## 2 Theory of Chatter Identification
### Principle of Chatter Identification  
Because of the inevitability of vibration in the cutting process, the displacement signals collected in the 3D-EVC processes are actually nonlinear and non-stationary signals. In the process of 3D-EVC, the selection of the chatter signal is significant for correctly identifying instability phenomena. In the process of the specific chatter identification experiment, the signal selected during the 3D-EVC process should follow the following two points: (1) the change of the state in the cutting process can be reflected from two aspects of the time domain and the frequency domain; and (2) the assembly of the sensor cannot affect the stability of the whole cutting system. In the traditional machining process, the common signals applied in chatter identification are mainly the tool tip vibration displacement signal, the cutting force signal, the sound signal and the cutting acceleration signal. However, 3D-EVC technology is an ultra-precision manufacturing technology, and the cutting process has high precision as the small cutting rate, high cutting speed, and low cutting depth. In addition, the displacement sensor is easy to assemble with the tool without affecting the whole cutting system characteristics with high resolution. Therefore, the displacement  
Figure 2: Flowchart of EMD algorithm. sensor is used to measure the vibration and displacement signals of the tool tip as the main signal for chatter identification in the process of 3D-EVC. In the process of 3D-EVC, the vibration frequency under a stable machining state is the frequency of the tool tip vibration set in the 3D-EVC apparatus. In order to avoid the phenomenon of over cutting or not cutting during the process, the vibration frequency of tool ellipse is relatively low. In this paper, the vibration frequency of the tool tip in the three directions is set as 40 Hz, so the chatter frequency is higher than the original vibration frequency of the tool. When the cutting process becomes unstable during processing, the phenomenon of frequency shift will appear and the unstable frequency of chatter phenomenon during the cutting process will exceed the elliptical vibration frequency applied to the diamond tool in the non-resonant 3D-EVC apparatus. On the other hand, another important factor in the chatter identification strategy is the selection of chatter characteristics. There are many methods for extracting the features of the chatter state in the time domain and frequency domain. The mean square frequency (MSF) is an ideal characteristic value which could reflect the changing situation of the cutting signal in the frequency domain. MSF represents the weighted mean of the square of the vibration frequency, and the weight is the amplitude of the power spectrum. In the cutting process, the machining frequency is not a constant value due to the chatter appearance as a result of the change of cutting parameters, tool wear, the properties of the work-piece material and the processing mechanism under different processing methods, and so on. The chatter phenomenon will appear and the frequency of chatter gradually increase and exceed the elliptical vibration frequency applied to the diamond tool in the non-resonant 3D-EVC apparatus with the change of the cutting parameter. The formula of MSF is shown as the following:  
\[MSF=\sum\limits_{i=2}^{N}{x_{i}^{2}}/{4\pi}{2}{\sum\limits_{i=1}^{N}{x_{i}^{2}}} \tag{5}\]  
In addition, the other parameter used for chatter identification is the one step self-correlation coefficient \(\rho_{1}\). In general, \(\rho_{1}\) has a consanguineous relationship with the variety of energy in vibration signals, the value of \(\rho_{1}\) will reduce suddenly when the energy concentrates around the chatter frequencies when the amplitude of vibration signals becomes uneven with the chatter frequency. The value of \(\rho_{1}\) can be expressed as:  
\[\rho_{1}=\cos{2\pi}{f_{i}}\Delta \tag{6}\]  
where \(\Delta\) is the sampling interval, the value of \(\rho_{1i}\) decreased with the increase of \(f_{i}\).