## 5 Results in Material Constants  
The values required for RM were measured using an impedance analyzer and then converted. With the APC-855 material as an example, we took the average of the results from the five specimens. Table 2 presents the resonant and anti-resonant frequencies measured by the impedance analyzer in the TE mode, which were substituted into Equations (7) through (10) to derive \(K_{t}\), \(C_{33}^{E}\), and \(C_{33}^{D}\). Table 3 presents the resonant and anti-resonant frequencies measured by the impedance analyzer in the TS mode, which were substituted into Equations (12) through (16) to derive \(K_{15}\), \(C_{44}^{E}\), and \(C_{44}^{D}\). Table 4 shows the capacitance values measured at 1 kHz in the TE and TS modes, which were substituted into Equations (10), (11), (17), and (18) to derive \(\varepsilon_{33}^{T}\) and \(\varepsilon_{11}^{T}\). Table 5 displays the resonant and anti-resonant frequencies measured by the impedance analyzer in the LE mode, which were substituted into Equations (19) through (21) to derive \(K_{33}\), \(S_{33}^{E}\), and \(S_{33}^{D}\). Table 6 contains the resonant and anti-resonant frequencies measured by the impedance analyzer in the LTE mode, which were substituted into Equations (22) through (24) to derive \(K_{31}\), \(S_{11}^{E}\), and \(S_{11}^{D}\). Table 7 shows the resonant and anti-resonant frequencies measured by the impedance analyzer in the RAD mode, which were substituted into Equations (25) through (27) to derive \(K_{p}\) and \(\sigma^{E}\). \begin{table}
\begin{tabular}{c c c c c c} \hline
**TE Mode** & \(f_{r}\) **(kHz)** & \(f_{a}\) **(kHz)** & \(k_{t}\) & \(\varepsilon_{33}^{E}\)(\(\times 10^{10}\)N/\(m^{2}\))** & \(c_{33}^{D}\)(\(\times 10^{10}\)N/\(m^{2}\))** \\ \hline Sample 1 & 1205 & 1875 & 0.7967 & 5.7156 & 15.6476 \\ Sample 2 & 1162.5 & 1857.5 & 0.8093 & 5.2983 & 15.3568 \\ Sample 3 & 1170 & 1875 & 0.8107 & 5.2763 & 15.3900 \\ Sample 4 & 1190 & 1855 & 0.7976 & 5.5726 & 15.3155 \\ Sample 5 & 1182.5 & 1850 & 0.8007 & 5.4972 & 15.2331 \\ \hline Avg. Value & – & – & 0.8030 & 5.4720 & 15.3886 \\ \hline \end{tabular}
\end{table} 
Table 2: Resonant and anti-resonant frequencies and the converted material constants in thickness extension (TE) mode. \begin{table}
\begin{tabular}{c c c c c c} \hline
**TS Mode** & \(f_{r}\) **(kHz)** & \(f_{a}\) **(kHz)** & \(k_{15}\) & \(c_{44}^{E}\)(\(\times 10^{10}\)N/\(m^{2}\))** & \(c_{44}^{D}\)(\(\times 10^{10}\)N/\(m^{2}\))** \\ \hline Sample 1 & 1735 & 2100 & 0.6028 & 2.3078 & 3.6251 \\ Sample 2 & 1720 & 2045 & 0.5804 & 2.2797 & 3.4377 \\ Sample 3 & 1720 & 2055 & 0.5867 & 2.2765 & 3.4714 \\ Sample 4 & 1760 & 2062.5 & 0.5607 & 2.3974 & 3.4968 \\ Sample 5 & 1745 & 2092.5 & 0.5913 & 2.3407 & 3.5992 \\ \hline Avg. Value & – & – & 0.5844 & 2.3204 & 3.5260 \\ \hline \end{tabular}
\end{table} 
Table 3: Resonant and anti-resonant frequencies and the converted material constants in thickness shear extension (TS) mode. Table 8 compares the material constants obtained using RM and those provided by the manufacturer for piezoelectric material APC-855 from APC International, Ltd., and Table 9 compares the material constants for piezoelectric material C-2 from the Fuji Ceramics Corporation. Some differences exist between the material constants obtained using RM in this study and those provided by the manufacturer or in literature for the two piezoelectric ceramic materials. Although the difference is not very large in piezoelectric \(d\)-form parameters, the finite element analysis produces calculating inaccuracy or it cannot obtain the results because of the missing terms in the manufacturer provided one. Due to transformation in calculation, the \(e\)-form parameters in the manufacturer provided one have a larger difference with those obtained by the resonance method in this study. We therefore also substituted these material constants into FEM to calculate the high-frequency in-plane low-frequency out-of-plane