Abstract
A definition of the modes of a nonlinear autonomous system was developed. The existence conditions and orbits' nature of modes are given by using the geometry theory of invariant manifolds that include stable manifold theorem, center maifold theorm and sub-center manifold theorem. The Taylor series expansion was used in order to approach the sub-manifolds of the modes and obtain the motions of the mods on the manifolds. Two examples were given to demonstrate the applications.
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References
R. M. Rosenberg, On nonlinear vibrations of systems with many degrees of freedom,Advances in Applied Mechanics,9 (1966), 155–242.
R. H. Rand, Nonlinear normal modes in two degree of freedom systems,Journal of Applied Mechanics,38, 2 (1971), 561.
S. W. Shaw and C. Pierre, Normal modes for nonlinear vibratory systems,Journal of Sound and Vabration,164, 1 (1993), 85–124.
S. W. Shaw and C. Pierre, Normal modes of vibration for nonlinear continuous systems,Journal of Sound and Vibration,169, 3 (1994), 319–347.
L. Jezequel and C. H. Lamarque, Analysis of nonlinear dynamical systems by the normal form theory,Journal of Sound and Vibration,149, 3 (1991), 429–459.
Liu Liansheng, Huo Quanzhong and Huang Kelei, A method of finding the principle modes of nonlinear vibration systems and their stabilites..Applied Mathematics and Mechanics (English Ed.),8, 6 (1987), 523–532.
Liu Dongshen and Huang Kelei, A method of modal analysis using nonlinear vibration systems,Acta Mechanca Sinica,20, 1 (1988), (in Chinese)
Chen Yushu,The Modern Methods in Non-Linear Dynamics, Science Press (1992). (in Chinese).
J. Carr,Applications of Centre Manifold Theory, Springer-Verlag, New York (1981).
Al Kelley, On the Lyapunov subcentre manifold,Journal of Mathematical Analysis and Applications,18, 3 (1967), 472–478.
M. Hirsch and S. Smale,Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York (1974).
V. I. Arnold,Geometrical Method in the Theory of Ordinary Differential Equations, Springer-Verlag, New York (1988).
J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York (1986).
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Communicated by Chen Zhida
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Guojing, Z., Jianguo, W. Invariant sub-manifolds and modes of nonlinear autonomous systems. Appl Math Mech 19, 687–693 (1998). https://doi.org/10.1007/BF02452377
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DOI: https://doi.org/10.1007/BF02452377