First Passage Probability for Nonlinear Oscillators

An approximate method of calculating the probability that a nonlinear oscillator will fail within a specified interval of time is developed, where failure is assumed to occur at the instant the response amplitude first exceeds a critical level. It is shown for oscillators driven by white noise that the energy envelope of the response process is well represented as a one-dimensional Markov process. From the appropriate Fokker-Planck equation of this process simple differential equations for the moments of the time to failure are derived, and integrated numerically in certain cases. In the case of an oscillator with linear damping but a nonlinear spring of the power law type, a complete analytical solution is found in terms of hypergeometric functions. A comparison with digital simulation results indicates that the proposed theory yields a lower bound from the mean time to failure which is close when the damping is very light.