3自由度单碰振动系统的Lyapunov指数谱和周期泡现象  

Spectrum of Lyapunov Exponents and the Phenomenon of Periodic Bubble for a 3-DOF Single Vibro-impact System

作　　者：王栋[1] 张艳龙[1] 王丽[2] WANG Dong;ZHANG Yanlong;WANG Li(School of Mechatronic Engineering,Lanzhou Jiaotong University,Lanzhou 730070,China;School of Mathematics,Lanzhou City University,Lanzhou 730070,China)

机构地区：[1]兰州交通大学机电工程学院,兰州730070 [2]兰州城市学院数学学院,兰州730070

出　　处：《噪声与振动控制》2018年第6期24-29,共6页Noise and Vibration Control

基　　金：国家自然科学基金资助项目(11302092)

摘　　要：给出了一类3自由度单碰振动系统运动方程和状态方程,引入局部映射得到Poincaré映射和Jacobi矩阵,通过Gram-Schmidt正交化和范数归一化得出该系统Lyapunov指数谱的计算方法。通过分岔图、相图和Lyapunov指数谱的表现形式,通过数值仿真分析该系统在一定参数下的动力学行为。结果表明,利用Lyapunov指数谱可以有效判断系统的稳定性,同时发现系统在一定参数下存在周期泡和混沌泡的现象,并且随着质量比的减小,系统运动由倍周期分岔序列进入混沌运动,导致周期泡和混沌泡现象的消失。Kinematic equations and state equations of a 3-DOF single vibro-impact system are given. Poincax6 map and Jacobi matrixes are obtained by introducing local mapping. The method of calculating the spectrum of Lyapunov exponents of the system is presented based on Gram-Schmidt orthogonalization and norm normalization. Under the certain parameters, the dynamical behaviors of the system are analyzed in the form of bifurcation diagrams, phase diagrams and the spectrum of Lyapunov exponents by means of numerical simulation. The results show that the spectrum of Lyapunov exponents can effectively judge stability of the system, and there are the phenomena of the periodic bubble and chaotic bubble under certain system parameters. With decreasing of the mass ratio, the motion of the system will enter the chaotic motion from the double periodic bifurcation sequence and lead to the disappearance of the periodic bubble and chaotic bubble phenomena.

关 键 词：振动与波 LYAPUNOV指数谱 局部映射 碰撞振动 周期泡 混沌泡 

分 类 号：O32[理学—一般力学与力学基础]