Virtual Experiments in Mechanical Vibrations Structural Dynamics and Signal Processing

The first book of its kind to explain fundamental concepts in both vibrations and signal processing using MATLAB virtual experiments

Students and young engineers with a strong grounding in engineering theory often lack the practical skills and knowledge required to carry out experimental work in the laboratory. Fundamental and time-consuming errors can be avoided with the appropriate training and a solid understanding of basic concepts in vibrations and/or signal processing, which are critical to testing new designs.

Virtual Experiments in Mechanical Vibrations: Structural Dynamics and Signal Processing is designed for readers with limited knowledge of vibrations and signal processing. The intention is to help them relate vibration theory to measurements carried out in the laboratory. With a hands-on approach that emphasizes physics rather than mathematics, this practical resource explains fundamental concepts in vibrations and signal processing. It uses the concept of a virtual experiment together with MATLAB to show how the dynamic properties of vibration isolators can be determined, how vibration absorbers can be designed, and how they perform on distributed parameter structures.

Readers will find that this text:

• Allows the concepts of experimental work to be discussed and simulated in the classroom using a physics-based approach
• Presents computational virtual experiments using MATLAB examples to determine the dynamic behaviour of several common dynamic systems
• Explains the rationale of virtual experimentation and describes typical vibration testing setups
• Introduces the signal processing tools needed to determine the frequency response of a system from input and output data
• Includes access to a companion website containing MATLAB code

Virtual Experiments in Mechanical Vibrations: Structural Dynamics and Signal Processing is a must-have resource for researchers, mechanical engineers, and advanced undergraduate and graduate students who are new to the subjects of vibrations, signal processing, and vibration testing. It is also an invaluable tool for universities where the possibilities of doing experimental work are limited.

Michael J Brennan, Department of Mechanical Engineering, Universidade Estadual Paulista, Brazil
Michael Brennan has extensive experience in teaching, research and consulting in the area of vibrations and signal processing. He was a Professor at the Institute of Sound and Vibration Research at Southampton University until September 2010, when he moved to the Department of Mechanical Engineering, Universidade Estadual Paulista (UNESP). He has taught vibrations and signal processing at undergraduate and post-graduate levels, and has supervised about 30 Doctoral students and about 20 MSc students to successful completion. He has authored or co-authored approx. 140 journal papers and approx. 150 conference papers.

