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THE SI SYSTEM OF UNITS1

CHAPTER 1 OSCILLATORY MOTION5

1.1 Harmonic Motion6

1.2 Periodic Motion9

1.3 Vibration Terminology11

CHAPTER 2 FREE VIBRATION16

2.1 Vibration Model16

2.2 Equation of Motion: Natural Frequency16

2.3 Energy Method20

2.4 Rayleigh Method: Effective Mass23

2.5 Principle of Virtual Work25

2.6 Viscously Damped Free Vibration27

2.7 Logarithmic Decrement31

2.8 Coulomb Damping35

CHAPTER 3 HARMONICALLY EXCITED VIBRATION49

3.1 Forced Harmonic Vibration49

3.2 Rotating Unbalance53

3.3 Rotor Unbalance56

3.4 Whirling of Rotating Shafts59

3.5 Support Motion63

3.6 Vibration Isolation65

3.7 Energy Dissipated by Damping67

3.8 Equivalent Viscous Damping70

3.9 Structural Damping72

3.10 Sharpness of Resonance74

3.11 Vibration-Measuring Instruments75

CHAPTER 4 TRANSIENT VIBRATION89

4.1 Impulse Excitation89

4.2 Arbitrary Excitation91

4.3 Laplace Transform Formulation94

4.4 Pulse Excitation and Rise Time97

4.5 Shock Response Spectrum100

4.6 Shock Isolation104

4.7 Finite Difference Numerical Computation105

4.8 Runge-Kutta Method112

CHAPTER 5 SYSTEMS WITH TWO OR MORE DEGREES OF FREEDOM126

5.1 The Normal Mode Analysis127

5.2 Initial Conditions131

5.3 Coordinate Coupling134

5.4 Forced Harmonic Vibration139

5.5 Finite Difference Method for Systems of Equations141

5.6 Vibration Absorber144

5.7 Centrifugal Pendulum Vibration Absorber145

5.8 Vibration Damper147

CHAPTER 6 PROPERTIES OF VIBRATING SYSTEMS163

6.1 Flexibility Influence Coefficients164

6.2 Reciprocity Theorem167

6.3 Stiffness Influence Coefficients172

6.4 Stiffness Matrix of Beam Elements176

6.5 Static Condensation for Pinned Joints176

6.6 Orthogonality of Eigenvectors177

6.7 Modal Matrix179

6.8 Decoupling Forced Vibration Equations181

6.9 Modal Damping in Forced Vibration182

6.10 Normal Mode Summation183

6.11 Equal Roots187

6.12 Unrestrained （Degenerate） Systems189

CHAPTER 7 LAGRANGE’S EQUATION199

7.1 Generalized Coordinates199

7.2 Virtual Work204

7.3 Lagrange’s Equation207

7.4 Kinetic Energy, Potential Energy,and Generalized Force in Terms of Generalized Coordinates q214

7.5 Assumed Mode Summation216

CHAPTER 8 COMPUTATIONAL METHODS227

8.1 Root Solving227

8.2 Eigenvectors by Gauss Elimination229

8.3 Matrix Iteration230

8.4 Convergence of the Iteration Procedure233

8.5 The Dynamic Matrix233

8.6 Transformation Coordinates （Standard Computer Form）234

8.7 Systems with Discrete Mass Matrix235

8.8 Cholesky Decomposition237

8.9 Jacobi Diagonalization242

8.10 QR Method for Eigenvalue and Eigenvector Calculation247

CHAPTER 9 VIBRATION OF CONTINUOUS SYSTEMS258

9.1 Vibrating String268

9.2 Longitudinal Vibration of Rods271

9.3 Torsional Vibration of Rods273

9.4 Vibration of Suspension Bridges276

9.5 Euler Equation for Beams281

9.6 System with Repeated Identical Sections286

CHAPTER 10 INTRODUCTION TO THE FINITE ELEMENT METHOD287

10.1 Element Stiffness and Mass287

10.2 Stiffness and Mass for the Beam Element292

10.3 Transformation of Coordinates（Global Coordinates）295

10.4 Element Stiffness and Element Mass in Global Coordinates297

10.5 Vibrations Involving Beam Elements302

10.6 Spring Constraints on Structure309

10.7 Generalized Force for Distributed Load311

10.8 Generalized Force Proportional to Displacement313

CHAPTER 11 MODE-SUMMATION PROCEDURESFOR CONTINUOUS SYSTEMS329

11.1 Mode-Summation Method329

11.2 Normal Modes of Constrained Structures335

11.3 Mode-Acceleration Method339

11.4 Component-Mode Synthesis341

CHAPTER 12 CLASSICAL METHODS351

12.1 Rayleigh Method351

12.2 Dunkerley’s Equation358

12.3 Rayleigh-Ritz Method363

12.4 Holzer Method366

12.5 Digital Computer Program for the Torsional System369

12.6 Myklestad’s Method for Beams371

12.7 Coupled Flexure-Torsion Vibration375

12.8 Transfer Matrices376

12.9 Systems with Damping378

12.10 Geared System380

12.11 Branched Systems381

12.12 Transfer Matrices for Beams383

CHAPTER 13 RANDOM VIBRATIONS395

13.1 Random Phenomena395

13.2 Time Averaging and Expected Value396

13.3 Frequency Response Function398

13.4 Probability Distribution401

13.5 Correlation407

13.6 Power Spectrum and Power Spectral Density411

13.7 Fourier Transforms417

13.8 FTs and Response424

CHAPTER 14 NONLINEAR VIBRATIONS436

14.1 Phase Plane436

14.2 Conservative Systems438

14.3 Stability of Equilibrium441

14.4 Method of Isoclines443

14.5 Perturbation Method445

14.6 Method of Iteration448

14.7 Self-Excited Oscillations451

14.8 Runge-Kutta Method453

APPENDICES462

A Specifications of Vibration Bounds462

B Introduction to Laplace Transformation464

C Determinants and Matirces469

D Normal Modes of Uniform Beams479

E Introduction to MATLAB？487

F Computer Programs492

G Convergence to Higher Modes501

ANSWERS TO SELECTED PROBLEMS506

INDEX519