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Table of Contents 1

1 Introduction to Structural Dynamics 1

1.1 Essential Characteristics and Basic Assumptions 1

1.1.1 Essential Characteristics 2

1.1.2 Basic Assumptions in Structural Dynamics of This Book 3

1.2 Missions of Structural Dynamics 5

1.2.1 Response Analysis 5

1.2.2 Inverse Problem of Type Ⅰ: System Identification 6

1.2.3 Inverse Problem Type Ⅱ: Load Identification 6

1.2.4 Vibration Control 6

1.3 Types of Dynamic Loads 7

1.3.1 Periodic Load 7

1.3.2 Impulsive Load 9

1.3.3 Random Load 9

1.4 Formulation of the Equations of Motion 9

1.4.1 Direct Equilibration Using d'Alembert's Principle 10

1.4.2 Variational Approach 10

1.5 Continuous and Discrete Structural Systems 13

References 15

2 Time-Domain Analysis of Continuous Systems 17

2.1 Free Transverse Vibration of Strings 18

2.2 Free Axial Vibration of Elastic Rods 20

2.3 Free Torsional Vibration of Cylinder Rods 22

2.4 Free Transverse Vibration of Euler-Bernoulli Beams 23

2.4.1 Simple Supported Beams 25

2.4.2 Cantilever Beams 26

2.4.3 Fixed-Fixed Beams 27

2.4.4 Free-Free Beams 28

2.5 Free Transverse Vibration of Rectangular Thin Plates 29

2.5.1 Kinematic Description 29

2.5.2 Equilibrium Equation 31

2.5.3 Boundary Conditions 33

2.5.4 Solutions of Rectangular Thin Plates with Simple-supported Edges 34

2.6 Some Properties of Natural Modes 36

2.6.1 Orthogonality of Mode Shapes 37

2.6.2 Modal Scaling 38

2.6.3 Expansion Theorem 40

2.6.4 Rayleigh Quotient 40

Problems 41

References 43

3 Time-Domain Analysis of SDOF Systems 44

3.1 From Continuous Systems to Generalized SDOF Systems 44

3.1.1 Historical Rayleigh's Method 44

3.1.2 An Improved Approach of Rayleigh's Method 48

3.2 Mathematical Modelling of Lumped-Parameter Systems 51

3.2.1 Direct Equilibration Modeling Using d'Alembert's Principle 52

3.2.2 Modeling Based on Principle of Virtual Displacements 53

3.3 Free Vibration of SDOF Systems 53

3.3.1 Free Vibration of Undamped SDOF Systems 53

3.3.2 Free Vibration of Viscous-Damped SDOF Systems 56

3.4 Dynamic Behavior of Undamped SDOF Systems under Harmonic Excitation 61

3.5 Viscous-Damped SDOF Systems to Harmonic Excitation 62

3.5.1 General Solution 62

3.5.2 Steady-State Response 64

3.5.3 Complex Expression of the Response 66

3.5.4 Resonance Response 66

3.5.5 Forced Vibration by Support Motion 69

3.5.6 Vibration Isolation 72

3.5.7 Motion Transducer 75

3.6 Expansion to Periodic Excitation via Fourier Series 79

3.6.1 Fourier Series for Arbitrary Periodic Functions 79

3.6.2 Steady-State Response under Arbitrary Periodic Excitations 80

3.7 Response to Impulsive Loading 82

3.7.1 Pulse Excitation 82

3.7.2 Shock Response Spectrum 89

3.7.3 Shock Isolation 91

3.8 Response of SDOF Systems in Case of the General Dynamic Excitation 92

3.8.1 Impulse Function 92

3.8.2 Impulse Response 93

3.8.3 Duhamel Integration 94

3.8.4 Arbitrary Support Motion 96

3.9 Damping 97

3.9.1 Damping Models in Structural Dynamics 98

3.9.2 Energy Losses and Equivalent Viscous Damping 104

3.9.3 Illustration of the Errors Due to the Equivalence 110

Problems 112

References 117

4 Time-Domain Analysis of MDOF Systems 118

4.1 Continuous Systems to MDOF Systems: Discretization Approaches 119

4.1.1 Direct Lumped-Parameter Methods 119

4.1.2 Generalized Displacements ＆ Rayleigh-Ritz's Methods 120

4.1.3 Assumed Mode Method: a Realization of General Rayleigh-Ritz Methods 125

4.1.4 Choosing the Shape Functions 128

4.1.5 Finite Element Method 129

