A simple solution for rolling of a cylinder with rate-dependent friction

It is well known that friction shows rate-dependent properties, but including this effect into the contact problem of a wheel under tractive rolling, i.e. the generalization of Carter’s solution, has so far been difficult. Indeed, early numerical solutions have found various instabilities or zig-zag in the predicted shear tractions, and more refined friction laws have been suggested to avoid these complications. With a “Winkler” model, which is used in the FASTSIM type of algorithms of Kalker, the relative velocity in the slip area develops a non-physical jump, which is reflected also in friction coefficient and shear tractions. This is due to the simplification of the elastic deformation with a purely “local” model. However, given the Winkler approximation is a strong one, it is suggested to avoid the full solution, and we develop instead an approximation which assumes a constant friction coefficient in the slip area, which leads to a nonlinear algebraic equation to find this average friction coefficient. This permits a quite simple solution which shows all the expected trends in terms of creepage vs tangential force ratio, and hence can be useful for estimating the effect of rate-dependency of friction. Comparison with experimental data is very satisfactory.