## 5 Conclusions  
In this paper, the centroids of each radial section of an end milling cutter with the same eccentricity and different structures are calculated and analyzed, and the dynamic characteristics of the cutters are studied using the finite element method. The main conclusions are as follows.  
Figure 13: Comparison curve of total displacement response of end milling cutter.  
(1) A new general mathematical model for calculating the centroid of the radial section of an end milling cutter is established, and the proposed mathematical model can be widely used other kinds of end milling cutters. The centroid of each section of the designed end milling cutter is calculated. According to the calculations, it is found that the centroid distribution of the type A-D end milling cutters with the same overall eccentricity are different: the position of the centroid of each section are different, the change trend of the abscissas and ordinates of the centroid are different, the eccentricity angles of the centroid position in each radial section are different, and the maximum eccentricities are different, which are 0.35180 mm, 0.05000 mm, 0.26616 mm, 0.067000 mm, respectively. In addition, the sections of the maximum eccentricity are different. For the A-type end milling cutter it is at the \(h\) = 15 mm section, for the C-type it is at the \(h\) = 13 mm section, and the eccentricity of each section of the B-type and D-type end milling cutters do not change much.  
(2) Combined with finite element modal analysis, all types of end milling cutters with the same eccentricity have the same modal shapes and similar natural frequencies in the same orders. Through finite element harmonic response analysis, the resonance frequency bands of various types of end milling cutters are obtained. Under the same dynamic excitation, each type of end milling cutter responds to displacement: A = 0.043800 mm \(<\) C = 0.072100 mm \(<\) B = 0.17930 mm \(<\) D = 0.22670 mm.  
(3) Through investigation, it is found that the dynamic characteristics of the end milling cutter with the same eccentricity but different structure are different. The dynamic performance of each type of end milling cutter with the same overall eccentricity and the same level of dynamic balance accuracy: the A-type end milling cutter is the best, followed by the C-type end milling cutter, and the B-type end milling cutter is better than the D-type end milling cutter. This shows that the centroid distribution may have an effect on the dynamic characteristics of the milling cutter. Without changing the overall eccentricity, it provides a new idea for the structural design with the goal of improving the dynamic characteristics of the end milling cutter, and provides a theoretical basis and reference for the high precision design of the end milling cutter.  
Conceptualization, H.Y. and M.Z.; methodology, H.Y. and M.Z.; software, H.Y. and W.L.; validation, H.Y., M.Z. and W.Z.; formal analysis, H.Y.; investigation, H.Y.; resources, W.N.; data curation, H.Y. and W.L.; writing--original draft preparation, H.Y.; writing--review and editing, H.Y. and M.Z. and W.Z.; visualization, W.L.; supervision, M.Z. and W.Z.; project administration, M.Z.; funding acquisition, M.Z. All authors have read and agreed to the published version of the manuscript.  
This work was funded by the High-level Talents Project (2020 GSP11) supported by the Central Government for the Reform and Development of Local Universities in Heilongjiang Province.  
Not applicable.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.