Elements of Structural Dynamics

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 One-Dimensional Wave Equation 22

2.2.2 An Euler-Bernoulli Beam - the One-Dimensional 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 Boundary-Value Problem (BVP) for the Wave Equation 42

2.4.3 Adjoint Operator and Self-Adjoint 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 Boundary-Value 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 Eigenfunction-Expansion 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 Nonself-adjoint Continuous Systems 69

2.9.1 Eigensolution of Nonself-adjoint 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 Well-Posedness 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 Element-Wise 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 A-Priori 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 Semi-Definite 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 multi-support excitation 198

6.6.1 Continuous system under multi-support excitation 199

6.6.2 MDOF systems under multi-support 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 Interval-Shifting 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 Skew-Symmetric 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 Newmark-ß Method - a Single-Step Implicit Method 289

8.4.1 Some Degenerate Cases of the Newmark-ß Method and Stability 292

8.4.2 Undamped Case - Amplitude and Periodicity Errors 295

8.4.3 Amplitude and Periodicity Errors 295

8.5 HHT-a and Generalized-a 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 Moment-Generating Functions 317

9.4 Conditional Probability, Independence and Conditional Expectation 317

9.4.1 Conditional Expectation 319

9.5 Some oft-Used 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(¿) 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