常磁场下铁磁矩形薄板的非线性固有振动

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摘要通过建立磁场中铁磁矩形板的力学模型，对其在常磁场下的非线性固有振动问题进行研究，并分析静载效应。根据哈密顿变分原理，得到磁场中铁磁矩形板的磁弹性非线性振动方程，给出磁化电磁力和涡流电磁力表达式。基于摄动展开法，确定静磁力作用下的静挠度和非线性扰动方程。应用伽辽金法与多尺度法，得出振动系统近似解析解和固有频率表达式。通过算例，给出了三种材料的矩形薄板固有频率随时间、磁场强度、初值、边长比等的变化规律特性曲线图。结果表明：固有频率随时间的增大，最终会趋于一定值，随上下表面磁场的变化，会呈现出对称的趋势，随边长比的增大，其频率会逐渐减小；系统呈现典型的非线性特征，解析解与数值解较为吻合。

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关键词 ：铁磁矩形板, 非线性固有振动, 常磁场, 静载效应, 多尺度法

Abstract：By establishing the mechanical model of the ferromagnetic rectangular plate in the magnetic field, the nonlinear natural vibration under the constant magnetic field is studied, and the static load effect is considered. According to the Hamiltonian variational principle, the magneto-elastic nonlinear vibration equation of the rectangular plate in magnetic field, the expressions of magnetizing electromagnetic and eddy current electromagnetic forces are given. Based on the perturbation expansion method, the static deflection and nonlinear perturbation equations under action of magnetostatic force are determined. The approximate analytical solution and natural frequency expression of the vibration system are obtained by means of Galerkin method and multi-scale method. Through numerical calculations, for the rectangular thin plates of three materials, characteristic curves of natural frequency with time, magnetic field strength, initial value, plate aspect ratios etc. are given. The results show that the natural frequency increases with time and eventually tends to a certain value. With the change of magnetic fields in top and bottom surface, it may show a symmetrical trend. As the aspect ratios increases, natural frequency may decrease gradually, which means the system presents a typical nonlinear characteristics. In addition, the analytical solution and the numerical solution obtained in paper are in good consistence.

收稿日期: 2021-11-22 出版日期: 2023-02-15

引用本文:

陶善泽1,2,胡宇达1,2. 常磁场下铁磁矩形薄板的非线性固有振动[J]. 振动与冲击, 2023, 42(3): 74-82.
TAO Shanze1,2, HU Yuda1,2. Nonlinear natural vibration of ferromagnetic rectangular thin plate under constant magnetic field. JOURNAL OF VIBRATION AND SHOCK, 2023, 42(3): 74-82.

链接本文:

http://jvs.sjtu.edu.cn/CN/ 或 http://jvs.sjtu.edu.cn/CN/Y2023/V42/I3/74

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