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Dynamics of gravity oriented axi-symmetric satellites with thermally flexed appendages Ng, Chun Ki Alfred 1986

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DYNAMICS OF GRAVITY ORIENTED A X I-SYMMETRIC SATELLITES WITH THERMALLY FLEXED APPENDAGES by CHUN KI ALFRED NG B . A . Sc. , University of British Columbia, 1984 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1986 © CHUN KI ALFRED NG, 1986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the 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 or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: November, 1986 ABSTRACT The equations of motion for a satellite with a rigid central body and a pair of appendages deforming due to thermal effects of the solar radiation are derived. The dynamics of the system is studied in two stages: (i) librational dynamics of the central body with quasi-steady thermally flexed appendages; (ii) coupled librational/vibrational dynamics of the spacecraft. Response of the system is investigated numerically over a range of system parameters and effect of the thermal deformations assessed. The study indicates that for a circular orbit, the flexible system can become unstable under critical combinations of system parameters and initial conditions although the corresponding rigid system continues to be stable. However, in eccentric orbits, depending on the initial conditions, thermally flexed appendages can stabilize or destabliIize the system. Attempt is also made to obtain an approximate closed-form (analytical) solution of the problem to quickly assess trends and gain better physical appreciation of response characteristics during the preliminary design. Comparisons with numerical results show approximate analysis to be of an acceptable accuracy for the intended objective. The closed-form solution can be used with a measure of confidence thus promising a substantial saving in time, effort, and computational cost. ii TABLE OF CONTENTS A B S T R A C T " LIST OF FIGURES v LIST OF SYMBOLS x 1. INTRODUCTION 1 1.1 Preliminary Remarks 1 1.2 Scope of the Investigation 11 2. FORMULATION OF THE PROBLEM 12 2.1 Preliminary Remarks 12 2.2 Position of the Satellite in Space 12 2.3 Solar Radiation Incidence Angles 15 2.4 Shape of the Thermally Flexed Appendage 18 2.5 Determination of Kinetic and Potential Energies 22 2.5.1 Assumptions 22 2.5.2 Coordinate system 23 2.5.3 Kinetic energy 26 2.5.4 Potential energy 28 2.6 Equations of Motion 30 3. LIBRATIONAL DYNAMICS OF SATELLITES WITH THERMALLY FLEXED APPENDAGES 32 3.1 Preliminary Remarks 32 3.2 Equilibrium Orientation 33 3.3 Stability in the Small .40 3.4 Motion in the Large .49 3.4.1 Variation of parameters method .49 3.4.2 Numerical method 55 iii 3.4.3 Discussion of results 57 4. COUPLED LIBRATIONAL/ VIBRATIONAL DYNAMICS OF SATELLITES WITH THERMALLY FLEXED APPENDAGES 69 4.1 Preliminary Remarks 69 4.2 Equilibrium Orientation 69 4.3 Numerical Analysis of the Nonlinear Equations 78 4.3.1 Appendage disturbance 78 4.3.2 Central body disturbance 92 4.3.3 Influence of thermal deformation on system stability (circular orbits) 102 4.3.4 Eccentric orbits 109 4.3.5 Influence of thermal deformation on system stability (eccentric orbits) 122 4.4 Analytical Solution 127 4.4.1 Variation of parameters method 127 4.4.2 Improved analytical solution 134 4.4.3 Discussion of results 140 5. CONCLUDING COMMENTS 158 5.1 Summary of Conclusions 158 5.2 Recommendations for Future Work 160 BIBLIOGRAPHY 162 APPENDIX I EVALUATION OF APPENDAGE KINETIC ENERGY 164 II EVALUATION OF APPENDAGE POTENTIAL ENERGY 166 I I I DETAILS OF THE EQUATIONS OF MOTION 168 IV MATRIX [M] 178 V HOMOGENEOUS SOLUTION OF EQUATION (4.7) 180 iv LIST OF FIGURES Figure Page 1-1 A schematic diagram of the Radio Astronomy Explorer Satellite with long flexible antennae and booms 2 1-2 The proposed Orbiter-based tethered subsatellite system scheduled to be launched in 1989 3 1-3 A schematic diagram of the European Space Agency's L-SAT expected to be launched in 1987 . 5 1-4 A schematic diagram of the proposed Mobile Satellite System (MSAT) 6 1-5 Artist's view of the Orbiter-based manufacture of structural components for construction of a space platform 7 1- 6 Contribution of environmentally induced torques for a typical satellite 8 2- 1 Spacecraft geometry and nominal equilibrium position 13 2-2 Orbital elements defining the position of center of mass of a satellite in space. 14 2-3 Modified Eulerian rotations yp, <pt and X defining an arbitrary orientation of the satellite in space 16 2-4 Solar radiation incidence angles <p* <f>*, and <p* 17 x y z 2-5 A comparison between Brereton's approach and the present approximate solution for the shape of a thermally flexed appendage: (a) Alouette I Appendage; 21 (b) Alouette II Appendage 21 2-6 Coordinates with respect to Xp,Yp,Zp-axes defining thermal deformation and vibration of flexible appendages 24 2- 7 Vectors r( and ru defining position vectors of mass elements on the lower and upper appendages, respectively 29 3- 1 An illustration of the orbit with p=w, i=0, showing the position of the sun relative to the spacecraft. 32 v 3-2 Effect of inertia parameters, orbit eccentricity, and thermal deformation on the equilibrium orientation; (a) slender central body with relatively light appendages; 35 (b) stubby central body with relatively light appendages; 36 (c) slender central body with relatively heavy appendages; 37 (d) stubby central body with relatively heavy appendages 38 3-3 Equilibrium orientations at 0 and ir-6_ in circular orbits .40 3-4 Effect of inertia parameters on stability of the system in circular orbits .44 3-5 Librational response of the system with K = 1.0, K. = -0.1 showing unstable motion caused by thermal deflection: (a) 0-5 orbits; 46 (b) 45-50 orbits; .47 (c) 55-60 orbits .48 3-6 Comparison of numerical and analytical solutions for different inertia parameters: (a) slender central body with relatively light appendages; 58 (b) stubby central body with relatively light appendages; 60 (c) slender central body with relatively heavy appendages; 62 (b) stubby central body with relatively heavy appendages 63 3-7 Comparison of numerical and analytical solutions for non-zero spin parameter 65 3-8 Comparison of numerical and analytical solutions for a disturbance across the orbital plane 66 3- 9 Comparison of numerical and analytical solutions for eccentric orbits; (a) e= 0.1; 67 (b) e= 0.2 68 4- 1 Effect of system parameters on the equilibrium orientation: (a) slender central body with small appendages; 72 (b) slender central body with large appendages; 73 (c) stubby central body with small appendages; 74 (d) stubby central body with large appendages 75 4-2 Variation of tip deflection at equilibrium with 8 showing the dominance of thermal effect 76 vi 4-3 Effect of appendage flexibility on its equilibrium position 77 4-4 System response with: (a) in-plane appendage disturbance; 80 (b) out-of-plane appendage disturbance .81 4-5 Typical response of the system for an impulsive appendage disturbance. . .83 4-6 A comparison of response for systems with one and two flexible appendages: (a) in-plane appendage disturbance; 84 (b) out-of-plane appendage disturbance 85 4-7 System response showing the effect of symmetric and asymmetric appendage disturbances; (a) symmetric in-plane disturbance; 87 (b) symmetric out-of-plane disturbance; 88 (c) asymmetric in-plane disturbance; 89 (d) asymmetric out-of-plane disturbance 90 4-8 System response showing the effect of appendage disturbance in the presence of thermal deformation 91 4-9 System response showing the effect of in-plane central body disturbance: (a) displacement disturbance; 93 (b) impulsive disturbance 94 4-10 System response showing the effect of central body disturbance: (a) out-of-plane disturbance; 95 (b) combined in-plane and out-of-plane disturbances 97 4-11 System response showing the effect of central body disturbance in the presence of thermal deformation 100 4-12 Typical response in circular orbits showing the effect of inertia parameters and thermal deformation: (a) slender central body with small appendages; 103 (b) stubby central body with small appendages; 104 (c) slender central body with large appendages 105 vii 4-13 System response in circular orbits showing the destabilizing influence of thermally deformed appendages: (a) librational response; 106 (b) vibrational response 107 4-14 System response in eccentric orbits: (a) in-plane appendage disturbance, e = 0.1; 110 (b) in-plane appendage disturbance, e= 0.2; 111 (c) out-of-plane appendage disturbance, e= 0.1; 112 (d) out-of-plane appendage disturbance, e= 0.2 114 4-15 System response showing the effect of eccentricity and central body disturbance: (a) librational response; 117 (b) vibrational response 118 4-16 System response showing the effect of eccentricity and thermal deformation: (a) appendage disturbance; 119 (b) central body disturbance, librational response; 120 (c) central body disturbance, vibrational response 121 4-17 System response in eccentric orbits showing the effect of thermal deformation: (a) an increase in libration amplitude; 123 (b) a decrease in libration amplitude 124 4-18 Typical response in eccentric orbits of spacecraft with thermally flexed appendages showing the influence of initial conditions: (a) destabilizing influence; 125 (b) stabilizing influence 126 4-19 A comparative study showing the deficiencies of the analytical solution: (a) librational response; 131 (b) vibrational response 132 viii 4-20 A comparison between numerical and improved analytical solutions for a small appendage disturbance: (a) analytically obtained results; 141 (b) numerical results 142 4-21 A comparison between numerically and improved analytically predicted responses with central body disturbance: (a) vibrational response; 143 (b) librational response 144 4-22 A comparison between numerical and improved analytical solutions in the presence of severe out-of-plane disturbance: (a) librational response; 146 (b) vibrational response 147 4-23 A comparison between numerical and improved analytical solutions showing the effect of inertia parameters on correlation: K g = 0.1, K j = 0.75; 149 4-24 A comparison between numerical and improved analytical solutions for a satellite with relatively large appendages and a slender central body: « a = 0.5. K f = 0.75 152 4-25 A comparison between numerical and improved analytical solutions showing the effect of a stubby central body: K = 0.1, K .= 0.25 155 a i ix LIST OF SYMBOLS 2a length of the central body, Fig. 2-1 a^ appendage radius aj length ratio, a/ a1* a2* a3 constants written in terms of p, CJ, and i, Eq. (2.2) appendage wall thickness b1* b2' b 3 components of u along X g, Y and Z g axes, respectively; Eq. (2.1) dj i= 1 5, coefficients in characteristic equations (V.2) and (V.5) f^, f ^ functions containing the zeroeth and first order derivative terms in <p and \p, Eq. (3.32) gj i= 1,..., 7, functions defining equilibrium state of the system, Eqs. (3.1) and (4.1) h angular momentum per unit mass of the system i inclination of the orbit with respect to the ecliptic plane, Fig. 2-2 ? , j , k unit vectors in the directions of X , Y , and Z -axes, p' V P P P P respectively ? , j , (c unit vectors in the directions of X , Y , and Z -axes s s s S' S' s respectively k^ thermal conductivity of the appendage l ^ appendage length, Fig. 2-1 I* thermal reference length of the appendage, Eq. (2.8) l x , ly, l z direction cosines of R with respect to Xp, Y and Zp-axes, respectively, Eq. (2.24) mass per unit length of the appendage n n^ appendage vibration frequencies in and across the orbital plane, respectively; Eq. (4.4) n^, n^ librational frequencies in and across the orbital plane, respectively; Eq. (3.21) np1' nq1' n01' n\p1 a t t e n u a t e d frequencies associated with n n^, n^, and n^, respectively; Eqs. (V.2) and (V.5) x 2 solar radiation intensity, 1360 W/m position vectors to mass elements on the lower and upper appendages, respectively, as measured from S time unit vector representing the direction of solar radiation, Eq. (2.1) lower appendage vibration in and out of the orbital plane, respectively; Eq. (2.15) upper appendage vibration in and out of the orbital, plane, respectively; Eq. (2.17) distance from the appendage attachment points along X| and X u-axes, respectively thermal deflections of the lower appendage in and out of the orbital plane, respectively; Eq. (2.14) thermal deflections of the upper appendage in and out of the orbital plane, respectively; Eq. (2.17) i=pl, ql, pu, qu, c6, i/>; j = 1, 2, 3; amplitudes of the sinusoidal functions in the analytical solution Young's modulus of the appendage material nonlinear functions, Eq. (3.15) approximation to and F2, respectively, with constraints; Eq. (3.28) generalized forces in the equations of motion i= 1,..., 6, non-linear functions, Eq. (4.7) i= 1 6, approximation to Hj with constraints, Eq. (4.8) I = I y = I z area moment of inertia of the appendage mass moment of inertia of the system with undeformed appendages about Yp or Zp-axes mass moment of inertia of the central body about X , P Yp, and Zp-axes, respectively; 1^= I z 3 3 inertia ratios r n ^ l ^ / I and m^l^/ 11, respectively xi inertia ratios 1—I x/ I and 1 - I x / I t . respectively length ratio, l ^ / I* mass of the central body coefficients in Mathieu equation, Eq. (3.16) generalized coordinates for the lower and upper appendage in-plane vibration, respectively dimensionless ratios P(/ l b and P / l b , respectively generalized coordinates for the lower and upper appendage out-of-plane vibration, respectively dimensionless ratios Q|/ l ^ and Q u/ l ^ , respectively position vector to S as measured from the centre of the Earth centre of mass of the system origins of the reference coordinate systems measuring deformation of the lower and upper appendages, respectively; Fig. 2-6 total kinetic energy of the system, appendages, and central body, respectively; T = ~fapp + b components of TQpp w ' t n subscripts representing libration, thermal deflection, and vibration contributions, respectively total potential energy of the system, appendages, and central body, respectively; U = ^3pp + ^ ^ 4 - U g total strain energy of the appendages components of u a p p with subscripts representing libration, thermal deflection, and vibration contributions, respectively intermediate axes during the Eulerian rotations, Fig. 2-3 principal coordinate system for the lower appendage with origin at S|, Fig. 2-6 inertial reference frame with Earth as the origin, Fig. 2-2 principal coordinates of the central body, Fig. 2-3 xii orbital frame with X g in the direction of the local vertical, Y along the local horizontal, and Z towards O S the orbit normal, Fig. 2-3 principal coordinates for the upper appendage origin at S u, Fig. 2-6 integration factor in Mathieu equation, Eq. (3.18) i= 1 4, , constants, Eq. (III.3) absorptivity and coefficient of thermal expansion, respectively, for the appendage material i=pl, ql, pu, qu, <p, i//; j = 1, 2, 3; phase angles in the sinusoidal functions of the analytical solution eccentricity emissivity of the appendage material functions of e and 6, Eq. (111.1) rotation across the orbital plane, Fig. 2-3 0e~ a solar radiation incidence angles, Eq. (2.5) rotation about the axis of symmetry of the satellite, Fig. 2-3 gravitational constant true anomaly angular velocity of the system at the perigee point longitude of the ascending node, Fig. 2-2 spin parameter, Eq. (3.7) Planck's constant argument of the perigee, Fig. 2-2 fundamental frequency of the appendage frequency ratio, cj^/d^ angular velocities of the system about X Yp, and Z -axes, respectively; Eq. (2.18) rotation in the orbital plane, Fig. 2-3 i//e~ a xiii * 1 fundamental mode of a cantilever beam, Eq. (2.16) Subscripts e equilibrium position 0 initial condition Dots and primes represent differentiation with respect to t and 8 respectively. The word "system" refers to the central body with appendages. xiv ACKNOWLEDGEMENT The author wishes to express his gratitude to Prof. V. J. Modi for the guidance and encouragement throughout the study and particularly for his help during the initial critical stage. The investigation reported in the thesis was supported in part by the Natural Sciences and Engineering Research Council, Grant No. A-2181. xv 1. INTRODUCTION 1.1 Preliminary Remarks In the early stages of space exploration, satellites tended to be relatively small, simple in design and essentially rigid. However, for a modern spacecraft with its lightweight, flexible, deployable members in the form of solar panels, antennae, and booms, it is no longer true. This point can be well-emphasized by several examples; (i) Ever increasing demand on power for operation of the on board instrumentation, scientific experiments, communications systems, etc., has been reflected in the size of the solar panels. The Canada/USA Communications Technology Satellite (CTS, Hermes) launched in January 1976 carried two solar panels, 1.14m x 7.32m each, to generate around 1.2kW of power. (ii) Use of large members may be essential in some missions. For example, Radio Astronomy Explorer (RAE) satellite used four 228.2m antennae to detect low frequency signals (Fig. 1-1). (iii) For identifying extraterrestrial radio sources, the Applied Physics Laboratory of the Johns Hopkins University once proposed a gravitationally stabilized Tethered Orbiting Interferometer (TOI) consisting of two spacecraft connected by a line 2-6km long. In fact, NASA has shown considerable interest in exploiting application of the Orbiter based tethered subsatellite system, extending to 100km, and has initiated, through contracts, preliminary studies to establish its feasibility (Fig. 1-2). (iv) Preliminary configurations of the next generation of satellites such as L-SAT (Large SATellite System, Olympus), DBS (Direct Broadcast 1 Figure 1-1 A schematic diagram of the Radio Astronomy Explorer Satellite with long flexible antennae and booms. 