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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.
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Keywords:
- vibroimpact system /
- P-bifurcation /
- stochastic parametric excitation
[1] 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
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[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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[1] 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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