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基于传输矩阵法的任意变厚度环型压电超声换能器

董宜雷 陈诚 林书玉

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基于传输矩阵法的任意变厚度环型压电超声换能器

董宜雷, 陈诚, 林书玉

Arbitrary variable thickness annular piezoelectric ultrasonic transducer based on transfer matrix method

Dong Yi-Lei, Chen Cheng, Lin Shu-Yu
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  • 变厚度环型径向振动压电超声换能器可以实现阻抗变换、能量集中, 具有辐射面积大、全指向性等优点, 在功率超声、水声等领域被广泛应用. 由于求解复杂变厚度金属圆环径向振动的波动方程比较困难, 本文使用传输矩阵法将变厚度金属圆环的径向振动转化为$ N $个等厚度金属圆环径向振动的叠加, 得到了任意变厚度金属薄圆环径向振动的等效电路图、共振频率方程和位移放大系数表达式, 分析了锥型、幂函数型、指数型、悬链线型金属圆环的位移放大系数与几何尺寸的关系. 在此基础上, 推导了由任意变厚度金属圆环和等厚度压电圆环复合而成的压电超声换能器径向振动的等效电路和共振频率方程. 为了验证理论结果的正确性, 使用有限元软件进行仿真, 所得一阶、二阶的共振频率和位移放大系数的数值解与理论解符合较好. 本研究给出了任意变厚度金属圆环径向振动的普适解, 为设计和优化径向压电超声换能器提供了理论指导.
    The variable thickness annular radial piezoelectric ultrasonic transducer can realize impedance transformation and energy concentration, has the advantages of large radiation area and full directivity, and is widely used in power ultrasound, underwater acoustic and other fields. Because solving complex variable thickness metal ring radial vibration wave equation is more difficult, in this paper, the radial vibration of metal rings with variable thickness is transformed into the superposition of the radial vibrations of N metal rings with equal thickness by using the transfer matrix method. The equivalent circuit diagram, the resonance frequency equation and the expression of the displacement amplification coefficient of the radial vibration of the metal thin ring with arbitrary thickness are obtained. The relationship between the displacement amplification coefficient and the geometric size of the cone, power function, exponential and catenary metal rings is analyzed. On this basis, the equivalent circuit and resonance frequency equation of radial vibration of piezoelectric ultrasonic transducer which is composed of a metal ring with variable thickness and a piezoelectric ring with equal thickness are derived. In order to verify the correctness of the theoretical results, the finite element software is used in simulation, and the numerical solutions of the first and second order resonance frequency and displacement amplification coefficients are in good agreement with the theoretical solutions. In this paper, the universal solution of radial vibration of metal ring with arbitrary variable thickness is given, which provides theoretical guidance for designing and optimizing the radial piezoelectric ultrasonic transducers.
      通信作者: 林书玉, sylin@snnu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11674206, 11874253, 12174240)资助的课题.
      Corresponding author: Lin Shu-Yu, sylin@snnu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11674206, 11874253, 12174240).
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    孙淑珍, 李俊宝 2019 声学学报 44 743Google Scholar

    Sun S Z, Li J B 2019 Acta Acust. 44 743Google Scholar

    [2]

    刘世清, 杨先莉, 张志良, 陈赵江 2013 声学学报 38 188Google Scholar

    Liu S Q, Yang X L, Zhang Z L, Chen Z J 2013 Acta Acust. 38 188Google Scholar

    [3]

    Jin O K, Jung G L 2007 J. Sound Vib. 300 241Google Scholar

    [4]

    Wang H M, Luo D S 2016 Appl. Math. Model. 40 2549Google Scholar

    [5]

    Hyun K L, Dabin J 2020 Optik (Stuttg) 222 165480Google Scholar

    [6]

    Luka P, Duje M, Fran M, Marko S, Marko B, Sven L 2022 IEEE Int. Ultrason. Symp. 124 106737Google Scholar

    [7]

