104 N. Pandiya et al. Pole 1: 19.105081 Hz Pole 4: -19.105081 Hz Group 1: Traditional Residue Group 2: Proposed Residue Group 1: Traditional Residue Group 2: Proposed Residue Group 1: Traditional Residue Group 2: Proposed Residue Group 1: Traditional Residue Group 2: Proposed Residue Group 2 Group 2 Group 2 Group 1 Group 1 Group 1 MAC Value Group 1 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Group 1 Group 1 Group 2 Group 2 Group 2 Group 1: Traditional Residue Group 2: Proposed Residue Group 1: Traditional Residue Group 2: Proposed Residue Pole 2: 23.432167 Hz Pole 5: -23.432167 Hz Pole 3: 26.930614 Hz Pole 6: -26.930614 Hz 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Fig. 10.6 MAC matrix for dependence of columns of the residue matrix obtained from traditional and the proposed process for the three poles (from the second-step least squares computation) in the pole selection phase is seen to improve the clarity of the stabilization chart [16]. To compare the models obtained from the two approaches, the residues were also computed using the second leastsquares step for the selected poles in Fig. 10.4. The two sets were compared using a modal assurance criterion approach. This is possible due to the structure of the residue matrix, shown in Equation 10.17. The column-wise MAC was expected to show complete linear dependence of the columns and this was indeed the case. Figure 10.6 shows a MAC plot of residue for each of the 6 poles (3 positive frequencies and 3 negative frequencies). The columns of the residues correspond to the long dimension (4 elements in 9 vectors) and show a unity MAC value throughout. The MAC plots also indicate that the residues from the proposed method are of unity rank since the residues from the traditional two-step process are forced to be of unity rank. The FRFs were then reconstructed using the residue from the traditional as well as the proposed approach, and plotted against the original FRFs (Fig. 10.7, for example). It may be seen from the reconstructed FRF curves that the proposed process identifies the same modal model as traditional process. Of course, a deviation between the two modal models is to be expected in presence of noise in the measurements. The residues, after being reshaped as vectors were also plotted in the complex plane (Fig. 10.8) and the mean phase correlations indicated that both the methods resulted in real modes. The low deviations from the mean phase for all the poles and the imaginary nature of the residues both indicate a correct estimation of the modal model. It is worthwhile to mention that there is no restriction on the complexity or scaling of the modal residue under the proposed scheme. There are various scaling strategies for the participation vectors that have been applied to clear up the stabilization
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