## 4 Multi-Objective Optimization of Frame Structures  
### Design of Experiments  
The frame structure is composed of interconnected components, and the size changes of each component have different effects on the modal and vibration response. Therefore, the thickness of the mounting plate P1, the inner diameter of the connecting tube P2, the outer diameter of the connecting tube P3, the height of the connecting tube P4, the outer diameter of the connection block P5, the thickness of the motor plate P6, and the width of the motor plate P7 were selected as the parameter variables of the design to study the correlation effects of each component parameter on the vibration characteristics. The range of the values is shown in Table 5. The inner diameter of connecting tube P2 in the variables is always smaller than the outer diameter of connecting tube P3. The selection of spatial sample points employs the optimal spatial filling design [30; 31; 32] and is refined using Latin hypercube design [33; 34]. Sample point sampling is shown in Figure 12.  
Figure 12: Sample points for experimental design. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Variable Ranges** \\ \hline Thickness of the mounting plate P1/mm & 2–5 \\ Inner diameter of the connecting tube P2/mm & 0–12 \\ Outer diameter of the connecting tube P3/mm & 10–19 \\ Height of the connecting tube P4/mm & 418–478 \\ Outer diameter of the connection block P5/mm & 35–60 \\ Thickness of the motor plate P6/mm & 4–8 \\ Width of the motor plate P7/mm & 330–450 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Initial values and value ranges for rack design variables. that increasing the thickness of the motor plate (P6) enhances the axial modal frequency and consequently reduces the axial vibration amplitude. The test calculation shows that the axial modal frequencies are consistent with the third- or fourth-order modal frequencies, as shown in Figure 13b. Since the axial wave modes of different structures are consistent with the third- or fourth-order modal frequencies, the third-order modal frequencies can be increased to avoid shaft frequency excitation and reduce vibration response. From the correlation, Figure 13a indicates that the mounting plate thickness P1, connecting tube inner diameter P2, connecting tube outer diameter P3, motor plate thickness P6, and motor plate width P7 are more correlated. So, four of them are selected for response surface analysis. Figure 14 shows the response surface of the interaction between the mounting plate thickness P1 and the inner diameter of the connecting tube P2. Figure 14a shows that the maximum axial amplitude increases first and then decreases with the increase in the installation plate thickness P1. That is because the mounting plate thickness P1 increases, and the axial modal frequencies approach the excitation frequency, resulting in resonance. As the thickness of the mounting plate P1 continues to increase, the axial modal frequencies increase, and the difference between the excitation frequency increases, thereby hindering the ability to stimulate the resonance. Figure 14: Response surface of axial maximum amplitude (**a**) and axial modal frequencies (**b**) by P1 and P2 fitting. Figure 13: (**a**) Parameter correlation analysis: (**b**) relationship between axial modal frequencies and third- and fourth-order modal frequencies. The inner diameter of the connecting tube P2 increases, and the axial maximum amplitude first decreases and then increases. That is because the connecting tube's inner diameter P2 changes in the connecting tube with the quality of weakening the structural rigidity of the impact of the increase, and the axial modal frequencies increase. The connecting tube inner diameter P2 continues to increase, the connecting tube thickness is too small, resulting in a lack of stiffness, and the axial modal frequency decreases, increasing the maximum amplitude of the axial component.