AF criterion γij ≤0.10 with 95% confidence. Taking the intersections of the results, only the first 8 elastic modes for random substructure 2 strictly pass all of the AF correlation criteria at the 95% confidence level. Uncertainties in the modal mass and stiffness for the first 20 elastic modes of substructure 2 are recovered from the testanalysis correlation uncertainty using Eqs. (10-12). Figure 2 shows the recovered RMS modal mass uncertainty, while Fig. 3 illustrates the corresponding RMS modal stiffness uncertainty. The technique described in Section 2.2 was then used to propagate the uncertainties in the substructure 2 modal matrices into the uncertainties in the corresponding CB representation mass and stiffness matrices. The covariance propagation approach outlined in Section 2.3 was then used to propagate the substructure uncertainty into the modal mass and stiffness uncertainties for the first 20 elastic system modes. Figure 4 shows the RMS system modal mass uncertainty, while Fig. 5 presents the corresponding RMS modal stiffness uncertainty. Equations (36-37) and (39) were then used to recover the uncertainties in the system correlation metrics for the first 20 free elastic modes from the propagated system modal matrix uncertainties. Figure 6 shows the recovered RMS system crossorthogonality matrix. In order to demonstrate the accuracy of the proposed covariance propagation procedure and the corresponding linear perturbation equations relating uncertainties in correlation metrics and modal matrices, a corresponding 10,000 iteration MC analysis was performed. The substructure 2 CB mass and stiffness matrices were randomized with 15% dispersion using the maximum entropy approach discussed previously [27]. As mentioned in Section 2.2, the substructure modal mass and stiffness matrices only contain information regarding the uncertainties due to the modes used in substructure test-analysis correlation. Only this uncertainty can be propagated from the free substructure test-analysis correlation results into the system level correlation uncertainty. In order to simulate this in the MC analysis, the randomized CB matrices must be filtered to the contribution from the 20 free substructure modes prior to the synthesis of the random system. This is accomplished by using the projector defined in Eq. (16), giving ΔMCBT 2 = P T 2T ΔMCB 2 P T 2 ΔK CBT 2 = P T 2T ΔK CB 2 P T 2 where ΔMCB 2 and ΔK CB 2 are the fully randomized substructure 2 CB mass and stiffness matrices, respectively. Table 2 compares the system correlation metric uncertainties computed using covariance propagation with those computed using the MC analysis results. One-sigma percent frequency uncertainties recovered using covariance propagation, σω COV , are all less than 1.5%, indicating that all 20 system modes pass the 3.0% allowable frequency error criterion 95% of the time. The frequency uncertainty values range between 0.02% and 1.43%, and are overall somewhat less than the values found at the substructure level. One-sigma percent frequency uncertainties recovered from MC analysis, σω MC , listed in Table 2, agree very well with the covariance values. The largest error is 7% for mode 6, which has the smallest uncertainty value. The average error magnitude is 2.48%. Table 2 also lists the 95th percentile for cross-generalized mass recovered using covariance propagation, γjj.05 COV . The results indicate that all system modes except two pairs, 11-12 and 19-20, which are closely spaced in frequency, pass the AF criterion for cross-orthogonality, γjj ≥0.95, with 95% confidence. The actual values recovered using Eq. (39) for modes 19 and 20 are not reported because they are not physically realizable, due to very poor correlation. Corresponding results recovered from MC analysis, γjj.05 MC , agree very closely with values produced using covariance propagation. The largest discrepancy is 8.7% for closely spaced mode pair 11-12, while the average error over the first 18 modes is 1.0%. The 95th percentile values predicted using MC for closely spaced mode pair 19-20 are so low that they indicate essentially no correlation at all between truth and FEM modes. This is consistent with the unrealizable values predicted using covariance propagation and the corresponding linear perturbation formulas. Maximum one-sigma off-diagonal cross-orthogonality terms computed using covariance propagation, σγij COV , and the coupled mode number are listed in Table 3. Only modes 1 through 10 pass the AF criterion γij ≤0.10 with 95% confidence. The MC statistics for off-diagonal cross-orthogonality are also listed in Table 3, as well as the identity of the corresponding coupled mode. Once again, the MC results agree very well with the covariance predicted results, except for the closely spaced modes, however the trend is the same. The coupling between modes is exactly the same in both analyses. Both methods also predict that only the first 10 system modes strictly pass all of the AF correlation at the 95% confidence level. Therefore, at least for this application, the MC analysis validates the accuracy of the covariance propagation approach and the corresponding linear perturbation equations relating the correlation metrics and the modal mass and stiffness matrices. While 144
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