By G. Schmidt

This e-book expounds the speculation of non-linear vibrations, an issue of serious curiosity at the moment due to its many functions to special fields in physics and engineering. After introducing chapters giving the fundamental thoughts for the research of non-linear platforms the authors advance intimately the speculation of chosen issues encountered of their personal paintings, featuring unique fabric, methods and result of research, and offering illustrations of worthy purposes.

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In this book, strange attractor problems are not treated ; attractors are always understood in the sense of simple attractors. 2. 1. The harmonic balance method This method is based on a very simple idea and that is why it may seem less rigorous mathematically and not exact enough in approach. However, when combined with experience and sound engineering judgment on the part of the analyst it can be turned into a very efficient tool for solving non-linear problems, sometimes even excelling procedures which are grounded more thoroughly in mathematics.

The term refers to a situation in which the component of the excited vibration fully synchronizes with the corresponding component of the self-excited vibration, and the frequency is controlled by the self-excitation frequency. Details are given in Chapter 5. 5. 5. 5, 1) satisfies the condition Iz(Z)I, Iz (z)I < v for allz>zoif Iz(zo)I, Iz'(zo)I < 17 holds with a positive n = r7(e), in other words, if the solutions keep a given maximum distance for all time when they have a certain positive maximum distance for an initial time z = zo.

The functions Z1, Z2 are periodic with period 27r. 5, 17) are called characteristic exponents, they, and the functions Z1, Z2, are in general non-real. 5,18) -s

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