## 3 Quantified Damage Analysis of Train Gearbox
### Cumulative Fatigue Damage Theory Based on Miner's Rule  
This study aims not to measure the lifespan of vehicle components but to evaluate the fatigue load on the train reduction gear in relation to the traveled distance. Assuming that repetitive torque variations are the primary failure factors for rotating drivetrain components, the cumulative fatigue damage \(D\) is quantified using the torque-revolutions (T-N) curve and Miner's Rule, as shown in Equations (14) and (15) [12; 13; 14; 15]. \[log(T)-log(S_{u})=\left(\frac{-1}{\gamma}\right)\cdot log(N)\lor N=\left[\frac {S_{u}}{T}\right]^{\gamma} \tag{14}\]  
\[D=\sum_{i=1}^{n}\frac{n_{i}}{N_{i}}=\sum_{i=1}^{n}\frac{n_{i}}{\left[\frac{S_{ u}}{T}\right]^{\gamma}}=\sum_{i=1}^{n}\frac{n_{i}\cdot T_{i}^{\gamma}}{S_{u}^{ \gamma}} \tag{15}\]  
Here, \(S_{u}\) denotes the ultimate tensile strength of the material and \(\gamma\) represents the fatigue damage exponent. Additionally, \(T_{i}\) is the torque and \(n_{i}\) represents the number of motor revolutions corresponding to \(T_{i}\). In Equation (15), assuming that the material properties of the components being compared are identical, \(S_{u}\) in the denominator can be disregarded, allowing the relative cumulative fatigue damage \(QD\) to be calculated as shown in Equation (16). \[QD=\sum_{i=1}^{n}n_{i}\cdot T_{i}^{\gamma} \tag{16}\]  
As shown in Figure 6, the method for calculating the relative cumulative fatigue damage \(QD\), as defined in Equation (16), involves using the measured rotational speed data \(\omega_{d}\) and torque data \(T_{d}\) from testing the rotating shaft. First, based on the range of the measured torque values, define torque intervals of a specific size (from \(T_{d(1)}\) to \(T_{d(5)}\)). If a maximum or minimum torque value exists within any given interval, this value is used to further divide the interval along the time axis. As a result of this process, the measured data are divided into \(n\) regions \((R_{1}\ R_{n})\) along the time  
Figure 6: Concept of the calculation of QD. axis, as illustrated in Figure 6. For any \(i\)-th time region \(R_{i}\), applying Equation (16) yields the following:  
\[ \begin{array}{c}QD=\sum\limits_{i=1}^{n}{{{n_{i}}}\cdot T_{i}^{\gamma}}=\frac{1 }{{2\pi}}\sum\limits_{i=1}^{n}\left({{T_{i}^{\gamma}}f_{i_{i-1}}^{t_{i}}\omega_ {d}(t)dt}\right)\\ \text{Here, }{{n_{i}}}=\frac{1}{{2\pi}}\int_{{{I_{i-1}}}}^{{{I_{i}}}}{{\omega_{d} (t)dt}}\end{array}  \tag{17}\]  
In Equation (17), \(T_{i}\) represents the reference torque value in the \(i\)-th time region \(R_{i}\). However, as shown in Figure 6, the measured torque values \(T_{d}\) within the region \(R_{i}\) may vary, making it challenging to determine a single reference torque value \(T_{i}\) for this region. To address this issue, the region \(R_{i}\)(where \(t_{i-1}\leq t\leq t_{i}\)) is divided into \(m_{i}\) equal intervals, with the time interval \(\left({\delta t}\right)_{i}\) and the discrete time points \(t_{ij}\) defined as follows:  
\[\left({\delta t}\right)_{i}=\left({{t_{i}}-{t_{i-1}}}\right)/{{m_{i}}},\ t_{ ij}={{t_{i-1}}}+{j}{\left({\delta t}\right)_{i}} \tag{18}\]  
Using Equation (18), Equation (17) can then be rearranged as follows:  
\[ \begin{array}{c}QD=\frac{1}{{2\pi}}\sum\limits_{i=1}^{n}\left({{T_{i}^{ \gamma}}f_{i_{i-1}}^{t_{i}}\omega_{d}(t)dt}\right)\\ \frac{1}{{2\pi}}\sum\limits_{i=1}^{n}\left[{\underset{{{m_{i}}=\infty}}{\lim} \sum\limits_{i=1}^{m_{i}}\left({{T_{ij}^{\gamma}}({{t_{ij}}})\cdot\omega_{d}( {{t_{ij}}})\cdot\left({\delta t}\right)_{i}}\right)}\right]\\ \frac{1}{{2\pi}}\sum\limits_{i=1}^{n}\int_{{{t_{i-1}}}}^{{{t_{i}}}}{{T_{i}^{ \gamma}}\omega_{d}(t)dt}=\frac{1}{{2\pi}}\int_{{{t_{0}}}}^{{{t_{n}}}}{{T_{i}^{ \gamma}}\omega_{d}(t)dt}\end{array}  \tag{19}\]  
Equation (19) expresses the relative cumulative fatigue damage \(QD\) as an integral with respect to time. For practical convenience, this equation can be converted into a form based on the number of revolutions. This concept is illustrated in Figure 7, and the transformed equation is shown in Equation (20). \[