In the multibody formulation of the contact problem, the kinematic contact constraint conditions are formulated in terms of the normal and tangents to the contact surfaces. Using the assumption of nonconformal contact, the second time derivatives of the contact constraints, which are required in the augmented Lagrangian formulation of the multibody equations, contain third derivatives of the position vectors of the contact points with respect to the surface parameters that describe the geometry of the contact surfaces. These derivatives must be accurately calculated in order to develop a robust numerical algorithm for solving the multibody differential and algebraic equations of the contact problem. An important application for the procedure developed in this paper is the wheel/rail interaction. In order to allow a general description for the wheel and rail profiles, the spline function representation is used. A multi-layer spline function algorithm is used in order to ensure accurate calculation of the third derivatives with respect to the surface parameters when a small number of nodal points is used. The problems of continuity of the derivatives and smoothness of these functions are addressed. The proposed method allows using wheel and rail profiles obtained from direct measurements. Numerical results show that this multibody approach, whic leads to accurate value of the normal force at the contact, can capture the coupling between the vertical and the lateral motion of the wheelset.

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