This book expounds the theory of non-linear vibrations, a topic of great interest at present because of its many applications to important fields in physics and engineering. After introducing chapters giving the basic techniques for the study of non-linear systems the authors develop in detail the theory of selected topics encountered in their own work, presenting original material, approaches and results of analysis, and providing illustrations of useful applications.
Author(s): G. Schmidt, A. Tondl
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 421
Tags: Механика;Теория колебаний;
Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Preface......Page 6
Contents ......Page 8
Introduction ......Page 10
1.1. Non-linear vibration, non-linear characteristics and basic definitions ......Page 13
1.2. Some examples of excited and self-excited systems ......Page 18
1.3. Basic features of excited systems ......Page 27
1.4. Basic features of self-excited systems ......Page 30
1.5. Stability ......Page 32
2.1. The harmonic balance method ......Page 36
2.2. The Van der Pol method ......Page 39
2.3. The integral equation method ......Page 42
2.4. Stability conditions ......Page 45
2.5. The averaging method ......Page 47
3.1. Characteristic features of auxiliary curves, particularly the backbone curves and the limit envelopes ......Page 49
3.2. Use of auxiliary curves for preliminary analysis ......Page 58
3.3. Use of auxiliary curves for preliminary analysis of parametrically excited systems ......Page 60
3.4. Auxiliary curves of higher-order systems ......Page 65
3.5. Use of auxiliary curves in analysis of systems with several degrees of free-dom ......Page 72
3.6. Identification of damping ......Page 75
4.1. Fundamental considerations ......Page 78
4.2. Practical solution of the phase portraits ......Page 91
4.3. Examples of systems of group (b) ......Page 92
4.4. Examples of systems of group (c) ......Page 97
4.5. An example of a system of group (a) ......Page 100
5.1. Amplitude equations ......Page 113
5.2. Resonance curves, extremal amplitudes, and stability ......Page 118
5.3. Non-linear damping ......Page 125
5.4. Forced and self-excited vibrations ......Page 130
5.5. Parametric and self-excited vibrations ......Page 139
5.6. Forced, parametric and self-excited vibrations ......Page 142
5.7. Non-linear parametric excitation. Harmonic resonance ......Page 147
5.8. Non-linear parametric excitation. Subharmonic resonance ......Page 153
6.1. Single and combination resonances ......Page 155
6.2. Stability of vibrations with many degrees of freedom ......Page 163
6.3. Vibrations in one-stage gear drives ......Page 167
6.4. Torsional gear resonance 170 ......Page 0
6.5. Combination gear resonances ......Page 174
6.6. Internal resonances in gear drives ......Page 179
6.7. Torsional vibrations in N-stage gear drives ......Page 185
6.8. Strong coupling between gear stages ......Page 194
6.9. Application of computer algebra ......Page 197
7.1. Fundamental considerations ......Page 200
7.2. Methods of investigating stability in the large for disturbances in the initial conditions ......Page 202
7.3. Investigation of stability in the large for not-fully determined disturbances ......Page 207
7.4. Examples of investigations concerning stability in the large for disturbances in the initial conditions ......Page 213
7.5. Investigation of stability in the large for other types of disturbances ......Page 233
7.6. Other applications of the results ......Page 236
7.7. Examples ......Page 237
8.1. Duffing system with a softening characteristic ......Page 248
8.2. Some special cases of kinematic (inertial) excitation ......Page 257
8.3. Parametric vibration of a mine cage ......Page 271
9.1. Basic considerations and methods of solution ......Page 279
9.2. Two-mass systems with two degrees of freedom ......Page 283
9.3. Chain systems with several masses ......Page 303
9.4. Example of a rotor system ......Page 314
10.1. Application of the quasi-static method ......Page 324
10.2. Application of the integral equation method. Probability densities ......Page 327
11.1. The amplitude probability density ......Page 337
11.2. Statistical properties of the vibrations ......Page 344
11.3. Non-stationary probability density, transition probability density and two-dimensional probability density ......Page 354
12.1. Basic properties ......Page 363
12.2. Internal resonance ......Page 371
12.3. Narrow-band random excitation ......Page 376
12.4. Broad-band random excitation ......Page 381
12.5. Fokker Planck Kolmogorov equation ......Page 385
12.6. Behaviour of the solution ......Page 392
12.7. Application of computer algebra ......Page 401
Appendix ......Page 402
Bibliography ......Page 406
Index ......Page 416
