period as affected by: (a) the solar parameter; (b) the i n e r t i a parameter 1.0 Tav, 0 0.5 1.0 0.5 20 max 0 40 Imax 20 e = o C =0.1 P =0.1 Kj= 1.0 $ = 0.5 <t> = 0 •Numerical "Butenin L inear (c) 0.1 Figure 3.3 0.2 0.3 0.4 0.5 1.0 0.5 1.0 0.5 20 0 40 20 e = o C =0.1 p = 0.1 Kj= 1 0 l|f = 0.5 $-0.5 g =0.5 -r-° -Numerical Butenin Linear " (d) . ' - — I 1 1 1 _ 90 p 180 270 360 System plots, showing the maximum amplitude and average period as affected by: (c) the aspect r a t i o ; (d) the solar aspect angle o 1.0 Tav,£ 0.5 1.0 Tav,^  0.5 40 IPI max 0 80 I max 40 e = 0 9 = 0.5 C=0.1 Ki = 1.0 P = o.i p'=o - 1 .0 <j>=0 Po = 0 - 5 — Numerical — Butenin — Linear (e) - 1.0 - 0 . 5 0 0.5 1.0 0.5 1.0 0.5 4 0 0 8 0 40 e = o 9 =o C=0.1 Kj = 1.0 0 Numerical l|j = 0 Butenin p = 0.5 Linear - i f ) \ i t / " i , . / i N I 1 -1.0 - 0 . 5 0 p; 0.5 1.0 Fiqure 3.3 System plots showing the maximum amplitude and average period as affected by: (e) the i n i t i a l disturbance ip Q' ; (f) the i n i t i a l disturbance g Q 1I l l (i) In the absence of the solar radiations ( i . e . , c=0), period - 1/n^ ( o r b i t a l time). ( i i ) For i n i t i a l conditions s a t i s f y i n g the r e l a t i o n s : the lp-period i s i d e n t i c a l to the o r b i t a l period. Interest-ingly, the i n i t i a l conditions leading to a periodic motion plot as a c i r c l e (Figure 3.4). A quick check through integration of the exact equations of motion using t h i s approximate input information showed the motion to be nearly periodic (Figure 3.5) with the period predicted by the approximate methods. At t h i s stage, i t may be appropriate to comment on the e f f e c t of e c c e n t r i c i t y . An observation of the equations of motion reveals that, i n the absence of any external disturbance, the periodic forcing function ( i . e . , 2e Sin 0) would excite only the pitching l i b r a t i o n s without a f f e c t i n g the r o l l degree of freedom. For an a r b i t r a r y disturbance, e c c e n t r i c i t y can have a substantial detrimental e f f e c t on the planar response, however, the cross-plane-motion would experience only the second order e f f e c t . 112 Figure 3.4 Approximate estimate of i n i t i a l conditions for periodic (period=2fr) l i b r a t i o n s 3.1.4 L i b r a t i o n a l S t a b i l i t y The s t a b i l i t y bounds fo r the general motion were established by analyzing the l i b r a t i o n a l response, over 40-50 o r b i t s , to the systematically varied i n i t i a l conditions ipQ' and 8 Q 1 for a range of the solar parameter values. The vast amount of information thus gathered i s condensed i n the form of s t a b i l i t y plots which indicate allowable impulsive disturbances (^ Q=8Q=0) for non-tumbling motion (Figure 3.6). I t may be noted that an increase i n the value of the solar parameter, i n general, leads to a reduction i n the region of s t a b i l i t y . This, once again, emphasizes the importance of the solar r a d i a t i o n pressure, s i g n i f i e d by c f i n obtaining the design data useful for gravity oriented s a t e l l i t e s . 3.2 L i b r a t i o n a l Damping and Spatial Attitude Control Using the Solar Radiation Pressure The s a t e l l i t e response, obtained over a wide range of i n i t i a l conditions and physical parameters showed substantial influence of the solar r a d i a t i o n pressure on the l i b r a t i o n a l dynamics of a gravity gradient system. In general, the e f f e c t i s d e s t a b i l i z i n g . However, the r e s u l t s suggest that through a judicious control of the generalized forces Q^, and , normally involving the solar parameter, the radiations may be able to provide an e f f e c t i v e damping torque to a t t a i n a desired attitude. Hence the f e a s i b i l i t y of using the solar pressure i n achieving general attitude control was explored. 3.2.1 Proposed Controller Model and the Equations of Motion The proposed c o n t r o l l e r model i s shown i n Figure 3.7. It e s s e n t i a l l y consists of two sets of plates and G2/ made of highly r e f l e c t i v e , l i g h t but r i g i d material (e.g. aluminized Mylar membrane) to govern the generalized forces through the regulated motion of the centre of pressure. The angular p o s i t i o n of these sets i s kept stationary with respect to the x 2,y 2,z^-axes, themselves being fixed at 45° to each other. The motion of the plates with respect to the s a t e l l i t e body i s controlled along the z 2-axis (£^) as well as the y 2 - a x i s (m^ ) whereas the set G 2 i s free to move along the z 2 ~ a