## 3 Experimental and Numerical Results
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Mode** & **FEM (Hz)** & \begin{tabular}{c} **AF-ESPI** \\ **(HZ)** \\ \end{tabular} & \begin{tabular}{c} **Difference** \\ **(\%)** \\ \end{tabular} & **LDV (Hz)** & \
\begin{tabular}{c} **Difference** \\ **(\%)** \\ \end{tabular} & **Impedance** &
\begin{tabular}{c} **Difference** \\ **(Hz)** \\ \end{tabular} \\ \hline \multirow{2}{*}{1} & Anti-phase & 21 & 20 & \(-\)4.76 & 20 & \(-\)4.76 & & \\  & In-phase & 26 & 22 & \(-\)15.38 & 20 & \(-\)23.08 & & \\ \hline \multirow{2}{*}{2} & Anti-phase & 151 & 170 & 12.58 & 140 & \(-\)7.28 & 138 & \(-\)8.61 \\  & In-phase & 183 & 175 & \(-\)4.37 & 150 & \(-\)18.03 & 150 & \(-\)18.03 \\ \hline \multirow{2}{*}{3} & Anti-phase & 499 & 490 & \(-\)1.80 & 420 & \(-\)15.83 & & \\  & In-phase & 603 & 495 & \(-\)17.91 & 430 & \(-\)28.69 & 480 & \(-\)20.40 \\ \hline \multirow{2}{*}{4} & Anti-phase & 834 & 800 & \(-\)4.08 & 890 & 6.71 & 945 & 13.31 \\  & In-phase & 941 & 965 & 2.55 & 920 & \(-\)2.23 & 1110 & 17.96 \\ \hline \multirow{2}{*}{5} & Anti-phase & 1148 & 1200 & 4.53 & 1290 & 12.37 & 1345 & 17.16 \\  & In-phase & 1332 & 1400 & 5.11 & 1380 & 3.60 & 1545 & 15.99 \\ \hline \multirow{2}{*}{6} & Anti-phase & 1245 & \multirow{2}{*}{} & & & & & \\  & In-phase & 1436 & & & & & & \\ \hline \multirow{2}{*}{7} & Anti-phase & 1905 & 1870 & \(-\)1.84 & 2030 & 6.56 & 1950 & 2.36 \\  & In-phase & 2120 & 2100 & \(-\)0.94 & 2060 & \(-\)2.83 & 2150 & 1.42 \\ \hline \multirow{2}{*}{8} & Anti-phase & 2227 & \multirow{2}{*}{} & & & & & \\  & In-phase & 2586 & & & & & & \\ \hline \multirow{2}{*}{9} & Anti-phase & 2890 & 2890 & 0.00 & 2980 & 3.11 & 2810 & \(-\)2.77 \\  & In-phase & 3176 & 3150 & \(-\)0.82 & 3150 & \(-\)0.82 & 3200 & 0.76 \\ \hline \multirow{2}{*}{10} & Anti-phase & 4203 & 4200 & \(-\)0.07 & & & & \\  & In-phase & 4559 & 4500 & \(-\)1.29 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Resonant frequencies of piezoelectric valveless micropumping element under water-coupled conditions. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Mode** & **FEM (Hz)** & \begin{tabular}{c} **AF-ESPI** \\ **(HZ)** \\ \end{tabular} & \begin{tabular}{c} **Difference** \\ **(\%)** \\ \end{tabular} & **LDV (Hz)** & \begin{tabular}{c} **Difference** \\ **(\%)** \\ \end{tabular} & **Impedance** & \
\begin{tabular}{c} **Difference** \\ **(Hz)** \\ \end{tabular} &
\begin{tabular}{c} **Difference** \\ **(\%)** \\ \end{tabular} \\ \hline \multirow{2}{*}{1} & Anti-phase & 79 & 87 & 10.13 & 80 & 1.27 & 79 & 0 \\  & In-phase & 79 & 87 & 10.13 & 80 & 1.27 & & \\ \hline \multirow{2}{*}{2} & Anti-phase & 491 & 510 & 3.87 & 520 & 5.91 & 480 & \(-\)2.24 \\  & In-phase & 492 & 519 & 5.49 & 500 & 1.63 & 470 & \(-\)4.47 \\ \hline \multirow{2}{*}{3} & Anti-phase & 1363 & & & 1380 & 1.25 & & & \\  & In-phase & 1366 & & & 1350 & \(-\)1.17 & & & \\ \hline \multirow{2}{*}{4} & Anti-phase & 1636 & 1660 & 1.47 & 1780 & 8.80 & 1720 & 5.13 \\  & In-phase & 1638 & 1770 & 8.06 & 1800 & 9.89 & 1790 & 9.28 \\ \hline \multirow{2}{*}{5} & Anti-phase & 2294 & 2410 & 5.06 & 2410 & 5.06 & 2425 & 0.62 \\  & In-phase & 2296 & 2430 & 5.84 & 2410 & 4.97 & 2390 & \(-\)1.67 \\ \hline \multirow{2}{*}{6} & Anti-phase & 2780 & 2865 & 3.06 & 2810 & 1.08 & & \\  & In-phase & 2783 & 2855 & 2.59 & & & & \\ \hline \multirow{2}{*}{7} & Anti-phase & 3443 & 3620 & 5.14 & 3280 & \(-\)4.73 & & & \\  & In-phase & 3446 & 3655 & 6.07 & 3270 & \(-\)5.11 & & & \\ \hline \multirow{2}{*}{8} & Anti-phase & 4589 & 4555 & \(-\)0.74 & & & 4320 & \(-\)5.44 \\  & In-phase & 4593 & 4560 & \(-\)0.72 & & & 4340 & \(-\)5.07 \\ \hline \multirow{2}{*}{9} & Anti-phase & 4957 & 5125 & 3.39 & & & 5010 & \(-\)2.29 \\  & In-phase & 4960 & 5150 & 3.83 & & & 5050 & \(-\)1.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Resonant frequencies of piezoelectric valveless micropumping element under air-coupled conditions. LDV: laser Doppler vibrometer.