## 5 Instantaneous Friction Energy Consumption of the Minor Flank of the Cutter Tooth
The angle between the vertical node and the minor flank of the cutter tooth and the \(c_{i}\) axis is \(\theta_{ci}\). \(\tau\) is the equivalent stress of the minor flank of the cutter tooth at the network node under the thermal coupling field. \(F_{1}\), \(F_{2}\) and \(F_{3}\) are the equivalent stresses in the direction of the tetrahedral area element, respectively. The equivalent stress and normal stress are represented by vectors in the cutter tooth coordinate system, as shown in Equations (37) and (38). \[\left\{ \begin{array}{l}\overset{\rightarrow}{F}_{1}=(a_{F1},b_{F1},c_{F1}) \\ \overset{\rightarrow}{F}_{2}=(a_{F2},b_{F2},c_{F2})\\ \overset{\rightarrow}{F}_{3}=(a_{F3},b_{F3},c_{F3})\end{array} \right. \tag{37}\]  
\[\overset{\rightarrow}{F}_{p}=\left(0,0,\sqrt{a_{i}^{2}+b_{i}^{2}+c_{i}^{2}}\right) \tag{38}\]  
From Equations (37) and (38), \(\theta_{k1}\), \(\theta_{k2}\), and \(\theta_{k3}\) can be expressed as:  
\[\left\{ \begin{array}{l}\theta_{k1}=\arccos(\frac{\overset{\rightarrow}{F} _{p}}{F_{1}})\\ \theta_{k2}=\arccos(\frac{\overset{\rightarrow}{F}_{p}}{F_{2}})\\ \theta_{k3}=\arccos(\frac{\overset{\rightarrow}{F}_{p}}{F_{3}})\end{array} \right. \tag{39}\]  
According to Equations (37)-(39), the normal stress was obtained as:  
\[F_{p}(a_{i},b_{i},c_{i})=F_{1}(a_{i},b_{i},c_{i})\cdot\cos\theta_{k1}+F_{2}(a_ {i},b_{i},c_{i})\cdot\cos\theta_{k2}+F_{3}(a_{i},b_{i},c_{i})\cdot\cos\theta_ {k3} \tag{40}\]  
\(F_{n}\) is the normal pressure of the friction contact area unit, and \(\mu\) is the friction coefficient. Then, the work performed by the unit friction force in the \(dt\) can be expressed as:  
\[dW=\mu(a_{i},b_{i},c_{i},t)\cdot F_{n}(a_{i},b_{i},c_{i},t)\cdot v_{m}(a_{i},b _{i},c_{i},t)\cdot dt \tag{41}\]  
It was assumed that during the frictional motion of the tool-worker interface, all the frictional work was converted into the thermal energy of the system. \[dW=dE \tag{42}\]  
From Equations (37)-(42), the friction coefficient distribution function is shown in Equation (43). \[\mu(a_{i},b_{i},c_{i},t)=\frac{2h\cdot v(a_{i},b_{i},c_{i},t)}{0.5a^{3}\cdot F _{p}(a_{i},b_{i},c_{i},t)\cdot(e^{\frac{h\cdot v}{kT(a_{i},b_{i},c_{i})}}-1)} \tag{43}\]  
Figure 12: Equivalent stress decomposition. The friction stress distribution function obtained from Equations (41)-(43) is shown in Equation (44). \[f_{p}(a_{i},b_{i},c_{i},t)=\mu(a_{i},b_{i},c_{i},t)\cdot F_{p}(a_{i},b_{i},c_{i},t) \tag{44}\]  
According to Equation (38), the friction coefficient distribution of the minor flank of the cutter tooth was calculated, as shown in Figure 12.  
From the above analysis, the friction coefficient was mainly affected by the friction speed and the normal stress. Comparing Figures 8 and 13, the friction speed and normal stress in the non-contact area between the minor flank of the cutter tooth and the machining transition surface of the workpiece was 0. So, the friction coefficient boundary was the same as the friction speed boundary. However, due to the different normal stress values in the frictional contact area, the instantaneous distribution of the friction coefficient on the minor flank and the distribution of the friction speed were different. Comparing the distribution of the friction coefficient of the minor flank of the three teeth, it can be seen that the distribution area of the friction coefficient of the minor flank of each tooth was completely different from the boundary shape. It showed that it was greatly affected by the cutter tooth error and milling vibration. According to Equation (44), the friction force distribution on the minor flank of the first cutter tooth was calculated, as shown in Figure 14.  
Figure 13: Instantaneous friction coefficient distribution on the minor flank of the cutter tooth of scheme 1 (\(\rho_{i}=52^{\circ}\)). (**a**) Cutter tooth 1; (**b**) Cutter tooth 2; (**c**) Cutter tooth 3.