3 Orb i t e r Figure 1-2 The proposed Orbiter-based tethered subsatellite system scheduled to be launched in 1989. 4 System), MSAT (Mobile SATellite System), etc., suggest a trend towards spacecraft with large flexible members (Figs. 1-3 and 1-4). (v) Space engineers are involved in assessing feasibility of constructing gigantic space stations which cannot be launched in their entirety from the earth, but have to be constructed in space through integration of modular subassemblies. In-orbit assembly of enormous orbiting station such as Space Operations Center (SOC) and futuristic design of Solar Power Station (SPS) suggest large space structures with an increasing role of structural flexibility in their dynamical and control considerations (Fig. 1-5). This being the case, flexibility effects on satellite attitude motion and its control has become a topic of considerable importance. It should be emphasized that prediction of satellite attitude motion is by no means a simple proposition, even Jf the system is rigid. Flexible character of the appendages makes the problem enormously complex. The presence of environmental forces such as the solar radiation (pressure and thermal effects), earth's magnetic field, free molecular forces, etc., which are capable of exciting elastic degrees of freedom, add to the challenge. Figure 1-6 shows contribution of several environmental forces as functions of altitude for a representative satellite GEOS-A 1. At low altitudes (<1000 km), as can be expected, the atmospheric effects are dominant. The gravity gradient contribution diminishes with altitude in the inverse square manner. Effect of the earth's magnetic field is several orders of magnitudes smaller than the solar pressure, which is essentially independent over the range of the earth's orbit. It is of particular significance that near the synchronous altitude, gravity and solar pressure contributions (even for this satellite of relatively small projected area; no 6 Figure 1-4 A schematic diagram of the proposed Mobile Satellite System (MSAT). 7 Figure 1-5 Artist's view of the Orbiter-based manufacture of structural components for construction of a space platform. 8 ,5 I n | | TT I I I 5 0 10 5 10 2 5 10 2 5 10 Altitude, km Figure 1-6 Contribution of environmentally induced torques for a typical satellite. 9 large solar panels) are essentially the same. Furthermore, the solar radiation leads to differential heating of the satellite, depending upon its attitude, resulting in thermal deflection of the flexible members mentioned before. Corresponding changes in inertia and elastic characteristics would naturally reflect on dynamics, stability, and control of the satellite. The contemporary interest in space science and technology has resulted in an enormous body of literature since launching of the first satellite, Sputnik, in 1957. Broadly speaking, the available literature in the area of satellite dynamics may be classified as follows: (a) dynamics, stability, and control of rigid satellites; (b) effect of environmental forces on the attitude dynamics of rigid satellites; (c) large, flexible system dynamics and control; (d) dynamics and control of flexible systems in the presence of environmental forces. Most of the available literature belongs to the first three categories 2-4 and has been reviewed at length by Modi, Shrivastava, and Tschann . On the other hand, behaviour of flexible satellites when exposed to free molecular environment, solar radiation, Earth's magnetic field, etc., remains virtually unexplored, except for some simplified preliminary studies by Modi, 5-15 Brereton, Kumar, Goldman, Yu, Bainum, and Krishna 5 Modi and Brereton studied librational dynamics of a free-free beam thermally flexed due to solar radiation. Using a quasi-steady representation for the deformed beam and the concept of integral manifold in phase space, stability charts were obtained. In general, flexibility of the satellite tended to reduce its stability for all positions of the sun (solar aspect 10 angle); however, the reduction was considered to be of no major concern. Using a similar approach, Modi and Kumar studied librational dynamics of a rigid satellite with thermally flexed plate-type appendages. The study indicated the thermoelastic behaviour of appendages to adversely affect the satellite performance. In general, flexibility of appendages caused an increase in the amplitude and average period of libration, and a decrease in the stability region. The inclusion of solar radiation pressure and eccentricity effects further deteriorated the stability of the satellite. Goldman^ used thermal deflection of appendages to explain the anomalous behaviour of Naval Research Satellite 164. However, as in the case of the previous study, he did not consider vibration of the flexible members. Yu studied thermally induced vibration of a beam connected to a rigid, orbiting body. The body was found to be stable if the beam pointed away from the sun but unstable when it pointed towards the sun. 9 10 However, the results are controversial as Augusti and Jordan obtained the opposite results using other approaches. More recently, Bainum and Krishna investigated the influecnce of solar radiation pressure on librational response of a free-free orbiting 1112 13 beam ' and a square plate in orbit. The main objective was to assess the effect of solar pressure on control laws to achieve the desired shape and orientation control. The simplified linear analyses were intended to provide only preliminary data indicative of trends. Subsequently, the authors 14 15 extended the analysis to account for thermal effects * . The results showed that librational response for a thermally deformed appendage to be an order of magnitude larger than that with the solar pressure effect alone. 11 1.2 Scope of the Investigation With this as background, the thesis studies librational dynamics of spacecraft having a central rigid body with two flexible beam-type appendages, nominally aligned along the local vertical (gravity gradient stabilized configuration), free to vibrate as well as deform under the influence of the solar radiation. The problem is analyzed in four stages. To begin with, thermal analysis of an orbiting cantilever beam is carried out and an expression for its deformed configuration as a function of the solar aspect angle obtained. This is followed by a detailed nonlinear formulation of the problem using the classical Lagrangian procedure. To better appreciate the dynamical response and physical behaviour of the system, it would be useful to isolate the effect of thermal deformation and vibration of the flexible appendage. To that end, vibration terms in the governing equations are purposely suppressed and a parametric analysis of equilibrium configuration and response carried out numerically. A simplified nonlinear analytical model is developed next and its validity assessed through comparison with numerical data. Finally, librational response of the spacecraft with thermally flexing and vibrating appendages is presented. As before, equilibrium configurations and response are studied parametrically. An improved closed-form solution to this general problem is developed and compared with the numerical results. Throughout the emphasis is on better appreciation of the complex interactions between attitude dynamics, vibration, and thermally induced deformations. The results should be particularly useful during the preliminary design phase of this class of satellites. 2. FORMULATION OF THE PROBLEM 2.1 Preliminary Remarks This chapter begins with a discussion on the position and orientation of a satellite in space, followed by the determination of solar radiation incidence angles and the shape of a thermally flexed appendage. The kinetic and potential energies of the system are then derived. Finally, using Lagrange's formulation, the equations of motion for librations of the central body and vibrational degrees of freedom for the appendages are determined. The satellite consists of a rigid, axi-symmetric cylinder of length 2a with two flexible beam-type appendages attached to its flat ends (Fig. 2-1). Each appendage, of length l ^ , is assumed to be a thin walled circular tube with constant mass density, flexural rigidity, and cross sectional area along its length. The joint between the satellite and the appendage is assumed to be rigid, i.e., no joint rotation is allowed. Since the satellite is stabilized by the gravity gradient, the system is aligned along the local vertical in nominal equilibrium position. The appendage pointing towards or away from the earth is designated as the lower or upper appendage, respectively. 2.2 Position of the Satellite in Space Consider a spacecraft with its centre of mass at S negotiating an arbitrary trajectory about the center of force coinciding with the homogeneous, spherical Earth's center. At any instant, the position of S is determined by the orbital elements p, i , co, e, R, and 6. In general, p, i , co, a n d e, are fixed while R and 9 are functions of time (Fig. 2-2). 12 13 Local Vertical Upper Appendage a Orbit a •Central Body Lower Appendage Earth Figure 2-1 Spacecraft geometry and nominal equilibrium position. 14 Figure 2-2 Orbital elements defining the position of center of mass of a satellite in space. 15 As the spacecraft has finite dimensions, i.e., it has mass as well as inertia. Hence, in addition to negotiating the trajectory, it is free to undergo librational motion about its center of mass. Let Xp.Yp.Zp be the principal body axes of the central body with their origin at S. On the other hand, X s , Y s , Z s represents moving coordinates along the local vertical, local horizontal, and orbit normal, respectively. Any spatial orientation of the satellite can be described by three modified Eulerian rotations: a pitch motion, \p, about the Z g - a x i s ; a roll motion, <p, about the Y-ax is ; and a yaw motion, X, about the X-axis (Fig. 2-3). 2.3 Solar Radiation Incidence Angles Position of the satellite with respect to the sun is defined by the solar radiation incidence angles, and They are defined as angles between the unit vector, u, representing the direction of solar radiation, and the X Y and Z axes, respectively (Fig. 2-4). With reference to the moving coordinate system 1 j k s > the unit vector u can be written as, A u = [—ai cos 9 + 0 2 sin 8]i, A A + [a\ sin 8 + ai cos 8]jt + a3k„ = M. + b2jt + bzk,, ••• (2.1) where; a\ = cos p cos tv + sin p cos » sin w ; a 2 = cos p sin u — sin p cos t cos u>; 03 = sinpsin». •••(2.2) Now, in terms of the principal body coordinates: Figure 2-3 Modified Eulerian rotations \p, c6, and X defining an arbitrary orientation of the satellite in space. 17 18 ^ A A t, = (cos tp cos 4>)ip + (cos ip sin <6 sin A - sin ip cos X)jp A + (cos ip sin d> cos A + sin ip sin A)fcp ; j , = (sin ^ cos c6)t'p + (sin ip sin sin A + cos ip cos A);'p + (sin tp sin cos A — cos ip sin X)kp ; A A A A fc, = - sin <}> ip + cos $ sin Xjp + cos 0 cos Afcp . • • • (2.3) Hence, substituting from Eq. (2.3) into (2.1), u can be rewritten as u = cos <f>*x t p + cos 4>l jp + cos <t>*z kp , . . . (2.4) where; t ^ ! cos <px = b\ cos tp cos <6 + fc2 sin ip cos 0 — 63 sin c6 ; cos rf>* = 61 (cos tp sin c6 sin A — sin ip cos A) + 62(sin tp sin c6 sin A + cos ip cos A) + 63 cos <p sin A ; cos c6* = b\(cos ^ 1 sin <p cos A + sin ip sin A) + 62(sin rp sin c6 cos A — cos ip sin A) 4- 63 cos <f> cos A . . . . (2.5) It can be seen that, in general, fc = fc{0>u>hPAA) , and <p) = <p*(9,u,i,p,\,<p,ip) , j = y or z . 2.4 Shape of the Thermally Flexed Appendage The plane in which centre line of the thermally flexed appendage lies is difficult to determine and changes with the librational motion. This problem is overcome by taking the projections of a thermally flexed 19 appendage on the x p , Y p a n d Xp,Z p-planes. Let the solar radiation intensity be' q g W/m2, then its components along Y and Z axes are given by q and q respectively, where: qv = q, cos 6*y ; g? = g,cost£*. •••(2.6) Using Eq. (2.6) together with Brereton's equation for the shape of the centre line of a thermally flexed appendage give 1: ^ = -ln[cos(j5-)] cos<£* ; £ = - l n [ c o s ( £ ) ] c o s ^ ; •••(2.7) where 6 , and 5, represent deflections of the centre line in the X Y and y z P P Xp,Zp planes, respectively. Here 77 is the distance measured along the undeformed appendage with n = 0 at the fixed end, and I* is called the thermal reference length given by, r . J i L ^ + ^ M L ) ! ^ ) , . ...(2.8) Note, Eq. (2.7) represents steady-state solution obtained by solving the differential heat balance relation for a thin-walled circular tube. The transient solution is not included here because its time constant is small 1. Goldman^ has obtained an approximate steady-state solution for a thermally flexed appendage as ^ = ^ ) 2 - s ^ [ l + ^ ) c o s ^ ] ; ^ = ^ ) 2 c o - « [ l + 5(p)coB^l ; - ( 2 . 9 ) 20 where = ...(2.10) The second term in Eq. (2.9) accounts for the effect of longitudinal expansion on appendage deflection. This is small and hence not considered by Brereton. A lso , the second term in Eq. (2.8) is small as compared to the first term; therefore, the I* values calculated from Eqs. (2.8) and (2.10) are approximately the same. Equations (2.7) and (2.9) are not convenient for integration especially if the integrand also has transcendental functions; hence, the approximation of Eq. (2.9) is used in the analysis: F = 2 (F } c o s ^ ; | = ^ ) 2 c o s ^ . ...(2.11) Equations (2.9) and (2.11) are approximately the same for small cos 0* or 7 j / l * . The differences between Eqs. (2.7) and (2.11) are not obvious. Thermal deflections of the appendages during the most critical condition (<p* or 0* = 0) as given by Eqs. (2.7) and (2.11) are compared in Figure 2-5. Physical properties correspond to appendages used on Alouette I and II satellites. The figure shows that the difference in 6 j / I* increases with TJ/I*; however, it is negligible for rj/|* less than 0.6. For steel and beryllium-copper appendages, rj/l*= 0.6 corresponds to 63m and 75m, respectively. Appendages of these lengths should be adequate for most satellites; hence, Eq. (2.11) can be used with confidence to represent the shape of a thermally flexed appendage with L* limited to between 0 and 0.6. L*= 0 can mean either there is no appendage (1^= 0) or the thermal deflection of the appendage is ignored (l*= » ) . 21 Alouette 1 Alouette II Unifs Material Sleei Beryllium copper Bending stiffness, E l b 144 6.4 N m 2 Mass / length , m b 0.102 0.021 k g / m Radius, a b 1.207 0.635 . cm Wall thickness, b b 0.017 0.005 cm Absorptivity, a t 0.900 0.450 -Emissivity, c b 0.800 0.250 -Thermal conductivity, k b 45 86.5 W / m ° c Coefficient of thermal expansion, a , 11.7x10 - 6 18.0x10 - 6 • c - ' Thermal reference length, 1* 105 126 m Figure 2-5 A comparison between Brereton's approach and the present approximate solution for the shape of a thermally flexed appendage: a) Alouette I Appendage; b) Alouette II Appendage. 22 2.5 Determinat ion of Kinetic and Potential Energ ies 2.5.1 A s s u m p t i o n s In order to gain better apprec ia t ion as to the p h y s i c a l behaviour of the s y s t e m in te rms of interact ions between l ibrat ion d y n a m i c s , f lex ib i l i ty , and thermal d e f o r m a t i o n s , severa l s i m p l i f y i n g a s s u m p t i o n s were introduced as indicated b e l o w . This made the s y s t e m d y n a m i c s more amenable to approximate c l o s e d - f o r m ana lys is and hence parametr ic eva lua t ion . (i) In genera l , centre of m a s s of the s y s t e m sh i f ts when the appendages are thermal ly f lexed or v ibrat ing. S i n c e the central b o d y m a s s is usual ly much greater than the appendage m a s s , the shift in the centre of m a s s is a s s u m e d to be smal l and ignored . (ii) Except for sate l l i tes in high latitude or alt i tude orb i t , a part of the trajectory is c o v e r e d by the Earth s h a d o w . O b v i o u s l y , in this reg ion , the thermal ly induced d e f o r m a t i o n s wil l be m i n i m a l . For the sake of s i m p l i c i t y , this e f fec t is ignored . (iii) C o n s i d e r i n g the m o t i o n of the f ixed end of the cant i l evered o appendage to be neg l ig ib le , Yu has s h o w n that the equat ion for t ransverse v ibrat ion of a thermal ly f lexed appendage is g iven b y , „ r 3 4 t y d2MT d2w E I b l & + -dx2- + m>>Wr = 0> - t 2 - 1 2 with boundary c o n d i t i o n s : to = dtv ~dx = 0 at X 0; EIb d2w + MT = EIb d*w 8MT 0. at X = If,. dx2 dxz dx Here M T is the thermal bending m o m e n t g iven by 23 MT = / EcxTTe(xa, ya,za)za dA, J Area where T e ( x a , y a , z g ) is the difference between the ambient temperature and the temperature at a point on the appendage with coordinates ( x a , y a , z a ) . The integral is over the cross sectional area of the appendage. Assuming My to be small, Eq. (2.12) simplifies to the usual Euler-Bernoulli beam equation, E l » - a * + m > W = 0> • ( 2 - 1 3 ) with boundary conditions: dw w = — — = 0 at i — 0; dx r,rd2w „rd3w „ E I b a ^ = E I b d ^ = 0 & t x = lb-In the present study, thermal deformations were obtained accounting for My, however, the mode shape used in vibration analysis corresponds to the conventional Euler-Bernoulli beam. /) The second and higher modes of vibration, longitudinal and torsional oscillations, and foreshortening effect of appendages are not considered. ) Other environmental disturbances, such as solar pressure, aerodynamic drag, and magnetic torques are neglected. 