## 4 Instantaneous Bias of the Main Section of the Milling Cutter under Vibration
\(\Delta r_{i}\) is the radial error of the \(i\)th cutter tooth. \(n\) is the rotational speed of the milling cutter. \(f_{z}\) is the feed rate per tooth. \(a_{p}\) is the cutting depth of the milling cutter. \(a_{e}\) is the cutting width of the milling cutter. Figure 5: Milling cutter structure and cutter tooth serial number. During experiments, the milling vibration acceleration signals were acquired, as shown in Figure 6. Where \(a_{x}(t)\), \(a_{y}(t)\), and \(a_{z}(t)\) are milling vibration acceleration signals of the milling cutter along the workpiece coordinate system \(x\), \(y\), and \(z\), respectively. According to the different variation shown by the time domain characteristic of milling vibration in Figure 6 and its corresponding time, the cutting process was divided into multiple cutting stages. Where \(t_{0}\) is the starting time of each cutting stage, \(t^{\prime}\) is the middle time of an each cutting stage, \(\Delta t\) is the time interval of the cutting stage. \[t=t_{0}+\Delta t/2 \tag{32}\]  
According to Table 2 and Figure 6, the instantaneous main cutting force energy consumption of cutter teeth were calculated by using Equation (30). The time-domain characteristic curve of the main cutting force energy consumption of five cutter teeth of the milling cutter were obtained, as shown in Figure 7.  
Where \(P_{t}\)\({}^{0}\) is main cutting force energy consumption of the \(i\)th cutter tooth without milling vibration. \(P_{i}\) is main cutting force energy consumption of the \(i\)th cutter tooth with milling vibration. According to Figure 7, the results of the energy consumption of the main cutting force of the cutter teeth showed that the energy consumption of the main cutting force of each cutter tooth was periodic. The waveforms of energy consumption distribution of main cutting force of each cutter tooth of milling cutter was different. It was mainly reflected in the different values and periodic of the main cutting force energy consumption of each cutter tooth. This was because the tooth errors and the milling vibration of each tooth were different. As a result, the distribution of main cutting force energy consumption of the milling cutter composed of main cutting force energy consumption of the cutter teeth with different waveforms had dynamic characteristics. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Cutter** & \multicolumn{5}{c}{**Cutter Tooth Number**} & \multicolumn{5}{c}{**Cutting Parameters**} \\ \cline{2-9}
**Tooth** & \(i=1\) & \(i=2\) & \(i=3\) & \(i=4\) & \(i=5\) & \(n\) & \(f_{z}\) & \(a_{p}\) & \(a_{e}\) \\
**Error** & \(i=1\) & \(i=2\) & \(i=3\) & \(i=4\) & \(i=5\) & (rpm) & (mm/z) & (mm) & (mm) \\ \hline \(\Delta z_{d}^{i}\) & 0.051 & 0.008 & 0.000 & 0.012 & 0.022 & & & & \\ (mm) & 0.000 & 0.004 & 0.001 & 0.010 & 0.018 & & 1576 & 0.073 & 10 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Milling experimental parameters. Figure 6: Milling vibration acceleration signal and selection of cutting stage. Identification Method for the Dynamic Characteristics of Energy Consumption of the Milling Cutter Main Cutting Forces  
In order to unveil the dynamic characteristics of the energy consumption of the main cutting force of the milling cutter, the instantaneous energy consumption of the main cutting force of the milling cutter was obtained by using Equations (30) and (31), as shown in Equation (33). \[\left\{ \begin{array}{l}P(t^{i})=\sum\limits_{i=1}^{N}P_{i}(t^{i})=\sum \limits_{i=1}^{N}\int_{m_{0}(t^{i})}^{m_{k}(t^{i})}dP_{i}(x_{i},y_{i},z_{i})\\ \\ t^{i}_{1}\leq t^{i}\leq t^{i}_{4}\end{array} \right. \tag{33}\]  
where \(N\) is the amount of milling cutter teeth. The energy consumption of the main cutting force of the milling cutter and the cutter teeth with time was solved by using Equation (33), as shown in Figure 8.  
\(P^{0}\) is main cutting force energy consumption of the milling cutter without milling vibration. \(P\) is main cutting force energy consumption of the milling cutter with milling vibration.