磁场中旋转运动圆板磁弹性超谐-组合共振
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摘要
针对磁场中的旋转运动导电圆板,基于弹性理论和电磁原理,给出旋转圆板的动能、应变能和电磁力表达式。应用Hamilton原理,推导出几何非线性下圆板的磁弹性横向振动方程。针对横向磁场中双频外载作用下圆板的轴对称振动问题进行分析,利用辽金积分法得到夹支约束下旋转板的无量纲化振动微分方程;应用多尺度法求解非线性方程,得到超谐-组合联合共振下系统的幅频响应方程,并对稳态解的稳定性进行分析,得出稳定解的判定条件。由算例给出了旋转运动圆板共振幅值随频率参数、磁场及力幅的变化规律曲线图,分析了转速、磁场等参量对共振稳态解的影响;通过分析所绘制动相平面图上奇点附近相轨迹的变化趋向,进一步阐明了稳态解的多值性和稳定性。
Aiming at a rotating conductive circle plate in magnetic field,based on the theory of elastic and electromagnetic principle,the kinetic energy,strain energy and electromagnetic forced expressions of rotating plate were obtained. Applied the Hamilton theory,the electroelastic transverse vibration equation of circular plate under the condition of geometric nonlinear was set up. The axisymmetric problem of a circular plate under the dual-frequency excitations in a transverse magnetic field was analyzed,and the non-dimensional differential equation of vibration of the rotating plate was obtained by the Galerkin integral method. The multiscale method was used to solve the nonlinear equation,and the amplitude frequency response equation of the system with ultraharmonic-combination resonance was received,and the stability of steady solution was analyzed. An example was given to show the curves of the resonance amplitude of the rotating circular plate with frequency parameters,magnetic field and force amplitude,and the effects of rotational speed and magnetic field on the steady-state solution were analyzed; by analyzing the trend of the phase trajectory near the singularity on the plot of the moving phase,the multivalue and stability of the steady state solution were further elucidated.
Aiming at a rotating conductive circle plate in magnetic field,based on the theory of elastic and electromagnetic principle,the kinetic energy,strain energy and electromagnetic forced expressions of rotating plate were obtained. Applied the Hamilton theory,the electroelastic transverse vibration equation of circular plate under the condition of geometric nonlinear was set up. The axisymmetric problem of a circular plate under the dual-frequency excitations in a transverse magnetic field was analyzed,and the non-dimensional differential equation of vibration of the rotating plate was obtained by the Galerkin integral method. The multiscale method was used to solve the nonlinear equation,and the amplitude frequency response equation of the system with ultraharmonic-combination resonance was received,and the stability of steady solution was analyzed. An example was given to show the curves of the resonance amplitude of the rotating circular plate with frequency parameters,magnetic field and force amplitude,and the effects of rotational speed and magnetic field on the steady-state solution were analyzed; by analyzing the trend of the phase trajectory near the singularity on the plot of the moving phase,the multivalue and stability of the steady state solution were further elucidated.
引文
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[20]JABBARI M,GHANNAD M,NEJAD M Z.Effect of thickness profile and fg function on rotating disks under thermal and mechanical loading[J].Journal of Mechanics,2016,32(1):35-46.
[2]郑晓静,刘信恩.铁磁导电梁式板在横向均匀磁场中的动力特性分析[J].固体力学学报,2000,21(3):243-250.ZHENG Xiaojing,LIU Xinen.Analysis on dynamic characteristics for ferromagnetic conducting plates in a transverse uniform magnetic field[J].Acta Mechanica Solida Sinica,2000,21(3):243-250.
[3]ZHENG X J,WANG X Z.A magnetoelastic theoretical model for soft ferromagnetic shell in magnetic field[J].International Journal of Solids and Structures,2003,40(24):6897-6912.
[4]HASANYAN D J,LIBRESCU L,AMBUR D R.Buckling and postbuckling of magnetoelastic flat plates carrying an electric current[J].International Journal of Solids and Structures,2006,43(16):4971-4996.