x i s {l^)• Equations (3.3) govern the motion of the s a t e l l i t e equipped with the proposed c o n t r o l l e r where the new expressions for the generalized forces are obtained to account for the solar torque provided by the plates. The general procedure adopted here f i r s t considers a given set of plates at an arb i t r a r y angular p o s i t i o n about the z 2 ~ a x i s with t h e i r e f f e c t i v e centre of pressure coordinates as n,m,£ with respect to the x 2,y 2,z 2~axes. Next, the contribution of the i n d i v i d u a l set i s reduced to the s p e c i f i c configura-Cy| ind r ica I sate l I i te 117 (a) (b) Figure 3.7 Schematic diagram of the s a t e l l i t e - c o n t r o l l e r : (a) equilibrium; (b) instantaneous control con-fig u r a t i o n ; e-<j>, ijj=g=x=io°, ^l=3'=x,=o.i 118 tions of the sets G 1 and G 2. Superposing the res u l t s y i e l d s the following expressions for the generalized forces: •+ p, Cos ce+nol.Jt^* (e+Hf-d>;+ p» Cos(e+ty)}| + U 2 C 0 5 P /VS } { ( l + -2.(>) (SLyi(.e+lk-4)J+P,Cos(0-Mli;) + £ , Cos (G+Hr) + (3, Sen p.Scyj (e+-HO + Cos p]| J :|&iSC*p 5c*(e+-tyJ-f*2Xos p>)-^.( Cos p Cost0+"i]r-(D) - P i Cos pSCv\(e+^)+fa.S£r»P>)}.|{StnCe+'^-<W +- p, Cos - (64^1 + U z \ j T ) { - ( \ - z \ > ) (stn(e+-iM>; Cos(e + i|r)) + 0 + 2f>)(Str»pCos(e+^-4>)-PiSc^. •SCA(G4^)- ^ C O S p) }({Scv>Ce+^-c»-Sty> Cos 119 Q A = - {SAu-t-P-tt/c'J wi, { S i » (e+ iM>) + P, Cos(e+V)^.' (3.16) On substituting the above r e l a t i o n s i n equation (3.3) the governing equations take the form: {(\+ e Co s 0 ) V - 2 e Sen 8 C i +-V;}- 2.0+e Cos 0) Ci \y') p' ta„ p> 1 3 3 { ^ C ^ V ; S c n P J / C o s p J = (\+e)/(i+€Cos0) {(i+eCos0) p>"-2eSm0 p'} + {(l+ eCos0)(i+ivO+ 3 Cos tyj. {*!- Ci+V') Scwp}. Cos p> = G|pO+e)/o + *Case). ( X - O + V ) SL^p } ] = C,% (\+ef/(i+eCos6.)3 (3.17) where 120 G^ = [ { (Cv C o s P+^ S c "W^ |{Stn(G + ^ -<fr)4.f,Cos(Q+^)}l]-{fcp/(/ff Cosp)} • -p, Sin p Sen CG+^J-p^CoS P)} . 1 {Sinte+^-fcJ- 5^ f> ^  (0+^$ + P,CosCe+^)t P ,SCnp5i^(9+^)+ pz Cos P i |J Gp = 2f>[c^{StnpCos(0+^-(t))-p, Sen p5cn (e+V;-p2CosP} "Cx { Cos p CosC0+^ -4 ) ) -p , Cos p&M(e+TW+ P>4*"frf| . |{Sm(©4-U>-<j>) + P,Cos(Q+ii^| + Cc^^.[-<i-2|3;-. [Scn(&UM>)+Pi<W^^ -p ( SmpScn(6f^ ; -p z Cos p5] .|fScn(04^)-SCnp. CosCe+tiJ-^+fi.CosCG+^ + ^SCnpSCnCO+^+P^plI The regulation of the plate movement governs the magnitude and sign of c. according to the r e l a t i o n s : with |c.|< c , k denoting an integer, and j=i[>,-B,A. The J — ITlctX integer n i s odd for U> and even for 6 and A . u . and v. 3 3 are the gains i n the 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 s and Yj r e f e r to the p o s i t i o n control angles. 3.2.2 Results and Discussion In the absence of a known closed form solution, the governing d i f f e r e n t i a l equations were integrated numerically, using the procedure discussed before, to obtain the damping and attitude control c h a r a c t e r i s t i c s of the system. Figure 3.8(a) shows a few t y p i c a l situations i n which the proposed solar c o n t r o l l e r attempts to provide a desired o r i e n t a t i o n to the s a t e l l i t e i n the e c l i p t i c plane. I t becomes evident that the c o n t r o l l e r i s success-f u l i n positioning the s a t e l l i t e p r e c i s e l y along the l o c a l v e r t i c a l as well as the l o c a l h o r i z o n t a l . Furthermore, through a suitable choice of the p o s i t i o n control parameters Y., intermediate orientations can also be attained. 3 Figure 3.8(b) displays c h i e f l y the same trends for geo-stationary o r b i t s . The c o n t r o l l e r continues to be e f f e c t i v e even i n an e l l i p t i c o r b i t as suggested by the plots i n Figure 3.8(c). The planar response exhibits l i m i t cycle behaviour, as expected, though r o l l and yaw motions can be damped com-p l e t e l y . However, for the general case of ar b i t r a r y o r ien-tation, l i m i t cycles would be present i n a l l the l i b r a t i o n a l modes. A parametric study was undertaken to aid the design of the control system as applied to synchronous s a t e l l i t e s . The r e s u l t i n g information i s summarized i n Figures 3.9, 3.10. Influence of the c o n t r o l l e r gains (y ^ # v_.) , the maximum available control torque (<* c = ) , and the s a t e l l i t e max i n e r t i a d i s t r i b u t i o n parameter (K^) on the time-index T^ i s shown i n Figure 3.9. In general, increase i n c o n t r o l l e r gains and ^ m a x a f f e c t the time-index favourably, to a point, beyond which the dependence becomes somewhat weak. Longer s a t e l l i t e s (large K^) withstand disturbances better and quickly regain the equilibrium p o s i t i o n compared to short and stubby designs. The semi-passive c o n t r o l l e r i s able to damp the l i b r a t i o n a l motion within a f r a c t i o n of an o r b i t , even when the system i s subjected to such a large disturbance. e = o , i = <}> = o ; ^ o = (B=X0 = 3o° , 4( = P0' = A'o = o P = 0 , Cmax = 1.5 Q - O r b i t s Figure 3.8 Typical examples of the general attitude control using the proposed solar c o n t r o l l e r i n : (a) a c i r c u l a r o r b i t , i=0 K|= 0.2 , i=23.5 , e = 0 P = ° . C m o x = 1 . 