# Investigation on Sensorless Estimating Method and Characteristics of Friction for Ball Screw System
## 4 Friction Characteristics Analysis
### Velocity-Dependent Friction Characteristics
#### 4.2.1 Effect of the Feed Velocity on the IAFVs
Figure 12: _Cont._  
Friction is usually expressed as a nonlinear function of velocity, such as in the Coulomb friction model, the Coulomb model with Stribeck friction, the LuGre and Leuven model, and the generalized Maxwell-slip friction model [9]. The most commonly used engineering models employ the static friction, Coulomb friction and viscous friction models. The relationship between the friction and the velocity for steady-state motion can be expressed as follows [8; 21]:  
\[F_{ss}(v)=F_{c}\text{sgn}(v)+(F_{s}-F_{c})^{-(v/v_{s})^{2}}\text{sgn}(v)+ \sigma_{2}v \tag{9}\]  
The MVOIAFs corresponding to the specific positions in Table 3 for the 18 velocities can be calculated by using the PPOCM to obtain six groups of data for the forward and backward feeds. The optimization function, fmin, which is provided by the MATLAB optimization toolbox [22], is used to identify the model parameters in Equation (9) to generate the friction characteristic curve (FCC). Figure 12: Variations in the MVOIAF with WP for the different feed velocities. (**a**) Position direction; (**b**) Negative direction. Figure 13: Variations in the MVOIAF with WP and the feed velocity. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Symbol** & **a-a** & **b-b** & **c-c** & **d-d** & **e-e** & **f-f** \\ \hline Positive (mm) & 20 & 60 & 125 & 135 & 200 & 300 \\ Negative (mm) & 20 & 40 & 110 & 125 & 220 & 300 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Position corresponding to the symbols a-a to f-f in Figure 12.  
Figure 14: Comparison of the experimental data with three model curves for the forward feed. **(a)** Position of 20 mm; **(b)** Position of 60 mm; **(c)** Position of 125 mm; **(d)** Position of 135 mm; **(e)** Position of 200 mm; **(f)** Position of 300 mm. Figure 15: _Cont._  
Comparing the spline interpolation curves with the position section curves in Figures 14 and 15 clearly shows that the DPFMOPV can accurately predict the variations in the MVOIAFs with the WP and the feed velocity. Comparing Figures 11 and 12 with Figures 14 and 15 reveals that the nonuniform pitch errors of the shaft have an impact on the repeatability of the IAFVs for the three  
Figure 15: Comparison of the experimental data with three model curves for the backward feed. (**a**) Position of 20 mm; (**b**) Position of 40 mm; (**c**) Position of 110 mm; (**d**) Position of 125 mm; (**e**) Position of 220 mm; (**f**) Position of 300 mm. experiments. There are especially large differences during the decline following the peak of the IAF curves resulting from the pitch errors that can be observed between the three experiments in Figure 14b,d and Figure 15c,d. The IAFVs exhibit good repeatability for Interval 3 and point b-b over the range of the experimental backward feed velocity, as shown in Figure 14b,e,f. For the other intervals, the repeatability of the IAFVs is better for feed velocities above 400 mm/min. The shapes of the friction characteristic curve vary with the WP and the feed direction, and the position sections fluctuate around the corresponding friction characteristic curves. For the interval between 60 and 125 mm, the friction characteristic curves decrease as the feed velocity increases, exhibiting nonconventional Stribeck-curve behavior, as shown in Figure 14b,c and Figure 15c,d. The first two intersection points of the friction characteristic curve with the position section at the end of low feed velocity are denoted by A and B. The velocities corresponding to points A and B are denoted by \(v_{A}\) and \(v_{B}\) and also vary with the WP. When the feed velocity is less than \(v_{A}\), the values of the friction characteristic curve are greater than the values of the position section, and for \(v_{A}<v<v_{B}\), the values of friction characteristic curve are less than the position section values.