Preface xiii

List of Abbreviations xv

List of Symbols xvii

About the Companion Website xxi

1 Introduction 1

1.1 Introduction 1

1.2 Typical Laboratory-Based Vibration Tests 3

1.3 Relationship Between the Input and Output for a SISO System 5

1.4 A Virtual Vibration Test 6

1.5 Some Notes on the Book 7

References 7

2 Fundamentals of Vibration 9

2.1 Introduction 9

2.2 Basic Concepts – Mass, Stiffness, and Damping 9

2.3 Single Degree-of-Freedom System 11

2.4 Free Vibration 11

2.5 Impulse Response Function (IRF) 13

2.6 Determination of Damping from Free Vibration 17

2.7 Harmonic Excitation 19

2.8 Frequency Response Function (FRF) 22

2.9 Other Features of the Receptance FRF 28

2.10 Determination of Damping from an FRF 29

2.11 Reciprocal FRF 33

2.12 Summary 35

References 37

3 Fourier Analysis 39

3.1 Introduction 39

3.2 The Fourier Transform (FT) 39

3.2.1 Example – SDOF system 44

3.3 The Discrete Time Fourier Transform (DTFT) 45

3.4 The Discrete Fourier Transform (DFT) 48

3.5 Inverse Fourier Transforms 53

3.6 Summary 57

References 58

4 Numerical Computation of the FRFs and IRFs of an SDOF System 61

4.1 Introduction 61

4.2 Effect of Sampling on the FRFs 61

4.2.1 Receptance 62

4.2.2 Mobility 66

4.2.3 Accelerance 71

4.3 Effect of Data Truncation 77

4.4 Effects of Sampling on the IRFs Calculated Using the IDFT 85

4.5 Summary 91

References 92

5 Vibration Excitation 93

5.1 Introduction 93

5.2 Vibration Excitation Devices 93

5.2.1 Electrodynamic Shaker 93

5.2.2 Instrumented Impact Hammer 94

5.3 Vibration Excitation Signals 96

5.3.1 Excitation at a Single Frequency 98

5.3.2 Excitation Using a Random Signal 104

5.3.3 Excitation Using a Chirp or Swept Sine 110

5.3.4 Excitation Using a Half-Sine Pulse 113

5.4 Summary 117

References 117

6 Determination of the Vibration Response of a System 119

6.1 Introduction 119

6.2 Determination of the Vibration Response 119

6.2.1 Convolution in the Time Domain 119

6.2.2 Calculation of the Response via the Frequency Domain 120

6.2.3 Numerical Integration of the Equation of Motion 121

6.3 Calculation of the Vibration Response of an SDOF System 121

6.3.1 Impulsive Force 122

6.3.2 Half-sine Force Impulse 122

6.3.3 Chirp (Swept Sine) Force Input 123

6.3.4 Random Force Input 125

6.4 Summary 129

References 130

7 Frequency Response Function (FRF) Estimation 131

7.1 Introduction 131

7.2 Transient Excitation 131

7.2.1 H1 and H2 Estimators 134

7.2.2 Coherence Function 135

7.2.3 Examples 137

7.3 Random Excitation 144

7.4 Comparison of Excitation Methods and Effects of Shaker–Structure Interaction 151

7.5 Virtual Experiment – Vibration Isolation 157

7.5.1 The Physics of Vibration Isolation 157

7.5.2 Experimental Determination of the Stiffness and Damping of a Vibration Isolator 159

7.5.3 Experiment to Investigate the Trade-off Between Decreasing the Response at the Resonance Frequency and Improving Vibration Isolation 163

7.6 Summary 167

References 168

8 Multi-Degree-of-Freedom (MDOF) Systems: Dynamic Behaviour 169

8.1 Introduction 169

8.2 Lumped Parameter MDOF System 169

8.2.1 Example – 3DOF System 170

8.2.2 Free Vibration 175

8.2.3 Resonance and Anti-resonance Frequencies 177

8.2.4 Modal Decomposition 181

8.2.5 Impulse Response Function (IRF) 188

8.3 Continuous Systems 193

8.3.1 Rod 193

8.3.1.1 Natural Frequencies and Mode Shapes 195

8.3.1.2 Impulse Response Function (IRF) 197

8.3.2 Beam 201

8.3.2.1 Natural Frequencies and Mode Shapes 202

8.3.2.2 Impulse Response Function (IRF) 203

8.4 Summary 210

References 213

9 Multi-Degree-of-Freedom (MDOF) Systems: Virtual Experiments 215

9.1 Introduction 215

9.2 Two Degree-of-Freedom System: FRF Estimation 215

9.2.1 Determination of a Modal Model 220

9.3 Beam: FRF Estimation 223

9.3.1 Determination of a Modal Model 229

9.4 The Vibration Absorber as a Vibration Control Device 234

9.4.1 Theory 234

9.4.2 Effect of a Vibration Absorber on an SDOF System 235

9.4.3 Vibration Absorber Attached to an SDOF System – Virtual Experiment 237

9.4.4 Vibration Absorber Attached to a Cantilever Beam – Virtual Experiment 251

9.5 Summary 256

References 258

Appendix A Numerical Differentiation and Integration 259

A.1 Differentiation in the Time Domain 259

A.2 Integration in the Time Domain 260

A.3 Differentiation and Integration in the Frequency Domain 262

Reference 262

Appendix B The Hilbert Transform 263

References 265

Appendix C The Decibel: A Brief Description 267

Reference 268

Appendix D Numerical Integration of Equations of Motion 269

D.1 Euler’s Method 269

D.2 The Runge–Kutta Method 271

References 273

Appendix E The Delta Function 275

E.1 Properties of the Delta Function 276

E.2 Fourier Series Representation of a Train of Delta Functions 277

Reference 277

Appendix F Aliasing 279

References 285

Appendix G Convolution 287

G.1 Relationship Between Convolution and Multiplication 291

G.2 Circular Convolution 296

References 299

Appendix H Some Influential Scientists in Topics Related to This Book 301

Index 307

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.