## 3 Experiment of Diamond Micro-Planing and Power Spectral Density Analysis of Surface Profile  
In this study, a self-developed three-axis machine tool was used as the platform, and a micro-planing experiment was carried out on the surface of oxygen-free copper using natural diamond tools, as shown in Figure 3. The diamond tool was clamped on the \(Z\)-axis, and the oxygen-free copper workpiece was clamped on the \(X\)-axis worktable by a vice. The size of the workpiece to be processed was 15 \(\times\) 22 mm. The tool parameters and basic parameters of planing are shown in Table 1. The selected tool-tip radius is large, suitable for rough and semi-finished machining, and the machining efficiency is relatively high. Diamond tools can generally process nonferrous metals, and oxygen-free copper, as a common optical mold material, was used as the material to be machined in this work. During the planing process, the \(X\)-axis was used as the planing direction, and the \(Y\)-axis was used as the feed direction. In the tool setting step, after the relative positions of the tool and the workpiece in the \(Z\) direction were determined, the \(Z\)-axis remained at the given target position and was always in the servo state. After a single planing action was completed, the \(X\) and \(Y\)-axis motions were used to complete the tool retraction and feed action. The G-code command of the WHILE cycle was used to complete the planing process for oxygen-free copper by the cyclic tool path processing method. The influence of the \(Z\)-axis on machining can be excluded. The tool path trajectory is shown in Figure 4. This processing method can ensure that the processing time for each groove is equal and the processing interval between two adjacent grooves is equal, and it is convenient for the subsequent analysis of the relative vibration between the tool tip and the workpiece time-domain characteristics. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Tool Rake Angle** & **Tool Clearance Angle** & **Tool Radius** & **Tool Sharp Corner** & **Single Feed—\(f\)** & **Depth of Cut** \\ (\({}^{\circ}\)) & (\({}^{\circ}\)) & **(mm)** & (\({}^{\circ}\)) & **(mm/Cycle)** & **(mm)** \\ \hline
0 & 15 & 1.61 & 60 & 0.008 & 0.005 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of diamond tool and workpiece. Figure 4: Planing cycle tool path diagram. Figure 3: Diamond planing experiment. According to the parameters in Table 2, it can be calculated that the single complete planing cycle T\({}_{0}\) is 18.000 s, the corresponding frequency \(f_{0}\) is 0.056 Hz, and the contact time \(t\) between the tool and the workpiece is 2.640 s. When the machine tool is processing, the relative vibration between the tool tip and the workpiece will be reflected on the surface of the workpiece, and its manifestations are spatial ripples of different amplitudes and wavelengths. To study the relationship between the relative vibration and the surface morphology of the workpiece, it is necessary to establish the relationship between the vibration frequency (time domain) and the spatial wavelength of the ripples on the workpiece surface (spatial domain). The surface morphology of planing is determined by the ratio of the vibration frequency \(f_{t}\) between the tool tip and the workpiece and the cycle tool path frequency \(f_{0}\):  
\[\frac{f_{t}}{f_{0}}=I+D \tag{7}\]  
In the formula, \(I\) is the integer part of the frequency ratio; \(D\) is the fractional part of the frequency ratio; and \(I\) represents the number of times that a relative vibration between the tool and the workpiece is completed per cycle of the tool path. The absolute value of \(|\,D\,|\) represents the phase lag between the tool and the workpiece that cannot complete the entire vibration cycle for each cycle of the tool path.