压电结构迟滞非线性与冲击特性的实验研究
|
摘要
压电材料以其较强和稳定的压电性能成为应用最广泛的智能材料,其中超声电机和压电发电就是压电材料在工程中的典型应用。由于压电材料本身存在迟滞非线性,给压电材料的应用带来一定困难。因此,研究压电材料的迟滞非线性等动力学特性对压电材料的有效应用具有重要意义。
本文围绕压电材料三次多项式迟滞模型进行了定性分析和改进,对压电材料的迟滞特性进行了实验研究;对超声电机中存在的非线性现象进行了实验测量,并对结果进行了分析;根据设计制作的冲击测试装置对压电陶瓷叠片的冲击响应特性开展了实验研究。主要工作和结论如下:
(1)简述了压电材料的发展、性质以及在超声电机和压电发电中的应用,并阐述了压电材料迟滞非线性、超声电机非线性动力学和压电结构冲击特性的实验研究进展。
(2)通过与他人建立的迟滞模型进行定性比较和实验研究,得出三次多项式迟滞模型的适用范围及影响迟滞的因素,并提出了改进的压电迟滞模型。
(3)用实验方法测量了旋转行波型超声电机的反馈电压和转速,揭示超声电机整机模型动力学响应的多解现象。
(4)通过分析和测量压电元件的尺寸、模态对发电量的影响,确定冲击测试装置整体的尺寸;并测量了冲击力及压电元件的最大承受力,以保证压电元件正常工作;最后测量了压电元件的发电功率,为在考虑压电迟滞下压电发电的理论研究做准备工作。
通过本文的研究,为三次多项式迟滞模型和超声电机整机模型的应用提供了依据,为理论分析压电结构的动力学响应和电学响应提供了基础。
Piezoelectric materials are widely used as smart materials for their stable piezoelectric properties. Ultrasonic motors and piezoelectric power generators are two typical applications of piezoelectric materials in engineering. As the undesired hysteretic nonlinearities being inherent in piezoelectric materials, it is important to research the hysteretic nonlinear dynamics of piezoelectric materials.
In this paper, the cubic polynomial model was analyzed and improved firstly, and some experiments on hysteresis of piezoelectric ceramic laminate were carried out. Secondly, the nonlinear behavior of ultrasonic motor was measured and analyzed. Finally, by using impact test equipment, some experimental researches were done to investigate the impact responses of the piezoelectric ceramic laminate. The major work of the paper was summarized as follows:
(1) The properties of piezoelectric materials and their applications in ultrasonic motors and power generator were introduced briefly. The development of hysteretic nonlinearities of piezoelectric materials, nonlinear dynamics of ultrasonic motors, as well as impact properties of piezoelectric structures is summarized.
(2) The applicable condition and related parameters of the cubic polynomial hysteresis model were analyzed qualitatively by comparing it with the other hysteresis model. Based on the above analysis, one improved piezoelectric hysteresis model was proposed.
(3) With rotary traveling wave ultrasonic motor’s feedback voltage and rotor speed measured, the dynamic responses of the stator and the rotor were measured and some nonlinear phenomena were found.
(4) The impact test device was invented by analyzing and measuring the size and mode of piezoelectric ceramic element. The impact responses of piezoelectric ceramic laminates and the maximum carrying capacity of piezoelectric ceramic elements were measured to ensure piezoelectric ceramic element work normally. The generating power of piezoelectric structure was tested. All these results would make a foundation for further theoretical research of piezoelectric power generators in future.
The results given in this thesis provide a basis for the applications of the cubic polynomial hysteresis model and the model of whole system of ultrasonic motor, and make a foundation for theoretical research of mechanical and electrical responses of piezoelectric structure.
