## 3 Milling Force Fluctuation Characteristics
### Uniformity Characteristics of Force Fluctuation  
There is a special milling situation in the milling process of a helical-fluent cutter. Namely, when the first cutting edge gradually exits the workpiece, then the next cutting edge gradually enters the workpiece. Also, the length and thickness of the cutting edge are not changed. Therefore, the milling force will be constant and \(F_{pp}\) will be zero. This phenomenon is called the uniformity milling process and it is presented in Figure 4. The uniformity milling process is greatly affected by the entrance and exit angles. According to Equation (13), the parameters that affect these two angles are \(A_{e}\), \(A_{p}\), \(R\), \(N_{t}\) and \(\beta\). As mentioned in Section 3.1, if the cutting depth of the milling process is \(A_{p}\), the projection range of the first edge on the virtual edge is from zero to \(A_{p}\) and the range of the second edge is from \(Z_{0}\) to \((Z_{0}+A_{p})\). Therefore, if \(A_{p}\) is equal to \(Z_{0}\), then the projection range of the virtual edge is continuous. This means that the milling force is stable and is no longer affected by the rotation angle and radial depth of the cut. Accordingly, \(A_{p}\) is defined as \(A_{poc}\), which represents one-cycle of axial depth of cut and is expressed as:  
\[A_{poc}=\frac{2\pi R}{N_{t}tan\beta}. \tag{14}\]The virtual-edge projection of a uniformity milling process is illustrated in Figure 4a. In order to prove the uniformity characteristics of the milling force, it is assumed that the axial cutting depth is \(A_{pac}\) and the radial cutting depth \(A_{e}\) is an arbitrary value. At this time, the calculation of the milling force can be divided into two cases. The first case is that the force projection area is on a single edge and the effective projection area is from \(Z_{1}\) to \((Z_{1}+\delta Z)\). The other case is that the force projection area exists on multiple edges and the effective projection area is from \(Z_{2}\) to \((Z_{2}+\delta Z)\). In Figure 4, the effective force projection area A of the first case is displayed in the color orange and the rotation angle is \(\theta_{Z1}\). In the first case, the milling force of the virtual-edge projection area \(F_{ve}\) can be calculated as follows:  
\[ \begin{array}{l}F_{A}\big{(}\theta_{Z_{1}},A_{proc}\big{)}=KC\sum\limits_{ i=Z_{1}+\Delta Z}^{Z_{1}+\delta Z}G\Big{(}\theta_{Z_{1}}-\frac{tang\beta}{R} \times(i-\Delta Z)\Big{)}\\ =KC\sum\limits_{i=\Delta Z}^{\delta Z}G\Big{(}\theta_{Z_{1}}-\frac{tang\beta} {R}\times Z_{1}-\frac{tang\beta}{R}\times(i-\Delta Z)\Big{)}\\ =F_{ve}\big{(}\theta_{0},A_{proc}\big{)}.\end{array}  \tag{15}\]  
In the second case, the first edge is cutting out of the workpiece and the second edge is cutting in the workpiece; the effective force projection area exists on multiple edges. Then, the effective force projection area B is displayed in the color red and the rotation angle is \(\theta_{Z2}\). \[ \begin{array}{l}F_{B}\big{(}\theta_{Z2},A_{proc}\big{)}=KC\Bigg{(}\sum \limits_{i=Z_{1}+\Delta Z}^{A_{rec}}G\Big{(}\theta_{Z_{2}}-\frac{tang\beta}{R} \times(i-\Delta Z)\Big{)}+\sum\limits_{i=Z_{2}+\Delta Z}^{A_{rec}}G\Big{(} \theta_{Z_{2}}-\frac{2\pi}{R_{i}}-\frac{tang\beta}{R}\times(i-\Delta Z) \Big{)}\Bigg{)}\\ =KC\Bigg{(}\sum\limits_{i=Z_{1}+\Delta Z}^{A_{rec}}G\Big{(}\theta_{Z_{2}}- \frac{tang\beta}{R}\times(i-\Delta Z)\Big{)}+\sum\limits_{i=Z_{2}+\Delta Z}^ {A_{rec}+\delta Z}G\Big{(}\theta_{Z_{2}}-\frac{2\pi}{R_{i}}+\frac{tang\beta }{R}\times A_{proc}-\frac{tang\beta}{R}\times(i-\Delta Z)\Big{)}\Bigg{)}\\ =KC\Bigg{(}\sum\limits_{i=Z_{1}+\Delta Z}^{A_{rec}}G\Big{(}\theta_{Z_{2}}- \frac{tang\beta}{R}\times(i-\Delta Z)\Big{)}\\ =F_{ve}\big{(}\theta_{0},A_{proc}\big{)}.\end{array}  \tag{16}\]  
Therefore,  
\[\left\{ \begin{array}{l}F_{A}\big{(}\theta_{Z_{1}},A_{proc}\big{)}=F_{B} \big{(}\theta_{Z_{2}},A_{proc}\big{)}=F_{ve}\big{(}\theta_{0},A_{proc}\big{)} =C_{0}\\ F_{pp}\big{(}A_{proc}\big{)}=0\end{array}