## 3 Dynamic Calculation of Bridge and Train Vibrations
### Analyses of Considered Steel Bridge in Frequency and Train Speed Domain  
A fast Fourier transform (FFT) analysis of the main acceleration results is performed for each calculated train speed step. Figure 4 depicts the results obtained with the FFT analysis of the calculated acceleration results in the frequency domain for the considered steel bridge at mid-span and at the quarter points. The diagram on the left illustrates the total considered  
Figure 3: Calculated acceleration results at \(v=135\) km/h: bridge and bogie frame accelerations in time-domain. frequency range from 0 to 50 Hz, and the diagram on the right shows the frequency range from 0 to 10 Hz in more detail. It should be noted that the amplitudes in the FFT illustrations in Figure 4 and further do not directly correspond to the calculated amplitudes in the time domain, which result from the superposition of all frequency proportions. The bridge's resonance frequency \(n_{d,1}\)--the measured first bending mode of the considered bridge, see Table 1 and Section 4.2, is marked in Figure 4 at 4.52 Hz, as well as the range of natural frequencies of the bogie frames \(n_{\rm b}\) and car bodies \(n_{\rm c}\), estimated for each train coach with the bogie frame mass \(m_{\rm b}\), car body mass \(m_{\rm c}\), and primary and secondary suspension stiffnesses \(k_{\rm p}\) and \(k_{\rm s}\) according to Equations (9) and (10), as given in ref. [53]:  
\[n_{\rm b}=\frac{1}{2\pi}\sqrt{\frac{2\,k_{\rm p}}{m_{\rm b}}} \tag{9}\]  
for the bogie frames, and  
\[n_{\rm c}=\frac{1}{2\pi}\sqrt{\frac{2\,k_{\rm s}}{m_{\rm c}}} \tag{10}\]  
for the car bodies. The dashed orange lines denote the excitation frequencies \(n_{\rm train,i}\) resulting from the regularly spaced axle loads. The primary excitation is caused by the superimposed action of both wheels of one Jacobs bogie in the distance of \(s=17.5\) m to the next bogie. The resulting excitation frequencies can be calculated as follows:  
\[n_{\rm train,i}=\frac{v\cdot i}{s}\ \,\ i=1,\ 2,\ldots \tag{11}\]  
Figure 4 shows that all relevant frequency shares of the structural accelerations are caused by the excitation frequencies calculated with \(i=1\) to \(i=8\) and calculated with \(i=18\) to \(i=21\), according to Equation (11). With this, the frequency which is closest to the first bending frequency \(n_{1}\) (\(\approx n_{d,1}\)) of the considered steel bridge causes the most prominent peak, illustrating the proximity of the examined train speed of \(v=135\) km/h to the bridge structures resonance speed \(v_{\rm crit,2}=17.5\) m \(\cdot\) 4.52 Hz/2 = 39.55 m/s = 142.4 km/h according to Equation (5). Also, the frequency shares close to the second and third natural frequencies \(n_{d,2}\) (\(\approx n_{\rm train,8}\)) and \(n_{d,3}\) (\(\approx n_{\rm train,19}\)) are slightly higher than those of the adjacent excitation frequencies. Furthermore, the contribution of the train masses in the system of equations of the dynamic multi-body system of the bridge structure and crossing train increases the modal masses of the bridge. This leads to a slightly lower resulting resonance frequency of the  
Figure 4: FFT analysis results from train-induced bridge vibrations at \(v=135\) km/h in the frequency domain for the considered steel bridge. bridge structure if the train is located on the bridge compared to the unloaded bridge (e.g., described in ref. [54]). Since the FFT analysis of the bridge vibrations only included the period during the train passage and not the free decay process, the results show mainly frequency shares due to the external excitation frequencies, making it challenging to identify the bridge structure's natural frequency. Generally, the more pronounced the acceleration peak close to \(n_{1}\) becomes, the closer the excitation frequency is to the natural frequency of the bridge structure.