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

doi:10.7498/aps.72.20222110

摘要

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.

关键词:

Abstract

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.

Keywords:

annular piezoelectric ultrasonic transducer /
radial vibration /
transfer matrix method /
equivalent circuit

作者及机构信息

陕西师范大学物理学与信息技术学院, 陕西省超声重点实验室, 西安　710119

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.

Authors and contacts

Shaanxi Provincial Key Laboratory of Ultrasound, School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710119, China

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).

文章全文 : translate this paragraph

参考文献

[1]	孙淑珍, 李俊宝 2019 声学学报 44 743 Google Scholar Sun S Z, Li J B 2019 Acta Acust. 44 743 Google Scholar
[2]	刘世清, 杨先莉, 张志良, 陈赵江 2013 声学学报 38 188 Google Scholar Liu S Q, Yang X L, Zhang Z L, Chen Z J 2013 Acta Acust. 38 188 Google Scholar
[3]	Jin O K, Jung G L 2007 J. Sound Vib. 300 241 Google Scholar
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[5]	Hyun K L, Dabin J 2020 Optik (Stuttg) 222 165480 Google Scholar
[6]	Luka P, Duje M, Fran M, Marko S, Marko B, Sven L 2022 IEEE Int. Ultrason. Symp. 124 106737 Google 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 2107 Google Scholar
[8]	Feng J J, Liu L, Ma X Hong, Yuan J H, Wang A J 2017 Int. J. Hydrogen Energy 42 2071 Google Scholar
[9]	Mohammad J B, Ammad M, Thomas L, Jochen B, Konrad K 2022 Bioresour Technol. 348 126785 Google Scholar
[10]	Liu H B, Wang X K, Qin S, Lai W J, Yang X, Xu S Y 2021 Sci. Total Environ. 789 147862 Google Scholar
[11]	Li S, Wang Y H, Wu S S, Niu W D, Yang S Q 2022 Appl. Math. Model. 109 455 Google Scholar
[12]	Liu Z Y, Miao K, Tan Z M 2022 Appl. Acoust. 187 108497 Google Scholar
[13]	王晓宇, 林书玉 2021 声学学报 46 271 Google Scholar Wang X Y, Lin S Y 2021 Acta Acust. 46 271 Google Scholar
[14]	刘垚, 李斌 2017 电加工与模具 03 61 Google Scholar Liu Y, Li B 2017 Electromach. Mould. 03 61 Google Scholar
[15]	李井, 丁艳红, 梁欣 2016 机械设计与制造 7 197 Google Scholar Li J, Ding Y H, Liang X 2016 Mach. Design Manufact. 7 197 Google Scholar
[16]	巩建辉, 王晨丰, 吴承启 2022 机械工程师 1 151 Google Scholar Gong J H, Wang C F, Wu C Q 2022 Mech. Engineer. 1 151 Google Scholar
[17]	Wang S, Shan J J, Lin S Y 2022 IEEE Int. Ultrason. Symp. 120 106640 Google Scholar
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[19]	Hu L Q, Wang S, Lin S Y 2022 Chin. Phys. B 31 508 Google Scholar
[20]	许龙, 李伟东 2019 声学学报 44 826 Google Scholar Xu L, Li W D 2019 Acta Acust. 44 826 Google Scholar
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[22]	刘世清, 林书玉, 郭建中 2006 压电与声光 03 347 Google Scholar Liu S Q, Lin S Y, Guo J Z 2006 Piezoelect. Acoustoop. 03 347 Google Scholar
[23]	王晓宇, 林书玉 2020 陕西师范大学学报(自然科学版) 48 107 Google Scholar Wang X Y, Lin S Y 2020 J. Shaanxi Normal Univ. (Nat. Sci. Ed.) 48 107 Google Scholar
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施引文献

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.

下载: 全尺寸图片幻灯片

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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	6	10	21894	21892	0.01	112380	110790	1.44
幂函数型	5	10	21679	21679	0	113330	111720	1.44
指数型	4	10	21401	21399	0.01	112770	111480	1.16
悬链线型	3	10	20968	20957	0.05	109700	108890	0.74

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	6	10	1.1985	1.1987	0.02	1.8304	1.8160	0.79
幂函数型	5	10	1.1997	1.2000	0.02	1.9876	1.9670	1.04
指数型	4	10	1.1992	1.1995	0.02	2.2501	2.2135	1.16
悬链线型	3	10	1.1959	1.1956	0.02	2.8118	2.7359	2.77

