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Transient response of nonlinear vibro-impact system under Gaussian white noise excitation through complex fractional moments

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Abstract

The approximate transient response of a nonlinear vibro-impact system under Gaussian white noise excitation is investigated by the methods of stochastic averaging and the Mellin transform. The Zhuravlev nonsmooth transformation is utilized to convert the nonlinear vibro-impact system into an equivalent system without velocity jumps by introducing an impulsive damping term. The Itô stochastic differential equation with respect to the amplitude response and the related Fokker–Plank–Kolmogorov (FPK) equation governing the amplitude response probability density of the vibro-impact system are derived with the stochastic averaging method. The Mellin transform is introduced to solve the FPK equation. The differential relations of complex fractional moments are obtained. The probability density function for transient response of this system constructed by solving a set of differential equations yields complex fractional moments. Two illustrative examples are examined to evaluate the effectiveness of the proposed solution procedure. The effects of restitution factors are investigated on the transient probability density distribution of the vibro-impact systems. At the same time, the convergence and the error analyses for different restitution factors are illustrated. Influences of the truncate term on the convergence and the error are further illustrated. The results obtained from the proposed procedure agree well with the results from Monte Carlo simulations.

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Authors and Affiliations

  1. School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, 030024, People’s Republic of China

    Xiufeng Xie, Junlin Li & Rong Guo

  2. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, People’s Republic of China

    Di Liu

Authors
  1. Xiufeng Xie
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  2. Junlin Li
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  3. Di Liu
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  4. Rong Guo
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Correspondence to Junlin Li.

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Xie, X., Li, J., Liu, D. et al. Transient response of nonlinear vibro-impact system under Gaussian white noise excitation through complex fractional moments. Acta Mech 228, 1153–1163 (2017). https://doi.org/10.1007/s00707-016-1761-8

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  • DOI: https://doi.org/10.1007/s00707-016-1761-8

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