## 4 Mechanistic Cutting-Force Model  
The method for calculating the cutting force in ellipse-based trochoidal milling is based on the mechanistic cutting-force model [37]. As shown in Figure 8, the cutter is divided into a finite number of small differential elements along the tool axis. For each cutting element, three orthogonal cutting-force components acting on the \(j\)th flute can be expressed as follows:  
\[\left[ \begin{array}{c}dF_{t,j}(\varphi_{j}(z))\\ dF_{r,j}(\varphi_{j}(z))\\ dF_{a,j}(\varphi_{j}(z))\end{array} \right]=\left[ \begin{array}{c}K_{te}dz+ K_{tc}h_{j}(\varphi_{j}(z))dz\\ K_{re}dz+K_{rc}h_{j}(\varphi_{j}(z))dz\\ K_{ae}dz+K_{ac}h_{j}(\varphi_{j}(z))dz\end{array} \right] \tag{21}\]  
where \(K_{te}\), \(K_{re}\), and \(K_{ae}\) are the edge force coefficients in tangential, radial, and axial directions, respectively. \(K_{tc}\), \(K_{re}\), and \(K_{ac}\) are the cutting-force coefficients in tangential, radial, and axial directions, respectively. \(dz\) is the axial differential height. \(h_{j}(\varphi_{j}(z))\) is the instantaneous undeformed chip thickness. \(\varphi_{j}(z)\) is the radial immersion angle of the cutting edge \(j\), and it can be expressed as follows:  
\[ \begin{array}{ll}\varphi_{j}(z)&=\varphi+(j-1)\varphi_{p}-\psi(z)\\ &=\frac{2\pi n}{60}t+(j-1)\frac{2\pi}{N}-\frac{2z\tan\beta}{D}\end{array}  \tag{22}\]  
where \(\varphi\) is the rotation angle of the cutting edge. \(\varphi_{p}\) is the pitch angle of the cutter. \(\psi(z)\) is the radial lag angle caused by the local helix angle. \(n\) is the spindle speed. \(t\) is the time parameter. \(N\) is the number of cutting flutes. \(\beta\) is the constant helix angle of the cutter. \(D\) is the cutter diameter. The feed direction can change the distribution of the cutting force in the \(x\)-axis and \(y\)-axis directions in the WCS. The radius of the curvature of the tool  
Figure 8: Cutting-force analysis in ellipse-based trochoidal milling. feed rates at each point of the cutting-edge circle [38; 39], as shown in Figure 9. Therefore, the instantaneous undeformed chip thickness is determined as follows:  
\[h_{j}(\varphi_{j}(z))=f_{l}(\varphi_{j}(z))\sin(\varphi_{j}(z)+\pi-\varphi_{1}), \text{ and }\varphi_{st}\leq\varphi_{j}(z)\leq\varphi_{ex} \tag{23}\]  
where  
\[f_{l}(\varphi_{j}(z))=\frac{f_{l}\big{[}R_{p}+R_{c}\cos(\varphi_{1}-\varphi_{j }(z))\big{]}}{R_{p}} \tag{24}\]  
where \(f_{l}(\varphi_{j}(z))\) is the feed rate at each point of the cutting-edge circle. \(\varphi_{1}\) is measured clockwise from the positive \(y_{c}\)-axis to the vector \(\overset{\rightarrow}{C\!P_{1}}\). \(\pi-\varphi_{1}\) denotes the angular change in the feed direction. \(f_{l}\) is the normal feed per tooth at the cutter center. \(R_{p}\) is the radius of curvature of the toolpath and can be expressed as follows:  
\[R_{p}=\frac{\left[1+\left(\frac{d\varphi_{JC}(\theta)}{d\varphi_{JC}(\theta)} \right)^{2}\right]^{\frac{3}{2}}}{\left|\frac{d^{2}\varphi_{JC}(\theta)}{d \varphi_{JC}^{2}(\theta)}\right|}=\frac{\left[1+\left(\frac{a\sin\theta}{2\pi }\right)^{2}\right]^{\frac{3}{2}}\left|\left(\frac{s}{2\pi}+b\cos\theta\right) ^{3}\right|}{\left|\frac{sa\cos\theta}{2\pi}+ab\right|} \tag{25}\]  
Once three orthogonal cutting-force components are obtained, they can be mapped into a Cartesian coordinate system. The tangential, radial, and axial forces are resolved in the feed (\(x\)), normal (\(y\)), and axial (\(z\)) directions using a transformation as follows:  
\[\left[ \begin{array}{c}dF_{x,j}(\varphi_{j}(z))\\ dF_{y,j}(\varphi_{j}(z))\\ dF_{z,j}(\varphi_{j}(z))\end{array} \right]=\left[ \begin{array}{ccc}-\cos \varphi_{j}(z)&-\sin\varphi_{j}(z)&0\\ \sin\varphi_{j}(z)&-\cos\varphi_{j}(z)&0\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}dF_{t,j}(\varphi_{j}(z))\\ dF_{r,j}(\varphi_{j}(z))\\ dF_{a,j}(\varphi_{j}(z))\end{array} \right] \tag{26}\]  
Then, the total cutting-force components for the rotational position \(\varphi\) can be evaluated by summing up the forces acting on all edges along the axial depth of the cut:  
\[