Structural Dynamic Analysis with Generalized Damping Models

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 xiNomenclature xvChapter 1. Introduction to Damping Models and Analysis Methods 11.1. Models of damping 31.1.1. Single-degree-of-freedom systems 41.1.2. Continuous systems 81.1.3. Multiple-degrees-of-freedom systems 101.1.4. Other studies 111.2. Modal analysis of viscously damped systems 131.2.1. The state-space method 141.2.2. Methods in the configuration space 151.3. Analysis of non-viscously damped systems 211.3.1. State-space-based methods 221.3.2. Time-domain-based methods 231.3.3. Approximate methods in the configuration space 231.4. Identification of viscous damping 241.4.1. Single-degree-of-freedom systems 241.4.2. Multiple-degrees-of-freedom systems 251.5. Identification of non-viscous damping 281.6. Parametric sensitivity of eigenvalues and eigenvectors 291.6.1. Undamped systems 291.6.2. Damped systems 301.7. Motivation behind this book 321.8. Scope of the book 33Chapter 2. Dynamics of Undamped and Viscously Damped Systems 412.1. Single-degree-of-freedom undamped systems 412.1.1. Natural frequency 422.1.2. Dynamic response 432.2. Single-degree-of-freedom viscously damped systems 452.2.1. Natural frequency 462.2.2. Dynamic response 472.3. Multiple-degree-of-freedom undamped systems 522.3.1. Modal analysis 532.3.2. Dynamic response 552.4. Proportionally damped systems 582.4.1. Condition for proportional damping 602.4.2. Generalized proportional damping 612.4.3. Dynamic response 652.5. Non-proportionally damped systems 802.5.1. Free vibration and complex modes 812.5.2. Dynamic response 872.6. Rayleigh quotient for damped systems 932.6.1. Rayleigh quotients for discrete systems 942.6.2. Proportional damping 962.6.3. Non-proportional damping 972.6.4. Application of Rayleigh quotients 1002.6.5. Synopses 1012.7. Summary 101Chapter 3. Non-Viscously Damped Single-Degree-of-Freedom Systems 1033.1. The equation of motion 1043.2. Conditions for oscillatory motion 1083.3. Critical damping factors 1123.4. Characteristics of the eigenvalues 1133.4.1. Characteristics of the natural frequency 1143.4.2. Characteristics of the decay rate corresponding to the oscillating mode 1183.4.3. Characteristics of the decay rate corresponding to the non-oscillating mode 1223.5. The frequency response function 1233.6. Characteristics of the response amplitude 1263.6.1. The frequency for the maximum response amplitude 1283.6.2. The amplitude of the maximum dynamic response 1373.7. Simplified analysis of the frequency response function 1413.8. Summary 144Chapter 4. Non-viscously Damped Multiple-Degree-of-Freedom Systems 1474.1. Choice of the kernel function 1494.2. The exponential model for MDOF non-viscously damped systems 1514.3. The state-space formulation 1534.3.1. Case A: all coefficient matrices are of full rank 1534.3.2. Case B: coefficient matrices are rank deficient 1584.4. The eigenvalue problem 1624.4.1. Case A: all coefficient matrices are of full rank 1624.4.2. Case B: coefficient matrices are rank deficient 1654.5. Forced vibration response 1664.5.1. Frequency domain analysis 1674.5.2. Time-domain analysis 1684.6. Numerical examples 1694.6.1. Example 1: SDOF system with non-viscous damping 1694.6.2. Example 2: a rank-deficient system 1704.7. Direct time-domain approach 1744.7.1. Integration in the time domain 1744.7.2. Numerical realization 1754.7.3. Summary of the method 1794.7.4. Numerical examples 1814.8. Summary 184Chapter 5. Linear Systems with General Non-Viscous Damping 1875.1. Existence of classical normal modes 1885.1.1. Generalization of proportional damping 1895.2. Eigenvalues and eigenvectors 1915.2.1. Elastic modes 1935.2.2. Non-viscous modes 1975.2.3. Approximations for lightly damped systems 1985.3. Transfer function 1995.3.1. Eigenvectors of the dynamic stiffness matrix 2015.3.2. Calculation of the residues 2025.3.3. Special cases 2045.4. Dynamic response 2055.4.1. Summary of the method 2075.5. Numerical examples 2085.5.1. The system 2085.5.2. Example 1: exponential damping 2105.5.3. Example 2: GHM damping 2135.6. Eigenrelations of non-viscously damped systems 2155.6.1. Nature of the eigensolutions 2165.6.2. Normalization of the eigenvectors 2175.6.3. Orthogonality of the eigenvectors 2195.6.4. Relationships between the eigensolutions and damping 2235.6.5. System matrices in terms of the eigensolutions 2255.6.6. Eigenrelations for viscously damped systems 2265.6.7. Numerical examples 2275.7. Rayleigh quotient for non-viscously damped systems 2305.8. Summary 234Chapter 6. Reduced Computational Methods for Damped Systems 2376.1. General non-proportionally damped systems with viscous damping 2386.1.1. Iterative approach for the eigensolutions 2396.1.2. Summary of the algorithm 2446.1.3. Numerical example 2466.2. Single-degree-of-freedom non-viscously damped systems 2476.2.1. Nonlinear eigenvalue problem for non-viscously damped systems 2506.2.2. Complex conjugate eigenvalues 2516.2.3. Real eigenvalues 2536.2.4. Numerical examples 2576.3. Multiple-degrees-of-freedom non-viscously damped systems 2596.3.1. Complex conjugate eigenvalues 2606.3.2. Real eigenvalues 2626.3.3. Numerical example 2636.4. Reduced second-order approach for non-viscously damped systems 2646.4.1. Proportionally damped systems 2666.4.2. The general case 2716.4.3. Numerical examples 2746.5. Summary 277Appendix 281Bibliography 299Author index 329Index 335