# Identification of Damage in Beams by Modal Curvatures Using Acoustic Beamformers
## 5 Conclusions  
This paper presented an approach to damage identification based on the use of modal curvatures as an observed response quantity. Three different estimators were compared: the classical objective function, based on comparison between numerical and experimental measurements, and two beamformers, which are Bartlett and MVDR, based on projection of measurements onto the replica vector. To date, these two beamformers have primarily been applied to inverse problems concerning source identification in acoustics. Instead, we applied them to structural vibration problems in beams. Modal curvatures are highly reputed for being locally sensitive to damage, especially at an early stage. In fact, it was made evident in this paper that the use of curvatures also has some drawbacks. These include small variations away from the damaged area, and the use of accelerometers in measurements, to enable curvature normalization and subsequent comparison between analytical and experimental results. We considered cases of free-free beams with different damage depths and locations (i.e., below a sensor and in between two sensors), both numerical and experimental. In all cases, the pattern of experimental curvatures was quite similar to the numerical one, although the experimental variations below the cut appeared to be smaller than the numerical ones, similar to what was observed for frequency variation. This hints at a possible overestimation of the effect of damage with the model presently in use. Figure 19: Contour plots of the estimators for damage scenario D4.B (**(a)** objective function, **(b)** Bartlett, **(c)** MVDR) (circle: correct damage parameters; cross: identified parameters). Figure 18: Contour plots of the estimators for damage scenario D3.B (**(a)** objective function, **(b)** Bartlett, **(c)** MVDR) (thick circle: correct damage parameters; thin concentric circles: identified parameters). All the three processors provided a unique solution even when using one single modal curvature, both using experimental and pseudo-experimental data. All the processors showed high sensitivity to the damage location but lower sensitivity to the damage depth, with MVDR showing sensitivity regarding both parameters. This is particularly relevant for the early-stage identification of damage. It was also evident that MVDR can determine the damage depth and location with an accuracy comparable to methods based on frequency variation. Overall, the MVDR beamformer appeared as the most robust estimator among those investigated. It provides a clear and stable indication of damage parameters, with very limited sensitivity to error, enabling to identify damage by employing modal curvatures. It then gives the opportunity to take advantage of the strengths of this modal quantity, among which are its limited sensitivity to environmental factors, and high local sensitivity to damage. This property is, however, twofold. In fact, on the one hand, it ensures large and easy-to-measure changes; on the other hand, it requires sensors in close proximity to the damage. Also in this regard, the MVDR beamformer has proven its effectiveness in the worst-case scenario of damage located in between two sensors. Issues that will have to be addressed in future work concern the minimum number of measurements needed to obtain a unique solution, more complex damage scenarios, including more than one notch, as well as a more detailed damage description, for instance, including the width. Conceptualization, A.P. ; Methodology, A.P. ; Software, U.E. ; Writing--original draft, A.P. ; Writing--review & editing, U.E. ; Visualization, U.E. All authors have read and agreed to the published version of the manuscript. This work was initiated when [PERSON] was Visiting Professor at the Department of Structural and Geotechnical Engineering of Sapienza University of Roma, whose support under the grant 'Professori visitatori 2020' CUP B82F20001090001 is gratefully acknowledged. Data will be made available under request. The authors declare no conflict of interest.