## 4 Results and Discussion
### Analysis with Damaged Structural Components
#### 4.3.2 Effect of Missing Claws
As expected, the sagging of the third rail increases with the increasing number of missing claws. However, it is interesting to note that the sagging increases much more as the number of missing claws increases. As the contact force on the collector shoes increases with the sagging, more than two missing claws can exert a considerably higher force than normal operating conditions. This will increase the possibility of bottoming out the current collector device's suspension and damaging the collector shoes. Figure 10 shows the damage and failure of the structural components and current collector shoes because of excessive contact force due to excessive sagg  
Figure 9: Deflection of the third rail due to failure of the support structure. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No of Missing Claws** & \multicolumn{4}{c}{**Low-Frequency Resonance below 10 Hz Due to Number of Missing Claws**} \\ \hline  & 1 st resonance, Hz & \multicolumn{2}{c}{2 nd resonance, Hz} & \multicolumn{2}{c}{3 rd resonance, Hz} \\  & Analytical & Numerical & Analytical & Numerical & Analytical & Numerical \\
1 & 4.54 & 4.8 & & & & \\
2 & 2.23 & 2.4 & 6.19 & 6.45 & & \\
3 & 1.27 & 1.35 & 3 & 3.6 & 7.32 & 7.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: List of Low-Frequency Resonance below 10 Hz Due to Number of Missing Claws. In practice, the third rail alignment can be measured using a linear variable displacement transducer (LVDT), which scans the movement of the collector shoes [30] and gives a rough idea about the sagging of the third rail and possible need for maintenance. The results from Figure 9 can be used to predict the number of missing claws and possible disruption in the operation. Thus, the analysis would help predictive maintenance as the results can be used to detect the possible deterioration or the failure of the structural components. The current collector device is often modeled as a cantilever arm with a spring and collector shoe attached to it. A simple method to estimate the contact force between the third rail and the collector shoe is to add a constant force (\(F_{a}\)) with the force proportional to the spring displacement. The force is applied to ensure proper contact between the collector shoe and the third rail. Similarly, deflection of the third rail increases the displacement of the spring, and the contact force increases. A typical representation of the current collector device and the contact force model is shown in Figure 11a. A typical value of applied constant force may range from 120 N to 145 N [31]. Assuming the contact shoe is always in contact with the third rail, the displacement of the spring (\(x\)) can be correlated with the deflection of the third rail (\(u\)). The total contact force between the collector shoe and the third rail is given by:  
\[Contact\ force(F_{c})=F_{a}+K_{spring}*\left(\frac{a}{b}\right)^{2}*u \tag{11}\]  
Here, the typical values for constant force (\(F_{a}\)), spring stiffness (\(K_{spring}\)), the distance between the pivot point and spring (a), and length of cantilever arm (b) are selected to be 120 N, 11300 N/m, 65 mm and 428 mm, respectively [8]. The total contact force considering the deflection of the third rail from Figure 9 was computed for a different number of missing claws. The results are shown in Figure 11b. Here, it should be noted that the model used for computing the contact force does not consider any structural irregularities such as misalignments, rough surface, and so on. However, this simple model successfully showed that the large deflection of the third rail due to missing claws could exert excessive forces on the current collector devices and cause structural damage. Figure 10: Nature of the damages and structural failures in third rail structure and current collector devices (CCD).