# Vibration Analysis of a Piezoelectric Ultrasonic Atomizer to Control Atomization Rate
## 6 Results and Discussion  
Numerical FEM simulations of the atomizer were performed, and the vibration modes and the resonant frequencies were determined using a forced harmonic analysis (as described in Appendix A). In particular, a numerical frequency sweep of the system using the physical properties described in Table 4 was carried out, and the mesh atomizer surface displacements with conical apertures corresponding to different frequencies were estimated and plotted in Figure 12. The first five resonant frequencies were estimated, and after a further analysis, the vibrational modes with the largest out-of-plane displacements were located at two resonance frequencies of about 110 kHz (\(f_{65}\)) and 140 kHz (\(f_{7}\)). Figure 11: Scheme of the atomization rate measurements. \begin{table}
\begin{tabular}{c c} \hline \hline
**Physical Property** & **Value** \\ \hline Outer diameter of PZT ring & \(D_{out}=16\) mm \\ Inner diameter of PZT ring & \(D_{in}=8\) mm \\ Diameter of stainless-steel disk & \(D_{out}=16\) mm \\ Diameter of apertures zone & \(D_{a}=3.6\) mm \\ Thickness of thin plate & \(h=0.05\) mm \\ Thickness of PZT ring & \(h_{ PZT}=0.63\) mm \\ Resonant frequency & \(f_{r}=110\) kHz \\ Number of apertures & \(n=551\) \\ Large/Small diameter of cone aperture & \(D_{l}=80\)\(\mu\)m/\(D_{s}=10\)\(\mu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Atomizer physical properties. In Figure 13, the numerical simulation findings (line plots) at various voltage levels (20-80 V) are compared with the experimental results of the atomization rate obtained by a frequency sweep over the range from 100 kHz to 150 kHz. The small discrepancies were expected, as the PZT ring was bonded to the metal mesh using epoxy and the wires were soldered to the system. Overall, the peak atomization rates matched closely with the resonant frequencies from the numerical simulations, with the maxima occurring at frequencies around 110 kHz and 141 kHz. From these results, it is possible to observe that the resonance frequency is not altered with the voltage change; only the amplitude of the central displacement is affected. The volume generated by the global deformation of the thin plate \((\Delta V_{s})\), as well as the additional contribution of the micro-aperture deformations \((\Delta V_{D})\), were also numerically estimated. The maximum deformation of the micro-aperture located near the center of the thin plate was measured directly from FEM at 110 kHz for the values of the voltage in the range [\(-\)80, 80] V, as shown in Figure 14. A symmetric cyclic behavior of the small (\(D_{s1}\)) and large (\(D_{l1}\)) diameters was found; however, these changes were very small (about 0.025 \(\upmu\)m) with respect to their initial value at rest. Thus, its contribution to the total value change \((\Delta V_{D})\) was not found to be significant. Figure 12: Numerical results of the frequency sweep for 20 V obtained by FEM. A schematic of the frequency mode cross-section is shown above every detected frequency. Figure 13: Comparison between the frequency sweep FEM results for different voltages and the experimental results of the atomization rate. Additionally, the global liquid volume displaced by the thin plate was calculated based on the FEM results (Figure 15). The distribution of the particle displacement at the surface of the thin plate as given by the FEM was calculated by using the shell method of integration [28]. An illustration of the integration method considering the limits (\(a_{n}\), \(b_{n}\)) is described in Figure 15. This method allows obtaining volumes generated by rotating an area between any two functions, \(f(r)\) and \(g(r)\), where \(f(r)\) is the displacement curve and \(g(r)\) is a threshold. According to our experimental findings, atomization was not detected below 20 V. Therefore, the displacement amplitude (or volume variation) at that value was used as a threshold (\(\Delta V_{S}^{th}\)).