高维复杂碰撞振动系统的概周期环面分岔与混沌
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摘要
建立了一类三自由度含间隙碰撞振动系统的力学模型,推导了系统周期运动的解析解及Poincaré映射。基于六维Poincaré映射方法研究了系统的Hopf-flip余维二分岔和倍化分岔。在Hopf-flip余维二分岔中先发生Flip分岔后发生Hopf分岔,并展现了由环面倍化和"贝壳形"概周期吸引子向混沌演化的两种非常规路线。其后分析了系统周期运动经倍化分岔向混沌的演化的过程中,存在着十分复杂的非常规转迁过程和精彩的动力学行为。
The mechanical model of a three-degree-of-freedom vibro-impact system with clearance is established,analytical solution and Poincaré mapping of the periodic motion in the system are deduced. Based on the six-dimensional Poincaré mapping method,the Hopf-flip codimension two bifurcation and doubling bifurcation are studied. During the process of Hopf-flip codimension two bifurcation,doubling bifurcation firstly takes place,and then the Hopf bifurcation occurs,two kinds of unconventional routes from torus doubling and"shell"quasi-periodic attractor to chaos are unfolded. Afterwards,in the process of periodic motion of the system via doubling bifurcation to chaos,there exists extremely complicated unconventional process and excellent dynamics behavior.
The mechanical model of a three-degree-of-freedom vibro-impact system with clearance is established,analytical solution and Poincaré mapping of the periodic motion in the system are deduced. Based on the six-dimensional Poincaré mapping method,the Hopf-flip codimension two bifurcation and doubling bifurcation are studied. During the process of Hopf-flip codimension two bifurcation,doubling bifurcation firstly takes place,and then the Hopf bifurcation occurs,two kinds of unconventional routes from torus doubling and"shell"quasi-periodic attractor to chaos are unfolded. Afterwards,in the process of periodic motion of the system via doubling bifurcation to chaos,there exists extremely complicated unconventional process and excellent dynamics behavior.
引文
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[8]尧辉明.一类三自由度系统的稳定性研究[J].机械强度,2005,27(1):006-011.YAO HuiMing.Stability analysis of a class of three-degree-offreedom vibro-impact system[J].Journal of Mechanical Strength,2005,27(1):006-011(In Chinese).
[9]李万祥.高维含间隙振动系统的分岔与混沌研究[J].机械强度,2004,26(5):479-483.LI WanXiang.Bifurcation and chaos of multi-degree-of-freedom system with clearance[J].Journal of Mechanical Strength,2004,26(5):479-483(In Chinese).
[2]Guanwei Luo.Jianhua Xie.Bifurcation and chaos in a system with impacts[J].Physica D,2001,148:183-200.
[3]Xie Jianhua.Codimension two bifurcation and hopf bifurcation of an impacting vibrating system[J].Applied Mathmatics and Mechanics,1996,17:65-75.
[4]丁旺才,谢建华.碰撞振动系统的一类余维二分岔及T2环面分岔[J].力学学报,2003,35(4):504-507.DING WangCai,XIE JianHua.A codimension-2 bifurcation andT2torus bifurcation cation of three-degree-of-freedom vibroimpact system[J].Mechanica Sinica,2003,35(4):504-507(In Chinese).
[5]张永祥,孔贵芹,俞建宁.振动筛系统的两类余维三分岔与非常规混沌演化[J].物理学报,2008,57(10):6182-6187.ZHANG YongXiang,KONG GuiQin,YU JianNing.Two codimension 3 bifurcations and non-typaical routes to chaos of a shaker system[J].Acta Physica Sinca,2008,57(10):6182-6187(In Chinese).
[6]张永祥,孔贵芹,俞建宁.振动筛系统的Hopf-Hopf-Flip分岔与混沌演化[J].工程力学,2009,26(1):233-237.ZHANG YongXiang,KONG GuiQin,YU JianNing.Hopf-Hopf-flip bifurcation and routes to chaos of a shaker system[J].Engineering Mechanics,2009,26(1):233-237(In Chinese).
[7]张永燕.多自由度碰撞系统的动力学研究[D].兰州:兰州交通大学,2012:086-113.ZHANG YongYan.Research on nonlinear dynamics of multipledegree-of-freedom in vibro-impact system[D].Lanzhou:Lanzhou Jiaotong University,2012:086-113(In Chinese).
[8]尧辉明.一类三自由度系统的稳定性研究[J].机械强度,2005,27(1):006-011.YAO HuiMing.Stability analysis of a class of three-degree-offreedom vibro-impact system[J].Journal of Mechanical Strength,2005,27(1):006-011(In Chinese).
[9]李万祥.高维含间隙振动系统的分岔与混沌研究[J].机械强度,2004,26(5):479-483.LI WanXiang.Bifurcation and chaos of multi-degree-of-freedom system with clearance[J].Journal of Mechanical Strength,2004,26(5):479-483(In Chinese).