## 5 Results and Discussion
### Optimization Results of Stress Distribution Considering Contact Force Fluctuations  
In the assembly, deviations manifesting as gear tooth jitters emerge due to the assembly clearances, subsequently leading to alterations in the contact forces. As stipulated by Equation (11), a rectification term, \(\cos^{3/2}\left(\frac{\pi-2\theta}{4}\right)\), is incorporated during the computation of the contact forces. As with the parameter configurations delineated in Section 5.4, the optimization trajectory is vividly portrayed in Figure 14.  
Figure 14: Variation of objective function model and minimum objective value considering contact force fluctuations. (**a**) Objective function model. (**b**) Minimum objective value vs. number of function evaluations.  
Figure 13: Evolution process of optimal fitness value.  
Ultimately, the performance metrics of the predictive model were elucidated as MSE = 6.8474 \(\times\) 10\({}^{-8}\), RMSE = 2.6168 \(\times\) 10\({}^{-4}\), and MAE = 4.6979 \(\times\) 10\({}^{-5}\), and the coefficient of determination, R\({}^{2}\), stood at 1. Collectively, these indices underscore the model's robust explanatory power concerning the data, as graphically elucidated in Figure 15.  
Upon the execution of the genetic algorithm, the optimal solution procured was [0.0645 mm, 0.0083 mm, 0.0819\({}^{\circ}\)]. This corresponds to an optimal fitness value (i.e., predictive error) of 0.4648. The trend of this optimal fitness value in relation to the number of generations can be visually represented in Figure 16.  
In the comprehensive analysis on the stress distribution in the cycloid-pin gear systems, two distinct methodologies were adopted: one that factored in the nuances of the contact force fluctuations and another that disregarded these fluctuations. Before incorporating the effect of the contact force fluctuations, the Bayesian-enhanced least squares boosted surrogate model exhibited an R-squared value of 0.9998, an MSE of 9.6527 \(\times\) 10\({}^{-7}\), an  
Figure 16: Evolution process of optimal fitness value under the influence of contact force fluctuations.  
Figure 15: Predicted value vs. actual value under the influence of contact force fluctuations.  
RMSE of 9.8248 \(\times\) 10\({}^{-4}\), and an MAE of 2.7699 \(\times\) 10\({}^{-4}\). This model predicted an optimal error configuration of [0.0641 mm, 0.0058 mm, 0.0584\({}^{\circ}\)] with a predictive error (or fitness value) of 0.4691.  
Upon the consideration of contact force fluctuations, brought about by gear tooth jitters due to assembly clearances, the model's performance metrics exhibited an even higher fidelity. The R-squared value reached perfection, standing at 1, while the MSE, RMSE, and MAE values were reduced to 6.8474 \(\times\) 10\({}^{-8}\), 2.6168 \(\times\) 10\({}^{-4}\), and 4.6979 \(\times\) 10\({}^{-5}\), respectively. The optimal error values in this scenario were [0.0645 mm, 0.0083 mm, 0.0819\({}^{\circ}\)], corresponding to a slightly improved predictive error of 0.4648.  
The observed improvements in the model's performance metrics when accounting for the contact force fluctuations underpin the significance of considering such dynamic effects in the optimization process. By ensuring a holistic view of the system behavior, this study provides a pivotal foundation for more accurate predictive modeling and optimization in gear systems.