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《山东大学学报(理学版)》 ›› 2022, Vol. 57 ›› Issue (3): 68-77.doi: 10.6040/j.issn.1671-9352.0.2021.418

• • 上一篇    

非光滑变换下随机碰撞系统的路径积分算法

王亮,景康康*,彭佳慧,徐伟   

  1. 西北工业大学数学与统计学院, 陕西 西安 710129
  • 发布日期:2022-03-15
  • 作者简介:王亮(1982— ),男,副教授,博士生导师,研究方向为随机动力学. E-mail:liangwang1129@nwpu.edu.cn*通信作者简介:景康康(1997— ),女,硕士研究生,研究方向为随机动力学. E-mail:jingkangkang@mail.nwpu.edu.cn
  • 基金资助:
    国家自然科学基金资助项目(11972289)

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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摘要： 碰撞系统由于物块在碰撞面速度的突变,导致不能使用传统的解析方法和数值算法直接求解此类系统,尤其是含有随机因素的碰撞振动系统。基于Ivanov非光滑变换方法将碰撞系统变换为连续系统,相较于Zhuravlev非光滑变换,Ivanov非光滑变换避免狄拉克函数的不连续性,然后结合Gauss-Legendre路径积分法,分别研究受加性或乘性高斯白噪声激励的自治与非自治碰撞振动系统的瞬态和稳态响应的概率密度函数。结果表明,对于非自治碰撞振动系统,当振幅逐渐增大时,系统发生随机P分岔现象。最后利用蒙特卡洛法验证该方法的有效性。

关键词: 碰撞振动系统, Ivanov非光滑变换, 路径积分法, 概率密度

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

中图分类号: 

  • O211.63
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