磁场中轴向运动铁磁梁固有振动与内共振的理论及数值模拟研究

Theoretical investigation and numerical simulation of the natural vibration and internal resonance of an axially moving ferromagnetic beam in magnetic field

该文章研究了在磁场环境中做轴向运动铁磁梁的非线性双向固有频率和内共振问题。给出了梁的动能、势能以及洛伦兹力和磁体力偶的表达式,根据哈密顿原理推导出磁场中轴向运动铁磁梁的磁弹性双向耦合非线性振动方程。利用多尺度法求解耦合方程,得到双向振动固有频率表达式。进一步研究梁两个振动方向的固有频率接近1:1时的内共振问题,得到了相互耦合的特征方程。通过算例,得到了梁固有频率与振动时间、磁感应强度和轴向速度的曲线图和系统发生内共振时共振幅值的能量交换时程响应图。在此基础上,利用ABAQUS有限元分析软件计算了梁的前12阶振动模态和对应的固有频率,数值模拟结果与理论值吻合较好。

The nonlinear two-way natural frequency and internal resonance of a ferromagnetic beam moving axially in a magnetic field were studied.The expressions of kinetic energy,potential energy,Lorentz force and magnetic force couple of the beam were given.The magnetoelastic two-way coupling nonlinear vibration equation of the axially moving ferromagnetic beam in the magnetic field was derived according to the Hamilton principle.The multi-scale method was used to solve the coupled equation,and the natural frequency of the bidirectional vibration was obtained.The internal resonance of the beam with the natural frequencies of the two vibration directions close to 1∶1,was analysed,and the characteristic equations of mutual coupling were achieved.Through calculation examples,the curves of the natural frequency of the beam against the vibration time,magnetic induction intensity and axial velocity were presented and the time history response diagram of the energy exchange of resonance amplitude during the system’s internal resonance was obtained.On this basis,the first twelve vibration modes and corresponding natural frequencies of the beam were calculated by using ABAQUS finite element analysis software.The numerical simulation results are in good agreement with the theoretical values.