Study on Vibration Characteristics of Functionally Graded Material Composite Spherical Piezoelectric Transducer

Abstract

Non-uniform composite structures for transducers exhibit considerable potential in enhancing impedance matching and efficiency. Here, a functionally graded material composite spherical piezoelectric transducer (FGM-cSPT) is proposed, and a three-port electromechanical equivalent circuit model is established. The correctness of the theoretical model is verified using the finite element method and experiment. Based on the theoretical model, the effects of the non-uniform coefficient and the geometric dimension of FGM-cSPT on the electromechanical vibration characteristics (resonance frequency, anti-resonance frequency, and effective electromechanical coupling coefficient) of the transducer are analyzed. The results show that the non-uniform coefficient and geometric dimension can effectively regulate the vibration characteristics of the FGM-cSPT, which can be used to guide engineering design. Our methodology will offer possibilities for designing FGM-cSPTs and may promote applications in various fields, such as marine exploitation, structural health detection, and energy collection.

1. Introduction

Spherical piezoelectric transducers are the optimal choice to realize omnidirectional radiation, which is widely used in marine exploitation, structural health detection, and energy collection [1,2,3]. Spherical piezoelectric transducers are usually composite metals to enhance structural strength and improve impedance matching. Wang et al. proposed a sandwiched spherical piezoelectric transducer to enhance the heat dissipation capacity of the spherical transducer through composite metal [4,5]; Tang et al. proposed a cascade spherical piezoelectric transducer to increase the electrical power of the spherical piezoelectric transducer [6,7]; and Kong et al. presented a novel spherical piezoceramic transducer called a spherical smart aggregate, which can be embedded in concrete structures for health monitoring [8,9].

On the other hand, non-uniform composite piezoelectric transducers have shown great potential in improving the performance of transducers in recent years. For example, Dong et al. designed a metagel impedance transformer to achieve broadband transmission, which is inspired by the gradient acoustic impedance distribution in the head of an Indo-Pacific humpback dolphin [10]. Bian et al. filled a specific binary mixture into the conical cavity structure produced by 3D precise printing to build a matching layer where the acoustic impedance decreases exponentially along the thickness direction, which can make the transmitter voltage response bandwidth reach 110% [11]. Ji et al. developed an acoustic impedance matching layer that achieves a gradient variation in equivalent sound impedance in the thickness direction, and its lateral structure is designed as a phononic crystal, which can effectively combine acoustic impedance matching and lateral vibration suppression [12]. Non-uniform composite structures can improve the performance of transducers in terms of impedance matching and efficiency. The material parameters of functionally graded material (FGM) show gradient changes that are the same as those of non-uniform graded materials mentioned in the literature. Therefore, using FGM as composite layers is also expected to achieve the function of non-uniform composite layers [13,14,15].

Nevertheless, research on non-uniform composite structures of spherical piezoelectric transducers, particularly their vibration characteristics, still needs to be completed. Consequently, there is a compelling need to develop theoretical models for acoustic vibration analysis of such transducers. Among the well-established methodologies, the finite element method (FEM) is a reliable and effective approach [16,17,18]. However, this method is computationally intensive, and extensive calculations are time-consuming. Therefore, it is of great significance to propose a simplified and convenient theoretical model. The electromechanical equivalent circuit method (EECM) has straightforward physical significance, and can predict the vibration characteristics of piezoelectric transducers. The elements based on the EECM can be connected by boundary conditions, which are a valuable tool for analyzing and optimizing transducer performance [19,20]. The EECM can analyze more complex vibration systems based on the characteristics linked by boundary conditions. Due to the principle of the electric force analogy, the electromechanical equivalent circuit method can analyze the impedance characteristics of piezoelectric transducers, and is often used to analyze the resonance frequency, anti-resonance frequency, and effective electromechanical coupling coefficient of piezoelectric transducers. The EECM can be used for the preliminary design of piezoelectric transducers [21,22,23].

In this study, the EECM is proposed to analyze vibrational characteristics of a functionally graded material composite spherical piezoelectric transducer (FGM-cSPT). Based on this model, the resonance frequency, anti-resonance frequency, and the effective electromechanical coupling coefficient of FGM-cSPT are investigated. The theoretical results are subsequently validated through experimental results, lending credibility to the analysis.

