# Effect of Track Irregularities on the Response of Two-Way Railway Tracks
## 4 Numerical Results and Discussions
### Effect of Train Speed and Track Irregularities  
After validating the results of the soil model against [PERSON]'s solution, the train is modeled as a quarter train model. Figure 4 presents the configuration of the train on one rail. Table 1 shows the properties of the quarter train model. Track with sleepers at a spacing of 0.7 m centers with dimensions 2.5 m \(\times\) 0.35 m \(\times\) 0.25 m is considered. The rail was modeled using a beam element for which the properties are given in Table 2. The track and soil were modeled as three-dimensional solid elements. The depth of the domain is taken to 5 m and the distance between the two tracks is 6.5 m. Table 3 summarizes the properties of the track and soil. Rail-pads are represented by a combination of a spring-damper system to combine the rails and sleepers. The profile of the analyzing domain is shown in Figure 5.  
Figure 4: Configuration of train wheels.  
Figure 3: Comparison of the numerical and analytical solution at the line of observation (0.5 m from surface) (**a**) longitudinal displacement (**b**) vertical displacement.  
Figure 2: Half-space subjected to the point load.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Axle load, P (kg) & 17,679.22 \\ Car mass, \(\overline{m}_{w1}\) (kg) & 55,963.86 \\ Bogie mass, \(\overline{m}_{w2}\) (kg) & 3312.313 \\ Wheel mass, \(m_{w3}\) (kg) & 1016.047 \\ Hertz contact Stiffness, \(K_{H}\left(GN/m^{1.5}\right)\) & 93.7 \\ Secondary suspension stiffness, \(k_{w1}\) (N/m) & \(1.31\times 10^{3}\) \\ Primary suspension stiffness, \(k_{w2}\) (N/m) & \(3.28\times 10^{3}\) \\ Secondary suspension damping, \(c_{w1}\) (Ns/m) & 90 \\ Primary suspension damping, \(c_{w2}\) (Ns/m) & 30 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Train properties [15].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Basic Modulus, E (GPa) & 210 \\ Density, \(\rho\) (kg/m\({}^{3}\) ) & 7897 \\ Bending stiffness around x, \(I_{x}\) (m\({}^{4}\)) & \(1.245\times 10^{-5}\) \\ Bending stiffness around z, \(I_{z}\) (m\({}^{4}\)) & \(4.526\times 10^{-6}\) \\ Torsional stiffness around y, \(J_{y}\) (m\({}^{4}\)) & \(6.554\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Rail properties [15].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline
**Sleeper (Concrete)** & \\ Elastic Modulus, E (GPa) & 10 \\ Poisson ratio, & 0.2 \\ Density, \(\rho\) (kg/m\({}^{3}\) ) & 2400 \\ Primary wave velocity, \(V_{p}\) (m/s) & 2151.6 \\ Secondary wave velocity, \(V_{s}\) (m/s) & 1317.6 \\ Rayleigh wave velocity, \(V_{r}\) (m/s) & 1201.2 \\
**Ballast** & \\ Elastic Modulus, E (GPa) & 0.13 \\ Poisson ratio, & 0.4 \\ Density, \(\rho\) (kg/m\({}^{3}\) ) & 1600 \\ Primary wave velocity, \(V_{p}\) (m/s) & 417.3 \\ Secondary wave velocity, \(V_{s}\) (m/s) & 170.3 \\ Rayleigh wave velocity, \(V_{r}\) (m/s) & 160.3 \\
**Soil** & \\ Elastic Modulus, E (GPa) & 0.025 \\ Poisson ratio, & 0.45 \\ Density, \(\rho\) (kg/m\({}^{3}\) ) & 1800 \\ Primary wave velocity, \(V_{p}\) (m/s) & 229.5 \\ Secondary wave velocity, \(V_{s}\) (m/s) & 69.2 \\ Rayleigh wave velocity, \(V_{r}\) (m/s) & 65.6 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Material properties of sleeper, ballast, and soil [15].  
In this paper, only one train car was considered to generalize the response of ground under loadings due to different levels of track irregularities on two tracks. Various experimental trials were carried out for different speeds at 30 m/s, 40 m/s, 50 m/s and 60 m/s, to study the effect of track quality on ground vibrations. Six (06) classes or levels of track irregularities were selected and calculated using the numerical generation method based on the PSD function. Table 4 shows the six track classes and the values of their parameters A, \(\phi_{1}\) and \(\phi_{2}\).