Derivation of the Exact Curvature Formulation for Gothic Arch Ball Screw Grooves

No exact formulations are present in literature to calculate the principal curvatures of the ball screw grooves, while approximate equations are commonly adopted due to their simplicity. However, they derive from the ball bearing theory and do not take into account ball screw’s characteristic main geometric parameters, such as the helix angle. This leads to approximation relative errors up to 50% for commercial off-the-shelf ball screws. In this paper a new exact explicit exact solution is proposed by means of a rigorous mathematical approach. The helix angle and a generic gothic arch shape of the raceway profile are taken into account. The results of the exact and literature formulations are presented and compared. The principal curvature is seen to be strongly dependent on the helix angle and, therefore, the proposed exact solution should be used in the design phase and, in general, when high accuracy is required, such as in high-fidelity models for prognostic activities.