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On the librational dynamics of damped satellites Tschann, Christian Aime 1970

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ON THE LIBRATIONAL DYNAMICS OF DAMPED SATELLITES by CHRISTIAN AIME TSCHANN Ingenieur CESTI, Paris, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of MECHANICAL ENGINEERING We accept this tresis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1970 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that publication, in part or in whole, or the copying of this thesis for financial gain shall not be allowed with-out my written permission. Christian A. Tschann Department of Mechanical Engineering The University of British Columbia Vancouver 8, Canada Date ABSTRACT The thesis examines diverse methods of damping the librational motion of earth-orbiting satellites. Starting with passive stabilization, two classical mechanisms for energy dissipation are studied, for performance comparison, when executing librations in the orbital plane. The first model, consisting of a sliding mass restricted to relative translational motion with respect to the main satellite body, establishes the suitability of various approaches to the problem in circular orbit. In this case, numerical and analog methods do not readily yield information on the influence of parameters and approximate methods are found to be particularly helpful. Butenin's method based on aver-aging techniques predicts the response of the satellite with good accuracy for small damping constant while the exact solution to the linearized equations provides optimum damper characteristics for motion in the small. A comparison of the sliding mass damper model with a damper boom mechanism involving only relative rotational displacements, is £hen performed for equal equilibrium inertias of the damping devices. It indicates that, for optimum transient tuning, the damper boom would have a better time-index while the sliding mass would lead to smaller steady-state amplitudes iii for low eccentricity orbits. A numerical example using GEOS-A satellite data illustrates the outcome of the study when applied to physical situations. A stability analysis is also included which uses Routh and Lyapunov approaches to determine the domain of parameters leading to asymptotic stability, as well as numerical methods to define the bounds on stable initial disturbances: it is found that for most practical applications, the stability contour in circular orbit is close to that of the undamped case. How-ever, for eccentric trajectory, the amount of damping critically affects asymptotic stability. The next model, which involves active stabilization, uses solar radiation pressure to achieve planar librational control of a satellite orbiting in the plane of the ecliptic. This is obtained by adjusting the position of the center of pressure with respect to the center of mass through a con-troller depending on a linear combination of librational velocity and displacement. The motion in circular orbit is; first investigated through the W.K.B. method. Although the approximate equation involves an infinity of turning points, only a few of them are required to evaluate the damped behaviour of the system. A comparison of the analytical results with a numerical integration of the exact equation of motion shows good agreement only over a limited range of parameters and,therefore, the latter is used to complete the study for circular and elliptic cases. The concept leads iv to great versatility in positioning a satellite at any angle with respect to the local vertical. Also, high transient ; performance is observed about local vertical and horizontal and the dichotomous property of good transient associated with poor steady-state inherent to passive damping can be avoided by selecting appropriate controller parameters. An example is included which substantiates the feasibility of the configuration. Finally, the attention is directed towards the in-fluence of gravity torques on the stability of damped axisymmetric dual-spin satellites. The nutation damper mounted on the slowly-spinning section is of the pendulum type. For this section rotating at orbital angular rate, application of the Kelvin-Tait-Chetaev theorem indicates that the asymptotic stability region reduces basically to the mainly positive stable spin region of the undamped case. However, some care is required depending upon the shape and natural frequency of the damper. If the damper section rotates at a much higher rate than the orbital one, torque-free motion need only be considered for short term pre-dictions. Stability charts corresponding to this case, given for comparison, emphasize the effect of gravity. TABLE OF CONTENTS CHAPTER PAGE 1 INTRODUCTION . . . 1 1.1 Preliminary Remarks 1 1.2 Literature Survey 3 1.3 Purpose and Scope of the Investigation . . 12 2 LIBRATIONAL DYNAMICS OF PASSIVELY DAMPED SYSTEMS 15 2.1 Preliminary Remarks 15 2.2 Formulation of the Problem 16 2.2.1 Sliding mass damper 16 2.2.2 Damper boom 21 2.3 Integration of the Exact Equations of Motion 25 2.3.1 Digital technique 25 2.3.2 Analog Simulation 25 2.4 Approximate Solutions in Circular Orbit 33 2.4.1 Linearized analysis . . . . . . . . 34 2.4.2 Butenin's method 48 2.5 Optimized Performance of Sliding Mass . Damper and Damper Boom Models 61 2.5.1 Preliminary remarks 61 2.5.2 Analysis 67 (a) Transient optimization 67 (b) Steady-state optimization 71 vi CHAPTER PAGE 2.5.3 Results and discussion 73 2.5.4 Numerical example 8 8 2.6 Stability Analysis 95 2.6.1 Circular orbit 96 (a) Routh-Hurwitz analysis 96 (b) Lyapunov analysis 102 (c) Zubov's method Ill (d) Numerical results 115 2.6.2 Elliptic orbit 127 2.7 Concluding Remarks 132 3 ATTITUDE AND LIBRATIONAL CONTROL OF A SATELLITE USING SOLAR RADIATION PRESSURE . . 138 3.1 Introductory Remarks 138 3.2 Formulation of the Problem . . 139 3.3 Motion in Circular Orbit 143 3.3.1 Approximate W.K.B. solution . . . . 143 (a) Analysis 144 (b) Discussion of results 153 3.3.2 Numerical results 157 3.4 Motion in Eccentric Orbit 178 3.5 • Feasibility Study 193 3.6 Concluding Remarks 200 4 INFLUENCE OF GRAVITY-TORQUES ON THE STABILITY OF DAMPED AXISYMMETRIC DUAL-SPIN SATELLITES . . . 204 4.1 Preliminary Remarks 204 vii CHAPTER PAGE 4.2 Formulation of the Problem 20 5 4.2.1 Torque-free motion 210 4.2.2 Motion in the gravity field 212 4.3 Analysis 216 4.3.1 Torque-free motion 216 4.3.2 Motion in the gravity field 221 (a) Damped case 221 (b) Undamped case 228 4.4 Concluding Remarks 230 5 CLOSING COMMENTS 231 5.1 Summary 2 31 5.2 Recommendations for Future Work 2 32 BIBLIOGRAPHY 234 LIST OF TABLES TABLE PAGE 2-1 Representative Gravity-Gradient Satellite Characteristics 88 2-2 Sliding Mass Damper Optimum Parameters 91 2-3 Damper Boom Optimum Parameters 95 LIST OF FIGURES FIGURE PAGE 1-1 Plan of study 14 2-1 Geometry of SMD model 17 2-2 Geometry of DB model 22 2-3 Analog programming of SMD model (a=0); e=0 28 2-4 Comparison of SMD model stability boundaries as obtained by analog and numerical techniques; e=0, K±=l 29 2-5 Influence of damper parameters on system re-sponse; e=0, K^=l, , = 0.5: (a) damping constant 31 (b) natural frequency 31 2-6 Influence of damper inertia on system response; e=0, Ki=l, <1^ =0.5 32 2-7 Variation of ai and a 2 as functions of damper frequency and inertia parameter: (a) Kd=0.01, l/x*m=0.5 37 lb) Kd=0.05, l/x*m=0.5 . . . 37 (c) Kd=0.01, l/x*m=1.4 38 (d) Kd=0.05, l/x*m=2.0 38 2-8 Variation of a^ and a^ as functions of damping constant and inertia parameter; o)*2 — 4 : sm (a) Kd=0.01 39 (b) Kd=0.05 40 2-9 Comparison between analytically and numerically predicted optimum damping constants; Kd=0.01: (a) K. =1. 0 42 l (b) Ki=0.6 42 X FIGURE PAGE 2-10 Comparison between analytically and numerically predicted responses; Kd=0.01: (a) K.=1.0, 0,52=6.25, lA*m=1.0; ^=0, • ; (b) K.=1.0, c^=6.25, lA s m=1.0; ^=30° , Tp  ' = 0 (c) K°=1.0, lA*m=1.2; ^ = 2 0 % K = 0 - 3 • • (d) K. =0.6 , go =4.0, 1/x* =1.2; \p =20°, 1 sm sm o • • , (e) K.-0.6, ^ = 4 . 0 , 1/^=2.0; ^=20°, ip'=  0.3 ro Representative variation of system character-istic frequencies with ' for low values of 1/T* ' sm Variation of Q^ and as a function of damper frequency and inertia parameter; 1/T* =0.5: sm (a) Kd=0.01 (b) Kd=0.05 2-13 Dynamical response of the satellite at low damping 2-14 Comparison between analytically and numerically predicted responses; OJsm2=4.0, l/r*m=0.5, Kd=0.01: (a) Ki=1.0 (b) Ki=0.6 2-15 Comparison between analytically and numerically predicted responses during larger disturbances; oa*2=4.0, 1/T* =0.5, K,=0.01: sm ' sm d (a) 1^=1.0; ^=0.2 (b) Ki=0.6; ^o=ll° , ip^ =0.2 (c) 1^=1.0; ipo=170 , 1^=0.3 2-11 2-12 43 44 45 46 47 50 56 56 57 58 58 59 59 60 xi FIGURE PAGE 2-16 Configurations of root-coalescence of quartic equations 6 2 2-17 Distribution of characteristic roots as a function of damper parameters: (a) DB model 6 4 (b) SMD model 65 2-18 Optimum SMD characteristics as a function of K. and a 75 x 2-19 Variation of SMD time-index as a function of K. and a . . . 76 1 « 2-20 Frequency-response of SMD in small eccentricity orbit 77 2-21 Steady-state amplitude of SMD corresponding to transient optimization: (a) K =0.01 79 d (b) a=7i/4 79 2-22 Optimum DB characteristics as a function of K. and 3 81 l 2-23 Variation of DB time-index as a function of K." and 3: l (a) H =0.01 82 d (b) Hd=0.05 82 2-24 Frequency response of DB in small eccentricity orbit 84 2-25 Steady-state amplitude of DB corresponding to optimum transient: (a) Hd=0.01 85 (b) 3=tt/2 85 2-26 Variation of SMD time-index as a function of equilibrium inertia of the damping mass; K =0.005 87 d 2-2-7 Comparison of SMD and DB time-indices as a function of inertia of the damper unit . . . . 89 xii FIGURE PAGE 2-28 Comparison of SMD and DB steady-state amplitudes corresponding to optimum trans-ient parameters, as a function of inertia of the damper unit 90 2-29 Optimum damper characteristics for GEOS-A satellite: (a) SMD 9 3 (b) DB 94 * 2 2-30 Variation of (w ) ... , as a function of , sm critical K. and a: l (a) K =0.01 99 d (b) Kd=0.05 99 2-31 Stable equilibria of optimized SMD configuration 100 * 2 2-32 Variation of (<JO,, ) ... , as a function of db critical K i and Hd(3=90°) 101 2-33 Domain of stability of SMD model as found from Hamiltonian: (a) a=00 . 108 .(b) a=90° 108 2-34 Distribution of critical saddle points for SMD model as a function of K. and a: l (a) Kd=0.01 109 (b) Kd=0.05 . . . . . 109 2-35 Numerical determination of boundary of stability for SMD model; a=0, K.=l, K =0.01: i d (a) z^=0 117 (b) zd=0 118 2-36 Numerical determination of boundary of stability for SMD model; a=0 . 6 , 1^=0 . 8 , Kd=0.01: (a) zd=0 . 119 (b) zd=0 120 2-37 Response of SMD model to large disturbance . . . 121 xiii FIGURE PAGE 2-38 Effect of hard-stops on the stability domain of SMD model 12 3 2-39 Influence of hard-stops on the response of SMD model 124 2-40 Numerical determination of boundary of stabil-ity of DB model (B=90°): (a) e^=0 125 (b) £d=0 126 2-41 Effect of hard-stops on the stability domain of DB model 12 8 2-42 Influence of hard-stops on the response of DB model 129 2-4 3 Charts of stable damper parameters in presence of eccentricity: (a) SMD 133 (b) DB 134 3-1 Geometry of satellite motion 140 3-2 Variation of the characteristic function p(0): (a) <j> = 0, K ± = 1, v c = 0, p c = 40 146 (b) regions of specific solutions 146 3-3 Influence of angular disturbance on the response of the satellite; <t>  = 2, K ± = 1, v c = 0, y c = 60, ^ = 0: (a) ipQ = 3°, ijJQ = 6° 155 (b) = 12° , ijjQ = -12° 156 (c) \j) = 23° , = 34° 156 3-4 Influence of impulsive disturbance on the response of the satellite 158 3-5 Influence of y on the correlation between analytically and numerically predicted responses 159 3-6 Influence of v on the response of the satellite 159 xiv FIGURE PAGE 3-7 Influence of K. on the response of the satellite , . * . . . 160 3-8 Optimization of controller parameters for Y = 0 : (a) C =2 ; 162 max (b) C =0.2 162 max (c) Comparison of steepest paths 163 3-9 Typical examples of satellite response to external disturbances; y = 0: (a)-(b) Effect of angular disturbances 165 (c)-(d) Effect of impulsive disturbances . . . . 165 3-10 Influence of controller parameters on satellite response for y = Q: (a) Variation of G 166 max (b) Variation of v c 166 Positioning of satellite away from the local vertical, y = 0 . 3 168 Influence of system parameters on transient performance; 0<Y<TT/2: (a) Variation of y and v for C = 1.4 . . . . 169 c c max (b) Variation of y and v for C = 0.2 . . . . 169 c c max (c) Variation of K. and C 170 l max (d) Variation of ^ and y 170 3-13 Optimization of controller parameters for Y = TT/2 : (a) K. = 0.2 173 l (b) K. = 0.6 173 i (c) Variation of K. 174 (d) Influence of C 175 max (e) Influence of initial conditions • 175 3-14 System plots for Y = IT/2: (a)-(b) Typical behaviour for two sets of con-troller parameters 176 3-11 3-12 XV FIGURE PAGE (c) Domain of stable controller parameters for 4>Q = 72° 177 (d) Domain of stable controller parameters for 4>Q = 55° 177 3-15 Influence of step-size on the numerical integration of equation of motion 179 3-16 Influence of controller parameters on steady-state amplitude; Y = 0,K^ = 1/<J> = 0: (a) vc=0, e=0.1 181 (b) C =1.5, e=0.2 181 max ' 3-17 Effect of eccentricity on satellite perform-ance , y = 0/K^ = 1, cf> = 0 r (a) Chart of y c optimum; v c = 0 183 (b) Steady-state amplitude; v = 0 183 (c) Chart of y c optimum; v c = 10 184 (d) Steady-state amplitude; v = 1 0 1 8 4 c 3-18 Steady-state performance as affected by inertia parameter; y = 0/ $ = 0 : (a) e = 0.1, C = 1.5, v = 10 186 max c (b) y = 80, C = 2, v = 0 186 c max c 3-19 Response plots; y = 0, K. = 1, cj> = 0, \p = 30? ^ ' = 0 , y = 8 0 , v = 0 : 1 Yo ' c ' c (a) e = 0.1 187 (b) e = 0.2 187 3-20 Performance of the satellite when stabilized about a canted axis; y = 0.6, <j> = 0, K^ = 0.2: (a) y = 3 0 188 (b) y = 5 0 188 c 3-21 Influence of y on the steady-state amplitude 190 3-22 Effect of eccentricity on a satellite stabilized along the local horizontal; TT . . v _ n 0 Y = 7 / < F > = 0 , K . = 0 . 2 , v = 10: FIGURE xvi PAGE (a) Chart of y c optimum 191 (b) Steady-state amplitude . . . . . 191 3-23 Amplitude of librations as affected by system parameters; £ c = 2 ' = Q e = 0 1 . Y 2 ' Snax u ' e U--L-(a) Controller Parameters 192 .(b) Inertia Parameter . . . . . . . 192 3-2 4 Critical effect of eccentricity on attitude and librational control; u C A n n . ' y=TT, y =v =50 194 2 c c 3-25 Representative satellite configuration 196 3-26 Schematic diagramme of controller servosystem . . 197 3-27 Variation of dynamical state'during control operation 199 3-28 Feasibility study: (a) Controller panel design data 201 (b) Influence of panel deployment on inertia parameter 201 4-1 Geometry of dual-spin satellite and Euler angles 206 4-2 Torque-free motion stability charts; S. = 0.01 220 d 4-3 Stability charts for motion in the gravity-field; co = w Q: y 8 (a) R d = 1; Q d = 0 . . 225 (b) R = 0; Q = 0.5 226 d d (c) R d = 1; Q d = 1 227 4-4 Stability chart for undamped satellite in the gravity-field . 229 ACKNOWLEDGEMENT The author acknowledges his indebtedness to his supervisor, Dr. V.J. Modi, who provided the subject of the thesis as part of a continuing research in the field of satellite dynamics. His guidance and encouragement throughout the study have been invaluable. The project 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 a.,b.(i=l,2) modulii of real and imaginary parts of 1 1 characteristic roots, respectively c dimensionless equilibrium of SMD about c.m. of satellite body, 1 + c' damping constant e eccentricity hg angular momentum per unit mass of satellite k spring constant 1 length of damper boom 1 ,1 ,1 direction cosines of the outward local x Y z vertical with respect to x,y,z axes m^ mass of satellite body m^ mass of SMD or tip mass of DB p (G),p(0) characteristic functions appearing in ° equations (3-12) and (3-14) , respectively r distance from center of force to c.m. of satellite r distance from center of force to perigee p of the orbit r distance from satellite center of mass to s element of area dA 2 2 s ,t slopes of tangents to p(9) at turning points, section 3.3.1 t time set of orthogonal coordinates with origin at S, z q along the outward local vertical x,y,z principal coordinates of satellite body (without damper), with origin at its center of mass XIX x,y ,z set of coordinates parallel to x,y,z and with origin at c.m. of the system consis-ting of satellite body and damper SMD off-set distance corresponding to unstretched position of the spring SMD displacement from the unstretched position Jd dimensionless SMD displacement about equilibrium position, Z - Z satellite area Aj ,Bj (j=l,...5) yy C C max C' F H Hd I I, constants of integration entering solution rij / eq. (3.14) first moment of area about y-axis solar parameter, r^l+p-x) S A^/yC 11 maximum value of solar parameter C speed of light dissipation function Hamiltonian parameter describing the inertia of DB, I /I ° yy inertia parameter of DS, ( Ik + I r)/ I>j inertia of DS slowly-rotating section with respect to spin-axis inertia of DS rotor with respect to spin-axis I ,1 ,1 xx yy z z transverse moment of inertia of DS 2 equivalent inertia of DB, M^l /(1+m^/m^) principal moments of inertia of satellite body "'"xd' "'"yd' "^ zd principal moments of inertia of DS nutation damper relative rotor inertia, 1^.7(1^+1^) z z 1 I I X K. inertia parameter. (I -I )/I i c xx z z yy K^ parameter describing the inertia of SMD in the unstretched position. M, z^/i ^ d o' yy K_ parameter describing the equilibrium inertia of SMD, c 2K, d M * T ^ equivalent sliding damper mass, m^/(1+m^/m^) Q^ generalized radiation forces on satellite Q,,Q9 characteristic constants of SMD system, 1 ^ eq. (2-28) Rd'^d'Pd parameters describing the inertias of DS nutation damper S system center of mass S solar constant S 1 solar energy per unit time incident upon a unit area inclined at angle a^, S|cosou| S^ parameter relating inertias of DS damper to that of satellite body, I x d / I T T kinetic energy T damping time-index DS kinetic energy per unit transverse moment ds of inertia U potential energy U* DS potential energy per unit transverse moment ds of inertia X parameter describing DS rotor momentum, 1+Ja or 1+Jfi r Z dimensionless displacement of SMD about unstretched position of the spring, z l / z o Z equilibrium position of SMD, measured from e unstretched position of the spring and expressed as a fraction of z xxi a inclination of SMD with respect to the least moment of inertia axis of satellite body, or gain in analog simulation a,3,y modified Euler angles describing librational attitude of DS system ou angle of incidence of solar radiation on satellite area dA, 0+^ -<j> 3 angular position of DB with respect to the least moment of inertia axis of satellite body in unstretched state of the spring, or gain in analog simulation Y position control angle in controller charac-teristic equation,or gain in analog simulation 6 relative angular displacement of DS rotor with respect to slowly rotating section, or gain in analog simulation e angular displacement of damper as measured from its unstretched position,or gain in analog simulation equilibrium position of DB e, angular displacement of DB with respect to the equilibrium position, e-e 6 Z*9 ri . ( j=l,. . . 5) transformed solution, exp ( / p d0) J ^o o 0 position angle of satellite in its orbit as measured from pericentre 0^ abscissa of turning points X large parameter, y /2 X positive constant entering the change in variable £ = Xt X? complex form of characteristic root y gravitational constant y ,v proportionality constants in controller c c characteristic relation XXI1 p P/T a T i> ^ GO 1 0J U), CO ,U) ,0} x' y • z x y z n independent variable, £=At distance from center of force to satellite element dm satellite reflectivity and transmissivity, respectively relative spin of DS rotor when the system does not librate damper parameter; m/c',1 /c', or ' dimensionless damper parameter, ujgT planar librational angle equilibrium position of satellite body as measured from local vertical librational angle of satellite body as measured from its equilibrium position, natural frequency dimensionless natural frequency, w/to0 orbital frequency angular velocities about x,y,z axes, respectively solar aspect angle as measured from perigee dimensionless angular velocities about x,y,z axes, respectively relative spin of DS rotor for torque-free motion, d6/d? Acronyms, Subscripts c.m. max o center of mass maximum initial condition T 3> xxiii DB,db damper boom SMD,sm sliding mass damper DS,ds dual-spin system Dots and primes indicate.differentiation with respect to t and 0 (Chapters 2 and 3) or t and £ (Chapter 4), respectively. 1. INTRODUCTION 1.1 Preliminary Remarks The motion of a spacecraft offers two aspects of interest: the determination of its mass center trajectory, which can be dealt with by making use of Keplerian relations, and the displacements about its mass center, generally referred to as librations. The latter is of practical importance for appropriate orientation of a satellite with respect to the earth, as would be the case in telecommuni-cation , meteorology or military missions. As a matter of fact, under external disturbances (micrometeorite impacts, atmosphere, solar radiation pressure, gravitational and magnetic fields) any satellite, although positioned correctly initially, tends to deviate in time from its preferred orien-tation, raising the possibility of mission failure. Fortun-ately, several methods of attitude control are available to counteract the resulting undesirable librations; they may be classified as passive or active techniques. Passive stabilization provides the necessary attitude control if the demand on pointing accuracy is not too severe (3-5°), without requiring any power supply aboard the space-craft. Stabilization is achieved by employing environmental forces in conjunction with the physical properties of the satellite. 