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Railroad track spirit level in place indicating 5 in (130 mm) of superelevation between the inside and outside rails of a curve along the Keystone Corridor near Narberth, Pennsylvania In curved track, it is usually designed to raise the outer rail, providing a banked turn , thus allowing trains to maneuver through the curve at higher speeds ...
The relationship between speed and tilt can be calculated mathematically. We start with the formula for a balancing centripetal force: θ is the angle by which the train is tilted due to the cant, r is the curve radius in meters, v is the speed in meters per second, and g is the standard gravity, approximately equal to 9.81 m/s²:
l = slope length α = angle of inclination. The grade (US) or gradient (UK) (also called stepth, slope, incline, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the slope, where zero indicates horizontality. A larger number ...
To compensate for this, the gradient should be a little less steep the sharper the curve is; the necessary grade reduction is assumed to be given by a simple formula such as 0.04 per cent per "degree of curve", the latter being a measure of curve sharpness used in the United States. On a 10-degree curve (radius 573.7 feet) the grade would thus ...
Ideally, the track should have sleepers (railroad ties) at a closer spacing and a greater depth of ballast to accommodate the increased forces exerted in the curve. At the ends of a curve, where the rails straighten out, the amount of cant cannot change from zero to its maximum immediately. It must change gradually in a track transition curve ...
The actual equation given in Rankine is that of a cubic curve, which is a polynomial curve of degree 3, at the time also known as a cubic parabola. 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, were the principles beginning to be applied in practice.
Sine, cosine, and versine of θ in terms of a unit circle, centered at O. The Hallade method, devised by Frenchman Emile Hallade, is a method used in track geometry for surveying, designing and setting out curves in railway track.
The third independent discovery occurred in the 1800's when various railway engineers sought a formula for gradual curvature in track shape. By 1880 Arthur Newell Talbot worked out the integral formulas and their solution, which he called the "railway transition spiral". The connection to Euler's work was not made until 1922. [2]