## 2 A Mechanistic Model for Bull-Nose End Mills
### Cutting Force Model  
A force model was developed to obtain an instantaneous cutting force magnitude for a given depth of cut. It is based on a mechanistic approach that assumes a relationship between forces and the uncut chip thickness by means of the cutting coefficients. Cutting models have been developed for ball-end mills that present similarities with the geometry of a bull-nose end mill. For instance, [PERSON] et al. [24] proposed a model based on an analytic representation of ball-shaped helical flute geometry. [PERSON] et al. reported the prediction of forces from the orthogonal cutting data [22] and for better detail, an oblique cut-off model presented by [PERSON] and [PERSON] [31] was evaluated in sections. However, the development of a generalized and accurate model for cutting coefficients for a bull-nose end mill has not been set. [PERSON] et al. [21] developed a model using averaged cutting coefficients in the toroid part, but they recognized that the flank of the tool presented a different behavior.  
The cutting force model is established by introducing two coefficients for the tangential, radial, and axial directions: one is associated with cutting (shearing) \(K_{c}\), which is directly related to the dynamic chip thickness \(h(\phi,\kappa)\) and the second is associated with the friction (rubbing) coefficient \(K_{e}\). As shown in Figure 2, the differential forces components \(dF_{t}\), \(dF_{r}\), and \(dF_{a}\) are orientated for each differential cutting edge according to the cutting edge angle \(\kappa(z)\). The cutting edge of the tool is discretized in disks with a thickness of \(b=1\) mm along the \(z\)-direction. With this discretized approach, it is possible to understand in detail the behavior of the cutting coefficients along the cutting edge in the toroid part. Figure 2 illustrates the cutting parameters of thin-floor machining: axial depth of cut \(a_{p}\) along the \(z\)-direction, radial immersion \(a_{e}\) of cut in the \(y\)-direction, and spindle speed \(n\).  
The mechanism of the bull-nose end mill is modeled using a mechanistic linear model. Here, the differential cutting forces on the local tool system follow the analytical approach in [32]:  
Figure 1: Composition of optical microscope photographs of the bull-nose end mill taken with an Alicona microscope: (**a**) bottom view (diameter measurement); (**b**) frontal view (nose radius measurement); (**c**) lateral view (helix angle measurement).  
Figure 2: Main cutting parameters in milling operation with a bull-nose end mill: axial discretization and differential cutting force components.  
\[\begin{array}{l}dF_{t}\big{(}\phi_{j}\big{)}=K_{tc}h\big{(}\phi_{j},\kappa\big{)} dz+\ K_{tc}dz\\ dF_{r}\big{(}\phi_{j}\big{)}=K_{rc}h\big{(}\phi_{j},\kappa\big{)}dz+\ K_{rc}dz \\ dF_{a}\big{(}\phi_{j}\big{)}=K_{ac}h\big{(}\phi_{j},\kappa\big{)}dz+\ K_{ac}dz \end{array} \tag{1}\]  
where \(dF_{t,r,a}\) are the differential force components, \(\phi_{j}(t)\) is the angle of each tooth on the cutter, which change over time, and \(h\big{(}\phi_{j},\kappa\big{)}\) is the instantaneous chip thickness as described by [PERSON] et al. in [33]:  
\[h\big{(}\phi_{j},\kappa\big{)}=f_{z}\sin\phi_{j}\sin\kappa. \tag{2}\]  
Here, \(f_{z}\) is the feed rate per tooth and \(\kappa\) is the cutting-edge angle for a given axial slice. By means of a rotation matrix, it is possible to transform the differential cutting forces to Cartesian coordinates  
\[\left\{\begin{array}{l}dF_{x}\big{(}\phi_{j},z\big{)}\\ dF_{y}\big{(}\phi_{j},z\big{)}\\ dF_{z}\big{(}\phi_{j},z\big{)}\end{array}\right\}=\left[\begin{array}{ccc} \cos\phi_{j}&\sin\phi_{j}\sin\kappa&-\sin\phi_{j}\cos\kappa\\ \sin\phi_{j}&-\cos\phi_{j}\sin\kappa&\cos\phi_{j}\cos\kappa\\ 0&-\cos\kappa&-\sin\kappa\end{array}\right]\left\{\begin{array}{l}dF_{t} \big{(}\phi_{j},z\big{)}\\ dF_{r}\big{(}\phi_{j},z\big{)}\\ dF_{a}\big{(}\phi_{j},z\big{)}\end{array}\right\}. \tag{3}\]