Analytical Dynamics

Cover......Page 1
S Title......Page 2
Title: ANALYTICAL DYNAMICS......Page 3
531 \'.11-dc21......Page 4
Dedicattion......Page 6
CONTENTS......Page 8
APPROACH......Page 14
CONTENTS......Page 15
A CKNOWLEDGMENTS......Page 17
1.2 SYSTEMS OF UNITS......Page 20
1.3 Review OF VECTOR ANALYSIS......Page 23
1.3.1 RECTILINEAR (CARTESIAN) COORDINATES......Page 24
1.3.2 CURVILINEAR COORDINATES......Page 27
1.4 NEWTONIAN PARTICLE MECHANICS......Page 45
1.5 DEGREES OF FREEDOM AND CONSTRAINTS......Page 52
1.6 IMPULSE AND MOMENTUM......Page 55
1.7 WORK AND ENERGY......Page 62
1.7.1 GRAVITATIONAL POTENTIAL ENERGY......Page 63
1.7.2 POTENTIAL ENERGY OF SPRINGS......Page 65
1.7.4 WORK-ENERGY RELATIONS......Page 67
1.8 EQUILIBRIUM AND STABILITY......Page 73
1.9 FREE RESPONSE OF LINEAR SYSTEMS......Page 77
1.10 RESPONSE TO HARMONIC EXCITATION......Page 81
1.11 FORCED RESPONSE OF LINEAR SYSTEMS......Page 87
1.12 FIRST INTEGRALS......Page 90
1.13 NUMERICAL INTEGRATION OF EQUATIONS OF MOTiON......Page 94
HOMEWORK EXERCISES......Page 96
2.1 INTRODUCTION......Page 106
2.2 MOVING COORDINATE FRAMES......Page 107
2.3 REPRESENTATION OF VECTORS......Page 110
2.4 TRANSFORMATION OF COORDINATES, FINITE ROTATIONS......Page 116
2.5 INFINITESIMAL ROTATIONS, ANGULAR VELOCITY......Page 126
2.6 RATE OF CHANGE OF A VECTOR, ANGULAR ACCELERATION......Page 136
2.7 RELATIVE VELOCITY AND ACCELERATION......Page 143
2.8 OBSERVATIONS FROM A MOVING FRAME......Page 152
Horwwoiu ExERcIsEs......Page 163
3.2 EQUATIONS OF MOTION......Page 172
3.3 LINEAR AND ANGULAR MOMENTUM......Page 176
3.4 WORK AND ENERGY......Page 180
3.5 IMPACT OF PARTICLES......Page 183
3.6 VARIABLE MASS AND MASS FLOW SYSTEMS......Page 189
3.7 CONCEPTS FROM ORBITAL MECHANICS: THE TWO BODY PROBLEM......Page 192
3.8 THE NATURE OF THE ORBIT......Page 196
3.9 ORBITAL ELEMENTS......Page 206
3.10 PLANE KINETICS OF RIGID BODIES......Page 212
3.11 INSTANT CENTERS AND ROLLING......Page 219
3.12 ENERGY AND MOMENTUM......Page 222
REFERENCES......Page 225
HOMEWORK EXERCISES......Page 226
4.1 INTRODUCTION......Page 234
4.2 GENERALIZED COORDINATES......Page 235
4.3 CONSTRAINTS......Page 238
4.3.1 HoLoNoMic CONSTRAINTS......Page 239
4.3.2 NONHOLONOMIC CONSTRAINTS......Page 242
4.4 VIRTUAL DISPLACEMENTS AND VIRTUAL WORK......Page 249
4.5 GENERALIZED FORCES......Page 256
4.6 PRINCIPLE OF VIRTUAL WORK FOR STATIC EQUILIBRIUM......Page 260
4.7 D\'ALEMBERT\'S PRINCIPLE......Page 264
4.8 HAMILTON\'S PRINCIPLES......Page 268
4.9 LAGRANGE\'S EQUATIONS......Page 272
4.10 LAGRANGE \'S EQUATIONS FOR CONSTRAINED SYSTEMS......Page 279
HOMEWORK EXERCISES......Page 284
5.2 NATURAL AND NONNATURAL SYSTEMS, EQUILIBRIUM......Page 292
5.3 SMALL MOTIONS ABOUT EQUILIBRIUM......Page 298
