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PL N R
MULTIBODY
DYN MICS
F o r m u l a t i o n ,
P r o g r a m m i n g ,
a n d A p p l i c a t i o n s
Parviz E. Nikravesh
r
öCYCRC Press
W / TaylorS Francis Group
Boca Raton London New York
CRC Press is an im p r in t of the
Taylor Francis Group, an i n f o r m a business
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Table of Contents
Preface
Author
Chapter
Introduction 1
1.1 Multibody Mechanical Systems 1
1.2 Ty pes of Analyses 2
1.3 M ethods of Formulation 3
1.4 Com puter Program ming 9
1.4.1 MATLAB® 10
1.5 Application Examples 11
1.5.1 Double A-Arm Suspension 12
1.5.2 MacPherson Strut Suspension 13
1.5.3 Film-Advancer Mechanism 16
1.5.4 Web-Cutter 16
1.5.5 Variable-Length Pendulum 16
1.6 Un it System 19
1.7 Rem arks 20
Chapter 2
Preliminaries 21
2.1 Reference Axes 21
2.2 Scalars and Vectors 21
2.2.1 Arrays 30
2.3 Matrices 32
2.3.1 Matrix Operations 33
2.4 Vector, Array, and M atrix Differentiation 36
2.4.1 Time Derivatives 36
2.4.2 Partial Derivatives 38
2.5 Equations and Expressions 41
2.5.1 Com pact and Expanded Forms 42
2.6 Remarks 43
2.7 Problems 43
Chapter 3
Fundamentals of Kinematics 47
3.1 A Particle 47
3.1.1 Kinem atics of a Particle 47
3.2 Kinem atics of a Rigid Body 48
3.2.1 Coordinates of a Body 49
3.2.2 Velocity of a Body 54
3.2.3 Acceleration of a Body 56
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3.3 Definitions 58
3.3.1 Array of Coordinates 60
3.3.2 Degrees of Freedom 61
3.3.3 Constraint Equations 62
3.3.4 Kinem atic Joints 65
3.4 Rem arks 68
3.5 Problems 68
Chapter 4
Fundam entals of Dynam ics 73
4.1 Newton s Laws of Motion 73
4.1.1 Dynam ics of a Particle 74
4.1.2 Dyn amics of a System of Particles 74
4.2 Dynam ics of a Body 78
4.2.1 Mo ment of a Force 79
4.2.2 Centroidal Equations of Motion 80
4.2.3 Noncen troidal Equations of Mo tion 85
4.3 Force Elements 88
4.4 Applied Forces 90
4.4.1 Gravitational Force 90
4.4.2 Point-to-Point Actuator 90
4.4.3 Point-to-Po int Spring 91
4.4.4 Point-to-Point Dam per 92
4.4.5 Combined Elements 92
4.4.6 Rotational Elements 95
4.4.7 Viscous Friction 96
4.5 Reac tion Force 97
4.5.1 Newton s Third Law 98
4.5.2 Method of Lagrange Multipliers 99
4.5.3 Coulomb Friction 99
4.6 Rem arks 100
4.7 Problems 100
Chapter 5
Point-Coordinates: Kinematics 105
5.1 Multipoint Representation 105
5.1.1 Double A-Arm Suspension 107
5.1.2 MacPherson Suspension 108
5.1.3 Filmstrip Advancer 110
5.2 Stationary and Primary Points 112
5.3 Constraints 113
5.3.1 Leng th Constraint 114
5.3.2 An gle Constraints 114
5.3.3 Simple Constraints 115
5.4 Velocity and Acceleration Constraints 118
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5.5 Secondary Points 122
5.6 Exam ple Programs 126
5.6.1 Double A-Arm Suspension 126
5.6.2 M acPherson Suspension 130
5.6.3 Filmstrip Advancer 133
5.7 Rem arks 134
5.8 Problem s 134
Chapter 6
Point-Coordinates: Dynam ics 141
6.1 System of Un constrained Particles 141
6.1.1 A Two-Particle System 141
6.1.2 Unconstrained Particles— General 144
6.2 System of Constrained Particles 144
6.2.1 A Two-Particle System 145
6.2.2 M acPherson Suspension 147
6.2.3 Constrained Particles— General 149
6.3 Force and M ass Distribution 151
6.3.1 Two Primary Points 153
6.3.2 Three Primary Points 156
6.3.3 M ass Collection 158
6.3.4 Double A-Arm Suspension 158
6.4 Exact M ass Distribution 160
6.4.1 Two Primary Points 161
6.4.2 Three Primary Points 165
6.4.3 M ass Addition 168
6.5 Rem arks 168
6.6 Problem s 168
Chapter 7
Body-C oordinates: Kinematics 177
7.1 Gene ral Procedure 177
7.2 Kinem atic Joints 181
7.2.1 Revolute (Pin) Joint 182
7.2.2 Translational (Sliding) Joint 183
7.2.3 Revo lute-Revolute Joint 183
7.2.4 Revolute-Translational Joint 184
7.2.5 Rigid Joint 185
7.2.6 Simple Constraints 187
7.3 Exam ples 187
7.3.1 Double A-Arm Suspension 188
7.3.2 M acPherson Suspension 190
7.3.3 Filmstrip Advancer 195
7.4 Velocity and Acceleration Constraints 196
7.4.1 Revolute Joint 198
7.4.2 Translationa l Joint 200
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7.4.3 Revolute-Revolute Joint 201
7.4.4 Revolute-Trans lational Joint 203
7.4.5 Simple Con straints 204
7.4.6 System Jacobian 204
7.5 Programm ing Note 206