2 For instance, the gravity-gradient across the space vehicle tends to make the long (minimum moment of inertia) axis point along the local vertical. Any residual motion may then be minimized or even totally arrested by, e.g., relative displacements of secondary components linked to-gether through dissipative mechanisms. Although the magnitude and direction of the earth's magnetic field change with the position in orbit, the pre-sence of ferromagnetic materials aboard a satellite results in hysteresis losses and damping. Moreover, at various positions over the earth's surface, the magnetic field approaches the local vertical and the creation of a dipole inside the spacecraft can help its capture in proper position (e.g., the Transit Research And Attitude Control (TRAAC) satellite). Aerodynamic forces may also prove useful in the orientation of a satellite relative to the orbital velocity vector. However, the presence of atmosphere causes orbit decay thus limiting any long-term application. The rate of change of momentum of photons carried by solar radiation exerts a pressure on the satellite surface. As a result, an asymmetric distribution of area with respect to the center of mass can create a moment which may be utilized to establish a preferred orientation. Active stabilization uses energy stored aboard the space vehicle and hence, pays penalty in terms of cost, 3 life-time, and reliability. However, it can maintain a specified orientation with a great degree of accuracy. Its most modern embodiment is the dual-spin concept, in which a principal axis of least or maximum moment of inertia is maintained perpendicular to the orbital plane. The momentum rigidity of a high spin rotor, together with a second body spinning at a lower rate, say, the orbital angular velocity, offers the interesting possibility of pointing constantly an antenna towards the earth. Motion of the spin axis about the initial momentum vector is damped through a passive nutation damper. As an example of efficiency, the MILitary COMmunication SATellite (MILCOMSAT)launched 9 February, 1969 has kept its pointing accuracy below 0.05°. 1.2 Literature Survey The dominant presence of gravity-gradient torques over a wide range of altitudes has led to considerable research related to its influence on the attitude stability of space-crafts. The first theoretical planar analyses for undamped 1 2 satellites were carried out by Klemperer (1960), Baker (1960) and Schechter3 (1964). Zlatousov et al4 (1964) and 5-11 Brereton and Modi (19 6 6-68) employed numerical methods involving the use of the stroboscobic phase-plane to study motion in the large for orbits of arbitrary eccentricity. 12-14 This work has been extended by Modi and Shrivastava (1969-70) to the case of coupled librations. 4 However, the bulk of the literature deals with ways of reducing the pointing error inherent to pure gravity-gradient stabilization, within an acceptably short period of time. A purely passive technique, which has received particular attention, is the one involving relative dis-placements between auxiliary bodies and the main satellite linked together through energy dissipative mechanisms. One of the early designs was the "Vertistat" proposed 15 by Kamm (1962) in which two rods are hinged to the main body perpendicular to one another. One rod in the orbital plane damps the pitch while the other acts upon the coupled roll-yaw motion. This basic configuration gave way to 1 s further investigation by Hartbaum et al. (1965) who optimized the damper parameters for both transient and steady-state 17 regimes. Another recent paper by Clark (1970) specialized on the effect of small eccentricity on the pitch motion of 18 such a system. Back in 1962, Zajac studied the planar motion of two bodies hinged at their center of mass: optimization indicated that the time to damp to half-ampli-19 tude could be as low as 0.137 of the orbital period. Etkm 20 21 (1962), Maeda (1963) and Hughes (1966) derived the govern-ing equations for different configurations of multi-rod damped satellites: the performance was investigated by optimizing the least damped mode with the help of steepest descent techniques. 5 In 196 3, Newton studied the dynamics of the TRAAC satellite the damping of which consisted of an extendible lossy spring connecting the main body to a tip mass. Simul-taneously, the general formulation involving 7 degrees of 23 freedom was derived by Vanderslice : it was found that damping would be effective in the plane of the orbit only. An attempt to remove this limitation was performed by 24 Buxton et al. (1965) with their Rice/Wilberforce concept involving "lossy" torsional motion of the spring through a viscous damper located inside the tip-mass of the previous model. In conjunction with the TRAAC project, Mobley and 2 5~ 2 6 Fischell (1963) studied the capture of this satellite by using the earth magnetic field, and improved its damping by adding magnetic hysteresis rods, the latter being found more efficient than the originally proposed damper. > 27 Almost at the same time as Newton, Paul proposed a configuration consisting of an extendible dumbbell satellite connected by a spring-dashpot mechanism: this system was frequency dependent in contrast with the TRAAC system where dissipation was a function of amplitude. The planar analysis in circular orbit was performed by linearization and some 2 8 numerical data was included. Pringle (1968) gave three degrees of freedom to this system and its stability was investigated' 29 by use of the Hamiltonian function. Modi and Brereton (1968) extended the original work to an arbitrarily shaped 6 body librating in the plane of an elliptic orbit and showed, numerically, that limit cycles associated with the system were identical to one of the periodic solutions of the undamped case. A simplification to the "Vertistat" was put forward 30 in 1964 by Tinling and Merrick who presented a model in which a single rod asymmetrically connected to the main satellite body achieved damping about three axes through coupling. A thorough study of the motion and stability of 31 such a system was later undertaken by Bainum (1967) in view of its application to the Department Of Defense Gravity Experiment (DODGE) satellite. Further, the merit of such a 32 damping scheme was established by Bainum and Mackison (1967) who analyzed and compared the torsion wire gimballed mechanism with a time-lag magnetic damping system: the former had a time constant less than three orbits while the latter, al-though it offered the same performance in roll and yaw failed to. damp the pitch motion efficiently. Later the same year, 33 Tinling et al. studied the effect of splitting the damper to avoid design constraints. Numerical results suggested that for equivalent weight ratios, greater attitude errors could be-expected compared to the original one-rod concept.. A. mechanism, involving viscous damping in con-junction with a torsion wire sustaining the moving part, 34 was investigated by Paul, West and Yu as early as 1963. 7 A bar magnet was mounted on the rotor and a disk of magnetic material attached to the stator gave rise to energy losses by hysteresis. The efficiency of such a system was proved, 35 later that year, by Fletcher et al. for a specific communication satellite configuration. However, constant torque hysteresis dampers have the dichotomous property of large hysteresis being associated with good transient and poor steady-state performance while the converse holds for 3 6 light hysteresis. Alper and O'Neill (1967) developed an amplitude dependent hysteresis damper where this limitation 37 was removed. Recently (1969) , Connell , dealing with the general problem of optimizing the performance of an arbitrar-ily shaped two-body satellite, also suggested the possibility of varying the shape of the auxiliary body according to the mode of response. For orbits above the earth's effective atmosphere (-500 miles), the major external disturbance appears to be solar radiation: this was pointed out successively by 38 39 Roberson (1958), Hall (1961) and a recent significant 40 study by Flanagan and Modi (1969), which showed that the only predominant force above 6,000 miles is direct solar radiation. In general, its effect is rather destabilizing and hence detrimental to the satellite performance. However, there have been several studies to explore utilization of this minute force to advantage through "solar sailing" for 41-44 interplanetary travel . In a similar manner, appropriate 8 control of the solar parameter can be utilized to maintain a satellite in a desired attitude. 45 Sohn (1959) proposed specific configurations for satellite stabilization with respect to the sun using, e.g., 46 black and white bodies. Galitskaya and Kiselev (1965) studied the principle of librational control of space probes about three axes. The qualitative study provided useful " information about design and control of vanes, although no attempt was made to solve the precise equations of motion. 47 Almost at the same time, Mallach presented a phase-plane analysis of a simplified model making use of average torques. 48 Modi and Flanagan (19 69) examined the planar attitude control of a gravity-oriented satellite in an ecliptic orbit using solar radiation pressure as a damping torque through c a controller of the form C a iji1 . A numerical study showed the motion to be highly damped even over a restricted domain of 49 maximum allowable controller torque. Crocker (19 70) studied the feasibility of maintaining the spin-axis of a satellite along the Sun-satellite line by means of controllable paddles. The gravity forces were neglected and an adequate nutation damper was assumed to be mounted on the spacecraft so that the angular momentum vector was close to the spin-axis. Spinning bodies have received, in the past decade, considerable attention owing to their particular stability properties. For rigid axisymmetric bodies under the influ-ence of gravity forces and with the axis of spin perpendicular 9 50 to. the orbital plane, Thomson (1962) presented a stability criterion using linearized analysis while Pringle5^(1964) investigated motion in the large employing the Hamiltonian as a Lyapunov function. Asymmetry was taken into account by 52 Kane and Shippy (196 3) applying Floquet theory. The same 53 method was used later by Kane and Barba to deal with motion in the small for arbitrary eccentricity. Wallace and 54 Meirovitch (1967) studied the same problem by an asymptotic analysis in conjunction with Lyapunov's direct method. 5 5 —5 8 Neilson and Modi (196 8) gave insight into the problem of stability in the large by making use of the integral manifold concept. According to classical mechanics, the stable rotational motion of a rigid body under no external forces is possible only if the axis of rotation is a principal axis of least or greatest inertia. If the body is not rigid and energy is dissipated by the cyclic forces acting on it while under nutation, then only the motion about the axis of maximum inertia is stable. It turns out that for slowly spinning rigid satellite, the internally dissipated energy is such as to overcome the stabilizing influence of gravity and the system ends up in a state of tumbling about the axis of maximum moment of inertia: the classical example is that 59 of Explorer I The constraint of "major axis spin rule" was subse-quently removed by the introduction of the dual-spin concept 10 which allows two sections to nominally rotate about a common axis at different rates relative to inertial space. Most of the research in this area has been performed for the case of torque-free environment and "this is somewhat less a matter of wholly academic concern than has been the case for simple spinning satellites, due to the domains of the dual-spin system parameter space which might prove to be of 71 practical interest." 6 0 An early paper by Roberson (19 57) had anticipated that torques generated by a disk rotating about an axis fixed in a rigid body could deeply affect its motion. However, the first fundamental contribution to the feasibility of dual-spin stabilization was by Landon and Stewart^ "*" (1964) : an energy-sink method indicated no constraints on inertias when energy dissipation took place on the slowly-rotating 6 2 part of the system. lorillo , a year later, extended this concept to the case where energy dissipation occurs on both bodies. 6 3 Likins (1967) developed, for a specific configura-tion involving an axisymmetric rotor and an asymmetric body containing a ball-in-tube damper constrained to move parallel to the rotor axis, an accurate stability criterion based on 64 Routh analysis. Mingori (1969) took a more general approach which involved two dissipative sections. Floquet analysis stressed the sensitivity of the system behaviour to the relative effectiveness of the sources of energy dissipation. 11 6 5 Pringle extended his theorems on Lyapunov stability to the case of dual-spin spacecrafts and gave a rigorous proof of 6 6 the "maximum moment of inertia spin-axis" rule. Cloutier investigated the stability and performance of a nutation damper consisting of mass shifting perpendicular to the spin-axis: again, it led to no restrictions on inertia ratios or damper size when dissipation occurs on a despun platform. 6 7 In another paper , the same author extended his work to a damper involving two degrees of freedom in a plane perpendic-ular to the spin axis. An approximate solution was derived for the nutation angle and its decay was optimized in terms 6 8 of system parameters. Sen (1970) studied a four mass nutation damper whose design constraints were not as severe 69 as in the case of Likins. Bainum et al. analyzed the stability and performance of a dual-spin Small Astronomy Satellite (SAS-A). It was found that asymmetry noticeably deteriorates the performance of the nutation damping system. Studies which take the gravity torque into account are rare but they offer, nevertheless, valuable results. Of . 70 particular interest is the conclusion by Kane and Mingori (1965) that the stability of undamped axisymmetric dual-spin satellites is equivalent to that of rigid spinning bodies. 71 White and Likins (19 69) extended the research to slightly asymmetric system by making use of asymptotic expansions and resonance lines. Finally, it is of importance to note 72 73 the work of Roberson et al. ' (1966, 1969) concerning the ( ; 12 equilibrium positions of a single rigid body containing a symmetric, constant speed, fixed axis rotor, also called gyrostat, in presence of gravity forces. One of the most spectacular results is that a specified rotor orientation within the body can lead to as many as 24 body equilibria. Although no stability study has yet been conducted, the special case involving coincidence between gyrostat and one of the principal axis of the main body, aligned with the 74 normal to the orbital plane has been examined by Yu (1969). The analysis includes infinitesimal, and asymptotic stability in presence of two types of damping (ball-in-tube and eddy-current) . The investigation was confined to specific dis-tributions of inertias inside the system. 1.3 Purpose and Scope of the Investigation As evident from the literature survey, a consider-able amount of information exists in the field of damping the librational motion of gravity-oriented satellites. In most cases, however, the analyses have been related to the feasibility and performance of specific configurations. The aim of Chapter 2 is to compare the performance of two passive damping mechanisms which involve relative displacements of a secondary body and appear to be basic components of all systems used to date. The analysis is restricted to planar motion which reduces the complexity of the problem. 13 Following the tracks of Modi and Flanagan, Chapter 3 examines the control of solar radiation torque as a way to orient and damp a satellite when librating in the plane of an ecliptic orbit. This new approach to the problem of attitude control differs fundamentally from the previously developed concepts and is effective both for circular and elliptic orbital motions. Finally, Chapter 4 establishes the effect of the gravity field on the stability of a damped axisymmetric dual-spin system when one of its sections rotates at orbital angular velocity. The influence of the shape of the damper is also studied. Figure 1-1 presents schematically the plan of study. Figure 1-1 Plan of study 2. LIBRATIONAL DYNAMICS OF PASSIVELY DAMPED SYSTEMS 2.1 Preliminary Remarks This chapter investigates the attitude dynamics of passively damped rigid satellites free to librate in the orbital plane. Before proceeding to analyze such systems, it is appropriate to briefly comment on the planar character 15 of the motion considered here. As shown by Kamm , Hartbaum 16 et al. and several other investigators, when a damper is moving in the plane of a circular orbit, small amplitude librations can be described by two uncoupled sets of equations. One of them represents the pitch motion and the other, the coupled roll-yaw behaviour. As a result, the effect of small eccentricity orbit is felt in the pitch degree of freedom only. Thus, the proposed study, despite its apparent limitation, is representative of any actual motion of a spacecraft with a damper in the pitch-plane, and should prove useful during preliminary design. The passive damping mechanisms considered are of two classical types, one involving translation, while the other representing rotation of the damper unit, relative to the main satellite body. They have been investigated together for the purpose of performance comparison. A discussion relative to the merits of numerical and analog solutions to the exact planar equations of motion, is 16 followed by a study of the suitability of approximate methods to the problem at hand, in circular orbit. Optimization of damper parameters is achieved by linearization and use of the root-coalescence method, after establishing its applic-ability for both proposed damper configurations. A comparison of performance based on equal inertia penalty is then con-sidered in cases of circular and small eccentricity orbital motions. Asymptotic stability in the small, and, the extent of the domain of disturbances leading to it, are discussed towards the end. Here, although large initial conditions would certainly lead to excitation of motion across the orbital plane, several studies'^ have proved that the stable domain, found by use of the planar equations of motion, is indeed conservative. Valuable information can thus be obtained with a relatively small amount of computation. 2.2 Formulation of the Problem 2.2.1 Sliding mass damper Consider a system represented by an arbitrarily shaped body with center of mass at G^ and a point mass m^ connected to it through a spring-dashpot device (Figure 2-1). Let the damper be constrained to move along an axis which makes an angle a with the axis of minimum moment of inertia of the body. The center of mass, S, of the system describes an elliptic orbit about the center of force, 0. During Figure 2-1 . Geometry of SMD model 18 planar librational motion, the expression for kinetic energy associated with the system can be written as ± ( ^ ( r / -h trvL Z The potential of the satellite is given by U  -It / ^ + 1 k sm I J  f>  Z ** (2.2) where p, the distance between the center of force and an element of mass of the system, can be written as l ^ - ^ rcosifj + ^V-S/Tt-^ -X.J + £ (2.3) As x,y,z are small quantities compared to r, it is convenient to expand -jj in terms of ^ by using the binomial theorem. 1 3 Neglecting terms of order higher than (—) , A - 1 (1-  S fzaosip  + f  ~ r I >" ( * ' V Zr z J. - Zyc.^.s/n.ij' Cos^j + ^sircij/J / 0 (2.4) Since S is the centre of mass of the system Jfc  cLht z Jy  dm.  r J^dM.-  ° 19 Also, J f j  f l  - I _ _ ) - - l [ l  +1 - I (z+*  \2fl-  CasZoi)} z i ;; » *v  z[ $ % ** & idi J 'ft J r ^ ' i i h ^ n Y - i l 1 . ** Substituting (2.4) and (2.5) into (2.3) leads to the following expression for potential energy i t s U ^ J i ' - ^ ' C ^ ' b i ^ ( 2 - 6 ) 1 . 2 Denoting the dissipating function as = — Cgmz-^/ using Lagrangian formulation and putting Ki  -- ^ t -- M a A . SHI  cL / sm. Z ' 20 s . . . .(2.7) the equations of motion in \p and Z degrees of freedom appear as [UK^O'ZY](e  + t) (ul)i(^f) + ^K; S'n.tf  Cosij/ + (l+zjsirt^+c^  Gos(<J/-f~x)J  =0 (2.8a) z * i - ( t * i ) ( e * f ) Jm. yt [i* ( 1 - 3 CoSZ + - O (2.8b) The orbital perturbations due to librational motion being 75 76 negligible ' the classical Keplerian relations /r3 1 + , r u z n. = constant (2.9) £ i+ e. cos 9 9 are valid. Using these relations and writing 0 in terms of the orbital angular velocity 0- z) it is possible to change the independent variable to 0. The equations of motion, then, take the form I — (2.10) 0 '• 2 V z 21 [i + Ktfl+zfJ  £ f*.  ("/)] 1 '  J  2(I+<LCOS9)  L ' + K d (lsin. 2 (if/tot^J s o (2.iia) a / z% [ J h £ L L  - zasi^e 7 Z ' L (t + ecose)* i+e-CosG  J L e.casG)*  j + e,cos9 = {</>'+*)*- 3 + { / 1 + ecosQ 2 . Z (2.lib) * * where 10 = /wfl and t: = co0 t o m . . . . . (2.12) sm sm 8 sm 8 sm 2.2.2 Damper boom The system consists (Figure 2-2) of an arbitrarily shaped body, as in the previous case, but with the damper mass m^ replaced by the tip mass of a boom (negligible axial moment of inertia) which is linked to the main body at G^ through a dissipative hinge mechanism. For such a system executing librational motion in the orbital plane, the ex-pressions for kinetic and potential energies are 22 Figure 2-2 Geometry of DB model 23 -it fa^j  - V ^ J (2.14) • 2 Introducing the dissipating function F ^ = 1/2 c d b e and using Lagrangian formulation lead to the following equations of motion: (8* f)(»»*)  + £ Hi*  ^ JlsCnZffr  + H^sZ^e)] 4 1 fJ^ sin. £ Cos Z if;  J = O (2.15a) 4 + Cd, <£ + Ik ab >7— - O (2.15b) where I , 0 ^ t ' / O * • % ) H * / 1 . . . . .(2.16) 24 The change to independent variable 6 is accomplished with the help of previously mentioned relations (equations (2.9), (2.10)), hence equations (2.15) take the form <£% 2 * s i K 6 f t *  ils* * ' ) T i+e. cos Q (  / + 1 f  A sin. Z ( £ +  (b 1+ &CosO , e.' + e J * * J i z ± ] L Tjlf^^f  * (UecosG/ - — £ = O (2.17b) where CO*-  CJ jL /0J„  and T*  = cdT  . . . .