2. Structure and Principle

2.1. Structure of FGM-cSPT

The FGM-cSPT is composed of a spherical piezoelectric ceramic and FGM spherical shell, as shown in Figure 1. The inner radius of the piezoelectric ceramic spherical shell is “a” and the outer radius is “b”. The FGM spherical shell is composed of a ceramic–hydrogel composite, and its mixing ratio varies exponentially along the radial direction. The inner radius is b, the outer radius is c, and the distribution law of Young’s modulus (E), density ($\rho$), and Poisson’s ratio (ν) is presented as follows,

$$E=E_p+\left(E_W-E_P\right){\left({\displaystyle \frac{r-b}{c-b}}\right)}^{\beta },$$

(1)

$$\rho ={\rho }_p+\left({\rho }_W-{\rho }_P\right){\left({\displaystyle \frac{r-b}{c-b}}\right)}^{\beta },$$

(2)

$$\nu ={\nu }_p+\left({\nu }_W-{\nu }_P\right){\left({\displaystyle \frac{r-b}{c-b}}\right)}^{\beta },$$

(3)

where $E_p$, ${\rho }_p$, and ${\nu }_p$ are, respectively, Young’s modulus, density, and Poisson’s ratio of ceramics, $E_W$, ${\rho }_W$, and ${\nu }_W$ are, respectively, Young’s modulus, density, and Poisson’s ratio of hydrogels, and β is non-uniform coefficients.

2.2. EECM for FGM-cSPT

Unlike the exponential change of a single material, obtaining an analytical solution for FGM with a transition change between two materials is difficult. An approximate laminate model is a common and effective method to simulate a functionally graded sphere with arbitrary radial material properties [24,25]. Therefore, the FGM spherical shell is divided into “m” sub-spherical shells (m layers), assuming each sub-spherical shell is isotropic, as shown in Figure 2.

Therefore, the material parameters of layer i are shown as follows,

$$E_i=E_p+\left(E_W-E_P\right){\left[{\displaystyle \frac{it}{c-b}}\right]}^{\beta },$$

(4)

$${\rho }_i={\rho }_p+\left({\rho }_W-{\rho }_P\right){\left[{\displaystyle \frac{it}{c-b}}\right]}^{\beta },$$

(5)

$${\nu }_i={\nu }_p+\left({\nu }_W-{\nu }_P\right){\left[{\displaystyle \frac{it}{c-b }}\right]}^{\beta },$$

(6)

where $t={\displaystyle \frac{c-b}{m}}$, i = 1,2……m.

The force and vibration velocity of each layer are continuous at the interface. According to the boundary conditions of each layer of the isotropic spherical shell at the interface, the EECM as shown below can be obtained.

The impedance expression of each layer is shown below [4],

$$\begin{gathered} \begin{array}{c}{Z}_{i1}=\left(2{\mathrm{c}}_{12\left(\mathrm{i}\right)}-{{\displaystyle \frac{1}{2}}\mathrm{c}}_{11\left(\mathrm{i}\right)}-\alpha (\mathrm{i}){\mathrm{c}}_{11\left(\mathrm{i}\right)}\right){\displaystyle \frac{4\mathsf{\pi }{r}_i\mathrm{j}}{\omega }}\\ \\ +{\displaystyle \frac{c_{11\left(\mathrm{i}\right)}{r}_i^{2}4\pi \mathrm{j}{q}_{\left(\mathrm{i}\right)}}{\omega {\mathsf{\Lambda }}_{1\left(\mathrm{i}\right)}}}\left[\begin{array}{c}{J}_{\alpha \left(\mathrm{i}\right)}\left(q_{\left(\mathrm{i}\right)}{r}_{i+1}\right)Y_{\alpha \left(\mathrm{i}\right)-1}\left(q_{\left(\mathrm{i}\right)}{r}_i\right)\\ -J_{\alpha \left(\mathrm{i}\right)-1}\left(q_{\left(\mathrm{i}\right)}{r}_i\right)Y_{\alpha \left(\mathrm{i}\right)}\left(q_{\left(\mathrm{i}\right)}{r}_{i+1}\right)\end{array}\right]\\ \\ -c_{11\left(\mathrm{i}\right)}\sqrt{r_i}\sqrt{r_{i+1}}{\displaystyle \frac{8\mathrm{j}}{\omega {\mathsf{\Lambda }}_{1\left(\mathrm{i}\right)}}},\end{array} \end{gathered}$$

(7)

$$\begin{array}{c}{Z}_{i2}=\left(-2{\mathrm{c}}_{12\left(\mathrm{i}\right)}+{{\displaystyle \frac{1}{2}}\mathrm{c}}_{11\left(\mathrm{i}\right)}+\alpha (\mathrm{i}){\mathrm{c}}_{11\left(\mathrm{i}\right)}\right){\displaystyle \frac{4\mathsf{\pi }{r}_{i+1}\mathrm{j}}{\omega }}\\ +{\displaystyle \frac{{\mathrm{c}}_{11\left(\mathrm{i}\right)}{r}_{i+1}^{2}4\mathsf{\pi }\mathrm{j}{q}_{\left(\mathrm{i}\right)}}{\omega {\mathsf{\Lambda }}_{1\left(\mathrm{i}\right)}}}\left[\begin{array}{c}{J}_{\alpha \left(\mathrm{i}\right)}\left(q_{\left(\mathrm{i}\right)}{r}_i\right)Y_{\alpha \left(\mathrm{i}\right)-1}\left(q_{\left(\mathrm{i}\right)}{r}_{i+1}\right)\\ -J_{\alpha \left(\mathrm{i}\right)-1}\left(q_{\left(\mathrm{i}\right)}{r}_{i+1}\right)Y_{\alpha \left(\mathrm{i}\right)}\left(q_{\left(\mathrm{i}\right)}{r}_i\right)\end{array}\right]\\ -{\mathrm{c}}_{11\left(\mathrm{i}\right)}\sqrt{r_i}\sqrt{r_{i+1}}{\displaystyle \frac{8\mathrm{j}}{\omega {\mathsf{\Lambda }}_{1\left(\mathrm{i}\right)}}},\end{array}$$

