Codimension two bifurcation and chaos of a vibro-impact forming machine associated with 1：2 resonance case  

作　　者：Guanwei Luo Jianning Yu Jianhua Xie 

机构地区：[1]School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China [2]Department of Engineering Machanics Southwest Jiaotong University, Chengdu 610031, China

出　　处：《Acta Mechanica Sinica》2006年第2期185-198,共14页力学学报（英文版）

基　　金：The project supported by the National Natural Science Foundation of China （10572055, 50475109） and the Natural Science Foundation of Gansu Province Government of China （3ZS051-A25-030（key item）） The English text was polished by Keren Wang.

摘　　要：A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1：2 resonance is obtained, Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion, but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are also discussed.A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1：2 resonance is obtained, Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion, but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are also discussed.

关 键 词：Vibration Impact 1：2 resonance . Codimension two bifurcation 

分 类 号：O34[理学—固体力学]