# Numerical Analyses of Ultrasonic Atomization Utilizing Acoustic Effects of a Beam Diaphragm
## 4 Results and Discussions  
Resonance frequencies obtained from the eigenvalue analysis are shown, with results from the experiments, in Table 3. Frequencies in the numerical analysis that are closest to the peak frequencies in the hydrophobic cases experiment with 3.1 W of transducer power were chosen in the table. This result shows that the body consisting of the hydrophobic diaphragm and water film in the experiment oscillated at resonance frequencies of the diaphragm. Modal shapes which correspond to each resonance frequency are shown in Figure 8. The modal shapes were basically bending modes without twist, and the number of nodes and anti-nodes increased with an increase in the frequency. It is noted that many modes with twists exists between the modes in the figure although they are not shown. This tendency indicates that the whole surface of the diaphragm has a large amount of acceleration in the out-of-plane direction, especially in higher-frequency domains, and can be a potential region where atomization occurs. Figure 9 shows the harmonic response from the numerical analysis using the hydrophobic diaphragm. Since the condenser microphone used in the experiments mainly senses acoustic pressure, the amplitude of the diaphragm in the analyses is plotted. The percentage of power transferred to the diaphragm from the transducer was unknown in the experiments. Therefore, the result that is important in this plot is not the absolute value of the amplitude but the relative variation with respect to frequency. Thus, the amplitude was normalized by a single average value of all results in the analyses. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Mode \#/Frequencies (Hz)**} \\ \hline
**Markers** & **a** & **b** & **c** & **d** & **e** & **f** \\ \hline Experiment & 6187 & 7406 & 8718 & 10,010 & 11,760 & 13,540 \\ Modal analysis & 26/6129 & 31/7658 & 33/8952 & 38/10,237 & 43/11,543 & 51/13,783 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Eigen frequencies obtained from FEA and corresponding peak frequency in the experiment. Figure 8: Modal shape corresponding to the eigen frequencies and peak frequencies in the experiment shown in Table 3. (**a–f**) correspond to the marker in Table 3.  
The red and blue curves in Figure 9 show results with 3.1 W and 6.0 W of transducer power, respectively. By increasing the power in the experiment, the height and width of the water pillar increased, and the peak frequency shifted in the direction of the lower frequency. This tendency was reproduced in the results of the analysis. Peak frequencies in the results of the experiments are marked as vertical dashed lines **a** to **f**. The peak frequencies of the experiments and analyses agreed well. As the main difference between models with 3.1 w and 6.0 W is the distribution of water thickness, this result indicates that the peak shift mainly occurred through an increase in the mass of the oscillating body that consists of the diaphragm and water column. Results from the numerical analyses using a hydrophilic diaphragm are shown in Figure 10. The green and brown curves correspond to results when the power of the transducer was 3.1 W and 6.0 W, respectively. Results with and without consideration of the interaction between the diaphragm and water film were compared by solid and dashed lines. Vertical dashed lines show the frequencies marked in the results of the experiments. The peak shift in the results of the experiment was reproduced in the numerical analyses, as well as in the case of the hydrophobic diaphragm. For markers **a** to **f**, the deformed shape of the diaphragm in this harmonic analysis was almost the same as that obtained in the eigenvalue analysis despite the shift in the frequencies. In the experiments  
Figure 10: Results of harmonic response analyses for the case with hydrophobic diaphragm. Figure 9: Results of harmonic response analyses for the case with the hydrophilic diaphragm.