## 4 Results and Discussion
Figure 2: The power spectral density of the carbody acceleration: (**a**) at the carbody centre; (**b**) against the front bogie; (**c**) against the rear bogie. To evaluate the influence of the vertical bending upon the of the vertical vibration behaviour of the carbody, the diagrams in Figure 3 are utilised, showing the power spectral density of the vertical acceleration at the velocity of 180 km/h for both the 'rigid carbody' model and the 'flexible carbody' model. For the 'flexible carbody' model, four values of the carbody bending frequency were considered, namely 6 Hz, 8 Hz, 10 Hz and 12 Hz. What can be noticed in the diagrams are the peaks for the four resonance frequency of bending and the ones corresponding to the vibration rigid modes--the resonance frequency of the carbody bounce and the resonance frequency of the carbody pitch. Apart from these resonance peaks, a series of anti-resonance frequencies can be identified due to the geometric filtering effect [21, 44, 45, 46, 47]. As shown in Figure 2, the bending vibration is visibly influencing the vibration behaviour of the carbody, mainly at its centre (diagram a) and less above the two bogies (diagrams b and c). The contribution of the bending vibration is all the more important as the carbody flexibility is higher, which can be clearly noticed at the carbody centre, where the spectrum of the power density of the acceleration is dominated by the bending vibration of the carbody at the frequency of 6 Hz. The more the b  
Figure 3: Influence of the vertical bending upon the power spectral density of the carbody acceleration: (**a**) at the carbody centre; (**b**) against the front bogie; (**c**) against the rear bogie. increases, the less important the weight of this vibration mode is, the dominant vibration mode being the bounce in the reference points of the carbody above the bogies and the spectrum of the power density of the accelerations dominated by the pitch vibration, both for the 'rigid carbody' and for the 'flexible carbody' models. In these points, the bending weight is very low (see diagrams (b') and (c')), irrespective of the frequency of this vibration mode. Based on the diagrams in Figure 4, the influence of the vertical bending upon the level of vertical vibrations in the carbody reference points in correlation with velocity is analysed. This analysis is performed by comparing the values of the root mean square of acceleration for the 'flexible carbody' model with those corresponding to the 'rigid carbody' model. The vertical bending causes the increase of the level of vibrations at the centre of carbody, with this increase being important under certain conditions--high flexibility of the carbody and high velocities (diagram a). This is visible in Figure 5, which features the percentage growth of the root mean square of acceleration for the 'flexible carbody' model versus the 'rigid carbody' model. As an example, the root mean square of acceleration for the flexible carbody goes up by 93.6% for \(f_{vb}=6\) Hz at a velocity of 200 km/h compared to the root mean square of acceleration for the rigid carbody. For higher values of the carbody bending frequency, the increase in the root mean square of acceleration does not exceed 20%. As seen in the diagrams (b) and (c), the carbody vertical bending does not sensibly affect the level of vibrations above the bogies, which is valid irrespective of the carbody bending frequency. Figure 4: Influence of the carbody vertical bending upon the root mean square of acceleration in correlation with the velocity: (**a**) at the carbody centre; (**b**) against the front bogie; (**c**) against the rear bogie. Another interesting aspect to be analysed is related to the manner in which the vertical bending changes the position of the point where the vibrations level is the highest, hereafter called the critical point of the carbody.