First-Passage Approximations for Simple Oscillators

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Abstract
Simple analytical formulas are presented for approximate computation of the first-passage probabilities for the random response of a linear single-degree-of-freedom system. The response is taken to be zero-start and the excitation is stationary white noise with a normal probability distribution. Both small-time (nonstationary) and large-time (stationary) responses are included. Certain attractive existing approximate procedures are shown to be sometimes very significantly in error. The new formulas to compute the two parameters of the stationary problem are based on empirical data and the results of an existing approximate procedure which is quite accurate but cumbersome to use. Given the parameters of the stationary problem, a simple procedure is presented for accurately approximating the nonstationary behavior. The limitation on the approximate results is that they do not apply if the crossing level is smaller than the root mean square value of the stationary response.

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