## 4 Results and Discussion  
### Relationship between Tool Chatter and Transient DOC in Micro-Milling  
Figure 3 illustrates the impact of tool chatter on the transient DOC in micro-milling. The origin of the XYZ coordinate system coincides with the center of the tool ball head. Specifically, the tool deformation accompanied by tool chatter can be decomposed into the deflection around the top of the tool insert axis and the translation in space. It is assumed that the known positional translation of the center of the tool ball head in the X, Y and Z direction is \(\Delta x\), \(\Delta y\) and \(\Delta z\), respectively. Therefore, the coordinate of the tool ball head center in the XYZ coordinate system before and after the milling tool deformation is (0, 0, 0)  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(d_{1}\) & \(d_{2}\) & \(d_{3}\) & \(d_{4}\) & \(d_{5}\) \\ \hline
0.54 & 4.89 & \(-3.03\) & 0.014 & 1.12 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameters of the J-C failure criterion algorithm for the Cu workpiece used in the FE model [18].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material Properties** & **Unit** & **Cu** &
\begin{tabular}{c} **Cemented** \\ **Carbide** \\ \end{tabular} & **Diamond** \\ \hline Thermal conductivity & W/(mm*k) & 386 & 90 & 2000 \\ Density & Kg/m3 & 8960 & 14,500 & 4250 \\ Elastic modulus & GPa & 124 & 580 & 1147 \\ Poisson ratio & — & 0.34 & 0.22 & 0.07 \\ Specific heat capacity & J/(Kg*K) & 383 & 220 & 502 \\ Thermal expansion & \(\text{coefficient}\) & \(\text{K}^{-1}\) & \(5\times 10^{-5}\) & \(5.4\times 10^{-6}\) & \(1.18\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Physical parameters of Cu [18], cemented carbide [19] and diamond [20].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**A(MPa)** & **B(MPa)** & **n** & **C** & **m** \\ \hline
90 & 292 & 0.31 & 0.025 & 1.09 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of the J–C constitutive law for the Cu workpiece used in the FE model [18].  
and (\(\Delta x\), \(\Delta y\), \(\Delta z\)), respectively. As shown in Figure 3, a UVW coordinate system is set at the tangent point between the tool ball head and the machined surface. Therefore, the UVW coordinate system can be regarded as the translation of the XYZ coordinate system through a rotation by \(\phi\) around the Z axis, which can be described by the vector of (\(-\)R, 0, 0). Then, the coordinates in the UVW coordinate system after the deformation of the center of the tool ball head can be expressed by Equations (4) and (5):  
\[T=\begin{bmatrix}\cos\phi&-\sin\phi&1&0\\ \sin\phi&\cos\phi&0&0\\ 0&0&0&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}1&0&0&-R\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} \tag{4}\]  
\[\begin{bmatrix}u\\ v\\ w\\ 1\end{bmatrix}=T^{-1}\begin{bmatrix}\Delta x\\ \Delta y\\ \Delta z\\ 1\end{bmatrix} \tag{5}\]  
where \(\phi\) is the pre-deflection angle of the milling tool with respect to the workpiece surface, and \(T\) is the transformation matrix for transforming the XYZ coordinate system to the UVW coordinate system. Therefore, the transient DOC \(t_{0}\) associated with the tool chatter in micro-milling can be expressed by Equation (6):  
\[t_{0}=t-u \tag{6}\]