[5]HASANYAN D J,KHACHATURYAN G M,PILIPOSYAN G T.Mathematical modeling and investigation of nonlinear vibration of perfectly conductive plates in an inclined magnetic field[J].Thin-Walled Structures,2001,39(1):111-123.
[6]TAKAGI T,TANI J.New numerical analysis method of dynamic behavior of a thin plate under magnetic field considering magnetic viscous damping effect[J].International Journal of Applied Electromagnetics in Materials,1993,4(1):35-42.
[7]胡宇达,胡朋.轴向运动导电板磁弹性非线性动力学及分岔特性[J].计算力学学报,2014,31(2):180-186.HU Yuda,HU Peng.Magneto-elastic nonlinear dynamics and bifurcation of axially moving current-conducting plate[J].Chinese Journal of Computational Mechanics,2014,31(2):180-186.
[8]HU Y D,LI J.The magneto-elastic subharmonic resonance of current-conducting thin plate in magnetic field[J].Journal of Sound and Vibration,2009,319(3/4/5):1107-1120.
[9]HU Y D,WANG T.Nonlinear free vibration of a rotating circular plate under the static load in magnetic field[J].Nonlinear Dynamics,2016,85(3):1825-1835.
[10]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 Yuda,WANG Tong.Nonlinear resonance of a conductive rotating circular plate in magnetic field[J].Journal of Vibration and Shock,2016,35(12):177-181.
[11]BAYAT M,RAHIMI M,SALEEM M.One-dimensional analysis for magneto-thermo-mechanicl response in a functionally graded annular variable-thickness rotating disk[J].Applied Mathematical Modelling,2014,38(19/20):4625-4639.
[12]ZENKOUR A.On the magneto-thermo-elastic responses of fg annular sandwich disks[J].International Journal of Engineering Science,2014,75(2):54-66.
[13]高原文,周又和,郑晓静.横向磁场激励下铁磁梁式板的混沌运动分析[J].力学学报,2002,34(1):101-108.GAO Yuanwen,ZHOU Youhe,ZHENG Xiaojing.Analysis of chaotic motions of gemetrically nonliear ferromagnetic beamplates exicited by transverse magnetic fields[J].Acta Mechanica Sinica,2002,34(1):101-108.
[14]王省哲,郑晓静.铁磁梁式板磁弹性初始后屈曲及缺陷敏感性分析[J].力学学报,2006,38(1):33-40.WANG Xingzhe,ZHENG Xiaojing.Analysis on magnetoelastic initial post-buckling and sensitivity to imperfection for ferromagnetic beam-plates[J].Chinese Journal of Theoretical and Applied Mechanics,2006,38(1):33-40.
[15]BHANDARI A,KUMAR V.Ferrofluid flow due to a rotating disk in the presence of a non-uniform magnetic field[J].International Journal of Applied Mechanics&Engineering,2016,21(1):273-283.
[16]RAD A B,SHARIYAT M.Thermo-magneto-elasticity analysis of variable thickness annular fgm plates with asymmetric shear and normal loads and non-uniform elastic foundations[J].Archives of Civil&Mechanical Engineering,2016,16(3):448-466.
[17]PEI Y C,CHATWIN C,HE L,et al.A thermal boundary control method for a flexible thin disk rotating over critical and supercritical speeds[J].Meccanica,2017,52(1/2):383-401.
[18]SHARMA S,SANEHLATA Y.Finite difference solution of elastic-plastic thin rotating annular disk with exponentially variable thickness and exponentially variable density[J].Journal of Materials,2013(5):809205.
[19]HUSSAIN S,AHMAD F,SHAFIQUE M,et al.Numerical solution for accelerated rotating disk in a viscous fluid[J].Applied Mathematics,2013,4(6):899-902.
[20]JABBARI M,GHANNAD M,NEJAD M Z.Effect of thickness profile and fg function on rotating disks under thermal and mechanical loading[J].Journal of Mechanics,2016,32(1):35-46.