Analysis of resonant characteristic changes depending on the thickness of the shell in a class IV flextensional transducer

Jungsoon Kim; Moojoon Kim

doi:10.7776/ASK.2025.44.3.217

Preview

Research Article

The Journal of the Acoustical Society of Korea. 31 May 2025. 217-222
https://doi.org/10.7776/ASK.2025.44.3.217

Analysis of resonant characteristic changes depending on the thickness of the shell in a class IV flextensional transducer

클래스 IV 플렉스텐셔널 트랜스듀서의 쉘 두께 변화에 따른 공진 특성 해석

Jungsoon Kim¹

Moojoon Kim²^*

김 정순¹

김 무준²^*

¹School of Electrical and Control Engineering, Tongmyong University

²Department of Physics, Pukyong National University

^{*Corresponding Author}

License (open-access, http://creativecommons.org/licenses/by-nc/4.0/):

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

This study analyzes the effect of changes in the shell thickness of a class IV flextensional transducer with an elliptical cylindrical shell on its resonance characteristics. A mathematical model was proposed, and both theoretical calculations and experimental results confirmed that the resonant frequency increases with shell thickness. The proposed model demonstrated high accuracy, providing a valuable foundation for optimizing transducer design and evaluating new designs.

Keywords

Flextensional transducer

Shell thickness

Resonant characteristic

Broadband receiver

본 연구는 클래스 IV 플렉스텐셔널 트랜스듀서의 타원형 원통 쉘 두께 변화가 공진 특성에 미치는 영향을 분석하였다. 수학적 모델을 제안하고, 이론 계산과 실험을 통해 쉘 두께가 증가할수록 공진 주파수가 상승함을 확인하였다. 제안된 모델은 높은 정확성을 보여 트랜스듀서 설계의 최적화 및 새로운 디자인 검토에 유용한 기초 자료를 제공하며, 수중 음향 송신기 및 광대역 수신기와 같은 응용 분야에 활용 가능성을 제시한다.

키워드

플렉스텐셔널 트랜스듀서

쉘 두께

공진 특성

광대역 수신기

MAIN

I. Introduction
II. Theoretical analysis
III. Experiment and results
IV. Conclusions

I. Introduction

Class IV flextensional transducers have been widely used as sound sources in underwater acoustics due to their excellent low frequency characteristics, high electro- acoustic conversion efficiency, and durability.^[1,2,3] Recently, they have also gained attention as broadband receivers for underwater applications.^[4] The structure of this transducer transmits the longitudinal vibration mode generated by a piezoelectric stack to an elliptical cylindrical shell, utilizing the amplification effect of flexural vibrations in the shell. As a result, conventional one dimensional analysis methods, such as equivalent circuit models, are inapplicable, rendering the mathematical analysis of characteristics considerably complex. While many studies have reported characteristic analyses using numerical methods, such as finite element analysis, to address this complexity, these approaches often require significant computational resources and time, making it challenging to optimize models or evaluate new designs efficiently.^[5,6,7] A mathematical model for analyzing the characteristics of class IV flextensional transducers was proposed by Brigham;^[8] however, its mathematical complexity has limited its practical applicability. Building upon this theory, Lam^[9] proposed a relatively simpler analytical model. A theoretical formula for predicting the transducer’s resonant frequency was derived through numerical analysis, and corresponding prediction results were reported. However, this analytical expression was not experimentally validated. Moreover, several errors were identified in its theoretical formulation, and the analysis lacks sufficient consideration of the effects of structure variations in class IV flextensional transducers. Although several studies have investigated how changes in shell thickness influence the resonant frequency of such transducers, as mentioned earlier, there is a scarcity of research deriving the full frequency spectrum of input admittance.^[8,10]

In the present study, we derive a corrected analytical expression for the mechanical impedance of an elliptical shell by conducting a mathematical analysis based on elliptical shell vibration theory.^[9] Furthermore, we focus on a simplified class IV flextensional transducer to examine how variations in the thickness of its elliptical cylindrical shell affect its resonance characteristics, and we present experimental validation of the results.

II. Theoretical analysis

The structure of the class IV flextensional transducer used for theoretical analysis is shown in Fig. 1.^[9]

https://cdn.apub.kr/journalsite/sites/ask/2025-044-03/N0660440305/images/ASK_44_03_05_F1.jpg

Fig. 1.