本文围绕压电材料三次多项式迟滞模型进行了定性分析和改进,对压电材料的迟滞特性进行了实验研究;对超声电机中存在的非线性现象进行了实验测量,并对结果进行了分析;根据设计制作的冲击测试装置对压电陶瓷叠片的冲击响应特性开展了实验研究。主要工作和结论如下:
(1)简述了压电材料的发展、性质以及在超声电机和压电发电中的应用,并阐述了压电材料迟滞非线性、超声电机非线性动力学和压电结构冲击特性的实验研究进展。
(2)通过与他人建立的迟滞模型进行定性比较和实验研究,得出三次多项式迟滞模型的适用范围及影响迟滞的因素,并提出了改进的压电迟滞模型。
(3)用实验方法测量了旋转行波型超声电机的反馈电压和转速,揭示超声电机整机模型动力学响应的多解现象。
(4)通过分析和测量压电元件的尺寸、模态对发电量的影响,确定冲击测试装置整体的尺寸;并测量了冲击力及压电元件的最大承受力,以保证压电元件正常工作;最后测量了压电元件的发电功率,为在考虑压电迟滞下压电发电的理论研究做准备工作。
通过本文的研究,为三次多项式迟滞模型和超声电机整机模型的应用提供了依据,为理论分析压电结构的动力学响应和电学响应提供了基础。
Piezoelectric materials are widely used as smart materials for their stable piezoelectric properties. Ultrasonic motors and piezoelectric power generators are two typical applications of piezoelectric materials in engineering. As the undesired hysteretic nonlinearities being inherent in piezoelectric materials, it is important to research the hysteretic nonlinear dynamics of piezoelectric materials.
In this paper, the cubic polynomial model was analyzed and improved firstly, and some experiments on hysteresis of piezoelectric ceramic laminate were carried out. Secondly, the nonlinear behavior of ultrasonic motor was measured and analyzed. Finally, by using impact test equipment, some experimental researches were done to investigate the impact responses of the piezoelectric ceramic laminate. The major work of the paper was summarized as follows:
(1) The properties of piezoelectric materials and their applications in ultrasonic motors and power generator were introduced briefly. The development of hysteretic nonlinearities of piezoelectric materials, nonlinear dynamics of ultrasonic motors, as well as impact properties of piezoelectric structures is summarized.
(2) The applicable condition and related parameters of the cubic polynomial hysteresis model were analyzed qualitatively by comparing it with the other hysteresis model. Based on the above analysis, one improved piezoelectric hysteresis model was proposed.
(3) With rotary traveling wave ultrasonic motor’s feedback voltage and rotor speed measured, the dynamic responses of the stator and the rotor were measured and some nonlinear phenomena were found.
(4) The impact test device was invented by analyzing and measuring the size and mode of piezoelectric ceramic element. The impact responses of piezoelectric ceramic laminates and the maximum carrying capacity of piezoelectric ceramic elements were measured to ensure piezoelectric ceramic element work normally. The generating power of piezoelectric structure was tested. All these results would make a foundation for further theoretical research of piezoelectric power generators in future.
The results given in this thesis provide a basis for the applications of the cubic polynomial hysteresis model and the model of whole system of ultrasonic motor, and make a foundation for theoretical research of mechanical and electrical responses of piezoelectric structure.
引文
[1]小西良弘辻俊郎.电子陶瓷基础和应用[M].王兴斌译.北京:机械工业出版社,1983.
[2]山东大学压电铁电物理教研室.压电陶瓷及其应用[M].山东:山东人民出版社,1974.
[3]陈欢欢.压电材料新应用让人边走边发电[EB/OL]. [2007-9-20]. http://www.sciencenet.cn/htmlnews/200792083924295190083.html?id=190083.
[4]叶甘露.会发电的公路.科海故事博览:百科论坛[J]. 2009,24:48.
[5]赵淳生.超声电机技术与应用[M].北京:科学出版社,2007.
[6] Aurelle N, Guyomar D, Richard D, et al. Non-linear behavior of an ultrasonic transducer[J]. Ultrasonics, 1996, 34: 187~191.
