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Research Papers

Stationary Response Probability Distribution of SDOF Nonlinear Stochastic Systems

[+] Author and Article Information
Lincong Chen

College of Civil Engineering,
Huaqiao University,
Xiamen, Fujian 361021, China;
Key Laboratory for Structural Engineering and
Disaster Prevention of Fujian Province,
Xiamen, Fujian 361021, China
e-mail: lincongchen@hqu.edu.cn

Jun Liu

College of Civil Engineering,
Huaqiao University,
Xiamen, Fujian 361021, China;
Key Laboratory for Structural Engineering and
Disaster Prevention of Fujian Province,
Xiamen, Fujian 361021, China
e-mail: 90liujun@163.com

Jian-Qiao Sun

Professor
School of Engineering,
University of California,
Merced, CA 95343
e-mail: jqsun@ucmerced.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 26, 2016; final manuscript received March 17, 2017; published online April 5, 2017. Assoc. Editor: Daining Fang.

J. Appl. Mech 84(5), 051006 (Apr 05, 2017) (8 pages) Paper No: JAM-16-1577; doi: 10.1115/1.4036307 History: Received November 26, 2016; Revised March 17, 2017

There has been no significant progress in developing new techniques for obtaining exact stationary probability density functions (PDFs) of nonlinear stochastic systems since the development of the method of generalized probability potential in 1990s. In this paper, a general technique is proposed for constructing approximate stationary PDF solutions of single degree of freedom (SDOF) nonlinear systems under external and parametric Gaussian white noise excitations. This technique consists of two novel components. The first one is the introduction of new trial solutions for the reduced Fokker–Planck–Kolmogorov (FPK) equation. The second one is the iterative method of weighted residuals to determine the unknown parameters in the trial solution. Numerical results of four challenging examples show that the proposed technique will converge to the exact solutions if they exist, or a highly accurate solution with a relatively low computational effort. Furthermore, the proposed technique can be extended to multi degree of freedom (MDOF) systems.

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Figures

Grahic Jump Location
Fig. 1

The stationary PDF of example 2 with parameters β0 = β1 = β3 = 0.1, ω = 1.0, and D1 = 0.1. (a) The closed-form PDF in Eq. (24). (b) The PDF obtained with Monte Carlo simulations.

Grahic Jump Location
Fig. 2

The stationary PDF of example 3 with parameters α = −0.1, β = 1.0, and Di = 0.25. (a) The closed-form PDF in Eq. (30). (b) The PDF obtained with Monte Carlo simulations. (c) The averaged PDF in Eq. (32). (d) The exponential polynomial approximation in Eq. (31).

Grahic Jump Location
Fig. 3

The marginal probability densities of p1(x) and p2(y) of example 3 obtained from the corresponding joint PDF. Symbols (∘, *) denote the Monte Carlo data. Solid line denotes the analytical solutions. (a) and (b): the closed-form PDF in Eq. (30). (c) and (d): the exponential polynomial approximation in Eq. (31). (e) and (f): the averaged PDF in Eq. (32).

Grahic Jump Location
Fig. 4

The stationary PDF of example 4 with parameters α1 = α5 = 1, α3 = −2.5, β1 = β2 = 0.1, and D1 = 0.1. (a) The closed-form PDF in Eq. (35). (b) The PDF obtained with Monte Carlo simulations. (c) The exponential polynomial solution in Eq. (36). (d) The iterated exponential polynomial approximation in Eq. (37).

Grahic Jump Location
Fig. 5

The marginal probability densities of p1(x) and p2(y) of example 4 obtained from the corresponding joint PDF. Symbols (∘, *) denote the Monte Carlo data. Solid line denotes the analytical solutions. (a) and (b): the closed-form PDF in Eq. (35). (c) and (d): the exponential polynomial solution in Eq. (36). (e) and (f): the iterated exponential polynomial approximation in Eq. (37).

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