## 3 Calculation Model of Instantaneous Cutting-Force Energy Efficiency and Specific Cutting Energy
The coordinate of \(o_{j}(x^{\alpha_{j}},y^{\beta_{j}},z^{\alpha_{j}})\) and \(m_{j}^{\prime}(x^{m_{j}^{\prime}},y^{m_{j}^{\prime}},z^{m_{j}^{\prime}})\) in the workpiece coordinate system were obtained using Equations (32)-(33). \[o_{j}(x^{\alpha_{j}},y^{\beta_{j}},z^{\alpha_{j}})\ =\ \left[x^{\alpha_{j}}\quad y^{ \beta_{j}}\quad z^{\alpha_{j}}\quad 1\right]^{T}\ =\ A_{3}A_{2}T_{3}T_{2}A_{1}T_{1}\cdot\left[x^{ \alpha_{j}}_{i}\quad y^{\alpha_{j}}_{i}\quad z^{\alpha_{j}}_{i}\quad 1\right]^{T} \tag{32}\]  
\[m_{j}^{\prime}(x^{m_{j}^{\prime}},y^{m_{j}^{\prime}},z^{m_{j}^{\prime}})\ =\ \left[x^{m_{j}^{ \prime}}\quad y^{m_{j}^{\prime}}\quad z^{m_{j}^{\prime}}\quad 1\right]^{T}\ =\ A_{3}A_{2}T_{3}T_{2}A_{1}T_{1}\cdot\left[x^{m_{j}^{ \prime}}_{i}\quad y^{m_{j}^{\prime}}_{i}\quad z^{m_{j}^{\prime}}_{i}\quad 1\right]^{T} \tag{33}\]  
Using Equations (27)-(33), the distribution functions of the instantaneous cutting-force energy consumption and instantaneous shear energy consumption of the milling micro-elements were expressed as  
\[p_{c}^{i}(z,t)\ =\ v_{c}^{i}(t)\cdot k_{t}\cdot\kappa_{c}(z,t)\cdot h_{D}^{i}(z,t)/\cos\beta \tag{34}\]  
Figure 4: Velocity decomposition of synthetic cutting motion of cutting edge feature points. \[p_{s}^{i}(z,t)\ =\ v_{s}^{i}(t)\cdot(\tau^{i}(z,t)\cdot h_{D}^{i}(z,t)/\cos\beta)/ \sin\theta_{s} \tag{35}\]  
where  
\[v_{s}^{i}(t)\ =\ v_{c}^{i}(t)\cdot\cos\theta_{s} \tag{36}\]  
\[\theta_{s}\ =\ \pi/4+\gamma_{0}-\arctan\mu \tag{37}\]  
\[\mu(x_{i},y_{i},z_{i})\ =\ \frac{2h\cdot v(x_{i},y_{i},z_{i})}{0.5a^{3}\cdot F _{N}(x_{i},y_{i},z_{i})\cdot(e^{\frac{h\cdot v}{kT(x_{i}y_{i}z_{i})}}-1)} \tag{38}\]  
and where \(k_{i}\) is the correction factor of tangential force, and its value range is 1.17-1.68, when the end mill is cutting titanium alloy. \(\kappa_{c}(z,t)\) is the unit cutting force. \(v_{s}^{i}(t)\) is the shear direction speed at the selected point of the \(i\)th cutter tooth. \(\tau^{i}(z,t)\) is the shear stress at any point of the ith cutter tooth. \(\theta_{s}\) is the shear angle. \(\mu\) is the friction coefficient, which can be obtained from the literature [23]. \(a\) is the lattice constant (2.9506 \(\times\) 10\({}^{-10}\) m). \(h\) is the Planck constant (\(l\) = 6.62607015 \(\times\) 10\({}^{-34}\) J-s). \(k\) is the Boltzmann constant (\(k\) = 1.380649 \(\times\) 10\({}^{-23}\) J/K). \(v\) is the frequency of atomic forced vibration. \(F_{N}\) is the normal stress at the characteristic point of the cutting edge. Thus, the instantaneous shear energy efficiency and specific cutting energy of the milling micro-element can be expressed as  
\[\eta_{s_{c}}^{i}(z,t)\ =\ \frac{p_{s}^{i}(z,t)}{p_{c}^{i}(z,t)}\ =\ \frac{v_{s}^{i} \cdot\tau^{i}(z,t)\cdot h_{D}^{i}(z,t)\cdot(dz/\cos\beta)/\sin\theta_{s}}{v_{ c}^{i}(t)\cdot k_{t}\cdot\kappa_{c}(z,t)\cdot h_{D}^{i}(z,t)\cdot(dz/\cos\beta^{i})}\ =\ \frac{v_{s}^{i}(t)\cdot\tau^{i}(z,t)}{v_{c}^{i}(t)\cdot\sin \theta_{s}\cdot k_{t}\cdot\kappa_{c}(z,t)} \tag{39}\]  
\[o_{j}(x^{\alpha_{j}},y^{\beta_{j}},z^{\alpha_{j}})\ =\ \left[x^{\alpha_{j}}\quad y^{ \beta_{j}}\quad z^{\alpha_{j}}\quad 1\right]^{T}\ =\ A_{3}A_{2}T_{3}T_{2}A_{1}T_{1}\cdot\left[x^{ \alpha_{j}}_{i}\quad y^{\alpha_{j}}_{i}\quad z^{\alpha_{j}}_{i}\quad 1\right]^{T} \tag{32}\]0  
\[o_{j}(x^{\alpha_{j}},y^{\beta_{j}},z^{\alpha_{j}})\ =\ \left[x^{\alpha_{j}}\quad y^{ \beta_{j}}\quad z^{\alpha_{j}}\quad 1\right]^{T}\ =\ A_{3}A_{2}T_{3}T_{2}A_{1}T_{1}\cdot\left[x^{ \alpha_{j}}_{i}\quad y^{\alpha_{j}}_{i}\quad z^{\alpha_{j}}_{i}\quad 1\right]^{T} \tag{32}\]1  
From Equations (39)-(41), due to the influence of milling vibration, cutter tooth error and other factors, the cutting attitude angle of the milling cutter and the cutting trajectory of the cutter tooth were in an unstable state. This unstable state caused the instantaneous contact relationship between the milling micro-element on the cutting edge of the cutter tooth and the workpiece to change frequently.