[7] Samal MK, Seshu P, Parashar SK, et al. Nonlinear longitudinal vibrations of transversally polarized piezoceramics: experiments and modeling[J]. Nonlinear Dynamics, 2004, 37: 51~73.
[8] Wagner UV, Hagedorn P. Nonlinar effects of piezoceramics excited by weak electric fields[J]. Nonlinear Dynamics, 2003,31: 133~149.
[9] Kamlaha M. Ferroelectric and ferroelastic piezoceramics modeling of electromechanical hysteresis phenomena[J]. Continuum Mech Thermidyn, 2001,13: 219~168.
[10]胡敏强,金龙,顾菊平.超声波原理与设计[M].北京:科学出版社,2005.
[11]邓习树,吴运新.国内外直线超声电机的研究现状与关键技术[J].微电机,2005,38(5):78-81.
[12] Popov EP. The theory of nonlinear automation control system[M]. Moscow: karl B DA, 1998.
[13] Popov EP, Poltov IP. Approximation methods for investigation of nonlinear automatic system[M]. Russian: Fizmatgiz, 1996.
[14] Yu Y, Naganathan N, Dukkipati R. Preisach modeling of hysteresis for piezoceramic autuator system[J]. Mechanism and Machine Theory, 2002,37: 49~59.
[15] Ball BL, Smith RC. A Stress-dependent hysteresis model for ferroelectric materials[J]. Journal of Intelligent Material Systems and Structures, 2007, 18(1): 69~88.
[16] Zhou X, Chattopadhyay A. Hysteresis behavior and modeling of piezoceramic actuators[J]. Journal of Applied Mechanics, 2001, 68: 270~277.
[17] Walter V, Delobelle P, Moal PL. A nonlinear electromechnical model for ferroelectric materials: Application to soft-PZT thick films screen-printed on alumina substrate[J]. IEEE Transactions on Ultrasonics, Ferrpelectrics and Frequency Control, 2003, 50: 471~479.
[18] Hall DA. Review Nonlinearity in piezoelectric ceramics[J]. Journal of Materials Science, 2001, 36:4575~ 4601.
[19] Von Wagner U, Hagedorn P. Nonlinear effects of piezoceramics excited by weak electric fields[J]. Nonlinear Dynamics, 2003, 31: 133~149.
[20] Parashar SK, Von Wagner U, Hagedorn P. A modified timoshenko beam theory for nonlinear shear-induced flexural vibrations of piezoceramic continua[J]. Nonlinear Dynamics, 2004, 37: 181~205.
[21] Tan P, Tong LY. A one-dimesional model for non-linear behaviour of piezoelectric composite materials[J]. Composite Structures, 2002, 58: 551~561.
[22]秦月霞,胡德金.压电驱动器的非线性建模[J].上海交通大学学报,2004,38(8):1334~1336.
[23]荣伟彬,曲东升,孙立宁,徐晶,蔡鹤皋.压电陶瓷微位移器件迟滞模型的研究[J].压电与声光,2003,25(1):22~25.
[24]崔玉国,孙宝元,董维杰等.压电陶瓷执行器迟滞与非线性成因分析[J].光学精密工程,2003,11(3):270~275.
[25]刘向东,修春波,刘承等.基于混沌神经网络的压电陶瓷迟滞模型[J].北京理工大学学报,2006,26(2):135~138.
[26]付耀东.考虑压电材料迟滞非线性时旋转行波超声电机动力学分析[D].天津:天津大学,2007.
[27] Clyde Jake Kendall. Parasitic Power Collection in Shoe Mounted Devices. Massachusetts: Massachusetts Institute of Technoligy[D], 1998. 17-20.
[28] Romain Guigon, Jean-Jacques Chaillout, Thomas Jager and Ghislain Despesse. Harvesting raindrop energy : experimental study[J]. Smart Materials and Structures, 2008,17:1-6.
[29]唐可洪,阚君武,朱国仁等.遥控器用压电发电装置的供电特性[J].光学精密工程,2008,16(01):92-96.
