## 3 Experimental Verification of Chatter Suppression Effect with TMDs
### Effect of Various Circumferential Arrangements of TMDs on Chatter Suppression  
According to the experimental results presented in Section 2.3, the shell mode with 6 nodes in the circumferential direction was the predominant chatter vibration mode for the workpiece wall thickness of 4 mm. In a previous study [27], the vibration amplitude during chatter generation was reduced when three TMDs with a mass ratio of 1.5% to the workpiece mass were attached to the workpiece tip at an equal pitch in the circumferential direction. In addition, prior research has demonstrated that the installation of an additional mass with a mass ratio of 1.5% could reduce chatter vibration. Therefore, it was suggested that the chatter was reduced by the added mass rather than the influence of the TMD, for a sufficiently large mass ratio. However, the negative impacts of centrifugal forces, such as increased runout and deformation of the workpiece, become a concern if a large mass is added to a thin-walled cylinder. In the present study, the influence of the mounting position and the tuning parameters of the TMDs on chatter was investigated for extremely small mass ratio of the TMD (mass ratio of approximately 0.2%). The circumferential installation pitch of the TMDs or the masses was defined as the mounting angle \(\theta_{1}\) and \(\theta_{2}\) based on the TMDs or the masses mounted on the \(y\)-axis, as depicted in Figure 8. In particular, the mounting position of the TMDs or the masses in the \(z\)-axis direction was defined as the distance \(l_{\rm d}\) from the workpiece tip. First, the chatter suppression effect was examined when three TMDs or three additional masses were installed at equally spaced positions in the circumferential direction the same as the mounting arrangement of the TMDs in the previous study [27]. The mounting angle of the TMDs or the masses in the circumferential direction were \(\theta_{1}=\theta_{2}=120^{\circ}\). The mounting positions of all the TMDs or all the masses along the \(z\)-axis were \(l_{\rm d}=35\) mm. The  
Figure 8: Mounting position of TMDs or additional masses along the \(z\)-axis and circumferential direction. Figure 7: Frequency response functions for each beam length TMD. natural frequencies of the adopted 3 TMDs were 1990 Hz close to the chatter frequency of (3,1) mode. The workpiece vibration waveforms measured by the eddy current sensor of #2 measuring point with and without the masses or TMDs are illustrated in Figure 9, which shows that the workpiece vibration amplitude during the chatter generation is reduced by adding the masses or TMDs; however, the chatter cannot be completely controlled. Figure 10 depicts the frequency analysis results of the workpiece vibration displacement with and without the masses or TMDs during cutting, 25 s after the initiation of cutting (at 7.6 mm from the workpiece tip). As observed from Figure 10b,c, the chatter frequency was observed upon attaching the masses or TMDs. Compared to the case of the mounted TMDs with a mass ratio of 1.5% arranged at an equal pitch in the circumferential direction in the previous study, the installation of the TMDs with a mass ratio of 0.2% did not reduce the chatter vibration very well. Figure 10: _Cont._  
Figure 9: Vibration displacement of workpiece at 2 measurement point during cutting: (**a**) Without and with masses (mounting arrangement: \(\theta_{1}=\theta_{2}=120^{\circ}\), \(l_{\text{d}}=35\) mm); (**b**) Without and with TMDs (mounting arrangement: \(\theta_{1}=\theta_{2}=120^{\circ}\), \(l_{\text{d}}=35\) mm). Figure 11 displays the results of enlarging the workpiece vibration waveforms at nine measuring points of the eddy current sensors during a single rotation cycle when the TMDs were installed. As indicated by the red-dashed arrows in Figure 11, the position corresponding to the vibration nodes at the nine measurement points moved forward by half a rotation in 0.08 s, which is half of the rotation period.