## 4 Structural Optimization
### Comprehensive Discussion of Optimization Model  
From the data analysis and modeling results of the above three sections, the coupling relationships between the three optimization objectives can be observed. The optimization of a single objective may lead to performance losses in other areas. Therefore, when several  
Figure 12: Minimize the Y-axis displacement: (**a**) Fitness values. (**b**) Optimized structure. (**c**) Comparison of output displacement before and after optimization.  
Figure 13: Maximum energy transfer efficiency: (**a**) Fitness values; (**b**) Optimized structure; (**c**) Comparison of energy transfer efficiency before and after optimization.  
single-objective functions to be optimized are incompatible, multi-objective optimization (MOO) arises, which simultaneously considers multiple interrelated or conflicting optimization objectives. In this case, there is usually no single optimal solution, and the optimal solution must be obtained through weighting or by applying additional constraints to the functions. Thus, under the various optimization objectives described in this section, the trade-offs between the three objective functions are made based on the rate at which different design variables influence the objective function trends and the actual application of the displacement reducer. While ensuring a certain reduction ratio and output efficiency, the goal is to maximize the maximum constant stiffness range. The objective function is set as  
\[\text{Min }R\big{(}a_{x},\ldots,h_{x},h_{y},w_{\text{ad}}, \ldots,w_{\text{ef}}\big{)} =\omega_{1}\rho+\omega_{2}R+\omega_{3}\eta,\quad i=0,1,2,\ldots,n,\] \[\omega_{1}+\omega_{2}+\omega_{3} =1, \tag{17}\]  
\(\omega\) is a weighted variable. In addition to the constraints shown in the table, the following additional constraints are specified: the reduction ratio \(R>8\) and the energy transfer efficiency \(\eta>20\%\).  
After 1000 generations of iteration, the fitness values are as shown in Figure 14. The results tend to converge after about 580 iterations, demonstrating the feasibility of the optimization algorithm. Compared to Section 3, the reduction ratio increased by 149.5%, which increased from 4.5 prior to optimization to 11.23 following optimization.The maximum stiffness is 0.85 N/\(\,\)um. Figure 15 shows a comparison of the structure before and after optimization, highlighting the effects of changes in geometric position. The dashed line represents the results before optimization, while the solid line represents the results after optimization. Figure 16 illustrates the relationship between input and output displacements before and after optimization.  
Figure 14: Fitness values.  
Additionally, as the stroke gradually increases, the accuracy of the analytical model diminishes due to significant beam deformation. These issues will be addressed in our future work, where we plan to conduct more in-depth research.  
Based on the final optimization results, modal simulations were conducted. Figure 17 shows the first four modal shapes of the reducer obtained from the finite element model, and the corresponding frequencies are listed in Table 4. It can be observed that the first mode corresponds to irregular lateral oscillations of the reducer in the X-Z plane. The second mode corresponds to an out-of-plane deflection mode. The third mode corresponds to the working mode of the reducer. The fourth mode corresponds to regular lateral oscillations in the X-Z plane.  
Figure 16: The input–output displacement relationship.  
Figure 15: Optimized structure.