压电材料矩形板的热振动分析
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摘要
为保证压电材料结构在高温环境中安全工作,以压电材料矩形板为研究对象,根据弹性力学有限变形基本理论推导出了其在外激励和恒定温度场共同作用下的振动方程和协调方程.利用Bubnov-Galerkin原理,并引入瑞利阻尼得到热振动的非线性动力学方程.进一步运用多尺度法求得矩形板主共振时的幅频响应方程和相频响应方程.用ANSYS软件进行模态、谐响应及瞬态动力学分析,讨论了温度对横向位移的影响,分析了速度、加速度和最大应力值的变化规律以及最大应力的出现位置,结果表明温度升高和长宽比减小都会使系统的固有振动频率减小,且前者使弯曲挠度增大,后者使弯曲挠度减小.
In order to ensure the safety of piezoelectric structure in high temperature environment,piezoelectric materials rectangular plate as the research object,the non-linear vibration equation and the coordination equation in a coupled environment of external excitation and temperature field were derived based on the basic theory of finite deformation of elastic mechanics.The non-linear dynamic equations of thermal vibration were obtained by introducing the Bubnov-Galerkin principle and Rayleigh damping.Further,the amplitude frequency response equation and the phase frequency response equation of rectangular plates were obtained by multi-scale method when the primary resonance occurred.The modal,harmonic response and transient dynamics response of system were analyzed by using ANSYS software.The effect of temperature on the transverse displacement response was analyzed.The variations of velocity and acceleration and the characteristics of maximum stress value,maximum stress position were also discussed.The results show that the natural vibration frequency of the rectangular plate decreases with the increase of temperature and the decrease of the length width ratio.The former makes the bending deflection increase but the latter makes it decrease.These conclusions are valuable to engineering practice.
In order to ensure the safety of piezoelectric structure in high temperature environment,piezoelectric materials rectangular plate as the research object,the non-linear vibration equation and the coordination equation in a coupled environment of external excitation and temperature field were derived based on the basic theory of finite deformation of elastic mechanics.The non-linear dynamic equations of thermal vibration were obtained by introducing the Bubnov-Galerkin principle and Rayleigh damping.Further,the amplitude frequency response equation and the phase frequency response equation of rectangular plates were obtained by multi-scale method when the primary resonance occurred.The modal,harmonic response and transient dynamics response of system were analyzed by using ANSYS software.The effect of temperature on the transverse displacement response was analyzed.The variations of velocity and acceleration and the characteristics of maximum stress value,maximum stress position were also discussed.The results show that the natural vibration frequency of the rectangular plate decreases with the increase of temperature and the decrease of the length width ratio.The former makes the bending deflection increase but the latter makes it decrease.These conclusions are valuable to engineering practice.
引文
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[15]贾菲.压电陶瓷[M].林声和,译.北京:科学出版社,1976.
