become indefinite, by ranking the experimental and transmission simulator modes based upon their contribution to the offending negative eigenvalues. The metrics were applied to two systems and were found to produce significant insight into the cause of the negative mass. The mass was found to become negative for two reasons: 1.) the system contains modes that are completely removed by the substructuring process leaving mass near zero or 2.) the modal model of the transmission simulator is inadequate to describe the motion of the transmission simulator in the C system, leading to inaccuracies that cause too much mass to be removed. The first problem can sometimes be addressed by removing certain modes from the C system, although this was found to degrade the substructuring predictions somewhat for the cylindrical system. To address the second problem, one must either increase the number of modes used to describe the transmission simulator A, or reduce the number of modes in C. The metrics presented here can also be used to determine which of the transmission simulator’s modes (A) contribute most to the negative eigenvalues. Once those modes have been identified, their modal mass can be reduced so that C minus A does not produce negative mass. One example was presented where the negative mass was eliminated by identifying which transmission simulator modes contributed most to the negative mass and increasing their modal scale factors by 58% (i.e. decreasing their modal mass to 40% of its original value). This was found to produce a physically realizable model, but such a large reduction in modal mass does not seem reasonable. This is just an initial effort into the idea of mass adjustment and further research is needed to understand when this is or is not a viable solution. The issue of the transmission simulator modes not spanning the space of the modes of C was also explored, and the proposed metrics were found to sometimes point to modes for which the span was inadequate. For the beam system, this problem was remedied by increasing the number of modes in the transmission simulator model or by decreasing the number of modes in C, although with a consequent reduction in the bandwidth of the model that was obtained for B. The authors suspect that there may be other ways of remedying the problem of negative mass that have not yet been explored. For example, one may find that the transmission simulator model is inaccurate and must be improved before adequate results can be obtained. In any event, the metrics presented in this work should help to guide the analyst to the source of problem so an appropriate solution can be found. 5. ACKNOWLEDGEMENT This material is based on work supported by Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. 6. REFERENCES [1] R. R. Craig and M. C. C. Bampton, "Coupling of Substructures for Dynamic Analysis," AIAA Journal, vol. 6, pp. 1313-1319, 1968. [2] M. S. Allen, R. L. Mayes, and E. J. Bergman, "Experimental Modal Substructuring to Couple and Uncouple Substructures with Flexible Fixtures and Multi-point Connections," Journal of Sound and Vibration, vol. 329, pp. 4891–4906, 2010. [3] R. L. Mayes and M. Arviso, "Design Studies for the Transmission Simulator Method of Experimental Dynamic Substructuring," in International Seminar on Modal Analysis (ISMA2010) Lueven, Belgium, 2010. 132
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