# Vibration and Derailment Analyses of Trains Moving on Curved and Cant Rails
## 2 Finite Element Simulation of Trains Moving on Curved Rails
### A frictional Contact Finite Element for Two Wheels Moving on Curved Rails  
Ju [16] developed a stiffness matrix at one node using the sticking, sliding, and separation modes. The major advantage of the formulation is that it is symmetric, even in the case of the sliding between the wheel and rail. The two-node stiffness matrix is then transformed to the 18 by 18 stiffness matrix of nodes B, I and J (Figure 2), where each node has three global translation and three global rotation DOFs for the contact of a wheel and rail. Since the above method was used to simulate the wheel moving on a straight line rail, we modified it and added the longitudinal stiffness to solve moving train problems on curved railways. [PERSON]'s variational method [17] is often used for rolling contact problems. This method follows the half-space assumption and [PERSON]'s friction law [18]. Thus, in this study, the [PERSON]'s friction law was used directly to obtain the actual contact stiffness of the wheel and rail based on the 3D finite element analysis. In the finite element analysis, we used the Newton-Raphson method to calculate the nonlinear contact between the rail and wheel. Thus, the current contact stiffness for a specific iteration can be written as follows [19]:  
\[\left\{ \begin{array}{c}df_{s}\\ df_{n}\\ df_{s1}\end{array} \right\}=\left[K_{\text{local}}\right]\left\{ \begin{array}[ ]{c}dU\\ dW\\ dV\end{array} \right\}, \tag{6}\]  
where \(\left[K_{\text{local}}\right]=\left[ \begin{array}{ccc}k&0&0\\ 0&k&0\\ 0&0&k\end{array} \right]\) for the sticking mode, \(\left[K_{\text{local}}\right]=\left[0\right]\) for separation mode, (7)  
\[\text{and }\left[K_{\text{local}}\right]=\left[ \begin{array}{ccc}\mu^{2}k& \mu k&0\\ \mu k&k&0\\ 0&0&k_{v}\end{array} \right]\text{ for the sliding mode}. \tag{8}\]  
\(df_{s}\), \(df_{n}\), \(df_{s1}\), \(dU\), \(dW\), and \(dV\) are incremental nodal contact forces and displacements in the local directions between two iterations, in which \(k\) and \(k_{v}\) are penalty constants, and \(\mu\) is the coefficient of friction, where it is positive for \(\Delta U<0\) and negative for \(\Delta U>0\), and \(\Delta U\) is the difference in the relative tangential displacement between the current and previous time steps. The force and displacement vectors are obtained between two iterations, and all the nodal coordinates of the wheel and rail are set to the last iteration. Furthermore, the direction of \(df_{s}\) and \(dU\) is set to the total frictional force direction of the last iteration. Thus, at the convergence of a time step, a fixed coordinate for each wheel and rail node can be obtained, so \(dU\) and \(dV\) will approach zero. Thus, even for the sliding mode, the [PERSON]'s friction theory is still followed. As shown in Figure 2, the possible contact region for the wheel and rail should be inputted as a number of nodes along the wheel and rail surfaces, respectively, where the wheel nodes are simulated as the cubic-splines to represent the contact surface. The proposed theory can be used to analyze multi contacts between the wheel and rail, where the node (rail nodes) to cubic-splines (wheel surface) contact method is used [20]. As shown in Figures 1 and 2, local direction 3 is set to the rail longitudinally tangential to the moving wheel direction, the local direction 1 is set in the cart direction to the center of the rail with a current radius \(R\), and \(3\times 1\) obtains local direction 2, which is a cart angle to the vertical direction. The stiffness matrices discussed above are expressed as contact coordinates. They can be transformed back into the local 1-2-3 coordinates as follows:  
\[[\text{K}_{\text{1-2-3}}]=[\text{T}_{3}]^{\text{T}}\ [\text{K}_{\text{local}}]\ [\text{T}_{3}], \tag{9}\]  
\[\text{where}\ [\text{T}_{3}]=\left[ \begin{array}{cc}\{\text{u}\}&\{\text{v}\}&\{ \text{w}\}\end{array}