Analytical approximation of dynamic responses of random parameter nonlinear systems based on stochastic perturbation-Galerkin method

The analytic approximation of dynamic responses is very significant for performing the optimization, parameter identification and reliability analysis of structural systems. However, the dynamic responses analysis of nonlinear random parameter systems is a challenging task due to the combined effects of randomness and strong nonlinearity. To address this problem, a new solution method based on the stochastic perturbation-Galerkin method is developed to obtain analytical solutions of the dynamic responses of single-degree-of-freedom nonlinear systems with random parameters. By combining the high-order perturbation and the Newmark-β method, the dynamic responses of systems are initially approximated using the power series expansions. Then a new approximation is defined for the Galerkin projection by utilizing the different orders of the power series expansion terms as trial functions. The employed Galerkin projection ensures the statistical minimization of the random error of the approximate series expansion. The numerical example shows that, for the first time, high-precision analytical expressions for the dynamic responses of a single-degree-of-freedom Duffing system with random parameters are obtained, even if the nonlinear coefficient reaches a value of 50. And it is found that the relationship between the dynamic responses and random variable is strongly nonlinear and constantly evolves over time, becoming increasingly complex along with the nonlinear coefficient. Numerical results further indicate that the new method owns superior computational accuracy compared with the perturbation method and the generalized polynomial chaos method of the same order, can better maintain convergence during long-time integration and has better efficiency than the direct Monte Carlo simulation method.