substructures. None of the work performed in this area addresses the test-analysis correlation component of model validation, at either the substructure or system level. The goal of this investigation is to develop a methodology for studying the effects of uncertainty on accepted metrics for testanalysis correlation of complex spacecraft that are validated on a substructure-by-substructure basis. The objective is to be able to quantify the level of accuracy required at the substructure level to produce acceptable correlation at the system level. Linear perturbation analysis is used to relate uncertainty in test-analysis correlation metrics to uncertainty in substructure modal mass and stiffness. A previously developed statistical model for modal based test-analysis correlation metrics [21] is discussed, which results in a corresponding covariance matrix. Linear covariance propagation is then used to propagate freeinterface modal correlation metric uncertainty in the substructures into the expected free-free correlation metric uncertainty for the system using CB based component mode synthesis and reduced order modeling. This work is of special interest because substructures are most often tested in a free-free configuration. Understanding substructure correlation requirements will positively impact the speed of the loads analysis process. A simple example is investigated, and the results are substantiated using Monte Carlo analysis. 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. Currently there is no uncertainty quantification performed or required by these agencies for test-analysis correlation in the low-frequency regime. The approach presented in this paper offers several advantages over other methods. A user can choose to propagate either an assumed level of test-analysis correlation uncertainty, or uncertainty derived from vibration test results, etc. It is not reliant on the knowledge of any specific 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. 2. THEORY 2.1 Quantification of Substructure Uncertainty In this work, uncertainty in the substructures is quantified in terms of uncertainty in the substructure test-analysis correlation metrics, modal frequency and cross-orthogonality. In this case, the substructures are assumed to have been tested in a freefree configuration. Uncertainty in substructure modal mass and stiffness must then be related to the uncertainty in modal frequency and cross-orthogonality. A detailed analysis can be found in Kammer et al. [21], but the results are summarized again here. The uncertainty is defined with respect to the nominal substructure FEM. The nominal substructure modes are assumed to be normalized with respect to mass, such that the nominal substructure modal mass and stiffness are m=I and k =Ω, where Ω is the matrix of nominal eigenvalues. The “truth” model of the substructure can also be represented in nominal modal coordinates as mT and kT . The uncertainty in the modal mass and stiffness can then be defined as Δm= mT −m= mT −I Δk = kT −k = kT −Ω (1) Uncertainty in substructure modes and eigenvalues can likewise be expressed as Δφ= φT − φ ΔΩ=ΩT −Ω (2) It is assumed that the truth modes can be well approximated as a linear combination of the nominal modes, such that φT ≈ φγ . The cross-orthogonality between the nominal and truth modes can then be written as φ T Mφ T = φ T Mφγ = γ. It is also assumed that the truth modes, φT , are normalized with respect to the nominal mass matrix, instead of the unknown truth mass matrix. This is done to be consistent with the definitions of cross-orthogonality used by both the United States Air Force [7] and NASA [6]. Therefore, for the jth mode φTj T Mφ Tj = γj T γ j =1 (3) in which the subscript j indicates the corresponding matrix column of γ . In an actual application of test-analysis correlation, the cross-orthogonality between test and FEM modes is calculated with respect to a reduced FEM mass matrix called a testanalysis model (TAM), instead of the full mass matrix M. However, if the truth modes are spanned by the nominal modes and an exact model reduction is used, such as the Modal TAM [22], then the cross-orthogonality results are the same. 137
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