First-passage probabilities for randomly excited systems: Diffusion methods

Methods of nonlinear random vibration analysis

“…Different versions of the method are available and are widely used in the problems of response prediction, stability analysis and the first passage and fatigue failure analyses. Extensive surveys of related literature have been published (Ibrahim 1985;Roberts & Spanos 1986;Roberts 1986a;Zhu 1988). …”

“…The details of the derivation of these equations along with a discussion on the initial conditions, boundary conditions, well posedness, eigenvalues and eigenfunctions and the existence, uniqueness and stability of stationary solutions can be found in the works of Bharucha Reid (1960), Stratonovich (1963), Caughey (1963a, 1971), Feller (1966), Fuller (1969. and Roberts (1986a). A comprehensive treatment of the FPK equation and its application in physical sciences is available in the monographs by Risken (1989) and Horsthemke & Lefever (1984).…”

“…These equations are referred to as the generalized Pontriagin-Vitt (GPV) equations in the literature. Although no exact analytical solution exists for finding Mk, several approximations are available and they have been reviewed by Roberts (1986a). These methods include method of weighted residuals (Spanos 1983), random walk models (Toland & Yang 1971;Roberts 1978), finite difference method (Roberts 1986b), finite element method (Spencer & Bergman1985) and cell mapping techniques (Sun & Hsu 1988).…”

“…Subsequently, the method has been generalized to include mdof systems and nonstationary inputs and widely used in random vibration studies (Roberts & Spanos 1986). The method has also formed the basis for the study of first passage failures (Roberts 1986a) and stability analysis (Ibrahim 1985). For problems wherein the time varying nature of the system such as deterministic excitations or nonstationary inputs needs to be preserved, Lin (1986) has proposed that the temporal averaging in Stratonovich's procedure may be dispensed with.…”

First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations

“…The combination of the classical stochastic averaging method and the diffusion process method for first-passage time has been applied by many researchers to SDOF stochastic systems [11][12][13]. Recently, the combination of stochastic averaging method for a quasi-Hamiltonian system and the diffusion process method for first-passage time has been applied to study the first-passage time of MDOF strongly nonlinear stochastic systems [14].…”