## 2 Methodology
Therefore, we use Formula (33) to determine the equivalent parameters of the track in the horizontal plane; the resulting values for the initial parameters are \(\left(E_{P}\right)_{PZ}=1195\) kN\(\cdot\)m\({}^{2}\), \(u_{\Gamma}=23000\) kN\(\cdot\)m\({}^{-2}\), \(f_{\Gamma}=51\) kN\(\cdot\)s\(\cdot\)m\({}^{-2}\), \(m_{V}=0.14\) Mg\(\cdot\)m\({}^{-1}\), \(\alpha_{s}^{V}=1.48\) m\({}^{-1}\)) \(K_{\
u}^{V}=31080\) kN\(\cdot\)m\({}^{-1}\), \(\beta_{\
u}^{V}=68.9\) kN\(\cdot\)s\(\cdot\)m\({}^{-1}\), and \(m_{\
u}^{V}=0.14\div 0.19\) Mg.  
To determine \(m_{res}^{V}\), Formulas (33) and (31) were used. The equivalent stiffness, \(K_{res}^{V}\), was determined based on the displacements of the rail's bending center. If it is determined based on the compression of the rail head, then \(K_{re}^{V}=\frac{1}{V\bar{v}(0)}=20900\) kN\(\cdot\)m\({}^{-1}\). Next, we consider the planar vibrations of a railcar moving along a track modeled as a system with lumped parameters. We will use the results presented analytically at the beginning of the article and in Table 1, where the rail is also represented as a system with lumped parameters. We write the differential relationship between the forces \(P_{1}\) and \(P_{2}\) from one bogie and the deflections of the rail under the first and second wheelsets of the first bogie (see Figure 1), neglecting the deflections under the wheelsets of one bogie caused by the interaction forces of the wheelsets of the other bogie and the rail (41):  
\[a_{4}\ddot{\omega}_{1}+a_{3}\ddot{\omega}_{1}+a_{2}\ddot{\omega}_{1}+a_{1} \dot{\omega}_{1}+a_{0}\omega_{1}=P_{1}+b_{3}\ddot{P}_{2}+b_{2}\bar{P}_{2}+b_{ 1}\dot{P}_{2}+b_{0}P_{2} \tag{42}\]  
And similar expressions are used for the rest of the deflections. In Equation (42), dots denote time derivatives. Substituting the expressions for deflections (1) into (42), after differentiating them beforehand, we obtain a system of differential equations. By performing simple transformations--pairwise addition and subtraction of the equations--we arrive at the following system of differential equations:  
\[ \begin{array}{l}2\big{(}a_{4}\dddot{\eta}_{3}+a_{3}\dddot{\eta}_{3}+a_{2} \ddot{\eta}_{3}+a_{1}\dot{\eta}_{3}+a_{0}q_{3}\big{)}=b_{3}\dddot{T}_{1}+b_{2} \dddot{T}_{1}+b_{1}\dddot{T}_{1}+(b_{0}+1)T_{1}+H_{1}+H_{2}\\ 2l_{1}\big{(}a_{4}\dddot{\eta}_{4}+a_{3}\dddot{\eta}_{4}+a_{2}\ddot{\eta}_{4}+ a_{1}\dddot{\eta}_{4}+a_{0}q_{4}\big{)}=b_{3}\dddot{T}_{2}+b_{2}\dddot{T}_{2}+b_{ 1}\dddot{T}_{2}+(b_{0}-1)T_{2}+H_{2}-H_{1}\\ 2\big{(}a_{4}\dddot{\eta}_{5}+a_{3}\dddot{\eta}_{5}+a_{2}\ddot{\eta}_{5}+a_{1 }\dddot{\eta}_{5}+a_{0}q_{5}\big{)}=b_{3}\dddot{T}_{3}+b_{2}\dddot{T}_{3}+b_{ 1}\dddot{T}_{3}+(b_{0}+1)T_{3}+H_{3}+H_{4}\\ 2l_{1}\big{(}a_{4}\dddot{\eta}_{6}+a_{3}\dddot{\eta}_{6}+a_{2}\ddot{\eta}_{6}+ a_{1}\dot{\eta}_{6}+a_{0}q_{6}\big{)}=b_{3}\dddot{T}_{4}+b_{2}\dddot{T}_{4}+b_{ 1}\dddot{T}_{4}+(b_{0}-1)T_{4}+H_{4}-H_{3}\end{array}  \tag{43}\]  
The following notations are introduced here:  
\[ \begin{array}{c}H_{n}=a_{4}\dddot{\eta}_{n}+a_{3}\dddot{\eta}_{n}+a_{2}\ddot {\eta}_{n}+a_{1}\dot{\eta}_{n}+a_{0}\eta_{n}\;(n=\overline{1.4}),\\ T_{1}=P_{1}+P_{2},\;T_{2}=P_{1}-P_{2},\;T_{3}=P_{3}+P_{4},\;T_{4}=P_{3}-P_{4} \end{array}  \tag{44}\]  
The differential equations of the wagon oscillations (8) can be written in the following form (excluding the cyclic coordinate):  
\[\left\{ \begin{array}{c}m\ddot{q}_{1}+S_{1}+S_{2}=0,\;\dot{q}_{2}+l(S_{2}-S_ {1})=0\\ m_{T}\ddot{q}_{3}-S_{1}+P_{1}+P_{2}=0,\;mr_{T}\ddot{q}_{5}-S_{2}+P_{3}+P_{4}=0 \\ J\tau\ddot{q}_{4}-l_{1}(P_{1}-P_{2})=0,\;J\tau\ddot{q}_{6}-l_{1}(P_{3}-P_{4})= 0\end{array} \right. \tag{45}\]  
Equation (45) can be derived directly from D'Alembert's principle [24]. In these equations, \(S_{1}\) and \(S_{2}\) are the forces arising in the spring assemblies of the central suspension system. For viscous damping, they are given by \(S_{i}=K\Delta_{i}+\beta\dot{\Delta}_{i}\;(i=1.2)\); and for Coulomb friction dampers, they are given by \(S_{i}=K\Delta_{i}+Fsign\Delta_{i}\), where \(F\) is the Coulomb friction force.