PSI - Issue 62

Israel Alejandro Hernández-González et al. / Procedia Structural Integrity 62 (2024) 879–886 Hernández-González et al./ Structural Integrity Procedia 00 (2019) 000 – 000 3 a data vector x (t) ∈ ℝ . Assuming that the system is linear time-invariant and a number l ∈ℕ ∗ of natural modes of vibration are excited, the system's response can be expanded using modal superposition as: ( ) = ( ) = ∑ =1 ( ), (2) where ∈ℂ is the modal matrix composed by l mode shape vectors ∈ℂ , and ( ) = [ 1 ( ), … . , ( )] T stands for the modal displacement vector. In order for SOBI to extract complex-valued mode shapes, the measured data ( ) can be augmented in its analytic form through the Hilbert transform ( ℋ ) (McNeill & Zimmerman, 2008), which imparts a phase shift of 90 degrees. That is ̌( ) = ( )+ ̃( ) , where ̃( ) = ℋ( ( )) , and denotes the imaginary unit. The fundamental hypothesis of BSS stipulates that ̌( ) can be described as a linear combination of ̌( ) as: ̌( ) = ̌( ) + ( ), (3) where ∈ℂ is the so-called mixing matrix, and ( ) represents a zero-mean temporally and spatially stationary white noise. The SOBI technique finds independent components that produce diagonal time-shifted covariance matrices, relying on second-order statistics. In this sense, it is convenient to first whiten the observed data using principal component analysis (PCA). This is carried out by eigenvalue decomposition of the covariance matrix of the observations, that is { ̌( ) ̌( ) T }= (0) = T , where and correspond to the diagonal eigenvalue and the orthogonal eigenvector matrices, respectively. On this basis, the whitened data is obtained as ( ) = m ̌( ) , with m = −1/2 T being called the whitening matrix. Making use of ( ) , the second step in the SOBI algorithm involves estimating an orthogonal matrix that approximately diagonalizes several time-shifted covariance matrices with time lags τ i ,1≤ ≤ . Using the joint approximate diagonalization (JAD) algorithm, this can be expressed mathematically through an optimization problem as: T ( ) ≈ diag[ 1 ( ), … , ( )] ⇒ mΨin ∑ off ( T ( ) ) = 1 ,1≤ ≤ , (4) where (τ) is the time-shifted covariance matrix of the whitened data, and the terms ( ) denote their autocovariances. Thus, the independent components and the mixing matrix and can be obtained as: = m−1 = 1/2 , and ̌ = ̌, (5) with = − denoting the de-mixing matrix. Recall that both the mixture matrix and the independent components are complex-valued, which allows one to write: = + , = + , with , ∈ℝ × , , ∈ℝ × , (6) where the “ R ” and “ C ” indexes correspond to the real and imaginary parts of the respective terms. Therefore, substituting Eq. (7) into (3), the system's response x can be this expressed as a linear combination of the real and imaginary parts of ( ) as: (t) = ( R + i C )( R (t) + i C (t))= R R (t)− C C (t). (7) In this work, to determine the frequencies and damping ratios from the independent components ( ) , the single degree of freedom (SDOF) Ibrahim Time-Domain (ITD) dynamic identification approach is adopted. Using this method, the free decay response for a certain component is first calculated by means the Natural Excitation Technique (NExT). 881