各种板边条件下大挠度圆板的全局分岔和混沌

GLOBAL BIFURCATION AND CHAOS OF A CIRCULAR PLATE FOR VARIOUS BOUNDARY CONDITIONS

计及几何非线性 ,在各种板边条件下 ,建立了圆板受迫振动时的非线性动力学方程 .采用 Galerkin法 ,将圆板的非线性动力学偏微分方程简化成三种标准类型的 Duffing方程 .利用 Melnikov函数法 ,确定了可能发生混沌的临界条件 .通过数值仿真得到了单谐波运动、亚谐波运动和混沌运动 ,给出了相平面轨迹、时程曲线 .由分岔图判定系统是否处于混沌状态 ,对系统的全局分岔进行了讨论 ,实验和数值计算都发现了突变 (或跳跃 )现象 .

The nonlinear dynamic equation of a circular plate for various boundary conditions is derived,with the geometric nonlinear effects taken into account.By using Galerkins method,the governing partial differential equation was reduced to three standard types of Duffings equation.The critical conditions which could lead to chaotic motion were given by Melnikovs function.The solution of simple harmonic motion,many subharmonic motions and chaotic motion were found numerically.A demonstrative example was discussed through phase portrait,time history and bifurcation diagram.Finally the global bifurcation was discussed.Phenomena of catastrophe (jump) were observed both experimentally and numerically.