[30]袁江波,谢涛,陈维山等.悬臂梁压电发电装置的实验研究[J].振动与冲击,2009,28(7):69-72.
[31]廖海洋,钟正青.压电陶瓷轮胎发电机的设计[J].光学精密工程,2009,17(6):1327-1332.
[32]丁庆军,姚志远,郑伟等.行波型超声电机定子摩擦材料的研制及其摩擦磨损性能研究[J].摩擦学学报,2007,27(6):578-582.
[33]朱华,陈超,赵淳生.行波型杆式超声电机的动力学分析与性能仿真[J].振动与冲击,2008,27(6):103-107.
[34]曾劲松,姚志远,赵淳生.超声电机中的非线性现象研究[J].中国机械工程.2006,17(10):1047-1050.
[35]赵淳生,黄卫清.超声电机的试验研究[J].振动、测试与诊断,2003,23(1):1-5.
[36]杨明,朱美玲,赵淳生.环形行波式超声马达的输出特性及实验研究[J].振动、测试与诊断,1997,17(4):12-17.
[37]高键.旋转行波超声电机的非线性动力学建模与机理分析[D].天津:天津大学,2007.
[38]许煜寰.铁电与压电材料.北京:科学出版社,1978.
[39] Wang C L, Jan C, Chen Y H. Piezomechanic using intelligent variable-stucture control[J]. IEEE Transcations on Industrial Electronic, 2001,48(1):47-59.
[40] Sherrita S, Wiedericka HD, Mukherjeea BK, et al. Field dependence of the complex piezoelectric, dielectric, and elastic constants of Motorola PZT 3203 HD ceramic[J]. Proceedings of SPIE Conference of Smart Materials Technologies, 1997: 99-109.
[41] Kamlah M, Jiang Q. A model for PZT ceramics under uni-axial loading[J]. Wissenschaftliche Berichte, FZKA-6211 Forschungszentrum Karlsruhe, IMF, 1998: 34-35.
[42] Tanimoto T, Yamamoto K, Morili T. Nonlinear Stress-strain behavior of piezoelectric ceramics under tensile loading[J]. Proceedings of the Ninth IEEE International Symposium, 1994: 394-397.
[43]杨绍普,申永军.滞后非线性系统的分岔与奇异性[M].北京:科学出版社,2003,7.
[44]高健,曹树谦.考虑压电材料非线性特性时旋转行波超声电机定子的主共振响应.非线性动力学报,2005,12:66-78.
[45] B.贾菲等.压电陶瓷[M].林声和,译.北京:科学出版社,1979.
[46]靳洪允.压电材料的结构及其性能研究[J].山东陶瓷,2005,28(4):9-14.
[47]王春雨. PbZrTiO3压电陶瓷迟滞特性及控制研究[D].吉林:长春理工大学,2009.
[48]曹树谦,张文德,萧龙翔.振动结构模态分析:理论、实验与应用.天津:天津大学出版社,2001,3.
[49]曹树谦,陈予恕.压电超声电机的非线性动力学研究进展.力学进展,2005,35(2):153-160.
[50]刘清.压电超声结构非线性动力学实验研究[D].天津:天津大学,2008.
[51] Hu Minqiang, Mo Yueping, Shi Bin, et al. Research of manufacture technology of traveling wave type [J]. Ultrasonic Motors, 2001, 29 (6):7-11. (in Chinese).
[52]孙慷,张学福.压电学(上,下册)[M].北京:国防工业出版社,1984.
[53] Minoru Kurosawa, Kentaro Nakamura, Takashi Okamoto. An ultrasonic motor using bending vibrations of a shot cylinder [J]. IEEE Transactions, 1989, 36 (5):517-521.
[54] Morita T, Kurosawa M, Higuchi T. An ultrasonic motor using a bending cylindrical transducer based on PZT thin film [J]. Sensors and Actuators, 1995, 50 (1 2), A: Physical: 75-80.
