## 3 Dynamic Calculation of Bridge and Train Vibrations
### Analyses of Considered Steel Bridge in Frequency and Train Speed Domain
Following the same principle as in Figures 6 and 7, Figure 8 displays the FFT results of one adjacent car body end (\(\hat{w}_{\text{c},2,\text{front}}\)) and the bridge vibrations at midspan (\(\hat{w}_{\text{b}}(x=L/2)\)). Figure 6: FFT analysis results of bogie frame \(b2\) accelerations for selected speed ranges: (**a**) 3D illustration for train speed range \(\Delta v_{l}\) = 60 km/h \(\ldots\) 420 km/h; (**b**) selected FFT profile at \(v\) = 135 km/h. Figure 7: FFT analysis results of bogie frame \(b2\) accelerations for selected speed ranges: (**a**) planar illustration for train speed range \(\Delta v_{1}\) = 60 km/h \(\ldots\) 420 km/h; (**b**) planar illustration for train speed range \(\Delta v_{2}\) = 100 km/h \(\ldots\) 200 km/h. Again, both the bridge structure and the examined car body experience their most prominent acceleration peak slightly below \(v_{\text{crit,1}}\) at 276 or 277 km/h (\(\approx\) 97% \(v_{\text{crit,1}}\)). In the case of the bridge structure (Figure 8c), this peak occurs again close to the first bending frequency, \(n_{1}\). Furthermore, the bridge's resonance frequency is also recognizable in the numerically obtained calculation results of bridge accelerations in the cropped train speed range \(\Delta v_{2}\) at again 138 km/h (\(\approx\) 97% \(v_{\text{crit,2}}\)), see Figure 8d, similar to the bogie frame accelerations, see Figure 7b. The car bodies' FFT results illustrated in Figure 8a,b, however, appear to be influenced by the bridge structure's natural frequency, particularly very close to the critical speeds \(v_{\text{crit,1}}\) and \(v_{\text{crit,2}}\), but are generally at significantly lower frequencies between 0 and 4 Hz, while they also scatter stronger than the results of the bridge structure and bogie frames. Figure 9 further illustrates the close relationship between the FFT results of the bridge structure's and the bogie frame's accelerations. It features the planar illustration of the FFT results of both, but they are, unlike in Figures 7 and 8, normalized to the maximum results at each train speed step. Additionally, the speed-dependent excitation speeds are marked with orange dashed lines according to Equation (11) and \(i\) = 1, 2,..., 6. Figure 8: Planar illustrations of FFT analysis results for selected speed ranges: (**a**) car body accelerations at \(\Delta v_{1}\) = 60 km/h... 420 km/h; (**b**) car body accelerations at train speed range \(\Delta v_{2}\) = 100 km/h... 200 km/h; (**c**) bridge structure accelerations at midspan at train speed range \(\Delta v_{1}\) = 60 km/h... 420 km/h; (**d**) bridge structure accelerations at midspan at train speed range \(\Delta v_{2}\) = 100 km/h... 200 km/h. The illustrations indicate that there is, in principle, very similar behavior of both frequency components, i.e., at the same crossing speeds of the train, similar frequency components dominate in the vibration response of the bridge structure or the regarded bogie frame. However, these are strongly dependent on the excitation frequency and tend to agree unambiguously with the structure's natural frequency only in the resonance case. In the case of both the bridge structure and the bogie frame, the frequency components due to the excitation frequencies that are not very close to the natural frequency of the structure \(n_{1}\) are much less pronounced between approximately 120 and 155 km/h--between the two cancellation speeds \(v_{\text{canc,3}}\) and \(v_{\text{canc,4}}\) and in the vicinity of the critical resonance speed \(v_{\text{crit,2}}\)--than in the rest of the speed range considered. In the case of the bogie frame accelerations, the respective maximum acceleration peaks in the frequency range appear as broader red or orange areas in the planar representation of Figure 8 and are, thus, less prominent than in the case of the bridge structure. In addition, a significant proportion of the velocity range also shows pronounced frequency components at 0 to 1.5 Hz, which are close to the natural frequency of the car bodies.