DYNAMICS OF GRAVITY ORIENTED A X I-SYMMETRIC THERMALLY FLEXED SATELLITES WITH APPENDAGES by CHUN KI ALFRED NG B . A . S c . , University of British Columbia, 1984 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E Department of We STUDIES Mechanical Engineering accept this thesis as conforming to the required THE UNIVERSITY OF BRITISH COLUMBIA November, © CHUN KI standard 1986 ALFRED N G , 1986 In presenting advanced Library agree degree shall that purposes this thesis at the The make representatives. be granted extensive by the not of Mechanical The University of British 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: November, 1986 be Head that for copying of Columbia the my copying of requirements Columbia, I agree reference allowed without Engineering of of British available It is understood for financial gain shall Department for fulfilment University it freely permission may in partial this and thesis Department or my study. publication written of that I for or by for an the further scholarly his or her this permission. thesis ABSTRACT The equations of motion for a satellite with a rigid a pair of appendages deforming due to thermal effects radiation are derived. The dynamics of the system stages: (i) librational thermally flexed of the solar is studied in two dynamics of the central body with appendages; (ii) coupled central body and quasi-steady librational/vibrational dynamics of the spacecraft. Response of the system system study parameters and effect is investigated of the thermal deformations conditions although of system the corresponding However, in eccentric flexed combinations rigid a range of assessed. The indicates that for a circular orbit, the flexible system unstable under critical can become parameters and initial system continues to be stable. orbits, depending on the initial conditions, thermally appendages can stabilize or destabliIize the system. Attempt made to obtain an approximate closed-form problem response numerical accuracy with numerically over to quickly assess (analytical) solution trends and gain better physical is also of the appreciation of characteristics during the preliminary design. Comparisons with results show approximate analysis to be of an acceptable for the intended objective. The closed-form a measure of confidence thus promising effort, and computational cost. ii solution a substantial can be used saving in time, T A B L E OF C O N T E N T S ABSTRACT " LIST OF FIGURES v LIST OF S Y M B O L S x 1. INTRODUCTION 1 2. 1.1 Preliminary 1.2 Scope of the 1 Investigation 11 FORMULATION OF THE PROBLEM 12 2.1 Preliminary 12 2.2 Position of 2.3 Solar Radiation 2.4 Shape of 2.5 Determination 2.6 3. Remarks Remarks the Satellite in Space 12 Incidence Angles 15 the Thermally Flexed Appendage of Kinetic and Potential 18 Energies 22 2.5.1 Assumptions 22 2.5.2 Coordinate system 23 2.5.3 Kinetic energy 26 2.5.4 Potential 28 Equations of energy Motion 30 LIBRATIONAL D Y N A M I C S OF S A T E L L I T E S WITH THERMALLY APPENDAGES Remarks FLEXED 32 3.1 Preliminary 3.2 Equilibrium Orientation 33 3.3 Stability .40 3.4 Motion in the Large in the 32 Small 3.4.1 Variation of 3.4.2 Numerical parameters .49 method method .49 55 iii 3.4.3 4. results 57 COUPLED LIBRATIONAL/ VIBRATIONAL DYNAMICS WITH THERMALLY FLEXED A P P E N D A G E S OF S A T E L L I T E S 69 4.1 Preliminary Remarks 69 4.2 Equilibrium Orientation 69 4.3 Numerical 4.4 5. Discussion of Analysis of the Nonlinear Equations 78 4.3.1 Appendage disturbance 78 4.3.2 Central 92 4.3.3 Influence of thermal (circular orbits) body disturbance 4.3.4 Eccentric orbits 4.3.5 Influence of thermal (eccentric orbits) Analytical deformation on system 102 109 deformation on system stability 122 Solution of stability 127 4.4.1 Variation parameters 4.4.2 Improved 4.4.3 Discussion of analytical method solution 127 134 results 140 CONCLUDING C O M M E N T S 158 5.1 Summary 158 5.2 Recommendations of Conclusions for Future Work 160 BIBLIOGRAPHY 162 APPENDIX I E V A L U A T I O N OF A P P E N D A G E KINETIC ENERGY II E V A L U A T I O N OF A P P E N D A G E POTENTIAL ENERGY III DETAILS OF THE E Q U A T I O N S OF MOTION 168 IV MATRIX 178 V H O M O G E N E O U S SOLUTION OF E Q U A T I O N (4.7) [M] iv 164 166 180 LIST OF FIGURES Figure 1-1 1-2 1-3 1-4 1-5 1- 6 Page A schematic diagram of the Radio Astronomy long flexible antennae and booms The proposed Orbiter-based be launched in 1989 tethered Explorer Satellite with 2 subsatellite system scheduled to 3 A schematic diagram of the European expected to be launched in 1987 Space A schematic diagram (MSAT) Mobile Satellite of the proposed Agency's L-SAT .5 System 6 Artist's view of the Orbiter-based manufacture of structural components for construction of a space platform 7 Contribution of environmentally induced torques for a typical satellite 8 2- 1 Spacecraft geometry 2-2 Orbital elements defining the position of center of mass of a satellite in space. 14 2-3 Modified Eulerian rotations yp, <p and orientation of the satellite in space 16 Solar radiation 2-5 A 3- 1 equilibrium comparison position X defining an 13 arbitrary incidence angles <p* <f>*, and <p* x y z between Brereton's approach and the present of a thermally flexed 17 approximate solution for the shape (a) Alouette I Appendage; 21 (b) Alouette II Appendage 21 appendage: Coordinates with respect to Xp,Yp,Zp-axes defining thermal deformation and 2- 7 nominal t 2-4 2-6 and Vectors r and lower and upper ( r vibration of flexible appendages u 24 defining position vectors of mass elements on the appendages, respectively A n illustration of the orbit with p=w, the sun relative to the spacecraft. v i=0, showing 29 the position of 32 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 36 central body with relatively light appendages; (c) slender central body with relatively heavy appendages; 37 (d) stubby central body with relatively heavy appendages 38 3-3 Equilibrium 3-4 Effect of inertia parameters on stability orbits 3-5 orientations (a) 0-5 3-7 3-8 3- 9 4- 1 4-2 and ir-6_ in circular orbits by thermal .40 of the system in circular .44 Librational response of the system with K = unstable motion caused 3-6 at 0 1.0, K. = -0.1 showing deflection: orbits; 46 (b) 45-50 orbits; .47 (c) 55-60 orbits .48 Comparison of numerical and analytical solutions parameters: for different inertia (a) slender central body with relatively light appendages; 58 (b) stubby with relatively light appendages; 60 (c) slender central body with relatively heavy appendages; 62 (b) stubby with relatively heavy appendages 63 central body central body Comparison of numerical parameter and analytical solutions Comparison of numerical across the orbital plane and analytical solutions Comparison of numerical and analytical solutions for non-zero spin 65 for a disturbance 66 for eccentric orbits; (a) e = 0.1; 67 (b) e = 0.2 68 Effect of system parameters on the equilibrium orientation: (a) slender central body with small appendages; 72 (b) slender central body with appendages; 73 (c) stubby central body with small appendages; 74 (d) stubby with 75 central body large large appendages Variation of tip deflection at equilibrium dominance of thermal effect vi with 8 showing the 76 4-3 Effect 4-4 System of appendage flexibility response on its equilibrium position 77 with: (a) in-plane appendage disturbance; (b) out-of-plane 4-5 4-6 4-7 appendage disturbance Typical response of the system disturbance. A comparison appendages: 80 for an .81 impulsive appendage . .83 of response for systems with one and two flexible (a) in-plane appendage disturbance; 84 (b) out-of-plane 85 appendage disturbance System response showing the effect appendage disturbances; of symmetric and asymmetric (a) symmetric in-plane disturbance; 87 (b) symmetric out-of-plane 88 disturbance; (c) asymmetric in-plane disturbance; 89 (d) asymmetric out-of-plane 90 disturbance 4-8 System response showing the effect presence of thermal deformation of appendage disturbance 4-9 System response showing the effect disturbance: of in-plane central (a) displacement (b) impulsive 4-10 System (a) out-of-plane 4-11 4-12 body disturbance; 93 disturbance response (b) combined in the 91 94 showing the effect of central body disturbance: disturbance; in-plane and 95 out-of-plane disturbances 97 System response showing the effect of central body disturbance in the presence of thermal deformation Typical response in circular orbits showing the effect parameters and thermal deformation: 100 of inertia (a) slender central body with small appendages; 103 (b) stubby central body with small appendages; 104 (c) slender central body with large appendages vii 105 4-13 4-14 4-15 4-16 4-17 System response in circular orbits showing of thermally deformed appendages: 106 (b) vibrational response 107 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 S y s t e m response disturbance: showing the effect of eccentricity and central body (a) librational response; 117 (b) vibrational response 118 System response showing thermal deformation: the effect of eccentricity and (a) appendage disturbance; 119 (b) central body disturbance, librational response; 120 (c) central body disturbance, vibrational response 121 System response deformation: in eccentric orbits showing the effect of thermal increase in libration amplitude; (b) a decrease 4-19 influence (a) librational response; (a) an 4-18 the destabilizing 123 in libration amplitude 124 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 A comparative solution: study showing the deficiencies of the analytical (a) librational response; 131 (b) vibrational response 132 viii 4-20 4-21 A comparison between for a small appendage numerical and improved disturbance: analytical solutions (a) analytically obtained results; 141 (b) numerical results 142 A comparison between numerically and improved responses with central body disturbance: (a) vibrational analytically predicted response; 143 (b) librational response 4-22 4-23 A comparison between numerical and improved analytical the presence of severe out-of-plane disturbance: 146 (b) vibrational 147 response A comparison between numerical and improved analytical showing the effect of inertia parameters on correlation: K = 0.1, K 0.75; j solutions 149 = A comparison between numerical and improved analytical solutions for a satellite with relatively large appendages and a slender central body: « = 0.5. K = 0.75 152 a 4-25 solutions in (a) librational response; g 4-24 144 f A comparison between numerical and improved showing the effect of a stubby central body: K = 0.1, K . = 0.25 a i ix analytical solutions 155 LIST OF 2a length a^ appendage aj length a 1* 2* 3 a of the central ratio, a/ constants written a 1* b 2' 3 components b body, Fig. 2-1 radius appendage wall b SYMBOLS in terms of p, CJ, and i , Eq. (2.2) thickness of u along X , Y g and Z g axes, respectively; Eq. (2.1) dj i= 1 (V.2) f^, f ^ 5, coefficients in characteristic equations and ( V . 5 ) functions containing derivative gj the zeroeth and first order terms in <p and \p, Eq. (3.32) i = 1,..., 7, functions defining equilibrium state of the system, Eqs. (3.1) and (4.1) h angular momentum i inclination per unit mass of the system of the orbit with respect to the ecliptic plane, Fig. 2-2 ? , j ,k p' V P unit vectors in the directions of X , Y , and Z -axes, P P P respectively ? , j , (c s s s unit vectors in the directions of X , Y , and Z -axes S' S' s respectively k^ thermal conductivity l^ appendage I* thermal reference length direction cosines of R with respect l , x ly, l z of the appendage length, Fig. 2-1 of the appendage, Eq. (2.8) to Xp, Y and Zp-axes, respectively, Eq. (2.24) mass per unit n n^ appendage length vibration of the appendage 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) n p1' n q 1 ' 0 1 ' \p1 n n a t t e n u ated frequencies associated with n n^, respectively; Eqs. ( V . 2 ) and ( V . 5 ) x n^, n^, and solar radiation position intensity, 1360 W/m 2 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 - a x e s , respectively u thermal of deflections the orbital thermal of of the lower appendage in and out plane, respectively; Eq. (2.14) deflections of the upper appendage in and out 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) and F2, respectively, with approximation to 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 undeformed of inertia of the system with appendages about Yp or Zp-axes mass moment of inertia of the central body Yp, and Zp-axes, respectively; 1^= I 3 inertia ratios r n ^ l ^ / I xi about X , P z 3 and m^l^/ 1 , respectively 1 inertia ratios 1 — I / I and 1 - x length I /I . x respectively t ratio, l ^ / I* mass of the central coefficients generalized appendage body in Mathieu equation, Eq. (3.16) coordinates for the lower and upper in-plane vibration, respectively dimensionless ratios P / l ( generalized and P / l , b respectively b coordinates for the lower and upper appendage out-of-plane vibration, respectively dimensionless ratios Q|/ l ^ position the and Q / l ^ , respectively u vector to S as measured from the centre of Earth centre of mass of the system origins of the reference coordinate systems deformation measuring of the lower and upper appendages, respectively; Fig. 2-6 total kinetic energy of the system, appendages, and central body, respectively; T = ~f pp b + a components of T pp Q w ' t n subscripts representing libration, thermal deflection, and vibration contributions, respectively total potential energy of the system, appendages, and central body, respectively; U = ^ pp 3 total strain energy components of p p u a + ^ ^ 4 - U g of the appendages with subscripts representing libration, thermal deflection, and vibration contributions, respectively intermediate axes during the Eulerian principal coordinate system rotations, Fig. 2-3 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 vertical, Y along g the in the direction of the local local horizontal, and Z towards S O the orbit normal, Fig. 2-3 principal coordinates for the upper appendage origin at S , Fig. 2-6 u integration i= factor 4, 1 in Mathieu equation, Eq. (3.18) , constants, Eq. ( I I I . 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 sinusoidal in the functions of the analytical solution eccentricity emissivity of the appendage material functions of rotation 0e~ e and 6, Eq. (111.1) across the orbital plane, Fig. 2-3 a solar radiation rotation Fig. incidence angles, Eq. (2.5) about the axis of symmetry of the satellite, 2-3 gravitational constant true anomaly angular velocity of the system longitude of the ascending spin at the perigee point node, Fig. 2-2 parameter, Eq. (3.7) Planck's constant argument of the perigee, Fig. 2-2 fundamental frequency angular frequency ratio, of the appendage cj^/d^ velocities of the system about Z -axes, respectively; Eq. (2.18) rotation i//e~ in the orbital plane, Fig. 2-3 a xiii X Yp, and * fundamental mode of a cantilever beam, Eq. (2.16) 1 Subscripts e equilibrium 0 initial position condition Dots and primes represent differentiation with respectively. The word respect to t and "system" refers to the central body appendages. xiv with 8 ACKNOWLEDGEMENT The the author wishes to express his gratitude to Prof. V. J . Modi for guidance and encouragement throughout the study his help during the initial critical The Natural investigation reported Sciences and Engineering and particularly for stage. in the thesis was supported in part by the 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 modern spacecraft with its lightweight, flexible, deployable form can (i) and essentially rigid. However, for a members of solar panels, antennae, and booms, it is no longer be well-emphasized Ever by several true. This point examples; increasing demand on power f o r operation of the on board instrumentation, scientific experiments, communications has been reflected (ii) 1.2kW Use of large members antennae to detect For identifying Laboratory may consisting fact, N A S A Explorer extraterrestrial (RAE) satellite used radio sources, the Applied Tethered of two spacecraft Physics Orbiting Interferometer (TOI) connected has shown considerable its feasibility L-SAT four 228.2m signals (Fig. 1-1). tethered by a line 2-6km subsatellite system, extending to 100km, studies to establish (Fig. 1-2). configurations of the next generation (Large long. In interest in exploiting application has initiated, through contracts, preliminary Preliminary missions. For of the Johns Hopkins University once proposed a of the Orbiter based (iv) 1.14m x 7.32m each, to be essential in some low frequency gravitationally stabilized and (CTS, Hermes) launched in of power. example, Radio Astronomy (iii) Satellite 1976 carried two solar panels, generate around systems, etc., in the size of the solar panels. The Canada/USA Communications Technology January in the S A T e l l i t e System, Olympus), DBS 1 of satellites such as (Direct Broadcast Figure 1-1 A schematic diagram of the Radio Astronomy Explorer Satellite with long flexible antennae and booms. 