# The Impact of Hollow Wear on the Stability of High-Speed Railway Vehicles
## 2 The Influence of Tread Hollow Wear on the Wheel-Rail Contact Relationship
### Dynamic Model  
To study the effect of hollow wear on the dynamics of high-speed vehicles, the dynamics model was developed using multibody system software SIMPACK 2018 (as shown in Figure 3). The model was established for a single vehicle, ignoring the asymmetry of the vehicle structure and loading, as well as the nonlinear factors between connecting components. Therefore, it may be simplified to a multi-degree-of-freedom system consisting of the vehicle body, frame, and wheelsets, with each component connected by linear springs and damping units. The relative motion of the vehicle system is shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Wheel Number** & **W1** & **W2** & **W3** & **W4** & **W5** \\ \hline Running mileage (\(\times 10^{4}\) km) & 0 & 4.5 & 9.5 & 18.5 & 26 \\ Wear width (mm) & 0 & 32 & 40 & 46 & 53 \\ Wear depth (mm) & 0 & 0.32 & 0.61 & 0.96 & 1.26 \\ Depth of hollow wear (mm) & 0 & 0.17 & 0.31 & 0.52 & 0.64 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Measurement results of wheel wear profiles in a re-profiling interval.  
Figure 3: The dynamics model.  
Figure 2: Field measurements for wheel profile.  
The vehicle model comprises a single carbody, a pair of bogie frames, four wheelsets, and associated gearboxes. Its primary suspension system features a steel spring, a vertical shock absorber, and a module for positioning the axle box arm. The secondary suspension is equipped with an air spring, an anti-roll torsion bar, a lateral shock absorber as well as both vertical and anti-hunting shock absorbers, in addition to a lateral stop. Key parameters of the vehicle are shown in Table 3.  
The calculation is based on full load conditions, utilizing a 60N rail and an LMC tread wheel. The gauge of the rail measures 1435 mm, with a cart of 1:40. Additionally, the inner spacing between the wheelset is 1353 mm. There is a certain deviation between the actual position and the ideal position of the rail on the railway line, which is called track irregularity and is the primary contributor to vibrations within vehicle systems. The spectrum of track irregularity measured along the Beijing Tianjin line in China (as shown in Figure 4) is used for the track excitation in SIMPACK software.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Component**} & \multicolumn{6}{c}{**Degree of Freedom**} \\ \cline{2-7}  & **Longitudinal** & **Lateral** & **Vertical** & **Pitch** & **Yaw** & **Roll** \\ \hline Carbody & \(x_{c}\) & \(y_{c}\) & \(z_{c}\) & \(\beta_{c}\) & \(\psi_{c}\) & \(\varphi_{c}\) \\ Bogie & \(x_{bi}\) & \(y_{bi}\) & \(z_{bi}\) & \(\beta_{bi}\) & \(\psi_{bi}\) & \(\varphi_{bi}\) \\ Wheelset & \(x_{wi}\) & \(y_{wi}\) & \(z_{wi}\) & \(\beta_{wi}\) & \(\psi_{wi}\) & \(\varphi_{wi}\) \\ Axle box & & & & \(\beta_{ai}\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: The relative motion of the vehicle system.  
Figure 4: The track irregularity spectrum.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter Type** & **Parameter** & **Numeric Value** & **Unit** \\ \hline Vehicle & Distance between front and rear bogies & 19 & m \\ parameters & Wheelbase & 2.7 & m \\  & Diameter of the wheel & 920 & mm \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} Quality \\ parameters \\ \end{tabular} } & Carbody mass & 42 & t \\  & Bogie mass & 2.7 & t \\  & Wheelest mass & 1.7 & t \\  & Axle box mass & 0.099 & t \\ \hline \multirow{4}{*}{
\begin{tabular}{c} Suspension parameters \\ \end{tabular} } & Vertical stiffness of primary spring & 1.2 & MN/m \\  & Longitudinal stiffness of primary spring & 13.7 & MN/m \\  & Lateral stiffness of primary spring & 4.99 & MN/m \\  & Vertical stiffness of secondary spring & 0.26 & MN/m \\  & Vertical damping of secondary spring & 20 & kN-s/m \\  & Lateral damping of secondary spring & 22 & kN-s/m \\  & Damping of anti-hunting absorber & 26 & kN-s/m \\ \hline \hline \end{tabular}
\end{table}
Table 3: The main parameters of a single vehicle.