## 4 Geometric Parameter Optimization Design for the Fixed Frog Area
### Optimization Design Method for the Nose Rail Height  
The plan and side schematic diagram of the nose rail in the fixed frog area is shown in Figure 13. The nose rail is located in the center of the two wing rails, symmetrically distributed relative to the gauge line, and continuously widened and heightened with the longitudinal direction of the frog from the theoretical point of frog, so as to meet the needs of the smooth crossing of wheels. Section A is the theoretical point of frog, the top width of the nose rail at Section B is 20 mm, and the top width of the nose rail at section H is 50 mm. One section is set every 5 mm from section B to section H, and the profile of the nose rail can be interpolated from the above 8 key sections. Set the height of a total of 6 sections from section B to section G to the variable value \(h_{\rm{F}},\ k=1,\ 2\ldots\ 6\), and set the height values of section A and section H to the fixed value h, respectively, \(h_{\rm{A}}\) and \(h_{\rm{H}}\). A series of nose rail profiles can be generated by continuously adjusting the nose rail height values of the 6 sections. Figure 11: On the stock rail side: (**a**) Wheel/rail lateral force; (**b**) Wheel/rail vertical force. In the same way, the wheel/rail force at the stock rail and the frog area is selected as the evaluation index of nose rail height. Therefore, the objective function of the optimization model is determined by Equation (12), and the total objective function is determined by Equation (13). To reduce the wheel/rail impact force generated when the wheel load transits from the wing rail to the nose rail, the height of the top surface of the nose rail should be gradually increased until it reaches the standard rail surface height. In order to make the optimized top surface of the nose rail smooth enough, the height value of each key section should be kept decreasing along the longitudinal direction of the frog, with the following constraints, as shown in Equation (20). \[h_{\rm A}>h_{\rm k}>h_{\rm k+1}>h_{\rm H} \tag{20}\]  
Similarly, the minimum height value \(h_{\rm min}\) shall be set according to the bearing capacity of the nose rail, and the height value at section B shall meet the following constraints:  
\[h_{\rm 1}>h_{\rm min} \tag{21}\]  
Figure 12: On the frog side: (**a**) Wheel/rail lateral force; (**b**) Wheel/rail vertical force. According to the technical manual of railway public works design, the nose rail height value of each fixed section is set as: the reduction value at Section A is \(h_{\rm A}=6\) mm, the minimum height value at Section B is \(h_{\rm min}=2\) mm, the height value at section H is \(h_{\rm H}=0\)[23]. At the same time, the population dimension in particle swarm optimization algorithm is set to \(D=6\), and other parameters are the same as the wing rail lifting optimization design example. The calculation process is similar to the wing rail optimization method and will not be repeated here. Therefore, the established optimization model of the nose rail height values is as follows. The objective function:  
\[\min:f(h_{1},\ h_{2},\ \ldots,\ h_{6})=f(h_{k}) \tag{22}\]  
The constraint condition:  
\[\left\{ \begin{array}{c}h_{\rm A}>h_{k}>h_{k+1}>h_{\rm H}\\ h_{1}>h_{\rm min}\\ \max\Big{(}\Big{|}\frac{F_{\rm rf}}{F_{\rm zul}}\Big{|},\Big{|}\frac{F_{\rm rf }}{F_{\rm zul}}\Big{|}\Big{)}<1\\ \max\Big{(}\Big{|}\frac{\mathbb{P}-F_{\rm rf}}{\mathbb{P}}\Big{|},\Big{|}\frac {\mathbb{P}-F_{\rm rf}}{\mathbb{P}}\Big{|}\Big{)}<0.6\end{array} \right. \tag{23}\]  
Figure 14 shows the evolution curve of the objective function value of the nose rail height optimization model solved by particle swarm optimization algorithm. It can be seen that when the number of iterations is about 85, the objective function value has tended to be stable. The optimization results of the wing rail heighting value are shown in Table 4.