Analysis of limit cycles and stochastic responses of a real-power vibration isolation system under delayed feedback control

In this paper, the dynamical properties of a real-power vibration isolation system with delayed feedback control subjected to deterministic and stochastic excitations are considered. According to the free vibration analysis, it is found that a large number of limit cycles may be existed for certain time delay and feedback gain. Then, the relationship of amplitude and frequency is derived for the undamped system. For the system with harmonic excitation, multi-valued phenomena are observed due to the existence of the limit cycles. In this respect, with the change of time delay, in every period the response is similar to time delay island, and the number of islands is different under different excitation frequency. Additionally, for analyzing the complex dynamic properties, the vibration isolation system with Gauss white noise excitation is explored by the largest Lyapunov exponent and the stationary probability density. The symmetrical period-doubling bifurcation phenomenon is found and verified. Finally, by using Monte Carlo simulation, the stationary probability density is explored from original system. The change of time delays can induce the occurrence of stochastic bifurcation and the response from two peaks becomes triple peaks.