Abstract
The 1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in 1:2 internal resonance were derived by using the direct method of normal form. In the normal forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the 4-dimension bifurcation equations were reduced to 3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on 4-dimension center manifolds.
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References
Nayfeb A H, Mook D T. Nonlinear Oscillations[M]. New York: John Wiley & Sons, 1979.
Langford W F, Zhan K, Dynamics of 1/1 resonance in vortex-induced vibration[A]. In: M P Paidoussis Ed. ASME Fundamental Aspects of Fluid-Structure Interactions[C]. PVP-Vol.247, Book,No G00728-1992.
Leblanc V G, Langford W F. Classification and unfoldings of 1:2 resonant Hopf bifurcation[J]. Arch Rational Mech Anal,1996,(136):305–357.
WU Zhi-qiang. Nonlinear normal modes and Normal Form direct method for nonlinear system having multi-degrees of freedom[D]. Ph D Thesis. Tianjin: University of Tianjin,1996. (in Chinese)
CHEN Fang-qi, WU Zhi-qiang, CHEN Yu-shu. The high codimensional bifurcations and universal unfolding problems for a class of elastic bodies under a periodic excitation[J]. Acta Mechanica Sinica,2001,33(3):286–293. (in Chinese)
CHEN Yu-shu, YANG Cai-xia. Dynamic model of a rigid-flexible coupled nonlinear system[J]. Chinese Space Sci & Tech,2000(3):7–12. (in Chinese)
CHEN Yu-shu, Leung A Y T. Bifurcation and Chaos in Engineering[M]. London: Springer-Verlag,1998.
LU Qi-shao. Bifurcation and Singularity[M]. Shanghai: Shanghai Science and Technology Press, 1995. (in Chinese)
CHEN Yu-shu. Theory of Bifurcation and Chaos in Nonlinear Vibration System[M]. Beijing: Higher Education Press,1993. (in Chinese)
LU Qi-shao. Qualitative Theory and Geometrical Methods of Ordinary Differential Equations[M]. Beijing: Beijing Aerospace University Press,1988. (in Chinese)
Arnold V I. Geometrical Methods in the Theory of Ordinary Differential Equations[M]. 2nd ed. New York: Springer-Verlag,1988.
Golubitsky M, Schaeffer D G. Singularities and Bifurcation Theory,Vol.1[M]. New York: Springer-Verlag,1985.
Chow S N, Hale S. Methods of Bifurcation Theory[M]. New York: Springer-Verlag,1992.
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Chen, Ys., Yang, Cx., Wu, Zq. et al. 1:2 Internal Resonance of Coupled Dynamic System with Quadratic and Cubic Nonlinearities. Applied Mathematics and Mechanics 22, 917–924 (2001). https://doi.org/10.1023/A:1016342326645
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DOI: https://doi.org/10.1023/A:1016342326645