# An Improved Cutting Force Model for Ultrasonically Assisted Grinding of Hard and Brittle Materials
## 4 Model Prediction and Experimental Verification  
### Model Prediction  
By substituting the values of volume correction parameters into the cutting force model, numerical simulation can be carried out. Then, the relations between input variables and the axial and feed direction forces are obtained, as shown in Figure 16.  
Figure 15: Influences of input variables on the value of \(K\) and \(K_{s}\): (**a**) spindle speed \(n_{r}\); (**b**) amplitude \(A\); (**c**) feed rate \(v_{F}\); (**d**) cutting depth \(a_{P}\); and (**e**) radius \(r_{c}\).  
The axial force decreases as the vibration amplitude increases, which is depicted in Figure 16b. The ultrasonic vibration can effectively reduce the axial force because the trend in decreasing force is large. In addition, the feed direction force decreases as the amplitude increases, which may be explained by an increased trajectory of the lateral abrasive grains.  
Figure 16d shows the predicted relation between the radius of the transition fillet and the cutting force. It can be seen that the axial force increases as the radius increases. The main reason is that the number of active diamond grains in the end face decreases as the radius of transition fillet increases. Correspondingly, the material removal by the actions of intermittent hammering decreases. This result is consistent with the fact that the cutting force is larger as the tool wear condition becomes serious. However, the number of active diamond grains in the lateral face decreases as the radius of the transition fillet increases, which contributes to the decrease in the feed force.  
Figure 16a shows that both the axial and feed forces decrease as the spindle speed increases. On the contrary, they increase as the feed rate increases as shown in Figure 16c, and the feed force increases as the cutting depth increases, as shown in Figure 16e.  
Figure 16: Predicted relation between the variables and the cutting force: (**a**) spindle speed \(n_{r}\); (**b**) amplitude \(A\); (**c**) feed rate \(v_{F}\); (**d**) radius \(r_{c}\); and (**e**) cutting depth \(a_{p}\).  
Based on the results of the experiment with the input variables in Table 4, the values of the correction parameters in the model of [PERSON] [25] and [PERSON] [28] can be recalculated. Then, the cutting forces can be predicted though their models. Since the model in this paper is based on the fillet radius of the grinding wheel, the selected grinding depth is larger than the fillet radius of the tool, so the grinding depth parameter is larger, but the range of other parameters is basically the same.  
The comparison between the results with the theoretical values of the presented model and the experimental results, as shown in Figure 19, show that the axial force varies with the different spindle speeds and amplitudes and the feed force varies with the different spindle speeds as examples to show the comparative results. It can be seen that the prediction accuracy of the presented model is superior to that of others, which also proves the reliability of the presented model.  
Figure 18: Comparison of the feed direction force between the experimental results and predicted results: (**a**) spindle speed \(n_{r}\); (**b**) amplitude \(A\); (**c**) feed rate \(v_{F}\); (**d**) cutting depth \(a_{p}\); and (**e**) radius \(r_{c}\).  
An interesting phenomenon was observed. The edge chipping of the slot was decreased through an increase in the radius of the transition fillet. This phenomenon might be caused by the decrease in the feed force. The cutting force model can also be applied to the grinding of other hard and brittle materials, such as ceramics.