Galerkin Method For First-Passage Problem Of Quasi-Hamiltonian Systems

The reliability problem of nonlinear structural systems is of great significance in both theory and engineering.For the system of engineering structures subjected to random excitations,once the quantity of state,such as displacement or energy exceed the safety domain for the first time,the system is failure.That is the definition of first-passage failure,while the probability of the quantity of state stay in safety domain is the reliability of the system.In prestent paper,based on the stochastic averaging method for quasi Hamiltonian system,the first-passage failure reliability of MDOF nonlinear system under Gauss white noise excitation is studied.Based on the integrability and resonance of Hamiltonian system,the quasi-Hamiltonian systems can be classified as five groups:quasi-non-integrable Hamiltonian system,quasi-partially-integrable and non-resonant Hamiltonian system,quasi-partially-integrable and resonant Hamiltonian system,quasi-integrable and non-resonant Hamiltonian system,quasi-integrable and resonant Hamiltonian system.By using the stochastic averaging method of quasi-Hamiltonian system,the backward Kolmogorov equation about Hamiltonian and the boundary conditions of the equation can be established.By solving the backward Kolmogorov equation,the conditional reliability function of the quasi-Hamiltonian system can be obtained.In this paper,by using Galerkin method,An approximate solution of the backward Kolmogorov equation is derived for the quasi-Hamiltonian system.In this approximate solution,the conditional reliability function is composed of a series of basis functions and time-varying coefficients.The basis function is composed of the combination of Kummer function and trigonometric function,and its form is related to the dimension and resonance of the quasi Hamiltonian system.By using the proposed method in this paper,some examples are calculated.Through contrast with Monte Carlo simulation results,the applicability of proposed method is veryfied.In addition,some factors that affect the conditional reliability of the quasi-Hamiltonian system,such as damping,excitation intensity and resonance,are studied.