100 N. Pandiya et al. The coefficients required in Equation 10.5 are then computed using the Equations 10.8 and 10.9. di =aNo,i (10.8) [α+i ]= No−1 c=0 i d=0 (−1)cbNo−1−c,i−d ∗trace([Bd,c]) (10.9) The eigen-solution using the companion matrix leads to the same poles as the roots of the characteristic equation [15]. In other words, the characteristic equations of a square polynomial matrix and the companion matrix constructed using its matrix-coefficients is identical. The monic denominator-polynomial can hence be used to to obtain the eigen-frequencies present in the frequency range of interest. The FRF matrix (or generally the transfer function matrix) is expressed using Equations 10.4 and 10.5 as: [H(s)]No×Ni = [ α+(s)]No×No[β(s)]No×Ni d(s)1×1 +[R(s)]No×Ni (10.10) The residues for each of the poles are computed by substituting the partial-fraction model (Equation 10.11) of the FRF in Equation 10.10. H(s) No ×Ni = Nr r=0 [Ar]No ×Ni s −λr + [Ar]∗ No ×Ni s −λ∗r + nu i=−nl [Ri]s i No ×Ni (10.11) In conjunction with Equation 10.17, UMPA uses this equation system for the second least-squares step for estimating the scaling factors and modal vectors [8]. Equation 10.12 is over-determined using the frequencies available according to the frequency range selected. H(s) No ×Ni = ⎡ ⎢⎣ Lr No ×2Nr % 1 s −λr & 2Nr ×2Nr ⎤ ⎥⎦ Qrψr H 2Nr ×Ni + nu i=−nl [Ri]s i No ×Ni (10.12) By multiplying the resulting equation throughout by the factor (s −λr), the equation for the residue [Ar] of the rth pole emerges (Equation 10.13). Ar No ×Ni = lim s→λr H(s) ∗(s −λr) = lim s→λr [α+(s)][β(s)] d(s) +[ R(s)] ∗(s −λr) (10.13) Since λr is a root of the polynomial represented by d(s), the limit takes an indeterminate form (the numerator and denominator both vanish at λr). However, since the denominator is now a monic polynomial, L’Hôpitals rule may easily applied to compute the limit for the residue. Ar No ×Ni = [α+(s)][β(s)] d (s) s=λr (10.14) Hence, the residue is calculated for each pole at each model order using Equation 10.15. This added information is beneficial in applying an additional filter to the stabilization chart during the pole selection stage to test for consistency of residues between successive model order iterations. Ar No ×Ni = (No−1)m i=0 [ α+i ] si m−1 i=0 [βi]s i Nom i=1 idis i−1 s=λr (10.15)
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