(8)

$$Z_{i3}={\mathrm{c}}_{11\left(\mathrm{i}\right)}\sqrt{r_i}\sqrt{r_{i+1}}{\displaystyle \frac{8\mathrm{j}}{\omega {\mathsf{\Lambda }}_{1\left(\mathrm{i}\right)}}},$$

(9)

where ${\mathsf{\Lambda }}_{1\left(\mathrm{i}\right)}=J_{\alpha \left(\mathrm{i}\right)}\left(q_{\left(\mathrm{i}\right)}{r}_{i+1}\right)Y_{\alpha \left(\mathrm{i}\right)}\left(q_{\left(\mathrm{i}\right)}{r}_i\right)-J_{\alpha \left(\mathrm{i}\right)}\left(q_{\left(\mathrm{i}\right)}{r}_i\right)Y_{\alpha \left(\mathrm{i}\right)}\left(q_{\left(\mathrm{i}\right)}{r}_{i+1}\right)$, ${\alpha }_{\left(\mathrm{i}\right)}=\sqrt{{\alpha }_{1\left(\mathrm{i}\right)}^2+1/4}$, $q_{\left(\mathrm{i}\right)}=\omega /c_{m\left(\mathrm{i}\right)}$, ${\alpha }_{1\left(\mathrm{i}\right)}=\sqrt{2(c_{22\left(\mathrm{i}\right)}+c_{23\left(\mathrm{i}\right)}-c_{12\left(\mathrm{i}\right)})/c_{11\left(\mathrm{i}\right)}}$, $c_{m\left(\mathrm{i}\right)}=\sqrt{c_{11\left(\mathrm{i}\right)}/{\rho }_{\left(\mathrm{i}\right)}}$, ${\mathrm{c}}_{11\left(\mathrm{i}\right)}={\mathrm{c}}_{22\left(\mathrm{i}\right)}={\displaystyle \frac{{\mathrm{E}}_{\left(\mathrm{i}\right)}\left(1-{\mathsf{\nu }}_{\left(\mathrm{i}\right)}\right)}{{\mathrm{k}}_{\left(\mathrm{i}\right)}}}$, ${\mathrm{c}}_{12\left(\mathrm{i}\right)}={\mathrm{c}}_{23\left(\mathrm{i}\right)}={\displaystyle \frac{{\mathrm{E}}_{\left(\mathrm{i}\right)}{\mathsf{\nu }}_{\left(\mathrm{i}\right)}}{{\mathrm{k}}_{\left(\mathrm{i}\right)}}}$, ${\mathrm{k}}_{\left(\mathrm{i}\right)}=\left(1+{\mathsf{\nu }}_{\left(\mathrm{i}\right)}\right)\left(1-2{\mathsf{\nu }}_{\left(\mathrm{i}\right)}\right)$.

The EECM in Figure 3 can be considered as a cascade of several two-terminal networks, as shown in Figure 4.

According to Figure 4, the relationship between force and vibration velocity can be expressed as

$$\left[\begin{array}{c}{F}_i^{\prime }\\ v_i^{\prime }\end{array}\right]=\left[\begin{array}{cc}{A}_i& B_i\\ C_i& D_i\end{array}\right]\left[\begin{array}{c}{F}_{i+1}^{\prime }\\ -v_{i+1}^{\prime }\end{array}\right],$$

(10)

where $A_i={\displaystyle \frac{Z_{11i}}{Z_{21i}}}$, $B_i={\displaystyle \frac{∆Z_i}{Z_{21i}}}$, $C_i={\displaystyle \frac{1}{Z_{21i}}}$, $D_i={\displaystyle \frac{Z_{22i}}{Z_{21i}}}$, $Z_{11i}=Z_{i1}+Z_{i3}$, $Z_{22i}=Z_{i2}+Z_{i3}$, $Z_{12i}=Z_{21i}=Z_{i3}$, $∆Z_i=\left|\begin{array}{cc}{Z}_{11i}& Z_{12i}\\ Z_{21i}& Z_{22i}\end{array}\right|$.

