Exact Formulation for the Curvature of Gothic Arch Ball Screw Profiles and New Approximated Solution Based on Simplified Groove Geometry

The correct evaluation of the curvatures of ball screw grooves allows the accurate design of the constructive parameters of this mechanism and enhancing its performance. The formulation commonly used in the literature, however, refers to ball bearing geometry, ignoring the shape of the section’s profile and the helix angle. In this paper, the exact formulae for calculating the principal curvature radii of the screw shaft and the nut grooves are analytically derived and presented. These equations, obtained through a rigorous differential geometry approach, consider the helix angle and a gothic arch profile. An approximated formulation is proposed simplifying the exact solution under the assumption of a circular groove profile. These new simple formulae accurately reproduce the exact curvature radii values with a mean relative error of approximatively 0.51% and 0.40%, respectively, for the screw shaft and nut grooves, against the value of more than 50% obtained by using the literature formulae for common off-the-shelf ball screws, especially those with high helix angles. Furthermore, they allow a computational time saving of 98%, making them suitable for being incorporated in high-fidelity dynamic models of ball. Finally, two MATLAB functions are provided to easily evaluate the complex curvature exact solution.