Modeling of Reverse Curves on a Railway Line Using the Analytical Design Method

The article deals with the issue of designing reverse curves in a railway track, i.e. a geometric system consisting of two circular arcs (usually with different radii), directed in opposite directions and directly connected to each other. It is also about being able to recreate (i.e. model) the existing geometric system with reverse arcs, so that it is then possible to correct the horizontal ordinates in the area where the circular arcs connect. An analytical method of designing track geometric systems was used, in which individual elements of these systems are described using mathematical equations. The design itself is carried out in the appropriate local Cartesian coordinate system, which is based on symmetrically arranged adjacent main directions of the route. The origin of this system is located at the point of intersection of adjacent main directions, whose coordinates in the global system are known. In the case of reverse curves, a third main direction appears, which significantly complicates the design procedure. The initial values ​​of the radii of the reverse arcs must correspond to the existing system of main directions. The introduction of transition curves causes these radii to decrease; their values ​​are determined iteratively. A set of formulas for creating a geometric system of reverse curves is presented. These formulas were used in the calculation example. A graph of the horizontal curvature of the track axis and a method for determining the possible train speed without the use of cant on an arc and with the use of cant are shown. The presented procedure is universal and can be applied to other geometric situations involving the design of reverse curves.