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Heavy-Tailed Response of Structural Systems Subjected to Stochastic Excitation Containing Extreme Forcing Events

[+] Author and Article Information
Han Kyul Joo, Mustafa A. Mohamad

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139

Themistoklis P. Sapsis

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
e-mail: sapsis@mit.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 17, 2017; final manuscript received February 2, 2018; published online July 26, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(9), 090914 (Jul 26, 2018) (12 pages) Paper No: CND-17-1377; doi: 10.1115/1.4039309 History: Received August 17, 2017; Revised February 02, 2018

We characterize the complex, heavy-tailed probability density functions (pdfs) describing the response and its local extrema for structural systems subject to random forcing that includes extreme events. Our approach is based on recent probabilistic decomposition-synthesis (PDS) technique (Mohamad, M. A., Cousins, W., and Sapsis, T. P., 2016, “A Probabilistic Decomposition-Synthesis Method for the Quantification of Rare Events Due to Internal Instabilities,” J. Comput. Phys., 322, pp. 288–308), where we decouple rare event regimes from background fluctuations. The result of the analysis has the form of a semi-analytical approximation formula for the pdf of the response (displacement, velocity, and acceleration) and the pdf of the local extrema. For special limiting cases (lightly damped or heavily damped systems), our analysis provides fully analytical approximations. We also demonstrate how the method can be applied to high dimensional structural systems through a two-degrees-of-freedom (TDOF) example system undergoing extreme events due to intermittent forcing. The derived formulas can be evaluated with very small computational cost and are shown to accurately capture the complicated heavy-tailed and asymmetrical features in the probability distribution many standard deviations away from the mean, through comparisons with expensive Monte Carlo simulations.

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Figures

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Fig. 1

(Top) Background stochastic excitation including impulsive loads in (vertical arrows) upward arrows. (Bottom) System response displacement.

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Fig. 2

Schematic representation of the PDS method for an intermittently forced system

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Fig. 3

(Severely underdamped case) Comparison between direct Monte Carlo simulation and the analytical pdf for the SDOF system 1. The pdfs for the envelope of each of the stochastic variables, displacement, velocity, and acceleration are presented. The dashed line indicates one standard deviation.

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Fig. 4

(Severely overdamped case) Comparison between direct Monte Carlo simulation and the analytical pdf for SDOF system 2. The pdf for the value of each stochastic process is shown. The dashed line indicates one standard deviation.

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Fig. 5

(Critically damped system) Comparison between direct Monte Carlo simulations and the semi-analytical pdf for SDOF system 3. Dashed lines indicate one standard deviation.

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Fig. 6

The considered TDOF system. The excitation is applied to the first mass.

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Fig. 7

(TDOF system) Comparison between direct Monte Carlo simulation and the semi-analytical approximation. The pdf for the value of the time series is presented. Dashed line indicates one standard deviation.

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Fig. 8

(Local extrema for TDOF) Comparison between direct Monte Carlo simulation and the semi-analytical approximation. The pdf for the local extrema of the response is presented. Dashed line indicates one standard deviation.

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