4.2 Modeling of Equations of Motion for MDOF Systems 137

4.2.1 Direct Equilibration Modeling Using d'Alembert's Principle 137

4.2.2 Modeling with Principle of Virtual Displacements 138

4.2.3 Modeling with Lagrange's Equations 139

4.3 Free Vibration of Undamped MDOF Systems 140

4.3.1 Eigenvalue Problem, Natural Frequencies and Mode Shapes 140

4.3.2 Orthogonality 144

4.3.3 Modal Scaling 147

4.3.4 Eigenvalue Separation Property 148

4.4 Rayleigh and Rayleigh-Ritz's Methods for MDOF Systems:Model Reduction 149

4.4.1 Rayleigh Quotient for MDOF Systems 149

4.4.2 Rayleigh's Method for MDOF Systems 150

4.4.3 Rayleigh-Ritz's Method for MDOF Systems 151

4.4.4 Assumed Mode Method for MDOF Systems 155

4.5 MDOF Systems with Rigid-Body Modes 155

4.5.1 Small Fictitious Stiffness 156

4.5.2 Eigenvalue Shifting 157

4.5.3 Constraining of Rigid-Body Modes 159

4.6 Damping in MDOF systems 160

4.7 Numerical Evaluation of Responses of MDOF Systems 160

4.7.1 Numerical Derivatives 161

4.7.2 Central Difference Method 161

4.7.3 Newmark-β Method 163

4.8 Dynamic Response of MDOF Systems: Mode Superposition Method 167

4.8.1 Transformation of Coordinates 167

4.8.2 Modal Damping 168

4.8.3 Initial Conditions in Modal Coordinates 169

4.8.4 Mode Superposition for Free Vibration of Undamped MDOF Systems 169

4.8.5 Mode Superposition of Free Vibration of Damped MDOF Systems 171

4.8.6 Mode Superposition of Forced Vibration of Undamped MDOF Systems 171

4.8.7 Mode Superposition of Forced Vibration of Damped MDOF Systems 172

4.8.8 Mode-Displacement Solution and Mode-Acceleration Solution 172

Problems 174

References 177

5 Frequency-Domain Analysis 178

5.1 Frequency-Domain Analysis of SDOF Systems 178

5.1.1 System Equations and Transfer Function 179

5.1.2 Poles, Natural Frequencies, Damping Ratio and Residues 179

5.1.3 Transfer Function Plots 181

5.1.4 Frequency Response Function and Impulse Response Function 181

5.1.5 Influence of Mass, Damping and Stiffness Changes 183

5.2 Frequency-Domain Analysis of MDOF Systems 185

5.2.1 System Equations and Transfer Function 185

5.2.2 Poles, Natural Frequencies and Damping Ratio 186

5.2.3 Modal Vectors and Residues 187

5.2.4 Modal Participation Factors 189

5.2.5 Frequency Response Function Matrix and Impulse Response Function Matrix 189

5.2.6 Undamped and Proportionally Damped Systems 191

5.2.7 Orthogonality 193

5.2.8 Modal Vector Scaling 197

5.2.9 Numerical and Experimental Approaches 198

References 200

6 Experimental Modal Analysis and Applications 201

6.1 Basic Modal Model Equations 201

6.1.1 Modal Model 202

6.1.2 State Space Model 203

6.1.3 Rational Fraction Polynomial Model 204

6.2 Modal Parameter Estimation 205

6.2.1 Basic Concept 206

6.2.2 SDOF Methods 210

6.2.3 MDOF Time-Domain Methods 213

6.2.4 MDOF Frequency-Domain Methods 217

6.2.5 Output-Only or Operational Modal Analysis 225

6.2.6 Conclusions 229

6.3 Modal Validation 231

6.3.1 Modal Scale Factor and Modal Assurance Criterion 232

6.3.2 Mode Participation 233

6.3.3 Reciprocity 235

6.3.4 Mode Complexity 235

6.3.5 Modal Phase Collinearity and Mean Phase Deviation 236

6.3.6 Modal Confidence Factor 237

6.3.7 Synthesis of Frequency Response Functions 238

6.3.8 Discussion 238

6.4 Applications of Modal Parameters 239

6.4.1 Forced Response Analysis 239

6.4.2 Sensitivity Analysis 241

6.4.3 Structural Dynamics Modification ＆ Assembly 246

6.5 Combining Numerical and Experimental Models 261

6.5.1 Model Updating 261

6.5.2 Pre-Test Analysis 275

References 282