A Study of the Effect of m and n Coefficients of the Hertzian Contact Theory on Railroad Vehicle Dynamics

For the wheel/rail contact problem, the Hertz theory for two elastic bodies in contact is commonly used to determine the shape and dimensions of the contact area and the local deformation of the wheel and rail surfaces at the contact region. The shape of the contact area is assumed to be elliptical. The ratio of the contact ellipse semi-axes is equal to the ratio of two non-dimensional contact area coefficients, known as m and n coefficients. Hertz presented a table of these two coefficients, determined as a function of an angular parameter, θ. Most railroad vehicle dynamic codes use this table with online interpolation to determine the contact ellipse semi-axes. Recently, it was found that this original table may be too coarse, and that more data points are needed within the table for solving the wheel/rail contact accurately. This paper discusses the effect of the accuracy of the m and n coefficients in solving for wheel/rail contact, and demonstrates this effect with two numerical examples that show the resulting differences in the dynamic behavior of railroad vehicles dependent on this accuracy. A new table with more data points is presented that is recommended for use in railroad vehicle dynamic codes that employ the Hertzian contact for solving the wheel/rail contact interaction. This modified table was originally derived by Jean-Pierre Pascal as a part of collaborative research between the Federal Railroad Administration (FRA) and the French Ministry of Transportation.