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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
[1] HONG Ling, XU Jianxue. Discontinuous bifurcations of chaotic attractors in forced oscillators by generalized cell mapping digraph(GCMD)method[J]. International Journal of Bifurcation and Chaos, 2001, 11(3):723-736.
[2] WANG Liang, HUANG Mei, XU Wei, et al. The suppression of random parameter on the boundary crisis of the smooth and discontinuous oscillator system[J]. Nonlinear Dynamics, 2018, 92(3):1147-1156.
[3] PANKAJ K, NARAYANANB S. Bifurcation analysis of a stochastically excited vibro-impact Duffing-Van der Pol oscillator with bilateral rigid barriers[J]. International Journal of Mechanical Sciences, 2017, 127:103-117.
[4] 魏立祥,张建刚,南梦冉,等. 具有时滞的磁通神经元模型的稳定性及Hopf分岔[J]. 山东大学学报(理学版),2021,56(5):12-22. WEI Lixiang, ZHANG Jiangang, NAN Mengran, et al. Stability and Hopf bifurcation of a flux neuron model with time delay[J]. Journal of Shandong University(Natural Science), 2021, 56(5):12-22.
[5] HU Rongchun, GU Xudong, DENG Zicheng. Stochastic response analysis of multi-degree-of freedom vibro-impact system undergoing Markovian jump[J]. Nonlinear Dynamics, 2020, 101(2):823-834.
[6] 朱位秋.非线性随机动力学与控制:Hamilton理论体系框架[M]. 北京: 科学出版社,2003. ZHU Weiqiu. Nonlinear stochastic dynamics and control: Hamilton theory framework[M]. Beijing: Science Press, 2003.
[7] ZHU Weiqiu. Nonlinear stochastic dynamics and control in Hamiltonian formulation[J]. ASME Applied Mechanics Reviews, 2006, 59(l/2/3/4/5/6):230-248.
[8] LI Chao, XU Wei, FENG Jinqian, et al. Response probability density functions of Duffing-Van der Pol vibro-impact system under correlated Gaussian white noise excitations[J]. Physica A, 2013, 392(6):1269-1279.
[9] LUTES L D. Approximate technique for treating random vibration of hysteretic systems[J]. The Journal of the Acoustical Society of America, 1970, 48(1):299-306.
[10] SHINOZHK M. Simulation of multivariate and multidimensional random processes[J]. The Journal of the Acoustical Society of America, 1972, 49(l):357-368.
[11] WEHNER M F,WOLFER W G. Numerical evaluation of path-integral solutions to Fokker-Planck equations[J]. Physical Review A, 1983, 27(5):2663-2670.
[12] NORMARK A B. Universal limit mapping in grazing bifurcations[J]. Physical Review E, 1997, 55(1):62-68.
[13] WEN Guilin, XIE Jianhua, XU Daolin. Onset of degenerate Hopf bifurcation of a vibro-impact oscillator[J]. Journal of Applied Mechanics, 2004, 71(4):579-581.
[14] LI Chao, XU Wei, YUE Xiaole. Stochastic response of a vibro-impact system by path integration based on generalized cell mapping method[J]. International Journal of Bifurcation and Chaos, 2014, 24(10):1450129.
[15] QIAN Jiamin, CHEN Lincong. Random vibration of SDOF vibro-impact oscillators with restitution factor related to velocity under wide-band noise excitations[J]. Mechanical Systems and Signal Processing, 2021, 147:107082.
[16] ZHU Haitao. Stochastic response of a vibro-impact Duffing system under external Poisson impulses[J]. Nonlinear Dynamics, 2015, 82(1/2):1001-1013.
[17] NAESS A, MOE V. Efficient path integration methods for nonlinear dynamic systems[J]. Probabilistic Engineering Mechanics, 2000, 15(3):221-231.
[18] WEHNER M F, WOLFER W G. Numerical evaluation of path-integral solutions to Fokker-Planck equations for the time and functionally dependent coefficients[J]. Physical Review A, 1987, 35(4):1795.
[19] NAESS A, JOHNSEN J M. Response statistics of nonlinear, compliant offshore structures by the path integral solution method[J]. Probabilistic Engineering Mechanics, 1993, 8(2):91-106.
[20] YU J S, CAI G Q, LIN Y K. A new path integration procedure based on Gauss-Legendre scheme[J]. International Journal of Nonlinear Mechanics, 1997, 32(4):759-768.
[21] SUN J Q, HAU C S. The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation[J]. Journal of Applied Mechanics, 1990, 57(4):1018-1025.
[22] PIRROTTA A, SANTORO R. Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method[J]. Probabilistic Engineering Mechanics, 2011, 26(1):26-32.
[23] BUCHER C, MATTEO A D, PAOLA M D, et al. First-passage problem for nonlinear systems under Levy white noise through path integral method[J]. Nonlinear Dynamics, 2016, 85(3):1445-1456.
[24] 潘鑫凯, 王小盾, 朱海涛. 简支梁非线性响应的三维路径积分法分析[J]. 计算力学学报, 2019, 36(3):395-400. PAN Xinkai, WANG Xiaodun, ZHU Haitao. Three dimensional path integration analysis nonlinear responses of simply supported beams[J]. Chinese Journal of Computational Mechanics, 2019, 36(3):395-400.
[25] ZHURAVLEV V F. A method for analyzing vibration-impact system by means of special functions[J]. Mechanics of Solids, 1976, 11(2):23-27.
[26] IVANOV A P. Impact oscillations: linear theory of stability and bifurcations[J]. Journal of Sound and Vibration, 1994, 178(3):361-378.
[27] HUANG Dongmei, XU Wei, XIE Wenxian, et al. Dynamical properties of a forced vibration isolation system with real-power nonlinearities in restoring and damping forces[J]. Nonlinear Dynamics, 2015, 81(1/2):641-658.
[28] DIMENTBERG M F, GAIDAI O. Random vibrations with strongly inelastic impact: response PDF by the path integration method[J]. International Journal of Nonlinear Mechanics, 2009, 44(7):791-796.
[29] REN Zhicong, XU Wei. Dynamic and first passage analysis of ship roll motion with inelastic impacts via path integration method[J]. Nonlinear Dynamics, 2019, 97(4):391-402.
[30] REN Zhicong, XU Wei, QIAO Yan. Local averaged path integration method approach for nonlinear dynamic systems[J]. Applied Mathematics and Computation, 2019, 344:67-77.
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