5 30 <})=n/4;|jj = V j = i O ; Y i = 0 - 30 - 3 0 90 30 - 3 0 <()=n/2;Ll j=V j = 3 0 ; Y f i = Y x = 0 ' 0 'A Y ^ T T / 2 (|> = TT/2;Llj = 30 , Vj = 5 ; y 0 = O , Y x = = 0.3 -30 90 30 -30 •=T/2;|lj=Vj =10;Yj =0 /^)=n / 2;|i j = V j =30 ; Y ^ = YX = TT/2 , Y,-O 4>=TT/2 ; |ij =30 Vj = 5 ; Yj = 0.3 4 0 9 -Orb i ts Figure 3.8 Typical examples of the general attitude control using the proposed solar c o n t r o l l e r i n : (b) a c i r c u l a r o r b i t , i=23.5° e = 0.1, i= 23.5° <|>=0 . l|)o = (J = X 0 = 30° P = 0 , Cmax =1.5 ; = B'= A'0 = 0 30 - 3 0 90 30 - 3 0 30 - 3 0 K i = 0 . 2 ; j l J = Vj=10 ; V = 0 - 30 •»L J -WW - - 3 0 K i = 0 . 2 ; J i j = V j = 3 0 J Y / , = Y X = 0 Y^ = n / 2 + 1 1 K,=0.2;Ll j =30,V j = 5 ; = 0 90 30 fr - 3 0 30 - 3 0 -Xj =0.5;Llj = V j = 10 ; Yj = 0 + K i = 0 . 2 ; | l j = V j = 3 0 ; Y ^ = Y x = T T / 2 Y^ =o JLJ_ + + Kj = 0.2; |lj =30 ,Vj = 5 ; Yj = ° - 3 1 1 4 0 0 -Orbi ts Figure 3.8 Typical examples of the general attitude control using the proposed solar c o n t r o l l e r i n : (c) an eccentric o r b i t , e=0.1, i=23.5° 4 2.0 0.5 p = o , 4>=TT/2 , e=o ; t = P; = ^o= ° 0 t i = 2 3 . 5 O , Y j = 0 t = P„ = *o = 3 0 X 1 1 TT 1 1 \ v V j = , ° r — r 1 1 1 T — V Llj = 10 X lit \ 1 \ \ ^ ^ ^ ^ ^ ^ ^ ^ •m \ \ ^^^^^^^^^ ICmax = 1.5,Ki = 0.6 i 2.0 1.5 1.0 0.5 •Mi 8 8 10 0 K; = 0.6 Cmax -1-5 |ij=10 . Vj = 10 2 Cmax 0.2 0.4 0.6 0.8 Figure 3.9 System plots showing damping cha r a c t e r i s t i c s of the s a t e l l i t e for attitude control, along the l o c a l vertical,. Y-j-=0 - - — - . -Similar results were also obtained for the alignment of the s a t e l l i t e along the l o c a l h o r i z o n t a l . The time-index v a r i a t i o n with c e s s e n t i a l l y followed the s i m i l a r trend. max 1 However, as can be expected, higher gains were required to capture the s a t e l l i t e i n t h i s configuration. Here, as the control moment has to overcome the gravity gradient torque, the shorter s a t e l l i t e s showed better performance. For intermediate orientations, where the s a t e l l i t e response shows l i m i t cycle behaviour, i t would be useful to record steady state amplitudes and average equilibrium positions. This i s presented i n Figure 3.10. Importance of the position control parameters Yj becomes apparent. Now the s a t e l l i t e i s able to change i t s or i e n t a t i o n i n o r b i t , thus extending the scope of i t s mission. In fa c t , a s u i t -able s e l e c t i o n of c o n t r o l l e r gains and c can lead to any max u desired s p a t i a l o r i e n t a t i o n . In general, the penalty for thi s v e r s a t i l i t y would be i n terms of higher l i m i t cycle amplitude. However, depending upon the mission, a judicious choice of parameters would normally l i m i t i t to an acceptable value. Success of the proposed c o n t r o l l e r presents an ex-c i t i n g p o s s i b i l i t y for station-keeping of communications s a t e l l i t e s and space stations of the future (Figure 3.11). The c o n t r o l l e r size i s rather modest and so i s the power consumption of the servosystem. So far as the size of 621 (°) (b) t—1 igure 3.11 Solar r a d i a t i o n control system for the next generation of space vehicles: ° (a) communications s a t e l l i t e ; (b) space station the c o n t r o l l e r for a given s a t e l l i t e i s concerned, i t i s s t r i c t l y related to c and the maximum allowable transa-•* max l a t i o n a l motion of the plates. For example, a solar c o n t r o l l e r for INTELSAT IV series of s a t e l l i t e s would require 2 the plate s i z e of 7 f t . with permissible movement of 1 f t . to obtain a c m a x o r 1-5. According to a preliminary estimate, this would enable the s a t e l l i t e to regain the preferred orientation within ±0.003° i n less than 25 seconds when -7 9 exposed to a micrometeorite impact of -5 x 10 slug f t . / s e c . The power required for the c o n t r