Stochastic reduced-order models for stable nonlinear ordinary differential equations

Two methods based on stochastic reduced-order models (SROM) are proposed to solve stochastic stable nonlinear ordinary differential equations. One general method available for the probabilistic characterization of the response of nonlinear systems subjected to random excitation is Monte Carlo (MC), wherein the response of the nonlinear system must be calculated for a large number of samples of the input, which can be very computationally demanding. Random vibration theory is also inadequate for calculating response statistics for both linear systems under non-Gaussian inputs and nonlinear systems subjected to any kind of excitation. The two methods proposed are based on SROM, i.e., stochastic models with a finite number of optimally selected samples. The first method uses a SROM model for the random input. The second method is based on a surrogate model for the response of the nonlinear system defined on a Voronoi tessellation of the input samples. The newly proposed methods are applied for stable nonlinear ordinary differential equations, with deterministic coefficients and stochastic input, that are used in engineering applications: single-degree-of-freedom Duffing and Bouc–Wen systems, and a two-degree-of-freedom nonlinear energy sink system. The numerical results suggest that SROMs are able to estimate statistics of the stochastic responses for these systems efficiently and accurately, results validated by the benchmark MC results.