## 3 Validation of the Theory
### Validation of the Theory through Experimental Test  
To verify the accuracy of the theoretical derivation, an experimental test was conducted on a trilaminar bending ring with unequal diameters prototype, as shown in Figure 5. The prototype consisted of a PZT-4 ceramic material and an aluminum alloy 6061 metal substrate. The structural dimensions of the prototype can be found in Table 4.  
The prototype was placed on soft foam to simulate the free inner and free outer boundary conditions, and its conductance curve was tested by using the impedance analyzer (Agilent 4294A). Simultaneously, a corresponding two-dimensional axisymmetric finite element model was developed using the \"Piezoelectric, Solid\" Module of the finite element software COMSOL. Figure 6 illustrates the configuration of the model, which encompasses 596 quadrilateral solid elements and 783 nodes. The polarization direction of both piezoelectric ceramic rings is parallel to the thickness direction (as indicated by the arrows in Figure 6). The bottom surface of the upper piezoelectric ceramic ring and the top surface of the lower piezoelectric ceramic ring are grounded (represented by the blue line in Figure 6). \begin{table}
\begin{tabular}{c c} \hline \hline
**Structural Dimensional Parameter** & **Value (mm)** \\ \hline \(r_{0}\) & 6 \\ \(r_{1}\) & 26 \\ \(r_{2}\) & 60 \\ \(h_{m}\) & 1.5 \\ \(h_{p}\) & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The structural dimensions of the finite element model. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline
**Boundary Condition** & \multicolumn{6}{c}{**Free Inner and**} & \multicolumn{6}{c}{**Fixed Inner and**} & \multicolumn{6}{c}{**Free Inner and**} & \multicolumn{6}{c}{**Free Inner and**} & \multicolumn{6}{c}{**Free Inner and**} \\  & \multicolumn{6}{c}{**Fixed Outer**} & \multicolumn{6}{c}{**Free Outer**} & \multicolumn{6}{c}{**Free Outer**} & \multicolumn{6}{c}{**Supportive Outer**} \\ \hline Resonant/Antiresonant frequency & \(f_{1}\) & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{1}\) & \(f_{4}\) & \(f_{5}\) & \(f_{6}\) & \(f_{1}\) & \(f_{4}\) & \(f_{5}\) & \(f_{3}\) & \(f_{41}\) & \(f_{4}\) & \(f_{5}\) & \(f_{63}\) & \(f_{41}\) & \(f_{5}\) & \(f_{3A simple harmonic alternating voltage of 1V is applied to the top surface of the upper piezoelectric ceramic ring and the bottom surface of the lower piezoelectric ceramic ring (represented by the red line in Figure 6). A hard connection exists between the ceramic and metal substrate. Importantly, no specific boundary conditions are imposed on the inner and outer circumferential surfaces of the model to simulate the free boundary (depicted by the green line in Figure 6). Subsequently, a harmonic response analysis is conducted on the model to obtain the simulated conductance curve. Figure 4: Deflection \(w\) distributions for the first-order and the third-order bending modes under different boundary conditions. (**a**) the free inner and fixed outer; (**b**) the fixed inner and free outer; (**c**) the free inner and free outer; (**d**) the free inner and supportive outer. Figure 5: Photograph of trilaminar bending ring with unequal diameters prototype. (**a**) the prototype; (**b**) the test settings. Figure 7 shows the comparison between the test results and the finite element simulation results. It is evident that the test result of the conductance curve is in good agreement with the simulation result. The frequencies corresponding to the two peaks of the conductance curve represent the first-order and third-order bending resonant frequencies of the prototype, respectively. Further comparison of the resonant frequencies is shown in Table 5, which includes the theoretical calculation results, finite element simulation results, and experimental test results.