## 2 A Mechanistic Model for Bull-Nose End Mills
### Experimental Procedure  
A total of 120 cuttings were performed for 100% radial immersion in aluminum 7075T6 during dry machining. The forces were recorded by using a dynamometer 9257B (Kistler Group, Winterthur, Switzerland) for the set of cutting conditions listed in Table 1. The spindle speed is set at 3000 rpm based on the dynamometer's natural frequency to avoid the amplification of milling forces. The force signals were acquired using a VibSoft-20 acquisition card (Polytec GmbH, Waldbronn, Germany) at a sample rate of 48 kHz and processed in a custom-made MATLAB app to remove drift and noise phenomena.  
Cutting forces data were collected for increments of 0.1 mm in the axial depth of cut with the aim of minimizing the model errors and obtaining detailed information of the cutting coefficients along the cutting edge in the toroid section. For a given axial depth of cut, four values of feed per tooth were tested as listed in Table 1.  
For a set of experiments with a specific axial depth of cut, the system of Equations (7) is solved to obtain the \(K_{c}\) and \(K_{c}\) cutting coefficients. The computed data were fitted by using a second-order polynomial function as listed in Table 2.  
Figure 3 displays the fitted model for shear coefficients in the tool coordinate system (tangential, radial, and axial directions). It can be seen from Figure 3 that the cutting coefficients, in the range of 0.1-1.5 mm, change significantly whereas for the section approaching the flank section of the cutting edge, these remain almost constant. This can be explained from Figure 3d, in which the cutting-edge angle starts from zero at the bottom of the cutting edge and increases to 90\({}^{\circ}\) at the flank tool section. In a square end mill, where the cutting-edge angle is 90\({}^{\circ}\) at any edge location, the cutting coefficients remain constant. However, for the bull-nose end mill, the cutting coefficient exhibits nonlinear behavior. For this reason, the proposed mechanistic model is important to predict accurately stability boundaries.  
\begin{table}
\begin{tabular}{c c} \hline \hline Spindle speed & 3000 rpm \\ Radial Immersion & 16 mm, down milling \\ Depth of cut (\(a_{p}\)) & 0.1-3.0 [mm] \\ Feed per tooth (\(f_{2}\)) & 0.05, 0.10, 0.15, 0.20 [mm/tooth] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Characterization test parameters.  
Figure 3: Experimental cutting coefficients (circle marks) vs. fitted model (solid lines) along the toroidal section: (**a**) tangential; (**b**) radial; (**c**) axial directions; (**d**) x angle along the toroid section.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Cutting Coefficients** & **Toroid [N/mm\({}^{2}\)]** & **Fank [N/mm\({}^{2}\)]** \\ \hline \(K_{tc}\) & \(138.89z^{2}-625z+1136.9\) & 434 \\ \(K_{rc}\) & \(132.98z^{2}-581.96z+785.4\) & 149.3 \\ \(K_{ac}\) & \(37.57z^{2}-159.06z+171.06\) & 2.8 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Cutting coefficients of the tool.  
Notice from Figure 3a-c that the cutting coefficients in all directions present high values for the toroid section: almost three times higher in the radial direction (see Figure 3b), and almost five times higher in the axial direction (see Figure 3c) compared to the flank section of the cutting edge, which could affect significantly the critical depth of cut of thin-floors machining.  
In order to validate the proposed model, slot milling tests were performed. Figure 4 shows the comparison between the proposed cutting force-fitted model and experimental forces for two sets of cutting conditions. The accuracy of the proposed model to fit experimental data is shown in Figure 4; it is evident that in both cases, the proposed forced model predicts the experimental data well.