(2.18) dt>  ' 9 cjL 0 Jl> 25 2.3 Integration of the Exact Equations of Motion 2.3.1 Digital technique In each case, the system is governed by a set of two second-order coupled, non-linear, non-autonomous differential equations which does not admit of any closed form solution. In this instance, numerical integration was found to be a powerful tool, although restricted, for a set of parameters, to mere description of the system response. Prior to the implementation of Adams-Bashforth pre-77 dictor corrector technique with a fourth-order Runge-Kutta starter the system had to be transformed into a set of four first-order differential equations X^  G  (X Y  e) G being a non-linear function of the state-variables x and the independent variable 0. The step-size of the integration was chosen equal to lp (non-autonomous) and 3° (autonomous). 7 8 As previously demonstrated by Brereton and Modi , this provides results of sufficient accuracy without involving excessive computer time. 2.3.2 Analog Simulation An attempt to establish the suitability of analog simulation to study the behaviour of the sliding mass damper was performed on an analog computer PACE 231-R5. To conform to its capacity, the integration had to be restricted to 26 the case of a circular orbit (e=0). For ease of application, and without any loss of generality, the configuration where a=0 , i.e., damper constrained to move along the minimum moment of inertia axis, was examined. For such a system the equations of motion deduced from (2.11) appear as /„ K d (t+ Z)z] f  + Z /£ (l* 2) z'(i + f) + 3 £ A'i 4- K^ ( U Z ) Z 1 Cosfj^O (2.19a) z % i S + (cj**-3Gosy-f' g-Ztl/Jz.* ljjx+ Zljs + 3 Gos*<fr t-sni (2 • 19b) Recognizing that the stable equilibrium position occurs for l/s.^.O  ; Z* ^ - s//ajf*-3}  (2.20) and making the change in variable Z = Z g + z^, the equations of motion about the equilibrium position can be written as ^ K / / f " + z ^ H J o* ( c t i f l ^  * t - ° J (2.21a) ft  * $ * Si where c = 1 + Z . e 27 Argument 0 was used as the independent variable of the com-puter and the equations were solved by the general method 79 of programming . Trigonometric functions of the dependent variable were generated by generalized integration technique, i . E . , y. f  j£y dtp : /dfd.6 uf  J dif T J df r For efficient operation of the system, in particular multi-pliers and dividers, amplitude-scaling was necessary. As shown in Figure 2-3, it was found convenient to use five constant parameters (a,3,Y/6/£) to control all voltages. The unit of time being equivalent- to 1 radian, the simulation is about 1,000 times faster than the actual system response and, thus, does not require any time-scaling. Analog results were found to be very helpful in understanding some basic properties of the damped system under consideration. Figure 2-4 indicates the difference between the undamped and damped stability boundaries. It can be seen that the introduction of damping does not alter the undamped boundary as long as the initial conditions are in the 1st, 3rd or 4th quadrants. However, damping results in a considerably larger stability region in the 2nd quadrant. Normally, satellites would be launched with damper locked in equilibrium position. Such an undamped configuration would be prone to tumbling motion when exposed to sufficient Figure 2-3 Analog programming of SMD model (a=0); e=0 29 Figure 2-4 Comparison of SMD model stability boundaries as obtained by analog and numerical techniques; e=0, K.=1 i 30 disturbance. But there is an interesting possibility of capturing the satellite by the gravity field in desired orientation merely by releasing the damper at the appropriate point; for example, if released in the stable part of the 2nd quadrant, the satellite would be captured in the right-side-up position. On the other hand, conditions outside the stability region may lead to an upside-down configuration. The figure clearly shows that analog simulation per-formed on the model agrees quite well with the numerical results. Several response curves are presented to further emphasize the influence of parameters involved in the system. * As expected, there exists optimum values of 1 / T s m (Figure * 2-5 (a) ) and to (Figure 2-5 (b) ) leading to fastest transient sm response. It appears difficult to optimize accurately these parameters, for sets of K^ and K^, by making use of analog computer without resorting to some more elaborate programming (say, method of steepest descent). However, a crude scanning enables one to predict, for small disturbances (\p < 30°, 4>1 < 0.5 rad./rad.), the values of damper parameters leading to fastest convergence. The results are indicated on Figure 2-6 which emphasize the preponderant influence of K^, i.e., the inertia of the damper mass with respect to the mass center of the body. Note that the maximum amplitude of librational response is not altered substantially by increas-ing K n, but the time to damp is reduced significantly, d 31 +30. l|l°0. "i 1:r-'HE' r— M LUL NFL/ SiTi r • T WY r RVV 1. J I/t'sO.I U)*= 6.0 sm sm i—i—i—i 5 o r b . +30-, -30. C 1/ T =0.6 sm CJ*= 6.0 sm 1 o r b . +30. -30. 1/T" 3.0 sm 2 CJ= 6.0 sm I—I—f-5 o r b . +30. lji° 0. - 3 0 . CJ*«4.0 1/T*. 1.0 sm sm H 1—1 l o r b . +30. f - 3 0 J 10 6.0 sm 1/T* 1.0 sm H 1 1 1 o rb . +30 . f 0. - 3 0 . BIBB M M CO*. 9.0 sm 1/T \ 1.0 sm I—i—i—i 5 o r b . (b) Figure 2-5 Influence of damper parameters on system response; e=0, Ki=l, ipo=0 , 4>Q=0 .5 : (a) damping constant; (b) natural frequency 32 I 1 1—H 1—I +30, - 30 . +1. Zd a - l i KysO.OI ; LJ*= 6.0 ; 1/Tt0.8 sm sm K , = 0.05 ; CJ=26jO ; 1/T2M.<5 a sm sm Figure 2-6 Influence of damper inertia on system response; e=0, 1^=1, ipQ=0, 4^=0.5 33 Based on these results, it can be concluded that the analog simulation technique can be used to advantage in studying satellite dynamics problems, namely, extensive simulation can be performed for a variety of specific situations, with relative ease. However, the dependence of the computer on manual control is a severe constraint during optimization and definitely asks for more sophisticated software. The problem of accuracy, common to all analog tech-niques, was not found to be critical in this particular example although it was experienced in many instances such as : (i) potentiometer settings; (ii) amplifier's drift over a period of time, due to unbalance; (iii) instability of the reference source. 2.4 Approximate Solutions in Circular Orbit A logical approach to test the influence of parameters was to obtain an: approximate closed-form solution. Again, the SMD model (a=0) served the purpose in this investigation which consists of a linearization procedure and an attempt to discover the effect of non-linearities in the case of small damping. 34 2.4.1 Linearized analysis This technique has been used widely to study the dynamics of damped systems, since, regardless of initial disturbances, any asymptotically stable system passes through a state of small magnitude motion. It is intended, here, to indicate the advantages and limitations of such a method in the particular case at hand. Linearization of equations (2.21) about the equilib-rium position leads to (2.22a) / (2.22b) with the characteristic equation * A -•7? (2.23) 35 In general, equation (2.23) has two pairs of complex conjugate roots r - ± i L t X*  = - ^  t . . .(2.24) Here a 1 and a 2 are positive quantities provided w > 3, which appears as the only restriction for stability in the Routh1s sense. Defining h X * 3 (<;+<? K) + 1 + C.*Kjl t + cTKk.  * (2.25a) + + UcTKi Z<z  Kl ^ (2.25b) the solution of (2.22) can be written as Q -«t 6 i ] s s ^ c o s f e e + 6 ) + ij/ c.* c o s f a e + t j (2.26a) 36 (2.26b) where , cj)^, <t>2 are determined from initial conditions'. The solution in (2.26) clearly shows the damped response of the system with increasing 8, i.e., in the phase-plane both degrees of freedom are represented by trajectories which spiral inwards to a stable focus. The convergence behaviour of the system can be studied quite effectively s through the plots of a^ and a 2 given in Figure 2-7. It i *2 apparent that for larger values of w g m , one of the modes of, the response is likely to damp more rapidly than the other. Further, Figure 2-8 shows that for given values of K^, K^ * 2 * and to there exists an optimum value of 1/x . It is given sm c sm by the value of damping constant which results in the maximum value of a^, since the contribution of the second mode associated with a 2 becomes negligible in a short time. Also, * 2 in the region of optimum damping, for given K^ and lower values of the inertia parameter K^ result in better conver-gence. As expected, the comparison of Figures 2-8(a) and (b) indicates an improvement of performance with an increase in the damper inertia, K^. As far as the frequency of the response is concerned, * it decreases with increasing 1/t and decreasing K^. .25, 37 •15L ' / /, ai»a2 •10L .05L .001 ) \ (a) \ A 1/Tto.5 sm K d = 0.01 K; = 1.0 _ 0 . 6 ... 0.2 l 8 10 sm Figure 2-7 Variation of a^ and a 2 as functions of damper frequency and inertia parameter: " 1 / T s m = 0 - 5 ; ( b ) K d = 0 - 0 5 ' V T ^ O . S (a) Kd=0.01, sm a1» a2 0.0 B 1 - 4 sm 38 sm Figure 2-7 Variation of a^ and a 2 as functions of damper frequency and inertia parameter: (c) K =0.01 l/x*m=1.4; (d) Kd=0.05, LA* m=2.0 Figure 2-8 Variation of a1 and a 2 as functions of damping constant and inertia parameter; co*2=4 : ; (a) 1^=0.01 s m Jo.5 a. sm Figure 2-8 Variation of a 1 and a 2 as functions of damping constant and inertia parameter; J"9 * 0) (b) Kd=0.05 :2= i sm To obtain a confirmation of these results, the exact equations of motion (2.21) were integrated numerically for *2 given inertia parameters of the system. By fixing and * varying 1 A systematically, it was possible to arrive at the damping constant corresponding to the fastest convergence. * Figure 2-9 compares the optimum 1/T as obtained by anal-ytical and numerical methods. To emphasize the validity of the theoretical approach, the response of the system predicted by the linearized equations was compared with that determined by a numerical integration of the exact equations of motion. The system parameters and initial conditions were varied systematically over a wide range; however, for conciseness, only a few of the representative plots are given here. Figure 2-10(a) shows that for an impulsive disturbance as large as 0.5 rad./rad., the analytically predicted response compares quite well with the numerical result, both in ampli-tude and phase. For the angular disturbance, (Figure 2-10 (b) ) although the response exhibits a good correlation in amplitude, small discrepancy in phase was observed, particularly for initial conditions higher than 30°. Even with an arbitrary disturbance involving initial displacement and velocity, i the correlation continues to remain acceptable as suggested by Figures 2-10 (c), (d), (e). Note that, in all cases, the disturbances used are quite large compared to the degree of Kj = 0.01 Kj = 1.0 analytical — +o-°i +osai numerical <K*°-5J 1/TT (a) 2.0 2.5 ur 2 _ analytical ! +o'=° 1 numerical 4*0-0; to-05-J KD:0.01 K: s0.6 \ (b) / 3.0 2.0 2.5 UJ 3.0 Figure 2-9 Comparison between analytically and numerically predicted optimum damping constants; Kd=0.01: (a) Ki=1.0; (b) Ki=0.6 to 2.4 1.6 0.8 0.0 approximate analytical exact numerical Figure 2-10 o rbits Comparison between analytically and numerically predicted responses; K d = 0.01: (a) K.=1.0, = 6.25, 1/T*m=1.0; 1^=0 , ^=0.5 U) 0 1 ft 1 1 \ 1 1 l\ 'A l \\ /A \ 1 \\ -1 7 M 1 \ '/\\ '/A '// A 1 A //A \ 11 A\ lA 1 * ' M 1 \ \JJ y | jl i i n ' \ M 11 * * exnrt numerifnl •A v / l -A ;A A 1 A / \\ i \\ /A \ /A ' /Vrv \ J I ** 1/ • i * / \ V/ r \v/ V/ U / K j = 1.0 Kd=0.01 CA)"*2=6.25 1/T*s 1.0 sm ( ( sm j 4 5 orbits 8 Figure 2-10 Comparison between analytically and numerically predicted responses; Kd=0.01: (b) K.=1.0, 0 3 * 2 = 6 . 2 5 , 1/T*m=1.0; ^=30°, ij^ =0 f 0 -30 A \ I n I f A 1 1 u A A . A 1 J \ 1 \ 1 \J \'J ,'j \ \ I v VI approximate analytical exact numerical 4 z A h A ? i \'/ \ / v/ 1. 0 K. a 1.0 K. • 0.01 i a U)*2* 4.0 1/T*r 1.2 sm | j sm j 30 0 -30 5. 4. a 2 1 d 1 7M \ 1 A \ 1 \\ \ 7 JJ  ^ — 4 '/ V / approximate analytical exact numerical 'A 1 \ ^ 1 } /*y \ 1 \ 1 I // \// \ / / ^ ~ \ II \ 1 -V Kj = 0.6 Kj =0.01 CJ*^ 4.0 1/T*«1.2 • . S m . i = 20° + 0 > a 3 0 2 4 5 orbits 8 9 cn Figure 2-10 Comparison between analytically and numerically predicted responses; K,=0.01: (d) K.=0.6, co*2=4.0, 1/t* =1.2; ip =20°, ^'=0.3 Ct 1 sm ' sm T r> r o rbits Figure 2-10 Comparison between analytically and numerically predicted responses; Kd=0.01: (e) K.=0.6, u)*2=4.0, l/x*m=2.0; ipQ=20° , 1^=0.3 station keeping normally required by communication, weather or military satellites. The analytically predicted features of the system performance were checked using similar comparison plots. It is of interest to note that, irrespective of the values of system parameters, the linearized analysis determined the response with considerable precision, over a wide range of initial conditions. 80 2.4.2 Butenin's method In the particular case of small damping constant, * 1/Tgn)/ the system seems to be well-suited to investigate the effect of non-linearities by an averaging technique which, in turn, offers the interesting possibility of by-passing the eigenvalue problem. After series expansion of non-linear terms, equations (2.21) can be expressed in the form where Lz 1,4....,7 The complementary solution of (2.27) is given by lj/ = ( k ^  + +• h xih, (k,G  + fig.) £ -r o( CL, Co5 (2.27b) (2.28a) f k t 9 i f i ) + o( zL (2.28b) where a ^ ,3^,32 a r e constants of integration to be determined from initial conditions and, k^ and k 2 are characteristic principal frequencies obtained from h4- k*[ 3 L Uc.*ki ( ^ J  1 + 4 3 Ki+ckk (to?-* St-) - o (2.29) It is apparent that for positive values of K^ and K^, real *2 roots of equation (2.29) only exist for oj > 3 . The * 2 variations of k^ and k 2 with cosm for the representative values of K^ and K^ are shown in Figure 2-11. The ratios of principal modes are given by k1?k2 sm Figure 2-11 Representative variation of system character-istic frequencies with w*2 for low values of 1/x * s m ' sm (2.30a) (2.30b) Consider now the solution of the system of equations (2.27) in the same form as (2.28) but with a,b,3-^,32 as functions of 9. To achieve this, we impose on the functions a#b,3^,32 conditions which permit one to take the derivatives of lp and z^ as if 3^,3-^,32 were constants. Differentiating ip and z^ with respect to 0 and using the equations of motion lead to the constraint relations a/sin.  + k'sinrj  + cos  ^ + CQS  tj  s O (2.31a) c^ 'cos <f  + ax 6 Coi y - O^ cl.^  sin.  ^ - oC^  sin.  J  s O (2 . 31b) cchi cos + cos tj - s , h ~ ? " V * ^ s ' n - k ) = -f-*' (2 .31c) - oCd ajkJ si*x. % - <X2 5«.n. ^ - o^ Oyrf/A-, c » s f Costy s w h e r e (2.31d) j - ^ ^a.sih. + bsin. j / cos ^ + hkz costj ^ a. Cos ^ + °<2 b co^ ^  / - auk^ sin ^ - o( zLkz s/'n.^ J £ * = ^ /'ec .s/n <* + fo Sih. IJ J  e x C o s ^ + b h z Cos ^  , dt ex. Cos ^ <=os ij t - 0( £ CL ki Sin. ^ -a( zlo hzs> in*jJ The variables a', b 1, b|3^  can now be determined from this system of four equations as / -c/, / cos + p* ,ott k z - oiz ht I Cos 9 * o4 f s i n ^  -t> / < * / " - — * * — f % n +  L a * co*h f  L 7 / <JtMt-Uz « 7 -I 1 - 4 V -o( zkz < 1 1 sm. f D Cos (2.32a) (2.32b) (2.32c) (2.32d) It follows that for sufficiently small values of functions * * f and g (i.e., for small values of the R.H.S. of equations (2.27), the derivatives ff" ' ' etc., are small. Consequently, a , b , a r e slowly changing functions of angle 6. It is assumed that the variations in these quantities are relatively small compared to the waves in the resulting dynamic system. Thus, the average of the R.H.S. of equations (2.32) over the periods 2-rr/k^  and 27T/k2 gives the approximate equations for the determination of the unknowns a, b, / * For example / CL . r o4 0^kz-C<2 k t f-* 1 C*L kL - k z (2.33a) L : f^ 4- L Q z I (2.33b) L c/j^- otj,^ ^ kt - kz J where ' - M r c o s * tffi**  ^ ^ r zi r  f 2j r 4? 1 1 Using orthogonality of the trigonometric functions gives, on integration, the expressions for a and b as -<p  G &L = CC <2 (2.34a) b = b e 2 (2.34b) where - A 1 <*, k t Q z = 1 *2 The variations of Q^ and Q 2 with o)smare shown m Figure 2-12 for several values of K. and K,. a„ and b A are determined l a U 0 from initial conditions. Similarly from equations (2.32c) and (2.32d) one can obtain the values of 3^ and 3 2 as z / x _ . , z , -z^e. A r A + A 1- £ + 6 i / l - e (2.35a) I 1 I * (  J J A - Zo + <= a* Z<?t 1 - <2 + 3) J L f l - e . -zc^e (2.35b) where A = _ J 2 * _ _ ^ & ? - £ ^ 3 £ 3 - — — / ^ a , * 1 ( c = - ( 4 a ; + w ** - ? s ? ) + ^ A; - 3 J ^ 1 H 7 The analytical solution in its final form can then be written as = a e ^ -s/>t ^  # / /^ jj / 4 ^ ^ i ^ / i /^y/ (2.36a) / / I / / / ) + dzL0 e cos i2(ep (2.36b) The analytical solution given by (2.36) shows that the convergence of the system to the null static equilib-rium state is directly linked to the coefficients Q^ and Q^ • Plots of these quantities are given in Figure 2-12. They appear to be identical to those of a 1 and a 2 presented in Figure 2-7 (a) and (b) and therefore, the same observations concerning the system behaviour could be drawn. However, Q^ and Q 2 being proportional to the damping constant, one would expect the rate of convergence to improve with in-* creasing 1/t . This is true for small values of this sm parameter for which the non-linear function g remains small, but no information can be gathered for higher values of * damping. As a result, the optimum 1 / T s m cannot be detected by use of this method. On the other hand, the 9-dependance of the phase-angle should insure a sufficiently good correl-ation with the exact solution. This assessment was verified by comparisons of the approximate response with a numerical integration of the exact equations of motion. For small disturbances, the method is able to predict the amplitude and frequency of the satellite librations quite accurately (Figures 2-13, 2-14). The discrepancies concerning the sliding mass damper motion are of little consequence, since the pointing accuracy of the spacecraft is of major concern. With larger disturbances the deviation from numerical re-sults is more pronounced (Figure 2-15). This stems from the q,,O2 gure 2-12 Variation of Q-j_ and Q 2 as a function of damper frequency and inertia parameter; 1/x* =0.5: (a) Kd=0.01; (b) Kd=0.05 sm - 8 Z 3 -exact numerical approximate analytical 4 5 orbits 7 8 Fiaure 2-13 Dvnamical resoonse of the satellite at low damn-inn U1 +8 0 A ,f\ \ A A A A A i f V V * / V ^ 1/ \ 7 < \ \ 7 V/ \ exacx numerical approximate analytical A A f\ /A r i r I I \ \ i \ '/ / y * J y ' V / l }jj -(a) Kj =i.o Kd-.o.oi 60**4.0 V%*-. 0.5 sm i am ).2 Z 3 + 8 <|»° o - 8 4 Z 3 2 f\ [  \ ffa ' \ '/ A ) i / v\ / \\ , \ 'f \ /v. 1 II \\ \ 1 " 1 VI VI * exact numerical approximate analytical A \ A f 7 1 1 I '/ V \ // V 1 1 I 1 V (b) K j -.0.6 Kd:0.01 CJ*4.0 V ^ O 5 sm i Him il -. 0° To ill': 0.2 To 1 0 1 2 3 4 5 6 7 8 9 orbits Figure 2-14 Comparison between analytically and numerically predicted responses; oj*2=4.0, 1/x* =0.5, K,=0.01: sm sm ' d (a) K.=1.0; (b) K.=0.6 1 l 0 1 2 3 4 5 6 7 8 9 f 0 -20 5 Z 3 1 ry \ / A / V/ V exact numerical approximate analytical f\ ! <\ p \ ft \ \y —fr V (b) Kj:0.6 K d : 0 . 0 1 GJ* 4.0 VZ* 0-5 sm | ^m +0-11° i - 0 2 i orbits Figure 2-15 Comparison between analytically and numerically predicted responses during larger disturbances; u*2 = 4.o, l/x*m=0.5, Kd=0.01: (a) K.=1.0, ^ 0 = H ° . 2 ; (b) Ki=0.6, ^ o=ll 0, ^=0.2 2 3 Figure 2-15 Comparison between analytically and numerically predicted responses during larger disturbances; w*2=4.0, 1 / t * =0.5, Kd=0.01: (c) K^l.O; 4^ = 1 7 ° , ^'=0.3 To 61 i initial expansion and the assumption of slowly-varying parameters. 2.5 Optimized Performance of SMD and DB Models 2.5.1 Preliminary Remarks It is intended here to optimize the performance of the passively damped gravity-gradient satellites under con-sideration. For motion in a circular orbit, the governing equations of motion are homogeneous and the system response reduces to a transient. However, the eccentricity of the orbit introduces a continuous excitation on the system, which tends to reach a steady-state. The optimization con-siderations in the two cases being different, it would be pertinent to briefly comment upon them. For transient condition, the optimization refers to the minimization of the time taken by the satellite to reach an asymptotic equilibrium when disturbed from this position. Normally,the satellites used for communications, weather forecasting and military scouting are permitted only small amplitude librations. Under this condition, the study of linearized equations of motion gives excellent approximation to the actual motion, as pointed out in Section 2.3.1. Inspection of the solutions obtained in (2.26) and (2.36) indicate that, for a stable linearized system, the negative reciprocal of the largest real part of the eigenvalues is a measure of the damping time to 1/e (generally referred to as the damping time-index). Therefore, for a given set of parameters, the problem is to minimize the maximum real part of the characteristic roots. This could be achieved by implementing, using digital programming, one of the minimization techniques. However, a more direct approach 18 was suggested by Zajac and later extended by Borelli and 16 Leliakov (as reported by Hartbaum et al. ) for character-istic equations of the damper boom type. It states that the greatest attainable root-coalescence (Figure 2-16), obtained through parameter combinations, leads to the fastest con-vergence , i.e., smallest time-index. Applicability of this result to the sliding mass damper model is now considered. (41 M J2> (a) fbj n; (3) —H— (  C) Figure 2-16 Configurations of roots coalescence of quartic equations The quadrupole configuration (Figure 2-16(a)) is generally out of practical reach due to the severe constraints it imposes on the design parameters: this was pointed out 16 by Hartbaum et al. for the damper boom case and a prelim-inary analysis of the SMD model showed, in addition to the restrictions on spring-dashpot characteristics, a damper inertia parameter K^ to be as high as 0.8. The attention was therefore focused on the 2/2 distribution (Figure 2-16(b)) which was later observed to be compatible with the range of parameters under consideration. An investigation of the characteristic roots (A^  = -a^ ± bj; j = 1,2) of the linearized damper boom equations, as a function of damper parameters leading to a stable con-figuration (i.e., a^ and a^ positive), shows that there * * exists a set of values and for which coalescence 2/2 occurs (Figure 2-17(a)). This particular situation corresponds to ai = a2 = :Qo" F o r o t ^ e r s e t s damper parameters, the real part of at least one of the roots is always greater than -Q , meaning thereby that one mode is damped slower than the mode corresponding to root coalescence. In the case of the sliding mass damper, much of the same can be said, except for a small region near the boundary of stability of the linearized system where better transient may be expected. This is indicated in Figure 2-17(b) by the segment of frequencies (3, to ) neighbouring the stability / v 64 i/4_=i.o A+ =-a2±b2 K: = 0.6 H^s 0.01 Figure 2-17 Distribution of characteristic roots as a function of damper parameters: (a) DB model Figure 2-17 Distribution of characteristic roots as a function of damper parameters: (b) SMD model * 2 * boundary u)gm = 3, where for values of 1 / T s m larger or equal to the critical damping, a.