## 4 Dynamic Characteristics Analysis
### Harmonic Response Analysis  
The harmonic response analysis is also called frequency sweep analysis. Through applying the load boundary condition to the opposite milling cutter, the amplitude of the end milling cutter in different natural frequency range is obtained. Two kinds of load boundary conditions are mainly used in frequency sweep analysis: (a) concentrated force is applied at the tool tip and (b) a linear load is applied at the cutting edge. Considering the actual working conditions of the end milling cutter, with reference to previous studies [30], in this paper, a single concentrated 2 \(n\) force was applied at the tip of the irregular tooth end milling cutter to replace the actual load. From the results of the Section 4.2 modal analysis, it is known that the frequency range of 1 to 6 modal orders is 1170.4 Hz-14,192 Hz. In order to ensure the accuracy of analysis, the analysis frequency range 0-16,000 Hz and frequency interval 10 Hz were set. The amplitude response curves of the type A-D end milling cutters in the directions of \(x\), \(y\), and \(z\) are shown in Figures 10-12. From the amplitude response curve, it can be seen that the maximum amplitude response of the end milling cutter appears near the first and second natural frequencies. According to the modal analysis results in the previous section, the frequency of the external excitation load range is 1000 Hz-1500 Hz, and the total displacement harmonic response curves of A, B, C and D end milling cutters are pertinently calculated, as shown in Figure 13.  
Figure 11: Amplitude response curve of (**A**–**D**)-type end milling cutter in \(y\) direction. Figure 12: Amplitude response curve of (**A**–**D**)-type end milling cutter in z direction. Figure 10: Amplitude response curve of (**A**–**D**)-type end milling cutter in \(x\) direction. It can be seen from the figure that when the milling frequency is lower than 1100 Hz and higher than 1300 Hz, the amplitude of the A-D type end milling cutters tends to be infinitesimal. When the frequency range is from 1100 Hz to 1300 Hz, the A-type end milling cutter resonance frequency band is from 1160 Hz to 1200 Hz, the resonance frequency band of the B-type end milling cutter is higher than that of the A-type end milling cutter in 1270 Hz-1310 Hz, the resonance frequency band of the C-type end milling cutter is similar to that of the A-type end milling cutter in 1150 Hz-1190 Hz, and the resonance frequency band of the D-type end milling cutter is in 1240 Hz-1280 Hz. When the frequencies of the type A-D end milling cutters are at 1180 Hz, 1290 Hz, 1170 Hz, 1260 Hz, the displacement of each type of end milling cutter reaches a peak at 0.043800 mm, 0.17930 mm, 0.072100 mm, and 0.22670 mm, respectively. The response displacement of each type of the end milling cutters under the same dynamic excitation: A < C < B < D. It can be known from the results that there is little difference in the frequency corresponding to the maximum response displacement of end milling cutters with different structures. Moreover, the range of their resonance frequency bands are similar. Thus, the dynamic performance of the type A end milling cutter is the best. Combined with the finite element modal analysis, the first six natural frequencies of the type A, B, C and D end milling cutters have little difference among the same orders, and the modal shapes are as follows: first- and second-order integral swing, third- and fourth-order integral torsion, fifth-order radial expansion and sixth-order uniform expansion. It is known that the vibration modes of the same order of the A-D end milling cutters are the same. From the analysis of the amplitude response curve of the end milling cutter, we can know the resonance frequency band and the maximum response displacement of each type of end milling cutter.