四边固定载流矩形薄板的磁弹性稳定性分析

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。

四边简支载流矩形薄板的磁弹性动力屈曲分析

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,导出四边简支载流矩形薄板在电磁场与机械载荷共同作用下的磁弹性动力屈曲方程。应用Galerkin原理将该屈曲方程整理为Mathieu方程的标准形式,并利用Mathieu方程解的稳定区域与非稳定区域的分界,即方程系数的本征值关系,得出磁弹性问题屈曲临界状态的判别方程。通过具体算例,给出四边简支矩形板的磁弹性动力屈曲方程以及屈曲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分析讨论。

Nonlinear Dynamic Responses of the Tensioned Tether under Parametric Excitations

The nonlinear dynamic responses of the tensioned tether subjected to combined surge and heave motions of floating platform are investigated using 2-D nonlinear beam model. It is shown that if the transverse-axial coupling of nonlinear beam model and the combined surge-heave motions of platform are considered, the governing equation is not Mathieu equation any more, it becomes nonlinear Hill equation. The Hill stability chart is obtained by using the Hill's infinite determinant and harmonic balance method. A parameter M, which is the function of tether length, the surge and heave amplitude of platform, is defined. The Hill stability chart is obviously different from Mathieu stability chart which is the specific case as M=0. Some case studies are performed by employing linear and nonlinear beam model respectively. It can be found that the results differences between nonlinear and linear model are apparent.