32 J. G. Kjeld and A. Brandt and the Modified Multiple-reference Ibrahim Time Domain method. Among other MPE methods, the MITD and MMITD method are presented in [4]. In [5] and [6], numerical methods for calculating the uncertainty on the modal parameters have been presented with focus on simulated and measured vibration data. 4.2 Theory The FRF matrix of a system, [H], can be described as the inverse of the system impedance matrix, [Z] which can be obtained by applying the Laplace transformation of Newton’s equation based on physical properties including the mass, [M], stiffness, [K], and damping matrices, [C]. For large systems, such as most civil structures, we rarely know the damping matrix and extraction of modal parameters based on the entire modal matrices is inefficient in terms of computation. Instead, the Laplace transformation of Newton’s equation can be rewritten using the mode shape matrix, [ ], and pole matrix, [S] [H(f)]=[ ][S−1][ T] (4.1) In time domain, this can be written as [h(t)]=[ ] esrt [L]T (4.2) where is the eigenvector matrix, esrt is the diagonal pole matrix and L is the modal participation factor (MPF) matrix. 4.2.1 Multiple-Reference Ibrahim Time Domain (MITD) Method The MITD method is an extension of the Ibrahim time domain (ITD) method presented in [7]. The MITD method allows for multiple references to be included in the parameter estimation and is applicable for Operational Modal Analysis (OMA) as it works on free decay measurements and thus correlation functions. The common equation given in Eq. (4.2) is also valid for the MITD method. Next step is to repeat this equation at different times, a total of mtimes in row direction andntimes in column direction [Hmn(t)]= ⎡ ⎢⎢ ⎢⎣ [h(t)] [h(t + t)] · · · [h(t +(n−1) t)] [h(t + t)] [h(t +2 t)] · · · [h(t +n t)] . . . . . . · · · . . . [h(t +(m−1) t)] [h(t +n t)] . . . [h(t +(m+n−2) t)] ⎤ ⎥⎥ ⎥⎦ (4.3) This equation is known as the block Hankel matrix (referred to as the Hankel matrix). By using the Hankel matrix, Eq. (4.2) can be expanded into [Hmn(t)]=[ ˜ ] e srt [ ˜L]T (4.4) The expanded mode shapes are given as ˜ = ⎡ ⎢⎢ ⎢⎣ [ ] [ ] esr t . . . [ ] esr(n−1) t ⎤ ⎥⎥ ⎥⎦ (4.5) and the extended MPF matrix ˜L = [L]T esr t [L]T · · · esr(n−1) t [ L]T (4.6)
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