V. D. Borisenko, S. A. Ustenko, I. V. Ustenko


Context. The problem of geometric modelling of transitional curves, which are placed between rectilinear and circular sections of railway tracks, can be considered solved sufficiently. However, there are a number of factors that contribute to the development of new methods for modelling these important sections of the railway tracks. The main of them are the increase in the speed of the train, the increase in their mass, the limited size of the territory on which the railway is built, etc. The importance of this issue is greatly increased when laying rails in a mountainous area, when trains have to overcome the ups and downs, bend around natural and artificial obstacles. Under these conditions, the transition curves acquire a spatial character.

Objective. Further development of the method of geometric modelling of spatial transition curves, which are placed between rectilinear and circular sections of railway tracks located in two parallel planes.

Method. Transitional sections of the railway track are modelled using parametric curves, in which the length of the curve arc is taken as the parameter. To close the mathematical model of the transition curves, it is assumed that the curvature of the curve is subject to a polynomial dependence of the fourth degree, and torsion to the second degree. The unknown coefficients of these polynomial dependencies, which are necessary for calculating the coordinates of the simulated transition curves, are determined by a numerical method, in particular, by minimization of the functional for which the deviation of the intermediate obtained final point of the transition curve from the given one is accepted.

Results. On the basis of the proposed theoretical proposition, a program code for calculating and visualizing spatial transition curves providing a smooth transition from rectilinear sections of a railway track to a circular one is developed, provided that both these sections are in parallel planes.

Сonclusions. A new method is proposed for modelling the spatial transition curves of railway tracks, which are laid on the terrain with a complex relief. Practical implementation of many variants of spatial transition curves placed between the rectilinear and circular sections of the railway track has proved the operability of the method of their geometric modelling.


Spatial transition curve; the railways; geometric modeling; natural parameterization; curvature; torsion.


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