1:1内共振条件下矩形薄板的全局分叉和多脉冲混沌动力学

李双宝; 张伟

doi:10.3879/j.issn.1000-0887.2012.09.002

1:1内共振条件下矩形薄板的全局分叉和多脉冲混沌动力学

doi: 10.3879/j.issn.1000-0887.2012.09.002

李双宝^1,,
张伟²

中国民航大学理学院, 天津 300300;

基金项目: 国家自然科学基金资助项目(10732020；11072008；11102226)；中央高校基本科研业务基金资助项目(ZXH2011D006;ZXH2012K004)

详细信息

通讯作者:
李双宝(1978—), 男,河北人,讲师,博士(联系人. E-mail:shuangbaoli@yeah.net).

中图分类号: O322; O302
计量
- 文章访问数: 1964
- HTML全文浏览量: 210
- PDF下载量: 940
- 被引次数: 0
出版历程
- 收稿日期: 2011-07-20
- 修回日期: 2012-05-06
- 刊出日期: 2012-09-15

Global Bifurcations and Multi-Pulse Chaotic Dynamics of a Rectangular Thin Plate With One-to-One Internal Resonance

LI Shuang-bao^1
,,
ZHANG Wei²

¹College of Science, Civil Aviation University of China, Tianjin 300300, P.R.China;²College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P.R.China

摘要

摘要: 首次利用广义Melnikov方法研究了一个四边简支矩形薄板的全局分叉和多脉冲混沌动力学．矩形薄板受面外的横向激励和面内的参数激励．利用von Krmn模型和Galerkin方法得到一个二自由度非线性非自治系统用来描述矩形薄板的横向振动．在1∶1内共振条件下，利用多尺度方法得到一个四维的平均方程．通过坐标变换把平均方程化为标准形式，利用广义Melnikov方法研究该系统的多脉冲混沌动力学，并且解释了矩形薄板模态间的相互作用机理．在不求同宿轨道解析表达式的前提下，提供了一个计算Melnikov函数的方法．进一步得到了系统的阻尼、激励幅值和调谐参数在满足一定的限制条件下，矩形薄板系统会存在多脉冲混沌运动．数值模拟验证了该矩形薄板的确存在小振幅的多脉冲混沌响应．
- 矩形薄板 /
- 全局分叉 /
- 多脉冲混沌动力学 /
- 广义Melnikov方法
Abstract: Global bifurcations and multipulse chaotic dynamics for a simply supported rectangular thin plate were studied using the extended Melnikov method for the first time. The rectangular thin plate was subjected to transversal and inplane excitations. A two-degree-of-freedom nonlinear non-autonomous system governing equations of motion for the rectangular thin plate was derived using the von Karman type equation and the Galerkin’s approach. The resonant case considered here is 1∶1 internal resonance. The averaged equation was obtained by the method of multiple scales. After transforming the averaged equation into a standard form, the extended Melnikov method was employed to show the existence of multi-pulse chaotic dynamics, which coudle be applied to explain the mechanism of modal interactions of thin plates. A skill for calculating the Melnikov function was given without the explicit analytical expression of homoclinic orbits. Furthermore, the restrictions on the damping, excitations and the detuning parameters were obtained, under which multi-pulse chaotic dynamics was expected. The results of numerical simulations are also given to indicate the existence of small amplitude multipulse chaotic responses for the rectangular thin plate.
- rectangular thin plate /
- global bifurcations /
- multi-pulse chaotic dynamics /
- extended Melnikov method

HTML全文

参考文献(23)

[1]	Wiggins S. Global Bifurcations and Chaos-Analytical Methods[M]. Berlin, New York: Springer-Verlag, 1988.
[2]	Kovacic G, Wiggins S. Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation[J]. Physica D, 1992, 57(1/2):185-225.
[3]	Kaper T J, Kovacic G. Multi-bump orbits homoclinic to resonance bands[J]. Transactions of the American Mathematical Society, 1996, 348(10):3835-3887.
[4]	Camassa R, Kovacic G, Tin S K. A Melnikov method for homoclinic orbits with many pulse[J]. Archive for Rational Mechanics and Analysis, 1998, 143(2):105-193.
[5]	Haller G, Wiggins S. Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forced nonlinear Schrdinger equation[J]. Physica D, 1995, 85(3):311-347.
[6]	Haller G. Chaos Near Resonance[M]. Berlin, New York: Springer-Verlag, 1999.
[7]	Hadian J, Nayfeh A H. Modal interaction in circular plates[J]. Journal of Sound and Vibration, 1990, 142(2):279-292.
[8]	Yang X L, Sethna P R. Local and global bifurcations in parametrically excited vibrations nearly square plates[J]. International Journal of Non-Linear Mechanics, 1991, 26(2):199-220.
[9]	Yang X L, Sethna P R. Non-linear phenomena in forced vibrations of a nearly square plate: antisymmetric case[J]. Journal of Sound and Vibration, 1992, 155(3):413-441.
[10]	Feng Z C, Sethna P R. Global bifurcations in the motion of parametrically excited thin plate[J]. Nonliner Dynamics, 1993, 4(4):389-408.
[11]	Chang S I, Bajaj A K, Krousgrill C M. Nonlinear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance[J]. Nonlinear Dynamics, 1993, 4(5):433-460.
[12]	Abe A, Kobayashi Y, Yamada G. Two-mode response of simply supported, rectangular laminated plates[J]. International Journal of Non-Linear Mechanics, 1998, 33(4):675-690.
[13]	Zhang W, Liu Z M, Yu P. Global dynamics of a parametrically and externally excited thin plate[J]. Nonlinear Dynamics, 2001, 24(3):245-268.
[14]	Zhang W. Global and chaotic dynamics for a parametrically excited thin plate[J]. Journal of Sound and Vibration, 2001, 239(5):1013-1036.
[15]	Anlas G, Elbeyli O. Nonlinear vibrations of a simply supported rectangular metallic plate subjected to transverse harmonic excitation in the presence of a one-to-one internal resonance[J]. Nonlinear Dynamics, 2002, 30(1):1-28.
[16]	Zhang W, Song C Z, Ye M. Further studies on nonlinear oscillations and chaos of a symmetric cross-ply laminated thin plate under parametric excitation[J]. International Journal of Bifurcation and Chaos, 2006, 16(2):325-347.
[17]	Zhang W, Yang J, Hao Y X. Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory.[J] Nonlinear Dynamics, 2010, 59(4):619-660.
[18]	Yu W Q, Chen F Q. Global bifurcations of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation[J]. Nonlinear Dynamics, 2010, 59(1/2):129-141.
[19]	Li S B, Zhang W, Hao Y X. Multi-pulse chaotic dynamics of a functionally graded material rectangular plate with one-to-one internal resonance[J]. International Journal of Nonlinear Sciences and Numerical Simulation, 2010, 11(5):351-362.
[20]	Zhang W, Li S B. Resonant chaotic motions of a buckled rectangular thin plate with parametrically and externally excitations[J]. Nonlinear Dynamics, 2010, 62(3):673-686.
[21]	Chia C Y. Non-Linear Analysis of Plate[M]. McMraw-Hill Inc. 1980.
[22]	Timoshenko S, Woinowsky-Krieger S. Theory of Plates and Shells[M]. New York: McGraw-Hill, 1959.
[23]	Nayfeh A H, Mook D T. Nonlinear Oscillations[M]. New York: Wiley-Interscience, 1979.