A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems

Lincong CHEN, J. Q. SUN

doi:10.1007/s10483-020-2614-7

Applied Mathematics and Mechanics >

2020 , Vol. 41 >Issue 6: 967 - 982

DOI: https://doi.org/10.1007/s10483-020-2614-7

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A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems

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1. College of Civil Engineering, Huaqiao University, Xiamen 361021, Fujian Province, China;
2. Key Laboratory for Intelligent Infrastructure and Monitoring of Fujian Province, Jimei Avenue 668, Xiamen 361021, Fujian Province, China;
3. Department of Mechanical Engineering, School of Engineering, University of California, Merced, CA 95343, U. S. A.

Received date: 2020-02-02

Revised date: 2020-03-09

Online published: 2020-06-08

Supported by

Project supported by the National Natural Science Foundation of China (Nos. 11672111, 11332008, 11572215, and 11602089), the Program for New Century Excellent Talents in Fujian Province's University, the Natural Science Foundation of Fujian Province of China (No. 2019J01049), and the Scholarship for Overseas Studies from Fujian Province of China

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Abstract

Analytical and numerical studies of multi-degree-of-freedom (MDOF) nonlinear stochastic or deterministic dynamic systems have long been a technical challenge. This paper presents a highly-efficient method for determining the stationary probability density functions (PDFs) of MDOF nonlinear systems subjected to both additive and multiplicative Gaussian white noises. The proposed method takes advantages of the sufficient conditions of the reduced Fokker-Planck-Kolmogorov (FPK) equation when constructing the trial solution. The assumed solution consists of the analytically constructed trial solutions satisfying the sufficient conditions and an exponential polynomial of the state variables, and delivers a high accuracy of the solution because the analytically constructed trial solutions capture the main characteristics of the nonlinear system. We also make use of the concept from the data-science and propose a symbolic integration over a hypercube to replace the numerical integrations in a higher-dimensional space, which has been regarded as the insurmountable difficulty in the classical method of weighted residuals or stochastic averaging for high-dimensional dynamic systems. Three illustrative examples of MDOF nonlinear systems are analyzed in detail. The accuracy of the numerical results is validated by comparison with the Monte Carlo simulation (MCS) or the available exact solution. Furthermore, we also show the substantial gain in the computational efficiency of the proposed method compared with the MCS.

Key words： stationary response; multi-degree-of-freedom (MDOF) nonlinear system; Fokker-Planck-Kolmogorov (FPK) equation; least square method

Cite this article

Lincong CHEN, J. Q. SUN . A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems[J]. Applied Mathematics and Mechanics, 2020 , 41(6) : 967 -982 . DOI: 10.1007/s10483-020-2614-7

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