## 4 Error Sensitivity Analysis of the Dynamic Performance Based on the PCE Model of the Gear System
### Sensitivity Analysis of System Dynamic Response to Random Manufacturing Errors  
By random sampling 20,000 times, the global sensitivities of four typical manufacturing errors to the system dynamic response were analyzed by using the Sobol\({}^{\prime}\) analysis method. The changes in sensitivity indices of each variable were compared by sampling in 5-, 6-, and 7-grade processing accuracies.  
The error sensitivity results of the gear system under different precisions are shown in Figure 17. In the figure, the red part is the main effect coefficient, while the white part is the total effect coefficient. It can be seen that the profile shape error had the greatest influence on the peak-to-peak value of the dynamic meshing force, followed by the shape error and slope error along the direction of the lead, and the profile slope error had the least influence. When the machining accuracy of the gear decreased, and the variation range of error expanded, the influence degree of the shape error and slope error along the direction of the lead increased gradually, forming three main influencing factors together with the profile shape error. The influence of the profile slope error was always minimal.  
Figure 17: Error sensitivity indices under different precisions: (**a**) grade 5 accuracy, (**b**) grade 6 accuracy, and (**c**) grade 7 accuracy.  
As can be seen from the error sensitivity results above, the main effect coefficients of all errors were smaller than the total effect coefficients, and the differences between them reflect the proportion of the high-order effect of the error. The total interaction effect between each error and other errors is shown in Table 4 and Figure 18. The total interaction effect between the shape error and other errors was greater than that between the slope error and other errors for the three accuracy levels. Compared with grade 5 accuracy, the total interaction effect between each error and other errors was significantly reduced for grade 6 accuracy. The total interaction effect between shape errors and other errors increased obviously when the gear precision grade was changed from 6 to 7.  
Through the comparative analysis above, it can be seen that the four typical manufacturing errors had an impact on the dynamic performance of the gear system. Gear precision significantly changes the degree of influence and interaction of each error. In the actual machining of the batch gears, the measurement items of the machined tooth surface can be selected according to the machining accuracy, so as to effectively control the machining quality and ensure the performance consistency of the batch gears. That is, under grade 5 accuracy, the tooth profile shape deviation is the primary control error. At 6 and 7 levels of accuracy, it is also necessary to pay attention to the deviation of the lead.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Accuracy Grade**} & \multicolumn{2}{c}{**Tooth Profile**} & \multicolumn{2}{c}{**Lead**} \\ \cline{2-5}  & **Shape Error, \(E_{f_{R}}\)** & **Slope Error, \(E_{Ha}\)** & **Shape Error, \(E_{f\beta}\)** & **Slope Error, \(E_{H\beta}\)** \\ \hline
5 & 0.2831 & 0.1714 & 0.3243 & 0.1951 \\
6 & 0.2095 & 0.1259 & 0.1783 & 0.1274 \\
7 & 0.2748 & 0.1044 & 0.2457 & 0.1241 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The comparison of the statistics of the gear meshing performance before and after modification optimization.  
Figure 18: The total interaction effect between each error and other errors.