# The Two-Parameter Bifurcation and Evolution of Hunting Motion for a Bogie System
## 2 Dynamic Model
### Wheel-Rail Interaction Force  
#### 2.1.1 Nonlinear, Non-Smooth Wheel-Rail Contact Relation  
Accurately describing the wheel-rail contact geometry relationship is crucial for calculating the wheel-rail force, especially in studying hunting motion with large lateral displacement. In this paper, RSGEO software version 4.9 [48] is used to calculate the wheel-rail contact geometry parameters, and the output is in the form of parameter tables. The summary of the calculation process was as follows:  
1. The CHN60 rail and LMA wheel profile (extensively adopted in Chinese railways) are obtained and imported into the software.
2. The parameters required for calculating the wheel-rail contact relationship are entered, as shown in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Description** & **Value** \\ \hline \(K_{2x}\) & Secondary longitudinal stiffness & 0.2 MN/m \\ \(K_{2y}\) & Secondary lateral stiffness & 0.2 MN/m \\ \(K_{2z}\) & Secondary vertical stiffness & 0.25 MN/m \\ \(C_{2x}\) & Secondary longitudinal damper & 0 N\(\cdot\)s-m\({}^{-1}\) \\ \(C_{2y}\) & Secondary lateral damper & \(3.4\times 10^{8}\) N\(\cdot\)s-m\({}^{-1}\) \\ \(C_{2z}\) & Secondary vertical damper & \(1.93\times 10^{5}\) N\(\cdot\)s-m\({}^{-1}\) \\ \(d_{w}\) & Half distance of the primary suspension & 1 m \\ \(d_{s}\) & Half distance of the secondary suspension & 1.2 m \\ \(l_{t}\) & Half of the axle distance & 1.2 m \\ \(a_{0}\) & Half of wheelset contact distance & 0.7465 m \\ \(r_{0}\) & Centered wheel rolling radius & 0.4575 m \\ \(k_{0}\) & Flange contact stiffness & 146 MN/m \\ G & Resultant shear modulus & 8.2677 \(\times\)\(10^{10}\) Pa \\ \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 1: A schematic diagram of the lateral dynamic model.  
3. The calculated wheel-rail contact geometry parameters are compiled into data tables about the lateral displacement of the wheelset, as shown in Table 3.  
Figure 2 illustrates the contact angle \(\delta_{(L,R)}\) and minor semi-axes \(a_{(L,R)}\) among the wheel-rail contact parameters. It can be observed that the contact angle \(\delta_{(L,R)}\) is close to the linear relation when the wheelset lateral displacement is small and semi-axes \(a_{(L,R)}\) have already exhibited significant nonlinear characteristics. In contrast, larger wheelset lateral displacement exhibits significant nonlinear and non-smooth characteristics, especially after being subjected to the flange force. It can be seen that the linear wheel-rail contact relationship can no longer accurately describe the wheel-rail contact relationship in the severe hunting motion of a vehicle system with worn-type wheels.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline [PERSON]’s ratio \(\
u\) & 0.3 & wheelset lateral displacement \(y_{w}\) (mm) & [\(-\)12, 12] \\ elastic modulus E (Pa) & \(2.1\times 10^{11}\) & the flange clearance \(\eta\) (mm) & 9.1 \\ normal force \(N\) (N) & 55,860 & \\ \hline \hline \end{tabular}
\end{table}
Table 2: The parameters used in RSGRO software calculations.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Unit** \\ \hline \(r_{(L,R)}\) & Rolling radius & m \\ \(\delta_{(L,R)}\) & Contact angle & rad \\ \(\phi_{iw}\) & Roll angle of the wheelset & rad \\ \(a_{(L,R)}\) & Major semi-axes of the contact spot & m \\ \(b_{(L,R)}\) & Minor semi-axes of the contact spot & m \\ \(C_{11(L,R)}\) & Longitudinal Kalker creepage coefficient & N \\ \(C_{22(L,R)}\) & Lateral Kalker creepage coefficient & N \\ \(C_{23(L,R)}\) & Lateral/spin Kalker creepage coefficient & N\(\cdot\)m \\ \(C_{33(L,R)}\) & Spin Kalker creepage coefficient & N\(\cdot\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Wheel–rail contact parameters.  
Figure 2: The data tables of (**a**) the contact angle \(\delta_{(L,R)}\); (**b**) minor semi-axes \(a_{(L,R)}\).