## 3 Experimental Verification of Chatter Suppression Effect with TMDs
### Effect of Various Equivalent Mass Ratios of TMD on Chatter Suppression  
The chatter suppression effect was investigated upon further reducing the equivalent mass ratio of the additional mass or the TMD. The chatter vibration mode of the cantilever-supported workpiece exhibited the largest amplitude at the tip and the smallest amplitude at the fixed end of the workpiece. Therefore, when the mounting position of the masses or TMDs was moved towards the vicinity of the fixed end, the equivalent mass ratio of the masses or TMDs to that of the workpiece became smaller. The equivalent mass ratio is defined as the ratio of the actual mass of the additional mass (5.0 g) to the equivalent mass of the workpiece without the masses or TMDs in this paper. The equivalent mass of the workpiece for each mounting position \(l_{\text{d}}\) was obtained by curve-fitting its frequency response functions at the tip when it was excited at \(l_{\text{d}}\) = 35 mm, 50 mm, and 70 mm from the tip using an impact hammer (PCB Piezotronics, Inc, 086C01 (Depew, NY, USA)). The chatter suppression effect was verified when the masses or TMDs were mounted in the vicinity of the fixed end of the workpiece. First, the presence or absence of chatter was examined with the masses attached at \(l_{\text{d}}\) = 50 mm or \(l_{\text{d}}\) = 70 mm from the workpiece tip. The circumferential mounting arrangement of the masses was unequally spaced at \(\theta_{1}\) = 110\({}^{\circ}\) and \(\theta_{2}\) = 220\({}^{\circ}\), similar to the case in Figure 12a, to compare with the case of \(l_{\text{d}}\) = 35 mm discussed in Section 3.2. The equivalent mass ratios were 0.61% and 0.36% when the masses were added at \(l_{\text{d}}\) = 50 mm and \(l_{\text{d}}\) = 70 mm from the workpiece tip, respectively (equivalent mass ratio is 1.1% in the case of \(l_{\text{d}}\) = 35 mm). The workpiece vibration waveforms measured by the eddy current sensor of number 2 measuring point for the masses attached at \(l_{\text{d}}\) = 50 mm  
Figure 12: Workpiece vibration displacement at number 2 measurement point during cutting. Comparison of (**a**) vibration waveforms with and without masses (mounting arrangements: \(\theta_{1}\) = 110\({}^{\circ}\), \(\theta_{2}\) = 220\({}^{\circ}\), \(l_{\text{d}}\) = 35 mm), (**b**) frequency analysis with and without masses (mounting arrangements: \(\theta_{1}\) = 110\({}^{\circ}\), \(\theta_{2}\) = 220\({}^{\circ}\), \(l_{\text{d}}\) = 35 mm), (**c**) vibration waveforms with and without TMDs (mounting arrangements: \(\theta_{1}\) = 110\({}^{\circ}\), \(\theta_{2}\) = 220\({}^{\circ}\), \(l_{\text{d}}\) = 35 mm), and (**d**) frequency analysis with and without TMDs (mounting arrangements: \(\theta_{1}\) = 110\({}^{\circ}\), \(\theta_{2}\) = 220\({}^{\circ}\), \(l_{\text{d}}\) = 35 mm). and \(l_{\rm d}=70\) mm from the tip are illustrated in Figure 13a,c respectively. The frequency analysis results of the time waveforms for the 0.05 s period in the initial 12.5 s after cutting (vertical dotted line in Figure 13a,c are presented in Figure 13b,d respectively. As observed from Figure 13a,b the chatter was mostly suppressed when the masses were attached at \(l_{\rm d}=50\) mm from the tip. In contrast, the chatter was generated when the masses were attached at \(l_{\rm d}=70\) mm from the tip. Therefore, the chatter could not be suppressed even if the masses with an equivalent mass ratio of less than 0.61% were unequally arranged in the circumferential direction to avoid coincidence with the pitch of the nodes of the chatter vibration mode. Next, the chatter suppression effect was examined for the TMDs mounted at \(l_{\rm d}=70\) mm from the workpiece tip. The circumferential mounting arrangement of the TMDs was unequally spaced at \(\theta_{1}=110^{\circ}\) and \(\theta_{2}=220^{\circ}\), similar to the case in Figure 12c, to compare with the \(l_{\rm d}=35\) mm case discussed in Section 3.2.