# Evaluation of Dynamics of a Freight Wagon Model with Viscous Damping
## 4 Simulation Study
### Conditions of the Study  
The study on the influence of the nature of damping on the dynamics of a rail vehicle was carried out on two S-curved track models, differing in length, radius values, and superelevation:  
* The first track: \(R=150\) m, \(\text{cant}\)\(h=0\) mm, total length = 220 m;
* The second track: \(R=320\) m, \(\text{cant}\)\(h=110\) mm, total length = 950 m.  
Plan views of the simulated tracks are presented in Figure 12.  
Figure 11: Comparison of vertical vibrations of a vehicle model body with dry friction and viscous damping: (**a**) Vehicle body free vibration; (**b**) Phase portrait. Geometric irregularities were implemented in both tracks, generated from the power spectral densities for the vertical (\(S_{zz}\)) and lateral (\(S_{yy}\)) directions [28], according to Formulas (11) and (12):  
\[S_{zz}=\frac{A_{v}\Omega_{c}^{2}}{(\Omega^{2}+\Omega_{r}^{2})(\Omega^{2}+\Omega_ {r}^{2})}, \tag{11}\]  
\[S_{yy}=\frac{A_{v}\Omega_{c}^{2}}{(\Omega^{2}+\Omega_{r}^{2})(\Omega^{2}+\Omega _{r}^{2})} \tag{12}\]  
where \(\Omega_{c}\) = 0.8246 rad/m, \(\Omega_{r}\) = 0.0206 rad/m, \(A_{v}\) = 1.08 \(\times\) 10\({}^{-6}\) m rad, \(A_{\text{A}}\) = 6.1 \(\times\) 10\({}^{-7}\) m rad. The values were adopted from ORE B 176 for a high level of irregularities [29]. Track irregularities belong to very important input parameters for a model. They influence the rail vehicle dynamics and its resulting dynamic response [30; 31; 32]. Based on the height of the irregularities and their standard deviation, a qualitative classification of the track geometric quality can be made. The height of the irregularities for the modeled tracks corresponds to the good maintenance condition (QN1) for tracks with a permissible speed \(\leq\) 80 km/h according to the UIC 518 leaflet [33]. In turn, in terms of the standard deviation of the irregularities, the track is classified as a track of medium maintenance condition (QN2). The values of statistical measures describing irregularities for a track of geometric quality QN1 are smaller than for QN2. The values of irregularities' measures based on UIC 518 for a given track geometric quality and the values of irregularities' measures implemented in the simulation model are shown in Table 3.  
In rail vehicle dynamics simulation, the track is often modeled as flexible [34]; however, for the purpose of this study, the tracks were assumed to be rigid in order to reduce the number of potential variables affecting the vehicle's dynamic responses. The vehicle speed on the first track was \(v\) = 50 km/h, while on the second track, with a larger radius of the curves, it was \(v\) = 80 km/h. Due to the high running speed in relation to the radius of the curves, the test scenario using the first track may correspond to a vehicle emergency (e.g., due to a faulty brake system) or driver error. The assumed geometrical parameters of the first track and the assumed running speed result in an unsustainable lateral acceleration of \(a_{y}\) = 1.29 m/s\({}^{2}\) and a cart deficiency of \(CD\) = 197 mm, which significantly exceed the permissible values used on railway lines worldwide. The second track, on the other hand, was designed such that for a running speed of \(v\) = 80 km/h, the unbalanced lateral acceleration \(a_{y}\) = 0.79 m/s\({}^{2}\), while the cart deficiency \(CD\) = 121 mm. The acceleration values were obtained using a simple point-mass vehicle model of mass \(m\) running on a slope (Figure 13). By applying D'Alembert principle to the \(x\)-axis, the following relation is obtained (13):  
\[mg\sin\alpha-m\frac{V^{2}}{R}\cos\alpha=0 \tag{13}\]  
Expressing sin\(\alpha\) using track cant (14):  
\[\sin\alpha=\frac{h}{s}g_{s}^{\frac{h}{2}}-\frac{V^{2}}{R}\sqrt{1-\left(\frac {h}{s}\right)^{2}}=0 \tag{14}\]