## 2 Model of the Railway Vehicle for Ride Comfort Assessment at Vertical Vibrations  
To evaluate the ride comfort during running at a constant velocity on a track with vertical irregularities, a model of the rigid-flexible coupled vehicle is being used. This model is featured in Figure 1[35; 43], and Table 1 contains the model parameters. In terms of the track model, the rigid track hypothesis is adopted. This simple approach is justified by the fact that the eigenfrequencies of the vertical vibrations modes of the vehicle carbody, relevant for the ride comfort, are much lower than the wheelset-track system frequencies. Because bogies and wheelsets have a reduced elastic deformation, they are represented through rigid bodies. The vibration modes of the two bogies in the vertical plan are bounce \(z_{b1,2}\) and pitch \(\theta_{b1,2}\). The vertical displacements of the four wheelsets \(z_{w1\ldots 4}\) are equal with the vertical track irregularities. \[z_{w1,2}=\eta(x+a_{c}\pm a_{b}),\ z_{w3,4}=\eta(x-a_{c}\pm a_{b}) \tag{1}\]  
where \(\eta(x)\) is the function of the vertical track irregularity and \(x=Vt\) is the coordinate for the carbody center. The vehicle carbody is represented by a free-free equivalent beam of constant section and uniformly distributed mass, Euler-Bernoulli type. This model gives the possibility to consider the vertical vibration modes of the carbody that are relevant for the ride comfort, \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{4}{c}{The Parameters of the Carbody and Bogie} \\ \hline Symbol & Definition & Symbol & Definition \\ \hline \(m_{c}\) & carbody mass & \(L_{c}\) & carbody length \\ \(m_{b}\) & bogie suspended mass & \(2a_{c}\) & carbody wheelbase \\ \(l_{c}\) & carbody inertia moment & \(2a_{b}\) & bogie wheelbase \\ \(J_{b}\) & bogie inertia moment & \(l_{1,2}=L_{c}/2\pm a_{c}\) & supporting points position of the \\  & & & \\ \hline \multicolumn{4}{c}{The parameters of the secondary suspension per bogie} \\ \hline \(2c_{zc}\) & vertical damping & \(2k_{zc}\) & vertical stiffness \\ \(2c_{xc}\) & longitudinal damping & \(2k_{xc}\) & longitudinal stiffness \\ \(2c_{0c}\) & angular damping & \(2k_{0c}\) & angular stiffness \\ \hline \multicolumn{4}{c}{The parameters of the primary suspension per wheelset} \\ \hline \(2c_{zb}\) & vertical damping & \(2k_{zb}\) & vertical stiffness \\ \hline \end{tabular}
\end{table} 
Table 1: The parameters of the model vehicle. Figure 1: The mechanical model of the railway vehicle. namely the rigid modes vibration--bounce \(z_{c}\), pitch \(\theta c\)--and the first vertical bending mode. The vertical displacement of the carbody medium fiber \(w_{c}(x,t)\) is the result of overlapping of the three vibration modes  
\[w_{c}(x,t)=z_{c}(t)+\left(x-\frac{L_{c}}{2}\right)\theta_{c}(t)+X_{c}(x)T_{c}(t) \tag{2}\]  
where \(T_{c}(t)\) is the coordinate of the carbody bending and \(X_{c}(x)\) represents the natural function of this vibration mode, described in the equation  
\[X_{c}(x)=\sin\beta x+\sinh\beta x-\frac{(\sin\beta L_{c}-\sinh\beta L_{c})( \cos\beta x+\cosh\beta x)}{\cos\beta L_{c}-\cosh\beta L_{c}} \tag{3}\]  
\[\text{with }\beta=\sqrt{\frac{\omega_{c}^{2}\rho_{c}}{EI}}\text{ and }\cos\beta L_{c}\cosh\beta L_{c}=1 \tag{4}\]  
where \(\omega_{c}\) is the natural angular frequency of the carbody bending, and \(\rho_{c}=m_{c}/L_{c}\)--beam mass per length unit. Each wheelset is connected to the bogie frame through a Kelvin-Voigt system modeling the primary suspension. The bogie is linked to the carbody through the secondary suspension, modeled via an ensemble of three Kelvin-Voigt systems, two for translation (vertical and longitudinal) and one for rotation.The Kelvin-Voigt system for translation in the longitudinal direction models the longitudinal traction system between carbody and bogie, while the elements in the Kelvin-Voigt system for rotation take over the relative angular displacement between carbody and bogie.