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横向磁场中大挠度金属薄板的混沌振动

薛春霞 张善元 树学锋

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横向磁场中大挠度金属薄板的混沌振动

薛春霞, 张善元, 树学锋

The chaotic vibration of a metal plate with large deflection under a transverse magnetic field

Shu Xue-Feng, Xue Chun-Xia, Zhang Shan-Yuan
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  • 利用Karman关于板的大挠度理论,考虑涡电流在板中引起的Lorentz力,导出了在横向磁场和横向载荷共同作用下薄板的非线性运动方程.借助Bubnov-Galerkin法将非线性偏微分方程转化为含三次非线性项的常微分方程.在定性分析的基础上,利用次谐轨道的Melnikov函数给出了发生Smale马蹄型混沌运动的阈值条件,进而数值计算了系统的分岔图、相应的相图、Poincaré映射和时程曲线,给出了混沌运动的数字特征.分析结果表明:磁感应强度和外载荷都会影响系统的振动特性.
    By using Karman’s plate theory of large deflection, the nonlinear equation of motion of a thin metal plate with the coaction of a transverse uniform magnetic field and a transverse load is established. These equations consider the magnetic Lorentz force induced by the eddy current. Based on the Bubnov-Galerkin method, the nonlinear partial differential equation is transformed into a third-order nonlinear ordinary differential equation. By using the sub-harmonic orbit Melnikov function method, the criterion of the Smale-horseshoe chaos is also acquired. Furthermore, the chaotic motion is numerically simulated with Matlab. The bifurcation diagram, the phase curve, the Poincaré map and the evolution curve are calculated. The digital characteristics of the chaotic motions are provided based on the analysis. The analysis results show that the magnetic induction intensity and the external load may affect the vibration of the system.
    • 基金项目: 国家自然科学基金(批准号:10772129)资助的课题.
    [1]

    Bai X Z 1996 Advances in mech. 26 389(in Chinese) [白象忠 1996 力学进展 26 389]

    [2]

    Moon F C, Holmes P J 1979 J. Sound Vib. 65 275

    [3]

    Takashi H, Toshiaki K 1996 Phys. Lett. A 21 29

    [4]

    Hasanyan D J, Khachaturyan G M, Piliposyan G T 2001 Thin-Walled Structures 39 111

    [5]

    Gaganidze E, Esquinazi P, Ziese M 2000 J. Magnetism Magnetic Materials 210 49

    [6]

    Gao Y W,Zhou Y H,Zheng X J 2002 Acta Mech. Sin. 34 101(in Chinese) [高原文、周又和、郑晓静 2002 力学学报 34 101]

    [7]

    Sheng D F,Cheng C J 2004 International Journal of Solids and Structures 41 7287

    [8]

    Samoylenko S B, Lee W K 2007 Nonlinear Dynamics 47 405

    [9]

    Yang X L, Xu W, Sun Z K 2006 Acta Phys. Sin. 55 1678(in Chinese) [杨晓丽、徐 伟、孙中奎 2006 物理学报 55 1678]

    [10]

    Lei Y M, Xu W 2007 Acta Phys. Sin. 56 5103(in Chinese) [雷佑铭、徐伟 2007 物理学报 56 5103]

    [11]

    Ji Y, Bi Q S 2009 Acta Phys. Sin. 58 4431 (in Chinese) [季颖、毕勤胜 2009 物理学报 58 4431]

    [12]

    Wang L, Xu W, Li Y 2008 Chin. Phys. B 17 2446

    [13]

    Li X C, Xu W, Li R H 2008 Chin. Phys. B 17 557

    [14]

    Xing Z C, Xu W, Rong H W, Wang B Y 2009 Acta Phys. Sin. 58 824(in Chinese) [邢真慈、徐 伟、戎海武、王宝燕 2009 物理学报 58 824]

    [15]

    Zhou P, Wei L J, Cheng X F 2009 Acta Phys. Sin. 58 5201(in Chinese) [周 平、危丽佳、程雪峰 2009 物理学报 58 5201]

    [16]

    Vol’emer A S 1959 Flexible plates and shells (Beijing: Science Press of China) (in Chinese) [沃耳密尔 A.S.著 卢文达等译 1959 柔韧板与柔韧壳(北京:中国科学出版社)]

    [17]

    Zeeman E C 1976 Bull. Inst. Math. Applic. 12 207

    [18]

    Holmes P J, Rand D A 1976 J. Sound Vib. 44 237

    [19]

