## 2 Milling Force Fluctuation Mechanism of Helical-Edge Cutter  
### Milling Process and Theory Preconditions  
A helical-edge cutter performs a unique milling process where every cutting edge removes material from a workpiece gradually and continuously. Therefore, the milling force caused by each cutting edge continuously fluctuates from minimum to maximum value. The milling geometry and mechanism of a helical-edge cutter are presented in Figure 1. The geometry of a down-milling process and related parameters are shown in Figure 1a. The machine tool and workpiece share the same \(x\)-\(y\)-\(z\) coordinate system, while the milling cutter uses the \(t\)-\(r\)-\(a\) coordinate system. The milling cutter feeds along the negative \(x\)-direction and rotates clockwise from zero to 360\({}^{\circ}\). The force generated by the rotation of the milling cutter is presented in Figure 1b. As mentioned previously, due to the unique milling mechanism caused by the helix angle, the force generated by each edge is continuous and periodic. Hence, the fluctuation force of the milling cutter obtained by the superposition of the forces generated by all edges is also periodic and continuous. This provides a theoretical possibility to reveal and summarize the force fluctuation characteristics \(F_{PP}\) (the peak-to-peak value of the milling force). In order to simplify the research process, this study adopts the following preconditions:  
* The milling process is general and simple, which means the influences of the tool wear, deformation, runout and chatter are not considered;
* The milling force model is widely accepted [9]. In this model, the tool path is considered as a circle when calculating the chip thickness. The cutting force caused by the size effect coefficients (\(K_{tc}\), \(K_{rc}\), \(K_{ac}\)) is considered as the main factor and the ploughing effect is ignored in side milling;
* The processing parameters are smooth and not extreme. Under this condition, the milling force coefficient is stable and can be considered as an average value. ### Milling Process and Theory Preconditions  
To define the general form of the force fluctuation model, the basic milling force model is used based on the oblique cutting theory. The total milling force of a helical-edge cutter is calculated by superimposing the forces of all its micro-elements. As shown in Figure 2, the cutting area is equally divided along the axial direction. According to the accuracy  
Figure 1: Milling geometry and mechanism of a helical-edge cutter: (**a**) process of down milling and (**b**) the distribution of milling forces. requirement, the height of each micro-element is set to \(\Delta Z\). Then, the force components of each micro-element in the three directions are expressed as follows [9]:  
\[\left\{ \begin{array}{l}F_{t}\big{(}\theta_{i,j}\big{)}=K_{tc}fsin(\theta_{i,j})g \big{(}\theta_{i,j}\big{)}\Delta Z\\ F_{r}\big{(}\theta_{i,j}\big{)}=K_{rc}fsin(\theta_{i,j})g\big{(}\theta_{i,j} \big{)}\Delta Z\\ F_{a}\big{(}\theta_{i,j}\big{)}=K_{ac}fsin(\theta_{i,j})g\big{(}\theta_{i,j} \big{)}\Delta Z\\ \end{array} \right. \tag{1}\]  
where, \(F_{t}\), \(F_{r}\), \(F_{a}\) are tangential, radial and axial cutting forces, respectively. \(K_{tc}\), \(K_{rc}\), \(K_{ac}\) are tangential, radial and axial cutting force coefficients, respectively. The variable \(f_{t}\) is the feed per tooth and \(g(\theta)\) is a judgment function to determine whether the micro-element is in milling. \[\Delta Z=\frac{A_{p}}{N_{d}}, \tag{2}\]  
\[g\big{(}\theta_{i,j}\big{)}= \begin{cases}1,&\theta_{st}\leq\theta_{i,j}\leq \theta_{ex}\\ 0,&\text{Others}\end{cases} , \tag{3}\]  
\[\theta_{st}=\left\{ \begin{array}{l}\pi\hskip 14.226378 pt,\hskip 14.226378 ptup \hskip 14.226378 pt\text{milling}\\ \pi-arccos(1-\frac{A_{c}}{R})\hskip 14.226378 pt,\hskip 14.226378 ptdown\hskip 14.226378 pt milling^{\prime}\end{array} \right. \tag{4}\]  
\[\theta_{ex}=\left\{