o l l e r operation i s e s s e n t i a l l y to overcome f r i c t i o n against the t r a n s a l a t i o n a l motion of the plates. Obviously, the higher power requirement would correspond to rapid changes i n the plate l o c a t i o n . Here, the semi-passive character of the system l i m i t s the peak power consumption to -5 watts. A comment concerning the earth shadow, which would render the c o n t r o l l e r i n e f f e c t i v e , i s appropriate here. For a geostationary o r b i t , influence of the shadow i s con-fined to a quarter of the s a t e l l i t e ' s l i f e - s p a n , and even here only during 5% of the o r b i t a l period (Figure 3.12). The . results showed the c o n t r o l l e r performance to remain v i r t u a l l y unaffected. Figure 3.12 Geometry of earth shadow to 133 3.3 Concluding Remarks The important conclusions based on the analysis may be summarized as follows: (i) The solar r a d i a t i o n pressure considerably a f f e c t s the coupled l i b r a t i o n a l response and s t a b i l i t y . This e f f e c t , p a r t i c u l a r l y s i g n i f i c a n t for i n -plane l i b r a t i o n s , i s generally detrimental to the s a t e l l i t e performance. ( i i ) The d i r e c t l i n e a r i z a t i o n and the non-linear Butenin methods can predict the l i b r a t i o n a l amplitude and frequency with a considerable accuracy,even for r e l a t i v e l y large disturbances (-20°). The main discrepancy i s i n the phase; fortunately, i t i s of l i m i t e d s i g n i f i c a n c e from p r a c t i c a l design considerations. A successful estimate of the periodic solutions by the methods represents a r e s u l t of far reaching importance, ( i i i ) The proposed solar c o n t r o l l e r has a p a r t i c u l a r l y outstanding performance when the s a t e l l i t e , with preferred orientation along the l o c a l horizontal or v e r t i c a l , i s i n a c i r c u l a r o r b i t . (iv) The e f f e c t of e c c e n t r i c i t y and intermediate orientations i s to introduce l i m i t cycles whose amplitude can be minimized by a judicious choice of the c o n t r o l l e r parameters. Even here, there i s a 134 p o s s i b i l i t y of completely eliminating t h i s steady state periodic motion by introducing a disturbance s e n s i t i v e control function, however, th i s aspect needs to be explored, (v) The semi-passive nature of the c o n t r o l l e r makes i t p a r t i c u l a r l y a t t r a c t i v e . I t promises a reduction i n cost and a substantial increase in l i f e - s p a n . 4. PLANAR LIBRATIONS OF A SATELLITE WITH FLEXIBLE APPENDAGES Having studied the l i b r a t i o n a l dynamics and attitude control of a r i g i d s a t e l l i t e accounting for r a d i a t i o n pressure, the next l o g i c a l step would be to explore the influence of solar heating. In the early stages of space exploration, (when spacecrafts were small, mechanically simple and quite r i g i d ) thermal deformations were r e l a t i v e l y i n s i g n i f i c a n t . On the other hand, a d i s t i n c t i v e feature of a modern vehicle i s the natural separation of the s t r u c t u r a l systems into two subgroups: e s s e n t i a l l y r i g i d central body,with one or more f l e x i b l e appendages. The characteriz-ation primarily stems from a widespread use of l i g h t weight deployable members, e.g., antennae, gravity gradient booms, solar panels, etc., which are inherently f l e x i b l e . The s i t u a t i o n i s further aggravated by the d i f f e r e n t i a l 5 solar heating. A NASA report (19 69) has presented an excellent review of the work i n t h i s area. The v i b r a t i o n c h a r a c t e r i s t i c s of several i n d i v i d u a l members of a spacecraft subjected to d i f f e r e n t i a l thermal heating has been analyzed i n considerable d e t a i l s However, influence of t h e i r f l e x i b i l i t y on l i b r a t i o n a l response and s t a b i l i t y has received r e l a t i v e l y less attention. The pertinent l i t e r a t u r e i s confined, i n general, to gravity 136 8 2 — 84 gradient systems with thermally flexed booms Ever increasing demand on power for the operation of the onboard instrumentation, s c i e n t i f i c experiments, communications systems, etc., has been r e f l e c t e d i n the size of the solar panels. I t has increased to a point where t h e i r f l e x i b i l i t y can no longer be neglected. For example, the proposed Canadian Communications Technology S a t e l l i t e (CTS), to be launched i n 19 74, i s designed to carry two solar panels, 3.75' x 24' each to generate 1.2 KW. This chapter reports a rather modest attempt at analyzing the dynamics of a s a t e l l i t e with such large f l e x i b l e appendages. The problem, i n general, i s extremely complex, how-ever, to gain some appreciation of the role played by f l e x i b i l i t y i n t h i s class of problems, a s i m p l i f i e d model i s considered. 