压电圆盘的轴对称振动：改进的双勒让德多项式方法

PDF(1247 KB)

振动与冲击 ›› 2023, Vol. 42 ›› Issue (21) : 169-175.

论文

周红梅，韩康乐，禹建功，张会端，王现辉，张小明

作者信息 +

Axi-symmetric vibration of piezoelectric disc based on improved double-Legendre polynomial

ZHOU Hongmei, HAN Kangle, YU Jiangong, ZHANG Huiduan, WANG Xianhui, ZHANG Xiaoming

Author information +

文章历史 +

摘要

压电陶瓷的力电耦合特性使其广泛应用于结构的形状控制、振动和噪声控制以及结构损伤监测等诸多领域。双勒让德多项式法通过在本构方程中添加两个方向上的矩形窗函数而自动满足压电结构的边界条件，成功应用于压电圆盘的振动特性研究。但该方法在求解高阶模态时面临大量的数值积分计算，时间成本剧增。本文在传统双勒让德多项式法振动分析基础上，利用勒让德多项式性质推导了其中积分的解析表达式，使得计算效率提高90%以上。并通过与已有文献结果比对验证了该方法的正确性。最后分析了压电圆盘固有频率与径厚比的关系，结果显示频率半径积随着径厚比增大趋于稳定，且低阶更快达到稳定值。

Abstract

With force electric coupling characteristics, piezoelectric ceramics are widely used in structural shape control, vibration and noise control and structural damage monitoring, and the other fields. Double Legendre polynomial method can automatically add to boundary conditions through the rectangular window function for the constitutive equation.Which automatically meet the boundary conditions of piezoelectric structure, successful application in the study of vibration characteristics of piezoelectric disc. But calculating process of the traditional method is slow. Higher-order data need more time to compute, so it limits the double Legendre polynomial method in the application of the vibration control equation of piezoelectric materials. Based on the traditional double Legendre polynomial method about vibration analysis, using nature properties of the Legendre polynomial, the analytical integral expression is deduced, which makes the calculation efficiency by more than 90%. By compared with the existing literature the presented method is correct and valid. At last the relationship between the natural frequency and radius-thickness ratio of the piezoelectric disc are analyzed. The results show that the frequency radius product tends to be stable with the increase of diameter-thickness ratio, and the lower order reaches a stable value faster.

导出引用

周红梅，韩康乐，禹建功，张会端，王现辉，张小明. 压电圆盘的轴对称振动：改进的双勒让德多项式方法[J]. 振动与冲击, 2023, 42(21): 169-175

ZHOU Hongmei, HAN Kangle, YU Jiangong, ZHANG Huiduan, WANG Xianhui, ZHANG Xiaoming. Axi-symmetric vibration of piezoelectric disc based on improved double-Legendre polynomial[J]. Journal of Vibration and Shock, 2023, 42(21): 169-175

