非线性弹性地基上四边自由矩形薄板的分岔与混沌运动分析

Bifurcation and chaos analysis of thin rectangular plate on nonlinear elastic foundation

研究了非线性地基上矩形薄板的分岔与混沌运动.运用Hamiltion能量变分原理,建立了非线性弹性地基上四边自由矩形薄板的非线性振动方程.应用分离变量法和Galerkin法对方程进行求解,得到仅以W(τ)为未知函数的Mathieu-Duffing型非线性参数振动方程.在数值分析中,分别对该方程取某一连续变化的参数为变量进行分析,分别作出系统运动的分岔图以及进入混沌运动的庞加莱映射图、相平面轨迹图和时间历程曲线波形图,以揭示地基板系统进入分岔与混沌运动的规律.

The bifurcation and chaos analysis of thin rectangular plate on nonlinear foun- dation is studied. Based on the Hamilton theory, the nonlinear vibration equations for the system is established by using Segregation variable method and Galerkin method,then a nonlinear parametric vibration equation, which is similar to Mathieu-Duffing equation is obtained and its unknown parametric is only W（r）. The map of bifurcation diagrams for the system and Poincare map are made. The chaotic motion of the phase plane trajectory and the time history curve is formed when the system enter chaos are made. Therefore, the characteristics are revealed when the system is fallen into the bifurcation and chaos motion.