3 Orbiter Figure 1-2 The proposed Orbiter-based to be launched in 1989. tethered subsatellite system scheduled 4 System), M S A T towards (v) (Mobile SATellite System), etc., suggest spacecraft with large flexible members (Figs. 1-3 and Space engineers are involved in assessing gigantic space cannot from a trend stations which feasibility be launched of constructing in their entirety the earth, but have to be constructed in space through integration of modular subassemblies. In-orbit assembly orbiting station of enormous such as Space Operations Center (SOC) and futuristic design of Solar Power Station (SPS) suggest large space with an increasing role of structural flexibility control considerations (Fig. structures in their dynamical be its control has become emphasized a topic a simple proposition, even J f the system the problem environmental forces such effects), earth's magnetic 1-6 shows enormously is by no means is rigid. Flexible character of the complex. The presence of as the solar radiation (pressure and thermal field, free molecular forces, etc., which are capable of exciting elastic Figure motion of considerable importance. It should that prediction of satellite attitude motion appendages makes and 1-5). This being the case, flexibility effects on satellite attitude and 1-4). degrees of freedom, add to the challenge. contribution of several environmental forces as functions of altitude for a representative satellite G E O S - A . A t low 1 altitudes (<1000 km), as can be expected, the atmospheric dominant. The gravity gradient contribution inverse square manner. Effect diminishes with altitude of the earth's magnetic field orders of magnitudes smaller than the solar pressure, which independent over the range effects are in the is several is essentially of the earth's orbit. It is of particular significance that near the synchronous contributions (even for this satellite altitude, gravity and solar pressure 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 0 10 I 5 n 10 | 2 | TT 10 5 I 2 I 5 I ,5 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 flexible members mentioned before. Corresponding elastic characteristics would naturally control of The resulted the dynamics, stability, interest in space science and enormous body satellite, Sputnik, in 1957. area of on and and satellite. contemporary in an reflect changes in inertia the of literature since Broadly speaking, the satellite dynamics may be classified dynamics, stability, and control of rigid (b) effect of forces on the environmental launching of available as (a) technology has the first literature in the follows: satellites; attitude dynamics of rigid satellites; (c) large, flexible system dynamics and (d) dynamics and control of environmental forces. Most of the available control; flexible systems in the literature belongs to the presence first three of categories 2-4 and has been reviewed at length the other hand, behaviour of by Modi, Shrivastava, and Tschann flexible satellites when exposed to . On free molecular environment, solar radiation, Earth's magnetic field, etc., remains virtually unexplored, except for some simplified preliminary 5-15 Brereton, Kumar, Goldman, Yu, Bainum, and Krishna 5 Modi and thermally flexed for the deformed Brereton due to beam space, stability charts studied librational dynamics of studies by a free-free Modi, beam solar radiation. Using a quasi-steady representation and in phase the concept of integral manifold were obtained. In general, flexibility of tended to reduce its stability for all positions of the sun the satellite (solar aspect 10 angle); however, the reduction was Using considered to be of no major a similar approach, Modi and Kumar concern. studied librational dynamics of a rigid satellite with thermally flexed plate-type appendages. The study affect an indicated the thermoelastic behaviour the satellite performance. In general, flexibility of appendages increase in the amplitude decrease and of appendages to adversely and average caused period of libration, and a in the stability region. The inclusion of solar radiation pressure eccentricity effects further deteriorated the stability of the satellite. Goldman^ used anomalous behaviour case thermal of Naval deflection of appendages to explain the Research Satellite 164. However, as in the of the previous study, he did not consider vibration of the flexible members. Yu studied thermally induced a rigid, orbiting body. The body was pointed away from vibration of a beam found to be stable the sun but unstable when connected to if the beam it pointed towards the sun. 9 However, the results are controversial the opposite results using other More recently, Bainum as Augusti 10 and Jordan obtained approaches. 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 and of solar pressure on control orientation control. The simplified provide only preliminary data laws to achieve the desired shape linear analyses were intended to indicative of trends. Subsequently, the authors 14 15 extended the analysis to account showed that librational response an for thermal effects * . The results for a thermally deformed appendage to be 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 appendages, nominally aligned along the local vertical stabilized begin with, thermal carried out and an formulation problem analysis of an using the classical better appreciate the dynamical of the system, it would and be analysis of equilibrium configuration and nonlinear analytical response model is developed of the spacecraft with thermally flexing parametrically. An comparison with response response and and numerical before, equilibrium improved physical of next results should be design phase of this class of and a parametric out numerically. A and its validity data. Finally, librational and vibrating response appendages are studied results. better appreciation of the interactions between attitude dynamics, vibration, and deformations. The thermal solution to this general problem is compared with the numerical is on behaviour that end, vibration carried configurations and closed-form Throughout the emphasis procedure. are purposely suppressed through developed a detailed nonlinear Lagrangian assessed is presented. A s in four stages. configuration as a function vibration of the flexible appendage. To equations under the cantilever beam is useful to isolate the effect terms in the governing simplified orbiting angle obtained. This is followed by of the problem deformation is analyzed expression for its deformed of the solar aspect To (gravity gradient configuration), free to vibrate as well as deform influence of the solar radiation. The To beam-type thermally complex induced particularly useful during the preliminary satellites. 2. FORMULATION 2.1 Preliminary This of and PROBLEM Remarks chapter begins with a discussion a satellite incidence OF THE in space, followed on the position by the determination angles and the shape of a thermally potential energies of the system flexed of solar radiation appendage. The kinetic are then derived. Finally, using Lagrange's formulation, the equations of motion body and orientation and vibrational degrees of freedom for librations of the central f o r the appendages are determined. The satellite consists of a rigid, axi-symmetric with two flexible beam-type Each appendage, of length with constant l ^ , is assumed mass density, flexural rigidity, and cross length. The joint between the satellite be rigid, i.e., no joint rotation by the gravity earth 2.2 Position is aligned a spacecraft determined i, along is assumed to is stabilized along the local vertical in towards or away from in Space with its centre of mass at S negotiating arbitrary trajectory about the center of force homogeneous, spherical area as the lower or upper appendage, respectively. of the Satellite Consider and the appendage position. The appendage pointing is designated 2-1). circular tube sectional is allowed. Since the satellite gradient, the system equilibrium the to be a thin walled 2a of length appendages attached to its flat ends (Fig. its nominal cylinder coinciding with the Earth's center. A t any instant, the position by the orbital elements an of S is p , i , co, e, R, and 6. In general, p , co, a n d e, are fixed while R and 9 12 are functions of time (Fig. 2-2). 13 Local Vertical Upper Appendage •Central Body a Orbit a Lower Appendage Earth Figure 2-1 Spacecraft geometry and nominal equilibrium position. 14 Figure 2-2 Orbital elements defining the position of satellite in space. center of mass of a 15 As the spacecraft has finite dimensions, i.e., it has mass as well inertia. Hence, in addition to undergo librational motion principal body axes of other hand, X , Y , Z s s satellite 2.3 the motion, X, about Solar Radiation Position of solar radiation between the X the Y its center central represents s the trajectory, g X-axis (Fig. satellite to the the S. On the local vertical, orientation of Eulerian rotations: a pitch Y - a x i s ; and a 2-3). Angles with respect to the and axes, respectively With reference origin at motion, <p, about the vector, u, representing the and Z to mass. Let Xp.Yp.Zp be modified incidence angles, unit is free normal, respectively. Any spatial Incidence the it moving coordinates along the Z - a x i s ; a roll the of body with their can be described by three motion, \p, about yaw about the local horizontal, and orbit the negotiating as (Fig. sun is defined They are direction of defined by the as angles solar radiation, and 2-4). moving coordinate system 1 j k s > the unit vector u can be written as, u = [—ai cos 9 + A 0 2 sin 8]i, A A + [a\ sin 8 + ai cos 8]j t + a k„ 3 = M . + b j + bzk,, 2 t ••• (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». Now, in terms of the principal body coordinates: •••(2.2) Figure 2-3 Modified Eulerian rotations \p, c6, and X defining an orientation of the satellite in space. arbitrary 17 18 ^ A A t, = (cos tp cos 4>)i + (cos ip sin <6 sin A - sin ip cos X)j p p A + (cos ip sin d> cos A + sin ip sin A)fc ; p j , = (sin ^ cos c6)t' + (sin ip sin sin A + cos ip cos A);' p + (sin tp sin A cos A A p cos ip sin X)k ; — p A A fc, = - sin <}> i + cos $ sin Xj + cos 0 cos Afc . p p • • • (2.3) p Hence, substituting from E q . (2.3) into (2.1), u can be rewritten as u = cos <f>* t + cos 4>l j x where; p p + cos <t>* k , z . . . (2.4) p t ^ ! cos <p = b\ cos tp cos <6 x +fc sin ip cos 0 — 2 63 sin c6 ; cosrf>*= 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 = and 2.4 fc{0>u>hPAA) , <p) = <p*(9,u,i,p,\,<p,ip) , Shape of the Thermally The plane is difficult problem Flexed Appendage in which centre to determine j = y or z . line of the thermally flexed and changes with the librational is overcome by taking the projections appendage motion. This of a thermally flexed lies 19 appendage on the , x p Let along Y Y a n d p Xp,Z -planes. p the solar radiation and Z intensity be' q axes are given by q W/m , 2 g and q then its components respectively, where: q = q, cos 6* ; y v g = g,cost£*. •••(2.6) ? Using Eq. (2.6) together with centre line of a thermally flexed ^ = Brereton's equation for the shape of the appendage g i v e : 1 -ln[cos(j5-)] cos<£* ; £ = -ln[cos(£)]cos^; where 6 , and 5, represent deflections y z •••(2.7) of the centre line in the X Xp,Zp planes, respectively. Here 77 is the distance measured undeformed thermal appendage with n= Y and P P along the 0 at the fixed end, and I* is called the reference length given by, r . J i L ^ +^ M L ) ! ^ ) , . Note, Eq. (2.7) represents steady-state solution the differential heat transient solution balance (2 obtained by solving relation for a thin-walled circular tube. The is not included here Goldman^ has obtained thermally flexed ... .8) because its time constant an approximate steady-state solution is s m a l l . 1 for a appendage as ^ = ^) -s^[l 2 + ^)cos^]; ^ = ^ ) c o - « [ l + 5(p)coB^l; 2 -(2.9) 20 where ...(2.10) = The second term expansion on appendage in E q . (2.9) accounts for the effect first term; therefore, the I* are approximately Equations if of the integrand longitudinal deflection. This is small and hence not considered by Brereton. A l s o , the second term the of in E q . (2.8) is small values calculated from as compared to E q s . (2.8) and (2.10) the same. (2.7) and (2.9) are not convenient also has transcendental for integration especially functions; hence, the approximation E q . (2.9) is used in the analysis: = 2 F ( F | = ^ ) Equations cos 0* o s ^ ; deflections condition (<p* or 0* = ...(2.11) c o s ^ . (2.9) and (2.11) are approximately or 7 j / l * . The differences obvious. Thermal 2 } c between the same for small E q s . (2.7) and (2.11) are not of the appendages during the most critical 0) as given by E q s . (2.7) and (2.11) are compared in Figure 2 - 5 . Physical properties I and II satellites. The figure correspond to appendages used on Alouette in 6 j / I* increases shows that the difference with TJ/I*; however, it is negligible for r j / | * less than beryllium-copper 0.6 corresponds to 63m and 75m, appendages, rj/l*= respectively. Appendages of these lengths 0.6. For steel and should be adequate for most satellites; hence, E q . (2.11) can be used with confidence to represent the shape of a thermally 0.6. L * = deflection flexed 0 can mean either of the appendage appendage with L* limited there is no appendage is ignored (l*= »). to between (1^= 0 and 0) or the thermal 21 Material Bending s t i f f n e s s , E l Mass/length, m Radius, a Wall t h i c k n e s s , b Absorptivity, a Emissivity, c b b b b t b Thermal conductivity, k Coefficient of thermal expansion, a , b T h e r m a l reference length, 1* Figure 2-5 Alouette 1 Alouette II Sleei 144 0.102 1.207 Beryllium c o p p e r 6.4 0.021 0.635 0.017 0.900 0.800 0.005 0.450 0.250 cm 45 86.5 W/m°c 11.7x10 105 -6 18.0x10 126 Unifs Nm kg/m . cm 2 - - -6 •c-' m A comparison between Brereton's approach and the present approximate solution for the shape of a thermally flexed appendage: a) Alouette I A p p e n d a g e ; b) Alouette II Appendage. 22 2.5 2.5.1 Determination of Kinetic order to gain system in and thermal deformations, indicated mass is are of for of is mass is simplicity, Considering of high the to be latitude the Earth is motion Since will and altitude shadow. amenable when appendage or introduced the the central mass, the body shift in ignored. orbit, a part O b v i o u s l y , in be to evaluation. shifts small of flexibility, were more parametric vibrating. behaviour dynamics, assumptions system or than physical libration hence the the dynamics deformations effect the and flexed by induced this system assumed in to between greater covered thermally the mass much as simplifying analysis satellites trajectory (iii) made thermally usually centre Except the several This centre appendages (ii) Energies appreciation interactions closed-form general, the of below. approximate In terms better the (i) Potential Assumptions In as and minimal. For this the of the region, sake of ignored. of the fixed end of the cantilevered o appendage to be negligible, transverse vibration of a „ 3 ty with boundary thermally dM r l& + -dx2- dw Here M T dx is flexed that the appendage equation is given 2 the + M T = EI thermal by, dw T >>Wr = > + m d*w b for 2 dtv = 0 ~dx 2 b shown 0 - t conditions: to = EI has 2 4 E I b Yu dx z bending at X 8M T dx moment 0; 0. given at X by = If,. 