    Kai X Z, Huang S M, Wu L, Ran T, Peng Y J, Mao Z M, Chen F, Li G R 2019 J. Mater. Sci. Technol. 35 2107Google Scholar

    [8]

    Feng J J, Liu L, Ma X Hong, Yuan J H, Wang A J 2017 Int. J. Hydrogen Energy 42 2071Google Scholar

    [9]

    Mohammad J B, Ammad M, Thomas L, Jochen B, Konrad K 2022 Bioresour Technol. 348 126785Google Scholar

    [10]

    Liu H B, Wang X K, Qin S, Lai W J, Yang X, Xu S Y 2021 Sci. Total Environ. 789 147862Google Scholar

    [11]

    Li S, Wang Y H, Wu S S, Niu W D, Yang S Q 2022 Appl. Math. Model. 109 455Google Scholar

    [12]

    Liu Z Y, Miao K, Tan Z M 2022 Appl. Acoust. 187 108497Google Scholar

    [13]

    王晓宇, 林书玉 2021 声学学报 46 271Google Scholar

    Wang X Y, Lin S Y 2021 Acta Acust. 46 271Google Scholar

    [14]

    刘垚, 李斌 2017 电加工与模具 03 61Google Scholar

    Liu Y, Li B 2017 Electromach. Mould. 03 61Google Scholar

    [15]

    李井, 丁艳红, 梁欣 2016 机械设计与制造 7 197Google Scholar

    Li J, Ding Y H, Liang X 2016 Mach. Design Manufact. 7 197Google Scholar

    [16]

    巩建辉, 王晨丰, 吴承启 2022 机械工程师 1 151Google Scholar

    Gong J H, Wang C F, Wu C Q 2022 Mech. Engineer. 1 151Google Scholar

    [17]

    Wang S, Shan J J, Lin S Y 2022 IEEE Int. Ultrason. Symp. 120 106640Google Scholar

    [18]

    陈诚, 林书玉 2021 物理学报 70 017701Google Scholar

    Chen C, Lin S Y 2021 Acta Phys. Sin. 70 017701Google Scholar

    [19]

    Hu L Q, Wang S, Lin S Y 2022 Chin. Phys. B 31 508Google Scholar

    [20]

    许龙, 李伟东 2019 声学学报 44 826Google Scholar

    Xu L, Li W D 2019 Acta Acust. 44 826Google Scholar

    [21]

    许龙, 范秀梅 2021 应用声学 40 878Google Scholar

    Xu L, Fan X M 2021 J. Appl. Acoust. 40 878Google Scholar

    [22]

    刘世清, 林书玉, 郭建中 2006 压电与声光 03 347Google Scholar

    Liu S Q, Lin S Y, Guo J Z 2006 Piezoelect. Acoustoop. 03 347Google Scholar

    [23]

    王晓宇, 林书玉 2020 陕西师范大学学报(自然科学版) 48 107Google Scholar

    Wang X Y, Lin S Y 2020 J. Shaanxi Normal Univ. (Nat. Sci. Ed.) 48 107Google Scholar

    [24]

    邓婷婷, 傅波 2017 机械工程师 1 36Google Scholar

    Deng T T, Fu B 2017 Mech. Engineer. 1 36Google Scholar

    [25]

    Lin S Y, Xu L 2012 IEEE Int. Ultrason. Symp. 52 103Google Scholar

    [26]

    Lin S Y 2007 Sens. Actuators A Phys. 134 505Google Scholar

  • 图 1  变厚度环型压电超声换能器的纵向截面示意图

    Fig. 1.  Schematic diagram of longitudinal section of variable thickness annular piezoelectric ultrasonic transducer.

    图 2  等厚度金属薄圆环径向振动的等效电路图

    Fig. 2.  Equivalent circuit of metal thin circular annular in radial vibration.

    图 3  变厚度金属薄圆环径向振动的等效电路图

    Fig. 3.  Equivalent circuit diagram of metal thin annular with variable thickness in radial vibration.

