no specific timing values are listed here, the covariance propagation method offers a significant advantage in required computation time over the MC analysis, especially as the system becomes large. 4.0 CONCLUSION As space structures become larger and more complex, ground based vibration tests of the entire spacecraft become either problematic or impossible. Instead, the spacecraft is tested and validated only at the substructure level. The substructure tests are usually performed in a simulated free-free configuration for simplicity and accuracy. A methodology has been presented here for studying the effects of uncertainty on metrics used for test-analysis correlation of complex spacecraft that are validated on a substructure-by-substructure basis. The objective is to quantify the level of accuracy required at the substructure level to produce acceptable accuracy at the system level. This is done by propagating uncertainty in test-analysis correlation metrics from free-free substructure vibration tests into uncertainty in the synthesized system correlation metrics. Previously derived results are used to relate uncertainty in accepted correlation metrics to substructure modal mass and stiffness uncertainties. The correlation uncertainty in each substructure can either be prescribed, for the purpose of numerical experimentation, or it may be available from existing substructure test data. Linear covariance propagation is used to propagate the substructure modal matrix uncertainty into the uncertainty in the corresponding Craig-Bampton substructure representations. Linear covariance propagation is then used again to propagate the substructure uncertainties into the full system modal matrices. The uncertainties in the system correlation metrics are then extracted from the modal matrix uncertainties to determine the impact of uncertainty at the substructure level. For illustration, the proposed method was applied to a simple two-substructure system shaped as a T-beam. A Monte Carlo analysis was performed in which only one of the substructures was randomized using the Maximum Entropy approach. This resulted in corresponding substructure correlation uncertainty statistics in which only the first 8 elastic free-free substructure modes passed all of the Air Force mandated correlation criteria at the 95% confidence level. The covariance propagation approach presented in this paper was then used to propagate the substructure uncertainty into the full system. The results showed that the first 10 elastic system modes passed all of the Air Force correlation criteria with 95% confidence. A full system Monte Carlo analysis was also performed to validate the accuracy of the covariance propagation method. The results of the two methods agreed very well, except in the case of closely spaced modes, but even then the error was less than 10%. The overall system correlation results with respect to which modes passed the required correlation criteria were identical. Organizations, such as NASA and the Air Force make critical decisions on spacecraft performance and survivability based on the results of test-analysis correlation metrics. In order to ensure the success of finite element model validation where there is no system level test, uncertainty in the substructures must be propagated into the system level correlation metrics. It is believed that the method presented in this paper offers a unique and efficient approach for the required uncertainty propagation. A user can choose to propagate either an assumed level of substructure test-analysis correlation uncertainty, or correlation uncertainty derived from vibration test results. The method is not reliant on the specification of uncertainty in individual model design parameters. It includes all forms of model uncertainty. It is fast, compared with Monte Carlo techniques, and it propagates uncertainty in the correlation metrics directly. Even if the spacecraft will be tested as a system prior to flight, an understanding of the required level of substructure correlation, and how the related uncertainties propagate into the system, will save a great deal of time, effort, and cost during the system level test and analysis. Future work will focus on studying and mitigating observed sensitivities in the method for closely spaced modes, as well as the application of the method to representative spacecraft models. Acknowledgment This material is based on work supported by the Air Force Office of Scientific Research under grant FA9550-09-1-0180. This funding is gratefully acknowledged. References [1] F. M. Hemez, S. W. Doebling, and M. C. Anderson, "A Brief Tutorial on Verification and Validation," presented at 22nd International Modal Analysis Conference, Dearborn, MI, 2004. [2] ASME, "Guide for Verification and Validation in Copmputational Solid Mechanics," ASME V&V 10-2006, P. T. C. C. 60, Ed.: ASME, 2006. [3] T. L. Paez, "Introduction to Model Validation," presented at 27th International Modal Analysis Conference, Orlando, FL, 2009. [4] R. L. Mayes, "Model Correlation and Calibration," presented at 27th International Modal Analysis Conference, Orlando, FL, 2009. 145
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