## 2 Methodology
### Rigid-Flexible Coupled Dynamics Model of Vehicle-Turnout System  
To accurately explore the effect of SRRV deviation on the dynamic response of vehicle passing through turnout, a rigid-flexible coupled dynamics model of vehicle-turnout system was firstly established. Based on [PERSON] principle and referencing the CRH380A model, a multi-rigid-body dynamic model of high-speed vehicle was established. The model incorporates a total of 42 degrees of freedom, including 6 degrees of freedom each for 1 body, 2 bogies, and 4 wheelsets in the longitudinal, lateral, heave, roll, pitch, and yaw directions. To simulate the damping effect of the vehicle suspension system, the model includes forces from primary suspension springs, secondary air springs, secondary lateral dampers, and secondary vertical dampers, as illustrated in Figure 2(a). In this study, the Craig-Bampton modal synthesis method from structural dynamics was utilized to describe the elastic deformation of turnout rails. This approach was integrated with the hybrid coordinate method, commonly used in multi-body system modeling, to effectively capture the coupling between rigid-body motion and elastic deformation, thereby reflecting the dynamic characteristics of the system. \[M\ddot{u}+Ku=F \tag{1}\]where \(M\) and \(K\) represent the mass matrix and stiffness matrix in the FE model; \(F\) is the load matrix; and \(u\) is the displacement matrix of DOF, divided into boundary DOF \(u_{B}\) and internal structural DOF \(u_{I}\):  
\[u=\left[ \begin{array}{c}u_{B}\\ u_{I}\end{array} \right]=\left[\phi_{C}\,\phi_{N}\right]\left[ \begin{array}{c}u _{B}\\ q_{m}\end{array} \right] \tag{2}\]  
where \(\phi_{C}\) and \(\phi_{N}\) represent a constraint mode set and a fixed boundary characteristic mode set; \(q_{m}\) is the internal DOF mode coordinate. Through the calculations above, the C-B mode set \(\phi_{CB}=\left[\phi_{C}\,\phi_{N}\right]\) was obtained. This was orthogonally transformed. Let  
\[ \begin{array}{c}\overline{M}_{CB}=\phi_{CB}^{T}M\phi_{CB}\\ \overline{K}_{CB}=\phi_{CB}^{T}K\phi_{CB}\end{array}  \tag{3}\]  
where \(\overline{M}_{CB}\) is a mass matrix established by regularization; \(\overline{K}_{CB}\) is a stiffness matrix established by regularization. Then, an eigenvalue analysis was carried out on the equation below:  
\[\left\{\overline{K}_{CB}-\omega^{2}\,\overline{M}_{CB}\right\}\varphi=0 \tag{4}\]  
Eigenvectors were worked out, and the mass matrix was used for regularization. After this, the eigenvectors were turned into a transformation matrix \(\overline{T}\). The following is the orthogonalized C-B mode set:  
\[\overline{\phi_{CB}}=\phi_{CB}\overline{T} \tag{5}\]  
The finite element model for turnout rails uses the Beam188 element, incorporating the actual structural components and constraints of turnouts. It takes into account the complex cross-sectional features of turnout rails within the switch and crossing areas, as shown in Figure 3b. A master node is assigned to each half-span rail to accurately capture higher-order rail deformations and to meet the modeling requirements for the forces on stock rail fasteners as well as for the close and top adhesion between switch rails (or point rails) and stock rails (or wing rails) [40; 41]. The lateral restraint stiffness of the rail used is  
Figure 3: (**a**) Vehicle dynamic model; (**b**) the FEM model of turnout rails. 20 kN/mm, and the vertical restraint stiffness is 10 kN/mm. The vertical damping of the rail used is 50 kN\(\cdot\)s/m [42]. The wheel-rail contact model was used to connect the high-speed turnout model and the vehicle model, including the geometric calculation of wheel-rail contact and the solution of wheel-rail rolling contact force.