4.1 Formulation of the Problem Consider a s a t e l l i t e with a r i g i d , c y l i n d r i c a l body carrying n rectangular appendages mounted i n the y-z plane. As shown i n Figure 4.1, the long axis of the i t n plate (appen-dage) makes an angle with respect to the z-axis. Let at a given instant, the deflected configuration of the plate (length assumed much larger than width and thickness) due to the temperature gradient across the thickness be as i n -dicated i n the diagram. Denoted by x,y,z are the p r i n c i p a l coordinate axes of the main body with o r i g i n at i t s centre ^X^Axis of the orbit Figure 4.1 Geometry of a ' s a t e l l i t e with f l e x i b l e appendages 138 of mass s. The o v e r a l l centre of mass i s represented by s^. Any deviation of the z-axis from the l o c a l v e r t i c a l defines the l i b r a t i o n a l angle ip. The k i n e t i c and poten t i a l energy expressions have to be obtained f i r s t for the c l a s s i c a l Lagrangian formulation. The k i n e t i c energy due to o r b i t a l , l i b r a t i o n a l and vibratory motion can be written as • (4.1) { (Z\ ft\ dtf^ h CoSPti * U dh) }\ {(£; j i ; d ^ f - ^ i i i cose* nTi de;f + ( S i J n . - S l n c X i w ( oie-,) j ] where i =1,2, ••• n, and integration extends over the length of the plate. Contribution to the poten t i a l energy arises due to the motion i n the g r a v i t a t i o n a l f i e l d and the f l e x i b i l i t y of the appendages: 139 SC-n o<j rv\j where <{K_. represents the j t h normal mode (j=l,2 ••• m) of the i t n appendage (i=l,2 ••• n) having amplitude A^ _. and the natural frequency . Using the above expressions and recognizing that the equations of motion i n and A^ _. degrees of freedom can be written as: 140 + ( Z i H i Cos a t i r } ] ( d + « ) + 2 [ ^ : 1 . / ( S . i 5 . + -^•J*];™,- Cos*,- atif j ] 5^ajxG>s^+C3/A/R 3J. [ O/^J ) (Zli IS," mi ( Z 1 J^jCos^i We ^ ( 0 -( X i I S i ^ i G > s * i w j d £ 0 ] C o s 2 i | r =, • • • -[Aij j*^ ^ati-O/Yvi,) (Z-.zjA;) £fej.fni dfc) Jcfcjfn,'d&]. (4.3) (4.4) 141 The r e s u l t s of Chapter 2 showed that the value of • • 3*2 ij» i s of the same order as 0. Also y/R =0 /(1+e Cos 0), hence i n equation (4.4) a l l terms due to the g r a v i t a t i o n a l f i e l d can be ignored i n view of the r e l a t i v e l y high natural frequency of vibrations (e.g., i t s values i n case of CTS solar panels -0.1 c.p.s.). This s i m p l i f i e s the equations of motion corresponding to the e l a s t i c degrees of freedom to A-.} J*tj_»ii dt\ - U ^ Z i H j j /kj 4UjriTi d(\ \d>^ . • • • • (4 • At t h i s stage information concerning the deformation of the panels caused by solar heating i s e s s e n t i a l to proceed further. The temperature d i s t r i b u t i o n i n the panels i s expected to show a c y c l i c behaviour modified by the transient terms. However, the thermal analysis of a boom ca r r i e d out 8 2 by Modi and Brereton has c l e a r l y established that the time constant i s rather small. This implies a d i r e c t dependence of Q^j on the p o s i t i o n of the s a t e l l i t e with respect to the sun, thermal lag being n e g l i g i b l e compared to the l i b r a t i o n a l and o r b i t a l periods. In general, i n t e r n a l d i s s i p a t i o n of the system would cause transients to damp out leaving only the forced motion of the panels. As periods of the general-142 ized forces and the o r b i t a l motion have the same order, the terms may also be neglected thus reducing (4.5) to ftij - Q'^l J dP-Cj YY\\ d l i ] (4-6) A system configuration can now be represented by the steady state d e f l e c t i o n of the panels corresponding to the s a t e l l i t e o r i e n t a t i o n . Hence, the governing equation of l i b r a t i o n a l motion becomes [ i% +Zi M cos *j) ™,- ati -o/^i) K z i j i ^ 10/*v*) { ^ ; Cos*,- w <Ujf]. St* f Cos iV + (ty/R 3) [(»/W(5 jf; * 10-Cos 2 ^ - Qi|r (4.7) This general equation i s applicable to s a t e l l i t e s having an a r b i t r a r y number of e l a s t i c appendages. However, 143 for better appreciation of the f l e x i b i l i t y e f f e c t s , a r e l a t i v e l y simple model with two i d e n