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Research Papers

Survival Probability Determination of Nonlinear Oscillators Subject to Evolutionary Stochastic Excitation

[+] Author and Article Information
Pol D. Spanos

L. B. Ryon Chair in Engineering,
Rice University,
MS 321, P.O. Box 1892,
Houston, TX 77251
e-mail: spanos@rice.edu

Ioannis A. Kougioumtzoglou

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

Manuscript received November 19, 2013; final manuscript received November 27, 2013; accepted manuscript posted December 9, 2013; published online January 15, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(5), 051016 (Jan 15, 2014) (9 pages) Paper No: JAM-13-1477; doi: 10.1115/1.4026182 History: Received November 19, 2013; Revised November 27, 2013; Accepted December 09, 2013

A novel approximate analytical technique for determining the survival probability and first-passage probability density function (PDF) of nonlinear/hysteretic oscillators subject to evolutionary stochastic excitation is developed. Specifically, relying on a stochastic averaging/linearization treatment of the problem, approximate closed form expressions are derived for the oscillator nonstationary marginal, transition, and joint-response amplitude PDFs and, ultimately, for the time-dependent oscillator survival probability. The developed technique exhibits considerable versatility, as it can handle readily cases of oscillators exhibiting complex hysteretic behaviors as well as cases of evolutionary stochastic excitations with time-varying frequency contents. Further, it exhibits notable simplicity since, in essence, it requires only the solution of a first-order nonlinear ordinary differential equation (ODE) for the oscillator nonstationary response variance. Thus, the computational cost involved is kept at a minimum level. The classical hardening Duffing and the versatile Preisach (hysteretic) oscillators are considered in a numerical examples section, in which comparisons with pertinent Monte Carlo simulations data demonstrate the reliability of the proposed technique.

Copyright © 2014 by ASME
Topics: Probability , Density
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References

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Figures

Grahic Jump Location
Fig. 1

Nonseparable excitation evolutionary power spectrum

Grahic Jump Location
Fig. 5

First-passage PDFs for the oscillator (S1 = 1,ω02 = π2,β0 = 0.0628,ɛ = 0.2) for various values of the parameter λ; comparisons with MCS (10,000 realizations)

Grahic Jump Location
Fig. 6

Time-dependent survival probabilities for the oscillator (S1 = 1,ω02 = π2,β0 = 0.0628,ɛ = 1) for various values of the parameter λ; comparisons with MCS (10,000 realizations)

Grahic Jump Location
Fig. 2

Equivalent natural frequencies of the oscillators with parameter values (S1 = 1,ω02 = π2,β0 = 0.0628,ɛ = 0.2) and (S1 = 1,ω02 = π2,β0 = 0.0628,ɛ = 1)

Grahic Jump Location
Fig. 3

Equivalent half-natural periods of the oscillators with parameter values (S1 = 1,ω02 = π2,β0 = 0.0628,ɛ=0.2) and (S1 = 1,ω02 = π2,β0 = 0.0628,ɛ = 1); comparisons with the ones corresponding to the discretized time domain

Grahic Jump Location
Fig. 4

Time-dependent survival probabilities for the oscillator (S1 = 1,ω02 = π2,β0 = 0.0628,ɛ = 0.2) for various values of the parameter λ; comparisons with MCS (10,000 realizations)

Grahic Jump Location
Fig. 7

First-passage PDFs for the oscillator (S1 = 1,ω02 = π2,β0 = 0.0628,ɛ = 1) for various values of the parameter λ; comparisons with MCS (10,000 realizations)

Grahic Jump Location
Fig. 8

Preisach hysteretic (relay) operator

Grahic Jump Location
Fig. 13

First-passage PDFs for the oscillator (S1 = 5,ω¯2 = π2,β0 = 0.0628,ψ = 1,ϕ = 2) for various values of the parameter λ; comparisons with MCS (10,000 realizations)

Grahic Jump Location
Fig. 14

Time-dependent survival probabilities for the oscillator (S1 = 5,ω¯2 = π2,β0 = 0.0628,ψ = 1,ϕ = 1) for various values of the parameter λ; comparisons with MCS (10,000 realizations)

Grahic Jump Location
Fig. 15

First-passage PDFs for the oscillator (S1 = 5,ω¯2 = π2,β0 = 0.0628,ψ = 1,ϕ = 1) for various values of the parameter λ; comparisons with MCS (10,000 realizations)

Grahic Jump Location
Fig. 9

Equivalent damping elements of the oscillators with parameter values (S1 = 5,ω¯2 = π2,β0 = 0.0628,ψ = 1,ϕ = 2) and (S1 = 5,ω¯2 = π2,β0 = 0.0628,ψ = 1,ϕ = 1)

Grahic Jump Location
Fig. 10

Equivalent natural frequencies of the oscillators with parameter values (S1 = 5,ω¯2 = π2,β0 = 0.0628,ψ = 1,ϕ = 2) and (S1 = 5,ω¯2 = π2,β0 = 0.0628,ψ = 1,ϕ = 1)

Grahic Jump Location
Fig. 11

Equivalent half-natural periods of the oscillators with parameter values (S1 = 5,ω¯2 = π2,β0 = 0.0628,ψ = 1,ϕ = 2) and (S1 = 5,ω¯2 = π2,β0 = 0.0628,ψ = 1,ϕ = 1); comparisons with the ones corresponding to the discretized time domain

Grahic Jump Location
Fig. 12

Time-dependent survival probabilities for the oscillator (S1 = 5,ω¯2 = π2,β0 = 0.0628,ψ = 1,ϕ = 2) for various values of the parameter λ; comparisons with MCS (10,000 realizations)

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