Structural Dynamic Analysis with Generalized Damping Models : Analysis

Since Lord Rayleigh introduced the idea of viscous damping in his classic work "The Theory of Sound" in 1877, it has become standard practice to use this approach in dynamics, covering a wide range of applications from aerospace to civil engineering. However, in the majority of practical cases this approach is adopted more for mathematical convenience than for modeling the physics of vibration damping.

Over the past decade, extensive research has been undertaken on more general "non-viscous" damping models and vibration of non-viscously damped systems. This book, along with a related book Structural Dynamic Analysis with Generalized Damping Models: Identification , is the first comprehensive study to cover vibration problems with general non-viscous damping. The author draws on his considerable research experience to produce a text covering: dynamics of viscously damped systems; non-viscously damped single- and multi-degree of freedom systems; linear systems with non-local and non-viscous damping; reduced computational methods for damped systems; and finally a method for dealing with general asymmetric systems. The book is written from a vibration theory standpoint, with numerous worked examples which are relevant across a wide range of mechanical, aerospace and structural engineering applications.

Contents

1. Introduction to Damping Models and Analysis Methods.
2. Dynamics of Undamped and Viscously Damped Systems.
3. Non-Viscously Damped Single-Degree-of-Freedom Systems.
4. Non-viscously Damped Multiple-Degree-of-Freedom Systems.
5. Linear Systems with General Non-Viscous Damping.
6. Reduced Computational Methods for Damped Systems

Author Notes

Sondipon Adhikari is Chair Professor of Aerospace Engineering at Swansea University, Wales. His wide-ranging and multi-disciplinary research interests include uncertainty quantification in computational mechanics, bio- and nanomechanics, dynamics of complex systems, inverse problems for linear and nonlinear dynamics, and renewable energy. He is a technical reviewer of 97 international journals, 18 conferences and 13 funding bodies. He has written over 180 refereed journal papers, 120 refereed conference papers and has authored or co-authored 15 book chapters.

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

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