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The analytical solution of vertical, pitching, yawing, lower rolling, and higher rolling frequency expressions for linear guideway type (LGT) recirculating rollers with arbitrarily crowned profiles was explored. Each roller inside LGT recirculating roller is divided into three parts: two crowned and one cylindrical. The superposition method is introduced to obtain the stiffness equation for rollers compressed between the carriage and profile rail, and a model of discrete normal springs is used to construct the stiffness equation for LGT recirculating roller. Five equations of motion were obtained using Lagrange's equation. The results reveal that the frequencies are affected by the value of preload and of stiffness. The natural frequency of the exponential profile is close to that of the forth power profile. Therefore, recirculating rollers with an exponential profile, a small crowned depth, and a large crowned length seem to be optimal, having a higher frequency of LGT recirculating roller and a lower edge stress concentration.

Analytical Solution of the Frequency Expressions for Rigid-Body Natural Vibration of a Carriage on Linear Guideway Type Recirculating Rollers