# Investigation on Sensorless Estimating Method and Characteristics of Friction for Ball Screw System
## 4 Friction Characteristics Analysis
### Discussion  
The variation in the IAFV with the WP at a fixed feed velocity depends on the torsional rigidity of the TTSOS, the pitch errors, and the incorrect alignment of the linear guides and the ball screw. The higher the torsional rigidity of the TTSOS is, the higher the IAFV. Pitch errors and incorrect contact cause a local increase in the IAFV and affect the repeatability of the IAFVs for the different feed tests. The local increase in the IAFVs decreases as the feed velocity increases. The larger the nonuniform pitch errors of the shaft are, the larger the differences between the IAFVs for the different tests at low feed velocities. The variation in the MVOLAFs with the feed velocity at a fixed position exhibits nonconventional Stribeck-curve behavior.  
The SFFTCs of the FFS consist of four groups. The fundamental frequency component is excited by the rolling tool marks, while the fundamental frequency and multiple components result from the pitch of the shaft and the refeeding of the balls within the screw-nut and the linear ball rail guides. At  
Figure 16: Effects of the feed velocity and direction on the FFT spectrum for FFS. (**a**) The component amplitudes of the FFS for the nine segments at the forward feed velocity of 560 mm/min; (**b**) The frequency spectrum over the WTDOW at the feed velocity of 560 mm/min; (**c**) The frequency spectrum over the WTDOW at the feed velocity of 400 mm/min; (**d**) The frequency spectrum over the WTDOW at the feed velocity of 280 mm/min; (**e**) The frequency spectrum over the WTDOW at the feed velocity of 200 mm/min; (**f**) The frequency spectrum over the WTDOW at the feed velocity of 80 mm/min.  
Figure 17: Amplitudes of FFT components for different segments.  
a low feed velocity, the high torsional rigidity of the TTSOS and local pitch errors cause the frequency spectrum of the FFS to become more continuous, whereas increasing the feed velocity results in discontinuities in the FFS frequency spectrum. The FOSFFTCs that are excited by the pitch of the shaft and the rolling tool marks do not vary with the feed velocity, and their amplitudes increase with the local pitch errors. The effect of local pitch errors on the amplitudes of the corresponding components decreases as the feed velocity increases. The amplitude of the fundamental frequency component resulting from the pitch of the shaft over the WTDOW for the backward feed is much greater than that for the forward feed, which is affected very little by the feed velocity. However, this difference decreases as the feed velocity increases.  
The balls within the working circulation of the screw-nut undergo pure rolling for feed velocities less than 120 mm/min, rolling and intermittent sliding for feed velocities ranging from 120 to 440 mm/min, and rolling and continuous sliding for feed velocities above 440 mm/min. In pure rolling, the FOSFFTCs that are excited by the refeeding of the balls remain constant, and the amplitudes of their components are not affected by the feed velocity and direction or the WP. The direction of sliding motion is opposite to that of the ball's center motion resulting from rolling, which produces a decrease in the fundamental frequency of the RFOBSMs. The feed direction has almost no effect on the amplitudes of the RFOBSM, which increase with the sliding velocity.