# Vibration and Derailment Analyses of Trains Moving on Curved and Cant Rails
## 4 Validations of the Proposed Finite Element Simulations  
In this section, a Taiwan high-speed train (SKS-700) carriage on curved rails to validate the proposed finite element method, where the carriage contains two bogies and four wheel sets, as shown in Figure 5, which provides the dimensions, dampers, springs, and masses. In the finite element model, the car bodies, bogies, and wheel sets are assumed to be rigid bodies, where master nodes are set to the mass center, and spring-damper elements set at the slave nodes are used to connect them. Four wheel set elements, as mentioned in Section 2, are arranged to connect to the curved rail with five segments, as shown in Figure 3, where the curve lengths are 500 m for the first straight line, 2000 m for the first clothoid curve, 10,908.3078 m for the circular curve, 2000 m for the second clothoid curve, and 800 m for the last straight line. Since the location of the wheel is known, the wheel and rail contact locations, nodes D (shown in Figure 2), can be determined. Then, using the input geometries of the wheel and rail, as shown in Figure 2, the program will automatically calculate the contact behavior, including the sticking, sliding, and separation modes of each wheel and rail using the equations in Section 2.3 and Reference [16]. Moreover, the deformation at any node will affect the contact result, since all the finite elements are coupled in the numerical analysis. In the finite element analysis, the [PERSON]'s direct integration method is used with a time step length of 0.01 s, while the nonlinear equation is solved by the Newton-Raphson method. Figure 6 shows the comparison of the normal and shear contact forces of the wheel and rail between the finite element result and the theoretical result, when the carriage speed is 80 m/s and cant angle constants _Ct_ are 200 and 900 m for two cases. It is noted that the rail curve radius is much larger than the carriage length, so the contact forces of each wheel from the finite element analysis are very similar. This figure indicates that the finite element and theoretical results are almost identical. It is noted that the theoretical solution can be easily obtained from Equations (25) and (26), but the  
Figure 4: Sketch drawing of moving axis element on curved track system (the element contains two rigid bodies contacting at the wheel and rail surfaces, where slave nodes SB for the wheel and slave nodes SC for the rail are obtained from the input data. Usually, 30 to 60 nodes for each slave node group are accurate enough to model the possible contact surface. Rails are modeled using a number of beam elements, where the program automatically find the active beam element that the wheel is contacting). undergoes complex calculations, where each time step often requires two to four iterations to reach the convergence of the nonlinear matrix equation. Figure 7 shows the beginning of the sliding between the wheel and rail at a certain curve radius, when the carriage moves on the curved rail with a constant speed. Each symbol in Figure 7 represents a whole finite element analysis, where Equation (24) are used to both theoretical and finite element results automatically to obtain the curve radius under the fixed values of the carriage speed, \(Ct\), and frictional coefficient. The sliding between the wheel and rail can be found in Figure 8, which was directly obtained from the finite element results for \(Ct=400\) m, \(V=180\) m/s, and \(\mu=0.2\). It is noted that the wheel and rail shapes, as shown in this figure, are inputted data. The contact is in the sticking mode for a shear contact force smaller than the normal contact force times of the frictional coefficient, and otherwise, the wheel will move toward the outer rail. Figure 7 indicates that the proposed method can accurately be used to simulate the sliding behavior between a wheel and rail under various carriage speed, rail can, and friction coefficient conditions for the wheel and rail.