ΔMSYS = φSYS T TT ΔMGT φSYS = φSYS T TiT ΔMCBT i Ti φ SYS i=1 nsub ∑ (31) where φSYS are the system modes of interest. Vectorizing the uncertainty matrices in Eq. (31) and employing elimination matrices gives vech ΔMSYS ( ) = φSYS T TiT ⊗φ SYS T TiT ⎡⎣ ⎤⎦vech ΔMCBT i ( ) i=1 nsub ∑ (32) or ΔpMSYS = W1 W2 Wnsub ⎡ ⎣ ⎤ ⎦ ΔpMCBT 1 ΔpMCBT 2 ΔpMCBT nsub ⎧ ⎨ ⎪ ⎪⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪⎪ ⎭ ⎪ ⎪ ⎪ =WΔpMCBT (33) in which matrices Wi = φ SYS T TiT ⊗φ SYS T TiT are of dimension n SYS ( ) 2 ×n CB i for n SYS system modes. Using the procedure outlined in the previous section, the system modal mass covariance matrix is then given by CΔMSYS =W CΔMCBT 1 0 0 0 CΔMCBT 2 0 0 0 0 0 CΔMCBT nsub ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ WT =WC MCBT WT (34) where it is assumed that the individual substructure uncertainties are uncorrelated with one another. A parallel analysis produces the same form for the system modal stiffness uncertainty CΔ KSYS =WCK CBT WT (35) The mean square values of the uncertainties in the system modal mass and stiffness matrices can then be recovered from the diagonal of the matrices CΔ MSYS and CΔ KSYS , respectively. Once the uncertainty in the system modal mass and stiffness is recovered, the uncertainty in the system modal correlation metrics can be determined using the methods presented in Ref. [21]. It was shown that the variance of the off-diagonal system cross-orthogonality terms can be expressed as E ΔγSYSij 2 ( ) = 1 ΩSYSi −ΩSYSj ( ) 2 E Δ KSYSij 2 ( ) +Ω SYSj 2 E Δ MSYSij 2 ( ) ⎡ ⎣ ⎤ ⎦ (36) where ΩSYSi is the ith system eigenvalue. The variance of the system natural frequencies is given by E ΔωSYSj 2 ( ) = 1 4ΩSYSj E Δ KSYSjj 2 ( ) +Ω SYSj 2 E Δ MSYSjj 2 ( ) ⎡ ⎣ ⎤ ⎦ (37) Using numerical experiments, it was shown that the off-diagonal uncertainty terms ΔγSYSij are not only zero mean, but normally distributed, and within each column, independent. Equation (4) then indicates that the term cj =1− γSYSjj 2 is the sum of the squares of nSYS −1 zero mean, normally distributed variables ΔγSYSij . If the terms ΔγSYSij all had unit variance, cj would be represented by a chi-square distribution with nSYS −1 degrees of freedom [3]. In the current case, the off142
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