2 - 1 2 23 Ecx T (x , y ,z )z MT = / e T a a a dA, a J Area where T (x ,y ,z ) e a a is the difference g and the temperature at a point ( x , y , z ) . The integral a a between on the appendage with is over the cross a the ambient sectional area temperature coordinates of the appendage. Assuming Euler-Bernoulli My to be small, E q . (2.12) simplifies to the usual beam equation, E with boundary » - a * l + m > W = > ,dw - 1 3 ) E b „ 3 r I at i — 0; „dw 2 r r a ^ In the present = E I b d^ = 0 & study, thermal t x= were Euler-Bernoulli modes of vibration, longitudinal oscillations, and foreshortening effect obtained shape used in vibration corresponds to the conventional The second and higher lb deformations accounting for My, however, the mode /) ( 2 conditions: dw w = —— = 0 dx analysis • 0 beam. and torsional of appendages are not considered. ) Other environmental drag, and magnetic 5.2 Coordinate disturbances, such as solar pressure, aerodynamic torques are neglected. system Figure 2 - 6 shows the coordinate e three sets of coordinate axes: system used in the analysis. There 24 X p-V p Z : X |> |- |; coordinates any point Y Z of the a n d X ' u' uY U Z on the centre linear these e X p> p' p Y Z line of the X|,Y|,Zj lower _ a x e s r e P r e s e n t principal appendages. The coordinates and upper appendages are u u system involves simple transformation. with coordinates appendage. Let thermal obtained Y first u (X|, 0, 0) on the deflection of the appendage center shift a position with coordinates (X|, yj, Z|) where y| and Z| can now Since of and X , Y , Z - a x e s , respectively. coordinates to Xp.Yp.Zp coordinate Consider a point lower h system with undeformed written with respect to the Referring T from to opposite direction, a negative Figure 2-6 the the point be Eq. (2.11): <p* is measured with respect have line of Coordinates with respect deformation and vibration the Yp-axis (Fig. 2-4) sign is introduced to of while Y| and in Eq. (2.14a). Xp,Yp,Zp-axes defining flexible appendages. thermal to 25 Transverse vibration of the appendage shifts the point to a new position with coordinates (Xj, Y | + V | , Since the Euler-Bernoulli beam Z|+Wj). equation is assumed to be valid and only the fundamental mode o f 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 o f 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), with ... (2.16) Pik= 1.875104; cri = 0.734095. The fundamental natural frequency o f transverse vibration, CJ,, is given by I EIh 2 wi = Wb) Finally, deformation m ll b at a point on the lower appendage is referred to the principal coordinate s y s t e m X Y Z H point are: (-( +a), - ( y X| | + V | H i.e., the new coordinates of the r* ) , z^w^. Similarly, the sequence f o r the movement of a point on the centre line o f the upper appendage is f r o m x deformation) and finally to x , Y + u vibration) where y , z u u v , and w u u V u , 0, 0 t o x z + u w u y z u (thermal (thermal deformation are given by: plus 26 *« = 2(77) cosrf),; ^ = Q,(<)$i(i.). •••(2.17) Finally, position of the point with respect to Xp.Yp.Zp-axes, ' given s by the coordinates (x +a, Y +v , z +w ). u u u u u 2.5.3 Kinetic energy The kinetic energy of an axi-symmetric rigid central body with I =I = I can be written as, z T . . = 1M[# + [Rd) ) + l[W 2 x c b + J ( W J + J)], W u = A — (0 + i>) sin<p ; where: x oj = 4> cos A + (0 + ^) cos <f> 8in A ; y tv = —d> sin A + (0 + ^) cos c4 cos A . • • • (2.18) z For the lower appendage, velocity at a point with coordinates (-( !+ ), -(Y|+|).|+ |) 9 x a v V {)<l z w i s i v e nb y. pp. = [V + u {zi + wi) + u.(yi + t>j)]s x p y + \ y ~ U:i t v + [V, - x + ) ~ u {zi + u>i) -yta x (J (yi + vi) + uj (xi + a) + x y i>i]j k P + vn]k, , • • • (2.19) where Vx Vy and V z represent components of the velocity of the centre of mass, S, in X Y and Z directions, respectively: 27 V = (R cos rp + x V = (i2 cos ip + y R9 sin ip) cos <p ; R9 sin sin <p sin A — (R sin ip — R9 cos V = (R cos \p + z + (jR Similarly, for a point (x + a, y u u+ v , V„, R9 sin rp) sin $ cos A sin ip — R9 cos rp) sin A . on the upper appendage with coordinates z +w ), u app u u . = [V + u (z x y + tu«) - w {y + v )]i u z u + [Vy + ^ ( x . + a) - u) (z x + [V, + u (y„ The kinetic energy TCVP. = ~ - of the derivation u u p + ty ) + y„ + i) ]j a B p + v ) - ojy(x + a) + i„ 4- w \k . x Details cos A ; u u n • • • (2.20) p of the appendages can now be calculated a s , { fj V£ + jfVl a p p app are given in Appendix I. dx*}. ••• (2.21) It shows that , in general, can be written as T p. = m l [R 2 ap b b + {R9) } + T + T + T + 2 t t v where the subscripts I, t, and v represent vibration, respectively. For instance, T| kinetic energy t 7} + )t T Uv + T, + t v libration, thermal deflection, and librational motion and thermal deflection. energy of the satellite, T , is given by T () represents the contribution to the due to a coupling between The total kinetic 7), „ , — Tc.6. + Tapy. • 2.5.4 Potential The energy 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 a s , " =-{f w *-"»3 f "•"*£!}• + •••<»•»> From Figure 2 - 7 : ri = \Rl - (a + xi)]i + \Rl - ( x p y + )]j yi Vl p + [Rl + (zi + Wi)]f% ; t r u = [Rl + (a + x )\i + [Rl + x + tt p y (y, + t, )]y a p + (z + xv )]k ; n where I l , and I represent x* y z w u . . . (2.23) p the direction cosines of R with X Y and p' p' Z p - a x e s , respectively, and are defined a s : l = cos ip cos <f>; x l = cos y sin 4> sin A — sin ^ cos A ; l = cos rp sin <f> cos A + sin if> sin A . •••(2.24) z Substituting order terms from (2.22) and ignoring 1/R and higher gives, U . = app Details E q . (2.23) into of the derivation + U -rU l l are explained +U v + U + Di,. + Di,. + Di,,. ltt in Appendix II. Figure 2-7 Vectors r ( lower and r u defining position vectors and upper appendages, respectively. of mass elements on the 30 Since the Euler-Bernoulli beam equation only the fundamental mode of vibration is assumed to be valid and is considered, the strain energy associated with the appendages can be written a s , 1 U = [(P,) + (Q/) + {Pu) + (Q„) ] • 2 e Thus, the total potential energy of the u = U +U cA 2.6 Equations of Using the can be obtained where q = 2 2 • • • (2.25) 2 s y s t e m , U, is given by, . + u. app e Motion Lagrangian formulation, the governing equations of motion from, R, 6, (f>, X, P , Q , P , Q , and F ( ( u u q represents the generalized forces. In general, the orbital motion effect of librational is small unless the and vibrational motions on the system dimensions are comparable to 1 fi 17 R ' . Hence, the orbit can be represented by the classical Keplerian relations: R H {\ + ecos0) 1 e R'0 = where h is the angular momentum the orbit. eccentricity of the h\ per unit mass of the system and e is 31 In satellite application, it as an independent variable is convenient to instead of -—^ dt~ dd' _ - dt ^. \ Pp 6 time. Using Eq. (2.26) and substituting: the remaining use the true anomaly 2 £_ _ W ~ 2 2£sinfl d l + ecos9d^'' seven equations corresponding to the p u . a n d 0- c u a n , generalized be obtained as explained coordinates in Appendix I I I 3. LIBRATIONAL D Y N A M I C S OF SATELLITES WITH THERMALLY FLEXED APPENDAGES 3.1 Preliminary Remarks The governing nonlinear, nonautonomous, and coupled equations motion do not admit appreciation of to make of any known c l o s e d - f o r m solution. To get the complex dynamics with thermal response of the system was explored setting vibrational coordinates zero (P| = an ecliptic orbit Q| = P = P- = with the perigee considered, i.e., p=cj and i =0 (Fig. point between decided step, librational generalized 0). Furthermore, a particular U some effects, it was the problem progressively complex. A s a first of the sun and the case of earth was 3-1). Solar Radiation Figure 3-1 An illustration of the orbit with p=co, i=0, of the sun relative to the spacecraft. 32 showing the position 33 To begin with, variation of equilibrium configurations of with the true anomaly is discussed. This is followed of limiting inertia Application of parameters the procedure of to obtain an approximate approximate (Kj and K ) for g variation of stable by the motion parameters the system determination in the small. is next illustrated solution of the problem. Finally, validity response is assessed by comparison with the of the 'exact' numerical solution. 3.2 Equilibrium Orientation Equilibrium orientation librational angles \j/ forces, librational Qt of the central body, represented by and X , 0 , e can be obtained by putting g velocities, and accelerations equal to dT dlp gityAA) = [0 2 ^ , <M) dT = [- <73(Vs<M) = [iO d\ K where q = ip\ d>' A', t + dT d\ } <f>", X" . dU-i di>\ 5=0n dUi 9=0« d<t>\ + the generalized zero: = 0 ; • -(3.1a) = 0 ; •••(3.16) dU-i dXl =o •••(3.1c) g Limiting appendage deflection to deformation only, i.e., neglecting appendage vibration (Ej = Q.\— £ = u thermal S u = 0), Eq. (3.1c) becomes u' = 0 •••(3.2) x giving <f> = 0. •••(3.3a) e Since X is the angle of rotation about the axis of it can assume any value without e into (yaw), affecting the equilibrium orientation. Let A = 0. Substituting Eq. (3.3) symmetry (3.1b) gives • • • (3.36) 34 <72(^M = 0,A = 0) = 0; e leaving only one equation 9i(ipe,<t>e = 0,A to e solve, = 0) e = -e + K e sin V cos %P - K cx { [-e s i n ( > + 9) - sinf> + Q) costy + 9) 2 c it g al 2 c + cos (xp + 9)] + e [1 s i n ( ^ + 9) cos(tf> + 9)- sin i> cos xp sin (V> + 9) 2 2 g s i n ip sm(xp + 9) cos(tf» + 6)] J 2 ...(3.4) where the coefficients Since the readily equation available defined solution of generally unstable in the as the satellites. The problem roots of several -7T/2 to of by functions or small. In gravitational minimizing is contrast, gradient the \p stabilized J instead, where ' (3.5) represents a constrained optimization available implementation in the UBC Computer of the problem Library is a modified quadratic for and this version approximation class of method of 18 19 W i l s o n , Han, and Powell particularly for orbits for with (L*= four ' , was used here problems with few Figure 3-2 shows the different e g\, problems. The NLPQLO subroutine, which Schittkowski's large is not approach. However, with trigonometric 7T/2 for is overcome subroutines are a numerical equation constraint Equation in character, its solution can be very J = with the turn to a nonlinear range in Appendix I I I (Eqs. 111.1 - 111.3). is transcendental and one has to numerical should be are sets variables variation of 0.6) appendage thermal of inertia \p e because of and efficiency constraints. with parameters deformation. its 8 in circular and without (L*= elliptic 0) and 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 f o r L*= 0 or 0.6, the maximum increases with an increase in eccentricity or a decrease hand, an increase in K g value of \\p \ e in Kj. On the other has insignificant effect on maximum f o r L* = 0 and shows only a slight increase f o r 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 of the deformed satellite at two locations 6= orientations 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 3-3 Figure 3.3 Stability are out using the linearized valid provide and in circular for small useful significantly in the small equations magnitudes information lower design of the system equations - V ' e { l + K a (b\ 2 at + iP{K e 2 2 at c + K g a t a 2 [~hb 2 for \p and <p degrees + e bib c at al -c c 2 z 2 + K ot {e \-{b\ al 2 c + 2e 6i + c 2 + b )} 2 2 c 2 + e [^b\ 2 g - ^"{^036263} + <i>'{K cx {e b + <p{K cx b \-2b at a cost. gives; rp"{\ + K a b } system is of motion. Obviously, the results in the preliminary computing of the dynamical of initial conditions. Such a study can Linearizing the librational it orbits. in the Small Investigation of the motion carried at t9 Equilibrium orientations - 2 ^e b }} g 2 + b\) + 6x63a]} hh)} - b\) - b\]}} of freedom 41 rp"{-K a b b } at 2 2 + 3 rp'{K cz e b b } al + tp{K Q b3[-€ bi 2 al + c g it z 2 e 2 at 2 at 2 2 c 2 + K oc [Zb\ + b\ + t hb g c + ±e (b 2 c + 3fcg - 2 g + 63) + g ^ M a + (1 - K )tr + K a {b e (bia it 2 4>'{-e + Katoc \bib - e(fc? + bl)}} + 2 + <(>{! + K e c ^e b ]} + <j>"{l + K ia b\} a 2 bl)]} - = 0; where • • • (3.6) bj's Appendix are given by Eq. (2.1) III parameter (Eqs. 111.1 which - and other 111.4). Here is determined = from the a coefficients is referred initial are to defined as the spin conditions. From Eq. (3.2), constant = ° • Equation (3.6) • • • (3.7) can be rewritten as: [^i(')]q = lA2(*)]q +M*)]; , where For (3.8) q = (x\>\ a circular in orbit and <p\ •••(3.8) <p) . T (i.e., non-spinning satellite), a=0 [A^] = 0 and Eq. becomes, q'^lWrVaWlq. Equation (3.9) nonautonmous represents differential periodicity, 27T, in the a set equations independent of four •••(3.9) linear, homogeneous, having coefficients variable, 6. of the same Hence from the Floquet 20 theory , it follows that there exists a basis of four vectors for the equation. Let A(t9)=[A (t9), . . 1 (3.9) , then ., A (0)] 4 be a basis for the solution of Eq. 42 A'(* + 2ir) = £ / } A t y ) where / ' = [fl, fl, fl, fl] , a n d t n vectors f e (3.10) (» = 1 , . . . , 4 ) , are independent. The basis of the solution for E q . (3.9), 0(0) = [ 0 ( 0 ) , . . 1 the given A ' ( 0 ) with the property be constructed from G(0 + 2 J T ) = .,0(0)], can that (3.11) n®(6). 4 Writing G(0) = ^c,A'(fl) 4 1= , Eqs. (3.10) and (3.11) give 4 1 4 3=1 i=l 4 4 Since A (0) are independent, the coefficients giving f\-p n fn ft nontrivial solution, the ft ~ ft Here For stable in E q . (3.12) must vanish, 4 ft ft - M ft J2 ft ft ft-n ft fi ft ft ft For (3.12) ^De^-^)A'(0) = O. or (3.13) } = ea determinant A* ft ft » ft ft 3 / ft ft ft ft ft ft ft - M ft = 0 3 3 ft - H M-j, . . ., M4, are called the characteristic motion. multipliers of E q . (3.9). The difficulty vectors f. Since / i j ' s 1 used to in applying the theorem determine are f. independent of A(0), a numerical of the method can be From E q . (3.10), 1 A*(2tf) = hence, if A(0) lies in determination is a unit y matrix, then A(27r) would give, A(2TT) = [ / The procedure for q(0)=A(0)=(!, 0, 0, 0 ) E/yA (°)» p 1 f* obtaining f * s now becomes obvious. Putting 1 and numerically T /*]. integrating E q . (3.9) over a period of 27T, / One can repeat Once the = A ( 2 r ) = q(27r). 1 1 J the process f ' s are known, j i j ' s 1 Figure 3-4 the inertia of thermal appendages for A(0)=(0, 1, 0, 0 ) and hence the shows variation of the parameters. K. and K . for deformation gravity them also increases; hence, the body by may is an additional Figure 3-5 stability gradient unstable region for compares the Kj< K. analysis L*= data 0 but librational equations of as presented in Fig. 3-4 unstable for L*= of restoring moment by response for predict the provided by for L*= and K > the parameters motion. The the although the 0. Furthermore less than 0.25 librational determined. bounds as affected 1.0. The results were obtained through numerical original nonlinear can be system remains stable be unstable for f , and so on. system in a circular orbit. Effects deformed and undeformed systems with inertia K.= a stability for are also indicated. Note, as inertia increases, the itself the T 0.6, central there 0. thermally Kj = integration -0.1 of and the linearized Floquet system to be stable 0.6. The prediction is substantiated by the for System Orbital Porometer Elements a, P u i = 90' = 90" = 0 € = 0 = 0.01 Spin Parameter a =0 44 1-T L* = 0.0 0.50- Unstable Region K: - -0.50 1L* = 0.6 0.50- Unstable Regions K: -0.50 Kr Figure 3-4 Effect of inertia parameters on stability of the s y s t e m in circular orbits. Note the presence of additional instability region with thermal deformation. 45 'exact' numerical results which show the become completely Merit of motion, from unstable the the after Floquet theory results of the equations at one orbit, whereas equations took fifty orbits to The main disadvantage with n degrees of followed large value of becomes clear. It numerical the at the the integration eigenvalues n, the amount of of of the integration of same to to the nonlinear involved may a system be obtained determine a 2n x 2n matrix are effort linearized is that for equations have to one orbit unstable conclusion. Floquet theory over predicted integration numerical arrive of L* = 0.6 orbits. freedom, n linearized by repeated vectors. Finally, the 55 \p response for basis of first 2n required. With a become prohibitive. System Porometers = 0.01 l = 1.00 =^0.10 Orbital Elements Initial Positions Initial Velocities 46 u = 90' = 90' = 0 = 0 Spin Porameter a =0 1> 0. X 0 = 5' = 5' = 0 =o X' = 0.096 0° £=0 L* = 0 L*.= 0.6 6 (No. of Orbits) Figure 3-5 Librational response of the system with K = showing unstable (a) 0-5 orbits. g motion caused by thermal 1.0, K. = deflection: -0.1 System Parameters a, = 0.01 K = 1.00 Kj =-0.10 0 Spin Parameter a =0 Orbital Elements P u i c = 90' = 90" = 0 = Initial Positions Initial Velocities f0. r 0 A o = 5= 5' =0 47 =o 0 V = 0.096 0 10 6 (No. of Orbits) Figure 3-5 Librational response of the system with K = showing unstable motion (b) 45-50 orbits. a caused by thermal 1.0, Kj = deflection: -0.1 System Parameters a, = 0.01 K = 1.00 K, =-0.10 Orbital Elements = 90' = 90* = 0 a Spin Porameter a =0 0 *° Initial Positions X 0 = 5* = 5' =0 Initial Velocities 48 = o #; = o X' = 0.096 = io. '<% (K\ iT\ ft V 1 j w \y w >W w w -5-10 1 1 10 e =0 L* = 0 L* = 0.6 55 57 58 59 60 6 (No. of Orbits) Figure 3-5 Librational response of the system with K = showing unstable (c) 55-60 orbits. motion caused by thermal 1.0, K.