## 3 Dynamic Responses for Ball End Milling
### Dynamic Cutting Forces  
[PERSON] and [PERSON] [14] introduced the linear cutting forces used in this study. The geometry of a helical ball end milling tool is given in Figure 2. Furthermore, the cutting forces, chip thickness, and area of cutting segment for ball end milling are plotted in Figure 3. \(\mathrm{d}F_{\mathrm{t}}\), \(\mathrm{d}F_{\mathrm{r}}\), and \(\mathrm{d}F_{\mathrm{a}}\) are the differential dynamic cutting force components acting on the cutter along the tangential, radial, and axial directions, respectively, and are given as:  
\[ \begin{array}{l}dF_{t}=K_{te}\ ds+K_{tc}\ t(\psi_{i},\hat{k})\ db\\ dF_{r}=K_{re}\ ds+K_{rc}\ t(\psi_{i},\hat{k})\ db\\ dF_{a}=K_{ae}\ ds+K_{ac}\ t(\psi_{i},\hat{k})\ db\end{array}  \tag{3}\]  
where \(K_{\mathrm{te}},\ K_{\mathrm{re}},\ K_{\mathrm{ae}}\) represent the edge force coefficients along the tangential, radial, and axial directions and \(\mathrm{d}s\) represents the differential segment length of the curved cutting edge. \(K_{\mathrm{tc}},\ K_{\mathrm{rc}},\ K_{\mathrm{ac}}\) represent the cutting force coefficients (or shear force coefficients) along the tangential, radial, and axial \(\mathrm{l}\) directions \(\hat{t}(\psi_{i},\hat{k})=h(\psi_{i})\sin\hat{k}\) denotes the uncut chip thickness normal to cutting edge, in which \(\psi_{i}\) denotes the instantaneous angular immersion in the global coordinate system, and \(\hat{k}\) represents the angle in a vertical plane between the \(z\)-axis and a point on the flute, \(h(\psi_{i})\) is the thickness of the instantaneous chip. \(\mathrm{d}\ b=\mathrm{d}\ z/\sin\hat{k}\) represents the differential length of the cutting edge, in which \(\mathrm{d}\ z\) is differential axial height. The differential curve length \(\mathrm{d}\ s\), the area of cutting segment \(\mathrm{d}\ A\), and the angular position \(\psi_{i}\) are derived as follows:  
\[ds=f_{1}(\varphi)R_{0}d\varphi \tag{4}\]  
\[dA=\hat{t}(\psi_{i},\hat{k})\mathrm{d}\ b=h(\psi_{i})dz=h(\psi_{i})\cot l_{0} R_{0}d\varphi \tag{5}\]  
Figure 1: The tool spindle modeled as a rotor-bearing system. \[\psi_{i}(z)=\theta+(i-1)\frac{2\pi}{N_{f}}-\varphi=\theta+(i-1)\frac{2\pi}{N_{f}}- \frac{z}{R_{0}}\tan I_{0} \tag{6}\]  
where \(R_{0}\) is the ball radius of the tool, \(\varphi=\frac{z}{R_{0}}\tan I_{0}\) represents the lag angle between the tip and a point on the helical flute at height \(z\), \(i\) is the number of flute, \(I_{0}\) is the helix angle, \(\theta\) is measured clockwise from \(+y\) axis, \(N_{f}\) is total number of flutes, and \(z\) is the Z-coordinate of a point located on the cutting edge. The coefficient \(f_{1}(\varphi)\) is listed in Appendix B. If the tool position does not vary much during the short cuts [16], the chip thickness becomes:  
\[h(\psi_{i})\approx\left[(f_{\mathrm{d}}+x(t_{\mathrm{j}})-x(t_{\mathrm{j-1}}) )\right]\sin\psi_{\mathrm{i}}+\left[y(t_{\mathrm{j}})-y(t_{\mathrm{j-1}}) \right]\cos\psi_{i} \tag{7}\]  
where \(f_{\mathrm{d}}\) represents feed per tooth. \(x(t_{\mathrm{j}})\) and \(y(t_{\mathrm{j}})\) represent the current tool-tip deflections. \(x(t_{\mathrm{j-1}})\) and \(y(t_{\mathrm{j-1}})\) represent the tool-tip deflections at the previous time. The radial, tangential, and axial forces are zero when the tool is not in cut. This can be modeled by multiplying the equations described as the force by a function \(W(\psi_{i})\), which is defined when a tooth is in or out of cut. Moreover, the tooth is in cut if \(\theta_{a}+2(n-1)\pi\leq\psi_{\mathrm{i}}\leq\theta_{b}+2(n-1)\pi\), where \(n\) is the number of spindle rotation, and \(\theta_{a}\) and \(\theta_{b}\) are the start and exit angles, respectively. This function is given by:  
\[W(\psi_{i})= \begin{cases}1,&\theta_{a}+2(n-1)\pi\leq\psi_{\mathrm{i}}\leq \theta_{b}+2(n-1)\pi\\ 0,&else\end{cases}  \tag{8}\]  
Figure 2: The geometry of a helical ball end mill. Substituting Equations (4), (5) and (7) into Equation (3) yields the following equation:  
\[