## 4 Case Study
### Numerical Model
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Designation** & \multicolumn{2}{c}{**Adopted Value**} & **Unit** \\ \hline \(c_{bas}\) & & Floor & 10.2 & mm \\ \hline \(c_{par}\) & Equivalent thickness & Walls & 10.3 & mm \\ \cline{2-5} \(c_{cvb}\) & & Roof & 8.8 & mm \\ \hline \multicolumn{5}{c}{**Bogies**} \\ \hline \(K_{p}\) & Primary suspension stiffness & 564 & kN/m \\ \hline \(c_{p}\) & Primary suspension damping & 18 & kNm/s \\ \hline \(K_{bls}\) & Axle-box connecting rod stiffness & Top & 6.5 & MN/m \\ \hline \(K_{bli}\) & \multicolumn{2}{c}{Bottom} & 25 & MN/m \\ \hline \(K_{rc}\) & \multicolumn{2}{c}{Stiffness of the wheel–rail contact} & \(1.5674\times 10^{9}\) & N/m \\ \hline \(\Delta M_{lc}\) & & \begin{tabular}{c} Girder \\ (central zone) \\ \end{tabular} & 42 & kg/m \\ \hline \(\Delta M_{la}\) & Additional mass & \begin{tabular}{c} Girder \\ (extremities) \\ \end{tabular} & 38 & kg/m \\ \hline \(\Delta M_{t}\) & & \
\begin{tabular}{c} Crossmember \\ \end{tabular} & 92 & kg/m \\ \hline \(\Delta M_{e}\) & &
\begin{tabular}{c} Axles \\ \end{tabular} & 271 & kg/m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 3: Mass elements in the finite element model: **(a)** carbody; **(b)** bogie. Figure 4 depicts the modal configurations and natural frequencies of the vehicle's carbody. Among the rigid-body modes, 1C is a rotation about the \(x\) axis, 2C is a translation along the \(y\) axis, and 3C is a rotation about the \(z\) axis. Due to the relevant contribution of the localized damping provided by the suspension elements, these rigid-body modes presented a high degree of complexity. Among the deformation modes, 4C and 6C are, respectively, the first and second torsional modes, and 5C is the first bending mode. Several local modes involving the bending of elements of the box, in particular the base, walls and roof, were also identified. As an example, Figure 5 illustrates the first two local modes (1L and 2L) that involve bending movements of the base of the carbody, with natural frequencies of the damped system equal to 8.85 Hz and 9.63 Hz, respectively. As can be seen in Figure 5, despite being clearly local, modes 1L and 2L contain global displacement components, which might be confused with global modes and create extra difficulties for an automatic mode-pairing algorithm. Particularly, in Mode 1L, these global movements are very similar, in a smaller scale, to the first global bending mode (5C). Figure 4: Numerical modes of the BBN vehicle’s carbody. Figure 5: Numerically obtained local vibration modes of the housing. The vehicle damping matrix was constructed as the sum of a Rayleigh damping matrix and the matrix resulting from the scattering of the matrices of the elements with localized damping, particularly the primary, secondary and anti-hunting dampers. The addition of the matrices from the localized dampers to the Rayleigh damping matrix makes it nonproportional to the stiffness and mass matrices, which results in complex modes. The Rayleigh constants were calculated setting damping coefficients equal to 2% for the 4C and 6C vibration modes. The evolution of the damping coefficient as a function of frequency for the Rayleigh damping portion is graphically represented in Figure 6.