# Vibration Analysis of a Piezoelectric Ultrasonic Atomizer to Control Atomization Rate
## 3 Finite Element Analysis
A forced harmonic analysis with a sinusoidal variation of the voltage was performed on this system as described in Figure 6. The resonant frequencies of the device and the maximum out-of-plane displacements of the mesh were recorded for the three micro-aperture shape geometries. Figure 8 shows a summary of results of the estimated out-of-plane displacement measured at the center of the thin plate against the voltage amplitude applied to the piezoelectric ring when the device was driven at the resonant frequency. The results indicate that the displacement increased linearly with the applied voltage for all three types of aperture shapes. The results show only a small change in the resonance frequency, which seems to be correlated to a loss of density due to the volume removed by micro-apertures in the thin plate, as predicted by the plate theory (Equation (5)). Thus, cylindrical openings with a constant diameter have the largest volume and, also, a resonance frequency, followed by the conical and pyramidal. However, in this work, our focus will be on the conical aperture because of the valveless pumping effect reported in the literature and its better atomization performance [23]. Figure 6: FEM model showing (**a**) a sectional view of the atomizer and boundary conditions and (**b**) mesh of the system. Figure 7: Geometrical shapes of the micro-apertures of (**a**) conical, (**b**) cylindrical, and (**c**) pyramidal holes. (**d**) Section view of a solid model of the atomizer disc with micro-apertures on the meshed thin plate. Table 3 shows the results of FEM analysis with a frequency sweep while applying 80 V to the PZT ring. A cross-section view of the vibration mode shape at the resonance frequency is also shown. These results were obtained using the model of the meshed thin plate with conical shape apertures. The results gave the first five vibration modes. The subscript \"s\" indicates that the mode was identified as axisymmetric. In all cases, the largest out-of-plane displacements were identified in the center of the meshed thin plate, with a maximum 110-kHz mode found, the value close to those reported for an ultrasonic atomizer, with similar materials reported in Reference [11]. Figure 9 provides details of the distribution of the displacement through the disk for the axisymmetric vibration mode \(f_{65}\) obtained at a frequency of 110 kHz and using an applied voltage in the range of 5 V to 80 V. The location of the largest displacement was around the center of the thin plate where the micro-apertures were located. This value of displacement corresponded to the frequency tuning value, with the best performance of the atomizer found in our experimental results (discussed in the Experimental section). The modeling methodology for the atomizer dynamic performance was then used to study the control of the atomization rate by correlating it to the maximum displacements as a function of frequency and voltage amplitudes. Figure 8: Displacement against the voltage amplitude. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \begin{tabular}{c} **Symmetric** \\ **Modes** \\ \end{tabular}  & \begin{tabular}{c} **Frequency** \\ **(kHz)** \\ \end{tabular}  & \begin{tabular}{c} **Shape** \\ \end{tabular}  & \ \begin{tabular}{c} **Max. Positive** \\ **Displacement** \\ **(\(\mu\)m)** \\ \end{tabular}  & \begin{tabular}{c} **Max. Negative** \\ **Displacement** \\ **(\(\mu\)m)** \\ \end{tabular}  \\ \hline \hline \(1\) & \(f_{15}=5\) & & \(1.18\) & \(-0.67\) \\ \hline \hline \(2\) & \(f_{25}=20\) & & \(2.09\) & \(-4.62\) \\ \hline \hline \(3\) & \(f_{38}=28\) & & \(6.71\) & \(-17.1\) \\ \hline \(4\) & \(f_{46}=62\) & & \(7.76\) & \(-18.