# Self-Tuning Algorithm for Tuneable Clamping Table for Chatter Suppression in Blade Recontouring
## 4 Chatter Suppressing Strategy for the TCT Architecture
### Excitation Frequency Dependent Tuning for the TCT  
At this point, there are different optimization procedures to maximize the negative part of (21)--or minimize its positive part. The traditional approach, in which constant device tuning is provided no matter what the frequency of the external excitation \(g\) is, benefits from the presence of three invariant points in (21). In this manner, a wide-bandwidth optimization of either negative or positive real parts can be achieved as follows: first, the two neighbouring invariants of the corresponding side of the curve are matched and, then, a device damping for a zero-tangent at the mentioned invariants is set. The application of this strategy to the TCT architecture leads to the analytical expressions  
\[f_{\text{o},\pm}=\sqrt{1-\mu\pm\frac{\sqrt{2\mu}}{2}}\quad\text{and}\quad \zeta_{1,\text{o},\pm}=\sqrt{\frac{3\mu}{8}}\sqrt{\frac{\sqrt{2}\pm\sqrt{\mu} }{\sqrt{2}\pm(3-2\mu)\sqrt{\mu}}}, \tag{22}\]  
for constant frequency ratio and device damping tunings, respectively. A more detailed explanation of the procedure for obtaining these expressions can be found in [23]. Constant parameter tuning strategies offer a good solution for fixed dampers or for dampers whose stiffness and damping cannot be externally altered in real-time. However, in semi-active devices like the TCT, the stiffness of the controlled mode can be externally controlled in real-time by means of a rotary spring driven by a motor and an encoder. Therefore, if the frequency of the external excitation \(g\) is known, a more efficient tuning strategy based on that frequency can be devised [29; 40]. Attending to (18), the arising dominant chatter frequency \(\omega_{\text{c}}\) can be considered as sampling on the real part of the receptance function. Thus, a general chatter frequency-dependent optimal tuning can be developed by optimising the negative or positive real part of \(h_{22}\) for every dimensionless frequency \(g=\omega_{\text{c}}/\omega_{2}\). This can be achieved by simply deriving \(\text{Re}\;h_{22}\) with respect to the frequency tuning \(f\) and setting it equal to zero as  
\[ \begin{split}\frac{\text{d}\;\text{Re}\;h_{22}(g,f)}{\text{d}f}= \\ \frac{-2 fu((f^{2}-g^{2})(g^{2}-1)+g^{2}\mu)^{2}+8 fg^{4}\mu\zeta_{ 1}^{2}(g^{2}-1)(g^{2}-\mu-1)}{((f^{2}-g^{2})(g^{2}-1)+g^{2}\mu)^{2}+4f^{2}g^{2 }\zeta_{1}^{2}(g^{2}-1)^{2})^{2}}\equiv 0\end{split}  \tag{23}\]  
for every value of the dimensionless frequency \(g\) and standalone table damping \(\zeta_{1}\). Solving (23) leads to computing five roots: one at \(f_{0}=0\) and four others symmetrically arranged with respect to \(f_{0}\). By just considering the positive tuning values, the dimensionless (chatter) frequency-dependent tuning formula \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Part/Structure**} \\ \hline nat. frequency & dimless frequency \\ \hline \(\omega_{2}:=\sqrt{\frac{k_{2}}{m_{2}}}\) & \(g:=\frac{\omega}{\omega_{2}}\) \\ \hline \multicolumn{3}{c}{**TCT/Tuning**} \\ \hline nat. frequency & frequency ratio & mass ratio & damping ratio \\ \hline \(\omega_{1}:=\sqrt{\frac{k_{1}}{m_{1}}}\) & \(f:=\frac{\omega_{1}}{\omega_{2}}\) & \(\mu=\frac{m_{2}}{m_{1}}\) & \(\zeta_{1}=\frac{c_{1}}{2m_{1}\omega_{1}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of the dimensionless dynamic and tuning parameters of the structure. \[f_{\frac{\mathrm{d}\mathrm{Re}\;h_{22}}{d/z}-0,+}(g)=\sqrt{\frac{g^{2}(g^{4}+1+\mu-( \mu+2)g^{2}\pm 2\zeta_{1}^{2}\sqrt{(g^{2}-1)^{3}(g^{2}-1-\mu)}}{(g^{2}-1)^{2}}} \tag{24}\]  
for positive (\(\beta_{0}>0\)) and negative (\(\beta_{0}<0\)) directional factors is attained. The expression (24) presents the tuning limits  
\[g_{\mathrm{lim},-}=1\quad\text{and}\quad g_{\mathrm{lim},+}=\sqrt{1+\mu}, \tag{25}\]  
outside of which the tuning can actually be applied. When machining close to the resonance of the standalone blade system, dimensionless chatter frequencies close to \(g=1\) are expected.