= deflection: -0.1 49 3.4 Motion in the Large In order to investigate large amplitude nonlinear equations of motion as presented Although these equations do not p o s s e s s solution, they approximate discussed 3.4.1 motion of the s y s t e m , the in Appendix I I I any exact known c l o s e d - f o r m can be solved in an approximate variation in this Variation of parameters must be used. manner. Two methods, an and an ' e x a c t ' numerical, are section. of parameters method 21 The method, which was suggested by Butenin differential , is intended for equations with small nonlinearities. Application of the procedure to the governing equations of (Eqs. the 111.5 - motion 111.7) required careful expansion of each term and truncating series at an appropriate order depending upon the relative the term. For example, ct^ being quite small, only the first were retained. Similarly, for terms independent order terms and e were retained. For e c magnitude of order of a ^ , fourth terms and higher third and higher order terms in e were neglected, i.e., e = 2esin0(1 — ecos0); c e = 3(1 - ecosfl + r ' c o s ^ ) . 2 g With this simplication, the equations of motion for \p and <j> degrees of freedom b e c o m e : rp" - 2esin 9(1 - ecos0)^' + ZK (l it = Fx (xp, <f>, - ecos 8 + e cos 8)4> 2 <f>", a) + 2e sin 8(1 - e cos 9); c6" - 2esin0(1 - ecos9)<f>' + (1 + 3K )(l it = F (rP,<p,iP',<t>',iP",<p",CT); 2 2 -ecos9 + <?cos 9}<f> 2 --.(3.14) 50 where: F i = 2 # ' ( 1 + if,') + ZK xp{4> + -rp )(\ 2 e cos 6) 2 2 + ^<f> ) + 2a [(6? - b\)xp - b,b + b b d>] + K a {a4>{l at - ecosfl + 2 it 2 2 2 2 2 3 + b\i>{l + 2tf>') + b [tp" - 2<p<p'{l + *P') - <f> xp" - a<f>' - t'rp 2 2 2 - xp - 2xpxp' 2e[a<p + tp') sin - + b \-<p xp" - 2 # ' ( 1 + rp') + 2 2 + bib [a + 2 2 F 2 2a<p - c<f>' - 2o-c*£sin0] 2<p4>' + 2o<pip' + 2<p \p'\ + 2 6 M ' e s i n 0 } ; 2 2 + xP' - \<p ) + 3K <p(xp + ^ ) ( 1 -ecosB = -<PW 2 2 2 2 it ( 6 + b <P(l + 2rP') + b [-a 2 2 + bl\<p" -a- 2 2 2 o 3 rp'{<j + M)] + 2& 6 (<r^ + rpxp'<fi + 1 2 - 3<p (l + rp') + a<f>xp' - 4 ( # ' 2 2 + 26 63^'esin0} . and ?2 t o 2 WeslnO) + VV-')"in 6} •••(3.15) 2 Ignoring 0) - rP'{(T + 2 3 2 2 + - [ ( 6 - b\)<p + 6 (&! + b xp)\ + rp'-Z-) 2 + bib [a + e cos 2 + K a {-a(l at 6] obtain the generating solution, E q . (3.14) can be rewritten as: xP" + PlW + P 46)rp 2 = 0 l + P 4>W = 0 ; <p" + PiW . . . (3.16) 2 22 in the Mathieu form and can be solved using the standard approach = x}e <P where a is the integration i a = a . Let: ; 4> = ; factor defined as • • • (3.17) r° 2J m ) d ( 0 = e(l-cos0)-y(l-cos20). ••(3.18) 51 Substituting from E q . (3.17), (3.16) takes j" + ^ " ~y&- ^ + - ^ the f o r m : = 2 e s i n - T ^ = 0 " ( 1 " € C 0 8 9 )5 - " • ( J U 9 ) Simplifying, i" + JV" = (! - ecos0)[ecos0(3ff, - 4- 2€sin0]; n t ci" + nl4> = (1 - ecos0)[ecos0(3iir, - 1)$ ; • • • (3.20) ( where; n j = 3K,- ; ( 4 = 1 + 3iT, . • • • (3.21) t The solution for E q . (3.20) can now be written as: J.-A Lvn( P i /? 1 +i (l-3g- - )ersin[(n^ + l)g+^] rP - A,{sm(n,9n + [ — t (1 -ZKu^e 2 t r-sin[(n + 2)0 + ^] sin[(n , - 2)0 + / ^ h j L (2r^ + l)( , + 1) (2n^ - l)(n^ - 1) J J v> 8 nv 2 1 L 2 8 where; </ L (2n^ + l)( n<k v 2n^ +1 (l-3ii: )n ,€ r-sm[(n^ + 2)0 4-^] tt sin[(n^ - 1)0 + fa _ j + l) 2n^ — 1 J sin[(n^ - 2)0 + ( 2 n ^ - l ) ( n ^ - l ) JJ ' l v>i = ~ 2 — r ; e ^ 2 The method of variation = ^ - of parameters now applied. In order to reduce the amount terms in E q . (3.22) were considered: as described by Butenin was of work, only the predominant ^ 52 ij> = Ay sm(n^O + 0$); sin(n^0 + 0+). 4> = The objective amplitude w a s to and phase obtain angle a solution functions of ••• (3.23) essentially harmonic but w i t h the 6: ^ = ^ ( 0 ) s i n ( f t y 0 + jfy(0)); i = A+{9) sin( V There They u n k n o w n s , A ^ , A ^ , 0^, a n d 0^ c a n be e v a l u a t e d equations first are f o u r of motion. derivative Differentiating of by introducing For example, • • • (3.24) + P^B)). logical obvious to constraints constraints E q . (3.24) t o be the s a m e a s that be a n d u s i n g the would of determined. be f o r the E q . (3.23). 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 i n ( n ^ + P+) + A h e n c e , the logical constraints would • • • (3.25) ^ + /fy) cos(n^ + P+); become: A'q sin t)^ + A^Plj, cos jty = 0 ; ji^sinify + j 4 ^ / f y c o s t f y where: = 0; • • (3.26) Vtl> = ^9 + p^,; V<t> = n<t>0 + P<t> • Differentiating E q . (3.25) and s u b s t i t u t i n g in E q . (3.14) l e a d s 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;=F {Ate*anti* ...,ff)i : l t F * = F (^e sin»7^,...,a). •••(3.28) a 2 2 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 p ^ to be slowly varying parameters, their 1 averages over a period are used: j /»2JT i»27r /«2jr j /»2JT *2x plx "*=Si" C C * Denoting E m,f(0)= e I m a /(£) di; , s i n i* >* "* dr d -'' ' d9 3 29 E q . (3.29) becomes: ^0 ^ = 0; 4> = 0 ; ^ i l r ^ l ^ ' ^ ^2{2al{2a E = 3 1 + - 2^i,co.»«l + 2ci[-2a r? 2 hcos20 2 - 2 £ W*1 + - a A E , 2 K 2 9 + a A E2 2 1)COs2( 2a a [2a 4^ 2 1 2 2 c o s 2 9 + ( n j + 5)^ + (r. + 7 - 4 a ) £ 2 2)SinW t C o a 2 0 + (n + 2 2 2 1)Sb2 1)C0S ,„ ,]}} ; .sin 20 5)E ^ 2 l a e 54 + °>l i^e) E + 2 a ^ ( 4 + n\- 2a )}} . ... (3.30) 2 M Using the initial conditions and Eqs. (3.17) and (3.22) gives; , (1 - 3Jfc)e I", < & = + \ n ^ _ 3 [i - l n l _H 4 ( •« £ n ^_ 1 } j} c o s ^ o + HI - A ^ , A ^ , / 3 ^ , and / ? ^ can now be determined Q Q ; and the analytical solution becomes: _ sin[(n^+^-l)g + ^ ] i 0 E in[(n +^-2)g + ^ 0 (2n, t< J 2n^ - 1 + (l-3/^ )n^e ] 8 - l)(n, - 1) JI ^ o n + 8mg ^ = | ^ | s m [ ( ^ + /3^)0 + ^ ] + 2^-1 p sm[(>V + + 2)0 + L [2n^ + l)(n* +1) 1 y» ~ - J 2 2n^ + l [ 8 I (2n^ + l)(n^ +1) sm[(n^ + ^ - 2 ) 0 4 - M ]))• + The corresponding analytical integrating E q . (3.7) with ^ ' -(3.3.) solution for X can be obtained by and <p substituted using E q . (3.31). 55 3.4.2 Numerical method Consider a set of first order differential y' = with the numerical solution at size used in the numerical obtain y : one-value f(v,0) denoted by y n—T complicated differential accurate and reliable equations, the results than the multi-value * multi-value y Q is obtained by a linear n interpolation corrector formula. If to uses only y ^ n y„. For to determine 'n method gives more until y _ to y denoted n method. The approximated the y - n m times methods processes; prediction and test with a user specified tolerance, then y =y is repeated step n-k correction. In the prediction process, the approximation process is the one-value approach. method consists of two then refined by the r procedures. The former latter uses k previous values, y„ „,...,y„ the A where integration. Basically, there are two and multi-value n while 8 =nr equations, value value satisfies the «. Otherwise the by is error correction ' n,i satisfies the error test and hence n,m Vi ^n,nr — The order of a multi-value method refers to the complexity correction formula. In general, higher the order, more accurate the at the expense of method was on the multi-value the solution a higher computing cost. Since the system's equations of multi-value of motion are rather complex, the preferred. A m o n g the numerous subroutines based method, the IMSL ; DGEAR subroutine was chosen. It is 23 adapted from (i) Gear's Accuracy. multi-value (ii) subroutine DIFSUB and has the The subroutine is based on the method with a variable order of following implicit advantages. Adam's up to twelve. This high order ensures the numerical solution to be accurate. Ease of Programming. In the correction process, the Jacobian of 56 the differential the user. In DGEAR Jacobian (iii) equations is required; hence, it should be coded in by subroutine, there is an option of evaluating the numerically. Economy. Based on the result integration of the error test step, the subroutine automatically at each changes the order correction formula or the step size, if necessary. This feature the (iv) subroutine efficient Additional feature. and economical in terms makes of computing cost. By changing the parameter subroutine, it can be used to solve stiff of the METH of the equations based on Gear's method . 2 3 In order to obtain a numerical solution, the equations of were first written as a set of first order differential equations. For example, the equations of motion for \p and <j> degrees of obtained earlier motion freedom can be written in matrix form as • • • (3.32) where f^ and f^ generally are functions of \p, <p, \p\ not constants, are defined 0 ' , a, and M j j ' s . M - ' s , in Appendix IV. The equation of motion for X degree of freedom was (Eq. 3.7), • • • (3.33) A' = o- + (l + V') ^> sin where a was given by the initial conditions, <* = (w*)o = Ao - (1 + 1>' )un <p . 0 Rewriting together E q . (3.32) as a set of four 0 first order differential with E q . (3.33) gives a set of five equations, first order equations differential 57 q' = (Mi]"*; (3.34) where: M M [M ) = 1 q = n M 2i M 0 0 0 (V', 2 2 0 0 0 0 0 0 0 10 0 1 0 0 <f>\ rP, <p, f=(/v» Equation n /*> (3.34) was tolerance set at 10" . The 0 0 0 0 1 A) ; T 4>', <T + {l + i>')sin<f>) . T solved by the IMSL : DGEAR subroutine matrix [M^] was with the inverted numerically at each iteration step. 3.4.3 Discussion of results Validity of the analytical solution was inertia parameters and assessed over a range of initial conditions. For conciseness, only a set of representative results useful in establishing trends are presented Figure 3-6 presents few here. the librational response for the satellite in circular orbits. It is significant to note that the analytical procedure is able to predict amplitude accuracy as well as frequency over a wide range of central body and However, the accuracy K g with an acceptable appendage configurations. drops o f f for small Kj (stubby (relatively heavy appendages). The degree of central body) or large error in the phase is cumulative increases with 6. Fortunately, phase does not constitute an and important parameter in the satellite dynamics analysis. In general, error in the amplitude prediction for X response was prediction may explained by be good at one the equation found to fluctuate randomly. The instant but poor at the other. This can used to evaluate the X response, be 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). 59 legend Nurnencd 10 11 a iKln of OrbVts) different al and - - • , . f numerical d y with Comparison of nu r • centra analytical s o l u t i o n . . ^ . ^ n 0 b o s Figure 3-6 inertia P " appendages 8 * * 8 0° , ^ l e n d Orbits), e U g n t Syslem Parameters a, = 0.01 K = 0.25 Kj = 0.25 L* = 0.60 G Orbital Clements = 90= 90* = 0 Initial Positions Initial Velocities f. V; <t>. X. = 30= 0 = 0 = o. <p' K o 60 = 0.50 = 0.5O = 0 Spin Porometer a =0 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 K =0.25 Kj = 0.25 L* = 0.60 a Orbital Elements Initial Positions = = = = f. = 30* 90' 90* 0 0 = 0 = 0 X Initial Velocities 61 = o.so =0.50 x: = o Spin Parameter a =0 50- A -25- U. -50 \1 _ — 20 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. Jo The error between (p at any instant one instant the numerical and analytical is accumulated in the integration . . . (3.35) solutions for \p' process. If and the error at cancels that at the other, the resulting discrepancy between numerical and analytical solutions is insignificant. On the discrepancy becomes significant if the other errors at different the hand, the instants do not cancel. Figure 3-7 the presence of shows that essentially the spin. In Fig. 3-8, solution fails to predict the excitation. The failure analytical 3-9 in-plane out-of-plane analytical solution for C6Q and 0 ' . N presents the response in non-circular orbits. Note, the solution is able to predict discrepancy, particularly It analytical motion caused by the is not unexpected because the quite well, although the magnitude some continues even in it can be seen that the \p depends on \//Q and I / / Q ' but not .Figure similar trend amplitude of modulation amplitude at higher in \p response and phase continue to show e. should be pointed out that validity of the c l o s e d - f o r m solution has been assessed here under most demanding conditions. In practice, scientific over and application satellites a specified limit (say 0.1° have their for analytical acceptable of accuracy at least controlled for 4 ° . Thus in practical application, solution is expected to predict the degree motion communications satellite; 2 ° weather satellite, etc.) ranging over 0.01the librational librations with an in the preliminary design stage. 65 System Poramelers a, = 0.01 K =0.25 K| = 0.75 L* = 0.60 c Orbital Elements Initial Velocities = 90' = 90' = 0 = 0 V; = 0.50 X' = 0.02 = o.so Spin Porometer a = 0.02 legend Numerical Analytical 1 2 3 4 5 6 (No. of Orbits) Figure 3-7 Comparison of numerical and analytical spin parameter. solutions for non-zero System Orbital Initial Initial Parameters Elements Positions Velocities = 90* = 90' *. = o t; = = X a, 0.01 = 0.25 = 0.75 = 0.60 = K K, L* c P u i £ Spin o 66 0 0 <t>. = = o 0 = o <p- = 0.50 c x; = o 0 Parameter =0. 4 legend Numerical Analytical 9 (No. of Orbits) Figure 3-8 Comparison of numerical and analytical across the orbital plane. solutions f o r a disturbance 67 System Porometers o, K = Orbitol Initiol Initial Elements Positions Velocities = 90* = 90" f t: = 0 . 5 <t>: = 0 . 5 = 0 A. = 0.1 0.01 c = 0.25 Kj = 0.75 L* = 0.60 0 = 0 = o = 0 x: =o Spin Porometer a = 0 45 30 15 ft 0 -15 "A - v A A j\/h in A A y v \ -30H -45 i i i i 30 AAAA/^ 10 <t>° 0 A A A A W W W \ - V V V VW \ / W yj V -30 60 s 1 I I I legend Numerical Analytical 1 2 3 4 d (No. of Orbits) Figure 3 - 9 Comparison of numerical orbits: (a) e = 0.1. and analytical solutions for eccentric 68 Orbital Elements System Porometers a, = 0.01 K„ = 0 . 2 5 Ki = 0.75 L* = 0.60 Initial Positions Initial Velocities = 90' = 90* V- 0. =o =o V; = 0.50 0; = 0.50 = 0 = 0.2 X. = 0 X" 0 = 0 Spin Porgmeler a =0 45 30 f\ •v y : 15 f 0 0 -15 A s\ } -30-45 30 — !• A A\ v \J —1 1 V 1 legend Numerical Analytical i 2 3 e (No. of Orbits) Figure 3-9 Comparison of numerical orbits: (b) e = 0.2. and analytical solutions for eccentric 4. COUPLED LIBRATION/VIBRATION THERMALLY 4.1 Preliminary D Y N A M I C S OF SATELLITES WITH FLEXED Remarks This chapter builds on the earlier the APPENDAGES analysis and considers dynamics of nonlinear system accounting for thermal vibration. Hence, all the seven equations of Appendix III constraint (Eqs. 111.5 is kept - deformation and appendage motion, presented in detail 111.11), are used in the the same as before, i.e.,. p = analysis. Orbital o> and i = 0. The chapter begins with a discussion on stable equilibrium of the central an 'exact' rigid body and the numerical analysis of flexible the orientation appendages. This is followed nonlinear s y s t e m . The phenomenon associated with the appendage vibration beat and the effect of eccentricity on system response are demonstrated. The attention is also directed towards thermal the condition of deformation and vibration closed-form analysis of variation parameters method of of the instability in the presence of the appendages. Next, an approximate nonlinear system is attempted method and its limitations is modified, an improved analytical using the discussed. Finally, the 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 in by 70 d 3T 9T \ — ( — \ - — 92(g) 1-^^50 03(g) 5c6 U^ar 9i(q) — ] - 0- dx\ '=o»=o~ + ar dQi 97 where The stable idB^apJ dQiiq'=q"=o~ dP dP J,'= »=o ~ + n u dQ + n 5 g dU { VdB^dQj 5 dUj \d_ dT_, _ dT_ (g) ; arj-i + r d , ^ T . _ oT_ 0e(g) ' dPl J q'=q"=0 ~ ' + •dd^dQi Q g dP[ ar _ d<t>lq'=q"=0 ~ " aA ide^dP/ 05(g) + dU] dQ lq'= "=o u (4.1) = 0 q <<1 = ip,<t ,^,Ej,Q ,E.u>Q solution to the above set of equations > i u is obtained by minimizing J , where «=l with constraints: -ir/2 < ip < ir/2 ; e K/2 ; -TT/2 <4> < e -TT/2 <A < JT/2 E - K P / e - K P „ e ;• < l ; < l ; (4.2) -1<S„<1Equation (4.2) is s o l v e d by the NLPQLO subroutine over a range o f inertia parameters (K , Kj), orbital eccentricity g ratio (cc>= u>^/ 6p\ co^ = fundamental r perigee) with thermal 4-3. (e), and frequency, 6p= appendage frequency orbital rate at effects. The results are summarized in Figs. 4-1 t o Since the orbit is taken t o be in the ecliptic plane, it is apparent that 71 the central deflecting body rotation only Effect orientation in the of the is confined to the pitch with the orbital inertia plane, i.e., X = parameters <> / = on the central is presented in Fig. 4 - 1 . For small K appendages attached eccentricity, effect orientation to a slender central of is virtually flexibility effect body = on the central as the central 0. equilibrium and large g (Fig. 4-1a). On the other becomes significant progressively stubby and the Q body), irrespective appendage flexibility negligible Q. = appendages Kj (i.e., small of body the orbital equilibrium hand, the body becomes appendages relatively heavy (Figs. 4-1b to 4-1d). Figure 4-2 flexible shows variation appendages with the any instant, the thermal As the flexibility its natural of inertia frequency, on the parameters variation of a> r orbital in the of remains as expected because the rate approaches the flexible the appendage flexibility, is taken to be at and P_ for deformation. is expected to rigid appendages. considered here, g deflection contributions spacecraft's equilibrium in its orbit (Eq. (3.21)). In contrast, P | the is larger than the limit for tip in circular or eccentric orbits. Note, at diminishes, difference studies effect the spacecraft the equilibrium satellite deflection decrease, vanishing in the Figure 4-3 of ue 6= fairly libration appendage r frequency. by configuration. Position 9 0 ° . For the values of constant with the frequency diverges as a> as reflected is held constant approaches 1, i.e., as 72 System Parameters Orbital Elements Equilibrium Orientations a, = 0.01 90' = 0 0.25 0.75 0.60 = 0. = = = p u = K K, L* X. =0 i = 90° 0 Q = 0 Qu. = 0 Qle 0.4 0.2- e = 0.2 cj = 2.0 r CJ, =20.0 0 0.25 0.50 0.75 l r 1.25 1.50 1.75 0/TT Figure 4-1 Effect of system parameters on the equilibrium (a) slender central body with small appendages. orientation: 73 System Parameters a, K Kj L* a = = = = 0.01 1.00 0.75 0.60 Orbital Elements Equilibrium Orientations p = 90* u <P. = = 90° 0 X, i = o =0 Qle = 0 flu. = 0 1 20 Figure 4 - 1 Effect of system parameters on the equilibrium (b) slender central body with large appendages. orientation: 74 System Orbital Equilibrium Parameters Elements Orientations a, K a = = 0.01 0.25 p a Kj L* = = 0.25 0.60 i = 90° 0, = = 90° 0 X. = 0 =0 Q,e = 0 Que = 0 0/TT Figure 4-1 Effect of system parameters on the equilibrium (c) stubby central body with small appendages. orientation: 75 System Parameters a, K K, = = 0.01 1.00 0.25 L* = 0.60 Q Orbital Elements Equilibrium Orientations p u 0. i = = = 90° 90' 0 X, = 0 =0 Qle = 0 Que = 0 2 0/TT Figure 4-1 Effect of system parameters on the equilibrium (d) stubby central body with large appendages. orientation: 76 Equilibrium Orientations £= 0 TOtal Thermal Flexible £ = 0 Total Thermal c o • Flexible s— (1) Q € = 0.1 Totol Thermal Flexible £=0.1 Totcl Thermal Flexible Figure 4-2 Variation of tip deflection at equilibrium dominance of thermal effect. with 8 showing the 77 Figure 4-3 Effect Note of the appendage divergence flexbility of Pj on and its P_ ue equilibrium at co = r 1. position. 