Abstract
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.
Highlights
Random vibration theory and numerical methods to date provide efficient solutions for second-moment properties of the states of arbitrary linear systems subjected to Gaussian random input
Two types of stochastic reducedorder models (SROM)-based solutions were presented, and numerical results were compared with the Monte Carlo (MC) estimates
The first method, that is, the SROM-based solution involves the construction of a SROM for the input process, whose samples are used to calculate the responses of the nonlinear systems
Summary
Random vibration theory and numerical methods to date provide efficient solutions for second-moment properties of the states of arbitrary linear systems subjected to Gaussian random input. The second method, referred to as the extended SROM method , i.e., the ESROM method, builds upon the SROM method to constructs an SROM-based surrogate model for the response of the dynamic system, defined on a Voronoi tessellation of the input samples Both methods proposed in this paper may be used as alternatives to MC for calculating response statistics for dynamic nonlinear equations, with just a fraction of the computational effort. Applications of the methods proposed are shown for simple second-order, oneand two-degree-of-freedom nonlinear dynamic ordinary differential equations (ODEs) with deterministic coefficients and stochastic input. Response statistics such as probability tail distributions and stochastic moments are compared with the reference MC results
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