## 4 Geometric Parameter Optimization Design for the Fixed Frog Area
### Optimization Design Method for the Wing Rail Lifting Values  
Figure 7 shows the plan view and side view of the wing rail, which deviates to both sides of gauge line with the longitudinal extension of frog. Section A is the throat of the frog, and section K is the section with the top width of the nose rail of 50 mm. Nine key sections are taken between section A and section K, then the wing rail can be interpolated from a total of 11 sections above. Section A is the starting point of deviation and lifting, so the lifting value of the wing rail at section A is set to zero (\(h_{\text{A}}=0\)). Based on wheel/rail contact characteristics of the fixed frog area mentioned above, the wheel load transfer has been completed at section K, so the lifting value of the wing rail at section K is also set to zero (\(h_{\text{k}}=0\)). The lifting value of the nine sections from section B to section J is set as a series of variable values \(h_{\text{k}}\) (k = 1\(\cdot\)9), a series of complete wing rail profiles can be obtained by optimizing the wing rail lifting values of the nine sections. As the object of optimization are wheel/rail forces at the stock rail and the frog area, the objective function of the optimization model can be determined by Equation (12):  
\[\left\{ \begin{array}{l}f_{1}(h_{k})=\max(|F_{\text{yri}}|)\\ f_{2}(h_{k})=\max(|F_{\text{zri}}|)\\ f_{3}(h_{k})=\max(|F_{\text{yri}}|)\\ f_{4}(h_{k})=\max(|F_{\text{zri}}|)\\ \end{array} \right.\quad i=1,\;2,\;3,\;4,\;k=1,\;2\ldots 9 \tag{12}\]  
where, \(F_{\text{yri}}\) and \(F_{\text{yri}}\) are wheel/rail lateral forces at the stock rail and the frog area, and \(F_{\text{zri}}\) and \(F_{\text{zri}}\) are wheel/rail vertical forces at the stock rail and the frog area. According to the order of magnitude and importance, the above four objective functions are weighted in turn, and the total optimization objective function is shown in Equation (13):  
\[f(h_{k})=\omega_{1}f_{1}(h_{k})+\omega_{2}f_{2}(h_{k})+\omega_{3}f_{3}(h_{k})+ \omega_{4}f_{4}(h_{k}) \tag{13}\]  
where, \(\omega_{1}\sim\omega_{4}\) are the weight coefficient corresponding to each objective function. In order to ensure the smooth transfer of wheelset load from the wing rail to nose rail, the height of the wing rail top surface should be gradually increased until the max-num value is reached. Therefore, the constraint conditions are set as Equation (14). \[0<h_{k}<h_{k+1}<h_{\max} \tag{14}\]  
Wheel/rail contact points jump near the section with the width of the nose rail top surface of 20 mm. When wheelset rolls from the wing rail to the nose rail from the throat of the frog, if the wing rail height value at section E is set unreasonably, the nose rail will bear severe wheel/rail impact load at the weak section with narrow top width, so the minimum height value \(h_{\min}\) shall be set, as shown in Equation (15). \[h_{4}>h_{\min} \tag{15}\]  
The optimized wing rail shall ensure the driving safety of the vehicle passing through the fixed frog area. Therefore, the derailment coefficient and wheel load reduction rate at the stock rail side and the frog side shall meet the following constraints, as shown in Equations (16) and (17):  
\[\max\left(\left|\frac{F_{\text{yri}}}{F_{\text{zri}}}\right|,\left|\frac{F_{ \text{yri}}}{F_{\text{zri}}}\right|\right)<1 \tag{16}\]  
\[\max\left(\left|\frac{\overline{\text{P}}-F_{\text{zri}}}{\overline{\text{P} }}\right|,\left|\frac{\overline{\text{P}}-F_{\text{zri}}}{\overline{\text{P} }}\right|\right)<0.6 \tag{17}\]  
where, \(\overline{\text{P}}\) is the average wheel load of the left and right wheels of the wheelset. Therefore, the established optimization model of wing rail lifting values is as shown in Equations (18) and (19). Figure 7: Plan and side view of the wing rail. The objective function:  
\[\min:f(h_{1},\ h_{2},\ \ldots,\ h_{9})=f(h_{k}) \tag{18}\]  
The constraint condition:  
\[\left\{