THE ELEMENTARY PART OF A TREATISE ON THE DYNAMICS OF A SYSTEM OF RIGID BODIES
CHAPTER 1: MATHEMATICAL MECHANICAL PRELIMINARIES 11 MATHEMATICS 111 ELLIPTIC INTEGRALS 112 ELLIPTIC FUNCTIONS 12 RIGID BODY KINEMATIC 121 ORTHOGONAL MATRICES 122 EULER ANGLES 123 KINEMATIC EQUATIONS 13 RIGID BODY DYNAMICS 131 KINETIC ENERGY OF RIGID BODY 132 CANONICAL VARIABLES 133 SERRET-ANDOYER CANONICAL VARIABLES 134 DYNAMIC EQUATIONS 14 CHAOTIC MOTION 141 POINCARÉ SECTIONS 142 MELNIKOV'S METHOD 143 HOLMS AND MARSDEN'S MODIFICATION OF MELNIKOV'S METHOD 144 WIGGINS AND HOLMES' MODIFICATION OF MELNIKOV'S METHOD REFERENCES CHAPTER 2: REENTRY ATTITUDE DYNAMICS 21 INTRODUCTION 22 AERODYNAMICS OF REENTRY VEHICLES 221 AERODYNAMIC FORCES AND MOMENTS 222 AERODYNAMIC COEFFICIENTS FOR REENTRY CAPSULES 23 THE EQUATIONS OF MOTION 231 GENERAL CASE 232 UNPERTURBED CASE 24 ANALYTICAL SOLUTIONS OF THE UNDISTURBED EQUATION FOR SINUSOIDAL AERODYNAMIC MOMENT 25 ANALYTICAL SOLUTIONS OF THE UNDISTURBED EQUATION FOR BIHARMONICAL AERODYNAMIC MOMENT 251 QUALITATIVE ANALYSIS OF THE UNPERTURBED SYSTEM PHASE PORTRAIT 252 ANALYTICAL SOLUTION FOR THE UNPERTURBED CASE 253 SOLUTION FOR SINUSOIDAL MOMENT AS A PARTICLE CASE OF BIHARMONICAL SOLUTION 26 QUASISTATIC SOLUTIONS FOR THE DISTURBED EQUATION OF MOTION 27 ADIABATIC INVARIANTS AND THE APPROXIMATE SOLUTION FOR THE DISTURBED MOTION 271 ACTION OF THE BODY WITH BIHARMONICAL RESTORING MOMENT 272 ACTION OF THE BODY WITH SINUSOIDAL RESTORING MOMENT 28 BIFURCATION AND WAYS OF ITS ELIMINATION AT THE DESCENT OF SPACECRAFT IN THE RAREFIED ATMOSPHERE 281 STABILITY OF PERTURBED MOTION 282 SEPARATION OF FAST AND SLOW VARIABLES 283 AVERAGED EQUATIONS 284 CALCULATION PROCEDURE 285 ELIMINATION OF THE BIFURCATION BY THE CHOICE OF THE REENTRY CAPSULE GEOMETRY 286 ELIMINATION OF THE BIFURCATION BY THE SELECTION OF INITIAL CONDITIONS OF MOTION 287 ELIMINATION OF THE BIFURCATION: NUMERICAL SIMULATION 29 CHAOTIC ATTITUDE MOTION OF REENTRY VEHICLE WITH AN INTERNAL MOVING MASS 291 PROBLEM FORMULATION 292 HOMOCLINIC ORBITS OF BIHARMONICAL SYSTEM 293 MELNIKOV'S CRITERION FOR THE PERTURBED MOTION 294 NUMERIC SIMULATION OF THE CHAOTIC MOTION 210 CHAOTIC BEHAVIOR OF BODIES IN A RESISTANT MEDIUM 2101 PROBLEM FORMULATION 2102 SPATIAL MOTION OF THE ASYMMETRIC BODY 2103 THE PLANAR MOTION OF THE BODY UNDER THE BIHARMONIC TORQUE AND SMALL DISTURBANCE TORQUE 211 CHAOTIC MOTION OF A REENTRY CAPSULE DURING DESCENT INTO THE ATMOSPHERE CHAPTER 3: DYNAMICS AND CONTROL OF COAXIAL SATELLITE GYROSTATS 31 INTRODUCTION 32 ATTITUDE MOTION EQUATIONS 321 EULER EQUATIONS 322 SERRET-ANDOYER VARIABLES 323 HAMILTONIAN EQUATIONS 33 INTEGRABLE CASES IN THE DYNAMICS OF COAXIAL GYROSTATS 331 UNPERTURBED MOTION AND STATIONARY SOLUTIONS 332 TYPES OF THE GYROSTATS AND SINGULAR POINTS IN THE PHASE SPACE 34 THE EXACT ANALYTICAL SOLUTIONS 341 INTEGRATION OF THE CANONICAL EQUATIONS BY QUADRATURE 342 ANALYTICAL SOLUTIONS FOR THE OBLATE GYROSTATS 343 ANALYTICAL SOLUTIONS FOR THE OBLATE-INTERMEDIATE GYROSTATS 344 