[55]夏长亮.行波型超声波电机及其驱动控制系统的研究[D] .杭州:浙江大学, 1995.
[56]赵向东,袁义坤,赵淳生.超声电机定子的有限元分析及其模态混叠现象[J].振动工程学报, 2004, 17 (S) : 866-868.
[57]魏守水.超声马达的驱动及其控制技术[D] .南京航空航天大学博士论文,2000.
[58] Kentaro Nakamura, Minour Kurosawa, Sadayuki Ueha. Design of a Hybrid Transducer Type Ultrasonic Motor [J]. IEEE Transaction on Ultrasonic and Frequency Control, 1993, 40 (4):395 - 401.
[59] Kentaro Nakamura, Minoru Kurosawa, Sadayuki Ueda. Characteristics of a Hybrid Transducer - Type Ultrasonic Motor [J]. IEEE Transaction on Ultrasonic and Frequency Control, 1998, 38 (3):188-193.
[60] Philip J Rayner, Roger W Whatmore. Travelling Wave Ultrasonic Motor Using the B08 Flexural mode of a Circle Membrane [J]. IEEE Transaction on Ultrasonic and Frequency Control, 2001, 48 (3):683-690.
[61] Nesbitt W, Hagood IV ,Andrew J ,et al. Modeling of a Piezoelectric Rotary Ultrasonic Motor [J]. IEEE Transaction on Ultrasonic and Frequency Control, 1995, 42 (2):210-224.
[62]赵向东,赵淳生.行波超声电机带齿定子固有频率解析[J].振动、测试与诊断,1998,18(4):243-247.
[2]山东大学压电铁电物理教研室.压电陶瓷及其应用[M].山东:山东人民出版社,1974.
[3]陈欢欢.压电材料新应用让人边走边发电[EB/OL]. [2007-9-20]. http://www.sciencenet.cn/htmlnews/200792083924295190083.html?id=190083.
[4]叶甘露.会发电的公路.科海故事博览:百科论坛[J]. 2009,24:48.
[5]赵淳生.超声电机技术与应用[M].北京:科学出版社,2007.
[6] Aurelle N, Guyomar D, Richard D, et al. Non-linear behavior of an ultrasonic transducer[J]. Ultrasonics, 1996, 34: 187~191.
[7] Samal MK, Seshu P, Parashar SK, et al. Nonlinear longitudinal vibrations of transversally polarized piezoceramics: experiments and modeling[J]. Nonlinear Dynamics, 2004, 37: 51~73.
[8] Wagner UV, Hagedorn P. Nonlinar effects of piezoceramics excited by weak electric fields[J]. Nonlinear Dynamics, 2003,31: 133~149.
[9] Kamlaha M. Ferroelectric and ferroelastic piezoceramics modeling of electromechanical hysteresis phenomena[J]. Continuum Mech Thermidyn, 2001,13: 219~168.
[10]胡敏强,金龙,顾菊平.超声波原理与设计[M].北京:科学出版社,2005.
[11]邓习树,吴运新.国内外直线超声电机的研究现状与关键技术[J].微电机,2005,38(5):78-81.
[12] Popov EP. The theory of nonlinear automation control system[M]. Moscow: karl B DA, 1998.
[13] Popov EP, Poltov IP. Approximation methods for investigation of nonlinear automatic system[M]. Russian: Fizmatgiz, 1996.
[14] Yu Y, Naganathan N, Dukkipati R. Preisach modeling of hysteresis for piezoceramic autuator system[J]. Mechanism and Machine Theory, 2002,37: 49~59.
[15] Ball BL, Smith RC. A Stress-dependent hysteresis model for ferroelectric materials[J]. Journal of Intelligent Material Systems and Structures, 2007, 18(1): 69~88.
[16] Zhou X, Chattopadhyay A. Hysteresis behavior and modeling of piezoceramic actuators[J]. Journal of Applied Mechanics, 2001, 68: 270~277.