^  and a 2 are simultaneously less than or equal to ~Q0« However, these configurations correspond to high values of i.e., the equilibrium position of the damper mass is very far from the center of mass of the main satellite body. This feature is undesirable for two reasons: (i) it restricts the domain of stable initial conditions about the equilibrium; (ii) it creates problems from the design point of view. It may be pointed out that, for higher values of * 1/T g m, real roots, among which at least one is greater than ~QQ/ appear in the vicinity of the stability boundary thus eliminating any possibilities of better transient in that * 2 * 1 area (e.g., w = 3.2, l/x = 1.5; = -0.36 3 ± 2.222; ^ ' sm ' sm ± X^ = -0.1760; X^ = -0.596). As a increases, the frequency segment shrinks and for a position of the damper along the local horizontal, the plots of characteristic roots are comparable to those of the damper boom model (3=90°). Therefore, the 2/2 configuration of roots provides the best possible transient consistent with design feasibil-ity. The mathematical treatment of the problem appears to be more complex to deal with in the case of SMD than for DB. However, conclusions drawn from the analysis of numer-67 ous plots such as those indicated in Figure 2-17 and from checks obtained by numerical integration of the exact differential equations, validate the root coalescence approach. In the case of small eccentricity orbit, the ampli-tude of the steady-state response remains small, particularly for a satellite with a large inertia parameter K^, and once more the linearized equations of motion can be used with a non-homogeneous part describing the action of gravitational forces. One may then obtain classical frequency responses or steady-state amplitude for optimum transient parameters determined above. 2.5.2 Analysis (a) Transient optimization (i) Sliding Mass Damper. For a satellite in circular orbit the governing equations of motion (2.11) reduce to [t*Ki.  MZJ f'"<i{»  z) f') * 2 £ sin. Ztjs + Kj (t+i) Xsin.Z (tf  - O (2.37a) 7 = + Z {js + 3 o o ^ Y ( 2 . 3 7 b ) o The stable equilibrium configuration corresponds to values of Z and ty which are roots of the equations KL am. Zij^ * A ^ (l+Z^Sih*  z fy ¥o(J  - o (2.38a) (2.38b) Introducing the change in variables denoting c = 1 + Z g and linearizing (2.37) about the equi-librium position lead to ^ + 5k V 5 - t * r ° fc '  ° < 2 - 4 0 a > where t - si*. * ( & * « ) / ( l + e ' K i ) £ - l A / ^ = 3<z sin. Z^ij^ + oSj ' • • .(2.41) Looking for coalescence of the type 2/2 yields the following constraints on the coefficients of the characteristic equation S ' z - S I / Z (2.42a) V 'Z  ' S 4-ZT  - - 0 ^ 7 ( 2 . 4 2 b ) ^ s Y = <r = £ (2.42c) ^ = ~ 2C £ ' (2 .4 2d) where S' and P1 are the sum and product of the two distinct double roots, respectively. For given K^ and K^ and taking equation (2.38) into * * account, it is possible to determine oj and 1/x , compatible ' c sm ' sm r with (2.42), by solving ar} 18th degree polynomial in c. (ii) Damper Boom. For e = 0, equations (2.17) reduce to COS  *(/**£.)] + (Zos Z Z ^ [Met •S'n- ° (2.43a) ' 7 (2.43b) with stable equilibrium position given by Z U * <1 - - ^ z ( f i t ^ ) K; + Hi Z ( f * £<-J (2.44a) e * I % fe)  * O (2.44b) Letting i <2-«> and linearizing equation (2.43) about the equilibrium position lead to T  *  K  T  + £ - L < 2 ' 4 6 a ) * \ £ ° L  * J < < - ° (2.46b) where ^ r j + /-/^  ; ^ a c 1 / Z * L . . . . (2.47) Root coalescence of the type 2/2 requires r -O^ \t /2> (2.48a) 2 S'P' = - X  \ u i I (2.48b) (2.48c) (2.48d) Equations (2.44) and (2.48) can now be solved to obtain the * * optimum values of 1 / t ^ and However, it may be pointed out that this optimization process is not associated with the roots of a. polynomial since equation (2.44b) is transcendental. (b) Steady-State Optimization Neglecting 2nd and higher order terms in eccentricity and system state variables, the linearized equations of motion for the sliding mass damper case can be written as The steady-state solution of such a system can be readily found through one of the classical methods of the theory of ordinary differential equations. Putting (2.49a) (2.49b) where JZ>r F z H ; . .(2.50) gives 4 u £ l»*x f j i , I MaX (  HM+ J Z \ J>*+ E z 1 z i 2 (2.51a) (2.51b) As far as the damper boom model is concerned, the linearized equations of motion appear in the form & K t * ^ (2.52a) where Putting (2.52b) h s 7 s : A-s 3 £ cJ„ J e. db * z A- 1-X* K - $ ( < * , - Y , ) *  % (p,  - \ ( f r K-- M + r ( / l i - s ' 1 ) ; L -V- ? (K  - * f> ( l -  ft)  J <2-53, 73 leads to B / s (XJLIL i 2 (2.54a) I Q 2 / J 2 (2.54b) 2.5.3 Results and Discussion (a) Sliding mass damper To start with, Laguerre1s method was used to determine roots of the 18th degree polynomial in c, i.e., 1 + the method does not require any particular specifi-cation of starting values as would be the case in Newton-Raphson approach. Furthermore, as against the Newton-Raphson method, which a-priori eliminates all the roots except the one to which the method converges depending upon the initial value, Laguerre's method provides all the roots at once. For any choice of parameters, it was found that only two real roots appear, thus reducing the number of alternatives. For higher values of a(a > 1), the polynomial becomes ill-conditioned and the method fails to converge. One is then forced to implement Newton-Raphson's method making use of the root given by Laguerre1 s method for a -*• 1 as the starting value. The coefficients of the above mentioned polynomial depend on squared trigonometric functions of angle 2a and thus suggest symmetries in the results about a = n where n is an integer. * Figure 2-18 shows the optimum values for w and sm * 1/t as functions of K. and a for given K,. For a increasing OLLL 1. CI *2 and K^ decreasing, is constantly decreasing. On the other hand, the damping constant increases with decreasing K^ and remains constant for a = tt/2. To get a better picture of the system performance it seemed appropriate to investigate the damping time-index (i.e., time to damp to 1/e of the initial disturbance) - * T g m = 4x s m, computed for optimum damper parameters as a function of a. As shown in Figure 2-19, for given K^ and K^, there exists a 0pti mum' c o r r e sP o n c^ :'- n9 t o a minimum damping time. It should be pointed out that a variation of a from zero to its optimum value does not alter the damping time substantially, whereas any increase beyond a . . causes . optimum a significant degradation of the performance. It is also observed that with decreasing K., a .. increases and 3 l optimum brings about an improvement in the response of the system It should be noted, however, that T is independent of K-sm c l * for a = tt/2, which is a consequence of (l/x ) = ' ^ ' sm optimum constant. The frequency response plots in Figure 2-20 show * 2 * that for small values of to and l/x , there exists a sm sm possibility of resonance. However, the magnification factor shows considerable reduction for larger values of these Figure 2-18 Optimum SMD characteristics as a function of K. and a a ( rad) Figure 2-19 Variations of SMD time-index as a function of K. and a Figure 2-20 Frequency response of SMD in small eccentricity • orbit *2 .^d, parameters. It can be shown that for large oo , | — | sm 6 mcix tends to an asymptotic value Ms ^ I * * * 3(«j Z+ ZK^Ji CosZo(  + K'tL*)*-l-Ki independent of the damping which increases for increasing zd a and decreasing K.. On the other hand, | — | asymptotic-1 3 max ally reaches zero, irrespective of the values of parameters. it  0  A However, it should be remembered that high a) and l/x are ^ sm / sm in conflict with transient optimization requirements. Figure 2-20 clearly emphasizes the fact that the introduction of damping improves the transient response only at the cost of steady-state performances. Irrespective of the values of the parameters involved, the magnification factor, at best, can approach 1. It would be desirable to study the steady-state amplitude attained using optimum transient parameters: this is shown in Figure 2-21. It is apparent that for given K^, and K. between 0.5 and 1.0, a has but a small influence on 1 ^d the magnification factor | — I (Figure 2-21(a)). Further-e max more, it is interesting to note, for K^ < 0.5, the existence of a peak amplitude the magnitude of which varies with a. The plots also emphasize the effect of satellite inertia on its steady-state performance: for large values of K^(K^ > 0.5), they clearly indicate the steady-state amplitude to Figure 2-21 Steady-state amplitude of SMD corresponding to transient optimization: (a) Kd=0.01, (b) a=7i/4 VD diminish with increasing K.. In Figure 2-21(b) are shown ^d 1 zd the variations of I — I and I — I in the reqion 1e 1 max 1e 1 max ^ 0.5 < K. < 1, for different K,. It is observed that for l d K^ > 0.6, the librational angle is increasing with the inertia of the damper, while the damping mass displacement varies in the opposite sense. (b) Damper Boom The transcendental equation in e e, obtained using * (2.44) and (2.48), which aims at obtaining optimum * and oo^ k r w a s solved using classical Newton-Raphson' s method. In general, it worked well although some care was required in selecting the starting values of the iterative process. The coefficients of the non-linear equation in being periodic in 6, lead to results which are symmetrical about 8 = (2n + 1) j . The roots obtained indicate that the 2/2 optimization is restricted to a limited domain of the parameters, beyond which one would have to consider a 3/1 type of root-coalescence. Figure 2-22 gives an example of * * optimized and I / t ^ a s functions of K^ and 8* Except TT * * in the neighbourhood of 8 = 2", and l A d b are decreasing with decreasing K^. As 8 decreases, it appears that the domain of optimization shrinks until it disappears completely. - * Plots of the damping time-index T ^ = 4Tdt)/(l+Hd) as a function of K^ and 8/ for optimum damper parameters (Figure Ki Figure 2-22 Optimum DB characteristics as a function of K. and 3 1 10„ 6L db HD = 0.01 Kj = 1.0 0 . 8 0.6 0.4 0.2 (a) 1.3 1.4 \ P frad) 1.5 P (rod) Figure 2-23 Variation of DB time-index as a function of K. and g: (a) H,=0.01; (b) Hd=0.05 1 d CO to 2-2 3), emphasize the privileged configuration where 8 = j: any variation A3, on each side of this position tends to increase the damping time. It can also be seen that an increase in H^ (Figure 2-23(b)) results in a decreasing possibility of achieving a 2/2 type of optimization although it decreases the damping time. The frequency response (Figure 2-24) exhibits, in general, the same behaviour as that of the previous model. As ' * lJj I — I tends to zero for large <JO*u, d approaches an 1e 'max db' -— max asymptotic value A/ = £ 1+^dL ^ 3 (Kf+ZH^  Ki  cosZfitHj'jV*-  1 -Moreover, the resonance amplitudes are increasing with de-creasing K^. The steady-state amplitude plots for optimum transient parameters show the same pattern as in the previous case. In particular, variations of 8 have but a small effect on I — I over the range 0.5 < K. < 1.0. Where optimization 'e 'max 3 l c is possible over a sufficiently large range of K^, Figure 2-25(a) shows again the existence of peak amplitudes occuring for K^ < 0.5. Increasing the inertia of the boom, increases the steady-state amplitude of the main body, as evident from Figure 2-25 (b). As far as the motion of the boom is con-cerned, the figure clearly shows that for K^ > 0.5, increasing the boom inertia results in reduction of its steady-state amplitude. 2 Figure 2-24 * H i b Frequency response of DB in small eccentricity orbit Figure 2-25 Steady-state amplitude of DB corresponding to optimum transient (a) Hd=0.01; (b) 3=TT/2 00 Ul The analysis shows that it is possible to optimize the damper parameters of the two models following the 2/2 configuration of characteristic roots. The shrinkage of the optimization domain observed for the damper boom model results from its inability to bring about the necessary change in inertia to meet the constraint requirements. In the case of the sliding mass damper, it is possible to modify the inertia of the damping unit over the whole range of K^, by properly choosing spring stiffness and damping constant. On the other hand, the inertia of the damper boom with respect to the center of mass of the body being fixed, the optimization is restricted to a limited range of K^. This distinction suggests a rational criterion for comparison based on equal equilibrium inertias of the passive damping devices. When the sliding damper mass reaches equilibrium, its dimensionless inertia, with respect to the center of mass of the body be-2 comes equal to K^ = c K^ which is comparable to H^. From the transient point of view, the variation of the sliding mass damper time-index with equilibrium inertia K^, is indicated in Figure 2-26. It can be seen that/ for a given K^, optimization with decreasing K^ is possible at the expense of an increasing damper inertia. Besides, by selecting, for each K^, the configuration leading to the optimum damping time-index, one can compare it to the corres-ponding damper boom performance: this is shown in Figure Figure 2-26 Variation of SMD time-index as a function of equilibrium inertia of the damping mass; K =0.005 2-27. It is apparent that for the same ratio of damper to satellite body inertias, one can expect, better performance from the damper boom system. However, from the steady-state magnification factors corresponding to an optimum transient configuration, it is found (Figure 2-28) that the sliding mass damper system leads to smaller amplitudes. 2.5.4 Numerical example In order to get better appreciation of the optimized damper models, the results were applied to the Geodetic Earth Orbiting Satellite (GEOS A). The physical properties of the satellite are summarized in Table 2-1. TABLE 2-1 REPRESENTATIVE GRAVITY-GRADIENT SATELLITE CHARACTERISTICS Satellite Geos A Moments of inertia I XX 615.3 slug x ft . I yy 617.0 slug x ft2. i zz 20.8 slug x ft . Mass 386 .6 lbs. mass Eccentricity 0 .07 Semi-major axis 8,000 km. T., ,T db ' sm db 0 Figure 2-27 K:=1.0 0.8 0.6 0.4 K^s constant H d > K f damper^unit°f ^ ^ ° B t i m e " i n d i c e s a s * function of inertia of the 00 vo Figure 2-2 8 Comparison of SMD and DB steady-state amplitudes corresponding to optimum transient parameters, as a function of inertia of the damper unit Table 2-2 indicates the values of parameters for optimum configuration of the sliding mass damper. For given K. = 0.96363, a series of values for K, can be selected to i d obtain corresponding optimum T g m and a for transient response (Figure 2-19). With damper inclination known, one can de-termine dimensionless natural frequency and damping constant using Figure 2-18. TABLE 2-2 SLIDING MASS DAMPER OPTIMUM PARAMETERS Kd 0.005 0.01 0.03 a(rad.) 0.5 0.51 0.57 T sm 6. 36 4.54 2.70 *2 CO sm 5.233 5.240 5.208 * 1 / t ' sm 0.628 0.881 1.478 Kf 0.016 0.032 0.092 ipe (deg.) -0.4 -0.8 -2.4 U 1 (deg.) |rd'max ^ 4.5 4.7 5.6 1 d1 max . 0.7 0.66 0.60 Equation (2.38) can now be solved for ip , thus giving 2 K f = (1 + Z g) K^. Alternatively, K f can be obtained directly using Figure 2-27. To complete the analysis, steady-state amplitude of the librational motion can be obtained using Figure 2-21. To facilitate comparison between the physical values involved, some of the parameters, such as, equilibrium position, damper mass, etc., are plotted as a function of Z q, i.e., the unstretched length of the spring (Figure 2-29 (a) ) • It can be seen that increasing z q results in a decreasing ; value of m^. However, it should be emphasized that the damper mass has an amplitude of motion, during transient, equal to z q for initial disturbances of the order of ipQ = 0.5 rad./rad. Therefore, noting that the introduction of stops would decrease the efficiency of optimum performance, one is bound to keep the value of z q under an acceptable limit. On the other hand, decreasing z q results in high values of m^ and a compromise has to be found. To establish a meaningful comparison with the damper boom unit, the inertia parameter H^ of the latter was set equal to the equilibrium inertia parameter K f of the sliding mass model. Table 2-3 indicates the values of parameters resulting from optimization procedure (Figures 2-22, 23, 25, 27, 28) in the case where 8=90°. As expected the damping time-index T ^ appears to be smaller and the steady-state amplitude larger than in the previous case. Moreover, very small tip mass is required as shown in Figure 2-29(b), even for small length of the boom. (xlO'^ n) Z Q ( * 1 0 " 2 c m ) K d = 0 . 0 0 5 0.01 rn, d ( l b s ) K d = 0 . 0 3 0.05 Z 0 ( » 1 0 " 2 c m ) Z „ ( . 1 0 c m ) sm o1 dynexsec ^ cm K:=0,96 Z 0 f « 1 0 cm) Figure 2-29 Optimum damper characteristics for GEOS-A satellite: (a) SMD IX) to 10 94 8 . 6 . md(lbs) 4 . 2 _ \ i \ • i \ \ \ \ \ \ \ \ X ' \ \ \ \ \ \ \ HdB= 0.016 0.032 0.092 0.148 \ \ \ Kj = 0.96 P = TT/2 \ 10 I (»«10"2cm) 20 30 Figure 2-29 Optimum damper characteristics for GEOS-A satellite: (b) DB TABLE 2-3 PAMPER BOOM OPTIMUM PARAMETERS o (3 = 90° ) Hd 0.016 0.032 0.092 *db 4.53 3.21 1.88 *2 wdb 5. 80 5.73 5.50 1 * Tdb 0.869 1.206 1.939 j dyne x cm db deg. 10.71 20.98 57.81 £, dyne x cm db deg/sec 1.816 x 103 5.007 x 103 23.046 x 103 1^ d1 max(deg) 4.6 5.0 7.4 1 £ I 1 d1max(deg) 14.0 14.0 14.0 2.6 Stability Analysis The analysis carried out in section (2.5) implied that all optimized configurations were asymptotically stable since the real part of the characteristic roots were always found to be negative. However, in cases where, for design purposes, one has to choose parameters even slightly outside the set leading to fastest transient, it is essential to test their stability. 96 > Linearized analysis does not provide any information on the bounds to be applied to disturbances, in order to keep the satellite within the region of attraction of a desired orientation, thus emphasizing the need for further investigation. Although Lyapunov analyses can lead to some approximate results, numerical integration of the exact equations of motion is indeed more efficient. As, for the sliding mass damper model, optimum inclin-ation of the damper varies constantly with the inertia parameters, K^ and K^, the full equations of motion are con-sidered. This, being not the case for the damper boom model, 8 is chosen equal to 90° in the subsequent investigation. 2.6.1 Circular orbit In this section, motion in the small is studied using Routh-Hurwitz as well as Lyapunov-Hamiltonian methods, for comparison. The latter is extended to approximate bounds on librations of the system under study. The limitations of the Zubov's method, in studying motion in the large, are explained. This is followed by accurate informations on stability bounds as provided by numerical procedure. (a) Routh-Hurwitz analysis 81 It is well established that, for a linear system with a constant coefficients characteristic equation in the form of a quartic A + <2- A + CL^X  + <3-2  \ + CL 4 z O the necessary and sufficient conditions for asymptotic stability are > ° y < Y > a J J C L s ( C L z - ^ ) > ^ y These conditions are now applied to test the stability of the systems under consideration. (i) Sliding mass damper Application of the criterion to the characteristic equation of system (2.40) reduces, after some manipulations, to the following expressions > ° (2.57a) T* Jfr) Kj  Cos Z + C^Kjl  Cos Z (o(+ > O (2.57b) + «L  ) > 0 • • ^ . K\  CoSZ^+C^KjiCoS (2.57c) * 2 Here, c and ip are functions of K. , K,, a and u , thus e l d sm' jeopardizing, due to strong coupling, the hope of obtain-* 2 ing the critical values of oos]m as a simple algebraic function of other parameters. However, the boundary of stable para-meter domain can be approximated by numerical integration of the exact equations of motion (2.11). Figure 2-30 show *2 plots of (to ) • , » , for different values of a and K.. c sm critical i *2 It is apparent that lower values of w s m are permissible when the line of action of the damper is away from the least moment of inertia axis. Substitution of parameters leading to optimum trans-ient (Section 2.5.2(a)) in inequalities (2.57) checked well with expected asymptotic stability. For reference, a plot of the stable equilibrium positions corresponding to optimum transient response is given in Figure 2-31. (ii) Damper-boom (8=9 0°) The asymptotic stability conditions corresponding to equations (2.43) appear in the form 1 X* > O (2.58a) Ki  > H, (2.58b) ^ J L  ? (2.58c) . A / - " * * 2 A plot of critical frequencies co^ versus other parameters is indicated in Figure 2.32. Again, substitution of optimum transient parameters in (2.58) checked asymptotic stability. ur Kd = 0.01 Q = ( i - l ) « ^ . radians «unstability occurs under the curves 0.2 0.4 0.6 K: 0.8 3 _ KJ = 0.05 a = ( i - l ) . n radians V ' 8 i =1 unstability occurs under the curves (b) 0.2 0.4 0.6 0.8 K; Figure 2-30 Variation of ) c r i t i c a l as a function of K^ and a: (a) Kd=0.01; sm (b) Kd=0.05 VD 3 3 2 2 5 0 Q ( rad ians) Figure 2-31 Stable equilibria of optimized SMD configuration CJ *2 db *2 Figure 2-32 Variation of (w,, ) ... .. as a function of <„ - » ~ > do critical K^ and Hd(3=90°) 102 (iii) Remarks Conditions (2.58) point out that, when H^ is close to * 2 Ki' wdb s h o u l c ^ b e l a r 9 e t o insure stability. However, results from optimization 2/2 indicate that optimum values *2 of never exceed 6. This may explain that with increas-ing H^, the possibility of optimizing is steadily decreasing, especially for low values of K^, and one cannot but associate the concepts of optimization and stability. It is important to recognize the absence of damping. constant in the stability conditions. As pointed out by 31 Bamum , the presence of positive damping, irrespective of> its magnitude, insures asymptotic stability. (b) Lyapunov analysis The method of Lyapunov applied to autonomous damped mechanical systems was first derived in rigorous terms by 82 8 3 84 Pringle ' whose work was based on results by Malkin , 8 5 and Lefschetz and La Salle . The outcome was the effective use of the Hamiltonian as a testing function for Lyapunov stability. A brief summary of the method is given below. Consider the expression for Hamiltonian of a mechanical system H = E M -  ~ L ( 2 , 5 9 ) L where L = Lagrangian q^ = generalized coordinates p^ = generalized momenta 103 The equations of motion in Hamilton's canonical form can be written as O H h i s - - — ^ ( 2 . 