Figure 5: Cumulative sum of eigenvalues of QCM versus number of modes included in the sum. The numerical values of eCM are shown in Table 3, in the same format that was used in the previous section. An additional column is also shown that gives the total contribution of each mode to all of the eigenvalues of QCM that are greater than one (all four that are shown in this case). One observes that mode 21 contributes the most to the large eigenvalues of QCM , contributing over 86% to the 3rd eigenvalue. The total contributions of modes 19, 20 and 22 make up a similar fraction of the other eigenvalues. Eigenvalues of QCM Mode of C Sum CM e for O1 = 2.140 CM e for O2 = 1.845 CM e for O3 = 1.838 CM e for O4 = 1.734 21 1.60 0 0.01 1.59 0 20 1.48 1.04 0.17 0 0.27 19 1.34 0.21 0.43 0 0.69 22 0.98 0.37 0.58 0 0.03 26 0.95 0.10 0.38 0 0.47 29 0.32 0.32 0 0 0 25 0.17 0.02 0.07 0 0.08 23 0.11 0 0.02 0 0.08 34 0.08 0 0 0.08 0 35 0.08 0 0.08 0 0 Table 3: Contribution, CM e , of each mode of C to the eigenvalues of QCM . These results suggest that if modes 19-22 are eliminated from system C, then QCM should be less than 1.0. However, when this is done, one of the eigenvalues remains above 1.0. The 26th mode must also be eliminated from the database to produce a positive definite mass matrix. However, one should not discard modes without cause since they may be important to the model for B and to the substructuring predictions obtained with it. To check whether these modes might be important to the model of the cylinder, each mode was plotted. For example, the 20th mode of the C substructure is shown in Fig. 6. This and the other mode shape plots reveal that all of the modes in question are dominated by motion of the transmission simulator. The ratio between the maximum displacement of the cylinder part of the structure to the maximum displacement of the transmission simulator was between 4.1% and 10% for these modes. Once the transmission simulator is subtracted, one would expect that this mode might contribute very little to the modal mass of B, so small errors and approximation due to the truncated modal basis could cause this mode to have negative mass after substructure uncoupling. 128
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