## 3 Results  
Calculation of Resonance and Antiresonance Frequencies for Fully Electroded Resonators, Model Verification  
The resonance frequency of the thickness oscillation will be first determined for a resonator whose thickness and diameter are given in Table 1, with fully electroded circular surfaces. The result will be compared with the analytical calculation and the experimentally determined value. The calculation will be performed on the finite element meshes shown in Figure 3. We will compare the results obtained on both types of finite element meshes. The material properties are given in Table 2. We consider the polarization direction to be homogeneous throughout the resonator volume, in the direction of the resonator thickness, i.e., along the z axis, as shown in Figure 3. The calculated frequency spectrum contains the oscillation mode shown in Figure 4.  
Figure 3: Finite element mesh corresponding to the symmetric halves of the resonators, (**a**) in the shape of circular segment, (**b**) in the shape of an intersection of circles. Half of the resonator cross section is shown, and the center of the resonator is on the left. The shape of the oscillation corresponds to the shape of the thickness oscillation in Section 5.3 in [1]. The calculated resonance frequencies are  
\[f_{ra}=988,\!961\,\mathrm{Hz}\]  
for mesh variant (a),  
\[f_{rb}=988,\!915\,\mathrm{Hz}\]  
for mesh variant (b). The calculated antiresonance frequencies are  
\[f_{aa}=1,\!115,\!829\,\mathrm{Hz}\]  
for mesh variant (a),  
\[f_{ab}=1,\!115,\!712\,\mathrm{Hz}\]  
for mesh variant (b). According to [1], the resonance frequency of the fundamental thickness oscillation can be expressed as  
\[f_{rt}=\frac{\eta}{2\pi}\sqrt{\frac{c_{33}^{D}}{\varrho}}, \tag{26}\]  
where \(\eta\) is the wavenumber, \(c_{33}^{D}\) is the element of the elastic modulus tensor at constant electric induction, and \(\varrho\) is the density. The wavenumber \(\eta\) must satisfy the relevant boundary conditions on the circular surfaces of the resonator and the transcendental equation applies to it:  
Table 2.0  
The electromechanical coupling coefficient for the thickness mode of the oscillation can be expressed as  
Table 2.1  
Let us reiterate that the polarization is assumed to be in the direction of the resonator thickness, i.e., in the direction perpendicular to the circular surfaces. Therefore, the components of material tensors with coefficients 33 appear in the relation (28). The resonance frequency of the fundamental mode of the thickness oscillation calculated for said resonator is  
Table 2.2  
The difference between the value of the resonance frequency \(f_{rt}\) calculated analytically and the numerically obtained values of \(f_{ra}\) and \(f_{rb}\) is 0.6%, which can be considered a very good agreement. We can conclude that the numerical model is thus verified. Table 2.3 Oscillation shape and frequency for mesh variant (**a**), for mesh variant (**b**). Using a similar procedure as with the resonance frequency, the antiresonance frequency can be analytically calculated, see [1]. The value is  
Table 2.4  
It can be noted that the values calculated by the finite element method (FEM) model are 0.6% lower in the case of both resonance and antiresonance frequencies. Since the calculation of both values is identical in principle and the same values of material parameters are used, a similar difference can be expected between the values obtained analytically and numerically. The experimentally determined value of the resonance frequency reported in [1] is  
Table 2.5  
Compared to \(f_{ra}\), \(f_{rb}\), it differs by approximately 1.3%. Let us now compare the difference between the frequencies \(f_{ra}\), \(f_{rb}\) that were calculated on different meshes, as documented in Table 2.6 The difference is in the order of 0.001%. The calculation performed on both variants of the meshes yields comparable results.