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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.
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Keywords:
- thin metal plate /
- large deflection /
- transverse magnetic field /
- chaotic vibration
[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 55 1678]
[10] Lei Y M, Xu W 2007 Acta Phys. Sin. 56 5103(in Chinese) [雷佑铭、徐伟 2007 56 5103]
[11] Ji Y, Bi Q S 2009 Acta Phys. Sin. 58 4431 (in Chinese) [季颖、毕勤胜 2009 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 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 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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[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 55 1678]
[10] Lei Y M, Xu W 2007 Acta Phys. Sin. 56 5103(in Chinese) [雷佑铭、徐伟 2007 56 5103]
[11] Ji Y, Bi Q S 2009 Acta Phys. Sin. 58 4431 (in Chinese) [季颖、毕勤胜 2009 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 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 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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