t i c a l panels has been considered. Here, the panels are mounted symmetrically about the s a t e l l i t e mass centre with t h e i r long axis along the z-coordinate.This s i m p l i f i e s equation (4.7) to t U ) U | R 3 ) [ i r » . - I | | - 2 J ( 2 N ^ ^ 1 d t 1 + •••• (4-8) U / ™ S ) ( J I , y y \ { cU . j ] SLY) I ^ C O S ^ -where 4.2 Thermal Deflection of the Appendages The d i f f e r e n t i a l equation governing the temperature v a r i a t i o n across the thickness of the panels (thickness assumed much smaller compared to the length and width of the panels) can be written as {£rj &f) - o .... (4.9) with boundary conditions given by 144 1= +a Here T^ and represent the steady state maximum and minimum temperature of the appendage, respectively (Figure 4.2(a)) and the absolute ambient temperature at the s a t e l l i t e a l t i t u d e i s taken to be zero. Furthermore, the problem which i s inherently conjugate i n character, i s s i m p l i f i e d by neglecting temperature-deflection i n t e r a c t i o n . Integrating equation (4.9) and introducing the above boundary conditions lead to: with temperature d i f f e r e n t i a l across the thickness given T = T 2 - < x t c r s » i C I - a ) / K by A T = T , - T 2 = 2 ^ ^ T 2 Through algebraic manipulations, these re l a t i o n s can also be put into the form: 145 '/4 ^2. =[(.o<a^/ o< ecrs)/ A T - 2 . 0 ^ 0 3 T ^ 4 a ( K which i s amenable to an i t e r a t i v e scheme providing the exact value of and hence the temperature difference AT. However, i t may be worthwhile to investigate the accuracy of the f i r s t and the second approximations during the i t e r a t i o n process. The f i r s t approximation y i e l d s (4.10) CAT), = * a < / a / K (4.11) while repeating the i t e r a t i v e cycle once more leads to (ATk= ( o^s'a-l K)/[ 1 + z{(°<«. Sa.\K)/(<^s/2«jc3) j (4.12) 146 A comparison of the approximate results with the exact AT i s presented i n Figure 4,3 for a few t y p i c a l s i t u a t i o n s . The approximations are found to predict the temperature difference with a considerable accuracy, thus suggesting a p o s s i b i l i t y of using (AT)^ i n analyzing thermal deflections of the panels. The s i m p l i f i c a t i o n leads to the expressions fpr curvature,induced by the temperature 92 gradient,and the corresponding 6 as (Figure 4.2(b)): ( l / f c ) ^ ' ( b / E I f ) J _ + S j d % : * t « < t r f a ^ 2 * ) (4.13) where With th i s information, integrals i n the equation of motion (4.8) can be evaluated: £ 147 Figure 4.2 (a) coordinate system for thermal analysis (b) thermal d e f l e c t i o n of the panel 148 — Exact Approximate - 1 Approximate - 2 a P - f t Figure 4.3 Solar radiation induced temperature difference across the f l a t plate appendage-thickness 149 where 9,(8) - { 3 $ +-(1/2.) Sin I S - U S(n^}fc^^) y s ; = {(§-sc*s)/s\2~ •••• (4.i4) The equation of motion then becomes + C 3 i u j R 3 ; L t x i L - r ^ + ^ ^ 1 ^ ; + ' 9 . 3 ( ^ + •••• ( 4 - 1 5 ) ^ j ^ ) 9^(^)5 ] ' s u i p Cos y - 0 - v where i x l . C'|D w ( > ^ I W Q/a) ^ 150 3 Dividing throughout by (yl /R ) and recognizing that the equation can be rewritten as • ^ ( & ) - ^ k C » } ] + 3 [ K i + I ^ , ( S J + ^ (4.16) Expanding the trigonometric functions appearing i n the expressions for g^,g2,g2 and retaining terms upto 7 0(6 ) gives: •9,(5) - a > o ) - ( $ V a ) ^ a 7 ^ ) - 0^1360;•• •••• <4-17> On substituting for g-^c^ a n (^ ^ 3 ^ r o m above and introducing the generalized force expression from (2.9), the equation 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 with f l e x i b l e panels becomes symmetrically mounted and undergoing only planar motion, do not provide any solar torque. 4.3 Results and Discussion 4.3.1 System Response The system behaviour was studied over a range of important parameters through numerical integration of the governing non-linear, non-autonomous d i f f e r e n t i a l equation. Figure 4.4 (a) compares the e f f e c t of f l e x i b i l i t y (Q ) on |ifj| and T over a range of the i n e r t i a parameter. [(!+-€ Cos &) ~ 2 eSCy) 0 ( [ I + i > £-(<#6<0 -(4. I t may be pointed out that the panels, being g =0.5 m*p= 0.8 ^0 = o >P= 2.5 <|) = 0 +0=0 .5 c = o , e =0 c = o , e = o.i c = o.i , e = o.i 1.0 0.5 1.0 0.5 1.0 0.5 40 30 o 'I max 20 10 1.0 h 0.5 1.0 0.5 1.0 0.5 40 30 20 0.5 0.6 0.7 v 0.8 0.9 1.0 10 0.5 0.6 r oT=i 0.7 K . 