Introduction to Mechanical Vibrations

Preface xi About the Companion Website xv 1 The Transition from Dynamics to Vibrations 1 1.1 Bead on a Wire: The Nonlinear Equations of Motion 2 1.1.1 Formal Vector Approach using Newton’s Laws 3 1.1.2 Informal Vector Approach using Newton’s Laws 5 1.1.3 Lagrange’s Equations of Motion 6 1.1.3.1 The Bead on a Wire via Lagrange’s Equations 7 1.1.3.2 Generalized Coordinates 9 1.1.3.3 Generalized Forces 9 1.1.3.4 Dampers – Rayleigh’s Dissipation Function 11 1.2 Equilibrium Solutions 12 1.2.1 Equilibrium of a Simple Pendulum 12 1.2.2 Equilibrium of the Bead on the Wire 13 1.3 Linearization 14 1.3.1 Geometric Nonlinearities 14 1.3.1.1 Linear EOM for a Simple Pendulum 15 1.3.1.2 Linear EOM for the Bead on the Wire 17 1.3.2 Nonlinear Structural Elements 18 1.4 Summary 19 Exercises 19 2 Single Degree of Freedom Systems – Modeling 23 2.1 Modeling Single Degree of Freedom Systems 23 2.1.1 Deriving the Equation of Motion 24 2.1.2 Equations of Motion Ignoring Preloads 27 2.1.3 Finding Spring Deflections due to Body Rotations 29 Exercises 34 3 Single Degree of Freedom Systems – Free Vibrations 39 3.1 Undamped Free Vibrations 39 3.2 Response to Initial Conditions 41 3.3 Damped Free Vibrations 44 3.3.1 Standard Form for Second-Order Systems 46 3.3.2 Undamped 47 3.3.3 Underdamped 48 3.3.4 Critically Damped 50 3.3.5 Overdamped 51 3.4 Root Locus 52 Exercises 53 4 SDOF Systems – Forced Vibrations – Response to Initial Conditions 59 4.1 Time Response to a Harmonically Applied Force in Undamped Systems 59 4.1.1 Beating 61 4.1.2 Resonance 63 Exercises 65 5 SDOF Systems – Steady State Forced Vibrations 67 5.1 Undamped Steady State Response to a Harmonically Applied Force 67 5.2 Damped Steady State Response to a Harmonically Applied Force 70 5.3 Response to Harmonic Base Motion 73 5.4 Response to a Rotating Unbalance 77 5.5 Accelerometers 82 Exercises 85 6 Damping 89 6.1 Linear Viscous Damping 89 6.2 Coulomb or Dry Friction Damping 93 6.3 Logarithmic Decrement 96 Exercises 97 7 Systems with More than One Degree of Freedom 101 7.1 2DOF Undamped Free Vibrations – Modeling 101 7.2 2DOF Undamped Free Vibrations – Natural Frequencies 104 7.3 2DOF Undamped Free Vibrations – Mode Shapes 106 7.3.1 An Example 107 7.4 Mode Shape Descriptions 110 7.5 Response to Initial Conditions 112 7.6 2DOF Undamped Forced Vibrations 115 7.7 Vibration Absorbers 116 7.8 The Method of Normal Modes 118 7.9 The Cart and Pendulum Example 123 7.9.1 Modeling the System – Two Ways 124 7.9.1.1 Kinematics 124 7.9.1.2 Newton’s Laws 125 7.9.1.3 Lagrange’s Equation 127 7.10 Normal Modes Example 129 Exercises 132 8 Continuous Systems 137 8.1 The Equations of Motion for a Taut String 137 8.2 Natural Frequencies and Mode Shapes for a Taut String 139 8.3 Vibrations of Uniform Beams 142 Exercises 151 9 Finite Elements 153 9.1 Shape Functions 153 9.2 The Stiffness Matrix for an Elastic Rod 155 9.3 The Mass Matrix for an Elastic Rod 161 9.4 Using Multiple Elements 164 9.5 The Two-noded Beam Element 167 9.5.1 The Two-noded Beam Element – Stiffness Matrix 168 9.5.2 The Two-noded Beam Element – Mass Matrix 171 9.6 Two-noded Beam Element Vibrations Example 173 Exercises 177 10 The Inerter 181 10.1 Modeling the Inerter 181 10.2 The Inerter in the Equations of Motion 184 10.3 An Examination of the Effect of an Inerter on System Response 186 10.3.1 The Baseline Case – p = 0 187 10.3.2 The Case Where the Inerter Adds Mass Equal to the Block’s Mass – p = 1 188 10.3.3 The Case Where p is Very Large 188 10.4 The Inerter as a Vibration Absorber 190 Exercises 193 11 Analysis of Experimental Data 195 11.1 Typical Test Data 195 11.2 Transforming to the Frequency Domain – The CFT 197 11.3 Transforming to the Frequency Domain – The DFT 200 11.4 Transforming to the Frequency Domain – A Faster DFT 202 11.5 Transforming to the Frequency Domain – The FFT 203 11.6 Transforming to the Frequency Domain – An Example 204 11.7 Sampling and Aliasing 207 11.8 Leakage and Windowing 212 11.9 Decimating Data 216 11.10 Averaging FFTs 225 Exercises 228 12 Topics in Vibrations 231 12.1 What About the Mass of the Spring? 231 12.2 Flow-induced Vibrations 233 12.3 Self-Excited Oscillations of Railway Wheelsets 238 12.4 What is a Rigid Body Mode? 249 12.5 Why Static Deflection is Very Useful 251 Exercises 254 Appendix A: Least Squares Curve Fitting 257 Appendix B: Moments of Inertia 261 B.1 Parallel Axis Theorem for Moments of Inertia 262 B.2 Moments of Inertia for Commonly Encountered Bodies 263 Index 265