# 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
### Instantaneous Cutting Thickness Model  
The calculation of instantaneous cutting thickness is the basis of the solving force model. [PERSON] [23] established an analytical model of instantaneous cumulative cutting thickness; however, the iterative solution requires a lot of computation. Accordingto the actual cutting trajectory of the cutter, [PERSON] et al. [24] built an instantaneous cutting thickness model considering the cutter runout by using microscopic geometric relations and obtained approximate solutions through Taylor series expansion. In this paper, based on the above modeling experience, an instantaneous cutting thickness model of micro-milling is established considering the cutter runout, the minimum cutting thickness, material elastic recovery and cutter vibration displacement, as shown in Figure 11.  
The instantaneous cutting thickness model considering cutter runout is calculated using Equation (34):  
\[t_{c}(k)=f_{z}(1-\frac{\psi}{\pi})\sin\theta_{k}+2R_{t}\cos\varphi_{k}-R_{t} \psi\sin\varphi_{k}+R\frac{\psi^{2}}{2} \tag{34}\]  
where, \(\theta_{k}=\theta+\pi(k-1)\), \(\theta\) is the tooth angle; \(\varphi_{k}=\varphi_{0}-\pi(k-1)\), \(\varphi_{0}\) is the cutter run-out angle, which is \(45^{\circ}\); \(\psi=(f_{z}\cos\theta_{k}+2R_{t}\sin\varphi_{k})/(R+f_{z}\cos\theta_{k}/\pi-R _{t}\cos\varphi_{k})\). When \(2R_{t}\cos\varphi_{k}>f_{z}\), single tooth cutting phenomenon occurs. If \(k-1\) tooth is truncated, the cutting thickness of \(k\) tooth needs to be corrected, as shown in Equation (35). \[t_{c}(k)_{\text{mod}}=t_{c}(k)+t_{c}(k-1) \tag{35}\]  
Because the processing scale is greatly reduced, the plough effect cannot be ignored. The plough effect causes elastic recovery of materials, and elastic recovery of materials is calculated using Equation (36). \[\left\{ \begin{array}{ll}\delta(k-1)=\delta_{\text{max}}=\frac{3\sigma_{c}}{4 \varepsilon}\cdot r_{c}\cdot\left[2\exp\left(\frac{H}{\sigma_{t}}-\frac{1}{2} \right)-1\right]&t_{c}(k-1)>\delta_{\text{max}}\\ \delta(k-1)=t_{c}(k-1)&t_{c}(k-1)\leq\delta_{\text{max}}\end{array} \right. \tag{36}\]  
If the elastic recovery of the \(k-1\) tooth workpiece material is \(\delta(k-1)\), the cutting thickness of the \(k\) tooth needs to be modified, as shown in Equation (37). \[t_{c}(k)_{\text{mod}}=t_{c}(k)+\delta(k-1) \tag{37}\]  
Figure 11: The schematic of the instantaneous cutting thickness. Under the excitation of micro-milling force, there exists relative vibration between the micro-milling cutter and the workpiece, which makes the actual cutting path of the cutter deviates from the theoretical cutting path. Vibration affects the instantaneous cutting thickness. The instantaneous cutting thickness model considering the cutter vibration displacement is divided into two cases:  
1. The blade does not run out of the cutting; Instantaneous cutting thickness in the milling process is shown in Figure 12. The cutter tooth immerses in the workpiece in tooth cutting period 2. Chatter mark is left on workpiece. The chatter mark in tooth cutting period 3 is corresponding to chatter mark in tooth cutting period 2. On the aspect of machined surface elastic recovery, in fact, machined surface elastic recovery caused in tooth cutting period 2 increases instantaneous cutting thickness in tooth cutting period 3. That is because machined surface elastic recovery caused in tooth cutting period 2 occurs after cutting. The effects of vibration and machined surface elastic recovery are independent. As a result, the instantaneous cutting thickness in tooth cutting period 3 should be: \[t_{c}(3)_{\text{mod}}=t_{c}(3)+(v_{2}-v_{3})\] (38) where, \(v_{2}\) and \(v_{3}\) represent vibration displacement corresponding to tooth cutting period 2 and 3, respectively. 2. Cutter tooth loses contact with the workpiece. It is assumed that the cutter tooth immerses in the workpiece in tooth cutting period 1. Instantaneous cutting thickness in the milling process is shown in Figure 13. The cutter tooth loses contact with the workpiece in tooth cutting period 2.