The forced chaotic and irregular oscillations of the nonlinear two degrees of freedom (2DOF) system

摘要

The behavior of a 2DOF nonlinear mechanical system excited by a pure harmonic force is studied by the first approximation (averaging and asymptotic) method and by numerical simulations. The response curves, instability domains, displacement versus time dependencies and phase plane portraits are determined. By means of the first approximation solution we find three instability domains, where jumps occur, or in which the beats, irregular and chaotic motions emerge. The numerical simulations confirm these properties, but show that there exist several new bifurcations and instability domains, in which the response on the pure harmonic excitation is chaotic. The response curve of the low damped system has in the first resonance the multifold solution. The corresponding oscillations depend on the history, i.e. on the way and the speed of the very slow exciting frequency variation, at which the system comes into the current state. The responses at increasing or decreasing frequencies differ, the hysteresis loops exist.