## 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  
In previous studies, the trajectory of the milling cutter was usually generated by the radial runout and tilt runout of the milling cutter [46; 47; 48]. As the milling cutter rotates, the position and trajectory of each point on the cutting edge of the milling cutter will deviate from the ideal value. In this study, the radial runout of the milling cutter is ignored since the milling cutter moves in the \(x\)-direction as a feed. For the tilt runout, it is attributed to the inclination between the milling cutter axis and the machine spindle due to the vibration of the milling cutter. Figure 24 shows the creation of surface topography. Before the creation of the surface topography, the tool coordinate system (O\({}_{\text{c}}\)-x\({}_{\text{c}}\)-y\({}_{\text{c}}\)-z\({}_{\text{c}}\)) and the workpiece coordinate system (O\({}_{\text{w}}\)-x\({}_{\text{w}}\)-y\({}_{\text{w}}\)-z\({}_{\text{w}}\)) are established. The origin of the tool coordinate system is set to the projection of the bottom center of the milling cutter on the workpiece. For easy calculation, it is necessary to discretize the milling cutter along the axial direction. Under ideal conditions, when \(z(j,k,t_{i})\leq a_{p}\), the coordinate values of the cutting edge at the moment \(t_{i}\) can be expressed as [49]:  
Figure 24: The creation of surface topography. \[\left\{ \begin{array}{c}x_{o}(j,k,t_{i})=V_{c}t+R\sin(\omega t+\frac{2j\pi}{N_{z}} -\frac{k\Delta z\tan\beta}{R})\\ y_{o}(j,k,t_{i})=R\cos(\omega t_{i}+\frac{2(\pi}{N_{z}}-\frac{k\Delta z\tan \beta}{R})\\ z_{o}(j,k,t_{i})=k\Delta z\end{array} \right. \tag{20}\]  
where \(j\) denotes the \(j\)th flute, \(k\) denotes the \(k\)th unit in the axial direction, \(V_{c}\) denotes the milling speed, and \(\omega\) denotes the spindle speed. \(N_{z}=4\), is the number of teeth of the milling cutter. \(R=6\) mm is the radius of the milling cutter. \(\Delta z\) denotes the thickness of the axial unit. \(\beta=45^{\lx@math@degree}\) is the helix angle of the flathead milling cutter, and \(a_{p}\) is the axial milling depth. For the axial coordinate values, \(z_{o}(k)\) is written as \(z_{o}(j,k,t_{i})\) for a uniform expression form. When the vibration responses of the milling cutter in the \(x\)- and \(y\)-directions are \(d(x,L,t_{i})\) and \(d(y,L,t_{i})\), respectively, the tilt angle \(\tau\) of the milling cutter can be expressed as:  
\[\tau=\tan^{-1}\left(\frac{\sqrt{d(x,L,t_{i})^{2}+d(y,L,t_{i})^{2}}}{L}\right) \tag{21}\]  
In this study, the position angle \(\phi\) of the tilt angle is defined as the angle between the tilt direction of the milling cutter and the \(x\)-direction in the \(x\)-o-\(y\) plane. According to the direction of the force on the milling cutter, the position angle \(\phi\) takes a range of \(\frac{\pi}{2}\leq\phi\leq\frac{3\pi}{2}\). The position angle \(\phi\) can be expressed as:  
\[\left\{ \begin{array}{c}\phi=\tan^{-1}\left(\frac{d(x,L,t_{i})}{d(y,L,t_{i})} \right)+\frac{\pi}{2}\,F_{y}(t_{i})>0\\ \phi=\tan^{-1}\left(\frac{d(y,L,t_{i})}{d(x,L,t_{i})}\right)+\pi\,F_{y}(t_{i}) \leq 0\end{array} \right. \tag{22}\]  
Once the tilt angle \(\tau\) and the position angle \(\phi\) of the milling cutter are determined, the coordinates of the cutting edge of the milling cutter under vibration response can be obtained by rotating (around the \(y\)-axis and \(z\)-axis) and translating the coordinate values in the ideal condition. The real coordinates of the cutting edge can be expressed as:  
\[\left[ \begin{array}{cccc}x(j,k,t_{i})&y(j,k,t_{i})&z(j,k,t_{i})&1\right]=[x_ {o}(j,k,t_{i})&y_{o}(j,k,t_{i})&z_{o}(j,k,t_{i})&1\end{array} \right] \tag{23}\]  
For simplicity of representation, the coordinate values of the cutting edge can be expressed as:  
\[\left\{ \begin{array}{c}x(j,k,t)=f_{x}(j,k,t)\\ y(j,k,t)=f_{y}(j,k,t)\\ z(j,k,t)=f_{z}(j,k,t)\end{array} \right.