Introduction to Mechanical Vibrations

DR. RONALD J. ANDERSON is a Professor in the Department of Mechanical and Materials Engineering, Queen's University at Kingston, Canada. He received his B.Sc.(Eng) from the University of Alberta in 1973, his M.Sc.(Eng) from Queen's University in 1974, and his Ph.D. from Queen's University in 1977. His doctoral research was in the field of road vehicle dynamics. From 1977 to 1979, he was a Defence Scientist with the Defence Research Establishment Atlantic where he was engaged in research on the dynamics of novel ships. From 1979 to 1981 he was Senior Dynamicist with the Urban Transportation Development Corporation where he worked on rail vehicle dynamics, particularly suspension design for steerable rail vehicles. He joined Queen's University in 1981 and, while conducting research into vehicle dynamics and multibody dynamics, has been teaching undergraduate courses on dynamics and vibrations and postgraduate courses on advanced dynamics and engineering analysis. Dr. Anderson has been the recipient of several departmental and faculty-wide teaching awards. He has also served the University in the academic administrative roles of Head of Department, Associate Dean (Research), and Dean of Graduate Studies.

• Preface xiAbout the Companion Website xv1 The Transition from Dynamics to Vibrations 11.1 Bead on a Wire: The Nonlinear Equations of Motion 21.1.1 Formal Vector Approach using Newton’s Laws 31.1.2 Informal Vector Approach using Newton’s Laws 51.1.3 Lagrange’s Equations of Motion 61.1.3.1 The Bead on a Wire via Lagrange’s Equations 71.1.3.2 Generalized Coordinates 91.1.3.3 Generalized Forces 91.1.3.4 Dampers – Rayleigh’s Dissipation Function 111.2 Equilibrium Solutions 121.2.1 Equilibrium of a Simple Pendulum 121.2.2 Equilibrium of the Bead on the Wire 131.3 Linearization 141.3.1 Geometric Nonlinearities 141.3.1.1 Linear EOM for a Simple Pendulum 151.3.1.2 Linear EOM for the Bead on the Wire 171.3.2 Nonlinear Structural Elements 181.4 Summary 19Exercises 192 Single Degree of Freedom Systems – Modeling 232.1 Modeling Single Degree of Freedom Systems 232.1.1 Deriving the Equation of Motion 242.1.2 Equations of Motion Ignoring Preloads 272.1.3 Finding Spring Deflections due to Body Rotations 29Exercises 343 Single Degree of Freedom Systems – Free Vibrations 393.1 Undamped Free Vibrations 393.2 Response to Initial Conditions 413.3 Damped Free Vibrations 443.3.1 Standard Form for Second-Order Systems 463.3.2 Undamped 473.3.3 Underdamped 483.3.4 Critically Damped 503.3.5 Overdamped 513.4 Root Locus 52Exercises 534 SDOF Systems – Forced Vibrations – Response to Initial Conditions 594.1 Time Response to a Harmonically Applied Force in Undamped Systems 594.1.1 Beating 614.1.2 Resonance 63Exercises 655 SDOF Systems – Steady State Forced Vibrations 675.1 Undamped Steady State Response to a Harmonically Applied Force 675.2 Damped Steady State Response to a Harmonically Applied Force 705.3 Response to Harmonic Base Motion 735.4 Response to a Rotating Unbalance 775.5 Accelerometers 82Exercises 856 Damping 896.1 Linear Viscous Damping 896.2 Coulomb or Dry Friction Damping 936.3 Logarithmic Decrement 96Exercises 977 Systems with More than One Degree of Freedom 1017.1 2DOF Undamped Free Vibrations – Modeling 1017.2 2DOF Undamped Free Vibrations – Natural Frequencies 1047.3 2DOF Undamped Free Vibrations – Mode Shapes 1067.3.1 An Example 1077.4 Mode Shape Descriptions 1107.5 Response to Initial Conditions 1127.6 2DOF Undamped Forced Vibrations 1157.7 Vibration Absorbers 1167.8 The Method of Normal Modes 1187.9 The Cart and Pendulum Example 1237.9.1 Modeling the System – Two Ways 1247.9.1.1 Kinematics 1247.9.1.2 Newton’s Laws 1257.9.1.3 Lagrange’s Equation 1277.10 Normal Modes Example 129Exercises 1328 Continuous Systems 1378.1 The Equations of Motion for a Taut String 1378.2 Natural Frequencies and Mode Shapes for a Taut String 1398.3 Vibrations of Uniform Beams 142Exercises 1519 Finite Elements 1539.1 Shape Functions 1539.2 The Stiffness Matrix for an Elastic Rod 1559.3 The Mass Matrix for an Elastic Rod 1619.4 Using Multiple Elements 1649.5 The Two-noded Beam Element 1679.5.1 The Two-noded Beam Element – Stiffness Matrix 1689.5.2 The Two-noded Beam Element – Mass Matrix 1719.6 Two-noded Beam Element Vibrations Example 173Exercises 17710 The Inerter 18110.1 Modeling the Inerter 18110.2 The Inerter in the Equations of Motion 18410.3 An Examination of the Effect of an Inerter on System Response 18610.3.1 The Baseline Case – p = 0 18710.3.2 The Case Where the Inerter Adds Mass Equal to the Block’s Mass – p = 1 18810.3.3 The Case Where p is Very Large 18810.4 The Inerter as a Vibration Absorber 190Exercises 19311 Analysis of Experimental Data 19511.1 Typical Test Data 19511.2 Transforming to the Frequency Domain – The CFT 19711.3 Transforming to the Frequency Domain – The DFT 20011.4 Transforming to the Frequency Domain – A Faster DFT 20211.5 Transforming to the Frequency Domain – The FFT 20311.6 Transforming to the Frequency Domain – An Example 20411.7 Sampling and Aliasing 20711.8 Leakage and Windowing 21211.9 Decimating Data 21611.10 Averaging FFTs 225Exercises 22812 Topics in Vibrations 23112.1 What About the Mass of the Spring? 23112.2 Flow-induced Vibrations 23312.3 Self-Excited Oscillations of Railway Wheelsets 23812.4 What is a Rigid Body Mode? 24912.5 Why Static Deflection is Very Useful 251Exercises 254Appendix A: Least Squares Curve Fitting 257Appendix B: Moments of Inertia 261B.1 Parallel Axis Theorem for Moments of Inertia 262B.2 Moments of Inertia for Commonly Encountered Bodies 263Index 265