from the associated left singular vector. The singular vector associated with these significant singular values is nominally a logical index vector (when scaled by the square root of the singular value) and is used to determine the location in the original pole weighted vector matrix for those pole weighted vectors that are identified as being in a consistent cluster. This is accomplished by multiplying the left singular vector by the square root of the singular value and retaining all positions (indexes) above a threshold (typically 0.9). The positions of the non-zero elements in this vector are the indexes into the pole weighted vector matrix for all vectors belonging to a single cluster. 4.8 Step 8: Evaluate Pole Cluster Effectively, Step 7 yields, cluster by cluster, the pole weighted vectors that will be used to identify a single consistent spatiotemporal set of data representing a single complex modal frequency, modal vector, and modal scaling. For each identified pole weighted vector cluster, a singular value decomposition (SVD) is performed on the identified set of pole weighted vectors. Note that the number of pole weighted vectors included in each cluster will in general not be the same. The significant left singular vector is the dominant (average) pole weighted vector. Use the zeroth order portion of this dominant vector to identify the modal vector and the relationship between the zeroth order and the first order portions of the dominant vector to identify the complex modal frequency. Complete the solution for modal scaling by using any MIMO process of your choice. A complete discussion of the choices can be found in the literature [48]. 4.9 Step 9: Complete Statistical Evaluation for Each Pole Cluster Since each cluster in Step 8 involves a number of estimates of complex modal frequency, modal vector, and modal scaling, computing statistics on the variation in the cluster is a natural part of this autonomous modal parameter estimation process. Tables 1 to 3 are examples of statistics that can be easily computed based upon these data clusters. Complete details about these statistics, their rationale and how each is computed can be found another paper [59]. Naturally, the statistics in these tables give the possibility, with experience, of developing an automatic threshold on any number of physical or statistical values to exclude solutions which are deemed physically unrealistic or statistically unacceptable. At this time, this is left to user interaction at the completion of the CSSAMI processing. Index Sample Size (N) Frequency (Hz) Damping (Hz) Frequency (Hz) Damping (Hz) Std. Dev. (Hz) (Mean) (Mean) 8 16 362.564 -3.1666 362.564 -3.1670 0.1677 19 16 363.860 -3.5650 363.860 -3.5650 0.1251 10 16 557.055 -2.8966 557.055 -2.8966 0.0795 12 16 761.224 -5.2229 761.224 -5.2229 0.1008 2 17 764.190 -2.5371 764.190 -2.5371 0.1529 16 16 1222.980 -4.0883 1222.980 -4.0883 0.0271 13 16 1224.055 -3.9540 1224.055 -3.9540 0.0419 4 17 1328.036 -6.6495 1328.036 -6.6507 0.0907 17 16 1328.803 -5.4761 1328.803 -5.4768 0.1871 6 17 2019.269 -8.2512 2019.271 -8.2511 0.2543 23 15 2023.802 -7.5597 2023.801 -7.5598 0.0642 26 10 2321.862 -3.9074 2321.860 -3.9070 0.1690 28 8 2322.335 -3.8356 2322.335 -3.8367 0.2244 22 15 2337.895 -3.7967 2337.895 -3.7971 0.2287 TABLE 1. Modal Frequency Statistics 375
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