# Feed Rate Variation Strategy for Semi-Conical Shell Workpiece in Ball Head End Milling Process
## 4 Experiment Results and Validation  
The milling experiment is conducted under the condition where the semi-conical shell workpiece is milled in a practical finish milling process with the cutting fluid condition and milling tool path based on the contour mentioned above, as shown in Figure 14a. Two workpieces of the same size are used, and the DHF ball nose endmill cutter is used, whose geometric parameters are given in Table 1. The milling parameters are described at the beginning of Section 2. Since the cutting fluid is used and the whole outer surface of the workpiece is milled, a triaxial accelerometer is mounted on the front  
Figure 13: Results of continuous feed rate variation (CFV), discrete feed rate variation (DFV) and improved discrete feed rate variation (IDFV) for each single contour: (**a**,**d**,**g**) feed rate; (**b**,**e**,**h**) radial cutting force; (**c**,**f**,**i**) vibration response; (**a**–**c**) contour at HP; (**d**–**f**) contour at MP; (**g**–**i**) contour at LP. end of the machine center spindle to collect and illustrate the vibration of the workpiece as shown in Figure 14b. The measured time-domain vibration signals with and without feed rate variation for the whole milling process can be seen in Figure 15. It can be seen that the consuming time of the whole milling process is reduced by 25% with the use of the proposed feed rate variation strategy. Since the finishing process is employed, the machining quality and precision are the priority of consideration while the time saving is not too large. The vibration amplitude will increase gradually for the first half period, and the increase rate will slow down at a high level when there is no feed rate variation. In contrast, the vibration amplitude will continue to increase for the whole milling process with the use of the variable feed rate. To verify the performance comprehensively and reasonably, time-domain vibration signals for a single contour line are extracted and compared, as shown in Figure 16. Three contour paths are selected as mentioned in Section 2.1, which are situated at the height of 120 mm (HP), 80 mm (MP) and 40 mm (LP). Since the consuming time is different between machining with and without IDFV, the vibration figures for the same contour are arranged in an adjacent position. From Figure 16, a common phenomenon can be observed that the vibration amplitude is large at the right end of the workpiece where the radius is large, and it will decrease at the left end where the radius is small, which are in accordance with the simulation results from Figure 12. With the use of DFV, there will be two sudden increases, which are in accordance with the simulation results from Figure 13.  
Figure 14: Photos of experimental setup (**a**) milling experiment; (**b**) ball head endmill cutter and accelerometer. Figure 15: Time-domain vibration signals for the whole milling process: (**a**) without and (**b**)with the feed rate variation. After the milling process, some outer surface positions of the workpiece are selected, and their surface roughness is measured by the Tesa Rugosurf 20. The selected measurement positions are distributed uniformly at 1/6, 2/6 and 3/6 of the total length of each contour tool path and three times are measured at each selected position. The measured average surface roughness \(R_{a}\) is shown in Figure 17, and the results are ranging between \(R_{a}\) 7 \(\upmu\)m and \(R_{a}\) 8 \(\upmu\)m. It should be pointed out that the practical milling studied in this paper is the finish milling. The workpiece will then be machined by turning and obtained the roughness requirement of \(R_{a}\) 6.3 \(\upmu\)m eventually. From Figure 17, the surface roughness from each measurement position are all increased at different levels with the use of DFV. The largest increase of the surface roughness occurred at the 2/6 position, where a higher feed rate is used. However, the rate of the increase is near 5%, which is acceptable.