**2. Modeling**

## *2.1. Basic Equations*

Figure 1 exhibits the schematic representation of the radially layered cylindrical piezoceramic/epoxy composite transducer. It consists of a solid epoxy disk, two epoxy rings, and two axially polarized piezoceramic rings. These components are arranged alternatively in the radial direction. The two piezoceramic rings are connected in parallel electrically, and are denoted as piezoceramic ring #1 and piezoceramic ring #2, respectively. Three epoxy layers are denoted as epoxy disk #1, epoxy ring #2 and epoxy ring #3, respectively. The geometric and material parameters of each layer are different. The radial location of the interface between each layer and the axial height of the transducer are defined as *Ri* (*i* = 1, 2, 3, 4, 5) and *h*, respectively.

**Figure 1.** Schematic representation of the radially layered cylindrical piezoceramic/epoxy composite transducer.

A harmonic form of voltage is used as the excitation source, which is expressed as:

$$V(t) = V\_0 e^{j\omega \cdot t},\tag{1}$$

where *V*0, *j* = √−1, *ω* = 2*π f* , *f* and *t* are the excitation amplitude, the imaginary unit, the circular frequency, the excitation frequency and the time, respectively.

Under the assumption of plane stress, the harmonic radial displacement *urP*(*i*), radial stress *<sup>σ</sup>rP*(*i*), electric potential *φ*(*i*) and electric displacement *Dz*(*i*) of the *i*-th piezoelectric layer (*i* = 1, 2) are expressed as follows [52,53]:

$$\mu\_{rP(i)} = [A\_{P(i)}f\_1(r,i) + B\_{P(i)}f\_2(r,i)]e^{j\omega \cdot t},\tag{2}$$

$$\sigma\_{rP(i)} = \left[A\_{P(i)}f\_3(r,i) + B\_{P(i)}f\_4(r,i) + \varepsilon\_{31(i)}(V\_0/h)\right]e^{j\omega\cdot t},\tag{3}$$

$$
\phi\_{(i)} = z(V\_0/h)e^{j\omega \cdot t},\tag{4}
$$

$$D\_{z(i)}(z) = \left[A\_{P(i)}e\_{\mathfrak{M}(i)}k\_{P(i)}\right] \wr \left(k\_{P(i)}r\right) + B\_{P(i)}e\_{\mathfrak{M}(i)}k\_{P(i)}\Upsilon\_{\mathfrak{l}}(k\_{P(i)}r) - \kappa\_{\mathfrak{z}\mathfrak{z}(i)}^{\varepsilon}(\mathcal{V}o/h)\left[e^{j\omega}\right] \tag{5}$$

In Equations (2)–(5), the functions from *f* 1(*r,i*) to *f* 4(*<sup>r</sup>*,*i*) can be expressed as following [52,53]:

$$f\_1(r, i) = f\_1(k\_{P(i)}r),\tag{6}$$

$$f\_2(r, i) = \mathcal{Y}\_1(k\_{P(i)}r),\tag{7}$$

$$f\_3(r, i) = c\_{11(i)}^E k\_{P(i)} l\_0(k\_{P(i)} r) + [(c\_{12(i)}^E - c\_{11(i)}^E) / r] l\_1(k\_{P(i)} r),\tag{8}$$

$$f\_4(r, i) = c\_{11(i)}^E k\_{P(i)} \chi\_0(k\_{P(i)}r) + [(c\_{12(i)}^E - c\_{11(i)}^E)/r] \chi\_1(k\_{P(i)}r),\tag{9}$$

