## 4 Case Study
### Mode Pairing  
A dynamic test, described in detail in [4], was performed based on a set of 14 accelerometers distributed along the floor of the carbody. The tests allowed for the identification of 5 experimental vibration modes, depicted in Figure 7, which clearly correspond to the numerical modes 1C to 5C previously presented in Figure 4. In Figure 7, for each vibration mode, the respective natural frequency and Mode Complexity Factor (MCF) [38] are also indicated. Figure 6: Rayleigh damping curve. Figure 7: Experimental carbody modes. To demonstrate the performance of the EMAC in automatically pairing these experimental modes to their numerical correspondents, the vehicle's FE model was divided into the four clusters presented in Figure 8. The carbody clusters, namely the floor, walls and roof, where each split into sub-clusters containing only the degrees of freedom associated with the vertical (\(y\)) and transverse (\(z\)) directions, respectively. Accordingly, seven separate clusters were obtained in total. In the \"other elements\" cluster, the remaining degrees of freedom of translation and rotation of the numerical model were included, encompassing, among others, the bogies and the seats. For each of these clusters, the relative modal strain energy (see Figure 9) was calculated through Equation (29) considering 80 vibration modes, obtained through a numerical modal analysis taking the damping into account. Figure 10 presents the MAC and EMAC correlation matrices between the 80 numerical modes and the 5 experimentally identified modes (see Figure 7). In this Figure, the 80 numerical modes correspond to 40 complex-conjugate pairs. The EMAC values for the 2C, 3C and 5C modes were obtained by weighting the MAC by the modal strain energy of the clusters 1, 3 and 5 (see Figures 8 and 9). The EMAC values for the 1C and 4C modes resulted of the weighting of the MAC values by the modal strain energy of the clusters 1, 4 and 6 (see Figures 8 and 9). Cluster 1 is representative of the pos  
Figure 8: Identification of the clusters used in the numerical model of the BBN vehicle. Figure 9: Relative MSE values for the different clusters and vibration modes, obtained based on the numerical model of the BBN vehicle. and the measurement direction used in the dynamic test of the carbody, since the sensors were installed on the vehicle's floor. The use of other clusters in the weighting of the MAC values, particularly in the rigid-body modes, allowed to highlight the interrelationship between the degrees of freedom of the base, walls and roof in the \(y\) and \(z\) directions. As can be clearly seen, the EMAC matrix is significantly cleaner compared to the MAC matrix. The application of the EMAC allowed for the proper pairing the between experimental and the numerical modes, as indicated by the little green arrows in Figure 10b. On the other hand, the use of the MAC parameter resulted in a high level of correlation  
Figure 10: Pairing of experimental and numerical modes of the BBN vehicle based on the (**a**) MAC and (**b**) EMAC parameters. between the experimental modes and several numerical modes, which made it impossible to establish a proper pairing between experimental and numerical modes. The EMAC parameter facilitated the pairing of the 1C and 5C experimental modes, especially. Figure 11 shows two numerical vibration modes, modes 7 and 50, which can be paired with experimental modes 1C and 5C, respectively, and which the use of the EMAC parameter allowed to exclude. Mode 7 is a transverse rotation mode of both bogies. It causes small-amplitude transverse-rotation movements of the box. Mode 50 is a global bending mode with local-box base movements. It is also important to highlight the significant correlation between vibration modes 1C and 4C. As can be seen from Figure 7, due to the limited number of points where the experimental information was available, the deformed modal configurations of these two modes, from the perspective of the instrumented points, are very similar.