中文摘要：

本文针对四边固定载流矩形薄板，利用Mathieu方程解的稳定性，研究其在电磁场与机械荷载共同作用下的磁弹性稳定性问题。首先在载流薄板的磁弹性非线性运动方程、物理方程、几何方程、洛仑兹力表达式及电动力学方程的基础上，导出了载流薄板在电磁场与机械荷载共同作用下的磁弹性动力稳定方程，然后应用Galerkin方法将稳定方程整理为Mathieu方程的标准形式，并将薄板的动力稳定性问题归结为对Mathieu方程的求解。利用Mathieu方程的稳定解区域与非稳定解区域的分界，即方程系数λ和η的本征值关系，得出了磁弹性问题失稳临界状态的判别方程。通过具体算例，给出了四边固定载流矩形薄板的磁弹性动力失稳临界状态与相关参量之间的关系曲线，并对计算结果及其变化规律进行了分析讨论。

英文摘要：

The magnetic-elasticity stability problem of a current plate clamped at each edge, which is under the action of mechanical load in magnetic field, was studied by using the stability of Mathieu equation＇s solution in this paper. At first, based on nonlinear magnetic-elasticity kinetic equations, physical equations, geometric equations, the expressions of Lorentz forces and electrodynamics equations, the magnetic-elasticity kinetic steady equation of the problem were derived. Then, the equation was changed into the standard form of the Mathieu equation using Galerkin method. So, the stability problem is transformed to solve a Mathieu equation. By discussing the eigenvalue relation of the coefficient ,λ and η in Mathieu equation, means determining the boundary lines between the steady and unsteady solution areas of Mathieu equation, the criterion equation of the problem is presented here. As an example, a current plate clamped at each edge was solved. The curves of the relations among the parameters when the plate is in the critical situation of steady are shown in the paper. The calculated answers and the regularity of parameters variation are also discussed.