## 4 Comparison of Different Wheel-Rail Contact Models
### Comparison under Seismic Conditions  
#### 4.2.1. Wheel-Rail Contact Dynamics  
The wheel-rail dynamic responses on the left side of the first wheelset calculated based on the different models were compared under seismic conditions firstly. The calculation results based on Models 1-4 are denoted by \(R_{1}\), \(R_{2}\), \(R_{3}\), and \(R_{4}\), respectively. To clarify the effect of considering multipoint contact versus single-point contact, the wheel-rail dynamic responses calculated on the basis of Models 1 and 2 are compared, as shown in Figures 14 and 15. The relative calculation error between the two models can be expressed as \((R_{1}-R_{2})/R_{2}\times 100\%\). Figure 14 shows that the differences in \(F_{Y}\) and \(F_{Z}\) that were calculated based on Models 1 and 2 mainly occur during the period when the contact force surges, i.e., when the wheel-flange contact occurs. Figure 15 shows the wheel-rail contact states obtained at a time of 3.80 s based on the two models. The wheel profile and rail profile are represented by red and blue curves respectively in Figure 15. In this case, the two-point contact state can be simulated based  
Figure 14: Time histories of wheel-rail contact forces based on Models 1 and 2: (**a**) Vertical wheel-rail force; (**b**) Lateral wheel-rail force. Figure 15: Wheel-rail contact state at time \(t=3.80\) s simulated based on Models 1 and 2: (**a**) Simulation result based on Model 1; (**b**) Simulation result based on Model 2.  
on Model 2 while the contact area existing in the wheel tread region is ignored when Model 1 is used. The solutions to \(F_{Z}\) based on Models 1 and 2 are 221.78 kN and 258.16 kN, respectively, with a calculated relative error of \(-\)14.09%, and the solutions to \(F_{Y}\) based on Models 1 and 2 are 130.20 kN and 76.98 kN, respectively, with a calculated relative error of 69.13%. Considering single point contact only will adversely affect the simulation of the wheel-rail contact state. Hence, multipoint contact should be considered under seismic conditions. To clarify the effect of different algorithms for the wheel-rail normal compression amount, the wheel-rail dynamic responses calculated based on Models 2 and 3 are compared, as shown in Figure 16. Both wheel-tread contact region and flange-root contact region are considered as wheel-rail contact potential areas, which are recorded as region 1 and region 2, respectively. From Figure 16a,b, it can be found that the difference in \(\delta_{c}\) based on the two models is mainly reflected in region 2, where the maximum values of \(\delta_{c}\) obtained based on Models 2 and 3 are 0.186 mm and 0.083 mm, respectively, and the former is 2.24 times that of the latter. From Figure 16c,d, it is observed that the trend in \(F_{N}\) is consistent with \(\delta_{c}\) since the same calculation formula for \(K_{nr}\) is adopted. The maximum values of \(F_{N}\) calculated based on Models 2 and 3, are 297.72 kN and 87.02 kN, respectively, in region 2, and the former is 3.42 times that of the latter. In addition, the wheel jumping amount, defined as the maximum vertical distance between the wheel and rail without wheel-rail contact occurring, was also introduced as an indicator of wheel rail dynamic response here. The responses of the wheel jump amount are given, as depicted in Figure 16e. Wheel jump is consistent with the occurrence of the contact in region 2 based on Model 2, which means that an unreasonable surge in the wheel-rail contact force will lead to an unreasonable wheel jump. Hence, the algorithm for the wheel-rail normal compression amount based on normal penetration should be introduced to avoid any unreasonable surge in the wheel-rail normal contact force.