Solutions of Symmetries for Piezoelectric Stack Actuators
-
摘要: 研究了压电堆叠作动器的对称性,并给出了系统存在的守恒量和对称性解.以轴向运动的压电堆叠作动器为研究对象,根据其结构特点,选取位移和磁链作为广义坐标,运用能量方法,建立了压电堆叠作动器的Lagrange(拉格朗日)方程.引入位移和磁链广义坐标的无限小群变换,分别研究了压电堆叠作动器的Noether对称性和Lie对称性,给出了广义Noether恒等式、广义Killing方程、广义Noether定理和Lie定理,计算了压电堆叠作动器存在的Noether对称性和Lie对称性的生成元,并给出了相应系统存在的守恒量.最后,利用得到的守恒量,给出了压电堆叠作动器对称性解,并计算了在控制电压变化的情况下位移和速度的动态响应曲线.Abstract: The symmetries of piezoelectric stack actuators were investigated, and the solutions of conserved quantities and symmetries were given. The piezoelectric stack actuator of axial movement was considered and its structural characteristics were analyzed, accordingly the displacement and the flux linkage were selected as the generalized coordinates, then the electromechanical coupling Lagrangian equations were established with the energy method. Through the infinitesimal transformation of the displacement and flux linkage coordinates, the Noether symmetries and Lie symmetries were studied respectively, in turn the generalized Noether identity, the generalized Killing equations, the generalized Noether theorm and the Lie theorm were presented. The generators of the Noether symmetries and the Lie symmetries for the piezoelectric stack actuator were calculated, and the corresponding conserved quantities were derived. At last, with the obtained conserved quantities the solutions of symmetries were got, and the dynamic response curves of the actuator’s displacement and speed were calculated under the changing control voltage.
-
Key words:
- piezoelectric stack /
- actuator /
- symmetry /
- conserved quantity
-
[1] Chopra I, Sirohi J.Smart Structures Theory [M]. New York: Cambridge University Press, 2014. [2] Kapuria S, Agrahari J K. Two dimensional shear lag solution for stress transfer between rectangular piezoelectric wafer transducer and orthotropic host plate[J].European Journal of Mechanics—A/Solids,2016,55: 181-191. [3] ZHAO Chun-sheng.Ultrasonic Motors Technologies and Application [M]. Beijing: Science Press, 2011. [4] HE Si-yuan, Chiarot P R, Park S. A single vibration mode tubular piezoelectric ultrasonic motor[J].IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control,2011,58(5): 1049-1061. [5] LIU Ying-xiang, CHEN Wei-shan, YANG Xiao-hui, LIU Jun-kao. A T-shape linear piezoelectric motor with single foot[J].Ultrasonics,2015,56: 551-556. [6] Royston T J, Houston B H. Modeling and measurement of nonlinear dynamic behavior in piezoelectric ceramics with application to 1-3 composites[J].The Journal of the Acoustical Society of America,1998,104(5): 2814-2827. [7] Elka E, Elata D, Abramovich H. The electromechanical response of multilayered piezoelectric structures[J].Journal of Microelectromechanical Systems,2004,13(2): 332-341. [8] Low T S, Guo W. Modeling of a three-layer piezoelectric bimorph beam with hysteresis[J].Journal of Microelectromechanical Systems,1995,4(4): 230-237. [9] Weinberg M S. Working equations for piezoelectric actuators and sensors[J].Journal of Microelectromechanical Systems,1999,8(4): 529-533. [10] deVoe D L, Pisano A P. Modeling and optimal design of piezoelectric cantilever microactuators[J].Journal of Microelectromechanical Systems,1997,6(3): 266-270. [11] C?té