## 4 Conclusions  
Through static analysis, we found the maximum deformation position by adding cutting force to different processing positions of the direct-drive turntable. At the maximum deformation position, the stress was far less than the yield limit of the material, and the deformation degree was also within the allowable range of processing conditions. The finite element modal analysis of the direct-drive turntable shows that the lower natural frequency mainly affects the C-axis part of the direct-drive turntable. To further investigate the relationship between the machining frequency and the dynamic deformation of the direct-drive turntable, harmonic response analysis was performed on the direct-drive turntable. Safety analysis of the dynamic characteristics of the direct-drive turntable was carried out. The machining frequency is far from the natural frequency, so a direct-drive turntable can avoid that resonance during machining. Combined with the variable density method, the mathematical topology optimization model was established, and topology optimization of the single and composite conditions was carried out, respectively. The topology optimization results were analyzed, and the mass redundancy of the swing frame was reduced by topology optimization. Size optimization was used for the other parts of the direct-drive turntable. The research results provide a theoretical basis for the multi-objective optimization of direct-drive turntables.  
Multi-objective optimization based on a genetic algorithm was proposed to realize the lightweight design of a direct-drive turntable under the condition that the dynamic and static performance of the turntable meets the machining requirements. The optimized size parameters were selected according to the direct-drive turntable's structural characteristics and topology optimization results. The optimized size parameters were screened via sensitivity analysis. Through the central composite experimental design, 45 test points were obtained, and the 45 points were fitted to the response surface model using the BP neural network. Taking the total mass, the first natural frequency, and the maximum deformation of the direct-drive turntable as the optimization objectives, NSGA-II was used for multi-objective optimization of the direct-drive turntable, and we obtained the optimal solution. The results show that the maximum deformation and the first-order natural frequency of the direct-drive turntable is optimized to a certain extent, and the mass of the direct-drive turntable is significantly reduced by 21.394%. The dynamic and static characteristics of the direct-drive turntable are improved. The correctness of the optimization results is verified via modal experiments, indicating that the multi-objective optimization method has reasonable engineering practicability.  
Conceptualization, K.L., B.T. and J.Z.; methodology, K.L., J.W. ([PERSON]) and J.Z.; software, B.H. and J.W. ([PERSON]); validation, K.L., B.T. and J.Z.; formal analysis, B.H.; investigation, B.H.; resources, B.H.; data curation, B.H. and J.W. ([PERSON]); writing--original draft preparation, B.T. and J.W. ([PERSON]); writing--review and editing, B.T.; visualization, J.Z.; supervision, K.L.; project administration, B.H. and J.W. ([PERSON]); funding acquisition, B.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Foundation of Artificial Intelligence Key Laboratory of Sichuan Province, grant number 2020 RY01, the Science & Technology Department of Sichuan Province, grant number 2021 YFG0050, and the Graduate Innovation Fund of Sichuan University of Science & Engineering, grant number Y2022047.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
All the authors are greatly acknowledged for their financial support in making this research possible.  
The authors declare no conflict of interest.