# Effect of Track Irregularities on the Response of Two-Way Railway Tracks
## 2 Train-Track-Soil Interaction Model
### Track-Soil Model  
To investigate the ground vibration due to a moving train, the parts of the track such as the sleeper and ballast, and soil domain are modeled as a homogeneous material. The rail is modeled as a 3D Euler-Bernoulli beam. Three-dimensional (3D) FEM is adapted to model the track and soil. The geometry, material damping, radiational damping and mesh size of the model are the critical  
Figure 1: Pictorial representation of the quarter train model. parameters. The track (i.e., sleeper and ballast) and soil are modeled as brick elements. The governing equation of motion for track and soil is given as:  
\[\mathbf{\overline{M}}\mathbf{\ddot{v}}(t)+\mathbf{\overline{C}}\mathbf{\dot{v}}( t)+\mathbf{\overline{K}}\mathbf{v}(t)=\mathbf{\overline{F}}(t), \tag{6}\]  
where \(\mathbf{\overline{M}}\), \(\mathbf{\overline{C}}\) and \(\mathbf{\overline{K}}\) are the total mass, damping and stiffness matrices of complete track and soil model which includes rail, sleepers, ballast and soil domain. \[\mathbf{\overline{M}}=\mathbf{M_{ts}}+\mathbf{M_{r}} \tag{7}\]  
\[\mathbf{\overline{K}}=\mathbf{K_{ts}}+\mathbf{K_{r}} \tag{8}\]  
\[\mathbf{\overline{C}}=\mathbf{C_{ts}}+\mathbf{C_{r}}+\mathbf{C_{s,p}}, \tag{9}\]  
where \(\mathbf{M_{ts}}\) and \(\mathbf{M_{r}}\) are the mass matrices of track-soil and rail, respectively; \(\mathbf{K_{ts}}\), \(\mathbf{K_{r}}\) are the stiffness matrices of track-soil and rail, respectively, and \(\mathbf{C_{ts}}\), \(\mathbf{C_{r}}\) are the material damping matrices of track/soil and rail respectively. Radiational damping coefficient is presented as \(\mathbf{C_{s,p}}\). \[\mathbf{M_{ts}}=\int_{\mathbf{V_{e}}}\rho\mathbf{N^{T}NdV} \tag{10}\]  
\[\mathbf{K_{ts}}=\int_{\mathbf{V_{e}}}\mathbf{B^{T}cBdA}, \tag{11}\]  
where \(\rho\) is the density of respective material and \(\mathbf{N}\) the shape function field. In the stiffness matrix, \(\mathbf{c}\) is the matrix of material constant and \(\mathbf{B}\) is the strain matrix. The Rayleigh damping is used to calculate the material damping, a linear combination of the mass matrix and stiffness matrix. \[\mathbf{C_{ts}}=\alpha_{ts}\mathbf{M_{ts}}+\beta_{ts}\mathbf{K_{ts}} \tag{12}\]  
\[\mathbf{C_{r}}=\alpha_{r}\mathbf{M_{r}}+\beta_{r}\mathbf{K_{r}}, \tag{13}\]  
where \(\alpha_{ts}\), \(\beta_{ts}\), \(\alpha_{r}\) and \(\beta_{r}\) are the Rayleigh damping coefficients. Material damping calculated by this method is frequency dependent and can be predicted using Equations (12) and (13), for the constant damping ratio, the scalar values of the coefficients in the linear combination can be calculated as given in [30]. The Lysmer and Kuhlmeyer model [29] for infinite media is used to formulate the artificial boundary. The Lysmer boundary is a way to avoid the full reflection of waves by the domain boundaries. This is critical because reflected waves may cause resonance in the domain that does not actually exist in reality. The implementation of this kind of boundary is very easy and compatible with finite element modeling as it is a simple connection of dampers to all degrees of freedom of soil boundary nodes and the other end is fixed. The radiational damping coefficient of these dampers can be calculated by Equation (14). \[\mathbf{C_{s,p}}=\rho\mathrm{AV_{s,p}}, \tag{14}\]  
where A is the cross-sectional area of the element, \(\rho\) is the mass density and \(\mathrm{V_{s,p}}\) are the wave velocity depending upon the type of wave (shear wave velocity \(\mathrm{V_{s}}\) or compressional wave velocity \(\mathrm{V_{p}}\)). A wide range of frequencies are excited in the wave propagation problems. To solve this problem easily is to select a high cutoff frequency to get an accurate solution. Selecting the mesh size of elements is very important for defining the proper time step, correct wave front propagation through space from one point to the other, and stability of the solution. The Courant-Friedrichs-Lewy (CFL) method has been used as in [31].