磁场中旋转圆板的磁弹性行波动力学分析
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摘要
研究磁场作用下导电旋转圆形薄板的行波动力学特性问题。根据哈密顿原理推导出磁场作用下旋转运动圆板的磁弹性振动控制方程,根据边界条件设定行波特性振型函数,应用伽辽金积分得到了行波动力学特征方程。通过算例分析旋转运动圆板在磁场作用下的前、后行波振动频率变化和各阶模态的临界转速与振动失稳问题,并得到了圆形薄板临界转速对应各阶模态的变化规律,分析了不同磁场强度对各阶模态振动频率的影响曲线和不同振动模态阻尼的变化曲线,以及相同磁场作用下旋转圆板厚度变化对振动频率和临界转速的影响曲线。结果表明:磁场、转速、板厚等参数对旋转圆板的行波振动有显著影响。
The dynamical characteristics of the traveling waves of a conductive rotating circular plate in magnetic fields are investigated.Firstly,the governing vibration equation of a thin circular rotating plate in magnetic field is derived based on Hamilton's principle.Then,the traveling wave dynamical equation is derived through assuming proper mode shape function according to the boundary condition and using Galerkin method.The variation of the vibration frequency of the forward traveling wave and the backward traveling wave,the critical rotational speed and the instability of each mode are then investigated with some examples.The relation curves between the magnetic strength and the vibration frequency as well as the damping of each mode,the relation curve between the thickness of the plate and its vibration frequency as well as the critical speed are also presented.The results show that the traveling wave vibration would be affected significantly by the magnetic field,the speed,the thickness and other system parameters.
The dynamical characteristics of the traveling waves of a conductive rotating circular plate in magnetic fields are investigated.Firstly,the governing vibration equation of a thin circular rotating plate in magnetic field is derived based on Hamilton's principle.Then,the traveling wave dynamical equation is derived through assuming proper mode shape function according to the boundary condition and using Galerkin method.The variation of the vibration frequency of the forward traveling wave and the backward traveling wave,the critical rotational speed and the instability of each mode are then investigated with some examples.The relation curves between the magnetic strength and the vibration frequency as well as the damping of each mode,the relation curve between the thickness of the plate and its vibration frequency as well as the critical speed are also presented.The results show that the traveling wave vibration would be affected significantly by the magnetic field,the speed,the thickness and other system parameters.
引文
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[2]Southwell R V.On the free transverse vibration of a uniform circular disk clamped at its center and on the effects of rotation[J].Proceeding of the Royal Society,1922,101:133—153.
[3]Gao Yuanwen,Xu Bang,Huh H.Electromagnetothermo-mechanical behaviors of conductive circular plate subject to time-dependent magnetic fields[J].Acta Mechanics,2010,210(1-2):99—116.
[4]Hu Yuda,Li Jing.Magneto-elastic combination resonances analysis of current-conducting thin plate[J].Applied Mathematic and Mechanics(English Edition),2008,29(8):1053—1066.
[5]Hu Yuda,Li Jing.The magneto-elastic subharmonic resonances of current-conducting thin plate in magnetic field[J].Journal of Sound and Vibration,2009,319(3-5):1107—1120.
[6]胡宇达,张金志.轴向运动载流导电板磁热弹性耦合动力学方程[J].力学学报,2013,45(5):792—796.Hu Yuda,Zhang Jinzhi.Magneto-thermo-elastic coupled dynamic equations of axially moving carry current plate in magnetic field[J].Acta Mechanica Sinica,2013,45(5):792—796.
[7]Allahverdizadeh A,Naei M H,Nikkhah B M.Nonlinear free and forced vibration analysis of thin circular functionally graded plates[J].Journal of Sound and Vibration,2008,310(4-5):966—984.
[8]Chen Y R,Chen L W.Vibration and stability of rotating polar orthotropic sandwich annular plates with a viscoelastic core layer[J].Composite Structures,2007,78(1):45—57.
[9]唐亮,黄迪山,朱晓锦.旋转圆环动态特性的行波解[J].应用力学学报,2011,28(3):259—265.Tang Liang,Huang Dishan,Zhu Xiaojin.The traveling wave solutions of dynamic characteristic of the rotating circle[J].Chinese Journal of Applied Mechanics,2011,28(3):259—265.
[10]李龙飞,王省哲,周又和.黏弹性夹芯层合旋转圆板的气动弹性动力稳定性分析[J].固体力学学报,2013,34(4):325—332.Li Longfei,Wang Xingzhe,Zhou Youhe.Analysis on aeroelastic dynamic stability of a rotating sandwich annular plate with viscoelastic core layer[J].Acta Mechanic Solida Sinica,2013,34(4):325—332.
[11]赵飞云,洪嘉振,刘锦阳,等.高速旋转柔性矩形薄板的动力学建模和近似算法[J].振动工程学报,2006,19(3):416—421.Zhao Feiyun,Hong Jiazhen,Liu Jinyang,et al.Dynamic modeling and modal truncation approach for a high-speed rotating thin elastic rectangular plate[J].Journal of Vibration Engineering,2006,19(3):416—421.
[12]王彤.旋转运动圆薄板的磁弹性非线性共振[D].秦皇岛:燕山大学,2015.Wang Tong.Magneto-elastic nonlinear vibration of rotating circular plate[D].Qinhuangdao:Yanshan University,2015.
[13]Huang X.Y,Wang X,Yap F F.Feedback control of rotating disk flutter in an enclosure[J].Journal of Fluids and Structures,2004,19(7):917—932.
[14]李龙飞,王省哲.旋转薄圆盘的行波动力学特性分析[J].力学季刊,2007,28(4):631—637.Li Longfei,Wang Xingzhe.Dynamic characteristics analysis of traveling waves for rotating thin disks[J].Chinese Quarterly of Mechanics,2007,28(4):631—637.