Introduction to Mechanical Vibrations

An in-depth introduction to the foundations of vibrations for students of mechanical engineering

For students pursuing their education in Mechanical Engineering, An Introduction to Mechanical Vibrations is a definitive resource. The text extensively covers foundational knowledge in the field and uses it to lead up to and include: finite elements, the inerter, Discrete Fourier Transforms, flow-induced vibrations, and self-excited oscillations in rail vehicles.

The text aims to accomplish two things in a single, introductory, semester-length, course in vibrations. The primary goal is to present the basics of vibrations in a manner that promotes understanding and interest while building a foundation of knowledge in the field. The secondary goal is to give students a good understanding of two topics that are ubiquitous in today's engineering workplace - finite element analysis (FEA) and Discrete Fourier Transforms (the DFT- most often seen in the form of the Fast Fourier Transform or FFT). FEA and FFT software tools are readily available to both students and practicing engineers and they need to be used with understanding and a degree of caution. While these two subjects fit nicely into vibrations, this book presents them in a way that emphasizes understanding of the underlying principles so that students are aware of both the power and the limitations of the methods.

In addition to covering all the topics that make up an introductory knowledge of vibrations, the book includes:

?          End of chapter exercises to help students review key topics and definitions

?          Access to sample data files, software, and animations via a dedicated website

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

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.