# Recurrence Quantification Analysis (RQA) of Toroidal End Tool Milling Process
## 3 Results and Analysis
### Recurrence Plots Analysis
This is the axial direction in which the stiffness of the tool is greatest. Hence, the RP diagram of the force component F\({}_{\text{a}}\) is much more uniform than that of the component F\({}_{\text{FN}}\). Both components F\({}_{\text{a}}\) and F\({}_{\text{FN}}\) repel the tool from the workpiece but in different directions. Therefore, it can be assumed that the differences between the graphs are due to the different tool stiffnesses in the radial and axial directions. The F\({}_{\text{FN}}\) component represents the force responsible for the radial displacement of the tool relative to the workpiece. The stiffness of the tool in the radial direction is low, resulting in vibrations in this direction. Therefore, scattered and dashed fields are observed in the plot of F\({}_{\text{f}}\). The force varies periodically, indicating that vibrations of the tool in the radial direction are probably occurring. This suggests that each cutting force component must be analyzed individually, as a single universal analysis method cannot be applied to all components. As the cutting parameters \(f_{z}\) and \(a_{p}\) vary, the cross-sectional area of the cutting layer also changes. This cross section is defined by the geometric parameters: width (\(b\)) and thickness (\(h\)). As the cross-sectional area increases, the magnitude of each cutting force component also increases, in accordance with the mechanistic model of cutting force components. In this model, the force components are represented by the following equations [48]:  
\[F_{f}=k_{\text{f}c}\ b\ h+k_{\text{f}e}\ b, \tag{13}\]  
\[F_{\text{FN}}=k_{\text{FN}c}\ b\ h+k_{\text{f}Ne}\ b, \tag{14}\]  
\[F_{a}=k_{\text{f}ac}\ b\ h+k_{\text{f}ac}\ b. \tag{15}\]  
These equations incorporate experimentally determined coefficients \(k\) as well as the cross-sectional parameters \(b\) and \(h\) of the cutting layer, depth of cut- and feed-dependent. Thus, an increase in feed per tooth \(f_{z}\) or depth of cut \(a_{p}\) should theoretically result in an increase in each cutting force component. However, in toroidal milling, the entering angle also varies with the depth of cut, which subsequently affects the relationship among the force components. Consequently, changes in cutting forces may not always be directly proportional to the cross-sectional area of the cutting layer. As the entering angle changes, the cutting conditions evolve, necessitating a detailed investigation of the dynamics in toroidal milling. The response patterns (RP) of each cutting force components were analyzed as a function of \(f_{z}\) and \(a_{p}\). For each force component, only the color intensity of the fields changed; their shape and distribution across the diagram remained consistent. The color within the fields slightly intensifies as the feed rate \(f_{z}\) increases, but not in all cutting tests. In contrast, changes in color intensity as a function of depth of cut \(a_{p}\) are not regular. These changes are insufficient to formulate definitive conclusions about variations in the dynamics of the milling process depending on the \(a_{p}\) and \(f_{z}\). Drawing concrete conclusions or identifying clear graph features indicative of milling dynamics remains a challenge. The application of artificial intelligence to detect subtle image changes in recurrence plots could be beneficial in this analysis.