The stability and guidance of a railway vehicle is provided in part by the contact geometry of the wheel-rail contact, and any changes to this geometry will effect the response of the railway vehicle. Typically the condition and shape of the wheel and rail are monitored separately and off-line, but it is the combination of the two in situ that really affects the dynamics. This paper outlines a practical approach to estimating a nonlinear function within a dynamic system by using a piecewise cubic functions. The parameters for the cubic functions are estimated with a least squared approach applied to the dynamic measurements taken from the system. A simplified plan-view wheelset and suspended mass model is introduced to use as an application of this technique. The nonlinear aspect of the contact is included in the form of a conicity function which varies with the relative lateral wheel-rail position. The conicity is successfully estimated using the least-squares method outlined in the paper. (6 pages)