78 4.3 Numerical Analysis of With the the inclusion of Nonlinear appendage vibration, complexity increases markedly, and the libration response may different compared to response in detail. The equations were better with, thermal problem be expected to be integrated using the system numerical (Section 3.4.2). appreciation deformation shows response of lower the Appendage disturbance For orbital of that obtained earlier. This section studies the procedure described earlier 4.3.1 Equations the plane to one of equal to the effect of appendage flexibility, to is purposely neglected (L*= 0). Figure start 4-4a system with an initial disturbance applied in the the appendages. The disturbance corresponds to appendage deformed deflection of initially 10% of its in its fundamental length. Several mode with the interesting features tip become apparent: (i) in-plane motion (ii) the disturbance of both the in vibrational appendage excites and librational pitch motion has a high frequency appendage (iii) the motion superposed on appendages exhibit eigenvalues, np and n eigenvalues p 1 contribution from the two closely spaced (Appendix V , Eq. V.2). Note, one of appendage and coupling with the a minor libration the frequency. This appendages, though structurally different eigenvalues due to field modes. resonance, due to r gravitational in-plane it. is identically c j , i.e., the suggests that the two a slightly beat only the identical, have difference in the motion. Response of one appendage thus acts as a forcing function for the other through 79 both a librational coupling closely corresponding to the third If initial the same in-plane as well disturbance is applied across the orbital as o u t - o f - p l a n e the motion with a moderate appendages exhibit yaw a strong beat and, through a weak coupling, in the lower appendage in the orbital disturbance represents an initial fundamental appendage mode with tip length plane plane the pitch. A s before, out-of-plane orbit motion as well. impulsive disturbance applied distribution corresponding to the product remains virtually amplitude has increased significantly in Fig. 4-4a the of unchanged. However, the (±0.6° to The rate, i.e., 21^0. Compared to Fig. 4 - 4 a , of the libration and ± 5 ° in 4-5). The fact that the a small difference response data Fig. 4 - 6 . for (Fig. 4-6b) closer look Section 4.4 (Eq. 4.10) response of frequencies a satellite the absence of A beat in their The appendage out-of-plane Note of is the is shown in Fig. 4 - 5 . velocity amplitude Fig. libration response in the velocity equal to twice and orbital vibration roll and a negligible The system response with an initial the plane, motions are excited as shown in Fig. 4 - 4 b . A s can be expected, the o u t - o f - p l a n e dominant eigenvalue, through by a comparison of appendages as presented an in-plane (Fig. 4-6a) fundamental in or an mode as before. response with a single appendage. at the approximate c l o s e d - f o r m solution as given in also suggested conditions when appendages through before, the is substantiated disturbance in the configuration will have no beat both the appendages is indeed with one and two is subjected to beat the response. With initial symmetric solution shows terms even two conditions applied or asymmetric corresponding to appendage beat to deflections as 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 = i - 0 0 Initial Positions % = 0 to = 0 *o = 0 Elo = 0 Initial Velocities f'0 = 0 9,o = Qio = Puo = o Qo = o P..-uo Quo = — <t>' K =0 0 =° Elo = o 0.05 0 0 0 U 0.01 1>°: o 1 2 0 (No. of Orbits) Figure 4-4 System response with: (b) out-of-plane appendage disturbance. Orbital Elements System Parameters a, *« K, L* u r 0.01 = 0.25 = 0.75 = 0 = 20.00 p u i c = = 90" = 90° = 0 = 0 Initial Positions *o = 0 <t>o =0 *o = 0 P|o = 0 Q'Io ,o = 0.05 Initial Velocities Euo = Quo P-uo 82 % =o <K = o K =o Qio Qo U =o =o =o 0.05 ro oI o.oo- ^X, ^ w v v ^ n QJ -0.0510050- 1 1 O T— V 0- X —* Ol -50-100 0.05 to I O ^ - ^ 0 ^ 0.00 ^ ^ - - V Y ^ ^ 3 QJ -0.05 100 501 1 o- o X 3 Ol -50-100 i 1 2 3 9 (No. of Orbits) Figure 4-4 System response with: (b) out-of-plane appendage disturbance. 83 Figure 4-5 Typical response of the system disturbance. for an impulsive appendage System Parameters K, L* = = = = = 0.01 0.10 0.75 0 20.00 Orbital Elements p = u= i = E = 90" 90° 0 0 Initial Positions 84 Initial Velocities *o = <f>o = *o = 05 Ei. = Q.o = K = p; = 0 Q\ = 0 Puo = Quo = Puo = Quo = Two A p p e n d a g e s 0.25 f ° 0.00 -0.25 \k W ,J l i it w 1 W w % -0.50 1 2 0 (No. of Orbits) Figure 4-6 A comparison of response for systems with one and two appendages: (a) in-plane appendage disturbance. flexible System Parameters a, K K| L* a 0.01 = 0.10 = 0.75 = 0 20.00 = Orbital Elements Initial Positions p = 90° u = 90° i = 0 t = 0 % = 0o = *o = P.o = Qio = P-uo = Quo — 0 0 0 0 0.05 0 0 Initial Velocities 0; = *o Pio Qio P-uo Quo = = = = 85 — 100 1 2 6 (No. of Orbits) Figure 4-6 A comparison of response for systems with one and two appendages: (b) o u t - o f - p l a n e appendage disturbance. flexible 86 to vanish. The numerically substantiated 0 Q|Q= two response results as given in Fig. 4-7 this conclusion. A s expected, symmetric disturbances (E| = motion obtained - P - 0 ' —10 U = —u0^ r e s u l t e d i n virtually (Figs. 4-7a and 4-7b). However, asymmetric - Q . ) amplified U U (Figs. 4-7c the libration no librational disturbances (P|Q= E r> response approximately u by a factor of and 4-7d). Even with the inclusion of thermal equilibrium appendage e f f e c t s , which would change configuration of the appendage in the orbit, the beat phenomenon continues to persist (Fig. 4-8). The most approximate valuable satisfying aspect of the analysis was the ability analytical insight solution, discussed of the later in Section 4.4, to provide into the dynamical behaviour of such a complex s y s t e m . 87 Figure 4-7 System response showing the effect of symmetric and asymmetric appendage disturbances: (a) symmetric in-plane disturbance. 88 System Parameters a, = 0.01 K = 0.25 K, = 0.75 L* = 0 a co, = 20.00 Orbital Elements p - 90° u = 90° i = 0 e = 0 Initial Positions Y'o = 0 to = 0 *o = 0 P.o = Qio = Euo = Quo o o..05 o 0.05 initial Velocities = Vo = K = Eio = Qio = Euo = Quo 0.01 Figure 4-7 System response showing the effect of symmetric and asymmetric appendage disturbances: (b) symmetric o u t - o f - p l a n e 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 o u t - o f - p l a n e disturbance. 91 Figure 4-8 System response showing the effect of the presence of thermal deformation. appendage disturbance in 92 4.3.2 Central body Consider disturbance the case with an initial disturbance body. To begin with, let the thermal applied to the central deflection of appendages be ignored as before. For an in-plane disturbance, Figure 4-9a shows that the central body is librating at frequency frequency n ^ while the appendages are vibrating at a) with the librational frequency r before, here n ^ and a> are eigenvalues r superposed on it. A s explained 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 gives the same libration amplitude which 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 s y s t e m response f o r 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 disturbance. This is because the out-of-plane libration is strongly with the in-plane libration, which in turn is coupled vibration. The figure also shows out-of-plane central body coupled to the in-plane the transfer of vibration energy f r o m the 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 Figure 4-11 shows the s y s t e m response when thermal disturbances. deflection of appendages is accounted for. This case corresponds to results given in 93 Figure 4 - 9 S y s t e m response showing the effect of in-plane central disturbance: (a) displacement disturbance. body 94 System Parameters a, K 0 K, fo = 0 = o x = 0 P.o= 0 Q.o = o <P 0 0 Eio = Qio = o o o o o 0 o =° K II I I Euo= Quo= K =o o = 90° a = 90" i = 0 £ = 0 Initial Velocities -3-3 r p Initial Positions 0-1 Ol L' w = 0.01 = 0.25 = 0.75 = 0 = 20.00 Orbital Elements -10 6 (No. of Orbits) Figure 4-9 System response showing the effect of disturbance: (b) impulsive disturbance. in-plane central body 95 Figure 4-10 S y s t e m response showing the effect of central (a) out-of-plane disturbance. body disturbance: System Porameters K CJ. = = = = = 0.01 0.25 0.75 0 20.00 Orbital Elements p = u = i = e = 90° 90* 0 0 Initial Positions % = K = *o = Elo = Qio = Euo = Quo Initial Velocities 0 5« 0 0 0 0 0 % =0 K =o Pio 9io Po Quo = = = = U 0.25-1 96 0 0 0 0 -0.25 0.25 0.00 -0.25 1 2 0 (No. of Orbits) Figure 4-10 System response showing the effect (a) o u t - o f - p l a n e disturbance. of central body disturbance: 97 Figure 4-10 System response showing the effect of central body (b) combined in-plane and o u t - o f - p l a n e disturbances. disturbance; 98 System Parameters -0.25 1 2 3 4 6 (No. of Orbits) Figure 4 - 1 0 System response showing the effect of central body (b) combined in-plane and o u t - o f - p l a n e disturbances. disturbance: 99 Fig. 4-10b with the thermal difference in the response except for amplitude. However, it effect neglected. There is no noticeable a slight increase in vibrational is significant that even in the presence of deformation, librational on appendage vibration. disturbance continues to have very little thermal influence Figure 4-11 System response showing the effect of the presence of thermal deformation. central body disturbance in System Parameters L* = = = = = 0.01 0.10 0.75 0.60 20.00 Orbital Elements Initial Positions p = 90° u - 90" i = 0 €= 0 to K Q.o Euo Quo s—V fO 5« 5« 0 0 0.05 0 0.05 Initial Velocities 101 % = K = K = Bio Q!o Euo Quo = = = = 0.5- 1 1 o 0 J X cu -0.5-1- 0.5I o 0- X Ol -0.5-110.5- 1 1 O 0- X 3 QJ -0.5-11- fO 0.5- o 0- 1 i v / 3 Ol -0.5-11 2 3 6 (No. of Orbits) Figure 4-11 System response showing the effect of central the presence of thermal deformation. body disturbance in 102 4.3.3 Influence of thermal deformation on system stability The results obtained in Sections 4.3.1 effect has minor influence on libration conclusion is further responses between deformations and 4.3.2 the systems with (L*= for three sets of inertia show that and vibration substantiated in Fig. 4-12. 0.6) (circular orbits) thermal amplitudes. This The figure compares and without (L*= parameters. The initial 0) thermal conditions in each case corresponds to combined central body and appendage disturbances. For a slender central deformation body with small appendages, thermal has negligible influence on response amplitude 4-12a). However, as the central body becomes stubby (Kj = or the appendages become large results in larger libration (K = a deformation and vibration amplitudes. Although not all found to be valid the irrespective initial conditions. With the results of conditions, thermally Fig. 4-12 flexed central body (Kj = 0.75) conditions correspond to The figure as background, at critical The inertia parameters with large appendages ( K = a severe central shows that without thermal of thermal system starts to tumble L*= phase and amplitude 0 and L*= 0.6. 0.75). The a slender initial deformation, the system remains (i//) that both the of represent the body and appendage disturbances. stable with pitch amplitude deformation, the initial appendages can be expected to destabilze system as shown in Fig. 4-13. for 0.25, Fig. 4-12b) 0.75, Fig. 4-12c), thermal results are shown here, this observation was of or phase (Fig. about 8 0 ° . However, with the inclusion of yaw and roll as \p> 9 0 ° . Note librations are also different Figure 4-12 Typical response in circular orbits showing the effect parameters and thermal deformation: (a) slender central body with small appendages. of inertia Initial Velocities % = 0.50 80400-40-80200f—v 100- 1 i oX 0- DJ -100- I o X - CL -100 I O 3 K> I o 3 0-1 1 2 6 (No. of Orbits) Figure 4-12 3 Typical response in circular orbits showing the effect of parameters and thermal deformation: (b) stubby central body with small appendages. inertia System Parameters Orbital Elements Initial Positions fo =300o = 105 Initial Velocities % =0.50 *' =0 0 0 60300-30-60200s—s | o 1000- X QJ ,— -100-200200- s •o 1 O 1000- X Q_l -100-200200100- | o X 3 Q_l 0-100- -200200s—>> IO 1001 O 0X 3 -100QJ -200 1 2 6 (No. of Orbits) Figure 4-12 Typical response in circular orbits showing the effect parameters and thermal deformation; (c) slender central body with large appendages. of inertia 106 Figure 4-13 System response in circular orbits showing the influence of thermally deformed appendages: (a) librational response. destabilizing System Porameters Orbital Elements a, K K, co, p = 90° a = 0.01 = 0.75 = 0.75 = 20.00 co = 90° i = 0 t = 0 Initial Positions = 30° = 0 <Po = *o = 0 P.o = 0.10 Q.o = 0.10 P-uo = 0 9 — 0 U 0 Initial Velocities 0.70 0 K = 0.25 K = 0 Pio = 0 9i = 0 Euo = 2.00 2.00 Quo 107 r= 0 — 200-1 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 influence of thermally deformed appendages: (b) vibrational response. destabilizing 109 4.3.4 Eccentric orbits This section attempts to assess the the system response. Note, eccentricity The thermal to deformation of presented earlier (Section 4.3.1 of orbital eccentricity constitutes an in-plane the flexible help isolate the eccentricity effect on disturbance. appendages is purposely neglected contribution. The circular orbit case results and 4.3.2) would serve as a reference for comparison. Figures 4-14a orbits, e = and 4-14b 0.1, 0.2, when the as before. Note, in-plane amplitude show the lower appendage eccentricity pitch oscillation ( ± 1 4 ° for a disturbance results in a e= e= case where \p was this in-plane the is subjected to eccentric induced excitation to the circular orbit large amplitude system response in two 0.1, ± 3 2 ° for 0.2) compared only ± 0 . 2 5 ° . However, even with librations, the in-plane vibrational amplitude appendages remain the same as before, with the characteristic phenomenon. On the other hand, the orbital frequency substantially. Interestingly, the value at perigee apparent and reaches a maximum stiffness terms containing large eccentricity vibration beat increases the frequency has a Obviously, 1/(1+ecosr5) reaches a maximum value at beat minimum at apogee. This is due to 1 /(1 + ecosd) quantity of the as coefficients. 0 = 7T resulting in frequency condensation. Character of remains the response to an o u t - o f - p l a n e essentially the same as above except three dimensional librational and vibrational motions are excited (Figs. 4-14c is discernable in roll as well. Turning to the body disturbance that now, due to coupling, and 4-14d). Note, the frequency condensation effect librations appendage subjected to system response in an eccentric orbit with the central a combined pitch and roll disturbance (Fig. 4-15), it can 110 Figure 4-14 System response in eccentric disturbance, e= 0.1. orbits: (a) in-plane appendage 111 Figure 4-14 S y s t e m response in eccentric orbits: (b) in-plane appendage disturbance, e— 0.2. 