7 Operational Modal Analysis with a 3D Laser Vibrometer Without External Reference 79 rescaling merging modal analysis PoSER PoGER PreGER Fig. 7.2 Order of steps for the three merging strategies for SSI cov/ref all measurements, are used. Mevel et al. [3] and Döhler et al. [1] introduce three different strategies for data merging of multisetup measurements using reference sensor data. These approaches mainly differ in the order of the steps: modal analysis, data merging and data re-scaling (see Fig. 7.2). The classic approach, termed PoSER (Post Separate Estimation Re-scaling), is the common practice in operational modal analysis. Each setup j D1: : : ns is processed individually to obtain the modal parameters f j r; j r and mode shapes j r for the onemode r. The separate, incomplete mode shapes are then combined and rescaled relative to measurement j s D 2 666 664 ref ;j r m;j r ˛j ;1 r m;1 r : : : ˛j ;ns r m;ns r i 3 777 775 with ˛ j ;j r D ref ;j r H ref ;j r 1 ref ;j r H ref ;j r ; (7.16) where ır and ım denote reference channels and moving channels, respectively and ıH denotes a conjugate transpose. Especially if a large number of DOFs need to be included, this approach requires many measurements to be processed individually. The second merging approach, termed PoGER (Post Global Estimation Re-scaling), aims at reducing the manual workload by processing all setups together. However this can only be done if the tested structure exhibits linear behavior (e.g. no frequency shifts due to temperature, loading amplitudes, etc.). The reference sensors, whose signals are used to compute the covariances are required to stay on the same location in all setups. The reference-based covariance matrices are stacked block-row-wise ORall i D 2 666 4 OR1 i OR2 i : : : ORns i 3 777 5 : (7.17) These total covariance matrices for different time lags i are then assembled into a block-Toeplitz-matrix, that is used for system identification and modal analysis, as described in the previous section. The resulting mode shapes j r come out stacked in the same manner. Each setup’s part has to be rescaled separately similarly to the PoSER approach, Eq. (7.16), and the reference channels are removed afterwards. To overcome the limitations regarding the linear behavior of the tested structures, a third approach, termed PreGER (Pre Global Estimation Re-scaling), was introduced. A detailed derivation of the procedure is given in [3], here it is briefly described. In contrast to the other approaches, rescaling is now applied to the covariance matrices, before merging the setups. The scaling matrices are obtained in state-space by means of a common controllability matrix, which reflects the differences in excitation between all setups. In order to compute this matrix, the covariance-matrices of only the reference channels ORj;ref i are built for all setups j and stacked block-column-wise ORall;ref i DhOR1;ref i : : : ORns;ref i i : (7.18)
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