ISBN: 3540423737
TITLE: Rigid Body Dynamics of Mechanisms
AUTHOR: Hubert Hahn
TOC:

Preface vii
1. Introduction 1
1.1 Tasks in multibody simulation, analysis, and control 1
1.2 Coordinates and frames 3
1.3 Formulation of the model equations 4
1.4 Prototype applications of rigid-body mechanisms 7
1.5 General-purpose rigid-body analysis programs 21
1.5.1 Design of an engineering model 22
1.5.2 Input and output data 24
1.6 Purpose of this monograph 25
2. Planar and spatial vectors, matrices, and vector functions 33
2.1 Planar vectors and matrices 33
2.1.1 Elementary vector and matrix operations 34
2.1.1.1 Geometric vectors 34
2.1.1.2 Algebraic vectors 37
2.1.2 Time derivatives of displacement vectors and orientation matrices 47
2.1.2.1 Velocities and angular velocities 48
2.1.2.2 Accelerations and angular accelerations 50
2.2 Spatial vectors and matrices 53
2.2.1 Displacement vectors, frames, and orientation matrices 54
2.2.1.1 Basis transformation 56
2.2.1.2 Coordinate transformation 59
2.2.1.3 Bryant angles 61
2.2.2 Time derivatives of displacement vectors and orientation matrices 65
2.2.2.1 Velocities and angular velocities 65
2.2.2.2 Accelerations and angular accelerations 67
2.2.2.3 Kinematic differential equation 67
3. Constraint equations and constraint reaction forces of mechanisms 75
3.1 Kinematics of planar and spatial rigid-body systems 75
3.1.1 Kinematics of planar mechanisms 75
3.1.1.1 Pure kinematic analysis of planar mechanisms 79
3.1.1.2 Regular and singular planar kinematics 81
3.1.1.2.1 Regular constraint Jacobian matrix 81
3.1.1.2.2 Singular constraint Jacobian matrix 82
3.1.1.3 Kinematics in planar dynamic analysis 83
3.1.2 Kinematics of spatial mechanisms 84
3.1.2.1 Pure kinematic analysis of spatial mechanisms 84
3.1.2.2 Kinematics in spatial dynamic analysis 86
3.1.3 Singularity analysis of a planar slider-trank mechanism 87
3.1.3.1 Identification of singularities by direct inspection 87
3.1.3.2 Local algebraic singularity analysis of the slider-crank mechanism 89
3.1.3.2.1 Local analysis of Case 1 (psi_i(t) = a_1(t)) 91
3.1.3.2.2 Local analysis of Case 2 (x^R_PO = -a_2(t)) 103
3.2 Constraint reaction forces and torques of mechanisms 120
3.2.1 Constraint reaction forces of planar mechanisms 120
3.2.2 Constraint reaction forces of spatial mechanisms 123
4. Dynamics of planar and spatial rigid-body systems 129
4.1 Linear momentum and angular momentum of a rigid body 129
4.1.1 Linear momentum 129
4.1.2 Angular momentum 131
4.1.3 Properties of the inertia matrix 134
4.1.3.1 Physical interpretation of J^L_P 134
4.1.3.2 Time dependence of J^L_P and J^R_P 135
4.1.3.3 Steiner-Huygens relation 135
4.2 Newton-Euler equations of an unconstrained rigid body 137
4.2.1 Force moments and couples 137
4.2.2 Newton's law 140
4.2.3 Euler's law 141
4.2.4 Newton-Euler equations of a rigid body under planar and spatial motion 143
4.2.4.1 Spatial motion 143
4.2.4.2 Planar motion 147
4.3 Equations of motion of planar and spatial rigid-body mechanisms 150
4.3.1 Equations of planar motion of unconstrained rigid bodies in DE form and of constrained rigid-body systems in DAE form 151
4.3.1.1 A single unconstrained rigid body 152
4.3.1.2 System of unconstrained rigid bodies 154
4.3.1.3 A single rigid body constrained with respect to the base 154
4.3.1.4 System of constrained rigid bodies 156
4.3.2 Equations of spatial motion of unconstrained rigid bodies in DE form and of constrained rigid-body mechanisms in DAE form 158
4.3.2.1 A single unconstrained rigid body 158
4.3.2.2 System of unconstrained rigid bodies 159
4.3.2.3 A single rigid body constrained with respect to the base 159
4.3.2.4 System of constrained rigid bodies 161
4.4 Numerical solution of DAEs - a brief discussion 162
4.4.1 Ideal situation 163
4.4.1.1 Algebraic aspects 163
4.4.1.2 Numerical integration step 165
4.4.2 More realistic situations 166
4.4.2.1 Singular matrix A 166
4.4.2.2 Constraint violation 166
5. Model equations of planar and spatial joints 171
5.1 Theoretical modeling of planar joints 173
