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Study of radial bending vibration and excitation of piezoelectric rings

Pan Rui Mo Xi-Ping Chai Yong Zhang Xiu-Zhen Tian Zhi-Feng

Citation:

Study of radial bending vibration and excitation of piezoelectric rings

Pan Rui, Mo Xi-Ping, Chai Yong, Zhang Xiu-Zhen, Tian Zhi-Feng
cstr: 32037.14.aps.73.20240887
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  • Piezoelectric ring transducer is one of the most common underwater transducers, and its radial vibration, bending vibration in-plane $r \text- \theta $ and out-of-plane $r \text- \theta $ have been widely studied. However, the current research on the bending vibration in the plane $r \text- z$ of the ring is insufficient, although it may have a noticeable influence on the applicability of the underwater transducers. In this study, mechanical analysis and related mathematical calculations of the bending vibrations in the plane $r \text- z$ are carried out by using the thin-shell theory. Herein, the following three aspects are studied: 1) free vibration theory solution, 2) forced vibration: multi-order modal excitation theory, and 3) related finite element calculations and experimental verification. In this study, the bending vibration equations under electrical short and electric open condition are derived, and the multi-order resonance frequency prediction formulas and shape functions for both conditions are obtained by analytical solution and function fitting. Using the finite element method, the influence of piezoelectric effect and the range of applicability of these two electrical conditions are analyzed. The non-homogeneous equations under forced vibration are solved. By utilizing the orthogonal completeness of the vibration mode function, an integral transformation with the vibration mode function can be defined as the basis vector, so that the equation is solved in a simple positive space, and the results reveal the relationship between the coefficients of the modes of different orders and the voltage distribution. By modal theory, the effects of electrical excitation conditions on the multistep bending vibration modes are investigated, and effective methods such as unimodal excitation, partial excitation and single-ended excitation acting on several different target modes are obtained. The proposed piezoelectric ring unimodal excitation and single-ended excitation methods successfully excite the target modes in the experiments: the unimodal excited ring excites only one of its corresponding bending modes, while the single-ended excitation method excites all the bending modes of the first five orders, and its modal strength characteristics are in accordance with the theoretical predictions. This study involves finite element simulation, experimental and theoretical comparative verification, which are in good agreement. The relevant conclusions can provide a theoretical basis for identifying the vibration modes of piezoelectric ring and the fine tuning of modal excitation.
      Corresponding author: Mo Xi-Ping, moxp@mail.ioa.ac.cn ; Chai Yong, chaiyong@mail.ioa.ac.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 62127801).
    [1]

    Butler J L, Sherman C H 2016 Transducers and Arrays for Underwater Sound (Cham: Springer International Publishing) p191

    [2]

    滕舵, 王龑, 解柯柯 2018 陕西师范大学学报(自然科学版) 46 30Google Scholar

    Teng D, Wang Y, Xie K K 2018 J. Shaanxi Norm. Uni. (Nat. Sci. Ed. ) 46 30Google Scholar

    [3]

    周利生, 许欣然 2021 声学学报 46 1250Google Scholar

    Zhou L S, Xu X R 2021 Acta Acust. 46 1250Google Scholar

    [4]

    张鸿磊 2018 硕士学位论文 (哈尔滨: 哈尔滨工程大学)

    Zhang H L 2018 M. S. Thesis (Harbin: Harbin Engineering University

    [5]

    柴勇, 莫喜平, 刘永平, 潘耀宗, 张运强, 李鹏 2017 声学学报 42 619Google Scholar

    Chai Y, Mo X P, Liu Y P, Pan Y Z, Zhang Y Q 2017 Acta Acust. 42 619Google Scholar

    [6]

    宋哲, 严由嵘 2018 水下无人系统学报 26 498

    Song Z, Yan Y R 2018 J. Unmanned Underw. Systems 26 498

    [7]

    林书玉 2023 声学学报 28 102Google Scholar

    Lin S Y 2023 Acta Acust. 28 102Google Scholar

    [8]

    穆廷荣 1979 声学学报 3 161Google Scholar

    Mu Y R 1979 Acta Acust. 3 161Google Scholar

    [9]