6 0 a ) '  ^fi Q u T-*  ^ (2-60b) where R^'s are generalized non-conservative dissipative forces. The equilibrium configurations of such a system are found for equation (2.60) identically equal to zero, i.e., ^ = 7?- S O ( 2 .61a ) ^ O nf,"  ( 2 ' 6 1 b ) The total time-derivative of H(autonomous system) is given by (2.62) A damped mechanical system is defined to be "pervas-ively" or "completely" damped if H = < 0, and H = 0 for all t' occurs if and only if the system is at an equilibrium point. 82 86 As proved by Pringle and Zajac , the equilibrium solution of a "completely" damped mechanical system is 1) asymptotically stable if H is a positive definite function 104 2) unstable if H can take on negative values for values of dependent variables arbitrarily close to the equilibrium point. Writing the Hamiltonian in the form , (2.63) where H^ is a positive definite function of the generalized velocities and H a function of generalized coordinates only, the test for sign-property of H is reduced to that of Hq. For two degrees of freedom systems the test corresponds to the sign property of the matrix ^PF* H XHI (2.64) Let the origin be a stable equilibrium position and H(0) = 0 . Let R be a small closed region about the origin of state-space, bounded by a surface H = H Q. The form of H suggests that H is a decreasing function of t and therefore, any motion started at instant t on the boundary of R will end at the origin. Increasing H q leads to nested hyper-surfaces up until the outer one passes through a singular • state of the system for which hypotheses regarding H or H are not satisfied. In the plane of generalized coordinates, 105 i.e., q = 0, the largest region R is represented by the lowest H=Hg=HQ separatrix curve passing through one of the. saddle-points neighbouring the equilibrium point under consideration. (i) Sliding mass damper The Hamiltonian is put in a form analogous to (2.63) where Such that ' ^ O H fa) --/ H  *  -q~*  OA * As for a damped system, 1/t has to be positive (also refer to condition (2.57(a)), the test for pervasiveness is as follows: suppose there exists a motion such that z' = 0 at all time, then the linearized equations (2.40) d reduce to 106 where z^ corresponds to a constant displacement of the damper. A sufficient condition for (2.65) to have = z, = 0 as the d a unique solution appears as and therefore, the system being pervasive where (2.66) is satisfied, one can apply Pringle's theorem. Positive definite-ness of H , as established through (2.64), leads to previously found conditions (2.57b) and (2.57c). of Lyapunov stable parameters, (2.66) is always satisfied, and therefore there is coincidence between Routh and Lyapunov results. However, a doubt always persists in implementation of Lyapunov analysis through the use of the Hamiltonian since stability might occur ever} though the system is not pervasive; this uncertainty does not exist in Routh1s analysis which gives the entire domain of stable parameters at once. requires investigation of singularities of the system. , They all lie in the plane - z^, and are obtained by solving the system of equations (2.61a). Their type is defined by the sign-property of (2.64). Two typical behaviours of the system are observed for extreme values of a. When the damper is aligned with the least moment of inertia axis, there (2.66a) (2.66b) It can be verified, by inspection, that in the domain The extent of asymptotic stability about the origin 107 *2 exists only one saddle point between two focii for all a) sm and therefore the separatrix passes through it (Figure 2-33(a)). Any motion starting inside the domain defined by the separatrix goes to the asymptotically stable equilibrium, but this does not imply that motion starting outside the separatrix is necessary unstable. Thus, the analysis leads to a conserv-ative domain. For the damper perpendicular to the least moment of inertia axis, there are four possible configurations for singularities as shown in Figure 2-33 (b). The transition in the distribution and type of singular points occurs at bifurcation points, which, for given K^ and K^ are defined by : &C* 3 jKj /K^/(l  , j J 3 Ji^/(/*/%-l) The boundary for guaranteed stability is found by selection of the saddle-point leading to lowest value for H^. The results for these two extreme positions of damper suggest a wide variation in the configuration of singular points. A study of saddle points which determine the boundary of stability in the case of optimum transient damper parameters indicates, as suggested by the size of the surrounding stability domain, the strength of the corres-ponding equilibrium position. Figures (2-34(a)) and (2-34(b)) show that for a up to 45° in the case of K, = 0.01 and 55° ingularity 0) a=90° Kj=0.81 Kd=0.01 •singularity y y \ asymptotically stnhlo region Figure 2-33 180 -180 (b) Domain of stability of SMD model as found from Hamiltonian (a) a=0 °; (b) a=90° o CO 109 9 0 - 9 0 (a) (b) Figure 2-34 Distribution of critical saddle-points SMD model as a function of K. and a: (a) Kd=0.01; (b) Kd=0.05 1 for 110 for K^ = 0.05, the inclination of the damper has relatively-small influence on the location of singularities and hence on stability. This is due to the appearance of a unique saddle point between asymptotically stable equilibria. But, above the stated values of a, bifurcations occur. In particular, for low values of K^, the critical saddle point is close to the origin and limits the amplitude of motion about the focus. For higher values of K^, although the location of singularity appears far from the origin, it suggests, according to the Lyapunov contour in if)^  - z^ plane, a shrinkage in the range of stable ip^  values. Although the study reported here is valid for 0 < a < 90° a simple geometrical similitude would lead to comparable results for any inclination of the damper. . (ii) Damper boom (8=90°) Here also, the Hamiltonian of the system can be put into the form (2.63) with H F  , SKISI*?}  - SHJSJSFFTE)  CJ* z 2 such that and o /Z •CL  <£. * Ill By a reasoning analogous to that employed for the previous model and taking > 0, the test for pervasiveness leads to ^ ^ K ' 4 ° (2.67b) Positive-definiteness of H results in (2.58) and over the region of stable parameters, (2.67) is satisfied. . It follows from conditions (2.58) that the point IT (ip=j + 2kiT, £ = 0) is the only saddle-point of the system and the separatrix in the xb-c plane has for equation (K^-H^). The stable domain is comparable to the one obtained for the sliding mass damper model at small a. (c). Zubov's method The study of stability in the large through Hamilton-ian is useful to the extent that it gives, analytically, preliminary bounds on the disturbances. However, the results are quite conservative, thus necessitating further investi-gation. Among the several methods available for improving the accuracy, Zubov's technique, which aims at building a Lyapunov V-function, was attempted. 87 8 8 According to Zubov ' , for a system given by s £ ( * • ) ' (2.68a) (o)  r O (2.68b) I 112 a Lyapunov V-function can be determined through the solution of the following differential equation Z  ¥  FSM  • - + (*•)(*-*) 5:1 dXs 1 ' where n is the number of state-variables and cf) (x) a positive definite function. When (2.69) can be solved in closed-form, the exact stability bound is directly given by V=l, otherwise, one has to resort to an approximate solution, e.g., in the form of a series V r V 2  + V s  + * I/*.... * 14 (2.70) Here V^ is a homogeneous form of M degree in the state-variables x g (S=l,...n), i.e., for any scalar y FA  *  (*„**,-*>.) The coefficients of V^ have to be computed numerically, but due to the limited capacity of the computer memory, the infinite series must be truncated, say, at degree N Vz * 1/ * + 1/ (2.72) The approximate asymptotic stability region is now determined by the minimum value of V(N) 113 VfN) s C-t * positive constant (2.73) • for which V(N) is tangent to the surface V(N) = 0. The important steps in the application of the method to the case of the sliding mass damper (a = 0) may be summarized as follows: (i) The system of equations (2.21) was put into the form oo (2.74) i-t m,  m,  *» * * I I I*'* linear part Non-linear terms expansion in power series where P g ( m ^ , m 2 / m 3 ) are constants corresponding to the differ-^ th ent combinations of m. ( £ m. = constant) in the S first-1 i=l 1 order differential equation. (ii) After checking asymptotic stability of the linear part ( a s^)/ a solution for V can be found through the recurrence formula 4 (2.75a) - tf  /*) (2.75b) A7-1 4 F  1 4 , «h FT KN-  R ( * ) ^ - i t Z (2.75c) 114 (iii) The minimum of V(N) subject to the constraint V(N) = 0 can be obtained through the penalty function technique 89 90 ' which involves minimization of J x  (2.76) H being the penalty coefficient. (iv) To get a representation of the stability domain, one would use iterative techniques to define cross-sections of the hypersurfaces defined by V = Numerous difficulties in implementing the proposed scheme become apparent. For a homogeneous form V , the number of terms involved is given by A/ = 4(4+1)  (4+m-t) M .I (2.77) meaning thereby that a truncated series V(10) will involve 996 terms I Although the use of an algorithm suggested by 91 Yu and Vongsuriya in solving (2.75) avoided solving sets of N q simultaneous linear equations, the truncated series had to be limited to degree 5 to keep implied computations, in step (ii), manageable. However, step (iii) involving 8 V the analytical evaluation of -5— turned out to be quite o x • 1 impractical, irrespective of the optimization methods con-92 93 94 • , sidered ' ' . Thus, although the method, in principle, can provide stability bounds to the desired degree of 115 accuracy, the truncation approximation imposed due to compu-tational restrictions, renders it of questionable value in the present situation. It should be pointed out that the method has been used in solving simple power-^engineering problems with a, measure of success. However, usefulness of the method in solving problems of complex multi-degrees of freedom systems appears to be doubtful indeed. Under these circumstances, it seemed appropriate to turn the attention towards the numerical integration technique which gave precise answers quite readily. (d) Numerical results Systematic variations of initial conditions about the equilibrium position corresponding to an optimum transient configuration was performed. Test for instability was the capture of the satellite in upside-down position. As the damping system is not the one which is excited in the first place, but merely responds to excitation due to motion of the spacecraft under angular and/or impulsive disturbances, and since we are primarily interested in the motion of the latter, it was considered appropriate to study the stability bound in the plane for different states of the damper. Besides, this form of representation permits comparison of the results with the stability bound of the 2 2 undamped satellite represented by ip ' - 3K.cos \p  = 0. 116 (i) Sliding mass damper Figure 2-35 shows the type of stability contour obtained for optimized configuration at a=0°. For the dis-turbance received at an instant when the damper is at its equilibrium position (zd=0), the contour remains close to nd the undamped case except for the 2 quadrant where the possibility of capture exists even for high impulsive dis-turbances. This property persists even for damper position away from its equilibrium (zd=0). The stability bound based on Hamiltonian (indicated by ) shows the conservative character of these results, especially in nd the 2 quadrant where it fails to indicate the capture opportunity for tumbling satellites. Similar observations can be made with reference to Figure 2-36 where the damper is inclined at angle a=0.6 radians with respect to the least moment of inertia axis. This angle is chosen since it occurs in a region where both optimization and stability are at their best. It is indicated that motion can still be stable for initial state of the damper far from its equilibrium, although the domain of stability of the spacecraft is greatly reduced. nd Consider now the behaviour of the system in the 2 quadrant. It can be seen that for initial conditions in that region, the damping is quite efficient but at the same time, displacements of the damper become significant (Figure 2-37). Usually, the design constraints would limit the damper ampli-tude to some maximum value Iz, I . This may be achieved by ' r i 1 m ^ v .. undamped boundary . Hamiltonian u _ numerical u 1/t" = 0.78 sm Figure 2-35 Numerical determination of boundary of stability for SMD model; a=0, K.=1, K,=0.01: (a) z'=0 l a a M M Figure 2-35 Numerical determination of boundary of stability a=0, 1^=1, Kd=0.01: (b) zd=0 for SMD model; i—1 00 Figure 2-36 Numerical determination of boundary of stability for SMD model; a=0.6, Ki=0.8, Kd=0.01: (a) z^=0 i-1 M vo Figure 2-36 Numerical determination of boundary of stability for SMD model; a=0.6, K.=0.8 , K,=0.01: (b) z,=0 l ' d d fo o -90 20 a = 0.6 rad K j=0 .8 K . = 0.01 a Wfm-2-14 1/T«m = 0.98 = -0.0196 rad 1 = 0.8512 e _L 5 o rbits <zd).= ° l^'JL-o Figure 2-37 Response of SMD model to large disturbance M fO i-1 122 the inclusion of "hard stops" on either side of the equilibrium position of the damper. For the mass hitting the hard-stop + at time t, zd(t ) = ~zd(t ). The change of momentum of the damper does not affect the librational motion of the space-vehicle since the trajectory of the damper passes through the center of mass of the whole system. As a first approx-imation, one can also neglect the perturbation of the orbit. The effects of introducing hard-stops on the stability bound of the damped satellite are shown in Figure 2-38. It can be seen that boundaries remain unchanged in all quadrants except the second where the capture property is reduced with decreasing|z^|max. Finally, although the stability boundary is not affected in other quadrants, the same cannot be said of the response (Figure 2-39). However, even for a severe limitation such a s l z < 5 l m a x < 2, the time to damp is quite acceptable. (ii) Damper boom (8=90°) Numerical investigation was performed for a value of 2 inertia H^ equal to the equilibrium inertia K^ = c K^ of the sliding mass damper, so as to compare the merits of both devices. The results are shown in Figure 2-40. Contrary to the previous case, there is no anomaly in the second quadrant and the domain of stability for small e q and e^ remains very similar to the one for the undamped'satellite. Again, the Hamiltonian provides a contour which is conservative. Figure 2-38 Effect of hard-stops on the stability domain of SMD model N) u> 90 -90 8 -4 / \ ^d'max2 0 0 a - 0.6 rad Kj =0.8 K = 0.01 a Cd* = 2.14 sm 1/T* =0.98 ' sm ,l|je =-0.0196 rad Z e = 0.8512 "v. ( t l= 5 7° f + d ' ^ 0 2 5 (zd)„=° -1 2 3 4 5 6 7 8 orbits Figure 2-39 Influence of hard-stops on the response of SMD model H* NJ it* undamped boundary Figure 2-40 Numerical determination of boundary of stability of DB model (8=90°): (a) e 1 =0 NJ Ln undamped boundary Figure 2-40 Numerical determination of boundary of stability of DB model (3=90°): (b) e=0 NJ cn 127 It should be noted that, in this case, the change of momentum due to impact with the hard-stop has a direct influ-ence on the librational motion, namely, if t is the time at which the boom strikes the hard-stop (lei = e ) then, ^ 1 1 max ' e(t'),-i(f)  (2.78a) f  (<0  • * W  * * W 1 + rf^  The influence of such constraints on the system appear to be very small so far as the stability domain is concerned (Figure 2-41). However, the response to large disturbances can be significantly affected (Figure 2-42). Moreover, even in absence of hard-stops, it appears that the time to damp is larger than that for the sliding mass damper model with |z d| m a x < 2 (Figure 2-39), suggesting that the SMD mechanism would be more effective in damping large disturbances. This is interesting since, for the parameters corresponding to optimum transient performance, the damper boom model seemed more efficient in damping small disturbances. 2.6.2 Elliptic Orbit Numerical study of the sliding mass damper system 29 was already performed by Modi and Brereton who showed that the limit cycles associated with the damped system are identical to the periodic solutions of the undamped case. undamped boundary numerical ijj(rad) Figure 2-41 Effect of hard-stops on the stability domain of DB model K; =0.8 K d =0.034 *2 U) d b = 5.28 1 / T . = 1.254 C0 = o Z'  =0 o max ro oo 0 1 2 3 4 5 6 7 Figure 2-42 Influence of hard-stops on the response of DB model 130 The analysis,performed in the previous section on circular orbits, has indicated that, for small damper displacements, the stability boundary in phase-plane remains close to that found for the undamped configuration. It, therefore, seemed logical to assume that the existing literature on undamped satellite in eccentric orbit^ provides sufficient design information. Rather, the attention was specifically directed towards the influence of damper parameters on motion in the small. Linearization of the sliding mass damper model equations about the static equilibrium position for e = 0, leads to lb" sin 6 Jy\ ZC. Kj l ^ ' * -3 Kj  Cos Z$4 * e?Kl CosZ ) ,/, "  1 + e.cosQ  '  1-K^ K JL  **  J+<Z* K L , 3 C.IC  si*. Z ^ Ze.s,n.Q + S 1 =S- r (2.79a) 1 *  eccs Q  D  + <=.*KI  ^ I  + E-COSV OJL LT*  (i+ec (l ccsQy  i+&cos9  J$L  L e-CosQ)*  l+e.C4 S 9  J^d. ~ Z<Z. LI/ + 0 JeL 1 + e.cos 9 ' c L cre cos 9  _ 3 c _ r M . 3c.  I c^-e- y  ± /  cvs z fa+  ,e (2.79b) Proceeding in the same manner for the damper boom model (3=90°) gives rise to the following set of equations / / * H.  £*  - Z e - S ' ^ G H.  €., 131 i + ecos0 3  HD £ U  -ZE.&IH.9 1 + EJCOSQ (2.80a) 2  &  Sin.  9 1 + e.cos6 I!' (I'*) 4) *z • Is-. I Zasin.0 l + e.cos9 Ze.s.it\ 9 7 (l+e.cose)*  1 + e.cos9 J^ 1 + e.cosQ l + e.cos9 (2.80b) The forcing terms on the right-hand side of the given equa-tions being bounded, stability of such systems depends only 95 96 on their homogeneous form ' and as (2.79) and (2.80) have coefficients periodic in 0 (period 2tt) , the Floquet theory is applicable. Without entering into the details 97 which can be found in Minorsky , it would be sufficient to mention here that the method involves the determination of a final condition matrix <J>(2ir)/ obtained by numerical integration of the equations of motion, from an initial condition matrix <p(0) made up of the following 4 sets of initial values: - S e £ # „ 1 z 3 4 r I 1 J " i o o o t ' -o 1 o o o o 1 o / / o o o 1 <j>(o) (2.81) 132 Evaluation of the eigenvalues X^ of <j> (2tt) yields the following stability criterion for the complete system: Results are indicated in Figure 2-43. For a given value of eccentricity and inertia parameters, it is apparent that the domain of stable damper parameters is considerably different from the circular orbit case: basically, the amount of damping plays a preponderant role. As a consequence, sets of parameters unstable in circular orbit become stable and vice-versa. As a rule, small damping is detrimental to stability especially at low values of inertia parameter K^. 2.7 Concluding Remarks ing remarks can be stated: (i) Analog simulation can present some advantages over the numerical techniques in cases where a large amount of computation has to be performed for a specific configuration. However, the limited capacity of the computer forbids its use in complex problems. The availability of more elaborate software by coupling with a digital computer (hybrid technique) (2.82a) (2.82b) Based on the results of previous sections, the follow-*2 U) sm • • • • • • a = o • • • • • • K i = 1 • . • • « • Kd = 0.01 • • • e = 0.2 • • unstable 1 • • • • » • unstable 2 0 1/T sm a =0 K. = 0.6 K - 0 . 0 1 a e = 0 . 2 Figure 2-43 Charts of stable damper parameters in presence of eccentricity: (a) SMD M u> u; (A) *2 db • • P = 9 0 ° • • • unstable K.= 1 i t • K. = 0 .04 d • • • • • e =0.2 * • • • • i 1 NS, Figure 2-4 3 Charts of stable damper parameters in presence of eccentricity: (b) DB ui 135 may help resolve several inherent limitations in logic and memory, (ii) Linearized analysis is well-suited for investi-gation over a wide range of initial conditions, not necessarily small. It also provides accurate information on the influence of damper parameters and yields an optimization criterion. Butenin's method offers, for small rates of damping, an interesting alternative to the eigen-value problem. The analysis is able to predict the amplitude and frequency of the response with sufficient accuracy to be useful in preliminary design. The condition of limited amplitude to which both methods are constrained, is normally satisfied by communication satellites requiring rigid attitude control. (iii) The study of optimization emphasized the difficulty to accommodate, for a given set of damper parameters, transient and steady-state performance. Taking into account that, for most communication satellites, orbital eccentricity remains small, a comparison between the SMD and DB models based on equal damper inertias shows that the DB model can damp a trans-ient faster, while, for optimum transient parameters, the SMD model offers a lower steady-state amplitude. 