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{*}}} $
9	10	22002	21989	0.06	22276	22269	0.03	0.156	0.158
6	10	21059	21042	0.08	21355	21348	0.03	0.166	0.169
3	10	19886	19856	0.15	20209	20192	0.08	0.178	0.182

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{*}}} $
9	10	95014	93401	1.73	96069	94814	1.32	0.148	0.172
6	10	98432	96383	2.13	99471	97922	1.58	0.144	0.177
3	10	105005	102020	2.93	106094	103830	2.18	0.143	0.186

下载: 导出CSV

map

[1]	孙淑珍, 李俊宝 2019 声学学报 44 743 Google Scholar Sun S Z, Li J B 2019 Acta Acust. 44 743 Google Scholar
[2]	刘世清, 杨先莉, 张志良, 陈赵江 2013 声学学报 38 188 Google Scholar Liu S Q, Yang X L, Zhang Z L, Chen Z J 2013 Acta Acust. 38 188 Google Scholar
[3]	Jin O K, Jung G L 2007 J. Sound Vib. 300 241 Google Scholar
[4]	Wang H M, Luo D S 2016 Appl. Math. Model. 40 2549 Google Scholar
[5]	Hyun K L, Dabin J 2020 Optik (Stuttg) 222 165480 Google Scholar
[6]	Luka P, Duje M, Fran M, Marko S, Marko B, Sven L 2022 IEEE Int. Ultrason. Symp. 124 106737 Google 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 2107 Google Scholar
[8]	Feng J J, Liu L, Ma X Hong, Yuan J H, Wang A J 2017 Int. J. Hydrogen Energy 42 2071 Google Scholar
[9]	Mohammad J B, Ammad M, Thomas L, Jochen B, Konrad K 2022 Bioresour Technol. 348 126785 Google Scholar
[10]	Liu H B, Wang X K, Qin S, Lai W J, Yang X, Xu S Y 2021 Sci. Total Environ. 789 147862 Google Scholar
[11]	Li S, Wang Y H, Wu S S, Niu W D, Yang S Q 2022 Appl. Math. Model. 109 455 Google Scholar
[12]	Liu Z Y, Miao K, Tan Z M 2022 Appl. Acoust. 187 108497 Google Scholar
[13]	王晓宇, 林书玉 2021 声学学报 46 271 Google Scholar Wang X Y, Lin S Y 2021 Acta Acust. 46 271 Google Scholar
[14]	刘垚, 李斌 2017 电加工与模具 03 61 Google Scholar Liu Y, Li B 2017 Electromach. Mould. 03 61 Google Scholar
[15]	李井, 丁艳红, 梁欣 2016 机械设计与制造 7 197 Google Scholar Li J, Ding Y H, Liang X 2016 Mach. Design Manufact. 7 197 Google Scholar
[16]	巩建辉, 王晨丰, 吴承启 2022 机械工程师 1 151 Google Scholar Gong J H, Wang C F, Wu C Q 2022 Mech. Engineer. 1 151 Google Scholar
[17]	Wang S, Shan J J, Lin S Y 2022 IEEE Int. Ultrason. Symp. 120 106640 Google Scholar
[18]	陈诚, 林书玉 2021 70 017701 Google Scholar Chen C, Lin S Y 2021 Acta Phys. Sin. 70 017701 Google Scholar
[19]	Hu L Q, Wang S, Lin S Y 2022 Chin. Phys. B 31 508 Google Scholar
[20]	许龙, 李伟东 2019 声学学报 44 826 Google Scholar Xu L, Li W D 2019 Acta Acust. 44 826 Google Scholar
[21]	许龙, 范秀梅 2021 应用声学 40 878 Google Scholar Xu L, Fan X M 2021 J. Appl. Acoust. 40 878 Google Scholar
[22]	刘世清, 林书玉, 郭建中 2006 压电与声光 03 347 Google Scholar Liu S Q, Lin S Y, Guo J Z 2006 Piezoelect. Acoustoop. 03 347 Google Scholar
[23]	王晓宇, 林书玉 2020 陕西师范大学学报(自然科学版) 48 107 Google Scholar Wang X Y, Lin S Y 2020 J. Shaanxi Normal Univ. (Nat. Sci. Ed.) 48 107 Google Scholar
[24]	邓婷婷, 傅波 2017 机械工程师 1 36 Google Scholar Deng T T, Fu B 2017 Mech. Engineer. 1 36 Google Scholar
[25]	Lin S Y, Xu L 2012 IEEE Int. Ultrason. Symp. 52 103 Google Scholar
[26]	Lin S Y 2007 Sens. Actuators A Phys. 134 505 Google Scholar

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