Ebook
Elements of Structural Dynamics: A New PerspectiveISBN: 9781118361214
448 pages
September 2012

Structural dynamics is a subset of structural analysis which covers the behavior of structures subjected to dynamic loading. The subject has seen rapid growth and also change in how the basic concepts can be interpreted. For instance, the classical notions of discretizing the operator of a dynamic structural model have given way to a settheoretic, functionspace based framework, which is more conducive to implementation with a computer. This modern perspective, as adopted in this book, is also helpful in putting together the various tools and ideas in a more integrated style.
Elements of Structural Dynamics: A New Perspective is devoted to covering the basic concepts in linear structural dynamics, whilst emphasizing their mathematical moorings and the associated computational aspects that make their implementation in software possible.
Key features:
 Employs a novel ‘top down’ approach to structural dynamics.
 Contains an insightful treatment of the computational aspects, including the finite element method, that translate into numerical solutions of the dynamic equations of motion.
 Consistently touches upon the modern mathematical basis for the theories and approximations involved.
Elements of Structural Dynamics: A New Perspective is a holistic treatise on structural dynamics and is an ideal textbook for senior undergraduate and graduate students in Mechanical, Aerospace and Civil engineering departments. This book also forms a useful reference for researchers and engineers in industry.
Preface xi
Acknowledgements xv
Introduction xvii
General Notations xxi
1 Structural Dynamics and Mathematical Modelling 1
1.1 Introduction 1
1.2 System of Rigid Bodies and Dynamic Equations of Motion 2
1.2.1 Principle of Virtual Work 2
1.2.2 Hamilton’s Principle 3
1.2.3 Lagrangian Equations of Motion 4
1.3 Continuous Dynamical Systems and Equations of Motion from Hamilton’s Principle 6
1.3.1 Strain and Stress Tensors and Strain Energy 7
1.4 Dynamic Equilibrium Equations from Newton’s Force Balance 11
1.4.1 Displacement–Strain Relationships 11
1.4.2 Stress–Strain Relationships 13
1.5 Equations of Motion by Reynolds Transport Theorem 13
1.5.1 Mass Conservation 15
1.5.2 Linear Momentum Conservation 16
1.6 Conclusions 17
Exercises 17
Notations 18
References 19
Bibliography 19
2 Continuous Systems – PDEs and Solution 21
2.1 Introduction 21
2.2 Some Continuous Systems and PDEs 22
2.2.1 A Taut String – the OneDimensional Wave Equation 22
2.2.2 An Euler–Bernoulli Beam – the OneDimensional Biharmonic Wave Equation 23
2.2.3 Beam Equation with Rotary Inertia and Shear Deformation Effects 27
2.2.4 Equations of Motion for 2D Plate by Classical Plate Theory (Kirchhoff Theory) 29
2.3 PDEs and General Solution 36
2.3.1 PDEs and Canonical Transformations 36
2.3.2 General Solution to the Wave Equation 38
2.3.3 Particular Solution (D’Alembert’s Solution) to the Wave Equation 38
2.4 Solution to Linear Homogeneous PDEs – Method of Separation of Variables 40
2.4.1 Homogeneous PDE with Homogeneous Boundary Conditions 41
2.4.2 Sturm–Liouville BoundaryValue Problem (BVP) for the Wave Equation 42
2.4.3 Adjoint Operator and SelfAdjoint Property 42
2.4.4 Eigenvalues and Eigenfunctions of the Wave Equation 45
2.4.5 Series Solution to the Wave Equation 45
2.4.6 Mixed Boundary Conditions and Wave Equation 46
