## 3 Fatigue Life Analysis Based on Load Spectrum  
### Obtaining Fatigue Load Spectrum  
Fatigue load is the repeated load that causes fatigue damage. Most parts are subjected to varying loads rather than constant loads in the working process. Fatigue load spectrum is divided into constant amplitude load spectrum, block spectrum, and random load spectrum [23]. The operating environment of urban buses is mainly in the urban areas of major cities, where traffic conditions are complex and random. The UDDS working condition is the urban road cycle condition formulated by the United States Environmental Protection Agency (the EPA), which simulates the urban road working condition of 12.07 km [16]. Compared with other working conditions, the UDDS working condition contains frequent braking conditions, and the average speed is 31.5 km/h, which is more in line with the actual operation of urban buses, and can be applied to simulate the bidirectional torque of the main reducer gear. Thus, the UDDS cycle is selected as the driving speed in the simulation, as shown in Figure 6.  
Urban buses are generally subjected to two kinds of forces during driving, namely vehicle driving resistance and driving force. Vehicle driving resistance includes rolling resistance, air resistance, acceleration resistance, and slope resistance. During the driving process, the driving resistance of the vehicle needs to maintain a balanced relationship with the driving force, which is called the vehicle driving equation [24]. \[\frac{T_{tq}i_{0}i_{S}\eta_{t}}{r}=mgfcos\alpha+\frac{C_{D}Au^{2}}{21.15}+mgsin \alpha+\delta m\frac{du}{dt} \tag{1}\]  
In the formula, \(T_{tq}\) is the output torque of the motor. \(i_{0}\) is the transmission ratio of the main reducer; \(i_{S}\) is the transmission ratio of the gearbox; \(\eta_{t}\) is the transmission efficiency; \(r\) is the wheel rolling radius; \(f\) is the road resistance factor; \(\alpha\) is road inclination; \(C_{D}\) is wind resistance coefficient; \(A\) is the windward area; \(u\) is instantaneous speed; and \(\delta\) is the rotation mass conversion coefficient. In this paper, vehicle parameters and motor characteristics are shown in Tables 2 and 3. According to the vehicle driving Equation (1), taking the UDDS cycle condition as the input of the driving speed, the corresponding main reducer input torque is calculated, as shown in Figure 7.  
It can be seen from Figure 7 that, during electromagnetic braking, the reverse torque generates in the driving pinion shaft. However, the friction braking does not produce reverse torque, so the reverse torque in Figure 7 is converted to positive direction, as shown in Figure 8.  
Figure 9 shows the torque-time history when the reverse braking torque is ignored. In this case, the reverse torque during electromagnetic braking is regarded as 0, which is  
Figure 6: UDDS speed condition diagram. In this paper, the fatigue life of gears is calculated by using the results of static analysis, so it is necessary to transform the torque-time history into the stress-time history, that is, the calculated torque load is transformed into the contact stress of the tooth surface and the bending stress of the tooth root, which can be realized by using Equations (2) and (3) [24]:  
\[\text{contact stress}:\ \sigma_{H}=\sqrt{\frac{4K_{H}T_{1}}{\phi_{R}(1-0.5\phi_{R })^{2}d_{1}^{3}u}}Z_{H}Z_{E}, \tag{2}\]  
\[\text{bending stress}:\ \sigma_{F}=\frac{K_{F}T_{1}Y_{Fa}Y_{Sa}}{\phi_{R}(1-0.5 \phi_{R})^{2}m^{3}z_{1}^{2}\sqrt{u^{2}+1}}, \tag{3}\]  
where, \(K_{H}\) and \(K_{F}\) are the load coefficients. The load coefficients include the use coefficient \(K_{A}\), the dynamic load coefficient \(K_{V}\), the longitudinal load distribution coefficient \(K_{\beta}\), and the load distribution coefficient among gear teeth \(K_{\alpha}\). The calculation formula is as follows:  
\[K=K_{A}K_{V}K_{\beta}K_{\alpha}, \tag{4}\]  
\(\phi_{R}\) is the tooth width coefficient, \(\phi_{R}=b/d_{1}\), \(b\) is the tooth width;  
\(d_{1}\) is the pitch circle diameter of the pinion.