Track transition curve

While railroad track geometry is intrinsically three dimensional, for practical purposes the vertical and horizontal components of track geometry are treated separately.

The design pattern for vertical geometry is typically a sequence of constant grade segments connected by vertical transition curves in which the local grade varies linearly with distance and in which the elelvation therefore varies quadratically with distance. Here grade refers to the tangent of the angle of rise of the track. A standard reference is Thomas F. Hickerson, "Route Location and Design", McGraw-Hill, 1926, 1953.

The design pattern for horizontal geometry is typically a sequence of straight line (aka tangent) and curve (i.e. circular arc) segments connected by transition curves. In a tangent segment the track bed roll angle is typically zero. In the case of railroad track the track roll angle is typically expressed as the difference in elevation of the two rails, a quantity referred to as the superelevation. A track segment with constant non-zero curvature will typically be superelevated in order to have the component of gravity in the plane of the track provide a majority of the centripetal acceleration inherent in the motion of a vehicle along the curved path so that only a small part of that acceleration needs to be accomplished by lateral force applied to vehicles and passengers or lading.

The change of superelevation from zero in a tangent segment to the value selected for the body of a following curve occurs over the length of a transition curve that connects the tangent and the curve proper. Over the length of the transition the curvature of the track will also vary from zero at the end abutting the tangent segment to the value of curvature of the curve body, which is numerically equal to one over the radius of the curve body.

The simplest and most commonly used form of transition curve is that in which the superelevation and horizontal curvature both vary linearly with distance along the track. Cartesian coordinates of points along this spiral are given by the Fresnel sin and cosine integrals. The resulting shape matches a portion of a Cornu spiral and is also referred to as a clothoid. However, as it causes the horizontal (centripetal) acceleration to ramp up linearly from zero to the value associated with the circular motion in the body of the curve, in a transportation context it may best be referred to as the linear spiral.

A transition curve can connect a track segment of constant non-zero curvature to another segement with constant curature that is zero or non-zero of either sign. Successive curves in the same direction are sometimes called progressive curves and successive curves in opposite directions are called reverse curves.

The linear spiral has two advantages. One is that it is easy for surveyors because the coordinates can be looked up in Fresnel integral tables. The other is that it provides the shortest transition subject to a given limit on the rate of change of the track superelevation (i.e. the twist of the track). However, as has been recognized for a long time, it has undesirable dynamic characteristics due to the large (conceptually infinite) roll acceleration and rate of change of centripetal acceleration at each end. Because of the capabilities of personal computers it is now practical to employ spirals that have dynamics better than those of the linear spiral. Detailed discussion may be found in http://www.arema.org/eseries/scriptcontent/custom/e_arema/comm/c17/research.htm