112 Figure 4-14 System response in eccentric disturbance, e = 0.1. orbits; (c) o u t - o f - p l a n e appendage Figure 4-14 System response in eccentric disturbance, e = 0.1. orbits: (c) o u t - o f - p l a n e appendage 114 Figure 4-14 S y s t e m response in eccentric orbits: (d) out-of-plane disturbance, e= 0.2. appendage 115 ^ UltaiiA." 0 j|ii||py^ ^yii|||||j|| -50 -100 Figure 4-14 III 0 1 6 (No. 2 of Orbits) S y s t e m response in eccentric disturbance, e = 0.2. orbits: (d) o u t - o f - p l a n e appendage 116 be concluded by comparison with the earlier that eccentricity affecting the effect overall orbits was attempts thermal studied earlier to distort local variations without other to study the effect with results presented at apogee to be the eccentricity. On the 0 (Fig. 4-10b) substantially of orbital deformation. Corresponding case for A g a i n , one notices amplification frequency e= trends. Finally, Fig. 4-16 in the presence of is confined to results for of in-plane distinctive hand, the effect the beat response. in Figs. 4 - 8 librations thermal circular and 4-11. and condensation of contributions of of eccentricity the orbital deformation is merely 117 Figure 4-15 System response showing the effect of eccentricity body disturbance: (a) librational response. and central Figure 4-15 System response showing the effect of eccentricity body disturbance: (b) vibrational response. and central 119 Figure 4-16 System response showing the effect of deformation: (a) appendage disturbance. eccentricity and thermal 120 Figure 4-16 System response showing the effect of eccentricity deformation: (b) central body disturbance, librational and thermal 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 As libration initial of pointed out deformation on system stability in Section 4.3.3, thermal and vibration amplitudes (eccentric deformation deformation in circular orbits, irrespective of a stubby central has negligible effect 0<= shown in Fig. 4-17 a 0.5, hC = applied, thermal deformation 0.6). T w o sets of in-plane results in higher depending on the or amplify the initial initial conditions, thermal deformation magnitude of deformation causes tumbling of initial can help to the libration offset eccentricity. Fig. 4-18 are not conditions are body disturbances. In Fig. 4 - l 8 a , conditions, the (Fig. 4 - l 8 b ) . conditions, in-plane same as before. The initial appendage and central vibration indicates lower deformation as background, the results of surprising. The system is the and deformation. This shows that, destabilizing influence due to With Fig. 4-17 in-plane thermal conditions are appendage disturbance appendage and central body disturbances, Fig. 4-17b in the presence of amplitudes as initial libration amplitudes. For a similar system under different amplitude show that deformation can libration and vibration compared here. In Fig. 4-17a, with only the on system response. However, for body with heavy appendages, thermal in an increase or decrease of orbits) increases the conditions. In eccentric orbits, results in Section 4.3.4 thermal result thermal s y s t e m . In contrast, under system is stabilized by thermal different thermal Figure 4-17 System response in eccentric orbits showing the effect thermal deformation: (a) an increase in libration amplitude. of Figure 4-17 System response in eccentric orbits showing the thermal deformation: (b) a decrease in libration amplitude. effect System Porometers = = = = Orbital Elements 0.01 0.50 0.60 20.00 90450- -45-90100y S 50- 1 i 0- O X QJ -50- -10010050- rO 1 O 0- X Q_l L* = 0.6 -50- -100100•—> 50- •O 1 O *x 3 QJ o-50- -100100S ro I N O 3 QJ 1 2 0 (No. of Orbits) Figure 4-18 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 4.4.1 Variation Solution of parameters method The method, as outlined analytical gradient solution for \p, <f>, P , Q j , P_ , and Q ( configuration u responses. For the gravity located on the local vertical, effect on the pointing sensitivity for the yaw degree u under consideration, acting as a communications satellite with the antenna librations in Chapter 3, is intended to obtain the of yaw is relatively small. Hence, the solution of freedom was obtained neglecting all the vibration terms, i.e., the same as the one presented in Section 3.4.2, E q . (3.35). Ignoring third and higher orders of e, the equations of motion <t>, E i , CL, E . . , a n d for \p t Q,. generalized coordinates can be written a s : 1>"-e ip' + K e rP = e + G {q); c it c g l <p -€ <p' + (l + Ki )<p = G {q)i n e t 2 Ei' - f c Z / + (w? - ecos6 + e cos 9)Pj = C? (g); 2 2 3 Qj' - e Qj + (w + 1 - ecos5 + e cos 2 2 = G (q); 2 c Ell ~ ecE!u +K 2 A - € cos 0 + e cos B)P = G {q); 2 2 U Q" - e Q' + (w + 1 - € cos 6 + e cos 2 c where Gj (i = 2 2 a 1, . . ., 6) are nonlinear h = G {q); 6 functions of generalized • • • (4.3) coordinates, velocities, and accelerations. The homogeneous set of E q . (4.3) represents the Mathieu equation 22 and can be solved using the standard procedure . Denoting: • • • (4-4) 128 the i homogeneous solution of E q . (4.3) can be written a s : - A , L n w + M I + ( - f " 1 ) 2 8 e [ s i n | ( n : + 1 ) 9+ ^ - •NOv-D' + 2n^, + 1 L I (2rty + l)(rty + 1) fti 2n^, — 1 J (2n^ - l)(n^ - 1) J) + e^i sin 0 - e^ sin 20 ; 2 (1 - 3K )e it = ^{sin(n^0 + /^) + rsin[(n^ + 1)0 + /fy] sin[(n^ - 1)0 + /fyi 2n^+ 1 2n^- 1 (1 - 3 i f ) n ^ r-sin[(r^ + 2)9+^] sin[{n+ - 2)0 + p+\ 2 t 7 8 + I (2n^ + l)(n^+l) + (2n^ - l)(r^ - 1) Pi = Apt sm(n 9 + P i)) p Qj = A qi p s'm(n 9 + p i); £.= A pu q q s'm(rip9 + p ); pu <3 = A s'm(n 9 q qu —a +P q d (4.5) ); q = qe" ; where with q= rp,(f>,P ,Q ,P ,Q ; i and i u u 1 /•* a = "2/ £ 0 c ( 0 ^ = e(l - c o s 0 ) - —(1 - cos 20) 4 The analytical method solution of variation is given b y : of parameters is now applied and the 129 1> =|^{sin[(n , + /?;)& + ^ ] + v (1 - 3K )e ^[{^ it sin[(r^ + Bin[(iy+i;-2)g + j f y ) ] * = f ^{sin[(n, { *• ^0 + + M + psin^ + 2 ( + + 2)0 + /fyp] (2n^ + l)(n^ + 1) , n + 1)6 + 2n^ + 1 1)9 + jfyo] ] ^ ( l - 3 / f , ) n ^ 2n,/, - 1 + ^ [ 0 . l ^ M i 2 1 a t M i n t + fl + W + L 2n^ + 1 M _ BinKn^ + Z f r - i y + fro^ ^ (1 - 3JST )n^e» r - B i n [ ( n , + ^ + 2)0 + a "J 2n^ - 1 + s\u\{n +^-2)9 <i> ^] (2fi^ + l)(n^+I) + 6^] ( 2 ^ - l)(n+ - 1) P = {^ in(n 0 + /? , )}e ; a / /S p p Qj = {A ism{n 9 + q u p )}e ; a q ql0 Pu = i p* s'm(n 9 + P )}e A p Q = {A sin(n 0 + /? u su 9 with j3^, and 0^ amplitudes ; a ptt0 )}e ; o guO (4.6) found to be the same as given in E q . (3.29). The and initial phase angles are determined from the initial conditions: = A {l- (l-3Jf )e 4*2-1 1+ 4>Q 3 r tt f (1 - 4( 3^)6 H -1 v h • a € (5n 2 2)el, " 4(n - 1) J 1 ° ^ ° 2 (l-3JJT - )e t t H- 1+ 1 (l-3JT )e ft H- 1 {5nl - 2)e] ^ C S + € ^ ~ ^ 2 5 130 Pj = Apt am Ppto ; Q E40 = Apin cosP IQ ; p P Qj — A i smp 0 q qlQ ; Q40 - A in cosP io ; q E.u0 = Ap q q sin^puo ; U P'uO — A n pu Q u0 Q' u0 cos P Q ; p ?U = A s'mp qtt ; qtlQ = -4gu"gCOS/? jO • 2t There presented are two limitations in E q . (4.6). The one frequency; hence, it averaging process solutions for fails to motion solution does not vibrational form of beat variation do not have approximate appendages to in Appendix I I I . of parameters at contain to method, the contributions solutions do not solution of The figure approximate the complete from vibrating librational librational response fails to estimate of is accurate. On the amplitudes analytical equations shows that although the libration predict the high frequency other terms, and response is indeed poor. The analytical to the peak-to-peak beat phenomenon, but a better of motion analytical answer. as given component, the prediction solution not the the solution for only also gives a poor estimate amplitudes. These shortcomings of a search for solution with hand, the vibrational prompted vibrate phenomenon. A l s o , due involve coupling between compares the numerical predict solution motion. Figure 4-19 'exact' predict involved in the librational this solution predicts the appendages. Simillarly, vibration i.e., the to analytical of of fails the solution 131 System Parameters legend Numerical Analytical 1 2 6 (No. of Orbits) Figure 4 - 1 9 A comparative study showing deficiencies solution: (a) librational response. of the analytical 19 A comparative study showing deficiencies of solution: (b) vibrational response. the analytical Figure 4-19 A comparative study showing deficiencies of solution: (b) vibrational response. the analytical 134 4.4.2 Improved analytical solution The deficiencies of the analytical solution obtained in the section may be attributed to the approximate the general equations of first out-of-plane to other is rewritten to degrees of vibrations predominantly affect in-plane librations, respectively, the coupling terms are reflect In include freedom. A s only the of The include any coupling terms. solution, Eq. (4.3) linear contributions from and o u t - o f - p l a n e retained equation does not improve the analytical order in Eq. (4.3) motion presented in Appendix I I I . homogeneous part of the order to representation last in-plane and appropriately this trend; •P" + 3K il> + K a,[(P}' + £'„') + 3(F, + F J ] = H^q); . 4>» + (1 + 3K )<p + K a [(g; 2 tt al it al - 3 + 4(Q, - QJ] = H {q) ; Fj' + u, Fj + a ( y + 3V) = H {q); 2 r z 3 QJ' + (u + 1)Q, + a (<f>" + 4<t>) = Ht(q); 2 3 ElL + " P + « (TP" + 3V) = H (q) ; 2 U Q^ + (u + l)Q z b u - a {<p" + 4<f>) = He{q) ; 2 where QpP-,,^', ?= ^ O Note, the homogeneous part of E q . (4.7) degrees of freedom omitted Eq. (4.7) freedom , shows coupling among the P|, and P_ ) and the u out-of-plane u was obtained using the Laplace transform Recognizing the fact other in-plane degrees of effects are purposely avoid non-autonomous character. The homogeneous solution of The method of from ^ (0, Q | , and Q ) . The eccentricity and thermal to • • • (4.7) 3 variation that for degrees of one can write; of parameters procedure (Appendix V). can now be applied. a given generalized coordinate, contribution freedom at different frequencies is relatively small, ^0) = ^i(0)sin[n^0+/? i(0)] 0 = j 4 ^ ( 0 ) f l i n t) ; n m^A^Wsmln^e = 4^(0) s i n ^ + P+M] i ; Pj{9) = A {0)am[n 9 pi3 3,( ) = 5 i 4 + p p i {e)) p 3 « » ( « ) 8 i n [ n ^ + /? | ('5)] a = ^ /3(^)sin^ 3 = A {0) sin fj ptt3 g pu3 3 ; ; Q (5) = > i g a 3 ( ^ ) s i n [ n ^ + ^ 3 ( ^ ) ] a a = Aq (9) s i n »7 ,3 . 3t u3 Using the procedure outlined derivatives of the amplitudes AL A'n = 3, the solution for the and phase angles can now be obtained irj cos 17^1; = X in Chapter H\ — cos 1701; TI* — £ T cos 17^3 ; n 1 A' IZ = P 3 p A' li = — # 4 cos T7 ; ; « 9 q 3 n rr* = —H$ cosn „3 ; n 1 A' z pu p p TJ* A' — —H cos 7 7 „ ; quZ — n 1 A 5 6 3 q H* ffpl = -—— A^in^i #1 = H 2 sin »7^i; s i n 77^1; -A^in^i #13 = - — - — s i n Apiztlpiz P' iz =-—-—H\ A izn i3 q q P' u3 ~ P 3 sin r? j ; 3 3 q —HI A Z nZ n pu T7p| ; P s i n ?7p 3 U 136 P'*** where H Assuming over a period A '*i i = i—I.— * ; H = H i{ n A s i n sin VtU • • • .<0 - -(4.8) the variation of parameters to be small, their averages are of interest: (2*)%i = {J *f*J Iff X 2 i* vti yn v$\ ^ H m d d dn Similar expressions are used for other A''s and j3''s. The average values were found to be as follows: t = rpl,<t>l,pl3,ql3,pu3,qu3 ; A[ = 0 j3[ » = pJ3, g/3, pu3, = 0 qu3 . Hence, the final solution for \p and <j> librations is given by: ~ ^ [{n - n ) - 2K a\{3 - n )] sm(n 9 + /^ )} 2 2 2 p K a {n + 3 dAn 2 at pl -n 2 \V * A 1 s MHi d - [Api sin(n 0 + p ) + A 2 pl 2 - 3) r 2 ai 2 pl pl pl2 ptt2 + A i am(n 9 + 0p!tl)] +M pu sm(n 0 + p )]} ; pl pK2 n 137 - ^ [ ( n - n ) - 2K a\(A ql K ctz(n - n )] m(n, J + ^ ) } 2 2 2 2 at ql 8 1 -4) ( 2 at r sm(n 9 + p )} J ; - [A i sm(n 9 + p ) - A q 2 ql ql2 qu2 ql ... (4.9) qn2 „ . -wftyitfox ^ 1 0 = tan (-^77-); where: /»^-t»->(=^S). ro Similarly for the vibration degrees of freedom; Ej = ~r~rT—^-{-A^ sin(n^ 0 1 + i( *>i d + /3^ ) + i4 , sin(n i0 + ^ ) } 1 1 v 2 p 2 - li) n n r-Jr |*3 (-T*$ +3X^-271$!+3) i ; a 1 x {^p/i [ _ v + P(2 I * P n 2 2_ 2 ) 'P 'V A (n 2 + R n P l ~ £J ( pl n - 2 a t n x { - j i p « i ( - n $ i + 3)(nJ - n ) smin^B + 2 pl + A {-n ptt2 + A {n puZ 2 pl - 3)(n pl 2 2 p pl p ) pui - n^) sin(n 0 + 0pn2) - n^) sin(n0 + /? „ )} 5 + 3)(nJ p 1 . , „ t P ~ »^iA"p "plJ i f a ( n 2J - 3 ) P <*i(»Jl - »Ji)(»J - »$i)("J - p i ) 2 2 p 3 „ n j + nJJ sin(n^^ + ^ x ) 8m n 0 + ^ 2 ) , 138 Q. = f * 3 ( ?\ { - A # s l n ^ e + fi#) + A* sm(n 9 * + K ql fa)} + -KataK-nl, + 4)(n - 2nJ + 4) 2 t - 4i ^ a l ( - 4 + 4)(n -24 +4) + n \ 2 r 1 at 1 A,1 " ^ »; " « ^ L {n -n' ) - ^ ^ ( ^ - 4 0 ( 4 - 4 ) 5 2 +n 2 - njl 8in(n itf + ^; ) * 2 ql B 2 q ql 1^3 2 v ]sin(n 0 + ^ )} g /3 + x {^uiC-n^ + 4 ) ( n 2 -nj )sin(n0 0+^,i) 1 - V2(- gi +)( g " W - A (n q sin(n i0 + /? n - 4)(n 2 qu3 4 2 1 1 g gtt2 - n\ ) sm(n 9 + p )} x q = TTT—^Tr{-^isin(n0i0 + ) 5 qtt3 + A^ sm{n i9 + fa)} * sin(n^0 + 2 p 1 + 2 x {A i 1 1 L 2 v p pttl 4-4 lKat<xl(-n +3)(n -2n +3) + ^p«2 *T"2—-yr v p i) 2 2 pl L 2 p re pl — n p +2 n i + n i - n j sm(n i0 + /? p p J (3 ) pnl pu2 ) 139 . + Ap - K l a a l ( n l *>' - n 2 p l ) ( - n 2 l 3) 2 . 2 p + ] (nj-nj^nj-nj,) , f J" " ^ + W J KgtczUnl - 3) ^ i ( ^ + 1 - n 2 j ( ^ - n 2 1 ) ( n 2 - n j x {-iljrfiC-nJj + 3)(nJ - n + A (n Qn = 2 pl TTT—^4{^i a n qi ~ 4>\) 1 n n + x j pl - njj s'm(n 9 + fa)} ; 2 p 2 p - 3)(n 2 plz pl p H 4>i n s K r-KataK-n^-r qu2 ~ ^sin{n i9 A q A){n\ - 2n\ +4) T „ > 2 2 x 2 + fa)} ' , nji + nj 8m(n^« + riif a (-n , +4)(n ^2n , +4) . -, ) ' ^ + n*! - 1 - 3K \ sm(n 9 \ q ~ ql) 2 at 2 2 ! 1 A it 2 n n 2 2 n 2 Kaialjn 2 + x +4 2 q 2 - A [n\ qlz The constants + fa ) 2 J n - 4) ) ( J ~ Jl) ' Kl* + /W n + 4){n W 2 ql ql J (n -n-)(n -n ) - ql) K £l)(*« 2 fa) 2 1 q ~ M - A i {-n +M e { 4r«i , + A + fa) pl + 3)(n - n ^) sin(n 0 + fa) 2 pl2 ) ) am(n^6 2 + A {-n 1 q 8 m - njj sin(n 0 + /fy) gl 2 - 4)(n* - njj sm(n 9 + fa)} . 1 appearing q in Eqs. (4.9) and (4.10) are defined . . . (4.10) in Appendix V . 140 4.4.3 Discussion of results To accuracy of check the compared with the equations of this results given by numerical motion (Appendix Response of compared with numerically closed-form vibrational to the system to displacement of the parameter. most computer of the exact parameters of first the mode is Note, the phenomenon, librational and body are presented results for in Fig. 4-21. is again excellent except for a course, is expected because assumed X solution. However, as explained satellites, X is not likely to be a critical X response can easily be improved by higher order terms, of course, at a cost in terms of time. the success, possible limitations solution must be recognized, particularly unusually severe pure o u t - o f - p l a n e 4-22. form in the in Fig. 4-20. central sets of Furthermore, accuracy of In spite of analytical length (X) response. This, of application including appropriate its obtained data the two simplified form before, for the was as well as frequencies very well. Similar results small discrepancy in yaw of of system a disturbance in the 20% of a large disturbance applied to the Correlation between integration solution, it 4-25). solution predicts the beat amplitudes analytical I I I ) over a range of and initial conditions (Figs. 4-20 appendage tip approximate Note, the presence of approximate out-of-plane Besides approximate analytical approximate in the presence of solution fails to predict in-plane of the ever in Fig. motion disturbance although the o u t - o f - p l a n e disturbances are hardly magnitude this disturbance. This is illustrated reasonably well correlated. T o be fair, this as such large of in the response is is an unusually demanding test encountered in practice. initial disturbances, accuracy of solution also depends on the inertia the parameters, K g 141 Figure 4-20 A comparison between numerical and improved solutions for a severe appendage disturbance: (a) analytically obtained response. analytical 142 Figure 4-20 A comparison between numerical and improved solutions for a severe appendage disturbance: (b) numerical results. analytical 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 Orbital Elements p- 90° CJ 90° i = 0 t = 0 = Initial Positions fo = 5« <t>o =5*o = 0 0 Q.o = 0 Euo = 0 Quo = 0 EIO = 144 Initial Velocities % =o = 0 X.; = 0.0872 Ei = o Qio = o E = o Quo = o 0 u o Numerical Solution -0.25 1 2 i 3 6 (No. of Orbits) Figure 4 - 2 1 A comparison between numerical and improved analytically predicted responses with central body disturbance: (b) vibrational response. 1 2 6 (No. of Orbits) Figure 4-21 A comparison between numerical and improved analytically predicted responses with central body disturbance: (b) vibrational response. 