## 3 Method for Solving Instantaneous Friction Wear and Energy Density on the Tooth Flank  
From Figures 1 and 2, the friction velocity and contact stress during the transient contact friction between the tool flank and the processed transition surface are shown in Figure 4.  
Figure 3: Instantaneous wear boundary of cutter tooth 1. From Figure 4, stress components along the transient friction velocity direction of the micro-element on the tool flank are obtained using the instantaneous thermal coupling field equivalent force in the workpiece coordinate system:  
\[\tau_{i}{}^{\prime}(t)=\sigma_{x}(t)cos\gamma_{x}+\sigma_{y}(t)cos\gamma_{y}+ \sigma_{z}(t)cos\gamma_{z} \tag{11}\]  
where \(\gamma_{x}(t)\),\(\gamma_{y}(t)\),\(\gamma_{z}(t)\) are the angles between \(\sigma_{x}(t)\), \(\sigma_{y}(t)\), \(\sigma_{z}(t)\) and the friction velocity, respectively. From Equations (7) and (11), the transient friction energy consumption, \(P_{i}(t)\), at any point on the cutter's face after the micro-element of the tool flank and the cumulative friction energy consumption, \(\Sigma E_{i}\), are:  
\[P_{i}(t)=\tau_{i}{}^{\prime}(t)\cdot\oint_{L}ds_{i}\cdot v_{i}{}^{\prime}(t), \sum E_{i}=\int_{t_{j}}^{t_{j+1}}P_{i}(t)\cdot dt \tag{12}\]  
where \(t_{j}\), \(t_{j+1}\) are the cutting start time and cutting end time of the cutter teeth, respectively. From Equations (11) and (12), the instantaneous friction wear energy density, \(\rho_{i}(t)\), at any point on the micro-element and the average friction wear energy density, \(\overline{\rho_{i}}\), during the cutting period are:  
\[\rho_{i}(t)=P_{i}(t)/V_{i}(t)=v_{i}{}^{\prime}(t)\cdot\tau_{i}{}^{\prime}(t)/ v_{hi}(t),\overline{\rho_{i}}=\sum E_{i}/\sum V_{i} \tag{13}\]  
where \(\Sigma V_{i}\) is the total volume of wear at any point on the tool flank after the micro-element. The above models could be used to solve for the instantaneous friction and wear energy density, in combination with the energy-efficient milling cutter and its cutting titanium alloy process conditions provided by the company. The milling method is dry-milling, the milling cutter is an indexable carbide milling cutter with a diameter of 32 mm, and the tooth error and processing parameters are listed in Table 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Tooth Error**} & \multicolumn{3}{c}{**Tooth Number**} & \(n\) & \(v_{f}\) & \(a_{p}\) & \(a_{e}\) \\ \cline{2-7}  & \(i=1\) & \(i=2\) & \(i=3\) & **(rpm)** & **(mm/min)** & **(mm)** & **(mm)** \\ \hline \(\Delta z_{i}\) (mm) & 0.000 & 0.001 & 0.003 & & & \\ \(\Delta r_{i}\) (mm) & 0.025 & 0.000 & 0.064 & 1143 & 500 & 0.7 & 16 \\ \hline \end{tabular}
\end{table} 
Table 2: Experiment scheme 1. Figure 4: Instantaneous friction velocity and friction force. Using the above scheme, the vibration acceleration signals were obtained by the experiments, as shown in Figure 5.  
The milling tool model in Figure 1 and its transient cutting behavior solution method were used to analyze the thermal coupling field of cutting titanium alloy in combination with the experimental parameters in Table 1, where the thermal coupling field distribution of the first tooth of the above experimental milling cutter at the 287 th cutting cycle with a contact angle of 45\({}^{\circ}\) is shown in Figure 6.  
Based on Table 1 and Figure 5, Equation (13) is used to solve for the instantaneous friction and wear energy density, where the instantaneous friction and wear energy density on the flank of the three cutter teeth varies in the 1 st milling cut cycle, the 143 rd milling cut cycle, the 287 th milling cut cycle, the 430 th milling cut cycle, and the 574 th milling cut cycle, as shown in Figure 7.  
In the figure, \(e_{1}\), \(e_{2}\), \(e_{3}\), \(e_{4}\), and \(e_{5}\) are the five characteristic points selected along the flank of the cutter tooth, whose \(x_{i}\) and \(y_{i}\) directional coordinates in the coordinate system of the cutter tooth are (3.635, 6.700), (3.650, 6.700), (3.675, 6.700), (3.700, 6.700), and (3.725, 6.700), respectively.