136 From design consideration, the DB model seems to be less constraining, (iv) Optimization by the root-coalescence method appears to be associated with asymptotic stability. For the SMD model, the inclination of the damper should remain below a certain critical value beyond which the appearance of a singularity, in the vicinity of the equilibrium state, reduces the stability domain. It should be pointed out that the range of a leading to the largest region of stability is also the one corresponding to the best damping time-index. The numerical study suggests the interesting possibility of capture, in proper orientation, of a satellite which, in absence of damping, would undergo tumbling motion. However, the prohibitive displacements of the sliding mass may impose the introduction of hard-stops. In that case, although the above-mentioned property of capture is adversely affected, the time to damp a transient is still quite acceptable. In the case of the DB model, the stability domain is comparable to that of the undamped satel-lite. Optimum configuration of this model, although leading to better transient performance for small amplitude motion, appears to be surpassed by SMD configuration with hard-stops, when subjected to ; high initial disturbances. 137 (v) It should be noted that positive damping appears as a necessary condition of asymptotic stability in circular orbital motion. In presence of eccentricity, however, the amount of damping requires careful examination. 3. ATTITUDE AND LIBRATIONAL CONTROL OF A SATELLITE USING SOLAR RADIATION PRESSURE 3.1 Introductory Remarks This chapter presents a generalization of the model 48 first proposed by Modi and Flanagan . The solar torque is controlled according to the librational displacement in addition to the velocity, through a relation of the form: c oc The implication of such solar torque control is apparent as, now, we have a possibility of not only libra-tional damping but also changing the satellite's preferred orientation in orbit. Obviously, this would extend the satellite versatility in accomplishing diverse missions and, perhaps, its life-time due to "semi-active" nature of the controller. The analysis describes the planar motion of an arbi-trarily-shaped satellite equipped with the proposed controller, in an ecliptic orbit. Apart from an attempt to obtain an approximate closed-form solution in circular orbit, the study is carried out using numerical methods. An investi- . gation of optimized performance and typical response plots are included for average equilibrium position of the satellite 139 between local vertical and horizontal. Practical feasibility is also considered through a simple arrangement involving an unfurlable membrane, to achieve the desired value of solar torque by varying the moment of area. Some numerical data are given about the order of magnitude of parameters involved. The results suggest an interesting possibility of stabilizing a gravity-gradient satellite along the local horizontal, a normally unstable configuration in pure gravitational field. 3.2 Formulation of the Problem Consider a rigid satellite of arbitrary shape with center of mass at S, executing planar librational motion while moving in an elliptic orbit about the center of force 0. As 40 48 pointed out by Flanagan and Modi ' , the satellite can be represented quite effectively by a flat plate with its center of pressure displaced from the center of mass (Figure 3-1). The angle between the local vertical, OS, and the z-axis in the sense of the orbital motion defines the libration angle ip. For such a system, the potential and kinetic energy expressions are given by u \ - r FA  - i - j w - 3 w 140 reflected \ Figure 3-1 Geometry of satellite motion 141 Using the Lagrangian formulation, the equations of motion corresponding to the three degrees of freedom can be written as F (3.1) J-clt + (3-2) (3.3) where Q^(i = r,0,ijj) are the generalized forces due to solar radiation. As the orbital perturbations due to librational 75 76 motion of a satellite are small ' , the solution of (3.1) 3 and (3.2) to 0(1/r ) leads to the classical Keplerian relations given in (2.9). Introducing the change of independent variable to 6 in equation (3.3), through (2.9) and (2.10), leads to (l+  (LCOS G) Ze. (l+fj)  sin.9 + 3 K; sih. ifcos^J^ Q. (3.4) y r n The solar radiation force on an element of area (specular reflection) is given by cLF ^'fj-p-^Js/nty  kJ.dA hence 142 r £ (l+f-Tl)  <X£ /cos *L /. A^ (3.5) Substituting in equation (3.4) leads to (  1 + e.cosOj Ljj"- Ze.(l+f'Jsin9  + 3 Kj si a. / cos jj . <=-( U &F  SI*.  (O*FTR</>).  LI*,  {E*  </>-4)L  (3.6) (j +e.GosQj 3 ' ' ' where (3.7) Normally, the effect of solar radiation is detrimen-tal as it substantially reduces the available regions of stability and drastically diminishes the value of critical 40 eccentricity for stable motion . However, C can be controlled in magnitude and sign by maneuvering the moment of area A so as to make the solar torque oppose the librational motion. In this analysis, the solar parameter is taken to be con-trolled according to the relation C z 4 (3.8) 143 max (3.9) where C •„ represents the limit on the solar parameter as IllClX imposed by the satellite design, and the position control parameter y is taken to be between 0 and tt/2. Therefore, equation (3-6) can be rewritten as (i -T-  ecos 9 J  (jj" - Ze.( ij sin. 9  + 3 A/ s/n.  GQS  ^  s L+E. E.COS9 sm. (3.10) 3.3 Motion in Circular Orbit :For circular orbit, equation (3.10) reduces to C Ma-X (3.11a) (3.11b) The governing non-linear, non-autonomous differential equation (3.11) does not possess any closed-form solution. One is thus forced to resort to either approximate analytical techniques or numerical integration. 3.3.1 Approximate W.K.B. Solution: Y=0, C =°° L c ' max It may be pointed out that, in the field of attitude control of satellites, several investigators^ have applied the classical W.K.B.J, method"*"^ "1" to the situation 144 governed by the differential equation, x + q(t)x = 0, where the function q(t) remains bounded from zero and admits only small amplitude waves about a sufficiently large mean value. However, in the present analysis, q(t) represents an oscillating function which introduces an infinity of turning points in the problem. Thus, the response changes drastically in nature from exponential to oscillatory. Fortunately, the efficient form of damping under study makes it unnecessary to derive solutions beyond a short transient period. integration of the exact equation of motion, show good agree-ment over a range of system parameters and may prove useful as a first approximation in design of a spacecraft. (a) Analysis The results, which are•checked against a numerical For small librational angle, equation (3.11) can be linearized about \p=0 and put into the form (3.12) where Cj0 , 3«; f  j^/V/d?-^ Introducing the classical change in variable ; ;; 145 equation (3.12) can be transformed into Putting X = y /2, equation (3.13) is rewritten in c the following familiar form YJ*  + /AFE)  J  *  O  (3.14) j-(o)=  - si^ (G-ty-  ± s^z(Q-4)  + ± (sX i + ]/csinz(9-<f>jj where is a periodic function of 9 (period tt) and X is a large parameter. As shown in Figure 3-2(a) the function p(9)is under the dominance of the term of order 1, while the terms of 2 order (1/X) and.(1/X ) force it to remain close to zero in its positive part. The sketch in Figure 3-2(b) shows period tt divided into five regions each corresponding to a specific solution for ri. The aim is to correlate these solutions to obtain piecewise continuity over one period. (i) Region I: R6 Substituting ZT = / ^ ^ and oJ^ z / - kfO) in equation (3.14) leads to the differential equation P[0] P[9] 146 Figure 3-2 Variation of the characteristic function p(8): (a) <£=0, K. =1, v =0, y =40; (b) regions of specific solutions 147 2 ^ - [ X - r N j O f O <3.15, where fz p M .  i E . i t l t J 4 jb* 16 ^ Now, provided i / 1 £ - £  JL  I  «  1 <3.16, £ I 4 f>'  H j* I a first approximation to m in (3.15) leads to a solution for r\ which has the form ' J/; > , / /•« i/z (3.17) (ii) Region II: 0 1 - e 1 < 0 < 9.^  As 0 approaches a root of p(0), obviously, condition (3.16) is no longer satisfied and one has to resort to an alternate form of solution. In the vicinity of 0 = 0 ^ one can approximate p(0) by [E-EJ  s z (3.i8) where • t ' M Letting U.FE-EJFA)  (3.19) 148 and substituting in equation (3.14) lead to the Airy equation ^LD  _ UH  = O  (3.20) d o 2 ' the solution of which can be expressed in terms of Airy functions of the first and second kind Vj z (0) s  Az  Ai(u)  *  B z  Bl(v)  (3.21) For u large, i.e., for X large, one can approximate Ai(u) and Bi(u) by their asymptotic expansion, therefore ^00  J  z fir  r {  3 J  fir  /  13  / On the other hand, as 0 tends to 0^, one can obtain an expression for n^ in terms of u, by making use of (3.18) and (3.19), in the form - A , (9),  U'V-S" 5 '  /  AI  (X  /*(-/>{$F  ^ - F  U*) R&  1 An identification of coefficients in (3.22) and (3.23) leads to the matching condition / (3.24) 149 B 2 . / T A i J B , (3-25) (iii) Region III: e x < 9 < 0 2 Here, the function p(9) remains close to zero and condition (3.16) is never satisfied over [9^, 0 t h u s elim-inating the trigonometric solution corresponding to the form given in (3.17) with a linking solution of the type (3.21). However, a method involving limited computation, 102 as given by Morse and Feschbach for the case where the roots of p(0) are close to each other, can be used to advantage. Approximating the function p(9) over [9^, 8 2] by a parabola, transforms equation (3.14) into ^L1 + FA-B  X?)  H  = O  (3.26) ( / / EL* where 10 3 Two independent solutions of (3.26) are of the form and *  ^ (.1  XH,F T  i , \A V 150 where 0< - 1 - A ( § - G ^ cL^'J and ^F^ is the confluent hypergeometric function. Unfortunately, it would be quite impractical to use such a function since it is not tabulated due to the number of arguments involved. However, the roots of p(x) being close together legitimates the use of the first terms of the series expansion of ^F^ about x = 0. Accompanied with the expansion of exponential terms, one arrives at the following solution for (3.25) I  *  ±L L1 Q where Vz <P(X)=  Z (4O(-LJZ. Z  + (TI^-FFX  + R)*. 4 '  .J The determination of A^ and B^ is obtained by matching ru and ri_ and their derivatives at 8 = 8., which corresponds 6 2 " 8 1 to u = 0 and x = = -x q, respectively. This gives the matching conditions as A, d?  I  s 49! = (Alff  A . & 1 „ „, J ^ L : 3 r(i) / W < 3 - 2 9 ) 151 (iv) Region IV: 02 < 0 < 0 2 + £2 The method, here, is similar to that used in region II. Approximating p(0) by (3.30) where t z - - k ' W and introducing the change in variable V.  (6-%)  (AT) */3 (3.31) lead to Airy equation. Therefore, (3.32) Determination of A^ and B^ is similar to that of A 3 and B^, i.e., solution n^ a n c^ must match at 6 = Q^, that is x = + x q and v = 0, respectively. The conditions to be satisfied are M A4  B4 , Z / 3 * ->1/6 AJFFA)  + B 3 <PFEJ (3.33) M M - + B± = A, DPF  , BA 4S\ DIE  LX-X 3 cfiL  L - r ^ L „ (3.34) <Z-xc 152 (v) Region V: 0 > 0 2 + e 2 In this region, condition (3.16) is satisfied again and the solution has the same form as that in Region I. Therefore %  M-[-M<)f  fa  -tt*  fwi J %)  /(-titf  A)I (3.35) The matching conditions between r)^  and n,- are obtained, as in section (ii), from h (e) r h (e) g i v i n g /La £± t 3 A (3-36) " FP i -1 . z l A± t 1 X 6 s zfi? ( 3 - 3 7 > It should be emphasized that: (i) The matching process between Regions I and II as well as IV and V can be justified rigorously by referring 104 to the theory of two-variable expansion method . A com-bination of inner and outer expansion insures continuity of the solution over an interval containing the matching region. 153 (ii) Representation of the function p(8) in Region III by a parabola destroys any attempt to deal with continuity at the end points, although the solutions are matched. The choice of a quartic as a sub-stitute for p(0) may eliminate this limitation, but at the cost of increased algebra, (iii) The parameter X must be large in all applications of the method. This is stressed by inequality (3.16) and by the matching conditions obtained through the two-variable expansion concept. (b) Discussion of results The success of the method depends on the accuracy of the solution in the region corresponding to initial conditions. Because p(0) is approximated over its positive part by a parabola and bounds on the interval leading to an Airy solution (Regions II and IV) are not precisely defined, it is desirable to start in regions where p(0)is negative and condition (3.16) satisfied. This can be achieved, in all cases, for a particular range of solar aspect angle $. From computational considerations, the following remarks can be made: (i) Constants of the type B 4 n + 2 ( n = 0/1/2...) can be ignored since they are of order exp (-Xf). (ii) Constants involving terms of the form exp ( + Xf) have to be decoupled in order to avoid building-up of coefficients and round-off error. -1/4 (iii) As X is large, it is possible to replace [—p(0)] , 154 (8 and y/^pTFT d£ i n t e r i n s o f their asymptotic expansions. This approximation was not used in the results presented here. However, it can be utilized to advantage where the computer is not readily available, (iv) Computation of the analytical solution was restric-ted to regions of the Type I and III since it is not easy to compute Airy functions for any argument. This does not represent a serious limitation since the solutions of Type II are valid only over a short interval. The accuracy of the method was checked against a numer-ical integration of the exact equations of motion. Adams-Bashforth predictor-corrector technique was used here with a step-size of one degree. It was observed that the solar controller can effec-tively damp the librational motion to an adequate level, in approximately one orbit (4 turning points). Figure 3-3 presents a few representative plots corresponding to initial angular disturbances. The accuracy of the analytical solution is quite acceptable for small excitation. However, with > 20°, the process of linearization ceases to be effective so far. as the details of the response are concerned. From; design consideration, this is of limited concern as the method still continues to predict the character of the re-155 Figure 3-3 Influence of angular disturbance on the response of the satellite; cj>=2, K.=l, v =0 , u =60, ^=0: (a) Figure 3-3 Influence of angular disturbance on the response of the satellite; <t>=2, K =1, v =0, y =60, ip' =0 : (b)ij; =12°, ^=-12°; (c) ^ =23° , Ln CT> 157 sponse and gives an estimate of the time to damp. The system appears to be quite insensitive to impulsive disturbances as indicated in Figure 3-4. The influence of the damping parameter u (i.e., 2A) is briefly shown in Figure 3-5. Although higher values of A are detrimental to the transient, they lead to better accuracy of the analytical solution. The two last figures corresponding to this section emphasize the influence of the parameters on the transient response without deteriorating the accuracy of the analytical solution, while Figure 3-7 points out the influence of the satellite inertias on the behaviour of the system. In conjunction with Figure 3-3 it shows improved damping characteristics with increasing K^. 3.3.2 Numerical Results Even though the analytical results are able to depict certain features of the system, they fail to give indications as to the effect of a limited C . Moreover, although max ^ . implementation of the W.K.B. method would still be possible n for y = 2' l t s application appears out of question for intermediate values of this parameter, due to the absence of static equilibrium position about which linearization could be performed. Therefore, numerical integration repre-sents the only alternative. The analysis was carried out for values of y corres-ponding to average equilibrium position of the satellite 158 K; = 1 4> = 2 nc=60 vc=o e = 0 numer ica l ana lyt ical 0> f _ - 0 . 5 f = 1.0 o 180 360 54( e° Figure 3-4 Influence of impulsive disturbance on the response of the satellite Figure 3-5 Influence of y c on the correlation Figure 3-6 Influence of v on the response between analytically and numeric- of the satellite ally predicted responses cn VD 160 e° Figure 3-7 Influence of K. on the response of the satellite 161 between local vertical and horizontal. Case 1: y = 0 In this particular case both gravity field and con-troller tend to align the minimum moment of inertia axis of the satellite with the local vertical. For given values of 0, K^ and C m a x / it appears tempting to optimize the values of constants y and v , i.e.. to determine conditions for c c' ' fastest transient response. Figures 3-8(a) /(b) give, over a range of y and v , the result of the investigation. The lines indicating the combination of parameters leading to the same time-index T^, i.e., the time (expressed in fraction of orbits) to damp to 1/10 of the initial disturb-ance, clearly show the existence of a steepest path. However, it is appropriate to point out that: (i) Along the steepest path, the variation of time-index is very small compared to the variation of y c and v c thus emphasizing the difficulty in applying one of the steepest-descent methods as a way to optimization. (ii) The steepest path appears to vary considerably with initial conditions as indicated in Figure 3-8 (c). (iii) Even for values of parameters y and v far from c c the steepest path (yc large and vc=0) the perform-ance of the system still appears to be excellent. 8 Pc 10 Figure 3-8 Optimization of controller parameters for y=0: (c) comparison of steepest paths This insensitivity of the transient performance to the constants u and v is indeed fortunate as c c it would permit large values of parameters for steady-state performance in elliptic orbit, if required. Figures 3-9(a) and 3-9(b) represent response of the satellite to rather large angular disturbances. Two impor-tant features are of interest here: (i) The time to damp does not seem to be significantly affected by the magnitude of the disturbance, (ii) The system being non-autonomous, one would expect the response to depend on the position of the satellite in the orbit, when the disturbance occurs. This is indeed the case as shown in Figure 3-9 (b). However, it is interesting to note that the non-autonomous character appears to affect the details of the response without substantially altering the damping time-index. Furthermore, the system being periodic in ir, it is sufficient to study the response over that interval. Response of the system to impulsive disturbances (Figures 3-9 (c) and 3-9(d)) essentially display the same features. Figure 3-10(a) summarizes the influence of max-imum available solar damping torque on the system behaviour. As can be expected, the time to damp is affected by C ; Figure 3-9 Typical examples of satellite response to external disturbances; Y=0: (a)-(b) Effect of angular disturbances; (c)-(d) Effect of impulsive disturbances Figure 3-10 Influence of controller parameters on satellite response for y=0: (a) Variation of C : (b) Variation of v max c M CTl however, the dependence appears to be quite weak. Even for Cmax a s l o W a S t i l e librational motion damps out in about two orbits. Figure 3-10 (b) emphasizes the same behaviour with reference to v . c Case 2: 0 < y < j From equation (3-11), it is apparent that, for this case, there does not exist any static equilibrium position. This may be explained by the fact that, although the gravity field tends to align the satellite along the local vertical, the controller attempts to position the satellite at an angle y with respect to the local vertical. The system being non-autonomous, one would expect it to reach a limit cycle. Figure 3-11 illustrates this behaviour of the system for y = 0.3 radians. Depending on the system parameters, the amplitude of oscillations and average position of the satellite can vary considerably. This is in sharp contrast with the case where y = 0. For higher values of C (Figure 3-12 (a)) it should be noted that in order to get max ^ ' a low amplitude, y c must be increased. On the other hand, an average position, significantly different from the local vertical, can only be obtained for high values of v c and at the cost of larger amplitude. For lower values of c m a x > the amplitude of motion reaches a stationary value very rapidly when \>c is large and so does the average position, which e = 0 Y = 0 . 3 4>=o il/ = o To C = 2 max orbits Figure 3-11 Positioning of satellite away from the local vertical, y=0.3 cr\ oo 15 10. 10 e . O Kj =0.6 Y = 0.3 < M c = 0.2 max Figure 3-12 Influence of system parameters on transient performance: 0<y<ir/2 (a) Variation of u and v for C = 1.4; (b) Variation of y and v for C =0.2 C m a x CTl vo 1 e > 0 y-0.6 <j> =0 . Ki =0.2 Kj =0.6 K ; = 1.0 0.5 1.0 max (C) 30 o> •S 20! a E o 10 0 100] *—> ? 80 TJ c O !