# Failure Life Prediction of Hub Bearing in Composite Tooling
## 2 Derivation of Loads on the Hub Bearing
### Derivation of Loads on the Hub Bearing  
Static load tests on the disk narrow assembly were performed to derive the equivalent load on the hub bearing. The static load test equipment was configured, as shown in Figure 5. The bolt fastening region of the disk narrow arm connected to the frame was fixed to the jig. An actuator was connected to the hub bearing mounting area. A 3000 N load was applied for each direction(X, Y, Z direction).  
Figure 4: Integrated strain history at 14–16 km/h.  
Figure 3: Strain gauge locations of disk narrow arm.  
The strain transformation matrix was calculated using the strain data obtained through the static load tests and cyclic loadings. Here, the strain transformation matrix refers to the sensitivity of the strain to cyclic load and is a matrix for transforming the strain data of field tests into the load history. Using the strain data obtained through field tests, as shown in Figure 4 and the strain transformation matrix obtained through the static load test, the load histories (Fx, Fy, Fz) of the disk narrow assembly were reconstructed as shown in Figure 6. The reconstructed load histories in the x, y, and z directions were calculated in the form of the load sizes (Figure 7) and the principal directions (Table 2) over time through the synthesis of load vectors.  
The load sizes and principal directions were converted into equivalent loads using Equation (1) presented in ISO 6336-6 [15] to predict the fatigue life of the hub bearing.  
\[F_{\alpha\text{j}}=\left(\sum_{i=1}^{k}\frac{n_{i}F_{i}^{x}}{n_{i}}\right)^{ \frac{1}{x}} \tag{1}\]  
where,  
\(F_{\alpha\text{j}}\): Equivalent load acting on the bearing  
\(n_{i\text{:}}\): Number of cycles for divided range i  
\(F_{i\text{:}}\): Load for divided range i  
\(x\): Slope of the Wohler curve (acceleration index)  
If it is assumed that the wheel and disk narrow of the tractor does not slip on the ground, when the tractor's working speed is 15 km/h, and the diameter of the disk narrow is 600 mm, the rotation speed of the disk narrow and hub bearing will be 132.6 rpm. The number of cycles, ni, was calculated using the rotation speed of the hub bearing and the total time of load history 224.3 s. Although the  
\begin{table}
\begin{tabular}{c c c c} \hline
**Direction** & \(\theta_{x}\) & \(\theta_{y}\) & \(\theta_{z}\) \\ \hline
**Angle**,\({}^{\circ}\) & 82.4 & 41.0 & 50.0 \\ \hline \end{tabular}
\end{table}
Table 2: Principal direction of disk narrow assembly.  
Figure 5: Static test of disk narrow assembly.  
slope of the Wohler curve varies with the parts and the processing method, three is generally applied in the case of ball bearings [11]. The load data in Figure 7 and derived by the equation (1) leads to an equivalent load at 1.69 kN. The equivalent load was split into loads in the x, y, and z directions using the angle of the principal direction of the disk narrow assembly like Table 2. The separated loads for x, y, and z directions were calculated as 0.22 kN, 1.27 kN, and 1.08 kN respectively as shown in Figure 8. The fatigue life of the hub bearing was analyzed using the results of the calculation of the loads.  
Figure 6: Load reconstruction for disk harmony assembly in all directions.