Probability of Failure in Random Vibration

Close approximations to the first-passage probability of failure in random vibration can be obtained by integral equation methods. A simple relation exists between the first-passage probability density function and the distribution function for the time interval spent below a barrier before out-crossing. An integral equation for the probability density function of the time interval is formulated, and adequate approximations for the kernel are suggested. The kernel approximation results in approximate solutions for the probability density function of the time interval and thus for the first-passage
probability density. As a result of the theory, an exact inclusion-exclusion series for the first passage density function in terms of unconditional joint crossing rates is obtained. The results of the theory agree well with simulation results for narrow-banded processes dominated by a single frequency, as well as for bimodal processes with two dominating frequencies in the structural response.