横向磁场中导电条形板的1:3内共振

On the 1:3 Internal Resonance of Current-Conducting Strip Plate in Transverse Magnetic Field

本文针对横向磁场中的导电条形板,给出横向恒定磁场环境下条形板的非线性磁弹性振动微分方程和所受电磁力的表达式.对于一边固定一边简支的条形板,通过位移模态展开,分离时间变量和空间变量,利用Galerkin积分法得到系统两自由度非线性内共振振动微分方程.采用多尺度法,得到系统1:3内共振情况下关于模态振幅和相位的调制方程.通过算例,得到了系统内共振时一阶模态和二阶模态幅值的时间历程响应图和相平面图,分别讨论了系统初值、板厚以及磁场强度对系统内共振特性的影响,结果表明系统呈现明显的非线性内共振特征,磁场强度对内共振有明显的抑制作用.

For a conductive strip plate in transverse magnetic field, the nonlinear magneto-elastic vibration differential equation and the expression of electromagnetic force are obtained. For a strip plate with one side fixed and another side simply supported, time variable and space variable are separated by means of the displacement modal expansion, and the nonlinear internal resonance vibration equations of two-degree-of-freodom systems are deduced by the Galerkin integral method. The amplitude-phase modulation equations are derived for the case of 1：3 internal resonance using the multi-scale method. Through numerical examples, the time history response diagrams and the phase diagrams of the I st and 2nd modes of intemal resonance amplitude are obtained. The effects of initial values, plate thickness and magnetic intensity on the internal resonance are respectively discussed. As the results indicate, the system shows obvious nonlinear internal resonance characteristics and the internal resonance is inhibited by the magnetic field.