# Effect of Track Irregularities on the Response of Two-Way Railway Tracks
## 3 Numerical Analysis Procedure  
The equation of motion for train Equation (1) and the equation of motion of track and soil Equation (6) are coupled by the loads of the train due to its weight and interaction forces \(\left\{F_{w/r}\right\}\). The interaction forces depend on the position of the wheel on the rail, track irregularity, and time-dependent displacement of the rail. To integrate the two sub-systems (i.e., train and track-soil), an iterative scheme has been used which considers the contact force equilibrium as given in [34]. This iterative method also incorporates the Newmark integration method to solve the coupled train-track-soil dynamic system equations. The complete algorithm is given as follows:  
Step 1: at first time step and iteration (\(i=1\)), train responses \({{}_{i}^{t}}[v_{w}],{{}_{i}^{t}}[\dot{v}_{w}]\) and\({{}_{i}^{t}}[\ddot{v}_{w}]\) and track responses \({{}_{i}^{t}}[v],{{}_{i}^{t}}[\dot{v}]\) and\({{}_{i}^{t}}[\ddot{v}]\) are assumed initially as zero.  
Step 2: after first time step, the second iteration (\(i+1\)) starts with the computation of the interaction force vector of all wheels \({{}_{i+1}^{t+A}}\)\(\left\{F_{w/r}\right\}\). The vertical contact forces of \(k\) -th wheel can be evaluated from the last known responses of the wheel and rail at the position of contact by using equation Equation (5).  
Step 3: this vertical contact forces with gravity loads of train given in Equation (2) are applied on rail as external loads. The track/soil responses at \((i+1)\) iteration with the time step \((t+\Delta t)\) are computed by the following equations.  
\[([K]+a_{1}[M]+a_{2}[C])^{t+\Delta t}\{v\}= \begin{array}{l}{{}^{t+\Delta t}\{F(t)\}+[M]\Big{(}a_{1}{}^{t} \{v\}+a_{3}{}^{t}\Big{\{}\dot{v}\}\Big{)}a_{4}{}^{t}\Big{\{}\ddot{v}\}\Big{)}} \\ {+[C]\Big{(}a_{2}{}^{t}\{v\}+a_{5}{}^{t}\Big{\{}\ddot{v}\}+a_{6}{}^{t}\Big{\{} \ddot{v}\}\Big{)}}\end{array} \tag{22}\] \[\begin{array}{l}{{}^{t+\Delta t}\Big{\{}\ddot{v}\}=a_{1}\Big{(}{}^{t+ \Delta t}\{v\}-{}^{t}\{v\}\Big{)}-a_{5}{}^{t}\Big{\{}\dot{v}\}-a_{6}{}^{t} \Big{\{}\ddot{v}\}}\end{array}\] (23) \[\begin{array}{l}{{}^{t+\Delta t}\Big{\{}\ddot{v}\}=a_{1}\Big{(}{}^{t+ \Delta t}\{v\}-{}^{t}\{v\}\Big{)}-a_{3}{}^{t}\Big{\{}\dot{v}\}-a_{4}{}^{t} \Big{\{}\ddot{v}\}}\end{array} \tag{24}\]  
Step 4: after getting the response of track/soil at a given time step, update the rail responses in Equation (5) at every position of wheels on rails defined as \(\begin{array}{l}{{}^{t+\Delta t}_{i+1}\{v\}}\end{array}\). Before moving to the next iteration, the convergence of the solution must be checked by the following equation.  
\[\begin{array}{l}\dfrac{\|{}^{t+\Delta t}_{i+1}\{v\}-{}^{t+\Delta t}_{i}\{v \}\|}{\|{}^{t+\Delta t}_{i+1}\{v\}\|}\leq\varepsilon,\end{array} \tag{25}\]  
where the \(\begin{array}{l}{{}^{t+\Delta t}_{i+1}\{v\}},\begin{array}{l}{{}^{t+ \Delta t}_{i}\{v\}}\end{array}\) are the displacements of track at present and previous iterations and \(\varepsilon\) is the specified tolerance. If the convergence criterion is satisfied, return to the initial iteration (or Step 2) for the next time step otherwise move to step 5.  
Step 5: if the convergence criterion is not satisfied, take the rail displacement at a given time \(\{v_{r}\}\) from \(\begin{array}{l}{{}^{t+\Delta t}_{i+1}\{v\}}\end{array}\) and calculate \(\begin{array}{l}{{}^{t+\Delta t}_{i+1}\{F_{w/r}\}}\end{array}\) by using Equation (5). Then, apply it on wheels as external force and obtained \(\begin{array}{l}{{}^{t+\Delta t}_{i+1}\{v_{w}\}},\begin{array}{l}{{}^{t+ \Delta t}_{i+1}\{\dot{v}_{w}\}}\end{array}\) and \(\begin{array}{l}{{}^{t+\Delta t}_{i+1}\{\ddot{v}_{w}\}}\end{array}\) by solving equations Equations (22)-(24). Now take \(\{v_{w3}\}\) from \(\begin{array}{l}{{}^{t+\Delta t}_{i+1}\{v_{w}\}}\end{array}\) and turn to step 2 for the next iteration.