# Improved Cutting Force Modelling in Micro-Milling Aluminum Alloy LF 21 Considering Tool Wear
## 3 An Improved Cutting Force Model in Micro-Milling LF 21 Considering Tool Wear
### Micro-Milling Force Model  
Tool wear will lead to increase in the arc radius of the cutting edge. [PERSON] et al. [22] proposed that the friction between the dead zone metal and the cutter tip under the arc radius of the cutting edge should be considered when establishing the force model. Based on the above research results, the authors built a micro-milling force model dominated by shear effects considering tool wear, as shown in Equation (10):  
\[\left\{ \begin{array}{l}F_{x}(t)=-(K_{tc}\cdot A_{c}+K_{tp}\cdot V_{c}+K_{tw }\cdot\Phi_{c})-(K_{fc}\cdot A_{s}+K_{fp}\cdot V_{s}+K_{fw}\cdot\Phi_{s})\\ F_{y}(t)=(K_{tc}\cdot A_{s}+K_{tp}\cdot V_{s}+K_{tw}\cdot\Phi_{s})-(K_{fc} \cdot A_{c}+K_{fp}\cdot V_{c}+K_{fw}\cdot\Phi_{c})\qquad t_{c}>t_{\text{min}} \\ F_{z}(t)=(K_{ac}\cdot A_{a}+K_{ap}\cdot V_{a})\\ \text{where,}\\ \\ \left\{\begin{array}{l}A_{c}=\frac{R}{\sin\varphi}\sum\limits_{k=0}^{K-1} \int_{\theta_{s}}^{\theta_{s}}t_{c}(t,k,\theta)\cos\theta d\theta\\ A_{s}=\frac{R}{\sin\varphi}\sum\limits_{k=0}^{K-1}\int_{\theta_{s}}^{\theta_{s }}t_{c}(t,k,\theta)\sin\theta d\theta\\ A_{a}=\frac{R}{\sin\varphi}\sum\limits_{k=0}^{K-1}\int_{\theta_{s}}^{\theta_{s }}t_{c}(t,k,\theta)d\theta\\ \end{array}\right.\\ \\ \left\{\begin{array}{l}V_{c}=\frac{R}{\sin\varphi}\sum\limits_{k=0}^{K-1} \int_{\theta_{s}}^{\theta_{s}}A_{p}\cos\theta d\theta\\ V_{s}=\frac{R}{\sin\varphi}\sum\limits_{k=0}^{K-1}\int_{\theta_{s}}^{\theta_{s }}A_{p}\sin\theta d\theta\\ V_{a}=\frac{R}{\sin\varphi}\sum\limits_{k=0}^{K-1}\int_{\theta_{s}}^{\theta_{s }}A_{p}d\theta\\ \end{array}\right.\\ \\ \end{array} \right.\\ \\ \end{array}\right.\\ \\ \end{array} \tag{12}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Number** & **Spindle Speed** & **Feed per Toth** & **Axial Cutting** & **Cutting** & **Simulation** & **Experimental** & **Relative Error** \\  & **n (rpm)** & **\(f_{x}\) (um/z)** & **Depth \(\pm_{p}\) (um)** & **Time t (min)** & **Output (\%)** & **Results (\%)** & **(\%)** \\ \hline
1 & 40,000 & 0.4 & 20 & 4 & 0.8467 & 1.1533 & 26.58 \\
2 & 40,000 & 0.6 & 30 & 6 & 2.1133 & 1.8867 & 12.01 \\
3 & 50,000 & 0.8 & 20 & 6 & 5.7200 & 8.0333 & 28.80 \\
4 & 60,000 & 0.4 & 40 & 6 & 5.5467 & 6.7266 & 17.54 \\
5 & 60,000 & 0.8 & 30 & 4 & 10.6400 & 11.0733 & 3.91 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The comparison of experimental and simulated reduction rate of the tool diameter. \[\left\{ \begin{array}{l}\Phi_{c}=\frac{R}{\sin\varphi}\sum\limits_{k=0}^{K-1} \theta_{\theta_{k}}^{\theta_{k}}\cos\theta d\theta\\ \\ \Phi_{s}=\frac{R}{\sin\varphi}\sum\limits_{k=0}^{K-1}\theta_{\theta_{k}}^{ \theta_{k}}\sin\theta d\theta\end{array} \right. \tag{13}\]  
\(F_{x}(t)\), \(F_{y}(t)\) and \(F_{z}(t)\) are the components of cutting force decomposition to \(x\), \(y\) and \(z\) directions. \(K_{fc}\), \(K_{tc}\) and \(K_{ac}\) are radial, tangential and axial shear force coefficients, respectively. The coefficients of plough force \(K_{fp}\), \(K_{tp}\) and \(K_{ap}\) are radial, tangential and axial, respectively. \(K_{fw}\) and \(K_{tw}\) are radial and tangential friction coefficients, respectively. \(t_{c}(t,k,z)\) is the instantaneous cutting thickness of the \(k\) cutting edge at the axial position \(z\) at time \(t\). \(A_{p}\) is the area of plough area. \(t_{\min}\) is the minimum cutting thickness. \(\theta\) is the tooth angle. \(\theta_{s}\) and \(\theta_{e}\) are the limit cutting-in and cutting-out angles, respectively. \(R\) is the radius of the micro-milling cutter. \(k\) is the number of cutter tooth. \(K\) is the total number of cutter tooth. As shown in Figure 10, the shaded area is the area of the plough area when the actual cutting thickness is less than the minimum cutting thickness. Taking the maximum elastic recovery \(\delta\) as the node, the plough area is solved in two cases, respectively.