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TECHNICAL PAPERS

The Girsanov Linearization Method for Stochastically Driven Nonlinear Oscillators

[+] Author and Article Information
Nilanjan Saha

Department of Civil Engineering, Indian Institute of Science, Bangalore, Karnataka 560 012, India

D. Roy1

Department of Civil Engineering, Indian Institute of Science, Bangalore, Karnataka 560 012, Indiaroyd@civil.iisc.ernet.in

1

Corresponding author.

J. Appl. Mech 74(5), 885-897 (Nov 03, 2006) (13 pages) doi:10.1115/1.2712234 History: Received April 10, 2006; Revised November 03, 2006

For most practical purposes, the focus is often on obtaining statistical moments of the response of stochastically driven oscillators than on the determination of pathwise response histories. In the absence of analytical solutions of most nonlinear and higher-dimensional systems, Monte Carlo simulations with the aid of direct numerical integration remain the only viable route to estimate the statistical moments. Unfortunately, unlike the case of deterministic oscillators, available numerical integration schemes for stochastically driven oscillators have significantly poorer numerical accuracy. These schemes are generally derived through stochastic Taylor expansions and the limited accuracy results from difficulties in evaluating the multiple stochastic integrals. As a numerically superior and semi-analytic alternative, a weak linearization technique based on Girsanov transformation of probability measures is proposed for nonlinear oscillators driven by additive white-noise processes. The nonlinear part of the drift vector is appropriately decomposed and replaced, resulting in an exactly solvable linear system. The error in replacing the nonlinear terms is then corrected through the Radon-Nikodym derivative following a Girsanov transformation of probability measures. Since the Radon-Nikodym derivative is expressible in terms of a stochastic exponential of the linearized solution and computable with high accuracy, one can potentially achieve a remarkably high numerical accuracy. Although the Girsanov linearization method is applicable to a large class of oscillators, including those with nondifferentiable vector fields, the method is presently illustrated through applications to a few single- and multi-degree-of-freedom oscillators with polynomial nonlinearity.

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Figures

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Figure 1

(a) Typical PDFs of Gaussian and log-normal distributions and (b) the cumulative product of lognormal random variables

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Figure 2

A realization of the process y(t) with and without the CF

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Figure 3

The HD oscillator (Eq. 18)—histories of second moments: (a) E[x12] and (b) E[x22]; C=5.0, K1=K2=100.0, σ=5.0, h=0.01, X0={0,0}T; solid black lines indicate exact stationary limits

Grahic Jump Location
Figure 4

The HD oscillator (Eq. 18)—histories of second moments through GLM-I and II: (a) E[x12] and (b) E[x22]; C=1.0, K1=10.0, K2=100.0, σ=5.0, h=0.01, X0={0,0}T; solid black lines indicate exact stationary limits

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Figure 5

The HD oscillator (Eq. 18)—E[x12] for different values of K2 using GLM-II; C=1.0, K1=10.0, σ=0.50, h=0.01, X0={0,0}T; solid black lines indicate exact stationary limits

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Figure 6

The HD oscillator (Eq. 18)—histories of E[x22] for different values of h using GLM-II; C=1.0, K1=10.0, K2=100.0, F(t)=0, σ=5.0, X0={0,0}T; the solid black line indicates the exact stationary limit. Inset shows that GLM-I terminates unsuccessfully before 1s for h=0.05.

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Figure 7

The HD oscillator (Eq. 25)—ε1=0.25, ε2=1.0, ε3=1.0, ε4=0.0005, h=0.01, X0={0,0}T; histories of (a) variance of displacement and (b) variance of velocity, using GLM-II, SHS, and SNM

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Figure 8

The HD oscillator (Eq. 25)—histories of (a) variance of displacement and (b) variance of velocity using GLM-II and SNM; ε1=0.25, ε2=1.0, ε3=42.0, ε4=1.0, h=0.01, X0={0,0}T

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Figure 9

The DH oscillator (Eq. 26)—histories of (a) variance of displacement and (b) phase plot of E[x2] and E[x1] using GLM-II, SNM and SHS; ε1=0.25, ε2=0.5, ε3=0.5, ε4=0.10, h=0.01, X0={0,0}T

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Figure 10

The DH oscillator (Eq. 26)—histories of variance of displacement using GLM-II, SNM and SHS; ε1=0.25, ε2=0.5, ε3=0.5, ε4=1.0, h=0.01, X0={0,0}T

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Figure 11

The 2-DOF oscillator (Eq. 27)—second moment histories using GLM-I, GLM-II, and SNM: (a) E[x22] and (b) E[x32]; C1=C2=5.0, K1=K2=K3=100.0, α=100.0, σ1=σ2=5.0, h=0.01, X¯0={0,0,0,0}T

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Figure 12

The 2-DOF oscillator (Eq. 27)—plots of E[x32] using GLM-I and GLM-II for step sizes (a) h=0.01 and (b) h=0.005; C1=C2=5.0, K1=K2=K3=100.0, α=100.0, σ1=σ2=5.0, X¯0={0,0,0,0}T

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