## 4 Results and Discussion  
This section analyses the influence of the vertical bending upon the vertical vibrations behaviour of the railway vehicle carbody using the results from the numerical simulations developed based on the two previously presented models--the 'rigid carbody' model and the 'flexible carbody' model. The basic characteristics of the vibration behaviour of the vehicle carbody are analysed using the power spectral density of the vertical acceleration in the reference points of the carbody--at the centre of the carbody and against the two bogies. The influence of the vertical bending upon the vertical vibrations level in the reference points of the carbody are analysed based on the root mean square of the vertical acceleration, in correlation with the velocity and damping ratio of the vehicle suspension, for four values of the flexibility of the carbody, corresponding to the bending frequencies of 6 Hz, 8 Hz, 10 Hz and 12 Hz. The reference parameters of the vehicle model used in the numerical simulations are shown in Table 1. The eigenfrequencies of the vertical vibration modes of the carbody corresponding to the reference parameters of the model are the bounce eigenfrequency \(f_{b}\) = 1.17 Hz, pitch eigenfrequency \(f_{p}\) =1.46 Hz and vertical bending eigenfrequency \(f_{vb}\) = 8 Hz. It is considered that the vehicle is running on a low-quality track characterised by the coefficient \(A\) = 1.080 \(\times\) 10\({}^{-6}\) radm. For the reference parameters of the 'flexible carbody' model, the diagrams in Figure 2 show the power spectral density of the vertical acceleration in the reference points of the carbody, depending on the velocity. The bending vibration is observed as mainly affecting the vibration behaviour at the carbody centre (diagram a) and less above the two bogies (diagrams b and c). At the centre carbody, the power spectral density of the acceleration is dominated by the bounce vibration (at 1.17 Hz), but the contribution of the bending vibration (at 8 Hz) is also noticeable at high velocities. In the reference points above the bogies, the vibration behaviour of the carbody is dominated by the pitch vibration (at 1.46 Hz), whereas the bending weight is not significant. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Definition** & **Symbol** & **Definition** \\ \hline \multicolumn{4}{c}{Joint parameters for the ‘rigid carbody’ model and the ‘flexible carbody’ model} \\ \hline \multicolumn{4}{c}{The parameters of the carbody and bogie} \\ \hline \(m_{c}\) = 34,000 kg & carbody mass & \(L_{c}\) = 26.4 m & carbody length \\ \hline \(m_{b}\) = 3200 kg & bogie suspended mass & \(2a_{b}\) = 19 m & carbody wheelbase \\ \hline \(J_{c}\) = 1,963,840 kg\(\cdot\)m\({}^{2}\) & carbody inertia moment & \(2a_{b}\) = 2.56 m & bogie wheelbase \\ \hline \(J_{b}\) = 2048 kg\(\cdot\)m\({}^{2}\) & bogie inertia moment & \(h_{c}\) = 1.3 m\(h_{b}\) = 0.2 m & elevations of the secondary suspension plane \\ \hline \multicolumn{4}{c}{The parameters of the secondary suspension per bogie} \\ \hline \(2c_{zc}\) = 34.28 kNs/m & vertical damping & \(2k_{zc}\) = 1.2 MN/m & vertical stiffness \\ \hline \(2c_{zc}\) = 50 kNs/m & longitudinal damping & \(2k_{zc}\) = 4 MN/m & longitudinal stiffness \\ \hline \(2c_{\text{Qc}}\) = 2 kNm & angular damping & \(2k_{\text{Qc}}\) = 256 kN/m & angular stiffness \\ \hline \multicolumn{4}{c}{The parameters of the primary suspension per wheelset} \\ \hline \(4c_{zb}\) = 52.21 kNs/m & vertical damping & \(4k_{zb}\) = 4.4 MN/m & vertical stiffness \\ \hline \multicolumn{4}{c}{Parameters specific to the ‘flexible carbody’ model} \\ \hline \(El\) = 3.158 \(\times\) 10\({}^{9}\) Nm\({}^{2}\) & bending modulus & \(c_{mc}\) = 53.117 kNm/s & carbody modal damping \\ \hline \(m_{mc}\) = 35,224 kg & carbody modal mass & \(k_{mc}\) = 88.998 MN/m & carbody modal stiffness \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reference parameters of the vehicle model.