# Recurrence Quantification Analysis (RQA) of Toroidal End Tool Milling Process
## 3 Results and Analysis
### Relations Between RQA and Cutting Parameters  
Since RPs do not provide sufficient insight into the dynamics of the toroidal milling process, a Recurrence Quantification Analysis (RQA) was performed. Seven RQA indices were determined for each cutting test and each cutting force component, with the corresponding values presented in Tables 5-7. Due to the complexity of the numerical data, the results are illustrated in graphical form in Figures 9-11. Given the large volume of data, the values of the variation in the RQA indicators are shown as a function of feed per tooth at the central value of the \(a_{p}\) and as a function of axial feed at the central value of \(f_{z}\). The variation in these RQA measures in relation to the cutting parameters was then analyzed. To better visualize the varying RQA indicator values, the graphs were plotted on a logarithmic scale. Upon analyzing the variation in the RQA measures as a function of and \(f_{z}\) (Figures 10-12), it is evident that the trends of change differ for each component of the cutting force. Additionally, the selected RQA indicators exhibit distinct trends when plotted against feed per tooth versus depth of cut. The indicator RR exhibits nonmonotonic behavior as a function of feed per tooth \(f_{z}\) for the two cutting force components F\({}_{\text{f}}\) and F\({}_{\text{IN}}\). Similarly, for the F\({}_{\text{f}}\) component in relation to the depth of cut \(a_{p}\), this parameter also behaves nonmonotonically. This may suggest that the RR indicator is not directly influenced by the cutting parameters. In contrast, for the F\({}_{\text{a}}\) force component, the trend of the RR indicator as a function of the cutting parameters shows an overall increase, although the functional relationship is variable. \begin{table}
\begin{tabular}{c 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 & 0.15 & 0.97 & 19.55 & 666 & 2.58 & 9.67 & 250 \\
0.05 & 1.5 & 0.25 & 0.98 & 21.28 & 1108 & 2.45 & 11.28 & 321 \\
0.05 & 2.5 & 0.18 & 0.99 & 21.44 & 771 & 2.38 & 10.62 & 282 \\ \hline
0.1 & 0.5 & 0.14 & 0.97 & 21.26 & 735 & 2.75 & 10.52 & 259 \\
0.1 & 1.5 & 0.19 & 0.98 & 20.98 & 1765 & 2.55 & 10.46 & 339 \\
0.1 & 2.5 & 0.26 & 0.99 & 27.55 & 808 & 2.42 & 16.61 & 348 \\ \hline
0.15 & 0.5 & 0.22 & 0.98 & 21.86 & 647 & 2.66 & 12.85 & 358 \\
0.15 & 1.5 & 0.31 & 0.99 & 28.28 & 684 & 2.59 & 18.45 & 400 \\
0.15 & 2.5 & 0.26 & 0.99 & 25.72 & 2240 & 2.42 & 14.34 & 356 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: RQA indicators of F\({}_{\text{IN}}\). \begin{table}
\begin{tabular}{c 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 & 0.10 & 0.64 & 3.00 & 80 & 1.15 & 3.73 & 186 \\
0.05 & 1.5 & 0.20 & 0.74 & 9.97 & 191 & 3.09 & 6.96 & 253 \\
0.05 & 2.5 & 0.16 & 0.89 & 16.08 & 460 & 3.48 & 12.02 & 247 \\ \hline
0.1 & 0.5 & 0.11 & 0.72 & 3.13 & 71 & 1.01 & 3.98 & 90 \\
0.1 & 1.5 & 0.24 & 0.92 & 30.65 & 511 & 3.67 & 21.70 & 448 \\
0.1 & 2.5 & 0.21 & 0.94 & 8.83 & 490 & 2.14 & 11.77 & 371 \\ \hline
0.15 & 0.5 & 0.13 & 0.71 & 3.08 & 95 & 1.19 & 3.89 & 207 \\
0.15 & 1.5 & 0.26 & 0.89 & 10.67 & 426 & 2.17 & 14.41 & 354 \\
0.15 & 2.5 & 0.20 & 0.94 & 10.93 & 461 & 2.39 & 15.16 & 305 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: RQA indicators of F\({}_{\text{a}}\).