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皇甫玉高 李群宏. 一类单侧碰撞悬臂振动系统的擦边分岔分析[J]. 力学学报, 2008, 40(6): 812-819. DOI: 10.6052/0459-1879-2008-6-2007-522
引用本文: 皇甫玉高 李群宏. 一类单侧碰撞悬臂振动系统的擦边分岔分析[J]. 力学学报, 2008, 40(6): 812-819. DOI: 10.6052/0459-1879-2008-6-2007-522
Grazing Bifurcation of a Cantilever Vibrating System with One-sided Impact[J]. Chinese Journal of Theoretical and Applied Mechanics, 2008, 40(6): 812-819. DOI: 10.6052/0459-1879-2008-6-2007-522
Citation: Grazing Bifurcation of a Cantilever Vibrating System with One-sided Impact[J]. Chinese Journal of Theoretical and Applied Mechanics, 2008, 40(6): 812-819. DOI: 10.6052/0459-1879-2008-6-2007-522

一类单侧碰撞悬臂振动系统的擦边分岔分析

Grazing Bifurcation of a Cantilever Vibrating System with One-sided Impact

  • 摘要: 与光滑动力系统不同,擦边分岔是非光滑动力系统中的一种特殊分岔行为. 局部不连续映射是研究非光滑动力系统擦边分岔的一种有力工具.对一类单侧弹性碰撞悬臂振动系统进行了擦边分岔分析. 首先建立了系统对应的局部不连续映射(ZDM)和全局Poincar\'e映射,进而在其他参数固定,碰撞间隙g为分岔参数时利用数值仿真的方法分别对原系统和对应的Poincar\'e映射进行擦边分岔分析,得到了该系统的两种不同类型的擦边分岔行为:周期1到周期2运动和周期1到混沌,这两种擦边分岔与刚性碰撞系统的情况是不相同的. 由分析可知,对于含高阶非线性项的非光滑动力系统的擦边分岔,同样可以利用局部不连续映射的方法进行研究.

     

    Abstract: Grazing bifurcation is a special bifurcation behavior ofnon-smooth dynamical systems. In the present paper, grazing bifurcations ofa cantilever system with one-sided impact is investigated. Firstly, thelocal discontinuity mapping and Poincar\'e mapping of the vibrating systemare developed. Secondly, numerical simulation with the impacting clearanceas the bifurcation parameter are carried out to obtain two types ofgrazing bifurcations of the system, which are different from those of rigidimpact systems. Finally, it is further shown that the local discontinuitymapping can be also used to investigate grazing bifurcation of non-smoothdynamical systems with high order nonlinear terms.

     

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