# Recurrence Quantification Analysis (RQA) of Toroidal End Tool Milling Process
## 3 Results and Analysis
### Relations Between RQA and Cutting Parameters
These indicators for the F\({}_{\text{a}}\) force component can be expressed by the following formulas:  
\[\text{DET}_{\text{Fa}}=0.6466+0.1153\;a_{p}, \tag{22}\]  
\[\text{ENT}_{\text{Fa}}=-0.626+4.03\;a_{p}-1.084\;{a_{p}}^{2}, \tag{23}\]  
\[\text{VMAX}_{\text{Fa}}=-34.8+459\;a_{p}-134.0\;{a_{p}}^{2}, \tag{24}\]  
\[\text{RR}_{\text{Fa}}=-0.0189+0.3142\;a_{p}-0.0917\;{a_{p}}^{2}, \tag{25}\]  
\[\text{LMAX}_{\text{Fa}}=18.2+194.2\;a_{p}, \tag{26}\]  
Figure 14: Experimental models of RQA indicators of the Fa cutting force component: (**a**) DET; (**b**) ENT; (**c**) VMAX; (**d**) RR; (**e**) LMAX. This distinction arises from the fact that \(\mathrm{F_{f}}\) and \(\mathrm{F_{IN}}\) act within the cutting plane, whereas \(\mathrm{F_{a}}\) is perpendicular to them, aligned with the tool axis. The axial cutting force component \(\mathrm{F_{a}}\) is a passive force that is primarily responsible for pulling the tool away from the workpiece in the axial direction. Consequently, the analysis of the \(\mathrm{F_{a}}\) component leads to conclusions that are different from those derived for the \(\mathrm{F_{f}}\) and \(\mathrm{F_{IN}}\) components. In particular, the RQA measure models for the \(\mathrm{F_{a}}\) force differ significantly from those for the \(\mathrm{F_{f}}\) and \(\mathrm{F_{IN}}\) forces, with all \(\mathrm{F_{a}}\) models depending solely on the parameter \(a_{p}\). The feed per tooth \(f_{z}\) does not affect any of the RQA indicators determined in the axial direction. Therefore, it can be inferred that each RQA indicator identified for the \(\mathrm{F_{a}}\) force component is useful for monitoring changes only in the depth of cut \(a_{p}\). Notably, the \(\mathrm{DET_{Fa}}\) and \(\mathrm{LMAX_{Fa}}\) indicators exhibit linear increases with an increasing depth of cut, indicating sensitivity exclusively to changes in the cross-sectional area of the cutting layer due to increases in \(a_{p}\). In contrast, the \(\mathrm{ENT_{Fa}}\), \(\mathrm{VMAX_{Fa}}\), and \(\mathrm{RR_{Fa}}\) indicators vary nonmonotonically, reaching a maximum at the midpoint of the \(a_{p}\) range tested. This suggests that these measures are responsive not only to changes in the cross-sectional area but also to variations in milling dynamics associated with changes in the cutter entering angle, to which the axial component of the cutting force \(\mathrm{F_{a}}\) is particularly sensitive [1]. None of the RQA indicators for the \(\mathrm{F_{a}}\) component are effective in monitoring changes in feed per tooth. Variations in feed rate \(f_{z}\) can only be tracked through certain RQA measures for the \(F_{f}\) and \(F_{IN}\) components. Statistically, all models for the \(\mathrm{F_{a}}\) component are significant, with elevated \(p\)-values observed only for the \(a_{p}\) factor in the VMAX model, as seen with the \(F_{f}\) and \(F_{fN}\) components. This suggests that VMAX has a lower likelihood of producing repeatable results, implying that the use of this RQA measure to assess the dynamics of the toroidal milling process may introduce some degree of error. Additionally, it can be observed that all RQA indicator models for the \(\mathrm{F_{a}}\) cutting force exhibit a relatively high fitting error. This is likely due to the fact that they are related to only one parameter, the cutting depth \(a_{p}\). The lack of a functional relationship with the feed per tooth \(f_{z}\) results in an increased fitting error of the models in the process. Nonetheless, in every case, the \(p\)-value of the obtained models is below 0.05, indicating the statistical significance of the equations (Figure 14). The influence of the dynamics of the toroidal milling process on RQA parameters requires further research, taking into account other variables such as the material, tool wear, tool overhang, and cutting speed. Additionally, it can be observed that some RQA indicator models exhibit non-monotonic behavior. In every case, the monotonicity of the function was dependent on the parameter \(a_{p}\).