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# Bifurcations and Chaotic Motions of a Class of Mechanical System With Parametric Excitations

[+] Author and Article Information
Liangqiang Zhou

Associate Professor
Department of Mathematics,
Nanjing University of Aeronautics and Astronautics,
Nanjing 210016, China
e-mail: zlqrex@sina.com

Fangqi Chen

Professor
Department of Mathematics,
Nanjing University of Aeronautics and Astronautics,
Nanjing 210016, China
e-mail: fangqichen@nuaa.edu.cn

Yushu Chen

Department of Mechanics,
Tianjin University,
Tianjin 300072, China
e-mail: yschen@tju.cn

1Corresponding author.

Manuscript received February 21, 2014; final manuscript received January 14, 2015; published online April 2, 2015. Assoc. Editor: Gabor Stepan.

J. Comput. Nonlinear Dynam 10(5), 054502 (Sep 01, 2015) (8 pages) Paper No: CND-14-1051; doi: 10.1115/1.4029620 History: Received February 21, 2014; Revised January 14, 2015; Online April 02, 2015

## Abstract

Bifurcations and chaotic motions of a class of mechanical system subjected to a superharmonic parametric excitation or a nonlinear periodic parametric excitation are studied, respectively, in this paper. Chaos arising from the transverse intersections of the stable and unstable manifolds of the homoclinic and heteroclincic orbits is analyzed by Melnikov's method. The critical curves separating the chaotic and nonchaotic regions are plotted. Chaotic dynamics are compared for these systems with a periodic parametric excitation or a superharmonic parametric excitation, or a nonlinear periodic parametric excitation. Especially, some new dynamical phenomena are presented for the system with a nonlinear periodic parametric excitation.

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## References

Winkler, E., 1867, “Die Lehre von der Elastizitat und Festigkeit,” Dominicus, Prague.
Pasternak, P. L., 1954, “On a New Method of Analysis of an Elastic Foundation by Means of Two Foundation Constants (in Russian), Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu I Arkhitekture, USSR,” Moscow.
Lenci, S., and Tarantino, A. M., 1996, “Chaotic Dynamics of an Elastic Beam Resting on a Winkler-Type Soil,” Chaos Solitions Fractals, 7(10), pp. 1601–1614.
Lenci, S., Menditto, G., and Tarantino, A. M., 1999, “Homoclinic and Heteroclinic Bifurcations in the Non-Linear Dynamics of a Beam Resting on an Elastic Substrate,” Int. J. Non-linear Mech., 34(4), pp. 615–632.
De, S. K., and Aluru, N. R., 2005, “Complex Oscillations and Chaos in Electrostatic Microelectromechanical Systems Under Superharmonic Excitations,” Phys. Rev. Lett., 94(20), p. 204101. [PubMed]
Ding, H., and Zu, J. W., 2013, “Periodic and Chaotic Responses of an Axially Accelerating Viscoelastic Beam Under Two-Frequency Excitations,” ASME Int. J. Appl. Mech., 5(2), p. 1350019.
Chen, X. W., Jing, Z. J., and Fu, X. L., 2014, “Chaos Control in a Pendulum System With Excitations and Phase Shift,” Nonlinear Dyn., 78(1), pp. 317–327.
Yao, M. H., and Zhang, W., 2013, “Multipulse Heteroclinic Orbits and Chaotic Dynamics of the Laminated Composite Piezoelectric Rectangular Plate,” Discrete Dyn. Nat. Soc., 2013(1–27), p. 958219.
Ng, L., and Rand, R., 2003, “Nonlinear Effects on Coexistence Phenomenon in Parametric Excitation,” Nonlinear Dyn., 31(1), pp. 73–89.
Bridge, J., Rand, R., and Sah, S. M., 2009, “Slow Passage Through Multiple Parametric Resonance Tongues,” J. Vib. Control, 15(10), pp. 1581–1600.
Wiggins, S., 1990, Introduction to Applied Non-Linear Dynamical Systems and Chaos, Springer, New York.
Guckenheimer, J., and Holmes, P. J., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer, New York.

## Figures

Fig. 2

The phase portraits for (a) p < pcr and (b) p > pcr

Fig. 1

Mechanical model

Fig. 3

The functions hhom and hhet for β =-0.95 in case of p < pcr for (a) linear parametric excitation, (b) superharmonic parametric exaction with n = 2, (c) superharmonic parametric exaction with n = 3, and (d) nonlinear parametric excitation

Fig. 4

The functions hhom and hhet for β =-0.05 in case of p < pcr for (a) linear parametric excitation, (b) superharmonic parametric exaction with n = 2, (c) superharmonic parametric exaction with n = 3, and (d) nonlinear parametric excitation

Fig. 7

The functions hhom and hhet for β = 0.95 in case of p < pcr for (a) linear parametric excitation, (b) superharmonic parametric exaction with n = 2, (c) superharmonic parametric exaction with n = 3, and (d) nonlinear parametric excitation

Fig. 9

The functions hhom and hhet for β = -0.05 in case of p > pcr for (a) linear parametric excitation, (b) superharmonic parametric exaction with n = 2, (c) superharmonic parametric exaction with n = 3, and (d) nonlinear parametric excitation

Fig. 10

The phase portraits of system (5) for the case of (a) p < pcr and (b) p > pcr

Fig. 5

The functions hhom and hhet for β = 0 in case of p < pcr for (a) linear parametric excitation, (b) superharmonic parametric exaction with n = 2, (c) superharmonic parametric exaction with n = 3, and (d) nonlinear parametric excitation

Fig. 6

The functions hhom and hhet for β = 0.05 in case of p < pcr for (a) linear parametric excitation, (b) superharmonic parametric exaction with n = 2, (c) superharmonic parametric exaction with n = 3, and (d) nonlinear parametric excitation

Fig. 8

The functions hhom and hhet for β =-0.95 in case of p > pcr for (a) linear parametric excitation, (b) superharmonic parametric exaction with n = 2, (c) superharmonic parametric exaction with n = 3, and (d) nonlinear parametric excitation

## Errata

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