F, Masson P, Mrad N, Cotoni V. Dynamic and static modelling of piezoelectric composite structures using a thermal analogy with MSC/NASTRAN[J].Composite Structures,2004,65(3/4): 471-484. [12] DONG Xing-jian, MENG Guang. Dynamic analysis of structures with piezoelectric actuators based on thermal analogy method[J].International Journal of Advanced Manufacturing Technology,2006,27(9): 841-844. [13] Hagood IV N W, McFarland A J. Modeling of a piezoelectric rotary ultrasonic motor[J].IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control,1995,42(2): 210-224. [14] Adriaens H J M T S, de Koning W L, Banning R. Modeling piezoelectric actuators[J].IEEE/ASME Transactions on Mechatronics,2000,5(4): 331-341. [15] Elka E, Elata D, Abramovich H. The electromechanical response of multilayered piezoelectric structures[J].Journal of Microelectromechanical Systems,2004,13(2): 332-341. [16] 梅凤翔. 李群和李代数对约束力学系统的应用[M]. 北京: 科学出版社, 1999.(MEI Feng-xiang.Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems[M]. Beijing: Science Press, 1999.(in Chinese)) [17] 秦孟兆, 王雨顺. 偏微分方程中的保结构算法[M]. 杭州: 浙江科学技术出版社, 2012.(QIN Meng-zhao, WANG Yu-shun.Structure-Preserving Algorithm for Partial Differential Equation[M]. Hangzhou: Zhejiang Science and Technology Publishing House, 2012.(in Chinese)) [18] Martins N, Torres D F M. Noether’s symmetry theorem for nabla problems of the calculus of variations[J].Applied Mathematics Letters,2010,23(12): 1432-1438. [19] FU Jing-li, CHEN Li-qun. On Noether symmetries and form invariance of mechanico-electrical systems[J].Physics Letters A,2004,331(3/4): 138-152. [20] FU Jing-li, CHEN Li-qun, CHEN Ben-yong. Noether-type theorem for discrete nonconservative dynamical systems with nonregular lattices[J].Science China: Physics, Mechanics & Astronomy,2010,53(3): 545-554. [21] Feng M, Fredlund D G. Hysteretic influence associated with thermal conductivity sensor measurements[C]//Proceedings of the 52nd Canada Geotechnical Conference and Unsaturated Soil Group from Theory to the Practice of Unsaturated Soil Mechanics. Regina, Saskatchewan, Canada:[s. n.], 1999:651-657. [22] LONG Zi-xuan, ZHANG Yi. Noether’s theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space[J].Acta Mechanica,2014,225(1): 77-90. [23] 翟晓洋, 傅景礼. 汽车车体振动系统的对称性与守恒量研究[J]. 应用数学和力学, 2015,36(12): 1285-1293.(ZHAI Xiao-yang, FU Jing-li. Study on symmetries and conserved quantities of cehicle body vibration systems[J].Applied Mathematics and Mechanics,2015,36(12): 1285-1293.(in Chinese)) [24] 杨绍普, 陈立群, 李韶华. 车辆-道路耦合系统动力学研究[M]. 北京: 科学出版社, 2012.(YANG Shao-pu, CHEN Li-qun, LI Shao-hua.Dynamics of Vehicle-Road Coupled System [M]. Beijing: Science Press, 2012.(in Chinese)) [25] Yang S P, Li S H, Lu Y J. Investigation on dynamical interaction between a heavy vehicle and road pavement[J].Vehicle System Dynamics,2010,48(8): 923-944. [26] Mei F X, Wu H B.Dynamics of Constrained Mechanical Systems [M]. Beijing: Beijing Institute of Technology Press, 2009. [27] 毛剑琴, 李琳, 张臻, 李超, 马艳华. 智能结构动力学与控制[M]. 北京: 科学出版社, 2013.(MAO Jian-qin, LI Lin, ZHANG Zhen, LI Chao, MA Yan-hua.Smart Structure Dynamics and Control [M]. Beijing: Science Press, 2013.(in Chinese))
点击查看大图
计量
- 文章访问数: 862
- HTML全文浏览量: 108
- PDF下载量: 771
- 被引次数: 0