Analytical Method for Modifying Compound Curves on Railway Lines

The aim of the research presented in the article is to develop a method for modifying compound curves, i.e., geometric systems composed of two (or more) circular arcs with different radii, directed in the same direction and directly connected to each other. These curves are used when connecting two directions of the railway route where one circular arc is impossible due to permanent terrain obstacles. To solve the problem, an analytical method of designing track geometric systems was used, in which individual elements of these systems are described using mathematical equations. The modification itself involves introducing appropriate transition curves between the connecting arcs. Three possibilities for such a connection were presented, resulting from the method of considering conditions related to horizontal curvature of the track axis. A comparative analysis of the obtained solutions was conducted using the developed geometric test system. The analysis was based on the curvature values determined for the considered transition curves, after assuming varying lengths of these curves. For the recommended solution to the problem, it was necessary to verify the practical feasibility of horizontal ordinate values, which could not be too small relative to the implementation error. As stated, to limit the effects of this error, the transition curve lengths should be adjusted to specific geometric situations and excessively short curves should be avoided. As a result of the conducted research, the transition curve determined with strict curvature conditions was determined to be the most advantageous. It maintains curvature continuity along its entire length, there are no abrupt changes in curvature at the edges, and the changes in curvature along the length are much smoother than in the other curves considered. Therefore, this curve should be recommended for practical use.