## 3 Validation of the Theory
### Validation of the Theory through Finite Element Method  
The accuracy of the theoretical derivation can be verified through the finite element method. In this case, we utilized the \"Piezoelectric, Solid\" Module of the finite element software COMSOL 6.0 to construct a two-dimensional axisymmetric model of a trilaminar bending ring with unequal diameters, as illustrated in Figure 3. The model consists of 804 solid elements and 928 nodes. The element type is quadrilateral. The piezoelectric ceramic ring and the metal substrate ring are hard connected. Furthermore, in this setup, no voltage excitation is applied to the piezoelectric ceramic ring. The model incorporates PZT-4 ceramic material for the ceramic component and aluminum alloy 6061 material for the metal substrate. The material parameters for PZT-4 are referenced from ref. [28], while the material parameters for aluminum alloy 6061 can be found in Table 1. The structural dimensions of the model are presented in Table 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material for the** & **Young’s Modulus** & **Density (kg/m\({}^{3}\))** & **Poisson’s Ratio** \\ \hline Aluminum alloy 6061 & 68.9 & 2800 & 0.33 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The material parameters for aluminum alloy 6061.  
Figure 3: Finite element model of a trilaminar bending ring with unequal diameters.  
Sequentially, various boundary conditions, including free inner and fixed outer, fixed inner and free outer, free inner and free outer, and free inner and supportive outer, are applied to the model. Subsequently, eigenfrequency analysis is conducted to determine the resonant frequencies and antiresonant frequencies of the first-order and third-order bending vibrations under different boundary conditions. Additionally, displacement distribution curves for the first-order and third-order bending vibrations of the model are extracted. Similarly, the resonant frequencies, antiresonant frequencies, and displacement distribution curves of the model are calculated using the theoretical formulas in the second part.  
Table 3 presents a comparison between the finite element results and theoretical calculation results for the resonant frequencies and antiresonant frequencies. In Table 3,\(f_{71}\) and\(f_{73}\) denote the first-order and third-order resonant frequencies, and\(f_{41}\) and \(f_{43}\) denote the first-order and third-order antiresonant frequencies, respectively. The data shows that the discrepancy between the two sets of results is minimal, providing confirmation of the accuracy of the proposed theory. Notably, the errors in the third-order resonant frequencies are larger than those of the first-order resonant frequencies. This variation can be attributed to the relatively shorter wavelength at higher frequencies, resulting in a more pronounced error stemming from [PERSON]'s hypotheses. Furthermore, Figure 4 illustrates a comparison of the displacement distribution curves between the finite element results and the theoretical calculation results. It provides visual evidence of a strong agreement between the two sets of results. The high agreement between the finite element simulation results and the theoretical calculation results verifies the correctness of the proposed theoretical derivation.