## 4 Case Study
### Numerical Model  
The numerical model of the vehicle (Figure 2) was developed based on shell, beam and spring-dashpot assemblies. Particularly, the beam elements were used for modeling the bogies, and the spring-dashpot assemblies were used to simulate the suspensions, the connecting rods and the tilting system. The shell elements were used to model the floor, the roof and the wall panels. The thickness of the shell elements was considered to match the cross-sectional area with one of the real panels. The real panels, however, are formed by an upper and lower plate connected by diagonal plates, thus presenting an orthotropic behavior to bending. To adequately represent this behavior, the inertia of the shell elements was corrected by the RMI (Ratio of the Bending Moment of Inertia) [37] parameter given by the ratio between the real inertia of the panel and that given by the shell element. Figure 1: BBN vehicle: (**a**) perspective; (**b**) elevation; (**c**) floor plan. The main parameters of the numerical model are depicted in Table 1, according to the information provided by the manufacturer of the BBN vehicle. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Designation** & \multicolumn{2}{c}{**Adopted Value**} & **Unit** \\ \hline \multicolumn{5}{c}{**Carbody**} \\ \hline \(K_{S1}\) & Vertical secondary suspension stiffness & Front bogie & & \\ \hline \(K_{S2}\) & \multicolumn{2}{c}{suspension stiffness} & Rear bogie & 256.4 & kN/m \\ \hline \(c_{S}\) & \multicolumn{2}{c}{Vertical secondary suspension damping} & 35 & kNm/s \\ \hline \(K_{ST}\) & \multicolumn{2}{c}{Transverse secondary suspension stiffness} & 2500 & kN/m \\ \hline \(c_{ST}\) & \multicolumn{2}{c}{Transverse secondary suspension damping} & 17.5 & kNm/s \\ \hline \(K_{pend}\) & \multicolumn{2}{c}{Rigidity of the pendulum system} & 0 (at rest) & kN/m \\ \hline \(c_{AL}\) & \multicolumn{2}{c}{Anti-hunting suspension damping} & 400 & kNm/s \\ \hline \(K_{b}\) & Stiffness of the tilting bolster–load bolster connection rod & 20,000 & kN/m \\ \hline \(\Delta_{alumn}\) & \multicolumn{2}{c}{Aluminum density} & 2700 & kg/m\({}^{3}\) \\ \hline \(E_{alumn}\) & \multicolumn{2}{c}{Aluminum deformability module} & Dir x & 70 & GPa \\ \hline \(RMI_{b}\) & \multicolumn{2}{c}{Dri z} & 54.2 & GPa \\ \hline \(RMI_{p}\) & \multicolumn{2}{c}{Floor} & 90 & - \\ \hline \(RMI_{p}\) & \multicolumn{2}{c}{Corrective factor of the moment of inertia} & Walls & 114 & - \\ \hline \(RMI_{c}\) & \multicolumn{2}{c}{Rof} & 386 & - \\ \hline \(\Delta M_{b}\) & \multicolumn{2}{c}{Floor} & 70 & \% \\ \hline \(\Delta M_{p}\) & \multicolumn{2}{c}{Additional mass} & Walls & 20 & \% \\ \hline \(\Delta M_{c}\) & \multicolumn{2}{c}{Rof} & 10 & \% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the BBN vehicle’s numerical model. Figure 2: Overview of the BBN vehicle’s numerical model. Additionally, concentrated mass elements were used to incorporate the mass of some non-structural components and equipment at the wagon's floor and at specific locations distributed along the bogies. The adequate positioning of these elements is essential for accurately representing the vehicle's modal behavior. The positioning and corresponding values of these masses are depicted in Figure 3.