PSI - Issue 62

E. Tomassini et al. / Procedia Structural Integrity 62 (2024) 903–910 Author name / Structural Integrity Procedia 00 (2019) 000–000

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2.1. Overview of the continuous SHM steps 2.1.1. Automated modal identification through the SSI-cov algorithm

The covariance-driven Stochastic Subspace Identification (SSI-cov) is a suitable choice for continuous SHM due to its easy automation, minimal parameter requirements, and capability to detect closely spaced frequency modes (Reynders, 2012). Let us consider a dynamic system represented by degrees freedom (DOFs) on which output signals are recorded at the sampling frequency � =1/Δ t , with Δt being the sampling time interval. The main steps of the modal identification through the SSI-cov algorithms are the following:  Definition of the discrete time-state space model. The discrete time-state space model, representing the dynamic equilibrium of the system at discrete time instant k Δ t , with k being an integer, reads: ��� = � + � , � = � + � , (1) where ∈ ℝ ��×�� is the discrete state matrix, ∈ℝ �×�� is the output matrix, � , ��� ∈ ℝ �� are the state vectors at discrete time instant Δ and ( + 1)Δ , � ∈ ℝ � is the observation vector and � ∈ℝ �� and ∈ℝ � represent white noise processes accounting for the effect of unknown inputs modeled as stochastic processes, model inaccuracies and measuring noise.  Computation of the Singular Value Decomposition (SVD) of the block Toeplitz correlation matrix. The block Toeplitz matrix is a square matrix containing the block output correlation matrices of the signals � ∈ ℝ �×� evaluated for positive time lags , = 1, … , (2 � −1) where � denotes the time-lag step. Therefore, the block-Toeplitz matrix reads: �|� � = ⎣ ⎢ ⎢ ⎡ R � � R � � �� R � � �� R � � … R � … R � ⋮ ⋮ R �� � �� R �� � �� …⋱ R ⋮ � � ⎦ ⎥ ⎥ ⎤ = � ∈ℝ � � �×� � � . (2) Given the factorization property of the correlation matrix, � = ��� = ��� E[ ��� �� ] , the following equations can be used to extract the observability and the controllability matrices : �|� � = , with = � ⋮ � � �� � = �/� , = [ � � �� … ] = �/� � . (3)  Modal identification. Considering a system model order � (i.e. number of retained singular values in Eq. (3)), the matrices and in Eq. (1) can be extracted from the matrices and by mean of Eq. (3). Therefore, the modal matrix ∈ℝ �×� � and the eigenvalues matrix ∈ℝ �×� � of the structure for the discrete model can be evaluated from the eigenvalue decomposition of the matrix as: , = eig( ) where = [ � … � � ] , = � � ⋱ � � � (4) The eigenvalues � , natural frequencies � , damping ratios � , and modal matrix ∈ℝ �×� � of the continuos system can be evaluated as: � = ln � Δ , � = � 2 , ζ � = ( � || ) � || , = . (5) The process must iterate over various model orders ∈ [ , ] to assess the stability of the resulting modes, discriminating structural modes from spurious ones. After removing complex conjugate modes, the remaining ones are gathered in a stabilization diagram. Stable poles are identified by applying stability