An analytical solution of the stiffness equation for linear guideway type recirculating rollers with arbitrarily crowned profiles

This article derives the stiffness equation for linear guideway type (LGT) recirculating rollers with arbitrarily crowned profiles. Recirculating rollers compressed between a carriage and a profile rail are simulated as rollers compressed between two plates, and each roller is divided into three parts: two crowned and one cylindrical. The superposition method is introduced to obtain the stiffness equation for a crowned roller, for which rollers with circular, quadratic, cubic, fourth-order power, and exponential profiles were analysed. A model of discrete normal springs is used to obtain the stiffness equation for LGT recirculating rollers. The results reveal that a higher-order power can make an LGT with higher stiffness. The stiffness of the exponential profile is close to that of the fourth power profile. Therefore, recirculating rollers with an exponential profile, a small crowned depth, and a large crowned length seem to be optimal, obtaining a higher LGT stiffness and a lower edge stress concentration.