    Been K, Nam S, Lee H, Seo H seon, Moon W 2017 J. Acoust. Soc. Am. 141 4740Google Scholar

    [10]

    Lee J, Been K, Moon W 2023 J. Acoust. Soc. Am. 154 A228Google Scholar

    [11]

    Rogers P H 2005 J. Acoust. Soc. Am. 77 S16Google Scholar

    [12]

    Butler J L 1976 J. Acoust. Soc. Am. 59 480Google Scholar

    [13]

    Sherman C H, Parke N G 2005 J. Acoust. Soc. Am. 38 715Google Scholar

    [14]

    Huang C H, Ma C C, Lin Y C 2005 IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 52 1204Google Scholar

    [15]

    Aronov B S 2013 J. Acoust. Soc. Am. 134 1021Google Scholar

    [16]

    胡久龄 2023 博士学位论文 (哈尔滨: 哈尔滨工程大学)

    Hu J L 2023 Ph. D. Dissertation (Harbin: Harbin Engineering University

    [17]

    滕舵 2016 水声换能器基础 第233页

    Teng D 2016 Fundamentals of Hydroacoustic Transducers (Northwestern Polytechnical University Press) p233

    [18]

    Timošenko S P, Woinowsky-Krieger S 1996 Theory of Plates and Shells (New York: McGraw-Hill) p580

    [19]

    曹志远 1989 板壳振动理论 (北京: 中国铁道出版社) 第462页

    Cao Z Y 1989 Plate and Shell Vibration Theory (Beijing: China Railway Press) p462

    [20]

    栾桂冬 2005 压电换能器和换能器阵 (北京: 北京大学出版社) 第479页

    Luan G D 2005 Piezoelectric Transducers and Transducer Arrays (Beijing: Peking University Press) p479

    [21]

    潘瑞, 柴勇, 莫喜平, 刘文钊, 张秀侦 2024 应用声学 43 719Google Scholar

    Pan R, Chai Y, Mo X P, Liu W Z, Zhang X Z 2024 Appl. Acoust. 43 719Google Scholar

    [22]

    张海澜 2012 理论声学(北京: 高等教育出版社) 第139页

    Zhang H L 2012 Theoretical Acoustics (Beijing: Higher Education Press) p139

    [23]

    单影, 许龙, 周光平 2022 压电与声光 44 316Google Scholar

    Dan Y, Xu L, Zhou G P 2022 Piezoelectr. Acoustoopt. 44 316Google Scholar

  • 图 1  圆环的几种弯曲振动模态 (a) r-θ面外横振动; (b) r-θ面内弯曲; (c) r-z面内振动

    Figure 1.  Several bending vibration modes of a circular ring: (a) Out-of-plane r-θ transverse vibration; (b) in-plane r-θ bending; (c) in-plane r-z vibration.

    图 2  薄壳模型 (a) 圆柱壳体; (b) 微元受力; (c) 弯曲示意

    Figure 2.  Thin shell model: (a) Cylindrical shell; (b) microelement forces; (c) bending schematic.

    图 3  近似解与高精度数值解的误差

    Figure 3.  Error between approximate solution and high precision numerical solution.

    图 4  前六阶的振型

    Figure 4.  The first six order mode shapes.

    图 5  电压函数的分段近似

    Figure 5.  Segmental approximation of the voltage function.

    图 6  局部等幅激励

    Figure 6.  Local equal amplitude excitation.

    图 7  区间[a, b]激励下不同阶模态权值Gn

    Figure 7.  Modal weights Gn of different orders under excitation in interval [a, b].

    图 8  单端激励长度对不同阶模态影响

    Figure 8.  Effect of single-ended excitation length on different order modes.

    图 9  压电圆环有限元模型及网格划分

    Figure 9.  Finite element model and meshing of piezoelectric ring.

    图 10  电学开路与电学短路理论误差对比(相较于有限元) (a)第一阶模态误差; (b)第三阶模态误差; (c)第六阶模态误差

    Figure 10.  Comparison of theoretical errors of electrical open circuit and electrical short circuit (compared to finite element results): (a) Error in the first mode; (b) error in the third mode; (c) error in the sixth mode.

