## 2 Methodology  
To facilitate the solution of the problem under consideration, this present study examines the planar oscillations of a wagon with elastic-viscous connections, moving along a beam supported by an elastic-viscous inertial foundation. Let the wagon move translationally with velocity C along the beam [29]. The computational model of the system consists of three rigid bodies--the car body and two bogies--an inertial deformable beam with doubled rail parameters, and an inertial elastic-viscous foundation (Figure 1). Let us choose the following values for the crew as generalized coordinates: \(q_{1}=z\), \(q_{3}=z_{1}\), \(q_{5}=z_{2}\)--vertical movements of the body, the first and second bodies; \(q_{2}=\varphi\), \(q_{4}=\varphi_{1}\), \(q_{6}=\varphi_{2}\)--the angles of rotation of the same bodies relative to the main central axis \(y\), perpendicular to the vertical plane in which the axis of the track lies; and \(q_{7}=x\)--the displacement of the center of mass of the system along the axis of the track. Let us denote the deflections of the beam by \(\omega\), and the current ordinates of the geometric irregularity of the unloaded rail by \(\eta\). Let us express the movements of all wheelsets in terms of the specified values (the numbering of wheelsets is given in Figure 1):  
\[ \begin{array}{l}z_{k1}=\omega_{1}+\eta_{1}=q_{3}-l_{1}q_{4},\;z_{k3}=\omega _{3}+\eta_{3}=q_{5}-l_{1}q_{6}\\ z_{k2}=\omega_{2}+\eta_{2}=q_{3}+l_{1}q_{4},\;z_{k4}=\omega_{4}+\eta_{4}=q_{5} +l_{1}q_{6}\end{array}  \tag{1}\]  
where \(2l_{1}\) is the bogie base. Using the Expressions (1), we get  
\[ \begin{array}{l}2q_{3}=\omega_{1}+\omega_{2}+\eta_{1}+\eta_{2},\;2l_{1}q_{4} =\omega_{2}-\omega_{1}+\eta_{2}-\eta_{1}\\ 2q_{5}=\omega_{3}+\omega_{4}+\eta_{3}+\eta_{4},\;2l_{1}q_{6}=\omega_{4}-\omega _{3}+\eta_{4}-\eta_{3}\end{array}  \tag{2}\]  
We formulate the differential equations of motion for the railway vehicle under the same assumptions as in [24,26], using [PERSON]'s equations of the second kind:  
\[\frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{i}}-\frac{\partial T}{\partial \dot{q}_{i}}+\frac{\partial R}{\partial\dot{q}_{i}}+\frac{\partial\Pi}{ \partial\dot{q}_{i}}=Q_{i} \tag{3}\]  
where \(T\) is the kinetic energy of the system, \(\Pi\) is the potential energy of the system, \(R\) is the scattering function, and \(Q_{i}\) is the generalized force corresponding to generalized coordinate \(q_{i}\). Kinetic energy has a diagonal form:  
\[ \begin{array}{l}T=\frac{1}{2}m\dot{q}_{1}^{2}+\frac{1}{2}J_{2}^{2}+\frac{1} {2}m_{T}\Big{(}\dot{q}_{3}^{2}+\dot{q}_{5}^{2}\Big{)}+\frac{1}{2}J_{T}\Big{(} \dot{q}_{4}^{2}+\dot{q}_{6}^{2}\Big{)}+\\ +\frac{1}{2}\Big{(}m+2m_{T}+\frac{4J_{K}}{r^{2}}\Big{)}\dot{q}_{7}^{2}\end{array}  \tag{4}\]  
Figure 1: Railway vehicle design diagram. where \(m\) and \(m_{T}\) are body and bogie weights; \(J,\ J_{T},\) and \(J_{K}\) are, respectively, the moments of inertia of the body, bogie, and wheelset relative to the main central axis \((y);\ \text{and}\ r\) is the wheel radius. The potential compression energy of spring sets is calculated using [PERSON]'s theorem:  
\[\Pi=\frac{1}{2}K\Big{(}\Delta_{1}^{2}+\Delta_{2}^{2}\Big{)}=\frac{1}{2}K\Big{[} (q_{1}-lq_{2}-q_{3})^{2}+(q_{1}+lq_{2}-q_{5})^{2}\Big{]} \tag{5}\]  
where \(K\) is the spring set stiffness; \(\Delta_{1}\) and \(\Delta_{2}\) are the spring sets' compression; and \(2l\) is the wheelbase of the railway vehicle. The scattering function is proportional to the potential energy:  
\[R=\frac{1}{2}\beta\bigg{(}\dot{\Delta_{1}^{2}}+\dot{\Delta_{2}^{2}}\bigg{)}= \frac{1}{2}\beta\Big{[}\big{(}\dot{q}_{1}-l\dot{q}_{2}-\dot{q}_{3}\big{)}^{2}+ \big{(}\dot{q}_{1}+l\dot{q}_{2}-\dot{q}_{5}\big{)}^{2}\Big{]} \tag{6}\]  
where \(\beta\) is the viscous friction coefficient of dampers. As the generalized forces, we consider the vertical dynamic forces, \(P_{n}\), acting from the rails on the wheelsets (\((n=\overline{I,4})\)).