## 2 3-DOF Chatter Stability Model
### Dynamic Cutting Force Modeling for Bull-Nose End Milling  
Figure 2 demonstrates a bull-nose end cutter, which is characterized by two main parameters: the fillet radius \(R_{f}\) and the cutter diameter \(D\).  
For the bull-nose end milling process, the chip thickness can be distributed between the static and dynamic components. The static component is disregarded because it has no influence on milling stability [37]. The dynamic component of cutting-edge \(j\) is determined as follows:  
\[h_{d}(\varphi_{j},\kappa)=\left[\sin\varphi_{j}\sin\kappa\cos\varphi_{j}\sin \kappa-\cos\kappa\right]\begin{bmatrix}\Delta x\\ \Delta y\\ \Delta z\end{bmatrix} \tag{2}\]  
where  
\[\begin{array}{c}\varphi_{j}(z)=\varphi+(j-1)\varphi_{p}-\psi(z)\\ =\frac{2\pi n}{60}t+(j-1)\frac{2\pi}{N}-\frac{2\pi\tan\beta}{D}\end{array} \tag{3}\]  
\[\kappa(z)=\arccos\frac{R_{f}-z}{R_{f}},\,(z<R_{f}) \tag{4}\]  
\[\begin{bmatrix}\Delta x\\ \Delta y\\ \Delta z\end{bmatrix}=\begin{bmatrix}x_{c}(t)-x_{c}(t-T)\\ y_{c}(t)-y_{c}(t-T)\\ z_{w}(t-T)-z_{w}(t)\end{bmatrix} \tag{5}\]  
where \(\varphi_{j}\) denotes the radial immersion angle of cutting-edge \(j\). \(\varphi\) denotes the rotation angle of the cutting edge. \(\varphi_{p}\) denotes the pitch angle of the cutter. \(\psi(z)\) denotes the radial lag angle caused by the local helix angle. \(n\) denotes the spindle speed. \(t\) denotes the time  
Figure 2: Geometry of a bull-nose end cutter.  
parameter. \(N\) denotes the number of cutting edges. \(\beta\) denotes the constant helix angle of the cutter. \(\kappa\) denotes the axial immersion angle. \(\Delta x\), \(\Delta y\), and \(\Delta z\) denote the \(x\)-axis, \(y\)-axis, and \(z\)-axis dynamic displacements in the CCS. \(x_{c}(t)\) and \(y_{c}(t)\) denote the dynamic displacements of the cutter at the present time \(t\). \(x_{c}(t-T)\) and \(y_{c}(t-T)\) denote the dynamic displacements of the cutter at previous tooth periods \(t-T\). \(z_{w}(t)\) denotes the dynamic displacement of the workpiece at the present time. \(t\). \(z_{w}(t-T)\) denotes the dynamic displacements of the workpiece at previous tooth periods \(t-T\).  
The dynamic cutting forces in the CCS can be established as follows:  
\[\begin{bmatrix}F_{x}\\ F_{y}\\ F_{z}\end{bmatrix}=\sum\limits_{j=1}^{N}\sum\limits_{l=1}^{M}g(\varphi_{j}) \mathbf{T}(\varphi_{j},\kappa)\begin{bmatrix}K_{t,l}\\ K_{t,l}\\ K_{a,l}\end{bmatrix}h_{d}(\varphi_{j},\kappa)\frac{dz}{\sin\kappa} \tag{6}\]  
where  
\[g(\varphi_{j})=\left\{\begin{array}{cc}1\ \varphi_{st,l}<\varphi_{j}<\varphi_{ ex,l}\\ 0\ \ \ \text{otherwise}\end{array}\right. \tag{7}\]  
\[\mathbf{T}(\varphi_{j},\kappa)=\left[\begin{array}{cc}-\cos\varphi_{j}&- \sin\varphi_{j}\sin\kappa&-\sin\varphi_{j}\cos\kappa\\ \sin\varphi_{j}&-\cos\varphi_{j}\sin\kappa&-\cos\varphi_{j}\cos\kappa\\ 0&\cos\kappa&-\sin\kappa\end{array}\right] \tag{8}\]  
where \(M\) denotes the number of axial cutting disk elements. \(g(\varphi_{j})\) denotes a unit step function to decide whether the flute is in or out of the cutting state. \(\varphi_{st,l}\) and \(\varphi_{ex,l}\) denote the start and exit angles of the cutting disk element \(l\), respectively. \(\mathbf{T}(\varphi_{j},\kappa)\) denotes the rotational transformation matrix. \(K_{t,l}\), \(K_{r,l}\), and \(K_{a,l}\) denote the layered tangential, radial, and axial cutting force coefficients on the cutting disk element \(l\), respectively.  
Equation (6) delineates a linear cutting force model where the dynamic chip thickness is not correlated with the feed rate [38]. It can be converted into the following matrix form:  
\[\begin{bmatrix}F_{x}\\ F_{y}\\ F_{z}\end{bmatrix}=\mathbf{H}(t)\begin{bmatrix}\Delta x\\ \Delta y\\ \Delta z\end{bmatrix}=\begin{bmatrix}h_{xx}&h_{xy}&h_{xz}\\ h_{yx}&h_{yy}&h_{yz}\\ h_{xz}&h_{zy}&h_{zz}\end{bmatrix}\begin{bmatrix}\Delta x\\ \Delta y\\ \Delta z\end{bmatrix} \tag{9}\]  
where  
\[\left\{\begin{array}{l}h_{xx}=\sum\limits_{j=1}^{N}\sum\limits_{l=1}^{M}g( \varphi_{j})\big{[