Dynamics of Multibody Systems

HALF-TITLE......Page 3
TITLE......Page 5
COPYRIGHT......Page 6
CONTENTS......Page 7
PREFACE......Page 11
1.1 MULTIBODY SYSTEMS......Page 13
1.2 REFERENCE FRAMES......Page 15
1.3 PARTICLE MECHANICS......Page 18
1.4 RIGID BODY MECHANICS......Page 23
1.5 DEFORMABLE BODIES......Page 27
1.6 CONSTRAINED MOTION......Page 30
1.7 COMPUTER FORMULATION AND COORDINATE SELECTION......Page 34
1.8 OBJECTIVES AND SCOPE OF THIS BOOK......Page 37
2 REFERENCE KINEMATICS......Page 40
2.1 ROTATION MATRIX......Page 41
2.2 PROPERTIES OF THE ROTATION MATRIX......Page 47
2.3 SUCCESSIVE ROTATIONS......Page 51
2.4 VELOCITY EQUATIONS......Page 59
2.5 ACCELERATIONS AND IMPORTANT IDENTITIES......Page 67
2.6 RODRIGUEZ PARAMETERS......Page 71
2.7 EULER ANGLES......Page 75
2.8 DIRECTION COSINES......Page 80
2.9 THE 4 × 4 TRANSFORMATION MATRIX......Page 84
2.10 RELATIONSHIP BETWEEN DIFFERENT ORIENTATION COORDINATES......Page 92
3 ANALYTICAL TECHNIQUES......Page 97
3.1 GENERALIZED COORDINATES AND KINEMATIC CONSTRAINTS......Page 98
3.2 DEGREES OF FREEDOM AND GENERALIZED COORDINATE PARTITIONING......Page 106
3.3 VIRTUAL WORK AND GENERALIZED FORCES......Page 114
3.4 LAGRANGIAN DYNAMICS......Page 127
3.5 APPLICATION TO RIGID BODY DYNAMICS......Page 135
3.6 CALCULUS OF VARIATIONS......Page 141
3.7 EULER’S EQUATION IN THE CASE OF SEVERAL VARIABLES......Page 147
3.8 EQUATIONS OF MOTION OF RIGID BODY SYSTEMS......Page 154
3.9 NEWTON—EULER EQUATIONS......Page 162
3.10 CONCLUDING REMARKS......Page 166
4 MECHANICS OF DEFORMABLE BODIES......Page 171
4.1 KINEMATICS OF DEFORMABLE BODIES......Page 172
4.2 STRAIN COMPONENTS......Page 176
4.3 PHYSICAL INTERPRETATION OF STRAINS......Page 180
4.4 RIGID BODY MOTION......Page 181
4.5 STRESS COMPONENTS......Page 184
4.6 EQUATIONS OF EQUILIBRIUM......Page 187
4.7 CONSTITUTIVE EQUATIONS......Page 190
4.8 VIRTUAL WORK AND ELASTIC FORCES......Page 195
5 FLOATING FRAME OF REFERENCE FORMULATION......Page 200
5.1 KINEMATIC DESCRIPTION......Page 201
5.2 INERTIA OF DEFORMABLE BODIES......Page 212
5.3 GENERALIZED FORCES......Page 225
5.4 KINEMATIC CONSTRAINTS......Page 231
5.5 EQUATIONS OF MOTION......Page 235
5.6 COUPLING BETWEEN REFERENCE AND ELASTIC DISPLACEMENTS......Page 240
5.7 APPLICATION TO A MULTIBODY SYSTEM......Page 243
5.8 USE OF INDEPENDENT COORDINATES......Page 253
5.9 DYNAMIC EQUATIONS WITH MULTIPLIERS......Page 256
5.10 GENERALIZED COORDINATE PARTITIONING......Page 260
5.11 ORGANIZATION OF MULTIBODY COMPUTER PROGRAMS......Page 263
5.12 NUMERICAL ALGORITHMS......Page 266
6 FINITE-ELEMENT FORMULATION......Page 279
6.1 ELEMENT SHAPE FUNCTIONS......Page 280
6.2 REFERENCE CONDITIONS......Page 288
6.3 KINETIC ENERGY......Page 290
6.4 GENERALIZED ELASTIC FORCES......Page 299
6.5 CHARACTERIZATION OF PLANAR ELASTIC SYSTEMS......Page 300
6.6 CHARACTERIZATION OF SPATIAL ELASTIC SYSTEMS......Page 306
6.7 COORDINATE REDUCTION......Page 312
6.8 THE FLOATING FRAME OF REFERENCE AND LARGE DEFORMATION PROBLEM......Page 316
7 THE LARGE DEFORMATION PROBLEM......Page 321
7.1 BACKGROUND......Page 322
7.2 ABSOLUTE NODAL COORDINATE FORMULATION......Page 326
7.3 FORMULATION OF THE STIFFNESS MATRIX......Page 330
7.4 EQUATIONS OF MOTION......Page 334
7.5 RELATIONSHIP TO THE FLOATING FRAME OF REFERENCE FORMULATION......Page 335
7.6 COORDINATE TRANSFORMATION......Page 337
7.7 CONSISTENT MASS FORMULATION......Page 340
7.8 THE VELOCITY TRANSFORMATION MATRIX......Page 343
7.9 LUMPED MASS FORMULATION......Page 344
7.10 EXTENSION OF THE METHOD......Page 347
7.11 COMPARISON WITH LARGE ROTATION VECTOR FORMULATION......Page 351
A.1 MATRIX ALGEBRA......Page 357
A.2 EIGENVALUE ANALYSIS......Page 361
A.3 VECTOR SPACES......Page 362
A.4 CHAIN RULE OF DIFFERENTIATION......Page 365
A.5 PRINCIPLE OF MATHEMATICAL INDUCTION......Page 366
REFERENCES......Page 369
INDEX......Page 381