0.8 0.9 1.0 Figure 4.4 System plots showing the maximum l i b r a t i o n a l amplitude and averaqe period as affected by: (a) the i n e r t i a parameter •p Figure 4.4 System plots showing the maximum l i b r a t i o n a l amplitude and average period as affected by: (b) the f l e x i b i l i t y and appendage-size parameter 154 It i s apparent that the presence of f l e x i b i l i t y , i n general, leads to an increase i n l i b r a t i o n a l amplitude. This i s more pronounced when i s small or the s a t e l l i t e i s i n an eccentric o r b i t . The solar r a d i a t i o n pressure further adds to the amplitude of motion. The period of l i b r a t i o n s * accounting for f l e x i b i l i t y (Q =1) seems to be s l i g h t l y * higher than that for the r i g i d s a t e l l i t e (Q =0). * It was now thought appropriate to vary Q systematic-a l l y and assess i t s influence on the s a t e l l i t e performance (Figure 4.4 (b)). The study e s s e n t i a l l y confirms the f i n d -ings of the e a r l i e r comparison. E f f e c t of the panel s i z e , * denoted by I , on the system behaviour i s also shown i n the figure. I t may be observed that increasing the appendage si z e , i n general, leads to an increase i n the \\b\ as ^ 1r'max well as the average period of the motion. Overall trends remain the same for s a t e l l i t e s i n eccentric o r b i t s and under influence of the solar r a d i a t i o n pressure. 4.3.2 S t a b i l i t y The governing d i f f e r e n t i a l equation with periodic co-e f f i c i e n t s i s of the form s i m i l a r to that of equation (2.10), hence the "numerical experiment" described before results i n int e g r a l manifolds. The usefulness of such a phase space representation i s f a m i l i a r to us. For given e c c e n t r i c i t y , solar parameter and s a t e l l i t e c h a r a c t e r i s t i c s i t provides a l l 155 possible combinations of disturbances to which the s a t e l l i t e can be subjected without causing i t to tumble. A convenient condensation of data may be affected by p l o t t i n g the intercept of the -axis with the phase space-cross sections at 0=0 (Figure 4.5). The r e s u l t i n g s t a b i l i t y charts provide valuable information concerning the maximum permissible e c c e n t r i c i t y consistent with s t a b i l i t y . A few t y p i c a l situations recorded here c l e a r l y reveal the d e s t a b i l i z i n g influence of f l e x i b i l i t y . Thermoelastic behaviour of the large panels causes a substantial reduction in the zone of s t a b i l i t y and the value of c r i t i c a l eccen-t r i c i t y for stable operation. The solar r a d i a t i o n pressure only enhances these reductions. 4.4 Concluding Remarks (i) The thermoelastic behaviour of large appendages adversely affects the s a t e l l i t e performance. The amplitude as well as average period, i n general, * * increase with f l e x i b i l i t y (Q ) and size (I ) of P the panels. ( i i ) The f l e x i b l e nature of the s a t e l l i t e causes a substantial reduction i n size of the s t a b i l i t y region. The d e s t a b i l i z i n g influence i s quite severe when the e f f e c t of the solar r a d i a t i o n pressure i s also considered (c^O). = 0 . 7 m*p = 0.8 <(> = 0 Q* = 0 g = 0.5 Ip = 2.5 +o = 0 - - - - - Q# = l 1.5 1.0 0.5 •« 0 - 0 . 5 --1.0 -1.5 - 2 . 0 1.5 1.0 0.5 0 - - 0 . 5 -1.0 -1.5 0.05 0.10 0.15 0.20 0.25 e - 2 . 0 J. c = 0.1 _L _L 3 0.05 0.10 0.15 0.20 0.25 e Figure 4.5 S t a b i l i t y charts showing e f f e c t of the solar parameter, the f l e x i b i l i t y parameter and e c c e n t r i c i t y H-1 157 ( i i i ) If the s t a b i l i t y diagram (Figure 4.5) i s i n t e r -preted as a measure of the disturbance which the s a t e l l i t e can toler a t e without becoming unstable, i t i s apparent that even moderate values of e c c e n t r i c i t y seriously a f f e c t i t s performance. The c r i t i c a l value of e c c e n t r i c i t y for stable motion i s adversely affected by the solar r a d i a t i o n pressure as well as f l e x i b i l i t y . 5. CLOSING COMMENTS 5.1 Summary of the Conclusions As stated at the outset, the main objective of the study has been to gain an insight into the attitude dynamics of s a t e l l i t e s i n high a l t i t u d e o r b i t s as affected by the solar r a d i a t i o n pressure. This has been achieved with some measure of success through the systematic procedure adopted here. The plan of study, which opted for combating d i f f i c u l t i e s i n stages, proved to be quite successful. The important conclusions based on the analysis may be l i s t e d as follows: (i) The solar r a d i a t i o n pressure, normally neglected i n the analysis of a gravity oriented system, can have a substantial