## 3 Solution Method of the Cutter Tooth Instantaneous Cutting Boundary under Vibration
Using the method for constructing the tooth cutting edge equation shown in Equation (11), the equation of machining transition surface \(\Sigma_{i-1}\) formed by the (\(i-1\))th cutter tooth could be expressed as  
\[\Sigma_{i-1}(x,y,z)=\left\{ \begin{array}{ll}E_{i-1}(x,y,z)=\left[\begin{array} []{cc}x(t^{i-1})&y(t^{i-1})&z(t^{i-1})&1\end{array}\right]^{T}\\ \\ t_{1}^{i-1}\leq t^{i-1}\leq t_{4}^{i-1}\end{array} \right. \tag{18}\]  
where \(E_{i-1}(x,y,z)\) is the cutting edge equation in the workpiece coordinate system from the period of \(t_{1}^{i-1}\) to \(t_{4}^{i-1}\) of cutter tooth \(i-1\), \(t^{i-1}\) is the moment when the cutting edge of cutter \(i-1\) tooth cuts the workpiece, and \(t^{i-1}\) is the initial moment when the cutting edge of the cutter tooth \(i-1\) cuts into the side elevation \(G_{L}\) of the transition surface, \(t_{4}^{i-1}\) is the moment when the cutting edge of cutter tooth \(i-1\) on the upper surface \(G_{H}\) of the workpiece cuts out of the transition surface. Thus, the characteristic moment could be expressed as  
\[m_{ik}^{H}(t_{4}^{i})=\left\{ \begin{array}{l}E_{i}(x,y,z)=\left[\begin{array}{ cc}x(t_{4}^{i})&y(t_{4}^{i})&z(t_{4}^{i})&1\end{array}\right]^{T}\\ E_{i}(x,y,z)=\Sigma_{i-1}(x,y,z)\\ z(t_{4}^{i})=z_{H}\end{array} \right. \tag{19}\]  
According to Equation (13), it could be obtained that during the process of the cutter tip of the \(i\)th cutter tooth cuts into the side elevation \(G_{L}\) of the workpiece to be machined until it cut out of the transition surface \(\Sigma_{i-1}\), the curve equation of the characteristic point of the lower boundary of the cutting edge could be expressed as  
\[m_{io}^{s}(t^{i})=\left\{ \begin{array}{l}s_{i}(x,y,z)=\left[\begin{array}{ cc}x(t^{i})&y(t^{i})&z(t^{i})&1\end{array}\right]^{T}\\ t_{1}^{i}\leq t^{i}\leq t_{3}^{i}\end{array} \right. \tag{20}\]  
where \(t_{i}^{3}\) is the characteristic moment when the cutter tip cuts out of the transition surface \(\Sigma_{i-1}\), which could be expressed as  
\[m_{io}^{s}(t_{3}^{i})=\left\{ \begin{array}{l}s_{i}(x,y,z)=\left[\begin{array} []{cc}x(t_{3}^{i})&y(t_{3}^{i})&z(t_{3}^{i})&1\end{array}\right]^{T}\\ s_{i}(x,y,z)=\Sigma_{i-1}(x,y,z)\end{array} \right. \tag{21}\]  
After the cutter tip separate from the transition surface \(\Sigma_{i-1}\), the lower boundary of the cutting edge could be expressed as  
\[m_{io}^{E}(t^{i})=\left\{ \begin{array}{l}E_{i}(x,y,z)=\left[\begin{array}{ cc}x(t^{i})&y(t^{i})&z(t^{i})&1\end{array}\right]^{T}\\ E_{i}(x,y,z)=\Sigma_{i-1}(x,y,z)\\ t_{3}^{i}\leq t^{i}\leq t_{4}^{i}\end{array} \right. \tag{22}\]  
According to Equations (11) and (22), it could be acquired that, the upper instantaneous cutting boundary \(m_{ik}(t^{i})\) of the cutting edge could be given as  
\[m_{ik}(t^{i})=\left\{ \begin{array}{l}m_{ik}^{L}(t^{i}),\quad t_{1}^{i}\leq t ^{i}\leq t_{2}^{i}\\ m_{ik}^{H}(t^{i}),\quad t_{2}^{i}\leq t^{i}\leq t_{4}^{i}\end{array} \right. \tag{23}\]  
The lower instantaneous cutting boundary of the cutting edge could be given as  
\[m_{io}(t^{i})=\left\{ \begin{array}{l}m_{io}^{s}(t^{i}),\quad t_{1}^{i}\leq t ^{i}\leq t_{3}^{i}\\ m_{io}^{E}(t^{i}),\quad t_{3}^{i}\leq t^{i}\leq t_{4}^{i}\end{array} \right. \tag{24}\]  
The cutting boundaries were not only related to the instantaneous cutting pose of the current cutter teeth, but also closely related to the machining transition surface formed by the previous cutter teeth. The results showed that the instantaneous cutting boundaries were in an unstable state, which was affected by the vibration, the cutter tooth error, edge shape, and instantaneous cutting pose of the two adjacent cutter teeth. This not only directly affects the instantaneous cutting layer parameters, but also changes the distribution of instantaneous main cutting force energy consumption on the cutter teeth.