## 3 Results and Discussion
### Dynamic Characteristic Analysis Results and Discussion
#### 3.2.3 Harmonic Response Analysis Results and Discussion  
(1) Harmonic response analysis of the upper-surface edge of the workpiece  
A total of 60 solution schemes were set in the analysis, and 50 Hz was counted once. The amplitude and frequency curves of the direct-drive turntable in the X, Y, and Z directions were analyzed, as shown in Figure 16. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Order** & **Frequency (Hz)** & **Vibration mode** \\ \hline
1 & 595.06 & C-axis and workpiece oscillating back and forth \\
2 & 599.20 & C-axis and workpiece oscillating left and right \\ \hline
3 & 1581.5 & C-axis and workpiece rotation \\
4 & 2317 & C-axis and workpiece oscillating up and down \\
5 & 2436.6 & C-axis, workpiece, and swing frame oscillating back and forth \\
6 & 2469.7 & C-axis, workpiece, and swing frame oscillating left and right \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The first six vibration modes and natural frequencies of direct-drive turntable under turning conditions. Figure 15: Nephogram of the first six vibration modes under turning conditions. We can find, by comparing the three graphs, that there is a peak in both the X and Y directions when the milling frequency is close to 600 Hz. When the machining frequency increases, the amplitude will return to normal. When the machining frequency of the direct-drive turntable reaches about 1600 Hz and 2300 Hz, the amplitude will rise again. The equivalent speed of the electric spindle driving the tool during milling is about 9000 r/min, i.e., 150 r/s, far from 600 Hz, 1600 Hz, and 2300 Hz. It is observed that the amplitude is small, even when superimposed with static characteristic deformation. It is much less than 10 \(\upmu\)m. When the working frequency is near 600 Hz, 1600 Hz, and 2300 Hz, the direct-drive turntable has larger deformation, which corresponds to the first six orders of inherent frequency of modal analysis so that it can prove the correctness of the modal analysis. Therefore, the selected machining frequency for the direct-drive turntable should be far away from these three values to avoid damage to the direct-drive turntable structure. Since the superposition of the external high-frequency interference and milling force may reach the first-order natural frequency, the point with the natural frequency of 600 Hz was analyzed. Dynamic stiffness is one of the evaluation parameters of the structure's resistance to interference. Each different working speed of the milling force corresponds to a different dynamic stiffness, and the smaller the dynamic stiffness, the worse the structure's resistance to interference. The dynamic stiffness of the direct-drive turntable differs in the three directions, so it needs to be further calibrated by dynamic stiffness. The calculation formula for dynamic stiffness is:  
\[\text{K}=\text{F}/\text{A} \tag{16}\]  
In the formula, K is the dynamic stiffness, F is the corresponding milling force in each direction, and A is the maximum amplitude in each direction. The maximum deformation  
Figure 16: Harmonic response curve of turntable under edge loading: (a) X-directional harmonic response curve; (b) Y-directional harmonic response curve; (c) Z-directional harmonic response curve. in the X direction is 1.606 \(\times\) 10\({}^{-3}\) mm, in the Y direction is 1.109 \(\times\) 10\({}^{-3}\) mm, and in the Z direction is 2.295 \(\times\) 10\({}^{-6}\) mm; then, the dynamic stiffness in the three directions is Kx = 4.745 \(\times\) 10\({}^{5}\) N/mm, Ky = 2.678 \(\times\) 10\({}^{5}\) N/mm, and Kz = 1.918 \(\times\) 10\({}^{8}\) N/mm. When the milling force is applied to the maximum diameter of the workpiece on the direct-drive turntable, the dynamic stiffness of the turntable surface in the Y direction is the minimum when the frequency of the milling force is around 600 Hz, so when the above constraints are satisfied, the impact on the Y-direction machining accuracy is the greatest.