detrimental e f f e c t on i t s l i b r a t i o n a l performance. It makes the consider-ation of area d i s t r i b u t i o n about the mass centre as important as the mass d i s t r i b u t i o n , ( i i ) The concept of i n t e g r a l manifold can be success-f u l l y employed to develop s t a b i l i t y charts for planar l i b r a t i o n s . Besides giving the range of permissible disturbances and e c c e n t r i c i t y , i t provides information concerning periodic solutions. However, i n the case of a non-autonomous,coupled system, the concept has an inherent l i m i t a t i o n due 159 to the p r a c t i c a l d i f f i c u l t y of presenting and interpreting r e s u l t s i n a hypersurface. In such a s i t u a t i o n one i s forced to adopt a rather tedious procedure of analyzing the response over a large number of o r b i t s , ( i i i ) The solar r a d i a t i o n pressure can be used quite e f f e c t i v e l y i n l i b r a t i o n a l damping and attitude co n t r o l . The planar c o n t r o l l e r provides a ca p a b i l i t y of damping pitching l i b r a t i o n s about i t s preferred orientation, while the general model extends the application of the rad i a t i o n pressure to enable s p a t i a l attitude c o n t r o l . The semi-passive character of the c o n t r o l l e r promises a saving i n onboard fuel requirements and a con-sequent increase i n the s a t e l l i t e l i f e - s p a n . I t presents an exc i t i n g p o s s i b i l i t y of c o n t r o l l i n g the next generation of communications s a t e l l i t e s and space stations of the future, (iv) The approximate a n a l y t i c a l solutions developed here should prove useful, p a r t i c u l a r l y , for the preliminary design purposes. For small amplitude motion, usually the case of in t e r e s t , they can predict the l i b r a t i o n a l amplitude and frequency with an acceptable accuracy. S i m i l a r l y , the approximate analyses of damped l i b r a t i o n s using l i n e a r i z a t i o n and Airy's approaches essen-t i a l l y present a correct picture. 160 (v) The f l e x i b i l i t y can have a strong d e s t a b i l i z i n g influence on the s a t e l l i t e attitude behaviour. 5.2 Recommendations for Future Work The investigation reported here suggests several topics for future exploration. Only some of the important problems are mentioned here: (i) A p o s s i b i l i t y of hybridization of the solar control with active methods for quick and precise s a t e l l i t e alignment presents a potential area for future investigation. Of considerable si g n i f i c a n c e would be the problem of eliminating the l i m i t cycle type of response i n general attitude control using a disturbance sen s i t i v e control function. Attention should also be directed towards optimization of the system parameters and the analysis involving an alternate solar c o n t r o l l e r design r e l y i n g on plate rotations. An investigation of the hardware design d e t a i l s and control c i r c u i t r y would also be of i n t e r e s t , ( i i ) Inspite 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 importance. Hence, the e f f o r t s should be made to improve upon t h e i r accuracy. Even an approximate 161 procedure for predicting the bounds of stable motion would constitute an important step forward. ( i i i ) Most s a t e l l i t e s are provided with some form of damping mechanism, i n addition to the i n t e r n a l damping, to minimize the transient response. L i b r a t i o n a l dynamics of a damped s a t e l l i t e has 94 been studied at length by Tschann . It would be of considerable i n t e r e s t to evaluate an improve-ment i n the c o n t r o l l e r ' s performance due to the presence of damping, (iv) Any deviation from the c l a s s i c a l Keplerian System for the near-earth o r b i t a l motion has been only of academic i n t e r e s t so f a r . However, with current emphasis i n the non-polluting solar power using near-earth multibody systems, co-orbiting s a t e l l i t e configurations are bound to gain importance. Attitude control problems associated with such multibody systems are l i k e l y to be quite challeng-ing . (v) The f i e l d of i n v e s t i g a t i o n l e f t open by the fourth chapter i s rather wide. The prime objective of the future e f f o r t s should be to analyze l i b r a t i o n a l dynamics of a s a t e l l i t e , with a r b i t r a r y o r i e n t a t i o n of appendages, executing general three dimensional motion. 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