{ φ}r = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ zv r {ψ}r . . . z2 r {ψ}r z1 r {ψ}r z0 r {ψ}r ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭r For comparison purposes, the actual mean value and standard deviation of the poles (as well as, the separate frequency and damping means and standard deviations) which were used in the computation of the weighted solution are computed. Since these results are unweighted by the vector characteristics, they will be somewhat different from the vector weighted solution and provide comparitive feedback about the pole. 4.2 Modal Vector Statistics In order to evaluate the quality of the resulting modal vectors, several different parameters (representing noise to signal ratios) are calculated. These ratios are evaluated for both the original normalized vectors and the pole weighted (state extended) vectors and are computed using the singular value decomposition of each of the set of vectors. 4.2.1 Normalized Modal Vector Residual (NMVR) The first modal vector parameter is evaluated by taking the total residual magnitude (the Forbenius norm of the residuals) divided by the magnitude of the principal vector magnitude. In other words, the square root of the sum of the squares of the residual singular values divide by the first (largest) singular value. This provides an indication of the consistency of the original contributing vectors. Small values tend to indicate greater consistency. Large values indicate greater variance or the possibility that more than one mode has been included in a cluster. ⎡ σ⎦ = SVD ⎛ ⎝ ⎡ ⎣ ψ1 ψ2 . . . ψN ⎤ ⎦ ⎞ ⎠ (1) NMVR1 = √⎯ N k=2 Σ σ2 k N σ1 (2) The second modal vector parameter is evaluated by taking the largest residual magnitude divided by the magnitude of the principal vector magnitude. In other words, the second singular value divided by the first singular value. This provides an indication of the consistency of the original contributing vectors. A small value tends to indicate random variance. A large value can indicate a consistent modal contamination of the original vectors, possibly caused by a second mode included in the cluster. NMVR2 = σ2 σ1 (3) 4.2.2 Normalized State Vector Residual (NSVR) The associated state vector parameters are calculated analogous to the above except that the complete pole weighted (state extended) vector is used. ⎡ σ⎦ = SVD ⎛ ⎝ ⎡ ⎣ φ1 φ2 . . . φ N ⎤ ⎦ ⎞ ⎠ (4) 388
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