CONTENTS
viii
The Coordinate Vector, £, 196
The Loading Vector, p, 198
The Mass Matrix, M, 199
The Stiffness Matrix, K, 201
The Damping Matrix, C, 203
8.2
Equations of Motion: Newton’s Method, 204
8.3
Equations of Motion: Lagrange’s Formulation, 208
System Energies, 208
Hamilton’s Principle, 209
Dérivation for a Simple System, 209
Comments, 212
8.4
Free, Undamped Motion, 214
Frequencies, 214
Modal Vectors and Normalization, 214
Orthogonality of the Modal Vectors, 215
8.5
Forced, Damped Motion, 216
The Mode Shape Matrix, X, 217
Modal Damping Matrix, C, 217
Uncoupling the Equations of Motion, 218
Steady State Solutions. 219
8.6
Summary of the Normal Mode Method, 219
Problems, 220
References, 221
9
Applications of Multi-Degree of Freedom Analysis
James F. Wilson
9.1
A Fixed Leg Platform: Time Domain Responses, 224
Mathematical Model, 224
Frequencies, 225
Modal Vectors and Normalization, 226
Response to a Harmonie Wave, 227
Response to Earthquake Excitation, 231
9.2 A Monopod Gravity Platform: Free Vibration and Stability, 233
Mathematical Model, 233
Ftequencies and Mode Shapes, 236
Dynamic Stability, 238
9.3 Structural Response Statistics for Wave Loading, 239
9.4 A Fixed Leg Platform: Statistical Responses, 242
Problems, 246
References, 246
10 Continuons Systems
James F. Wilson
10.1 Modeling Beams and Cables, 250
Goveming Equations, 250
223
248