## 2 Chatter Generated during Turning of a Thin-Walled Cylinder
### Clutter Vibration Mode  
The present study focuses on the chatter generated during the finishing process of the external turning of a thin-walled cylindrical workpiece. The turning test in this study was performed on a CNC lathe (Takisawa Machine Tool Co., Ltd., TAC-510 (Okayama, Japan)). The cutting conditions in which the chatter could occur were determined from the preliminary cutting tests with varying cutting conditions. The width of cut was 0.025 mm, the cutting speed was 150 m/min, and the feed rate was 0.05 mm/rev for all the turning tests in this study. The cutting range was selected from the tip of the workpiece to 10 mm from the tip during feeding in the \(z\)-axis direction. To measure the workpiece vibration displacement in the radial direction during chatter generation, 9 eddy current sensors (Applied Electronics Corporation, PU-05 (Kanagawa, Japan)) were arranged at a regular pitch of 20\({}^{\circ}\) in the circumferential direction of the workpiece, as depicted in Figure 4a. The sensors were positioned at 5 mm in the axial direction from the tip of the workpiece, as shown in Figure 4b. In particular, sensor position #6 corresponds to the cutting point of the triangular insert (Tungaloy Corporation, TNMG160404-SS GH330 (Fukushima, Japan)), and the tool specifications are listed in Table 2. The occurrence of chatter was determined when the peak amplitude of the chatter frequency exceeded the threshold value at any one of the nine measurement points. The threshold value in the present work was defined as 3 um, that is, ten times the resolution of the eddy current sensor. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode Number** & **Natural Frequency** & **Damping Ratio** \\ _(m, n)_ & **(Hz)** & **(\%)** \\ \hline \hline \((2,1)\) & 1610 & 0.31 \\ \((3,1)\) & 1943 & 0.093 \\ \((4,1)\) & 3268 & 0.049 \\ \((1,1)\) & 3620 & 0.51 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Natural frequencies and damping ratio of workpiece for a 4.0 mm wall thickness of the workpiece. Figure 3: Frequency response functions: (**a**) Workpiece frequency response function for a 4.0 mm wall thickness of the workpiece; (**b**) Tool frequency response function in \(x,y,\) and \(z\) directions. The waveforms of the workpiece vibration displacement in measuring points 1 to 9 during chatter generation are shown in Figure 5a for a 4.0 mm wall thickness of the workpiece. The frequency analysis of the workpiece vibration displacement during chatter generation (Figure 5a) indicated that the chatter frequency was 1995 Hz, which was approximately equal to the natural frequency of the (3, 1) mode of the workpiece (Figure 3a; Table 1). As observed from Figure 5a, the phase was almost inverted in comparison to the vibration waveforms of measuring points 2 and 4 or those of 5 and 7. The chatter vibration mode displayed in Figure 5b was obtained by interpolating the 9 measured vibration displacements, considering the symmetry of the workpiece vibration mode. The chatter vibration mode displayed in Figure 5b corresponds well with the (3, 1) mode exhibited in Figure 3a. The blue arrows in Figure 5a,b represent the rotational direction of the workpiece, whereas the red-dashed arrow in Figure 5a plots the position of the antinode of the workpiece vibration at each measurement point. As indicated in Figure 5a, the antinode positions travel in the direction opposite to the workpiece rotation with time. Therefore, the chatter vibration mode was observed to travel in the direction opposite to the workpiece rotation, as indicated by the red arrow in Figure 5b. In particular, the chatter vibration mode was not fixed in space. The waveforms of the workpiece vibration displacement in measuring points 1 to 9 during chatter generation are shown in Figure 5c,d for a 2.6 mm and a 1.5 mm thick wall, respectively. As observed from Figure 5c,d, the chatter vibration modes for the other 2 thick walls rotate asynchronously with the rotation of the workpiece, the same as the case of a 4 mm thick wall, displayed in Figure 5a.