# Self-Tuning Algorithm for Tuneable Clamping Table for Chatter Suppression in Blade Recontouring
## 4 Chatter Suppressing Strategy for the TCT Architecture
### Comparison with Equivalent Constant Parameter Tuning Strategy  
The effect of the proposed tuning strategy can be appreciated by plotting \(\mathrm{Re}\;h_{22}\) optimally tuned for every value of \(g\) according to \(f_{0}(g)\) in (27), namely \(\mathrm{Re}\;h_{22}(f_{0}(g),g)\), and comparing it to the case with constant tuning \(f_{0,\pm}\) from (22) as in Figure 6a. Here, for comparison purposes, the optimal damping ratios from the corresponding constant tuning strategy in (22) have been set. The excitation frequency dependent strategy further optimizes the wide-bandwidth tuning strategy proposed in [23], except for a single value of \(g\) where \(f_{0}(g)\) and \(f_{0,\pm}\) obviously coincide.  
Figure 5: Optimal frequency of the TCT for \(\mu=5\,\%\) and \(\zeta_{1}=10\,\%\). The tuning ratio \(f\) of the two frequency dependent tunings (\(\frac{4\,\text{Be}\,h_{2}}{df}=0\) and \(\text{Re}\,h_{2}=0\)) are shown in dashed lines, as well as the combined tuning in the thick orange line. Constant frequency tuning ratios \(f_{o,\pm}\) are also provided in the horizontal thin lines for the sake of comparison.  
Figure 6: Comparison of the broad-bandwidth tuning and self-tuning strategy for positive and negative directional factor cases. (**a**) Dimensionless acceptance functions. (**b**) Zeroth order algorithm (ZOA) critical curves. (**c**) Dimensionless frequencies associated to stability asymptotes.  
The effect of the introduced strategy on the milling stability can be observed by computing stability limit given by the parametric curves of the dimensionless ZOA case as  
\[\sigma(g):=\frac{Z\,K_{tc}\,\beta_{0}}{k_{2}}\,a_{\text{lim}}(\omega_{2}\,g)= \frac{2\pi}{\text{Re}\,h_{22}(g)}\quad\text{and}\quad\
u(g):=\frac{Z}{\omega_{2} }\Omega(\omega_{2}\,g), \tag{28}\]  
where \(\sigma(g)\) and \(\
u(g)\) are respectively the dimensionless depth of cut and spindle speed. Figure 6b shows the dimensionless stability limits given by the piecewise smooth function \(\text{Re}\,\,h_{22}(f_{\text{o}}(g),g)\) and the ones subjected to constant tuning \(\text{Re}\,\,h_{22}(f_{\text{o},\pm},g)\). As can be observed, the stability is enhanced for both positive and negative directional factor cases over the whole range of spindle speeds, except for the aforementioned points where the two strategies converge. As noticed in [29], the 'double lobe' shape of the stability diagram becomes a'single lobe' structure.  
Finally, apart from the mentioned increase of the stability limit, a variation with respect the location of the maximum stability asymptotes \(g_{\text{a}}\) is also noticed. These stability asymptotes are directly linked to the dimensionless spindle speeds at which the so-called'sweets spots' take place by the relation \(\
u_{\text{a}}=g_{\text{a}}/l\), being \(l=1,2,3,...\) the lobe number. These dimensionless resonant frequencies can be numerically computed by simply \(\text{Re}\,\,h_{22}(g_{\text{a}})=0\). The curves of the optimal dimensionless frequency for the frequency dependent strategy presented in this paper are shown in Figure 6c for increasing mass ratio. Accordingly, by implementing the presented tuning and machining at spindle speeds calculated by \(g_{\text{a}}\), the maximum stability is guaranteed.