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The industry standard method used to validate finite element models involves correlation of test and analysis mode shapes using reduced Test-Analysis Models (TAMs). Some organizations even require this model validation approach. Considerable effort is required to choose sensor locations and to create a suitable TAM so that the test and analysis mode shapes will be orthogonal to within the required tolerance. This work uses a probabilistic framework to understand and quantify the effect of small errors in the test mode shapes on test-analysis orthogonality. Using the proposed framework, test-orthogonality is a probabilistic metric and the problem becomes one of choosing sensor placement and TAM generation techniques that assure that the orthogonality has a high probability of being within an acceptable range if the model is correct, even though the test measurements are contaminated with random errors. A simple analytical metric is derived that is shown to give a good estimate of the sensitivity of a TAM to errors in the test mode shapes for a certain noise model. These ideas are then applied to a generic satellite system, using TAMs generated by the Static, Modal and Improved Reduced System (IRS) reduction methods. Experimental errors are simulated for a set of mode shapes and Monte Carlo simulation is used to estimate the probability that the orthogonality metric exceeds a threshold due to experimental error alone. For the satellite system considered here, the orthogonality calculation is highly sensitive to experimental errors, so a set of noisy mode shapes has a small probability of passing the orthogonality criteria for some of the TAMs. A number of sensor placement techniques are used in this study, and the comparison reveals that, for this system, the Modal TAM is twice as sensitive to errors on the test mode shapes when it is created on a sensor set optimized for the Static TAM rather than one that was optimized specifically for the Modal TAM. These findings are evaluated in light of previously published studies of TAM sensitivity, and special attention is given to Gordis's theory, which suggest that TAM sensitivity is related to the natural frequencies of the structure when all measurement points are fixed. Some aspects of TAM sensitivity are problem dependent, so this one work cannot achieve a conclusive ranking of all of the available methodologies. Instead, this work focuses on presenting a set of tools and a probabilistic framework that can be used to correctly quantify TAM sensitivity and demonstrating the approach for one dynamic system and for a particular probabilistic model for the errors contaminating the test mode shapes. |
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