The parameters of the two-terminal network are related as follows,

$$T=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]=T_1{T}_2\dots T_m,$$

(11)

where $T_i=\left[\begin{array}{cc}{A}_i& B_i\\ C_i& D_i\end{array}\right]$.

Therefore, the EECM of FGM can be simplified as shown in Figure 5, where $Z_{S1}={\displaystyle \frac{A}{C}}-{\displaystyle \frac{∆T}{C}}$, $Z_{S2}={\displaystyle \frac{D}{C}}-{\displaystyle \frac{∆T}{C}}$, $Z_{S3}={\displaystyle \frac{∆T}{C}}$, $∆\mathrm{T}=\left|\begin{array}{cc}A& B\\ C& D\end{array}\right|$.

Based on the boundary conditions at the interfaces, the electromechanical equivalent circuits of FGM shells and spherical piezoelectric ceramics can be connected in cascaded, parallel series. Similar operations can also be applied when analyzing FGM spherical shells in conjunction with other vibrating systems, which demonstrates the simplicity of using electromechanical equivalent circuit models in processing vibrating systems. According to the continuity of force and vibration velocity at the interface between the FGM spherical shell and the spherical piezoelectric transducer, the EECM of the FGM-cSPT can be obtained, as shown in Figure 6.

Since the derivation process of the EECM of the piezoelectric ceramic spherical shell has been given in detail in reference [5], it will not be repeated in this paper. The impedance expressions of the piezoelectric ceramic spherical shell in the figure are as follows,

$$\begin{gathered} \begin{array}{c}{Z}_{p1}={\displaystyle \frac{\mathrm{j}4\mathsf{\pi }ab{M}_b^2}{\omega }}\left({\mathrm{H}}_1-{\displaystyle \frac{1}{2}}{\mathrm{H}}_2-{\mathrm{H}}_2\mathsf{\mu }\right)\\ \\ +{\displaystyle \frac{\mathrm{j}{4\mathsf{\pi }{a}^{2}b{M}_b^2\mathrm{H}}_2\lambda }{\omega {\Delta }_1}}\left[\begin{array}{c}{J}_{\mu }\left(\lambda b\right)Y_{\mu -1}\left(\lambda a\right)\\ -Y_{\mu }\left(\lambda b\right)J_{\mu -1}\left(\lambda a\right)\end{array}\right]\\ -{\displaystyle \frac{8\mathrm{j}{\mathrm{H}}_{2}ab{M}_a{M}_b}{\omega {\Delta }_1}},\end{array} \end{gathered}$$

(12)

$$\begin{array}{c}{Z}_{p2}={\displaystyle \frac{\mathrm{j}4\mathsf{\pi }ab{M}_a^2}{\omega }}\left({-\mathrm{H}}_1+{\displaystyle \frac{1}{2}}{\mathrm{H}}_2+{\mathrm{H}}_2\mathsf{\mu }\right)\\ +{\displaystyle \frac{\mathrm{j}{4\mathsf{\pi }{b}^{2}a{M}_a^2\mathrm{H}}_2\lambda }{\omega {\Delta }_1}}\left[\begin{array}{c}{J}_{\mu }\left(\lambda a\right)Y_{\mu -1}\left(\lambda b\right)\\ -Y_{\mu }\left(\lambda a\right)J_{\mu -1}\left(\lambda b\right)\end{array}\right]\\ -{\displaystyle \frac{8\mathrm{j}{\mathrm{H}}_{2}ab{M}_a{M}_b}{\omega {\Delta }_1}},\end{array}$$

(13)

$$Z_{p3}={\displaystyle \frac{8\mathrm{j}{\mathrm{H}}_{2}ab{M}_a{M}_b}{\omega {\Delta }_1}},$$

(14)