[17] Walter V, Delobelle P, Moal PL. A nonlinear electromechnical model for ferroelectric materials: Application to soft-PZT thick films screen-printed on alumina substrate[J]. IEEE Transactions on Ultrasonics, Ferrpelectrics and Frequency Control, 2003, 50: 471~479.
[18] Hall DA. Review Nonlinearity in piezoelectric ceramics[J]. Journal of Materials Science, 2001, 36:4575~ 4601.
[19] Von Wagner U, Hagedorn P. Nonlinear effects of piezoceramics excited by weak electric fields[J]. Nonlinear Dynamics, 2003, 31: 133~149.
[20] Parashar SK, Von Wagner U, Hagedorn P. A modified timoshenko beam theory for nonlinear shear-induced flexural vibrations of piezoceramic continua[J]. Nonlinear Dynamics, 2004, 37: 181~205.
[21] Tan P, Tong LY. A one-dimesional model for non-linear behaviour of piezoelectric composite materials[J]. Composite Structures, 2002, 58: 551~561.
[22]秦月霞,胡德金.压电驱动器的非线性建模[J].上海交通大学学报,2004,38(8):1334~1336.
[23]荣伟彬,曲东升,孙立宁,徐晶,蔡鹤皋.压电陶瓷微位移器件迟滞模型的研究[J].压电与声光,2003,25(1):22~25.
[24]崔玉国,孙宝元,董维杰等.压电陶瓷执行器迟滞与非线性成因分析[J].光学精密工程,2003,11(3):270~275.
[25]刘向东,修春波,刘承等.基于混沌神经网络的压电陶瓷迟滞模型[J].北京理工大学学报,2006,26(2):135~138.
[26]付耀东.考虑压电材料迟滞非线性时旋转行波超声电机动力学分析[D].天津:天津大学,2007.
[27] Clyde Jake Kendall. Parasitic Power Collection in Shoe Mounted Devices. Massachusetts: Massachusetts Institute of Technoligy[D], 1998. 17-20.
[28] Romain Guigon, Jean-Jacques Chaillout, Thomas Jager and Ghislain Despesse. Harvesting raindrop energy : experimental study[J]. Smart Materials and Structures, 2008,17:1-6.
[29]唐可洪,阚君武,朱国仁等.遥控器用压电发电装置的供电特性[J].光学精密工程,2008,16(01):92-96.
[30]袁江波,谢涛,陈维山等.悬臂梁压电发电装置的实验研究[J].振动与冲击,2009,28(7):69-72.
[31]廖海洋,钟正青.压电陶瓷轮胎发电机的设计[J].光学精密工程,2009,17(6):1327-1332.
[32]丁庆军,姚志远,郑伟等.行波型超声电机定子摩擦材料的研制及其摩擦磨损性能研究[J].摩擦学学报,2007,27(6):578-582.
[33]朱华,陈超,赵淳生.行波型杆式超声电机的动力学分析与性能仿真[J].振动与冲击,2008,27(6):103-107.
[34]曾劲松,姚志远,赵淳生.超声电机中的非线性现象研究[J].中国机械工程.2006,17(10):1047-1050.
[35]赵淳生,黄卫清.超声电机的试验研究[J].振动、测试与诊断,2003,23(1):1-5.
[36]杨明,朱美玲,赵淳生.环形行波式超声马达的输出特性及实验研究[J].振动、测试与诊断,1997,17(4):12-17.
[37]高键.旋转行波超声电机的非线性动力学建模与机理分析[D].天津:天津大学,2007.
[38]许煜寰.铁电与压电材料.北京:科学出版社,1978.
[39] Wang C L, Jan C, Chen Y H. Piezomechanic using intelligent variable-stucture control[J]. IEEE Transcations on Industrial Electronic, 2001,48(1):47-59.
