# Static and Modal Models of Automotive Wheel Hubs Based on Lightweight Design and Cost-Effectiveness
## 3 Results and Discussion
### Modal Analysis of the Wheel Hub  
Based on the static analysis results of the wheel hub, a six-mode modal analysis was conducted. According to the distribution of the two positions of the supporting force on the wheel hub, the modal analysis is also divided into two cases. Table 1 presents the modal analysis results for these two scenarios. After comparing the results, it was  
Figure 4: Deformation at different loading positions. (**a**) The deformation result when the support force is acting directly on the wheel spoke; (**b**) The deformation result when the support force is acting on the ventilation opening. Figure 5: Equivalent stress at different loading positions. (**a**) Equivalent stress when the support force is acting directly on the wheel spoke; (**b**) Equivalent stress when the support force is acting on the ventilation opening. found that the mode analysis results were not significantly affected by the different support force conditions on the wheel hub. Therefore, for analysis purposes, the scenario with the support force acting on the ventilation opening can be selected. The results of the six-mode modal analysis for this scenario are shown in Figure 6. The colours are deformations. Red means the largest deformation and blue means the smallest. Figure 6a-f correspond to the six modes from Mode 1 to Mode 6, respectively. Modal analysis can be used to determine the natural frequencies of the wheel hub. These natural frequencies can be used to evaluate the response characteristics and stability of the wheel hub under different vibration modes. Modal analysis results showed that the frequency values of the first four modes in the six modes are close to each other. Their mode shapes were also very similar, with only slight differences in certain vibration directions. Through the analysis of the mode shapes, it can be observed that there were primarily four types of mode shapes. The first type included Mode 1 and Mode 2, which exhibited predominantly vertical deformation vibrations. The second type included Mode 3 and Mode 4, where twisting deformation vibrations occurred in the wheel rim and spoke regions. The third type, represented by Mode 5, involved axial deformation vibrations of the wheel hub. The fourth type, represented by Mode 6, demonstrated triangular deformation vibrations in the wheel rim region as the mode order increases. Through the analysis of these modes, it can be seen that in order to extend the longevity of the wheel hub, the strength of the wheel rim and inner wheel spoke \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Modes Order**} & **Frequency (Hz)** & **Frequency (Hz)** \\  & **(Support Force Acting on the** & **(Support Force Is Acting on** \\  & **Wheel Spoke)** & **the Ventilation Opening)** \\ \hline
1 & 511.31 & 511.27 \\
2 & 511.46 & 511.63 \\
3 & 575.35 & 575.43 \\
4 & 575.56 & 575.66 \\
5 & 671.32 & 671.48 \\
6 & 723.55 & 723.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of modal frequencies for two different cases. Figure 6: Modal analysis of various modes when the support force is aligned with the ventilation hole. (**a**) Mode 1; (**b**) Mode 2; (**c**) Mode 3; (**d**) Mode 4; (**e**) Mode 5; (**f**) Mode 6.  
region should be appropriately reinforced; that is, the thickness of these positions should be increased. External excitations on the wheel hub primarily include road surface excitation frequency, tire-induced imbalance excitation frequency, transmission system-induced imbalance excitation frequency, and engine vibration frequency. Based on engineering experience, road surface excitation frequencies are generally below 3 Hz for highways and well-maintained urban roads, while excitation frequencies for uneven road surfaces are generally below 11 Hz. For a typical gasoline engine passenger vehicle, the stable idle speed of a cold four-cylinder engine is usually between 1000 rpm and 1200 rpm. After the engine is warmed up, the idle speed typically ranges from 700 rpm to 800 rpm.