## 4 Geometric Parameter Optimization Design for the Fixed Frog Area
### Optimization Design Method for the Wing Rail Lifting Values
\begin{array}{c}0<h_{k}<h_{k+1}<h_{\max}\\ h_{4}>h_{\min}\\ \max\left(\left|\frac{F_{\mathrm{F}ii}}{F_{\mathrm{z}ii}}\right|,\left|\frac{F_ {\mathrm{F}ii}}{F_{\mathrm{z}ii}}\right|\right)<1\\ \max\Bigl{(}\left|\frac{\mathrm{\bar{P}}_{-\mathrm{F}ii}}{\mathrm{\bar{P}}} \right|,\left|\frac{\mathrm{\bar{P}}_{-\mathrm{F}ii}}{\mathrm{\bar{P}}} \right|\Bigr{)}<0.6\end{array} \right. \tag{19}\]  
Figure 8 is the flow chart of the wing lifting values optimization algorithm based on PSO. The optimization process is as follows:  
1. Set the basic parameters of particle swarm optimization algorithm and initialize the population, and randomly generate \(N\) groups of wing rail lifting values according to the constraints. 2. A complete set of frog profiles is generated according to each lifting value and the profile of each key section, and the vehicle dynamics are calculated. 3. If the dynamic calculation results conform to safety conditions, the individual optimal solution and group optimal solution shall be updated according to the quality of the objective function value, and then judge whether the current iteration times have reached the maximum iterations. If the maximum iterations have been met, the optimization results shall be output. 4. If the dynamic calculation results do not meet the safety requirements or the current iterations number does not reach the maximum iterations, obtain a new group \(N\) lifting values according to Formula (14) and (15), and repeat the above calculation steps until the optimal lifting values are obtained. To verify the effectiveness of the wing rail lifting optimization algorithm, the dynamic performance of a metro vehicle passing through No. 12 fixed frog in the main direction at the speed of 80 km/h is calculated, and the effects of the non-optimized wing rail and the optimized wing rail on the dynamic performance are compared. See Table 2 for the main calculation parameters. According to the technical manual of railway public works design [23], the maximum height value of wing rail \(h_{\max}\) is set to 5 mm, the minimum height value of key section E is set to \(h_{\min}=2\) mm. The basic parameters of the PSO algorithm are set as follows according to the recommended values in literature [24]: population dimension \(D=9\), population size \(N=10\) and maximum flight speed \(v_{\max}=1\), individual learning factor \(c_{1}=1.5\), social learning factor \(c_{2}=1.5\), maximum iterations \(T_{\max}=100\), maximum inertia weight \(\omega_{\max}=0.8\), and minimum inertia weight \(\omega_{\min}=0.4\). Figure 9 shows the evolution curve of the objective function value of the wing rail lifting optimization model solved by particle swarm optimization algorithm. It can be seen that when the number of iterations is about 40, the objective function value has tended to be stable. The optimization results for the wing rail lifting values are shown in Table 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Car body mass (kg) & 41,910 \\ Frame mass (kg) & 4060 \\ Wheelset mass (kg) & 1670 \\ Gauge (m) & 1.435 \\ Nominal rolling circle radius (m) & 0.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Principal parameters. Figure 8: Flow chart of the wing rail optimization algorithm based on the PSO. Figure 9: Evolution curve of the objective function value. Figure 10 shows the change curve of wheelset vertical displacement of the vehicle passing through the fixed frog before and after optimization. Before wing rail optimization, the maximum vertical displacement of wheelset is 0.78 mm. After the wing rail optimization, the maximum vertical displacement of the wheelset is 0.51 mm, which is reduced by 34.6%, indicating that the optimized wing rail could greatly decrease the vertical vibration amplitude of the wheelset when passing through the frog and improve the running stability of the vehicle. Figures 11 and 12 show the wheel/rail force curve of the vehicle passing through the fixed frog area in main directions.