# Recurrence Quantification Analysis (RQA) of Toroidal End Tool Milling Process
## 3 Results and Analysis
### Relations Between RQA and Cutting Parameters

Table 8: Fitting level of experimental models and their factors. \[\text{ENT}_{\text{FNN}}=2.728-0.1302\ a_{p}, \tag{20}\]  
\[\text{VMAX}_{\text{FNN}}=136.5+860\,f_{z}+153.3\ a_{p}-44.5\ a_{p}{}^{2}. \tag{21}\]  
The most significant difference is found in the model for the ENT indicator. For the F\({}_{\text{f}}\) component, the ENT model depends on both milling parameters, \(f_{z}\) and \(a_{p}\); for the F\({}_{\text{FN}}\) component, however, the ENT\({}_{\text{FNN}}\) model is a function of only the depth of cut \(a_{p}\). This difference between the two ENT models is evident when analyzing the graphs in Figures 12(b) and 12(b). For the F\({}_{\text{f}}\) component, the ENT\({}_{\text{Ff}}\) model exhibits a nonmonotonic trend, reaching a maximum at the midpoint of the depth of cut range. This nonmonotonic behavior in the ENT\({}_{\text{Ff}}\) function is likely related to the entering angle variations that occur with changes in the depth of cut. Similar nonmonotonic trends are also observed in other RQA indicator models (Figures 13-14). The differences between the ENT models of the F\({}_{\text{f}}\) and F\({}_{\text{FN}}\) forces are also visible in the equations. The ENT\({}_{\text{Ff}}\) model (17) has a quadratic factor and two linear factors. The ENT\({}_{\text{FNN}}\) model (20) is very simple and has only one linear factor. The DET models for the two force components F\({}_{\text{f}}\) and F\({}_{\text{FN}}\) likewise differ. Although both models incorporate the variables \(f_{z}\) and \(a_{p}\), the DET\({}_{\text{FNN}}\) model (19) is more complex, including quadratic and 2-way factors, whereas the DET\({}_{\text{Ff}}\) model (16) consists only of linear factors. For the F\({}_{\text{f}}\) component, the DET\({}_{\text{Ff}}\) indicator increases linearly as the cross-sectional area of the cutting layer increases in the feed direction. In contrast, the VMAX indicator for both force components presents a more intricate form. For the F\({}_{\text{FN}}\) component, VMAX\({}_{\text{FNN}}\) depends on both \(f_{z}\) and \(a_{p}\), while the VMAX\({}_{\text{Ff}}\) varies nonmonotonically solely as a function of \(a_{p}\) and is insensitive to changes in feed per tooth \(f_{z}\). Both VMAX indicator models are very similar, differing only in the occurrence of the feed rate. However, it can be observed that for the F\({}_{\text{f}}\) component of the cutting force, the VMAX\({}_{\text{Ff}}\) model exhibits a relatively high error level of 26.41%. This is likely due to the fact that the VMAX parameter depends solely on the cutting depth \(a_{p}\), and its relationship with the feed \(f_{z}\) cannot be established or functionally described. Therefore, using the VMAX\({}_{\text{Ff}}\) indicator to assess the dynamics of the milling process in the feed direction may be subject to slight error. The analysis of VMAX indicator variability in the feed direction of milling requires further research. Nonetheless, the obtained VMAX\({}_{\text{Ff}}\) model is characterized by a low \(p\)-value, which indicates the high  
Figure 13: Experimental models of RQA indicators of the FFN cutting force component: (**a**) DET, (**b**) ENT; (**c**) VMAX. statistical significance of the model. In most of the models analyzed, satisfactory fits were achieved for the F\({}_{\text{f}}\) and F\({}_{\text{fN}}\) components, with low \(p\)-values for the model factors, indicating strong statistical significance. Elevated \(p\)-values were found only in the VMAX models, primarily for the \(a_{p}\) factor, suggesting lower statistical significance for this component. However, the overall \(p\)-value for each model is below 0.05, which is considered statistically significant. The RQA indicators for the axial component of the cutting force F\({}_{\text{a}}\) exhibit completely different variations compared to those for the F\({}_{\text{f}}\) and F\({}_{\text{fN}}\) components (Figure 14).