كتاب Non-Linear Vibrations

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Non-Linear Vibrations
G. Schmidt, A. Tondl

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Contents
Introduction . 9
1. Basic properties and definitions 12
1.1. Non-linear vibration, non-linear characteristics and basic definitions . 12
1.2. Some examples of excited and self-excited systems . 17
1.3. Basic features of excited systems 26
1.4. Basic features of self-excited systems . 29
1.5. Stability 31
2. Methods of solution 35
2.1. The harmonic balance method . 35
2.2. The Van der Pol method . 38
2.3. The integral equation method . 41
2.4. Stability conditions . 44
2.5. The averaging method . 46
3. Auxiliary curves for analysis of non-linear systems . 48
3.1. Characteristic features of auxiliary curves, particularly the backbone curves
and the limit envelopes . 48
3.2. Use of auxiliary curves for preliminary analysis . 57
3.3. Use of auxiliary curves for preliminary analysis of parametrically excited
systems 59
3.4. Auxiliary curves of higher-order systems 64
3.5. Use of auxiliary curves in analysis of systems with several degrees of free- dom . 71
3.6. Identification of damping . 74
4. Analysis in the phase plane 77
4.1. Fundamental considerations 77
4.2. Practical solution of the phase portraits 90
4.3. Examples of systems of group (b) 91
4.4. Examples of systems of group (c) 96
4.5. An example of a system of group (a) 99
5. Forced, parametric and self-excited vibrations 112
5.1. Amplitude equations . 112
5.2. Resonance curves, extremal amplitudes, and stability 117
5.3. Non-linear damping 124
5.4. Forced and self-excited vibrations . 129
5.5. Parametric and self-excited vibrations 138
5.6. Forced, parametric and self-excited vibrations . 141
5.7. Non-linear parametric excitation. Harmonic resonance . 146
5.8. Non-linear parametric excitation. Subharmonic resonance . 1528 Contents
6. Vibrations of systems with many degrees of freedom 154
6.1. Single and combination resonances . 154
6.2. Stability of vibrations with many degrees of freedom . 162
6.3. Vibrations in one-stage gear drives . 166
6.4. Torsional gear resonance 1.70
6.5. Combination gear resonances 173
6.6. Internal resonances in gear drives 178
6.7. Torsional vibrations in N-stage gear drives 184
6.8. Strong coupling between gear stages . 193
6.9. Application of computer algebra . 196
7. Investigation of stability in the large 199
7.1. Fundamental considerations 199
7.2. Methods of investigating stability in the large for disturbances in the initial
conditions 201
7.3. Investigation of stability in the large for not-fully determined disturbances . 206
7.4. Examples of investigations concerning stability in the large for disturbances
in the initial conditions . 212
7.5. Investigation of stability in the large for other types of disturbances . 232
7.6. Other applications of the results . 235
7.7. Examples . 236
8. Analysis of some excited systems . 247
8.1. Duffing system with a softening characteristic . 247
8.2. Some special cases of kinematic (inertial) excitation 256
8.3. Parametric vibration of a mine cage . 270
9. Quenching of self-excited vibration . 278
9.1. Basic considerations and methods of solution 278
9.2. Two-mass systems with two degrees of freedom 282
9.3. Chain systems with several masses . 302
9.4. Example of a rotor system 313
10. Vibration systems with narrow-band random excitation 323
10.1. Application of the quasi-static method . 323
10.2. Application of the integral equation method. Probability densities . 326
11. Vibration systems with broad-band random excitation . 336
11.1. The amplitude probability density 336
11.2. Statistical properties of the vibrations 343
11.3. Non-stationary probability density, transition probability density and twodimensional probability density . 353
12. Systems with autoparametric coupling 362
12.1. Basic properties . 362
12.2. Internal resonance . 370
12.3. Narrow-band random excitation 375
12.4. Broad-band random excitation 380
12.5. Fokker Planck Kolmogorov equation . 384
12.6. Behaviour of the solution . 391
12.7. Application of computer algebra . 400
Appendix . 401
Bibliography 405
Index

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