ANALYTICAL SOLUTIONS FOR THE INTERMEDIATE GYROSTATS 345 ANALYTICAL SOLUTIONS FOR THE PROLATE-INTERMDIATE GYROSTATS 346 ANALYTICAL SOLUTIONS FOR THE PROLATE GYROSTATS 347 DETERMINATION OF ANGULAR VELOCITY AND EULER ANGLES 35 DYNAMICS AND CHAOS CONTROL OF THE GYROSTATS 351 MELNIKOV METHOD 352 CHAOS CONTROL 353 NUMERICAL EXPERIMENT 36 DYNAMICS AND CONTROL OF DUAL-SPIN GYROSTAT SPACECRAFT WITH CHANGING STRUCTURE 361 GYROSTAT STABILIZATION 362 NUMERALS SAMPLES UNCONTROLLED MOTION OF THE PROLATE GYROSTAT WITH THE AXISYMMETRIC ROTOR (CASES 1 AND 2) CONTROLLED MOTION OF THE PROLATE GYROSTAT WITH AXISYMMETRIC ROTOR (CASES 3 AND 4) CONTROLLED MOTION OF THE GYROSTAT WIDTH SMALL ASYMMETRY ROTOR (CASES 5 AND 6) 37 ADIABATIC INVARIANTS IN THE DYNAMICS OF AXIAL GYROSTATS 371 ADIABATIC INVARIANT 372 OBLATE GYROSTAT 373 PROLATE GYROSTAT 374 INTERMEDIATE GYROSTAT CHAPTER 4: DEPLOYMENT, DYNAMICS, AND CONTROL OF A TETHER-ASSISTED RETURN MISSION OF A REENTRY CAPSULE 41 INTRODUCTION 42 MATHEMATICAL MODEL OF A SATELLITE WITH A TETHERED PAYLOAD 421 TETHER-ASSISTED RETURN MISSION 422 GENERAL CASE 423 CASE OF DEPLOYED TETHER SYSTEM 43 ANALYTICAL SOLUTION IN THE CASE OF A SLOW CHANGING OF THE PARAMETERS 431 EQUATION OF MOTION 432 THE ACTION INTEGRAL AND APPROXIMATE SOLUTIONS 433 APPROXIMATE SOLUTION FOR THE CASE OF SMALL OSCILLATIONS OF THE SATELLITE AXIS AROUND THE TETHER LINE 434 APPROXIMATE SOLUTION FOR THE CASE OF THE SATELLITE OSCILLATIONS NEAR LOCAL VERTICAL 437 NUMERICAL SIMULATION OF THE SATELLITE OSCILLATIONS 438 ESTIMATION OF THE MICROACCELERATIONS ON BOARD THE SATELLITE 44 OSCILLATIONS OF THE SATELLITE WITH A VERTICAL ELASTIC TETHER 441 EQUATION OF THE SATELLITE MOTION ON THE CIRCULAR ORBIT 442 BIFURCATION DIAGRAM 443 ANALYTICAL SOLUTIONS FOR THE SEPARATRICES 444 CHAOTIC MOTION OF THE SATELLITE WITH THE VERTICAL TETHER 445 DAMPING CHAOTIC OSCILLATIONS OF THE SATELLITE WITH THE VERTICAL TETHER 45 OSCILLATIONS IN THE CASE OF AN ELLIPTIC ORBIT 451 OSCILLATIONS OF SPACE TETHER SYSTEM ON AN ELLIPTIC ORBIT 452 OSCILLATIONS OF THE SATELLITE WITH THE VERTICAL TETHER ON AN ELLIPTIC ORBIT 46 SWING PRINCIPLE FOR DEPLOYMENT OF A TETHER-ASSISTED RETURN MISSION OF A REENTRY CAPSULE 461 MATHEMATICAL MODEL 462 AVERAGED EQUATION AND ANALYTICAL SOLUTION 463 NUMERIC ANALYSES 464 THE CONTROL LAW STABILITY 47 TETHER-ASSISTED RETURN MISSION FROM AN ELLIPTICAL ORBIT TAKING INTO ACCOUNT ATMOSPHERIC STAGE OF REENTRY 471 EQUATIONS OF MOTION 472 CALCULATION ALGORITHM FOR CHOOSING THE SEPARATION POINT 473 RESULTS OF NUMERICAL SIMULATIONS CHAPTER 5: REMOVAL OF LARGE SPACE DEBRIS BY A TETHER TOW 51 INTRODUCTION 52 DYNAMICS OF ORBITAL DEBRIS CONNECTED TO SPACECRAFT BY A TETHER IN A FREE SPACE 521 THE EQUATIONS OF MOTION 522 EQUATIONS OF THE SPATIAL MOTION OF THE AXISYMMETRIC PASSIVE SATELLITE RELATIVE TO THE TETHER 523 VALIDATION OF THE MATHEMATICAL MODEL 524 NUMERICAL RESULTS A CASE 1 B CASE 2 C CASE 3 D CASE 4 53 DYNAMICS OF LARGE ORBITAL DEBRIS REMOVAL USING TETHERED SPACE TUG IN THE EARTH'S GRAVITATIONAL