## 3 Experimental and Numerical Results
between anti-phase and in-phase modes (compared to the LDV experiment results). This may explain the discrepancies in the frequencies obtained using FEM and LDV. Figures 14-16 respectively present the impedance variation curves (anti-phase and in-phase motion) obtained under air-coupled, water-coupled, and glycerine-coupled conditions. It is also  
Figure 11: Laser Doppler vibrometer (LDV) displacement spectrum of air-coupled (**a**) anti-phase vibration and (**b**) in-phase vibration. Figure 12: LDV displacement spectrum of water-coupled (**a**) anti-phase vibration and (**b**) in-phase vibration. Figure 13: LDV displacement spectrum of glycerine-coupled (**a**) anti-phase vibration and (**b**) in-phase vibration. Figure 17 present the air-coupled and water-coupled displacements caused by vibration at the tip of the free edge, as measured by LDV at various voltages. Note that only the first four modes are presented in this comparison. Displacement was shown to increase rapidly with voltage until eventually converging at a constant value, as indicated in the figures. This means that the input excitation voltage could be controlled to within a range that would prevent damage to the pumping element from excessive voltage. Under air-coupled conditions, the displacement of the first mode (approximately 50 \(\upmu\)m) far exceeded that of the other three modes. In fact, 10 volts was sufficient to cause the piezoelectric bimorph to vibrate at its maximum displacement. Under water-coupled conditions, the maximum displacement at the mid-point along the free edge of the first mode was approximately 150 \(\upmu\)m, and 10 volts was required to excite the bimorph to vibrate at the maximum displacements of each mode. The results of the flow rate and the flow velocity of the pumping element for anti-phase and in-phase vibration are listed in Tables 6 and 7. As indicated in these data, both the gain of flow rate and the flow velocity increment are larger for anti-phase vibration than for in-phase vibration. For both average flow rate and velocity, the efficiency of low frequencies (first modes to third mode; that is, 20 Hz, 170 Hz, and 490 Hz) in anti-phase vibration is better because of the larger displacement, as shown in Tables 6 and 7. The gain of the second mode of anti-phase motion is relatively large compared  
Figure 16: Impedance variation curve of glycerine-coupled (**a**) anti-phase vibration and (**b**) in-phase vibration. Figure 17: Variations in displacement with excitation voltage for first four modes under (**a**) air-coupled and (**b**) water-coupled conditions, as measured using LDV. to the other modes. The flow velocity of the pumping element with a single bimorph is also measured through the experiment, as shown in Table 8. More experimental and theoretical results about a single bimorph can be found in the published paper written about our previous study [25]. The optical experiment has much more difficulties for the dual bimorph, since one of the bimorphs is fixed in the lower position, and is obstructed by another bimorph. Comparing the results of these two studies, it is indicated that the mode shapes of a single bimorph and dual bimorph are similar, and there is a slight difference in the resonant frequencies between the single and dual bimorphs. The resonant frequencies decrease both in single and dual-bimorph piezoelectric pumping elements when the similar modes that were excited from air changed to water or glycerine. As shown in Figure 18, the pumping efficiency is presented related to increments of velocity from the values listed in Tables 7 and 8. It is shown that the anti-phase vibration of the pumping element with the dual bimorph can clearly provide a better feature than that with a single bimorph due to the compression effect by two bimorphs, which deforms in opposite directions. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Frequency** & **1 min (\(\mu\)L/s)** & **2 min (\(\mu\)L/s)** & **3 min (\(\mu\)L/s)** &