MODELING THE TRANSITION CURVE ON A LIMITED TERAIN
DOI:
https://doi.org/10.15802/stp2017/99942Keywords:
geometric modelling, transition curve, limited area, the curvature of the curve, a cubic dependence, curvature distributionAbstract
Purpose. Further development of the method of geometric modelling of transition curves, which are placed between rectilinear and circular sections of railway tracks and are created in localities, the relief of which causes certain restrictions on the size of the transition curves of the railway track. Methodology. The equation of the transition curve is taken in parametric form, in which the length of the arc of the modelled curve is used as a parameter. As initial data in the modelling of the transition curve, the coordinates of its initial point and the angle of inclination in it are tangent, the radius of the circumference of the circular section and the parameter that is used as a constraint when placing a section of the railway track. The transition curve is modelled under the condition that the distribution of its curvature from the length of the arc - the natural parameter - is described by a cubic dependence. This dependence contains four unknown coefficients; the unknown is also the length of the arc. The coefficients of the cubic dependence and the length of the arc of the transition curve, the coordinates of its end point, the angle of inclination in it of the tangent are determined during the simulation of the transition curve. The application of boundary conditions and methods of differential geometry with respect to the distribution of the slope angle of the tangent to the simulated curve from the initial to the end points of the transition curve and the calculation of the coordinates of the end point of the curve allows us to reduce the problem of modelling the transition curve to determine the arc length of this curve. Directly the length of the transition curve is in the process of minimizing the deviation of the circumference of the circular path from its current value obtained when searching for the arc length. Findings. As a result of the computational experiment, the possibility of modelling a transition curve between a rectilinear and circular rail track in a region of a limited size has been proved. Originality. A method for geometric modelling of transition curves between a rectilinear and circular section of a railway track is developed in conditions of limited terrain size, on which rails are laid. The transition curve is represented in the natural parameterization, using the cubic dependence of the curvature distribution on the length of its arc. Practical value. The proposed method of modelling the transition curves in conditions of limited terrain size allows obtaining these curves with a high accuracy in a wide range of geometric parameters of rectilinear and circular sections of the railway track and a parameter that acts as a constraint in the modelling of the transition curve. The method can be recommended in the practice of building railways.
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