## 4 Results and Discussion
### Analysis with Damaged Structural Components
#### 4.3.2 Effect of Missing Claws  
Another effect of such deterioration in the third rail structure is the failure of the supporting structures. In normal conditions, the third rail is clamped at a certain height above the railway track by the claw structures. Periodic clamping arrangement limits the downward deflection of the third rail due to its self-weight. However, exposure to strong vibration could induce high stress at the claw insulator supports and rail joints, leading to eventual failure of structural components or unfastening of claws [25]. Any broken or  
Figure 6: Effect of structural deterioration on harmonic response of third rail. Figure 7: Typical resonance mode (\(\sim\)20 Hz) where whole third rail bounces on support. missing claws or supporting components have adverse effect on the dynamic response of the third rail. Furthermore, missing claws increase the sagging of the third rail, which drastically increases the contact force between the rail and the collector shoes. The effects of missing claws or supporting structures on the performance of third rail are discussed in this section. The most crucial change in the structural parameter due to the damaged or missing claws is the change of the span length. When a single claw structure is broken, the span length between two consecutive support structures becomes double at that particular section, which affects the overall dynamic response of the third rail. For example, a single missing or broken claw structure can be modelled by removing the spring support \(n_{1}\) in Figure 3. When \(n_{1}\) is removed, the span length between two consecutive spring supports, i.e., \(n_{0}\) and \(n_{2}\), becomes 2*d. This change in span length was incorporated in dynamic Equations (7)-(9). Thus, by changing the span length of two consecutive spring supports, the vibration characteristics of the third rail with missing or broken claw structures can be analyzed. The results from both analytical and numerical approaches are shown in Figure 8. The results present a harmonic response in terms of mobility amplitude of the third rail for a single missing claw (\(missing\)\(claws=n_{1}\) missing) to three missing claws (\(missing\)\(claws:\)\(n_{1}\)\(to\)\(n_{3}\)). Figure 8: Harmonic response of the multi-span third rail with missing claws: (**a**) number of missing claws = 1; (**b**) number of missing claws = 2; and (**c**) number of missing claw = 3. A damaged supporting component or missing claw resulted in low-frequency resonance below 10 Hz. Furthermore, as the number of missing claws increased, the resonances occurred at even lower frequencies than 10 Hz. Interestingly, the number of resonance that appeared below 10 Hz is equal to the number of missing claws. Moreover, the harmonic responses obtained from both the analytical and numerical approaches are very close to each other. For each number of missing claws, the resonance frequencies are listed in Table 5.  
Missing claws or supporting components increase the sagging of the third rail and, therefore, exert excessive contact force on the current collector shoes. Excessive contact forces could cause the twisting of the collector shoes, leading to loss of contact with the third rail and interrupting the power supply to the train. The sagging of the third rail and the contact force depends on the number of dropped or missing claws. The deflection of the third rail for a number of missing claws was simulated in ANSYS. The results are presented in Figure 9. With no missing claws, the maximum deflection of the third rail was less than 1.2 mm, which corroborated with the design criteria of the SMRT third rail to limit the third rail deflection below 1.5 mm [28; 29] For more than two damaged or missing claw structures, the deflection increased sharply and was found to be about 170 mm in the case of four missing claw structures. Similar results were reported in an investigation conducted by SMRT to inquire the causes of power disruption in train services [28].