摘要 研究在温度场、机械场与电磁弹性耦合作用下四边固定支撑矩形薄板的分岔与混沌运动问题。考虑温度场的影响,推导出在横向稳恒磁场和载荷共同作用下的四边固支矩形薄板的非线性磁弹性耦合振动方程。运用Melnikov函数方法,求出该问题在Smale马蹄映射下发生混沌运动的条件解。并对其进行了数值仿真,给出了该系统的分岔图、Lyapunov指数图、位移波形图、相平面轨迹图以及Poincare截面图,讨论了温度及电磁场强度对系统运动状态的影响。由仿真结果可知,通过变化温度场和电磁参数可以使系统进入混沌运动状态,或者避免混沌运动,以实现对系统振动特性的控制。
Abstract The problem of bifurcation and chaos of a four-side fixed thermo-electromagnetic-mechanic coupling rectangular thin plate is studied. Considering the effect of temperature, the vibration equations of the rectangular thin plate under the action of mechanic field, the temperature and steady transverse magnetic field are derived. By Melnikov function method, the chaos condition and judging criterion of the system under the condition of Smale horseshoe map are given. The vibration equations are simulated numerically. By some examples, the bifurcation diagram, the Lyapunov exponents diagram, the displacement wave diagram, the phase diagram and the Poincare section diagram of the system are obtained. The influences of electromagnetic field and the temperature on system vibration properties are analyzed. From the simulation, we can control the characteristics of the vibration system and can make the system become chaos motion or avoid chaos motion by changing the temperature and electromagnetic parameters.