1. INTRODUCTION Prior to the flight of a spacecraft, a test-validated finite element model (FEM) must be developed to provide accurate loads analysis and controls simulation. The FEM validation process is comprised of several activities, such as determining the model’s fidelity with respect to test data, quantification of uncertainty, and determining predictive accuracy [1-3]. Determining the fidelity-to-data is an exercise of fundamental importance. This process consists of comparing test and analysis predictions, also called test-analysis correlation, and then determining optimum changes in parameters that will update, calibrate, or tune the model [4]. Model correlation is usually performed in modal space, where the accuracy of the FEM is determined by comparing modal parameters derived from vibration tests with those predicted by the analytical model. Frequencies are compared directly, while corresponding mode shapes are compared using metrics based on orthogonality and cross-orthogonality of the modes with respect to a reduced analytical mass matrix [5]. The use of these metrics, and the required values for acceptable test-analysis correlation, are dictated by agencies such as NASA [6] and the United States Air Force. The requirements differ, depending on the agency. The Air Force, for example, requires testanalysis frequency errors less than or equal to 3.0%, cross-generalized mass values greater than 0.95, and coupling terms between modes of less than 0.10 in both cross-orthogonality and orthogonality [7]. Recently, work in the structural dynamics community on analytical model validation has focused on the quantification of model uncertainty within large numerical simulations, and its propagation into predicted results [8-10]. The concept of model uncertainty is the reality of design and construction. An engineer may design a single structure based on drawings, analysis, and experiments, but the item produced is one of a statistical population due to variations and uncertainties in geometry, material parameters, construction, etc. This leads to random populations of frequencies and mode shapes. There is a corresponding uncertainty and error in the measured test data. In the low frequency regime of modal-based test-analysis correlation and model updating, it is common practice to ignore the effects of both model and test uncertainty. However, if one does not examine the agreement between measurements and predictions relative to uncertainty, very erroneous and dangerous decisions can be made regarding the models ability to make accurate predictions within untested regimes [11]. Hasselman and his coauthors have produced a large body of work in the area of structural dynamics uncertainty quantification [12-14]. They have used and compared several techniques for propagating uncertainty through structural dynamic simulations, such as linear covariance propagation, the Vertex Method for fuzzy variables, Monte Carlo analysis, etc. As space structures become larger and more complex, ground based vibration tests of the entire spacecraft become either problematic or impossible due to lack of structural integrity, cost, complexity of the test, or simply lack of time. Instead, the spacecraft is validated on a substructure-by-substructure basis. Unavoidable uncertainty in substructure models and testing will have large, and possibly negative, impact on this new paradigm for model validation. To insure the success of this new model validation approach, several key questions must be addressed. For example, what level of accuracy or correlation do the substructures need to exhibit to have a required level of correlation at the system level? More specifically, how does uncertainty and error within each substructure propagate into, and affect model validation at the system level? 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. Researchers have recently started to investigate the effects of substructure uncertainty on synthesized system response using component mode synthesis techniques (CMS). The CMS approach has been used for years to solve large structural dynamics problems, and is built into many standard finite element analysis codes. Hinke et al. [15] consider uncertainty in the form of experimental measured variability, or noise, in substructure free-free modes. They use linear perturbation theory to determine the sensitivities of both fixed interface substructure, and global system modal parameters in terms of unconstrained substructure eigenvalues, based on the Craig-Bampton (CB) substructure representation [16]. Free substructure eigenvalue variance is propagated into fixed CB eigenvalue statistics using inverse linear covariance propagation, which can be subject to numerical ill-conditioning. In addition, they only consider uncertainty due to substructure eigenvalues. Mace and Shorter [17] also consider the effects of substructure model uncertainty on the system modal parameters and resulting frequency response. They use the CB formulation and linear perturbation theory to determine system modal parameters in terms of the random substructure eigenvalues. Uncertainty in the substructure eigenvalues is propagated into system modal parameters using a decoupled Monte Carlo approach. Uncertainty in the system level frequency response can then be recovered. De Klerk and Voormeeren [18] also consider substructure uncertainty in the form of experimental noise, but instead, it is in the frequency response measured during the substructure vibration test. They use a frequency domain CMS approach [19]. Linear perturbation theory is used to propagate the substructure uncertainty into the global response. Voormeeren and Rixen [20] use the same approach, but study the effects of uncertainty on the decoupling of a system into its component 136
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