5.2 Coordinate system Figure 2-6 shows the coordinate system used in the analysis. There e three sets of coordinate axes: 24 X p - V Z p : X |> Y | - Z | ; a n d X U ' Y u ' Z u - T h e X p > Y p ' Z p _ a x e s r e P r e s e n t principal coordinates of the system with undeformed appendages. The coordinates of any point on the centre line of the lower and upper appendages are first written with respect to the X|,Y|,Zj and X u , Y u , Z u - a x e s , respectively. Referring these coordinates to Xp.Yp.Zp coordinate system involves simple linear transformation. Consider a point with coordinates (X|, 0, 0) on the center line of the lower appendage. Let thermal deflection of the appendage shift the point to a position with coordinates (X|, yj, Z|) where y| and Z| can now be obtained from Eq. (2.11): Since <p* is measured with respect to the Yp-axis (Fig. 2-4) while Y| and Y have opposite direction, a negative sign is introduced in Eq. (2.14a). Figure 2-6 Coordinates with respect to Xp,Yp,Zp-axes defining thermal deformation and vibration of flexible appendages. 25 Transverse vibration of the appendage shifts the point to a new position with coordinates (Xj, Y | + V | , Z | + W j ) . Since the Euler-Bernoulli beam equation is assumed to be valid and only the fundamental mode of vibration is considered, V | and W | can be written as: «, = f|(o*i(*i); u» = ; •••(2.15) where Pj(t) and Q|(t) represent the amplitudes of vibration in Xp.Yp a r , d X ,Z -planes, respectively, $,(x) is the fundamental mode shape of a P P cantilever beam given by, $1(1) = cosh/?!Z - cos/?,x - a^sinhftx - sin/3iz), ... (2.16) Pik= 1.875104; with cri = 0.734095. The fundamental natural frequency of transverse vibration, CJ,, is given by wi = Wb) 2 I EIh mbll Finally, deformation at a point on the lower appendage is referred to the principal coordinate system X Y Z i.e., the new coordinates of the H H r* point are: (-( X |+a), - ( y | + V | ) , z^w^. Similarly, the sequence for the movement of a point on the centre line of the upper appendage is from x 0, 0 to x y z u (thermal deformation) and finally to x u, Y u + V u , z u + w u (thermal deformation plus vibration) where y u, z v u , and w u are given by: 26 *« = 2(77) cosrf),; ^ = Q,(<)$i(i.). •••(2.17) Finally, position of the point with respect to Xp.Yp.Zp-axes, 's given by the coordinates (xu+a, Yu+vu, zu+wu). 2.5.3 Kinetic energy The kinetic energy of an axi-symmetric rigid central body with I = I = I z can be written as, Tc.b. = 1M[# + [Rd)2) + l[Wx + J ( W J + WJ)], where: ux = A — (0 + i>) sin<p ; ojy = 4> cos A + (0 + ^ ) cos <f> 8in A ; tvz = —d> sin A + (0 + ^ ) cos c4 cos A . • • • (2.18) For the lower appendage, velocity at a point with coordinates (-(x!+a), -(Y|+v|). z|+w|) i s 9 i v e n by. V { ) < lpp. = [Vx + uy{zi + wi) + u.(yi + t>j)]sp + \vy ~ U:ixt + a) ~ ux{zi + u>i) -yt- i>i]jP + [V, - (Jx(yi + vi) + ujy(xi + a) + k + vn]k, , • • • (2.19) where V V and V represent components of the velocity of the centre x y z of mass, S, in X Y and Z directions, respectively: 27 Vx = (R cos rp + R9 sin ip) cos <p ; Vy = (i2 cos ip + R9 sin sin <p sin A — (R sin ip — R9 cos cos A ; Vz = (R cos \p + R9 sin rp) sin $ cos A + (jR sin ip — R9 cos rp) sin A . Similarly, for a point on the upper appendage with coordinates (x u + a, y u + v u , z u + w u ) , V„, a p p. = [Vx + uy(zu + tu«) - wz{yu + vu)]ip + [Vy + ^ ( x . + a) - u)x(zu + tyB) + y„ + i)a]jp + [V, + ux(y„ + vu) - ojy(xu + a) + i„ 4- wn\kp . • • • (2.20) The kinetic energy of the appendages can now be calculated as, TCVP. = ~ - { fj V£app + jfVl a p p dx*}. ••• (2.21) Details of the derivation are given in Appendix I. It shows that , in general, can be written as Tapp. = mblb[R2 + {R9)2} + Tt + Tt + Tv + 7})t + TUv + Tt,v + 7),()„ , where the subscripts I, t, and v represent libration, thermal deflection, and vibration, respectively. For instance, T| t represents the contribution to the kinetic energy due to a coupling between librational motion and thermal deflection. The total kinetic energy of the satellite, T, is given by T — Tc.6. + Tapy. • 2.5.4 Potential energy The gravitational potential energy of an axi-symmetric central body is given b y 1 , Now, the gravitational potential energy of the appendages can be written as, " w = - { f *-"»3+f "•"*£!}• •••<»•»> From Figure 2 -7: ri = \Rlx - (a + xi)]ip + \Rly - (yi + Vl)]jp + [Rlt + (zi + Wi)]f% ; r u = [Rlx + (a + xtt)\ip + [Rly + (y, + t,a)]yp + + (zn + xvu)]kp ; . . . (2.23) where I l w , and I represent the direction cosines of R with X Y and x* y z p' p' Zp-axes, respectively, and are defined as: lx = cos ip cos <f>; ly = cos sin 4> sin A — sin ^  cos A ; lz = cos rp sin <f> cos A + sin if> sin A . •••(2.24) Substituting from Eq. (2.23) into (2.22) and ignoring 1/R and higher order terms gives, Uapp. = + Ul-rUl + Uv + Ultt + Di,. + Di,. + Di,,. Details of the derivation are explained in Appendix I I . Figure 2-7 Vectors r( and ru defining position vectors of mass elements on the lower and upper appendages, respectively. 30 Since the Euler-Bernoulli beam equation is assumed to be valid and only the fundamental mode of vibration is considered, the strain energy associated with the appendages can be written a s 1 , Ue = [(P,)2 + (Q/)2 + {Pu)2 + (Q„)2] • • • • (2.25) Thus, the total potential energy of the system, U, is given by, u = UcA + Uapp. + ue. 2.6 Equations of Motion Using the Lagrangian formulation, the governing equations of motion can be obtained from, where q= R, 6, (f>, X, P ( , Q ( , P u , Q u , and F q represents the generalized forces. In general, the effect of librational and vibrational motions on the orbital motion is small unless the system dimensions are comparable to 1 fi 17 R ' . Hence, the orbit can be represented by the classical Keplerian relations: R He{\ + ecos0) 1 R'0 = h\ where h is the angular momentum per unit mass of the system and e is the eccentricity of the orbit. 31 In satellite application, it is convenient to use the true anomaly 6 as an independent variable instead of time. Using Eq. (2.26) and substituting: - — ^ dt~ dd' _ -2 £_ _ 2£sinfl d dt2 ~ W l + ecos9d^'', the remaining seven equations corresponding to the generalized coordinates ^ . \ Pp p u . a n d 0-u c a n be obtained as explained in Appendix I I I 3. LIBRATIONAL DYNAMICS OF SATELLITES WITH THERMALLY FLEXED APPENDAGES 3.1 Preliminary Remarks The governing nonlinear, nonautonomous, and coupled equations of motion do not admit of any known closed-form solution. To get some appreciation of the complex dynamics with thermal effects, it was decided to make the problem progressively complex. As a first step, librational response of the system was explored setting vibrational generalized coordinates zero (P| = Q | = P = P - U = 0). Furthermore, a particular case of an ecliptic orbit with the perigee point between the sun and the earth was considered, i.e., p=cj and i =0 (Fig. 3-1). Solar Radiation Figure 3-1 An illustration of the orbit with p=co, i=0, showing the position of the sun relative to the spacecraft. 32 33 To begin with, variation of equilibrium configurations of the system with the true anomaly is discussed. This is followed by the determination of limiting inertia parameters (Kj and K g ) for stable motion in the small. Application of the procedure of variation of parameters is next illustrated to obtain an approximate solution of the problem. Finally, validity of the approximate response is assessed by comparison with the 'exact ' numerical solution. 3.2 Equilibrium Orientation Equilibrium orientation of the central body, represented by the librational angles \j/Qt 0 e , and Xg, can be obtained by putting generalized forces, librational velocities, and accelerations equal to zero: gityAA) = [-dT dU-i dlp + di>\ n = 0 ; 5=0 • -(3.1a) 02^, <M) = [-dT dUi d<t>\ « = 0 ; 9=0 •••(3.16) <73(Vs<M) = [-iOKd\} dT dU-i d\ + dXl g=o •••(3.1c) where q = ip\ d>'t A', <f>", X" . Limiting appendage deflection to thermal deformation only, i.e., neglecting appendage vibration (Ej = Q.\— £ u = S u = 0), Eq. (3.1c) becomes u'x = 0 •••(3.2) giving <f>e = 0. •••(3.3a) Since X is the angle of rotation about the axis of symmetry (yaw), it can assume any value without affecting the equilibrium orientation. Let A e = 0. • • • (3.36) Substituting Eq. (3.3) into (3.1b) gives 34 <72(^ Me = 0,Ae = 0) = 0; leaving only one equation to solve, 9i(ipe,<t>e = 0,Ae = 0) = -ec + Kiteg sin V cos %P - Kalcx2 { [-ec s in 2 (> + 9) - sinf> + Q) costy + 9) + cos2(xp + 9)] + eg [1 s i n (^ + 9) cos(tf> + 9)- sin i> cos xp sin2(V> + 9) s i n 2 ip sm(xp + 9) cos(tf» + 6)] J ...(3.4) where the coefficients are defined in Appendix I I I (Eqs. 111.1 - 111.3). Since the equation is transcendental in character, its solution is not readily available and one has to turn to a numerical approach. However, the numerical solution of a nonlinear equation with trigonometric functions is generally unstable as the roots can be very large or small. In contrast, \pe should be in the range of -7T/2 to 7T/2 for gravitational gradient stabilized satellites. The problem is overcome by minimizing J instead, where J = g\, with the constraint ' Equation (3.5) represents a constrained optimization problem and several subroutines are available in the UBC Computer Library for this class of problems. The NLPQLO subroutine, which is a modified version of Schittkowski's implementation of the quadratic approximation method of 18 19 Wilson, Han, and Powell ' , was used here because of its efficiency particularly for problems with few variables and constraints. Figure 3-2 shows the variation of \pe with 8 in circular and elliptic orbits for four different sets of inertia parameters without (L*= 0) and with (L*= 0.6) appendage thermal deformation. 35 Figure 3-2 Effect of inertia parameters, orbit eccentricity, and thermal deformation on the equilibrium orientation: (a) slender central body with relatively light appendages. Figure 3-2 Effect of inertia parameters, orbit eccentricity, and thermal deformation on the equilibrium orientation: (b) stubby central body with relatively light appendages. Figure 3-2 Effect of inertia parameters, orbit eccentricity, and thermal deformation on the equilibrium orientation; (c) slender central body with relatively heavy appendages. Figure 3-2 Effect of inertia parameters, orbit eccentricity, and thermal deformation on the equilibrium orientation: (d) stubby central body with relatively heavy appendages. 39 It is apparent that for L*= 0 or 0.6, the maximum value of \\pe\ increases with an increase in eccentricity or a decrease in Kj. On the other hand, an increase in K g has insignificant effect on maximum for L* = 0 and shows only a slight increase for L* = 0.6. Note, as expected, \p& is zero in a circular orbit and in the absence of thermal deformation. However, with the thermal deforamtion of relatively heavy appendages attached to a stubby central body (Fig. 3-2d), the equilibrium configuration can vary between ±1° with a period n which is apparent from Eq. (3.4). Figure 3-3 illustrates typical equilibrium orientations of the deformed satellite at two locations 6= 6V, it-6 in a circular orbit. a* a In elliptic orbits and in absence of thermal deformation, the equilibrium configuration has a period of 2TT, the same as that of the functions e c and C g in Eq. (3.4). The effect of thermal deformation as represented by L*= 0.6 is to superpose an additional small amplitude contribution at a period it. From Fig. 3-2 one may conclude that orbital eccentricity (e) and the central body inertia ratio (Kj) are the dominant parameters governing satellite's equilibrium orientation. The effect of appendage inertia ratio and thermal deformation is relatively small. 40 Figure 3 - 3 Equilibrium orientations at t9 and in circular orbits. 3.3 Stability in the Small Investigation of the motion in the small of the dynamical system is carried out using the linearized equations of motion. Obviously, the results are valid for small magnitudes of initial conditions. Such a study can provide useful information in the preliminary design of the system at a significantly lower computing cost. Linearizing the librational equations for \p and <p degrees of freedom gives; rp"{\ + Kata2b22} - V ' e c{l + Kata2(b\ + b22)} + iP{Kiteg + K a t a 2 [~hb2 + ecbib2 + eg[^b\ - b\) - b\]}} - ^"{^036263} + <i>'{Katcx2{ecb2 - h h ) } + <p{Kalcx2bz\-2b2 + 2ec6i + ^egb2}} - c c + Kalot2{ec\-{b\ + b\) + 6x63a]} 41 rp"{-Kata2b2b3} + rp'{Kalcz2ecb2bz} + tp{KalQ2b3[-€cbi + ^egb2]} + <j>"{l + Kaia2b\} + 4>'{-ee + Katoc2\bib2 - ec(fc? + bl)}} + <(>{! + Kiteg + Katoc2[Zb\ + b\ + tchb2 + ±eg(b2 + 3fcg - bl)]} + (1 - Kit)tr + Kata2{b2ec(bia + 63) + g ^ M a -= 0 ; • • • (3.6) where b j ' s are given by Eq. (2.1) and other coefficients are defined in Appendix I I I (Eqs. 111.1 - 111.4). Here a is referred to as the spin parameter which is determined from the initial conditions. From Eq. (3.2), = constant = ° • • • • (3.7) Equation (3.6) can be rewritten as: [^i(')]q, = lA2(*)]q +M*)]; •••(3.8) where q = (x\>\ <p\ <p)T . For a circular orbit and a=0 (i.e., non-spinning satellite), [A^] = 0 and Eq. (3.8) becomes, q ' ^ l W r V a W l q . •••(3.9) Equation (3.9) represents a set of four linear, homogeneous, nonautonmous differential equations having coefficients of the same periodicity, 27T, in the independent variable, 6. Hence from the Floquet 20 theory , it follows that there exists a basis of four vectors for the equation. Let A(t9)=[A1(t9), . . ., A 4(0)] be a basis for the solution of Eq. (3.9) , then 42 A'(* + 2ir) = £ / } A t y ) (» = 1,. . . ,4), where / ' = [fl, fl, fl, fl] , a n d t n e vectors f are independent. The basis of the solution for Eq. (3.9), 0(0) = [0 1 (0), . . be constructed from the given A ' (0) with the property that G(0 + 2 J T ) = n®(6). 4 Writing G(0) = ^ c , A ' ( f l ) , Eqs. (3.10) and (3.11) give (3.10) . ,0(0)], can (3.11) or 4 4 4 1 = 1 3=1 i=l 4 4 ^ D e ^ - ^ ) A ' ( 0 ) = O . (3.12) Since A (0) are independent, the coefficients in Eq. (3.12) must vanish, giving f\-p ft n n ft ft - M ft f4 J2 ft ft ft-n ft fi ft ft ft For nontrivial solution, the determinant } = ea (3.13) ft ~ A* ft ft ft ft ft ft ft » ft ft 3 / 3 3 - M ft ft ft ft - H ft = 0 Here M-j, . . ., M4, are called the characteristic multipliers of Eq. (3.9). For stable motion. The difficulty in applying the theorem lies in determination of the vectors f1. Since / i j ' s are independent of A(0), a numerical method can be used to determine f1. From Eq. (3.10), A*(2tf) = E / y A y ( ° ) » hence, if A(0) is a unit matrix, then A(27r) would give, A(2TT) = [ / 1 p f* /*]. The procedure for obtaining f1 * s now becomes obvious. Putting q(0)=A(0)=(!, 0, 0, 0) T and numerically integrating Eq. (3.9) over a period of 27T, / 1 = A 1 (2 J r) = q(27r). One can repeat the process for A(0)=(0, 1, 0, 0) T for f , and so on. Once the f 1 ' s are known, j i j 's and hence the stability can be determined. Figure 3-4 shows variation of the stability bounds as affected by the inertia parameters. K. and K . for the system in a circular orbit. Effects of thermal deformation are also indicated. Note, as inertia of the appendages increases, the gravity gradient restoring moment provided by them also increases; hence, the system remains stable although the central body by itself may be unstable for Kj< 0. Furthermore for L*= 0.6, there is an additional unstable region for K. less than 0.25 and K > 0. Figure 3-5 compares the librational response for the thermally deformed and undeformed systems with inertia parameters Kj = -0.1 and K . = 1.0. The results were obtained through numerical integration of the a original nonlinear librational equations of motion. The linearized Floquet analysis data as presented in Fig. 3-4 predict the system to be stable for L*= 0 but unstable for L*= 0.6. The prediction is substantiated by the System Porometer a , = 0.01 Spin Parameter a = 0 Orbital Elements P u i € = 90' = 90" = 0 = 0 1-T L* = 0.0 0 . 5 0 -K: -Unstable Region - 0 . 5 0 1-L* = 0.6 44 0 . 5 0 -K: Unstable Regions - 0 . 5 0 Kr Figure 3-4 Effect of inertia parameters on stability of the system in circular orbits. Note the presence of additional instability region with thermal deformation. 45 'exact ' numerical results which show the \p response for L* = 0.6 to become completely unstable after 55 orbits. Merit of the Floquet theory becomes clear. It predicted unstable motion, from the results of the numerical integration of the linearized equations at one orbit, whereas the numerical integration of the nonlinear equations took fifty orbits to arrive at the same conclusion. The main disadvantage of the Floquet theory is that for a system with n degrees of freedom, n linearized equations have to be obtained first followed by repeated integration over one orbit to determine basis of 2n vectors. Finally, the eigenvalues of a 2n x 2n matrix are required. With a large value of n, the amount of effort involved may become prohibitive. 0° System Porometers = 0.01 = 1.00 =^0.10 u l Spin Porameter a = 0 Orbital Elements = 90' = 90' = 0 = 0 Initial Positions 1>0 = 5' 0. = 5' X = 0 Initial Velocities = o X' = 0.096 46 £ = 0 L* = 0 L*.