## 4 Multi-Objective Optimization of Frame Structures
### Optimization Results and Analysis  
Considering the frame structure size, weight, and manufacturing cost, the third group of optimal solutions is selected for rounding processing, and the comparison results are shown in Table 8. There is no significant change in vibration characteristics before and after rounding, and they are within a reasonable range.  
As can be seen from Figure 17, compared with the original model, the axial modal frequency and maximum axial amplitude of the frame structure after the round optimization are greatly optimized. The axial modal frequency (the third order) increased from 42.2 Hz to 52.3 Hz; at this time, the maximum axial amplitude decreased from 5.38 m/s to  
\begin{table}
\begin{tabular}{c c c c} \hline
**Design Variables** & **1** & **2** & **3** \\ \hline P1/mm & 4.49 & 4.72 & 4.33 \\ P2/mm & 6.62 & 10.61 & 9.22 \\ P3/mm & 17.63 & 18.16 & 17.22 \\ P4/mm & 443.70 & 449.99 & 450.03 \\ P5/mm & 59.94 & 52.07 & 37.82 \\ P6/mm & 7.69 & 7.84 & 7.68 \\ P7/mm & 449.74 & 449.58 & 350.45 \\ \hline Axial Maximum & Predicted & 1.41 & 1.44 & 2.10 \\ Amplification/(mm/s) & Simulated & 1.90 & 1.92 & 2.09 \\ \hline Axial Modal Frequency/Hz & Predicted & 53.99 & 54.85 & 50.40 \\ Simulated & 52.67 & 53.51 & 52.73 \\ \hline Maximum deformation/mm & Predicted & 0.348 & 0.56 & 0.47 \\ Simulated & 0.355 & 0.53 & 0.45 \\ \hline Maximum equivalent stress/MPa & Predicted & 51.35 & 48.91 & 42.12 \\ Simulated & 48.83 & 43.44 & 44.93 \\ \hline \end{tabular}
\end{table}
Table 7: Three optimal solutions.  
\begin{table}
\begin{tabular}{c c c} \hline
**Design Variables** & **After Optimization** & **Rounding Scheme** \\ \hline P1/mm & 4.33 & 4 \\ P2/mm & 9.22 & 9 \\ P3/mm & 17.22 & 17 \\ P4/mm & 450.03 & 450 \\ P5/mm & 37.82 & 38 \\ P6/mm & 7.68 & 7.5 \\ P7/mm & 350.45 & 350 \\ Axial Maximum & 2.09 & 2.15 \\ Axial Modal Frequency/Hz & 52.73 & 52.3 \\ Maximum deformation/mm & 0.45 & 0.34 \\ Maximum equivalent stress/MPa & 44.93 & 38.29 \\ \hline \end{tabular}
\end{table}
Table 8: Rounding of design variables for the optimal solution in Group 3.  
2.15 mm/s, and the maximum three-way amplitude under this working condition is below 2.5 mm/s.  
According to the \"frequency ratio = modal frequency/excitation frequency\" equation, the frequency ratio exceeds 1.2. Although the ideal vibration isolation condition is not reached, the amplitude can be effectively attenuated, indicating that structural optimization effectively improves the vibration characteristics of the frame. The maximum deformation and maximum equivalent stress vary greatly, and the structural strength meets the requirements.  
Figure 18 shows the deformation and stress results of the optimized structure. Compared with before optimization, the strength after optimization is more reliable. From Figure 18a, since the fan frame is a cantilever structure, the place with the greatest deformation is at the free end, where the motor plate and the connecting tube are. Due to the increased thickness of the optimized connecting tubes and mounting plates, the strength of the cantilever structure has increased, resulting in a significant reduction in deformation from 0.73 mm to 0.34 mm.  
Figure 18b shows that the stress after optimization was reduced from 48.44 MPa to 38.3 MPa, and the maximum equivalent stress of the structure always occurs on the mounting plate. That is because the mounting plate is a fixed end, and ultimately, all forces will transmit to the fixed installation position. The connecting tube is  
Figure 17: Comparison of vibration characteristics before and after optimization: (**a**) modal frequency; (**b**) vibration response at 2600 rpm.  
Figure 18: Optimized static results: (**a**) stress cloud map; (**b**) deformation cloud map.  
which resists deformation at the bend, and the connection is placed under the influence of moments, so the stress is relatively prominent.