Singular Analysis of Bifurcation of Nonlinear Normal Modes for a Class of Systems With Dual Internal Resonances

LI Xin-ye; CHEN Yu-shu; WU Zhi-qiang

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2002 > 23(10): 997-1007

LI Xin-ye, CHEN Yu-shu, WU Zhi-qiang. Singular Analysis of Bifurcation of Nonlinear Normal Modes for a Class of Systems With Dual Internal Resonances[J]. Applied Mathematics and Mechanics, 2002, 23(10): 997-1007.

Citation:

PDF( 385 KB)

Singular Analysis of Bifurcation of Nonlinear Normal Modes for a Class of Systems With Dual Internal Resonances

Department of Mechanics, Tianjin University, Tianjin 300072, P R China

Received Date: 2001-05-08
Rev Recd Date: 2002-05-10
Publish Date: 2002-10-15

Abstract

Abstract

The nonlinear normal modes(NNMs) associated with internal resonance can be classified into two kinds:uncoupled and coupled. The bifurcation problem of the coupled NNM of systems with 1:2:5 dual internal resonance is in two variables. The singular analysis of it is presented after separating the two variables by taking advantage of Maple algebra,and some new bifurcation patterns are found. Different from the NNMs of systems with single internal resonance,the number of the NNMs of systems with dual internal resonance may be more or less than the number of the degrees of freedom. At last,it is pointed out that bifurcation problems in two variables can be conveniently solved by separating variables as well as using coupling equations.
- dual internal resonance,
- nonlinear normal mode,
- mode coupling,
- mode bifurcation,
- the singularity theory

FullText(HTML)

References(17)

References

[1]	Rosenberg R M. On normal vibration of a general class of nonlinear dual-mode systems[J]. Journal of Applied Mechanics,1961,28:275-283.
[2]	Atkinson C P,Beverly T. A Study of nonlinearly related modal solutions of coupled nonlinear systems by supersition technique[J]. Journal of Applied Mechanics,1965,32:359-373.
[3]	Jonson T L,Rand R H. On the existence and bifurcation of minimal modes[J]. International Journal of Nonlinear Mechanics,1979,14:1-12.
[4]	Anand G V. Nature mode of a coupled nonlinear system[J]. International Journal of Nonlinear Mechanics,1972,7:81-91.
[5]	Yen D. On the normal modes of nonlinear dual-mass systems[J]. International Journal of Nonlinear Mechanics,1974,9:45-53.
[6]	Shaw S W,Pierre C. Normal modes for nonlinear vibrating systems[J]. Journal of Sound and Vibration,1993,164(1):85-124.
[7]	Nayfeh A,Lacabbonara W,Chin Char-Ming. Nonlinear normal modes of buckled beams:three-to-one and one-to-one internal resonances[J]. Nonlinear Dynamics,1999,18:253-273.
[8]	Nayfeh A H,Chin C,Nayfeh S A. On nonlinear normal modes of systems with internal resonance[J]. Journal of Vibration and Acoustics,1994,118:340-345.
[9]	Caughey T K,Vakakis A F,Sivo J M. Analytical study of similar normal modes and their bifurcation in a class of strongly nonlinear system[J]. International Journal of Nonlinear Mechanics,1990,25(5):521-533.
[10]	King M E,Vakakis A F. An Energy-based formulation for computing nonlinear normal modes in undamped continuous systems[J]. Journal of Vibration and Acoustics,1994,116:332-340.
[11]	Vakakis A F,Rand R H. Normal modes and global dynamics of a two-degree-of-freedom nonlinear system-Ⅰ Low energies[J]. International Journal of Nonlinear Mechanics,1992,27(5):861-874.
[12]	Vakakis A F,Vakakis R H,Rand R H. Normal modes and global dynamics of a two-degree-of-freedom nonlinear system-Ⅱ High energies[J]. International Journal of Nonlinear Mechanics,1992,27(5):875-888.
[13]	李欣业. 多自由度内共振系统的非线性模态及其分岔[D]. 博士论文.天津:天津大学,2000,49-59.
[14]	CHEN Yu-shu,Andrew Y T Leung. Bifurcation and Chaos in Engineering[M]. London:Springer-Verlag London Limited,1998,220-234.
[15]	唐云. 对称性分岔理论基础[M]. 北京:科学出版社,1998,135-141.
[16]	刘济科,赵令诚,方同. 非线性系统的模态分岔与模态局部化现象[J]. 力学学报,1995,27(5):614-617.
[17]	Rand R H. Nonlinear normal modes in a two degrees of freedom system[J]. Journal of Applied Mechanics,1971,38:561-573.