# Key Factors of the Initiation and Development of Polygonal Wear in the Wheels of a High-Speed Train
## 3 Simulation Model  
### Rigid-Flexible Coupling Dynamical Model of Vehicle-Track  
Figure 7 shows the rigid-flexible coupling dynamical model of vehicle/track [17; 18; 19; 20; 21; 22; 23]. The vehicle system model includes a car body, two bogies, and four wheelsets. The car body and frames of the two bogies are modelled as rigid bodies, and they each have five degree of freedoms. Wheelset structural flexibility is modelled using the finite element method and solved using the mode superposition method. The primary and secondary suspension systems of the vehicle are modelled with three-dimensional spring-damper elements. The track system is modelled as a monolithic track-bed system that includes rails, fasteners, slabs, and roadbed. The rails are modelled as Timoshenko beams with elastic discrete point support. Slab structural flexibility is considered using the finite element method and solved by using the mode superposition method [19]. The fastener is modelled using three-dimensional spring-damper elements. The roadbed support layer is modelled with equally distributed spring-damper elements. The parameters of the track and vehicle system are listed in Table 1.  
Figure 6: Decrease in order with a decrease in wheel diameter. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Vehicle System** & \multicolumn{2}{c}{**Track System**} \\ \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Car body mass (kg) & 44,039 & Young’s modulus of rail (N/m\({}^{2}\)) & \(2.059\times 10^{11}\) \\ \hline Bogie mass (kg) & 2439 & Shear modulus of rail (N/m\({}^{2}\)) & \(7.9\times 10^{10}\) \\ \hline Wheelset mass (kg) & 1881 & Density of rail (kg/m\({}^{3}\)) & 7860 \\ \hline Density of wheelset (kg/m\({}^{3}\)) & 7860 & Cross-sectional area (m\({}^{2}\)) & \(7.745\times 10^{-3}\) \\ \hline Primary suspension vertical stiffness (MN/m) & 9 & Bending moment about \(y\)-axis (m\({}^{4}\)) & \(3.217\times 10^{-5}\) \\ \hline Primary suspension vertical damping (kN-s/m) & 10.5 & Bending moment about \(z\)-axis (m\({}^{4}\)) & \(5.26\times 10^{-6}\) \\ \hline Primary suspension lateral stiffness (MN/m) & 5.45 & Polar moment of inertia (m\({}^{4}\)) & \(3.741\times 10^{-5}\) \\ \hline Primary suspension lateral damping (kN-s/m) & 5.0 & Lateral shear coefficient & 0.4570 \\ \hline Secondary suspension vertical stiffness (MN/m) & 0.24 & Vertical shear coefficient & 0.5329 \\ \hline Secondary suspension vertical damping (kN-s/m) & 25 & Density of slab (kg/m\({}^{3}\)) & 2800 \\ \hline Secondary suspension lateral stiffness (MN/m) & 0.13 & Young’s modulus of slab (N/m\({}^{2}\)) & \(3.6\times 10^{10}\) \\ \hline Secondary suspension lateral damping (kN-s/m) & 15 & Rail pad vertical stiffness (MN/m) & 40 \\ \hline Distance between two bogie frames (m) & 8.9 & Rail pad vertical damping (kN-s/m) & 30 \\ \hline Wheelbase (m) & 2.5 & Sleeper spacing (m) & 0.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values of the vehicle and track subsystem. Figure 7: Vehicle–track rigid–flexible coupling dynamical model. According to the solid finite element theory, the differential equation of vibration of wheelset is written as  
\[\mathbf{M}_{i}\big{[}\ddot{\mathbf{u}}_{i}(t)\big{]}+\mathbf{K}_{i}\{\mathbf{u} _{i}(t)\}=\{\mathbf{Q}_{i}(t)\}\ \ \ (i=1,2,3,4\cdots) \tag{2}\]  
where \(\ddot{\mathbf{u}}_{i}(t)\) is the acceleration vector of the nodes on the \(i\)th wheelset at time \(t\), \(\mathbf{u}_{i}(t)\) is the velocity vector of the nodes on the \(i\)th wheelset at time \(t\), \(\mathbf{M}_{i}\) is the mass matrix of the \(i\)th wheelset, \(\mathbf{K}_{i}\) is the stiffness matrix of the \(i\)th wheelset, and \(\mathbf{Q}_{i}(t)\) is the force vector of the nodes on the \(i\)th wheelset at time \(t\), and it contains wheel-rail forces and primary suspension forces.