where $M_a={\displaystyle \frac{e_{31}\sqrt{\lambda }}{{\epsilon }_{33}{∆}_1}}\left\{\begin{array}{c}-{\displaystyle \frac{4}{\pi }}{s}_{-{\displaystyle \frac{3}{2}},\mu }\left(\lambda b\right)+2a\lambda {∆}_1\left(\mu +{\displaystyle \frac{5}{2}}\right)s_{-{\displaystyle \frac{5}{2}},\mu +1}\left(\lambda a\right)\\ +2a\lambda \left[\begin{array}{c}{J}_{\mu +1}\left(\lambda a\right)Y_{\mu }\left(\lambda b\right)\\ -Y_{\mu +1}\left(\lambda a\right)J_{\mu }\left(\lambda b\right)\end{array}\right]s_{-{\displaystyle \frac{3}{2}},\mu }\left(\lambda a\right)\end{array}\right\}-{\displaystyle \frac{e_{33}}{{\epsilon }_{33}}}{a}^{-{\displaystyle \frac{1}{2}}}$, $M_b={\displaystyle \frac{e_{31}\sqrt{\lambda }}{{\epsilon }_{33}{∆}_1}}\left\{\begin{array}{c}-{\displaystyle \frac{4}{\pi }}{s}_{-{\displaystyle \frac{3}{2}},\mu }\left(\lambda a\right)+2b\lambda {∆}_1\left(\mu +{\displaystyle \frac{5}{2}}\right)s_{-{\displaystyle \frac{5}{2}},\mu +1}\left(\lambda b\right)\\ +2b\lambda \left[\begin{array}{c}{J}_{\mu +1}\left(\lambda b\right)Y_{\mu }\left(\lambda a\right)\\ -Y_{\mu +1}\left(\lambda b\right)J_{\mu }\left(\lambda a\right)\end{array}\right]s_{-{\displaystyle \frac{3}{2}},\mu }\left(\lambda b\right)\end{array}\right\}-{\displaystyle \frac{e_{33}}{{\epsilon }_{33}}}{b}^{-{\displaystyle \frac{1}{2}}}$, $\kappa ={\displaystyle \frac{1}{P}}$, $P=-{\displaystyle \frac{1}{{\epsilon }_{33}^s}}\left({\displaystyle \frac{1}{b}}-{\displaystyle \frac{1}{a}}\right)-\mathsf{\eta }{\Delta }_4+{\displaystyle \frac{{\Delta }_2{\Delta }_5}{{\Delta }_1}}-{\displaystyle \frac{{\Delta }_3{\Delta }_6}{{\Delta }_1}}$, ${\Delta }_4=2{\displaystyle \frac{{\mathrm{e}}_{31}}{{\mathsf{\epsilon }}_{33}^{\mathrm{s}}}}\sqrt{\lambda }S+{\displaystyle \frac{{\mathrm{e}}_{33}}{{\mathsf{\epsilon }}_{33}^{\mathrm{s}}}}\sqrt{\lambda }\left[b^{-{\displaystyle \frac{1}{2}}}{s}_{-{\displaystyle \frac{3}{2}},\mu }\left(\lambda b\right)-a^{-{\displaystyle \frac{1}{2}}}{s}_{-{\displaystyle \frac{3}{2}},\mu }\left(\lambda a\right)\right]$, ${\Delta }_5=2{\displaystyle \frac{{\mathrm{e}}_{31}}{{\mathsf{\epsilon }}_{33}^{\mathrm{s}}}}\sqrt{\lambda }J+{\displaystyle \frac{{\mathrm{e}}_{33}}{{\mathsf{\epsilon }}_{33}^{\mathrm{s}}}}\sqrt{\lambda }\left[b^{-{\displaystyle \frac{1}{2}}}{J}_{\mu }\left(\lambda b\right)-a^{-{\displaystyle \frac{1}{2}}}{J}_{\mu }\left(\lambda a\right)\right]$, ${\Delta }_6=2{\displaystyle \frac{{\mathrm{e}}_{31}}{{\mathsf{\epsilon }}_{33}^{\mathrm{s}}}}\sqrt{\lambda }Y+{\displaystyle \frac{{\mathrm{e}}_{33}}{{\mathsf{\epsilon }}_{33}^{\mathrm{s}}}}\sqrt{\lambda }\left[b^{-{\displaystyle \frac{1}{2}}}{Y}_{\mu }\left(\lambda b\right)-a^{-{\displaystyle \frac{1}{2}}}{Y}_{\mu }\left(\lambda a\right)\right]$, $S=\left.