## 4 Dynamic Response of the Carbody to Vertical Vibrations
If the frequencies of geometric filtering equal the resonance frequency of one of the vibration modes of the vehicle carbody, then its impact is greatly reduced. The geometric filtering effect manifests itself more intensely at the carbody center, where the geometric filtering effects given by the bogie wheelbase and the carbody wheelbase are cumulated. Above the bogies and the support points of the carbody on the secondary suspension, the geometric filtering effect has smaller impact because the geometric filtering effect, given by the carbody wheelbase, is missing. A final observation, highlighted in Figures 5 and 6, refers to the vibration behavior of the carbody above the two bogies. It is observed that for the undamped system, the vibration behavior of the carbody at points \(B_{1}\) and \(B_{2}\) is symmetrical (Figure 5), and when damping is taken into account, the response of the carbody at the same points is different (Figure 6). Figure 7 shows the dynamic response at the relevant points of the carbody, expressed as the unweighted PSD acceleration, for the damped mechanical model. These diagrams highlight several important properties of the vibration behavior of the carbody. Figure 5: Carbody acceleration for the undamped vehicle–track system: (**a**) for 0–50 Hz; (**a**) for 0–10 Hz. Figure 6: Carbody acceleration for the damped vehicle–track system: (**a**) for 0–50 Hz; (**a**) for 0–10 Hz. First, the peaks corresponding to the resonance frequencies of the vibration modes of the carbody are noted: at 1.17 Hz for the bounce vibrations; at 1.53 Hz for pitch vibrations; and at 8 Hz for vertical bending vibrations. At the carbody center (diagram a), the PSD of the acceleration is dominated by bounce vibrations, but at speeds higher than 200 km/h the weight of vertical bending vibrations also becomes important. Above the bogies (diagrams b and c), the dominant mode of vibration is the pitch, but bounce also has an important contribution. Then, it is observed that, regardless of the velocity, the carbody vibration level is higher above the bogies than in the middle of the carbody, and that the vibration level above the two bogies is asymmetric. It is also noted that at all relevant points of the carbody, the vibration level increases with velocity, but not monotonous. The PSD has several minima, which appear in the form of anti-resonance frequencies, and which are due to the geometric filtering effect. It is observed that the geometric filtering effect is more efficient at the resonance frequency of the vertical bending of the carbody. The results presented above, which highlight the most important features of the carbody vibration, are in agreement with the theoretical or experimental results presented in the scientific literature [35; 36; 37]. At the carbody center, at velocities greater than 200 km/h, vertical bending can become the dominant vibration mode for flexible carbodies, and this is evident in the diagrams in Figure 8. Diagram (a) shows the unweighted PSD of the acceleration for the carbody bending frequency of 6 Hz. It is observed that at velocities above 200 km/h, the frequency peak of the vertical bending dominates the PSD. Figure 7: Unweighted PSD of the acceleration, in (m/s\({}^{2}\))/(1/s): (**a**) at point \(M\); (**b**) point \(B_{1}\); (**c**) and point \(B_{2}\). Diagrams (b), (c), and (d) show the unweighted PSD of the acceleration at the middle of the carbody for five values of bending frequency at velocities of 200 km/h, 220 km/h, and 240 km/h. Table 2 shows the values of the bending stiffness of the carbody corresponding to the eigenfrequencies of the vertical bending analyzed in Figure 8.  
According to diagram (b), at the velocity of 200 km/h, for the resonance frequency of 6 Hz, vertical bending is the dominant mode of vibration. At the velocity of 220 km/h, the vertical bending with the eigenfrequencies of 6 Hz and 7 Hz, respectively, dominates the PSD of the acceleration.