(Color available online) Structure of the class IV flextensional transducer used for theoretical analysis.

The piezoelectric stack is modeled using a T-type equivalent circuit as illustrated in Fig. 2.^[11] The mechanical impedance of the elliptical shell, $Z_{m} / 2$ /2, is represented by connecting it to both acoustic terminals of the equivalent circuit.

https://cdn.apub.kr/journalsite/sites/ask/2025-044-03/N0660440305/images/ASK_44_03_05_F2.jpg

Fig. 2.

Equivalent circuit of the class IV flextensional transducer.

The impedance elements of the piezoelectric stack in the equivalent circuit are as follows.^[11]

(1)

Z_{1} = j Z_{e} \tan \frac{p k_{e} l_{c}}{2}, Z_{2} = - j \frac{Z_{e}}{\sin p k_{e} l_{c}},

Here, the cross-sectional area of the piezoelectric stack is $A = π {(\frac{ϕ_{c}}{2})}^{2}$ . When the sound velocity and density of the piezoelectric material are $c_{c}$ and $ρ_{c}$ respectively, the characteristic mechanical impedance of the piezoelectric material is $Z_{0} = ρ_{c} c_{c} A$ , and the effective characteristic impedance for longitudinal effects is $Z_{e} = Z_{0} \sqrt{1 - k_{3} 3^{2}}$ . The parameter $k_{33}$ represents the electromechanical coupling coefficient for longitudinal effects, and $p$ denotes the number of piezoelectric elements in the stack.

The effective wavenumber and sound velocity of the piezoelectric elements are given as follows.

(2)

k_{e} = \frac{ω}{c_{1} \sqrt{1 - k_{33}^{2}}}, c_{c} = \frac{1}{\sqrt{s_{33}^{D} ρ_{c}}} .

Additionally, the electrical capacitance $c_{0}$ and the turns ratio of the transformer 𝜙, as shown in the equivalent circuit, are calculated as follows:

(3)

C_{0} = \frac{ε_{33}^{2} A}{l_{c}}, ϕ = \frac{- d_{33} (1 - k_{33}^{2}) A}{s_{33}^{D} l_{c}} .

Meanwhile, for the insert with a length $l_{s}$ shown in Fig. 1, the quantities are represented in the equivalent circuit with the subscript $s$ . The impedance elements, characteristic mechanical impedance, wavenumber, and sound velocity are as follows.

(4)

Z_{s 1} = j Z_{s} \tan \frac{k_{s} l_{s}}{2}, Z_{s 2} = - j \frac{Z_{s}}{\sin k_{s} l_{s}}, Z_{s} = ρ_{s} c_{s} A, k_{s} = \frac{ω}{c_{s}}, c_{s} = \sqrt{\frac{E_{s}}{ρ_{s}}},

Here, $ρ_{s}$ and $E_{s}$ represent the density and Young’s modulus, respectively. The mechanical impedance, $\frac{Z_{m}}{2}$ , connected to both mechanical terminals of the equivalent circuit arises from the metal elliptical shell containing the piezoelectric stack, as shown in Fig. 1. Based on the previously reported findings, it can be derived as follows.

(5)

Z_{m} = \frac{F_{c}}{u_{c}} = \frac{1}{j ω \sum_{i} W_{2 i} \int_{- θ_{c}}^{θ_{c}} \frac{\cos θ \cos 2 i θ + \frac{\sin θ \sin 2 i θ}{2 i}}{h_{c}} \{\frac{{(a^{2} \sin^{2} θ + b^{2} \cos^{2} θ)}^{3 / 2}}{a b}\} d θ},

Here $i, j = 1, 2, 3, \dots, m, θ_{c} = 2 \tan^{- 1} \frac{b - (a - d)}{b + (a - d)}$ .

The $W_{2 i}$ in Eq. (5) is determined by the following matrix.

(6)

[\begin{matrix} W_{2} \\ W_{4} \\ ⋮ \\ W_{2 i} \\ ⋮ \end{matrix}] = {[\begin{matrix} K M_{2, 2} K M_{2, 4} & \dots & K M_{2, 2 j} & \dots \\ K M_{4, 2} K M_{4, 4} \\ ⋮ & ⋱ \\ K M_{2 i, 2} & K M_{2 i, 2 j} \\ ⋮ & ⋱ \end{matrix}]}^{- 1} [\begin{matrix} P M_{2} \\ P M_{4} \\ ⋮ \\ P M_{2 i} \\ ⋮ \end{matrix}]

The elements of the matrix are given as follows.

