横向磁场中轴向变速运动矩形板的参数振动

PARAMETRIC VIBRATION OF AXIALLY ACCELERATING RECTANGULAR PLATE IN TRANSVERSE MAGNETIC FIELD

针对磁场环境中轴向变速运动导电矩形薄板的磁弹性参数振动问题进行研究。在给出薄板运动的动能、应变能以及电磁力表达式基础上,应用哈密顿变分原理,推得轴向运动矩形薄板的磁弹性参数振动方程。针对横向磁场中四边简支边界约束下轴向变速运动矩形板的参数振动问题,通过位移函数的设定并应用伽辽金积分法,得到包含两个变系数项的马蒂厄振动方程。基于弗洛凯理论并应用平均法,对参数振动系统周期解的稳定性进行分析,得到稳定性判别条件。通过数值算例,给出参数振动系统周期解的稳定性图和振动响应曲线图,分析轴向速度等参量对薄板参数振动响应以及解的稳定性的影响。结果表明,稳定解区域对应的响应曲线呈现周期或概周期运动形式,不稳定解区域对应的响应曲线呈现发散形式。

The magneto-elastic parametric vibration problem of an axially moving current-conducting rectangular thin plate with pulsating speed in magnetic field was investigated. Based on the expressions of kinetic energy, strain energy and electromagnetic forces, the magneto-elastic parametric vibration equations of an axially moving rectangular thin plate were deduced by using Hamilton principle. Considering the problem of parametric vibration of an axially moving rectangular thin plate on simple supports with pulsating speed in transverse magnetic field, based on the displacement mode hypothesis, Mathieu equation contains two variable coefficients was obtained by using Galerkin method. Based on Floquet theory, by using averaging method, the stability of periodic solution of parametric vibration system was analyzed and the critical condition of stability was determined. By the numerical examples, the stability diagram and the vibration response diagram of parametric vibration system were obtained. The influence of axially moving velocity parameter on the vibration response and the stability of solution were analyzed. The results show that response curves correspond to the region of stable solution present periodic motion or quasi-periodic motion, while the region of unstable solution presents divergence form.