[16]杨学良,王轲.薄板的热振动特性分析[J].工业建筑,2013,43(1):334-336.Yang Xueliang,Wang Ke.Analysis of thermal vibration characteristics of thin plates[J].Industrial Constrol,2013,43(1):334-336.(in Chinese)
[17]张霄峙,陈丽华,张伟.微型压电层合板结构的振动特性研究[J].动力学与控制学报,2016,14(5):429-437.Zhang Xiaozhi,Chen Lihua,Zhang Wei.Study on the vibration characteristics of the structure of micro piezoelectric laminates[J].Journal of Dynamics and Control,2016,14(5):429-437.(in Chinese)
[2]田振国,白象忠.载流环形板的热磁弹性分析[J].中北大学学报(自然科学版),2009,23(1):8-15.Tian Zhenguo,Bai Xiangzhong.Analysis of thermalmagneto-elasticity for the current-carrying annular plate[J].Journal of North University of China(Natrual Science Edition),2009,23(1):8-15.(in Chinese)
[3]王书鹏.弹性支承梁在均布冲击荷载下的振动解耦[J].中北大学报(自然科学版),2015,37(3):67-72.Wang Shupeng.Coupled fibration of elastically supported beam under uniform impact load[J].Journal of North University of China(Natrual Science Edition),2015,37(3):67-72.(in Chinese)
[4]李世荣,苏厚德,程昌钧.热环境中粘贴压电层功能梯度材料梁的自由振动[J].应用数学和力学,2009,30(8):907-918.Li Shirong,Su Houde,Cheng Changjun.Free vibration of piezoelectric laminates in the thermal environment[J].Applied Mathematics and Mechanics,2009,30(8):907-918.(in Chinese)
[5]苏厚德,樊建领,俞树荣,等.粘贴压电层功能梯度材Euler梁热屈曲前自由振动的半解析数值法[J].航空材料学报,2014,34(6):90-97.Su Houde,Fan Jianling,Yu Shurong,et al.Pasted piezoelectric layer functional gradient material:the semianalytical numerical method of the free vibration of the pre-buckling of the Euler beam[J].Journal of Aviation Materials,2014,34(6):90-97.(in Chinese)
[6]王涛,秦荣,李双蓓.压电智能简支梁力电耦合性能分析[J].材料科学与工程学报,2007(6):961-964.Wang Tao,Qin Rong,Li Shuangbei.Analysis of mechanical and electrical coupling properties of piezoelectric smart simple supported beam[J].Journal of Materials Science and Engineering,2007(6):961-964.(in Chinese)
[7]Lee H J.Layerwise analysis of thermal shape control in graded piezoelectric beams[C]∥ASME International Mechanical Engineering Congress and Exposition,2003,21:1-9.
[8]叶文强,薛春霞.纵向冲击下弹簧连接的压电层合杆波动分析[J].中北大学学报(自然科学版),2014,35(5):530-535.Ye Wenqiang,Xue Chunxia.Wave analysis of piezoelectric laminated rods connected by springs under longitudinal impact[J].Journal of North University of China(Natrual Science Edition),2014,35(5):530-535.(in Chinese)
[9]Kapuria S,Alam N.Nonlinear Zigzag theory for buckling of hybrid piezoelectric rectangular beams under electrothermomechanical loads[J].Journal of Engineering Mechanics,2005,131(4):367-376.
[10]薛春霞,张善元,树学锋.横向磁场中大挠度金属薄板的混沌振动[J].物理学报,2010,59(9):6599-6605.Xue Chunxia,Zhang Shanyuan,Shu Xuefeng.Chaotic vibration of a thin metal plate with large deflection in transverse magnetic field[J].Acta Physica Sinica,2010,59(9):6599-6605.(in Chinese)
[11]Haozhong G,Aditi C,Jingmei L,et al.A higher order temperature theory for coupled thermo-piezoelectric-mechanical modeling of smart composites[J].International Journal of Solids and Structures,2000,37(44):6479-6497.
[12]褚亦清,李翠英.非线性振动分析[M].北京:北京理工大学出版社,1996.
[13]徐芝纶.弹性力学(下册)[M].第4版.北京:高等教育出版社,1990.
[14]王矜奉,苏文斌,王春明,等.压电振动理论及应用[M].北京:科学出版社,2011.
[15]贾菲.压电陶瓷[M].林声和,译.北京:科学出版社,1976.
[16]杨学良,王轲.薄板的热振动特性分析[J].工业建筑,2013,43(1):334-336.Yang Xueliang,Wang Ke.Analysis of thermal vibration characteristics of thin plates[J].Industrial Constrol,2013,43(1):334-336.(in Chinese)
[17]张霄峙,陈丽华,张伟.微型压电层合板结构的振动特性研究[J].动力学与控制学报,2016,14(5):429-437.Zhang Xiaozhi,Chen Lihua,Zhang Wei.Study on the vibration characteristics of the structure of micro piezoelectric laminates[J].Journal of Dynamics and Control,2016,14(5):429-437.(in Chinese)