    图 4  4种变厚度金属圆环一阶、二阶径向共振频率与$ {h}_{{\rm{a}}}/{h}_{{\rm{b}}} $的关系 (a)一阶径向共振; (b)二阶径向共振

    Fig. 4.  The relationship between the first and second order radial resonance frequencies of four kinds of variable thickness metal rings and thickness ratio $ {h}_{{\rm{a}}}/{h}_{{\rm{b}}} $: (a) First-order radial resonance; (b) second-order radial resonance.

    图 5  4种变厚度环型聚能器一阶和二阶径向共振位移放大系数随$ {h}_{{\rm{a}}}/{h}_{{\rm{b}}} $的变化曲线 (a)一阶径向共振; (b)二阶径向共振

    Fig. 5.  The relationship between the first and second order radial resonance displacement amplification coefficient and thickness ratio $ {h}_{{\rm{a}}}/{h}_{{\rm{b}}} $ of four kinds of variable thickness metal rings: (a) First-order radial resonance; (b) second-order radial resonance.

    图 6  锥型金属圆环的一阶、二阶径向共振频率与N的关系 (a)一阶径向共振; (b)二阶径向共振

    Fig. 6.  The relationship between the first and second order radial resonant frequencies of conical metal rings and N: (a) First-order radial resonance; (b) second-order radial resonance.

    图 7  压电陶瓷圆环径向振动的等效电路图

    Fig. 7.  Equivalent circuit diagram of piezoelectric ceramic annular in radial vibration.

    图 8  环型压电超声换能器径向振动的等效电路图

    Fig. 8.  Equivalent circuit diagram of circular piezoelectric ultrasonic transducer in radial vibration.

    图 9  锥型变厚度环型压电超声换能器一阶、二阶径向共振振型图 (a)一阶径向共振; (b)二阶径向共振

    Fig. 9.  First-order and second-order radial resonance mode shapes of conical variable thickness annular piezoelectric ultrasonic transducer: (a) First-order radial resonance; (b) second-order radial resonance.

    图 10  锥型变厚度环型压电超声换能器一阶、二阶径向共振频率和反共振频率与$ {h}_{{\rm{a}}}/{h}_{{\rm{b}}} $的关系 (a)一阶共振和反共振; (b)二阶共振和反共振

    Fig. 10.  The relationship between the first and second order radial resonance frequency and the anti-resonance frequency and the thickness ratio $ {h}_{{\rm{a}}}/{h}_{{\rm{b}}} $ of a conical variable thickness annular piezoelectric ultrasonic transducer: (a) The first-order radial resonance and anti-resonance; (b) the second-order radial resonance and anti-resonance.

    表 1  变厚度金属圆环径向一阶、二阶共振频率

    Table 1.  Radial first and second order resonance frequencies of metal rings with variable thickness.

    $ {h}_{{\rm{b}}}/{\rm{m}}{\rm{m}} $$ {h}_{{\rm{a}}}/{\rm{m}}{\rm{m}} $$ {f}_{{\rm{r}}1}/{\rm{H}}{\rm{z}} $${ {f}^{ {\rm{*} } }_{ {\rm{r} }1} }/{\rm{H} }{\rm{z} }$${\varDelta }_{ {f}_{ {\rm{r} }1} }/{\rm{\%} }$$ {f}_{{\rm{r}}2}/{\rm{H}}{\rm{z}} $${ {f}^{*}_{r2} }/{\rm{H} }{\rm{z} }$${\varDelta }_{ {f}_{ {\rm{r} }2} }/{\rm{\%} }$
    锥型61021894218920.011123801107901.44
    幂函数型510216792167901133301117201.44
    指数型41021401213990.011127701114801.16
    悬链线型31020968209570.051097001088900.74
    下载: 导出CSV

    表 2  变厚度金属圆环径向一阶、二阶共振位移放大系数

    Table 2.  Radial first and second order resonance displacement amplification coefficients of metal rings with variable thickness.

