## 4 Discussion  
The experimental results in Section 3 suggest that the chatter suppression effect can be enhanced by arranging the TMDs or the additional masses to avoid coincidence with the spacing of the nodes of the chatter vibration mode. This section considers the reasons for this improvement in the chatter suppression effect when the circumferential installation pitch of the TMDs is not aligned with the pitch of the nodes of the chatter vibration mode. First, the chatter suppression effect of the mounting arrangement of the TMDs is discussed by comparing the chatter vibration modes with and without the TMDs. As depicted in Figure 5a,c,d the observation results of chatter vibration mode of the workpiece without the additional masses or the TMDs suggested that the chatter vibration mode was the shell mode with circumferential nodes and that it rotated asynchronously with respect to the workpiece rotation. As the cylindrical workpiece is symmetrical without the additional masses or the TMDs, the workpiece has two degenerate eigenmodes that are out-of-phase by 90\({}^{\circ}\). In the occurrence of chatter without the masses or TMDs, the chatter vibration of the (3,1) mode, \(R_{m,s}(\theta,t)\), can be expressed as a superposition of two degenerate modes as follows:  
\[R_{m,s}(\theta,t)=C_{m,s}\cos\omega t\cos m\theta+D_{m,s}\sin\omega t\sin m\theta \tag{1}\]  
where \(t\) denotes the time variable, \(\theta\) represents the angular coordinate on the workpiece based on the mounting angle position of TMD 1, \(C_{m,s}\) and \(D_{m,s}\) represent the amplitude of the cosine and sine modes of the (\(m\),\(s\)) mode, respectively, and \(\omega\) denotes the angular frequency of chatter. According to Equation (1), the chatter vibration mode rotates asynchronously with respect to the workpiece rotation when the amplitudes of the two modes are nearly equal. In contrast, the chatter vibration mode rotates synchronously with the workpiece rotation, as observed in Figure 11, when the circumferential installation pitch of the TMDs is equally distributed to ensure coincidence with the node position of the chatter vibration mode. The reason for this synchronous rotation with the workpiece with equally spaced TMDs can be inferred from Equation (1), as follows. The mode in which the mounting position of the TMDs coincided with the antinode position of the degenerate mode was suppressed, whereas the mode in which the mounting position coincided with the node position of the degenerate mode was amplified directly into the self-excited vibration. The mode with the reference TMD1 mounting position as a node corresponds to the sine mode in Equation (1), whereas the mode with the TMD1 mounting position as an anti-node corresponds to the cosine mode in Equation (1). Thus, the amplitude of the cosine mode was reduced by the TMDs owing to the anti-node positions of the TMDs. However, the amplitude of the sine mode was not reduced because the TMDs were located at the node  
Figure 19: Workpiece vibration displacement at number 2 measurement point for mounting arrangement C: Comparison of (**a**) vibration waveforms with and without TMDs, (**b**) frequency analysis with and without TMDs. positions, that is, the TMDs influenced only one of the degenerate modes. Eventually, we concluded that the chatter vibration mode can rotate synchronously with the workpiece rotation because only the sine mode was amplified as the chatter vibration mode. Based on the considerations mentioned above, we conclude that the TMDs can influence both the degenerate modes and can effectively control chatter when the circumferential installation pitch of the TMDs is not equal to the pitch of the nodes of the chatter vibration mode.