## 5 Results and Validation
### Cutting-Force Analysis and Experimental Validation  
The cutting-force coefficients are identified by a series of milling tests on a VMC-850 machining center. The experimental setup is shown in Figure 14. The cutting forces are collected by a three-dimensional dynamometer Kistler 9257B. The workpiece is mounted on the dynamometer through a vise fixture. The milling tests are conducted in the type of down-milling mode and dry cutting operation. The machining parameters are listed in Table 4. According to the identification approach proposed by [PERSON] [40], the tangential, radial, and axial cutting and edge coefficients for the selected combination of cutter and workpiece materials are obtained. The cutting-force coefficients \(K_{tc}\), \(K_{rc}\), and \(K_{ac}\) are 2082.5, 625.1, and 947.8 N/mm\({}^{2}\) as calculated by means of the shearing action in the tangential, radial, and axial directions, respectively. The edge force coefficients \(K_{te}\), \(K_{re}\), and \(K_{ac}\) are 11.0, 24.4, and 2.0 N/mm, respectively.  
Figure 12: Variation of the engagement angle with the revolution angle for different compression ratios.  
Figure 13: Variation of the engagement angle with the revolution angle for different cutter radii.  
Figure 15 shows the ellipse-based trochoidal milling of straight slots. The cutting forces are measured by the dynamometer Kistler 9257B. For ellipse-based trochoidal milling, the cutting force in one cycle varies with the cutter location. Therefore, it is necessary to determine the cutting force at different revolution angles. According to the mechanistic cutting-force model presented in Section 4, the cutting force in ellipse-based trochoidal milling is predicted, as shown in Figure 16. The spindle speed is 2000 r/min. The feed rate is 480 mm/min. The semi-major and semi-minor axes of the fundamental ellipse are 15 and 11.25 mm. Therefore, the compression ratio is 75%. The cutter radius is 5 mm. The trochoidal step is 0.6 mm. It can be found that the predicted and measured cutting forces have the same profile in \(x\)-, \(y\)-, and \(z\)-axes, respectively, and the cutting-force amplitudes correspond well.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Spindle Speed** & **Axial Depth of Cut** & **Radius Depth of Cut** & **Feed Rate per Tooth** \\
**(rpm)** & **(mm)** & **(mm)** & **(mm/tooth)** \\ \hline  & & & 0.03 \\
2000 & 6 & 0.6 & 0.06 \\  & & & 0.09 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Machining parameters used in flank milling tests.  
Figure 14: Experimental setup for the milling tests.  
Figure 15: Ellipse-based trochoidal milling tests. (**a**) Machining coordinate system; (**b**) Straight slots in process, for the upper slot, \(a=15\) mm, \(b=7.5\) mm, for the lower slot, \(a=15\) mm, \(b=3.75\) mm; (**c**) Machined straight slots.  
Figure 17 shows the predicted total cutting forces involved in the trochoidal milling with different toolpath parameters. Comparing the total cutting forces with the engagement angles, it can be seen that the profile of the total cutting force corresponds to that of the engagement angle. As the compression ratio decreases, the trend of the total cutting force is the same as the trend of the engagement angle. Especially when the compression ratio decreases from 50% to 25%, the total cutting force increases sharply in the initial stage. This will severely damage the cutter as well as the machine tool and should be avoided.  
Figure 16: Predicted and measured cutting forces for ellipse-based trochoidal milling. \(A=15\) mm, \(b=11.25\) mm, \(\rho=75\%\), \(R_{c}=5\) mm, \(s=0.6\) mm. (**a**) Simulated cutting force in \(x\)-axis; (**b**) Measured cutting force in \(x\)-axis; (**c**) Simulated cutting force in \(y\)-axis; (**d**) Measured cutting force in \(y\)-axis; (**e**) Simulated cutting force in \(z\)-axis; (**f**) Measured cutting force in \(z\)-axis.