$K M_{2 i, 2 j} = {[Q_{2 i, 2 j}^{N}]}^{- 1} K_{2 i, 2 j}^{N} + {[Q_{2 i, 2 j}^{T}]}^{- 1} K_{2 i, 2 j}^{T} + ω^{2} \{{[Q_{2 i, 2 j}^{N}]}^{- 1} K_{2 i, 2 j}^{N} + {[Q_{2 i, 2 j}^{T}]}^{- 1} K_{2 i, 2 j}^{T}\}$

$P M_{2 i, 2 j} = {[Q_{2 i, 2 j}^{N}]}^{- 1} P_{2 i}^{N} + {[Q_{2 i, 2 j}^{T}]}^{- 1} P_{2 i}^{T},$

$Q_{2 i, 2 j}^{N} = \int_{0}^{2 π} \frac{\cos 2 j θ \cos 2 i θ}{R} d θ$

$Q_{2 i, 2 j}^{T} = \int_{0}^{2 π} \frac{2 j \sin 2 j θ \sin 2 i θ + α \cos 2 j θ \sin 2 i θ}{R} d θ$

$K_{2 i, 2 j}^{N} = \frac{E h^{3}}{12} \int_{0}^{2 π} \frac{(2 j \cos 2 j θ - α \sin 2 j θ) (α \sin 2 i θ - 2 i \cos 2 i θ)}{R^{3}} \frac{(4 j^{2} - 1)}{j} i d θ$

$K_{2 i, 2 j}^{T} = \frac{E h^{3}}{12} \int_{0}^{2 π} \frac{(2 j \cos 2 j θ - α \sin 2 j θ) \cos 2 i θ}{R^{2}} \frac{(4 j^{2} - 1)}{j} i d θ$

$M_{2 i, 2 j}^{N} = ρ h \int_{0}^{2 π} R \cos 2 j θ \cos 2 i θ d θ,$

$M_{2 i, 2 j}^{T} = ρ h \int_{0}^{2 π} R \sin 2 j θ \sin 2 i θ d θ,$

$P_{2 i}^{N} = \int_{0}^{2 π} R P_{N} \cos 2 i θ d θ,$

$P_{2 i}^{T} = \int_{0}^{2 π} R P_{T} \sin 2 i θ d θ, P_{N} = P_{c} \cos^{2} θ,$

$P_{T} = P_{c} \cos θ \sin θ, P_{c} = - \frac{F_{c}}{L_{0} h_{c}}, α = \frac{1}{R} \frac{d R}{d θ},$

$d s = R d θ,$

Here $Q$ , $K$ , $M$ , and $P$ are matrices that arise during the derivation of the determinant of the equation of motion. Although they do not have exact physical interpretations, their roles can be inferred from the governing equations. Specifically, $Q$ is associated with the out-of-plane shear force coefficients, $K$ corresponds to in-plane force coefficients, $M$ relates to rotational moments, and $P$ represents external force coefficients. Superscript ‘N’ and ‘T’ denote components normal and tangential to the shell surface, respectively. Here, $F_{c}$ : Force applied to the cross-section of the piezoelectric stack, $R$ : Local radius of curvature, $p$ : Number of piezoelectric element.

III. Experiment and results

To validate the theoretical calculation model, an aluminum elliptical cylindrical shell resembling the structure in Fig. 1 was fabricated. As shown in Fig. 3(a), a semicircular groove was created to secure the piezoelectric stack. The elliptical cylindrical shell was manufactured using Computer Numerical Control (CNC) milling, and 4 types with thicknesses of 4 mm, 5 mm, 6 mm, and 8 mm were produced. Photo of one of these shells is shown in Fig. 3(b). A single piezoelectric stack was inserted into each shell, and the electrical input admittance was measured for each case. The length of the piezoelectric stack was measured to be 72.18 mm, but the distance between the semicircular grooves designed for securing the stack was 71 mm, as shown in Fig. 3(a). This was done to ensure good contact with the aluminum shell by applying prestress. However, the effect of this prestress was not considered in the theoretical calculations. During the insertion process, the aluminum shell was clamped and deformed using a vise, the piezoelectric stack was inserted, and the vise was then released. The physical properties of the piezoelectric vibrators used in the experiment are listed in Table 1. As shown in Fig. 4, 6 piezoelectric vibrators were used, with 2 units each bonded face to face in opposing polarization directions. These pairs were then connected lengthwise, and power leads were attached.

https://cdn.apub.kr/journalsite/sites/ask/2025-044-03/N0660440305/images/ASK_44_03_05_F3.jpg

Fig. 3.

