## 4 Simulation and Discussion
### Example 1: Efficiency Optimization Based on Control Parameters  
According to the design parameters and TCA analysis, the range of the main three control parameters could be determined: the values of the design contact ratio are between 1.0 and 2.0; the values of the radius of the semi-major axis of the contact ellipse are between 2 mm and 10 mm; the values of amplitudes of the meshing transition point are between 4\({}^{\prime\prime}\) and 10\({}^{\prime\prime}\) (second of arc). The optimization parameters are defined as shown in Section 3.  
After the optimization calculation based on Figure 3, the best solutions and detailed machining parameters are obtained (shown in Table 2). The transmission efficiency could achieve 97.10% and the optimal parameters are as follows: the design contact ratio value is 1.19, the radius of the semi-major axis of the contact ellipse is 2.01 mm, the amplitude of meshing transition point is 8.26\({}^{\prime\prime}\).  
According to the simulation, the changing values of key performance indexes during the gear meshing process are obtained, including the entertainment speed, the relative sliding velocity, the load distribution, the friction coefficient, the transmission efficiency, and fitness values, as shown in Figure 4. The entertainment speed would increase during the meshing process while the relative sliding velocity will decrease first and then increase (as shown in Figure 4a,b). Both entertainment speed and relative sliding velocity are significantly affected by control parameters. The curve of transmission efficiency (Figure 4d) showed that transient efficiency in a different meshing position would be totally different, and the changing range could be up to a maximum of nearly 7%, which is affected by load,the entertainment speed, the relative sliding velocity, lubrication conditions, and many other parameters. Figure 4e showed that the PSOGSA algorithm has a good convergence speed, in which the third step made transmission efficiency more than 97%. It should be pointed out that a minus sign was put on the transmission efficiency value in order to calculate the minimum more conveniently.  
Figure 4: \(Cont\).  
Figure 5: Key performance indexes based on machining parameter optimization. (**a**) Entertainment speed; (**b**) Relative sliding velocity; (**c**) Friction coefficient; (**d**) Transmission efficiency; (**e**) Fitness values of PSOGSA.  
According to the comparisons between Figures 4 and 5, the entertainment speed increased, and the relative sliding velocity decreased after optimization (as shown in Figure 5a,b), which would help to increase transmission efficiency. The changing of entertainment speed and relative sliding velocity directly leads to the decrease in the friction coefficient (as shown in Figure 5c). Therefore, the transmission efficiency increased from 97.10% to 98.78%, an increase of 1.68 percent. The fitness value curve shows that the PSOGSA algorithm has a good convergence speed with the sixth step close to the convergence value, as shown in Figure 5e.  
According to Figures 4 and 5, the hybrid PSOGSA algorithm shows good convergence, in which the third and the sixth step nearly achieve the best solutions, respectively. Actually, the good performance of the PSOGSA algorithm has already been proven in former studies [44; 45] in other fields. The good convergence and stability of the hybrid PSOGSA algorithm has been confirmed again in mechanical engineering problems. Moreover, the friction coefficient and transmission efficiency also obey distribution rules and value ranges proposed by former studies, as in references [12; 13; 14; 15; 16] etc.