Analysis for probability density of response with jump phenomenon of a duffing oscillator to narrow band random excitation

An approximate analytical method is developed for calculating the probability density of the response of a Duffing oscillator subjected to narrow band random excitation when the stochastic jump phenomenon occurs. The stochastic jump phenomenon is characterized by the existence of multiple states of amplitude and sudden jumps between the states in the sample function of response. In this paper, the trigger for the stochastic jump is considered, and the probabilities of large and small amplitude states of the response are evaluated. The local probability density function of each state is separately assumed using the equivalent linearization method, and the probability density function of the overall response is derived by sum of these local probability density functions weighted with the probability of each state. In the illustrative example, the calculated probability density functions are compared with the Monte Carlo simulation results. It is shown that the present method well expresses the significant characteristic of the stochastic jump phenomenon.