## 3 Research Method
### Load Spectrum Analysis and the Equivalent Transformation  
When the vehicle is driving, the wheel-side electric drive system is subjected to the alternating loads. For high cycle loads (with a failure life longer than \(10^{3}\) cycles), the nominal stress method is usually used in conjunction with the S-N curve to evaluate the durability of the system. According to the speed variation of the vehicle under typical operating cycle conditions [29], the PMSM follows the vehicle speed to obtain the corresponding motor output characteristic curve, including the timing records of output torque and output speed, as shown in Figure 9. \begin{table}
\begin{tabular}{c c c c c} \hline  & **1\# Stage Driving** & **1\# Stage Driven** & **2\# Stage Driving** & **2\# Stage Driven** \\  & **Gear** & **Gear** & **Gear** & **Gear** \\ \hline Tooth Number & \(z_{11}=31\) & \(z_{12}=78\) & \(z_{21}=29\) & \(z_{22}=76\) \\ Tooth Modulus (mm) & \(m_{n1}=2.1\) & \(m_{n1}=2.1\) & \(m_{n2}=2.3\) & \(m_{n2}=2.3\) \\ Helix Angle (\({}^{\circ}\)) & \(\beta_{n1}=34\) & \(\beta_{n1}=34\) & \(\beta_{n2}=22\) & \(\beta_{n2}=22\) \\ Displacement Coefficient & \(x_{n11}=0.3\) & \(x_{n12}=0.1\) & \(x_{n21}=0.31\) & \(x_{n22}=-0.03\) \\ Pressure Angle (\({}^{\circ}\)) & \(\alpha=20\) & \(\alpha=20\) & \(\alpha=20\) & \(\alpha=20\) \\ Addendum Height Coefficient & \(h_{a1n}=1.3\) & \(h_{a1n}=1.3\) & \(h_{a2n}=1.24\) & \(h_{a2n}=1.24\) \\ Width (mm) & \(b_{11}=20\) & \(b_{12}=18\) & \(b_{21}=25\) & \(b_{22}=23\) \\ Base Circle Diameter (mm) & \(d_{b11}=71.901\) & \(d_{b12}=180.911\) & \(d_{b21}=66.964\) & \(d_{b22}=175.491\) \\ Pitch Circle Diameter & \(d_{p11}=78.995\) & \(d_{p12}=198.763\) & \(d_{p21}=72.289\) & \(d_{p22}=189.446\) \\ Center Distance (mm) & \(a_{1}=139\) & \(a_{1}=139\) & \(a_{2}=131\) & \(a_{2}=131\) \\ End-face Coincidence & \(\varepsilon_{a1}=1.629\) & \(\varepsilon_{a1}=1.629\) & \(\varepsilon_{a2}=1.812\) & \(\varepsilon_{a2}=1.812\) \\ Axial Coincidence & \(\varepsilon_{g1}=1.526\) & \(\varepsilon_{g1}=1.526\) & \(\varepsilon_{g2}=1.192\) & \(\varepsilon_{g2}=1.192\) \\ Total Coincidence & \(\varepsilon_{r1}=3.155\) & \(\varepsilon_{r1}=3.155\) & \(\varepsilon_{r2}=3.004\) & \(\varepsilon_{r2}=3.004\) \\ Reduction Ratio & \(i_{1}=2.516\) & \(i_{1}=2.516\) & \(i_{2}=2.621\) & \(i_{2}=2.621\) \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of the gear train. Figure 9: Typical operating cycle conditions of vehicle. (**a**) Vehicle speed curve in time domain. (**b**) Motor torque curve in time domain. (**c**) Motor speed curve in time domain. Based on the typical operating cycle conditions, we use the rain flow methodology to count the working conditions of the basic version, NVH version, and durability version, respectively. The histograms of working condition frequency are presented in Figure 10.  
When studying the limit value of load extrapolation, it is generally considered that \(10^{3}\) cycles of the over value cumulative cycle is a sufficiently representative order of magnitude [30]. The mathematical expression is as follows:  
\[1-F(T_{max})=10^{-6} \tag{22}\]  
where \(F(T)\) is the cumulative distribution function and \(T_{max}\) is the maximum torque of load sample statistics. Based on this, the fatigue strength index can be expressed as follows:  
\[b=\frac{lgS_{1000}-lgS_{be}}{lg10^{3}-lg10^{6}}=-\frac{S_{1000}}{3S_{be}} \tag{23}\]  
where \(b\) is the fatigue strength index and \(S_{1000}\) is the fatigue strength at 1000 cycles. \(S_{be}\) is the fatigue strength of the sample. The criterion for developing the extrapolated load spectrum is to omit the small load cycles that fall below the material fatigue limit [31]. We set the filtering threshold to be 55% of the fatigue limit of the smooth specimen. The corresponding stress amplitude is 98.2 Mpa, with the load below this amplitude being omitted.