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Vibro-impact dynamics near a strong resonance point

ISSN号：0567-7718
期刊名称：《力学学报：英文版》
时间：0
分类：O3[理学—力学]
作者机构：[1]School of Mechatronic Engineering,Lanzhou Jiaotong University,Lanzhou 730070, China, [2]Department of Engineering Mechanics,Southwest Jiaotong University, Chengdu 610031, China, [3]School of Mathematics, Physics and Software Engineering,Lanzhou Jiaotong University, Lanzhou 730070, China
相关基金：The project supported by National Natural Science Foundation of China （50475109, 10572055）, Natural Science Foundation of Gansu Province Government of China （3ZS061-A25-043（key item））. The English text was polished by Keren Wang.

作者： Guanwei Luo, Yanlong Zhang, Jianhua Xie, Jiangang Zhang

关键词：振动, 撞击, 共振, 动力学, Vibration, Impact, Resonance, Periodicmotion, Bifurcation

中文摘要：

有影响的二个典型振动系统被考虑其一是 two-degree-of-freedom 影响非强迫的僵硬身体的振动系统，另外的影响一个僵硬振幅站。如此的模型与重复影响在机械系统的动力学的研究起一个重要作用。在 vibro 影响系统的固定的点的二参数的分叉，把强壮的回声与 1:4 联系了，被为地图使用歧管的中心和正常表格方法分析。单个影响的周期的运动和 vibro 影响系统的 Poincar é地图是导出的经分解。稳定性和一个单个影响的周期的运动的本地分叉被使用 Poincar é地图分析。一个中心歧管定理技术被使用把 Poincar é地图归结为二维的，和正常形式地图因为 1:4 回声被获得。在为 1:4 回声的分叉点附近，二个 vibro 影响系统的本地行为被学习。在那里为 1:4 在分叉附近削尖强壮的回声存在时期一单个影响的运动和正切(褶层) 的 Neimark Sacker 分叉时期 4 四影响的运动的分叉，等等。从模拟的结果显示出 vibro 影响系统的动力学的一些有趣的特征：也就是，“圆由与僵绳的稳定、不稳定的 separatrices 一致形成了的异种诊所”， T _在, T _在和 T _外面类型正切(褶层)分叉，与时期四联系的伪周期的影响轨道四影响并且时期八八影响的运动，等等。到混乱的时期 4 四影响的运动的不同线路被数字模拟， vibro 影响系统在展出很复杂的伪周期的影响运动获得。

英文摘要：

Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1 ： 4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map for 1：4 resonance is obtained. Local behavior of two vibro-impact systems, near the bifurcation points for 1：4 resonance, are studied. Near the bifurcation point for 1：4 strong resonance there exist a Neimark-Sacker bifurcation of period one single-impact motion and a tangent （fold） bifurcation of period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact systems： namely, the ＂heteroclinic＂ circle formed by coinciding stable and unstable separatrices of saddles, Tin, Ton and Tout type tangent （fold） bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the vibro-impact systems exhibit very complicated quasiperiodic impact motions.

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