44 S. D. R. Amador et al. . H0(za) =Ф(Iza −Λ)−1 ΨT H0(zb) =Ф(Izb −Λ)−1 ΨT (4.10 ) Isolating. ФT in both equations and combining the obtained expressions yield .Ψ−1 [H1(zb) −H1(za)] T =ΛΨ−1 [H0(zb) −H0(za)] T (4.11 ) with .H1(z) =⎡zH(z) z 2H(z) · · · zn+1H(z)⎤ ∈CNo×(n+1)Ni (4.12 ) Writing down (4.11) for all the. Nf discrete frequency lines available in the frequency band, i.e., for . ωa an d. ωb ranging , respectively, from. ω0 to .ωNf−1 an d fro m. ω1 to .ωNf , and combining the equations corresponding to each pair of evaluated frequency values in a single matrix equation yield .Ψ−1BT = ΛΨ−1AT (4.13 ) wher e . A= ⎡ ⎢ ⎢ ⎢ ⎣ H0(z1) −H0(z0) H0(z2) −H0(z1) . . . H0(zNf ) −H0(zNf−1) ⎤ ⎥ ⎥ ⎥ ⎦ ∈CNo(Nf−1)×(n+1)Ni B= ⎡ ⎢ ⎢ ⎢ ⎣ H1(z1) −H1(z0) H1(z2) −H1(z1) . . . H1(zNf ) −H1(zNf−1) ⎤ ⎥ ⎥ ⎥ ⎦ ∈CNo(Nf−1)×(n+1)Ni (4.14 ) An eigenvalue problem can be formulated from (4.13) using the Doubl e Leas t Squares approach (DLS) [6, 1], giving . 1 2 (BTA∗ (ATA∗)−1 +BTB∗ (ATB∗) −1) =ΨΛΨ−1 (4.15 ) Once the eigenvalue problem Eq. (4.15) is solved, the modal participation factors can be retrieved as the 1st . No row s o f . Ψ , and the natural frequencies and damping ratios are computed from diagonal elements of . Λ . The eigenvalue problem as in Eq. (4.15) gives eigenvalues and eigenvectors not occurring in complex conjugate pairs. If modal participation factor vectors and continuous time poles occurring in conjugate pairs are desired, the following eigenvalue problem should be used instead. . 1 2 (Re(BTA∗)(Re(ATA∗))−1 +Re(BTB∗)(Re(ATB∗)) −1) =ΨΛΨ−1 (4.16 ) where.Re(•) stands for the real part of a complex number. 4.4 Application to the Multi-dataset Vibration Test of the Riga TV and Radio Broadcast Towe r In order to illustrate the practical application of the pCF-MM described in the previous section, a case study consisting of the multi-dataset output-only vibration test of the Riga TV and Radio Broadcast Tower is presented. The vibration test campaign to measure the vibration responses in 6 different datasets was carried out over the course of two days, a period within which mild wind velocities were observed. The acceleration response time series of each dataset were acquired for one hour and
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