= 0.6 6 (No. of Orbits) Figure 3-5 Librational response of the system with K g = 1.0, K. = -0.1 showing unstable motion caused by thermal deflection: (a) 0-5 orbits. System Orbital Initial Initial Parameters Elements Positions Velocities a, = 0.01 K 0 = 1.00 P = 90' f0 = 5- r0 = o Kj =-0.10 u = 90" 0. = 5' i = 0 A o = 0 V = 0.096 Spin Parameter c = 0 a = 0 47 10 6 (No. of Orbits) Figure 3-5 Librational response of the system with K a = 1.0, Kj = -0.1 showing unstable motion caused by thermal deflection: (b) 45-50 orbits. System Parameters a, = 0.01 K a = 1.00 K, =-0.10 Spin Porameter a = 0 Orbital Elements = 90' = 90* = 0 = io. Initial Positions = 5* = 5' X 0 = 0 Initial Velocities = o #; = o X' = 0.096 48 *° 0 -5--10 10 '<% (K\ iT\ ft V j w \y w1 1 1 >W w w e = 0 L* = 0 L* = 0.6 55 Figure 3-5 57 58 59 60 6 (No. of Orbits) Librational response of the system with K = 1.0, K.= -0.1 showing unstable motion caused by thermal deflection: (c) 55-60 orbits. 49 3.4 Motion in the Large In order to investigate large amplitude motion of the system, the nonlinear equations of motion as presented in Appendix I I I must be used. Although these equations do not possess any exact known closed-form solution, they can be solved in an approximate manner. Two methods, an approximate variation of parameters and an 'exact ' numerical, are discussed in this section. 3.4.1 Variation of parameters method 21 The method, which was suggested by Butenin , is intended for differential equations with small nonlinearities. Application of the procedure to the governing equations of motion (Eqs. 111.5 - 111.7) required careful expansion of each term and truncating the series at an appropriate order depending upon the relative magnitude of the term. For example, ct^ being quite small, only the first order terms were retained. Similarly, for terms independent of a^, fourth and higher order terms were retained. For e c and e third and higher order terms in e were neglected, i.e., ec = 2esin0(1 — ecos0); eg = 3(1 - ecosfl + r ' cos 2 ^). With this simplication, the equations of motion for \p and <j> degrees of freedom become: rp" - 2esin 9(1 - ecos0)^' + ZKit(l - ecos 8 + e2 cos2 8)4> = Fx (xp, <f>, <f>", a) + 2e sin 8(1 - e cos 9); c6" - 2esin0(1 - ecos9)<f>' + (1 + 3Kit)(l -ecos9 + <?cos2 9}<f> = F2(rP,<p,iP',<t>',iP",<p",CT); --.(3.14) 50 where: F i = 2#'(1 + if,') + ZKitxp{4>2 + -rp2)(\ - ecosfl + e2cos26) + Kata2{a4>{l + ^<f>2) + 2a2[(6? - b\)xp - b,b2 + b2b3d>] + b\i>{l + 2tf>') + b22[tp" - 2<p<p'{l + *P') - <f>2xp" - a<f>' - t'rp - xp - 2xpxp' - 2e[a<p + tp') sin 6] + b2\-<p2xp" - 2#'(1 + rp') + 2<p4>' - c<f>' - 2o-c*£sin0] + bib2[a2 + 2a<p + 2o<pip' + 2<p2\p'\ + 26 2 M ' es in0} ; F2 = -<PW + xP'2 - \<p2) + 3Kit<p(xp2 + ^ 2 ) ( 1 -ecosB + e2cos2 0) ( 62 + Kata2{-a(l + rp'-Z-) + 2 o- 2[(6 2 - b\)<p + 63(&! + b2xp)\ + b2<P(l + 2rP') + b22[-a - rP'{(T + + bl\<p" -a- rp'{<j + M)] + 2&162(<r^ + rpxp'<fi + WeslnO) + bib3[a2 - 3<p2(l + rp') + a<f>xp' - 4 ( # ' + VV-')"in 6} + 26263^'esin0} . •••(3.15) Ignoring and ?2 t o obtain the generating solution, Eq. (3.14) can be rewritten as: xP" + PlW + P246)rp = 0 l <p" + PiW + P24>W = 0 ; . . . (3.16) 22 in the Mathieu form and can be solved using the standard approach . Let: <P = x}ea ; 4> = ; • • • (3.17) where a is the integration factor defined as i r° a = 2 J0 m ) d ( = e ( l - c o s 0 ) - y ( l - c o s 2 0 ) . ••(3.18) 51 Substituting from Eq. (3.17), (3.16) takes the form: j" + ~y&- = 2 e s i n " ( 1 " € C 0 8 9 ) 5 ^ " + ^ - ^ - T ^ = 0 - " • ( J U 9 ) Simplifying, i" + nJV" = (! - ecos0)[ecos0(3ff,t - 4- 2€sin0]; ci" + nl4> = (1 - ecos0)[ecos0(3iir,( - 1)$ ; • • • (3.20) where; nj = 3K,-( ; 4 = 1 + 3iT,t . • • • (3.21) The solution for Eq. (3.20) can now be written as: J.-A Lvn(n P i /? 1 i (l-3g t- t)ersin[(n^ + l )g+^] sin[(n^ - 1)0 + fa rP - A,{sm(n,9 + + - [ — _ j (1 -ZKu^e2 r-sin[(nv> + 2)0 + ^] sin[(nv, - 2)0 +/^h j 8 L (2r^ + l)( n v , + 1) (2n^ - l)(n^ - 1) J J 1 2 L 2n^ +1 2n^ — 1 J (l-3ii: t t)n < / ,€ 2r-sm[(n^ + 2)0 4-^] sin[(n^ - 2)0 + 8 L (2n^ + l)( n < k + l) (2n^-l) (n^-l ) JJ ' l ^ where; ev>i = ~ 2 — r ; ^ 2 = ^ -The method of variation of parameters as described by Butenin was now applied. In order to reduce the amount of work, only the predominant terms in Eq. (3.22) were considered: 52 ij> = Ay sm(n^O + 0$); 4> = sin(n^0 + 0+). ••• (3.23) The ob jec t ive w a s to obtain a so lut ion essent ia l ly harmonic but with the ampl i tude and phase angle funct ions of 6: ^ = ^ (0)sin(fty0 + jfy(0)); i = A+{9) sin( V + P^B)). • • • (3.24) There are four unknowns , A ^ , A ^ , 0^, and 0^ to be de te rmined . T h e y can be eva luated by introducing logical constra in ts and using the equat ions of m o t i o n . For example , o b v i o u s constra in ts w o u l d be for the f irst der ivat ive of E q . (3.24) to be the s a m e as that of E q . (3.23). D i f ferent ia t ing E q . (3.24) g i v e s ; 4>' = A'f sin(»ty0 + Pf) + A^n^ + /fy) cos{n^9 + P+); 4>' = A\ s in(n^ + P+) + A ^ + /fy) cos(n^ + P+); • • • (3.25) hence , the log ica l constra ints w o u l d b e c o m e : A'q sin t)^ + A^Plj, cos j ty = 0 ; ji^sinify + j4^/fycostfy = 0 ; • • (3.26) where : Vtl> = ^ 9 + p^,; V<t> = n<t>0 + P<t> • Di f ferent ia t ing E q . (3.25) and subst i tut ing in E q . (3.14) leads to ; A'^nj, cos r]^ — A^n^Plp sin rj^, = F{ ; cos »fy - A^n^fy sin 17^  = ^2*; • • • (3.27) 53 w h e r e : F;=Fl{Ate*anti*t...,ff)i F2* = F 2 (^e a s in»7^,. . . ,a) . •••(3.28) Solving the four algebraic equations in (3.26) and (3.27) gives: AL = — cosrty ; FX A'x = — cos r?^; ^ n<t. of Ft* Assuming A ^ , A ^ , /3^, and p1^ to be slowly varying parameters, their averages over a period are used: j /»2JT i»27r /«2jr j /»2JT *2x plx "*=Si" C C * s i n i* dr>* d"*d9- - '3'29' Denoting Em,f(0)= I e m a / ( £ ) d i ; , Eq. (3.29) becomes: ^0 ^ = 0 ; 4> = 0 ; ^ = i l r^l^ 3' 1^ + K^2{2al{2a2Ehcos20 - a 2 A 2 E 2 , c o s 2 9 + (n2 + 5)El^a2e - 2^i,co.»«l + 2ci[-2a 2r? 1 ) C O s 2 (9 + a 2 A 2 E 2 t C o a 2 0 + ( n j + 5)^1)C0S,„ - 2 £ W * 1 + 2a1a2[2a24^2)SinW + (r.2 + 7 - 4a 2)£ 1 ) S b 2,]}} ; .sin 20 54 + °>lEi^e) + 2a ^ M ( 4 + n\- 2a2)}} . ... (3.30) Using the initial conditions and Eqs. (3.17) and (3.22) gives; , < (1 - 3Jfc)e I", 3 n l £ _ H • « & = + \ n ^ _ l [i - 4 ( n ^ _ 1 } j} cos^o + HI - ; A ^ , A ^ , / 3 ^ Q , and / ? ^ Q can now be determined and the analytical solution becomes: _ s in[ (n ^ + ^ - l )g + ^ 0 ] i (l-3/^t<)n^e2 p sm[(>V + + 2)0 + 2n^ -1 J 8 L [2n^ + l)(n* +1) E i n [ ( n 0+^-2 ) g + ^ o ] n 1 + (2n, - l)(n, - 1) JI + ^  8 m g ~ y » ^ = | ^ | s m [ ( ^ + /3^ )0 + ^ ] + - [ 2n^ + l 2 ^ - 1 J 8 I (2n^ + l)(n^  +1) sm[(n^  + ^ - 2 ) 0 4 - M ]))• + -(3.3.) The corresponding analytical solution for X can be obtained by integrating Eq. (3.7) with ^ ' and <p substituted using Eq. (3.31). 55 3.4.2 Numerical method Consider a set of first order differential equations, y' = f(v,0) with the numerical solution at 8 =nr denoted by y where r is the step size used in the numerical integration. Basically, there are two methods to obtain y n: one-value and multi-value procedures. The former uses only y n ^ while the latter uses k previous values, y„ „,...,y„ to determine y„. For n—T * n-k 'n complicated differential equations, the multi-value method gives more accurate and reliable results than the one-value approach. A multi-value method consists of two processes; prediction and correction. In the prediction process, the approximation to y n denoted by y n Q is obtained by a linear interpolation method. The approximated value is then refined by the corrector formula. If the y - value satisfies the error test with a user specified tolerance, then y =y «. Otherwise the correction n n,i ' process is repeated m times until y _ satisfies the error test and hence n,m Vi—^n,nr The order of a multi-value method refers to the complexity of the correction formula. In general, higher the order, more accurate the solution at the expense of a higher computing cost. Since the system's equations of motion are rather complex, the multi-value method was preferred. Among the numerous subroutines based on the multi-value method, the IMSL ; DGEAR subroutine was chosen. It is 23 adapted from Gear's subroutine DIFSUB and has the following advantages. (i) Accuracy. The subroutine is based on the implicit Adam's multi-value method with a variable order of up to twelve. This high order ensures the numerical solution to be accurate. (ii) Ease of Programming. In the correction process, the Jacobian of 56 the differential equations is required; hence, it should be coded in by the user. In DGEAR subroutine, there is an option of evaluating the Jacobian numerically. (iii) Economy. Based on the result of the error test at each integration step, the subroutine automatically changes the order of the correction formula or the step size, if necessary. This feature makes the subroutine efficient and economical in terms of computing cost. (iv) Additional feature. By changing the parameter METH of the subroutine, it can be used to solve stiff equations based on Gear's m e t h o d 2 3 . In order to obtain a numerical solution, the equations of motion were first written as a set of first order differential equations. For example, the equations of motion for \p and <j> degrees of freedom obtained earlier can be written in matrix form as where f^ and f^ are functions of \p, <p, \p\ 0 ' , a, and Mj j 's . M - ' s , generally not constants, are defined in Appendix IV. The equation of motion for X degree of freedom was (Eq. 3.7), where a was given by the initial conditions, <* = (w*)o = Ao - (1 + 1>'0)un <p0. Rewriting Eq. (3.32) as a set of four first order differential equations together with Eq. (3.33) gives a set of five first order differential equations, • • • (3 .32 ) A' = o- + (l + V')sin^> • • • (3 .33) 57 q' = (Mi]"*; ( 3 . 3 4 ) where: [M1) = Mn Mn 0 0 0 M2i M 2 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 q = (V', <f>\ rP, <p, A) T; f = ( / v » /*> 4>', <T + {l + i>')sin<f>)T . Equation (3.34) was solved by the IMSL : DGEAR subroutine with the tolerance set at 10" . The matrix [M^] was inverted numerically at each iteration step. 3.4.3 Discussion of results Validity of the analytical solution was assessed over a range of inertia parameters and initial conditions. For conciseness, only a set of few representative results useful in establishing trends are presented here. Figure 3-6 presents the librational response for the satellite in circular orbits. It is significant to note that the analytical procedure is able to predict amplitude as well as frequency with an acceptable degree of accuracy over a wide range of central body and appendage configurations. However, the accuracy drops off for small Kj (stubby central body) or large K g (relatively heavy appendages). The error in the phase is cumulative and increases with 6. Fortunately, phase does not constitute an important parameter in the satellite dynamics analysis. In general, error in the amplitude prediction for X response was found to fluctuate randomly. The prediction may be good at one instant but poor at the other. This can be explained by the equation used to evaluate the X response, legend Numericol Analytical 6 (No. of Orbits) Figure 3-6 Comparison of numerical and analytical solutions for different inertia parameters: (a) slender central body with relatively light appendages (0-5 Orbits). 5 9 10 11 legend Nurnencd a iK ln of OrbVts) different al and analytical solution. . ^ . ^ U g n t Figure 3 - 6 . n 0 f numerical - - • , b o d y with Comparison of nu s l e n d e r • centra inertia P " 8 * * 8 , ^ Orbits), appendages 0 ° Syslem Parameters a, = 0.01 K G = 0.25 Kj = 0.25 L* = 0.60 Spin Porometer a = 0 Orbital Clements = 90-= 90* = 0 = o. Initial Positions f. = 30-<t>. = 0 X. = 0 Initial Velocities V; = 0.50 <p'o = 0.5O K = 0 60 legend Numericol Analytical 2 3 4 5 e (No. of Orbits) Figure 3-6 Comparison of numerical and analytical solutions for different inertia parameters: (b) stubby central body with relatively light appendages (0-5 Orbits). System Porometers a, = 0.01 Ka =0.25 Kj = 0.25 L* = 0.60 Spin Parameter a =0 Orbital Elements = 90' = 90* = 0 = 0 Initial Positions f. = 30* = 0 X = 0 Initial Velocities = o.so =0.50 x: = o 61 50-- 2 5 -- 5 0 20 A U. _ \1 — legend Numerical Analytical 12 13 14 6 (No. of Orbits) Figure 3-6 Comparison of numerical and analytical solutions for different inertia parameters: (b) stubby central body with relatively light appendages (10-15 Orbits). 62 Figure 3 - 6 Comparison of numerical and analytical solutions for different inertia parameters: (d) stubby central body with relatively heavy appendages 64 A = / [(1 + ^'(0) sm HO + <r]dt. . . . (3.35) J o The error between the numerical and analytical solutions for \p' and (p at any instant is accumulated in the integration process. If the error at one instant cancels that at the other, the resulting discrepancy between the numerical and analytical solutions is insignificant. On the other hand, the discrepancy becomes significant if the errors at different instants do not cancel. Figure 3-7 shows that essentially the similar trend continues even in the presence of spin. In Fig. 3-8, it can be seen that the analytical solution fails to predict the in-plane motion caused by the out-of-plane excitation. The failure is not unexpected because the analytical solution for \p depends on \//Q and I / / Q ' but not C6Q and 0 N ' . .Figure 3-9 presents the response in non-circular orbits. Note, the analytical solution is able to predict amplitude modulation in \p response quite well, although the magnitude of amplitude and phase continue to show some discrepancy, particularly at higher e. It should be pointed out that validity of the c losed-form solution has been assessed here under most demanding conditions. In practice, scientific and application satellites have their librational motion controlled over a specified limit (say 0.1° for communications satellite; 2 ° for weather satellite, etc.) ranging over 0.01- 4 ° . Thus in practical application, the analytical solution is expected to predict the librations with an acceptable degree of accuracy at least in the preliminary design stage. System Poramelers a, = 0.01 K c =0.25 K| = 0.75 L* = 0.60 Spin Porometer a = 0.02 Orbital Elements = 90' = 90' = 0 = 0 Initial Velocities V; = 0.50 = o.so X' = 0.02 65 Figure 3-7 legend Numerical Analytical 1 2 3 4 5 6 (No. of Orbits) Comparison of numerical and analytical solutions for non-zero spin parameter. S y s t e m O r b i t a l I n i t i a l I n i t i a l P a r a m e t e r s E l e m e n t s P o s i t i o n s V e l o c i t i e s a , = 0.01 K c = 0.25 P = 90* *. = o t; = o K, = 0.75 u = 90' <t>. = o <p-c = 0.50 L* = 0.60 i = 0 X = 0 0 x; = o £ = 0 S p i n P a r a m e t e r o =0. 66 4 legend Numerical Analytical 9 (No. of Orbits) Figure 3-8 Comparison of numerical and analytical solutions for a disturbance across the orbital plane. ft 45 30 15 0 -15 -30H -45 30 10 <t>° 0 -30 60 Sys tem P o r o m e t e r s o , = 0.01 K c = 0.25 Kj = 0.75 L* = 0.60 Sp in P o r o m e t e r a = 0 Orbi to l E lements = 90* = 90" = 0 = 0.1 Initiol P o s i t i o n s f0 = 0 = o A . = 0 Initial Ve loc i t ies t: = 0 . 5 <t>: = 0 . 5 x: =o 1 2 3 4 d (No. of Orbits) 67 "A A A A j\/h in A - v v y \ i i i i AAAA/^  s A A A A W W W \ - V V V V W \ / W yj V I 1 I I legend Numerical Analytical Figure 3 - 9 Comparison of numerical and analytical solutions for eccentric orbits: (a) e = 0.1. 68 f 0 45 30 15 0 -15 -30--45 30 System Porometers a, = 0.01 K„ =0.25 Ki = 0.75 L* = 0.60 Spin Porgmeler a = 0 Orbital Elements = 90' = 90* = 0 = 0.2 Initial Positions V-0 = o 0. = o X. = 0 Initial Velocities V; = 0.50 0; = 0.50 X" = 0 :A s\ f\ A A\ • v y} — !• — 1 \J v V 1 1 legend Numerical  Analytical i 2 3 e (No. of Orbits) Figure 3-9 Comparison of numerical and analytical solutions for eccentric orbits: (b) e = 0.2. 4. COUPLED LIBRATION/VIBRATION DYNAMICS OF SATELLITES WITH THERMALLY FLEXED APPENDAGES 4.1 Preliminary Remarks This chapter builds on the earlier analysis and considers dynamics of the nonlinear system accounting for thermal deformation and appendage vibration. Hence, all the seven equations of motion, presented in detail in Appendix I I I (Eqs. 111.5 - 111.11), are used in the analysis. Orbital constraint is kept the same as before, i.e.,. p = o> and i= 0. The chapter begins with a discussion on stable equilibrium orientation of the central rigid body and the flexible appendages. This is followed by an 'exact ' numerical analysis of the nonlinear system. The beat phenomenon associated with the appendage vibration and the effect of eccentricity on system response are demonstrated. The attention is also directed towards the condition of instability in the presence of thermal deformation and vibration of the appendages. Next, an approximate closed-form analysis of the nonlinear system is attempted using the variation of parameters method and its limitations discussed. Finally, the method is modified, an improved analytical solution for circular orbits obtained, and its accuracy assessed by comparison with the 'exact ' numerical solution. 