## 5 Conclusions  
The third rail system comprising a long conductor rail and the supporting structures such as brackets, insulators, and claws are modeled to analyze the dynamic performance of the third rail. Using the structural design information and material properties, a 3D Finite Element Model was developed to closely resemble the SMRT third rail system, and numerical analysis was performed using ANSYS simulation. To verify the accuracy of the FEM model, the ANSYS results for single-span third rail were compared with experimental results from the laboratory setup. Thereafter, a multi-span finite element model was developed, and the results were compared with the analytical approach by modeling the third rail as an infinite Euler-Bernoulli beam on periodic spring supports. The modal analysis and harmonic response in terms of mobility analysis were conducted. Furthermore, the analysis was performed to understand the effect of structural failure on third rail vibration characteristics. The study was further extended to include the sagging of the third rail due to structural failure.  
The results from both the analytical and FEM analysis approaches showed a good agreement for the pin-to-pin mode of resonance frequencies. The third rail was found to exhibit resonance at a very low frequency starting from 10 Hz. The first five pin-to-pin mode of vibrations were identified below 300 Hz. The analysis showed that structural deterioration decreases the resonance frequencies; as a result, the third rail could exhibit resonance at even low frequencies, below 10 Hz. In fact, the effect of damaged or missing supporting structures such as claws was found to have a more adverse effect on lowering the resonance frequencies. It was found that the third rail with a single missing or damaged claw could occur at a frequency as low as 5 Hz. The resonance frequency reduced  
Figure 11: (**a**) Simplified model for current collector device; (**b**) contact force between collector shoe and third rail due to missing claws.  
further below 5 Hz with the increased number of consecutive missing or damaged claws. Furthermore, the results suggested that the collector shoe contact force could drastically increase due to the increase in third rail deflection because of the structural failure. The analysis could be further extended to include a moving load exerted by the collector shoe on third rail. Nonetheless, we believe that the current analysis serves as a tool to examine the third rail condition and perform predictive maintenance.  
Conceptualization, M.P and L.J.L.; methodology, M.P., L.J.L. and F.F.Y.; software, M.P. and L.J.L.; validation, M.P. and L.J.L.; formal analysis, M.P.; investigation, M.P.; resources, K.K.; data curation, M.P. and L.J.L.; writing--original draft preparation, M.P.; writing--review and editing, F.F.Y. and K.K.; supervision, F.F.Y.; project administration F.F.Y. and K.K.; funding acquisition, F.F.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Singapore Mass Rapid Transit (SMRT), grant number: EIRRA-RP5 and research works were carried out at SMRT-NTU Urban Rail Corporate Laboratory, NTU Singapore.  
Not applicable.  
The authors declare no conflict of interest.