## 5 Conclusions  
This paper presents a novel approach to the problem of automatic pairing complex vibration modes through the expansion of the MAC criterion weighted by the modal strain energy (EMAC), for its application in problems involving complex mode shapes. To enable the application of the EMAC to problems with complex modes, an expression for the modal strain energy was derived on the basis of the orthogonality conditions of a state-space formulation. The derived formulation allows to quantify the relative modal strain energy, used for the weighting of the MAC, in cases where the orthogonality condition between the modes and the stiffness matrix is not satisfied. Consequently, this made it possible to expand the criterion to more general applications. Subsequently, the effectiveness of the proposed criterion was demonstrated through a case study involving the pairing of modes of a BBN-type passenger vehicle. A detailed numerical model of the vehicle was developed in ANSYS(r) based on shell, beam, spring-dashpot assemblies and concentrated mass elements. Particularly, the shell elements were used to represent the panels of the carbody, the beam elements were used to model the bogies, the spring-dashpot assemblies were used to model the suspension components, and the mass elements were used to represent the non-structural equipment and components of the vehicle. The localized damping effect introduced by the dampers led to a non-proportional damping matrix and, consequently, to complex modes. A numerical modal analysis was performed, in which it was possible to identify the main rigid-body modes of the carbody in addition to the first structural modes associated  
Figure 11: Numerical vibration modes likely to pair with (**a**) experimental mode 1C, and (**b**) experimental mode 5C. with the bending and torsional movements of the carbody. In addition to these global modes, it was possible to identify several local modes, mainly for bending the carbody panels. These modes presented global components of small amplitude with a format very similar to some of the global modes, which imposes extra difficulties in the pairing process. The numerical modes were paired with five experimental modes, from which information was available on the modal ordinate amplitudes of the 14 points that were instrumented on the vehicle floor. To demonstrate the efficiency of the proposed criterion, the pairing process was performed by the MAC and EMAC, with the aim of comparing the results obtained. After applying both criteria, the EMAC proved to be much more assertive in establishing the correct correspondences between numerical and experimental modes, even in challenging situations, due to the reduced number and positioning constraints of the sensors and the complexity of the numerical model. The weighting by the modal deformation energy, used in the EMAC, was able to significantly reduce the erroneous correspondences between the experimental global modes and the purely local modes verified when applying the MAC. In addition, even more assertive results are expected in situations with a higher number of sensors and larger spatial distribution. In conclusion, the new formulation proposed for the application of the EMAC criterion to complex modes has proved to be very promising and represents an advance for future applications involving the updating of numerical models in the presence of non-proportional damping conditions. Conceptualization, D.R. ; Methodology, M.B. and V.Z. ; Software, D.R. and C.B. ; Validation, D.R. ; Writing--original draft, D.R. ; Writing--review & editing, C.B., M.B. and V.Z. ; Supervision, R.C. All authors have read and agreed to the published version of the manuscript. This work was financially supported by Base Funding--UIDB/04708/2020 and Programmatic Funding--UIDP/04708/2020 of the CONSTRUCT--Instituto de I&D em Estruturas e' construcoes, funded by national funds through the FCT/MCTES (PIDDAC). The authors also wish to express their gratitude to Eng.