5.4 RAYLEIGH\'S DISSIPATION FUNCTION......Page 305
5.5 EIGENVALUE PROBLEM FOR LINEARIZED SYSTEMS......Page 308
5.6 ORTHOGONALITY AND NORMALIZATION......Page 312
5.7 MODAL EQUATIONS OF MOTION AND RESPONSE......Page 315
5.8 GENERALIZED MOMENTUM, FIRST INTEGRALS......Page 319
5.9 ROUTH \'s METHOD FOR IGNORABLE COORDINATES......Page 321
5.10 IMPULSIVE MOTION......Page 323
5.11 HAMILTON \'S EQUATIONS......Page 327
5.12 COMPUTATIONALCONSIDERATION:  ALTERNATE DESCRIPTIONS OF THE MOTiON EQUATIONS......Page 330
5.13 ADDITIONAL DIFFERENTIAL VARIATIONAL PRINCIPLES......Page 333
HOMEWORK EXERCISES......Page 336
6.2 CENTER OF MASS......Page 342
6.3 MASS MOMENTS OF INERTIA......Page 345
6.4 TRANSFORMATION PROPERTIES......Page 353
6.4.1 TRANSLATION OF COORDINATES......Page 354
6.4.2 ROTATION OF COORDINATE AXES......Page 356
6.5 PRINCIPAL MOMENTS OF INERTIA......Page 362
6.6 INERTIA ELLIPSOID......Page 366
HOMEWORK EXERCISES......Page 370
7.2 BASIC KINEMATICS OF RIGID BODIES......Page 374
7.2.2 PURE ROTATION......Page 375
7.2.3 COMBINED TRANSLATION AND ROTATION......Page 378
7.3 EULER\'S AND CHASLES\'S THEOREMS......Page 379
7.4 RELATION BETWEEN DIRECTION  COSINES AND ANGULAR VELOCITY......Page 384
7.5 EULER ANGLES......Page 386
7.5.1 EULER ANGLE SEQUENCES......Page 387
7.5.2 ANGULAR VELOCITY AND ACCELERATION......Page 388
7.5.3 AXISYMMETRIC BODIES......Page 391
7.6 ANGULAR VELOCITLES AS QUASI-VELOCITIES(GENERALIZED SPEEDS)......Page 397
7.7 EULER PARAMETERS......Page 399
7.7.1 RELATING THE DIRECTION COSINES TO THE EULER PARAMETERS......Page 401
7.7.2 RELATING THE EULER PARAMETERS TO ANGULAR VELOCITIES......Page 403
7.7.3 RELATING TILE EULER PARAMETERS TO ThE DIRECTION COSINES......Page 405
7.7.4 RELATING THE EULER PARAMETERS TO THE EULER ANGLESAND VICE VERSA......Page 406
7.7.5 OTHER CONSIDERATIONS......Page 407
7.8.1 ROTATIONAL CONTACT, JOINTS......Page 410
7.8.2 TRANSLATIONAL MOTION AND SLIDING CONTACT......Page 414
7.8.3 COMBINED SLIDING AND ROTATION......Page 415
7.9 ROLLING......Page 419
HOMEWORK EXERCISES......Page 431
8.1 INTRODUCTION......Page 440
8.2 LINEAR AND ANGULAR MOMENTUM......Page 441
8.3.1 TRANSLATION OF COORDINATES......Page 444
8.3.2 ROTATION OF COORDINATES......Page 445
8.4 RESULTANT FORCE AND MOMENT......Page 448
8.5 GENERAL EQUATION OF MOTION......Page 450
8.6 ROTATION ABOUT A FIXED Axis......Page 462
8.7 EQUATIONS OF MOTION IN STATE FORM......Page 464
8.8 IMPULSE-MOMENTUM RELATIONSHIPS......Page 467
8.9 ENERGY AND WORK......Page 470
8.10.1 VIRTUAL WORK AND GENERALIZED FORCES......Page 475
8.10.2 GENERALIZED COORDINATES......Page 477
8.10.3 lAGRANGE\'S EQUATIONS IN TERMS OF THE EULER ANGLES......Page 481
8.10.4 LAGRANGE\'S EQUATIONS IN TERMS OF EULER PARAMETERS......Page 482
8.10.5 LAGRANGE\'S EQUATIONS AND CONSTRAINTS......Page 483
