# An Improved Cutting Force Model for Ultrasonically Assisted Grinding of Hard and Brittle Materials
## 5 Conclusions  
A predictive cutting force model was developed with the consideration of the influence of diamond grains in the transition fillet surface of the tool for predicting cutting forces in UAG. During the calculation of the cutting force, volume correction parameters \(K\) and \(K_{s}\) are used for the consideration of the overlapping of the removed volume of material and various protrusive heights. Through the slot grinding experiments, the values of \(K\) and \(K_{s}\) were obtained, and the influence of input variables and the radius of the transition filled on the cutting forces have been studied both theoretically and experimentally. Some conclusions can be summarized from this study,  
(1) Due to the variations in the normal and the tangential directions of the ultrasonic vibration amplitude, the cutting force of an abrasive grain in the transition fillet surface is a sum of the cutting forces produced by the former two modes of actions. In this paper, the cutting force model is established with the simultaneous consideration of the influence on the cutting force produced by the diamond grains in the end face, lateral face and transition fillet surface of the tool. Through the slot grinding experiments with different input parameters, the trend in the cutting force determined by the model agrees well with that of the experiments. The error of the proposed cutting force model is 10.3%, and the experimental results verify the correctness of the force model. The improved cutting force model can be used for research into the ultrasonically assisted grinding of hard and brittle materials.  
(2) All the input parameters (\(n_{r}\), \(A\), \(v_{F}\), \(a_{p}\), \(r_{c}\)) have a minor effect on the value of \(K_{s}\), which is basically a constant with a value of 0.166., but only \(n_{r}\), \(v_{F}\) and ap have minor effects on \(K\), which is basically a constant with a value of 0.37. However, vibration amplitude  
Figure 19: Comparison of the presented model and others’ models: (**a**) axial force \(F_{a}\)—spindle speed \(n_{r}\); (**b**) axial force \(F_{a}\)—amplitude \(A\); and (**c**) feed force \(F_{f}\) –feed rate \(v_{F}\).  
and radius \(r_{c}\) have a strong effect on the values of \(K\). The value of \(K\) increases as amplitude \(A\) increases but decreases as radius \(r_{c}\) increases. An equation is used to express the relationship between \(K\) and the input variables \(A\) and \(r_{c}\).  
(3) As the fillet radius increases from 0.2 to 1.2 mm, the cutting force increases by 139.6% in the axial direction and decreases by 70% in the feed direction. As the spindle speed and ultrasonic vibration amplitude increase, both the axial and feed forces decrease. Furthermore, the effect of ultrasonic vibration amplitude on the axial force is gradually reduced as the vibration amplitude becomes larger but is increased on the feed force. The cutting force increases as the feed rate and cutting depth increase.  
Conceptualization, R.K. and Z.D.; methodology, Z.D. and J.L.; software, F.Z.; validation, Y.B., F.Z. and J.D.; formal analysis, J.L.; investigation, J.D.; resources, R.K.; data curation, J.L.; writing--original draft preparation, R.K. and J.L.; writing--review and editing, Y.B.; visualization, F.Z.; supervision, Z.D.; project administration, R.K.; funding acquisition, R.K. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Key Basic Research Project, grant number 2020-JC**-ZD-176-01, China Postdoctoral Science Foundation, grant number 2020T130076 and Doctoral Scientific Research Fund of NSFL, grant number 2019-BS-053.  
Not applicable.  
The authors declare there is no conflict of interest regarding the publication of this paper.