More recently, as part of a U.S. Army funded SBIR Phase 2 activity, the new method was applied to a study of electronic card cage dynamic characteristics (Figure 4). This study represented the first application of the new sensitivity analysis method employing a component mode synthesis finite element model[9]. Results of the card cage study revealed important sensitivities to “Wedge-lock” card support and screw-down cover plate boundary stiffness uncertainties. Figure 4: Card Cage Assembly indicating Six Component Substructures 6. CONCLUSIONS AND RECOMMENDATIONS This paper provides a systematic derivation and assessment of two methods for trial vector definition that enhance the accuracy of Rayleigh-Ritz approximations for eigenvalue and mode shape sensitivity analysis. The first method, utilizing classical quasi-static residual vector augmentation, was first applied in 1980 to the study of offshore oil platform structural damage sensitivity. Classical quasi-static residuals proved to be effective and accurate in that study, since damage scenarios were localized. In subsequent activities, including (1) the P-5 Short Spacer International Space Station component vibration mode survey test (2000) and (2) an investigation of typical card cage dynamic sensitivities, system uncertainties (2007) were quite dispersed (rather than localized). The new approach for trial vector augmentation, described in this paper proved to be quite effective and accurate in these more recent applications. At the heart of the new sensitivity method is reduction of model order by several orders of magnitude, while preserving accuracy of altered system eigenvalues and mode shapes. This efficiency permits calculation of many parametric alterations in a timely manner using modest computer resources. Due to the presence of uncertainties in as-built systems and systems-in-service (especially as they age), parametric sensitivity analysis should be a standard part of engineering evaluations that determine peak structural loads, stability margins and other safety related metrics. Moreover, efficient parametric sensitivity models should always be available during vibration mode tests to facilitate reliable test-analysis reconciliation activities. The new method featured in this paper appears appropriate for addressing these needs. REFERENCES [1] “International Space Station P5 Modal Survey: Test Planning through FEM Reconciliation”, R. Coppolino, Proceedings of the 20th International Modal Analysis (IMAC) Conference, Feb 2002 [2] “Rates of Change of Eigenvalues and Eigenvectors”, R. I. Fox and M. P. Kapoor, AIAA Journal, Vol. 6, Dec. 1968, pp. 2426-2429 [3] “Simplified Calculation of Eigenvector Derivatives”, R. B. Nelson, AIAA Journal, Vol 14, Sept. 1976, pp. 1201-1205 [4] "Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik”, W. Ritz, J. reine angew. Math. 135, 1-61, 1908. [5] “Twenty Years of Structural Dynamic Modification-A Review, Proceedings of the 20th International Modal Analysis (IMAC) Conference, Feb 2002 [6] “A Hybrid Method for Component Mode Synthesis”, R. MacNeal, Computers and Structures, Vol. 1, pp 581-601, Dec 1971 [7] “Structural Mode Sensitivity to Local Modification”, R. Coppolino, SAE Aerospace Conference and Exposition, Paper 811044, Oct 1981 [8] “Coupling of Substructures for Dynamic Analysis”, R. Craig and M. Bampton, AIAA Journal, Vol. 6, July 1968 [9] “Efficient Modal Sensitivity Analysis of Complex Structural Assemblies”, R. Coppolino, Spacecraft and Launch Vehicle Dynamic Environments Workshop, June 2008 382
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