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TECHNICAL PAPERS

Nonstationary Response Envelope Probability Densities of Nonlinear Oscillators

[+] Author and Article Information
P. D. Spanos

 Rice University, 6100 Main Street, MS 321, Houston, TX 77005spanos@rice.edu

A. Sofi1

Dipartimento di Arte, Scienza e Tecnica del Costruire,  Università “Mediterranea” di Reggio Calabria, Via Melissari Feo di Vito, I-89124 Reggio Calabria, Italyalba.sofi@unirc.it

M. Di Paola

Dipartimento di Ingegneria Strutturale e Geotecnica,  Università di Palermo, Viale delle Scienze, 90128, Palermo, Italydipaola@diseg.unipa.it

1

Visiting Scholar at Rice University, Houston, TX.

J. Appl. Mech 74(2), 315-324 (Feb 06, 2006) (10 pages) doi:10.1115/1.2198253 History: Received June 01, 2005; Revised February 06, 2006

The nonstationary random response of a class of lightly damped nonlinear oscillators subjected to Gaussian white noise is considered. An approximate analytical method for determining the response envelope statistics is presented. Within the framework of stochastic averaging, the procedure relies on the Markovian modeling of the response envelope process through the definition of an equivalent linear system with response-dependent parameters. An approximate solution of the associated Fokker-Planck equation is derived by resorting to a Galerkin scheme. Specifically, the nonstationary probability density function of the response envelope is expressed as the sum of a time-dependent Rayleigh distribution and of a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. These functions are the eigenfunctions of the boundary-value problem associated with the Fokker-Planck equation governing the evolution of the probability density function of the response envelope of a linear oscillator. The selected basis functions possess some notable properties that yield substantial computational advantages. Applications to the Van der Pol and Duffing oscillators are presented. Appropriate comparisons to the data obtained by digital simulation show that the method, being nonperturbative in nature, yields reliable results even for large values of the nonlinearity parameter.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Time evolution of the series coefficients ci(t), i=0,1,…,7 (Van der Pol oscillator Eq. 52)

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Figure 2

Nonstationary PDF of the response envelope evaluated at different time instants by the proposed method setting N=19 (continuous line) and by MCS (symbols) (Van der Pol oscillator Eq. 52)

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Figure 3

Proposed approximation of the nonstationary PDF of the response envelope evaluated at t=5s retaining N series terms contrasted with MCS data (Van der Pol oscillator Eq. 52)

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Figure 4

Proposed approximation of the stationary PDF of the response envelope for different values of N compared to the exact solution (Eq. 15) and MCS data (Van der Pol oscillator Eq. 52)

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Figure 5

Proposed estimates of the first- (a) and second-order (b) statistical moments of the response envelope obtained by applying Eqs. 49,50 for different values of N compared to MCS data (Van der Pol oscillator Eq. 52)

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Figure 6

Time evolution of the series coefficients ci(t), i=0,1,…,7 (Duffing oscillator Eq. 59): (a) ε=0.5; (b) ε=2

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Figure 7

Nonstationary PDF of the response envelope evaluated at different time instants by the proposed method setting N=19 (continuous line) and by MCS (symbols) (Duffing oscillator Eq. 59): (a) ε=0.5; (b) ε=2

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Figure 8

Proposed approximation of the nonstationary PDF of the response envelope evaluated at t=20s retaining N series terms contrasted with MCS data (Duffing oscillator Eq. 59): (a) ε=0.5; (b) ε=2

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Figure 9

Proposed approximation of the stationary PDF of the response envelope for different values of N compared to the exact solution (Eq. 15), the solution pertaining to ε=0 and MCS data (Duffing oscillator Eq. 59): (a) ε=0.5; (b) ε=2.

Grahic Jump Location
Figure 10

Proposed estimate of the first-order statistical moment of the response envelope obtained by applying Eq. 49 for different values of N compared to MCS data (Duffing oscillator Eq. 59): (a) ε=0.5; (b) ε=2

Grahic Jump Location
Figure 11

Proposed estimate of the second-order statistical moment of the response envelope obtained by applying Eq. 50 for different values of N compared to MCS data (Duffing oscillator Eq. 59): (a) ε=0.5; (b) ε=2

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