对边简支载流矩形薄板在电磁场中的稳定性分析

MAGNETIC-ELASTISITY STABILITY CRITERION OF THIN CURRENT PLATE SIMPLY SUPORTED AT EACH EDGE

针对对边简支、另一对边固定载流矩形薄板,利用Mathieu方程解的稳定性,研究在电磁场与机械荷载共同作用下的磁弹性稳定性问题。在导出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的基础上,应用Galerkin原理将稳定方程整理为Mathieu方程的标准形式,并将其归结为对Mathieu方程的求解问题。利用Mathieu方程系数λ、η的本征值关系,得出载流薄板磁弹性动力稳定临界状态的判别方程,并给出当η为小激励时的稳定区域图,以及Mathieu方程稳定解区域和非稳定解区域的分界。最后通过具体数值算例,给出该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。研究结果表明,变化电磁场和通电电流的参数,可以改变电磁力的状态,从而达到控制载流薄板的变形,应力、应变状态及其稳定性的目的。

For a current carrying rectangular plate which is simply supported at two opposite boundaries and the other two are fixed, the magnetic-elasticity steady problem is studied. Based on deriving the magnetic-elasticity dynamic buckling equation of the plate applied mechanical load in a magnet field, the buckling equation is changed into the standard form of the Mathieu equation by using Galerkin method. Thus, the buckling problem comes down to solve the Mathieu equation. The criterion equation of the plate at the critical state of magnetic elasticity buckling is obtained with the analysis on the eigen value relations between the coefficients λ and η in the Mathieu equation. The map and the boundary lines of the steady areas of the Mathieu equation are shown when η is small exciter. At last, the curves of the relations among the critical state of magnetic elasticity dynamic buckling problem of the plate and the relative parameters are drawn out through a calculating example. The conclusions show that the electrical and magnetic forces may be controlled by changing the parameters of the current and the magnetic field so that the aim for controlling the deformation, stress, strain and the stability of the current carrying plate is achieved.