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随机参激下Duffing-Rayleigh碰撞振动系统的P-分岔分析

徐伟 杨贵东 岳晓乐

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随机参激下Duffing-Rayleigh碰撞振动系统的P-分岔分析

徐伟, 杨贵东, 岳晓乐

P-bifurcations of a Duffing-Rayleigh vibroimpact system under stochastic parametric excitation

Xu Wei, Yang Gui-Dong, Yue Xiao-Le
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  • 基于等效非线性系统方法和突变理论,分析了随机参激下Duffing-Rayleigh碰撞振动系统的P-分岔.首先,借助非光滑变换和狄拉克函数将原碰撞振动系统转化为一个不含速度跳的新系统;接着,利用等效非线性系统方法得到了系统的稳态概率密度函数;然后,应用突变理论,得到了随机P-分岔发生的临界参数条件的解析表达式.最后,通过典型概率密度函数曲线和图像验证了结果的正确性.
    Vibroimpact dynamics has been widely studied by experts and scholars in the fields of physics, engineering and mathematics. Most of the researches focus on vibroimpact systems under deterministic excitations by using numerical methods. However, random excitation often exists in vibroimpact system, whose roles cannot be neglected, sometimes may be quite important. Stochastic bifurcation is one of the most critical parts of stochastic dynamics, but the relevant researches about vibroimpact system are rarely seen so far due to the fact that the analytical method has its inherent difficulty. This paper aims to investigate the P-bifurcations of a Duffing-Rayleigh vibroimpact system under stochastic parametric excitation based on an equivalent nonlinear system method and the catastrophe theory. Firstly, the original Duffing-Rayleigh vibroimpact system is transformed into a new system without velocity jump by using the nonsmooth transformation method and Dirac function. Then, the equivalent nonlinear system method is introduced to obtain the stationary probability density of the response. Finally, the explicit parameter conditions for stochastic P-bifurcations are derived based on the catastrophe theory. Besides, the effect of stochastic parametric excitation on the system response is also discussed.
      通信作者: 徐伟, weixu@nwpu.edu.cn
    • 基金项目: 国家自然科学基金(批准号:11472212,11532011,11302170)资助的课题.
      Corresponding author: Xu Wei, weixu@nwpu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China(Grant Nos. 11472212, 11532011, 11302170).
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    Rong H W, Wang X D, Xu W, Meng G, Fang T 2005 Acta Phys. Sin. 54 2557(in Chinese)[戎海武, 王向东, 徐伟, 孟光, 方同2005物理学报54 2557]

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    Xu Y, Gu R C, Zhang H Q, Xu W, Duan J Q 2011 Phys. Rev. E 83 056215

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    Gu R C, Xu Y, Hao M L 2011 Acta Phys. Sin. 60 060513(in Chinese)[顾仁财, 许勇, 郝孟丽2011物理学报60 060513]

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    Hao Y, Wu Z Q 2013 Chin. J. Theor. Appl. Mech. 43 257(in Chinese)[郝颖, 吴志强2013力学学报43 257]

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    Wu Z Q, Hao Y 2013 Sci. Sin.:Phys. Mech. Astron. 43 524(in Chinese)[吴志强, 郝颖2013中国科学:物理学力学天文学43 524]

    [14]

    Wu Z Q, Hao Y 2004 Nonlinear Dynam. 36 229

    [15]

    Feng J Q, Xu W, Rong H W, Wang R 2009 Int. J. Non-Linear Mech. 44 51

    [16]

    Zhao X R, Xu W, Yang Y G, Wang X Y 2015 Commun. Nonlinear Sci. 35 166

    [17]

    Li C, Xu W, Wang L, Li D X 2013 Physica A 392 1269

    [18]

    Li C, Xu W, Yue X L 2014 Int. J. Bifurcat. Chaos 24 1450129

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    Zhuravlev V F 1976 Mech. Solids 2 23