    $ {h}_{{\rm{b}}}/{\rm{m}}{\rm{m}} $$ {h}_{{\rm{a}}}/{\rm{m}}{\rm{m}} $$ {M}_{{\rm{r}}1}^{{\rm{*}}} $$ {M}_{{\rm{r}}1}^{{\rm{*}}{\rm{*}}} $${\varDelta }_{ {M}_{ {\rm{r} }1}^{*} }$/%$ {M}_{{\rm{r}}2}^{{\rm{*}}} $$ {M}_{{\rm{r}}2}^{{\rm{*}}{\rm{*}}} $${\varDelta }_{ {M}_{ {\rm{r} }2}^{*} }/{\rm{\%} }$
    锥型6101.19851.19870.021.83041.81600.79
    幂函数型5101.19971.20000.021.98761.96701.04
    指数型4101.19921.19950.022.25012.21351.16
    悬链线型3101.19591.19560.022.81182.73592.77
    下载: 导出CSV

    表 3  锥型变厚度环型压电换能器一阶共振频率和反共振频率

    Table 3.  The first-order resonant frequency and anti-resonant frequency of conical variable thickness annular piezoelectric transducer.

    $ {h}_{{\rm{b}}}/{\rm{m}}{\rm{m}} $$ {h}_{{\rm{a}}}/{\rm{m}}{\rm{m}} $$ {f}_{{\rm{r}}1}/{\rm{H}}{\rm{z}} $${ {f}^{*}_{ {\rm{r} }1} }/{\rm{H} }{\rm{z} }$${\varDelta }_{ {f}_{ {\rm{r} }1} }/$%$ {f}_{{\rm{a}}1}/{\rm{H}}{\rm{z}} $${ {f}^{*}_{ {\rm{a} }1} }/{\rm{H} }{\rm{z} }$${\varDelta }_{ {f}_{ {\rm{a} }1} }/$%$ {K}_{{\rm{e}}{\rm{f}}{\rm{f}}1} $$ {K}_{{\rm{e}}{\rm{f}}{\rm{f}}1}^{{\rm{*}}} $
    91022002219890.0622276222690.030.1560.158
    61021059210420.0821355213480.030.1660.169
    31019886198560.1520209201920.080.1780.182
    下载: 导出CSV

    表 4  锥型变厚度环型压电换能器二阶共振频率和反共振频率

    Table 4.  The second-order resonant frequency and anti-resonant frequency of conical variable thickness annular piezoelectric transducer.

    $ {h}_{{\rm{b}}}/{\rm{m}}{\rm{m}} $$ {h}_{{\rm{a}}}/{\rm{m}}{\rm{m}} $$ {f}_{{\rm{r}}2}/{\rm{H}}{\rm{z}} $${ {f}^{*}_{ {\rm{r} }2} }/{\rm{H} }{\rm{z} }$${\varDelta }_{ {f}_{ {\rm{r} }2} }/$%$ {f}_{{\rm{a}}2}/{\rm{H}}{\rm{z}} $${ {f}^{*}_{ {\rm{a} }2} }/{\rm{H} }{\rm{z} }$${\varDelta }_{ {f}_{ {\rm{a} }2} }/$%$ {K}_{{\rm{e}}{\rm{f}}{\rm{f}}2} $$ {K}_{{\rm{e}}{\rm{f}}{\rm{f}}2}^{{\rm{*}}} $
    91095014934011.7396069948141.320.1480.172
    61098432963832.1399471979221.580.1440.177
    3101050051020202.931060941038302.180.1430.186
    下载: 导出CSV
  • [1]

    孙淑珍, 李俊宝 2019 声学学报 44 743Google Scholar

    Sun S Z, Li J B 2019 Acta Acust. 44 743Google Scholar

    [2]

    刘世清, 杨先莉, 张志良, 陈赵江 2013 声学学报 38 188Google Scholar

    Liu S Q, Yang X L, Zhang Z L, Chen Z J 2013 Acta Acust. 38 188Google Scholar

    [3]

    Jin O K, Jung G L 2007 J. Sound Vib. 300 241Google Scholar

    [4]

