procedure previously outlined, the contributions, AM e , of each of the modes of A to those eigenvalues are shown in Table 4. Modes 9 and 7 contribute significantly to the first and fourth of the eigenvalues of QAM , but there are a total of seven modes that contribute at least moderately to the negative eigenvalues. There is no justification for removing any of these modes from the model for A, but it is possible that the transmission simulator model is more massive than it should be due to an inaccurate value for its density. To explore this, the modal mass of the seven dominant modes in Table 4 was reduced to attempt to eliminate the negative eigenvalues. Trial and error revealed that the modal mass of these modes had to be reduced to 40% of the original value to obtain a positive definite mass matrix. There does not seem to be a physical justification for such a large reduction; this is a topic of ongoing research. However, it is interesting to note that the substructuring predictions for this system were still very accurate (identical to those shown above and in [2]) even after reducing the modal mass of the transmission simulator so dramatically. Eigenvalues of QAM Mode of A Sum AM e for O1=2.140 AM e for O2=1.845 AM e for O3=1.838 AM e for O4=1.734 9 1.42 1.41 0 0 0 7 1.35 0 0.01 0 1.34 6 0.90 0.15 0.47 0.29 0 1 0.88 0.16 0.48 0.23 0 10 0.80 0 0.48 0.32 0 11 0.79 0 0.30 0.49 0 2 0.79 0.37 0 0.42 0 12 0.37 0 0.01 0 0.35 13 0.11 0 0.05 0.03 0.02 14 0.09 0 0.03 0.06 0 Table 4: Contribution, AM e , of each mode of A to the eigenvalues of AM Q . 3.2.3. Ranking modes of C with respect to stiffness The final application considered is the ranking of the C structure modes based on the stiffness approximation. The case of 18 transmission simulator modes and 100 C modes is again examined. Applying the previously presented procedure, the eigenvalues of QCK WK WK T are computed. In contrast with the mass analysis, all of the eigenvalues of are less than 1.0, therefore the stiffness approximation for substructure B will be positive semi-definite. This is consistent with the authors’ experience that the mass approximation of B is usually the most restrictive. 4. CONCLUSION The Modal Constraints for Fixture and Subsystem method of component mode synthesis has recently been introduced as a means of deriving experimental models of substructures. The experimental substructure is tested in a free-free configuration while the interface is exercised by attaching a flexible fixture or transmission simulator. An analytical representation of the transmission simulator is then used to subtract its effects to produce the desired experimental model of the base structure. It has been observed that indefinite mass and (possibly) stiffness matrices can be obtained in this process. This paper presented simple metrics that can be used by the analyst to determine which modes of each of the subcomponents causes the mass matrix to 131
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