Finding Contact Constraints between Wheel and Rail using Geometrical Method

Solving the wheel/rail contact problem is one of most important issues in a rail vehicle dynamic simulation. For instance, the wear rate and pattern of the wheel/rail profiles can affect the vehicle stability. This effect is usually considered virtually in computer simulations rather than a real life condition. Also, obtaining an optimal wheel profile is mainly carried out using dynamic simulation packages. At the first stage, the wheel and rail contact point location is found, and then followed by its contact point shape and size. For this purpose, researchers simulate a quasi-static condition of a 2-bogie coach during its curving by the use of measured rail and wheel profiles to extract the required parameters.The authors of this paper describe various contact theories, and the geometrical contact equation within the wheel and rail interface is then presented. The results of a code developed in this study are compared with the output data of a commercial software tool such as ADAMS/Rail.1. Wheel/Rail Contact TheoriesA convenient method to find the wheel/rail contact point location is to assume that the contact patch is rigid (a non-elastic mode). Using this theory finds the minimum vertical distance between the wheel and rail profiles, and nominates corresponding points within the profiles as the possible contact point. In this method the wheel profiles moves laterally against the rail profile and contact point locations are found as a function of the wheelset lateral movement. After finding the contact points, the curvature of the profiles within the contact points is calculated and the contact location is achieved using Hertz formula. The outputs discontinuity is the major deficiency of the rigid contact model. Since the results of contact model will later be used in the software tool, it leads to a numerical instability of the dynamic simulation. This discontinuity occurs in the presence of a multi point contact model within the contact patch. The smoothing process of such discontinuities is one of the solutions by use of the splines. For instance, ADAMS/Rail smoothes the output results using Bezier-Splines.To address this problem, multi point contact models were developed by the researchers. By taking into account the deformation of the first contact point, further contact points will come into the simulation process. Meanwhile, the deformation of the profiles at the first contact points make it possible to approaching two profiles and appearing new contact points. The contact points will be obtained in such a manner to satisfy the force equilibrium. 2. Wheel/Rail Geometrical Contact Equations in a Rigid One-Point ContactContact equations were given for an unconstrained wheelset in a curve. Using these equations result in four final relationships as follow: , ,, where ξw and ηw are the x and y coordinates of the wheel profile, and ξr & ηr stand for x and y coordinates of the rail profiles, is the second index for the left profiles, and is the second index for the right profiles. is the gradient of the profile at the contact point. If the shape of profiles is known as mathematical functions, the Newton-Raphson equation can be used to solve these relationships. In most cases the shape of profiles is given by the x and y values, and the Newton-Raphson method is not thereafter applicable. Note that a geometrical solving method was used in this study.For this reason, the profiles are initially estimated using 3rd order splines so the curvature of the profiles at any point of the contact can be calculated. It is possible to smooth the profiles in this manner to decrease the discontinuity of the output results. A MATLAB code to solve these equations was developed. This code is capable to take into account the horizontal and vertical irregularities of rails as inputs into the simulation process. The output results of the code are contact points, the gradient within contact points, rolling radius of wheels and the contact areas. The profiles can have different shapes. The generality character of the code makes it possible to simulate wheelset movement along a track with a variable rail gauge, and the rail and wheel profiles.To find a solution, the code moves the wheel profiles laterally to a specified magnitude and then generates some artificial points on the wheel and rail profiles to obtain a better convergence rate and accuracy. The code then finds the shortest distance between the wheel and rail profiles. If the differential distance lies within the given range, the points will be recorded as contact points. Nevertheless, the wheel profiles will be rotated. These stages will be followed until the given range will be satisfied.The simulation results were compared with the two well-known commercial software tools such as ADAMS/Rail and VAMPIRE. All comparisons show an excellent agreement between the code developed in this study and the mentioned software outputs.