## 2 Dynamic Modeling of Bridge Structure and Crossing Train
### Mechanical Model of the Bridge Structure  
The bridge structure is modeled as a simply-supported single-span girder (a shear-rigid Euler-Bernoulli beam) with a span length of \(L\), where the beam's properties summarize all structure properties, including the track. The load-distributing effect of the track itself and the interaction dynamics between the structure and the rails, e.g., possible energy dissipation capacities and the restraining effect at the ends of the structure, are neglected. The beam's bend line \(w_{\text{B}}\)(\(x\),\(t\)) can be approximated by modal analysis for \(n\) considered vibration modes as follows:  
\[w_{\text{B}}(x,t)\approx\sum_{i=1}^{n}q_{i}(t)\cdot\phi_{i}(x),n=5. \tag{1}\]  
Five vibration modes with corresponding shape functions \(\phi_{i}(x)=\sin(i\ \pi\ x/L)\) and generalized displacements \(q_{i}(t)\) are considered in the computational investigations. The system of linear equations of motion for the generalized displacements \(q_{i}(t)\) of the bridge structure then follows:  
\[M_{\text{B}}\ddot{q}+C_{\text{B}}\dot{q}+K_{\text{B}}q=p_{\text{B}}. \tag{2}\]  
The modal mass, damping, and stiffness matrices \(M_{\text{B}}\), \(C_{\text{B}}\), and \(K_{\text{B}}\) on the left side of Equation (2) contain information on the bridge's span \(L\), bending stiffness \(EI\), mass distribution per unit length \(\mu\), and Lehr's damping factor \(\zeta\) (by applying Rayleigh damping), the latter two including the contribution of the (in the case of the investigations described) ballasted track. This mechanical model allows only consideration of vertical bending deformations due to the generalized external load vector \(p_{\text{B}}\). Shear deformations and horizontal and torsional vibrations are omitted; the latter would require applying a three-dimensional model of the bridge structure. As previously stated, verifying the compliance of occurring vertical bridge deck accelerations with the normative limits is mandatory for new and existing constructions, whereby the accelerations often become a decisive criterion. This article focuses on the dynamic assessment of a specific existing steel deck bridge in Austria (see Sections 3.1 and 4.1). Its properties, on which the following evaluations are based, are summarized in Table 1. The resonance frequency \(n_{d,1}\) of the first bending mode, and the structural damping factor \(\zeta\) were determined by executing in situ measurements described in the following Section 4.2. In the first step, the bending stiffness \(EI\) was estimated from the structure's cross-section using technical drawings and additional geometric site measurements. The bending stiffness was adequately adjusted in the second step to obtain a good agreement between the measured calculated first resonance frequency and the calculated frequency. Finally, the mass distribution per unit length \(\mu\) was recalculated with the relation \(\mu=EI\left(\frac{\pi}{2\ \eta_{1}\ L^{2}}\right)^{2}\), which can be applied to simply-supported beam structures, assuming the natural frequency \(n_{1}\) is approximately the resonance frequency \(n_{d,1}\) for small structural damping factors of \(\zeta<10\%\). The comparison of \(\mu\) with the mass distribution \(\mu\)', which denotes only the mass of the steel structure and additional mounted elements, e.g., the cantilever steel construction with fastened noise barrier, allows an estimation of the mass distribution of the track \(\mu_{\text{track}}=3601\) kg/m (rails, sleepers, and ballast bed). Assuming a mass distribution of 120 kg/m for the rails, 300 kg of concrete sleepers every 0.6 m (corresponding to 500 kg/m), and a ballast bed cross-section of 2.07 m\({}^{2}\) (at a ballast bed height of 0.55 m of the standardized superstructure in Austria), this results in an approximate ballast density of 1440 kg/m\({}^{3}\).