What is the cause of the motor shaft  problem causing the bearing to overheat?

First, the shaft problem of the motor: the most common motor, the deep groove ball bearing and the cylindrical roller bearing. When installed, one end is axially positioned and the other end is axially floated. The incitement you said, first of all I think you should check, how is your axial positioning done? Is the positioning reliable? If reliable, for deep groove ball bearings, its axial yaw should be its axial clearance. Generally not too big, but it depends on the radial clearance you choose. For cylindrical rod bearings, for the N and NU series, it cannot be used as a locating bearing. If you use them for positioning, it must be too large.

Second: You said that the shaft is moving, I think, if the positioning bearing is subjected to excessive axial load, it will cause the bearing to burn. Therefore, when choosing a locating bearing, look at how much the axial load is. Whether the bearing you choose can withstand it. In the case of the NJ series of cylindrical rod bearings, this axial load is completely absorbed by the sliding part, so it is not acceptable. For deep groove ball bearings, its axial capacity is up to a quarter of the radial, which varies from bearing to bearing.