    图 11  有限元计算与理论预测谐振频率(前六阶) (a) ${r_0}/h = $$ 5$; (b) ${r_0}/h = 25$

    Figure 11.  Finite element calculation and theoretical prediction of resonance frequency (first six orders): (a) ${r_0}/h = 5$; (b) ${r_0}/h = 25$

    图 12  压电圆环样品

    Figure 12.  Piezoelectric ring sample.

    图 13  单模态激励示意图(三维图红色为正极, 蓝色为负极)

    Figure 13.  Schematic of conformal excitation (red is positive and blue is negative in 3D).

    图 14  单端激励(a)与常规激励(b)示意图

    Figure 14.  Schematic diagram of single-ended excitation (a) and conventional excitation (b).

    图 15  电导纳及激光测振系统

    Figure 15.  Admittance and laser vibration measurement system.

    图 16  单端激励的激光测振图 (a) f = 19656 Hz; (b) f = 24656 Hz; (c) f = 36094 Hz; (d) f = 52594 Hz; (e) f = 72813 kHz; (f) f = 18750 Hz; (g) f = 45500 Hz; (h) 平均振速幅频响应

    Figure 16.  Vibration with single-ended excitation: (a) f = 19656 Hz; (b) f = 24656 Hz; (c) f = 36094 Hz; (d) f = 52594 Hz; (e) f = 72813 kHz; (f) f = 18750 Hz; (g) f = 45500 Hz; (h) mean amplitude-frequency response of the vibration rate.

    图 17  单端激励实验电导曲线及谐振频率 (a) 单端激励、全激励电导曲线与理论计算弯曲谐振频率; (b) 理论计算、有限元仿真与实验的谐振频率对比

    Figure 17.  Experimental conductance curves and resonance frequencies for single-ended excitation: (a) Single-ended excitation, full excitation conductance curves and theoretically calculated bending resonance frequencies; (b) comparison of theoretically calculated, finite element simulation and experimental resonance frequencies.

    图 18  单模态激励圆环的电导纳曲线 (a)电导; (b)电纳

    Figure 18.  Admittance curves of a conformally excited circular ring: (a) Conductance; (b) susceptance.

    图 19  周向不均匀导致的双峰 (a) f = 50750 kHz; (b) f = 52781 kHz; (c)平均振速幅频曲线

    Figure 19.  Double peak due to circumferential inhomogeneity: (a) f = 50750 kHz; (b) f = 52781 kHz; (c) mean amplitude-frequency response of the vibration rate.

    表 1  n阶模态对应的第m个节点位置z/l

    Table 1.  The m-th node positions z/l corresponding to n-th order modes.

    n m
    1234567
    10.22500.7788
    20.13210.50000.8677
    30.09440.35580.64410.9076
    40.07350.27680.49970.72980.8749
    50.06010.22650.40910.59090.77270.9545
    60.05090.19160.34620.50000.65380.80770.9615
    DownLoad: CSV

    表 2  单端激励激发出前n阶模态对应激励长度

    Table 2.  Excitation lengths of the first n order modes with single-ended excitation.

    n12345
    $ b/{l_{\max }} $0.50190.29040.20770.16160.1322
    DownLoad: CSV

    表 3  PZT-46压电陶瓷的材料参数

    Table 3.  Material parameters of PZT-46 piezoelectric ceramics.

    $ s_{11}^{\text{E}}/{\text{P}}{{\text{a}}^{ - {1}}} $ $ \rho /({\text{kg}} {\cdot} {{\text{m}}^{ - {3}}}) $ $ \varepsilon _{{33}}^{\text{T}}/({\text{F}} {\cdot} {{\text{m}}^{ - {1}}}) $ $ {d_{31}}/\left( {{\text{C}} {\cdot} {{\text{N}}^{ - {1}}}} \right) $ $ {d_{15}}/ ({{\text{C}} {\cdot} {{\text{N}}^{-{1}}}}) $ ${v_{12}}$ ${k_{31}}$
    14.31×10–12 7750 11.51×10–9 –1.56×10–10 4.96×10–10 0.33 0.365
    DownLoad: CSV

    表 4  前五阶弯曲模态谐振频率对比

    Table 4.  Comparison of the first five orders resonance frequencies of bending modes.

