## 3 Dynamics Modeling of Vehicle/Fixed Frog Interaction
### Rail Vibration System Model  
According to the different problems studied, the rail dynamics models selected are different, mainly including the massless rail, the inertial rail, and the flexible track. Among them, the inertial rail model can be for simulation of complex wheel/rail contact: railway track dynamic performance in the switches and turnouts, contacts between wheel-back and rail, vehicle derailment, etc., so the inertial rail model is selected for dynamic calculation in the paper. Assuming wheel/rail contact of a certain section on one side, the rail is simplified as a mass block with 3 DOFs: lateral shift, vertical shift and torsion, and the mass block is connected with the foundation through equivalent spring and damping. Selecting right rail as an illustration, the dynamic modeling of the rail vibration system is carried out as shown in Figure 4.  
The coordinate of wheel/rail contact point is \((y_{rr},z_{rr})\) in the rail coordinate system. P is the lateral component, \(Q\) is the vertical component, and \(M_{x}\) is the roll moment, respectively.  
Figure 4: Equivalent model of track vibration model.  
Figure 3: Vehicle dynamics model based on Simulink.  
\(K_{ry}\) and \(C_{ry}\) are the lateral support stiffness and damping of sleeper, fastener, and other basic components to the rail. \(K_{rz}\) is the vertical foundation stiffness of sleeper, fastener, and other basic components to the rail, and \(C_{rz}\) is the damping. Therefore, the fulcrum reaction of the rail is shown in Equation (1).  
\[\begin{cases}F_{ql}=-K_{rz}\big{(}z_{r}-\phi_{r}l_{ry}\big{)}-C_{rz}(\dot{z}_{r} -\dot{\phi}_{r}l_{ry}\big{)}\\ F_{qr}=-K_{rz}\big{(}z_{r}+\phi_{r}l_{ry}\big{)}-C_{rz}(\dot{z}_{r}+\dot{\phi}_ {r}l_{ry}\big{)}\\ F_{pl}=-K_{ry}(y_{r}-\phi_{r}l_{rz})-C_{ry}(\dot{z}_{r}-\dot{\phi}_{r}l_{rz})\\ F_{pr}=-K_{ry}(y_{r}-\phi_{r}l_{rz})-C_{ry}(\dot{z}_{r}-\dot{\phi}_{r}l_{rz}) \end{cases} \tag{1}\]  
The torsional torque of the rail is shown in Equation (2):  
\[M_{xx}=-K_{qr}\dot{\phi}_{r}-C_{\phi r}\dot{\phi}_{r} \tag{2}\]  
where, \(K_{\phi r}\) is the rail torsional stiffness; \(C_{\phi r}\) is the rail torsional damping.  
Further, the vibration expression for the rail system can be obtained as shown in Equation (3):  
\[\begin{cases}m_{r}\ddot{y}_{r}&=P+F_{pl}+F_{pr}\\ m_{r}\ddot{z}_{r}&=Q+F_{ql}+F_{qr}+m_{r}g\\ I_{rx}\dot{\phi}_{r}&=M_{x}+Py_{rr}-Qz_{rr}+M_{xt}-F_{pl}l_{rz}-F_{pr}l_{rz}+F_{ qr}l_{ry}-F_{ql}l_{ry}\end{cases} \tag{3}\]  
where, \(m_{r}\) is the equivalent mass of rail, \(y_{r}\), \(z_{r}\), and \(\phi_{r}\) are lateral, vertical shift, and torsional angle of the rail.  
The inner back of the wheelset impacts the side of the guard rail, resulting in additional wheel/rail impact force. Due to the short action time and large amplitude of this wheel/rail impact force, it can be considered as the lateral impact force between wheel-back and guard rail [16].  
To simulate the instantaneous impact load between wheel-back and guard rail, a contact condition is equivalent to the spring with great stiffness. Therefore, the wheel/rail elastic compression equation is:  
\[\varepsilon=l_{wc}-l_{tc}+y_{rail}+0.009+d_{a}-\Delta_{a}(x) \tag{4}\]  
where, \(k_{a}\) is the equivalent spring stiffness, \(l_{wc}\) is the lateral distance between wheel/rail contact point and the origin in wheelset translation coordinate system, \(l_{tc}\) is the lateral distance between wheel/rail contact point and the origin in the rail coordinate system, \(y_{rail}\) is the dynamic traverse of stock rail, \(d_{a}\) is the wheel flange thickness, \(\Delta_{a}(x)\) is the wheel flange groove width at current position of the wheelset. Further, the wheel/rail impact force is:  
\[F_{a}=\left\{\begin{array}{ll}k_{a}\varepsilon,&\varepsilon>0\ \ (\text{Contact state})\\ 0,&\varepsilon\leq 0\ \ (\text{Non contact state})\end{array}\right. \tag{5}\]