4.2 Equilibrium Orientation The equilibrium state of the system is determined from the governing nonlinear equations by setting the generalized velocities and accelerations equal to zero: 69 70 92(g) 03(g) 9i(q) 05(g) 0e(g) 97 ( g ) d 3T 9T dU] _ Q \ — ( — \ - — — ] - 0-1 -^^50 5c6 + d<t>lq'=q"=0 ~ ' U^ar " a A + dx\g'=o»=o~ ; ide^dP/ dP[ + dPl J q'=q"=0 ~ ' ar ar arj-i •dd^dQi dQi + dQiiq'=q"=o~ 5 r d , ^ T . _ oT_ dUj idB^apJ dPn + dPuJ,'=g»=o ~ 5 \d_{dT_, _ dT_ dU VdB^dQj dQn +dQulq'=q"=o = 0 (4.1) where <<1 = ip,<t>,^,Ej,Qi,E.u>Qu The stable solution to the above set of equations is obtained by minimizing J, where «=l with constraints: -ir/2 < ipe < ir/2 ; -TT/2 <4>e< K/2 ; -TT/2 < AE < JT/2 ;• - K P / e < l ; - K P „ e < l ; -1<S„<1- (4.2) Equation (4.2) is solved by the NLPQLO subroutine over a range of inertia parameters (K g, Kj), orbital eccentricity (e), and appendage frequency ratio (cc>r= u>^/ 6p\ co^  = fundamental frequency, 6p= orbital rate at perigee) with thermal effects. The results are summarized in Figs. 4-1 to 4-3. Since the orbit is taken to be in the ecliptic plane, it is apparent that 71 the central body rotation is confined to the pitch with the appendages deflecting only in the orbital plane, i.e., X = </> = Q. = Q = 0. Effect of the inertia parameters on the central body equilibrium orientation is presented in Fig. 4-1. For small K g and large Kj (i.e., small appendages attached to a slender central body), irrespective of the orbital eccentricity, effect of appendage flexibility on the central body equilibrium orientation is virtually negligible (Fig. 4-1a). On the other hand, the flexibility effect becomes significant as the central body becomes progressively stubby and the appendages relatively heavy (Figs. 4-1b to 4-1d). Figure 4-2 shows variation of the equilibrium tip deflection for flexible appendages with the satellite in circular or eccentric orbits. Note, at any instant, the thermal deflection is larger than the flexible deformation. As the flexibility diminishes, difference in the contributions is expected to decrease, vanishing in the limit for rigid appendages. Figure 4-3 studies effect of the appendage flexibility, as reflected by its natural frequency, on the spacecraft's equilibrium configuration. Position of the spacecraft in its orbit is taken to be at 6= 9 0 ° . For the values of inertia parameters considered here, remains fairly constant with the variation of a>r as expected because the libration frequency is held constant (Eq. (3.21)). In contrast, P | g and P_ue diverges as a>r approaches 1, i.e., as the orbital rate approaches the appendage frequency. 7 2 System Orbital Equilibrium Parameters Elements Orientations a, = 0.01 p = 90' 0. = 0 K Q = 0.25 u = 90° X . = 0 K, = 0.75 i = 0 Q l e = 0 L* = 0.60 Qu. = 0 0.4 0 . 2 -0 Figure 4-1 e = 0.2 cjr = 2.0 CJ, =20.0 0.25 0.50 0.75 0/TT l r 1.25 1.50 1.75 Effect of system parameters on the equilibrium orientation: (a) slender central body with small appendages. 73 System Orbital Equilibrium Parameters Elements Orientations a, = 0.01 p = 90* <P. = o K a = 1.00 u = 90° X , = 0 Kj = 0.75 i = 0 Qle = 0 L* = 0.60 flu. = 0 1 20 Figure 4-1 Effect of system parameters on the equilibrium orientation: (b) slender central body with large appendages. 74 System Orbital Equilibrium Parameters Elements Orientations a, = 0.01 p = 90° 0, = 0 K a = 0.25 a = 90° X . = 0 Kj = 0.25 i = 0 Q,e = 0 L* = 0.60 Que = 0 0/TT Figure 4-1 Effect of system parameters on the equilibrium orientation: (c) stubby central body with small appendages. 75 System Orbital Equilibrium Parameters Elements Orientations a, - 0.01 p = 90° 0 . = 0 K Q = 1.00 u = 90' X , = 0 K, = 0.25 i = 0 Qle = 0 L* = 0.60 Que = 0 2 0/TT Figure 4-1 Effect of system parameters on the equilibrium orientation: (d) stubby central body with large appendages. c o • s— (1) Q Equilibrium Orientations 76 £ = 0 TOtal Thermal Flexible £ = 0 Total Thermal Flexible € = 0.1 Totol Thermal Flexible £ = 0 . 1 Totcl Thermal Flexible Figure 4-2 Variation of tip deflection at equilibrium with 8 showing the dominance of thermal effect. 77 Figure 4-3 E f f ec t of appendage f lexbi l i ty on its equi l ibr ium p o s i t i o n . Note the d ivergence of Pj and P_ u e at co r = 1. 78 4.3 Numerical Analysis of the Nonlinear Equations With the inclusion of appendage vibration, complexity of the problem increases markedly, and the libration response may be expected to be different compared to that obtained earlier. This section studies the system response in detail. The equations were integrated using the numerical procedure described earlier (Section 3.4.2). 4.3.1 Appendage disturbance For better appreciation of the effect of appendage flexibility, to start with, thermal deformation is purposely neglected (L*= 0). Figure 4-4a shows response of the system with an initial disturbance applied in the orbital plane to one of the appendages. The disturbance corresponds to lower appendage deformed initially in its fundamental mode with the tip deflection equal to 10% of its length. Several interesting features become apparent: (i) in-plane disturbance of the appendage excites only the in-plane motion both in vibrational and librational modes. (ii) the pitch motion has a high frequency contribution from the appendage motion superposed on it. (iii) the appendages exhibit beat resonance, due to two closely spaced eigenvalues, np and n p 1 (Appendix V , Eq. V.2). Note, one of the eigenvalues is identically c j r , i.e., the appendage frequency. This suggests that the two appendages, though structurally identical, have a slightly different eigenvalues due to a minor difference in the gravitational field and coupling with the libration motion. Response of one appendage thus acts as a forcing function for the other through 79 a librational coupling closely corresponding to the third eigenvalue, If the same initial disturbance is applied across the orbital plane, both in-plane as well as out-of-plane motions are excited as shown in Fig. 4-4b. A s can be expected, the out-of -plane roll libration is the dominant motion with a moderate yaw and a negligible pitch. As before, the appendages exhibit a strong beat response in the out-of-plane motion and, through a weak coupling, in the plane of the orbit as well. The system response with an initial impulsive disturbance applied to the lower appendage in the orbital plane is shown in Fig. 4-5. The disturbance represents an initial velocity distribution corresponding to the fundamental mode with tip velocity equal to twice the product of appendage length and orbital rate, i.e., 21^0. Compared to Fig. 4-4a, the amplitude of vibration remains virtually unchanged. However, the libration amplitude has increased significantly (±0 .6° in Fig. 4-4a and ± 5 ° in Fig. 4-5). The fact that the beat response of the appendages is indeed through a small difference in their frequencies is substantiated by a comparison of response data for a satellite with one and two appendages as presented in Fig. 4-6. The appendage is subjected to an in-plane (Fig. 4-6a) or an out-of-plane (Fig. 4-6b) disturbance in the fundamental mode as before. Note the absence of beat response with a single appendage. A closer look at the approximate c losed-form solution as given in Section 4.4 (Eq. 4.10) also suggested conditions when even two appendage configuration will have no beat response. With initial conditions applied to both the appendages through symmetric or asymmetric deflections as before, the solution shows terms corresponding to beat envelope frequency 80 Figure 4-4 System response with: (a) in-plane appendage disturbance. 81 System Porometers ° l L* 0.01 0.25 0.75 0 20.00 Orbital Elements p = 90° u - 90° i = 0 i - 0 Initial Positions % = to = *o = Elo = 9,o = 0.05 P..- = -uoQuo — 0 0 0 0 0 0 Initial Velocities f'0 = 0 <t>'0 = 0 K = ° Elo = o Qio = 0 Puo = o QUo = o 0.01 1>°: o 1 2 0 (No. of Orbits) Figure 4-4 System response with: (b) out-of-plane appendage disturbance. System Parameters a, = *« = K, = L* = ur = 0.01 0.25 0.75 0 20.00 Orbital Elements p = 90" u = 90° i = 0 c = 0 Initial Positions 0 0 0 0 *o = <t>o = *o = P|o = Q , o = 0.05 'Io Euo = Quo Initial Velocities % = o <K = o K =o Qio = o P-uo =o QUo = o 82 0.05 ro I o X^, o.oo-QJ -0.05-100-50-1 1 O T— 0-X V —* O l - 5 0 -^ w v v ^ n -100 0.05 to I O 3 Q J -0.05 100 0.00 - ^ 0 ^ ^ ^ ^ - - V Y ^ ^ 1 50-1 o o-X 3 O l - 5 0 --100 1 2 9 (No. of Orbits) i 3 Figure 4-4 System response with: (b) out-of-plane appendage disturbance. 83 Figure 4-5 Typical response of the system for an impulsive appendage disturbance. 0.25 f ° 0.00 -0.25 -0.50 System Parameters K , L * = 0.01 = 0.10 = 0.75 = 0 = 20.00 Orbital Elements p = 90" u = 90° i = 0 E = 0 Initial Posit ions *o = <f>o = *o = E i . = Q.o = Puo = Quo = 05 Initial Velocit ies K = p;0 = Q\0 = Puo = Quo = 84 Two Appendages \k , J l i it W w 1 w W % 1 2 0 (No. of Orbits) Figure 4-6 A comparison of response for systems with one and two flexible appendages: (a) in-plane appendage disturbance. 100 System Parameters a, = K a = K| = L* = 0.01 0.10 0.75 0 20.00 Orbital Elements p = 90° u = 90° i = 0 t = 0 Initial Positions % = 0o = *o = P .o = Q i o = P-uo = Quo — 0 0 0 0 0.05 0 0 Initial Velocities 85 0; = *o = Pio = Qio = P-uo = Quo — 1 2 6 (No. of Orbits) Figure 4-6 A comparison of response for systems with one and two flexible appendages: (b) out-of-plane appendage disturbance. 86 to vanish. The numerically obtained response results as given in Fig. 4-7 substantiated this conclusion. As expected, symmetric appendage disturbances (E|0= - P - U 0' —10 = —u0^  r e s u l t e d i n virtually no librational motion (Figs. 4-7a and 4-7b). However, asymmetric disturbances (P|Q= Eur> Q|Q= - Q . U U ) amplified the libration response approximately by a factor of two (Figs. 4-7c and 4-7d). Even with the inclusion of thermal effects, which would change equilibrium configuration of the appendage in the orbit, the beat phenomenon continues to persist (Fig. 4-8). The most satisfying aspect of the analysis was the ability of the approximate analytical solution, discussed later in Section 4.4, to provide valuable insight into the dynamical behaviour of such a complex system. 87 Figure 4-7 System response showing the effect of symmetric and asymmetric appendage disturbances: (a) symmetric in-plane disturbance. 88 System Parameters a, = K a = K, = L* = co, = 0.01 0.25 0.75 0 20.00 Orbital Elements p - 90° u = 90° i = 0 e = 0 Initial Positions Y'o = to = * o = P . o = Q i o = Euo = Quo 0 0 0 o o. o 0.05 .05 initial Velocities = Vo = K = E i o = Q i o = Euo = Quo 0.01 Figure 4-7 System response showing the effect of symmetric and asymmetric appendage disturbances: (b) symmetric out-of-plane disturbance. 89 Figure 4-7 System response showing the effect of symmetric and asymmetric , appendage disturbances: (c) asymmetric in-plane disturbance. 90 Figure 4-7 System response showing the effect of symmetric and asymmetric appendage disturbances: (d) asymmetric out-of-plane disturbance. 9 1 Figure 4-8 System response showing the effect of appendage disturbance in the presence of thermal deformation. 92 4.3.2 Central body disturbance Consider the case with an initial disturbance applied to the central body. To begin with, let the thermal deflection of appendages be ignored as before. For an in-plane disturbance, Figure 4-9a shows that the central body is librating at frequency n ^ while the appendages are vibrating at frequency a) r with the librational frequency superposed on it. As explained before, here n ^ and a>r are eigenvalues obtained through an approximate analytical procedure explained in Section 4.4. Note, the absence of beat response is consistent with the appendage initial disturbance criterion mentioned earlier. Of particular interest is the fact that the central body's libration in pitch with amplitude as large as 5° results in virtually imperceptible appendage vibration. Even with an impulsive disturbance which gives the same libration amplitude as before, the appendage vibration is hardly excited (Fig. 4-9b). This is in sharp contrast to the appendage disturbance induced response studied in the previous section, which resulted in a significant librational motion of the central body. Figure 4-10a shows the system response for an initial out-of-plane central body disturbance. In contrast to Figure 4-4b, the in-plane vibration and libration are strongly excited by the out-of-plane central body disturbance. This is because the out-of-plane libration is strongly coupled with the in-plane libration, which in turn is coupled to the in-plane vibration. The figure also shows the transfer of vibration energy from the out-of-plane mode to the in-plane mode and vice versa. This energy transfer phenomenon is more pronounced in Fig. 4-10b where the central body is subjected to a combined in-plane and out-of-plane disturbances. Figure 4-11 shows the system response when thermal deflection of appendages is accounted for. This case corresponds to results given in 93 Figure 4 -9 System response showing the effect of in-plane central body disturbance: (a) displacement disturbance. 94 System Orbital Initial Initial Parameters E lements Posi t ions f o = 0 Veloci t ies a , = 0.01 p = 90° <P0 = o K = o K 0 = 0.25 a = 90" x 0 = 0 K =° K, = 0.75 i = 0 P . o = 0 Eio = o L ' = 0 £ = 0 Q.o = o Qio = o w r = 20.00 E u o = 0 Q u o = o o o II II o o -3-3 0-1 Ol -10 6 (No. of Orbits) Figure 4-9 System response showing the effect of in-plane central body disturbance: (b) impulsive disturbance. 95 Figure 4-10 System response showing the effect of central body disturbance: (a) out-of-plane disturbance. 0.25-1 -0.25 0.25 0.00 -0.25 System Porameters K CJ. = 0.01 = 0.25 = 0.75 = 0 = 20.00 Orbital Elements p = 90° u = 90* i = 0 e = 0 Initial Positions % = K = *o = E l o = Q i o = Euo = Quo 0 5« 0 0 0 0 0 Initial Velocities % = 0 K =o Pio = 0 9io = 0 PUo = 0 Quo = 0 96 1 2 0 (No. of Orbits) Figure 4-10 System response showing the effect of central body disturbance: (a) out-of-plane disturbance. 97 Figure 4-10 System response showing the effect of central body disturbance; (b) combined in-plane and out-of-plane disturbances. System Parameters 98 -0.25 1 2 3 4 6 (No. of Orbits) Figure 4 -10 System response showing the effect of central body disturbance: (b) combined in-plane and out-of-plane disturbances. 99 Fig. 4-10b with the thermal effect neglected. There is no noticeable difference in the response except for a slight increase in vibrational amplitude. However, it is significant that even in the presence of thermal deformation, librational disturbance continues to have very little influence on appendage vibration. Figure 4-11 System response showing the effect of central body disturbance in the presence of thermal deformation. System Parameters L* = 0.01 = 0.10 = 0.75 = 0.60 = 20.00 Orbital Elements p = 90° u - 90" i = 0 € = 0 Initial Posit ions to K Q.o Euo Quo 5« 5« 0 0 0.05 0 0.05 Initial Velocit ies % = K = K = Bio = Q!o = Euo = Quo = 101 s—V 0 . 5 -fO 1 1 o 0 J X cu - 0 . 5 -- 1 -I 0 . 5 -o 0 -X O l - 0 . 5 -- 1 -1-0 . 5 -1 1 O 0 -X 3 QJ - 0 . 5 -- 1 -1-fO 1 0 . 5 -i o 0 -v / 3 O l - 0 . 5 -- 1 -1 2 3 6 (No. of Orbits) Figure 4-11 System response showing the effect of central body disturbance in the presence of thermal deformation. 102 4.3.3 Influence of thermal deformation on system stability (circular orbits) The results obtained in Sections 4.3.1 and 4.3.2 show that thermal effect has minor influence on libration and vibration amplitudes. This conclusion is further substantiated in Fig. 4-12. The figure compares responses between the systems with (L*= 0.6) and without (L*= 0) thermal deformations for three sets of inertia parameters. The initial conditions in each case corresponds to combined central body and appendage disturbances. For a slender central body with small appendages, thermal deformation has negligible influence on response amplitude or phase (Fig. 4-12a). However, as the central body becomes stubby (Kj = 0.25, Fig. 4-12b) or the appendages become large (K a = 0.75, Fig. 4-12c), thermal deformation results in larger libration and vibration amplitudes. Although not all the results are shown here, this observation was found to be valid irrespective of initial conditions. With the results of Fig. 4-12 as background, at critical initial conditions, thermally flexed appendages can be expected to destabilze the system as shown in Fig. 4-13. The inertia parameters represent a slender central body (Kj = 0.75) with large appendages (K a = 0.75). The initial conditions correspond to severe central body and appendage disturbances. The figure shows that without thermal deformation, the system remains stable with pitch amplitude (i//) of about 8 0 ° . However, with the inclusion of thermal deformation, the system starts to tumble as \p> 9 0 ° . Note also that both the phase and amplitude of yaw and roll librations are different for L*= 0 and L*= 0.6. Figure 4-12 Typical response in circular orbits showing the effect of inertia parameters and thermal deformation: (a) slender central body with small appendages. 80-40-0--40--80-200-f—v 1 100-i o 0-X DJ -100-I o X CL - -100 I O 3 K> I o 3 0-1 Figure 4-12 Initial Velocities % = 0.50 1 2 3 6 (No. of Orbits) Typical response in circular orbits showing the effect parameters and thermal deformation: (b) stubby central body with small appendages. of inertia 60-30-0--30--60-200-s—s | 100-o 0-X QJ -100--200-200-, — s •o 100-1 O 0-X -100-Q_l -200-200-| 100-o 0-X 3 -100-Q_l -200-200-s—>> IO 1 100-O 0-X 3 -100-QJ -200 Figure 4-12 System Parameters Orbital Elements Initial Positions fo =30-0o = 0 Initial Velocities % =0.50 *' 0 =0 105 1 2 6 (No. of Orbits) Typical response in circular orbits showing the effect of inertia parameters and thermal deformation; (c) slender central body with large appendages. 106 Figure 4-13 System response in circular orbits showing the destabilizing influence of thermally deformed appendages: (a) librational response. 200 - 1 System Porameters a, = 0.01 K a = 0.75 K, = 0.75 co, = 20.00 Orbital Elements p = 90° co = 90° i = 0 t = 0 Initial Positions <Po = *o = P.o = Q.o = P-uo = 9 U 0 — = 30° = 0 0 0.10 0.10 0 0 Initial Velocities r0 = 0.70 K = 0.25 K = 0 Pio = 0 9 i 0 = 0 Euo = 2.00 Quo — 2.00 107 1 2 6 (No. of Orbits) Figure 4-13 System response in circular orbits showing the destabilizing influence of thermally deformed appendages: (b) vibrational response. Figure 4-13 System response in circular orbits showing the destabilizing influence of thermally deformed appendages: (b) vibrational response. 