8.11 D\'ALEMBERT\'S PRiNCIPLE FOR RIGID BODIES......Page 493
8.12 IMPACT OF RIGID BODIES......Page 496
HOMEWORK EXERCISES......Page 500
9.1 INTRODUCTION......Page 508
9.2 MODIFIED EULER\'S EQUATIONS......Page 509
9.3 MOMENT EQUATIONS ABOUT AN ARBITRARY POINT......Page 517
9.4 CLASSIFICATION OF THE MOMENT EQUATIONS......Page 522
9.5 QUASI-COORDINATES AND QUASI-VELOCITIES(GENERALIZED SPEEDS)......Page 523
9.6 GENERALIZED SPEEDS AND CONSTRAINTS......Page 529
9.7 GIBBS-APPELL EQUATIONS......Page 536
9.8 KANE\'S EQUATIONS......Page 543
9.9 THE FUNDAMENTAL EQUATIONS AND CONSTRAINTS......Page 550
9.10 RELATIONSHIPS BETWEEN THE FUNDAMENTAL EQUATIONS AND LAGRANGE\'S EQUATIONS......Page 551
9.11 IMPULSE-MOMENTUM RELATIONSHIPS FOR GENERALIZED SPEEDS......Page 558
HOMEWORK EXERCISES......Page 561
10.1 INTRODUCTION......Page 566
10.2 TORQUE-FREE MOTION OF INERTIALLY SYMMETRIC BODIES......Page 567
10.3 GENERAL CASE OF TORQUE-FREE MOTION......Page 575
10.4 POLHODES......Page 576
10.5 MOTION OF A SPINNING TOP......Page 582
10.6 ROLLING DISK......Page 592
10.7.1 FREE GYROSCOPE......Page 599
10.7.2 GYROCOMPASS......Page 601
10.7.3 SINGLE-AXIS GYROSCOPE......Page 604
HOMEWORK EXERCISES......Page 606
11.1 INTRODUCTION......Page 610
11.2.1 KINEMATICS AND GEOMETRY......Page 611
11.2.2 KINETIC AND POTENTIAL ENERGY FOR BEAMS......Page 616
11.2.3 KINETIC AND POTENTIAL ENERGY FOR TORSION......Page 618
11.2.4 OPERATOR NOTATION......Page 620
11.3.1 EXTENDED HAMILTON\'S PRINCIPLE......Page 623
11.3.2 BOUNDARY CONDITIONS......Page 625
11.3.3 SIMPLIFICATION......Page 627
11.3.4 OPERATOR NOTATION......Page 630
11.4 EIGENSOLUTION AND RESPONSE......Page 635
11.5.1 APPROXIMATION TECHNIQUES AND TRIAL FUNCTIONS......Page 645
11.5.2 METHOD OF ASSUMED MODES......Page 647
11.5.3 CONVERGENCE ISSUES......Page 651
11.6 KINEMATICS OF COMBINED ELASTIC AND LARGE ANGLE RIGID BODY MOTION......Page 656
11.7 DYNAMICS OF COMBINED ELASTIC AND LARGEANGLE RIGID BODY MOTION......Page 664
11.7.1 EQUATIONS OF MOTION IN HYBRID FORM......Page 665
11.7.2 EQUATIONS OF  MOTION IN DISCRETIZED FORM......Page 668
11.8 ANALYSIS OF THE EQUATIONS OF MOTION......Page 670
11.9 CASE 3, ROTATING SHAFTS......Page 678
11.10  HYBRID PROBLEMS......Page 682
HOMEWORK EXERCISES......Page 683
A.2 HISTORICAL EVOLUTION OF DYNAMICS......Page 688
A.3 BIOGRAPHES OF KEY CONTRIBUTORS TO DYNAMICS......Page 697
REFERENCES......Page 704
B.2 STATIONARY VALUES OF A FUNCTION......Page 706
B.3 STATIONARY VALUES OF A DEFINITE INTEGRAL......Page 712
B.4 THE VARIATIONAL NOTATION......Page 715
B.5 APPLICATION OF TIlE VARIATIONAL NOTATION TO DYNAMICS PROBLEMS......Page 718
HOMEWORK EXERCISES......Page 719
APPENDIX C: COMMON INERTIA PROPERTIES......Page 720
INDEX......Page 727
Errata......Page 738