{\displaystyle \frac{1}{4\sqrt{\lambda }r\left(\frac{1}{2}\mu +\frac{1}{4}\right)\left(\frac{1}{2}\mu -\frac{1}{4}\right)}}{2_{2}F}_3\left(\left[-{\displaystyle \frac{1}{2}},1\right],\left[{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}\mu +{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}\mu +{\displaystyle \frac{3}{4}}\right],-{\displaystyle \frac{1}{4}}{r}^2{\lambda }^2\right)\right|\begin{array}{c}b\\ \\ a\end{array}$, $J={\displaystyle \frac{\sqrt{2}}{4}}\sqrt{\lambda }\left.\left\{{\displaystyle \frac{\sqrt{2}{r}^{-\frac{1}{2}}{\lambda }^{-\frac{1}{2}}\left[4r^2{\lambda }^2+\left(2\mathsf{\mu }+1\right)\left(3+2\mathsf{\mu }\right)\right]J_{\mu }\left(\lambda r\right)}{\left(\frac{1}{2}\mathsf{\mu }-\frac{1}{4}\right)\left(2\mathsf{\mu }+1\right)\left(3+2\mathsf{\mu }\right)}}-{\displaystyle \frac{2\sqrt{2}{r}^{\frac{1}{2}}{\lambda }^{\frac{1}{2}}{J}_{\mu +1}\left(\lambda r\right)}{\left(\frac{1}{2}\mathsf{\mu }-\frac{1}{4}\right)\left(2\mathsf{\mu }+1\right)}}-{\displaystyle \frac{4\sqrt{2}r\lambda J_{\mu }\left(\lambda r\right)s_{\frac{3}{2},\mu }\left(\lambda r\right)}{\left(\frac{1}{2}\mathsf{\mu }-\frac{1}{4}\right)\left(2\mathsf{\mu }+1\right)\left(3+2\mathsf{\mu }\right)}}+{\displaystyle \frac{2\sqrt{2}r\lambda J_{\mu +1}\left(\lambda r\right)s_{\frac{1}{2},\mu }\left(\lambda r\right)}{\left(\frac{1}{2}\mathsf{\mu }-\frac{1}{4}\right)\left(2\mathsf{\mu }+1\right)}}\right\}\right|\begin{array}{c}b\\ \\ a\end{array}$, $Y={\displaystyle \frac{\sqrt{2}}{4}}\sqrt{\lambda }\left.\left\{{\displaystyle \frac{\sqrt{2}{r}^{-\frac{1}{2}}{\lambda }^{-\frac{1}{2}}\left[4r^2{\lambda }^2+\left(2\mathsf{\mu }+1\right)\left(3+2\mathsf{\mu }\right)\right]{\mathrm{Y}}_{\mathsf{\mu }}\left(\mathsf{\lambda }\mathrm{r}\right)}{\left(\frac{1}{2}\mathsf{\mu }-\frac{1}{4}\right)\left(2\mathsf{\mu }+1\right)\left(3+2\mathsf{\mu }\right)}}-{\displaystyle \frac{2\sqrt{2}{r}^{\frac{1}{2}}{\lambda }^{\frac{1}{2}}{Y}_{\mu +1}\left(\lambda r\right)}{\left(\frac{1}{2}\mathsf{\mu }-\frac{1}{4}\right)\left(2\mathsf{\mu }+1\right)}}-{\displaystyle \frac{4\sqrt{2}r\lambda Y_{\mu }\left(\lambda r\right)s_{\frac{3}{2},\mu }\left(\lambda r\right)}{\left(\frac{1}{2}\mathsf{\mu }-\frac{1}{4}\right)\left(2\mathsf{\mu }+1\right)\left(3+2\mathsf{\mu }\right)}}+{\displaystyle \frac{2\sqrt{2}r\lambda Y_{\mu +1}\left(\lambda r\right)s_{\frac{1}{2},\mu }\left(\lambda r\right)}{\left(\frac{1}{2}\mathsf{\mu }-\frac{1}{4}\right)\left(2\mathsf{\mu }+1\right)}}\right\}\right|\begin{array}{c}b\\ \\ a\end{array}$, ${\mathrm{H}}_1=2{\mathrm{c}}_{13}+2{\mathrm{e}}_{31}{\mathrm{e}}_{33}/{\mathsf{\epsilon }}_{33}^{\mathrm{s}}$, ${\mathrm{H}}_2={\mathrm{c}}_{33}+{\mathrm{e}}_{33}^2/{\mathsf{\epsilon }}_{33}^{\mathrm{s}}$, $n_1=\sqrt{b}{M}_b$, $n_2=\sqrt{a}{M}_a$, $n_0=4\pi \sqrt{a}\sqrt{b}{M_{a}M}_b\kappa$, $C_0=4\pi \kappa$, _mF_n ([a₁…a_m], [b₁…b_m], x) is the generalized hypergeometric function which is given by hypergeometric series.

