## 4 Results and Discussions
### Spindle Vibration under Different Cutting Parameters  
Figure 9 illustrates the variation of machining vibration with varying cutting parameters under constant feed of 0.05 and 0.075 mm/tooth. The response surface graphs clearly indicate that the vibration level induced in machining is clearly affected by the combined effect spindle speed and cutting depth. Roughly speaking, vibration amplitude increases with an increase in the spindle speed and cutting depth, as found in Figure 9. It is noted that machining with different cutting depths and at a constant feed 0.075 mm/tooth, vibration apparently increases with an increase in the spindle speed from 3000 to 8000 rpm, with a correlation coefficient (r = 0.92-0.96). For machining at a constant feed 0.05 mm/tooth, the correlation between roughness and spindle speed is not significant (r = 0.3-0.66) depending on the cutting depth. In particular, when the cutting depth is greater than 1.6 mm, the vibration level increases with an increasing speed and reaches the maximum value at a spindle speed of 6000 rpm; it then decreases with an increasing speed.  
Figure 8: Variation of surface roughness under different feed rates and specific cutting depths.  
Figure 7: Variation of surface roughness with varying spindle speed and cutting depth at lower and higher feed rate, respectively.  
Figure 10 illustrates the variation of the vibration level generated under different feed rates and cutting depths, in which the feed rates range from 600 to 2400 mm/min, corresponding to the spindle speed from 3000 to 8000 rpm and feed of 0.05 and 0.075 mm/tooth. The extent of the effect of feed rate on the vibration varies with a cutting depth in machining. Within the entire testing data, there is no significant correlation between the vibration level and the feed rate. However, as observed in Figure 10, at a lower cutting depth (Z \(<\) 0.8 mm), machining vibration is slightly increases with increasing the feed rate, with a correlation coefficient (r = 0.6-0.7). In addition, within the testing conditions (Z \(>\) 1.0 mm, F = 1800-2000 mm/min, and S = 6000-8000 rpm), the machining vibration apparently increases with an increase in the feed rate, spindle speed, and cutting depth, which shows a significant influence on vibration (r \(>\) 0.8), and therefore a negative influence on the surface equality, as shown in Figure 10.  
The effect of machining vibration on surface roughness can be observed from Figure 11, which shows that they are moderately correlated: r = 0.58 and 0.71 for feed of 0.05 and 0.075 mm/tooth, respectively. Evidently, the vibration level induced in machining affects the roughness of the machined surface to a different extent, which depends on the interactive effects of cutting parameters. As illustrated in Figure 9, machining vibrations are relatively lower (\(<\) 0.02 g) under a specific combination of feed rate and spindle speed, such as 1200 mm/min and 4000 rpm and 1500 mm/min and 5000 rpm, respectively. Consequently, finer surfaces with less roughness are obtained. However, if the feed rate and spindle speed are 1000 mm/min and 5000 rpm or 1600 mm/min and 8000 rpm, higher level vibrations are generated (\(>\)0.03 g) to deteriorate surface precision with high roughness. Therefore, the appropriate  
Figure 10: Variation of vibration level under different feed rates and specific cutting depths.  
Figure 9: Variation of vibration levels with varying spindle speed and cutting depth at lower and higher feed rate, respectively.  
selection of the cutting parameters can lessen the vibration in the machining operation to obtain better surface quality.