## 5 Conclusions  
This paper analyses the effects of the first vertical bending mode upon the vibration behaviour of the railway vehicle carbody. This is done by comparing the results from the numerical simulations developed based on a 'flexible carbody' vehicle model with the ones from the 'rigid carbody' model. The basic characteristics of the vibration behaviour of the carbody for various carbody flexibilities have been analysed by means of the power spectral density of the vertical acceleration in three reference points of the carbody situated at the carbody centre and against the two bogies. The bending vibration has been shown to influence the vibration behaviour of the carbody, mainly at its centre and less against the two bogies. At the carbody centre, the contribution of this vibration mode is more important at high velocities, to the point when the power spectral density of the vertical acceleration dominates in the case of very flexible carbodies. In the carbody reference points above the bogies, the power spectral density of the vertical acceleration is dominated by the pitch vibration, the bending weight being very low, even for a high flexibility in the carbody. To highlight the influence of the vertical bending upon the vibrations level in the carbody reference points, the values of the root mean square of vertical acceleration for the various flexibility of the carbody in the 'flexible carbody' model have been compared with the ones in the 'rigid carbody' model, in correlation with the velocity and the damping ratios of the two suspension stages of the vehicle. The results correlated with the velocity have shown an important increase in the vibrations level at the carbody centre due to the bending vibrations in the range of high velocities. For the most adverse conditions--high flexibilities of the carbody and high velocities--the root mean square of acceleration is almost double the root mean square of acceleration in the rigid carbody. Still, based on the results correlated with the velocity, it has been shown the bending vibrations do not change the position of the critical point of the vibrations level of the carbody. Both for the 'rigid carbody' model and the 'flexible carbody' model, the critical point of the carbody is located above one of the bogies, and its position modifies as a function of velocity. In accordance with the results correlated with the damping ratio of secondary suspension, the value of the damping ratio minimizing the root mean square of acceleration at the carbody centre has been proven to be lower for the flexible carbody than the rigid carbody. This value decreases along with the bending frequency of the carbody. In the points above the two bogies, the value of the damping ratio minimizing the root mean square of acceleration is the same for both models. For both models, the results correlated with the damping ratio of primary suspension have showed that the level of vibrations continuously decreases with the increase of the damping of primary suspension, yet the influence of the carbody vertical bending is visible in the decrease rate of the root mean square of acceleration in the carbody reference points. In case of the rigid carbody, the decrease rate of the root mean square of acceleration is higher above the two bogies than at the carbody centre. Instead, for the flexible carbody, the decrease rate of the root mean square of acceleration is higher at the carbody centre. Conceptual approach, M.D. ; methodology, M.D. ; software, M.D. and I.I.D. ; validation, M.D. ; formal analysis, M.D. and I.I.D. ; writing--original draft, M.D. ; writing--review and editing, M.D. and I.I.D. Both authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. * [PERSON] et al. (2015) [PERSON]; [PERSON]; [PERSON] Reduction of carbody flexural vibration by the high-damping elastic support of under-floor equipment. _QR RTRI_**2015**, _56_, 262-267. [CrossRef]
* [PERSON] et al.