内共振条件下轴向运动梁磁弹性超谐共振研究

以载流导线激发的磁场中轴向运动梁为研究对象，同时考虑外激励力作用，推导出梁的磁弹性非线性振动方程．通过位移函数的设定和伽辽金积分法，将非线性振动方程离散为常微分方程组．采用多尺度法得到系统的近似解析解．应用Matlab 和Mathematica 软件求解幅频响应方程，并对稳态解进行稳定性判定．通过具体算例得到前两阶假设模态的响应幅值随不同参数的变化规律．结果发现：系统在内共振条件下发生超谐波共振时，二阶假设模态幅值明显小于一阶；随着外激励的增大，多值解区域范围明显缩小；随着电流强度增加，振动幅值减小，表明载流导线能够起到控制共振的作用．

Study on the Magnetoelastic Superharmonic Resonance of an Axially Moving Beam with Internal Resonance

For an axially moving beam in the magnetic field excited by current-carrying wires, the magnetoelastic nonlinear forced vibration equation of the beam under external excitation is derived, which is discretized into a group of ordinary differential equations by using displacement function and the method of Galerkin. The approximate analytical solutions of the system are obtained by the multiple-scale method. The amplitude-frequency response equations are solved by Matlab and Mathematica software, and the stability of the steady-state solutions are analyzed. Through numerical examples, the variation law of the first-twoorder assumed amplitudes with different parameters is obtained. The results show that the second order amplitude is obviously smaller than the first order. Moreover, with the increase of external excitation, the range of multi-valued solution decreases obviously. As the current intensity increases, the vibration amplitude of the first-two mode decreases, indicating that the current can control the resonance.