# Key Factors of the Initiation and Development of Polygonal Wear in the Wheels of a High-Speed Train
## 3 Simulation Model
### Method for Online Searching Wheel-Rail Contact Points  
The bending deformation of the wheelset indicates that the wheel rim has hardly any deformation compared to the wheel tread when the wheelset generates bending deformation within the range of 800 Hz. This feature can be used to help solve the interaction between the geometrical position of the wheel-rail. \"The method for online searching wheel-rail contact points\" is used to determine the contact geometry relation between a flexible wheelset and a pair of rails [21]. The process is shown in Figure 9. The motion of the flexible wheelset includes a rigid motion and flexible deformation (Figure 9a,b). Then, a dummy wheelset is used to simulate the flexible wheelset (Figure 9c,d). The centre of the dummy wheelset's coordinates needs to be identified, and the contact geometry relation between the dummy wheelset and the two rails can then be solved by using the trace method. Therefore, solving the problem of the flexible wheelset in rolling contact with the two rails can be transformed effectively into solving the motion of the rigid wheelset. Figure 8: Bending modes of wheelset (0–800 Hz): (**a**) First bending mode at 87 Hz; (**b**) Second bending mode at 143 Hz; (**c**) Third bending mode at 284 Hz; (**d**) Fourth bending mode at 576 Hz. The most important part of solving the flexible wheelset motion is determining the centre of the dummy wheelset's coordinates. From Figure 9e, the displacement vector of the centre of the dummy wheelset is written as:  
\[\overrightarrow{O_{1}O_{DR1}}=\overrightarrow{O_{1}A_{DR1}}+\overrightarrow{A _{DR1}O_{DR1}} \tag{4}\]  
where \(\overrightarrow{O_{1}A_{DR1}}\) denotes the displacement vector of the nominal contact point in the natural system of coordinates after the wheelset generates rigid motion and flexible deformation. \(\overrightarrow{A_{DR1}O_{DR1}}\) denotes the displacement vector from the centre of the dummy wheelset to the nominal contact point. \[\overrightarrow{O_{1}A_{DR1}}=\left(\left[ \begin{array}{c}0\\ -l_{0}\\ -R_{0}\end{array} \right]^{\text{T}}\!\!\left[ \begin{array}{c}\mathbf{i}\\ \mathbf{j}\\ \mathbf{k}\end{array} \right]+\overrightarrow{\mathbf{u}_{A}}\right)\left[ \begin{array}{ccc}\cos\psi_{r}&\sin\psi_{r}&0\\ -\cos\phi_{r}\sin\psi_{r}&\cos\phi_{r}\cos\psi_{r}&\sin\phi_{r}\\ \sin\phi_{r}\sin\psi_{r}&-\sin\phi_{r}\cos\psi_{r}&\cos\phi_{r}\end{array} \right] \tag{5}\]  
where \(l_{0}\) is the horizontal distance between the centre of the dummy wheelset and the nominal rolling circle, \(R_{0}\) is the radius of the nominal rolling circle, \(\overrightarrow{\mathbf{u}_{A}}\) is the displacement vector of the contact point after the wheelset produces flexible deformation, and it can be solved using the method described in Section 3.1. \(\phi_{r}\) denotes the angle of roll after the wheelset generates rigid motion and \(\psi_{r}\) indicates the angle of yaw. The two parameters can be solved using a dynamic explicit integral algorithm. In addition, the transformation matrix between the natural system of coordinates and the wheelset system of coordinates is built using \(\phi_{r}\) and \(\psi_{r}\). \(\overrightarrow{A_{DR1}O_{DR1}}\) is written as  
\[\overrightarrow{A_{DR1}O_{DR1}}=\left[ \begin{array}{c}0\\ -l_{0}\\ -R_{0}\end{array} \right]^{\text{T}}\!\!\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\phi_{DR1}&\sin\phi_{DR1}\\ 0&-\sin\phi_{DR1}&\cos\phi_{DR1r}\end{array} \right]\left[ \begin{array}{c} \mathbf{i}\\ \mathbf{j}\\ \mathbf{k}\end{array} \right] \tag{6}\]  
Figure 9: Modelling flexible wheelset: (**a**) Static wheelset; (**b**) The wheelset after rigid motion; (**c**) The wheelset after flexible deformation; (**d**) dummy wheelsets; (**e**) the contact geometry relation between the dummy wheelset and the two rails. where \(\phi_{DR1}\) indicates the angle included between the axis of the wheelset and the horizontal line after the wheelset generates rigid motion and flexible deformation.