    Wang H M, Luo D S 2016 Appl. Math. Model. 40 2549Google Scholar

    [5]

    Hyun K L, Dabin J 2020 Optik (Stuttg) 222 165480Google Scholar

    [6]

    Luka P, Duje M, Fran M, Marko S, Marko B, Sven L 2022 IEEE Int. Ultrason. Symp. 124 106737Google Scholar

    [7]

    Kai X Z, Huang S M, Wu L, Ran T, Peng Y J, Mao Z M, Chen F, Li G R 2019 J. Mater. Sci. Technol. 35 2107Google Scholar

    [8]

    Feng J J, Liu L, Ma X Hong, Yuan J H, Wang A J 2017 Int. J. Hydrogen Energy 42 2071Google Scholar

    [9]

    Mohammad J B, Ammad M, Thomas L, Jochen B, Konrad K 2022 Bioresour Technol. 348 126785Google Scholar

    [10]

    Liu H B, Wang X K, Qin S, Lai W J, Yang X, Xu S Y 2021 Sci. Total Environ. 789 147862Google Scholar

    [11]

    Li S, Wang Y H, Wu S S, Niu W D, Yang S Q 2022 Appl. Math. Model. 109 455Google Scholar

    [12]

    Liu Z Y, Miao K, Tan Z M 2022 Appl. Acoust. 187 108497Google Scholar

    [13]

    王晓宇, 林书玉 2021 声学学报 46 271Google Scholar

    Wang X Y, Lin S Y 2021 Acta Acust. 46 271Google Scholar

    [14]

    刘垚, 李斌 2017 电加工与模具 03 61Google Scholar

    Liu Y, Li B 2017 Electromach. Mould. 03 61Google Scholar

    [15]

    李井, 丁艳红, 梁欣 2016 机械设计与制造 7 197Google Scholar

    Li J, Ding Y H, Liang X 2016 Mach. Design Manufact. 7 197Google Scholar

    [16]

    巩建辉, 王晨丰, 吴承启 2022 机械工程师 1 151Google Scholar

    Gong J H, Wang C F, Wu C Q 2022 Mech. Engineer. 1 151Google Scholar

    [17]

    Wang S, Shan J J, Lin S Y 2022 IEEE Int. Ultrason. Symp. 120 106640Google Scholar

    [18]

    陈诚, 林书玉 2021 物理学报 70 017701Google Scholar

    Chen C, Lin S Y 2021 Acta Phys. Sin. 70 017701Google Scholar

    [19]

    Hu L Q, Wang S, Lin S Y 2022 Chin. Phys. B 31 508Google Scholar

    [20]

    许龙, 李伟东 2019 声学学报 44 826Google Scholar

    Xu L, Li W D 2019 Acta Acust. 44 826Google Scholar

    [21]

    许龙, 范秀梅 2021 应用声学 40 878Google Scholar

    Xu L, Fan X M 2021 J. Appl. Acoust. 40 878Google Scholar

    [22]

    刘世清, 林书玉, 郭建中 2006 压电与声光 03 347Google Scholar

    Liu S Q, Lin S Y, Guo J Z 2006 Piezoelect. Acoustoop. 03 347Google Scholar

    [23]

    王晓宇, 林书玉 2020 陕西师范大学学报(自然科学版) 48 107Google Scholar

    Wang X Y, Lin S Y 2020 J. Shaanxi Normal Univ. (Nat. Sci. Ed.) 48 107Google Scholar

    [24]

    邓婷婷, 傅波 2017 机械工程师 1 36Google Scholar

    Deng T T, Fu B 2017 Mech. Engineer. 1 36Google Scholar

    [25]

    Lin S Y, Xu L 2012 IEEE Int. Ultrason. Symp. 52 103Google Scholar

    [26]

    Lin S Y 2007 Sens. Actuators A Phys. 134 505Google Scholar

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出版历程
  • 收稿日期:  2022-11-03
  • 修回日期:  2022-11-26
  • 上网日期:  2022-12-17
  • 刊出日期:  2023-03-05

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