磁场中轴向运动导电板磁弹性主共振分析

Analysis of Magneto-elastic Principal Resonance of Axially Moving Current-conducting Plate in Magnetic Field

给出轴向运动薄板动能、应变能以及电磁力虚功的表达形式。应用哈密顿变分原理,推得横向磁场中轴向运动条形导电薄板的非线性磁弹性振动方程。针对对边简支边界约束条件,通过位移函数的设定并应用伽辽金积分法,得到三阶位移展开形式下轴向运动板的非线性振动微分方程组。利用多尺度法对系统的主共振问题进行求解,分别得到三种频率关系条件下关于稳态解的幅频响应方程。依据李雅普诺夫稳定性理论对解的稳定性进行分析,得到相应的稳定性判别式。通过数值算例,得到轴向速度、磁感应强度、激励力幅值及板厚不同时的振幅变化规律曲线图,分析不同参量对共振幅值和非线性特征的影响,并对不同频率关系进行比较。

Based on the expressions of kinetic energy, strain energy and virtual work done by electromagnetic forces, the nonlinear magneto-elastic vibration equations of an axially moving strip thin plate in transverse magnetic field are deduced by using Hamilton principle. Based on displacement mode hypothesis, by using Galerkin method, vibration differential equations in the form of three orders displacement mode of axially moving thin plate with two simply supported are obtained. Principal resonance problem solved by using multiple scales method and amplitude-frequency response functions of steady solution in three different frequency relationships are obtained. Based on Liaypunov theory, stability of solution is analyzed and the critical condition of stability is determined. By the numerical examples, variation of amplitude curves in the cases of different axially moving velocity, magnetic induction intensity and thick of thin plate are obtained. The influence of different parameters on resonance amplitude and nonlinear characteristics are analyzed and different frequency relationship is compared.