Virtual Experiments in Mechanical Vibrations

Michael J. Brennan is a former professor in the Department of Mechanical Engineering, São Paulo State University, Brazil, and also in the Institute of Sound and Vibration Research, University of Southampton, UK. He has extensive experience in teaching, research, and consulting in vibrations and signal processing. He has authored or co-authored more than 225 journal papers and approximately 220 conference papers. Bin Tang is a professor in the School of Energy and Power Engineering, Dalian University of Technology, China. He is the author of more than 60 publications in national and international journals, including Journal of Sound and Vibration, Applied Mechanics Reviews, and Soft Robotics

• Preface xiiiList of Abbreviations xvList of Symbols xviiAbout the Companion Website xxi1 Introduction 11.1 Introduction 11.2 Typical Laboratory-Based Vibration Tests 31.3 Relationship Between the Input and Output for a SISO System 51.4 A Virtual Vibration Test 61.5 Some Notes on the Book 7References 72 Fundamentals of Vibration 92.1 Introduction 92.2 Basic Concepts – Mass, Stiffness, and Damping 92.3 Single Degree-of-Freedom System 112.4 Free Vibration 112.5 Impulse Response Function (IRF) 132.6 Determination of Damping from Free Vibration 172.7 Harmonic Excitation 192.8 Frequency Response Function (FRF) 222.9 Other Features of the Receptance FRF 282.10 Determination of Damping from an FRF 292.11 Reciprocal FRF 332.12 Summary 35References 373 Fourier Analysis 393.1 Introduction 393.2 The Fourier Transform (FT) 393.2.1 Example – SDOF system 443.3 The Discrete Time Fourier Transform (DTFT) 453.4 The Discrete Fourier Transform (DFT) 483.5 Inverse Fourier Transforms 533.6 Summary 57References 584 Numerical Computation of the FRFs and IRFs of an SDOF System 614.1 Introduction 614.2 Effect of Sampling on the FRFs 614.2.1 Receptance 624.2.2 Mobility 664.2.3 Accelerance 714.3 Effect of Data Truncation 774.4 Effects of Sampling on the IRFs Calculated Using the IDFT 854.5 Summary 91References 925 Vibration Excitation 935.1 Introduction 935.2 Vibration Excitation Devices 935.2.1 Electrodynamic Shaker 935.2.2 Instrumented Impact Hammer 945.3 Vibration Excitation Signals 965.3.1 Excitation at a Single Frequency 985.3.2 Excitation Using a Random Signal 1045.3.3 Excitation Using a Chirp or Swept Sine 1105.3.4 Excitation Using a Half-Sine Pulse 1135.4 Summary 117References 1176 Determination of the Vibration Response of a System 1196.1 Introduction 1196.2 Determination of the Vibration Response 1196.2.1 Convolution in the Time Domain 1196.2.2 Calculation of the Response via the Frequency Domain 1206.2.3 Numerical Integration of the Equation of Motion 1216.3 Calculation of the Vibration Response of an SDOF System 1216.3.1 Impulsive Force 1226.3.2 Half-sine Force Impulse 1226.3.3 Chirp (Swept Sine) Force Input 1236.3.4 Random Force Input 1256.4 Summary 129References 1307 Frequency Response Function (FRF) Estimation 1317.1 Introduction 1317.2 Transient Excitation 1317.2.1 H1 and H2 Estimators 1347.2.2 Coherence Function 1357.2.3 Examples 1377.3 Random Excitation 1447.4 Comparison of Excitation Methods and Effects of Shaker–Structure Interaction 1517.5 Virtual Experiment – Vibration Isolation 1577.5.1 The Physics of Vibration Isolation 1577.5.2 Experimental Determination of the Stiffness and Damping of a Vibration Isolator 1597.5.3 Experiment to Investigate the Trade-off Between Decreasing the Response at the Resonance Frequency and Improving Vibration Isolation 1637.6 Summary 167References 1688 Multi-Degree-of-Freedom (MDOF) Systems: Dynamic Behaviour 1698.1 Introduction 1698.2 Lumped Parameter MDOF System 1698.2.1 Example – 3DOF System 1708.2.2 Free Vibration 1758.2.3 Resonance and Anti-resonance Frequencies 1778.2.4 Modal Decomposition 1818.2.5 Impulse Response Function (IRF) 1888.3 Continuous Systems 1938.3.1 Rod 1938.3.1.1 Natural Frequencies and Mode Shapes 1958.3.1.2 Impulse Response Function (IRF) 1978.3.2 Beam 2018.3.2.1 Natural Frequencies and Mode Shapes 2028.3.2.2 Impulse Response Function (IRF) 2038.4 Summary 210References 2139 Multi-Degree-of-Freedom (MDOF) Systems: Virtual Experiments 2159.1 Introduction 2159.2 Two Degree-of-Freedom System: FRF Estimation 2159.2.1 Determination of a Modal Model 2209.3 Beam: FRF Estimation 2239.3.1 Determination of a Modal Model 2299.4 The Vibration Absorber as a Vibration Control Device 2349.4.1 Theory 2349.4.2 Effect of a Vibration Absorber on an SDOF System 2359.4.3 Vibration Absorber Attached to an SDOF System – Virtual Experiment 2379.4.4 Vibration Absorber Attached to a Cantilever Beam – Virtual Experiment 2519.5 Summary 256References 258Appendix A Numerical Differentiation and Integration 259A.1 Differentiation in the Time Domain 259A.2 Integration in the Time Domain 260A.3 Differentiation and Integration in the Frequency Domain 262Reference 262Appendix B The Hilbert Transform 263References 265Appendix C The Decibel: A Brief Description 267Reference 268Appendix D Numerical Integration of Equations of Motion 269D.1 Euler’s Method 269D.2 The Runge–Kutta Method 271References 273Appendix E The Delta Function 275E.1 Properties of the Delta Function 276E.2 Fourier Series Representation of a Train of Delta Functions 277Reference 277Appendix F Aliasing 279References 285Appendix G Convolution 287G.1 Relationship Between Convolution and Multiplication 291G.2 Circular Convolution 296References 299Appendix H Some Influential Scientists in Topics Related to This Book 301Index 307