## 5 Reliability-Driven Optimization Design for Key Structural Elements  
### Optimization Design Based on NSGA-II Genetic Algorithm  
Based on the above reliability calculation method to redesign the key geometric features of the large aeronautical planetary mechanism, the reliability-sensitive structural elements of large thin-walled parts in the system are screened and multi-objective dimensional optimization analysis is executed. Using the rated working conditions of the system as the load boundary conditions of the simulation model, the uncertainty effect of the load is ignored in reliability analysis processes, so the load distribution function in Equation (21) will be given in the form of the determined tooth root stress peak. The simulation analysis shows that the ring gear rim plays a main supporting role for the planetary gear system, and their core structural parameters largely determine the meshing quality of the whole planetary gear system. Additionally, insufficient rigidity of the ring gear rim will lead to excessive bending deformation for gear teeth, and increased risk of fatigue tooth breakage \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Components** & **Angular Velocity** & **Relative Angular Velocity** & **Single Tooth Meshing Times** \\ \hline Sun gear & \(\omega_{\rm S}\) & \(p\cdot\omega_{\rm S}/(1+p)\) & \(n_{\rm S}(t)=p\cdot\omega_{\rm S}\cdot t\cdot k_{\rm P}/(1+p)\) \\ Planet gear & \(\omega_{\rm S}/(1-p)\) & \(2p\cdot\omega_{\rm S}/(1-p^{2})\) & \(n_{\rm PS}(t)=n_{\rm PR}(t)=2p\cdot\omega_{\rm S}\cdot t/\left(p^{2}-1\right)\) \\ Ring gear & 0 & \(-\omega_{\rm S}/(1+p)\) & \(n_{\rm R}(t)=\omega_{\rm S}\cdot t\cdot k_{\rm P}/(1+p)\) \\ Planet carrier & \(\omega_{\rm S}/(1+p)\) & 0 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Kinematic parameter of planetary gear train. in the planetary mechanism. A dimensionless internal gear rim thickness parameter \(\Lambda\) is defined as the ratio of the rim thickness to the tooth height as  
\[\Lambda=\frac{d_{out}-d_{root}}{d_{root}-d_{minor}} \tag{22}\]  
where \(d_{out}\), \(d_{root}\), and \(d_{minor}\) are the outside, root, and minor diameters of the internal gear, respectively. The ring gear rim thickness dimensional characteristics are core factors in the lightweight design of a large aviation planetary system, and its dimensional growth contributes significantly to the weight growth for the planetary mechanism; therefore, the fatigue reliability and the ring gear mass are optimization goals. In addition, based on the NSGA-II genetic algorithm, a best stiffness matching result of the thickness of the rim and base plate that jointly meets the requirements of reliability and lightweight indicators is sought. A high-fidelity mechanical simulation model is constructed by the hierarchical finite element method. Accordingly, the tooth root hazard stress histories of the tooth root of the system at different rim thicknesses is calculated to provide effective load input variables for the system fatigue reliability assessment model, and 6 points are used in the range of 10-35 mm for the ring gear rim thickness and 5 mm for the point interval. The fatigue reliability at different rim thicknesses can be obtained by Equation (21). The fitting polynomial \(Y_{1}(x)\) for the ring gear rim thickness and the fatigue reliability and \(Y_{2}(x)\) for the ring gear rim thickness and the mass of the ring gear are obtained respectively by MATLAB programming. Due to the increase in rim thickness, the overall mass of the planet carrier and system fatigue reliability are on the rise. To facilitate the selection of the optimization algorithm, the objective function \(Y_{1}(x)\) is rewritten as Equation (22). \[Y_{1}(x)=4.519\cdot 10^{(-5)}\cdot x^{3}+0.0006571\cdot x^{2}-0.1636\cdot x-96.8 \tag{23}\]  
\[Y_{2}(x)=\ 0.00464\cdot x^{2}-4.53\cdot x+19.1 \tag{24}\]  
where \(x\) is the thickness of the rim in mm.