Chaotic dynamics of simple vibrational systems

Abstract

This study discusses the development of a technique for analysis of the dynamical regimes of complex mechanical systems consisting of a rotor motor coupled to a system with multi-degrees-of-freedom. To understand the possible qualitatively different dynamical regimes in such systems, a simple mechanical system is considered of the "rotator-oscillator" type with a finite power source. This system has four degrees-of-freedom and is defined in four-dimensional cylindrical phase space with 12 parameters. Near the main resonance the original system is reduced to the Lorenz system with four parameters defined in a three-dimensional Cartesian phase space. This is done with the help of a special change of variables, parameters, and employing an averaging method. Studying the latter system, the existence of one of the chaotic attractors, namely of Lorenz attractor is established. Also established is the Feigenbaum attractor and the alternation. Chaotic limit sets define chaotic behavior of the instantaneous frequency of rotation of the asynchronous motor. The Poincare mappings are presented to show the correspondence of the original 4 dof and averaged 3 dof systems. The qualitative rotational characteristics for different values of the system parameters are obtained. In particular, the system can possess normal Sommerfeld effect, doubled Sommerfeld effect and a so-called scattering of the torque curve. The scattering of the torque curve (which is a known effect in micro-electronics) is likely to be a new effect in mechanics. In contrast to the Sommerfeld effect, when frequency or amplitude jumps occur instantaneously (once the unstable point of the characteristic is reached), the jump to a next stable point may take a certain time, even infinite one. Such chaotic mistuning of the motor frequency would result in random vibrations leading to system wear and damage.