## 4 Numerical Modal Analysis
### Modal Analysis Results for the Base Model  
The resonance frequencies of the sunroof glass determined by the modal analysis were 15.673 Hz, 23.069 Hz, and 31.864 Hz for the first, second, and third orders, respectively. Figures 18-20 shows 1 st, 2 nd, and 3 rd mode shapes of sunroof glass. \begin{table}
\begin{tabular}{c c c c} \hline
**Property** & **Symbol** & **Unit** & **Value** \\ \hline Young’s modulus & \(E\) & GPa & 72 \\ Poisson’s ratio & \(\
u\) & - & 0.23 \\ Thermal expansion & \(\alpha\) & \(10^{-6}\) K\({}^{-1}\) & 9.5 \\ coefficient & \(f_{t}\) & MPa & 165 \\ Compressive strength & \(f_{c}\) & MPa & 300 \\ Density & \(\rho\) & kg/m\({}^{3}\) & 2500 \\ \hline \end{tabular}
\end{table} 
Table 6: Material properties of tempered glass. Figure 17: Sunroof model. (**a**) CATIA model with dimensions, (**b**) side view of CATIA model, (**c**) side view #1, (**d**) side view #2, (**e**) bottom view, (**f**) ISO view of ANSYS base model. Figure 19: 2 nd Mode shape of sunroof glass (23.069 Hz): (**a**) side view, (**b**) top view. Figure 20: 3 rd Mode shape of sunroof glass (31.864 Hz): (**a**) side view, (**b**) top view. Table 7 presents the comparative analysis results of resonant frequencies obtained from sunroof glass modal analysis and the ODS test, which are based on existing experimental data. The first resonance frequency of the sunroof glass was 15.851 Hz, with an error rate of approximately 1.6% compared to the experimental results. In addition, the second resonance frequency mode was confirmed to be 23.298 Hz, with an error rate of approximately 1.7%. Furthermore, starting with the third mode, the error rate tended to increase gradually because of environmental factors in the test conditions and differences in boundary conditions between testing and analysis. An additional aspect to consider is the qualitative comparison of mode shapes in addition to natural frequencies, as illustrated in Figure 21. Comprehensively analyzing the mode shapes of EMA, ODS, and analysis results, the first mode showed a slight deformation of the front part due to the influence of the sunroof front support, but the rear sunroof glass showed a bending mode shape that bent up and down. In addition, in the second and third modes, a torsional mode shape was shown in which the left and right sides were bent along the axis connecting the front and rear of the sunroof. In the case of the third mode, from the numerical analysis, this mode shape is different from the others. It is thought that there is another mode shape in addition to EMA and ODS which sits between the second and third mode shapes. This shape is caused by an unknown condition in the experimental setup and cannot be replicated by the numerical analysis. If there is a discrepancy between the two results, there are two ways to reduce it: (1) The finite element model should be delicately tuned (material properties, boundary conditions, element type and size, and so on) so that its modal properties are made as close as possible to the physical model. (2) The experiment should be done again with more accurate settings, such as changing the accelerometer location to accurately measure the natural frequency, and adjusting the excitation of an appropriate impact hammer. Before examining the influence of each factor, this study assessed whether the modal analysis outcomes of the sunroof glass aligned with the frequency range relevant to the bolt loosening phenomena. A prior investigation into bolt loosening issues in vehicle sunroofs highlighted the significant impact of the initial bolt axial force loss caused by repetitive external vibrations or impact loads. A direct correlation was observed between axial displacement and its influence on bolt loosening, with greater displacement yielding greater impact. Thus, the current focus was to verify the resonant frequency range that induces the maximum axial displacement, as confirmed by the ODS test evaluations.