JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (3): 68-77.doi: 10.6040/j.issn.1671-9352.0.2021.418

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Path integration method for the stochastic vibro-impact system under the non-smooth transformation

WANG Liang, JING Kang-kang*, PENG Jia-hui, XU Wei   

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xian 710129, Shaanxi, China
  • Published:2022-03-15

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Abstract: As for the vibro-impact system, the velocity will change suddenly when the impact occurs. As a result, it is not possible to employ the traditional analytical method and the numerical algorithm to solve such systems directly, particularly for the system with stochastic factors. Based on the Ivanov non-smooth transformation method, the vibro-impact system can tranform into the continuous system. Compared with the Zhuravlev non-smooth transformation, the Ivanov non-smooth transformation avoids the discontinuity of the Dirac delta function. Then, the Gauss-Legendre path integration method is proposed to calculate the probability density function of the autonomous and the non-autonomous vibro-impact system, which excited by the additive and multiplicative Gaussian white noise. The result shows that when the amplitude increases, the non-autonomous vibro-impact system occurs stochastic P-bifurcation phenomenon. Finally, it demonstrates that the path integration solutions agree well with MC simulations.

Key words: vibro-impact, Ivanov non-smooth transformation, path integration method, probability density function

CLC Number: 

  • O211.63
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