5.1.1 Absolute constraints 174
5.1.1.1 Position constraints between a body and the base 174
5.1.1.1.1 Partial-position constraint (massless revolute-translational link) 174
5.1.1.1.2 Complete-position constraint (revolute joint) 179
5.1.1.2 Orientation constraint (massless translational link) 181
5.1.1.3 Orientation and partial-position constraint (translational joint) 181
5.1.1.4 Combined orientation/partial-position constraint 183
5.1.1.5 Constant-distance constraint (massless revolute-revolute link) 184
5 1 2 Relative planar joints between two bodies 186
5.1.2.1 Position constraints 186
5.1.2.1.1 Partial-position constraint (massless revolute-translational link) 186
5.1.2.1.2 Complete-position constraint (revolute joint) 190
5.1.2.2 Orientation constraint (massless translational link) 192
5.1.2.3 Relative orientation and partial position constraint (translational joint) 193
5.1.2.4 Combined orientation/partial-position constraint 196
5.1.2.5 Constant-distance constraint (massless revolute-revolute link) 196
5.1.3 Pseudo-joint and force/torque elements 198
5.1.3.1 Example of a translational spring element 198
5.1.3.2 Example of a torsional spring 198
5.2 Theoretical modeling of spatial joints 200
5.2.1 Building blocks of joint models 200
5.2.1.1 Common-point constraint (BB1; three constrained translational DOFs) 201
5.2.1.2 Parallel-axes constraint (BB2; two constrained rotational DOFs) 204
5.2.1.3 Straight-line-point-follower constraint(BB3; two constrained translational DOFs) 208
5.2.1.4 Rotation-blocker constraint (BB4; one constrained rotational DOF) 212
5.2.1.5 Constant-distance constraint (BB5; one constrained translational DOF) 218
5.2.2 Theoretical models of common joints 220
5.2.2.1 Spherical joint (BB1; constrains three translational DOFs) 220
5.2.2.2 Massless spherical-spherical link (BB5; constrains one translational DOF) 222
5.2.2.3 Translational joint (BB2, BB4; constrains three rotational DOFs) 223
5.2.2.4 Universal joint (BB1, BB4; constrains three translational and one rotational DOF) 226
5.2.2.5 Revolute joint (BB1, BB2; constrains three translational and two rotational DOFs) 228
5.2.2.6 Cylindrical joint (BB2, BB3; constrains two translational and two rotational DOFs) 231
5.2.2.7 Prismatic joint (BB2, BB3, BB4; constrains three rotational and two translational DOFs) 234
6. Constitutive relations of planar and spatial external forces and torques 239
6.1 Constitutive relations of planar external forces and torques 239
6.1.1 Gravitational force (weight) 241
6.1.2 Applied forte and moment 241
6.1.3 Translational force elements between two bodies 243
6.1.3.1 Translational spring 246
6.1.3.2 Translational damper 247
6.1.3.3 Actuator 250
6.1.3.4 Torsional spring and damper 250
6.1.3.5 Torque generated by a motor 250
6.2 Constitutive relations of spatial external forces and torques 251
A. Appendix 255
A.1 Special vector and matrix operations used in mechanics 255
A.1.1 Euclidean vector space 255
A.1.2 Scalar product and cross product of planar vectors 258
A.1.3 Cross product of spatial vectors 262
A.1.4 Time derivatives of planar orientation matrices and of planar vectors in different frames 266
A.1.5 Time derivatives of spatial orientation matrices and of spatial vectors in different frames 273
A.1.6 Derivatives of vector functions 282
A.2 Lagrange formalism of a rigid body under spatial motion 290
A.2.1 Kinetic energy of an unconstrained rigid body 291
A.2.2 Spatial equations of motion of an unconstrained rigid body for P = C 294
A.2.3 Spatial equations of motion of a constrained rigid body 296
A.3 Model equations of planar and spatial mechanisms 298
A 4 Constraint equations of a general universal joint 302
A.4.1 Notation and abbreviations 303
A.4.2 Computation of constraint equations 305
A.4.2.1 First constraint equation 305
A.4.2.2 Second constraint equation 309
A.4.2.3 Third constraint equation 313
A.4.2.4 Fourth constraint equation 316
A.4.3 Computation of the shortest distance between two rotationaxes 319
References 321
Index 329
List offigures 333
END