    Guckenheimer J, Holmes P J 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (New York: Springer-Verlag)

    [20]

    Thompson J M T, Stewart H B 2002 Nonlinear Dynamics and Chaos (2nd edition) (New York: John Wiley & Sons)

    [21]

    Ueda Y 1980 In New Approaches to Nonlinear Problems in Dynamics p311 SIAM: Philadelphia

    [22]

    Zhang S Y,Liu Z F, Lu G Y 2009 Acta Mech. Solida Sin. 22 287(in Chinese) [张善元、刘志芳、路国运 2009 固体力学学报 22 287]

    [23]

    Liu Z R 1994 Perturbation Criteria for Chaos(Shanghai: Scientific and Technological Education Publishing House) p28 (in Chinese)[刘曾荣 1994 混沌的微扰判据(上海:上海科技教育出版社)第28页]

  • [1]

    Bai X Z 1996 Advances in mech. 26 389(in Chinese) [白象忠 1996 力学进展 26 389]

    [2]

    Moon F C, Holmes P J 1979 J. Sound Vib. 65 275

    [3]

    Takashi H, Toshiaki K 1996 Phys. Lett. A 21 29

    [4]

    Hasanyan D J, Khachaturyan G M, Piliposyan G T 2001 Thin-Walled Structures 39 111

    [5]

    Gaganidze E, Esquinazi P, Ziese M 2000 J. Magnetism Magnetic Materials 210 49

    [6]

    Gao Y W,Zhou Y H,Zheng X J 2002 Acta Mech. Sin. 34 101(in Chinese) [高原文、周又和、郑晓静 2002 力学学报 34 101]

    [7]

    Sheng D F,Cheng C J 2004 International Journal of Solids and Structures 41 7287

    [8]

    Samoylenko S B, Lee W K 2007 Nonlinear Dynamics 47 405

    [9]

    Yang X L, Xu W, Sun Z K 2006 Acta Phys. Sin. 55 1678(in Chinese) [杨晓丽、徐 伟、孙中奎 2006 物理学报 55 1678]

    [10]

    Lei Y M, Xu W 2007 Acta Phys. Sin. 56 5103(in Chinese) [雷佑铭、徐伟 2007 物理学报 56 5103]

    [11]

    Ji Y, Bi Q S 2009 Acta Phys. Sin. 58 4431 (in Chinese) [季颖、毕勤胜 2009 物理学报 58 4431]

    [12]

    Wang L, Xu W, Li Y 2008 Chin. Phys. B 17 2446

    [13]

    Li X C, Xu W, Li R H 2008 Chin. Phys. B 17 557

    [14]

    Xing Z C, Xu W, Rong H W, Wang B Y 2009 Acta Phys. Sin. 58 824(in Chinese) [邢真慈、徐 伟、戎海武、王宝燕 2009 物理学报 58 824]

    [15]

    Zhou P, Wei L J, Cheng X F 2009 Acta Phys. Sin. 58 5201(in Chinese) [周 平、危丽佳、程雪峰 2009 物理学报 58 5201]

    [16]

    Vol’emer A S 1959 Flexible plates and shells (Beijing: Science Press of China) (in Chinese) [沃耳密尔 A.S.著 卢文达等译 1959 柔韧板与柔韧壳(北京:中国科学出版社)]

    [17]

    Zeeman E C 1976 Bull. Inst. Math. Applic. 12 207

    [18]

    Holmes P J, Rand D A 1976 J. Sound Vib. 44 237

    [19]

    Guckenheimer J, Holmes P J 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (New York: Springer-Verlag)

    [20]

    Thompson J M T, Stewart H B 2002 Nonlinear Dynamics and Chaos (2nd edition) (New York: John Wiley & Sons)

    [21]

    Ueda Y 1980 In New Approaches to Nonlinear Problems in Dynamics p311 SIAM: Philadelphia

    [22]

    Zhang S Y,Liu Z F, Lu G Y 2009 Acta Mech. Solida Sin. 22 287(in Chinese) [张善元、刘志芳、路国运 2009 固体力学学报 22 287]

    [23]

    Liu Z R 1994 Perturbation Criteria for Chaos(Shanghai: Scientific and Technological Education Publishing House) p28 (in Chinese)[刘曾荣 1994 混沌的微扰判据(上海:上海科技教育出版社)第28页]

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  • PDF下载量:  670
  • 被引次数: 0
出版历程
  • 收稿日期:  2009-09-09
  • 修回日期:  2009-12-30
  • 刊出日期:  2010-09-15

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