## 4 Results and Discussions  
### Initial Identification  
In the cutting process, there are two main characteristics to identify undesired vibrations: (1) the vibration amplitude increases gradually in the time domain as a result of the vibration energy increases; and (2) the frequency shift phenomenon will appear with the change of cutting the state in the frequency domain. This is one of the most significant signs to determine the occurrence of undesired vibration. The time domain signal of three directions in the cutting process is collected by the displacement sensor, when the depth of cut is set to 5 \(\upmu\)m, as shown in Figure 5. It can be seen from the time domain signals of three directions on the cutting tool that the vibration amplitudes along the Z direction have greater values compared to the other two directions. This is because that the oscillation generated on the tool tip of 3D-EVC apparatus in Z direction contains more capacity. Consequently, vibration signals in Z direction are considered as the research object and further analysis in this paper. In order to obtain the different cutting states in 3D-EVC process, the original signals of vibration in Z direction are obtained under various cutting conditions (the depth of cut was 5 \(\upmu\)m, 10 \(\upmu\)m and 20 \(\upmu\)m, respectively). The time domain signals in three kinds of cutting depth are shown in Figure 6a-c, the amplitude of vibration signals in frequency domain obtained by FFT are shown in Figure 7a-c, respectively. Three typical states (stable cutting, transient state, and chatter state) in the cutting process of 3D-EVC can be obtain based on the variation of frequency with the varied of cutting depth. In the stable condition of the cutting process, the amplitude of vibration signal is relatively small and there is no larger peak or trough in Figure 6a; the frequency domain spectrum obtained by FFT is shown in Figure 7a. It is observed that the characteristics of the peaks appear at the frequency about 40 Hz corresponding to the natural frequency of 3D-EVC apparatus. In the condition of a transient state, some smaller fluctuations appeared in the time domain, and the amplitude of vibration grows, with a small change in Figure 6b, while the frequency domain spectrum has a significant change which appears as two peaks in 150 Hz and 200 Hz, as shown in Figure 7b. But the amplitude is relatively small \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \
\begin{tabular}{c} **Tensile Strength Rm** \\ **(N/mm\({}^{2}\))** \\ \end{tabular} &
\begin{tabular}{c} **Elongation to** \\ **Failure/(\%)** \\ \end{tabular} & **Vickers Hardness** & **Density (g/cm\({}^{3}\))** \\ \hline Copper & 335–370 & 24 & 85–128 & 8.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties. Figure 5: The time domain signal of three directions in 3D-EVC process (spindle speed was 90 rpm/min, and the feed rate was 10 mm/min, slotting. and there is no obvious change in the time domain. In the condition of a chatter state, the amplitude of vibration increases obviously, the change of amplitude has an effect on the three-dimensional elliptic trajectory of the tool-tip in Figure 6c. Moreover, the frequencies of the characteristic peaks at 150 Hz and 200 Hz are formed by the major components in the frequency spectrum, as shown in Figure 7c. In this stage, the cutting process is accompanied by harsh noise, and the work-piece appears obvious vibration mark. This is caused by the extrusion pressure between the tool-tip and the work-piece increasing with the rise of cutting depth. On the other hand, the cutting condition also affects the quality of the finished surface. Figure 8 shows quality of the finished surface under the three different conditions during the 3D-EVC process, respectively.