109 4.3.4 Eccentric orbits This section attempts to assess the effect of orbital eccentricity on the system response. Note, eccentricity constitutes an in-plane disturbance. The thermal deformation of the flexible appendages is purposely neglected to help isolate the eccentricity contribution. The circular orbit case results presented earlier (Section 4.3.1 and 4.3.2) would serve as a reference for comparison. Figures 4-14a and 4-14b show the system response in two eccentric orbits, e= 0.1, 0.2, when the lower appendage is subjected to a disturbance as before. Note, in-plane eccentricity induced excitation results in a large amplitude pitch oscillation ( ± 1 4 ° for e= 0.1, ± 3 2 ° for e= 0.2) compared to the circular orbit case where \p was only ± 0 . 2 5 ° . However, even with this large amplitude in-plane librations, the in-plane vibrational amplitude of the appendages remain the same as before, with the characteristic beat phenomenon. On the other hand, the orbital eccentricity increases the beat frequency substantially. Interestingly, the vibration frequency has a minimum value at perigee and reaches a maximum at apogee. This is due to the apparent stiffness terms containing 1 /(1 + ecosd) quantity as coefficients. Obviously, 1/(1+ecosr5) reaches a maximum value at 0 = 7T resulting in frequency condensation. Character of the response to an out-of-plane appendage disturbance remains essentially the same as above except that now, due to coupling, three dimensional librational and vibrational motions are excited (Figs. 4-14c and 4-14d). Note, the frequency condensation effect is discernable in roll librations as well. Turning to the system response in an eccentric orbit with the central body subjected to a combined pitch and roll disturbance (Fig. 4-15), it can 110 Figure 4-14 System response in eccentric orbits: (a) in-plane appendage disturbance, e= 0.1. 111 Figure 4-14 System response in eccentric orbits: (b) in-plane appendage disturbance, e— 0.2. 112 Figure 4-14 System response in eccentric orbits; (c) out-of-plane appendage disturbance, e = 0.1. Figure 4-14 System response in eccentric orbits: (c) out-of-plane appendage disturbance, e= 0.1. 114 Figure 4-14 System response in eccentric orbits: (d) out-of-plane appendage disturbance, e= 0.2. 115 ^ 0 -50 -100 0 UltaiiA." III j|ii||py^ ^yii|||||j|| 1 2 6 (No. of Orbits) Figure 4-14 System response in eccentric orbits: (d) out-of-plane appendage disturbance, e = 0.2. 116 be concluded by comparison with the earlier results for e = 0 (Fig. 4-10b) that eccentricity effect is confined to local variations without substantially affecting the overall trends. Finally, Fig. 4-16 attempts to study the effect of orbital eccentricity in the presence of thermal deformation. Corresponding case for circular orbits was studied earlier with results presented in Figs. 4-8 and 4-11. Again, one notices amplification of in-plane librations and condensation of frequency at apogee to be the distinctive contributions of the orbital eccentricity. On the other hand, the effect of thermal deformation is merely to distort the beat response. 117 Figure 4-15 System response showing the effect of eccentricity and central body disturbance: (a) librational response. Figure 4-15 System response showing the effect of eccentricity and central body disturbance: (b) vibrational response. 119 Figure 4-16 System response showing the effect of eccentricity and thermal deformation: (a) appendage disturbance. 120 Figure 4-16 System response showing the effect of eccentricity and thermal deformation: (b) central body disturbance, librational response. Figure 4-16 System response showing the effect of eccentricity and thermal deformation: (c) central body disturbance, vibrational response. 122 4.3.5 Influence of thermal deformation on system stability (eccentric orbits) A s pointed out in Section 4.3.3, thermal deformation increases the libration and vibration amplitudes in circular orbits, irrespective of the initial conditions. In eccentric orbits, results in Section 4.3.4 show that thermal deformation has negligible effect on system response. However, for a stubby central body with heavy appendages, thermal deformation can result in an increase or decrease of libration and vibration amplitudes as shown in Fig. 4-17 0<a= 0.5, hC = 0.6). Two sets of initial conditions are compared here. In Fig. 4-17a, with only in-plane appendage disturbance applied, thermal deformation results in higher libration and vibration amplitudes. For a similar system under different initial conditions, in-plane appendage and central body disturbances, Fig. 4-17b indicates lower libration amplitude in the presence of thermal deformation. This shows that, depending on the initial conditions, thermal deformation can help to offset or amplify the destabilizing influence due to eccentricity. With Fig. 4-17 as background, the results of Fig. 4-18 are not surprising. The system is the same as before. The initial conditions are in-plane appendage and central body disturbances. In Fig. 4 - l 8 a , thermal deformation causes tumbling of the system. In contrast, under different magnitude of initial conditions, the system is stabilized by thermal deformation (Fig. 4- l8b) . Figure 4-17 System response in eccentric orbits showing the effect of thermal deformation: (a) an increase in libration amplitude. Figure 4 -17 System response in eccentric orbits showing the effect thermal deformation: (b) a decrease in libration amplitude. 90-45-0--45--90 -100-y S 1 50-i O 0-X - 5 0 -Q J -100-100-rO 50-1 O 0-X - 5 0 -Q_l -100-100-•—> •O 1 50-O o -*x 3 - 5 0 -Q J -100-100-S N ro I O 3 Q J Figure 4-18 System Porometers = 0.01 = 0.50 = 0.60 = 20.00 Orbital Elements L* = 0.6 1 2 0 (No. of Orbits) Typical response in eccentric orbits of spacecraft with thermally flexed appendages showing the influence of initial conditions: (a) destabilizing influence 18 Typical response in eccentric orbits of spacecraft with thermally flexed appendages showing the influence of initial conditions: (b) stabilizing influence 127 4.4 Analytical Solution 4.4.1 Variation of parameters method The method, as outlined in Chapter 3, is intended to obtain the analytical solution for \p, <f>, P ( , Qj, P_u, and Q u responses. For the gravity gradient configuration under consideration, acting as a communications satellite with the antenna located on the local vertical, effect of yaw librations on the pointing sensitivity is relatively small. Hence, the solution for the yaw degree of freedom was obtained neglecting all the vibration terms, i.e., the same as the one presented in Section 3.4.2, Eq. (3.35). Ignoring third and higher orders of e, the equations of motion for \pt <t>, E i , CL, E . . , a n d Q,. generalized coordinates can be written as: 1>"-ecip' + KitegrP = ec + Gl{q); <pn-€e<p' + (l + Kit)<p = G2{q)i Ei' - f cZ / + (w? - ecos6 + e2 cos2 9)Pj = C?3(g); Qj' - ecQj + (w2 + 1 - ecos5 + e2 cos2 = GA(q); Ell ~ ecE!u + K 2 - € cos 0 + e2 cos2 B)PU = Gh{q); Q" - ecQ'a + (w2 + 1 - € cos 6 + e2 cos2 = G6{q); • • • (4.3) where Gj (i = 1, . . ., 6) are nonlinear functions of generalized coordinates, velocities, and accelerations. The homogeneous set of Eq. (4.3) represents the Mathieu equation 22 and can be solved using the standard procedure . Denoting: • • • (4-4) 128 the homogeneous solution of Eq. (4.3) can be written as: i - A , L n w + M + ( 1 - f " ) e [ s i n | ( n : + 1 ) 9 + ^ - • N O v - D ' + fti I 2 L 2n ,^ +1 2n ,^ — 1 J 8 I (2rty + l)(rty + 1) (2n^ - l)(n^ - 1) J) + e i^ sin 0 - e 2^ sin 20 ; (1 - 3Kit)e rsin[(n^ + 1)0 + /fy] sin[(n^ - 1)0 + /fyi = ^{sin(n^0 + /^) + 2n^+ 1 2 n ^ - 1 (1 - 3 i f t 7 ) n ^ 2 r-sin[(r^ + 2)9+^] sin[{n+ - 2)0 + p+\ + 8 I (2n^ + l)(n^ +l) + (2n^ - l)(r^ - 1) Pi = Apt sm(np9 + Ppi)) Qj = Aqi s'm(nq9 + pqi); Apu s'm(rip9 + ppu); Aqus'm(nq9 + P q d ) ; £.= <3 = —a where with and q = qe" ; q = rp,(f>,Pi,Qi,Pu,Qu; 1 /•* a = " 2 / 0 £ c ( 0 ^ (4.5) = e(l -cos0) - —(1 - cos 20) 4 The method of variation of parameters is now applied and the analytical solution is given by: 129 1> = |^{sin[(n v , + /?;)& + ^ 0] + (1 - 3Kit)e ^[{^ + ^ + 1)6 + [ 2n^ + 1 sin[(r^ 1)9 + jfyo] ] ^  ( l - 3 / f , ( ) n ^ 2 p s i n ^ + + 2)0 + /fyp] 2n,/, - 1 + (2n^ + l)(n^ + 1) + Bin[(iy+i;-2)g + jfy)] n , . 1 a * = f ^{sin[(n, + ^ 0 + M + l ^ M i t M i n t + fl + W + M { *• 2 L 2n^ + 1 _ BinKn^ + Z f r - i y + fro^ ^ (1 - 3JSTa)n^e» r-Bin [ (n , + ^ + 2)0 + ^ ] (2fi^  + l)(n^+I) + 2n^ - 1 "J s\u\{n<i> +^-2)9 + 6^] (2^ - l)(n+ - 1) P / = {^ / Sin(n p0 + /?p,u)}ea; Qj = {Aqism{nq9 + pql0)}ea; Pu = iAp* s'm(np9 + Pptt0)}ea ; Qu = {A s usin(n 90 + /?guO)}eo; with j3 ,^ and 0^ found to be the same as given in Eq. (3.29). The amplitudes and initial phase angles are determined from the initial conditions: (4.6) 4>Q = A f { l -( l-3Jf t t )e 4 * 2 - 1 4( 1 + (1 - 3 ^ ) 6 H -1 (l-3JJTt-t)e H-1 1 + (l-3JT f t )e H-1 3 r v € h • a (5n2 - 2)el, " 4(n2 - 1) J 1 C ° S ^ ° + € ^ ~ ^ 2 5 {5nl - 2)e] ^  130 PjQ = Apt am Ppto ; E40 = Apinp cosPPIQ ; Qj0 — Aqi smpqlQ ; Q40 - Aqinq cosPqio ; E.u0 = ApU sin^puo ; P'uO — Apunp cos P?UQ ; Qu0 = Aqtts'mpqtlQ ; Q' u 0 = -4gu"gCOS/?2tjO • There are two limitations to this form of approximate solution presented in Eq. (4.6). The solution predicts the appendages to vibrate at one frequency; hence, it fails to predict beat phenomenon. A lso , due to averaging process involved in the variation of parameters method, the solutions for librational motion do not have contributions from vibrating appendages. Simillarly, vibration solutions do not contain librational terms, i.e., the solution does not involve coupling between librational and vibrational motion. Figure 4-19 compares the approximate analytical solution with the 'exact ' numerical solution of the complete equations of motion as given in Appendix I I I . The figure shows that although the analytical solution for libration response fails to predict the high frequency component, the estimate of amplitudes is accurate. On the other hand, the prediction of vibrational response is indeed poor. The analytical solution not only fails to predict the beat phenomenon, but also gives a poor estimate of the peak-to-peak amplitudes. These shortcomings of the analytical solution prompted a search for a better answer. 131 System Parameters legend Numer ica l Analyt ica l 1 2 6 (No. of Orbits) Figure 4 - 1 9 A comparative study showing deficiencies of the analytical solution: (a) librational response. 19 A comparative study showing deficiencies of the analytical solution: (b) vibrational response. Figure 4-19 A comparative study showing deficiencies of the analytical solution: (b) vibrational response. 134 4.4.2 Improved analytical solution The deficiencies of the analytical solution obtained in the last section may be attributed to the approximate representation in Eq. (4.3) of the general equations of motion presented in Appendix I I I . The homogeneous part of the equation does not include any coupling terms. In order to improve the analytical solution, Eq. (4.3) is rewritten to include first order linear contributions from other degrees of freedom. A s in-plane and out-of-plane vibrations predominantly affect only the in-plane and out-of-plane librations, respectively, the coupling terms are appropriately retained to reflect this trend; •P" + 3Kttil> + Kala,[(P}' + £'„') + 3(F, + FJ] = H^q); . 4>» + (1 + 3Kit)<p + Kala3[(g; - + 4(Q, - QJ] = H2{q) ; Fj' + u,r2Fj + a 3 ( y + 3V) = Hz{q); QJ' + (u2 + 1)Q, + a3(<f>" + 4<t>) = Ht(q); ElL + "2PU + «z(TP" + 3V) = Hb(q) ; Q^ + (u2 + l)Qu - a3{<p" + 4<f>) = He{q) ; • • • (4.7) where ? = Q p P - , , ^ ' , ^ O ^ , Note, the homogeneous part of Eq. (4.7) shows coupling among the in-plane degrees of freedom P|, and P_u) and the out-of-plane degrees of freedom (0, Q| , and Q u ) . The eccentricity and thermal effects are purposely omitted to avoid non-autonomous character. The homogeneous solution of Eq. (4.7) was obtained using the Laplace transform procedure (Appendix V). The method of variation of parameters can now be applied. Recognizing the fact that for a given generalized coordinate, contribution from other degrees of freedom at different frequencies is relatively small, one can write; ^ 0 ) = ^ i ( 0 ) s i n [ n ^ 0 + / ? 0 i ( 0 ) ] = j 4 ^ ( 0)flin t)n ; m^A^Wsmln^e + P+M] = 4^(0) s i n ^ i ; Pj{9) = Api3{0)am[np9 + ppi3{e)) 3,(5) = i 4 « » ( « ) 8 i n[n^ + /?a|3('5)] = ^ g / 3 ( ^ ) s i n ^ 3 ; = Apu3{0) sin fjptt3 ; Q a(5) = > i g a 3 ( ^ ) s i n [ n ^ + ^ a 3 ( ^ ) ] = Aqu3(9) s i n »73 t,3 . Using the procedure outlined in Chapter 3, the solution for the derivatives of the amplitudes and phase angles can now be obtained ALX = irj cos 17^1; A'n = — H\ cos 1701; A'PIZ = 1 TI* — £ T 3 cos 17^3 ; n p A'qli = — # 4 cos T79;3 ; n « A'puz = 1 r r * —H$ cosnp„3 ; n p A' — AquZ — 1 TJ* —H6 cos 7 7 5 „ 3 ; nq ffpl = -—— H* sin » 7^i; A ^ i n ^ i # 1 = H2 s i n 77^1; -A^in^i #13 = - — - — s i n T7p| 3 ; Apiztlpiz P'qiz = -—-—H\ s i n r?3j3 ; Aqiznqi3 P'Pu3 ~ —HI s i n ?7pU3 ApuZnPnZ 136 P'***= i —I.— H* s i n ; where Hi = Hi{An s i n VtU • • • .<0 - -(4.8) Assuming the variation of parameters to be small, their averages over a period are of interest: A'*i = ( 2 * ) % i X{J2*f*J Iff Hi*mvtidyndv$\dn^ Similar expressions are used for other A''s and j3''s. The average values were found to be as follows: A[ = 0 t = rpl,<t>l,pl3,ql3,pu3,qu3 ; j3[ = 0 » = pJ3, g/3, pu3, qu3 . Hence, the final solution for \p and <j> librations is given by: ~ ^2[{n2p - n2pl) - 2Kata\{3 - n2pl)] sm(npl9 + /^2)} Kaia3{n2 - 3) r + dAn2 -n2 \VA*1 sMHid + M + Apui am(nn9 + 0p!tl)] - [Api2 sin(n p l0 + ppl2) + Aptt2 sm(npl0 + ppK2)]} ; 137 - ^ 2 [ ( n 2 - n2ql) - 2Kata\(A - n2ql)] 8m(n, 1J + ^ ) } Katctz(n2 -4) (r - [Aqi2 sm(nql9 + pql2) - Aqu2 sm(nql9 + pqn2)} J ; ... (4.9) where: „ . -wftyitfox ^ 1 0 = tan (-^77-); /»^-t»->(=^S). ro Similarly for the vibration degrees of freedom; Ej = ~r~rT—^-{-A^1sin(n^10 + /3^1) + i4v,2sin(npi0 + ^ 2 ) } + d i ( n * > i - nli) ; r - J r a | * 3 i ( - T * $ 1 + 3 X ^ - 2 7 1 $ ! + 3 ) 2 2 1 . , „ „ , x {^p/i [ v _ n 2 P * n j t + nJJ sin(n^^ + ^x) + AP(2 I (n2_n2 ) + R Pl ~ n£J 8m(npl0 + ^ 2 ) 'P 'V i f a t a 2 ( n J - 3 ) P ~ » i^A"p - "plJ 2 P <*i(»Jl - »Ji)(»J - »$i)("J - n p i ) x { - j i p « i ( - n $ i + 3)(nJ - n2pl) smin^B + ppui) + Aptt2{-n2pl + 3)(nJ - n^) sin(npl0 + 0pn2) + ApuZ{n2p - 3)(n2pl - n2 )^ sin(np0 + /?p„3)} 5 138 Q. = f * 3 ( * K?\ { - A # s l n ^ e + fi#) + A* sm(nql9 + fa)} + -KataK-nl, + 4)(n2 - 2nJ t + 4) - 4 i + n \ r^ a tal ( -4 1 + 4)(n 2-24 1+4) ]sin(n g 0 + ^ / 3)} A5,21 " ^ »; 2 " « ^ + n2ql - njl 8in(nBitf + ^;2) L {n2q-n'ql) v* 1^3 - ^ ^ ( ^ - 4 0 ( 4 - 4 ) -+ x {^uiC-n^ +4)(n 2 -nj 1 )s in ( n 0 1 0 + ^ , i ) - V2(-Wgi + 4)(ng " sin(n gi0 + /?gtt2) - Aqu3(n2q - 4)(n 2 1 - n\x) sm(nq9 + pqtt3)} 5 = T T T — ^ T r { - ^ i s i n ( n 0 i 0 + + A^2 sm{npi9 + fa)} 1 + 2 i 1 1 L 2 x { Apttl v p * + n2 sin(n^0 + (3pnl) 4-4 lKat<xl(-n2pl+3)(n2p-2n2pl+3) i + ^p«2 * T " 2 — - y r + npi - nj sm(n pi0 + /?p u 2) L v r e p — npi) J 139 . l - K a l a l ( n 2 p l - n 2 l ) ( - n 2 p + 3)2] . f , + Ap*>' ( n j - n j ^ n j - n j , ) J " " ^ + W J KgtczUnl - 3)  + ^ i ( ^ 1 - n 2 j ( ^ - n 2 1 ) ( n 2 - n j 1 ) x {-iljrfiC-nJj + 3)(nJ - n2pl) am(n^6 + fa) + Apl2{-n2pl + 3)(n2p - n2^) sin(npl0 + fa) + Aplz(n2p - 3)(n2pl - njj s'm(np9 + fa)} ; Qn = TTT—^4{^i sHn4>ie + M ~ A^sin{nqi9 + fa)} annqi ~ n4>\) K ' 1 + j r-KataK-n^-r A){n\ - 2n\x +4) , x { 4r«i T 2 „2 > 2 nji + nj 8m(n^« + fa) , riif a ta 2 !(-n 2, 1+4)(n 2^2n 2, 1+4) . -, + Aqu2 q) 2 A ' ^ + n*! - 1 - 3Kit\ sm(nql9 + fa2) \nq ~ nql) J ~ M (n2-n2)(n2-n2) J + K - n£l)(*« - nql) Kaialjn2 - 4) x + 4 ) ( n J ~ W Jl) 8 ' m Kl* + /W - Aqi2{-n2ql + 4){n2q - njj sin(ngl0 + /fy2) - Aqlz[n\ - 4)(n*1 - njj sm(nq9 + fa)} . . . . (4.10) The constants appearing in Eqs. (4.9) and (4.10) are defined in Appendix V. 140 4.4.3 Discussion of results To check the accuracy of this approximate analytical solution, it was compared with the results given by numerical integration of the exact equations of motion (Appendix III) over a range of system parameters and initial conditions (Figs. 4-20 to 4-25). Response of the system to a disturbance in the form of the appendage tip displacement of 20% of its length in the first mode is compared with numerically obtained data in Fig. 4-20. Note, the c losed-form solution predicts the beat phenomenon, librational and vibrational amplitudes as well as frequencies very well. Similar results for a large disturbance applied to the central body are presented in Fig. 4-21. Correlation between the two sets of results is again excellent except for a small discrepancy in yaw (X) response. This, of course, is expected because of the simplified form of the assumed X solution. However, as explained before, for most application satellites, X is not likely to be a critical parameter. Furthermore, accuracy of X response can easily be improved by including appropriate higher order terms, of course, at a cost in terms of computer time. In spite of the success, possible limitations of this approximate analytical solution must be recognized, particularly in the presence of unusually severe pure out-of-plane disturbance. This is illustrated in Fig. 4-22. Note, the approximate solution fails to predict in-plane motion in the presence of out-of-plane disturbance although the out-of-plane response is reasonably well correlated. To be fair, this is an unusually demanding test as such large disturbances are hardly ever encountered in practice. Besides magnitude of the initial disturbances, accuracy of the approximate analytical solution also depends on the inertia parameters, K g 141 Figure 4-20 A comparison between numerical and improved analytical solutions for a severe appendage disturbance: (a) analytically obtained response. 