## 4 Numerical Modal Analysis
### Modal Analysis According to Changes in Design Factors  
Adjustments will be made to the following three primary design factors to propose an optimal design to avoid resonance frequencies of the sunroof glass and reduce vibrations: 3D modeling design elements, Young's modulus, glass thickness, and bracket position. First, focusing on Young's modulus, a coefficient reflecting displacement changes relative to stress in elastic materials, this study examined its impact on the resonance frequency. The tempered glass used in sunroofs typically comprises 45-75% SiO\({}_{2}\), 10-30% Al\({}_{2}\)O\({}_{3}\), 0-20% B\({}_{2}\)O\({}_{3}\), and 10-25% Na\({}_{2}\)O, with potential enhancements in mechanical strength achievable through modifications of its composition. Chemical strengthening methods, such as precise temperature control, can induce a compressive stress layer on the glass surface, enhancing its mechanical properties. In particular, Al\({}_{2}\)O\({}_{3}\), found in tempered glass, contributes to the ion exchange performance and can influence the Young's modulus depending on its concentration [28]. Given its pivotal role in product design and its adaptability to manufacturing improvements, the Young's modulus is a critical design factor for optimization. The Young's modulus was modified by increasing or decreasing by  
Figure 21: Mode shape comparison: 1 st mode shape from (**a**) EMA, (**b**) ODS, (**c**) numerical analysis. 2 nd mode shape from (**d**) EMA, (**e**) ODS, (**f**) numerical analysis. 3 rd mode shape from (**g**) EMA, (**h**) ODS, (**i**) numerical analysis. 0.05 GPa and repeating the analysis five times each; the results of the 3D modeling analysis on the sunroof glass are as follows. Table 8 lists the variation in natural frequency with the changes in [PERSON]'s modulus. The natural frequency increases as the [PERSON]'s modulus increases, and vice versa. For example, with a standard change of 0.05 GPa, the first mode experiences an approximate 0.055 Hz increase, whereas the difference mode displays a resonance frequency shift of approximately 0.08 Hz. The thickness of the glass is a crucial design factor because of its ease of application and modification during product design. Moreover, variations in mass resulting from changes in thickness affect the vibration characteristics and manufacturing costs. The effects of thickness alterations on the sunroof glass were examined by repeating the analyses ten times with thickness adjustments of \(\pm\)0.2 mm. The ensuing analysis results are outlined below. An increase in glass thickness corresponds to a rise in resonant frequency (Table 9), whereas a decrease in thickness leads to a proportional decrease in resonance frequency. Unlike the linear relationship observed with the changes in Young's modulus, changes in glass thickness result in varying rates of resonant frequency change. In particular, as the thickness increased, the rate of change in the first resonant frequency increased from approximately 0.05 Hz to 0.08 Hz, while a thinning of the glass resulted in a decrease in this rate from 0.04 Hz to 0.01 Hz. For example, with a maximum analysis thickness of 6.0 mm, the first mode experienced an approximate 0.08 Hz increase, while the second mode increased by approximately 0.2 Hz, reaching 15.995 Hz and 23.928 Hz, respectively. By contrast, with a minimum glass thickness of 4.0 mm, the analysis yielded frequencies of 15.581 Hz and 22.603 Hz for the first and second modes, respectively, showing an approximately 0.002 Hz and 0.06 Hz change for the first and second modes, respectively.