## 3 Experimental and Numerical Results  
Figures 8a,b, 9a,b and 10a,b respectively present the vibration characteristics of a valveless pumping element (operating in air, water and glycerine) under structure-fluid coupling, as derived from AF-ESPI experiments and FEM computation. Tables 3-5 present a detailed comparison of the resonant frequencies obtained using AF-ESPI and numerical analysis. The mode shapes and the corresponding resonant frequencies are presented for anti-phase as well as in-phase motions. The AF-ESPI results clearly show fringes that are indicative of mode shapes. In this case, we measured only out-of-plane motion due to its relatively large displacement, compared to in-plane motion. The brightest fringes represent the nodal lines of the piezoceramic bimorph in resonance. The remaining fringes represent the constant displacement contours. Figure 7: Configuration in electrical connections of dual piezoelectric bimorphs to drive in (**a**) anti-phase motion and (**b**) in-phase motion. The results in Figures 8-10 indicate that the mode shapes of the anti-phase and in-phase motions are similar to the vibrations of a piezoelectric bimorph coupled with air. When the fluid effect was more pronounced (water-coupled and glycerine-coupled cases), the differences between the anti-phase and in-phase mode shapes was more obvious. In the glycerine-coupled case, the anti-phase and in-phase motions differed considerably in terms of mode shape and resonant frequency due to the effects of viscosity. Under all three conditions, the mode shapes and resonant frequencies were similar at low frequencies (both upper and lower plates). The resonant frequencies decreased with density and an increase in the viscosity of the fluid. Figure 8: AF-ESPI and FEM results obtained from pumping element coupled with air (**a**) upper and (**b**) lower piezoelectric plate. In the air-coupled case, mode 3 did not appear in the AF-ESPI experiment below 6 kHz. In the water-coupled and glycerine-coupled cases, modes 6 and 8, and modes 3, 5, 6, and 8 did not appear, respectively. In the water-coupled case, mode 3 appeared in the AF-ESPI measurements; however, the mode shape was slightly altered due to the effects of density, sound velocity, and viscosity. Mode 3 cannot be measured by ESPI, as shown in Figure 8, which might be since there are two bimorphs vibrating in the small space (the distance between the two plates is very small), and the frequencies of mode 3 and 4 are relatively close. The mode shapes will be affected by both the interactions between the two plates. Figure 9: AF-ESPI and FEM results obtained from pumping element coupled with water in the (**a**) upper and (**b**) lower piezoelectric plates. two bimorphs and two modes. The similar reason might explain the results of modes 5, 6, 7, and 8 in the water-coupled case, as shown in Figure 9. However, the modes with more vibration did not appear in glycerine-coupled case, as shown in Figure 10, since the fluid property of glycerine might have a more dramatic influence on the piezoelectric elements. Table 3 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 a differences of less than 11%. Under water-coupled and glycerine-coupled conditions, the differences between the experimental and numerical results were less pronounced at higher frequencies (Figures 8 and 9). The  
Figure 10: AF-ESPI and FEM results obtained from pumping element coupled with glycerin on the (**a**) upper and (**b**) lower piezoelectric plates. differences between the two (in terms of frequency) were less than 10% for all of the modes, as listed in Tables 4 and 5.