146 System Parameters Orbital Elements = K = K, = L* = u = p = 90" v — 90* i = 0 c = 0 a, e r 0.01 0.10 0.75 0 20.00 Initial Positions i> 0 =o = 0 x = 0 E.o= 0 Q.o = o P-uo = 0 Quo = o L . 0 O 0 Initial Velocities v -;, =oi _. 0 ; =1-00 K = • P-io = 0 I 9i = 0 P-uo = Oj Quo = o! 0 1 0 1 20 0 Figure 4-22 1 2 0 (No. of Orbits) 3 A comparison between numerical and improved analytical in the presence, of severe o u t - o f - p l a n e disturbance: (a) librational response. solutions 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 i (Figs. 4-23 to K a and smaller K. affect 4-25). Note, with K = g appendages with a slender central well amplitudes body (K- = the earlier. A l s o , the newer body), the responses compare g reasonably 0.5, Fig. 4-24) or a correlation between the two limitations of the proposed solution were solution is applicable only to of autonomous s y s t e m s , i.e., deflection of situations discussed of practical appendages. importance with in circular or near-circular orbits. Furthermore, the development materials of appear to class of solution appendage (K = in circular orbits with no thermal applicability large (i.e., small 0.25, Fig. 4-25), although frequencies and Fortunately, there are a number satellites 0.75 ( is poor due to discrepancies in phase. Some of satellites and K = are reasonably well predicted, the responses not 0.1 (Fig. 4-23). However, with a large stubby central the accuracy adversely i promises to reduce thermal the approach to deformations. Thus the autonomous systems in circular orbits be a serious restriction. T o put satellites in circular orbit it differently, stages of the control Furthermore, it resulted 1/3 in a significant the time of there is a the solution during the system design cannot be questioned. the A M D A H L 470-V8 s y s t e m , a typical approximately does where the proposed c l o s e d - f o r m is applicable. In any case, usefulness of preliminary of reduction in computational run for the analytical time. For solution takes the corresponding numerical analysis. 149 5- 0° A ,f\ A W -10 20 i 7 i legend 10- Numerical Analytical -20 1 2 6 (No. of Orbits) Figure 4-23 A comparison between numerical and improved analytical showing the effect of inertia parameters on correlation; K = 0.1, K.= 0.75. a solutions System Parameters Orbital Elements Initial Positions f = 30' <P = 0 A = 0 P = 0.10 Initial Velocities fo = 0 50 <f>'o =° 25 Q, = Euo= o-io Quo 0 Q!o Puo = Quo = 0 L* = = = = = 0.01 0.10 0.75 0 20.00 p = 90° u = 90' i = c = 0 0 0 0 | o 0 o 1 0 K p; 0 150 =o =o = o 2 00 2 00 2 0 (No. of Orbits) -23 A comparison showing the between effect K = 0.1, K = 0.75. a : of numerical inertia and improved parameters on analytical correlation: solutions System Parameters L* = 0.01 = 0.10 = 0.75 = 0 = 20.00 Orbital Elements Initial Positions — 30° xn« p = 90° v = 90° 0o i = £ = Eio 0 0 Ko 9io E u o Quo o 0 0.10 0.10 0 0 151 Initial Velocities % = 0.50 0; = 0.25 K =0 Elo = 0 Qio = 0 Euo = 2.00 Quo ~ 2.00 I o X QJ I O X 67 I O rO I O Analytical Solution 3 Ol 1 2 0 (No. of Orbits) Figure 4-23 A comparison between numerical and improved analytical showing the effect of inertia parameters on correlation; K = 0.1, K.= 0.75. a ' solutions 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 153 System Parameters Numerical Solution 1 2 6 (No. of Orbits) 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 I System Parameters Orbital Elements a, K K| L* u p = 90° u = 90" c r = 0.01 = 0.50 = 0.75 = 0 = 20.00 i = e = 0 0 Initial Positions Vo = 30 «o = 0 0 H.O = 0 10 Q.o = 0 Euo = 0 9 — 0.10 K = U 0 Initial Velocities 0.50 0 0; = 0.25 0 Pio = 0 9 i o = 2.00 p' = 2.00 0 -uo Q* =»uo = 154 r = K = 200-1 rO I o X QJ -200200100I O Oi -100200200100- ro i I 0- O *x 3 QJ --100- 200200fO 100- O 0- 1 1 'x OI -100-200( 1 2 6 (No. of Orbits) 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 1 155 Figure 4-25 A comparison between numerical and improved analytical solutions showing the effect of a stubby central body: < = 0.1, Kj = 0.25. a System Parameters 156 Orbital Elements K) I o X I O I o dT fO I o QJ 3 K> I o Numerical Solution 3 Ol 0 1 2 6 (No. of Orbits) Figure 4-25 A c o m p a r i s o n b e t w e e n n u m e r i c a l and i m p r o v e d a n a l y t i c a l s o l u t i o n s s h o w i n g the e f f e c t o f a s t u b b y c e n t r a l b o d y : K = 0.1, K. = 0.25. 3 I 15075- 1 o o- X QJ -75-150-J 15075- 1 o 0 J X v—s 6 7 -75-150 15075- ro 1 1 O 0- X 3 Q_l -75-150150- •O • 75- I O T— 0- OI -75-150 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 C O M M E N T S 5.1 Summary of A Conclusions relatively simple model body and a pair of important flexible step forward problems. It is particularly (i) following A relevant the general are the orient inertia gravitational 0.75, K = 3 satellite at a larger 0.25, (iv) e= 0.1, CJ = deflection of central eccentricity body) and a larger e. eccentricity in equilibrium with its axis inclined to angle. So far as the flexible appendages compared to 2.0, and L * = 0.6 gave tfr = the about orbital 7° the with the 6 50% of its length. rate approaching the appendage equilibrium configuration diverges indefinitely local vertical Application of natural from its alignment. Floquet theory to the nonautonomous, linear suggests that small K. and K thermally induced deformations additional small strip of of Kj and the orbital i frequency, the system equilibrium contributions. For stable equilibrium configurations, Kj = A s expected, with the nominal of the satellite's deflections are dominant • boom tip of large appendages. Based on the parameter Kj (stubby the local vertical rigid conclusions can be made: are concerned, thermal (iii) to the next generation with relatively smaller value of tends to a central appendages, studied here represents an The most significant factors affecting orientation (ii) a satellite, consisting of in understanding dynamics of this class communications satellites analysis of of the promote instability. The appendages lead to an unstable region extending over a wide range K and K.. a i Even a small difference in gravitational 158 field experienced by the two 159 appendages a beat is sufficient phenomenon. Oscillation of excitation natural parameters of solution developed on the deformations an increase in orbtal is to coupling. The principle of is able to predict the beat thermal frequency leading to a given appendage serves as an for the other through a librational approximate effect to change their variation response rather distort the eccentricity tends to of accurately. The beat response, while increase the beat frequency. (v) A s expected, the modulate virtually This the no effect librational (vi) in-plane is related central effect to the the is to (vii) increase and or out-of-plane. frequencies, even large rotations appendages only by a small most in of the amount. interesting contribution of the orbital cause frequency condensation in all degrees of response. This is attributed in governing equations of increases stiffness of the flexible In circular orbits, the effect increase both librational initial is to fact that, due to a large difference freedom, including the beat 1 / ( 1 + e c o s 0 ) terms eccentricity pitch librations. However, surprisingly, it has and vibrational Perhaps one of orbital on appendage vibration, in-plane body excite eccentricity of of and vibrational the motion. In effect, appendage at thermal to apogee. deformations amplitudes conditions. In general, an increase in K g it (L*= 0.6) is to for a given set of or a decrease in Kj enhances this trend. In fact, under a critical combination of parameters, the system can become unstable although corresponding undeformed system (L* = (viii) In contrast to the conditions circular orbit the 0) may be stable. case, for et and system parameters, thermally 0, depending on flexed initial appendages may 160 stabilize (ix) The the system, analytical solution to the Butenin method, appears librational quite promising. In and vibrational general, it using the predicts frequencies with surprising accuracy. The solution devleoped here satellites complex problem, obtained is applicable to autonomous in circular orbits with no thermal s y s t e m s , i.e., deformation of appendages. 5.2 Recommendations for Future The thesis represents developing field structures to for mid-1990's, this study of the to problem parameters is restricted affecting to the spacecraft orbit. solution obtained considerable scope for its applicability closed-form only to solution to s u c c e s s f u l , would are control avenues a few of introduced to one them orbiting apply the in the help though effective formulation major autonomous systems. A search a coupled nonautonomous breakthrough to presents Perhaps the a major space problems improvement. lead to rapidly dynamics and its physical be useful to in any arbitrary satellites purposely appreciation. Now, it would The approximate of be fruitful, however, only plane. This assumption was on key and this U.S. class of are a number below: (ii) forces. With the c o m e . There likely of flexible a long time to touched upon focus large dynamicists are ecliptic of attention of are The by in exploration occupy the can pursue which (i) a beginning the environmental a space station going to engineers only at design and control in presence of commitment certainly aimed Work limitation is for s y s t e m , if in solution of such a 161 complex nonlinear s y s t e m . Furthermore, modified averaging procedure using other yaw solution through retention likely to be than of 2n implementation period, and improvement coupling and nonlinear terms f o c u s s e d on librational Rendering the and vibrational are should be control. appendages as well as the central the analysis applicable to a large class of Orbiter-based construction of space structural body flexible future will satellites, the components, and the proposed station. Addition of plate-type solar panels will add to the versatility of model. Inclusion of slewing motion sun-tracking maneuver more in successful. With dynamics well predicted and understood, the attention make of realistic for for the flexible appendages and the solar panels will render the and hence further add to its usefulness. model the 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 M o d i , 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. M o d i , V . J . , "Attitiude Dynamics of Satellites with Flexible A p p e n d a g e s A Brief Review," Journal of Spacecrafts and Rockets, V o l . 11, No. 11, Nov. 1974, pp. 743-751. 4. Shrivastava, S.K., and M o d i , V . J . , "Satellite Attitude Dynamics and Control in the Presence of Environmental Torques - A Brief review," Journal of Guidance, Control, and Dynamics, V o l . 6, No. 6, N o v . - D e c . 1983, pp. 461-471. 5. Modi, 6. Modi, V . J . , and Brereton, R.C., "Planar Librational Stability of a Long Flexible Satellite," Al AA Journal, V o l . 6, No. 3, March 1968, pp. 511-517. V . J . , and Kumar, K., "Librational Dynamics of a Satellite with Thermally Flexed Appendages," Journal of the Astronautical Sciences, V o l . 25, No. 1, J a n - M a r . , 1977, pp. 3-20. 7. 8. Goldman, R.L., "Influence of Thermal Distortion Stabilization," Journal of Spacecrafts and July 1975, pp. 406-413. Yu, Y.Y., "Thermally Induced Vibration on Gravity Gradient Rockets, V o l . 12, No. and Flutter of a Flexible Boom," Journal of Spacecrafts and Rockets, V o l . 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, V o l . 8, No. 2, Feb. 1971, pp. 202-204. 10. Jordan, P.F., "Comment on 'Thermally Induced Vibration and Flutter a Flexible B o o m ' , " Journal of Spacecrafts and Rockets, V o l . 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 of 8, 7, 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 Dumbbell Satellite," ARS Journal, V o l . 31, No. 8, A u g . 1961, pp. 1089-1096. 17. Yu, E.Y., "Long-term Coupling Effects Between the Librational and Orbital Motions of a Satellite," Al AA Journal, V o l . 2, No. 3, 1964, pp. 553-555. a Mar. 18. Schittkowski, K., "The and Powell with Function. Part 1: Vol. 38, Fasc 1, 19. Schittkowski, K., "The Nonlinear Programming Mehtod of W i l s o n , Han, and Powell with an Augmented Lagrangian-Type Line Search Function. Part 2: A n Efficient Implementation with Linear Least Square Subproblems," Numerische Mathematik, V o l . 38, Fasc 1, 1981, pp. 115-128. 20. Minorsky, N., Nonlinear Princeton, 1962, pp. 21. Butenin, N.V., Co., New Nonlinear Programming Mehtod of W i l s o n , Han, an Augmented Lagrangian-Type Line Search Convergence Analysis," Numerische Mathematik, 1981, pp. 83-114. Oscillation, D. Van Nostrand Co. Inc., 127-133. Elements of Non-linear Oscillations, Blaisdell Publishing York, 1965, pp. 102-137, 201-217. 22. Struble, R A . , Nonlinear Differential Equations, McGraw-Hill Inc., New York, 1962, pp. 220-234. 23. Gear, Book Co., C.W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall Inc., Englewood C l i f f s , 1971, pp. 158-166, 209-228. APPENDIX I Substituting - E V A L U A T I O N OF A P P E N D A G E from app ENERGY E q s . (2.19) and (2.20) into E q . (2.21) gives, J " {[V T . = ^ KINETIC + oj {zi + w ) + u (yi + x y w (xj + + [V - tv)] 2 z a) - u (z 2 y { x t>j] 2 + wi)-yi- t 4- [V - u (yi + v{) + (j} (xi + a) + z\ + u>;] } dx 2 z x ~YJ + y {l * + "A * + *)~Uz(yu Z v Q + [Vy + u (x z t + a) - u (z n + V )] W x 2 U + w) + y + H v B [V + w (y + v ) - u,j(x + a) + z z Since relations x the shift tt u n in the centre n rl + tv„] } dx . 2 n ••• (7.1) of mass is negligible, the following rh b (y* + v )dx t u = 0; u Jo fh fh / (z\ + wi) dxi + / (z + w )dx Jo Jo u fh / (yz + Jo f \z, l Jo Substituting z , v , and w u 2 v are valid: / (yi + vi)dx Jo u v] u =0; u fh v/) dxi -I (y„ + t ) „ ) dx = 0; •'o n + ti;,) dxi + I " ( i , + w ) dx = 0 . Jo a for V j , from u Z j , v , and w ( ( u from lb Jo f (*i(z))dz; b <6 Jo i $13 = r y' / 6 ($i( )) x s 1 Z"' 6 164 d z (1.2) E q s . (2.14), (2.15) and for y E q . (2.17), denoting: $12 = r • • ; 165 where ^(x) is given by Eq. (2.16), the expression for appendage kinetic energy (Eq. 1.1) takes the form, T . = m l {R app + (R9) ) + T + T + T 2 2 b b L + T[j + Tj + T , + Ti )C t v t ittV e ; where; 7} = m l {a + al + jj/3)(wj + u\) ; 2 b b b T,, = ^ ^ { ^ ( c o s t 2 41 + cos <f>\) + ( 2 Wf co.tf - u, cos^) + 2u [cos <p* (cos c6!) - cos cf>\ (cos c>*)]} ; x ri,. = y ^{*n[(i? + J?)(«2 + »l) + (Qi + Ql)i<4 + «J) + (P/Qi - P . Q « ) » * + u {PiQi - P Q, + P Q W W x + = ^.i,. = ( a $ 1 2 m 6 ^ 6 { tJ *M)w*[-w»(J i D + _ ( ^^{^l(Pi / } _ F t t )^ ( c o s - P.)cosfi Z u + ^«) + f>z{Qi n - P Q )} u a - Qu))} ; ^ ) + $ + Q.jlfcostf)} ; ( f + (Q, + Q.Jcos^; + w [-(P, - P„) cos c>; + (Q, + <?„) cos 2 + (w cos 0* - w cos <j>y) [{Pi - P„)u + {Qt + Q )uy]} . y z z u 2 APPENDIX II - E V A L U A T I O N OF A P P E N D A G E POTENTIAL ENERGY 3 Expanding E q . (2.23) and ignoring |n| = R{1 + | [-(a + x )l t 1/R and higher - (y, + vi)l + (z, + vn)l ] x y g + ^ 2 K + * 0 + (» + vi) + (* + 2 a order terms gives: «*) ]} ; 2 a 1/2 |r.| = i2{l + |[(a + z„)I + (y„ + v.)/, + (ar. + w,)/,] x + ^2 K +«) + (»• +«) + (*• +«)]} ' a z 2 u 2 w 2 1/2 4 Substituting from higher order terms, U a p E q . ( 11 . 1) into E q . (2.22) and ignoring p j f 1 U . = app - 2^2 + 2(o 1/R and can be written as, - i[-(o + x,% - (y, + )l + Vl [(« + *l? + (» + v,) + + w*) ] 2 + x ){yi + vi)U t 2 - 2{a + s + wM v x,)(2j + w,)l l x z - 2(yz + vi)(zi + wi)lyl ] J dz, s - /Q " - -[(a + + (y« + v )l + (z. + u y [(« + *«) + (y. + *>«) + (*. + ^«) ] 2 + ^ K« + 2 *«) ' + (y. + 2 2 2 v»)/J + (* + 2 wjil + 2(a + z«)(y» + v )l l + 2(a + x )(z + w„)l l u x y v + 2(y« + v )(z + w )lyl ]} dx . u n tt z n 166 n x z ... (J/.2) •• Finally, using Eqs. (1.2a) and (1.2b), the expression (Eq. 11.2) appendage can be written in the 2m l ft u . = --^ R b b form, l 11^ + 11^ + t energy +u +u + u e apP + U,, + potential t 1! 1;^; where: U i ^ t = { a 2 + a l b ^» ^ = C032 + l l / m _ 3 l l ) . ^ +COS2 ); - (P, + Pl)l] - (Q? + Q)/- + 2 ( - P . Q . 