PSI - Issue 62

906 E. Tomassini et al. / Procedia Structural Integrity 62 (2024) 903–910 Author name / Structural Integrity Procedia 00 (2019) 000–000 criteria to each pair and of frequencies, damping ratios and mode shapes belonging to different system model orders and − . In particular, two different strategies are typically employed: � � � − � ��� � max� � � , � ��� � < � , � � � − ���� � max� � � , � ��� � < � , � � � , ���� �> � , ( � � − � ��� ) max ( � � , � ��� ) +1− � � � , ���� �< � . (6) where the first one includes different thresholds for frequencies � , damping ratios � and mode shapes � , while the second one includes a single threshold for frequencies and MACs � .  C lustering. The full automation of the algorithm requires to also involve an automated clustering operation for grouping the stable poles. Among the many clustering techniques, the Hierarchical Clustering Algorithm proposed by Magalhães (2009) is used in this work. The HC is a machine learning technique able to cluster the stable poles as a function of a distance metric in terms of frequencies and MAC values. The outcomes of the clustering are the modes selected of the structure. 2.1.2. Frequency tracking, statistical pattern recognition and SHM The parameters established at each stage of the modal identification procedure can be applied to automatically identify modes within a training set of acceleration records. Subsequently, the outcomes can be compared with the reference modal results to trace each mode during a specified training period. Literature commonly acknowledges that environmental effects (i.e. temperature and humidity), influence modal features, especially frequencies. Then, the removal of these effects from the damage-sensitive features is essential for effective damage detection. To achieve this, it is imperative to formulate statistical models capable of reproducing the pattern of modal damage-sensitive features. In this approach, damage detection involves examining the residuals , which are defined as the disparity between the training set of damage-sensitive features and those predicted from the statistical model . When the structure is in a healthy state, matrix exclusively reflects the modeling error. Conversely, in the event of damage, which affects the matrix, encompasses the variance induced by the damages. A common approach for damage detection is through the use of statistical process control charts. This paper advocates for the application of Hotelling’s T 2 control charts , which define the statistical distance metric as: � = � − � � � Σ ��� � − � �, (7) where ∈ is the group size of identifications, is the mean of the residuals in the group, and � and Σ � are the mean values and the covariance matrix of the residuals estimated in the training period. 2.2. Methodology The bottleneck in SSI-cov algorithms arises from the Singular Value Decomposition (SVD) of the Toeplitz matrix (Eq. (3)). Specifically, the computational burden grows with the matrix size, which is quadratically increasing with the number of measurement channels. In the case of densely instrumented structures, even considering a low time lag, the size of the Toeplitz matrix becomes significant, compromising the computational efficiency of the SVD. This work aims to present a methodology capable of handling a large number of channels, reducing the comprehensive computational effort. The methodology's core involves dividing the whole-bridge analysis into many different subgroups to alleviate the SVD computational burden by decreasing the number of channels. However, recent research by Tomassini et al. (2023) suggests that, despite low modal coupling between different parts of a structure, conducting data-driven damage detection/localization with channel subgroups may obscure or confuse damage identification. If damage occurs in a specific span with modal coupling between subgroups, its impact may be noticeable not only in the control chart for the subgroup containing sensors in that span but also in the coupled ones. Therefore, the definition of subgroups must be done carefully to minimize the risk of damage misclassifications, considering both the static scheme of the bridge and the modal coupling between different portions of the bridge. An efficient way to qualify the modal coupling between the different portions of the structure is by performing a preliminary whole bridge identification. This can be done manually through the Frequency Domain Decomposition (FDD), which is a well-known frequency domain method based on the SVD of the Power Spectral Density (PSD) 4