# Vibration Analysis of a Piezoelectric Ultrasonic Atomizer to Control Atomization Rate
## 7 Conclusions  
In this research, three main aspects were considered; first, a FEM dynamic modeling of an ultrasonic atomizer was developed; second, a study of a higher vibrational mode analysis as a function of the aperture shape of a meshed thin plate was investigated; and third, a numerical approximation of the atomization rate based on a combination of the experimental and numerical results was proposed. The finite element analysis provided a method to determine the vibration modes of the complete system (meshed thin plate coupled to a PZT) in response to an applied harmonic voltage, which is a complex problem to solve analytically. The behavior of the atomizer system for three types of apertures was analyzed. The results showed that the resonant frequency was only slightly affected by the shape of the aperture, with a larger response for the conical aperture when compared with a pyramidal shape. In all cases, the maximum out-of-plane displacement showed a linear behavior with the applied voltage. Numerical simulations of out-of-plane displacement distribution were used to obtain an approximation of the volume change generated by the dynamic oscillations at the resonant frequency. The experimental results also demonstrated that a minimum threshold voltage was required to achieve atomization. A prediction of the atomization rate was determined considering the effect of the mass transfer, vibration analysis, and geometry of the aperture. The results were compared with the experimental results, and a good correlation was found. This study indicated that FEM provides correct information of the mechanical behavior occurring during the ultrasonic atomizing process. The results could allow to optimize the atomization parameters by finding the optimum driving frequency and voltage for a specific design of the meshed thin-plate coupled to a PZT. The next step in our research could be to implement the results for soft actuation optimization of a soft robotic system.  
Conceptualization, E.G.-B., K.J.L., H.-J.L. and A.B.; methodology, E.G.-B.; validation, E.G.-B. and H.-J.L.; formal analysis, E.G.-B. and A.B.; writing--original draft preparation, E.G.-B., A.B. and K.J.L.; and writing--review and editing, E.G.-B., A.B. and K.J.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by 2019 UC MEXUS-CONACYT, grant number CN-19-153 and partial support was also provided by the U.S. National Science Foundation under grant numbers CMMI-1762530 and CMMI-2032021 (principal investigator: [PERSON]).  
Not applicable.  
The datasets generated for this study are available on request by contacting the corresponding authors.  
Figure 17: Total atomization rate vs. voltage using \(\xi_{a}\), with the result used as a reference threshold in the calculation of the displaced volume.  
**Acknowledgments:** This project was supported by the 2019 UC MEXUS-CONACYT grant no. CN-19-153. Partial support was also provided by the U.S. National Science Foundation under grant numbers CMMI-1762530 and CMMI-2032021 (principal investigator: [PERSON]).  
**Conflicts of Interest:** The authors declare no conflict of interest.  
Appendix A Pseudocode for the modal and harmonic analysis implemented in ANSYS to solve the single thin plate and the meshed thin plate coupled to the PZT ring system problems.