# Recurrence Quantification Analysis (RQA) of Toroidal End Tool Milling Process
## 3 Results and Analysis
### Relations Between RQA and Cutting Parameters
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(f_{z}\) & \(a_{p}\) & RR & DET & L & LMAX & ENT & TT & VMAX \\ \hline
0.05 & 0.5 & 0.22 & 0.93 & 8.85 & 527 & 2.26 & 6.82 & 189 \\
0.05 & 1.5 & 0.29 & 0.95 & 14.75 & 493 & 2.79 & 11.91 & 372 \\
0.05 & 2.5 & 0.17 & 0.96 & 12.58 & 576 & 2.66 & 6.31 & 164 \\ \hline
0.1 & 0.5 & 0.21 & 0.94 & 12.02 & 558 & 2.46 & 8.82 & 219 \\
0.1 & 1.5 & 0.22 & 0.95 & 15.17 & 530 & 3.30 & 7.20 & 297 \\
0.1 & 2.5 & 0.20 & 0.97 & 13.76 & 630 & 2.67 & 7.46 & 188 \\ \hline
0.15 & 0.5 & 0.23 & 0.95 & 13.80 & 475 & 2.66 & 9.52 & 296 \\
0.15 & 1.5 & 0.38 & 0.98 & 30.53 & 576 & 3.28 & 26.24 & 450 \\
0.15 & 2.5 & 0.30 & 0.98 & 20.47 & 670 & 3.18 & 13.41 & 221 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: RQA indicators of F\({}_{\text{f}}\). For the DET indicator, the variations in value are minimal, making it challenging to clearly determine its relationship with the cutting parameters. However, a distinct trend is observed in the DET function for the F\({}_{\text{a}}\) force component. The TT and L indicators exhibit clearly nonmonotonic trends, particularly for the F\({}_{\text{f}}\) and F\({}_{\text{a}}\) components, suggesting a probable lack of correlation between these parameters and increasing cutting conditions. The indicators LMAX and VMAX display a monotonic trend only for the F\({}_{\text{a}}\) force component, while for other components, both indicators exhibit highly variable behavior. The ENT indicator is difficult to interpret, as its dependence on the cutting parameters cannot be clearly identified for any of the cutting force components. Figure 11: Changes in F\({}_{\text{RQA}}\) indicators in the function of: (**a**) depth of cut; (**b**) feed per tooth. Figure 9: Changes in F\({}_{\text{f}}\) RQA indicators in the function of: (**a**) depth of cut; (**b**) feed per tooth. The analysis of graphs depicting the variation in RQA values as a function of cutting parameters proved insufficient to accurately determine their relationships. Therefore, a response surface methodology (RSM) was employed to analyze these relationships in greater detail. For each RQA indicator and each cutting force component, an attempt was made to fit a mathematical model. However, in many cases, the model fit was either insufficiently robust or unattainable. A threshold criterion was established by which models with a fit level below 70% were discarded from further analysis. Table 8 presents the fitting results for the experimental models that met this criterion. For the F\({}_{\text{f}}\) and F\({}_{\text{FN}}\) components, the same three RQA indicators, DET, ENT, and VMAX, are related to the milling parameters. These indicators for the F\({}_{\text{f}}\) cutting force component are expressed by the following equations:  
\[\text{DET}_{\text{ff}}=0.90799+0.2197f_{z}+0.01777\;a_{p}, \tag{16}\]  
\[\text{ENT}_{\text{FF}}=1.300+4.73\;f_{z}+1.611\;a_{p}-0.4736\;{a_{p}}^{2}, \tag{17}\]  
\[\text{VMAX}_{\text{FF}}=45.4+459\;a_{p}-160.2\;{a_{p}}^{2}, \tag{18}\]  
and for the F\({}_{\text{FN}}\) force component:  
\[\text{DET}_{\text{FN}}=0.95018+0.1483\;f_{z}+0.02507\;a_{p}-0.00420\;{a_{p}}^{ 2}-0.0560\;f_{z}\;a_{p}, \tag{19}\]  
Figure 12: Experimental models of RQA indicators of the F\({}_{\text{f}}\) cutting force component: (**a**) DET, (**b**) ENT; (**c**) VMAX. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & \multicolumn{3}{c}{**R\({}^{2}\), \%**} & \\ \cline{2-7}  & **RR** & **DET** & **L** & **LMAX** & **ENT** & **TT** & **VMAX** \\ \hline \multirow{2}{*}{\(F_{f}\)} & - & 92.6 & - & - & 91.5 & - & 73.6 \\  & - & (\(a_{p}\),\(f_{z}\)) & - & - & (\(a_{p}\),\(f_{z}\)) & - & (\(a_{p}\)) \\ \hline \multirow{2}{*}{\(F_{fN}\)} & - & 97.5 & - & - & 77.5 & - & 87.4 \\  & - & (\(a_{p}\),\(f_{z}\)) & - & - & (\(a_{p}\)) & - & (\(a_{p}\),\(f_{z}\)) \\ \hline \multirow{2}{*}{\(F_{a}\)} & 84.2 & 76.1 & - & 74.9 & 73.2 & - & 71 \\  & (\(a_{p}\)) & (\(a_{p}\)) & - & (\(a_{p}\)) & (\(a_{p}\)) & - & (\(a_{p}\)) \\ \hline \hline \end{tabular}
\end{table}