旋转运动导电圆板的磁弹性参数振动分析

On the Parametric Vibration of a Rotating Circular Plate in Transverse Magnetic Field

针对磁场中旋转运动圆板,在动能、应变能表达式基础上,根据哈密顿原理导出圆板的磁弹性振动方程.应用伽辽金积分法,得到横向磁场中旋转变速运动圆板的轴对称参数振动微分方程.通过坐标变换得到包含两个变系数项的马蒂厄振动方程.应用弗洛凯理论和平均法对系统的参数振动问题进行求解.通过数值计算得到周期稳定图、对应的振动响应特性图和相轨迹图.结果表明:在稳定区域内,系统的幅频曲线呈现为周期或概周期变化形式;在不稳定区内,系统的幅频响应曲线呈现为发散变化形式.

The magneto-elastic parametric vibration equations of a conductive thin circular plate rotating in magnetic field were deduced by using Hamilton principle based on the kinetic energy and strain energy expressions. The Axisymmetric parameter vibration differential equations of the rotating circular plate with varying velocity in transversal magnetic field were obtained through the Galerkin integral method. Through the coordinate transformation, the Mathieu equation containing two varying coefficients was obtained. The problem of parametric vibration system was solved by using the averaging method and the Floquet theory. Through numerical calculation, the stability, vibration response and phase trajectory curves of parametric vibration system were obtained. The results show that the amplitude-frequency curves corresponding to the stable region reveal periodic motion or quasi-periodic motion characteristics, while those corresponding to the unstable region exhibit divergence form.