[40] Sherrita S, Wiedericka HD, Mukherjeea BK, et al. Field dependence of the complex piezoelectric, dielectric, and elastic constants of Motorola PZT 3203 HD ceramic[J]. Proceedings of SPIE Conference of Smart Materials Technologies, 1997: 99-109.
[41] Kamlah M, Jiang Q. A model for PZT ceramics under uni-axial loading[J]. Wissenschaftliche Berichte, FZKA-6211 Forschungszentrum Karlsruhe, IMF, 1998: 34-35.
[42] Tanimoto T, Yamamoto K, Morili T. Nonlinear Stress-strain behavior of piezoelectric ceramics under tensile loading[J]. Proceedings of the Ninth IEEE International Symposium, 1994: 394-397.
[43]杨绍普,申永军.滞后非线性系统的分岔与奇异性[M].北京:科学出版社,2003,7.
[44]高健,曹树谦.考虑压电材料非线性特性时旋转行波超声电机定子的主共振响应.非线性动力学报,2005,12:66-78.
[45] B.贾菲等.压电陶瓷[M].林声和,译.北京:科学出版社,1979.
[46]靳洪允.压电材料的结构及其性能研究[J].山东陶瓷,2005,28(4):9-14.
[47]王春雨. PbZrTiO3压电陶瓷迟滞特性及控制研究[D].吉林:长春理工大学,2009.
[48]曹树谦,张文德,萧龙翔.振动结构模态分析:理论、实验与应用.天津:天津大学出版社,2001,3.
[49]曹树谦,陈予恕.压电超声电机的非线性动力学研究进展.力学进展,2005,35(2):153-160.
[50]刘清.压电超声结构非线性动力学实验研究[D].天津:天津大学,2008.
[51] Hu Minqiang, Mo Yueping, Shi Bin, et al. Research of manufacture technology of traveling wave type [J]. Ultrasonic Motors, 2001, 29 (6):7-11. (in Chinese).
[52]孙慷,张学福.压电学(上,下册)[M].北京:国防工业出版社,1984.
[53] Minoru Kurosawa, Kentaro Nakamura, Takashi Okamoto. An ultrasonic motor using bending vibrations of a shot cylinder [J]. IEEE Transactions, 1989, 36 (5):517-521.
[54] Morita T, Kurosawa M, Higuchi T. An ultrasonic motor using a bending cylindrical transducer based on PZT thin film [J]. Sensors and Actuators, 1995, 50 (1 2), A: Physical: 75-80.
[55]夏长亮.行波型超声波电机及其驱动控制系统的研究[D] .杭州:浙江大学, 1995.
[56]赵向东,袁义坤,赵淳生.超声电机定子的有限元分析及其模态混叠现象[J].振动工程学报, 2004, 17 (S) : 866-868.
[57]魏守水.超声马达的驱动及其控制技术[D] .南京航空航天大学博士论文,2000.
[58] Kentaro Nakamura, Minour Kurosawa, Sadayuki Ueha. Design of a Hybrid Transducer Type Ultrasonic Motor [J]. IEEE Transaction on Ultrasonic and Frequency Control, 1993, 40 (4):395 - 401.
[59] Kentaro Nakamura, Minoru Kurosawa, Sadayuki Ueda. Characteristics of a Hybrid Transducer - Type Ultrasonic Motor [J]. IEEE Transaction on Ultrasonic and Frequency Control, 1998, 38 (3):188-193.
[60] Philip J Rayner, Roger W Whatmore. Travelling Wave Ultrasonic Motor Using the B08 Flexural mode of a Circle Membrane [J]. IEEE Transaction on Ultrasonic and Frequency Control, 2001, 48 (3):683-690.
[61] Nesbitt W, Hagood IV ,Andrew J ,et al. Modeling of a Piezoelectric Rotary Ultrasonic Motor [J]. IEEE Transaction on Ultrasonic and Frequency Control, 1995, 42 (2):210-224.
[62]赵向东,赵淳生.行波超声电机带齿定子固有频率解析[J].振动、测试与诊断,1998,18(4):243-247.