Forced Nonlinear Oscillations of an Autoparametric System—Part 2: Chaotic Responses

Abstract

Chaotic oscillations arising in forced oscillations of a two degree-of-freedom autoparametric system are studied. Statistical analysis of the numerically integrated nonperiodic responses is shown to be a meaningful description of the mean square values and the frequency contents of the responses. Some qualitative experimental results are presented to substantiate the necessity of performing the statistical analysis of the responses even though the system and the input are deterministic.

Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

“…Miles [10] showed that these amplitude and phase modulated motions can themselves bifurcate to chaotic amplitude modulations. No such complex solutions of the amplitude or the averaged equations were found for the subharmonic resonance case investigated by Hatwal et al [6,7].…”

“…The autoparametric system for the present study is shown in Figure 1, and its nondimensional equations of motion are given as follows [6,7]: …”

“…Hatwal etal. [6,7] used the harmonic balance method and direct numerical integration to study a variant of this system at moderately higher levels of excitations, and observed that over some forcing frequencies and amplitudes, the system response was an amplitude and phase modulated harmonic motion. For even higher excitation levels, the response was found to be chaotic.…”

Harmonic motions of a weakly forced autoparametric vibrating system

“…As can be seen from [10,14], the values of ζ 1 , ζ 2 , , 0 are considered to be sufficiently small. Therefore, we introduce a small parameter ϵ(0 < ≪ 1) and make transform , , , , , ,…”

“…The early results of system (1.1) with 0 0  F were traced to [10,14] by Hatwal, Mallik and Ghosh based on the harmonic balance method and direct numerical integration. At moderately higher levels of excitations, they observed that over some forcing frequencies and amplitudes, the system response was amplitude and phase modulated harmonic motion, and for even higher excitation levels, the response was found to be chaotic.…”