2 flu.. = 3 f , c y 2 i a lW - P«)-V - 2 Wi + + PiQi)lyh); Q.)M('»co8«; +1, a**;) APPENDIX All respect terms to III - DETAILS OF THE E Q U A T I O N S OF MOTION in the equations 1^, where of motion It = I + 2m ll\(a/l ) 2 b In the expression for strain energy U b g are nondimensionalized + with (o// ) + 1/3] • t (Eq. 2.25), the fundamental frequency co.j is nondimensionalized a s , § where 0 p is the angular addition, the following p U + ecos0 velocity notations rn of the system (///.la) at the perigee are used: 2e sin 0 1+ point. In (///.16) € COS 0 ' RH 2 1 (I I Lie) 1 + €COS0 ' Kat = m l /It; b b KH = i - 1 Jit ; (///.2) rV 1 « 2 = ^ ( a 3 = v - $ i 2 + -j—; F ) ; '6 lb - (///.3) «4=2( )(-T2-)' F l vt Uyz = ly cos t6* + l cos ; z — 0J ci = y cos — uz cos 0* ; bi cos i/> sin <p + b sin ip sin 0 + 63 cos <p ; 2 C2 = 61 sin ip — 62 cos ip ; 168 169 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 + 6 sin ^) cos A . • • • (IIIA) 2 The equations o f motion corresponding to each o f the generalized coordinates can be obtained from dVdq' dq + dq~ where q = yp, <6, X, P,, Q,, P , and Q u ' q u The individual term appearing in the above equation are evaluated below: \p - Degree of Freedom (Pitch): I9 2 t dt drp K j [oj' sin A + u' cos A + A'(w cos A — y y z UJ Z sin A)] cos <f> - (u/ys'm A + u cosA)<^'sin<^| z - (1 — Kit)(uj' sin<£ + u d>'cos<p) x + x + K^ail-iPtQZ go, + F Q ; - P ' j Q J sin U j - (u' sin<P + UJ <P'cos<P)(P] + Q^ + Pl + Q ) 2 X x tt - 2u s i n I ( P , P j + + P„P' + Q Q' ) tt x + (-PfS; + EiQt + P Q! n n a tt ~ f . f t , ) ( 6 e s i n <4 - ci' cos 4) + (EjUz + Quiy) [Pj cos A + QJ sin A + A ' ( - P , sin A + + (Pj cos A + + (P Uz u sin A ) ( P > + Q^ojy + PJOJ' + 2 ~ Q Uy) u + (P cos X-Q u n Z Q^J,) cos A)] cos <p cos <p [P« cos A - Q' sin A - A'(P„ sin A + Q a u cos A)] cos <p sin A ) ( P > , - g w , + P „ ^ - Qu' ) cos 0 y - <f>' sin 4> [{Pju + + QjUy^Pj z - Q u ){P u y cos A + 0, sin A) cos A - Q sin A)]} n u + 2 { wj,(C2 COS *) + W ( c COS a y i 2 - {u)' sin * + oj d)' cos *)(cos ** + cos <p* ) 2 x 2 x z - 2u sin4[cos**(cos**)' + cos<^(cos<f>*)'] x z - *'cos* [cos** (cos <p* )' - cos **;(cos**)'] z - sin*[cos**(cos**J" - cos^cos**)"] + c [w cos **, + w (cos **)' + (cos ft)"] + c\ [UJ cos 4* + (cos **)'] 4 x X x v + c [ - t v cos # - w (cos 41)' + (cos **,)"] + c' [-u z 6 x b x cos 41 + (cos 4* )']} y + <* {{P}' + Pi) cos 4 cos A + (QJ - ( g ) cos * sin A 3 + (P4 + E! ) [~ c cos 4 cos A — 4' sin * cos A — A' cos 4 sin A e u - u sin A cos 4 + y sin *] UJ x + (QJ — Q^) [e cos 4 sin A — *' sin * sin A + A' cos 4 cos A c + UJ X cos A cos 4 + {E4 — Wz + E-u) Wy s m - u \' cos A cos 4 + x + (0/ — s i 4] n $+ Vyf cos * Ux4' s i Q„) [ * sin 4 — —w Uz4' n — w * s i ^ cos 4 n ^ sin 0] cos * + u)' cos A cos * x - u ) \ ' sin A cos 4 — x4' cos A sin 4]} (jJ x + « 4 { ( - £ { ' + £i')(sin * cos * ; + c ) + (QJ' + < £ ) { - sin * cos **, + c ) 5 4 + ( ~ i j + E!u) [~ c(sin 4 cos **. + cs) - 2u sin 4 cos ** e x + cos 4{4' cos **. — W j , ) cos A — U C2 cos 4 + U C4,] 2 Z X + (Qj + Q'J [e (sin * cos ** - c ) - 2 u sin 4 cos **. c x 4 + cos 4{-<f>' cos ** + u ) sin A + w C2 cos 4 - w cs] yz y x 171 + {-Et K ( - 2 sin * cos ** + c ) + En) 4 + oj (-2<p' cos*cos** - 2sin *(cos**)' + c') x 4 — *' cos *(cos **.)' — sin *(cos **.)" + u {4> sin * cos A + A' cos * sin A) yz — U)' COS * COS A — (J C2 cos <f> — u (^2 cos *)'] yz Z z + (2i + £ J K(~2sin *cos ** - c ) 5 —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 * + ui (c2 cos *)']} > ; y 1 dT ( + ^( c o s *^( + « {(-Pj 4 c o s 3 3 * ^ ) ' - COS*! (COS * )')] + y + £'J[-^^(cos*;) + 3 d UyzJj(Vyz) A(OB*;)')] C ( + (-£, + Z J K K ^ ( c o s * ) + _((cos *:)')) y + u ^ K ) ] } } ; 3 172 1 8U i e dxp 2 = Kueg cos £ 2 J. • , , <p sin ip cos •/» t + Q +P + K e ^ai{-(P] at 2 g + <£) c o s <> / sin y cos i> 2 2 1 d & + a { - [cos 0 ; — ( c o s <f>*) + cos f —(cos 2 f )] - z + a {-(£/ + 3 + Z„)[" z ^ / y J —(/j„)} sin cos + / — ( y ] V r x + £.)[i£<"»*;) - '.-£«.> - ^<Wl 0 - Degree of Freedom (Roll): 1 '<£> = : { O! {(Pjcv + Pju' + g^Uy + QbJyK-Pu sin A + + 2 + (P' uz + P _ X - Q X ) ( - P „ sin X-Q - Q>v u + (Pjw* + ^ W y ) [ - P j sin A + + (P.w, - Q Wj,) a u cos A) cos A - A'(P/ cos A + Q sin A)] [-P'„ sin A - + oc2{cioj' + c\tjy yz cos A) z cos A + A'(-P„ cos X + Q sin A)] } u 2 + ( c s i n A ) ' [ - w c o s * + (cos </»*)'] + (c cos X)'[u c o s * + (cos^*)'] 3 x r 3 + c sin A [-wj. cos <f>* - w (cos 3 z x + (cos 0*)"] + c cos A [u' cos * + w (cos <j>*y)' + (cos 3 x r x x } r 173 + as{-(Pi + E l l ) sin A + - Q") cos A + (Pj + £„)(<* sin A - w cos A) + (Qj - Q'J(-e cos A - w sin A) x c x + (Ej +Z )(-w cosA + w A'sinA) a I x - (Qj - Q ){u) s'm\ + a; A'cosA)} a x x + « {(-rj' + Z")c sin A + (<g + Q^c cos A 4 3 3 + (-Pj + Ptt)(-e c sin A + u c cos A + u c 3 x 3 yz + (Q^ + Q [ , ) ( c 3 cos A - w C3 sin A + u x 2 u ci) cos A + c - e sin A - w ci) yz y + {-Pj + P„)[w C3 COS A + U J ( C 3 cos A)' x X + u) \' cos A + u>' sin A — u>' ci — UJ C'I\ yz yz [ + (fil + _ w Z z * 3 sin A - w x (c 3 sin A)' c - u X' sin A + wj, cos A + oj' ci + Wyc'x] } > ; yz -L— 2 y = - ( l + V')[(l + V ' ) s i n r C O S + (l-ir, )a; cos ] t r x r I 9 9<f> 2 t + K + u { a {-(1 + V') cos r [(-££[ + PJQ, + x at ( £ ? x + Q 2 4 + E l u u tt Q )] + 2 u - (1 + V') sin 0[(P,w, + QjUyKPu cos A + + ( E " z - Quy)(E £.g, - P'£J cos A - sin A) sin A)]} + a {-(1 + v') cos <i[a; (cos c6* + cos c**) + cos *(cos c>*)' - cos c6^(cos <f>* )'] 2 2 2 x + u l [cos r ^ ( c o s c4*) + cos r ^ ( C O S <£!)] + [ ( c o s : ) ' ^ ( c o s ^ ) - (cos *)'^(cos^)] Wx + r r [(c68^y-W,C08^]~((c08^) ) + [(cos f + u cos f ] — ((cos 0*)') + w x y y2 y 174 - a ( 1 + 1>') {[E4 3 + E!u) sin * cos A + (Qj - (Ei + £«)( y os* + + « {(£; 4 sin * sin A) + c w -£L)[-(i + (fi! + V'')cos*cos*; Q/J sin * sin A - -Qj(w 2 cos* + u sin* cos x + u, -^(cos<t>:) x A( (cog A)} ^)')] + V'')cos*cos* + ^A(cos*;) + ^((cos**)')] y + [-E4 + Eu) [ - ( ! + • / ' ) ( , 3 cos ^( + c o s * 2 w 3 c o s 0 c o - u s yz sin * cos A) 3 + 0^(0;*—(cos**,) + — ((cos*!)')) - w,— + (Qj + Q ) [ ~ ( + $')[-cos 1 <f>[cos d>* )' + 2 C J tt d d x z 2 C O S * C C S * * . + 0J yz d - —((cos*;)')) + u [u —[cosf ) x y + u y — ; — —— = Ku g cos */» sin * cos * I 9 <?* e : 2 t + Kateg { oi {-[Ej + Q + £ + Q ) cos i/> sin * cos * 2 2 2 2 + [Ejlz + QjlyKEj cos A + Qj sin A) cos * cos ip + [EJz - QJy)[P-n cos A - 1 sin A) cos * cos ip} 3 3 3 + a-2 { - [cos * ^ ( c o s **,) + cos **. ^ ( c o s **)] - l — [l )} y + a${-[Ei + Z«)(-^sin* + / + (Qj + z — Q )[~lz s i n * + l tt 1 3 x X 3 yz cos* sin A)cosi/» cos* cos A) cost/;} a {[-Ei +£,)[3^(cos*;) 4 x yz 3 -/j, cos^cos*sinA-/ —((„,)] 2 y 3 ] sin* sin A) 175 X - Degree o f Freedom —•—; = (1 - (Yaw): Kit)(jJ, + # e {<*. { ( - £ , 0 , " a + Fj'g, + £ - eeC-fig; + £Ja + £„fi'„ + 2W,(£,P| + t t g -EllQJ - £'«fij + u/,(£? + Q? + El 3 £«) K ( - £ / + + (S + fiJ K 2 c o s COS ^ £ + + W (C08 <py)' + ( C O S X "x(cos ;y - ( r cos r fi>4 ! ) " ] <g"] + ( - £ | + £ L ) [ w . C 0 8 ^ + (cOS^)T + (Sl + 2L)[wxco ^-(cos^y)]}}; 8 + a {(£i + + (fi! + (£.«. - S. ») [ £ - ^ ( « ) w 3 fi'J^K) fi. Aw]} w ["(£, + En)J^M + ( f i , - <?")^M } + + «4{(£| -£L)[C^(C0S^) - ^((COS^)')] + (fil+fi'jK^( ^) + ^((COS^)')] cos d d d + (~Ej + P ) [ w — ( c o s ^ ) + a , — ( ( c o s ^ ) ' ) - w„«—(w,)] 2 n x + (fi, + f i j [ % ( c o s c i : ) + w —((cos^)') + w^^K)]}} ; w x + (£.'.-fi.',)[£.^('.)-a|f(W] +«3M-(£i+£•) j ^ w + ( a + ) + £„£'„ + + * { - ( £ , + £ . K - (£| + £L)«, + {Qj- £ , K + (fi! + <*4 { +Q 0{(-a + £.)[i±(c08^) 4 +(a+fi«)[^( c o s ^) fi.)|f('*)} ly ^(ly)} Z - '»* Jx('*>]}} • ••• ( J// - ) 7 176 P, - In-plane Vibration of the Lower Appendage; + on{u)' cos (fc + w (cos <pV)' - (cos <p* )"} } ; x — = Kl x ai{-Q\u at + £ , ( « * + UJ ) + Qp uj } z y z y J t ^ = K a t i a i ® - ^ { - ' u auu 3 x y x E { z +/T^P,. Qj - Out-of-plane Vibration of the Lower W t X yz -a {icos* -y ,}} y v + UJ (cos<f>* )'}} ; - uu 2 4 - 2 x - a {w cos* 4 y •••(J/J.8) Appendage: - W x £ l } + azuJ '> + <* {UJ' cos 4> + UJ (cos * ) ' + (cos * ; ) " } } ; 4 -^U- X = at{<*l{P}Vz K y X y + 0,{"l + UJ ) + PjUJyUJ;} 2 + (X UJ UJ 3 X Z + a { u j c o s * * . + ujyUJyz + w (cos **,)'}} ; 2 4 x + a { i cos # - J,/ } } 4 yi + e^Q,. • • • (J//.9) P - In-plane Vibration of the Upper Appendage: = *-{«»{£: -<Q +«a«i V - a { w cos # + w (cos <p* )' - (cos <6*,)"} } ; 4 ^ ^ r x x = K a t { < x i + 2 £„(a; + wj) - Q ^ c , } - 2 a w, + a4{cj cos**, - w w^ + Wx(cos*;)'}| ; 2 2 + a { ^ cos * - lyly } J 4 Q u - Out-of-plane Z y Vibration of the Upper + K eP at v . n Apendage: + a { a ^ cos 0* + w ( c o s <f>y)' + (cos * * ) " } } ; / 02 ao„ t 4 x «. — + o r { w cos fc + a/j,Wj, + a;* (cos **,)'}} ; 2 4 z + a { ^ C O S c 6 * - Izlyz}} 4 +KatCvQ . u 3 APPENDIX In general, [M] coefficients of the coordinates. Its is a 7x7 IV - MATRIX symmetric matrix representing second order derivatives elements are listed [M] of the seven the generalized below; M n = cos d> + (1 — Ku) sin i> 2 2 + K { at + El + Ql) s i n * {(£? + ai 2 + (Pj cos A + Qj sin A ) cos <p + (P„ cos A 2 sin A ) cos d>} 2 2 2 + a {(C2 cos <f>) + s i n <j> (cos <f>* + cos c£*) + c\ 4- c 2 2 2 2 2 2 y — 2 sin c/>(c4 cos <£* — C5 cos <£*)} +a z sin 2 {(Pj + r E„) sin A - (Q, - ) cos A } + 2 a 4 { ( - P j + P„)[-sin (-sin<£cos <p* + C4) — c cos d> cos A] 2 r + (Qj + M12 = K 2 y [ - sin <p(-sin cos c/>* + C5) + c cos <f> sin A]} | ; 2 r 2 \ ari{(PjCosA + Q sinA)(—P^sinA 4- Q. cos A) cos <f> / at + (E cos A - Q sin A ) ( - P sin A cos A) cos 0} + + a sin <f>{(Ej + En) cos A + (Q, - Q J sin A} a u u a C i c cos<p 2 2 3 + OCA{(-EJ + P ) ( n - C i cos <p cos A + c cos <f> sin A - c sin <j> cos A) 2 + (Qj + Q J ( c i cos Mi3 = sin A + c cos ^ cos A + c sin 0 sin A)} > ; 2 -(1 - Ku) sin d> + K j - a q f j P + Q] + E l + < £ ) sin <6 2 at + a c o s { - ( P j + P j s i n A + (Q - Q J c o s A } 3 r i + « 4 { ( - P / + P„)(-sin^cos * + Ci) r M\i, = K \ &iEj sin <f> + a cos (/> sin A at 3 3 178 3 179 Afi6 = K j^lfia ^ ^+ 3 S at n a c o s $ c o s ^ 4- a4(sin <f> cos <£* + C5) j ; Mn = K | — a i P „ sin <f> — a cos at z cos A + a^-sin^cosc^* 4- c )j ; 4 M = l + K t { a i { ( - P / s i n A + Q cosA) + (P ,sinA + Q cosA) } 2 2 2 a / 2 t ji 4 - ct {c\ 4 - 4 } 2 + 2 a c , {{-Pj + P J sin A + (Q, + Q J cos A } } ; 4 M 2 3 = i f { a { - ( P / + P J cos A - (Q, - QJsinA} a t 3 + « c {(-£, 4 - P J cos A - (<?, + QJ sin A} } ; 4 Af 2 4 3 = i f | — a s i n A — ar c sin A} ; 3 a ( M25 = K at 4 3 | c*3 cos A 4 - or c cos A } ; 4 3 M 6 = -K"at | - a sin A 4 - a c sin A} ; 2 3 4 3 A/27 = K t j — a cos A 4 - « c cos A} ; a 3 4 3 M 3 3 = (1 - i f * ) + Kat<*i{P] + g M 3 4 = JTataiOj 5 M 3 5 = -K ,ctiPj M 3 6 = -AT ,aiQ ; 2 a a M37 = K aiP . u 2 2 ; a al + P 4- Q ) ; For i, j=4, . . .,7, The Mj.'s in Eq. (3.32) corresponds t o a particular case with P = ( Q,= E = u Q =o. u APPENDIX V - H O M O G E N E O U S SOLUTION OF E Q U A T I O N (4.7) Taking Laplace transform of the homogeneous form of E q . (4.7) gives; ' s + 3K 2 K a {s at 3 Q(3(s -l-3) s 2 a (s + 3) 2 it + u 2 + 3) 2 3 K a (s 3 0 2 0 at 3 + CU S 2 s t p + Vo +' K a [s(Pj 0 + 3) ' 2 al + P ) 0 + Pj u0 sEio + fio + Mo ' s + 1 + 3K K a (s 2 a (s 3 f *o S at 2 3 2 + fl s s s ^o & + ^„o 5 + 1 + u> 2 + Q; Q J 0 Q^ ] 0 1 (7.16) s *) a sets of - Q ( ^ + #>) a 3 + + 4) ' 2 3 ^'uO- 3(^o + + The above two 3 (V.la) « (s a t 2 if ta [ (^ - 0 o0 0 0 <f>' + = < -tf 2 + 4) £' ] Vo) s + 1 + UJ + 4) 2 3 + 4) 2 it ct {s + "aCs^o + u0 + 0 + #>) a sEuO + £ ' 2 0 equations are uncoupled, which can be solved independently. Equations in ( V . 1 a ) are solved first. The of 3x3 matrix on the the left hand side of the equation can be written as. determinant = d i ( s + UJ )(S + a^s + 2 di(s where: di + n^ 2 1 - 2K d = UzK di dz = ±-[3K di 2 n 2 4 2 i a t t u 2 = \{d2 + + 2 ; z -\2K a\)\ 2 2 r al -18K a al); l tpi = \id2-\ld\-Adz); n]i 3 pl 2 +u t i a d) 2 + n )(s 2 \ld -Ad ); 2 3 180 determinant n ); 2 p (V.2) 181 and ti>i < n i < n . n p The equations p in (V.1a) can now be written a s : { # o + Vo + # « « a 3 [ * ( £ i o + £ . 0 ) + £lo + £ « o ] } iT ett fl 2 rfi(»Ji-»Ji)4* + nJi) 2 { 2 a ( 5 ^ o + Vo) + 3 _ (3 - f f l ) , f(3~n i) 3 2 «(£»o + £ . 0 ) r(-n t t 8 (- + "Ji)J 2 1 + + PJO + £ ' « o } 5 3) {*Vo + Vo + ^ |Qf[a(£,o + £ . 0 ) + £Jo + £ « o ] } a 3 1 f(-" i + 2 »S,)t ra2 +"1)1 ) (^ +^i) J (« +»ji) a 2 { s £ , 0 + £ / ' o + O r 3 ( # 0 + Vo)} K al f(-n?M+3) (n ,--n 2 at >i-»li)("?-»Ji)(»J-»Ji)l (* + » J i ) 2 (-»*; , ( - ^ ++ 3a ) ^^ - - n^ ^, )) , ( - n ^ + S ^ - n ^ ) (s + n ) 2 a 2 {,* (* + + n}) n) + 2 pl J 2 {^(£/o-£ o) + £/o-£'uo}; u P -" f(-^i+ ) (-^i+3)l 3 l J <*i(»?,i - n j j t {s V o +V + 0 1 {s£ a 0 (s + n J J 2 tfW*(£jO (s' + ift) + £.0) f(-" i + n ) 2 2 + £ « 0 + "3(sVO + Vo)} J + £lo + £«<>] } (-n^ + n ) | 2 a t l ) 182 «Ji)(p - »Ji)(nJ " "Ji) t (-nj, + 3)*(nJ - nj,) (-^ + 3 ) d i( li R - n (5 2 + nji) ^ - ^ ) , {«(-£f + £d»)-£io + £ i o } - ••(^•3) 0 W D e n 0 t l n 9 : « + <f>'i n^,i ti \/*8 + (^) ; 2 ^s = Ai=tan^(^); *o A . 2= t a n -i !^io ( ) ; *0 A =tan-i(»); 3 *0 where i p i , pu. The inverse Laplace transform and P of Eq. ( V . 3 ) gives the solution for \p, P degrees of freedom as; ^ = -J-TT "U» i 1 —M^ \ vi) n p + ^ [(-(n 2 + ^ d B t t ~ l i ) ~ 2if««a (3 - n j j ] sm(n 0 + n t d i n P 3 ^ W p 2 2 01 - nJO + 2 i f a i ( 3 - n^)] s'm(n Q + /^ )} a l 2 ^ ~ vJ n n 1 f [^p/i s i n pl ( ^ i + Ppti) n - [A I s'm[n id + p ) + A P 2 Pi = ? ? ( 2 1 p pt2 PU2 5 + sin(n i0 + p )}} ; p pu2 i\(M^i* + M + ^ 2 sin(n i0 + p 1 + P*I A K " 2 /fy )} 2 Ppui)} ( 183 r-K tai -^ +3)(n;-2n a f 2 1 1 +3) , x j n 4 rKataK-n^+Z)^* A -Katalin^-nlJi-nl ir a l - 2nJ + 3) t + V^ Ji + 2 2 n ^ s m ( 0i* + W n i . , ai(nJ-3) rfi(»Ji-»Ji)( p- Ji)( 5- Ji) n n x {-Apuii-nli + V2(- n + 3)(n - n ) s i n ( n 0 + /? „i) 2 l 01 p ~ li) sm(n i0 + n p ) n p 3)(nji - 4i) s i n I- £ . = ? ? a ~ 2\ 2 x z " K n p i + )( l + 1 n siQ (^ n pu2 ( p * + /W)} J g n + n +M A s i n K* g + flw) } 1 + Mrfi J A X - n\ ) x r-^3(-^i+3)(nJ-2n2 +3) [ \2 _ a j \ P»i n r ^ g Q t ^ I n 3(-";i + 3)(n;-2n; +3) _ 1 + ^p«2 + , . » J i + »JJ sin(n 0 + / ? ) 1 2 A 7~2—-XT * p vH 2 + n pi' W-»Ji)(«J-"Jt) p l J + x {-^p/iC-nJx + 3 ) ( n J - n ) i n ( n , 0 + ^ 2 1 ^2(-nJi + 3)(nJ - nJO s v 1 sin(n 0 + £ Fl ^ p t t 2 ) / p Similar procedure ) p/2 2 1 p / 1 sin(n i0 + /? ) + ^ p / 3 ( n j - 3 ) ( n - 4 ) s i n ( n / + /? )} ; obtain the response i ~ 3) Kai4(nl + - pol 1 p/3 • • • (V.4) is applied to the set of equation in ( V . 1 b ) to in <t>, Q | , and Q. generalized coordinates. Determinant u 184 of the matrix on the left hand side of the equation can be written a s , determinant = d (s + 1 + w ) ( s + <* s + ds) 2 2 4 2 4 x (V.5) where; di = j-[{l+ 3K ) ds = j-[{l+ 3K ){l it + (1 + <4 ) - 16K a ] 2 at 3 + u> )- 32K a ] 2 it 2 al 3 Ifa-y/dl-ids); „2 <t>l n nl = q n = 1 + w? ; 2 and Rewriting the equations a s : (s + nn ))l L ql rfi(n;i-»Ji)4« + »Ji) 2 (-"Si+4) *(";i-»j,) (« +»ji) l 2 { ^o + r o + ^ . i a 5 2 2 2 - "^l) 3 Wfio - (* + ^ i ) 2 ( - » J i + 4) ( • + »nj i )) i 2 2 Q ) +& tt0 " SLol} ; ; rfiKi-»Ji)i (« + »Ji) (* + »i) J a di{n] x - nJJCn} - 2 2 x ql 2 +4)^-n ^ 2 (« + »5i) (* + " ) 2 2 [-(a 0 - QJ -i- J 2 & - Q'J} f(-^i +) K-^i) »Ji)(»J - »Ji)(«5 - "i) * (« + »Ji) (-n 3)(n^ - n,) ^ (^ + ni) (* + »J) ' 4 Kat^l di(n i 2 q 2 2 ~ 2 (-nfr+^n'-n',) 2 4- 2 2 2 2 2 and taking inverse Laplace transform gives the solutions for <p, Qj, and degrees of freedom as: ^ = A 1 „ 2 J ^ i KJ n + ^[(-(nJ - n ) 2 2 ql K tOtz{n — 4) r 2 a 2 + nJ - n ) 1 n\ ){n\ (-^i+^K-n,) + ro + 2 r n 2 1 ) - 2/f«,ol(4 - n j j ] sin(n^0 + + 2K a\{A at - n )] s\n{n e + /^ )} 2 ql ql 2 186 - [A s'm(n 9 + (5 ) - A ql qt2 ql2 3a (l-^ ) f 3 g t . n P )]} ; sin(n i0 + qn2 qa2 o + } + g . n ( Q + A fi 1 + , -K a {-n +4)(n 2 r 2 x{^J ; I L 2 + 4) 2 i + sm(n^« ( «- Ji) n J + 4)(n -2n +4) 2 at A n [K al{-n A - 2n 2 at 2 ql 2 ql 2 i JST.io§(n; - 4) - A {-n qu2 - + 4)(n - njj) sin(n 0 + /? „ ) 2 2 ql al -Ajfi3(»} - 4)(n 2 1 1^—%T{^l d l i q l ~ 4,l) n n + M ql g 2 - njj) sin(n 0 + /?««3)} 5 g srafn^fl + p4>i) - A& sin(n 0 + fll fa)} * K ~% l ) n v K q <f>\) L n n [^^^t^)" "^^ + * - 1 - H -M* I* ^2) 2 V+ 2 + (-j-W-ny " J s i n ( v + M s /f a (n;-4) 2 t flt ^l(^l x - ll)( q {-Aj-iC-nj! + 4){n\ - - A {-n ql2 2 qX ~ n n \\) n n ) sin(n^0 + qX qlx + 4)(nJ - n j j sin(n 0 + /fy ) ol 2 - i4j£(»; - 4)(nj! - n ) sin(n 0 + p )} 2 3 p ) 2 x g ql3 ; • • • (V.7) with: An = \k + (^-) ; 2 V A <s>\ n i2 Aiz Pa •o = ta,-( "•«<•): A3 = where i=0, q l qu. ; t a n - ' ( f ) ;
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Dynamics of gravity oriented axi-symmetric satellites...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Dynamics of gravity oriented axi-symmetric satellites with thermally flexed appendages Ng, Chun Ki Alfred 1986
pdf
Page Metadata
Item Metadata
Title | Dynamics of gravity oriented axi-symmetric satellites with thermally flexed appendages |
Creator |
Ng, Chun Ki Alfred |
Publisher | University of British Columbia |
Date Issued | 1986 |
Description | 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. |
Subject |
Thermal stresses Artificial satellites -- Dynamics Artificial satellites -- Orbits |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0097219 |
URI | http://hdl.handle.net/2429/26727 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1987_A7 N45_2.pdf [ 8.32MB ]
- Metadata
- JSON: 831-1.0097219.json
- JSON-LD: 831-1.0097219-ld.json
- RDF/XML (Pretty): 831-1.0097219-rdf.xml
- RDF/JSON: 831-1.0097219-rdf.json
- Turtle: 831-1.0097219-turtle.txt
- N-Triples: 831-1.0097219-rdf-ntriples.txt
- Original Record: 831-1.0097219-source.json
- Full Text
- 831-1.0097219-fulltext.txt
- Citation
- 831-1.0097219.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0097219/manifest