## 2 Dynamic Modeling of Bridge Structure and Crossing Train
### Mechanical Model of Train  
The dynamic excitation is generated by the articulated train moving over the bridge at a constant speed. It is modeled as a two-dimensional multi-body system of eleven coaches interconnected through Jacobs bogies, see Figure 2. Two driving directions were considered, hereafter referred to as driving directions toward Vienna or Salzburg. It is noted that the first car of the considered vehicle in the Vienna driving direction is always denoted \(c1\). Therefore, depending on the driving direction of the train, the order in which the cars and bogies arrive on the bridge changes, i.e., in the case of the Vienna train driving direction, car \(c1\) and bogie \(b1\) arrive first, and in the case of the Salzburg driving direction, car \(c11\) and bogie \(b12\) arrive first on the bridge. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **\(\mathbf{L}\) [m]** & \(\mathbf{EI}\) [Nm\({}^{2}\)] & \(\mathbf{\zeta}\) [\%] & \(\mathbf{n_{d,1}}\) [Hz] & \(\mathbf{\mu}\) [kg/m] & \(\mathbf{\mu}\) [kg/m] \\ \hline
1 & 11.0 & 5.70 \(\times\) 10\({}^{9}\) (\({}^{3}\)) & 1.63 (\({}^{2}\)) & 11.51 (\({}^{1}\)) & 7253 (\({}^{1}\)) \\
2 & 5.0 & 1.64 \(\times\) 10\({}^{9}\) (\({}^{3}\)) & 2.55 (\({}^{2}\)) & 23.17 (\({}^{1}\)) & 12,040 (\({}^{1}\)) \\ \hline \hline \end{tabular} \({}^{(1)}\) from analyses of the catalog of existing bridges [39; 44; 45]; \({}^{(2)}\) from EN 1991-2:2003 [46]; \({}^{(3)}\) recalculated from the relation: \(EI=\mu\left(\frac{2\,n_{1}L^{2}}{\pi}\right)^{2}\).  
\end{table} 
Table 2: Bridge properties of two additional fictitious bridge structures for dynamic calculation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**\(\mathbf{L}\) [m]** & \(\mathbf{EI}\) [Nm\({}^{2}\)] & \(\mathbf{\zeta}\) [\%] & \(\mathbf{n_{d,1}}\) [Hz] & \(\mathbf{\mu}\) [kg/m] & \(\mathbf{\mu}\) [kg/m] \\ \hline
33.3 (\({}^{1}\)) & 73.18 \(\times\) 10\({}^{9}\) (\({}^{1}\)) & 1.88 (\({}^{2}\)) & 4.52 (\({}^{2}\)) & 7188 (\({}^{3}\)) & 3587 (\({}^{1}\)) & 3601 \\ \hline \hline \end{tabular} \({}^{(1)}\) from planning documents and in situ geometric cross-section measurements; \({}^{(2)}\) from in situ vibration measurements (forced vibration tests with shakers, bandwidth analysis), see Section 4.2; \({}^{(3)}\) recalculated from the relation: \(\mu=EI\left(\frac{\pi}{2n_{1}L^{2}}\right)^{2}\) assuming \(n_{1}\approx n_{\text{d,1}}\) for \(\zeta<\) 10\(\%\).  
\end{table} 
Table 1: Bridge properties of the considered existing steel deck bridge required for dynamic calculation. Car bodies, bogie frames, and wheelsets are modeled as rigid bodies with mass inertia and one vertical and one rotational degree of freedom (DOF) each (besides the wheelsets with only vertical DOFs). All train components are coupled with linear spring-damper elements representing the train's primary and secondary suspension system. In addition to Equation (2), the following second equation of motion for the kinematics of the vehicle multi-body system must be solved in each time step:  
\[\mathbf{M}\mathbf{v}\ddot{\mathbf{u}}+\mathbf{C}\mathbf{v}\dot{\mathbf{u}}+\mathbf{K} \mathbf{v}\mathbf{u}=\dot{\mathbf{v}}+\mathbf{F}\mathbf{w}. \tag{3}\]  
The properties of the car bodies, bogies, and primary and secondary suspensions (see Figure 1) are included in the mass, damping, and stiffness matrices \(\mathbf{M}\mathbf{V}\), \(\mathbf{C}\mathbf{v}\), and \(\mathbf{K}\mathbf{v}\). The vector \(\mathbf{u}\) and its time derivatives \(\dot{\mathbf{u}}\) and \(\ddot{\mathbf{u}}\) consist of the previously mentioned DOFs of all trains' car bodies and bogie frames. The DOFs of the wheelsets, i.e., their vertical displacement \(w_{\mathbf{W}j}(t)\) at any time \(t\) of the train crossing (cf. Figure 2), can be extracted from the equation of motion of the train by equating them with the vertical bridge displacement \(w_{\mathbf{B}}(x_{j},t)\) at the respective wheel contact point.