## 2 The Mechanical Model of the Railway Vehicle-Track System
 \tag{18}\]  
The equations that describe the motion of the bogies are as follows:  
\[m_{b}\ddot{z}_{b1}=\sum_{j=1}^{2}F_{zbj}-F_{zx1}, \tag{20}\]  
\[m_{b}\ddot{z}_{b2}=\sum_{j=3}^{4}F_{zbj}-F_{zx2}, \tag{21}\]  
\[J_{b}\ddot{\theta}_{b1}=a_{b}\sum_{j=1}^{2}(-1)^{j+1}F_{zbj}-h_{b}F_{x1}, \tag{22}\]  
\[J_{b}\ddot{\theta}_{b2}=a_{b}\sum_{j=3}^{4}(-1)^{j+1}F_{zbj}-h_{b}F_{x2}, \tag{23}\]  
in which \(F_{zbj}\) are the forces applied by components of the primary suspension:  
\[F_{zb1,2}=-2c_{zb}(\dot{z}_{b1}+a_{b}\dot{\theta}_{b1}-\dot{z}_{w1,2})-2k_{zb}( z_{b1}\pm a_{b}\theta_{b1}-z_{w1,2}), \tag{24}\]  
\[F_{zb3,4}=-2c_{zb}(\dot{z}_{b2}\pm a_{b}\dot{\theta}_{b2}-\dot{z}_{w3,4})-2k_{ zb}(z_{b2}\pm a_{b}\theta_{b2}-z_{w3,4}), \tag{25}\]  
where \(a_{b}\) represents the semi-wheelase of the bogie. After processing, Equations (20)-(23) become the following:  
\[ \begin{array}{l}m_{b}\ddot{z}_{b1}+2c_{zb}[\dot{z}_{b1}-(\dot{z}_{w1}+\dot{ z}_{w2})]+2k_{zb}[z_{b1}-(z_{w1}+z_{w2})]+\\ +2c_{zc}(\dot{z}_{b1}-\dot{z}_{c}-a_{c}\dot{\theta}_{c}-\dot{\epsilon}T_{c})+2 k_{zc}(z_{b1}-z_{c}-a_{c}\theta_{c}-\epsilon T_{c})=0\\ \\ m_{b}\ddot{z}_{b2}+2c_{zb}[\dot{z}_{b2}-(\dot{z}_{w3}+\dot{z}_{w4})]+2k_{zb}[ z_{b2}-(z_{w3}+z_{w4})]+\\ +2c_{zc}(\dot{z}_{b2}-\dot{z}_{c}+a_{c}\dot{\theta}_{c}-\dot{\epsilon}T_{c})+2 k_{zc}(z_{b2}-z_{c}+a_{c}\theta_{c}-\epsilon T_{c})=0\\ \\ J_{b}\ddot{\theta}_{b1}+2c_{zb}a_{b}[2a_{b}\dot{\theta}_{b1}-(\dot{z}_{w1}- \dot{z}_{w2})]+2k_{zb}a_{b}[2a_{b}\theta_{b1}-(z_{w1}-z_{w2})]+\\ +2c_{xc}h_{b}[h_{b}\dot{\theta}_{b1}+h_{c}(\dot{\theta}_{c}+\lambda T_{c})]+2 k_{xc}h_{b}[h_{b}\theta_{b2}+h_{c}(\dot{\theta}_{c}+\lambda T_{c})]=0\\ \\ J_{b}\ddot{\theta}_{b2}+2c_{zb}a_{b}[2a_{b}\dot{\theta}_{b2}-(\dot{z}_{w3}- \dot{z}_{w4})]+2k_{zb}a_{b}[2a_{b}\theta_{b2}-(z_{w3}-z_{w4})]+\\ +2c_{xc}h_{b}[h_{b}\dot{\theta}_{b2}+h_{c}(\dot{\theta}_{c}-\lambda T_{c})]+2 k_{xc}h_{b}[h_{b}\theta_{b2}+h_{c}(\dot{\theta}_{c}-\lambda T_{c})]=0\\ \end{array}  \tag{26}\]  
On each wheelset, the vertical force from the primary suspension, \(F_{zbj}\) (see Equations (24) and (25)), and the vertical dynamic force, \(F_{j}\), with \(j=1\ldots 4\), act as follows:  
\[F_{j}=k_{H}(z_{wj}-z_{rj}-\eta_{j}). \tag{30}\]  
The equations of motion of the four wheelsets have the following general form:  
\[m_{w}\ddot{z}_{wj}=2F_{j}-F_{zbj}. \tag{31}\]  
After replacing the expressions of the forces \(F_{j}\) and \(F_{zbj}\), the equations of motion (31) can be rewritten as follows:  
\[ \begin{array}{l}m_{w}\ddot{z}_{w1,2}+2c_{zb}(\dot{z}_{w1,2}-\dot{z}_{b1}\mp a _{b}\dot{\theta}_{b1})+2k_{zb}(z_{w1,2}-z_{b1}\mp a_{b}\dot{\theta}_{b1})+\\ +2k_{H}(z_{w1,2}-z_{r1,2})=2k_{H}\eta_{1,2}\end{array}  \tag{32}\]\[m_{w}\ddot{z}_{u3,A}+2c_{zb}(\dot{z}_{u3,A}-\dot{z}_{b2}\mp a_{b}\dot{\phi}_{b2})+2 k_{zb}(z_{u3,A}-z_{b2}\mp a_{b}\theta_{b2})+. \tag{33}\]  
On each rail, the vertical dynamic force, \(F_{j}\) (Equation (30)), and the vertical force, \(F_{zrj}\), are given by the following relation:  
\[F_{zrj}=-2c_{zr}\dot{z}_{rj}-2k_{zr}z_{rj}. \tag{34}\]  
The equations of motion of the rails corresponding to the wheelset \(j\) are of the form  
\[m_{r}\ddot{z}_{rj}=F_{zrj}-2F_{j}, \tag{35}\]  
which are brought to the following form:  
\[m_{r}\ddot{z}_{r1,2}+2c_{r}\dot{z}_{r1,2}+2k_{r}z_{r1,2}+2k_{H}(z_{r1,2}-z_{w1,2})=-2k_{H}\eta_{1,2}, \tag{36}\]  
\[m_{r}\ddot{z}_{r3,A}+2c_{r}\dot{z}_{r3,A}+2k_{r}z_{r3,A}+2k_{H}(z_{r3,A}-z_{w3,A})=-2k_{H}\eta_{3,A}. \tag{37}\]  
The 15 equations of motion of the mechanical model of the passenger vehicle can be organized into a matrix equation:  
\[\mathbf{M}\ddot{\mathbf{p}}+\mathbf{C}\dot{\mathbf{p}}+\mathbf{K}\mathbf{p}= \mathbf{R}\dot{\mathbf{n}}+\mathbf{Q}\mathbf{n}, \tag{38}\]  
where \(\mathbf{p}\) is the vector of the displacements and \(\mathbf{\eta}\) is the vector of the inhomogeneous terms. The matrices \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the inertia, damping, and stiffness matrices of the system.