[1]    王德石,张恺.压电换能器设计原理[M].武汉:武汉理工大学出版社,2016:40-56.

Wang Deshi,Zhang Wei.Design Principle of Piezoelectric Transducer[M].Wuhan:Wuhan University of Technology Press,2016:40-56.

[2]    Stefan Johann Rupitsch.Piezoelectric Sensors and Actuators[M].Berlin:Springer,2018:261-339

[3]    张翊东.基于机电耦合法油井增产超声波换能器结构参数优化设计[D].哈尔滨工业大学,2014.

Zhang Yidong.Optimization design of structural parameters of ultrasonic transducers for oil well stimulation based on electromechanical coupling method[D].Harbin Institute of Technology,2014.

[4]    滕舵,陈航,朱宁.宽频带径向极化压电圆管水声换能器研究[J].压电与声光,2008,30(04):411-413.

Teng Ru, Chen Hang, Zhu Ning. Research on wide-band radial polarization piezoelectric tube underwater acoustic transducer [J].Piezoelectric and Sound and Light, 2008,30(04): 411-413.

[5]    周心一,黄建平,谢可夫.压电圆管换能器振动特性的薄壳理论分析[J].应用声学,2003,22(04):11-15.

Zhou Xinyi,Huang Jianping,XIE Kefu.Theoretical Analysis of Vibration of Piezoelectric Circular Tube Transducer by Thin Shell Theory[J].Applied Acoustics,2003,22(04):11-15.

[6]    Feilong Feng,Jianzhong Shen,Jingjun Deng.A 2D equivalent circuit of piezoelectric ceramic ring for transducer design[J].Ultrasonics,2006,44(22):723-726.

[7]    Lin Shuyu.The radial composite piezoelectric ceramic transducer[J].Sensors and Actuators A:Physical,2008,141(15):136-143.

[8]    Liu Shiqing,Yao Wei.Radiating Vibration Characteristics of Composite Tube Power Ultrasonic Piezoelectric Transducer[J].Journal of Mechanical Engineering,2008,44(10):239-244.

[9]    Liu Shiqing,Lin Shuyu.The analysis of the electro-mechanical model of the cylindrical radial composite piezoelectric ceramic transducer[J].Sensors and Actuators A:Physical,2009,155(1):175-180.

[10] 林书玉,王帅军,付志强等.径向极化压电陶瓷长圆管复合超声换能器的径向振动[J].声学学报,2013,38(03):354-363.

Lin Shuyu,Wang Shuaijun,Fu Zhiqiang,et al.Radiation Vibration of a Radially Polarized Piezoelectric Ceramic Long Circular Tube Composite Ultrasonic Transducer[J].Acta Acoustics Sinica,2013,38(03):354-363.

[11] 李陆化,刘世清,李丹.镶拼式复合压电超声换能器的径向振动特性[J].机械工程学报,2016,52(16):22-26.

[12] Longyang Jia,Guangbing Zhang,Xiaofeng Zhang,et al.Study on tangentially polarized composite cylindrical piezoelectric transducer with high electro-mechanical coupling coefficient[J].Ultrasonics,2017,74:204-210.

[13] Guo Li,Jihui Gong,Ting Wang,et al.Study on the broadband piezoelectric ceramic transducer based on radial enhancedcomposite structure[J].Ceramics International,2018,44:250-253.

[14] Shuyu Lin.Coupled vibration of isotropic metal hollow cylinders with large geometrical dimensions[J].Journal of Sound and Vibration,2007,305(7):308-316.

[15] 米海珍.弹性力学[M].北京:清华大学出版社,2016:115-120.

Mi Haizhen.Elasticity [M].Beijing: Tsinghua University Press,2016:115-120.

[16] A Ya Grigorenko,W H Müller,R Will,et al.Nonaxisymmetric vibrations of radially polarized hollow cylinders made of functionally gradient piezoelectric materials[J].Continuum Mech. Thermodyn,2012,24:515-524.

[17] 周福洪.水声换能器及基阵[M].北京: 国防工业出版社,1984:84-98.