On the analysis of the tristable vibration isolation system with delayed feedback control under parametric excitation

In this paper, the delayed feedback control of a tristable vibration isolation system (TVIS) under the stochastic parametric excitation is investigated. Firstly, theoretical solutions of the frequency response equation that governs the dynamical performance of the TVIS are derived. Meanwhile, the stability conditions are obtained. The controlled performance and response properties induced by the time delay, the feedback gains and the excitation amplitude are discussed. The existence of the time delay can considerably suppress the vibration amplitude. Particular phenomena induced by the time delay are observed. In addition, the first-order and second-order steady-state moments of the TVIS are obtained based on the Itô stochastic differential equations under the stochastic parametric excitation. The phenomenon of the time delay island about steady-state moments is found. The stationary probability density is discussed for the parametric resonance by means of the finite difference method. The variation of the noise bandwidth and the time delay can induce the transition of the stationary probability density between trivial solutions and non-trivial solutions. Overall, the response mechanism of the TVIS under the stochastic parametric excitation is revealed.