# Dynamic Performance Optimization of Circular Sawing Machine Gearbox
## 3 Gearbox Structure Optimization
### Multi-Objective Optimization  
According to the structural characteristics of the sawing gearbox, the multi-objective genetic algorithm was selected based on the Kriging model to fit the response surface function to the multi-objective optimization of the five main design variables. Taking the design variable as the input and the modal frequency as the output, the sample space was generated, the response surface model was constructed, the input value range was set, the multi-objective optimization of the design variables was completed, and the optimal solution of the design variables of the gearbox was obtained. A comparative analysis of the main design parameters before and after optimization is shown in Table 3.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Design Variable** & \(\mathbf{\alpha_{k}}\) & \(\mathbf{\beta_{k}}\) \\ \hline P1 & 1207.0035 & 0.03484 \\ P2 & 1206.4120 & 1.2158 \\ P3 & 1206.0075 & \(-\)1.2737 \\ P4 & 1205.9913 & \(-\)2.2569 \\ P5 & 1206.0577 & \(-\)1.4159 \\ P6 & 1206.1296 & 0.064 \\ P7 & 1206.5736 & 1.3852 \\ P8 & 1206.9093 & \(-\)2.0469 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Regression coefficient values for structure optimization.  
Figure 10: Relationship between the size variations and modals.  
Modal numerical simulation was performed on the optimized gearbox. This was followed by the application of a fixed constraint at the contact position between the gearbox and slider, and the addition of a mass unit at the bearing installation position. The calculation model was then solved, and the first ten steps modal of the optimized gearbox were obtained. The modal frequency is shown in Table 4.  
By comparing the relevant data in Tables 1 and 4, it can be determined that the optimization of the gearbox at the natural frequency was significantly enhanced. Moreover, the first-order natural frequency increased by approximately 82%, which is far from the possible excitation frequency range in the cutting process.  
According to the numerical results shown in Figure 11, the maximum principal stress of the gearbox is 164.74 kPa and the maximum deformation is 0.27454 \(\upmu\)m, which is lower than the ultimate compressive strength and tensile strength, and satisfies the design requirements.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Order** & **Frequency** & **Order** & **Frequency** \\ \hline
1 & 892.7 & 6 & 2458.6 \\
2 & 1295.2 & 7 & 2635.1 \\
3 & 1701.3 & 8 & 2880.4 \\
4 & 2048.9 & 9 & 2989.5 \\
5 & 2307.2 & 10 & 3106.7 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Gearbox modal frequency after structural optimization.  
Figure 11: Cloud map of gearbox after optimization. (**a**) Maximum principal stress and (**b**) total deformation.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Original Values/mm** & **Optimized Values/mm** & **Size Change/\%** \\ \hline P3 & 35 & 37 & 5.7 \\ P4 & 30 & 32 & 6.7 \\ P5 & 40 & 42 & 5 \\ P7 & 10 & 11 & 10 \\ P8 & 11.11 & 15 & 45 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparative results of gearbox design variables before and after optimization.