参考文献

[1] Su Z, Jin G, Ye T. Electro-mechanical vibration characteristics of functionally graded piezoelectric plates with general boundary conditions[J]. International Journal of Mechanical Sciences, 2018, 138-139:42-53.
[2] P.Deepak, K.Jayakumar, Satyananda Panda. Nonlinear free vibration analysis of piezoelectric laminated plate with random actuation electric potential difference and thermal loading[J].Applied Mathematical Modelling, 2021, 95:74-88.
[3] Kim D, Jin O K, Jung S I. Vibration characteristics of a piezoelectric open-shell transducer[J]. Journal of Sound & Vibration, 2012, 331(9):2038-2054.
[4] Li Y. D., Kang Y. L., Zhang N. A generalized hypergeometric function method for axisymmetric vibration analysis of a piezoelectric actuator[J]. European Journal of Mechanics, 2012, 31(1):110-116.
[5]Wenxiang Ding, M. Bavencoffe, M. Lethiecq.Accurate coupled vibration analysis of a piezoelectric ceramic cylinder by the superposition method[J]. Ultrasonics, 2021,115:106474
[6]张晓日,仲政.功能梯度压电圆板自由振动问题的三维精确分析[J].力学季刊,2005,26(1):81-86.
Zhang XiaoRi, Zhong Zheng.Three DimensionaI Exact Solution for Free Vibration of Functionally Gradient Piezoelectric Circular Plate[J].CHINESE QUARTERLY OF MECHANICS ,2005, 26(1):81-86.
[7]Ding Haojiang, Xu Rongqiao. Exact solutions for free vibrations of transversely isotropic circular plates[J]. Acta Mechanica Solida Sinica, 2000,13(2):105-111.
[8] Wang Sha, Lin Shuyu .An Exact and Practical Analyzing Model for Radial Vibration of Piezoelectric Spherical Transducers With Arbitrary Wall Thickness[J]. IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS,AND FREQUENCY CONTROL, 2021, 68(3):760-766.
[9] Razavi H, Babadi A F, Beni Y T. Free vibration analysis of functionally graded piezoelectric cylindrical nanoshell based on consistent couple stress theory[J]. Composite Structures, 2016, 160:1299-1309.
[10] Henrique Santos, Cristóvão M. Mota Soares, Carlos A. Mota Soares, et.al.A finite element model for the analysis of 3D axisymmetric laminated shells with piezoelectric sensors and actuators[J].Composite Structures,2006, 75:170-178.
[11] 张超,董世民,刘天明,等.压电陶瓷复合超声换能器径向振动特性的仿真研究[J].振动与冲击,2020, 39(21):217-225.
Zhang Chao, Dong ShiMin, Liu Tianming, et al. Simulation of radial vibration characteristics of piezoelectric ceramic composite ultrasonic transducer[J].Journal of Vibration and Shock, 2020, 39(21):217-225.
[12] Lezgy-Nazargah M, Vidal P, Polit O. An efficient finite element model for static and dynamic analyses of functionally graded piezoelectric beams[J]. Composite Structures, 2013, 104(10):71-84.
[13] Dae Jong Kim,Se Hwan Oh, Jin Oh Kim. Measurements of Radial In-plane Vibration Characteristics of Piezoelectric Disk Transducers[J]. Transactions of the Korean Society for Noise and Vibration Engineering, 2015, 25(1):13-23.
[14] E. Carrera, M. Boscolo.Classical and mixed finite elements for static and dynamic analysis of piezoelectric plates[J].International Journal for Numerical Methods in Engineering,2007,70(10): 1135-1181
[15] Grigorenko A Y, Efimova T L, Loza I A. Free vibrations of axially polarized piezoceramic hollow cylinders of finite length[J]. International Applied Mechanics, 2010, 46(6):625-633.
[16] Jan Sladek,Vladimir Sladek, Peter Stanak, et al. Analysis of the bending of circular piezoelectric plates with functionally graded material properties by a MLPG method[J]. Engineering Structures, 2013,47 :81-89.
[17] 林书玉.压电陶瓷薄圆环振子径向振动的机电等效电路及其分析[J].应用声学,2005,24(3):140-146.
Lin ShuYu. The electro-mechanical equivalent circuit of a piezoelectric ceramic thin ring in radial vibration and its analysis[J].Applied Acoustics, 2005, 24(3) :140-146.
[18]Lefebvre JE, Zhang V, Gazalet J, et al. Legendre polynomial approach for modeling free-ultrasonic waves in multilayered plates[J].Journal of Applied Physics, 1999, 85(7): 3419-3427.
[19] L Elmaimouni, J E Lefebvre, V Zhang,et al. A polynomial approach to the analysis of guided waves in anisotropic cylinders of infinite length[J]. Wave Motion, 2005, 42(2):177-189.
[20] A Raherison, F E Ratolojanahary, J E Lefebvre, et al. Legendre polynomial modeling of composite bulk acoustic wave resonators[J]. Journal of Applied Physics 2008, 104:014508.
[21]Elmaimouni L, Lefebvre E, Ratolojanahary F E, et al. Modal analysis and harmonic response of resonators: an extension of a mapped orthogonal functions technique[J]. Wave Motion, 2011, 48(1):93-104.
[22] N. Guo, The vibration characteristics of piezoelectric discs[D], Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, London, 1989.
[23] Cherif Othmani, Farid Takali, Anouar Njeh.Theoretical study on the dispersion curves of Lamb waves in piezoelectric- semiconductor sandwich plates GaAs-FGPM AlAs: Legendre polynomial series expansion[J]. Superlattices and Microstructures, 2017,106:86-101