## 3 Dynamic Calculation of Bridge and Train Vibrations  
### Analyses of Considered Steel Bridge in Time Domain  
The dynamic system's vibrations are numerically calculated considering the speed range of the trains between 60 and 420 km/h, discretized in 1 km/h steps. The coupled systems of equations of motion of bridge structure and train are numerically integrated using the pre-implemented differential equation solvers in MATLAB 2022 [50], specifically with the algorithm ode15 s for stiff problems, which is based on the numerical differentiation formulas (NDFs), see for instance in ref. [51]. Critical and, therefore, design-relevant accelerations of the bridge mainly occur when the external excitation of the crossing train causes resonance vibrations. The characteristic resonance speeds \(v_{\text{crit},i}\), at which the excitation speed of regularly spaced axle loads with the spacing \(s=17.5\) m being defined as the car length of the considered vehicle, see Figure 2, equals the resonance frequency \(n_{d,1}\) of the examined bridge structure, can be determined as follows:  
\[v_{crit,i}=\frac{s\cdot n_{d,1}}{i}\ \,\ i=1,\ 2,\ \ldots \tag{5}\]The bridge vibration responses induced by a crossing load can also be canceled out and thus feature significantly lower acceleration amplitudes than at other train speeds when the span- and natural frequency-dependent cancellations speeds \(v_{\text{canc},i}\) according to [52] are reached:  
\[v_{canc,i}=\frac{2\cdot L\cdot n_{d,1}}{2i-1}\;\;,\;i=1,\;2,\;\ldots \tag{6}\]  
Resonance accelerations induced at critical speeds \(v_{\text{cnt},i}\) close to a cancellation speed \(v_{\text{canc},i}\) are likely to be suppressed by the dominating cancellation effect.  
From the dynamic calculations performed, the bridge accelerations at midspan, and the bogie frame and car body accelerations in proximity to three train couplings at the front (bogie frame \(b2\)--wheelset \(w3\) and \(w4\)), end (bogie frame \(b11\)--wheelset \(w21\) and \(w22\)) and one middle (bogie frame \(b6\)--wheelset \(w11\) and \(w12\)) Jacobs bogie are read out as computation results. The bogie frame and car body vibrations are evaluated at the same distance from the bogie axes as in the in situ measurements (exemplarily shown in Figure 2), which means that the acceleration results include vibration components from vertical accelerations at each body's center of gravity and rotational accelerations as defined below:  
\[\ddot{w}_{\text{c},\text{k,near/front}}=\ddot{w}_{\text{c},k}\pm\left(\frac {d}{2}\pm s_{\text{c},k}-1.04m\right)\ddot{\varphi}_{\text{c},k} \tag{7}\]  
for the car body, and  
\[\ddot{w}_{\text{b},k,\text{near/front}}=\ddot{w}_{\text{b},k}\pm\left(\frac {b}{2}-0.30m\right)\ddot{\varphi}_{\text{b},k} \tag{8}\]  
for the bogie frame. Figure 3 shows exemplarily the numerically obtained acceleration results \(\ddot{w}_{\text{b},k,\text{front}}\) and \(\ddot{w}_{\text{b},k,\text{near}}\) of the three instrumented Jacobs-bogie frames (in black \(b11\), in blue \(b6\), and in red \(b2\)) at the location of the installed sensors (see Section 4.1) in the time domain. These time-dependent accelerations result from a calculation performed for one particular train speed \(v=135\) km/h and the Salzburg driving direction, i.e., the instrumented bogie \(b11\) enters the bridge first. The calculated bridge accelerations at midspan are underlaid as a grey curve for better clarity. The highlighted time ranges \(b11\), \(b6\), and \(b2\) denote the time series for the subsequent applied analyses in the frequency domain. It includes the period of time for both wheelsets of the considered bogie and the first wheelset of the subsequent bogie to enter and leave the bridge (cf. Figure 2).