## 3 Force Density Function of Variable Density Micro-Texture Ball-End Milling Cutter  
### Milling Force Model of Micro-Textured Ball-End Milling Cutter  
The high-speed milling of titanium alloy by micro-textured ball-end milling cutter is intermittent cutting. With the change of the micro-textured cutter from cutting in to cutting out the workpiece, the magnitude and direction of the instantaneous milling force also change, which affects the stress field distribution on the front of the cutter. Therefore, it is necessary to solve the milling cycle \(T\), the angle \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number**Cutting Parameters** & **Cutting Depth** & **Feed per Tooth** & **Cutter-Chip Contact** & **Cutter-Chip Contact** \\  & \(a_{p}\) **(mm)** & \(f_{z}\) **(mm)** & **Width \(I_{w}\) (mm)** & **Length \(I_{f}\) (mm)** \\ \hline
1 & 0.3 & 0.04 & 0.768 & 0.5302 \\
2 & 0.3 & 0.06 & 0.775 & 0.5613 \\
3 & 0.3 & 0.08 & 0.784 & 0.5888 \\
4 & 0.3 & 0.1 & 0.788 & 0.6031 \\
5 & 0.5 & 0.06 & 1.017 & 0.5691 \\
6 & 0.5 & 0.04 & 1.008 & 0.5405 \\
7 & 0.5 & 0.1 & 1.029 & 0.6124 \\
8 & 0.5 & 0.08 & 1.021 & 0.5911 \\
9 & 0.7 & 0.08 & 1.318 & 0.5928 \\
10 & 0.7 & 0.1 & 1.327 & 0.6103 \\
11 & 0.7 & 0.04 & 1.326 & 0.5445 \\
12 & 0.7 & 0.06 & 1.315 & 0.5968 \\
13 & 0.9 & 0.1 & 1.658 & 0.6179 \\
14 & 0.9 & 0.08 & 1.651 & 0.5946 \\
15 & 0.9 & 0.06 & 1.636 & 0.5675 \\
16 & 0.9 & 0.04 & 1.628 & 0.5494 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: 0.09–0.09–0.09 contact length and width of micro-textured cutter. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Milling Force in** & \multicolumn{8}{c}{**A Milling Cycle (s)**} \\ \cline{2-10}
**Directions** & **0.0004** & **0.0008** & **0.0012** & **0.0016** & **0.002** & **0.0024** & **0.0028** & **0.0032** & **0.0036** \\ \hline X(N) & 3.15 & 50.46 & 122.33 & 235.85 & 305.37 & 226.94 & 78.26 & 9.34 & 0 \\ Y(N) & \(-14.92\) & \(-77.98\) & \(-156.15\) & \(-220.8\) & \(-284.28\) & \(-229.1\) & \(-156.22\) & \(-12.44\) & 0 \\ Z(N) & 17.87 & 65.51 & 103.94 & 136.53 & 147.88 & 126.03 & 79.28 & 10.63 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Variation of milling force with time for micro-textured cutter. of cutting into workpiece \(\psi_{in}\), the time of cutting into the workpiece \(t_{i}\) and the cutting time \(t_{0}\) of the micro-textured ball-end milling cutter, respectively. The solution process is as follows:  
\[T=\frac{60}{n\cdot z}, \tag{2}\]  
\[\psi_{in}=180^{\circ}-\arccos\left(\frac{R-a_{p}}{R}\right), \tag{3}\]  
\[t_{i}=T\cdot\frac{\psi_{in}}{360^{\circ}}, \tag{4}\]  
\[t_{0}=T-t_{i}, \tag{5}\]  
where \(n\) is the spindle speed (r/min), \(z\) is the number of teeth on the tool edge, and \(R\) is the tool edge radius. The main factors of cutting parameters affecting milling force are the cutting depth and feed per tooth. Therefore, an empirical formula model of milling force was established by using multiple linear regression method. The empirical formula model of the milling force is as follows:  
\[F_{j}=C\cdot a_{p}^{x_{1}}\cdot f_{z}^{x_{2}}. \tag{6}\]  
Taking the logarithm of both sides is  
\[\lg F_{j}=\lg C+x_{1}\lg a_{p}+x_{2}\lg f_{z}. \tag{7}\]  
Let \(f=\lg F_{j}\), \(a_{0}=\lg C\), \(a_{1}=\lg a_{p}\), \(a_{2}=\lg a_{f}\), the linearization of Equation (7) is  
\[f=a_{0}+a_{1}x_{1}+a_{2}x_{2}. \tag{8}\]  
The calculation and fitting were performed using MATLAB software, and the milling force test data collected from the \(X\), \(Y\) and \(Z\) directions of nine variable density combination cutters and non-textured cutters were substituted into the calculation. Taking a micro-textured cutter with a texture density combination of 0.09-0.09-0.09 as an example, the milling forces in the \(X\), \(Y\) and \(Z\) directions measured by orthogonal tests are shown in Table 5. The experimental data in the table were substituted into MATLAB for calculation and solution, and the coefficients and exponentials of the empirical formula for milling forces were obtained.