# Recurrence Quantification Analysis (RQA) of Toroidal End Tool Milling Process
## 3 Results and Analysis
### Recurrence Plots Analysis  
The analysis of the statistical parameters of the cutting force components has so far yielded results that are difficult to interpret and lack clarity. Therefore, a recurrence analysis of the force signal was performed. To preserve all relevant information, the signal was analyzed without applying low-pass or high-pass filtering. For each recurrence analysis, 2000 samples were selected, corresponding to approximately three blade cycles at the selected high sampling frequency, which provides a representative signal for a single cutting test. Delay values (_d_) were calculated using the mutual information function [40; 41] for each signal to minimize the interaction between points within the measured time series. Embedding dimension values (_m_) were also determined using the nearest-neighbor methodology [40; 41]. The final parameter required for recurrence analysis was the recurrence threshold (\(\epsilon\)), which was referenced to the standard deviation of the analyzed signal to ensure consistent conditions for comparative analysis. The threshold values were determined experimentally and are presented in Table 4.  
Recurrence plots (RP) were generated using the estimated \(d\), \(m\), and \(\epsilon\) parameters, resulting in 27 RP diagrams in the three cutting force components for each of the nine cutting tests. RP analysis yielded several insightful observations. In particular, each cutting force component displays a distinct RP pattern (Figure 8). The dark blue squares visible in each graph correspond to the intervals between cuts. This is the time between each cutting cycle of the blade. On the other hand, the boxes at the corners of the squares, i.e., at the intersection of the colored vertical and horizontal stripes, represent the cutting. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{F_{f}}\) & \(\mathbf{F_{fN}}\) & \(\mathbf{F_{a}}\) \\ \hline
0.2\% SD & 0.2\% SD & 0.05\% SD \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Values of the recurrence threshold \(\epsilon\) for each force component. Figure 7: Changes in statistical parameters of \(\mathrm{F_{a}}\) cutting force component: (**a**) Mean Value; (**b**) Standard Deviation; (**c**) Skewness; (**d**) Kurtosis. It can be seen that the diagram for each of the cutting force components is different. The shape of the field in the diagram and the degree to which it is filled with color is different. These differences indicate completely different dynamics of the cutting process in each of the cutting directions examined. The cutting process is performed mainly in the feed direction, with the F\({}_{\text{f}}\) component acting along it. Therefore, the F\({}_{\text{f}}\) recurrence plot has a time-varying shape. The increasing shape of the field in the F\({}_{\text{f}}\) diagram is probably due to the change in the cross-sectional area of the cutting layer, which is superimposed on the radial stiffness of the tool. In the milling process, the cross-sectional area of the cutting layer is expressed by the following relation:  
\[A_{D}=b\ h. \tag{11}\]  
Parameter \(b\) represents the width of the chip and can be assumed to be equal to the cutting depth \(a_{p}\). The thickness of the cutting layer, \(h\), is influenced by the feed per tooth \(f_{z}\), as described by the following equation:  
\[h=f_{z}\sin(\phi), \tag{12}\]  
where \(\phi\) is the cutter rotation angle. This means that the thickness \(h\) changes at each cutting moment as the angle \(\phi\) of the blade in the material changes. This causes the thickness \(h\) and, at the same time, the field \(A_{D}\) to increase as the angle \(\phi\) increases during the concurrent cutting process. After reaching a maximum \(h\), the field rapidly decreases and reaches zero as the blade leaves the material. This phenomenon may be the main reason for this shape of the RP of force F\({}_{\text{f}}\). The F\({}_{\text{a}}\) component operates within the plane that is perpendicular to the cutting velocity vector \(v_{c}\).