## 2 Simulation Model Development
### Fatigue Life Evaluation Method
#### 2.5.2 RFC and Load Spectrum  
As the torque data measured in the field fluctuated irregularly, it was converted into a 3D map of mean, range, and number of cycles using the RFC [29; 30; 31; 32]. To quantify the spectrum load magnitude, the equivalent torque was calculated, based on the Smith-Watson-Topper (SWT) model, using the mean and range in the 3D map obtained from RFC, as shown in Equation (3) [33; 34]. In this study, the load data was measured for a relatively short time (55-90 s) during field operations, and the load data cannot be measured during the entire life cycle of the tractor. Nevertheless, in order to evaluate the fatigue life of a tractor spiral bevel gear, it is necessary the load history for the entire life cycle. In this case, generally, the load data measured in a short time can be extended by considering the entire life of the tractor, as shown in Equation (4) [3; 18]. Data of a previous study, obtained through the survey of the actual conditions of users in Korean agricultural operational times, were used to extend the number of cycles [35]. The tractor's service life was applied as 2736 h [9; 35].  
\[T_{e}=\sqrt{(t_{a}+t_{m})t_{a}} \tag{3}\]  
where \(T_{e}\) is the equivalent torque (Nm), \(t_{a}\) is the range of torque, and \(t_{m}\) is the mean torque (Nm).  
\[N_{T}=3600 NLh \tag{4}\]  
where \(N_{T}\) is the extended number of cycles of the load (cycles), \(N\) is the number of cycles of measured load data (cycles/s), \(L\) is the service life of the used tractor (years), and \(h\) is the annual usage time of the tractor (h/year).  
The modes of gear fatigue failure can be divided into tooth bending fatigue and contact fatigue. In general, failure of the tooth root is caused by bending stress in the gear, while wearing of the tooth surface is caused by contact stress [36]. In particular, although fitting is known as the most common failure mode, it is nevertheless difficult to determine the exact failure time, and power transmission is possible even if fitting occurs. Therefore, Korean agricultural machinery manufacturers regard bending stress as the more important major mode of fatigue failure [37]. The bending fatigue failure causes gradual damage to the gear teeth, which may result in the failure of the entire transmission. Therefore, in this study, the bending fatigue of the gear was regarded as a factor influencing the fatigue life and the fatigue life was evaluated based on the bending stress. For fatigue analysis of the spiral bevel gear, the measured torque was converted into bending stress, based on the specifications of the gears and major parameters presented Table 3 using ISO 10300-1:3, as shown in Equations (5)-(7) [38; 39; 40]:  
\[F_{mt}=\frac{2000\mathrm{T}}{d_{m}} \tag{5}\]  
Figure 4: Procedures of fatigue life evaluation for tractor transmission spiral bevel gear using field torque data.  
where \(F_{mt}\) is the nominal tangential force at mid-face width of the reference cone (N), T is the torque at the driving and driven gear (Nm), and \(d_{m}\) is the mean pitch diameter of the driving and driven gear (mm);  
\[\sigma_{F0}=\frac{F_{mt}}{bm_{mm}}Y_{Fa}Y_{Sa}Y_{\varepsilon}Y_{K}Y_{LS} \tag{6}\]  
where \(\sigma_{F0}\) is the nominal tooth-root stress (N/mm\({}^{2}\)), \(b\) is the gear face width (mm), \(m_{mm}\) is the mean normal module (mm), \(Y_{Fa}\) is the tooth form factor, \(Y_{Sa}\) is the stress correction factor, \(Y_{\varepsilon}\) is the contact ratio factor, \(Y_{K}\) is the bevel gear factor, and \(Y_{LS}\) is the load distribution coefficient;  
\[\sigma_{F}=\sigma_{F0}K_{A}K_{V}K_{F\beta}K_{F\alpha} \tag{7}\]  
where \(\sigma_{F}\) is the tooth-root stress (N/mm\({}^{2}\)), \(K_{A}\) is the application factor, \(K_{V}\) is the dynamic factor, \(K_{F\beta}\) is the face load factor, and \(K_{Fa}\) is the transverse load factor.