Galerkin scheme based determination of first-passage probability of nonlinear system response

An approximate analytical approach for determining the first-passage probability of the response of a class of lightly damped nonlinear oscillators to broadband random excitations is presented. Markovian approximations of both the response amplitude envelope and the response energy envelope are considered. This approach leads to a backward Kolmogorov equation, which governs the evolution of the survival probability of the system. This equation is solved approximately by employing a Galerkin scheme. A convenient set of confluent hypergeometric functions is used as an orthogonal basis for this scheme. The reliability of the derived analytical solution is demonstrated by comparisons with data derived by pertinent Monte Carlo simulations.