# Self-Tuning Algorithm for Tuneable Clamping Table for Chatter Suppression in Blade Recontouring
## 5 Iterative Tuning Algorithm for the TCT  
The excitation frequency dependent tuning strategy leads to a theoretical increase of the stability limits by optimising the system dynamics for each chatter frequency \(\omega_{\text{c}}\). Obviously, the direct application of this tuning would imply that the chatter frequency is known beforehand for every value of the spindle speed \(\Omega\). In practice, the tuning will only take place once the process is unstable, and for a certain spindle speed \(\Omega\) the arising unstable frequency \(\omega_{\text{c}}\) will vary with the tuning \(f\), since the dynamics of the TCT-blade assembly vary with every tuning. Consequently, the 'final' \(\omega_{\text{c}}\) cannot be known in advance. The practical realization of the previously developed tuning strategy must therefore be attained by an algorithm in which the optimal tuning is sought by iteratively measuring the chatter frequency and performing the tuning until the convergence is achieved.  
The flowchart of the algorithm proposed for the TCT is shown in Figure 7, which is implemented in a real-time controller. Regardless of the machining parameters, the TCT is initially tuned according to the broad-bandwidth tuning strategy in (22). Once the machining starts, the blade undergoes forced vibration even for the stable cutting situation due to the unavoidable stationary part of (7). The part vibration can be estimated by microphones or other contactless solutions, avoiding sensors that could interfere with the cutting process. Then, the vibration measurement is fed to the real-time controller for the continuous computation of the frequency spectrum. This forced vibration, which manifests itself in the form of harmonics at the tooth passing frequency \(\omega_{Z}=2\pi/T_{Z}\), does not usually entail a real hazard for an acceptable part quality, except in finishing passes where high surface quality is required. When regenerative vibrations arise, a dominant high amplitude peak at a frequency higher or lower than the standalone natural frequency \(\omega_{2}\) (for \(\beta_{0}>0\) and \(\beta_{0}<0\) cases, respectively) can be noticed, leading to deep marks and bad surface quality.  
Therefore, the real-time controller detects the onset of chatter when the frequency of the maximum vibration peak is not an integer of the tooth passing frequency \(\omega_{Z}\). If this is the case, the optimal natural frequency of the table \(\omega_{1}\) is determined according to the measured chatter frequency \(\omega_{\rm c}\) by setting \(\omega_{1}=f_{\rm o}(\omega_{\rm c}/\omega_{2})\,\omega_{2}\), with \(f_{\rm o}(g)\) from the expression in (27). The position of the rotary spring is then modified according to (1) to match the required natural frequency. This procedure is iteratively carried out until the machining becomes stable or when a limit of six iterations is reached, if the process cannot be stabilized at all.  
However, under certain cutting parameters, the tuning may oscillate and not converge to its optimal value even for a theoretically stabilizable point, as the changes on the assembly dynamics produced by the tuning lead to too large chatter frequency jumps. In these instances, the convergence of the algorithm can be improved as follows: after a threshold of three tuning iterations, the ideal objective tuning is calculated with the expression in (27), but in spite of directly tuning the device to that value, the device is tuned to the mean value of the actual tuning and the calculated ideal tuning, achieving a better convergence.