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Curve resistance (railroad)
In railroad engineering, curve resistance is the additional rolling resistance a train must overcome when travelling on a curved section of track.〔Hay p.142〕 Curve resistance is typically measured in per mille, with the correct physical unit being Newton per kilo-Newton or N/kN. Older texts still use the wrong unit of kilogram-force per tonne or kgf/t, which mixes an (outdated) unit of force and a unit of mass. Sometimes also kg/t was used, which confused the resisting force with a mass.
Curve resistance depends on various factors, the most important being the radius and the superelevation of a curve. Since curves are usually banked by superelevation, there will exist some speed at which there will be no sideways force on the train and where therefore curve resistance is minimum. At higher or lower speeds, curve resistance may be a few (or several) times greater.
==Approximation formulas==
Formulas typically used in railway engineering in general compute the resistance as inversely proportional to the radius of curvature (thus, they neglect the fact that the resistance is dependent on both speed and superelevation). For example, in the USSR, the standard formula is Wr (curve resistance in parts per thousand or kgf/tonne) = 700/''R'' where ''R'' is the radius of the curve in meters. Other countries often use the same formula, but with a different numerator-constant. For example, the US used 446/''R'', Italy 800/''R'', England 600/''R'', China 573/''R'', etc. In Germany, Austria, Switzerland, Czechoslovakia, Hungary, and Romania the term ''R - b'' is used in the denominator (instead of just ''R''), where ''b'' is some constant. Typically, the expressions used are "Röckl's formula", which uses 650/(''R'' - 55) for ''R'' above 300 meters, and 500/(''R'' - 30) for smaller radii. The fact that, at 300 meters, the two values of Röckl's formula differ by more than 30% shows that these formulas are rough estimates at best.
The Russian experiments cited below show that all these formulas are inaccurate. At balancing speed, they give a curve resistance a few times too high (or worse).〔Астахов p.115 Fig. 5.2; p.229, Fig. 5.6〕 However, these approximation formulas are still contained in practically all standard railway engineering textbooks. For the US, AREMA (American Railway Engineering ..., PDF, p.57 ) claims that curve resistance is 0.04% per degree of curvature (or 8 lbf/ton or 4 kgf/tonne). Hay's textbook also claims it is independent of superelevation.〔Hay, 1982. On p. 142: "experiments have shown no appreciable change in resistance with changes in superelevation" but he cites no reference.〕 For Russia in 2011, internet articles use 700/R.〔See (blog ) where it's erroneously claimed that the "удельного дополнительного сопротивления от радиуса кривой" (specific additional resistance due to the curve radius): wr = 700/Д. (where Д is the radius).〕〔See (ОПРЕДЕЛЕНИЕ СОПРОТИВЛЕНИЯ В КРИВОЙ ОТ ТРЕНИЯ ГРЕБНЯ КОЛЕСНОЙ ПАРЫ ) (Finding the resistance in a curve due to flange friction of the wheel pair)by к.т.н. Довбня Н. П., к.т.н. Бондаренко Л. Н., Кислый Д. Н. (к.т.н. stands for "candidate of technical sciences") of Dnepropetrovsk national technical university of railroad transportation named ...〕〔Also the Russian wikipedia uses the old approximation formulas.〕 German textbooks contain Röckl's formulas.〔See e.g. "Bahnbau" by V.Matthews, Teubner, 2007〕
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