(Color available online) Structure of the elliptical cylindrical shell (a) and photo of elliptical cylindrical shell with the piezoelectric stack (b).

Table 1.

Parameters used in the calculation.

Constant	Value	Unit
Density $ρ_{s}$	7,600	kg/m³
Diameter $ϕ_{c}$	10	mm
Length $l_{c}$	12	mm
Capacitance $C_{0}$	65.5	pF
Compliance $s_{33}^{D}$	6.2	pm²N
Compliance $s_{33}^{E}$	0.1	pm²N
Coupling constant $k_{33}$	0.8
Dielectric constant $ϵ_{33}^{T}$	1,130
Piezoelectric constant $d_{33}$	2.8	pC/N

https://cdn.apub.kr/journalsite/sites/ask/2025-044-03/N0660440305/images/ASK_44_03_05_F4.jpg

Fig. 4.

(Color available online) Mechanical and electrical structure of the piezoelectric stack.

To analyze the characteristics of the piezoelectric stack, the input admittance was measured at the electrical terminal of the piezoelectric stack, as shown in Fig. 5. The results, along with the theoretical calculations, are presented in the figure. The red dots represent the measured values, while the blue line represents the theoretical calculations. A strong agreement between the measured and calculated results can be observed. In the theoretical calculations, the equivalent circuit depicted in Fig. 2 was used, excluding the shell’s mechanical impedance ( $Z_{m}$ ), with both mechanical terminals of short circuit. The material properties of each piezoelectric element were based on the values listed in Table 1. As shown in the figure, the electrical input admittance was measured after inserting the piezoelectric stack into the elliptical cylindrical shell. The results, illustrated in Fig. 6(a), indicate that as the thickness of the metal shell increases from 4 mm to 8 mm, the resonance frequency of the transducer shifts from approximately 6 kHz to 8 kHz. Corresponding theoretical calculations are presented in Fig. 6(b), which also exhibit a similar trend of resonance frequency shift. It was further observed that the resonance frequency of the system appears in a lower frequency range compared to the resonance frequency of approximately 17 kHz for the piezoelectric stack alone, as shown in Fig. 5. In the theoretical calculations, the input admittance in the electrical terminals of the equivalent circuit was calculated with the shell’s mechanical impedance in both mechanical terminals. The resonance frequency is determined by $K M_{2 i, 2 j}$ in Eq. (6), where the terms $K_{2 i, 2 j}^{N}$ , and $K_{2 i, 2 j}^{T}$ , which form the frequency dependent coefficients $M_{2 i, 2 j}^{N}$ , and $M_{2 i, 2 j}^{T}$ , have a higher order dependency on the shell thickness $h$ . Therefore, as the thickness $h$ increases, the mechanical impedance results in a higher resonance frequency.

https://cdn.apub.kr/journalsite/sites/ask/2025-044-03/N0660440305/images/ASK_44_03_05_F5.jpg

Fig. 5.

(Color available online) Conductance (a) and susceptance (b) at the electrical input terminal of the piezoelectric stack (red dots represent measured values, and the blue line represents calculated values).

https://cdn.apub.kr/journalsite/sites/ask/2025-044-03/N0660440305/images/ASK_44_03_05_F6.jpg

Fig. 6.

(Color available online) Changes in resonance frequency with variations in the thickness of the elliptical cylindrical shell (a) measured values and (b) calculated values.

