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Research Papers

Nonlinear Oscillator Stochastic Response and Survival Probability Determination via the Wiener Path Integral

[+] Author and Article Information
Yuanjin Zhang

Institute for Risk and Uncertainty, University of Liverpool, Liverpool L69 3GH, UKe-mail: ylzhyj@liverpool.ac.uk

Ioannis A. Kougioumtzoglou

Department of Civil Engineering and Engineering Mechanics, The Fu Foundation School of Engineering and Applied Science, Columbia University, New York, NY 10027e-mail: ikougioum@columbia.edu

Manuscript received September 1, 2014; final manuscript received January 20, 2015; published online April 20, 2015. Assoc. Editor: Athanasios Pantelous.

ASME J. Risk Uncertainty Part B 1(2), 021005 (Apr 20, 2015) (15 pages) Paper No: RISK-14-1047; doi: 10.1115/1.4029754 History: Received September 01, 2014; Accepted February 05, 2015; Online April 20, 2015

A Wiener path integral (WPI) technique based on a variational formulation is developed for nonlinear oscillator stochastic response determination and reliability assessment. This is done in conjunction with a stochastic averaging/linearization treatment of the problem. Specifically, first, the nonlinear oscillator is cast into an equivalent linear one with time-varying stiffness and damping elements. Next, relying on the concept of the most probable path, a closed-form approximate analytical expression for the oscillator joint transition probability density function (PDF) is derived for small time intervals. Finally, the transition PDF in conjunction with a discrete version of the Chapman–Kolmogorov (C–K) equation is utilized for advancing the solution in short-time steps. In this manner, not only the nonstationary response PDF but also the oscillator survival probability and first-passage PDF are determined. In comparison with existing numerical path integral schemes, a significant advantage of the proposed WPI technique is that closed-form analytical expressions are derived for the involved multidimensional integrals; thus, the computational cost is kept at a minimum level. The hardening Duffing and the bilinear hysteretic oscillators are considered as numerical examples. Comparisons with pertinent Monte Carlo simulation (MCS) data demonstrate the reliability of the developed technique.

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Figures

Grahic Jump Location
Fig. 1

Nonstationary response variance c(t) of a Duffing oscillator under white noise excitation with parameter values S0=0.0637, ω02=1, β0=0.2, ε=0.2 (Case 1) and S0=0.0637, ω02=1, β0=0.2, ε=1 (Case 2); comparison with pertinent Monte Carlo simulations (10,000 realizations).

Grahic Jump Location
Fig. 2

Time-varying equivalent linear natural frequency ωeq(t) for a Duffing oscillator under white noise excitation with parameter values S0=0.0637, ω02=1, β0=0.2, ε=0.2 (Case 1) and S0=0.0637, ω02=1, β0=0.2, ε=1 (Case 2).

Grahic Jump Location
Fig. 3

Response displacement PDF for a Duffing oscillator under white noise excitation with parameter values S0=0.0637, ω02=1, β0=0.2, ε=0.2 (Case 1) for various time instants; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Grahic Jump Location
Fig. 4

Response displacement PDF for a Duffing oscillator under white noise excitation with parameter values S0=0.0637, ω02=1, β0=0.2, ε=1 (Case 2) for various time instants; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Grahic Jump Location
Fig. 5

Survival probability for a Duffing oscillator under white noise excitation with parameter values S0=0.0637, ω02=1, β0=0.2, ε=0.2 (Case 1) for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Grahic Jump Location
Fig. 6

First-passage PDF for a Duffing oscillator under white noise excitation with parameter values S0=0.0637, ω02=1, β0=0.2, ε=0.2 (Case 1) for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Grahic Jump Location
Fig. 7

Survival probability for a Duffing oscillator under white noise excitation with parameter values S0=0.0637, ω02=1, β0=0.2, ε=1 (Case 2) for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Grahic Jump Location
Fig. 8

First-passage PDF for a Duffing oscillator under white noise excitation with parameter values S0=0.0637, ω02=1, β0=0.2, ε=1 (Case 2) for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Grahic Jump Location
Fig. 9

Nonstationary response variance c(t) of a bilinear hysteretic oscillator under white noise excitation with parameter values S0=0.0637, a=0.6, β0=0.1, ω0=1, xy=1; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Grahic Jump Location
Fig. 10

Time-varying equivalent linear natural frequency ωeq(t) for a bilinear hysteretic oscillator under white noise excitation with parameter values S0=0.0637, a=0.6, β0=0.1, ω0=1, xy=1.

Grahic Jump Location
Fig. 11

Time-varying equivalent linear damping βeq(t) for a bilinear hysteretic oscillator under white noise excitation with parameter values S0=0.0637, a=0.6, β0=0.1, ω0=1, xy=1.

Grahic Jump Location
Fig. 12

Response displacement PDF for a bilinear oscillator under white noise excitation with parameter values S0=0.0637, a=0.6, β0=0.1, ω0=1, xy=1 for various time instants; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Grahic Jump Location
Fig. 13

Survival probability for a bilinear hysteretic oscillator under white noise excitation with parameter values S0=0.0637, a=0.6, β0=0.1, ω0=1, xy=1 for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Grahic Jump Location
Fig. 14

First-passage PDF for a bilinear hysteretic oscillator under white noise excitation with parameter values S0=0.0637, a=0.6, β0=0.1, ω0=1, xy=1 for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations).

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