## 4 Results and Discussion
### Analysis of Multi-Span Third Rail Model  
Building on the experimental verification of the finite element model, the analysis was extended to a 35 m multi-span third rail. Here, the results from the validated finite element model for multi-span third rail were compared against the results from the mathematical model developed in Section 3. The parameters listed in Table 3 for SMRT third rail were used to calculate the analytical solution for Equations (9) and (10). Figure 5a compares the numerical solution for the spring-supported multi-span third rail model and the analytical solution of the mathematical model. The analytical solution is for the point mobility of the third rail due to a unit harmonic force at mid-span, i.e., \(x_{c}=d/2\). It should be noted that no damping properties of either the third rail or the supporting structures/springs were considered in this analysis. The major difference between assuming continuous support and discrete periodic support is the presence of pin-to-pin vibration modes. Here, pins represent the discrete spring support points. Pin-to-pin vibration resonance occurs at specific frequencies when the wavelength of the vibration propagation is comparable to the distance between the two supports, \(d\). Pin-to-pin vibration resonance appears in one basic mode (first mode) and several higher modes [26]. The first pin-to-pin vibration occurs when the wavelength of the propagation is double the distance between two supports. Similarly, the second pin-to-pin vibration occurs when the wavelength is equal to the distance between two supports, and so on. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**Resonance Frequencies (Hz)**} & \multirow{2}{*}{**Difference (\%)**} \\  & **Experimental** & **FEM** & \\ \hline
1 & 11.3 & 11.15 & 1.35 \\
2 & 38.8 & 38.77 & 0.07 \\
3 & 67.5 & 69.84 & 3.35 \\
4 & 91.3 & 101.7 & 10.2 \\
5 & 128.0 & 150.36 & 14.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Natural frequencies for first five modes of vibration obtained from experimental and FEM analysis. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Parameters**} & **Units** & **Values** \\ \hline Effective Young’s modulus (\(E\)) & N/m\({}^{2}\) & 8.485 \(\times\) 10\({}^{10}\) \\ Second moment of inertia (\(I\)) & m\({}^{-4}\) & 6.45 \(\times\) 10\({}^{-6}\) \\ Distance between two supports (\(d\)) & m & 4.9 \\ Mass per unit length of beam (\(m_{r}^{\prime}\)) & kg/m & 17.8025 \\ Equivalent spring stiffness of the support (\(K\)) & MN/m & 2.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Equivalent structural parameters for analytical solution. The peaks of the mobility magnitude vs. frequency graph show vibration resonance of different modes. It is interesting to note that the first five pin-to-pin modes of vibration for the third rail occur below 300 Hz compared with the typical first pin-to-pin mode for railway structure, which occurs above 1000 Hz [10]. For SMRT third rail parameters, the first or basic pin-to-pin mode was found at 11.94 Hz based on the analytical solution. Similarly, the numerical solution from ANSYS simulation predicts the same mode at 11.74 Hz. A lower bending stiffness, lower mass per unit length, and a longer span length between supports are the reasons for the low pin-to-pin resonant frequency. It should also be noted that, while the mobility amplitude from the mathematical model mostly shows pin-to-pin modes, the ANSYS simulation includes many other non-pin-to-pin modes of vibration, especially at frequencies above 100 Hz. Although an FEM analysis gives more mode shapes compared with the analytical approach, it requires considerable computation time. As such, it is more convenient to use an analytical model for performing parametric analysis to analyze different scenarios such as the effect of a change in third rail material properties, structural parameters, and so on. An accurate analytical model can be used for a quick analysis of a given third rail system and structural properties.