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。
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.
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
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] TWINKLE C M, PITCHAIMANI J, RAJAMOHAN V. Free vibration modes of rectangular plate under non-uniform heating an experimental investigation[J]. Structures, 2020, 28: 1802-1817.
[2] SAMIR D, WAFIA B, HAMID S. Modal analysis of orthotropic thin rectangular plate based on analytical and finite element approaches[J]. Revue des Composites et des Matériaux Avancés-Journal of Composite and Advanced Materials, 2020, 30(5-6): 217-225.
[3] CHEN X C, CHEN L T, HUANG S B, et al. Nonlinear forced vibration of in-plane bi-directional functionally graded materials rectangular plate with global and localized geometrical imperfections[J]. Applied Mathematical Modelling, 2020, 93: 443-466.
[4] JAVANI M, KIANI Y, ESLAMI M R. Geometrically nonlinear free vibration of FG-GPLRC circular plate on the nonlinear elastic foundation[J]. Composite Structures, 2021, 261: 113515.
[5] SASADHAR D. Forced vibration of a thin non-homogeneous circular plate having a central hole[J]. Applied Mathematics and Mechanics, 1969, 11(1): 84-92.
[6] 胡宇达, 包海军. 热环境中旋转运动功能梯度圆板的强非线性固有振动[J]. 固体力学学报, 2020, 41(3): 258-272.
HU Yu-da, BAO Hai-jun. Strong nonlinear natural vibration of the functionally graded rotating circular plate in the thermal environment[J]. Chinese Journal of Solid Mechanics, 2020, 41(3): 258-272.
[7] 陈万吉, 任鹤飞. 基于新修正偶应力理论的Mindlin层合板自由振动分析[J]. 工程力学, 2016, 33(12), 31-37, 43.
CHEN Wan-ji, REN He-fei. Free vibration analysis of a laminated composite mindlin plate based on new modified couple streess theory[J]. Engineering Mechanics, 2016, 33(12): 31-37, 43.
[8] ELHAM T, NARIMAN A K, ALI I. Nonlinear vibration behavior of a carry current ferromagnetic beam plate under magnetic fields and thermal loads[J]. Journal of Vibration and Control, 2020, 26(15-16): 1276-1285.
[9] GOLUBEVA T N, KOROBKOV Y S, KHROMATOV V E. Influence of a longitudinal magnetic field on the vibration frequencies of ferromagnetic plates[J]. Russian Electrical Engineering, 2013, 84(3): 155-159.
[10] 李哲, 胡宇达. 横向磁场中旋变运动导电圆板的参强联合共振[J]. 振动与冲击, 1998, 19(3): 53-58.
LI Zhe, HU Yu-da. Magnetoelastic resonance of a conductive circular plate rotating with varying velocity under combined parametric and forced excitations[J]. Journal of Vibration and Shock, 2017, 36(23): 75-82.
[11] 王省哲, 薛标, 何宝明. 斜磁场中矩形铁磁薄板的几何非线性磁弹性行为分析[J]. 固体力学学报, 2008, 29(4): 333-340.
WANG Xing-zhe, XUE Biao, HE Bao-ming. Magnetoelastic behavior of rectangle ferromagnetic plates with geometrical nonlinearity subjected to oblique magnetic fields[J]. Chinese Journal of Solid Mechanics, 2008, 29(4): 333-340.
[12] KOU Y, WAMG L S, JIN K, et al. Theoretical and experimental investigations on the resonance frequency shift characteristic of a ferromagnetic plate[J]. European Journal of Mechanics / A Solids, 2015, 50: 112-119.
[13] HU Y D, MA B B. Magnetoelastic combined resonance and stability analysis of a ferromagnetic circular plate in alternating magnetic field[J]. Applied Mathematics and Mechanics, 2019, 40(7): 935-942.
[14] GAO Y W. Analysis on the magneto-elastic-plastic buckling/ snapping of cantilever rectangular ferromagnetic plates[J]. Acta Mechanica Solida Sinica, 2007, 20(2): 180-188.
[15] 胡宇达, 王彤. 磁场中导电旋转圆板的磁弹性非线性共振 [J]. 振动与冲击, 2016, 35(12): 177-181.
HU Yu-da, WANG Tong. Nolinear resonance of a conductive rotating circular plate in the magnetic field[J], Journal of Vibration and Shock, 2016, 35(12): 177-181.
[16] 胡宇达, 秦晓北. 磁场中旋转运动圆板磁弹性超谐-组合共振[J]. 振动与冲击, 2018, 37(12): 167-173+192.
HU Yu-da, QIN Xiao-bei. Magnetoelastic ultraharmonic- combination resonance of a rotating circil plate in magnetic field[J]. Journal of Vibration and Shock, 2018, 37(12): 167-173+192.
[17] HU Y D, Li W Q. Magnetoelastic axisymmetric multi-modal resonance and Hopf bifurcation of a rotating circular plate under aerodynamic load[J]. Nonlinear Dynamics, 2019, 97(2): 1295-1311.
[18] IRAZU L, ELEJABARRIETA M J. Vibration attenuation of conductive beams by inducing eddy currents[J]. Journal of Physics: Conference Series, 2016, 744(1): 012077.
[19] 周又和, 郑晓静. 电磁固体力学[M]. 北京: 科学出版社, 1999.
ZHOU You-he, ZHENG Xiao-jing. Electromagnetic solid mechanics [M]. Beijing: Sciences Press, 1999.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}
PDF(2921 KB)
