Track transition curve
From Wikipedia, the free encyclopedia
A Track transition curve, or spiral easement, is a mathematically calculated curve on a section of highway, or railroad track, where a straight section changes into a curve. It is designed to reduce the effects of centrifugal force experienced by users. In plan (i.e., the horizontal curve) the start of the transition is at infinite radius and at the end of the transition it has the same radius as the curve itself, thus forming a very broad spiral. At the same time, in the vertical plane, the outside of the curve is gradually raised until the correct degree of bank is reached.
If such easement were not applied the centripetal force needed to change the direction of a rail vehicle would be applied instantly at one point: the tangent point where the straight track meets the curve, with undesirable results. With a road vehicle the driver naturally applies the steering alteration in a gradual manner and the curve is designed to permit this, using these principles.
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[edit] History
On early railroads, because of the low speeds and wide-radius curves employed, the surveyors were able to ignore any form of easement but in 1835 Charles Vignoles published an exact, although complex, equation for the vertical component. In 1837 the vertical and horizontal components were combined in William Froude’s curve of adjustment (a cubic parabola), based on the theoretical calculations of William Gravatt. In the UK, only from 1845 when legislation and land costs began to constrain the laying out of rail routes and tighter curves were necessary, did the principles start to be applied in practice.
[edit] Geometry
While railroad track geometry is intrinsically three-dimensional, for practical purposes the vertical and horizontal components of track geometry are usually treated separately.
The overall design pattern for the 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 elevation therefore varies quadratically with distance. Here grade refers to the tangent of the angle of rise of the track. The design pattern for horizontal geometry is typically a sequence of straight line (i.e., a tangent) and curve (i.e. a 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 (cant or camber) 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 sine 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 segment with constant curvature 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.
[edit] References
- Simmons, Jack; and Biddle, Gordon (1997). The Oxford Companion to British Railway History. Oxford University Press. ISBN 0-19-211697-5.
- Biddle, Gordon (1990). The Railway Surveyors. Chertsey, UK: Ian Allen. ISBN 0-7110-1954-1.
- Hickerson, Thomas Felix (1967). Route Location and Design. New York: McGraw Hill. ISBN 0-07-028680-9.
- Cole, George M; and Harbin, Andrew L (2006). Surveyor Reference Manual. Belmont, CA: Professional Publications Inc, page 16. ISBN 1-59126-044-2.
- Railway Track Design pdf from The American Railway Engineering and Maintenance of Way Association, accessed 4 December 2006.
