LI Shuang-bao, ZHANG Wei. Global Bifurcations and Multi-Pulse Chaotic Dynamics of a Rectangular Thin Plate With One-to-One Internal Resonance[J]. Applied Mathematics and Mechanics, 2012, 33(9): 1043-1055. doi: 10.3879/j.issn.1000-0887.2012.09.002
 Citation: LI Shuang-bao, ZHANG Wei. Global Bifurcations and Multi-Pulse Chaotic Dynamics of a Rectangular Thin Plate With One-to-One Internal Resonance[J]. Applied Mathematics and Mechanics, 2012, 33(9): 1043-1055.

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

##### doi: 10.3879/j.issn.1000-0887.2012.09.002
• Rev Recd Date: 2012-05-06
• Publish Date: 2012-09-15
• 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.
•  [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.

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