7.5.1 Com mon Scripts 207
7.6 Exam ple Programs 211
7.6.1 Double A-A rm Suspension 212
7.6.2 MacPherson Suspension 216
7.6.3 Filmstrip Advancer 221
7.6.4 Variable-Length Pendulum 222
7.7 Other Types of Joints 223
7.8 Rem arks 224
7.9 Problem s 224
Chapter 8
Body-Co ordinates: Dynam ics 233
8.1 System of Unconstrained Bod ies 233
8.1.1 A Two-Body System 233
8.1.2 Unconstrained Bod ies— General 236
8.2 System of Constrained Bodies 237
8.2.1 Constrained Bod ies— General 237
8.2.2 A Two-Body System 239
8.2.3 Reac tion Forces 241
8.3 Exam ple Program s 246
8.3.1 Double A-Arm Suspension 246
8.3.2 M acPherson Suspension 247
8.3.3 Variable-Length Pendulum 249
8.4 Rem arks 251
8.5 Problems 252
Chapter 9
Joint-Coordinates: Kinematics 259
9.1 Vector-Loop M ethod 259
9.2 Joint Coordinate M ethod 261
9.3 Open-Chain Systems 262
9.3.1 Variable-Length Pendulum 264
9.3.2 A Three-Body System 267
9.3.3 A Floating System 269
9.3.4 General Form ulation 271
9.4 Open-Chain Example Programs 272
9.4.1 Variable-Length Pendulum 272
9.5 Closed-Chain Systems 274
9.5.1 Slider-Crank M echanism 275
9.5.2 General Formulation 277
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9.6 Closed-Chain Exam ple Programs 278
9.6.1 Double A-Arm Suspension 279
9.6.2 MacPherson Suspension 283
9.6.3 Filmstrip Advancer 290
9.7 Rem arks 290
9.8 Problems 291
Chapter 10
Joint-Coordinates: Dynam ics 297
10.1 Open-Chain Systems 297
10.1.1 Variable-Length Pendu lum 298
10.2 Open-Chain Exam ple Program 300
10.3 Closed-Chain Systems 300
10.4 Closed-Chain Exam ple Programs 302
10.4.1 Double A-Arm Suspension 302
10.4.2 M acPherson Suspension 302
10.5 Rem arks 303
10.6 Problem s 303
Chapter 11
Kinematic Analysis 305
11.1 Uncon strained Formulation 305
11.2 Constrained Form ulation 306
11.2.1 Driver Constraints 307
11.3 Solution Proced ures 309
11.3.1 Coordinate Partitioning M ethod 310
11.3.2 Append ed Constraint M ethod 312
11.4 Linear Algebraic Equations 314
11.5 Nonlinear Algebraic Equations 317
11.5.1 Newton-Raphson Method for One Equation in One Unknown 317
11.5.2 Newton-Raphson Method for
Equations in
Unknowns 319
11.6 Body Coordinate Form ulation 321
11.6.1 Filmstrip Advancer 323
11.6.2 Web-Cutter 324
11.7 Point Coo rdinate Form ulation 327
11.7.1 Filmstrip Advancer 328
11.8 Joint Coo rdinate Form ulation 329
11.8.1 Filmstrip Advancer 330
11.9 Rem arks 331
U.lOProblems 331
Chapter 12
Inverse Dynam ic Analysis 339
12.1 Unconstrained Form ulation 339
12.1.1 General Procedure 340
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12.2 Constrained Formulation 341
12.2.1 General Procedure 342
12.3 Different Jacobian Matrices 345
12.4 Body Coordinate Formulation 346
12.4.1 Filmstrip Advancer 347
12.4.2 Web-Cutter 349
12.5 Point Coordinate Formulation 350
12.5.1 Filmstrip Advancer 350
12.6 Joint Coordinate Formulation 350
12.7 Rem arks 351
12.8 Problems 351
Chapter 3
Forward Dynam ic Analysis 355
13.1 Unconstrained Formulation 355
13.1.1 Initial-Value Problems 356
13.1.2 Runge-Kutta Algorithms 358
13.1.3 Variable Step Size 360
13.1.4 General Proced ure 363
13.2 Constrained Formulation 364
13.2.1 Initial Conditions 366
13.2.2 General Procedure 367
13.3 Body Coordinate Formulation 369
13.3.1 Double A-Arm Suspension 371
13.4 Joint Coordinate Formulation 373
13.4.1 Variable-Length Pendulum 375
13.5 Point Coordinate Formulation 376
13.6 Constraint Violation 376
13.6.1 Constraint Violation Stabilization M ethod 376
13.6.2 Coordinate Partitioning M ethod 379
13.7 Rem arks 382
13.8 Problems 382
Chapter 4
Com plementary Analyses 387
14.1 Static Analysis 387
14.2 Static Equilibrium 389
14.3 Initial Condition Correction 392
14.3.1 Body -Coordinate Formulation 396
14.4 Three Com bined Analyses by Integration 398
14.5 Redun dant Constraints 399
14.6 Friction 400
14.7 Deformable Body 403
14.8 Rem arks 405
14.9 Problems 405
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Chapter 5
Projects 409
15.1 Windshield Wiper Mechan ism 409
15.2 Internal Com bustion Engine 412
15.3 Sled Test and Belted Dum my 414
15.4 Head and Neck 418
15.5 M ountain Bike 421
15.6 Motorcycle 425
15.7 Elliptical Exerc ise M achine 427
15.8 Swing 429
Appendix
M ass Center and M oment of Inertia 433
References
439
Index 441