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Preface xi

About the Companion Website xiii

1 Single Degree of Freedom Systems 1

1.1 Introduction 1

1.2 Spring-Mass System 1

1.3 Spring-Mass-Damper System 6

1.4 Forced Response 10

1.5 Transfer Functions and Frequency Methods 17

1.6 Complex Representation and Impedance 23

1.7 Measurement and Testing 25

1.8 Stability 28

1.9 Design and Control of Vibrations 31

1.10 Nonlinear Vibrations 35

1.11 Computing and Simulation in MatlabTM 38

Chapter Notes 43

References 44

Problems 46

2 Lumped Parameter Models 49

2.1 Introduction 49

2.2 Modeling 52

2.3 Classifications of Systems 56

2.4 Feedback Control Systems 57

2.5 Examples 59

2.6 Experimental Models 64

2.7 Nonlinear Models and Equilibrium 65

Chapter Notes 67

References 68

Problems 68

3 Matrices and the Free Response 71

3.1 Introduction 71

3.2 Eigenvalues and Eigenvectors 71

3.3 Natural Frequencies and Mode Shapes 77

3.4 Canonical Forms 86

3.5 Lambda Matrices 91

3.6 Eigenvalue Estimates 94

3.7 Computation Eigenvalue Problems in Matlab 101

3.8 Numerical Simulation of the Time Response in Matlabtm 104

Chapter Notes 106

References 107

Problems 108

4 Stability 113

4.1 Introduction 113

4.2 Lyapunov Stability 113

4.3 Conservative Systems 116

4.4 Systems with Damping 117

4.5 Semidefinite Damping 118

4.6 Gyroscopic Systems 119

4.7 Damped Gyroscopic Systems 121

4.8 Circulatory Systems 122

4.9 Asymmetric Systems 123

4.10 Feedback Systems 128

4.11 Stability in the State Space 131

4.12 Stability of Nonlinear Systems 133

Chapter Notes 137

References 138

Problems 139

5 Forced Response of Lumped Parameter Systems 143

5.1 Introduction 143

5.2 Response via State SpaceMethods 143

5.3 Decoupling Conditions and Modal Analysis 148

5.4 Response of Systems with Damping 152

5.5 Stability of the Forced Response 155

5.6 Response Bounds 157

5.7 Frequency Response Methods 158

5.8 Stability of Feedback Control 161

5.9 Numerical Simulations in Matlab 163

Chapter Notes 165

References 166

Problems 167

6 Vibration Suppression 171

6.1 Introduction 171

6.2 Isolators and Absorbers 172

6.3 OptimizationMethods 175

6.4 Metastructures 179

6.5 Design Sensitivity and Redesign 181

6.6 Passive and Active Control 184

6.7 Controllability and Observability 188

6.8 Eigenstructure Assignment 193

6.9 Optimal Control 196

6.10 Observers (Estimators) 203

6.11 Realization 208

6.12 Reduced-Order Modeling 210

6.13 Modal Control in State Space 216

6.14 Modal Control in Physical Space 219

6.15 Robustness 224

6.16 Positive Position Feedback Control 226

6.17 Matlab Commands for Control Calculations 229

Chapter Notes 233

References 234

Problems 237

7 Distributed Parameter Models 241

7.1 Introduction 241

7.2 Equations of Motion 241

7.3 Vibration of Strings 247

7.4 Rods and Bars 252

7.5 Vibration of Beams 256

7.6 Coupled Effects 263

7.7 Membranes and Plates 267

7.8 Layered Materials 271

7.9 Damping Models 273

7.10 Modeling Piezoelectric Wafers 276

Chapter Notes 281

References 281

Problems 283

8 Formal Methods of Solutions 287

8.1 Introduction 287

8.2 Boundary Value Problems and Eigenfunctions 287

8.3 Modal Analysis of the Free Response 290

8.4 Modal Analysis in Damped Systems 292

8.5 Transform Methods 294

8.6 Green’s Functions 296

Chapter Notes 300

References 301

Problems 301

9 Operators and the Free Response 303

9.1 Introduction 303

9.2 Hilbert Spaces 304

9.3 Expansion Theorems 308

9.4 Linear Operators 309

9.5 Compact Operators 315

9.6 Theoretical Modal Analysis 317

9.7 Eigenvalue Estimates 318

9.8 Enclosure Theorems 321

Chapter Notes 324

References 324

Problems 325

10 Forced Response and Control 327

10.1 Introduction 327

10.2 Response by Modal Analysis 327

10.3 Modal Design Criteria 330

10.4 Combined Dynamical Systems 332

10.5 Passive Control and Design 336

10.6 Distributed Modal Control 338

10.7 Nonmodal Distributed Control 340

10.8 State Space Control Analysis 341

10.9 Vibration Suppression using Piezoelectric Materials 342

Chapter Notes 344

References 345

Problems 346

11 Approximations of Distributed Parameter Models 349

11.1 Introduction 349

11.2 Modal Truncation 349

11.3 Rayleigh-Ritz-Galerkin Approximations 351

11.4 Finite Element Method 354

11.5 Substructure Analysis 359

11.6 Truncation in the Presence of Control 361

11.7 Impedance Method of Truncation and Control 369

Chapter Notes 371

References 371

Problems 372

12 Vibration Measurement 375

12.1 Introduction 375

12.2 Measurement Hardware 376

12.3 Digital Signal Processing 379

12.4 Random Signal Analysis 383

12.5 Modal Data Extraction (Frequency Domain) 387

12.6 Modal Data Extraction (Time Domain) 390

12.7 Model Identification 395

12.8 Model Updating 397

12.9 Verification and Validation 398

Chapter Notes 400

References 401

Problems 402

A Comments on Units 405

B Supplementary Mathematics 409

Index 413