Subcomponents System B Mode fn,A fn,C MS est. fn,B Actual fn,B % Error 4 4326.5 652.3 1083.3 1081.6 0.2% 5 11926.4 1453.6 0+i*1334.2 - - 6 16853.0 2924.7 2996.4 2981.6 0.5% 7 - 3090.5 5903.3 5845.1 1.0% 8 - 5751.6 8421.5 8422.1 0.0% 9 - 7285.6 0+i*9157.5 - - 10 - 9251.0 9824.0 9662.5 1.7% 11 - 12615.3 14832.6 14434.8 2.8% 12 - 13919.7 16965.1 16868.9 0.6% 13 - 14950.0 18066.0 20162.6 -10.4% 14 - 16853.0 0+i*20213 25365.3 - 15 - 19507.0 33706.2 26847.1 25.5% Table 1: Elastic Natural Frequencies (Hz) of Subsystems C and A, and those that result from using MCFS to compute B=C-A. The actual natural frequencies of the FEA model for B are also shown. The eigenvalues of the mass matrix of B that was estimated with the modal substructuring procedure were found and the lowest five were: -0.012, -0.00038, 3.9e-016, 0.0077, 0.074. Two of these are negative and one is practically zero, indicating that the model for B is not physically realizable. A model such as this is incompatible with certain solvers in FEA packages (which require positive definite mass), so one would prefer to find a physically realizable approximation to this model. In order to investigate this further, the source of these negative eigenvalues was investigated using the metrics developed in this paper. The matrix W was formed and CM Q was found to have two eigenvalues that were slightly greater than one and a third that was almost exactly equal to one. The EFI procedure was used to compute the contribution of each of the modes of C to the eigenvalues of the mass matrix. The elements of CM e corresponding to the modes that contributed most to those eigenvalues are shown in the table below. Each column heading gives the corresponding eigenvalue of QCM (recall that the eigenvalues of ˆBm are one minus the eigenvalues of CM Q ), and the CM e values give the contribution of each mode to that eigenvalue. Contributions below 0.01 have been shown with zeros to improve readability. Contribution to Eigenvalues of CM Q Mode of C CM e for O = 1.012 CM e for O = 1.00038 CM e for O = 1 14 0 0 1 6 0.77 0 0 5 0 0.38 0 4 0 0.37 0 3 0.08 0.01 0 7 0 0.08 0 9 0.07 0 0 11 0 0.06 0 2 0.02 0.03 0 12 0.04 0 0 Table 2: Contribution, CM e , of each mode of C to the eigenvalues of CM Q . Several interesting observations can be made. First, mode 14 is entirely responsible for the zero eigenvalue in ˆBm . Visual inspection reveals that mode 14 involves purely axial motion 124
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