## 2 Characterization of Instantaneous Cutting Energy Transmission and Conversion of Milling Cutter  
The milling cutter structure, cutting parameters and milling vibration directly determined the instantaneous cutting behavior of the milling cutter and cutter teeth, and affected the energy input and output of the instantaneous cutting of the milling cutter. Among them, the instantaneous cutting posture, energy composition and distribution of the end mill and its cutter teeth under vibration are shown in Figure 1.  
In Figure 1, _o-xyz_ is the workpiece coordinate system. _o-x_0y_0z_0 is the milling cutter's cutting coordinate system without vibration. _o-x_p_y_z_0 is the milling cutter's cutting coordinate system with vibration. _o-x_d_y_d_z_d is the milling cutter structure coordinate system. _o-x_q_y_z_i is the cutter tooth coordinate system. \(n\) is the spindle speed. _v_f is the feed velocity of milling cutter. _P_M(_t_) is torque input energy of the milling cutter. _P_x(_t_), _P_y(_t_) and _P_z(_t_) are the energy input to the milling cutter along the \(x\), \(y\) and \(z\) directions, respectively. Figure 1: Instantaneous cutting posture and energy of milling cutter under vibration. \(P_{x}^{i}(t)\), \(P_{y}^{i}(t)\) and \(P_{z}^{i}(t)\) are the energy consumption of the \(i\)th cutter tooth along the \(x\), \(y\) and \(z\) directions, respectively. \(P_{xz}(t)\), \(P_{xy}(t)\) and \(P_{xz}(t)\) are the input energy of the vibration of the milling cutter along the x, y and z directions, respectively. \(P_{c}^{i}(t)\) is the tangential-force energy efficiency of the \(i\)th cutter tooth. \(P_{s}^{i}(t)\) is the shear energy efficiency of the \(i\)th cutter tooth. \(P_{sV}^{i}(t)\) is the energy efficiency of milling volume of the \(i\)th cutter tooth. \(P_{sh}^{i}(t)\) is the heat generated during the shearing process of the cutter teeth. \(P_{f}^{i}(t)\) is the friction energy consumption of the \(i\)th cutter tooth. \(P_{fu}^{i}(t)\) is the wear energy consumption of the \(i\)th cutter tooth. \(P_{fu}^{i}(t)\) is heat generated by the friction and wear of the cutter teeth. According to Figure 1, the trajectory equation of any point of the cutting edge of the milling cutter in the workpiece coordinate system can be written as:  
\[ \begin{bmatrix}x(t)&y(t)&z(t)&1\end{bmatrix} ^{T}\ =\ A_{3}A_{2}T_{3}T_{2}A_{1}T_{1}[x_{ i}y_{i}z_{i}\ 1]^{T} \tag{1}\]  
where (\(x_{i}\), \(y_{i}\), \(z_{i}\)) is the coordinates of any point of the cutting edge in the cutter tooth coordinates system. The cutting edge equation was constructed based on the initial helical edge structure. The cutting edge equation is shown in Equation (2), which ignores the influence of tooth wear on the helical edge shape. \(A_{1}\), \(A_{2}\) and \(A_{3}\) are translation matrices, and \(T_{1}\), \(T_{2}\), and \(T_{3}\) are rotation matrices, as shown in Equations (3)-(8). \[\left\{ \begin{array}{lcl}x_{i}\ =\ r_{i}\sin(-\zeta_{i})\\ y_{i}\ =\ r_{i}\cos(-\zeta_{i})\\ z_{i}\ =\ (\zeta_{i}/360)\cdot 2\pi r_{i}\cot\beta\\ r_{i}\ =\ r_{1}-\Delta r_{i}\end{array} \right. \tag{2}\]  
where \(\Delta r_{i}\) is the radial error of the \(i\)th cutter tooth. \[A_{1}\ =\ \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&\Delta c_{i}\\ 0&0&0&1\end{bmatrix}  \tag{3}\]  
\[A_{2}\ =\ \begin{bmatrix}1&0&0&x_{p}(t)\\ 0&1&0&y_{p}(t)\\ 0&0&1&z_{p}(t)\\ 0&0&0&1\end{bmatrix}  \tag{4}\]  
\[A_{3}\ =\ \begin{bmatrix}1&0&0&y_{f}\cdot t-x_{0}\\ 0&1&0&W+r_{1}-a_{e}\\ 0&0&1&H-a_{p}\\ 0&0&0&1\end{bmatrix}  \tag{5}\]  
\[T_{1}\ =\ \begin{bmatrix}\cos\varphi_{si}&-\sin\varphi_{si}&0&0\\ \sin\varphi_{si}&\cos\varphi_{si}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{6}\]  
\[T_{2}\ =\ \left[ \begin{array}{ccc}\cos\varphi_{q}(t)&-\sin\varphi_{q}(t)&0&0\\ \sin\varphi_{q}(t)&\cos\varphi_{q}(t)&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right] \tag{7}\]  
\[T_{3}\ =\