# An Innovative and Efficient Method for Reverse Design of Wheel-Rail Profiles
## 5 Application Example
### Dynamic Performance Analysis  
In this paper, a dynamic vehicle-track simulation model is established based on the Simpack multi-body dynamic simulation software, where the vehicle parameters of high-speed multiple unit (MU) CRH2 are adopted for the vehicle model. As shown in Reference [23], according to their structural characteristics, high-speed bogies without considering side bearing, swing bolsters and swing platforms are used in order to establish a four-axle locomotive dynamic model. During the establishment of the model, the main structural components of the vehicle are simplified as rigid bodies, including the wheelset, bogie and vehicle body etc. The secondary suspension between the vehicle body and two bogies and the primary suspension between the two bogies and four wheelsets are simulated with springs and damping elements. Each suspension point has three degrees of freedom (DOF, longitudinal, lateral and vertical degree of freedom). The main motion mode of the wheelset is rolling, so except for the dive motion of the four wheelset rigid bodies, 5 DOFs with regard to lateral displacement, side rolling, up-and-down, dive and yaw are considered. Thus, there are 7 rigid bodies and 31 DOFs to consider with regard to the whole vehicle model. The dynamic performance of CHN60 rail profiles, optimized wheel profiles and LMa wheel profiles are compared based on three points i.e., the nonlinear critical vehicle speed, vehicle running stability and curve negotiation capability. (1) Critical speed  
Critical speed is the basic parameter for measuring the running stability of high-speed trains. In this paper, the nonlinear critical speed of high-speed EMUs is calculated based on the time domain response method. First, a section of irregularity is exerted at random on a track in order to excite the vibration of the vehicle system and then the speed of the vehicle on this track is reduced at a certain deceleration and the lateral wheel-rail displacement is observed for convergence to judge its critical speed. When the lateral wheelset displacement is less than 0.1 mm, it can be considered convergent and the corresponding speed at this time can be considered the critical speed of the train. Figure 22 shows the simulated results. The critical speed is 543 km/h when the CHN60 rail matches LMa and 602 km/h when the CHN60 rail matches optimized wheels. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Wheel Type**} & \multicolumn{3}{c}{**Max. Contact Stress (MPa)**} \\ \cline{2-4}  & **Lateral Displacement 0 mm** & **Lateral Displacement 3 mm** & **Lateral Displacement 6 mm** \\ \hline LMa & 1183.6 & 850.92 & 1123.6 \\ Optimized wheel & 1170.9 & 1008.4 & 939.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Maximum wheel-rail contact stress. Figure 22: Critical speed. (2) Vehicle stability  
The vehicle running stability is evaluated according to the Sperling stability indexes. In this paper, the vehicle speeds are set at 200, 250, 300 and 350 km/h. The U.S. Grade 6 railways track irregularity is adopted. Figure 23 shows the comparative analysis of the lateral stability results for two types of wheels based on the Sperling stability indexes. It can be seen that the lateral stability of the optimized wheel at the 4 speed levels is better than that of the LMa profile and the running stability is improved by about 7%. (3) Curve negotiation capability  
The speed of the vehicle passing the curve is set as 250 km/h. The parameters of the curve track are: circular curve length--500 m, radius length--3500 m, front/rear transition curve length--420 m, superelevation setting of outer rail--120 mm. The curve outlet and inlet are provided with 100 m straight lines. Figure 24 shows the simulated results. For the maximum dynamic response values, see Table 2.  
As shown in Figure 24 and Table 2, the lateral displacement of the first wheelset with optimized wheels on the curve line is greater than that of the LMa profile.