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Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation

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Abstract

The nonstationary probability densities of system response of a single-degree-of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied. Using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. According to the Galerkin method, the time-dependent coefficients can be solved from a set of first-order linear differential equations. Then, the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained. A van der Pol-Duffing oscillator subject to modulated white noise is given as an example to illustrate the proposed procedures. The effects of the system parameters, such as the linear damping coefficient and the nonlinear stiffness coefficient, on the system response are discussed.

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Authors and Affiliations

  1. Department of Mechanics, Zhejiang University, Hangzhou, 310027, P. R. China

    Xiao-ling Jin  (金肖玲) & Zhi-long Huang  (黄志龙)

  2. Department of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong, P. R. China

    Y. T. Leung  (梁以德)

  3. Department of Electronic Engineering, City University of Hong Kong, Hong Kong, P. R. China

    Xiao-ling Jin  (金肖玲)

Authors
  1. Xiao-ling Jin  (金肖玲)
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  2. Zhi-long Huang  (黄志龙)
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  3. Y. T. Leung  (梁以德)
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Corresponding author

Correspondence to Zhi-long Huang  (黄志龙).

Additional information

Project supported by the National Natural Science Foundation of China (No. 11025211), the Zhejiang Provincial Natural Science Foundation of China (No. Z6090125), and the Special Fund for National Excellent Ph. D. Dissertation and Research Grant Council of Hong Kong City (No.U115807)

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Jin, Xl., Huang, Zl. & Leung, Y.T. Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation. Appl. Math. Mech.-Engl. Ed. 32, 1389–1398 (2011). https://doi.org/10.1007/s10483-011-1509-7

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  • DOI: https://doi.org/10.1007/s10483-011-1509-7

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2010 Mathematics Subject Classification

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