# 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 Considering Tool Wear  
As the cutting process goes on, the arc radius of the cutting edge increases, the friction between the cutter and the workpiece also increases, which will lead to the increase in micro-milling force. Therefore, the influence of tool wear should be considered when building force model. As the radius of the tool tip arc increases, a small, stable chip will  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & **Spindle Speed** & **Axial Cutting** & **Feed per Tooth \(f_{\mathbf{z}}\)** & **F\({}_{\mathbf{x}}\) (N)** & **F\({}_{\mathbf{y}}\) (N)** & **F\({}_{\mathbf{z}}\) (N)** \\  & **n (rpm)** & **Depth \(a_{\mathbf{p}}\) (\(\mu\)m)** & **(\(\mu\)m/2)** & **F\({}_{  
exist between the tool tip arc and the workpiece and accumulate at this location for a long time, forming a built-up edge, hypothetically an ideal rigid plastic material, which is called the metal dead zone [25]. Part of the workpiece material in the dead zone flow out as chips along the rack face of the cutter, and the other part extrudes and rubs with cutter and remains on the surface, as shown in Figure 19. We propose a model of force coefficients mainly caused by the increase in arc radius of the cutting edge considering cutting dead zone, as shown in Equation (44):  
\[\left\{\begin{array}{l}K_{tw}=\tau r_{e}(\tau\sin\theta_{1}\tan\theta_{1}+ \frac{2\theta_{1}}{\cos\theta_{1}})\\ \\ K_{fw}=\tau r_{e}(2\sqrt{3}\sin\theta_{1})\end{array}\right. \tag{44}\]  
where, \(K_{tw}\) and \(K_{fw}\) are the coefficients of tangential and normal force in the friction action caused by the increase in the arc radius of the cutting edge. \(r_{e}\) is arc radius of the cutting edge; \(\theta_{1}\) is the stagnation angle, which is 14\({}^{\circ}\). \(\tau\) is the shear stress of materials. The additional friction force caused by tool wear can be expressed as:  
\[\left\{\begin{array}{l}F_{x}(t)=-(K_{tw}\cdot\Phi_{c})-(K_{fw}\cdot\Phi_{s} )\\ F_{y}(t)=(K_{tw}\cdot\Phi_{s})-(K_{fw}\cdot\Phi_{c})\\ F_{z}(t)=0\end{array}\right. \tag{45}\]  
in which:  
\[\left\{\begin{array}{l}\Phi_{c}=\frac{R}{\sin\varphi}\sum\limits_{k=0}^{K-1 }\int_{\theta_{s}}^{\theta_{c}}\cos\theta d\theta\\ \\ \Phi_{s}=\frac{R}{\sin\varphi}\sum\limits_{k=0}^{K-1}\int_{\theta_{s}}^{\theta _{c}}\sin\theta d\theta\end{array}\right. \tag{46}\]  
The validity of the built force model considering tool wear in micro-end milling of LF 21 process is testified by micro-slot milling experiments. Cutting parameters are as follows: n = 40,000 rpm, ap = 20 \(\upmu\)m and fz = 0.8 \(\upmu\)m/z. The prediction curve of micro-milling force considering tool wear and the force curve measured in experiments are shown in Figure 20.  
Figure 19: The interaction between the metal dead zone and the arc radius of the cutting edge.  
Figure 20: Experimental verification of three-dimensional dynamic force model: **(a)** Forces in X-direction; **(b)** Forces in Y-direction; **(c)** Forces in Z-direction.  
Figure 20 shows that the predicted peak values of micro-milling forces in X, Y and Z directions are basically consistent with the experimental results, and the changing rules of cutting force are essentially identical. Whereas the predicted results of X cutting force are generally greater than the experimental measured values. There exist primary and secondary peak forces in a cutting cycle. It is verified that the phenomenon of uneven loading of two teeth happens because of the cutter runout in micro-milling process. It is proved that the built micro-milling force model of aluminum alloy LF 21 is effective.