Curve resistance (railroad): Difference between revisions

▲==Erroneous formulas==

▲Prior to the [[#Russian experiments|Russian experiments]] in the 1960s, formulas for curve resistance neglected the fact that it's highly dependent on both speed and [[Cant (road/rail)|superelevation]] and used formulas that in general claimed it was only inversely proportion to the radius of curvature.<ref>Астахов p.113</ref> For example, in the [[USSR]] they claimed 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]] they still use the term ''R - b'' 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.

▲Today (2012) in both the [[US]] and [[Russia]] these erroneous formulas are presented on the Internet as though they were still correct. For the US, [[American Railway Engineering and Maintenance-of-Way Association|AREMA]] [http://www.arema.org/eseries/scriptcontent/custom/e_arema/Practical_Guide/PGChapter2.pdf American Railway Engineering ..., PDF, p.57] claims that curve resistance is 0.04% per [[degree of curvature]] (or 8&nbsp;[[lbf]]/[[ton]] or 4&nbsp;[[kgf]]/[[tonne]]). Hay's textbook also claims it is independent of superelevation.<ref>Hay, 1982. On p. 142: "experiments have shown no appreciable change in resistance with changes in superelevation" but he cites no reference.</ref> For Russia in 2011, internet articles claim it's still the old 700/R.<ref>See [http://build-master.blogspot.com/2011/02/blog-post_7955.html 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).</ref><ref>See [http://www.rusnauka.com/8_DNI_2009/Tecnic/43842.doc.htm 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 Sciences|"candidate of technical sciences"]]) of [[:ru:Днепропетровский национальный университет железнодорожного транспорта|{{ru icon}}Dnepropetrovsk national technical university of railroad transportation named ...]]</ref><ref>Even the [[:ru:Тяговые расчёты#Сопротивление движению|{{ru icon}} Russian wikipedia]] has it wrong.</ref>

▲===Formulas which try to account for superelevation (cant)===

▲Астахов proposed the use of a formula which when plotted<ref>Астахов p.119, Fig. 5.6</ref> is in substantial disagreement with the experimental results curves previously [[#Speed and cant dependence per Russian experiments|mentioned]]. His formula for curve resistance (in [[kgf]]/[[tonne]]) is the sum of two terms, the first term being a conventional k/R term (R is the curve radius in [[meters]]) with k=200 instead of 700. The second term is directly proportional to (1.5 times) the absolute value of the unbalanced acceleration in the plane of the track and perpendicular to the rail, such lateral acceleration being equal to the [[centrifugal force|centrifugal acceleration]] <math>\frac{v^2}{R} cos(\theta)</math>, minus the gravitation component opposing this acceleration: g·tan(θ), where θ is the angle of the banking due to superelevation and v is the train velocity in m/s.<ref>This formula is found in Астахов, bottom of p. 118. Since theta is a small angle, he assumes that cos theta is equal to unity. He approximates "tan theta" by h/S where h is the height of the superelevation of the outside rail and S is the distance between the centers of the pair of rails (something like the rail gauge, but slightly wider). His plot using this formula does show a minimum at the balancing speed (as it should) but the plotted curves of curve resistance suddenly change [[derivative|slope]] here from negative to positive, in contrast to the experimental curves which are smooth and nearly flat at this point. As velocity increases beyond the balancing speed ([[cant deficiency]]), his curves rise much faster than they do for velocities less than the balancing speed, while the experimental curves do just the opposite. Thus one must conclude that his proposed formula is quite erroneous.</ref>