€ 60 o a 8> 401 20 1 1 / 1 K j = 0.2 K j = 0.6 _ Kj =1.0 _ —-— e = 0 <j> = 0 max / =^80 Vc m 26 1 1 1.5 0.5 Y (d) 1.0 1.5 Figure 3-12 Influence of system parameters on transient performance: 0<y<tt/2 (c) Variation of K. and C ; (d) Variation of K. and y i max i -j o remains close to the local vertical (Figure 3-12(b)). The influence of C m a x and K^ on limit cycle ampli-tude and average position is shown in Figure 3-12(c) where high gains have been chosen to. attain small amplitude far from the local vertical. Apparently, low values of K^ and high values of C would lead to better performance. The max next logical step was to vary the position control parameter 7T Y systematically over the entire range [0, a n <^ ascertain the influence of system parameters y , v , C on average c c max position and amplitude of limit cycle around it (Figure 3-12(d)). It is interesting to note that a satellite with K^ as low as 0.2, which has poor stability in the gravity gradient field, can be positioned as far as 40° from the local vertical with a librational amplitude of - 3°. Further-more, increasing y tends to create more difficulty for the system to compromise between the gravity field and the solar controller. As a result, the amplitude of limit cycle increases. However, in the vicinity of y = j , there appears to be a possibility of static equilibrium which for the chosen system parameters is approached by the satellite with a low K^. An appropriate adjustment of y c and v c would show the same behaviour for higher values of K^. Case 3: y = TJ-7T In this case, ^ = j represents a static equilibrium position both from gravity field and controller point of view. However, it should be remembered that the controller has to provide a sufficient torque in order to counteract the destabilizing gravity gradient effect. An optimization analysis of the gains y and v , o c to achieve fastest transient response, was performed as in Case 1. A steepest path was found which is difficult to follow by a steepest-descent method (Figures 3-13(a) and 3-13 (b)). As expected a minimum value of \>c is required to ensure stability and, for a given value of y , this value of v increases with K.. Figures 3-13 (c), (d), (e) , indicate C X variation of steepest paths with K^, initial conditions and C m a x , respectively. In general the pattern is more regular than that for the case of y = 0 and shows only slight dependence on initial conditions. This information can be used to advantage in adjusting the gains for better transient characteristic. However, the main problem is to locate the domain , of parameters for capture along the local horizontal. Figure 3-14 shows two distinct possibilities of system behaviour. When the controller is not strong enough, even a small disturbance causes the system to be captured by the gravity field (Figure 3-14(a)) and results in large amplitude satel-lite oscillations about the local vertical. It should be noted, however, that no matter how large the amplitude of libration, the satellite never tumbles. In Figure 3-14(b), on the contrary, we have an asymptotic approach to the Pc Figure 3-13 Optimization of controller parameters for y=?-: (c) variation of K. 2 8 _ 6 _ 4 _ 2 _ J L He (e) K:=0.2 C = 1 max J L 10 Figure 3-13 Optimization of controller parameters for y=y : (d) Influence of C (e) Influence of initial conditions max Ui Figure 3-14 System plots for y ^ : (a)-(b) Typical behaviour for two sets of controller parameters H-1 100 80 60 40 20 -i 1 1 r Mc-5 15 1 r 100 \ \ \ \ e = 0 Y 4> =0 n 2 C = 2 max 25 35 45 stabi l i ty occurs between respective curves J L *o = 7 2 ° .2 60 20 •Mc" 5 15 25 35 45 e = 0 Y -IL C = 2 i ^ r max ()> =0 \ \ \ \ \ stabi l i ty occurs between respective curves // /> J L .2 Figure 3-14 System plots for Y=y: (c) Domain of stable controller parameters for ip =72°; (d) Domain of stable controller parameters for ib =55° o o i—1 horizontal equilibrium position. The controller is now strong enough to stabilize the satellite along the inher-ently unstable position in gravity gradient sense. Figures 3-14 (c) and 3-14(d) show for two values of disturbances, the domain of parameters which insures capture in the equilibrium position. It may be pointed out that for given value of y , there exist a lower and an upper value of v c c between which asymptotic stability is guaranteed. 3.4 Motion in Eccentric Orbit An approximate approach could conceivably be used, at least for small eccentricity and y=0, by linearization and use of the W.K.B. method which would result in adding a right-hand side to equation (3.12): the particular solution could be included in the analysis by employing any of the classical available techniques (e.g., harmonic balance). However, the amount of computation involved would hardly justify the results of limited validity. Therefore, here again, a numerical procedure seems more appropriate. The integration was performed in con-junction with a varying step-size (0.5-3°). Figure 3-15 shows the effect of the step-size on numerical results obtained.for representative sets of system parameters. As a consequence, the occurrence of odd phenomena had to be checked by lowering the value of step-size until no change in the response was observed. Again the analysis was carried He Figure 3-15 Influence of step-size on the numerical inte-gration of equation of motion out for values of y corresponding to an average equilibrium configuration of the satellite between local vertical and horizontal. Case 1: y = 0 With eccentricity of the orbit introducing a forcing function on the system, one is concerned about the steady-state amplitude of the resulting motion. This magnitude is taken as a criterion to test the influence of system para-meters. Figure 3-16 (a) presents the variations of amplitude as a function of damping constant y and damping design c constraint C , in absence of angular control (v = 0). For max 3 c low values of C there exist low optimum values of y . ITlciX C However, for larger C , it is desirable to increase y . ^ max c In this region, although the parameters are not on the steepest path for optimum transient response, the time to reach the equilibrium configuration was not substantially affected, as seen in the circular case (time to damp within two orbits). Furthermore, for a given value of system parameters, i.e., e, K^, <j>, y, v c the performance does not improve beyond a certain value of C m a x - Finally it is of interest to recognize that the device can reduce the steady-state motion to as low as - 0.5° even when orbit eccentricity is 0.1! With increasing vc» the performance, for given C m a x , is generally improved as can be seen in Figure 3-16 (b) , 20 40 60 80 (a) 10 20 30 V (b) Figure 3-16 Influence of controller parameters on steady-state amplitude; y=0, K.=l, <J>=0: (a) v =0, e=0.1; (b) C =1.5, e=0.2 1 c max 00 especially at lower values of y . However, for optimum steady-state response corresponding to large values of y , c v has but a small effect. c -t Looking at the effect of eccentricity on the perform-ance of the system, it is found, as expected, that the steady-state amplitude is quite sensitive at lower C . Figure max ^ 3-17(a) shows the variation of optimum y c as a function of orbit eccentricity and available C . For values of C max max less than 0.5, there exists, for a given eccentricity, an optimum value of y c corresponding to a minimum amplitude. However, for larger values of C , large values of y lead max c to small librations, irrespective of the values of eccentricity 48 (Figure 3-17 (b)). The results of Modi and Flanagan are indicated to emphasize the discrepancies due to their larger values of step-size (= 5°) in the numerical integration of the equation of motion. It would be pertinent to explore the influence of v o n the system behaviour in situations where large values of y or C cannot be attained due to , 3 c max some practical limitation. In this case, the introduction of angular dependence of the controller can be used to advantage as shown in Figures 3-17 (c) and 3-17(d). One of the significant parameters in this study is the inertia parameter K^, which is related to the shape and distribution of mass of the satellite. It is found that the steady-state amplitude is lower for higher K^ (natural (He 0.5 1.5 0.5 max (a) max (b) Figure 3-17 Effect of eccentricity on satellite performance; y=0 , K.=l, 4>=0, v =0 (a) Chart of u c optimum; (b) steady-state amplitude 1 c 1.5 00 <jj 40 30 20 opt 10 0.5 K;=l Y = 0 4> = o V = 10 c "max (C) 20 10 1.5 e=0.25 K. = 1 i Y =o 4> = o V = 10 c l i = ( L l ) or 80 »c " c ' o p t Figure 3-17 Effect of eccentricity on satellite performance; Y=0 , Ki=l, <P=0, v =10 (c) Chart of y c optimum; (d) Steady-state amplitude CO physical property observed for all satellites), although this dependence is substantially reduced as the damping parameter is increased (Figure 3-18 (a)). As a matter of fact, for high values of y c (Figure 3-18 (b)), K^ has no influence at all on the amplitude which depends only on eccentricity. Response-curves shown in Figure 3-19 further emphas-ize the outstanding performance of the damping associated with the device. It is interesting to note that as against the passive damping mechanisms studied in the pre-vious chapter, where the optimized steady-state amplitude can attain a definite asymptotic value depending upon e and K^ only, here the librational motion can be damped almost completely, for C as low as 0.6. In addition, the trans-c max ient performance remains excellent. Case 2: 0 < Y < TT/2 In this case, even in circular orbit, the system cannot reach any asymptotic equilibrium and hence eccentricity tends to increase the amplitude of steady-state librations. Characteristic plots of the performance of the system as a function of controller parameters are given in Figure 3-20. As pointed out in the circular orbit case, high values of Cm&x a r e desirable for lower amplitude of steady-state and significantly large value of mean position. However, in-creasing v , while leading to larger average equilibrium position, tends to increase the magnitude of librations. (a) 0.8 0.6 1° max 0 4 0.2 0 . 3 . 0 . 2 5 . 0 . 2 . 0.15. 0 . 1 . e = 0 .05_ Y = o (J) = o V c = 0 Hc=80 C = 2 max 0.2 0.4 0.6 0.8 Figure 3-18 (b) Steady-state performance as affected by inertia parameter; y=0, <j)=0 (a) e=0.1, C =1.5, v =10; (b) y =80, C =2, v =0 max c c max ' c e = 0.1 Ki=l <f>= 0 Mc=80 Vc=° Figure 3-19 Response plots: (a) e=0.1; (b) e=0.2 00 P-iQ C ti CD UJ I K> O ii o CTL I  o CD ^ Hi o l-i 3 fli o CD * O P- hti II, O rt • tr to CD w tu — rt £U CD t O P-ft II CD u> o s CD tr w —• rt PJ -c cr o P-I  H U1 P-O N CD Dj P) cr o c rt P> O PJ rt CD Pi 0) X P-(/) average position (deg.) o amplitude (deg.) NJ o w o w o •N O Hi 881 -< J* II II o o i> K> A comparison of Figures 3-20(a) and 3-20 (b) also indicated the advantage of using higher values of damping constant yc' Larger eccentricities adversely affect the perform-ance as, even for the best choice of controller parameters, the drastic decrease of mean average position and increase in amplitude of the steady-state cannot be avoided (Figure 3-21). Case 3: y = tt/2 As shown in the case of a circular orbit, there exists, after proper arrangement of the controller parameters, the possibility of stable motion about the local horizontal. Even under the influence of eccentricity, this property is preserved. Figure 3-22 shows the satellite performance at different eccentricities, for a set of controller parameters. For small values of eccentricity and higher values of C , 2 3 max' it is still desirable to have high values of y . But, when e is large, an optimum value of y is present over a wide v range of c m a x / thus limiting the performance. Higher values of C are imperative here, max The predominant parameter is V c since it governs the restoring torque designed to act against the gravity field. Figure 3-23 points out that for lower values of y it is advantageous to have high values of v while the e Figure 3-21 Influence of y on the steady-state amplitude Figure 3-22 Effect of eccentricity on a satellite stabilized along the local horizontal; y=J, <£=0, K.=0.2, vc=10 : (a) Chart of y optimum; (b) Steady-state amplitude C c (a) 1 0 0 -Figure 3-2 3 Amplitude of librations as affected by system parameters; y=j, C =2, c}>=0, e=0.1: (a) Controller parameters; (b) Inertia parameter m a X M VD NJ 19 3 reverse is true when y is sufficiently large. However, o high v is a warrant for stability insuring thereby the mean equilibrium position to remain on the local horizontal. Besides, for satellites with large K^, which are quite sensitive to gravity field, high values of v are required to' c maintain attitude along the local horizontal. All previous comments are true for small values of eccentricity (< 0.1), but when e is increased, it is advisable to be very selec-tive in the controller constants since, as shown in Figure 3-24, sets of y c and v quite efficient for all types of satellites at e = 0.1, fail to fulfill the purpose when ex-posed to higher eccentricity. 3.5 Feasibility Study The results presented in the preceding sections clearly emphasize rather outstanding performance of such a solar radiation controller in effectively damping librational motion. The capability of the device to stabilize the system at any inclination with respect to the local vertical further enhances its importance. Its ability to stabilize along the local horizontal is indeed remarkable. However, a ques--tion may arise as to the practical feasibility of such a controlling device. This section briefly describes a simple arrangement for realizing the desired value of solar parameter through specified variation of moment of area with respect to the center of mass. It also gives some numerical data about the order of magnitude of the parameters involved. 0.2 0.4 0.6 0.8 K-Figure 3-24 Critical effect of eccentricity on attitude and librational control; v=2- c =2, d>=0, u =v = ca 2. max T c c Consider a cylindrical satellite of mass 'm ' and radius R with symmetrically located solar panels and framed structure as shown in Figure 3-25. The mass and geometry of the components involved are indicated on the diagram. Each frame supports a roll of highly reflective unfurlable material (e.g., aluminized Mylar membrane) at distance 'p1 from the axis of the main cylindrical body. The membrane of width 'b' and mass 'm' can be deployed at the same rate in opposite directions. The arrangement enables one to vary the moment of area without displacing the center of mass of the system and with a minimum change in inertia parameter K^. Such a model effectively represents the system described in equation (3.10) since the cylindrical body, with center of mass and center of pressure along the axis, is not affected by radiation pressure in planar librations. The deployment control of unfurlable material may be achieved through a servosystem, the schematic of which is shown in Figure 3-26. Suitable sensors aboard the satellite can generate signals which, once amplified, summed and appropriately directed would drive a positioning system commanding the deployment of the unfurlable material. A limiting circuit is required so that maximum current corres-ponds to maximum permissible deployment of the membrane. It should be noted that crossing of the solar radiation direction by the panels necessitates a sudden change in the value of C (from +C to -C ) . This is accomplished through <b C Figure 3-26 Schematic diagramme of controller servosystem KD -J the switch situated between the limiting circuit and the ' positioning system. Fortunately, in this region, the damp-ing torque itself is small thus permitting sufficient time before regaining an appreciable magnitude. This is clearly shown in Figure 3-27 where, for the case considered, the change in sign occurs approximately over 10 degrees. Keep-ing in mind that a 24-hour communication satellite describes 1 degree of arc in approximately 4 minutes, it is apparent that this would provide sufficient time for the positioning system to reach the desired configuration. A condition on the values of gains involved, i.e., K^, K^, > is readily deduced: The energy necessary to actuate the whole system may be delivered by the solar panels. A satellite of given mass and inertia will have, depending on the mission, demands on its pointing accuracy established. The response charts given earlier can be o r b i t s Figure 3-27 Variation of dynamical state during control operation used to arrive at a suitable value of C to meet this re-max quirement. Figure 3-28(a) can then be utilized to determine various combinations of controller parameters that would meet the purpose. A question concerning the variation of inertia parameter K^, due to the motion of unfurlable material, which is not accounted for in the analysis, remains to be answered. Figure 3.28(b), showing the maximum variation of K^ due to the deployment of the membrane, provides information on this point. It is apparent that, over a wide range of parameters, the perturbing effect of deployment is likely to be negligible. 3.6 Concluding Remarks The study suggests quite clearly the possibility of libratiohal control of a satellite by means of solar radiation pressure. The proposed controller seems feasible and would confer to the satellite the following properties: (i) It is possible to align a satellite along the local vertical quite effectively for almost any combinations of physical parameters as the libra-tional motion, in presence of the proposed con-troller, would always be asymptotically stable. As a matter of fact, the satellite never tumbles irrespective of the magnitude of disturbance and hence the extent of the resulting librational m / b ( s l u g / f t ) Feasibility study: (a) Controller panel design data; of panel deployment on inertia parameter Influence motion. For given C , the time required to damp max is substantially independent of the magnitude of the disturbance. Even for C as low as 0.2. the max ' librational motion damps out in about 2 orbits. As opposed to the classical damping mechanisms studied in Chapter 2, the device can reduce the steady-state motion to much lower amplitudes (0.5° for e = 0.1) and still maintain, for the same set of controller parameters, an excellent trans-ient performance (time to damp, under 2 orbits). (ii) For intermediate values of position control para-7T meter (0<y<j) there is no possibility of static equilibrium and the satellite will always oscillate about an average position. This property can be used to advantage as it provides means of orient-ing the satellite away from the local vertical. Moreover, a proper adjustment of controller para-meters can reduce the amplitude of limit cycle, especially for satellites having low K^ (3° ampli-tude at 40° to local vertical for K.=0.2). x However, the strong effect of eccentricity is reflected through an increase in the amplitude of limit cycles. (iii) The controller is capable of stabilizing a satel-lite along the local horizontal which represents an unstable configuration in the gravity gradient field. Although this is possible for satellite of any inertia, the domain of parameters leading to such a configuration is less restricted for satellites with a small K^. Steady-state ampli-tude less than 9° for eccentricity as large as 0.2 can be obtained without affecting the trans-ient response. This is in sharp contrast to a pure gravity-gradient system which stabilizes only along the local vertical and whose performance deteriorates rapidly with a decrease in inertia and increase in eccentricity. Although the study was restricted to the plane of . the ecliptic, it is conceivable to devise on the same prin-ciple, a more sophisticated mechanism which would damp 3 degrees of freedom even on an inclined orbit. The analysis of this simple model promises excellent performance for the solar damper as compared to the more classical purely gravity-gradient activated mechanisms. 4. INFLUENCE OF GRAVITY TORQUES ON THE STABILITY OF DAMPED AXISYMMETRIC DUAL-SPIN SATELLITES 4.1 Preliminary Remarks In the case of an undamped axisymmetric dual-spin 70 71 satellite, Kane and Mingori and White and Likins have shown the stability equivalence with a rigid spinning body. 50 — 58 Pertinent literature being available , any further dis-cussion is unnecessary. However, in presence of damping, the parallel cannot be established since constraints, like the major-axis spin rule, are removed. Moreover, if the slowly-spinning body, on which the damper is mounted, rotates at a velocity comparable to orbital rate, one can expect interaction of the system with the gravity field. This situation would arise if an antenna were to be pointed constantly towards the earth. The basic model chosen for this investigation is of 6 8 the type presented by Sen , where the four-mass nutation damper (or wheel) has been replaced by an arbitrarily shaped pendulum in order to test the influence of its inertia on the stability of the proposed configuration. Results of stability in the small are presented for torque-free motion where Sen's configuration becomes a particular instance. Applicability of the Kelvin-Tait-Chetaev theorem to the gravity case with damping is established and stability bounds in the parameter domain presented. Loci of the points indicating a breakdown of the sufficient conditions for pervasiveness are also included. The results are compared to those obtained for the undamped case. 4.2 Formulation of the Problem Consider a dual-spin satellite with center of mass at S and moving in a circular orbit about the center of force 0 (Figure 4-1). It consists of three elements: two right circular cylinders with y-axis as a common spin-axis and a damper linked to body I through a dissipative restor-ing hinge at S. Body II spins at a rate 6 with respect to body I. The principal axes of the damper are coincident with those of body I when the system is at rest, while the damper is restricted to rotate about the x-axis only. Let XQ,yQ,Zg be a set of orthogonal coordinates with origin at S but orientated such that y^-axis is normal to the orbital plane and Zq lies along the extension of the radius-vector r. The coordinates y,(3 and a are modified Euler angles defining the attitude of body I relative to the non-inertial frame x 0,y 0,z Q. The first rotation y about the local horizontal, xg-axis, is referred to as roll, the second rotation 3 about the z^-axis, represents yaw, while the third rotation a, about the axis of symmetry is the spin. Again, the librational motion is assumed to have a 55 negligible influence on the motion of the center of mass and energy expressions need only involve terms related to nutation damper b o d y ! (rotor) 206 o r b i t Figure 4-1 Geometry of dual-spin satellite and Euler angles I librations. Let I T be the total transverse moment of inertia of bodies I and II with respect to the set of coordinates x,y,z and , I the moments of inertia with respect to y-axis of bodies I and II, respectively. With I x d / I z cj denoting the principal moments of inertia of the damper, the librational kinetic energy of the satellite can be written as Js z T  ( X 5-y z t> # Z r I n J (4.1) <  i  ft*  K*(<4)*(fy  - J ] where the angular velocities are given by cJ^= ^CJg Sin p <x + cos y si^^S + Y Cos/zj Cos o( 4 cdy = <5 - Jf  .s/'hy6 + O)q COS  ^ COS /3 - - Cxi& Sih, f j  Cos at + ^ cJf f Cos ft  s i ^  + X Cos/^J s i n - * . . . .(4.2) Now, the distance of a mass element from the centre of force can be written as 1 where 1 , 1 , 1 are the direction cosines of the outward local x y z vertical with respect to the x,y,z axes, i.e., Cos o< = - cos y o< + .s/n. ^  cos o< = .svVl ]f  cos J. =. Cos % cos* + fi  ^fK.  (X  . . . .(4.4) 0 < The potential energy of the satellite is given by x, y, z being small compared to r, the expression for can be expanded, using the binomial theorem, in terms of —. 1 3 Neglecting terms of order higher than (—) t 1 ~ v <? o * ' i ry j j As S is the centre of mass / r f j - j = j gel— O therefore, the librational potential energy can be written as <L - £ / < f f ^ J J Now, ( , (WrQ-l  A 'Jjd« - i ' X ^ t - h i ) . j fair  f ^ - ^ zy cf*>i  = J 7L j clh - ° / / j = ( z ^ -Substituting these expressions into (4.6) leads to 12 i L j ' I ^ (4.7) As the satellite is considered to move in a circular orbit, y 2 — 3 = Ug . Dividing (4.1) and (4.7) by I T and putting r I = ^ • Jz ^ ' S s Iz* IT  ' ~ I k 4- I? ' * I T . . .(4.8) h i ; 1-Q+ ; ^ - h i s h i ; . h i IxJL Ix-A ^xd. the expressions for kinetic and potential energies take the form T*-.  1 (a) 2 + oL' ) * 1 IcJ*+ 1 J.I f?cJ f  * /*) Js Z (  * Jr J 2 ; Z < / J •b 1 S, f/u)  +€.)*+ Q, fcdu + ) +• "RJ  COJT<f  -h ui sin. £ ) Z ^ *• s ^ ' a U I % 7 J J )7 (4.9) 210 *  % I* 4  1 S.  CjJj  £*  + <zch.siz .Kt (4.10) j d Js The associated dissipative function can easily be shown to be C=  I 5 1 * -Ws where rds = % ^ ± (4.11) / ^ ^ 4.2.1 Torque-free motion In this particular case, it is easier to formulate part of the problem in terms of Lagrangian quasi-coordinates 105 to , X 10 0) i.e. ^ h o ) , ct r q j t ^ ^ a ) / 3 J /qz;) J t ( q y - ^ /23?) * 1 ^ / * 6JW f ^ t 7 ( o - O - 6J f H u 1 . / O t f ) - T . CA, y 1 f^klJs  ) u CAL , ZJZ* / - O (4.12a) (4.12b) (4.12c) z With the two remaining equations in e and 6 obtained by the usual Lagrangian formulation / ( T Z J . i s - n z * . , o M . 1 2 a ) - T (4.12e) ( / " 2 * 1 2 2 Here U, reduces to S,o), e since the terms of order ds 2 d ds 2 u)Q are neglected. T^ and T^ are the generalized forces corresponding to bearing friction and motor reaction torque controlling the relative spin of the two rotating bodies. It is assumed that these quantities are either small or 6 3 compensated , in order to achieve linearization in the following analysis. Introducing the change in variable , \ t (4.13a) with X a positive constant and putting XI = ^ • J^L = J -T2. s g* (4.13b) * ^ * X 9 > the system of equations (4.12) can be written as i l l s ' * J s^xin.zej+sz^jcz; (1  + S4 (c+k+^LS^ejj + i ° (4.14a) + + 4 c^L s ° (4.14b) ( 1 + (Qjl* 7** si'fe-jj  + jf  -St7?* ^sifLZe.  €>'Sf  fy sin ze ((p^^JL  CO^JJ SJL^cL Z€.J r O (4.14c) _r2/ + a" - rz^ cos * 1 - si*, z £.J + CJ ds + 1 <£. - o T* (4.14d) j.I + r j - o (4.14e) where CJ ds = 2 / T z ex and primes denote differentiation with respect to E, . 4.2.2 Motion in the gravity field The stationary satellites requiring the spin rate of body I as close as possible to the orbital rate can attain this condition in two different ways: (i) Keep = Wg in inertial space. However, here the orientation of the spacecraft would tend to drift away from an earth-pointing position as a result of librational motion. This is not a serious problem as it could be corrected by, say, pulsing jets. (ii) Keep a = 0, i.e., control the spin rate of the spacecraft with respect to its principal y-axis. In that case, to take full advantage of a gravity-anchor for the damper, a is taken initially equal to zero. Both situations were studied and were found to give the same equations of motion in the small. The first case, however, is non-holonomic and, hence, it is more convenient to present the formulation for the second case. in variable described in (4.13) , where X = and £ = 0 yields the desired equations of motion in dimensionless form. In particular, the generalized momentum corresponsing to 6 degree of freedom is Lagrangian formulation in conjunction with the change and " Q J o consistent with the assumption of negligible bearing friction and motor reaction torque make appear 6 as a cyclic coordinate. r ' l For C T = i I ( 4 . 1 5 ) {ZmYzcC.O 12 + s cr+ 4 (4.16) <7 Putting = y "co^i" * <9^ , s - f  sih.fi  (sin-f+fi'J  Cos f = - f  "sin.fi  * , - " fs/ruftfi'J  Cos f ^ = , = -f'cosf three equations in the remaining 3/Y/E degrees of freedom can be obtained in the form Z-"*^*. f j ' -- * (4.17a) 4 D z f , £/) = O (4.17b) ^f+Bsf'-KOsiT"  Y,f,e,e.')*o (4.170 where ^ ^ ^ USUJ cos2/ 4 s i K / Z ^ r ( l - j J +  % C°s*£.^ ~'Z - SJL CoS/" ^3--° / S 3 " " A > ^ CJ C-7 - j Cos /£ I J . [i+Sj.  fe+7i Lsi'L*€.)J <9y+ I S t ^ fy sin.26. + COS/4 * Cos f s i ^ j j l f l -  Jj + StfQt/% Cbs^j^ + Jl((T+l)  + - siH.% (l-l)  + Su^ci + « * * / / J + + + Cos (sih.fi  siK f y1 ' j S . f h .^J  f l  + ^ l) Sol ^'J f  Casf  -^'cos^J £ I (1-JJ * | / cosjf^Kf  / ^ ^ - K L ) s u sin. - ^ c o s ^ e j J = - n^  CvsZeL  i-  ( j s i n .  £  Case. -J (zcosze  ^ ^ f^'^fi + cJ. €: + <£ * o a r * 4.3 Analysis In this section, stability conditions for the given dual-spin spacecraft are considered. It was seen, in section 2.5.1(b), that for passive gravity-oriented satellites, both Routh and Lyapunov-Hamiltonian approaches gave the same regions of parameter space leading to asymptotic stability. However, for the torque-free motion of a dual-spin spacecraft, 106 it has been demonstrated by Likins and Mingori that the system fails to be "completely" damped with respect to the attitude angles, although it fulfills that property with 8 2 respect to the angular rates. As a consequence, Pringle's theorem cannot be used and one has to resort to Routh analysis for necessary and sufficient asymptotic stability conditions. In the situation where the gravity field has influence 10 7 R 6 on the system, the Kelvin-Tait-Chetaev (KTC) theorem ' related to Pringle's theorem on Lyapunov stability can be applied, thus avoiding the use of Routh's method and associated tedious computations. 4.3.1 Torque-free motion By inspection of equations (4.14), a solution appears as ft y 6 ' and ft x = constant = constant = ft r ft = e = 0 . . . (4.18) z Putting \ s co^ i + j.r.2r and linearizing (4.14) about the equilibrium state (4.18), lead to the following set of equations: A l + + ^ ( l - Z X - ^ J . ) - o (4.19a) -Sit fa"  (lt'1  + 5. fa-^j'-O ( 4 . 1 9 b ) 4s The characteristic equation is obtained in the form a.o X * eLt + clz A* + ct3 X + <x4 s O (4.20) and the Routh-Hurwitz conditions for asymptotic stability were examined for three basic shapes of damper. In all cases, s 1 * SJL > ° provided the damping parameter 1 / T d s > 0 . Case (i): R d = 1; Q d = 0 This configuration corresponds to a dumbbell shaped damper aligned with the z-axis of body I in a state of rest. The conditions for asymptotic stability appear in the form ^(jX'rf-Su.  ( l * ' 1 ) * fa'St)*(*£+*)  - S J . ( J + ^ ) > ° ( 4 - 2 1 a ) •S^  (ix-ljflx+s^j(ix-z)* > o (4.21b) (lX-l)[(lX-1-S^)  - S d L J > 0 (4.21c) *2 As (4.21a) is always satisfied for ^ ^ 0> the conditions reduce to I * < - S o L (4.22a) IX > i (4.22b) IX ^ Z (4.22c) Case (ii): R d = 0; Q d = 1/2 This is the case of a damper-wheel similar to that 6 8 of Sen . However, it may be pointed out that Sen's formu-lation does not isolate the influence of damper inertia on stability. Here the coefficients of the characteristic equation must satisfy: ^)(IX-1) Z(IX-1-  f ( 4 . 2 3 b ) < ( I X ' J - f  ) ( I X ' J J > ° (4.23C) Inequalities (b) and (c) provide the boundaries of stability IX > 1 + (4.24a) IX < - (4.24b) since the roots of (4.23a) are always -Sd/2 and 1. confined between Case (iii): R, = -1; Q, = 1 d d Here the dumbbell damper is aligned with the spin axis when in a state of rest. The stability requirements on the parameters are: S U I*X*+  ^ (T  + ^ T X + ^ O * ^ ) * * / * ' - ^ / ^ ' ' ]  >°(4.25a) sd (TX -1-S^ J >O  (4.25b) (ix-l  -  S^)^(IX-1  J  FCJJF-  JJ-  STJ  > O (4.25C) by * 2 Forco > 1 the regions of stability are described ds IX  <0  ( 4 ' 2 6 A ) IX > 1 4-Sj 4- S < h 4- 2 — (4.26b) < - t Plots given in Figure 4-2 illustrate the stability criteria for the three shapes of damper. 220 jn -2 unstable 0 .8 1.2 1.6 I Figure 4-2 Torque-free motion stability charts; Sd=0.01 4.3.2 Motion in the gravity-field (a) Damped case Linearization of equations (4.17) about {v} lead to N M ' [ > J M - M M > [ « ] h  < ° 3 Y eJ = 0 (4.27) where [A] . H-S& O o 4 [ G ] , o o o -z-Sti'l-^j + lfl+JrJ ('-%) Q O Sj ds J M -/ o o I (i * Joa) * 4(sj_ 7?u -j) 4S, % ** ) 222 Zajac 107 In this instance the KTC theorem as presented by may apply provided the system is pervasively (com-pletely damped). To test this property, one asks the q u e s t i o n d o e s there exist a motion different from the null solution, such that e' is identically equal to zero for all 6? In other words, one tries to find a motion which would fail to excite the damper at all time. If it was so, e1 e 0 would imply e" = 0 and e E e , and the system would obey the following equations of motion: — / f"  * + t 3 f + =0 (4.28a) (4.28b) (4.28c) where constants can be identified by comparison to (4.27). Provided tz-B^t3fio (4.29a) a combination of equations (4.28 b, c) where U s h B * " % + t, B and leads to OjBz-u* B 3 t, - tsB>z Substituting in (4.28a) for y gives /J" * - o \ x where /\ under the constraint Ai- cfAU 40 (4.29b) Replacing the solutions found for 3 and y in, for instance, (4.28c), and identifying each independent term to zero so as to make the null solution of (4.27) the only solution, leads to the following constraints Therefore, (4.29) constitutes a set of sufficient conditions whereby the system under consideration is pervasive. Before proceeding further, it would be appropriate to present a brief statement of the KTC theorem as extended by Zajac: Consider the system given in equation (4.27) where [A] = symmetric positive definite matrix [D] = symmetric positive semi-definite damping matrix [G] = skew-symmetric gyroscopic matrix [K] = symmetric stiffness matrix. If the system is pervasive, a necessary and sufficient condition for asymptotic stability of the null solution is for K to be positive definite. Accordingly, Sylvester's criterion was implemented to yield the following constraints: K * ° (4.29c) (4.29d) (4.29e) IX > 1-S+ fa-Tk)  (4.30a) IX > (4.30b) I(X^)(4R cL + <*£)+* fa % (4.30c) where X * IT J<R These conditions, when analyzed for each configuration of the damper, can be reduced to the following forms: Case (i): R d = 1, Q d = 0; (Figure 4-3(a)) x* i f X > i - 4-Sjl Ciijs X > J 3 J 1 . . (4.31) if I < 1 - X  > ± (I _ ^ ** ) - 3 Case (ii) : R d = 0, Q d = 0.5; (Figure 4-3 (b)) if I > 1 - Si . X > 1 (l+ £ J J ' 2 J . .(4.32) if I < 1 - , X > £ - 3 £ ^ I Case (iii): R d = -1, Q d = 1; (Figure 4-3 (c)) cd* > 4 cts if T  > 1+ Sji x > l t s A . .(4.33) i f J <- i * §J. . x > ± f t  + Su U*f s I t c o g — J<J 0 - 2 -4 unstable J / damped boundary undamped » locus of points where the-- sys tem may not be pervasive 0 .5 1 5 M NJ U1 Figure 4-3 Stability chart for motion in the gravity field; oo =oj. : (a) R,=l, Q.=0 v 0 d d Figure 4-3 Stability chart for motion in the gravity field; to =u)Q : (b) R, = 0, 0^=0.5 y o d d 2 _ JO 0 - 2 - 4 unstable damped boundary undamped >» locus of points where t h e -•system may not be pervasive I I .5 1 1.5 to to Figure 4-3 Stability chart for motion in the gravity field; oj coQ : (c) R,=-l, Q^ "*" y u Q 228 Each of the plots of Figure 4-3 contains locii of points corresponding to failure of the system parameters to satisfy the sufficient conditions derived in (4.29) (b) Undamped case By putting S^ = 0, in Equation (4.27), the system reduces to l(l+J<r)j  f ' -  f l ( l  + J<r)-i)M =o (4.34a) (2  + lfuJir)J  + (31-  4 + f , o (4.34b) These equations are identical to those investigated by 55 Neilson in the case of an axisymmetric spinning rigid satellite, where the rate of spin has been replaced by Jo. The stability criteria based on a sign analysis of the coefficients of the characteristic equation are as follows: if  T>,  1 ' if  \T<1 and (3 I-+J**  Z^I2X Z- 4IX + (31-4-) + fl*X*~  4X 3Xs*-£T*X Z- 8IX+31 > O (4.35c) / i A plot of the stability region thus found is given in Figure 4-4. X > 1 J or X < 3 I (4.35a) x< i J or X > ± ,3 I (4.35b) 230 4.4 Concluding Remarks The torque-free analysis is justified in cases where the angular rate of the slowly-rotating section of the dual-spin spacecraft is much higher than the orbital angular velocity. For such a situation where >> u)g, the damper is driven by the coning motion of the spacecraft and gravity has but a small perturbing effect. If the slowly-rotating platform spins at the angular orbital rate and the damper moves perpendicular to the plane of the orbit, the asymptotic stability region reduces basically to the mainly positive (rotor) spin portion of the undamped case. The shape of the nutation damper is of little con-sequence when its inertia is small compared to that of the satellite body. However, for the pendulum in equilibrium along the spin-axis, the natural frequency of the damping mechanism close to its critical value may cause a reduction of the stability domain. For motion under the influence of the gravity field, this is essentially true for satellites spinning about a minimum moment of inertia axis. For slowly-rotating sections spinning at intermediate rates, there exists a wide variation in stability which can be assessed by the Floquet theory. i 5. CLOSING COMMENTS 5.1 Summary Although the study of passive damping mechanisms is limited to motion in the plane of the orbit, the information 14 thus obtained may be relevant to a more general situation Of particular interest are the features listed below: (i) It is difficult to adapt damper parameters simul-taneously to transient and steady-state modes of response. (ii) The damper boom model represent simplicity of design and better transient performance compared to the sliding mass system although the latter would lead to better steady-state response when tuned to optimum transient, (iii) There appears to be some advantage in using the sliding mass configuration to damp large disturb-ances . (iv) As against the case of circular orbit, the amount of damping plays a significant role in the stability of the systems in eccentric trajectory. The limitation in (i) was overcome by making use of the solar radiation torque to achieve librational control: fast transient and small steady-state amplitude (well below the passively damped systems) were obtained by proper adjust-ment of a set of controller parameters. Maintaining a satellite at any angle with respect to the local vertical with small pointing error and stabilization along the local horizontal represent significant features of this type of mechanism. The study of the dual-spin system rotating at orbital angular rate indicated a reduction in the domain of stable rotor spin involving the elimination of the negative spin region of the undamped case. 5.2 Recommendations for Future Work A logical extension to the second chapter would be the study of the coupled roll-yaw motion for both passive configurations. Root-coalescence might then be used"^ after th examination of the distribution of roots of a 6 order characteristic equation, for lack of an available rigorous mathematical treatment. Also of interest, would be the effect of damping on the domain of stable initial disturbances for elliptic orbital motion. The field of investigation left open by the third chapter is rather wide and would involve the analysis of two or three degrees of freedom motion in and out of the ecliptic plane. The form of the undamped governing equations may suggest desirable characteristics for the controller. The configuration of the dual-spin model could be made more general by, e.g., shifting the damper hinge to some arbitrary position inside the slowly-rotating section. A performance analysis could also be included. BIBLIOGRAPHY 1. Klemperer, W.B. , "Satellite Librations of Large Ampli-tude," ARS Journal, Vol. 30, No. 1, Jan. 1960, pp. 123-124. 2. Baker, R.M.L., Jr., "Librations on a Slightly Eccentric Orbit," ARS Journal, Vol. 30, No. 1, Jan. 1960, pp. 124-126. 3. Schechter, Hans B., "Dumbbell Librations in Elliptic Orbits," AIAA Journal, Vol. 2, No. 6, June 1964, pp. 1000-1003. 4. Zlatousov, V.A., Okhotsimsky, D.E., Sarghev, V.A., and Torzhevsky, A.P., "Investigation of a Satellite Oscillations in th^Plane of an Elliptic Orbit," Pro-ceedings of the xi International Congress of Applied Mechanics, Gorther, Henry, ed., Springer-Verlag, Berlin, 1964, pp. 436-439. 5. Brereton, R.C., and Modi, V.J., "On the Stability of Planar Librations of a Dumbbell Satellite in an Elliptic Orbit," Journal of the Royal Aeronautical Society, Vol. 70, No. 12, 1966, pp. 1098-1102. 6. Brereton, R.C., and Modi, V.J., "Stability of the Planar Librational Motion of a^gatellite in an Elliptic Orbit," Proceedings of the xvii International Astronautical Congress, Vol. IV, Gordon and Breach, Inc., New York, 1967, pp. 179-192. 7. Modi, V.J., and Brereton, R.C., Planar Librational Stability of a Long Flexible Satellite," AIAA Journal, Vol. 6, No. 3, March 1968, pp. 511-517. 8. Modi, V.J., and Brereton, R.C., "On the Periodic Solutions Associated with the Gravity-Gradient Oriented System: Part I - Analytical and Numerical Determination," AIAA Journal, Vol. 7, No. 7, July 1969, pp. 1217-1225. 9. Modi, V.J., and Brereton, R.C., "On the Periodic Solutions Associated with the Gravity-Gradient Oriented System: Part II - Stability Analysis," AIAA Journal, Vol. 7, No. 8, August 1969, pp. 1465-1468. 10. Modi, V.J., and Brereton, R.C., "Stability of a Dumbbell Satellite in a Circular Orbit during Coupled Librational Motions," Proceedings of the x v i i i t h International Astronautical Congress, Pergamon Press, London, 1968, pp. 109-120. 11. Brereton, R.C., "A Stability Study of Gravity-Oriented Satellites," Ph.D. dissertation, University of British Columbia, Nov. 1967. 12. Modi, V.J., and Shrivastava, S.K., "Coupled Librational Motion of an Axisymmetric Satellite in a Circular Orbit," The Aeronautical Journal of the Royal Aeronautical Society, Vol. 73, No. 704, August 1969, pp. 674-680. 13. Shrivastava, S.K., and Modi, V.J., "Effects of Atmosphere on Attitude Dynamics of Axisymmetric Satellites," Pro-ceedings of the xxth Congress of the International Astro-nautical Federation, in press. 14. Modi, V.J., and Shrivastava, S.K., "Effects of Inertia on Coupled Librations of Axisymmetric Satellites in Circular Orbits," presented for publication. 15. Kamm, L.J., "'Vertistat': An Improved Satellite Orientation Device," ARS Journal, Vol. 32, No. 6, June 1962, pp. 911-913. 16. Hartbaum, H., Hooker, W., Leliakov, I., and Margulies, G., "Configuration Selection for Passive Gravity-Gradient Satellites," Paper presented at the Symposium on Passive Gravity-Gradient Stabilization, Ames Research Center, Moffet Field, California, May 10-11, 1965. 17. Clark, J.P.C., "Response of a Two-Body Gravity-Gradient System in a Slightly Eccentric Orbit," Journal of Space-craft and Rockets, Vol. 7, No. 3, March 1970, pp. 294-298. 18. Zajac, E.E. , "Damping of a Gravitationally Oriented Two-Body Satellite," ARS Journal, Vol. 32, No. 12, Dec. 1962, pp. 1871-1875. ' 19, Etkin, B.,"Attitude Stability of Articulated Gravity-Oriented Satellites. Part I - General Theory and Motion in Orbital Plane," Report No. 89. University of Toronto, Institute of Aerophysics, Nov.19 62. 20". Maeda, H., "Attitude Stability of Articulated Gravity-Oriented Satellites. Part II - Lateral Motion," Report No. 93, University of Toronto, Institute of Aerophysics, June 1963. 21. Hughes, P.C., "Optimized Performance of an Articulated Gravity-Gradient Satellite at Synchronous Altitude," Report No. 118, University of Toronto, Institute of Aerospace Studies, Nov. 1966. 22. Newton, R.R., "Damping of a Gravitationally Stabilized ,Satellite," AIAA Journal, Vol. 2, No. 1, Jan. 1964, pp. 20-2 5. 23. 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II, "Attitude Control of a Sun-Pointing Spinning Spacecraft by Means of Solar Radiation Pressure," Journal of Spacecraft and Rockets," Vol. 7, No. 3, March 1970, pp. 357-359. 50. Thomson, W.T., "Spin Stabilization of Attitude against Gravity Torques," The Journal of the Astronautical Sciences, Vol. 9, No. 1, Jan. 1962, pp. 31-33. 51. Pringle, R., Jr., "Bounds on the Librations of a Symmet-rical Satellite," AIAA Journal, Vol. 2, No. 5, May 1964, pp. 908-912. 52. Kane, T.R., and Shippy, D.J., "Attitude Stability of a Spinning Unsymmetrical Satellite in a Circular Orbit," The Journal of the Astronautical Sciences, Vol. 10, No. 4, Winter 1963, pp. 114-119. 53. Kane, T.R., and Barba, P.M., "Attitude Stability of a Spinning Satellite in an Elliptic Orbit," Journal of Applied Mechanics, June 1966, pp. 402-405. 54. Wallace, F.B., Jr., and Meirovitch, L., "Attitude Insta-bility Regions of a Spinning Symmetric Satellite in an Elliptic Orbit," AIAA Journal, Vol. 5, No. 9, Sept. 1967, pp. 1642-1650. 55. Neilson, J.E., "On the Attitude Dynamics of Slowly-Spinning Axisymmetric Satellites under the Influence of Gravity-Gradient Torques," Ph.D. dissertation, University of British Columbia, Nov. 1968. 56. Modi, V.J., and Neilson, J.E., "Attitude Dynamics of Slowly-Spinning Axisymmetric Satellites under the In-fluence of Gravity-Gradient Torques," presented at the xx^h International Astronautical Congress, Argentina, 1969 . 57. Modi, V.J., and Neilson, J.E., "Roll Dynamics of a Spinning Axisymmetric Satellite in an Elliptic Orbit," Journal of the Royal Aeronautical Society, Vol. 72, No. 696, Dec. 1968, pp. 1061-1065. 58. Modi, V.J., and Neilson, J.E., "On the Periodic Solutions of Slowly-Spinning Gravity-Gradient Systems," to be pre-sented at the xxist International Astronautical Congress, Constance, West Germany, 19 70. 59. Likins, P.W., "Effects of Energy Dissipation on the Free-Body Motions of Spacecrafts," No. 32-860, Jet Propulsion Laboratory Technical Report, 19 66. 60. Roberson, R.E., "Torques on a Satellite Vehicle from Internal Moving Parts," Journal of Applied Mechanics, Vol. 25, pp. 196-200 (1958). 61. Landon, V.D., and Stewart, B., "Nutational Stability of an Axisymmetric Body Containing a.Rotor," Journal of Spacecraft and Rockets, Vol. 1, No. 6, December 1964, pp. 682-684. 62. lorillo, A.J., "Nutation Damping Dynamics of Axisymmetric Rotor Stabilized Satellites," American Society of Mechan-ical Engineers, Winter Meeting, Nov. 19 65. 63. Likins, P.W., "Attitude Stability Criteria for Dual-Spin Spacecraft," Journal of Spacecraft and Rockets, Vol. 4, No. 12, Dec. 1967, pp. 1638-1643. 64. 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