142 Figure 4-20 A comparison between numerical and improved analytical solutions for a severe appendage disturbance: (b) numerical results. 143 Figure 4-21 A comparison between numerical and improved analytically predicted responses with central body disturbance: (a) librational response. System Parameters Kj L* £J, = 0.01 = 0.10 = 0.75 = 0 = 20.00 -0.25 Orbital Elements p - 90° CJ = 90° i = 0 t = 0 Initial Posit ions f o = <t>o = *o = EIO = Q . o = Euo = Quo = 5« 5-0 0 0 0 0 Initial Velocit ies % =o = 0 X.; = 0.0872 E i 0 = o Qio = o E u o = o Quo = o 144 Numerical Solut ion 1 2 6 (No. of Orbits) i 3 Figure 4 -21 A comparison between numerical and improved analytically predicted responses with central body disturbance: (b) vibrational response. Figure 4-21 1 2 6 (No. of Orbits) A comparison between numerical and improved analytically predicted responses with central body disturbance: (b) vibrational response. 146 System Orbital Initial Initial Parameters Elements Positions Velocities i>0 = o v-;, =oi _. a, = 0.01 p = 90" 0 O = 0 0; =1-00 Ke = 0.10 v — 90* x 0 = 0 K = 0 1 • K, = 0.75 i = 0 E.o= 0 P-io = 0 I L* = 0 c = 0 Q.o = o 9 i 0 = 0 1 u r = 20.00 P-uo = 0 P-uo = Oj Quo = o L. Quo = o! 20 0 1 2 3 0 (No. of Orbits) Figure 4-22 A comparison between numerical and improved analytical solutions in the presence, of severe out-of-plane disturbance: (a) librational response. Figure 4-22 A comparison between numerical and improved analytical solutions in the presence of severe out-of-plane disturbance; (b) vibrational response. 148 and K.. In general, larger K and smaller K. affect the accuracy adversely i a i (Figs. 4-23 to 4-25). Note, with K g = 0.1 and K ( = 0.75 (i.e., small appendages with a slender central body), the responses compare reasonably well (Fig. 4-23). However, with a large appendage (K g = 0.5, Fig. 4-24) or a stubby central body (K- = 0.25, Fig. 4-25), although frequencies and amplitudes are reasonably well predicted, the correlation between the two responses is poor due to discrepancies in phase. Some of the limitations of the proposed solution were discussed earlier. A lso , the solution is applicable only to autonomous systems, i.e., satellites in circular orbits with no thermal deflection of appendages. Fortunately, there are a number of situations of practical importance with satellites in circular or near-circular orbits. Furthermore, the development of newer materials promises to reduce thermal deformations. Thus the applicability of the approach to autonomous systems in circular orbits does not appear to be a serious restriction. To put it differently, there is a large class of satellites in circular orbit where the proposed closed-form solution is applicable. In any case, usefulness of the solution during the preliminary stages of the control system design cannot be questioned. Furthermore, it resulted in a significant reduction in computational time. For the AMDAHL 470-V8 system, a typical run for the analytical solution takes approximately 1/3 the time of the corresponding numerical analysis. 149 5-0° -10 20 A ,f\ A W 7 i i 10--20 legend Numerical Analyt ica l 1 2 6 (No. of Orbits) Figure 4-23 A comparison between numerical and improved analytical solutions showing the effect of inertia parameters on correlation; K a = 0.1, K.= 0.75. 0 -23 System Parameters L* = 0.01 = 0.10 = 0.75 = 0 = 20.00 Orbital Elements p = 90° u = 90' i = 0 c = 0 Initial Positions f 0 = 30' <P0 = 0 A 0 = 0 P | o = 0.10 Q , 0 = o-io E u o = o 0 Quo Initial Velocities f o =0 50 <f>'o =° 25 K =o p ; 0 = o Q!o = o Puo = 2 00 Quo = 2 00 150 1 2 0 (No. of Orbits) A c o m p a r i s o n between numer ical and i m p r o v e d analyt ical so lu t ions s h o w i n g the e f fec t of inertia parameters on cor re la t ion : K a = 0.1, K := 0.75. I o X QJ I O X 67 I O rO I O 3 Ol Figure 4-23 System Parameters L* = 0.01 = 0.10 = 0.75 = 0 = 20.00 Orbital Elements p = 90° v = 90° i = 0 £ = 0 Initial Positions 0o K — x n « o E i o 9 i o E u o Quo 30°o 0 0.10 0.10 0 0 Initial Velocities % = 0.50 0 ; = 0.25 K = 0 Elo = 0 Qio = 0 Euo = 2.00 Quo ~ 2.00 151 Analytical Solution 1 2 0 (No. of Orbits) A comparison between numerical and improved analytical solutions showing the effect of inertia parameters on correlation; K = 0.1, a ' K.= 0.75. 152 Figure 4-24 A comparison between numerical and improved analytical solutions for a satellite with relatively large appendages and a slender central body: K = 0.5, K. = 0.75. a l 24 System Parameters 153 N u m e r i c a l So lu t i on 1 2 6 (No. of Orbits) A comparison between numerical and improved analytical solutions for a satellite with relatively large appendages and a slender central body: K = 0.5, K. = 0.75. a I rO I o X QJ I O 200-1 - 2 0 0 -200-100-O i -100-200-200-ro i 100-I O 0-*x 3 -QJ -100-200-200-fO 1 100-1 O ' x OI 0--100--200-( Figure 4-24 System Parameters a, = 0.01 K c = 0.50 K| = 0.75 L* = 0 ur = 20.00 Orbital Elements p = 90° u = 90" i = 0 e = 0 Initial Positions V o = « o = K = H.O = Q . o = Euo = 9 U 0 — 30 0 0 0 0 0 0.10 10 Initial Velocities r0 = 0; = K = Pio = 9io = p ' = -uo Q* = =»uo 0.50 0.25 0 0 2.00 2.00 0 154 1 2 6 (No. of Orbits) A comparison between numerical and improved analytical solutions for a satellite with relatively large appendages and a slender central body: K = 0.5, K. = 0.75. a 1 155 Figure 4-25 A comparison between numerical and improved analytical solutions showing the effect of a stubby central body: < a = 0.1, Kj = 0.25. K) I o X I O I o dT fO I o 3 Q J K> I o 3 Ol System Parameters Orbital E lements 156 N u m e r i c a l So lu t ion 0 1 2 6 (No. of Orbits) Figure 4-25 A c o m p a r i s o n between numerical and i m p r o v e d analyt ical so lu t ions s h o w i n g the e f fec t of a s tubby central b o d y : K = 0.1, K. = 0.25. 3 I 150-75-1 o o-X QJ -75 --150-J 150-75-1 o 0 J X v—s 6 7 - 7 5 --150 150-r o 1 75-1 O 0-X 3 Q_l - 75 --150-150-•O • 75-I O T — 0-OI - 7 5 --150 1 2 6 (No. of Orbits) Figure 4-25 A comparison between numerical and improved analytical solutions showing the effect of a stubby central body: K = 0.1 K. = 0.25. 3 I 5. CONCLUDING COMMENTS 5.1 Summary of Conclusions A relatively simple model of a satellite, consisting of a central rigid body and a pair of flexible appendages, studied here represents an important step forward in understanding dynamics of this class of problems. It is particularly relevant to the next generation of communications satellites with relatively large appendages. Based on the analysis following general conclusions can be made: (i) The most significant factors affecting the satellite's equilibrium orientation are the inertia parameter Kj and the orbital eccentricity e. A smaller value of Kj (stubby central body) and a larger eccentricity tends to orient the satellite in equilibrium with its axis inclined to the local vertical at a larger angle. So far as the flexible appendages are concerned, thermal deflections are dominant compared to the gravitational contributions. For stable equilibrium configurations, Kj = 0.75, K = 0.25, e = 0.1, CJ = 2.0, and L*= 0.6 gave tfr = 7 ° with the 3 • i 6 boom tip deflection of about 50% of its length. (ii) As expected, with the orbital rate approaching the appendage natural frequency, the equilibrium configuration diverges indefinitely from its nominal local vertical alignment. (iii) Application of the Floquet theory to the nonautonomous, linear system suggests that small K. and K promote instability. The thermally induced deformations of the appendages lead to an additional small strip of unstable region extending over a wide range of K and K.. a i (iv) Even a small difference in gravitational field experienced by the two 158 159 appendages is sufficient to change their natural frequency leading to a beat phenomenon. Oscillation of a given appendage serves as an excitation for the other through a librational coupling. The approximate solution developed on the principle of variation of parameters is able to predict the beat response rather accurately. The effect of thermal deformations is to distort the beat response, while an increase in orbtal eccentricity tends to increase the beat frequency. (v) As expected, the effect of orbital eccentricity is to increase and modulate the in-plane pitch librations. However, surprisingly, it has virtually no effect on appendage vibration, in-plane or out-of -plane. This is related to the fact that, due to a large difference in librational and vibrational frequencies, even large rotations of the central body excite appendages only by a small amount. (vi) Perhaps one of the most interesting contribution of the orbital eccentricity is to cause frequency condensation in all degrees of freedom, including the beat response. This is attributed to the 1/(1+ecos0) terms in governing equations of motion. In effect, it increases stiffness of the flexible appendage at apogee. (vii) In circular orbits, the effect of thermal deformations (L*= 0.6) is to increase both librational and vibrational amplitudes for a given set of initial conditions. In general, an increase in K g or a decrease in Kj enhances this trend. In fact, under a critical combination of parameters, the system can become unstable although the corresponding undeformed system (L* = 0) may be stable. (viii) In contrast to the circular orbit case, for et 0, depending on initial conditions and system parameters, thermally flexed appendages may 160 stabilize the system, (ix) The analytical solution to the complex problem, obtained using the Butenin method, appears quite promising. In general, it predicts librational and vibrational frequencies with surprising accuracy. The solution devleoped here is applicable to autonomous systems, i.e., satellites in circular orbits with no thermal deformation of appendages. 5.2 Recommendations for Future Work The thesis represents only a beginning in exploration of this rapidly developing field aimed at design and control of large flexible space structures in presence of the environmental forces. With the U.S. commitment to a space station by mid-1990's, this class of problems are certainly going to occupy the attention of dynamicists and control engineers for a long time to come. There are a number of avenues one can pursue which are likely to be fruitful, however, only a few of them are touched upon below: (i) The study of the problem is restricted to satellites orbiting in the ecliptic plane. This assumption was purposely introduced to help focus on key parameters affecting the dynamics and its physical appreciation. Now, it would be useful to apply the formulation to spacecraft in any arbitrary orbit. (ii) The approximate solution obtained though effective presents considerable scope for improvement. Perhaps the major limitation is its applicability only to autonomous systems. A search for c losed- form solution to a coupled nonautonomous system, if successful , would lead to a major breakthrough in solution of such a 161 complex nonlinear system. Furthermore, modified implementation of averaging procedure using other than 2n period, and improvement in yaw solution through retention of coupling and nonlinear terms are likely to be successful. With dynamics well predicted and understood, the attention should be focussed on librational and vibrational control. Rendering the appendages as well as the central body flexible will make the analysis applicable to a large class of future satellites, the Orbiter-based construction of structural components, and the proposed space station. Addition of plate-type solar panels will add to the versatility of the model. Inclusion of slewing motion for the flexible appendages and sun-tracking maneuver for the solar panels will render the model more realistic and hence further add to its usefulness. BIBLIOGRAPHY 1. Brereton, R.C., "A Stability Study of Gravity Oriented Satellites," Ph. D. dissertation, University of British Columbia, Nov. 1967. 2. Shrivastava, S.K., Tschann, C., and Modi , V.J., "Librational Dynamics of Earth Oriented Satellites- A Brief Review," Proceedings of 14th Congress on Theoretical and Applied Mechanics, Kurukshetra, India, 1969, pp. 284-306. 3. Modi, V.J. , "Attitiude Dynamics of Satellites with Flexible Appendages-A Brief Review," Journal of Spacecrafts and Rockets, Vol . 11, No. 11, Nov. 1974, pp. 743-751. 4. Shrivastava, S.K., and Modi, V.J. , "Satellite Attitude Dynamics and Control in the Presence of Environmental Torques - A Brief review," Journal of Guidance, Control, and Dynamics, Vol . 6, No. 6, Nov.-Dec. 1983, pp. 461-471. 5. Modi, V.J. , and Brereton, R.C., "Planar Librational Stability of a Long Flexible Satellite," Al AA Journal, Vol . 6, No. 3, March 1968, pp. 511-517. 6. Modi, V.J. , and Kumar, K., "Librational Dynamics of a Satellite with Thermally Flexed Appendages," Journal of the Astronautical Sciences, Vol . 25, No. 1, Jan-Mar . , 1977, pp. 3-20. 7. Goldman, R.L., "Influence of Thermal Distortion on Gravity Gradient Stabilization," Journal of Spacecrafts and Rockets, Vol . 12, No. 7, July 1975, pp. 406-413. 8. Yu, Y.Y., "Thermally Induced Vibration and Flutter of a Flexible Boom," Journal of Spacecrafts and Rockets, Vol . 6, No. 8, Aug. 1969, pp. 902-910. 9. Augusti, G., "Comment on 'Thermally Induced Vibration and Flutter of a Flexible B o o m ' , " Journal of Spacecrafts and Rockets, Vol . 8, No. 2, Feb. 1971, pp. 202-204. 10. Jordan, P.F., "Comment on 'Thermally Induced Vibration and Flutter of a Flexible B o o m ' , " Journal of Spacecrafts and Rockets, Vol . 8, No. 2, Feb. 1971, pp. 204-205. 11. Krishna, R., and Bainum, P.M., "Effect of Solar Radiation Disturbance on a Flexible Beam in Orbit," Al AA 21st Aerospace Sciences Meeting, Reno, Nevada, January 1983, Paper No. 83-0431. 12. Krishna, R., and Bainum, P.M., "Orientation and Shape Control of an Orbiting Flexible Beam Under the Influence of Solar Radiation Pressure," AASIAIAA Astrodynamics Conference, Lake Placid, N.Y., August 1983, Paper No. 83-325. 162 163 13. Bainum, P.M., and Krishna, R., "Control of an Orbiting Flexible Square Platform in the Presence of Solar Radiation," 14th Iinternational Symposium on Space Technology and Science, Tokyo, Japan, May-June 1984, Paper No. i -2-1. 14. Krishna, R., and Bainum, P.M., "Dynamics and Control of Orbiting Flexible Beams and Platforms Under the Influence of Solar Radiation and Thermal Effects," AIAAIAAS Astrodynamics Conference, Seattle, Washington, 1984, Paper No. 84-2000. 15. Krishna, R., and Bainum, P.M., "Environmental Effects on the Dynamics and Control of an Orbiting Large Flexible Antenna System," 35th Internationa/ Astronautical Congress, Lausanne, Switzerland, 1984, Paper No. IAF-84-358. 16. Moran, J.P., "Effects of Planar Librations on the Orbital Motion of a Dumbbell Satellite," ARS Journal, Vol . 31, No. 8, Aug. 1961, pp. 1089-1096. 17. Yu, E.Y., "Long-term Coupling Effects Between the Librational and Orbital Motions of a Satellite," Al AA Journal, Vol . 2, No. 3, Mar. 1964, pp. 553-555. 18. Schittkowski, K., "The Nonlinear Programming Mehtod of Wilson, Han, and Powell with an Augmented Lagrangian-Type Line Search Function. Part 1: Convergence Analysis," Numerische Mathematik, Vol . 38, Fasc 1, 1981, pp. 83-114. 19. Schittkowski, K., "The Nonlinear Programming Mehtod of Wilson, Han, and Powell with an Augmented Lagrangian-Type Line Search Function. Part 2: An Efficient Implementation with Linear Least Square Subproblems," Numerische Mathematik, Vol . 38, Fasc 1, 1981, pp. 115-128. 20. Minorsky, N., Nonlinear Oscillation, D. Van Nostrand Co. Inc., Princeton, 1962, pp. 127-133. 21. Butenin, N.V., Elements of Non-linear Oscillations, Blaisdell Publishing Co., New York, 1965, pp. 102-137, 201-217. 22. Struble, R A . , Nonlinear Differential Equations, McGraw-Hill Book Co., Inc., New York, 1962, pp. 220-234. 23. Gear, C.W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall Inc., Englewood Cl i f fs , 1971, pp. 158-166, 209-228. APPENDIX I - EVALUATION OF APPENDAGE KINETIC ENERGY Substituting from Eqs. (2.19) and (2.20) into Eq. (2.21) gives, Tapp. = ^ J " {[Vx + ojy{zi + w{) + uz(yi + tv)]2 + [Vy - w2(xj + a) - ux(zt + wi)-yi- t>j]2 4- [Vz - ux(yi + v{) + (j}y(xi + a) + z\ + u>;]2} dxt + ~YJQ {lv* + "AZ* + W*)~Uz(yu + VU)]2 + [Vy + uz(xn + a) - ux(zH + wv) + y B + vv]2 [Vz + wx(ytt + vu) - u,j(xn + a) + zn + tv„]2} dxn . ••• (7.1) Since the shift in the centre of mass is negligible, the following relations are valid: rlb rh / (yi + vi)dxt- (y* + vu)dxu = 0; Jo Jo fh fh / (z\ + wi) dxi + / (zu + wu)dxu =0; Jo Jo fh fh / (yz + v/) dxi -I (y„ + t )„) dxn = 0; Jo •'o fl\z, + ti;,) dxi + I " ( i , + wa) dxu = 0 . • • (1.2) Jo Jo Substituting for V j , Z j , v ( , and w ( from Eqs. (2.14), (2.15) and for y z u , v u , and w u from Eq. (2.17), denoting: lb Jo $12 = r fb(*i(z))dz; <6 Jo i y' 6 $13 = r / x ( $ i ( s ) ) d z ; 1 Z"'6 164 165 where ^(x) is given by Eq. (2.16), the expression for appendage kinetic energy (Eq. 1.1) takes the form, Tapp. = mblb{R2 + (R9)2) + TL+ Tt + Te + T[j + Tj )C + Tt,v + TiittV ; where; 7} = mblb{a2 + alb + jj/3)(wj + u\) ; T,,t = ^ ^ { ^ ( c o s 2 41 + cos2 <f>\) + ( W f co.tf - u, cos^) 2 + 2ux [cos <p*y (cos c6!) - cos cf>\ (cos c>*)]} ; r i , . = ^ { * n [ ( i ? + J?)