## 3 Milling Force Fluctuation Characteristics  
### Virtual-Edge Projection Method of Force Equation Transformation  
In order to identify the force behavior under different milling positions and rotation angles, a virtual-edge projection method is proposed to reveal the force relationship between each cutting edge. This method is based on the physical phenomenon that if the cutter edge is long enough, each cutting edge can be replaced by the previous edge above it, as shown in Figure 3. In Figure 3a, two milling cutters denoted as A and B have different milling positions and their axial cutting depth is \(A_{p}\). The difference between cutters A and B is that the milling position of cutter A is from zero to \(A_{p}\) and that of cutter B is from \(Z_{0}\) to \((Z_{0}+A_{p})\), where \(Z_{0}\) is an axial offset reference position, defined as follows:  
During the milling process, if an edge is in contact with the workpiece, then move the milling cutter downwards axially until the previous edge reaches the same position, the axial offset of this downward movement is defined as \(Z_{0}\).  
The milling forces caused by a single edge of the two cutters are analyzed for comparison. The milling force expression of the second cutting edge of cutter A can be calculated by Equation (8) as follows:  
Figure 3: Transformation of the milling force: (**a**) milling at different positions with equal axial cutting depth and (**b**) the virtual-edge projection.  
\[F_{A2}\big{(}\theta_{0},A_{p}\big{)}=KC\sum\limits_{i=\Delta Z}^{A_{p}}G\bigg{(} \theta_{0}-\frac{2\pi}{N_{t}}-\frac{tan\beta}{R}\times(i-\Delta Z)\bigg{)} \tag{10}\]  
At the same time, the corresponding milling force expression of the first edge of cutter B is calculated by:  
\[F_{B1}\big{(}\theta_{0},A_{p}\big{)}=KC\sum\limits_{i=Z_{0}+\Delta Z}^{Z_{0}+ Ap}G\bigg{(}\theta_{0}-\frac{tan\beta}{R}\times(i-\Delta Z)\bigg{)} \tag{11}\]  
Considering that in Equations (10) and (11) the rotation reference point \(\theta_{0}\) is the same, the following Equation (12) can be obtained:  
\[\begin{array}{l}F_{A2}\big{(}\theta_{0},A_{p}\big{)}=KC\sum\limits_{i= \Delta Z}^{A_{p}}G\bigg{(}\theta_{0}-\frac{2\pi}{N_{t}}-\frac{tan\beta}{R} \times(i-\Delta Z)\bigg{)}\\ =KC\sum\limits_{i=\Delta Z}^{A_{p}}G\bigg{(}\theta_{0}-\frac{2\pi}{N_{t}}+ \frac{tan\beta}{R}\times Z_{0}-\frac{tan\beta}{R}\times(i-\Delta Z)-\frac{tan \beta}{R}\times Z_{0}\bigg{)}\\ =KC\sum\limits_{i=Z_{0}+\Delta Z}^{A_{p}}G\bigg{(}\theta_{0}-\frac{2\pi}{N_{t} }+\frac{tan\beta}{R}\times Z_{0}-\frac{tan\beta}{R}\times(i-\Delta Z)\bigg{)} \\ =F_{B1}\Big{(}\theta_{0}-\frac{2\pi}{N_{t}}+\frac{tan\beta}{R}\times Z_{0},A_{ p}\Big{)}.\end{array} \tag{12}\]  
Therefore, based on Equation (12), the following conclusion can be drawn. The movement of the axial milling position can be compensated by the rotation angle of a cutter. If Equation (12) is transformed, then the following equations can be obtained:  
\[\left\{\begin{array}{l}F_{A2}\big{(}\theta_{0},A_{p}\big{)}=F_{B1}\big{(} \theta_{0},A_{p}\big{)}\\ Z_{0}=\frac{2\pi R}{N_{t}tan\beta}\end{array}\right.. \tag{13}\]  
Equation (13) illustrates that the milling force of each edge from zero to \(A_{p}\) can be replaced by the force of the previous edge from \(Z_{0}\) to \(Z_{0}+A_{p}\). Therefore, as shown in Figure 3b, if there is an infinitely long virtual edge along the first edge, the milling force of the other edges can always be represented by a specific part of the virtual edge. This virtual-edge projection method provides a more intuitive graphical representation for the mathematical expression of the milling force.