# Self-Tuning Algorithm for Tuneable Clamping Table for Chatter Suppression in Blade Recontouring
## 3 Unidirectional Milling Model with Tuneable Clamping Table
### Regenerative Milling Force  
For simplicity, a cylindrical and straight fluted end mill with \(Z\) regularly spaced teeth rotating at a constant spindle speed \(\Omega\) is considered. Assuming a cutting force model with linear characteristics with respect to the chip thickness \(h\), the specific cutting force vector \(\mathbf{f}=\mathbf{F}/a\) in the tangential-radial plane \((t,r)\) can be defined as  
\[\mathbf{f}_{t,r,i}(t)=-(\mathbf{K}_{\mathbf{c}}+\mathbf{K}_{\mathbf{c}}\,h_{i} (t)), \tag{4}\]  
where \(\mathbf{K}_{\mathbf{c}}=[K_{\text{te}},\,K_{\text{re}}]^{\intercal}\) and \(\mathbf{K}_{\mathbf{c}}=[K_{\text{fc}},\,K_{\text{rc}}]^{\intercal}\) are respectively the so-called edge and cutting coefficient vectors resulting from the linear fitting to the experimentally measured mean cutting forces [41]. Due to the regenerative effect, each cutting edge cuts the wavy surface left by the vibration of the previous tooth. Hence, the momentary chip thickness can be calculated by approximating the tool edge kinematics to a circular motion as  
\[h_{i}(t)\approx(f_{Z}\sin\varphi_{i}(t)+((x_{2}(t)-x_{2}(t-\tau))\cos\varphi_{ i}(t)), \tag{5}\]  
where \(\tau\) is the regenerative delay, which is equal to the tooth passing period \(T_{Z}=\frac{2\pi}{Z\Omega}\) in this case, and where  
\[\varphi_{i}(t)=\Omega t+2\pi\frac{i-1}{Z} \tag{6}\]  
stands for the angular position of the \(i\)th tooth at the instant \(t\). The total cutting force \(F_{2}\) to which the blade is subjected to for an axial depth of cut \(a\) can be hence calculated by summing the projections onto the flexible direction of all the \(Z\) teeth as  
\[F_{2}(t)=-a\sum_{i=1}^{Z}g(\varphi_{i}(t))((K_{\text{te}}+K_{\text{fc}}\,h_{i} (t))\sin\varphi_{i}(t)-(K_{\text{re}}+K_{\text{rc}}\,h_{i}(t))\cos\varphi_{i}( t)), \tag{7}\]  
where \(g(\varphi)\) represents the screen function  
\[g(\varphi)=\begin{cases}1,&\text{if}\quad\varphi_{\text{en}}<\varphi\bmod 2 \pi<\varphi_{\text{ex}},\\ 0,&\text{otherwise},\end{cases} \tag{8}\]  
which determines whether the tooth is in or out of cut.  
The total cutting force can be divided into two terms as \(F_{2}(t):=F_{2}(t,x_{2}(t),x_{2}(t-\tau))=G(t)+\Delta F(t,x_{2}(t),x_{2}(t- \tau))\). The former is the time-periodic stationary forcing term  
\[\begin{split} G(t)\equiv G(t+T_{Z})=-a\sum_{i=1}^{Z}g(\varphi_{ i}(t))((K_{\text{te}}+K_{\text{fc}}f_{Z}\sin\varphi_{i}(t))\sin\varphi_{i}(t) \\ -(K_{\text{re}}+K_{\text{rc}}f_{Z}\sin\varphi_{i}(t))\cos\varphi_{i}( t)),\end{split} \tag{9}\]  
which represents the cutting forces of the equivalent undeformed milling model. This term induces an unavoidable forced vibration \(\mathbf{x}_{\text{s}}(t)\). The latter is the state-dependent variational term of the cutting force. It arises due to the regenerative nature of the cutting force and is calculated as  
\[\Delta F(t,x_{2}(t),x_{2}(t-\tau))=-a\,K_{\text{fc}}\,B(t)((x_{2}(t)-x_{2}(t- \tau)), \tag{10}\]  
where  
\[\begin{split} B(t)=B(t+T_{Z}):=\\ \sum_{i=1}^{Z}g(\varphi_{i}(t))\bigg{(}\frac{K_{\text{rc}}}{K_{ \text{fc}}}\cos^{2}\varphi_{i}(t)-\sin\varphi_{i}(t)\cos\varphi_{i}(t)\bigg{)} =\sum_{i=1}^{Z}g(\varphi_{i})b(\varphi_{i})\end{split} \tag{11}\]is the time-periodic directional milling coefficient, which concentrates the projection of the cutting force onto the mode direction and the projection of the vibration onto the chip thickness direction.  
The presence of the regenerative--or dynamic--term of the cutting force makes the governing equation a time-periodic delay differential equation (DDE [42]). Under a certain set of cutting parameters, the DDE can undergo Hopf or period-doubling (flip) instabilities, which actually correspond to the large amplitude chatter vibrations.