## 3 Quantified Damage Analysis of Train Gearbox
### Analysis of Motor Operating Range Based on Simulation Results  
Using the relative damage analysis results for a travel distance of approximately 700,000 km, we analyzed changes in the motor operating range and the resulting energy conversion efficiency with varying gear ratios. Letting \(P_{bat}\) represent the battery output power and \(\eta_{m}\) represent the instantaneous motor efficiency, the average motor efficiency \(\eta_{m,avg}\) for the driving mode shown in Figure 9 is calculated using Equation (21).  
\[\eta_{m,avg}=\frac{\int P_{m}dt}{\int P_{bat}dt},\left(\eta_{m}=\frac{P_{m}}{P_ {bat}}=\frac{T_{m}\cdot\omega_{m}}{P_{bat}}\right) \tag{21}\]  
As the reduction gear ratio changes, the load torque on the motor changes, causing the motor's operating point to shift. Figure 15 illustrates how the motor's operating points vary when the gear ratio changes within the range of five to eight.  
With a gear ratio of five, the motor operating range is the narrowest, primarily occurring in the medium torque and medium speed regions, which are associated with high efficiency. The average efficiency at this ratio is 89.21%. At a gear ratio of six, the motor operating range slightly expands, but still mostly occupies the high-efficiency region of medium torque and medium speed. The average efficiency here is 89.27%.  
Figure 14: Relative damage of input and output shafts based on gear ratio.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Shaft** & \(\mathbf{r_{G}}\)**:5** & \(\mathbf{r_{G}}\)**:6** & \(\mathbf{r_{G}}\)**:7** & \(\mathbf{r_{G}}\)**:8** \\ \hline Input Shaft & 0.091 & 0.064 & 0.048 & 0.037 \\ \hline Output Shaft & 3.86 & 4.18 & 4.44 & 4.68 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Relative damage for different gear ratios. (Unit of QD: \(10^{20}\mathbf{rev}\cdot Nm^{3.33}\)).  
When the gear ratio increases to seven, the operating range further broadens. As a result, the proportion of operation in the high-efficiency region of medium torque and medium speed decreases, while the proportion of operation in low-efficiency regions, such as low speed with high torque or medium speed with low torque, increases. The average efficiency is 88.92%, which is lower than the efficiencies at gear ratios five and six.  
At a gear ratio of eight, the operating range is the widest, with a notably increased proportion in the low-efficiency region of high speed and low torque. Although the motor operates across a broad range of torque and speed, most operating points fall within low-efficiency regions. The average efficiency is the lowest at 88.51%.  
Using the motor operating point distribution results from Figure 15, the average energy efficiency for different gear ratios was calculated, with the results displayed in Table 4 and Figure 16 [16].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\mathbf{r_{G}\text{:}5}\) & \(\mathbf{r_{G}\text{:}6}\) & \(\mathbf{r_{G}\text{:}7}\) & \(\mathbf{r_{G}\text{:}8}\) \\ \hline Energy Efficiency (\%) & 89.21 & 89.27 & 88.92 & 88.51 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Energy efficiency based on gear ratio.  
Figure 15: Motor operation area: (**a**) gear ratio: 5, (**b**) gear ratio: 6, (**c**) gear ratio: 7, and (**d**) gear ratio: 8.