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