# Identification of Damage in Beams by Modal Curvatures Using Acoustic Beamformers
## 1 Introduction  
Non-destructive damage evaluation gathers several techniques based on the observation of structural response following applied stimuli. The tests that are presently in use can be of a static [1, 2] or dynamic nature, employing modal analysis. Over the years, the detection and identification of damage in structures by the inspection of modal quantities has proven to be a very effective strategy for structural health monitoring [3]. Indeed, the literature in this field covers a wide range of applications, ranging from mechanical to aerospace and civil structures. Applications to rotating machinery, roller bearings, large-span bridges, aqueducts and monuments are present [4, 5, 6, 7]. The accuracy of the identification procedure depends on several factors, among which are the quality of measurements, the effectiveness of the mathematical representation of the damaged structure, and the adequacy of the means that provide the descriptive parameters of damage [8]. The former has prompted researchers to search for the modal quantities which are the least sensitive to environmental effects, as it may not always be possible to perform the tests in laboratory conditions [9]. A review of different techniques and new developments in vibration-based identification studies may be found in [10]. Among the different modal quantities used in damage identification procedures, natural frequencies are universally recognized as easily and reliably measurable, although they have limited sensitivity to damage at an early stage. In fact, temperature and humidity can affect frequencies with changes that are of the same order of magnitude as those due to damage. Modal curvatures have recently attracted a great deal of interest from researchers who claimed that curvatures are a highly effective option because of their limited sensitivity to environmental factors. Indeed, it has been shown that they can be used successfully inreal engineering structures to localize damage in operational conditions [11; 12; 13; 14]. In fact, they are local quantities which have remarkable sensitivity to damage [11; 15], especially in the vicinity of the damage itself. However, employing curvatures for full damage identification (i.e., define location and quantification) poses a number of problems, among which is the need for a high density of sensors to ensure measurement in close proximity to damage [16; 17]. This is due to curvature variation spreading to undamaged parts, especially in the case of non-localized damage [18; 19]. This calls for an investigation to better exploit the properties of curvatures while minimizing their drawbacks [20]. Indeed, contrasting views are presented in the literature, concerning the use of modal curvatures to quantify damage. On the one hand, according to some research, modal curvatures are not a reliable quantity [21], and it is argued that the identification and quantification of damage should be treated separately. This is because an index of localization may not perform well in quantification and vice versa [22]. On the other hand, theoretical evidence of the possibility of using modal curvatures for quantifying damage has been reported [23]. The solution of the related inverse problem is yet another crucial aspect of damage identification and quantification procedures. The most established approaches rely on the minimization of an objective function, which measures the difference between the chosen modal quantities and the same quantities provided by a mathematical or numerical model. These are, in turn, functions of descriptive damage parameters. With the necessary and sufficient number of observations, free of any type of experimental errors and a perfect numerical model, a unique solution for the set of damage parameters is expected, as it is observed in many instances [24; 25]. However, in case of applications to real structures, not only can the uniqueness of the inverse solution be lost but also the estimated damage parameters may be highly misleading due to uncertainties.