# Self-Adaptive Differential Evolution with Gauss Distribution for Optimal Mechanism Design
## 5 Numerical Simulation
### Modified Differential Evolution for Real-World Problem
#### 5.4.4 Optimization Problem of Gear Reducer
Minimize:  
\[f(x)=0.7854x_{1}x_{2}^{2}(3.3333x_{3}^{2}+14.9334x_{3}-43.0934)- 1.508x_{1}(x_{6}^{2}+x_{7}^{2})\\ +7.4777(x_{6}^{3}+x_{7}^{3})+0.7854(x_{4}x_{6}^{2}+x_{5}x_{7}^{2}) \tag{38}\]  
Subject to  
\[g_{1}\ =\ 27x_{1}^{-1}x_{2}^{-2}x_{3}^{-1}-1\leq 0; \tag{39}\]  
\[g_{2}\ =\ 397.5x_{1}^{-1}x_{2}^{-2}x_{3}^{-2}-1\leq 0; \tag{40}\]  
\[g_{3}\ =\ 1.93x_{4}^{3}x_{2}^{-1}x_{3}^{-1}x_{6}^{-4}-1\leq 0; \tag{41}\]  
\[g_{4}\ =\ \frac{1.93x_{5}^{3}}{x_{2}x_{3}x_{7}^{4}}-1\leq 0; \tag{42}\]  
\[g_{5}\ =\ \frac{1}{(110x_{6}^{3})\sqrt{(\frac{245x_{4}}{2x_{3}})^{2}+16.9 \times 10^{6}}}-1\leq 0; \tag{43}\]  
\[g_{6}\ =\ \frac{1}{(85x_{7}^{3})\sqrt{(\frac{745x_{5}}{2x_{3}})^{2}+157.5 \times 10^{6}}}-1\leq 0; \tag{44}\]  
\[g_{7}\ =\ \frac{x_{2}x_{3}}{40}-1\leq 0; \tag{45}\]  
\[g_{8}\ =\ \frac{5x_{2}}{x_{1}}-1\leq 0; \tag{46}\]  
\[f(x)=0.7854x_{1}x_{2}^{2}(3.3333x_{3}^{2}+14.9334x_{3}-43.0934)- 1.508x_{1}(x_{6}^{2}+x_{7}^{2})\\ +7.4777(x_{6}^{3}+x_{7}^{3})+0.7854(x_{4}x_{6}^{2}+x_{5}x_{7}^{2}) \tag{38}\]0  
\[f(x)=0.7854x_{1}x_{2}^{2}(3.3333x_{3}^{2}+14.9334x_{3}-43.0934)- 1.508x_{1}(x_{6}^{2}+x_{7}^{2})\\ +7.4777(x_{6}^{3}+x_{7}^{3})+0.7854(x_{4}x_{6}^{2}+x_{5}x_{7}^{2}) \tag{38}\]1  
\[f(x)=0.7854x_{1}x_{2}^{2}(3.3333x_{3}^{2}+14.9334x_{3}-43.0934)- 1.508x_{1}(x_{6}^{2}+x_{7}^{2})\\ +7.4777(x_{6}^{3}+x_{7}^{3})+0.7854(x_{4}x_{6}^{2}+x_{5}x_{7}^{2}) \tag{38}\]2  
Tables 9 and 10 show the comparison of the our optimal solution with previous research which consists of the artificial immune system genetic algorithm (AIS-GA) [27], real-coded steady-state genetic algorithm (APM\({}^{rc}\)) [28], and EOBLDE. From Table 9, the best solution of G-ISADE is better than those of the remain algorithms. In addition, the standard deviation in statistical results of G-ISADE is \"0\", that verified the ability of convergence  
\[f(x)=0.7854x_{1}x_{2}^{2}(3.3333x_{3}^{2}+14.9334x_{3}-43.0934)- 1.508x_{1}(x_{6}^{2}+x_{7}^{2})\\ +7.4777(x_{6}^{3}+x_{7}^{3})+0.7854(x_{4}x_{6}^{2}+x_{5}x_{7}^{2}) \tag{38}\]3 Single stage spur gear reducer. of the proposed method in solving minimal weight problem of single stage spur gear reducer box. Problem 2: Two-stage planetary gear reducer  
Planetary gear reducer plays an important role in transmission systems and are commonly applied in industry. It has many pros in comparison with normal transmission systems, such as weight reduction, compact dimensions, wide range of speed ratios, and high efficiency. This section considers a two-stage planetary gear reducer, as described in \[f(x)=0.7854x_{1}x_{2}^{2}(3.3333x_{3}^{2}+14.9334x_{3}-43.0934)- 1.508x_{1}(x_{6}^{2}+x_{7}^{2})\\ +7.4777(x_{6}^{3}+x_{7}^{3})+0.7854(x_{4}x_{6}^{2}+x_{5}x_{7}^{2}) \tag{38}\]4 This problem involves nine design variables, including the number of teeth on the primary sun gear (1) and the secondary sun gear (1\({}^{\prime}\)), which are \(x_{1}\) and \(x_{6}\), respectively; the gear module, \(x_{2}\), \(x_{7}\); the face width, \(x_{3}\), \(x_{8}\); the sun gear's modification coefficient, \(x_{4}\), \(x_{9}\); and the first-stage transmission ratio, \(x_{5}\). To minimize the weight of the planetary gear reducer, the total volume of all sun gears and planetary gears is set as the objective function and displayed in Equation (50) with 10 constraint functions, which include stress limitations, and the adjacent condition is expressed in Equations (51)-(60). The details of the problem can be found in [29]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & **Best** & **Mean** & **Worst** & **Standard Deviation** \\ \hline AIS-GA & 2996.3483 & 2996.3501 & 2996.3599 & \(7.45\times 10^{-3}\) \\ APM\({}^{rc}\) & 2996.3482 & 2997.4728 & 3051.4556 & \(7.87\) \\ EOBLDE & 2993.6132 & 2993.6165 & 2993.6296 & \(0.33\times 10^{-2}\) \\ G-ISADE & 2993.61 & 2993.61 & 2993.61 & \(0.00\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Statistical results for single stage spur gear reducer problem. Figure 27: Two-stage planetary gear reducer.