## 2 Characterization of Instantaneous Cutting Energy Transmission and Conversion of Milling Cutter
\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{8}\]where \(a_{p}\) is the cutting depth. \(a_{e}\) is the cutting width. \(\Delta c_{i}\) is the axial error of the \(i\)th cutter tooth. \(\beta\) is the helix angle of the cutter tooth. \(\varphi_{si}\) is the angle between the cutter teeth. \(\varphi_{q}(t)\) is the instantaneous angle between the \(y_{d}\) axis and \(y_{v}\) axis in the \(x_{v}o_{v}y_{v}\) plane. \[\varphi_{q}(t)\ =\ \varphi_{q}(t_{0})+2\pi n\cdot t-\left\lfloor\frac{\varphi_{q} (t_{0})+2\pi n\cdot t}{2\pi}\right\rfloor\cdot 2\pi \tag{9}\]  
According to Figure 1, during the milling process, the energy input from the main drive system and feed system of the machine tool was transmitted to the milling cutter and workpiece through the cutter body, the functional interface of each cutter tooth and the cutting layer. The energy output of the milling cutter not only included the cutting energy consumption of the cutter tooth, but also included a variety of additional energy consumption caused by the intermittent cutting of the cutter tooth. There were a variety of transmission and conversion relationships between the instantaneous energy input and output of the milling cutter, as shown in Figure 2.  
In Figure 2, \(P(t)\) is the input energy. \(P_{c}(t)\) is the tangential-force energy consumption. \(P_{ik}(t)\) is the internal energy of the milling cutter. \(P_{cc}(t)\) is the energy consumption of the centrifugal force vibration of the milling cutter. \(P_{c}^{v}(t)\) is the milling cutter's cutting-force vibration energy consumption. From Figures 1 and 2, the energy input of milling cutter instantaneous cutting can be expressed as  
\[P(t)\ =\ P_{M}(t)+P_{1}(t)+P_{2}(t)+P_{3}(t) \tag{10}\]  
\[P_{1}(t)\ =\ P_{x}(t)+P_{y}(t)+P_{z}(t) \tag{11}\]  
\[P_{2}(t)\ =\ P_{vx}(t)+P_{vy}(t)+P_{yz}(t) \tag{12}\]  
where \(P_{1}(t)\) is the input energy of the machine tool feed system. \(P_{2}(t)\) is the additional vibration energy of the feed system. Additionally, \(P_{3}(t)\) is the additional energy provided by the machine tool for the energy consumption of the milling cutter. The energy output of the milling cutter instantaneous cutting can be written as  
\[P_{E}(t)\ =\ P_{ik}(t)+P_{cc}(t)+P_{c}(t)+P_{d}(t) \tag{13}\]  
Figure 2: Energy transfer and transformation in milling process of milling cutter. \[P_{c}(t) = P_{c}^{v}(t)+\sum\limits_{i\ =\ 1}^{Z}P_{s}^{i}(t)+\sum\limits_{i\ =\ 1}^{Z}P_{f}^{i}(t) \tag{14}\] \[P_{s}^{i}(t) = P_{sV}^{i}(t)+P_{sh}^{i}(t)\] (15) \[P_{f}^{i}(t) = P_{fw}^{i}(t)+P_{fh}^{i}(t) \tag{16}\]  
where \(P_{d}(t)\) is the additional energy consumption of the milling cutter energy consumption. Using Equations (10)-(16), analyzing the energy allocation of the milling cutter in the cutting process, the efficiency of converting the instantaneous input energy of the milling cutter into the instantaneous tangential-force energy can be solved, as shown in Equation (17). Among them, the additional energy provided by the machine tool in the energy consumption of the milling cutter was small and could not be calculated quantitatively, so it was ignored. Figure 2.0  
From Equation (14), the shear deformation energy of the milling cutter acting on the cutting deformation area of the workpiece was the effective cutting energy to form the machined surface. The efficiency of converting the instantaneous tangential-force energy of the milling cutter into instantaneous shear energy can be derived as shown in Equation (18). Among them, the instantaneous tangential-force energy consumption of the milling cutter was composed of the tangential-force energy consumption of different cutter teeth involved in cutting.