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Research Papers

Stochastic Harmonic Function Representation of Stochastic Processes

[+] Author and Article Information
Jianbing Chen

Associate Professor
State Key Laboratory of Disaster
Reduction in Civil Engineering and
School of Civil Engineering,
Tongji University,
1239 Siping Road,
Shanghai 200092, P. R. C.
e-mail: chenjb@tongji.edu.cn

Weiling Sun

Ph.D. Student
School of Civil Engineering,
Tongji University,
1239 Siping Road,
Shanghai 200092, P. R. C.
e-mail: wlsun0228@gmail.com

Jie Li

Distinguished Professor
State Key Laboratory of Disaster
Reduction in Civil Engineering and
School of Civil Engineering,
Tongji University,
1239 Siping Road,
Shanghai 200092, P. R. C.
e-mail: lijie@tongji.edu.cn

Jun Xu

Ph.D. Student
School of Civil Engineering,
Tongji University,
1239 Siping Road,
Shanghai 200092, P. R. C.
e-mail: 86xujun@tongji.edu.cn

1Corresponding author.

Manuscript received December 11, 2010; final manuscript received May 8, 2012; accepted manuscript posted June 6, 2012; published online July 31, 2012. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 80(1), 011001 (Jul 31, 2012) (11 pages) Paper No: JAM-10-1448; doi: 10.1115/1.4006936 History: Received December 11, 2010; Revised May 08, 2012; Accepted June 06, 2012

An approach to represent a stochastic process by the combination of finite stochastic harmonic functions is proposed. The conditions that should be satisfied to make sure that the power spectral density function of the stochastic harmonic function process is identical to the target power spectral density are firstly studied. Then, two kinds of stochastic harmonic functions, of which the distribution of the amplitudes and the random frequencies are different, are discussed. The probabilistic characteristics of the two kinds of stochastic harmonic functions, including the asymptotic distribution, the one-dimensional probability density function, and the rate of approaching the asymptotic distribution, etc., are studied in detail by theoretical treatment and numerical examples. Responses of a nonlinear structure subjected to strong earthquake excitation are investigated. The studies show that the proposed approach can capture the target power spectral density exactly with any number of components. The reduction of the components provides flexibility and reduces the computational cost. Finally, problems that need further investigations are discussed.

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References

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Figures

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Fig. 1

Overlapping of supports of ω˜i’s

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Fig. 2

One-dimensional PDF of SHFs: (a) one-dimensional PDF of SHF-I; (b) one-dimensional PDF of SHF-II (λ = 1.63); (c) one-dimensional PDF of SHF-II (λ = 1.96)

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Fig. 3

Typical time history of ground motion acceleration

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Fig. 4

The PSD of the SHF-II, the spectral representation, and the target PSD

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Fig. 5

Autocorrelation function of the SHF-II and the spectral representation

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Fig. 6

Relative entropy between the one-dimensional PDF of SHFs and normal distribution. (a) One-dimensional PDF of SHF-I; (b) one-dimensional PDF of SHF-II (λ = 1.6); (c) one-dimensional PDF of SHF-II (λ = 1.9).

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Fig. 7

A nine-story shear frame structure

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Fig. 8

Typical input and corresponding restoring force versus inter-story drift for linear and nonlinear structures: (a) input accelerogram; (b) restoring force curves

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Fig. 9

Standard deviations of structural response (Std.D denotes standard deviation): (a) standard deviation of bottom drift of the linear structure; (b) standard deviation of inter-story shear of the linear structure; (c) standard deviation of bottom drift of the nonlinear structure; (d) standard deviation of inter-story shear of the nonlinear structure; (e) standard deviation of bottom inter-story drift of the nonlinear structure

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