PSI - Issue 62

Israel Alejandro Hernández-González et al. / Procedia Structural Integrity 62 (2024) 879–886 Hernández-González/ Structural Integrity Procedia 00 (2019) 000 – 000

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2.2. SOBI-based neural network for OMA of structures The architecture of the proposed MTL-DNN, outlined in Fig. 1, encapsulates the fundamentals of the BSS formulation overviewed above. From the findings after inspecting Eq. (8), the main objective of the proposed network is to concurrently predict the real and imaginary parts of the independent components by means of two distinct branches, resulting in the mixing matrix (mode shapes) in complex form.

Fig. 1. Architecture of the proposed MTL-DNN for automated operational modal analysis of bridges. The input of the network corresponds to the acceleration time series ( ) defining different numbers of filtered signals depending on the number of independent components to be determined. These filtered signals correspond to different band-pass filters applied to the raw records, covering different frequency ranges of interest (∆ ) . Such a consideration represents an initial attempt to adapt the classical determined SOBI algorithm to the undetermined problem, in such a way that a larger number of components compared to the number of sensors can be estimated. The output of the network is the reconstructed signal ′( ) produced by the network after applying Eq. (3) with the estimated mixing matrix. To accommodate the sequential characteristics of the input time series, a time lag is introduced between the observational data ( ) and the network's outputs, namely ( ) and ′( ) . The number of input neurons is equal to the number of sensors in the structure ( ) multiplied by the number of defined frequency broadbands ( ). Subsequently, a series of dense layers are employed to emulate both the whitening process and the JAD technique in BSS. The following part of the network is divided into two dense branches of ℎ dense layers and number of neurons, except for the last layers of these two branches that contain neurons, equal to the number of independent components of interest. This division serves the primary purpose of separating the real and complex parts of the independent components into layers R and C , respectively. From Eq. (4), it can be obtained that R = , C = , and = + . Therefore, the weights coming from layers and are considered as the real and imaginary parts of the mode shapes , and , respectively. Along with the reconstructed signals, the output of the branches and are also monitored. In this way, the loss function for the MTL-DNN is defined as: = 1 [∑∑( , − ̂ ) 2 +∑∑( , − ̂ ) 2 +∑∑( , − ̂ ) 2 =1 × =1 =1 =1 =1 =1 ], (8) where , and , represent the real and complex part of the modal response obtained by SOBI, respectively; ̂ and ̂ represent the real and complex parts of the modal response predicted by the MTL-DNN, respectively; , and ̂ represent the input and reconstructed data, respectively. Note that the subscript is employed to indicate the