    阶数n实验/HzFEM/Hz理论/HzFEM误差/%理论误差/%
    11965619423197951.190.71
    22465624162241922.001.88
    33609435537348811.543.36
    45259452182519210.781.28
    57281372685746230.182.49
    DownLoad: CSV

    表 A1  径厚比${r_0}/h$对电学开路与电学短路理论的影响

    Table A1.  Influence of diameter to thickness ratio on the theory of electrical open circuit and electrical short circuit.

    阶数
    ${r_0}/h$
    5.0 6.3 7.9 10.0 12.6 15.8 19.9 25.1 31.5 39.7 50.0 62.9 79.2
    1有限元/Hz23427223192159621154209052077820726207172073120754207812080720830
    短路频率/Hz25053236262267822060216612140521242211382107321031210052098920978
    开路频率/Hz24276235252303822725225262239922319222682223622215222022219422189
    短路误差/%6.945.855.014.283.623.022.492.031.651.341.080.870.71
    开路误差/%3.635.406.687.437.757.807.687.487.267.046.846.676.52
    2有限元/Hz40544356963159728337258882413922944221562165321340211492103720973
    短路频率/Hz43504368273190428361258782418023044222982181421503213042117821098
    开路频率/Hz35260310812812626090247192381423225228452260222447223492228722248
    短路误差/%7.303.170.970.080.040.170.440.640.740.760.730.670.60
    开路误差/%13.0312.9310.997.934.511.351.223.114.385.195.675.946.08
    3有限元/Hz66373580895026043279373783263829010263492447823211223682182321477
    短路频率/Hz77602629435159342915363893158528135257212407422974222522178421484
    开路频率/Hz58123480954051034885308132793925963246352375923189228222258722438
    短路误差/%16.928.362.650.842.653.233.022.381.651.020.520.180.03
    开路误差/%12.4317.2119.4019.3917.5614.4010.506.512.940.092.033.504.47
    4有限元/Hz95852853737440663912544004619139426340733001627075249832355522608
    短路频率/Hz1252791003248070165355534524432737442323522868026098243292314322362
    开路频率/Hz91537739496026549730417373578731458283902627024839238922327522877
    短路误差/%30.7017.518.462.261.744.035.035.054.453.612.621.751.09
    开路误差/%4.5013.3819.0022.1923.2822.5220.2116.6812.488.264.371.191.19
    5有限元/Hz12755011539310196988402754706375253614451683842233276293892659324652
    短路频率/Hz18569414805111828894818763866199750865423633597931287279242557623976
    开路频率/Hz1345071076898659570092572844745540031345353056327765258452455723707
    短路误差/%45.5928.3016.007.261.212.755.136.216.365.984.983.822.74
    开路误差/%5.456.6815.0820.7124.1025.5625.3323.5420.4516.5612.067.663.83
    6有限元/Hz158793146882131735115583995378442970872591464943041762356903112327819
    短路频率/Hz2585512057691639431308461047128414468040555264590738625332182929826528
    开路频率/Hz18661614884611899895479770296265351559431143680232189289022662025073
    短路误差/%62.8240.0924.4513.215.200.344.006.127.137.516.935.874.64
    开路误差/%17.521.349.6717.3922.6125.7927.2527.1125.5522.9219.0214.479.87
    DownLoad: CSV
  • [1]

    Butler J L, Sherman C H 2016 Transducers and Arrays for Underwater Sound (Cham: Springer International Publishing) p191

    [2]

    滕舵, 王龑, 解柯柯 2018 陕西师范大学学报(自然科学版) 46 30Google Scholar

    Teng D, Wang Y, Xie K K 2018 J. Shaanxi Norm. Uni. (Nat. Sci. Ed. ) 46 30Google Scholar

    [3]

    周利生, 许欣然 2021 声学学报 46 1250Google Scholar

    Zhou L S, Xu X R 2021 Acta Acust. 46 1250Google Scholar

    [4]

    张鸿磊 2018 硕士学位论文 (哈尔滨: 哈尔滨工程大学)

