## 2 Instantaneous Posture of the Milling Cutter and the Minor Cutting Edge
\right]^{T} \tag{6}\]  
where \(\vartheta\) is the conversion relationship between the cutter tooth coordinate system and the workpiece coordinate system. Figure 3: Motion state of the milling cutter and the cutter tooth under vibration. The angle \(\varphi_{q}(t)\) between the \(V\) axis and \(Y\) axis in the \(UVW\) plane could be expressed as:  
\[\varphi_{q}(t)=\varphi_{g}(t_{0})+\varphi_{i}+2\pi n(t-t_{0})-\left\lfloor\frac{ \varphi_{g}(t_{0})+\varphi_{i}+2\pi n(t-t_{0})}{2\pi}\right\rfloor\cdot 2\pi \tag{7}\]  
In the equation, \(\varphi_{g}(t_{0})\) could be expressed as:  
\[\varphi_{g}(t_{0})=\arccos(\frac{a_{e}-r_{\max}}{r_{\max}}) \tag{8}\]  
The instantaneous position coordinates (\(x_{00}(t)\), \(y_{00}(t)\), \(z_{00}(t)\)) of the coordinate origin \(\alpha_{0}\) of the milling cutter cutting coordinate system in the workpiece coordinate system _o-xyz_ without vibration could be derived as:  
\[\left\{ \begin{array}{ll}x_{o0}=v_{f}t-r_{\max}\\ y_{o0}=W_{0}-a_{e}\\ z_{o0}=H_{0}-a_{p}\end{array} \right. \tag{9}\]  
The instantaneous angle \(\theta_{1}\)(t) between the projection of the \(W\) axis on the \(v_{00}w\) plane and the \(w\) axis and the instantaneous angle \(\theta_{2}\)(t) between the projection of the \(W\) axis on the \(u_{00}w\) plane and the \(w\) axis could be written as:  
\[\theta_{1}(t)=\arctan(\frac{A_{y}(t)}{l-A_{z}(t)}),\theta_{2}(t)=\arctan( \frac{A_{x}(t)}{l-A_{z}(t)}) \tag{10}\]  
The instantaneous offset angle \(\theta(t)\) of the milling cutter cutting coordinate system with vibration could be calculated as:  
\[\theta(t)=\arctan\frac{\sqrt{A^{2}{}_{x}(t)+A^{2}{}_{y}(t)}}{l-A_{z}(t)} \tag{11}\]  
The titanium alloy milling experiment was carried out by using the three-tooth carbide shoulder milling cutter with a diameter of 25 mm provided by the cooperative enterprise. The milling experiment was carried out on a three-axis milling machining center (VDL-1000E).The cutting stroke was 2.5 m. The size of the titanium alloy specimen was 250 mm \(\times\) 100 mm \(\times\) 20 mm. Its material composition is shown in Table 2.  
In order to verify the effectiveness of the calculation method for the instantaneous posture of the milling cutter and the minor cutting edge of the cutter teeth, change the milling cutter speed but with the cutting efficiency unchanged, and carry out the milling experiment on titanium alloy. In order to eliminate the influence of the three cutter tooth error changes caused by the replacement of cutter teeth on the friction and wear boundary of the minor flank and also to obtain the vibration data during the cutter tooth wear process, new cutter teeth were used for each experimental scheme during the experiment, and the milling vibration signal was measured in real time in the experiments. The experimental protocol is shown in Table 3. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline
**Element** & **Al** & **V** & **Fe** & **O** & **Si** & **C** & **N** & **H** & **Others** & **Ti** \\ \hline Contents (\%) & 5.5–6.8 & 3.5–4.5 & 0.3 & 0.2 & 0.15 & 0.1 & 0.05 & 0.01 & 0.5 & margin \\ \hline \end{tabular}
\end{table} 
Table 2: Titanium alloy workpiece material composition. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Experiment**} & \multicolumn{3}{c}{\(\mathbf{\Delta r_{i}}\) (mm)} & \multicolumn{3}{c}{\(\mathbf{\Delta Z_{i}}\) (mm)} \\ \cline{2-7}  & **Cutter Tooth 1** & **Cutter Tooth 2** & **Cutter Tooth 3** & **Cutter Tooth 1** & **Cutter Tooth 2** & **Cutter Tooth 3** \\ \hline Scheme 1 & 0 & 0.012 & 0.029 & 0 & 0.024 & 0.012 \\ Scheme 2 & 0.017 & 0 & 0.003 & 0.007 & 0.013 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Radial and axial errors of cutter teeth. Figure 4: Milling vibration time domain signals. (**a**) Experimental scheme 1; (**b**) Experimental scheme 2.