# Cutting Force and Cutting Quality during Tapered Milling of Glass Magnesium Board
## 3 Results and Discussion  
### Research on Cutting Force Based on Range Analysis  
Table 4 shows the results of the resultant force via range analysis. The larger R values indicate a greater contribution of the factor on the results [20,21]. In this work, the solution shows R\({}_{\rm h}\) = 33.637 \(>\) R\({}_{\rm X}\) = 11.563 \(>\) R\({}_{\rm n}\) = 2.123. Therefore, factor h (cutting depth) had the greatest influence on the cutting force, followed by factor \(\lambda\) (taper angle), and the factor n (spindle speed) had the least influence on the cutting force. In addition, taking the cutting force as the evaluation index, according to the value of K\({}_{\rm i}\), the optimal combination of the three factors with the lowest cutting force could be selected. It can be concluded that the combination \(\lambda_{1}\)n\({}_{3}\)h\({}_{1}\) was the optimal cutting parameter, in which the taper angle was 15\({}^{\circ}\), the spindle speed was 6000 rpm (43.96 m/s cutting speed, Equation (2)), and the cutting depth was 0.5 mm. The effects of the taper angle and cutting parameters on the cutting force are displayed in Figure 3. It was found that the cutting force was enhanced with the increase of the taper angle, decreased slightly with the increase of the spindle speed, and increased with the increase of the cutting depth. Figure 3: Trends of cutting force under different cutting conditions. Figure 2: Schematic diagram of the cutting experiment: (**a**) tapered cutting, (**b**) dynamometer, (**c**) cutting force diagram, (**d**) surface profiler, (**e**) machined surface. During the tapered milling, shown in Figure 4a,b, the cutting edge BC was rotated around the OO' axis to form a tapered surface. By cutting the tapered surface with a plane (KM) perpendicular to the machined surface, the gyration radius, diameter of a point on the cutting edge, and cutting speed can be expressed as Equations (2)-(4), respectively. \[\rho=S\cdot\tan(90^{\circ}-\lambda) \tag{2}\]  
\[d=S\cdot\cos\lambda \tag{3}\]  
\[V_{c}=\frac{\pi dn}{6\times 10^{4}} \tag{4}\]  
where \(\rho\) (mm) is the cutting radius of gyration at any point on the cutting edge BC, \(S\) (mm) is defined as the distance from the apex of the cone to the intercepted point, \(\lambda\) (\({}^{\circ}\)) is the taper angle, \(d\) (mm) is the diameter of a point on the cutting edge BC, and \(V_{c}\) (m/s) is the cutting speed of a point on the cutting edge BC. An illustrative sketch of feed per tooth and chip thickness is displayed in Figure 4c. During cutting, the unwanted material above the cutting layer was removed by the rake angle and formed into chips. The chip volume, namely, removal quantity, was directly determined by feed per tooth and cutting thickness, as expressed by Equations (5) and (6) [15]. \[U_{Z}=\frac{1000U}{n\cdot z} \tag{5}\]  
where \(U_{z}\) (mm) is the feed per tooth, \(U\) is the feed rate with a constant of 15 m/min, \(n\) (rpm) is the spindle speed, and \(z\) is the number of teeth, which was a constant of six in this work. \[a_{av}=\frac{1}{2}a_{max}=U_{z}\cdot\sqrt{\frac{h}{D}} \tag{6}\]  
where \(a_{av}\) (mm) is the average cutting thickness, \(a_{max}\) (mm) is the maximum cutting thickness, \(h\) (mm) is the cutting depth, and \(D\) is the tool radius with a constant of 140 mm. According to above analysis, the increased taper angle led to a decreased gyration radius and increased cutting speed. The glass MGO board used was an inhomogeneous composite material, having many hard particles. When the cutting edge with a higher speed bit into the glass MGO board, the greater impact force would act on the cutter from the hard particles. Therefore, the cutting force showed a growing trend with the increase of the taper angle of the cutters. Meanwhile, the feed per tooth and average cutting thickness increased with decreasing the spindle speed and increasing the cutting depth. Thus, the cutting quantity per time during milling decreased with the increase of spindle speed, and increased with increasing cutting depth.