## 2 Methodology
The inelastic component of the forces, \(S_{i}\) (denoted as \(S_{i}^{H}\)), is expressed by an analytical function that approximates the \"signature\" behavior but is itself a smooth function. \[S_{i}^{H}=Fsign\dot{\Delta}_{i}\approx F\cdot th\dot{\Delta}_{i} \tag{46}\]  
where \(a=0.025\frac{\varsigma}{\text{M}}\) is a coefficient characterizing the steepness of the characteristic slope at low displacement velocities. When using expressions (46), the integration of the differential equations proceeds without any peculiarities. From Equations (43) and (45), we eliminate the forces, \(T_{n}\) (Equation (44)). As a result, we obtain the following system of differential equations:  
\[\left\{ \begin{array}{l}m\ddot{q}_{1}+S_{1}+S_{2}=0,\;J\ddot{q}_{2}+l(S_{2}-S_{ 1})=0\\ b_{3}m_{T}\,\ddot{q}_{3}+(b_{2}m_{T}+2a_{4})\,\ddot{q}_{3}+(b_{1}m_{T}+2a_{3}) \,\ddot{q}_{3}+(b_{0}m_{T}+m_{T}+2a_{2})\,\ddot{q}_{3}+\\ +2a_{1}\dot{q}_{3}+2a_{0}q_{3}=b_{3}\,\ddot{S}_{1}+b_{2}\ddot{S}_{1}+b_{1}\dot{ S}_{1}+(b_{0}+1)S_{1}+H_{1}+H_{2}\\ l_{1}[b_{3}m_{T}^{\prime}\,\ddot{q}_{4}+(b_{2}m_{T}^{\prime}-2a_{4})\,\ddot{q}_{4 }+(b_{1}m_{T}^{\prime}-2a_{3})\,\ddot{q}_{4}+(b_{0}m_{T}^{\prime}-m_{T}^{ \prime}-2a_{2})\,\ddot{q}_{4}-\\ -2a_{1}\dot{q}_{4}-2a_{0}q_{4}\,\right]=H_{1}-H_{2}\\ b_{3}m_{T}\,\ddot{q}_{5}+(b_{2}m_{T}+2a_{4})\,\ddot{q}_{5}+(b_{1}m_{T}+2a_{3}) \,\ddot{q}_{5}+(b_{0}m_{T}+m_{T}+2a_{2})\ddot{q}_{5}+\\ +2a_{1}\dot{q}_{5}-2a_{0}q_{5}=b_{3}\,\ddot{S}_{2}+b_{2}\ddot{S}_{2}+b_{1}\dot{ S}_{2}+(b_{0}+1)S_{2}+H_{3}+H_{4}\\ l_{1}[b_{3}m_{T}^{\prime}\,\ddot{q}_{6}+(b_{2}m_{T}^{\prime}-2a_{4})\,\ddot{q}_{6 }^{\prime}+(b_{1}m_{T}^{\prime}-2a_{3})\,\ddot{q}_{6}+(b_{0}m_{T}^{\prime}-m_{T }^{\prime}-2a_{2})\,\ddot{q}_{6}^{\prime}-\\ -2a_{1}\dot{q}_{6}-2a_{0}q_{6}\,\right]=H_{3}-H_{4}\end{array} \right. \tag{47}\]  
To integrate Equation (47), the time derivatives of the function \(S_{i}^{H}\) are required. These will be calculated as derivatives of the composite function (46):  
\[\dot{S}_{i}^{H}=\frac{dS_{i}^{H}}{dt}=\frac{dS_{i}^{H}}{d\dot{\Delta}_{i}}\frac {d\dot{\Delta}_{i}}{dt}=Fa\frac{1}{ch^{2}a\dot{\Delta}_{i}} \tag{48}\]  
Differentiating the resulting expression as a product, we find  
\[ \begin{array}{l}\ddot{S}_{i}^{H}=-2 Fa^{2}\ddot{\Delta}_{i}^{2}\frac{sha\dot{ \Delta}_{i}}{ch^{2}a\dot{\Delta}_{i}}+Fa\frac{1}{ch^{2}a\dot{\Delta}_{i}}\, \ddot{\Delta}_{i}\\ \\ \ddot{S}_{i}^{\prime H}=-2 Fa^{3}\ddot{\Delta}_{i}^{-3}\frac{ch^{2}a\dot{\Delta} _{i}-3 ch^{2}a\dot{\Delta}_{i}}{ch^{2}a\dot{\Delta}_{i}}-6 Fa^{2}\,\ddot{\Delta} _{i}\ddot{\Delta}_{i}\frac{sha\dot{\Delta}_{i}}{ch^{2}a\dot{\Delta}_{i}}+Fa \,\ddot{\Delta}_{i}^{\prime}\frac{1}{ch^{2}a\dot{\Delta}_{i}}\end{array}  \tag{49}\]  
The system of differential Equation (47) can be integrated. It is a system of six equations of the 24 th order. The results of its integration, considering only the real part of the frequency characteristic in expression (36) (\(\overline{b}_{1}=\overline{b}_{3}=0\), \(\overline{b}_{0}=0.1848\), and \(\overline{b}_{2}=0.11413\); the \(\overline{a}_{n}\) parameters are taken from the fourth row of Table 1), are presented in the third row of Table 2.  
From the comparison of the amplitude values of the spring suspension forces, \(S\), and the interaction forces, \(S_{B}\), in the third and first cases, it is evident that all forces are greater when Coulomb friction dampers are used. In the case where the influence of the adjacent wheelset is neglected, all \(b_{n}=0\) should be set in system (47). Then, the system order is reduced to 20, and the right-hand sides of the third and fifth equations no longer contain the time derivatives of the forces, \(S_{i}\). The corresponding results are presented in the fourth row of Table 2, from which it can be seen that adjacent wheelsets under Coulomb friction damping have a greater effect on all forces (up to 21%) compared to viscous damping (the discrepancy does not exceed 10%, and only in the interaction forces).