# Investigation on Sensorless Estimating Method and Characteristics of Friction for Ball Screw System
## 4 Friction Characteristics Analysis
### Position-Dependent Friction Characteristics
#### 4.1.2 Spectra of Friction Fluctuating Signals  
Subtracting the IAFVs plotted as solid lines of the estimated friction yields the friction fluctuating signal (FFS). To investigate the variation in the spectral characteristics of the friction with the WP, Intervals 1 and 2 are each divided into two identical segments, and Interval 3 is divided into five segments. Fast Fourier transform (FFT) is applied to the spectra to analyze the frequency characteristics of the FFS over the nine intervals. The FFT results at a feed velocity of 5 mm/min for the forward and backward feeds are shown in Figures 9 and 10, respectively. As shown in Figures 9b and 10b, the significant FFT components (SFFTCs) can be classified into four groups. Group 1 has only one component, 0.56; Group 2 consists of the components 1, 2, 3 and 4; Group 3 is composed of the components 8, 16, and 24; and Group 4 is composed of the components 9.5, 19 and 28.5. The amplitude of component 0.56 is attributed to the rolling tool. This component was reported in the experimental results of ball screw preloading [12]. The pitch errors of the ball screw are a significant performance index for screw manufacturing and wear in use. Variations in the leads of the screw cause nonuniform compression, resulting in an impact between the balls and races that can lead to single and multiple friction oscillations of the shaft rotating frequency, such as those observed for the amplitudes of the component in Group 2 and reported in friction studies [10] and measurements of ball and screw preloading [12]. During the motion of the feed drive servomechanism, the balls refeeding within the screw-nut and the linear ball rail guides cause an impact that leads to high frequency signals during ball and screw preloading [12; 18; 19]. The balls asynchronously enter and leave working circulation and slide within their recycling paths. In this way, the refeeding of the balls excites a component of the fundamental frequency as well as the components of double and triple frequencies. The fundamental frequency of the ball refeeding within the screw-nut per revolution of the shaft is expressed as  
\[f_{tr\_s}=\frac{1}{2}z(d-d_{b}\cos\beta)/d \tag{7}\]  
where \(d\) denotes the diameter of the lead screw; \(d_{b}\) denotes the ball diameter; \(\beta\) denotes the contact angle between the raceway and the ball; and \(z\) denotes the number of balls that form a circle in a raceway. For this servomechanism, \(f_{\text{re\_s}}\approx 8\).  
The following values are used for the linear guide parameters in Equation (7). The diameter of the raceway of the guideway \(d\) goes to infinity; the ball diameter \(d_{b}\) is 6 mm; the contact angle \(\beta\) is 0; and the number of balls in the path of the raceway, \(z\), is 19. Thus, \(f_{\text{re\_p}}=9.5\). The amplitudes of the components for Groups 3 and 4 correspond to the friction oscillation that is excited by the refeeding of the balls within the screw-nut and the linear ball rail guides, respectively.  
As shown in Figures 9a-c and 10a-c, as the worktable approaches the servomotor, the amplitude spectrum of the FFS becomes more continuous, and the spectral components become more complex. These results show that the balls move more randomly within the working circulation due to the pitch errors of the shaft. As shown in Figures 9d and 10d, the WP and the feed direction have very small effects on the amplitudes of components 8 and 19; they significantly affect the amplitudes of components 0.56, 1 and 2 over Intervals 1 and 2, and have small effects over Interval 3. Thus, the incorrect alignment of the linear guides and the ball screw and pitch errors can excite fluctuations in friction due to the rolling tool and the shaft lead.  
Figure 9: Variations in the friction frequency spectrum with WP at a forward feed of 5 mm/min. (**a**) FFS of all segments; (**b**) FFS of segment No. 9; (**c**) FFS of segment No. 1; (**d**) Amplitude of all segments for significant frequency components.