CONTENTS Preface vii The SI system of Units xi 1 Oscillatory Motion 1 1.1 Harmonic Motion 2 1.2 Periodic Motion 4 1.3 Vibration Terminology 7 2 Free Vibration of Single Degree-of-Freedom Systems 12 2.1 Vibration Model 12 2.2 Equations of Motion: Natural Frequency 12 2.3 Viscously Damped Free Vibration 16 2.4 Logarithmic Decrement 20 2.5 Coulomb Damping 23 3 Harmonically Forced Vibration of Single Degree-of-Freedom Systems 34 3.1Forced Harmonic Vibration 34 3.2 Vibration Isolation 38 3.3 Rotating Unbalance 40 3.4 Rotor Unbalance 43 3.5 Whirling of Rotating Shafts 46 3.6 Support Motion 49 3.7 Energy Dissipated by Damping 51 3.8 Equivalent Viscous Damping 54 3.9 Structural Damping 55 3.10 Sharpness or Resonance 57 3.11 Vibration-Measuring Instruments 58 4 Transient Vibration of Single Degree-of-Freedom Systems71 4.1 Impulse Excitation 71 4.2 Arbitrary Excitation 73 4.3 Laplace Transform Formulation 76 4.4 Pulse Excitation and Rise Time 79 4.5 Shock Response Spectrum 81 4.6 Shock Isolation 85 4.7 Finite Difference Numerical Computation 86 4.8 Runge-Kutta Method 92 5 Multi-Degree-of-Freedom System Vibrations 106 5.1 The Normal Mode Analysis 107 5.2 Initial Conditions 110 5.3 Coordinate Coupling 114 5.4 Forced Harmonic Vibration 118 5.5 Orthogonality of Eigenvectors 120 5.6 Modal Matrix P 121 5.7 Decoupling Forced Vibration Equations 123 5.8 Modal Damping in Forced Vibration 125 5.9 Normal Mode Summation 126 5.10 Equal Roots 129 5.11 Unrestrained (Degenerate) Systems 131 5.12 Finite Difference Method for Systems of Equations 133 5.13 Vibration Absorber 135 5.14 Centrifugal Pendulum Vibration Absorber 137 5.15 Vibration Damper 138 6 Energy-Based Approaches 158 6.1 Energy Method 158 6.2 Rayleigh Method: Effective Mass 161 6.3 Generalized Coordinates 162 6.4 Virtual Work 168 6.5 Lagrange's Equation 172 6.6 Kinetic Energy, Potential Energy, and Generalized Force in Terms of Generalized Coordinate q 179 6.7 Assumed Mode Summation 180 7 Computational Methods 193 7.1 Root Solving 193 7.2 Eigenvectors by Gauss Elimination 195 7.3 Matrix Iteration 196 7.4 Convergence of the Iteration Procedure 198 7.5 The Dynamic Matrix 199 7.6 Transformation of Coordinates (Standard Computer Form) 199 7.7 Systems with Discrete Mass Matrix 200 7.8 Cholesky Decomposition 202 7.9 Jacobi Diagonalization 208 7.10 QR Method for Eigenvalue and Eigenvector Calculation 213 8 Vibration of Continuous Systems 222 8.1 Vibrating String 222 8.2 Longitudinal Vibration of Rods 225 8.3 Torsional Vibration of Rods227 8.4 Suspension Bridge as Continuous System 230 8.5 Euler Equation for Beams 235 8.6 System with Repeated Identical Sections 239 9 Mode-Summation Procedures for Continuous Systems (Beams) 250 9.1 Mode-Summation Method 250 9.2 Normal Modes of Constrained Structures 256 9.3 Mode-Acceleration Method 261 9.4 Component-Mode Synthesis 262 CHAPTER 10 Classical Methods 272 10.1 Rayleigh Method 272 10.2 Dunkerley's Equation 279 10.3 Rayleigh-Ritz Method 283 10.4 Holzer Method 286 10.5 Digital Computer Program For The Torsional System 289 10.6 Myklestad's Method for Beams 291 10.7 Coupled Flexure-Torsion Vibration 294 10.8 Transfer Matrices 295 10.9 Systems with Damping 296 10.10 Geared System 299 10.11 Branched Systems 300 10.12 Transfer Matrices for Beams302 11 Introduction to the Finite Element Method 313 11.1 Flexibility Influence Coefficients 313 11.2 Reciprocity Theorem 317 11.3 Stiffness Influence Coefficients 318 11.4 Stiffness Matrix of Beam Elements 322 11.5 Static Condensation for Pinned Joints 325 11.6 Axial Finite Element Stiffness and Mass Matrices 327 11.7 Stiffness and Mass Matrices for the Beam Finite Element 331 11.8 Transformation of Coordinates (Global Coordinates) 335 11.9 Element Stiffness and Element Mass in Global Coordinates 337 11.10 Vibrations Involving Beam Elements 342 11.11 Spring Constraints on Structure349 11.12 Generalized Force for Distributed Load 352 11.13 Generalized Force Proportional to Displacement 353 12 Random Vibrations 371 12.1 Random Phenomena 371 12.2 Time Averaging and Expected Value 372 12.3 Frequency Response Function 374 12.4 Probability Distribution 377 12.5 Correlation 382 12.6 Power Spectrum and Power Spectral Density 386 12.7 Fourier Transforms 391 12.8 FTs and Response 398 13 Nonlinear Vibrations 410 13.1 Phase Plane 410 13.2 Conservative Systems 412 13.3 Stability of Equilibrium 414 13.4 Method of Isoclines 416 13.5 Perturbation Method 418 13.6 Method of Iteration 421 13.7 Self-Excited Oscillations 424 13.8 Runge-Kutta Method 426 Appendices A Specifications of Vibration Bounds 434 B Introduction to Laplace Transformation 436 C Determinants and Matrices 441 D Normal Modes of Uniform Beams 451 E Introduction to MATLAB 459 F Computer Programs464 G Convergence to Higher Modes 473 Answers to Selected Problems 478 Index 491