## 3 Dynamic Responses for Ball End Milling
### Numerical Results and Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Spindle Section** & \(E\) **(N/mm\({}^{2}\))** & \(\rho\) **(kg/mm\({}^{3}\))** & \(d_{\text{e}}\) **(mm)** & \(d_{\text{I}}\) **(mm)** & **Length** \\ \hline I & \(2.14\times 10^{5}\) & \(7.833\times 10^{-6}\) & 100 & 60 & \(a=216\) mm \\ \hline II & \(2.14\times 10^{5}\) & \(7.833\times 10^{-6}\) & 160 & 60 & \(b=157\) mm \\ \hline III & \(2.14\times 10^{5}\) & \(1.45\times 10^{-5}\) & 6 & 0 & \(c=30\) mm \\ \hline Bearing A: \(K_{\text{xx}}^{b}=K_{\text{yy}}^{b}=1.575\times 10^{5}\) N\(/\)mm, \(C_{\text{xx}}^{b}=C_{\text{yy}}^{b}=15.75\) N\(/\)mm, \(K_{\text{yy}}^{b}=K_{\text{yx}}^{b}=C_{\text{yy}}^{b}=0\) \\ Bearing B: \(K_{\text{xx}}^{b}=K_{\text{yy}}^{b}=1.950\times 10^{5}\) N\(/\)mm, \(C_{\text{xx}}^{b}=C_{\text{yy}}^{b}=19.50\) N\(/\)mm, \(K_{\text{yy}}^{b}=K_{\text{yx}}^{b}=C_{\text{yy}}^{b}=C_{\text{yx}}^{b}=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Configuration data of rotor-bearing system in Figure 1b. Figure 4: Natural frequency vs. spindle speed at various modes for flexible rotor system. Figure 5: The mode shapes for the rotor-bearing system at the critical speeds. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mode** & **Type** & **Flutes** & \(N_{\sigma}\)**(rpm)** & \(f_{n}\)**(rad/s)** \\ \hline \multirow{4}{*}{1} & \multirow{2}{*}{Backward Mode} & 2 & 1323 & 277 \\ \cline{3-5}  & & 3 & 965 & 303 \\ \cline{3-5}  & & 2 & 19,953 & 4179 \\ \cline{3-5}  & & 3 & 2161 & 679 \\ \hline \multirow{4}{*}{2} & \multirow{2}{*}{Backward Mode} & 2 & 6040 & 1265 \\ \cline{3-5}  & & 3 & 4412 & 1386 \\ \cline{3-5}  & & 2 & - & - \\ \cline{3-5}  & & 3 & 9858 & 3097 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Critical speed vs. natural frequency for various modes. \(N_{t}=2\): The numerical studies for \(N_{t}=2\) at different spindle speed are shown in Figures 6-8. Figure 6a shows the displacement of the tool-tip with two flutes in the \(x\) direction, at the spindle speed of 1320 rpm, which is close to the critical speed of the first backward mode. Figure 6b also shows the \(x\)-\(y\) plane orbit of the tool-tip. The orbit is an encircled trajectory which expands clockwise around the origin in a stable loop. This mode is verified to be a backward mode. The fast Fourier transformation diagram (FFT diagram) for the tool-tip response in the \(x\) direction, at the spindle speed of 1320 rpm, is shown in Figure 6c. The tool is with two flutes. It is found that only one major peak, with the frequency of 276.5 rad/s, dominates the response of the rotor system. Figure 7a shows the displacement in the \(x\) direction of the tool-tip with two flutes, at the spindle speed of 6040 rpm, which is close to the critical speed of the second backward mode. It is found that the tool oscillates with the cutting-force frequency combined with a longer cycle vibration. The period of the long cycle is about 0.056 s, which explains how the workpiece surface is often found to have a lot of small waves inside the long cycle wave in the practical milling. From Figure 4, it can be seen that the lower first backward mode with the natural frequency of 112.1 rad/s is excited at the spindle speed of 6040 rpm. Figure 7b plots the tool-tip orbit. It can be seen from the figure that the track of the tip of the tool circles around, and there are many small rings inside. The large ring is excited by the second backward mode, and the small ring corresponds to the first backward mode. This phenomenon can be more clearly observed from the FFT plot shown in Figure 7c. At the spindle speed of 6040 rpm with two flutes, there are two main peaks in the \(x\)-response of the tool-tip. The two frequencies governing the behaviors of tool-tip correspond to the first and second backward modes, respectively. The obvious low frequency peak verifies the long cycle wave phenomenon. A very small peak near the right hand of cutting-force frequency is also found. It can be seen, from Figure 4, to coincide with the first forward mode at the spindle speed of 6040 rpm.