    Zhang H L 2018 M. S. Thesis (Harbin: Harbin Engineering University

    [5]

    柴勇, 莫喜平, 刘永平, 潘耀宗, 张运强, 李鹏 2017 声学学报 42 619Google Scholar

    Chai Y, Mo X P, Liu Y P, Pan Y Z, Zhang Y Q 2017 Acta Acust. 42 619Google Scholar

    [6]

    宋哲, 严由嵘 2018 水下无人系统学报 26 498

    Song Z, Yan Y R 2018 J. Unmanned Underw. Systems 26 498

    [7]

    林书玉 2023 声学学报 28 102Google Scholar

    Lin S Y 2023 Acta Acust. 28 102Google Scholar

    [8]

    穆廷荣 1979 声学学报 3 161Google Scholar

    Mu Y R 1979 Acta Acust. 3 161Google Scholar

    [9]

    Been K, Nam S, Lee H, Seo H seon, Moon W 2017 J. Acoust. Soc. Am. 141 4740Google Scholar

    [10]

    Lee J, Been K, Moon W 2023 J. Acoust. Soc. Am. 154 A228Google Scholar

    [11]

    Rogers P H 2005 J. Acoust. Soc. Am. 77 S16Google Scholar

    [12]

    Butler J L 1976 J. Acoust. Soc. Am. 59 480Google Scholar

    [13]

    Sherman C H, Parke N G 2005 J. Acoust. Soc. Am. 38 715Google Scholar

    [14]

    Huang C H, Ma C C, Lin Y C 2005 IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 52 1204Google Scholar

    [15]

    Aronov B S 2013 J. Acoust. Soc. Am. 134 1021Google Scholar

    [16]

    胡久龄 2023 博士学位论文 (哈尔滨: 哈尔滨工程大学)

    Hu J L 2023 Ph. D. Dissertation (Harbin: Harbin Engineering University

    [17]

    滕舵 2016 水声换能器基础 第233页

    Teng D 2016 Fundamentals of Hydroacoustic Transducers (Northwestern Polytechnical University Press) p233

    [18]

    Timošenko S P, Woinowsky-Krieger S 1996 Theory of Plates and Shells (New York: McGraw-Hill) p580

    [19]

    曹志远 1989 板壳振动理论 (北京: 中国铁道出版社) 第462页

    Cao Z Y 1989 Plate and Shell Vibration Theory (Beijing: China Railway Press) p462

    [20]

    栾桂冬 2005 压电换能器和换能器阵 (北京: 北京大学出版社) 第479页

    Luan G D 2005 Piezoelectric Transducers and Transducer Arrays (Beijing: Peking University Press) p479

    [21]

    潘瑞, 柴勇, 莫喜平, 刘文钊, 张秀侦 2024 应用声学 43 719Google Scholar

    Pan R, Chai Y, Mo X P, Liu W Z, Zhang X Z 2024 Appl. Acoust. 43 719Google Scholar

    [22]

    张海澜 2012 理论声学(北京: 高等教育出版社) 第139页

    Zhang H L 2012 Theoretical Acoustics (Beijing: Higher Education Press) p139

    [23]

    单影, 许龙, 周光平 2022 压电与声光 44 316Google Scholar

    Dan Y, Xu L, Zhou G P 2022 Piezoelectr. Acoustoopt. 44 316Google Scholar

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    [15] Shen Hui-Jie, Wen Ji-Hong, Yu Dian-Long, Wen Xi-Sen. Flexural vibration property of periodic pipe system conveying fluid based on Timoshenko beam equation. Acta Physica Sinica, 2009, 58(12): 8357-8363. doi: 10.7498/aps.58.8357
    [16] Liu Yan-Zhu. Planar vibration of a thin elastic rod with circular cross section in viscous medium. Acta Physica Sinica, 2005, 54(11): 4989-4993. doi: 10.7498/aps.54.4989
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Metrics
  • Abstract views:  822
  • PDF Downloads:  25
  • Cited By: 0
Publishing process
  • Received Date:  27 June 2024
  • Accepted Date:  03 August 2024
  • Available Online:  04 September 2024
  • Published Online:  05 October 2024

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