10 A Single Step Modal Parameter Estimation Algorithm: Computing. . . 101 The residue in Equation 10.15 is evaluated in the Z-domain and it must be scaled back to the complex-domain of the original FRFs so that it may be used to reconstruct FRFs. The scaling factor is derived in Equation 10.16 and is dependent on the Z-domain pole. Ar (jω) = Ar (s) sr ∗ t (10.16) It is also noteworthy that the residue is of unity rank. The rows of the residue matrix are the modal vectors scaled by appropriate entries from the participation vectors. Therefore, the row-wise (or column-wise) MAC is expected to be unity, indicating complete linear dependency of the residue matrix on each row/column. The structure of the residue matrix is shown in Equation 10.17. Ar No ×Ni =Qr[Lr]No ×2Nr [ψr] H 2Nr ×Ni (10.17) Here, Qr is the modal scaling factor, [Lr] is the matrix of modal participation vectors and [ψr] is the matrix of mode shape vectors. Hdenotes the Hermitian (complex-conjugate transpose) operation on a matrix. The comparison of the residue matrices obtained from the proposed methodology and the “traditional” process can hence be accomplished using MAC at each pole. Additionally, MPC and MPD metrics [16] that are applied to modal vectors can also be applied to the residue matrix after reshaping it into a vector. A summary of the current modal parameter estimation process, alongside the proposed method, is shown in Fig. 10.1. Fig. 10.1 The basic modal parameter estimation algorithm along with proposed modifications to bypass the second stage
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