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Research Papers

A Spectral Method for Describing the Response of a Parametrically Excited System Under External Random Excitation

[+] Author and Article Information
L. Bachelet

 LaMCoS, CNRS UMR 5259, INSA-Lyon F-69621, Francelucie.bachelet@insa-lyon.fr

N. Driot

 LaMCoS, CNRS UMR 5259, INSA-Lyon F-69621, Francenicolas.driot@insa-lyon.fr

J. Perret-Liaudet

 LTDS, Ecole Centrale de Lyon, 36 Avenue Guy de Collongue, 69134 Ecully Cedex, Francejoel.perret-liaudet@ec-lyon.fr

J. Comput. Nonlinear Dynam 3(1), 011008 (Nov 26, 2007) (10 pages) doi:10.1115/1.2815333 History: Received January 09, 2007; Revised July 23, 2007; Published November 26, 2007

This paper describes an original approach for computing the stationary response of linear periodic time variant MDOF systems subjected to stationary stochastic external excitation. The proposed method is derived in the frequency domain, is purely numerical, and provides the explicit power spectral density (PSD) of the response. Its implementation first requires expressing the PSD response as a function of the bilinear Fourier transform of the so-called bitemporal impulse response. Then, the spectral method is used to compute the bispectrum function. The efficiency of this spectral process is demonstrated by comparison with Monte Carlo simulations on three parametrical systems. The computational time required and the accuracy are very satisfactory.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Characteristics of the random external excitation: (a) PSD of the white noise used; (b) one temporal sample

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Figure 2

Stability map of the 1-DOF system: Gray areas=unstable areas

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Figure 3

PSD response of the 1-DOF system. Spectral method, dotted line; Monte Carlo simulation, solid line.

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Figure 5

The 5-DOF system studied

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Figure 6

Stability card of the 5-DOF system: Gray areas=unstable areas

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Figure 7

Response PSD of the 5-DOF system. Spectral method, dotted line; Monte Carlo simulation, solid line. (a) Sy1y1; (b) Sy5y5.

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Figure 9

5-DOF response PSD versus μ. μ=0, solid line; μ=0.2, dashed line; μ=0.5, dotted line; μ=0.8, dashed dotted line. (a) broad band display; (b) narrow band display.

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Figure 10

Response PSD of the gyroscopic system. Spectral method, dotted line; Monte Carlo simulation, solid line. (a) Sy1y1; (b) Sy1y2; (c) Sy2y2; (d) Narrow band view of Sy1y1.

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Figure 4

1-DOF response PSD versus μ. μ=0, solid line; μ=0.1, dashed line; μ=0.2, dotted line; μ=0.25, dashed dotted line. (a) Broad band display; (b) narrow band display.

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Figure 8

Response cross-PSD of the 5-DOF system. Spectral method, dotted line; Monte Carlo simulation, solid line. (a) Real part of Sy1y2; (b) imaginary part of Sy1y2; (c) real part of Sy4y5; (c) imaginary part of Sy4y5.

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