## 2 Materials and Methods
The next stage of the study includes the calculation of the main indicators of the wagon dynamics, which are used to estimate its motion. The vertical dynamics of the wagon when it moves over a joint irregularity are investigated. The design diagram of the wagon is shown in Figure 9.  
The equation of motion of the wagon can be written as [28]. \[ \begin{split}& M_{1}\cdot\frac{d^{2}}{dt^{2}}q_{1}+2\cdot k_{b} \cdot q_{1}-k_{b}\cdot q_{3}-k_{b}\cdot q_{5}=\\ &=-F_{FR}\cdot\Big{(}sign\Big{(}\frac{d}{dt}\delta_{1}\Big{)}+ sign\Big{(}\frac{d}{dt}\delta_{2}\Big{)}\Big{)},\end{split}  \tag{3}\]  
\[ \begin{split}& M_{2}\cdot\frac{d^{2}}{dt^{2}}q_{2}+\big{(}2\cdot l ^{2}\cdot k_{b}\big{)}\cdot q_{2}+(1\cdot k_{b})\cdot q_{3}-(l\cdot k_{b}) \cdot q_{5}=\\ &=F_{FR}\cdot l\cdot\Big{(}sign\Big{(}\frac{d}{dt}\delta_{1} \Big{)}+sign\Big{(}\frac{d}{dt}\delta_{2}\Big{)}\Big{)},\end{split}  \tag{4}\]  
\[ \begin{split}& M_{3}\cdot\frac{d^{2}}{dt^{2}}q_{3}-k_{b}\cdot q_{1}+( l\cdot k_{b})\cdot q_{2}+(k_{b}+2\cdot k_{1})\cdot q_{3}+2\cdot\beta_{1} \cdot\frac{d}{dt}q_{3}=\\ &=F_{FR}\cdot sign\Big{(}\frac{d}{dt}\delta_{1}\Big{)}+k_{1}( \eta_{1}+\eta_{2})+\beta_{1}\Big{(}\frac{d}{dt}\eta_{1}+\frac{d}{dt}\eta_{2} \Big{)},\end{split}  \tag{5}\]  
\[ \begin{split}& M_{4}\cdot\frac{d^{2}}{dt^{2}}q_{4}+\big{(}2\cdot a ^{2}\cdot k_{1}\big{)}\cdot q_{4}+2\cdot\beta_{1}\cdot\frac{d}{dt}q_{4}=\\ &=-k_{1}(\eta_{1}-\eta_{2})-\beta_{1}\cdot a\cdot\Big{(}\frac{d}{ dt}\eta_{1}-\frac{d}{dt}\eta_{2}\Big{)},\end{split}  \tag{6}\]  
\[ \begin{split}& M_{5}\cdot\frac{d^{2}}{dt^{2}}q_{5}-k_{b}\cdot q _{1}-(l\cdot k_{b})\cdot q_{2}+(k_{b}+2\cdot k_{1})\cdot q_{5}+2\cdot\beta_{1 }\cdot\frac{d}{dt}q_{5}=\\ &=F_{FR}\cdot sign\Big{(}\frac{d}{dt}\delta_{2}\Big{)}+k_{1}( \eta_{3}+\eta_{4})+\beta_{1}\Big{(}\frac{d}{dt}\eta_{3}+\frac{d}{dt}\eta_{4} \Big{)},\end{split}  \tag{7}\]  
\[ \begin{split}& M_{6}\cdot\frac{d^{2}}{dt^{2}}q_{6}+\big{(}2\cdot a ^{2}\cdot k_{1}\big{)}\cdot q_{6}+2\cdot\beta_{1}\cdot\frac{d}{dt}q_{6}=\\ &=-k_{1}\cdot a\cdot(\eta_{3}-\eta_{4})-\beta_{1}\cdot a\cdot \Big{(}\frac{d}{dt}\eta_{3}-\frac{d}{dt}\eta_{4}\Big{)},\end{split}  \tag{8}\]  
where \(M_{1}\), \(M_{2}\) are the mass coefficients of the body (mass and moment of inertia) during translational and angular displacements, respectively; \(M_{3}\), \(M_{4}\) are the mass coefficients (mass and moment of inertia) of the first bogie facing the engine during translational and angular displacements, respectively; \(M_{5}\), \(M_{6}\) are the mass coefficients (mass and moment of inertia) of the second bogie facing the engine during translational and angular displacements, respectively; \(k_{b}\) is the stiffness of the springs forming spring suspension; \(k_{1}\) is the track stiffness; \(\beta_{1}\) is the damping coefficient of the track; \(a\) is half the bogie; \(q_{i}\) is the generalized coordinates corresponding to translational and angular displacements of the wagon; \(F_{FR}\) is the frictional force in the spring suspension; \(\delta\) is the deformation of the spring suspension; \(\eta_{i}\) is the joint irregularity set as the harmonious law of motion. The design diagram includes three bodies: the wagon body and two bogies. It is taken into account that the oscillatory system has 6 degrees of freedom, which characterize translational displacements (bouncing oscillations) and angular displacements (galloping oscillations). Bouncing oscillations in Figure 9 are indicated by vertical arrows, and galloping by circular arrows. Figure 9: Design diagram of the wagon. Equations (3), (5) and (7) describe the translational displacements of the body and two bogies, respectively, and Equations (4), (6) and (8) describe their angular movements. The interaction of the body and the unsprung masses is described through an elastic connection that forms the spring suspension of the bogies. A track irregularity is included as a disturbing action, which generates the oscillatory process, and, thus, the frictional forces in the spring suspension of the wagon.