横向磁场中矩形导电薄板的内共振特性分析

ANALYSIS OF THE INTERNAL RESONANCE CHARACTERISTICS OF RECTANGULAR CURRENT-CONDUCTING THIN PLATE IN TRANSVERSE MAGNETIC FIELD

在给出薄板的电磁弹性运动基本方程及电磁力表达式的基础上,得到横向磁场作用下矩形薄板的磁弹性振动方程。针对一边固定三边简支的矩形薄板,通过位移模态展开并利用Galerkin法分离时间和空间变量,得到两自由度内共振非线性振动微分方程。采用多尺度法对模态方程组进行求解,得到了系统1∶3内共振情况下前两阶振动模态相互耦合的特征方程。通过算例,得到了关于系统内共振幅值的时程响应图和相图,分别讨论了无阻尼情况下系统初值、板厚以及有阻尼情况下磁场强度对内共振特性的影响,结果表明系统呈现明显的非线性内共振特征。

Based on the basic equation of electromagnetic elastic motion and the expression of electromagnetic force,the electromagnetic vibration equation of the rectangular thin plate in transverse magnetic field is obtained. According to a rectangular plate with one side fixed and three other sides simply supported,time variable and space variable are separated by the displacement modal expansion and the method of Galerkin,and the two-degree-of-freedom internal resonance nonlinear vibration differential equations are obtained. By the method of multiple scales,the coupled characteristic equations of the first two order vibration modes are obtained under the case of 1∶3 internal resonance. By an example,the time history response diagrams and phase diagrams of the internal resonance amplitude are got. The effects of internal resonance characteristics with initial value and the thickness for undamped system and also the magnetic field intensity for damping system are discussed respectively. The results show that the system presents obvious nonlinear internal resonance characteristics.