## 3 Quantified Damage Analysis of Train Gearbox
### Optimal Gear Ratio Selection Based on Simulation Results  
Using simulation analysis, we have obtained results for the relative damage of the input and output shafts with varying gear ratios (Figure 14) and the average energy efficiency (Figure 16). In this study, a space was constructed using these three results as coordinates to define a utopia point representing the most ideal and optimal performance. The gear ratio corresponding to the point with the minimum Euclidean distance from this utopia point was selected as the optimal gear ratio. This optimization process is illustrated in Figure 17.  
Figure 16: Energy efficiency at different gear ratios.  
Figure 17: Diagram for identifying pareto optimal points.  
To identify the Pareto optimal point, we used the relative damage on the input shaft of the reduction gear \((QD)\), the relative damage on the output shaft of the reduction gear \((QD_{Out})\), and the average energy efficiency \(\eta_{m,avg}\) obtained from the simulation results. The Pareto optimal point represents points that are not outperformed by any other point in terms of performance. To identify these points, each data point was compared, and those that were not dominated by others were selected as Pareto optimal points. The specific process for finding the Pareto optimal point was as follows [17]:  
First, data points are generated based on the simulation results. Then, for each data point \(i\), every other data point \(j\) is sequentially compared to \(i\). For example, if \(j\) is better than \(i\) in all performance criteria, \(j\) is considered to dominate \(i\). If \(j\) dominates \(i\), then \(i\) is removed from the list of Pareto optimal candidates, and the comparison stops. If \(j\) does not dominate \(i\), the comparison continues to the next \(j\). After all comparisons are complete, the Pareto optimal points are output and visualized in a graph.  
Once the Pareto optimal points are identified, the Euclidean distance to the utopia point, representing the most ideal values for all performance criteria, is calculated to determine the optimal gear ratio. In this study, the utopia point is defined as the point where the input shaft damage \((QD)\) and output shaft damage \((QD_{Out})\) are minimized, and the average energy efficiency \(\eta_{m,avg}\) is maximized. The Euclidean distance is calculated as follows [18].  
\[d=\sqrt{\left(\frac{QD-QD_{utopia}}{QD_{norm}}\right)^{2}+\left(\frac{QD_{Out }-QD_{Out,utopia}}{QD_{Out,norm}}\right)^{2}+\left(\frac{Eff-Eff_{utopia}}{ Eff_{norm}}\right)^{2}} \tag{22}\]  
The normalized values \(QD_{norm}\), \(QD_{Out,norm}\), and \(Eff_{norm}\) were calculated using the Linear Max-Min (LMM) method. The LMM method is calculated as follows.  
\[QD_{norm}=QD_{max}-QD_{min} \tag{23}\]  
\[QD_{Out,norm}=QD_{Out,max}-QD_{Out,min} \tag{24}\]  
\[Eff_{norm}=Eff_{max}-Eff_{min} \tag{25}\]  
The relative damage on the input and output shafts of the reduction gear, the average energy efficiency, and the Euclidean distances defined by Equation (22), obtained through the procedure outlined in Figure 17, are summarized in Table 5 and Figure 18 [17; 19; 20; 21; 22; 23].  
Figure 18: Euclidean distance by gear ratio.  
Based on the above results, considering both the relative damage on the input and output shafts and the average energy efficiency, the gear ratio with the smallest Euclidean distance is \(r_{G}=6\). This value can be considered as the optimal gear ratio.