## 3 Dynamic Responses for Ball End Milling
### Numerical Results and Discussion  
The vibration responses of the tool-tip for ball end milling are solved numerically using the GPEM for the flexible rotor-bearing system in this study. The used cutting conditions and the cutting parameters are given in Table 1[14; 18]. The cutting force coefficients \(K_{\text{ac}}\), \(K_{\text{tc}}\), \(K_{\text{rc}}\) are given in Appendix C. Table 2 gives the configuration data for the rotor-bearing system modeled in Figure 1b. \(E\) and \(\rho\) represent the shaft elastic modulus and density, respectively. \(d_{\text{O}}\) and \(d_{\text{I}}\) represent the shaft outer and inner diameters, respectively. The authors in [18] proposed the approximated critical speeds of the spindle \(N_{\text{cr}}\) (rpm) as:  
\[N_{\text{cr}}=\frac{f_{\text{n}}}{2\pi\cdot(m_{\text{k}}\cdot N_{\text{f}})} \times 60,\;\;m_{\text{k}}=1,\;2,\;3,\;\ldots \tag{16}\]  
where \(f_{\text{n}}\) (rad\(/\)s) represents the natural frequency of the tool-tip, \(N_{\text{f}}\) represents the flute number, and \(m_{\text{k}}\) represents the vibration cycle number of the tool from one flute to another during cutting. Different combinations of \(m_{\text{k}}\) and \(N_{\text{f}}\) lead to different excitation lines in the whirl map of the flexible rotor-bearing system, which will give different relationships between natural frequency and critical speed. Figure 4 is the whirl map which shows that the forward and backward modes are excited in the flexible rotor system under spindle speeds. The natural frequencies of forward mode increase as spindle speeds increase, but those of backward mode decrease as spindle speeds increase. The parameters \(N_{\text{f}}\) and \(m_{\text{k}}\) determine the excitation line in the whirl map. In this study, \(m_{\text{k}}=1\), and two cases including \(N_{\text{f}}=2\), \(3\), are used. Thus, the two whirl ratios are \(1/2\) and \(1/3\). The excitation line may intersect the natural mode curves at some corresponding spindle speeds, called the critical speeds. The natural frequencies and critical speeds of various modes can be expressed in Table 3 to show that the critical speeds of forward mode are higher than those of backward mode. Their associated mode shapes of forward and backward modes at critical speeds, with \(2\) and \(3\) flutes of the tool, are expressed in Figure 5. It is shown that the eigenvector magnitude for the mode \(1\) is the largest at tool-tip C, but those for the mode \(2\) is relatively small. In addition, when the spindle rotates at a higher critical speed, the frequency difference between the forward mode and backward mode becomes larger. This phenomenon is mainly due to the gyroscopic effect of the flexible rotor. Therefore, some lower natural modes will be significantly excited to become the main components of the tool-tip response. This will be verified in this section. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cutting Condition** & \\ \hline \multicolumn{3}{c}{**Carbide Ball End Mills**} \\ \hline Tool radius \(R_{0}\) & R3 & Workpiece material: Ti\({}_{6}\)Al\({}_{4}\)V Cutting coefficient \\ Tool length \(c\) & 30 mm & Shear stress \(\tau=613\) MPa \\ Axial depth of cut \(\hat{a}\) & 1 mm & Friction angle \(\beta=19.1+0.29\)\(\alpha\) \\ Feed per tooth \(f_{\text{d}}\) & 0.3 mm (2 flutes) & Chip compression ratio \(\gamma=ph^{\text{d}}\) \\ Rake angle \(\alpha\) & \(5^{\circ}\) & \(p=1.755-0.028\)\(\alpha\), \(q=0.331-0.0082\)\(\alpha\) \\ Helix angle \(l_{0}\) & \(30^{\circ}\) & Edge force coefficients: \\ \hline \(\rho=1.45\times 10^{-5}\) kg\(/\)mm\({}^{3}\) & \(K_{\text{te}}=24\) N\(/\)mm, \(K_{\text{re}}=43\) N\(/\)mm, \\ \(E=2.14\times 10^{5}\) N\(/\)mm\({}^{2}\) & \(K_{\text{ae}}=-3\) N\(/\)mm \\ \(N_{\text{f}}=2\) and \(3\) & Cutting force coefficients: given in Appendix C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cutting condition and cutting parameters.