# Noise Identification for an Automotive Wheel Bearing
## 5 Simulation for Structural Defect  
A simulation of the finite element analysis was performed to determine the reason for structural defects inside the wheel bearing. A simulation model of the assembled wheel bearing is constructed as shown in Figure 10a. A commercial software, ANSYS, was used for the simulation. As shown in Figure 10a, the finite element model is composed of an outer ring, dust shield, knuckle, and three bolts connecting them. The other components \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(f_{\text{FTF}}\) & \(f_{\text{BPFO}}\) & \(f_{\text{BPFI}}\) & \(f_{\text{BSF}}\) \\ \hline Inboard row & 0.42\(X\) & 6.29\(X\) & 8.71\(X\) & 2.47\(X\) \\ \hline Outboard row & 0.42\(X\) & 6.79\(X\) & 9.21\(X\) & 2.65\(X\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristic frequencies (defect frequencies) of the wheel bearing (\(X\) is the rotating frequency). Figure 9: Magnified waterfall plot for the outer ring vibration of Figure 5c. (hub, inner ring, balls, and retainer) were excluded from the finite element model for simplification and efficiency of the analysis. In the finite element model, the outer ring, dust shield, knuckle, and bolt had 74,249, 48,653, 13,8673, and 8023 nodes, respectively, and 44,808, 24,246, 87,833, and 4528 elements, respectively. Furthermore, steel, which has a Young's modulus of 206 GPa and Poisson's ratio of 0.3, was used for each component. The boundary and load conditions applied to the analysis model are shown in Figure 10b. As shown in this figure, specific parts of the knuckle that connect to the vehicle body were applied under fixed conditions. Next, given that the outer ring flange, dust shield, and knuckle were fastened using bolts, a fastening force was applied to the bolts. The bolt-clamping force was obtained using the following equation:  
\[F=T/(KD) \tag{5}\]  
where \(F\) denotes the bolt clamping force, \(T\) denotes the bolt-tightening torque, \(K\) denotes the torque coefficient, and \(D\) denotes the bolt diameter. In this study, the bolt tightening torque \(T\) was 100 Nm, bolt diameter \(D\) was 0.012 m, and torque coefficient \(K\) was 0.15. By substituting these values into Equation (5), the bolt clamping force \(F\) was calculated as 55,556 N, and the obtained value was applied to the analysis model. The effect of the weight of the wheel bearing was also considered by applying gravity. The radial deformation of the assembled wheel bearing is analyzed in the radial direction, as shown in Figure 11. Specifically, the nodes of the outer ring deformed in the outward radial direction are displayed in red (or positive values), while the nodes deformed in the inward radial direction are displayed in blue (or negative values). In Figure 11, the radial deformation is relatively large in the outer ring, and almost no radial deformation occurs in the other components. After removing other components that slightly affect radial deformation, the radial deformations are solely reanalyzed for the outer ring raceway of the inboard/outboard row, and these results are illustrated in Figure 12. In Figure 12a, the outer ring raceway of the inboard row is deformed in the inward direction near the flange and deformed in the outward direction between the flanges. Conversely, in Figure 12b, the outer ring raceway of the outboard row was deformed in the outward direction near the flange and in the inward direction between the flanges. This phenomenon occurs because the outer ring flange is bent in the inboard row direction (the axial direction) when the bolt is fastened as shown in the enlarged figure in Figure 11. Generally, the bending deformation of the outer ring flange occurs when the bolts are tightened. Thus, the outer ring of the inboard row is compressed near the flange, while the outer ring of the outboard row is tensioned near the flange. Figure 10: Finite element model of the assembled wheel bearing: (**a**) meshed model and (**b**) boundary conditions.