## 2 Initial Design and Finite Element Analysis of an Ultrasonic Cutting Horn
### Finite Element Analysis  
We developed a finite element model using ANSYS software to analyze the characteristics of the proposed sonotrode model. The convergence of the developed finite element model was secured, and it had 20,155 hexahedral elements with a size of 2.5 mm. Figure 2 shows the developed finite element model and boundary conditions. For the boundary conditions, the displacement of the shaft was constrained along the \(x\)- and \(y\)-axes. The  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & **Material** & **Young’s Modulus [GPa]** & **Density [kg/m\({}^{3}\)]** & **Poisson’s Ratio** \\ \hline Horn & Al7075 & 71.9 & 2802 & 0.34 \\ Blade & WC & 500 & 13,834 & 0.2 \\ Shaft & SCM435 & 205 & 6287 & 0.29 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of a sonotrode.  
natural frequencies and mode shapes were calculated by modal analysis. The frequency isolations are defined as follows:  
\[f_{\text{1 stisolation}}=f_{\text{axial}}-f_{\text{axial}-1} \tag{2}\]  
\[f_{\text{2 nd isolation}}=f_{\text{axial}+1}-f_{\text{axial}} \tag{3}\]  
where \(f_{\text{axial}}\), \(f_{\text{axial}-1}\), \(f_{\text{axial}+1}\), \(f_{\text{1 st isolation}}\) and \(f_{\text{2 nd isolation}}\) represent the axial mode frequency, one lower mode frequency, one higher mode frequency, the 1 st frequency isolation and the 2 nd frequency isolation. Figure 3 shows the mode shape of the axial mode and one lower and higher mode. In the axial mode, the horn vibrates along the axial direction. The one lower and higher mode of the sonotrode are the bending motion of the horn lower part and blade. The one lower mode shape is the 2 nd bending mode, where both ends of the blade deform in the \(y\)-axis direction, and the one higher mode shape is the 3 rd bending mode where both ends and the center of the blade deform in the \(y\)-axis direction. The axial frequency, the 1 st frequency isolation and the 2 nd frequency isolation were 28,520 Hz, 1588 Hz and 864 Hz, respectively.  
Figure 3: Mode shape and natural frequency of the (**a**) one lower mode (\(f_{\text{axial}-1}\)), (**b**) axial mode (\(f_{\text{axial}}\)) and (**c**) one higher mode (\(f_{\text{axial}+1}\)).  
Figure 2: Finite element model of the initial sonotrode.  
The maximum displacement at the horn lower face, minimum displacement at the horn lower face, gain and displacement uniformity were calculated by harmonic analysis. The gain and displacement uniformity were defined as follows:  
\[\alpha=\frac{\left(u_{\text{low}}\right)_{\text{min}}}{\left(u_{\text{low}} \right)_{\text{max}}}.100 \tag{4}\]  
\[g=\frac{\left(u_{\text{low}}\right)_{\text{max}}}{\left(u_{\text{upp}}\right)_ {\text{max}}} \tag{5}\]  
where \(a\), \(g\), \(\left(u_{\text{low}}\right)_{\text{min}}\), \(\left(u_{\text{low}}\right)_{\text{max}}\) and \(\left(u_{\text{upp}}\right)_{\text{max}}\) represent the displacement uniformity, gain, minimum displacement at the horn lower face, maximum displacement at the horn lower face and maximum displacement at the horn upper face, respectively. Under the analysis conditions, the displacement of the shaft was constrained along the \(x\)- and \(y\)-axes, and the axial mode frequency was used as the excitation frequency. Figure 4 shows the calculated displacement along the horn lower face. The maximum displacement, minimum displacement uniformity and gain were 16.80 \(\upmu\)m, 13.76 \(\upmu\)m, 1.07 and 81.9%, respectively. These dynamic characteristics (natural frequency, frequency isolation, displacement) are changed with changes in the design variables, so it is very hard to decide the specific values of design variables using the trial and error method. Therefore, we used the optimal design method to derive the best horn model.