## 3 Experimental and Numerical Results
Tables 3-5 show that under air-coupled conditions, the resonance frequency and mode shape obtained in the AF-ESPI experiments were in good agreement with the FEM results, with differences of less than 11%. Under water-coupled and glycerine-coupled conditions, the differences between experimental and numerical results were less pronounced at higher frequencies (Figures 9 and 10). The differences between the two (in terms of frequency) were less than 10% for all of the modes except for the first three. The disturbances were more pronounced at lower frequencies due to the large vibration displacement. It was easier to obtain AF-ESPI measurements from glycerol-coupled vibrations than from water-coupled vibrations thanks to the stability of the frequency response in higher viscosity liquids. Overall, in the AF-ESPI experiments, the degree to which the density, sound velocity, and viscosity coefficient influenced the vibration characteristics (particularly the mode shapes of the piezoelectric element) increased with an increase in frequency. At lower frequencies (below 2 kHz), the difference between AF-ESPI and numerical results (in terms of resonance frequency) were less pronounced under glycerine-coupled than air-coupled and water-coupled conditions. Figures 11-13 respectively present frequency spectrum diagrams derived from LDV measurements of air-coupled, water-coupled, and glycerine-coupled vibrations. Tables 3-5 respectively present the frequency values for air-coupled, water-coupled, and glycerine-coupled cases, as well as the degree to which they differed from the FEM results. It is clear that the differences were more pronounced under water-coupled and glycerine-coupled conditions than under air-coupled conditions. This may be because the laser light must penetrate the acrylic as well as the fluids in the LDV experiment, resulting in the consumption of some of the luminous energy, resulting in errors. The frequency difference between anti-phase and in-phase LDV measurements were more pronounced under water-coupled and glycerine-coupled conditions than under air-coupled conditions. The theoretical results for water-coupled and glycerine-coupled cases presented larger differences \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multirow{2}{*}{**FEM (Hz)**} & **AF-ESPI** & **Difference** & \multirow{2}{*}{**LDV (Hz)**} & **Difference** & **Impedance** & **Difference** \\  & & & **(Hz)** & & **(\%)** & & **(\%)** & **(Hz)** & **(\%)** \\ \hline \multirow{2}{*}{1} & Anti-phase & 19 & 19 & 0 & 20 & 5.26 & & \\  & In-phase & 24 & 24 & 0.00 & 30 & 25.00 & & \\ \hline \multirow{2}{*}{2} & Anti-phase & 136 & 136 & 0.00 & 140 & 2.94 & & \\  & In-phase & 166 & 155 & \(-\)6.63 & 130 & \(-\)21.69 & & \\ \hline \multirow{2}{*}{3} & Anti-phase & 452 & & & 470 & 3.98 & & \\  & In-phase & 550 & & & 620 & 12.73 & & \\ \hline \multirow{2}{*}{4} & Anti-phase & 766 & 720 & \(-\)6.01 & 730 & \(-\)4.70 & 775 & 1.17 \\  & In-phase & 871 & 900 & 3.33 & 830 & \(-\)4.71 & 1000 & 14.81 \\ \hline \multirow{2}{*}{5} & Anti-phase & 1048 & & & 1050 & 0.19 & & \\  & In-phase & 1230 & & & 1080 & \(-\)12.20 & & \\ \hline \multirow{2}{*}{6} & Anti-phase & 1143 & & & 1150 & 0.61 & & \\  & In-phase & 1324 & & & & & \\ \hline \multirow{2}{*}{7} & Anti-phase & 1758 & 1850 & 5.23 & 1850 & 5.23 & & \\  & In-phase & 1971 & 1950 & \(-\)1.07 & 1770 & \(-\)10.20 & & \\ \hline \multirow{2}{*}{8} & Anti-phase & 2040 & & & & & & \\  & In-phase & 2389 & & & & & \\ \hline \multirow{2}{*}{9} & Anti-phase & 2676 & 2450 & \(-\)8.45 & 2450 & \(-\)8.45 & & \\  & In-phase & 2963 & 2650 & \(-\)10.56 & 2650 & \(-\)10.56 & 3170 & 6.99 \\ \hline \multirow{2}{*}{10} & Anti-phase & 3906 & 3906 & 0.00 & & & & \\  & In-phase & 4267 & 4420 & 3.59 & & & \\ \hline \multirow{2}{*}{11} & Anti-phase & 4944 & 4950 & 0.12 & & & 5200 & 5.18 \\  & In-phase & 5343 & 5340 & \(-\)0.06 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Resonant frequencies of piezoelectric valveless micropumping element under glycerine-coupled conditions.