According to Figure 6, the input impedance of the transducer is

$$Z_{eq}={\displaystyle \frac{V_r}{I_r}}={\displaystyle \frac{n_0^2-j\omega C_0{Z}_m}{{\omega }^2{C}_0^2{Z}_m}},$$

(15)

where $Z_m=Z_{p3}+{\displaystyle \frac{Z_{p1}{Z}_f}{Z_{p1}{+Z}_f}}$ and $Z_f=Z_{p2}+n_2^2(Z_{S1}+{\displaystyle \frac{Z_{S2}{Z}_{S3}}{Z_{S2}{+Z}_{S3}}})$.

When the input impedance tends to zero, the resonance frequency equation can be written as

$$n_0^2-\mathrm{j}\omega C_0{Z}_m=0.$$

(16)

When the input impedance tends to infinity, the antiresonance frequency equation can be obtained as

$${\omega }^2{C}_0^2{Z}_m=0.$$

(17)

3. Results and Discussion

3.1. Vibration Modes and Admittance

Since Equations (15) and (16) are transcendental equations, whose analytic solutions cannot be obtained directly, Wolfram Mathematica 11.3 is used to solve the frequency equation in the following analysis. In addition, the numerical simulations were carried out using the FEM based on the solid mechanics and electrostatic modules of the commercial software COMSOL Multiphysics 6.2. Figure 7 shows the impedance and the first radial vibration mode, where the geometric dimensions of the FGM-cSPT are a = 30 mm, b = 35 mm, and c = 36 mm, and the non-uniformity coefficient β = 1. The materials used are shown in

Table 1. Material parameters of PZT-5A, ceramic, and hydrogel.

Parameters	Value
	PZT-5A
$\rho (\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)$	7750
$c_{11}^E (\times {10}^{10} \mathrm{N}/{\mathrm{m}}^2)$	12.1
$c_{12}^E (\times {10}^{10} \mathrm{N}/{\mathrm{m}}^2)$	7.54
$c_{13}^E (\times {10}^{10} \mathrm{N}/{\mathrm{m}}^2)$	7.52
$c_{33}^E (\times {10}^{10} \mathrm{N}/{\mathrm{m}}^2)$	11.1
$e_{31}\mathrm{N}\mathrm{m}/\mathrm{V}$	−5.4
$e_{33}\mathrm{N}\mathrm{m}/\mathrm{V}$	15.8
${\epsilon }_{33}^s (\times {10}^{-9} \mathrm{C}/\mathrm{m})$	7.35
Parameters	Value
	Ceramic	Hydrogel
$\rho (\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)$	7700	1070
E (N/m2)	53.2 × 109	790
σ	0.26	0.44

. The materials set up in COMSOL Multiphysics 6.2 are shown in Appendix A. The black line and red dashed line represent the results of the FEM and EECM, respectively, and the admittance obtained using the FEM is in good agreement with the EECM results. The first-mode resonance frequency and anti-resonance frequency of the FGM-cSPT obtained using the EECM are 23,794 Hz and 28,808 Hz, respectively. The first-mode resonance frequency and anti-resonance frequency of the FGM-cSPT obtained using the FEM are 23,798 Hz and 29,376 Hz, respectively. The first radial vibration mode calculated using the FEM is shown in the illustration, which is also known as the breathing vibration mode.

The proposed EECM is effective for analyzing radial vibrations of spherical piezoelectric transducers. However, when more coupled vibrations are involved, more complex models are necessary for accurate analysis.

3.2. Effect of FGM on Vibration Characteristics of FGM-cSPT

As a composite part of a spherical piezoelectric transducer, the FGM layer inevitably affects the overall vibration characteristics of the transducer. As shown in Equations (1)–(3) in Section 2, the non-uniform coefficient β affects the composition of the material. Hence, the effect of the FGM layer regarding thickness and the non-uniformity coefficient (β) on the vibration characteristics of the FGM-cSPT is investigated based on the EECM and FEM, as shown in Figure 8. Here, a dimensionless quantity(τ) is introduced, where τ$={\displaystyle \frac{b-a}{c-a}}$.

The resonance frequency and anti-resonance frequency obtained using the EECM and FEM are consistent in different geometric dimensions and non-uniform coefficients. As τ increases, the resonance frequency and anti-resonance frequency of the FGM-cSPT increase. In other words, the increase in the wall thickness of the FGM layer leads to a decrease in both the resonance frequency and anti-resonance frequency of the FGM-cSPT, which is consistent with the change in the resonance frequency and anti-resonance frequency of the composite spherical piezoelectric transducer when the outer layer thickness increases. Furthermore, by comparing the resonance frequency and anti-resonance frequency at β = 0.5, 1, and 2, it can be found that as β increases, both the resonance frequency and anti-resonance frequency decrease. Accordingly, the non-uniform coefficient should be considered in engineering design.

As the value of τ rises, the effective electromechanical coupling coefficient also ascends, as shown in Figure 9.

As the wall thickness of the FGM layer grows, the effective electromechanical coupling coefficient declines. Furthermore, by comparing β = 0.5, 1, and 2, it can be observed that as the β increases, the effective electromechanical coupling coefficient decreases. Therefore, when designing the FGM-cSPT, it is necessary to balance the effective electromechanical coupling coefficient and other performance indicators in the engineering design process. The effective electromechanical coupling coefficient obtained using the EECM and FEM are consistent in different geometric dimensions and non-uniform coefficients.

The FGM-cSPT is anticipated to be utilized in ocean exploration and structural health monitoring due to its omnidirectional radiation and gradient-matching properties. In addition, the great potential of non-uniform composite layers in acoustic impedance matching is also expected to improve the bandwidth of spherical piezoelectric transducers. However, the realization of FGMs that precisely adhere to theoretical specifications presents substantial manufacturing challenges. The theoretical model can be used as a guide for device design, such as the fabrication of heterogeneous models of structural gradient composites rather than materials that strictly conform to the density, Young’s modulus, and Poisson’s ratio functional gradient changes in Equations (1)–(3).

