一类含非黏滞阻尼的Duffing单边碰撞系统的激变研究

刘莉; 徐伟; 岳晓乐; 韩群

doi:10.7498/aps.62.200501

摘要

以一类含非黏滞阻尼的Duffing单边碰撞系统为研究对象, 运用复合胞坐标系方法, 分析了该系统的全局分岔特性. 对于非黏滞阻尼模型而言, 它与物体运动速度的时间历程相关, 能更真实地反映出结构材料的能量耗散现象. 研究发现, 随着阻尼系数、松弛参数及恢复系数的变化, 系统发生两类激变现象: 一种是混沌吸引子与其吸引域内的混沌鞍发生碰撞而产生的内部激变, 另一种是混沌吸引子与吸引域边界上的周期鞍(混沌鞍)发生碰撞而产生的常规边界激变(混沌边界激变), 这两类激变都使得混沌吸引子的形状发生突然改变.

关键词:

Abstract

In this paper studied is the crises in the Duffing vibro-impact oscillator with non-viscously damping by the composite cell coordinate system method. It is assumed that the non-viscously damping depends on the past history of the velocities other than the instantaneous generalized velocities. The energy dissipation behaviors of real structural materials can be preferably represented in the non-viscously damping models. Numerical simulations show that as the damping coefficient or the relaxation parameter or the recovery coefficient is varied, there appear two kinds of crises: one is the interior crisis, which results from the collision between a chaotic attractor and a chaotic saddle on the basin boundary, and the other is the regular/chaotic boundary crisis, which is due to the collision of a chaotic attractor with a periodic/chaotic saddle on its basin boundary. All the crises result in a sudden change in size and shape of the attractor.

Keywords:

non-viscously damping /
Duffing vibro-impact oscillator /
crisis /
the composite cell coordinate system method

作者及机构信息

1.
西北工业大学应用数学系, 西安 710129

基金项目: 国家自然科学基金(批准号: 11172233, 10932009, 11202160, 11302170)和西北工业大学基础研究基金(批准号: JC201266)资助的课题.

Authors and contacts

1.
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710129, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11172233, 10932009, 11202160, 11302170) and the Northwestern Polytechnical University Foundation for Fundamental Research, China (Grant No. JC201266 ).

参考文献

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[4]	Sieber J, Wagg D J, Adhikari S 2008 J. Sound Vib. 314 1
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[8]	Rong H W, Wang X D, Xu W, Fang T 2008 Acta Phys. Sin. 57 6888 (in Chinese) [戎海武, 王向东, 徐伟, 方同 2008 57 6888]
[9]	Su M B, Rong H W 2011 Chin. Phys. B 20 060501
[10]	Grebogi C, Ott E, Yorke J A 1983 Physica D 7 181
[11]	Grebogi C, Ott E, Yorke J A 1986 Phys. Rev. Lett. 57 1284
[12]	Szemplińka-Stupnicka W, Zubrzycki A, Tyrkiel E 1999 Nonlinear Dynam. 19 1936
[13]	Feng J X, Xu W 2011 Acta Phys. Sin. 60 080502 (in Chinese) [冯进钤, 徐伟 2011 60 080502]
[14]	Yue X L, Xu W, Wang L 2013 Commun. Nonlinear Sci. Numer. Simul. 18 3567
[15]	Hsu C S 1980 J. Appl. Mech. 147 931
[16]	Xu W, Sun C Y, Sun J Q 2013 Adv. Mech. 43 91 (in Chinese) [徐伟, 孙春艳, 孙建桥 2013 力学进展 43 91]
[17]	Hong L, Xu J X 1999 Phys. Lett. A 262 361
[18]	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 52 1365]
[19]	Jiang J, Xu J X 1994 Phys. Lett. A 188 137
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[22]	Li S, He Q 2011 Chin. J. Theor. Appl. Mech. 43 579 (in Chinese) [李爽, 贺群 2011 力学学报 43 579]
[23]	Yue X L, Xu W, Zhang Y 2012 Nonlinear Dynam. 69 437
[24]	Biot M A 1955 Phys. Rev. 97 14
[25]	Wagner N, Adhikari S 2003 AIAA J. 41 951

施引文献

[1]	Rayleigh J W S 1877 Theory of Sound (Vol.2) (New York: Dover Publications) pp135-216
[2]	Ruzziconi L, Litak G, Lenci S 2011 J. Vibroeng. 13 2238
[3]	Rossikhin Y A, Shitikova M V 2010 Appl. Mech. Rev. 63 010801
[4]	Sieber J, Wagg D J, Adhikari S 2008 J. Sound Vib. 314 1
[5]	Jin D P, Hu H Y 1999 Adv. Mech. 29 155 (in Chinese) [金栋平, 胡海岩 1999 力学进展 29 155]
[6]	Ding W C, Xie J H 2005 Adv. Mech. 35 513 (in Chinese) [丁旺才, 谢建华 2005 力学进展 35 513]
[7]	Lei H, Gan C B, Xie C Y 2010 Eng. Mech. 27 105 (in Chinese) [雷华, 甘春标, 谢潮涌 2010 工程力学 27 105]
[8]	Rong H W, Wang X D, Xu W, Fang T 2008 Acta Phys. Sin. 57 6888 (in Chinese) [戎海武, 王向东, 徐伟, 方同 2008 57 6888]
[9]	Su M B, Rong H W 2011 Chin. Phys. B 20 060501
[10]	Grebogi C, Ott E, Yorke J A 1983 Physica D 7 181
[11]	Grebogi C, Ott E, Yorke J A 1986 Phys. Rev. Lett. 57 1284
[12]	Szemplińka-Stupnicka W, Zubrzycki A, Tyrkiel E 1999 Nonlinear Dynam. 19 1936
[13]	Feng J X, Xu W 2011 Acta Phys. Sin. 60 080502 (in Chinese) [冯进钤, 徐伟 2011 60 080502]
[14]	Yue X L, Xu W, Wang L 2013 Commun. Nonlinear Sci. Numer. Simul. 18 3567
[15]	Hsu C S 1980 J. Appl. Mech. 147 931
[16]	Xu W, Sun C Y, Sun J Q 2013 Adv. Mech. 43 91 (in Chinese) [徐伟, 孙春艳, 孙建桥 2013 力学进展 43 91]
[17]	Hong L, Xu J X 1999 Phys. Lett. A 262 361
[18]	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 52 1365]
[19]	Jiang J, Xu J X 1994 Phys. Lett. A 188 137
[20]	He Q, Xu W, Li S 2008 Acta Phys. Sin. 57 743 (in Chinese) [贺群, 徐伟, 李爽 2008 57 743]
[21]	He Q, Xu W, Li S 2008 Acta Phys. Sin. 57 4021 (in Chinese) [贺群, 徐伟, 李爽 2008 57 4021]
[22]	Li S, He Q 2011 Chin. J. Theor. Appl. Mech. 43 579 (in Chinese) [李爽, 贺群 2011 力学学报 43 579]
[23]	Yue X L, Xu W, Zhang Y 2012 Nonlinear Dynam. 69 437
[24]	Biot M A 1955 Phys. Rev. 97 14
[25]	Wagner N, Adhikari S 2003 AIAA J. 41 951

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