Four-point contact between ball and raceways is common in machine elements like ball bearings and ball screws. The ideal four-point-contact machine element is designed with pure rolling (i.e., no sliding at contact points) to minimize friction. However, this ideal may not always be achieved, leading to sliding and higher frictional forces. In this paper, a simplified analytical model for rolling/sliding behavior and friction in four-point contact is developed, based on Coulomb friction model and rigid body assumption. It is found that pure rolling is only possible when the contact-point geometry satisfies a certain relationship. When pure rolling condition fails to hold, the sliding contact point(s) can be determined analytically as a function of contact forces and contact angles. Case studies are presented to demonstrate how the proposed model could elucidate the roles of misalignments, manufacturing errors and loading conditions on rolling/sliding behavior and friction.

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