Fast and Stable Iterative Algorithm for Searching Wheel–Rail Contact Point Based on Geometry Constraint Equations

This paper proposes a fast and stable iterative algorithm for wheel–rail contact geometry based on constraint equations, which can be implemented in dynamic wear simulations that real-time profile updating is needed. Further, critical factors that determine convergence and iteration stability are analyzed. A B-spline is adopted for wheel–rail profile modeling because it does not contribute to changes in the global shape of curves. It is found that the smoothness of the first and second derivative curves significantly affects the numerical stability of the Jacobian matrix, which determines the increments in iterations. Moreover, a damped Newton's iteration formula with a scaling factor of 0.5 is proposed considering the convergence rate and out-of-bound issues for the updated step. The influence of the initial iteration parameters on the convergence is studied using Newton fractals. The range within ±3 mm, centered on the target contact point, is found to be an unconditionally stable domain. The proposed method could achieve convergence within 10 and 30 steps under thread and flange contact conditions, respectively.