## 3 Experimental Verification of Chatter Suppression Effect with TMDs
### Design for TMDs in Case of Multiple Modes of Chatter Generation  
Table 3 in Section 2.3 revealed that chatter in both the (3,1) and (4,1) modes can occur simultaneously for a workpiece wall thickness of \(t=2.6\) mm. This section describes the chatter suppression effect of the TMDs mounting arrangements on multiple chatter modes. First, the suppression effect of multiple chatter modes was examined by installing three TMDs on the workpiece. The natural frequencies and circumferential installation pitch of the three TMDs are listed in Table 5. The natural frequencies of the TMDs were tuned to be close to the (3,1) mode and (4,1) mode chatter frequencies occurring for a wall thickness of 2.6 mm (Table 3). The mounting position of each TMD in the \(z\)-axis was \(l_{\text{d}}=35\) mm from the workpiece tip. Figure 15a,b display the circumferential mounting arrangements of the TMDs superimposed on the (3,1) and (4,1) modes of chatter. The natural frequencies of the 3 TMDs in mounting arrangement A were tuned to the (3,1) mode chatter frequency for a single TMD and to the (4,1) mode chatter frequency for the others. The circumferential installation pitch of the two TMDs tuned to the (4,1) mode was determined such that it did not coincide with the pitch of the nodes of the (4,1) mode. The natural frequencies of the three TMDs in mounting arrangement B were tuned to the (4,1) mode chatter frequency for a single TMD and to the (3,1) mode chatter frequency for the others. The circumferential installation pitch of the two TMDs tuned to the (3,1) mode was determined such that it did not coincide with the pitch of the nodes of the (3,1) mode. A comparison of the workpiece vibration waveforms measured by the eddy current sensor of #2 measuring point with and without the TMDs during the turning process for mounting arrangement A is presented in Figure 16a. In addition, the root mean square (RMS) values of the workpiece vibration displacement with and without the TMDs from the start time to the end time of cutting are displayed in Figure 16a. The frequency analysis of the time waveform for the 0.05 s period in the initial 28 s after cutting (vertical dotted line in Figure 16a) is presented in Figure 16b. The results presented in Figure 16 suggest that the (4,1) mode chatter can be suppressed even when the number of the TMDs with natural  
Figure 15: Mounting arrangement of three TMDs: (**a**) Mounting arrangement A; (**b**) Mounting arrangement B. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Mounting Arrangement A**} & \multicolumn{2}{c}{**Mounting Arrangement B**} \\ \cline{2-5}  & **Natural Frequency** & **Position (\({}^{\circ}\))** & **Natural Frequency** & **Position (\({}^{\circ}\))** \\  & **(Hz)** & & **(Hz)** & **(\({}^{\circ}\))** \\ \hline TMD 1 & 1600 & 0 & 1600 & 0 \\ TMD 2 & 2490 & 140 & 2490 & 110 \\ TMD 3 & 2490 & 250 & 1600 & 270 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Natural frequencies and circumferential installation pitch of each of three TMDs. A comparison of the workpiece vibration waveforms measured by the eddy current sensor of 2 measuring point and the RMS values with and without the TMDs during the turning process for mounting arrangement B is presented in Figure 17a. The frequency analysis of the time waveform for the 0.05 s period in the initial 28 s after cutting (vertical dotted line in Figure 17a) is presented in Figure 17b. The results presented in Figure 17 suggest that the (3,1) mode chatter can be largely suppressed even when the number of the TMDs with natural frequencies close to the (3,1) mode chatter frequency is 2. On the other hand, the (4,1) mode chatter vibration cannot be sufficiently suppressed when only 1 TMD with a natural frequency is close to the (4,1) mode chatter frequency is used. The results presented in Figures 16 and 17 suggest that at least two TMDs with natural frequencies close to the chatter frequency of each mode are required to suppress the chatter of each mode.