3.3. Experiment

Due to the difficulty of manufacturing FGM in practice, the correctness verification of theoretical models for FGM is carried out by degrading them to the theoretical models of isotropic materials to verify their validity [26,27,28]. Therefore, we degenerated the theoretical model for FGM to that for an isotropic material to verify the correctness.

Here, aluminum is selected as the composite layer. Because it is difficult to manufacture the whole spherical shell, the composite spherical piezoelectric transducer comprises two hemispheres. The prototype used in the experiment is shown in Figure 10. An impedance analyzer (BEIJING BAND ERA CO., LTD, PV520A-V, Beijing, China) was used to test the impedance frequency response curve. The results of the theoretical model and FEM were compared with the experimental test results, as shown in

Table 2. Comparison of experimental, FEM, and theoretical results.

Experiment		FEM				EECM
fre (Hz)	fae (Hz)	frf (Hz)	Δf1%	faf (Hz)	Δf2%	frt (Hz)	Δt1%	fat (Hz)	Δt2%
105,333	119,717	109,980	4.41	125,080	4.48	109,854	4.29	125,500	4.83

. The piezoelectric ceramic is PZT-4, with an outer diameter of 19.5 mm and an inner diameter of 16.5 mm. The spherical shell of the composite layer is made of aluminum, with an outer diameter of 22.5 mm and an inner diameter of 19.5 mm. The material parameters of PZT-4 and aluminum are shown in

Table 3. Material parameters of PZT-4 and aluminum.

Parameters	Value
	PZT-4
$\rho (\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)$	7500
$c_{11}^E (\times {10}^{10} \mathrm{N}/{\mathrm{m}}^2)$	13.9
$c_{12}^E (\times {10}^{10} \mathrm{N}/{\mathrm{m}}^2)$	7.78
$c_{13}^E (\times {10}^{10} \mathrm{N}/{\mathrm{m}}^2)$	7.43
$c_{33}^E (\times {10}^{10} \mathrm{N}/{\mathrm{m}}^2)$	11.5
$e_{31}\mathrm{N}\mathrm{m}/\mathrm{V}$	−5.2
$e_{33}\mathrm{N}\mathrm{m}/\mathrm{V}$	15.1
${\epsilon }_{33}^s (\times {10}^{-9} \mathrm{C}/\mathrm{m})$	5.62
Parameters	Value
	Aluminum
$\rho (\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)$	2700
E (N/m2)	69 × 109
σ	0.33

, where ${\Delta }_{f1}=\left|f_{re}-f_{rf}\right|/f_{re}$, ${\Delta }_{f2}=\left|f_{ae}-f_{af}\right|/f_{ae}$, ${\Delta }_{t1}=\left|f_{re}-f_{rt}\right|/f_{re}$, and ${\Delta }_{t2}=\left|f_{ae}-f_{at}\right|/f_{ae}$.

The resonance and anti-resonance frequencies for the composite spherical piezoelectric transducer are 105,333 Hz and 119,717 Hz based on the experiment, 109,980 Hz and 125,080 Hz based on the FEM, and 109,854 Hz and 125,500 Hz based on the EECM. Comparing the results for the experiment, FEM, and EECM, the results are consistent, which further explains the correctness of the theoretical model. The errors between the FEM and the experimental results are 4.41% for the resonance frequency and 4.48% for the anti-resonance frequency. In comparison, the errors between the EECM and the experimental results are 4.29% for the resonance frequency and 4.83% for the anti-resonance frequency. There are two main reasons for the errors. First, the material parameters used in the theoretical model and FEM model are based on standard values, while the piezoelectric material used in the experiment may have slight variations from these standard values, leading to experimental errors. Second, the theoretical model and FEM model assume a complete spherical shell, whereas the actual model consists of two hemispheres. This difference reduces the overall stiffness and contributes to the experimental error.

4. Conclusions

Here, an FGM-cSPT is proposed, and a theoretical model is established. The correctness of the theoretical model is verified using the FEM and by experiment. Based on the theoretical model, the electromechanical vibration characteristics of the FGM-cSPT are analyzed. The results show that the non-uniform coefficient of FGM can realize the regulation of the vibration characteristics of spherical piezoelectric transducers. The increase in the wall thickness of the FGM layer leads to a decrease in both the resonance frequency and anti-resonance frequency of the spherical piezoelectric transducer. When designing an FGM-cSPT, it is necessary to balance the effective electromechanical coupling coefficient and other performance indicators. In addition, the theory is universal and suitable for analyzing typical composite spherical piezoelectric transducers. The theory proposed in this paper can effectively guide the design of an FGM-cSPT and promote the application of the transducer in marine exploitation, structural health detection, and energy collection.