## 4 Dynamic Characteristics Analysis
### Modal Simulation Analysis of End Milling Cutter  
In this section, the finite element modal analysis method is used to solve the eigenvalues and eigenvectors of the system, and the eigenvalues and eigenvectors obtained are the natural frequencies and vibration modes of the end milling cutter. According to the actual milling speed range of the end milling cutter, the first six natural frequencies and modal shapes of the type A-D end milling cutters are calculated in this section. The finite element model established in Section 3.1 was imported into finite element software, and the material properties of end milling cutter were set, as shown in Table 2.  
According to the actual clamping situation of milling, the end of end milling cutter handle was set to full constraint. After the analysis types and options were specified, the first six-order natural frequencies of the type A-D end milling cutters were obtained, as shown in Table 3. The natural frequency is the main factor considering the stability of the milling process. When the external excitation frequency is the same as the natural frequency of the end milling cutter, resonance will occur and vibrations will increase. From the first six natural frequencies of the solution, it can be found that the difference between the first and second natural frequencies of theA-type and B-type end milling cutters is only 0.1 Hz. The first-order and second-order natural frequencies of the C-type and D-type end milling cutters are equal. For each type of end milling cutter, there is little difference between the third- and fourth-order natural frequencies, while the fifth- and sixth-order natural frequencies varies a lot. The third- and fourth-order natural frequencies of the A-type end milling cutter are near 6746.0 Hz, which are higher than those of the B-type and C-type end milling cutters and lower than the D-type end milling cutter. The \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Elastic Modulus (GPa)** & **Poisson’s Ratio** & **Density (g/cm\({}^{3}\))** \\ \hline \hline Cemented carbide & 640 & 0.22 & 14.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties of end milling cutter. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Order** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \multirow{4}{*}{Type} & A & 1182.1 & 1182.2 & 6746.3 & 6747.0 & 8758.8 & 13,391.0 \\  & B & 1288.8 & 1288.9 & 6465.6 & 6482.8 & 9622.1 & 14,192.0 \\ \cline{1-1}  & C & 1170.4 & 1170.4 & 6717.0 & 6717.9 & 8678.5 & 13,312.0 \\ \cline{1-1}  & D & 1258.1 & 1258.1 & 6676.8 & 6682.4 & 9345.9 & 13,947.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: First six-order natural frequencies (Hz) of each end milling cutters. fifth natural frequency of A-type end milling cutter is lower than that of the B-type end milling cutter, slightly higher than the C-type end milling cutter, and smaller than the D-type end milling cutter. The sixth natural frequency of the A-type end milling cutter is lower than that of the B-type end milling cutter, similar to the C-type end milling cutter, and slightly lower than the D-type end milling cutter. On the whole, the frequency of each order of the type A-D end milling cutters changes: the first- and second-order frequencies are similar, the third- and fourth-order frequencies are not much different, and the fifth- and sixth-order frequencies are far apart. The frequency values of each order, among each end milling cutter, are roughly the same. The results show that the tool structure has little effect on the natural frequency of different orders. In addition, the range of frequencies that can be used as the basis for harmonic response analysis is shown in the following passage. The first six mode shapes of the type A-D end milling cutters are shown in Figure 9. From the modal diagram, it can be seen that the first and second vibration modes of the type A-D end milling cutters are similar to each other, and they all swing in the XZ plane, but the swing directions are slightly different.