## 2 3-DOF Chatter Stability Model
### Identification of Layered Cutting Force Coefficients  
For the toroidal surface of a bull-nose end cutter, the diameter varies continuously along the cutter's axis direction. The cutting force coefficients are identified by the fact that the measured average cutting forces per tooth are equal to the predicted ones [41; 42]. The predicted average cutting forces acting on the \(l\)th cutting disk element are given as follows:  
\[\overline{F}_{q,l}=\frac{1}{\phi_{p}}\int_{\phi_{q,l}}^{\phi_{q,l}}\int_{z_{1,l} ^{2}}^{z_{2,l}}dF_{q,l}(\varphi,z)d\varphi,\,(q=x,y,z) \tag{11}\]  
where \(z_{1,l}\) and \(z_{2,l}\) are the contact boundaries of the \(l\)th cutting disk element within the cut.  
In the bull-nose end milling experiments, the measured cutting forces are obtained at various axial depths of the cut. The differences in average cutting forces between two adjacent axial depths of cut are calculated so as to acquire the average cutting forces corresponding to each cutting disk element:  
\[\overline{F}_{q,l}=\overline{F}_{q}(z_{l})-\overline{F}_{q}(z_{l-1}),\ \ \ (q=x,y,z;\ l=1,2,3,\ldots,N_{l}) \tag{12}\]  
where \(z_{l}\) denotes the \(l\)th axial depth of cut and \(z_{0}=0\), representing non-cutting. \(N_{l}\) is the number of layers.  
The instantaneous cutting forces generated on the \(l\)th axial cutting disk element at the radial immersion angle \(\varphi\) are expressed as follows:  
\[\begin{bmatrix}F_{x,l}\\ F_{y,l}\\ F_{z,l}\end{bmatrix}=\frac{f_{t}}{2}\mathbf{A}\begin{bmatrix}u_{1}\\ u_{2}\\ u_{3}\end{bmatrix} \tag{13}\]  
where  
\[\mathbf{A}=\begin{bmatrix}-K_{t,l}\sin 2\varphi_{j}&-2K_{r,l}\sin^{2} \varphi_{j}&-2K_{a,l}\sin^{2}\varphi_{j}\\ 2K_{t,l}\sin^{2}\varphi_{j}&-2K_{r,l}\sin 2\varphi_{j}&-K_{a,l}\sin 2\varphi_{j}\\ 0&-2K_{a,l}\sin\varphi_{j}&2K_{r,l}\sin\varphi_{j}\end{bmatrix} \tag{14}\]  
\[u_{1}=z\left|\begin{array}{c}z_{2,l}\\ z_{1,l}\end{array}\right. \tag{15}\]  
\[u_{2}=-\frac{R_{f}}{2}\Big{(}E\sqrt{1-E^{2}}+\arcsin E\Big{)}\left|\begin{array} []{c}z_{2,l}\\ z_{1,l}\end{array}\right. \tag{16}\]  
\[u_{3}=\frac{1}{R_{f}}\left(R_{f}z-\frac{z^{2}}{2}\right)\left|\begin{array}{c }z_{2,l}\\ z_{1,l}\end{array}\right. \tag{17}\]  
\[E=\frac{R_{f}-z}{R_{f}} \tag{18}\]  
where \(f_{t}\) is the normal feed per tooth.  
The instantaneous cutting forces are averaged within the radial immersion interval and expressed as follows:  
\[\begin{bmatrix}\overline{F}_{x,l}\\ \overline{F}_{y,l}\\ \overline{F}_{z,l}\end{bmatrix}=\frac{f_{t}}{\varphi_{p}}\mathbf{B}\begin{bmatrix} K_{t,l}\\ K_{r,l}\\ K_{a,l}\end{bmatrix} \tag{19}\]  
where  
\[\mathbf{B}=\begin{bmatrix}v_{3}u_{1}&(v_{2}-v_{1})u_{2}&(v_{2}-v_{1})u_{3}\\ (v_{1}-v_{2})u_{1}&v_{3}u_{2}&v_{3}u_{3}\\ 0&-v_{4}u_{3}&v_{4}u_{2}\end{bmatrix} \tag{20}\]  
\[v_{1}=\frac{1}{2}\varphi\left|\begin{array}{c}\varphi_{ex,l}\\ \varphi_{st,l}\end{array}\right. \tag{21}\]\[v_{2}=\frac{1}{4}\sin(2\varphi)\left|\begin{array}{c}\varphi_{ex,l}\\ \varphi_{st,l}\end{array}\right. \tag{22}\]  
\[v_{3}=\frac{1}{4}\cos(2\varphi)\left|\begin{array}{c}\varphi_{ex,l}\\ \varphi_{st,l}\end{array}\right. \tag{23}\]  
\[v_{4}=\cos\varphi\left|\begin{array}{c}\varphi_{ex,l}\\ \varphi_{st,l}\end{array}\right. \tag{24}\]  
Therefore, the cutting force coefficients of the \(l\)th axial cutting disk element are determined as follows:  
\[\left\{\begin{array}{l}K_{t,l}=\frac{2\pi}{f_{t}N\text{u}_{1}}-\frac{v_{3} \mathbb{F}_{x,l}-(v_{2}-v_{1})\mathbb{F}_{y,l}}{v_{2}^{2}+(v_{2}-v_{1})^{2}} \\ K_{r,l}=\frac{2\pi}{f_{t}N(\text{u}_{2}^{2}+u_{3}^{2})}\cdot\left\{\begin{array} []{c}\frac{u_{2}[(v_{2}-v_{1})\mathbb{F}_{x,l}+v_{3}\mathbb{F}_{y,l}]}{v_{2}^ {2}+(v_{2}-v_{1})^{2}}-\frac{u_{3}\mathbb{F}_{x,l}}{v_{4}}\\ K_{a,l}=\frac{2\pi}{f_{t}N(\text{u}_{2}^{2}+u_{3}^{2})}\cdot\left\{\begin{array} []{c}\frac{u_{3}[(v_{2}-v_{1})\mathbb{F}_{x,l}+v_{3}\mathbb{F}_{y,l}]}{v_{3}^ {2}+(v_{2}-v_{1})^{2}}+\frac{u_{2}\mathbb{F}_{x,l}}{v_{4}}\right\}\end{array} \right\}\end{array}\right. \tag{25}\]