2.4.7 Sturm–Liouville BoundaryValue Problem for the Biharmonic Wave Equation 48
2.4.8 Thin Rectangular Plates – Free Vibration Solution 53
2.5 Orthonormal Basis and Eigenfunction Expansion 56
2.5.1 Best Approximation to f(x) 57
2.6 Solutions of Inhomogeneous PDEs by EigenfunctionExpansion Method 59
2.7 Solutions of Inhomogeneous PDEs by Green’s Function Method 64
2.8 Solution of PDEs with Inhomogeneous Boundary Conditions 68
2.9 Solution to Nonselfadjoint Continuous Systems 69
2.9.1 Eigensolution of Nonselfadjoint System 69
2.9.2 Biorthogonality Relationship between L and L∗ 70
2.9.3 Eigensolutions of L and L∗ 73
2.10 Conclusions 74
Exercises 75
Notations 75
References 77
Bibliography 77
3 Classical Methods for Solving the Equations of Motion 79
3.1 Introduction 79
3.2 Rayleigh–Ritz Method 80
3.2.1 Rayleigh’s Principle 84
3.3 Weighted Residuals Method 85
3.3.1 Galerkin Method 86
3.3.2 Collocation Method 91
3.3.3 Subdomain Method 93
3.3.4 Least Squares Method 94
3.4 Conclusions 95
Exercises 95
Notations 96
References 97
Bibliography 97
4 Finite Element Method and Structural Dynamics 99
4.1 Introduction 99
4.2 Weak Formulation of PDEs 101
4.2.1 WellPosedness of the Weak Form 103
4.2.2 Uniqueness and Stability of Solution to Weak Form 104
4.2.3 Numerical Integration by Gauss Quadrature 107
4.3 ElementWise Representation of the Weak Form and the FEM 111
4.4 Application of the FEM to 2D Problems 113
4.4.1 Membrane Vibrations and FEM 113
4.4.2 Plane (2D) Elasticity Problems – Plane Stress and Plane Strain 115
4.5 Higher Order Polynomial Basis Functions 118
4.5.1 Beam Vibrations and FEM 118
4.5.2 Plate Vibrations and FEM 120
4.6 Some Computational Issues in FEM 121
4.6.1 Element Shape Functions in Natural Coordinates 122
4.7 FEM and Error Estimates 124
4.7.1 APriori Error Estimate 124
4.8 Conclusions 126
Exercises 126
Notations 127
References 129
Bibliography 129
5 MDOF Systems and Eigenvalue Problems 131
5.1 Introduction 131
5.2 Discrete Systems through a Lumped Parameter Approach 132
5.2.1 Positive Definite and SemiDefinite Systems 134
5.3 Coupled Linear ODEs and the Linear Differential Operator 135
5.4 Coupled Linear ODEs and Eigensolution 136
5.5 First Order Equations and Uncoupling 142
5.6 First Order versus Second Order ODE and Eigensolutions 143
5.7 MDOF Systems and Modal Dynamics 145
5.7.1 SDOF Oscillator and Modal Solution 146
5.7.2 Rayleigh Quotient 153
5.7.3 Rayleigh–Ritz Method for MDOF Systems 155
5.8 Damped MDOF Systems 156
5.8.1 Damped System and Quadratic Eigenvalue Problem 157
5.8.2 Damped System and Unsymmetric Eigenvalue Problem 158
5.8.3 Proportional Damping and Uncoupling MDOF Systems 159
5.8.4 Damped Systems and Impulse Response 160
5.8.5 Response under General Loading 161
5.8.6 Response under Harmonic Input 161
5.8.7 Complex Frequency Response 163
5.8.8 Force Transmissibility 165
5.8.9 System Response and Measurement of Damping 167
5.9 Conclusions 173
Exercises 173
Notations 175
References 177
Bibliography 177
6 Structures under Support Excitations 179
6.1 Introduction 179
6.2 Continuous Systems and Base Excitations 181
6.3 MDOF Systems under Support Excitation 185
6.4 SDOF Systems under Base Excitation 191
6.4.1 Frequency Response of SDOF System under Base Motion 192
6.5 Support Excitation and Response Spectra 196
6.5.1 Peak Response Estimates of an MDOF System Using Response Spectra 197
6.6 Structures under multisupport excitation 198
6.6.1 Continuous system under multisupport excitation 199
