## 3 Experimental Verification of Chatter Suppression Effect with TMDs
### Specifications of TMD  
Based on the actual operating vibration measurement results during chatter generation, the shell mode with circumferential nodes was the dominant workpiece vibration mode. In the present study, the TMDs were attached to the inner surface of the rotating workpiece using double-sided tape for the metal. The dimensions of the TMDs are presented in Figure 6a. In general, TMDs are cantilevered beams with varying natural frequencies based on the beam length. The cantilever beam length \(l\) in Figure 6a was set to \(l=19\), 21.5, 24 mm. Moreover, an additional mass of approximately 5.0 g is depicted in Figure 6b almost equal to that of the TMD. The TMDs and the masses are made of carbon steel (C45). The masses are attached to the workpiece instead of the TMDs to compare the vibration characteristics of the workpiece and the chatter suppression effect. To install the TMDs or the masses inside the cylindrical workpiece, the curvature of the bonded part of the TMDs and the masses were altered to match the inner diameter of the workpiece, as portrayed in Figure 6a,b. The picture of the mass and the TMDs are presented in Figure 6c. An impact test of the TMDs, fixed to the ring jig with double-sided tape as depicted in Figure 6d, was conducted using an impact hammer (PCB Piezotronics, Inc, 086E80 (Depew, NY, USA)) under the boundary condition of the cantilevered beam to identify the modal parameters of the TMDs. An accelerometer (Ono Sokki, NP-2106 (Kanagawa, Japan)) was attached to the tip of the TMD with an adhesive. The natural frequencies and the damping ratios of the TMDs were obtained by curve fitting the measured frequency response function for each beam length of the TMD presented in Figure 7. The beam length \(l\) of the TMD, natural frequency \(f_{\text{d}}\), damping ratio \(\zeta_{\text{d}}\), and actual mass \(m\) of each TMD are presented in Table 4. A picture of the TMDs attached to the workpiece is presented in Figure 6e. The number of TMDs or masses was set to 3 or 4 to minimize the imbalance and deformation of the workpiece resulting from the centrifugal force. The TMDs with the same beam length \(l\) were manufactured such that their mass and natural frequency were equal. The mass of the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Wall Thickness** **(mm)** & **Chatter Frequency** **(Hz)** & **Mode** & **Chatter Frequency** **(Hz)** & **Mode** \\ \hline
4.0 & 1995 & \((3,1)\) & – & – \\
2.6 & 1606 & \((3,1)\) & 2409 & (4, 1) \\
1.5 & 1458 & \((4,1)\) & – & – \\ \hline \hline \end{tabular}
\end{table}
Table 3: Chatter frequencies resulting from variations in wall thickness of the cylindrical workpiece.  
workpiece was 2097 g. The mass ratio of the TMD or the additional mass to the workpiece was approximately 0.2%.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**TMD**} & **Length of Beam \(I\)** & **Natural Frequency \(f_{\mathbf{d}}\)** & **Damping Ratio \(\zeta_{\mathbf{d}}\)** & **Mass \(m\)** \\  & **(mm)** & **(Hz)** & **(\%)** & **(g)** \\ \hline (a) & 19 & 2490 & 7.0 & 4.0 \\ (b) & 21.5 & 1990 & 4.8 & 4.3 \\ (c) & 24 & 1600 & 4.5 & 4.5 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Beam length, natural frequency, damping ratio, mass of TMDs.  
Figure 6: TMD and additional mass: (**a**) TMD dimensions; (**b**) Additional mass dimensions; (**c**) Picture of additional mass and TMDs; (**d**) Impact test of TMD; (**e**) Picture of TMDs attached to the workpiece.