## 3 Calculation Model of Instantaneous Cutting-Force Energy Efficiency and Specific Cutting Energy  
The instantaneous cutting behavior of the milling cutter and its cutter teeth under the combined action of milling vibration and cutter tooth error changed the instantaneous cutting contact relationship between the cutter teeth and the workpiece. It made the instantaneous cutting boundary and cutting volume of the cutter tooth cutting edge variable, as shown in Figure 3.  
According to Figure 3, the upper boundary (\(m_{k}\)) and lower boundary (\(m_{0}\)) of the cutting edge, which participate in cutting instantaneously under the influence of milling vibration and cutter tooth error, can be written as  
\[m_{k}\;=\; \begin{cases}r_{i}\cdot(\theta_{i}(t)-\theta_{st})\cdot\cot\beta&, \;\theta_{st}\leq\theta_{i}(t)<\theta_{p}\\ a_{p}&,\;\theta_{p}\leq\theta_{i}(t)\leq\theta_{et}\\ 0&,\;otherwise\end{cases}  \tag{22}\]  
\[m_{0}\;=\; \begin{cases}\Delta c_{i}+z_{0}^{i}(t)&,\;\theta_{st}\leq\theta_{i }(t)<\theta_{wt}\\ r_{i}\cdot(\theta_{i}(t)-\theta_{wt})\cdot\cot\beta&,\;\theta_{wt}\leq\theta_{ i}(t)\leq\theta_{et}\\ 0&,\;otherwise\end{cases}  \tag{23}\]  
\[\theta_{st}\;=\;3\pi/2-\arccos[(r_{i}-a_{e})/r_{i}] \tag{24}\]  
\[\theta_{et}\;=\;3\pi/2+(a_{p}-\Delta c_{i}+z_{p}^{i}(t))\cdot tan\beta/r_{i} \tag{25}\]  
\[\theta_{p}\;=\;\theta_{st}+(a_{p}-\Delta c_{i}+z_{p}^{i}(t))\cdot tan\beta/r_ {i} \tag{26}\]  
where \(\theta_{i}(t)\) is the instantaneous position angle of the tool tip. \(\theta_{st}\) is the initial cutting angle of the cutter teeth. \(\theta_{wt}\) is the initial cutting angle of the cutter teeth, which is \(3\pi/2\). \(\theta_{et}\) is the position angle when the cutter tooth completely cuts out the workpiece. \(\theta_{p}\) is the position angle when the cutting edge initially reaches the cutting depth. The instantaneous cutting-layer parameters and cutting volume per unit time of micro-elements milled on the cutting edge can be expressed as  
\[h_{D}^{i}(z,t)\;=\;f_{z}^{i}\cdot\sin\Bigl{(}\theta_{i}(t)-z^{i}\cdot\tan \beta/r_{i}\Bigr{)} \tag{27}\]  
Figure 3: Instantaneous cutting boundary of cutting edge and instantaneous cutting volume. (**a**) Instantaneous cutting boundary of cutting edge; (**b**) Instantaneous cutting volume. \[V_{i}(z,t)/dt\ =\ f_{z}^{i}\cdot\sin\left(\theta_{i}(t)-z^{i}\cdot\tan\beta/r_{i} \right)\cdot v_{c}^{i}(t)\cdot dz \tag{28}\]  
where \(f_{z}^{i}\) is the feed per tooth of the \(i\)th cutter tooth. \(v_{c}^{i}(t)\) is the component of the instantaneous velocity of the synthetic cutting motion of any point of the \(i\)th cutter tooth relative to the workpiece, considering the vibration velocity in the direction of the main motion velocity of the milling cutter. The \(v_{c}^{i}(t)\) of the milling micro-element is shown in Figure 4.  
In Figure 4, \(v_{c}^{i\alpha}\) is the component of \(v_{c}^{i}\) on the \(x\)-axis. \(v_{c}^{i\psi}\) is the component of \(v_{c}^{i}\) on the \(y\)-axis. \(v_{c}^{i\alpha}\) is the component of \(v_{c}^{i}\) on the \(z\)-axis. \(v_{c}^{\psi}\) is the component of the vibration velocity on the \(y\)-axis. \(v_{c}^{\psi}\) is the component of the vibration velocity on the \(z\)-axis. From Equations (1)-(9) and Figure 4, the instantaneous velocity of the synthetic cutting motion and instantaneous cutting velocity of the milling micro-element can be solved as  
\[v(t)\ =\ \sqrt{v_{x}^{2}(t)+v_{y}^{2}(t)+v_{z}^{2}(t)} \tag{29}\]  
\[ \begin{cases}v_{x}(t)\ =\ \partial x(t)/\partial t\\ v_{y}(t)\ =\ \partial y(t)/\partial t\\ v_{z}(t)\ =\ \partial z(t)/\partial t\end{cases}  \tag{30}\]  
\[v_{c}^{i}(t)\ =\ v(t)\cdot\cos\theta_{c}(t) \tag{31}\]  
where \(\theta_{c}(t)\) is the angle between \(v(t)\) and \(v_{c}^{i}(t)\) in the workpiece coordinate system. The direction of \(v_{c}^{i}(t)\) is consistent with the \(o_{j}m_{j}^{\prime}\) vector direction.