## 2 Wind-Train-Track Coupling Dynamic Model
### Modelling of Train  
A train usually comprises a series of carriages (e.g., 8 or 16 carriages are widely used in China). Typical carriages consist of a carriage body, suspension systems, bogies, pivots, wheelsets, and other components [30]. In the present study, the carriage was modelled as seven rigid bodies, i.e., one carriage body, two bogies, and four wheelsets (see Figure 2). The whole carriage model had 31 independent DOFs. The response vector of a carriage is expressed as follows. The values of the main parameters in Equations (1)-(23) are given in Table 1[8,10]. The matrix equation of the train motion subsystem can be expressed as:  
\[\mathbf{M_{v}}\ddot{\mathbf{X_{v}}}+\mathbf{C_{v}}\dot{\mathbf{X_{v}}}+\mathbf{ K_{v}}\mathbf{X_{v}}=\mathbf{F_{v}} \tag{1}\]  
where \(\mathbf{M_{v}}\), \(\mathbf{C_{v}}\), and \(\mathbf{K_{v}}\) are the mass, damping, and stiffness matrices, respectively; \(\ddot{\mathbf{X_{v}}}\), \(\dot{\mathbf{X_{v}}}\), and \(\mathbf{X_{v}}\) are the acceleration, velocity, and displacement vector, respectively; \(\mathbf{F_{v}}\) is the vector of forces on the train and can be expressed as:  
\[\mathbf{F_{v}}=\left[ \begin{array}{cc}\mathbf{F_{vc}}&\mathbf{F_{vb1}}& \mathbf{F_{vb2}}&\mathbf{F_{vw1}}&\mathbf{F_{vw2}}&\mathbf{F_{vw3}}&\mathbf{ F_{vw4}}\end{array} \right]^{T} \tag{2}\]  
where \(\mathbf{F_{vc}}\) is the vector of forces on the car body; \(\mathbf{F_{vb1}}\) and \(\mathbf{F_{vb2}}\) are the vector of forces on the first and second bogies, respectively; \(\mathbf{F_{vw1}}\), \(\mathbf{F_{vw2}}\), \(\mathbf{F_{vw3}}\), and \(\mathbf{F_{vw4}}\) are the vector of forces on the four wheelsets. \[\mathbf{F_{vc}}=\left[ \begin{array}{cc}F_{wy}&m_{c}g-F_{wz}&M_{wx}&M_{wy}&- M_{wz}\end{array} \right]^{T} \tag{3}\]  
\[\mathbf{F_{vb1}}=\mathbf{F_{vb2}}=\left[ \begin{array}{cccccc}0&m_{b}g&0&0&0 \end{array} \right]^{T} \tag{4}\]  
Figure 2: High-speed vehicle model: (**a**) side elevation, (**b**) plan view, and (**c**) end view. \[\mathbf{F_{wwj}}=\left[ \begin{array}{cccc}F_{yj}^{irr}&m_{w}g+F_{zj}^{irr}&M_{xj}^ {irr}&M_{zj}^{irr}\end{array} \right]^{T}(j=1,2,3,4) \tag{5}\]  
where \(F_{wy}\), \(F_{wz}\), \(M_{uxr}\), \(M_{uyr}\), and \(M_{uzz}\) are the excitations of aerodynamic loads, namely the side forces, lift forces, rolling, pitching, and yawing moments on the car body, respectively (details are shown in Section 3.2); \(m_{c}g\), \(m_{b}g\), and \(m_{w}g\) are the gravity on the car body, bogies, and wheelsets, respectively; \(F_{y}^{irr}\), \(F_{z}^{irr}\), \(M_{x}^{irr}\), and \(M_{z}^{irr}\) are the forces and moments induced by the track irregularity excitation (Section 2.2); \(j\) represents the serial number of the wheelset. The displacement vector \(\mathbf{X_{v}}\) of the train is expressed as:  
\[\mathbf{X_{v}}=\left[ \begin{array}{cccc}\mathbf{X_{vc}}&\mathbf{X_{vb1}}& \mathbf{X_{vb2}}&\mathbf{X_{vw1}}&\mathbf{X_{vw2}}&\mathbf{X_{vw3}}&\mathbf{X_ {vw4}}\end{array} \right]^{T} \tag{6}\]  
where, \(\mathbf{X_{vb1}}\), \(\mathbf{X_{vb2}}\), \(\mathbf{X_{vw1}}\), \(\mathbf{X_{vw2}}\), \(\mathbf{X_{vw3}}\), and \(\mathbf{X_{vw4}}\) are the displacement vector of the car body, the first and second bogies, and the four wheelsets, respectively. \[\mathbf{X_{vc}}=\left[ \begin{array}{cccc}Y_{c}&Z_{c}&\phi_{c}&\beta_{c}&\psi _{c}\end{array} \right]^{T} \tag{7}\]  
\[\mathbf{X_{vb1}}=\mathbf{X_{vb2}}=\left[ \begin{array}{cccc}Y_{b}&Z_{b}&\phi _{b}&\beta_{b}&\psi_{b}\end{array} \right]^{T} \tag{8}\]  
\[\mathbf{X_{vw1}}=\mathbf{X_{vw2}}=\mathbf{X_{vw3}}=\mathbf{X_{vw4}}=\left[ \begin{array}{cccc}Y_{w}&Z_{w}&\phi_{w}&\psi_{w}\end{array}