First-passage time of an inverted pendulum subject to high frequency harmonic and Gaussian white noise excitations

The first-passage time of an inverted pendulum subject to a combination of high frequency harmonic excitation and Gaussian white noise excitation is investigated. The high frequency harmonic excitation term is simplified to an equivalent autonomous nonlinear stiffness term by using the method of direct partition of motions. Then, the equations of motion of the equivalent system are reduced to an averaged I t o ˆ stochastic differential equation by using the stochastic averaging method of energy envelope. After that, a backward Kolmogorov equation governing the conditional reliability function of first-passage time is established by using the averaged I t o ˆ equation. The conditional reliability function and the conditional probability density of first-passage time from numerical solution of the backward Kolmogorov equation agree well with those from digital simulation of the equivalent system. The effects of system parameters on the conditional reliability function and the conditional probability density of the system are discussed.