Characterization and quantification of railroad spiral-joint discontinuities

Abstract

In this study, a new approach is proposed for the characterization and quantification of the spiral transitions in railroad vehicle system algorithms. Definitions of the super-elevation and balance speed are provided in order to have a better understanding of their variations within the space curve of the spiral segment. In order to develop this understanding, three different curves which have fundamentally different geometries are considered; a super-elevated constant-curvature curve with zero twist and zero vertical elevation, a vertically-elevated helix curve with a constant curvature and twist and zero super-elevation, and a spiral curve with non-zero varying-curvature, twist, super-elevation, and vertical elevation. The curve equations are developed in terms of Euler angles used by the rail industry to describe the track geometry in the computer simulations. Because the geometry of the spiral space curve can be completely defined in terms of two Euler angles only, the horizontal-curvature and the vertical-development angles; a third Euler angle referred to as the Frenet bank angle is written in terms of these two angles using an algebraic equation. The fact that, for given curvature and elevation angles, the Frenet bank angle cannot be treated as an independent geometric parameter is used to obtain accurate quantification of the spiral-intersection discontinuities. The severity of the twist and elevation discontinuities at the spiral intersections with the tangent and curve segments demonstrates the need for the adjustments used in practice by the rail engineers to achieve a higher degree of smoothness. In order to properly define the direction of the centrifugal force, a distinction is made between the super-elevation of a surface and the bank angle of a curve on the surface.