## 1 Introduction

Table 1: Meaning of milling cutter structure and cutting state parameters. where \(A_{1}\), \(A_{2}\), and \(A_{3}\) are translation matrices, and \(T_{1}\), \(T_{2}\), and \(T_{3}\) are rotation matrices, as shown in Equations (2)-(4):  
\[A_{1}= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&Z_{i}\\ 0&0&0&1\end{bmatrix} ,\ A_{2}= \begin{bmatrix}1&0&0&A_{x}(t)\\ 0&1&0&A_{y}(t)\\ 0&0&1&A_{z}(t)\\ 0&0&0&1\end{bmatrix} ,\ A_{3}= \begin{bmatrix}1&0&0&v_{f}\cdot t-x_{g}\\ 0&1&0&y_{g}^{(W)}+r_{1}-a_{e}\\ 0&0&1&z_{g}^{(H)}-a_{p}\\ 0&0&0&1\end{bmatrix}  \tag{2}\]  
\[T_{1}= \begin{bmatrix}\cos\varphi_{si}&-\sin\varphi_{si}&0&0\\ \sin\varphi_{si}&\cos\varphi_{si}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} ,\ T_{2}= \begin{bmatrix}\cos\varphi_{q}(t)&-\sin\varphi_{ q}(t)&0&0\\ \sin\varphi_{q}(t)&\cos\varphi_{q}(t)&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{3}\]  
\[T_{3}= \begin{bmatrix}\cos\theta_{2}(t)&\sin\theta_{1}(t)\sin\theta_{2}(t)& \cos\theta_{1}(t)\sin\theta_{2}(t)&0\\ -\sin\theta_{2}(t)&\sin\theta_{1}(t)\cos\theta_{2}(t)&\cos\theta_{1}(t)\cos \theta_{2}(t)&0\\ 0&\sin\theta_{1}(t)&\cos\theta_{1}(t)&0\\ 0&0&0&1\end{bmatrix}  \tag{4}\]  
According to the cutter structure and instantaneous cutting behavior, combined with the cutter radius \(r_{i}\) corresponding to each cutter tooth and the position angle corresponding to the cutting, the cutting trajectory of any cutter tip of the cutter can be obtained. After the three-dimensional coordinate transformation, the trajectory in the workpiece coordinate system can be obtained. Further considering the overall offset of the cutter tooth position caused by the vibration in the milling process, the cutter tooth positional offset model under the action of vibration can be constructed, as shown in Figure 2.  
Where \(l\) is the overhang of the milling cutter, while \(\theta_{1}(t)\) and \(\theta_{2}(t)\) are the projection angles of the attitude angle \(\theta\) on the _aoc_ and _boc_ surfaces, respectively, which are calculated as shown in Equation (5):  
\[\theta_{1}(t)=\arctan(\frac{A_{y}(t)}{l-A_{z}(t)}),\ \theta_{2}(t)=\arctan( \frac{A_{x}(t)}{l-A_{z}(t)}) \tag{5}\]  
In Equation (6), at time \(t\), the angle \(\varphi_{q}(t)\) between the structure coordinate system of the milling cutter and the cutting coordinate system with milling vibration can be expressed as follows:  
\[\varphi_{g}(0)=\arcsin(r_{g}-a_{e})/r_{g}+\pi/2 \tag{6}\]  
Figure 2: Offset model of cutter tooth position under vibration. \[\varphi_{q}(t)=\varphi_{g}(0)+\varphi_{sg}+2\pi mt-\left\lfloor\frac{\varphi_{g}(0)+ \varphi_{sg}+2\pi mt}{2\pi}\right\rfloor\cdot 2\pi \tag{7}\]  
The milling cutter tooth cutting edge equation is shown in Equation (8):  
\[S_{l}(x_{g},y_{g},z_{g})=\left\{ \begin{array}{c}\left[\begin{array}{c}x_{g}( t)\\ y_{g}(t)\\ z_{g}(t)\\ 1\end{array}\right]=\psi_{i}\cdot\left[\begin{array}{c}-r_{i}\cdot\sin\zeta_ {i}\\ r_{i}\cdot\cos\zeta_{i}\\ (\zeta_{i}/360)\cdot 2\pi r_{i}\cdot\cot\beta\\ 1\end{array}\right]\\ r_{i}=r_{\max}-\Delta r_{i},\ \ \ \zeta_{i}\geq 0^{\circ}\end{array} \right. \tag{8}\]  
where \(\psi_{i}=A_{3}A_{2}T_{3}T_{2}A_{1}T_{1}\). \(\zeta_{i}\) is the hysteresis angle of any point of the cutting edge with respect to the tool tip point, and \(\beta\) is the projection angle of the cutting edge in the z-direction. The change in the contact angle of the milling cutter teeth caused by the above vibration and the equation of the cutting edge of the cutter teeth can be brought into Equation (1) to obtain the cutting edge trajectory of the milling cutter teeth under vibration conditions.