Contents

Preface viii

1 Introduction To Vibration and the Free Response 1

1.1 Introduction to Free Vibration 2

1.2 Harmonic Motion 13

1.3 Viscous Damping 21

1.4 Modeling and Energy Methods 31

1.5 Stiffness 46

1.6 Measurement 58

1.7 Design Considerations 63

1.8 Stability 68

1.9 Numerical Simulation of the Time Response 72

1.10 Coulomb Friction and the Pendulum 81

Problems 95

MATLAB Engineering Vibration Toolbox 115

Toolbox Problems 116

2 Response To Harmonic Excitation 117

2.1 Harmonic Excitation of Undamped Systems 118

2.2 Harmonic Excitation of Damped Systems 130

2.3 Alternative Representations 144

2.4 Base Excitation 151

2.5 Rotating Unbalance 160

2.6 Measurement Devices 1662.7 Other Forms of Damping 170

2.8 Numerical Simulation and Design 180

2.9 Nonlinear Response Properties 188

Problems 197

MATLAB Engineering Vibration Toolbox 214

Toolbox Problems 214

3 General Forced Response 216

3.1 Impulse Response Function 217

3.2 Response to an Arbitrary Input 226

3.3 Response to an Arbitrary Periodic Input 235

3.4 Transform Methods 242

3.5 Response to Random Inputs 247

3.6 Shock Spectrum 255

3.7 Measurement via Transfer Functions 260

3.8 Stability 262

3.9 Numerical Simulation of the Response 267

3.10 Nonlinear Response Properties 279

Problems 287

MATLAB Engineering Vibration Toolbox 301

Toolbox Problems 301

4 Multiple-Degree-of-Freedom Systems 303

4.1 Two-Degree-of-Freedom Model (Undamped) 304

4.2 Eigenvalues and Natural Frequencies 317

4.3 Modal Analysis 331

4.4 More Than Two Degrees of Freedom 339

4.5 Systems with Viscous Damping 355

4.6 Modal Analysis of the Forced Response 361

4.7 Lagrange’s Equations 368

4.8 Examples 376

4.9 Computational Eigenvalue Problems for Vibration 388

4.10 Numerical Simulation of the Time Response 406

Problems 414

MATLAB Engineering Vibration Toolbox 432

Toolbox Problems 432

5 Design for Vibration Suppression 433

5.1 Acceptable Levels of Vibration 434

5.2 Vibration Isolation 440

5.3 Vibration Absorbers 453

5.4 Damping in Vibration Absorption 461

5.5 Optimization 469

5.6 Viscoelastic Damping Treatments 477

5.7 Critical Speeds of Rotating Disks 483

Problems 489

MATLAB Engineering Vibration Toolbox 499

Toolbox Problems 499

6 Distributed-Parameter Systems 500

6.1 Vibration of a String or Cable 502

6.2 Modes and Natural Frequencies 506

6.3 Vibration of Rods and Bars 517

6.4 Torsional Vibration 523

6.5 Bending Vibration of a Beam 530

6.6 Vibration of Membranes and Plates 542

6.7 Models of Damping 548

6.8 Modal Analysis of the Forced Response 554

Problems 564

MATLAB Engineering Vibration Toolbox 570

Toolbox Problems 570

7 Vibration Testing and Experimental Modal Analysis 571

7.1 Measurement Hardware 573

7.2 Digital Signal Processing 577

7.3 Random Signal Analysis in Testing 582

7.4 Modal Data Extraction 586

7.5 Modal Parameters by Circle Fitting 589

7.6 Mode Shape Measurement 594

7.7 Vibration Testing for Endurance and Diagnostics 604

7.8 Operational Deflection Shape Measurement 607

Problems 609

MATLAB Engineering Vibration Toolbox 613

Toolbox Problems 614

8 Finite Element Method 615

8.1 Example: The Bar 617

8.2 Three-Element Bar 623

8.3 Beam Elements 628

8.4 Lumped-Mass Matrices 636

8.5 Trusses 639

8.6 Model Reduction 644

Problems 647

MATL