## 3 Force Density Function of Variable Density Micro-Texture Ball-End Milling Cutter
### Establishment of Force Density Function for Variable Density Micro-Textured Cutter  
When micro-textured ball-end milling cutter cuts titanium alloy, the instantaneous milling forces in the three directions vary with time in the cutter-chip contact area, and the distributions along the cutter-chip contact length and width are uneven. Therefore, the second-order mixed partial derivative of the instantaneous milling force model was used to solve the force density function of the micro-textured ball-end milling cutter, and the instantaneous cutting force variation at a point on the cutter can be obtained by solving the force density function. By calculating the second-order mixed partial derivative of Equation (11), the force density functions of a micro-textured cutter with a texture density combination of 0.09-0.09-0.09 in the coordinate system of the machine tool can be obtained as follows:  
\[ \begin{array}{l}f_{x}=\frac{\partial F_{x}}{\partial l_{w}\partial l_{f}}=17 1.634\left(l_{w}-0.3174\right)^{-0.5315}\cdot\left(l_{f}-0.4898\right)^{-0.5699} \\ f_{y}=\frac{\partial F_{y}}{\partial l_{w}\partial l_{f}}=141.565\left(l_{w} -0.3174\right)^{-0.3119}\cdot\left(l_{f}-0.4898\right)^{-0.6664}\\ f_{z}=\frac{\partial F_{z}}{\partial l_{w}\partial l_{f}}=28.301\left(l_{w} -0.3174\right)^{-0.3175}\cdot\left(l_{f}-0.4898\right)^{-0.8193}\end{array} \, \tag{12}\]  
w \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Number** & \(\begin{array}{c}a_{p}\\ \text{(mm)}\end{array}\) & \(\begin{array}{c}f_{z}\\ \text{(mm/z)}\end{array}\) & \(\begin{array}{c}F_{x}\\ \text{(N)}\end{array}\) & \(\begin{array}{c}F_{y}\\ \text{(N)}\end{array}\) & \(\begin{array}{c}F_{z}\\ \text{(N)}\end{array}\) & \(\begin{array}{c}L_{g}(F_{x})\\ \text{(N)}\end{array}\) & \(\begin{array}{c}L_{g}(F_{y})\\ \text{(N)}\end{array}\) & \(\begin{array}{c}L_{g}(F_{y})\\ \text{(N)}\end{array}\) & \(\begin{array}{c}L_{g}(F_{z})\\ \text{($g$\)}(F_{z})\end{array}\) & \(\begin{array}{c}L_{g}(a_{p})\\ \text{($g$\)}(f_{z})\end{array}\) \\ \hline
6 & 0.5 & 0.04 & 230.43 & 227.16 & 139.25 & 2.36 & 2.36 & 2.14 & \(-0.3\) & \(-1.4\) \\
7 & 0.5 & 0.1 & 416.26 & 389.46 & 135.73 & 2.62 & 2.59 & 2.13 & \(-0.3\) & \(-1\) \\
8 & 0.5 & 0.08 & 222.35 & 216.19 & 99.25 & 2.35 & 2.33 & 2.00 & \(-0.3\) & \(-1.1\) \\
9 & 0.7 & 0.08 & 305.37 & 284.28 & 147.88 & 2.48 & 2.45 & 2.17 & \(-0.15\) & \(-1.1\) \\
10 & 0.7 & 0.1 & 408.36 & 296.41 & 162.39 & 2.61 & 2.47 & 2.21 & \(-0.15\) & \(-1.1\) \\
11 & 0.7 & 0.04 & 238.45 & 208.29 & 110.56 & 2.38 & 2.32 & 2.04 & \(-0.15\) & \(-1.4\) \\
12 & 0.7 & 0.06 & 348.95 & 315.82 & 164.58 & 2.54 & 2.50 & 2.22 & \(-0.15\) & \(-1.22\) \\
13 & 0.9 & 0.1 & 355.73 & 333.41 & 196.27 & 2.55 & 2.52 & 2.29 & \(-0.05\) & \(-1\) \\
14 & 0.9 & 0.08 & 328.63 & 304.18 & 200.08 & 2.52 & 2.48 & 2.30 & \(-0.05\) & \(-1.1\) \\
15 & 0.9 & 0.06 & 293.05 & 302.13 & 178.88 & 2.47 & 2.48 & 2.25 & \(-0.05\) & \(-1.22\) \\
16 & 0.9 & 0.04 & 280.62 & 293.06 & 152.83 & 2.45 & 2.47 & 2.18 & \(-0.05\) & \(-1.4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Cont. where \(l_{w}\in(1.072,1.929)\), and \(l_{f}\in(0.5905,0.6680)\). The force density functions above of the micro-textured ball-end milling cutter are solved in the coordinate system of the machine tool. However, the magnitude and direction of the instantaneous cutting force of the micro-textured ball-end milling cutter change with the rotation of the cutter from cutting in to cutting out the workpiece. The schematic diagram of the cutter from cutting in to cutting out the workpiece is shown in Figure 4. Setting XYZ as the workpiece coordinate system and XcYcZc as the tool coordinate system, Figure 4a shows the process from cutting in to cutting out of the micro-textured ball-end milling cutter, and Figure 4b shows the relationship between the coordinate systems of the tool and the machine tool during the cutting process of the micro-textured cutter.