# Identification of Damage in Beams by Modal Curvatures Using Acoustic Beamformers
## 1 Introduction
Therefore, the application of a robust estimator (or processor) to the data obtained experimentally may improve the quality of the solution to the inverse problem, especially when in the presence of disturbance due to operating and environmental conditions, noisy data, and modeling errors. A beamformer is an operator which enables the comparison of a set of data received by an array of sensors to model data, called the replica vector. In ocean acoustics, the operation is named acoustic beamforming, in short, beamforming, and is used to spatially filter the data set to estimate the location of the source of the signal, providing the look direction. Dating back to the end of the first half of the 20 th century, beamforming received much attention and has been applied to many different fields, from neurology to diagnostics in mechanics [26]. Interested readers can refer to review papers [27; 28], which cover the history and evolution of beamforming algorithms. In this paper, our focus is the application of beamformers, which have been successfully used in ocean acoustics, to identify damage using modal curvatures. This is because these algorithms were formulated to detect sources in space, which makes them good candidates to be applied in solving problems where local response changes have to be detected. Apart from the seminal work by [PERSON] and [PERSON] [29], beamforming has received limited attention to date in the field of structural identification, despite its success in a variety of applications. In this paper, we investigate possible enhancements in the accuracy of damage localization and quantification in beam-type structures by applying beamforming algorithms, using modal longitudinal strains measured on the surface of the beam. These response quantities directly relate to curvatures and do not require the use of finite differences in calculating curvatures--one of the main sources of errors, especially in quantification problems. In this paper, we do not address issues related to the number and location of the sensors, which are distributed uniformly along the beam axis and are in a fixed number of seven. We compare the performance of beamformers against a traditional algorithm used in damage identification, based on the minimization of an objective function. The beamforming algorithms considered are Bartlett and minimum variance distorsionless response (MVDR), and are introduced in Section 2. We utilize a finite element method (FEM) software package to build a data set of replica vectors and to investigate the sensitivity of modal quantities to damage. The determination of the modal quantities is referred to as a direct or forward problem. An investigation of the changes due to damage is addressed in Section 3, with a comparison between the numerical and experimental frequencies, mode shapes and modal curvatures. In Section 4, by means of a numerical study, we show that the MVDR processor is superior in sensitivity to damage parameters when compared to estimators based on the simple minimization of the difference between the model and measured data. In particular, this applies when dealing with critical cases, such as slight damage and damage located between two sensors. To further validate the present approach, we carry out a series of experiments on free-free beams with different damage intensities and locations, including cases of damage below a sensor and in-between two sensors. The proposed approach can be applied to other boundary conditions, provided that the corresponding replica vector is employed.