where *<sup>c</sup>E*11(*i*) = *<sup>s</sup>E*11(*i*)/(*sE*11(*i*)*<sup>s</sup>E*11(*i*) − *<sup>s</sup>E*12(*i*)*<sup>s</sup>E*12(*i*)), *<sup>c</sup>E*12(*i*) = <sup>−</sup>*sE*12(*i*)/(*sE*11(*i*)*<sup>s</sup>E*11(*i*) − *<sup>s</sup>E*12(*i*)*<sup>s</sup>E*12(*i*)), *<sup>e</sup>*31(*i*) = *<sup>d</sup>*31(*i*)/(*sE*11(*i*) + *<sup>s</sup>E*12(*i*)), *<sup>κ</sup><sup>ε</sup>*33(*i*) = *<sup>κ</sup>σ*33(*i*) − <sup>2</sup>*<sup>d</sup>*231(*i*)/(*sE*11(*i*) + *<sup>s</sup>E*12(*i*)); *<sup>c</sup>E*11(*i*), *<sup>c</sup>E*12(*i*), *<sup>e</sup>*31(*i*), and *<sup>κ</sup><sup>ε</sup>*33(*i*) are the effective elastic, piezoelectric and dielectric constants of the *i*-th piezoceramic layer, respectively. *kP*(*i*) = *<sup>ω</sup>*/*VrP*(*i*) and *VrP*(*i*) = \$*cE*11(*i*)/*ρP*(*i*) are the radial wave number and sound speed, respectively; *ρP*(*i*) is the density of the piezoceramic. *J*0(*kP*(*i*)*<sup>r</sup>*) is the Bessel function of the first kind, and *<sup>Y</sup>*0(*kP*(*i*)*<sup>r</sup>*) is the Bessel function of the second kind.

Similarly, the harmonic radial displacement *urE*(*i*) and radial stress *<sup>σ</sup>rE*(*i*) of the *i*-th elastic layers (*i* = 1, 2, 3) are expressed as follows [52–54]:

$$u\_{r\to(i)}(r) = \left[A\_{\to(i)}f\_{\mathbb{5}}(r,i) + B\_{\to(i)}f\_{\mathbb{6}}(r,i)\right]e^{i\omega\cdot t},\tag{10}$$

$$
\sigma\_{r\to(i)}(r) = [A\_{\to(i)}f\_7(r,i) + B\_{\to(i)}f\_8(r,i)]e^{\imath\omega\cdot t}.\tag{11}
$$

In Equations (10) and (11), the functions from *f* 5(*<sup>r</sup>*,*i*) to *f* 8(*<sup>r</sup>*,*i*) can be expressed as following [52–54]:

$$f\_5(r, i) = f\_1(k\_{E(i)}r),\tag{12}$$

$$f\_6(r, i) = \mathcal{Y}\_1(k\_{E(i)}r),\tag{13}$$

$$f\_{\mathcal{T}}(\mathbf{r}, \mathbf{i}) = [(\overline{\mathcal{E}}\_{(\mathbf{i})} k\_{\mathcal{E}(\mathbf{i})}) / (1 - \mu\_{(\mathbf{i})}^2)] \{ f\_0(k\_{\mathcal{E}(\mathbf{i})} r) + [(\mu\_{(\mathbf{i})} - 1) / (k\_{\mathcal{E}(\mathbf{i})} r)] f\_1(k\_{\mathcal{E}(\mathbf{i})} r) \},\tag{14}$$

$$f\_8(r, i) = [(\overline{\mathbb{E}}\_{(i)} k\_{\mathbb{E}(i)}) / (1 - \mu\_{(i)}^2)] \{ \mathbf{Y}\_0(k\_{\mathbb{E}(i)} r) + [(\mu\_{(i)} - 1) / (k\_{\mathbb{E}(i)} r)] \mathbf{Y}\_1(k\_{\mathbb{E}(i)} r) \},\tag{15}$$

where *kE*(*i*) = *<sup>ω</sup>*/*VrE*(*i*), *<sup>V</sup>*2*rE*(*i*) = *<sup>E</sup>*(*i*)/[*ρE*(*i*)(<sup>1</sup> − *<sup>μ</sup>*<sup>2</sup>(*i*))]; *ρE*(*i*), *<sup>E</sup>*(*i*) and *μ*(*i*) are the density, Young's modulus and Poisson's ratio of epoxy, respectively.
