## 4 Instantaneous Bias of the Main Section of the Milling Cutter under Vibration  
According to Figure 1 and Equations (6)-(11), the cutting edge of cutter tooth participating in cutting was affected by milling vibration, cutter tooth error, blade shape, and instantaneous cutting pose. An instantaneous cutting speed and main cutting force of cutting edge feature points of cutter teeth participating in cutting are shown in Figure 4.  
The position coordinates \(x_{i}^{mi}\), \(y_{i}^{mi}\), and \(z_{i}^{mi}\) of the characteristic point \(m_{i}(t_{i})\) on the cutting edge of the \(i\)th cutter tooth that participate in cutting instantaneously in the cutter tooth coordinate system \(o_{i}\)-\(x_{i}y_{i}z_{i}\) could be expressed as  
\[\left\{ \begin{array}{l}E_{i}(x,y,z)=\Omega_{i}.\left[\begin{array}{cc}x_{i }^{m_{i}}&y_{i}^{m_{i}}&z_{i}^{m_{i}}&1\end{array}\right]^{T}\\ \\ m_{i_{0}}(t^{i})\leq m_{i}(t^{i})\leq m_{ik}(t^{i})\end{array} \right. \tag{25}\]  
As shown in Figure 4, the instantaneous pose of the \(z_{i}\) axis in the workpiece coordinate system was obtained by Equation (11), which was in the instantaneous cutting main section \(Gom\) of the point \(m_{i}(t^{i})\), use \(o_{im}(t^{i})\) which is the intersection of \(z_{i}\) axis and main section \(Gom\) and points \(m_{i}(t^{i})\) construct linear equation \(h_{m}(t^{i})\). The instantaneous position coordinates \(x^{d}m\), \(y^{d}m\) and \(z^{d}m\) of the intersection \(q_{m}(t^{i})\) could be expressed as  
\[\Sigma_{i-1}(x,y,z)=h_{m}(t^{i}) \tag{26}\]  
Figure 4: Instantaneous cutting speed and main cutting force of the cutter tooth cutting edge. The instantaneous position coordinates of \(q_{m}(t^{i})\) in the cutter tooth coordinate system \(o_{i}\)-\(x_{i}y_{i}z_{i}\) could be expressed as  
\[\left[ \begin{array}{ccccc}x_{i}^{q_{m}}&y_{i}^{q_{m}}&z_{i}^{q_{m}}&1\end{array}  \right]^{T}=\Omega_{i}{}^{-1}.\left[ \begin{array}{ccccc}x^{q_{m}}&y^{q_{m}}&z ^{q_{m}}&1\end{array} \right]^{T} \tag{27}\]  
Then, in the cutter tooth coordinate system \(o_{i}\)-\(x_{i}y_{i}z_{i}\), the instantaneous cutting layer thickness \(h_{Dj}(x_{i},y_{i},z_{i})\) of the point \(m_{i}(t^{i})\) could be expressed as  
\[h_{Dj}(x_{i},y_{i},z_{i})=\sqrt{\left(x_{i}^{m_{i}}-x_{i}^{q_{m}}\right)^{2}+ \left(y_{i}^{m_{i}}-y_{i}^{q_{m}}\right)^{2}+\left(z_{i}^{m_{i}}-z_{i}^{q_{m} }\right)^{2}} \tag{28}\]  
In the cutter tooth coordinate system \(o_{i}\)-\(x_{i}y_{i}z_{i}\), the instantaneous principal motion speed of the point \(m_{i}(t^{i})\) could be calculated as  
\[v_{ci}(x_{i},y_{i},z_{i})=\frac{n\cdot 2\pi\cdot r_{i}}{60\times 1000} \tag{29}\]  
According to the Figure 4, Equations (28) and (29), the energy consumption distribution function of the instantaneous main cutting force of the cutter tooth was given by  
\[dP_{ti}(x_{i},y_{i},z_{i})=v_{ci}(x_{i},y_{i},z_{i})\cdot k_{t}\cdot p\cdot h _{Dj}(x_{i},y_{i},z_{i})\cdot(dz_{i}/\cos\beta) \tag{30}\]  
where \(p\) is the unit cutting force and \(k_{t}\) is the main cutting force correction coefficient. According to the Equations (23), (24), and (30), the instantaneous main cutting force energy consumption of the cutter tooth could be derived as  
\[P_{i}(t^{i})=\int_{m_{0}(t^{i})}^{m_{k}(t^{i})}dP_{ti}(x_{i},y_{i},z_{i}) \tag{31}\]  
In order to verify and further investigate the dynamic characteristics of energy consumption of milling cutter main cutting force, high speed milling experiment were carried out. The workpiece material was titanium alloy TC4. The milling cutter was an integral cemented carbide end milling cutter with diameter of 20 mm. The cutter tooth 1 is the longest cutter tooth with the bottom edge, and the other cutter teeth are sorted according to the follow-up cutting sequence, as shown in Figure 5.  
In order to eliminate the influence of cutting fluid on the accuracy of cutting force measurement, dry cutting was used in the experiment. The milling parameters and cutter teeth errors are shown in Table 2.  
\(\Delta z^{i}_{d}\) is the axial error of the \(i\)th cutter tooth.