diagonal terms ΔγSYSij will have different non-unit variances. Therefore, cj follows a generalized chi-square distribution. In general, the number of degrees of freedom is not nSYS −1 because many of the terms ΔγSYSij are small due to weak coupling between the nominal FEM modes and the jth truth mode. In terms of standard normal variables zi , cj can be expressed in the form cj =1− γSYSjj 2 = σ i 2zi 2 i=1 i≠j nSYS ∑ (38) in which σi 2 = E Δγ SYSij 2 ( ) are the off-diagonal cross-orthogonality variances recovered for the jth column using Eq. (36). A distribution for cj can then be constructed by taking a linear combination of single degree of freedom chi-square distributions χ2 1( ) using cj = σi 2 χ2 1( ) i=1 i≠j nSYS ∑ (39) The 1− α ( ) th percentile for cj can easily be computed and then the corresponding value for the jth cross-generalized mass, given by γSYSjj α = 1−cj α , can be compared to designated correlation metric criteria. 3.0 NUMERICAL EXAMPLE A simple example is considered to illustrate the application of the proposed uncertainty propagation technique. The system consists of two steel beams attached in the shape of a T. The cross member, substructure 1, is 12.0 in. long with a 0.5 by 0.5 in. cross section, while substructure 2 is 48 in. long with a 0.5 by 0.35 in. cross section. Both substructures are constrained to plane motion. The rotational degrees of freedom perpendicular to the plane of motion were statically reduced out, except at the substructure interface nodes, the midpoint of substructure 1 and the endpoint of substructure 2. Substructure 1 has nine nodes with 27 degrees of freedom, while substructure 2 has 49 nodes and 99 degrees of freedom. All degrees of freedom are retained in the CMS analysis, therefore substructure 1 has 24 fixed interface modes and 3 constraint modes, while substructure 2 has 96 fixed interface modes and 3 constraint modes in their respective CB representations. For illustration, only substructure 2 is assumed to have uncertainty. The uncertainty is quantified in terms of test-analysis correlation metrics, as discussed in Section 2.1. The uncertainty can either be specified by the analyst, as a part of a numerical experiment, or it can be based on available test-analysis correlation results. In this example, it is assumed that correlation uncertainty is available for substructure 2 from a free-free test for the first 20 elastic modes with frequencies between 45.4 Hz. and 6355.9 Hz, as listed in Table 1. The test results were simulated using Monte Carlo (MC) analysis. At each iteration, the nominal CB mass and stiffness matrices for substructure 2 were randomized using the Maximum Entropy approach developed by Soize [27]. A dispersion level is selected that can be thought of as being analogous to the global fractional uncertainty believed to exist in the matrix, and then the matrix is randomized subject to the constraints of maintaining symmetry and positive definiteness for the mass matrix, and positive semi-definiteness for the stiffness matrix. In this example, a dispersion level of 15% was selected for both mass and stiffness, and 10,000 iterations were performed. Details of the process can be found in Soize’s paper [27]. The advantage of this nonparametric approach, over the usual parameter sensitivity or perturbation methods, is that this randomization process automatically accounts for uncertainties that are not easily described by model parameters, such as model form, geometry, joints, etc. The resulting test-analysis correlation statistics for substructure 2 are listed in Table 1. All 20 elastic modes have one-sigma frequency uncertainties, σω, less than 1.50%, therefore they would pass the Air Force’s (AF) 3.0% allowable frequency error criterion 95% of the time. One-sigma percent uncertainty in generalized mass σΔM is approximately 2.0% for all the modes. The RMS cross-orthogonality matrix is illustrated in Fig. 1. The 95th percentile for cross-generalized mass, γ jj.05 , indicates that modes 10, 11, 15, 16, 19, and 20 do not pass the AF criterion for cross-orthogonality, γjj ≥0.95, with 95% confidence. Maximum one-sigma off-diagonal cross-orthogonality terms listed in Table 1 indicate that only modes 1 through 8 pass the 143
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