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Research Papers

Probability Distribution of Peaks for Nonlinear Combination of Vector Gaussian Loads

[+] Author and Article Information
Sayan Gupta1

Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, Indias.gupta@tudelft.nl

P. H. van Gelder

Department of Civil Engineering, Technical University of Delft, Stevinweg 1, P.O. Box 5048, 2600 GA Delft, The Netherlandsp.h.a.j.m.vangelder@tudelft.nl

1

Corresponding author.

J. Vib. Acoust 130(3), 031011 (Apr 08, 2008) (12 pages) doi:10.1115/1.2890404 History: Received July 14, 2006; Revised January 31, 2007; Published April 08, 2008

The problem of approximating the probability distribution of peaks, associated with a special class of non-Gaussian random processes, is considered. The non-Gaussian processes are obtained as nonlinear combinations of a vector of mutually correlated, stationary, Gaussian random processes. The Von Mises stress in a linear vibrating structure under stationary Gaussian loadings is a typical example for such processes. The crux of the formulation lies in developing analytical approximations for the joint probability density function of the non-Gaussian process and its instantaneous first and second time derivatives. Depending on the nature of the problem, this requires the evaluation of a multidimensional integration across a possibly irregular and disjointed domain. A numerical algorithm, based on first order reliability method, is developed to evaluate these integrals. The approximations for the peak distributions have applications in predicting the expected fatigue damage due to combination of stress resultants in a randomly vibrating structure. The proposed method is illustrated through two numerical examples and its accuracy is examined with respect to estimates from full scale Monte Carlo simulations of the non-Gaussian process.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 10

Probability density functions for peaks of Von Mises stress, estimated using the proposed method, with sample size 1×106, example 2

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Figure 9

Probability density functions for peaks of Von Mises stress, estimated using the proposed method, with sample size 1×104, example 2

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Figure 8

Effect of discretization on the overall accuracy of the estimated pdf of the peaks of Von Mises stress, example 2

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Figure 7

Probability density functions for the peaks of Von Mises stress, example 2

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Figure 6

Probability density functions for the peaks of Von Mises stress, example 2

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Figure 5

Power spectral density functions for the stress components at the root of the cantilever, example 2

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Figure 4

Schematic diagram of the support for the fire-water system in a nuclear power plant, all dimensions are in mm; example 2

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Figure 3

Effect of shape factor of SVV(ω) on pdf for peaks for V(t), example 1

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Figure 2

Probability density functions for peaks for V(t), example 1

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Figure 1

Schematic diagram for numerical algorithm for evaluating multidimensional integrals: q(x1,x2)=0 is the limit surface in the X1−X2 random variable space, and hY1(y1) and hY2(y2) are two importance sampling pdf’s; two design points at distance β from the origin

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