## 2 Establishment of the Rigid Vehicle-Flexible Track Coupled Dynamics Model
### Establishment of the Flexible Track Model
#### 2.2.2 Substructure Analysis of FE Models  
The flexible track structures contain a large number of nodes and elements. Thus, many degrees of freedom (DOFs) are introduced to the MBD simulation, and the computation time will increase greatly. To shorten the calculation time, the substructure analysis of FE models is conducted based on [PERSON]'s reduction theory, which is a process of condensing a group of elements into a super element by matrix [34; 35]. The dynamic equilibrium equation of the dynamic model is:  
\[M\ddot{u}+D\dot{u}+Ku=F, \tag{1}\]  
where \(M\), \(K\) and \(D\) are the mass matrix, stiffness matrix and damping matrix, respectively, and \(u\), \(i\) and \(F\) are the displacement matrix, velocity matrix and the load matrix, respectively. Nodes of FE models, which are selected to represent all node information, are denoted as master nodes. The others are called slave nodes [36]. Assuming that there are \(m\) DOFs of master nodes and \(s\) DOFs of slave nodes, the matrix form of Equation (1) is:  
\[\left( \begin{array}{cc}M_{mm}&M_{ms}\\ M_{sm}&M_{ss}\end{array} \right)\left( \begin{array}{cc}\ddot{u}_{mm}\\ \ddot{u}_{ss}\end{array} \right)+\left( \begin{array}{cc}D_{mm}&D_{ms}\\ D_{sm}&D_{ss}\end{array} \right)\left( \begin{array}{cc}\dot{u}_{mm}\\ \dot{u}_{ss}\end{array} \right)+\left( \begin{array}{cc}K_{mm}&K_{ms}\\ K_{sm}&K_{ss}\end{array} \right)\left( \begin{array}{cc}u_{mm}\\ u_{ss}\end{array} \right)=\left( \begin{array}{cc}F_{mm}\\ 0\end{array} \right), \tag{2}\]  
where \(K_{mm}\), \(K_{ms}\), \(K_{sm}\) and \(K_{ss}\) are the stiffness matrices with sizes of \(m\times m\), \(m\times s\), \(s\times m\) and \(s\times s\) in turn, \(M\) and \(D\) are the corresponding mass matrix and damping matrix, respectively, and \(F_{mm}\) is the load vector acting on the master nodes. [PERSON]'s reduction theory is a static transformation based on \(F=K\times u\). Then, we can get:  
\[u_{ss}=-{K_{ss}}^{-1}K_{sm}u_{mm}, \tag{3}\]  
Figure 4: FE model of slab track structure. (**a**) Sketch of the FE models of the structural system. (**b**) Detailed parameters of the mesh elements. Equation (4) can be obtained by calculating the first and second derivative of Equation (3):  
\[\left\{ \begin{array}{l}\dot{u}_{ss}=-{K_{ss}}^{-1}{K_{sm}}\dot{u}_{mm}\\ \ddot{u}_{ss}=-{K_{ss}}^{-1}{K_{sm}}\ddot{u}_{mm}\end{array} \right., \tag{4}\]  
Putting Equation (4) into Equation (2) to get Equation (5) for \(u_{ss}\):  
\[u_{ss}=-{K_{ss}}^{-1}\times\Big{[}\Big{(}M_{sm}-M_{ss}{K_{ss}}^{-1}{K_{sm}} \Big{)}\ddot{u}_{mm}+\Big{(}{C_{sm}}-{C_{ss}}{K_{ss}}^{-1}{K_{sm}}\Big{)}\dot{u}_ {mm}+{K_{sm}}u_{mm}\Big{]}, \tag{5}\]  
To balance Equations (3) and (5), coefficients of \(\dot{u}_{ss}\) and \(\ddot{u}_{ss}\) should be zero. That is:  
\[\left\{ \begin{array}{l}M_{sm}=M_{ss}{K_{ss}}^{-1}{K_{sm}}\\ {C_{sm}}={C_{ss}}{K_{ss}}^{-1}{K_{sm}}\end{array} \right., \tag{6}\]  
Whether Equation (6) can be satisfied or not depends on the selection of master nodes of the FE models. Considering the significance of track structural shapes, the mounting locations of fasteners and the connection positions between slab and mortar layer as well as between mortar layer and base, simultaneously shortening the computational time, a total of 602, 600, 560 and 448 master nodes of the rails, slabs, mortar layer and concrete base are selected with the length of flexible track being 25.95 m.  
The modal analysis of FE models of track structures before and after substructure analysis was conducted. The 7-14 order modal frequencies of FE models and the corresponding reduced models are listed in Table 1. It clearly shows that the frequency differences between the unreduced FE models and reduced models are negligible. Therefore, the master nodes of the track structures selected in this paper are appropriate. The modal superposition method is adopted to account for the contributions of the track structural modes to the vibration behavior of the coupled system, as reported in many studies [37; 38].