Numerical calculation for first excursion probabilities of linear systems

A numerical method is proposed to calculate the first excursion probability that the linear system response to non-stationary random excitation exceeds in magnitude its respective threshold level within a specified time interval. Its derivation is carried through three steps. First, the numerical expression for the first excursion probability, based on the modification of the level crossing rates, is developed. Secondly, the time domain expressions for the first three order spectral moments of the response are given, from which the level crossing rates can be computed. Thirdly, it is then demonstrated how Simpson’s rule of numerical integration can be applied to the approximate computation of the first three order spectral moments. It is shown by examples that both numerical accuracy and efficiency of this method are apparently very high.