# Self-Tuning Algorithm for Tuneable Clamping Table for Chatter Suppression in Blade Recontouring
## 3 Unidirectional Milling Model with Tuneable Clamping Table
### Zeroth-Order Milling Stability  
The asymptotic linear stability of the stationary solution of (2) can be assessed by introducing a perturbation \(\mathbf{p}(t)\) around the stationary solution \(\mathbf{x}(t)=\mathbf{x}_{\mathrm{s}}(t)+\mathbf{p}(t)\). For the case that the perturbed solution \(\mathbf{p}(t)\) is not influenced by the stationary cutting solution, the stability of (2) can be evaluated by studying the stability of the perturbed equation:  
\[\mathbf{M}\,\ddot{\mathbf{p}}(t)+\mathbf{C}\,\dot{\mathbf{p}}(t)+\mathbf{K}\, \mathbf{p}(t)=\Delta\mathbf{F}(t,p_{2}(t),p_{2}(t-\tau)). \tag{12}\]  
Numerical methods such as semidiscretization [43] or full-discretization [44] are well-established and efficient techniques for assessing the stability properties of time-periodic DDEs. However, like other numerical methods, they often hide the direct relations between process parameters and system stability that could serve as a core function for a stability maximization algorithm. Instead, the stability of (12) can be studied in the frequency domain by considering the marginal case where the system has a critical dominant vibration (chatter) frequency \(\omega_{\mathrm{c}}\) and its modulations \(\omega_{\mathrm{c}}+k\,\omega_{Z}\) related to the tooth passing frequency \(\omega_{Z}=2\pi/T_{Z}\) as  
\[\mathbf{p}(t)=\sum_{k=-\infty}^{\infty}\mathbf{p}_{k}\,\mathrm{e}^{\mathrm{i}( \omega_{\mathrm{c}}+k\,\omega_{Z})t}, \tag{13}\]  
together with the Fourier decomposition of the time-periodic milling coefficient  
\[B(t)=\sum_{k=-\infty}^{\infty}B_{k}\,\mathrm{e}^{\mathrm{i}\,k\,\omega_{Z}\,t }:=\frac{Z}{2\pi}\sum_{k=-\infty}^{\infty}\beta_{k}\,\mathrm{e}^{\mathrm{i}\,k \,\omega_{Z}\,t},\quad\text{where}\quad\beta_{k}=\int_{\varphi_{\mathrm{en}}} ^{\varphi_{\mathrm{ex}}}b(\varphi)\,\mathrm{e}^{\mathrm{i}\,k\,Z\,\varphi}\, \mathrm{d}\varphi. \tag{14}\]  
In case of only being interested in Hopf-type instabilities, (13) can be truncated to \(k=0\) and the directional milling coefficient approximated by is averaged value \(B(t)\approx B_{0}:=\frac{Z}{2\pi t}\beta_{0}\), where  
\[\beta_{0}=\int_{\varphi_{\mathrm{en}}}^{\varphi_{\mathrm{ex}}}b(\varphi)\, \mathrm{d}\varphi=\frac{1}{4}\biggl{[}\cos 2\varphi+\frac{K_{\mathrm{rc}}}{K_{ \mathrm{rc}}}(2\varphi+\sin 2\varphi)\biggr{]}_{\varphi_{\mathrm{en}}}^{ \varphi_{\mathrm{ex}}} \tag{15}\]  
is the mean directional factor [45]. This approximation--commonly known as the zeroth-order algorithm (ZOA [46])--leads to an eigenvalue problem from which the critical stability curves can be parametrically computed. ZOA provides accurate results in non-interrupted milling processes where only Hopf bifurcations are expected and the effect of machining mode coupling is negligible [47].  
In a unidirectional milling case like this, the eigenvalue problem is reduced to the following scalar characteristic equation:  
\[p_{2,0}=a\,K_{\mathrm{rc}}\,\beta_{0}\frac{Z}{2\pi}H_{22}(\omega_{\mathrm{c}} )(\mathrm{e}^{-\mathrm{i}\,\omega_{\mathrm{c}}\,\tau}-1)\,p_{2,0}, \tag{16}\]  
where \(H_{22}\) refers to the direct receptance function at the blade tip originated from  
\[\mathbf{H}(\omega)=(-\omega^{2}\,\mathbf{M}+\mathrm{i}\,\omega\,\mathbf{C}+ \mathbf{K})^{-1}=\bigl{[}H_{ij}(\omega)\bigr{]},\quad\text{where}\quad i,j=1,2. \tag{17}\]In this way, a semi-analytical expression of the parametric root crossing (\(\Omega_{l}(\omega_{\rm c})\), \(a(\omega_{\rm c})\)) curves can be raised as  
\[a(\omega_{\rm c})=-\frac{2\pi}{Z\;K_{tc}\;\beta_{0}\;{\rm Re}(H_{22}(\omega_{\rm c }))}\quad\text{and}\quad\Omega_{l}(\omega_{\rm c})=\frac{2\pi}{Z(2\psi_{22}( \omega_{\rm c})+(2l+3)\pi)}, \tag{18}\]  
where \(l=1,2,3,...\) stands for the lobe number and \({\rm Re}\;H_{22}\) and \(\psi_{22}\) are, respectively, the real part and phase of the aforementioned direct receptance function at the blade tip.