## 2 Experiment and Measurements
### Signal Processing  
In this experiment, the vibration accelerations at spindle and vise were measured in the three directions. The relative vibration accelerations between the spindle and the vise were also calculated. The current values of spindle were recorded synchronously during the cutting process. Since all the signals contain the measurements at different stages of milling process, such as idling, initial cutting, stable cutting and tool retracting, only the measured data of 14 s within the stable cutting were utilized for the steps of signal processing and feature extraction thereafter to enhance the surface roughness prediction of milling process. In general, the vibration and current measurements may inevitably contain complicated composition at different frequencies, including the uncorrelated signal components, disturbances and noises. During the cutting process, the vibration behavior may present the nonstationary manners, such as amplitude/frequency modulation or intermittency, while the blades of cutter move forward and try to remove the material from work-piece. On the other hand, the spindle currents are normally varying with time because the servo control system of spindle tries to overcome the various cutting resistance and then keeps the spindle running at a constant rotation speed. Since the vibration measurements and current signals normally present the complicated nonstationary behaviors, all the time series signals were first separated through the empirical mode decomposition (EMD) method [17], which can be formulated as  
\[x(t)\ =\ \sum_{k=1}^{m}c_{k}(t)+r_{m}(t) \tag{1}\]  
Figure 1: Two tri-axial accelerometers on spindle (**left**) and vise (**right**). \begin{table}
\begin{tabular}{c c} \hline Cutting speed (m/min) & 30, 35, 40, 45 \\ Feed per tooth (mm/tooth) & 0.05, 0.12, 0.18, 0.25 \\ Cutting depth (mm) & 0.5, 0.8, 1.2, 1.5 \\ Clamping force of vise (N) & 9807, 26,479, 45,112, 62,765 \\ Accumulated volume removal per cutter (mm\({}^{3}\)) & 0–7584 \\ Accumulated operation time per cutter (second) & 0–266.4 \\ \hline \end{tabular}
\end{table} 
Table 2: Cutting parameters, accumulated volume removal and operation time. where \(x(t)\) represents the measured complicated signal, \(c_{k}(t)\) is the \(i\)-th intrinsic mode function (IMF) and \(r_{m}(t)\) represents the signal residue or trend. Figure 2 illustrates the IMF1 to 16 of one of the vibration signals. According to the spindle speeds, the IMFs that contain the frequency components of two to four times of the spindle rotating frequencies were collected to synthesize the associated reconstructed signals. In such a way, the essential features related to the milling process can be reserved and those uncorrelated signal components, such as noise and signal trend, can be removed. In the cutting process, the tool blades cyclically impact the work-piece and hence the vibration behaviors normally present periodically amplitude-modulation phenomenon in spindle and work-piece. Meanwhile, the spindle current signal may also reveal the amplitude modulation behavior since the cutter encounters the periodic resistance from the work-piece and hence the servo control system tries to make the spindle at a constant rotation speed through varying the current. Therefore, the envelope analysis is capable of demodulating the measurements and is one of the useful techniques to extract the signal features that are correlated to the amplitude modulation characteristics. The envelope of  
Figure 2: Illustration of vibration signal intrinsic mode functions (IMFs) through empirical mode decomposition (EMD). measurement can be determined by means of the analytical signal method. The complex analytical signal \(z(t)\) of measurement \(x(t)\) is computed through the Hilbert transform [17],  
\[z(t)\ =\ x(t)+jH[x(t)]\ =\ A(t)e^{i\phi(t)} \tag{2}\]  
where \(x(t)\) is the measured signal, \(H[\bullet]\) represents the Hilbert transform operator and \(A(t)\) is the envelope of signal \(x(t)\).