Moment stability of linear vibro-impact system to boundary random parametrical excitation

The resonance response and moment stability of a single-degree-of-freedom linear vibro-impact oscillator with a one-sided barrier to a boundary random parametrical excitation are investigated. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts or velocity jumps, thereby permitting the applications of asymptotic averaging over the period for slowly varying random process. By using the Itô's differential rule, differential equations ruling the time evolution of the first and second order response moments are obtained. The necessary and sufficient conditions of stability in the moments are that the coefficients matrix of the differential equations ruling the moments have complex eigenvalues with negative real parts. The analytical expression of the stability condition in the first order moments is obtained, while results of the second order moments are given numerically. Some numerical simulations and graphs are presented for representative cases. It is founded that when the amplitude of the parametrical excitation increase, the stability regions will reduce whether in the first order moments or the second order moments. The stability regions will reduce to the minimum value in the principal resonance case. The stability regions based on different order moments will become identical when the intensity of the random disorder increases to zero. The stochastic excitation stabilizes the system in some cases.