## 4 Numerical Example  
The parameters of the SBG system are shown in Tables 1-3. The load is 2300 N-m, and the input speed is 8000 rpm. Figure 4 is the normal deviation of pinion surface before and after optimization, in which the blue line and the red line represent the surface of the SBG system before and after optimization based on the active precontrol strategy of vibration, respectively. The meshing performances of the SBG system are analysed based on TCA, and shown in Figure 5, in which the red line and the blue line represent the optimized SBG system and the preset SBG system, respectively. It is found that the design contact ratio of optimized gear transmission is bigger than that of the preset gear transmission. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Pinion** & **Gear** \\ \hline Number of teeth & 27 & & 74 \\ Module/mm & & 3.85 & \\ Pressure angle/(°) & & 20 & \\ Mean spiral angle/(°) & & 30 & \\ Hand of spiral & LH & RH \\ Shaft angle/(°) & & 87 & \\ Face width/mm & & 40 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Gear parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Preset** & **Optimized** \\ \hline Radial setting/mm & 115.269 & 113.106 \\ Initial cardiac angle setting/(°) & 43.535 & 43.111 \\ Roll ratio & 2.884 & 2.834 \\ Vertical offset/mm & \(-\)3.584 & \(-\)5.580 \\ Increment of machine centre to back/mm & \(-\)2.063 & \(-\)3.226 \\ Sliding base feed setting/mm & 0.070 & 0.446 \\ Machine root angle/(°) & 18.897 & 18.897 \\ Second-order coefficients & 0.059 & 0.039 \\ Third-order coefficients & 0.003 & 0.019 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Processing parameters of pinion. MS and LTE are the SBG vibration excitation sources. When the load is 2300 N-m, the LTE and MS of gears are analysed by LTCA and listed in Figures 6 and 7, respectively. By comparing analyses, it can be determined that the LTE of the preset SBG transmission is 11.080\({}^{\prime\prime}\) and that the LTE of the optimized SBG transmission is 3.278\({}^{\prime\prime}\), which is 70.42% lower than that of the preset SBG transmission. The MS of the preset SBG transmission is 4.3896 \(\times\) 10\({}^{8}\) N/m; the MS of the optimized SBG transmission is 4.912 \(\times\) 10\({}^{8}\) N/m, which is 10.63% bigger than that of the preset SBG transmission; and the MS amplitude of the optimized SBG transmission is also obviously lower than the preset SBG transmission. Based on the active precontrol strategy for shape and performance, the SBG meshing qualities can be significantly improved. Figure 4: The normal deviation of tooth surface. Figure 5: The TCA results. Figure 6: The LTE of gears based on LTCA. To further prove the strong points of the optimized gears in MS, the MSs of gears obtained through LTCA are compared with those obtained through FEA. To ensure the accuracy of the FEA results, first of all, the accuracy of the FEA model should be guaranteed. The accuracy of the FEA model can be divided into the following two parts: tooth surface accuracy and gear assembly accuracy. To ensure tooth surface accuracy, based on the tooth surface equation, the finite element mesh model of tooth surface is built by rotating and arraying the position coordinates of discrete points of the tooth surface, and the tooth surface obtained by this method can meet the requirements of high accuracy. The solid model of the gear and the pinion is constructed with the discrete points, and then the gear and the pinion are placed in the same meshing coordinate system based on the principle of coordinate transformation to complete the assembly of the SBG system. The seven-tooth FEA model of the SBG system, shown in Figure 8, is obtained, and the assembly accuracy meets the requirements. According to the FEA model, the FEA results of gears are carried out and shown in Figures 9 and 10.