**3. Validation**

In this section, an ANSYS numerical simulation and an experimental study were conducted to validate the reliability of the theoretical solution. The geometric dimensions of the transducer are given as: *R*1 = 10 mm, *R*2 = 15 mm, *R*3 = 20 mm, *R*4 = 25 mm, *R*5 = 30 mm, and *h* = 5.63 mm. Materials of the piezoceramic layers were selected as piezoceramic material type (PZT-5H), of which material parameters are listed in Table 1. Three different epoxy materials were chosen for epoxy layers, which have the same Poisson's ratio, approximately equal density and a certain difference in their Young's modulus. The material parameters of these three different epoxy materials can be found in Table 2, which are numbered as -1 , -2 , and -3 , respectively. In the following analysis, these geometric dimensions and material parameters will be adopted, unless otherwise stated.

**Table 1.** Material parameters of piezoceramic materials [52,56–58].


Permittivity of free space: *ε*0 = 8.85 × 10−<sup>12</sup> F/m.

**Table 2.** Material parameters of three types of epoxy materials.


#### *3.1. ANSYS Numerical Simulation*

In this section, a finite element analysis based on the software ANSYS R17.1 was performed to compare with the theoretical results. A three-dimensional model of one-twelfth of the transducer was created because of the structural symmetry, as shown in Figure 2. In the simulation, the elements, Solid 185 and Solid 5, were used for the epoxy parts and the piezoelectric parts, respectively. The total amounts of elements and nodes were set as 2460 and 3258, respectively, to guarantee the computational precision. All the voltage degrees of freedom (DOFs) of the positive electrodes were coupled together, and the electrical condition *V*0 = 1 V was applied. All the voltage DOFs of the negative electrodes were also coupled together, and the electrical condition *V*0 = 0 V was applied. The harmonic analysis type was selected, and the frequency range was from 10 kHz to 40 kHz. The simulated impedance–frequency curve is plotted in Figure 3. In addition, the theoretical impedance–frequency relation is also plotted in Figure 3. It can be found that the results from theoretical analysis and finite element analysis agree reasonably well with each other. Further, the theoretical and simulated first resonance and anti-resonance frequencies are compared in Table 3. The relative errors between the theoretical

values and the simulated ones for the first resonance frequency and the first anti-resonance frequency are −1.31% and −1.89%, respectively. The above comparative results validate the reliability of the theoretical solution.

**Figure 2.** Three-dimensional finite element model of one-twelfth of the transducer.

**Figure 3.** Theoretical and simulated impedance spectra.

**Table 3.** Comparisons between the calculated, simulated and experimental frequencies.

