## 4 Numerical Modal Analysis
### Modal Analysis According to Changes in Design Factors
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Nat. Freq**} & \multicolumn{6}{c}{**Young’s Modulus (GPa)**} \\ \cline{2-7}  & **7.2 (base)** & **7.15** & **7.1** & **7.05** & **7.0** & **6.95** \\ \hline Mode \#1 & 15.673 & 15.619 & 15.564 & 15.509 & 15.454 & 15.399 \\ Mode \#2 & 23.069 & 22.989 & 22.908 & 22.828 & 22.747 & 22.665 \\ Mode \#3 & 31.632 & 31.522 & 31.411 & 31.301 & 31.189 & 31.078 \\ \hline \multirow{2}{*}{**Nat. Freq**} & \multicolumn{6}{c}{**Young’s Modulus (GPa)**} \\ \cline{2-7}  & **7.2 (base)** & **7.25** & **7.3** & **7.35** & **7.4** & **7.45** \\ \hline Mode \#1 & 15.673 & 15.728 & 15.782 & 15.836 & 15.890 & 15.943 \\ Mode \#2 & 23.069 & 23.149 & 23.229 & 23.308 & 23.387 & 23.466 \\ Mode \#3 & 31.632 & 31.741 & 31.851 & 31.960 & 32.068 & 32.176 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Parametric study with the change in Young’s modulus. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Nat. Freq**} & \multicolumn{6}{c}{**Glass Thickness [mm]**} \\ \cline{2-7}  & **5.0(base)** & **4.8** & **4.6** & **4.4** & **4.2** & **4.0** \\ \hline Mode \#1 & 15.631 & 15.635 & 15.607 & 15.588 & 15.579 & 15.581 \\ Mode \#2 & 23.069 & 22.943 & 22.833 & 22.739 & 22.662 & 22.603 \\ Mode \#3 & 31.632 & 31.448 & 31.282 & 31.133 & 31.003 & 30.893 \\ \hline \hline \multirow{2}{*}{**Nat. Freq**} & \multicolumn{6}{c}{**Glass Thickness [mm]**} \\ \cline{2-7}  & **5.0(base)** & **5.2** & **5.4** & **5.6** & **5.8** & **6.0** \\ \hline Mode \#1 & 15.673 & 15.721 & 15.777 & 15.842 & 15.915 & 15.995 \\ Mode \#2 & 23.069 & 23.211 & 23.369 & 23.541 & 23.727 & 23.928 \\ Mode \#3 & 31.632 & 31.830 & 32.043 Subsequently, this study examined the frequency change characteristics resulting from variations in the bracket supporting the sunroof glass within the vehicle, as shown in Table 10. Such changes involve numerous design considerations, including bracket area, shape, and mounting location. In this study, a consistent bracket area and shape were maintained while altering the support point of the bracket. Figure 17 shows the configuration of the sunroof glass support bracket. A front bracket adjacent to the front of the sunroof and a rear bracket positioned near the center were installed symmetrically along the front-to-rear axis of the sunroof. Given the implications of bracket movement on factors, such as visibility and structural integrity, the position of the front bracket was fixed, and the support point of the rear bracket was varied. By moving the rear bracket support point forward and backward by 5 mm relative to the reference point, located 220 mm apart from the front brackets, this study analyzed the resulting changes in resonance frequency across ten support point configurations. Similar to the change in [PERSON]'s modulus and glass thickness, the resonant frequency increased as the distance between the brackets increased, and vice versa. According to a 5 mm change, which is the standard change in distance between brackets, the first and second modes showed an approximately 0.271 Hz and 0.507 Hz change in resonance frequency, respectively. Compared to the change in resonance frequency due to the change in [PERSON]'s modulus, the first mode showed 4.9- and two-fold changes in resonance frequency, respectively. In car mode, the amount of change was relatively large, at 6.3 times. The resonance frequency characteristics of sunroof glass were examined according to changes in three major design factors to propose an optimal design: [PERSON]'s modulus, glass thickness, and bracket position, which are design elements of 3D modeling. The results are summarized in Figure 22.