FIELD 531 MOTION EQUATIONS 532 ORBITAL MOTION 533 ATTITUDE MOTION OF THE SPACE DEBRIS 534 NUMERICAL SIMULATION AND ANALYSIS CASE1 CASE2 CASE3 CASE4 CASE5 CASE6 CASE7 MODEL CORRECTNESS 54 BEHAVIOR OF TETHERED DEBRIS WITH FLEXIBLE APPENDAGES 541 LAGRANGE FORMALISM AND GENERALIZED COORDINATES 542 SYSTEM'S KINEMATIC 543 KINETIC ENERGY OF THE SYSTEM 544 GENERALIZED FORCES 545 ANALYSIS AND NUMERICAL EXAMPLES PARAMETERS OF THE SYSTEM AND INITIAL CONDITIONS CASE1: MOTION OF THE DEBRIS WITH FORE-MOUNTED FLEXIBLE APPENDAGES CASE2: MOTION OF THE SPACE DEBRIS WITH FORE-MOUNTED FLEXIBLE APPENDAGES CASE3: MOTION OF THE SPACE DEBRIS WITH AFT-MOUNTED FLEXIBLE APPENDAGES 55 DYNAMICS, ANALYTICAL SOLUTIONS AND CHOICE OF PARAMETERS FOR TOWED SPACE DEBRIS WITH FLEXIBLE APPENDAGES 551 THE KINETIC ENERGY AND THE POTENTIAL ENERGY 552 LAGRANGE EQUATIONS AND NONPOTENTIAL GENERALIZED FORCES 553 A SIMPLIFIED MATHEMATICAL MODEL OF THE PLANE MOTION 554 NUMERICAL ANALYSIS AND SIMULATIONS 56 THE MOTION OF TETHERED TUG-DEBRIS SYSTEM WITH FUEL RESIDUALS 561 MATHEMATICAL MODEL KINEMATICS OF THE SYSTEM KINETIC ENERGY AND LAGRANGE EQUATIONS 562 SIMPLIFIED EQUATIONS 563 MOTION OF THE SYSTEM NEAR THE STATIONARY POINT: STATIONARY SOLUTIONS 564 LINEARIZED EQUATIONS 565 NUMERICAL EXAMPLE 57 DYNAMICS OF TOWED LARGE SPACE DEBRIS TAKING INTO ACCOUNT ATMOSPHERIC DISTURBANCE 571 MATHEMATICAL MODEL 572 CALCULATION OF AERODYNAMIC COEFFICIENTS 573 EFFECT OF MOMENTS ON THE DESCENT TRAJECTORY 574 ANALYSIS OF THE ATTITUDE MOTION OF THE SPACE DEBRIS GENERAL CASE THE ACTION OF THE MOMENT OF THE TETHER TENSION THE ACTION OF THE GRAVITATIONAL MOMENT THE ACTION OF THE PITCHING MOMENT 575 NATURE OF THE SPIN-UP OF THE SPACE DEBRIS 576 THRUST CONTROL FOR THE TETHER TENSION 577 DYNAMICS OF THE CONTROLLED MOTION OF THE TETHER SYSTEM 58 CHAOS BEHAVIOR OF SPACE DEBRIS DURING TETHERED TOW 581 EQUATIONS OF THE MOTION AND THE EQUILIBRIUM POSITIONS 582 LONGITUDINAL OSCILLATIONS OF THE TETHER 583 MELNIKOV METHOD AND HOMOCLINIC SOLUTIONS 584 POINCARÉ SECTIONS FOR THE GENERAL CASE CHAPTER 6: ORIGINAL TASKS OF SPACE MECHANICS 61 INTRODUCTION 62 GRAVITATIONAL STABILIZATION OF THE SATELLITE WITH A MOVING MASS 621 THE EQUATION OF MOTION OF A SATELLITE WITH A MOVABLE MASS 622 CHOICE OF THE CONTROL AND EQUATION OF CONTROLLED MOTIONS OF A SATELLITE 623 SWINGING AND REORIENTATION OF THE SATELLITE 63 THE DYNAMICS OF THE SPACECRAFT OF VARIABLE COMPOSITION 631 EQUATIONS OF MOTION OF A BODY WITH VARIABLE COMPOSITION 632 APPROXIMATE SOLUTIONS 633 MOTION OF THE CENTER OF MASS AND CALCULATION OF ERRORS OF APPLYING THE BRAKING IMPULSE 64 RESTORATION OF ATTITUDE MOTION OF SATELLITE USING SMALL NUMBERS OF TELEMETRY MEASUREMENTS 641 RESTORATION OF A HIGH-FREQUENCY MOTION USING A SMALL NUMBER OF MEASUREMENTS 642 ROTATIONAL MOTION OF THE FREE BODY ON AN ORBITAL PHASE OF ITS TRAJECTORY 643 MOTION OF THE BODY ON AN ORBITAL PHASE UNDER THE ACTION OF GRAVITY GRADIENT TORQUE 644 ROTATIONAL MOTION OF THE BODY DURING DESCENT INTO THE ATMOSPHERE