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    Zhu W Q 1998 Random Vibration (Beijing:Science Press) p334(in Chinese)[朱位秋1998随机振动(北京:科学出版社)第334页]

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    Ling F H 1987 Catastrophe Theory and its Applications (Shanghai:Shang Hai Jiao Tong University Press) p4(in Chinese)[凌复华1987突变理论及其应用(上海:上海交通大学出版社)第4页]

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    Zhu W Q 2003 Nonlinear Stochastic Dynamics and Control:Hamilton Theory System Frame (Beijing:Science Press) p280(in Chinese)[朱位秋2003非线性随机动力学与控制––Hamilton理论体系框架(北京:科学出版社)第280页]

    [2]

    Liu X B, Chen Q 1996 Adv. Mech. 26 437(in Chinese)[刘先斌, 陈虬1996力学进展26 437]

    [3]

    Xu W, He Q, Rong H W, Fang T 2003 Acta Phys. Sin. 52 1365(in Chinese)[徐伟, 贺群, 戎海武, 方同2003物理学报52 1365]

    [4]

    Arnold L 1998 Random Dynamical Systems (Berlin, Berlin Heidelberg, New York:Springer) p1

    [5]

    Namachchivaya N S 1990 Appl. Math. Comput. 38 101

    [6]

    Huang Z L, Zhu W Q 2002 J. Sound. Vib. 2 245

    [7]

    Chen L C, Zhu W Q 2010 Chin. J. Appl. Mech. 3 517(in Chinese)[陈林聪, 朱位秋2010应用力学学报3 517]

    [8]

    Rong H W, Wang X D, Xu W, Meng G, Fang T 2005 Acta Phys. Sin. 54 2557(in Chinese)[戎海武, 王向东, 徐伟, 孟光, 方同2005物理学报54 2557]

    [9]

    Rong H W, Wang X D, Meng G, Xu W, Fang T 2006 Chin. J. Appl. Mech. 27 1373(in Chinese)[戎海武, 王向东, 孟光, 徐伟, 方同2006应用数学和力学27 1373]

    [10]

    Xu Y, Gu R C, Zhang H Q, Xu W, Duan J Q 2011 Phys. Rev. E 83 056215

    [11]

    Gu R C, Xu Y, Hao M L 2011 Acta Phys. Sin. 60 060513(in Chinese)[顾仁财, 许勇, 郝孟丽2011物理学报60 060513]

    [12]

    Hao Y, Wu Z Q 2013 Chin. J. Theor. Appl. Mech. 43 257(in Chinese)[郝颖, 吴志强2013力学学报43 257]

    [13]

    Wu Z Q, Hao Y 2013 Sci. Sin.:Phys. Mech. Astron. 43 524(in Chinese)[吴志强, 郝颖2013中国科学:物理学力学天文学43 524]

    [14]

    Wu Z Q, Hao Y 2004 Nonlinear Dynam. 36 229

    [15]

    Feng J Q, Xu W, Rong H W, Wang R 2009 Int. J. Non-Linear Mech. 44 51

    [16]

    Zhao X R, Xu W, Yang Y G, Wang X Y 2015 Commun. Nonlinear Sci. 35 166

    [17]

    Li C, Xu W, Wang L, Li D X 2013 Physica A 392 1269

    [18]

    Li C, Xu W, Yue X L 2014 Int. J. Bifurcat. Chaos 24 1450129

    [19]

    Zhuravlev V F 1976 Mech. Solids 2 23

    [20]

    Zhu W Q 1998 Random Vibration (Beijing:Science Press) p334(in Chinese)[朱位秋1998随机振动(北京:科学出版社)第334页]

    [21]

    Ling F H 1987 Catastrophe Theory and its Applications (Shanghai:Shang Hai Jiao Tong University Press) p4(in Chinese)[凌复华1987突变理论及其应用(上海:上海交通大学出版社)第4页]

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出版历程
  • 收稿日期:  2016-06-03
  • 修回日期:  2016-07-25
  • 刊出日期:  2016-11-05

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