Basic of Mechanical Vibration 機(jī)械振動(dòng)基礎(chǔ)

Contents

Chapter 1 Introduction 1

1.1 Background  1

1.2 Study of Vibration  3

1.3 Basic Concepts of Vibration  6

1.3.1 Degrees of freedom  7

1.3.2 Newton?s laws 8

1.3.3 Vibration classification  8

1.4 Organization of the Book  9

Questions  10

Chapter 2 Free Vibration of Single-Degree-Of-Freedom(SDOF) Systems 11

2.1 The Basic Mechanical Components  11

2.2 Free Vibration of UndampedSystems  12

2.2.1 Modelling of undamped SDOF systems  12

2.2.2 Simple harmonic motion  15

2.2.3 MATLAB examples 15

2.2.4 Solution for undamped SDOF systems 18

2.3 Rotary Systems  20

2.4 Springs Connected in Series or in Parallel  22

2.5 Modelling Using Energy Method  23

2.6 Viscously Damped SDOF Systems  29

2.6.1 Case 1: Overdamped motion (ζ> 1) 31

2.6.2 Case 2: Underdamped motion (ζ< 1)  33

2.6.3 Case 3: Critically damped motion (ζ=1)  34

2.7 Evaluating Damping Ratio from Measurements (Logarithmic Decrement) 36

2.8 Summary: the Effects of Damping on an Unforced Mass-spring System 39

2.9 MATLAB Examples for Free Vibration of SDOF Systems  39

2.9.1 Free vibration for undamped SDOF systems  39

2.9.2 Free vibration for underdamped SDOF systems  40

2.9.3 Free vibration for critical damped SDOF systems  42

2.9.4 Free vibration for overdamped SDOF systems  43

2.9.5 The GUI program for free vibration of SDOF system 44

Questions  45

Chapter 3 Harmonic Excitation of SDOF Systems 50

3.1 Harmonic Excitation  50

3.2 Complex Analysis  53

3.3 Undamped SDOF System with Harmonic Excitation 54

3.3.1 Excitation frequency ≠ natural frequency  54

3.3.2 Excitation frequency=natural frequency (resonant condition) 56

3.3.3 Response ratio for undamped SDOF system  57

3.4 Damped SDOF System with Harmonic Excitation  59

3.4.1 Response for damped SDOF system with harmonic excitation  59

3.4.2 Dynamic magnification factor for damped SDOF system  60

3.4.3 Response ratio for β=1  63

3.5 Harmonic Base Excitation  65

3.5.1 Relative motion  66

3.5.2 Absolute motion  67

3.6 Transmissibility of Vibration 73

3.6.1 Motion transmissibility  74

3.6.2 Force transmissibility  75

3.7 Rotating Unbalance  77

3.8 MATLAB Examples for Forced SDOF System  81

3.8.1 Harmonic excitation of undamped SDOF systems  81

3.8.2 Rotating unbalance vibration of SDOF systems  83

Questions  85

Chapter 4 Vibration of SDOF Systems under General Excitation 92

4.1 The Impulse Response  92

4.2 The Principle of Superposition  95

4.3 Response of SDOF Systems under a General Periodic Force 97

4.4 System?s Response to General Excitation by Convolution  104

4.5 System?s Response to General Excitation by the Laplace Transform 108

4.6 The Transfer Function  112

4.7 Composite Function Excitation  113

Questions  116

Chapter 5 Vibration of Multiple-Degree-Of-Freedom (MDOF) Systems 119

5.1 Free Vibration of Structures with Two-Degree-of-Freedom  119

5.1.1 Equations of motion for free vibration of two-degree-of-freedom 120

5.1.2 Free vibration analysis  121

5.1.3 Free vibration responses  124

5.2 Modelling of a System with n Degrees of Freedom  125

5.2.1 Governing equations  125

5.2.2 Coordinate coupling  130

5.3 Influence Coefficient Method  133

5.3.1 The stiffness influence coefficients  134

5.3.2 Flexibility influence coefficients  136

5.4 The Lagrange?sEquation  137

5.4.1 Derivation of Lagrange?s equations in Cartesian coordinates  138

5.4.2 Extension to general coordinate systems  140

5.4.3 Application of Lagrange?sequation  140

5.5 Solving the Equations of Motion of Undamped Systems UsingMATLAB  144

5.6 Vibration of Undamped MDOF System  148

5.6.1 Free response of undamped MDOF systems  148

5.6.2 Harmonic excitation of undampedsystems  156

5.7 Vibration of Underdamped MDOF Systems  158

5.7.1 Free vibration of underdamped MDOF systems  158

5.7.2 Steady-state forced vibration response for MDOF systems 162

5.8 The Dynamic Vibration Absorber  166

Questions  169

Chapter 6 Vibration of Continuous Systems  175

6.1 Transverse Vibration of Cables  175

6.2 Transverse Vibration of Beams  181

6.2.1 The governing equation of motion for beams  181

6.2.2 The natural frequencies and mode shapes for beams  183

6.2.3 MATLAB examples  186

6.3 The Structural Modes for Plates  190

Questions  193

Appendix A Mathematical Background  195

A.1 Mathematical Relations  195

A.2 Ordinary Differential Equations (ODE) 195

Appendix B Basic of MATLAB 199

B.1 How to Read this MATLAB Tutorial  199

B.2 Making Matrix  200

B.2.1 The colon operator  201

B.2.2 Matrix manipulations  202

B.2.3 The end keyword  203

B.2.4 Transposing a matrix  203

B.2.5 Element-wise operations  204

B.3 Functions  205

B.4 Graphics and Plotting  205

B.5 Programming in MATLAB  208

B.5.1 The M-files 208

B.5.2 Repeating with “for” loops  208

B.5.3 “if” statements  209

B.5.4 Writing function subroutines  210

B.6 Saving and Loading  210

B.7 The Help Menu  211

B.8 Set the Display Format for Output 212

B.9 Closing Remarks and References 212

Appendix C Properties of Laplace Transform and Laplace Transform Pairs  214

Appendix D Technical Terms  216

References  228