Uncertainty in the cross-orthogonality matrix, Δγ, can be defined using the expression γ = I +Δγ. The constraint on the jth column of γ in Eq. (3) can then be expressed as Δγij 2 i=1 i≠j nq ∑ =1− γjj 2 =−Δγ jj 2 −2Δγ jj (4) or Δγij 2 i=1 nq ∑ =−2Δγjj (5) where nq is the number of nominal FEM and truth modes being correlated. Equation (4) indicates that within each column of the cross-orthogonality matrix γ , there is a constraint between the diagonal term and the off-diagonal terms, and Eq. (5) indicates that the uncertainty Δγjj is always negative. Following the approach presented in Ref. [21], using linear perturbation theory, uncertainty in the correlation metrics Δγ and ΔΩ can be related to uncertainty in the substructure modal mass and stiffness Δm and Δk . The first order representation of the substructure modal mass uncertainty is given by Δmjj = ΔMjj (6) Δmij =−Δγij −Δγji i ≠ j (7) where ΔMjj , defined by ΔMjj = φTj T MT φTj −1, represents the uncertainty in the jth generalized mass with respect to the truth modal space. The corresponding first order representation of the modal stiffness uncertainty has the form Δkjj =2ωj Δωj +ΔMjjΩj (8) Δkij =−ΔγijΩi −ΔγjiΩj i ≠ j (9) Note that the terms ΔMjj can be determined through careful modal testing as discussed in Ref. [23]. One of the main contributions of Ref. [21] was the derivation of the form of the substructure modal mass and stiffness covariance matrices in terms of test- or truth-analysis correlation uncertainty using analytical and numerical experimentation results. It was assumed that the expected values of the uncertainty in the substructure physical mass and stiffness matrices, E ΔM ( ) and E ΔK ( ) , are both zero. The same is then true for the modal matrices, E Δm ( ) =0 and E Δk ( ) =0. This then implies that the expected value of the uncertainty in the generalized masses, E ΔMjj ( ) =0, and the expected value of uncertainties in natural frequencies, E Δωj ( ) =0. Assuming that uncertainty in mass and stiffness are independent, it was shown that the variance of the off-diagonal substructure modal mass uncertainty terms is given by E Δmij 2 ( ) = Ωj −Ωi Ωj +Ωi E Δγij 2 ( ) −E Δγ ji 2 ( ) ⎡ ⎣ ⎤ ⎦ (10) The variance of the diagonal modal stiffness uncertainty can be computed using E Δkjj 2 ( ) = 4Ω j E Δωj 2 ( ) −Ω j 2E Δ Mjj 2 ( ) (11) while the variance of off-diagonal terms Δkij is given by E Δkij 2 ( ) = Ωj −Ωi Ωj +Ωi Ωj 2E Δγ ji 2 ( ) −Ω i 2E Δγ ij 2 ( ) ⎡ ⎣ ⎤ ⎦ (12) 138
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