## 3 Dynamic Calculation of Bridge and Train Vibrations
### Analyses of Considered Steel Bridge in Frequency and Train Speed Domain
If excitation frequencies and natural frequencies of the bridge structure differ more than in the example shown here, further analysis methods, for instance, a magnitude-squared coherence analysis of the output signal (the structural acceleration) and the input signal (the external excitation by the trains axle loads) can help to identify the natural frequency reliably from the calculation results (for example, described in ref. [55]). Figure 5 shows the FFT results (limited from 0 to 10 Hz) obtained for the acceleration results at \(v=135\) km/h of the three selected bogie frames mentioned above (Figure 5a), and the adjacent car bodies, their front and rear end, respectively (Figure 5b). It can be observed that the bogie frame vibrations feature a large frequency proportion in proximity to \(n_{1}\), while the majority of frequency shares of the car bodies lie significantly below the resonance frequency of the bridge. The influence of the excitation frequencies appears to be smaller than in the case of the bridge accelerations, whereas there is a discernible influence of the natural frequency ranges of the car bodies at approximately \(n_{\rm c}=0.76\) to 1.0 Hz and the bogie frames at \(n_{\rm b}=4.7\) to 5.4 Hz on the acceleration shares of both train components. The investigations of the calculated train-induced acceleration results of the considered bridge structure and selected bogie frames and car bodies of the train shown in Figures 4 and 5 are subsequently carried out for all calculated train-speed steps. These FFT analysis results are exemplarily displayed in Figure 6 for the accelerations \(\ddot{w}_{\rm b,2,front}\), the vertical vibrations at the front of the bogie \(b2\) which enters the bridge structure analogous to the executed in situ measurements (see Section 4.3) second to last, as a three-dimensional illustration for the entire range of considered train speeds \(\Delta v_{1}=60\) km/h \(\ldots\) 420 km/h, cropped to a frequency range of 0 Hz \(\leq n\leq 10\) Hz. The illustrations in Figure 5a correspond to one selected profile of FFT results at \(v=135\) km/h, as also shown in Figure 6b. Figure 5: Selected FFT analysis results at \(v=135\) km/h in the frequency domain: (**a**) bogie frame accelerations; (**b**) car body accelerations. Both subfigures of Figure 7 illustrate the same results in the train speed/frequency plane, whereby the colors representing the acceleration amplitudes are scaled to the maximum result for the considered speed ranges \(\Delta v_{1}\) (Figure 7a), and a more detailed investigation in \(\Delta v_{2}\) = 100 km/h \(\ldots\) 200 km/h respectively (Figure 7b). Again, the natural frequencies of the considered bridge \(n_{1}\), the bogie frames \(n_{\text{b}}\), and the car bodies \(n_{\text{c}}\) are marked in Figures 6b and 7, as well as the characteristic resonance speeds \(v_{\text{crit},i}\), at which the excitation speed of the regularly spaced axle loads with a spacing of \(s\) = 17.5 m (defined as car length, see Figure 2) equals \(n_{1}\), and the primary cancellation speeds \(v_{\text{canc},i}\), at which bridge vibrations induced by crossing loads can be canceled out, according to [52] and Equation (6). The most prominent peaks in Figures 6a and 7 emerge close to critical speeds \(v_{\text{crit},1}\) at 277 km/h (\(\approx\)97% \(v_{\text{crit},1}\)) and \(v_{\text{crit},2}\) at 138 km/h (\(\approx\)97% \(v_{\text{crit},2}\)), where, as described above, the slight undercutting of the critical speeds can be explained by the increase in the modal mass of the bridge due to the consideration of the unsprung wheelset masses of the multi-body model of the train. While the prominent peak at 277 km/h is associated with frequency shares closer to the bogie frames' natural frequencies \(n_{b}\), the second highest peak at 138 km/h appears, as seen in Figure 6, very close to the bridge structure's first bending frequency, \(n_{1}\).