# Self-Tuning Algorithm for Tuneable Clamping Table for Chatter Suppression in Blade Recontouring
## 6 Validation of the Concept
### Initial Value Time Domain Simulations with Fly-Over Effect  
So far, an ideal linear milling model has been considered in order to devise the tuning algorithm and validate it through semidiscretization calculations. However, the real milling cases are subdued to the fly-over non-smoothness [51], effect by which the cutting edge jumps out of the cut due to the vibration. Apart from being the cause of the vibration amplitude reaching a threshold instead of growing indefinitely, this strong non-linearity also induces a multi-regenerative effect than can make the chatter frequencies diverge from the ideal linear case [52] and, thus, compromise the effectiveness of the tuning strategy. In order to assess the robustness of the self-tuning algorithm against these deviations, initial value time domain simulations of (2) have been performed together with the self-tuning algorithm. With this purpose, (2) is transformed onto its modal state-space form and numerically solved through an exponential time differencing forward Euler scheme [53]. For each discrete time step, the cutting force in (7) is calculated by considering a total of 20 prior tooth passes for the computation of the chip thickness and switching it off whenever fly-over takes place (\(h_{i}(t)<0\)). For each \(\Omega,a\) point, the chatter detection algorithm is run every 100 tool rotation periods, of which the last 60 periods are taken for the FFT calculation. Chatter is detected and its dominant frequency \(\omega_{c}\) measured if any peak is noticed at frequencies not matching the tooth passing frequency \(\omega_{Z}\) and if its amplitude is higher than the 30% of the largest tooth passing harmonic. If this is the case, the tuning of the device \(\omega_{1}\) is computed according to Figure 7 based on the measured chatter frequency \(\omega_{c}\), and the system matrices corresponding to the new tuning \(k_{1}=m_{1}\,\omega_{1}^{2}\) are recalculated. Here, it is considered that the tuning mechanism (rotary spring) is agile enough to perform the system retuning without affecting the system stability. Therefore, the dynamics of the tuning mechanism are omitted and the tuning is assumed to take place instantaneously. The process is iteratively repeated until the stability is reached or the tuning goes into an oscillatory situation (machining stability is not possible). The results of the time domain simulations for the A, B, C and D points from the stability chart in Figure 8 are shown in Figure 9. As predicted by the semidiscretization calculations, the algorithm successfully stabilizes the initially Hopf unstable cases. Point B (see Figure 9b), chosen according to the stability asymptote in Figure 6c, confirms that large stability gains can be attained with the TCT by integrating the self-tuning algorithm and adjusting the spindle speed to the stability asymptote. The capacity of the algorithm to enhance the stability in period doubling zones is also demonstrated in Figure 9c, as one case of flip chatter was successfully suppressed by the tuning algorithm. An 'unstabilizable' case can be seen in Figure 9d, where an oscillatory behavior between two unstable tunings is reached. Finally, once the capability of the algorithm to maximize the stability of the system has been demonstrated, it would also be important to study the robustness aspects of the algorithm itself. One of the main hazards for all the mode tuning strategies is the uncertainty with respect to the actual dynamic parameters of the system, namely, initial deviations of the part dynamics or deviations due to the material removal. In this sense, an in-depth analysis of the uncertainty and variability of the system parameters and their effect on the tuning robustness would be decisive for a proper design of devices based on the TCT architecture. Figure 9: Results of the initial value time domain simulations with fly-over effect for the milling case in Table 2. (i) Blade tip displacement \(x_{2}\). (ii) standalone TCT tuning frequency \(\omega_{1}\).