IV. Conclusions

A mathematical model was proposed for the characteristic analysis of the class IV flextensional transducer, focusing on a simplified structure to investigate how changes in the thickness of its elliptical cylindrical shell affect its resonance characteristics. The analysis showed that as the shell thickness increased from 4.0 mm to 8.0 mm, the resonance frequency rose from approximately 6 kHz to 8 kHz. This indicates that increasing the thickness enhances the shell’s mechanical impedance, leading to a higher resonance frequency. The admittance measurements at the transducer’s electrical input terminal aligned well with theoretical calculations, validating the reliability of the proposed model. This demonstrates its applicability for optimizing shell designs and evaluating new transducer designs. The findings provide a design guide for tuning resonance frequencies by adjusting structural parameters such as shell thickness. These results are significant for optimizing the performance of transducers in applications like underwater acoustic transmitters and broadband receivers.

Acknowledgements

This work was supported by KRIT (Korea Research Institute for Defense Technology Planning and Advancement) - Grant funded by Defense Acquisition Program Administration(DAPA) (KRIT-CT-23-026, 22-305-B00- 001, 2023) to the Integrated Underwater Surveillance Research Center for Adapting Future Technologies.

References

Y. Roh and K. Lang, "Analysis and design of a flextensional transducer by means of the finite element method," Jpn. J. Appl. Phys. 47, 3997 (2008).

10.1143/JJAP.47.3997

H. Kim and Y. Roh, "Optimal design of a barrel stave flextensional transducer," Jpn. J. Appl. Phys. 48, 07GL07 (2009).

10.1143/JJAP.48.07GL07

T. Zhou, Y. Lan, Q. Zhang, J. Yuan, S. Li, and W. Lu, "A conformal driving class IV flextensional transducer," Sensors, 18, 2102 (2018).

10.3390/s1807210229966344PMC6069496

G. Kim, D. Kim, and Y. Roh, "Design of wideband flextensional hydrophone," Sensors, 24, 4941 (2024).

10.3390/s2415494139123988PMC11314701

A. Feeney, A. Tweedie, A. Mathieson, and M. Lucas, "A miniaturized class IV flextensional ultrasonic transducer," Phys. Procedia, 87, 10-15 (2016).

10.1016/j.phpro.2016.12.003

J. Zheng, S. Li, and B. Wang, "Design of low- frequency broadband flextensional transducers based on combined particle swarm optimization and finite element method," Smart Mater. Struct. 30, 105002 (2021).

10.1088/1361-665X/ac1b3c

S. Chen and Y. Lan, "Finite element analysis of flextensional transducer with slotted shell," Appl. Mech. Mater. 187, 151-154 (2012).

10.4028/www.scientific.net/AMM.187.151

G. A. Brigham, "Analysis of the class‐IV flextensional transducer by use of wave mechanics," J. Acoust. Soc. Am. 56, 31-39 (1974).

10.1121/1.1903229

Y. W. Lam, "Mathematical model of a class IV flextensional transducer and its numerical solution," Appl. Acoust. 36, 123-144 (1992).

10.1016/0003-682X(92)90066-2

K. Kang and Y. Roh, "A study on the frequency characteristics of a class IV flextensional transducer" (in Korean), J. Acoust. Soc. Kr. Suppl.1(s), 18, 375- 378 (1999).

G. E. Martin, "On the theory of segmented electrommechanical systems," J. Acoust. Soc. Am. 36, 1366- 1370 (1964).

10.1121/1.1919209

The Journal of the Acoustical Society of KoreaISSN:1225-4428(Print) 2287-3775(Online)한국음향학회

Preview

Analysis of resonant characteristic changes depending on the thickness of the shell in a class IV flextensional transducer

ABSTRACT

MAIN

Fig. 1.

(Color available online) Structure of the class IV flextensional transducer used for theoretical analysis.

Fig. 2.

Equivalent circuit of the class IV flextensional transducer.

(1)

(2)

(3)

(4)

(5)

(6)

Fig. 3.

(Color available online) Structure of the elliptical cylindrical shell (a) and photo of elliptical cylindrical shell with the piezoelectric stack (b).

Table 1.

Parameters used in the calculation.

Fig. 4.

(Color available online) Mechanical and electrical structure of the piezoelectric stack.

Fig. 5.

(Color available online) Conductance (a) and susceptance (b) at the electrical input terminal of the piezoelectric stack (red dots represent measured values, and the blue line represents calculated values).

Fig. 6.

(Color available online) Changes in resonance frequency with variations in the thickness of the elliptical cylindrical shell (a) measured values and (b) calculated values.

Acknowledgements

References