Preface xi
Nomenclature xv
Chapter 1. Introduction to Damping Models and Analysis Methods 1
1.1. Models of damping 3
1.1.1. Single-degree-of-freedom systems 4
1.1.2. Continuous systems 8
1.1.3. Multiple-degrees-of-freedom systems 10
1.1.4. Other studies 11
1.2. Modal analysis of viscously damped systems 13
1.2.1. The state-space method 14
1.2.2. Methods in the configuration space 15
1.3. Analysis of non-viscously damped systems 21
1.3.1. State-space-based methods 22
1.3.2. Time-domain-based methods 23
1.3.3. Approximate methods in the configuration space 23
1.4. Identification of viscous damping 24
1.4.1. Single-degree-of-freedom systems 24
1.4.2. Multiple-degrees-of-freedom systems 25
1.5. Identification of non-viscous damping 28
1.6. Parametric sensitivity of eigenvalues and eigenvectors 29
1.6.1. Undamped systems 29
1.6.2. Damped systems 30
1.7. Motivation behind this book 32
1.8. Scope of the book 33
Chapter 2. Dynamics of Undamped and Viscously Damped Systems 41
2.1. Single-degree-of-freedom undamped systems 41
2.1.1. Natural frequency 42
2.1.2. Dynamic response 43
2.2. Single-degree-of-freedom viscously damped systems 45
2.2.1. Natural frequency 46
2.2.2. Dynamic response 47
2.3. Multiple-degree-of-freedom undamped systems 52
2.3.1. Modal analysis 53
2.3.2. Dynamic response 55
2.4. Proportionally damped systems 58
2.4.1. Condition for proportional damping 60
2.4.2. Generalized proportional damping 61
2.4.3. Dynamic response 65
2.5. Non-proportionally damped systems 80
2.5.1. Free vibration and complex modes 81
2.5.2. Dynamic response 87
2.6. Rayleigh quotient for damped systems 93
2.6.1. Rayleigh quotients for discrete systems 94
2.6.2. Proportional damping 96
2.6.3. Non-proportional damping 97
2.6.4. Application of Rayleigh quotients 100
2.6.5. Synopses 101
2.7. Summary 101
Chapter 3. Non-Viscously Damped Single-Degree-of-Freedom Systems 103
3.1. The equation of motion 104
3.2. Conditions for oscillatory motion 108
3.3. Critical damping factors 112
3.4. Characteristics of the eigenvalues 113
3.4.1. Characteristics of the natural frequency 114
3.4.2. Characteristics of the decay rate corresponding to the oscillating mode 118
3.4.3. Characteristics of the decay rate corresponding to the non-oscillating mode 122
3.5. The frequency response function 123
3.6. Characteristics of the response amplitude 126
3.6.1. The frequency for the maximum response amplitude 128
3.6.2. The amplitude of the maximum dynamic response 137
3.7. Simplified analysis of the frequency response function 141
3.8. Summary 144
Chapter 4. Non-viscously Damped Multiple-Degree-of-Freedom Systems 147
4.1. Choice of the kernel function 149
4.2. The exponential model for MDOF non-viscously damped systems 151
4.3. The state-space formulation 153
4.3.1. Case A: all coefficient matrices are of full rank 153
4.3.2. Case B: coefficient matrices are rank deficient 158
4.4. The eigenvalue problem 162
4.4.1. Case A: all coefficient matrices are of full rank 162
4.4.2. Case B: coefficient matrices are rank deficient 165
4.5. Forced vibration response 166
4.5.1. Frequency domain analysis 167
4.5.2. Time-domain analysis 168
4.6. Numerical examples 169
4.6.1. Example 1: SDOF system with non-viscous damping 169
4.6.2. Example 2: a rank-deficient system 170
4.7. Direct time-domain approach 174
4.7.1. Integration in the time domain 174
4.7.2. Numerical realization 175
4.7.3. Summary of the method 179
4.7.4. Numerical examples 181
4.8. Summary 184
Chapter 5. Linear Systems with General Non-Viscous Damping 187
5.1. Existence of classical normal modes 188
5.1.1. Generalization of proportional damping 189
5.2. Eigenvalues and eigenvectors 191
5.2.1. Elastic modes 193
5.2.2. Non-viscous modes 197
5.2.3. Approximations for lightly damped systems 198
5.3. Transfer function 199
5.3.1. Eigenvectors of the dynamic stiffness matrix 201
5.3.2. Calculation of the residues 202
5.3.3. Special cases 204
5.4. Dynamic response 205
5.4.1. Summary of the method 207
5.5. Numerical examples 208
5.5.1. The system 208
5.5.2. Example 1: exponential damping 210
5.5.3. Example 2: GHM damping 213
5.6. Eigenrelations of non-viscously damped systems 215
5.6.1. Nature of the eigensolutions 216
5.6.2. Normalization of the eigenvectors 217
5.6.3. Orthogonality of the eigenvectors 219
5.6.4. Relationships between the eigensolutions and damping 223
5.6.5. System matrices in terms of the eigensolutions 225
5.6.6. Eigenrelations for viscously damped systems 226
5.6.7. Numerical examples 227
5.7. Rayleigh quotient for non-viscously damped systems 230
5.8. Summary 234
Chapter 6. Reduced Computational Methods for Damped Systems 237
6.1. General non-proportionally damped systems with viscous damping 238
6.1.1. Iterative approach for the eigensolutions 239
6.1.2. Summary of the algorithm 244
6.1.3. Numerical example 246
6.2. Single-degree-of-freedom non-viscously damped systems 247
6.2.1. Nonlinear eigenvalue problem for non-viscously damped systems 250
6.2.2. Complex conjugate eigenvalues 251
6.2.3. Real eigenvalues 253
6.2.4. Numerical examples 257
6.3. Multiple-degrees-of-freedom non-viscously damped systems 259
6.3.1. Complex conjugate eigenvalues 260
6.3.2. Real eigenvalues 262
6.3.3. Numerical example 263
6.4. Reduced second-order approach for non-viscously damped systems 264
6.4.1. Proportionally damped systems 266
6.4.2. The general case 271
6.4.3. Numerical examples 274
6.5. Summary 277
Appendix 281
Bibliography 299
Author index 329
Index 335
