Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

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

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 ) ;  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

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>
                        
                    
IIIF logo 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

Comment

Related Items