## 4 Validation of Surface Roughness Model and Process Parameter Optimization
### Establishment of Surface Roughness Model Considering Milling Cutter Vibration
#### 4.1.2 Surface Roughness Model Considering Milling Cutter Vibration
\tag{24}\]  
Combined with the previous studies, the undeformed cutting thickness can be derived from two adjacent cutting-edge trajectories (see Figure 25) [36,49]. Suppose there exist two points \(\mathrm{P}\big{(}x_{p}(j,k,t_{1}),\,y_{p}(j,k,t_{1})\big{)}\) and \(\mathrm{Q}\big{(}x_{q}(j-1,k,t_{2}),\,y_{q}(j-1,k,t_{2})\big{)}\). The point \(\mathrm{P}\) is any point on the \(j\)-th flute trajectory. The point \(\mathrm{Q}\) is the intersection of the \(j\)-th flute trajectory and the line from the tool center \(\mathrm{O}\) to the point \(\mathrm{P}\). \(t_{1,2}\) denote the time corresponding to the cutting edge passing through the points \(\mathrm{P}\) and \(\mathrm{Q}\), respectively. The undeformed cutting thickness considering the vibration of the milling cutter during milling can be expressed as:  
\[h_{c}(j,k,t_{1})=\sqrt{\left[x_{P}(j,k,t_{1})-x_{Q}(j-1,k,t_{2})\right]^{2}+ \left[y_{P}(j,k,t_{1})-y_{Q}(j-1,k,t_{2})\right]^{2}} \tag{25}\]  
The coordinates of the point P can be expressed using Equation (26):  
\[\left\{ \begin{array}{l}x_{P}(j,k,t_{1})=f_{x}(j,k,t_{1})\\ y_{P}(j,k,t_{1})=f_{y}(j,k,t_{1})\end{array} \right. \tag{26}\]  
For the point Q, the coordinates can be expressed as:  
\[\left\{ \begin{array}{l}x_{Q}(j-1,k,t_{2})=f_{x}(j-1,k,t_{2})\\ y_{Q}(j-1,k,t_{2})=f_{y}(j-1,k,t_{2})\end{array} \right. \tag{27}\]  
Since the time \(t_{1}\) corresponding to point P is the current time, the coordinates corresponding to point P can be obtained by Equation (26). For the coordinates of point Q, they can be calculated by the position relationship between point O\({}_{t1}\), point P, and point Q. Since point O\({}_{t1}\), point P, and point Q are located in a straight line, the coordinate relationship between the three points can be expressed as:  
\[\tan\varphi(j,k,t_{1})-\frac{f_{x}(j,k,t_{1})-f_{x}(j-1,k,t_{2})}{f_{y}(j,k,t_ {1})-f_{y}(j-1,k,t_{2})}=0 \tag{28}\]  
By iterating the Equation (28), the coordinates of point Q can be obtained where \(\varphi(j,k,t_{1})\) denotes the contact angle of the \(j\)-th flute of the milling cutter, which can be expressed as [36]:  
\[\varphi(j,k,t_{1})=\omega t_{1}+\frac{2j\pi}{N_{z}}-\frac{k\Delta z\tan\beta} {R} \tag{29}\]  
The minimum cutting thickness in the milling process is an important factor affecting the surface roughness. Based on the geometric parameters of the surface roughness and the minimum cutting thickness, the surface roughness model considering the vibration of the milling cutter is established. When the undeformed cutting thickness \(h_{c}\) is less than  
Figure 25: Analysis of the cutting-edge trajectory. the minimum cutting thickness \(h_{min}\), the material deformation of the workpiece surface is dominated by the plowing effect under the extrusion of the cutting edge of the milling cutter. When the undeformed cutting thickness gradually increases to the minimum cutting thickness, the milling process is dominated by the shear effect and the elastic response of the workpiece is negligible. According to the radius of the cutting edge of the milling cutter and the material properties, the minimum cutting thickness is expressed as [36]:  
\[h_{min}=\lambda_{e}r_{c} \tag{30}\]  
where, \(\lambda_{e}\) is the minimum cutting thickness proportionality constant, and \(r_{c}\) is the radius of the cutting edge. Based on the relationship between the minimum cutting thickness \(h_{min}\) and the undeformed cutting thickness \(h_{c}\) on the milling mechanism, the actual undeformed cutting thickness can be expressed as [49]:  
\[h_{c}(j,k,t)=\left\{ \begin{array}{c}p_{e}h_{c}(j,k,t)\ h_{c}(j,k,t)\leq h_{ min}\\ h_{c}(j,k,t)\ h_{c}(j,k,t)>h_{min}\end{array} \right. \tag{31}\]  
where \(p_{e}\) denotes the elastic recovery rate. Combining the above derivation, the actual cutting-edge trajectory considering the vibration response of the milling cutter and the minimum cutting thickness of the workpiece can be derived. When \(h_{c}(j,k,t)>h_{min}\), the actual cutting-edge trajectory can be expressed as:  
\[\left\{