# Self-Tuning Algorithm for Tuneable Clamping Table for Chatter Suppression in Blade Recontouring
## 2 Semi-Active Tuneable Clamping Table: Concept and Benefits  
Similarly to TMDs, the tuneable clamping table proposed for blade recontouring is based on mode tuning. When tuning the controlled mode close in the frequency band to the lowly-damped critical mode of the blade, the two modes interact and the shapes of the neighboring modes are combined. Due to the existence of the damping of the device, a partial transfer of damping between the ideal (stand-alone) modes is achieved, which can be used to increase the dynamic stiffness of the system. Nevertheless, the classical TMD and TCT concepts present significant differences. Figure 2 shows the differences in architecture between the well-known tuned mass damper and the tuneable clamping table. As opposed to the TMDs, where the controlled mode is placed in-parallel to the excitation force flow (Figure 2a), the TCT approach adopts an in-series configuration (Figure 2b), clamping the blade from below. In this way, mode tuning and the subsequent increase of the dynamic stiffness at the machining point can be achieved without physically attaching any device to the blade near the machining zone that could interfere with the cutting process. Moreover, this permits the development of a universal clamping system for the entire blade set without the need for an array composed by specific fixtures or dampers for every blade geometry. In order to cope with a wide set of blades with different dynamics, the TCT concept is implemented in a semi-active form in this paper. Semi-active damping devices, first introduced in [24] for chatter suppression, require external energy to achieve the tuning, but unlike active devices, the dissipation of vibration energy takes place without any external energy input. The alteration of the stiffness, damping or mass properties of the device mode can be achieved by multiple means. Purely mechanical assemblies based on preloaded elastomers [24], position-adjustable moving masses [25; 26] or rotary springs [27; 28; 29; 30] have been proposed to control the stiffness of the damper. Regarding the damping, eddy current modules have been proven to be a reliable alternative for obtaining a nearly-ideal and adjustable viscous damping [18; 28; 29]. The use of electrorheological and magnetorheological fluids to simultaneously control the stiffness and damping of the device has also been investigated [31], with application over vibration attenuation in thin walled part milling as well [32; 33]. Finally, it is also worth mentioning the use of piezoelectric shunts for semi-active damping purposes [34; 35; 36]. Indeed, regardless of how the mode is controlled, semi-active devices require a control strategy to achieve the tuning of the device mode to the critical mode to be damped. In this sense, velocity [34] or force [37] feedback control strategies, as well as fuzzy control algorithms [38; 39], can be found in the literature. Nonetheless, when it comes to suppressing regenerative chatter --rather than general vibration problems--iterative algorithms based on the measured chatter frequency show a higher degree of effectiveness [29; 40]. In the case of the semi-active TCT presented here, the controlled mode is materialized as a dominant translational mode with a fixed moving mass \(m_{1}\) guided by flexures of stiffness \(k_{\text{f}}\). The table features a rotary spring (see Figure 3) with unequal stiffnesses \(k_{\text{s,1}}\) and \(k_{\text{s,2}}\) in its two main directions and driven by a motor-encoder assembly. This permits externally modifying the stiffness \(k_{1}\) and, consequently, the natural frequency \(\omega_{1}\) of the device with the angular position of the spring \(\theta\) as  
\[k_{1}^{-1}=k_{\text{f}}^{-1}+\Big{(}k_{\text{s,1}}\cos^{2}\theta+k_{\text{s,2 }}\sin^{2}\theta\Big{)}^{-1}. \tag{1}\]  
In this manner, blade modes within the frequency range \([\omega_{1,\text{min}},\ \omega_{1,\text{max}}]\), as shown in Figure 3, can be tuned with the TCT.