(«2 + »l) + (Qi + Ql)i<4 + «J) + (P/Qi - P .Q«) W » W * + ux{PiQi - PZQ, + PuQn - PuQa)} + ( a $ 1 2 + *M)w*[-w»(JDi + ^ «) + f>z{Qi - Qu))} ; = m 6 ^ 6 { _ ( / } _ F t t ) ^ ( c o s ^ ) + ($ f + Q.jlfcostf)} ; ^ . i , . = tJ^^{^l(Pi - P.)cosfi + (Q, + Q.Jcos^; + w2[-(P, - P„) cos c>; + (Q, + <?„) cos + (wy cos 0* - wz cos <j>y) [{Pi - P„)u z + {Qt + Qu)uy]} . APPENDIX II - EVALUATION OF APPENDAGE POTENTIAL ENERGY 3 Expanding Eq. (2.23) and ignoring 1/R and higher order terms gives: |n| = R{1 + | [-(a + xt)lx - (y, + vi)ly + (z, + vn)lg] + ^ 2 Ka + * 0 2 + (» + vi)2 + (* + «*)a]}1/2; |r . | = i2{l + |[(a + z„)Ix + (y„ + v.)/, + (ar. + w,)/,] + ^ 2 Ka + z«)2 + (»• + u«)2 + (*• + w«)2]}1/2 ' • • * (m) 4 Substituting from Eq. ( 11 . 1) into Eq. (2.22) and ignoring 1/R and higher order terms, U a p p can be written as, Uapp. = - j f 1 - i [ - ( o + x,% - (y, + Vl)lv + + wM - 2^2 [(« + *l? + (» + v,)2 + + w*)2] + 2(o + xt){yi + vi)Us - 2{a + x , )(2j + w,)lxlz - 2(yz + vi)(zi + wi)lyls] J dz, - / Q - -[(a + + (y« + vu)ly + (z. + " [(« + *«) 2 + (y. + *>«)2 + (*. + ^«) 2] + ^  K« + *«) 2 ' 2 + (y. + v»)2/J + (* + w j i l + 2(a + z«)(y» + vu)lxly + 2(a + xv)(zn + w„)lxlz + 2(y« + vu)(zn + wtt)lylz]} dxn. ... (J/.2) 166 Finally, using Eqs. (1.2a) and (1.2b), the appendage potential energy expression (Eq. 11.2) can be written in the form, uapP. = --^ + ul + ut + u1! 2mblbfte R + U,,t+ 11^ + 11^ + 1;^; where: U i = ^ t { a 2 + a l b + l l / m _ 3 l l ) . ^ = »C032^ +COS2^ ); - (P,2 + Pl)l] - (Q? + Q2)/2- + 2 ( - P . Q . + PiQi)lyh); flu.. = 3 f , c y i a l W - P«)-V - Wi + Q . ) M ( ' » c o 8 « ; + 1 , a**;) APPENDIX I I I - DETAILS OF THE EQUATIONS OF MOTION All terms in the equations of motion are nondimensionalized with respect to 1^, where It = I + 2mbll\(a/lb)2 + (o//t) + 1/3] • In the expression for strain energy U g (Eq. 2.25), the fundamental frequency co.j is nondimensionalized as, § p U + ecos0 rn (///.la) where 0 p is the angular velocity of the system at the perigee point. In addition, the following notations are used: 2e sin 0 Kat = 1 + € COS 0 ' RH2  1 1 + €COS0 ' mblb/It; KH = i -1 Jit ; 1 r V « 2 = ^ ( F ) ; a 3 = v-$i2 + -j—; '6 lb « 4 = 2 ( F ) ( - T 2 - ) ' lvt = ly cos t6* + lz cos ; Uyz — 0Jy cos — uz cos 0* ; (///.16) (I I Lie) (///.2) - ( / / / . 3 ) c i = bi cos i/> sin <p + b2 sin ip sin 0 + 63 cos <p ; C2 = 61 sin ip — 62 cos ip ; 168 1 6 9 C3 = (61 cos t/» + 62 sin ip) cos d> — 63 sin <p ; C4 = -(61 sin •/» - 62 cos ip) sin <f> cos A + (61 cos •/> + 62 sin y ) sin A ; C5 = —(61 sin tp - 62 cos t/i) sin # sin A — (61 cos ip + 62 sin ^ ) cos A . • • • (IIIA) The equations of motion corresponding to each of the generalized coordinates can be obtained from dVdq' d q + d q ~ q ' where q= yp, <6, X, P,, Q,, P u , and Q u The individual term appearing in the above equation are evaluated below: \p - Degree of Freedom (Pitch): j [oj'y sin A + u'z cos A + A'(wy cos A — UJZ sin A)] cos <f> - (u/ys'm A + uz cosA)<^'sin<^| - (1 — Kit)(uj'xsin<£ + uxd>'cos<p) + K^ail-iPtQZ + go, + F U Q ; - P ' j Q J sin j - (u'xsin<P + UJX<P'cos<P)(P] + Q^ + Pl + Q2tt) - 2ux s in I(P,P j + + P„P' t t + Q aQ' t t) + (-PfS; + EiQt + PnQ!n ~ f . f t , ) ( 6 e s i n <4 - ci' cos 4) + (EjUz + Quiy) [Pj cos A + QJ sin A + A ' ( -P , sin A + cos A)] cos <p + (Pj cos A + sin A ) ( P > 2 + Q^ojy + PJOJ'Z + Q^J,) cos <p + (PuUz ~ QuUy) [P« cos A - Q'a sin A - A'(P„ sin A + Qu cos A)] cos <p + (Pu cos X-Qn sin A ) ( P > , - g w , + P „ ^ - Qu'y) cos 0 It92 dtKdrp - <f>' sin 4> [{Pjuz + QjUy^Pj cos A + 0, sin A) + - Quuy){Pn cos A - Qu sin A)]} + a 2 { wj,(C2 COS *) + W y i ( c 2 COS - {u)'x sin * + ojxd)' cos *)(cos2 ** + cos2 <p*z) - 2ux sin4[cos**(cos**)' + cos<^ (cos<f>*z)'] - *'cos* [cos** (cos <p*z)' - cos **;(cos**)'] - sin*[cos**(cos**J" - cos^cos**)"] + c4 [wx cos **, + wx(cos **)' + (cos ft)"] + c\ [UJX cos 4*v + (cos **)'] + c6 [-tv z cos # - wx(cos 41)' + (cos **,)"] + c'b [-ux cos 41 + (cos 4*y)']} + <*3 {{P}' + Pi) cos 4 cos A + (QJ - ( g ) cos * sin A + (P4 + E!u) [~ec cos 4 cos A — 4' sin * cos A — A' cos 4 sin A - ux sin A cos 4 + UJy sin *] + (QJ — Q )^ [ec cos 4 sin A — *' sin * sin A + A' cos 4 cos A + UJX cos A cos 4 — Wz s i n 4] + {E4 + E-u) Wy s m $ + Vyf cos * — w * si n ^ cos 4 - ux\' cos A cos 4 + Ux4' s i n ^  sin 0] + (0/ — Q„) [—w* sin 4 — Uz4' cos * + u)'x cos A cos * - u ) x \ ' sin A cos 4 — (jJx4' cos A sin 4]} + « 4 { ( - £ { ' + £i')(sin * cos *; + c5) + (QJ' + < £ ) { - sin * cos **, + c4) + (~ij + E!u) [~ ec(sin 4 cos **. + cs) - 2ux sin 4 cos ** + cos 4{4' cos **. — W j , 2 ) cos A — UZC2 cos 4 + UXC4,] + (Qj + Q'J [ec(sin * cos ** - c4) - 2 u x sin 4 cos **. + cos 4{-<f>' cos ** + uyz) sin A + wyC2 cos 4 - wxcs] 171 + {-Et + En) K(-2 sin * cos ** + c4) + ojx(-2<p' cos*cos** - 2sin *(cos**)' + c'4) — *' cos *(cos **.)' — sin *(cos **.)" + uyz{4> sin * cos A + A' cos * sin A) — U)'yz COS * COS A — (JZC2 cos <f> — uz (^2 cos *)'] + (2i + £ J K(~2sin *cos ** - c5) + —2*' COS * COS **. — 2 sin *(cCS **.)' — C5) + *'cos*(cos **)' + sin*(cos **)" + w,,.^-*'sin* sin A + A'cos* cos A) + {J cos * sin A + w 'C2 cos * + uiy(c2 cos *)']} > ; 1 dT ( 3 3 + ^ ( c o s * ^ ( c o s * ^ ) ' - COS*! (COS * y)')] + UyzJj(Vyz) + « 4 { ( - P j + £ 'J[-^^(cos*;) + A((COB*;)')] 3 d 3 + (-£, + Z J K K ^ ( c o s * y ) + _((cos *:)')) -+ u ^ K ) ] } } ; 172 1 8U 2 J. • , , = Kueg cos£ <p sin ip cos •/» ite2 dxp + Kateg^ai{-(P] + Q 2 + P 2 + <£) cos 2 </> sin y cos i> 1 d & ^ + a 2 { - [cos 0;—(cos <f>*) + cos fz—(cos f z ) ] - / y J — ( / j „ ) } + a 3 {-(£/ + Z„)[" sin V cos r + /x — ( y ] + + £.)[i£<"»*;) - '.-£«.> - ^ <Wl 0 - Degree of Freedom (Roll): 1 ':<£> = + { O! {(Pjcv2 + Pju'z + g^Uy + QbJyK-Pu sin A + cos A) + (P'uuz + P_ X - Q>v - Q X ) ( - P „ sin X-Qu cos A) + (Pjw* + ^ W y ) [ -P j sin A + cos A - A'(P/ cos A + Q sin A)] + (P.w, - Q a W j , ) [-P'„ sin A - cos A + A'(-P„ cos X + Qu sin A)] } + oc2{cioj'yz + c\tjy2 + (c 3 s inA) ' [ -w x cos r * + (cos </»*)'] + (c 3cos X)'[ux cos r * + (cos^*)'] + c 3 sin A [-wj. cos <f>*z - wx(cos + (cos 0*)"] + c 3 cos A [u'x cos r * + wx(cos <j>*y)' + (cos } 173 + as{-(Pi + E l l ) sin A + - Q") cos A + (Pj + £„)(<* sin A - wx cos A) + (Qj - Q'J(-ec cos A - wx sin A) + (Ej +Z a)(-w IcosA + wxA'sinA) - (Qj - Qa){u)xs'm\ + a;xA'cosA)} + «4{(-rj' + Z" )c 3 sin A + (<g + Q^c3 cos A + (-Pj + Ptt)(-ecc3 sin A + u xc 3 cos A + uyz sin A - w 2ci) + (Q^ + Q[,)( - ec c 3 cos A - wxC3 sin A + uyz cos A + uyci) + {-Pj + P„)[wxC3 COS A + UJ X(C3 cos A)' + u)yz\' cos A + u>'yz sin A — u>'zci — UJZC'I\ + ( f i l + [ _ w * c 3 sin A - wx(c3 sin A)' - uyzX' sin A + wj,2 cos A + oj'yci + Wyc'x] } > ; - L — = - ( l + V')[(l + V ' ) s i n r C O S r + (l-ir, t)a; xcos r] It92 9<f> + Kat { a x {-(1 + V') cos r [(-££[ + PJQ, + £.g, - P'tt£J + u x ( £ ? + Q 4 2 + E l + Q2u)] - (1 + V') sin 0[(P,w, + QjUyKPu cos A + sin A) + ( E u " z - Quy)(Eu cos A - sin A)]} + a2 {-(1 + v') cos <i[a;x(cos2 c6* + cos2 c**) + cos r*(cos c>*)' - cos c6^ (cos <f>*y)'] + u l [cos r ^ ( c o s c4*) + cos ^ ( C O S <£!)] + W x[(cos r:)'^(cos^) - (cos r*)'^(cos^)] + [ ( c 6 8 ^ y - W , C 0 8 ^ ] ~ ( ( c 0 8 ^ ) f ) + [(cos + ux cos fy] — ((cos 0*)') + wy2 174 - a 3 ( 1 + 1>') {[E4 + E!u) sin * cos A + (Qj - Q/J sin * sin A - (Ei + £«)(wycos* + sin * sin A) + - Q j ( w 2 cos* + ux sin* cos A)} + « 4{(£; - £ L ) [ - ( i + V'')cos*cos*; + u,x-^(cos<t>:) - A( ( c o g^)')] + (fi! + V'')cos*cos*y + ^ A(cos*;) + (^(cos**)')] + [-E4 + Eu) [ - ( ! + •/ , ' ) ( c o s ^ ( c o s + 2 w * c o s 0 c o s - uyz sin * cos A) 3 3 3 + 0^(0;*—(cos**,) + — ((cos*!)')) - w,— + (Qj + Q t t ) [~( 1 + $')[-cos <f>[cos d>*y)' + 2C J 2 C O S * C C S ** . + 0Jyz sin* sin A) d d d + ux[ux—[cosfz) - —((cos*;)')) + u y — ; —:—— = Ku eg cos */» sin * cos * It92 <?* + Kateg { oi {-[Ej + Q2 + £ 2 + Q2) cos2i/> sin * cos * + [Ejlz + QjlyKEj cos A + Qj sin A) cos * cos ip + [EJz - QJy)[P-n cos A - sin A) cos * cos ip} 1 3 3 3 + a-2 { - [cos *y ^ ( c o s **,) + cos **. ^ ( co s **z)] - lyz — [lyz)} + a${-[Ei + Z«)(-^sin* + /x cos* sin A)cosi/» + (Qj — Qtt)[~lz s i n* + lx cos* cos A) cost/;} 1 3 3 + a4{[-Ei +£,)[3^(cos*;) -/j, 2cos^cos*sinA-/ y—((„,)] X 3 3 ] 1 7 5 X - Degree of Freedom (Yaw): —•—; = (1 - Kit)(jJ, + #ae {<*. { ( - £ , 0 , " + Fj 'g, + £ t t g -EllQJ - eeC-fig; + £Ja + £„fi'„ - £ ' « f i j + u/,(£? + Q? + El + Q 2 ) + 2W , ( £ ,P | + + £„£ ' „ + + * 3 { - ( £ , + £ . K - (£| + £L)«, + {Qj- £ , K + (fi! - fi>4 + <*4 { ( - £ / + £ « ) K C O S ^ + WX(C08 <py)' + ( C O S r ! ) " ] + (S + fiJ K c o s £ + "x(cos r;y - (cos <g"] + (-£|+£L)[w.C08^ + (cOS^)T + (Sl + 2L)[wxco8^-(cos^y)]}}; + (£.«. - S.w») [ £ - ^ ( w « ) - fi. Aw]} + a3{(£i + + (fi! - fi'J^K) + ["(£, + En)J^M + ( f i , - <?")^M } + « 4 { ( £ | - £ L ) [ C ^ ( C 0 S ^ ) - ^ ( ( C O S ^ ) ' ) ] + (fil+fi'jK^(cos^) + ^((COS^)')] d d d + (~Ej + P n ) [ w 2 — ( c o s ^ ) + a , x — ( ( c o s ^ ) ' ) - w„«—(w,)] + (fi, + f i j [ w % ( c o s c i : ) + wx—((cos^)') + w^^K)]}} ; + (£.'.-fi.',)[£.^ ('.)-a|f(W] +«3M-(£i+£•) j^w+(a - fi.)|f('*)} + 04{(-a + £.)[i±(c08^) - lyZ^(ly)} + ( a + f i « ) [ ^ ( c o s ^ ) - '»* Jx('*>]}} • • • • (J / /-7) 1 7 6 P, - In-plane Vibration of the Lower Appendage; + on{u)'x cos (fc + wx(cos <pV)' - (cos <p*y)"} } ; — = Katl ai{-Q\ux + £ , ( « * + UJ2z) + Qpvujt} - a3uxuy - a 4 {w 2 cos* y - uzuyz + UJX(cos<f>*z)'}} ; - a 4 { i c o s * y - y y , } } + / T ^ P , . • • • ( J / J . 8 ) Qj - Out-of-plane Vibration of the Lower Appendage: W J t ^ = K a t i a i { ® - ^ - u ' x E { - W x £ l } + azuJ'> + <*4 {UJ'X cos 4>y + UJX (cos * y ) ' + (cos *;)"}} ; -^ U- = Kat{<*l{P}Vz + 0,{"l + UJ2) + PjUJyUJ;} + (X3UJXUJZ + a 4 { u j 2cos**. + ujyUJyz + w x(cos **,)'}} ; + a4 { i cos # - J,/ y i} } + e^Q,. • • • (J//.9) P - In-plane Vibration of the Upper Appendage: = *-{«»{£: -<QV +«a«i - a 4 { w x cos # + w x(cos <p*2)' - (cos <6*,)"} } ; ^ ^ r = K a t { < x i + £„(a;2 + wj) - Q ^ c , } - a3w, + a4{cj 2cos**, - w 2w^ + W x(cos*;)'}| ; + a 4 { ^  cos * y - lylyZ} J + KatevPn . Q u - Out-of-plane Vibration of the Upper Apendage: + a 4 { a ^ cos 0* + w x(cos <f>y)' + (cos **)"}} ; / t 02 ao„ «. — + or 4{w 2 cos fc + a/j,Wj,z + a;* (cos **,)'}} ; + a 4{^COSc6* - Izlyz}} +KatCvQu. APPENDIX IV - MATRIX [M] In general, [M] is a 7x7 symmetric matrix representing the coefficients of the second order derivatives of the seven generalized coordinates. Its elements are listed below; M n = cos2 d> + (1 — Ku) s in 2 i> + Kat { a i { (£? + + El + Ql) s in 2 * + (Pj cos A + Qj sin A) 2 cos2 <p + (P„ cos A - sin A) 2 cos 2 d>} + a 2 {(C2 cos <f>)2 + sin 2 <j> (cos2 <f>*y + cos2 c£*) + c\ 4- c 2 — 2 sin c/>(c4 cos <£* — C5 cos <£*)} + az sin 2 r {(Pj + E„) sin A - (Q, - ) cos A } + 2a4{( -P j + P„)[-sin r(-sin<£cos <p*y + C4) — c 2 cos2 d> cos A] + (Qj + [- sin <p(-sin r cos c/>* + C5) + c 2 cos2 <f> sin A]} |; M 1 2 = Kat \ ari{(PjCosA + Q /sinA)(—P^sinA 4- Q. cos A) cos <f> + (Ea cos A - Q u sin A ) ( - P u sin A - cos A) cos 0} + a 2 C i c 2 cos<p + a 3 sin <f>{(Ej + En) cos A + (Q, - Q J sin A} + OCA{(-EJ + P n ) ( - C i cos <p cos A + c 2 cos <f> sin A - c 3 sin <j> cos A) + (Qj + Q J ( c i cos sin A + c 2 cos ^  cos A + c 3 sin 0 sin A)} > ; M i 3 = -(1 - Ku) sin d> + Kat j -aqf jP 2 + Q] + E l + < £ ) sin <6 + a 3 c o s r { - ( P j + P j s i n A + (Q i - Q J c o s A } + «4{(-P/ + P„)(-sin^cos r* + Ci) M\i, = Kat \ &iEj sin <f> + a 3 cos (/> sin A 178 179 Afi6 = Kat j ^ l f i a S^ n ^  + a 3 c o s $ c o s ^  4 - a4(sin <f> cos <£* + C5) j ; Mn = Kat | —aiP„ sin <f> — az cos cos A + a^-s in^cosc^* 4 - c 4 ) j ; M 2 2 = l + K a t{ai{(-P/sinA + Q /cosA) 2 + (P t,sinA + Q j icosA)2} 4 - ct2{c\ 4 - 4 } + 2a 4c, {{-Pj + P J sin A + (Q, + Q J cos A } } ; M 2 3 = if a t { a 3 { - ( P / + P J cos A - (Q, - QJsinA} + « 4c 3 {(-£, 4 - P J cos A - (<?, + QJ sin A} } ; A f 2 4 = i f a ( | — a 3 s i n A — ar4c3sin A} ; M25 = Kat | c*3 cos A 4 - or4c3 cos A } ; M 2 6 = -K"at | - a 3 sin A 4 - a 4 c 3 sin A} ; A/27 = Kat j — a 3 cos A 4 - « 4 c 3 cos A} ; M 3 3 = (1 - i f * ) + Kat<*i{P] + g2 + P 2 4 - Q 2 ) ; M 3 4 = JTataiOj 5 M 3 5 = -Ka,ctiPj ; M 3 6 = -AT a , a iQ a ; M 3 7 = KalaiPu . For i, j=4, . . .,7, The Mj.'s in Eq. (3.32) corresponds to a particular case with P( = Q , = E u = Q u = o . APPENDIX V - HOMOGENEOUS SOLUTION OF EQUATION (4.7) Taking Laplace transform of the homogeneous form of Eq. (4.7) gives; S2 + C U 2 ' s2 + 3Kit Kata3{s2 + 3) Kala3(s2 + 3) ' Q(3 (s 2 - l -3 ) s2 + u 2 0 a3(s2 + 3) 0 s t p 0 + Vo +' Kata3[s(Pj0 + Pu0) + Pj 0 + £' o 0 ] sEio + fio + aMo + #>) sEuO + £ ' u 0 + "aCs^ o + Vo) (V.la) ' s 2 + 1 + 3Kit Kata3(s2 + 4) - t f a t « 3 ( s 2 + 4) ' ct3{s2 + 4) s2 + 1 + UJ2 0 s2 + 1 + u>2 a 3 ( s 2 + 4) 0 f S*o + <f>'0 + if f l t a 3 [ 5 ( ^ 0 - Q J + Q ; Q - Q^ 0 ] 1 s^o + & + a 3 ( s ^ + #>) s ^„o + ^'uO- a3 (^o + *0) = < (7 .16) The above two sets of equations are uncoupled, which can be solved independently. Equations in (V.1a) are solved first. The determinant of the 3x3 matrix on the left hand side of the equation can be written as. where: determinant = d i ( s 2 + UJ2 )(S4 + a^s 2 + d3) 1 - 2 K a t a 2 z ; di(s2 + n2^2 + n2pl)(s2 + n2p); (V.2) di d2 = dz = ntpi = n]i = U z K i t + u 2 r -\2Kala\)\ di ± - [ 3 K i t u 2 - 1 8 K a l a l ) ; di \id2-\ld\-Adz); \{d2 + \ld2-Ad3); 180 181 and nti>i < npi < np . The equations in (V.1a) can now be written as: { # o + Vo + #««a3 [*(£io + £ . 0 ) + £lo + £ « o ] } iT f lett3 f ( 3 ~ n 2 i ) (3 - f f l ) , rfi(»Ji-»Ji)4*2 + nJi) (-2 + "Ji)J { 2 a 3(5^ o + Vo) + «(£»o + £ . 0 ) + PJO + £ ' « o } 5 _ t t 8 r ( - n 2 1 + 3 ) {*Vo + Vo + a^|Qf3[a(£,o + £ . 0 ) + £Jo + £ « o ] } 1 f ( - " 2 i + ra2) +"1)1 » S , ) t ( « a + » j i ) (^ 2+^i) J { s £ , 0 + £ / ' o + O r 3 ( # 0 + Vo)} Katal f ( - n?M + 3 ) 2 ( n a , - - n a t l ) > i - » l i ) ( " ? - » J i ) ( » J - » J i ) l (*2 + »Ji) , ( - ^ + 3 ) ^ - ^ , ) , (*2 + n2) J { ^ ( £ / o - £ u o ) + £ / o - £ ' u o } ; ( - n ^ + S ^ - n ^ ) -»*; + a ^ - n ^ ) (s2 + n2pl) + {,  n}P f ( - ^ i + 3 ) ( - ^ i + 3 ) l - " l J <*i(»?,i - n j j t (s 2 + n J J (s' + ift) J { s V o + V 0 + tfW*(£jO + £ . 0 ) + £lo + £«<>] } 1 f ( -" 2 i + n2) (-n^ + n 2 ) | { s £ a 0 + £ « 0 + "3(sVO + Vo)} 182 di(nli - «Ji)(Rp - »Ji)(nJ " "Ji) t ( 5 2 + nji) (-nj, + 3)*(nJ - nj,) (-^  + 3 ) ^ - ^ ) , { « ( - £ f 0 + £d » ) - £ i o + £ i o } - ••(^•3) D e n 0 t l n 9 : W « + <f>'i n^,i ti ^ s = \/*8 + (^)2; Ai=tan^(^); *o A. 2 = t a n - i ( ! ^ i o ) ; * 0 A3=tan-i(»); * 0 where i pi, pu. The inverse Laplace transform of Eq. (V.3) gives the solution for \p, P( and P degrees of freedom as; ^ = -J-TT1 —M^n\ ~ n l i ) ~ 2if««a 2(3 - n j j ] sm(n010 + "U» p i - n vi) + ^ 2 [ ( - ( n 2 - nJO + 2 i f a l a i ( 3 - n^ )] s'm(nplQ + /^2)} + ^ B t t t 3 ^ W p 2 ^ f [^p/i s i n ( n ^ i 5 + Ppti) + AP*I + Ppui)} ddnPi ~ nvJ 1 - [API2 s'm[npid + ppt2) + APU2 sin(npi0 + ppu2)}} ; Pi = ? ? ( 2 1 " Ki\(M^i* + M + ^ 2 sin(npi0 + /fy2)} 1 183 f r-Katai - ^ 1 + 3 ) ( n ; - 2 n 2 1 + 3 ) , x j n J i + n ^ s m ( n 0 i * + W 4 rKataK-n^+Z)^* - 2nJ t + 3) 2 2 i A -Katalin^-nlJi-nl + V^ . , i r a l a i ( n J - 3 ) rfi(»Ji-»Ji)(np-nJi)(n5-nJi) x {-Apuii-nli + 3)(n 2 - n 2 x ) sin(n 0 1 0 + /? p„i) + V2 ( - n p i + z)(nl ~ nli) sm(n pi0 + ppu2) + " 3)(nji - 4i) s i n ( n p * + /W)} J £ . = ? ? a1 ~ K2l\ I - s i Q( n^ g +M + A n s i n K * g + flw) } 1 + Mrfi - n\x) J A r - ^ 3 ( - ^ i + 3 ) ( n J - 2 n 2 1 + 3 ) , . X \AP»i [ \n2 _ n a j 2 »Ji + »JJ sin(n v H0 + /? p o l ) r g ^ Q t 3 ( - " ; i + 3 ) ( n ; - 2 n ; 1 + 3 ) _ 2 i + ^p«2 7 ~ 2 — - X T + n p l - sin(n F l 0 + £ p t t 2 ) * p pi' + ^ I W - » J i ) ( « J - " J t ) J + ^ / Kai4(nl ~ 3)  x {-^p/iC-nJx + 3 ) ( n J - n 2 1 ) s i n ( n v , 1 0 + ^ p / 1 ) + ^2(-nJi + 3)(nJ - nJO sin(n pi0 + /? p / 2) + ^ p/3 ( n j - 3 ) ( n 2 1 - 4 1 ) s i n ( n / + /? p / 3)} ; • • • (V.4) Similar procedure is applied to the set of equation in ( V . 1 b ) to obtain the response in <t>, Q| , and Q. u generalized coordinates. Determinant 184 of the matrix on the left hand side of the equation can be written as, where; determinant = dx(s2 + 1 + w 2 ) (s 4 + <*4s2 + ds) di = j-[{l+ 3Kit) + (1 + <4 ) - 16Kata23] ; ds = j-[{l+ 3Kit){l + u>2)- 32Kala23] ; (V.5) and „2 n<t>l nql = Ifa-y/dl-ids); n 2 = 1 + w? ; Rewriting the equations as: - "^l) L n2 )l (s2 + n2ql) rfi(n;i-»Ji)4«2 + »Ji) (*2 + ^ i ) ( - " S i+4) ( - »J i + 4) n 2 ) i * ( " ; i - » j , ) l ( « 2 + » j i ) ( • 2 + » j i ) { 5 ^o + ro + ^ . i a 3 Wfio - Q t t 0 ) + & " SLol} rfiKi-»Ji)i («a + »Ji) (*2 + »2i) J di{n]x - nJJCn} - n\x){n\ - n2ql) 1 + n 2J (-^ i+^ K-n2,) +4)^-n2^ (« 2 + »5i) (*2 + " 2) J + ro + [-(a0 - Q J - i - & - Q'J} Kat^l f(-^i + 4)2K-^i) di(n2qi ~ »Ji)(»J - »Ji)(«5 - "2i) * («2 + »Ji) ( - n f r + ^ n ' - n ' , ) (-n2 4- 3)2(n^  - n2,) ^ (^2 + n2i) (*2 + »J) ' and taking inverse Laplace transform gives the solutions for <p, Qj, and degrees of freedom as: ^ = A 1 „ 2 J ^ i KnJ - n 2 1 ) - 2/f«,ol(4 - n j j ] sin(n^0 + + ^2[(-(nJ - n2ql) + 2Kata\{A - n2ql)] s\n{nqle + /^2)} KatOtz{n2 — 4) r r186 - [Aqt2 s'm(nql9 + (5ql2) - Aqn2 sin(noi0 + Pqa2)]} ; 3 a 3 ( l - ^ t ) f g . n + } + g . n ( Q + fi A 1 + , r-Kata2{-n2+4)(n2 - 2n2 + 4) i x { ^ J ; 2 A + sm(n^ « I L ( n « - n J i ) J A [Katal{-n2ql + 4)(n2-2n2ql+4) 2 i JST.io§(n; - 4) - Aqu2{-n2ql + 4)(n2 - njj) sin(nal0 + /? g„ 2) - - A j f i 3 ( » } - 4)(n21 - njj) sin(ng0 + /?««3)} 5 1^— % T { ^ l srafn^fl + p4>i) - A& sin(nfll0 + fa)} d l i n q l ~ n4,l) K * + Mnql ~ % l ) v L Knq n<f>\) + V 2 [^^^t^)" 2"^^ + * - 1 - H -M* I* + ^ 2) " ( - j - W - n y J s i n ( v + M s t / f f l t a 2 (n;-4)  ^ l ( ^ l - - nll)(nq ~ n\\) x {-Aj-iC-nj! + 4){n\ - n2qX) sin(n^0 + pqlx) - Aql2{-n2qX + 4)(nJ - n j j sin(nol0 + /fy2) - i4j£3(»; - 4)(nj! - n2x) sin(ng0 + pql3)} ; • • • (V.7) with: An = \k + (^-)2; V n<s>\ Ai2 Aiz Pa •o = ta,-( "•«<•): A 3 = t a n - ' ( f ) ; where i=0, q l ; qu. 

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