4.2 Step 2: Find Scaled Modal Vectors If the UMPA method is high order (coefficient matrices of size NS ×NS), solve for the complete vector (function of NL for all roots, structural and computational). −1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real Imaginary Mode # 1: 362.30 Hz 0.877% zeta MP U:−1.0o MPD U:4.87% | MP S :88.9o MPD S :6.08% −1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real Imaginary Mode # 1: 362.36 Hz 0.887% zeta MP U:−0.3o MPD U:9.04% | MP S :−91.3o MPD S :5.95% Figure 6. Typical Complex Plot of Unscaled and Scaled Modal Vectors Figure 6 shows plots of a complex valued modal vector originating from PTD (left) and ERA (right) methods. The green circle symbols represent the normalized modal vector associated with the consistency diagram for one of the poles of one of the high order solutions. The blue x symbols represent the scaled (in terms of residue) full length vectors that can be computed, mode by mode, before continuing. The purpose in calculating the complete, scaled vector is two-fold. The first is the additional discrimination of the algorithm that results from the longer vector length (PTD example on the left). The second is the availability of the modal scaling information, used to calculate Modal A. This scaling information can be useful at the end of the solution process in order to asses the quality of the results (magnitude and phase of Modal A, for example). Even long basis vectors should be scaled so that this assessment information is available for all cases (ERA example on the right). 4.3 Step 3: Determine Pole Surface Density Clusters Based upon the pole surface density threshold, identify all possible pole densities above some minimum value. This will be a function of the number of possible solutions represented by the consistency diagram. The pole density is defined for each pole as the number other poles within a specified complex tolerance radius. This range can be defined in terms of either absolute or relative frequency. Just as in Step 1, restricting the number of poles reduces both the time and the memory required for solution, but it also reduces the amount of information available to the algorithm. As a result, an overly restrictive threshold may be counterproductive. Figures 7 and 8 show the pole consistency plots (showing pole locations in the second quadrant of the s-plane) that are evaluated in this step. 371
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