针对面内轴向运动铁磁矩形薄板，研究静磁力作用且具有不同边界约束非线性系统的固有振动问题。根据电磁理论给出铁磁矩形板在外加磁场环境下所受的磁化力；基于动能和应变能的表达形式，应用哈密顿变分原理，推得轴向运动铁磁薄板的磁弹性非线性振动方程。考虑四边简支、对边简支对边自由、对边简支对边夹支的三种不同边界约束类型，通过伽辽金法进行离散，得到横向常磁场作用下薄板的非线性常微分振动方程，确定静磁力作用下板的静挠度。应用KBM法求解，得出非线性自由振动系统的位移解析解和固有频率表达式。应用Matlab软件进行数值计算，绘制了固有振动随轴向速度、磁场强度、初值等的变化规律，并进行了对比分析。结果表明：固有振动频率随轴向速度和磁场强度的增加而减小；振动频率与初值有关且随初值的增加而增大，非线性特征明显；不同材料和不同边界条件直接影响着板所受的静磁力和静挠度。
Abstract
The natural vibration of a nonlinear system with different boundary constraints under the action of static magnetic force is studied for a ferromagnetic rectangular plate moving axially in plane. The magnetization force of ferromagnetic rectangular plate under external magnetic field is given according to electromagnetic theory. Based on the expressions of kinetic energy and strain energy, the nonlinear magneto-elastic vibration equation of axially moving ferromagnetic thin plate is deduced by using Hamiltonian variational principle. Considering three different boundary constraint types of simply supported by four sides, simply supported by opposite sides and free by opposite sides, and simply supported by opposite sides and clipped by side, the nonlinear ordinary differential vibration equation of thin plate under transverse constant magnetic field was obtained by Galerkin method, and the static deflection of thin plate under static magnetic force was determined. Using KBM method, the displacement analytical solution and natural frequency expression of nonlinear free vibration system are obtained. Matlab software was used for numerical calculation, and the variation of natural vibration with axial velocity, magnetic field intensity, initial value and so on was drawn, and the comparative analysis was carried out. The results show that the natural vibration frequency decreases with the increase of axial velocity and magnetic field intensity. The vibration frequency is related to the initial value and increases with the increase of the initial value. Different materials and boundary conditions directly affect the static magnetic force and deflection of the plate.
关键词
铁磁薄板 /
非线性固有振动 /
轴向运动 /
静磁力 /
KBM法
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Key words
Ferromagnetic thin plate /
Nonlinear natural vibration /
Axial motion /
Static magnetic force /
KBM method
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