6.6.2 MDOF systems under multisupport excitation 202
6.7 Conclusions 203
Exercises 204
Notations 205
References 206
Bibliography 206
7 Eigensolution Procedures 209
7.1 Introduction 209
7.2 Power and Inverse Iteration Methods and Eigensolutions 210
7.2.1 Order and Rate of Convergence – Distinct Eigenvalues 212
7.2.2 Shifting and Convergence 213
7.2.3 Multiple Eigenvalues 215
7.2.4 Eigenvalues within an IntervalShifting Scheme with Gram–Schmidt Orthogonalisation and Sturm Sequence Property 216
7.3 Jacobi, Householder, QR Transformation Methods and Eigensolutions 220
7.3.1 Jacobi Method 220
7.3.2 Householder and QR Transformation Methods 224
7.4 Subspace Iteration 231
7.4.1 Convergence in Subspace Iteration 232
7.5 Lanczos Transformation Method 233
7.5.1 Lanczos Method and Error Analysis 235
7.6 Systems with Unsymmetric Matrices 237
7.6.1 SkewSymmetric Matrices and Eigensolution 245
7.6.2 Unsymmetric Matrices – A Rotor Bearing System 246
7.6.3 Unsymmetric Systems and Eigensolutions 253
7.7 Dynamic Condensation and Eigensolution 260
7.7.1 Symmetric Systems and Dynamic Condensation 262
7.7.2 Unsymmetric Systems and Dynamic Condensation 264
7.8 Conclusions 268
Exercises 268
Notations 269
References 272
Bibliography 273
8 Direct Integration Methods 275
8.1 Introduction 275
8.2 Forward and Backward Euler Methods 281
8.2.1 Forward Euler Method 281
8.2.2 Backward (Implicit) Euler Method 284
8.3 Central Difference Method 286
8.4 Newmarkβ Method – a SingleStep Implicit Method 289
8.4.1 Some Degenerate Cases of the Newmarkβ Method and Stability 292
8.4.2 Undamped Case – Amplitude and Periodicity Errors 295
8.4.3 Amplitude and Periodicity Errors 295
8.5 HHTα and Generalizedα Methods 297
8.6 Conclusions 303
Exercises 305
Notations 305
References 306
Bibliography 307
9 Stochastic Structural Dynamics 309
9.1 Introduction 309
9.2 Probability Theory and Basic Concepts 311
9.3 Random Variables 312
9.3.1 Joint Random Variables, Distributions and Density Functions 314
9.3.2 Expected (Average) Values of a Random Variable 315
9.3.3 Characteristic and MomentGenerating Functions 317
9.4 Conditional Probability, Independence and Conditional Expectation 317
9.4.1 Conditional Expectation 319
9.5 Some oftUsed Probability Distributions 319
9.5.1 Binomial Distribution 320
9.5.2 Poisson Distribution 320
9.5.3 Normal Distribution 321
9.5.4 Uniform Distribution 322
9.5.5 Rayleigh Distribution 322
9.6 Stochastic Processes 323
9.6.1 Stationarity of a Stochastic Process 323
9.6.2 Properties of Autocovariance/Autocorrelation Functions of Stationary Processes 325
9.6.3 Spectral Representation of a Stochastic Process 325
9.6.4 SXX(λ) as the Mean Energy Density of X(t) 327
9.6.5 Some Basic Stochastic Processes 328
9.7 Stochastic Dynamics of Linear Structural Systems 331
9.7.1 Continuous Systems under Stochastic Input 331
9.7.2 Discrete Systems under Stochastic Input – Modal Superposition Method 337
9.8 An Introduction to Ito Calculus 338
9.8.1 Brownian Filtration 340
9.8.2 Measurability 340
9.8.3 An Adapted Stochastic Process 340
9.8.4 Ito Integral 341
9.8.5 Martingale 342
9.8.6 Ito Process 343
9.8.7 Computing the Response Moments 352
9.8.8 Time Integration of SDEs 357
9.9 Conclusions 360
Exercises 361
Notations 363
References 365
Bibliography 366
Appendix A 367
Appendix B 369
Appendix C 375
Appendix D 379
Appendix E 387
Appendix F 391
Appendix G 393
Appendix H 399
Appendix I 407
Index 413