CΔKCBT i = RiC Δk i RiT (27) The diagonal terms in the covariance matrices correspond to the mean square values, or variances, of the mass and stiffness uncertainties in the corresponding vector ΔpX . Through extensive numerical experimentation, it was found that the covariance matrices for both the modal mass and stiffness are diagonal, meaning that the terms within each of the vectors Δpm i and Δp k i are uncorrelated. By either specifying, or being given, the variances of the correlation metrics E Δω j i ( ) 2 ⎡ ⎣⎢ ⎤ ⎦⎥ , E ΔMjj i ( ) 2 ⎡ ⎣⎢ ⎤ ⎦⎥ = E mjj i ( ) 2 ⎡ ⎣⎢ ⎤ ⎦⎥ , and E γij i ( ) 2 ⎡ ⎣⎢ ⎤ ⎦⎥ , the substructure modal uncertainty covariance matrices for each substructure can be easily constructed using Eqs. (10) through (12). 2.3 Propagation of Substructure Uncertainty into System The system matrices can be synthesized from the CB substructure representations by applying the appropriate constraints at the substructure interfaces. The uncoupled system displacement vector uG can be related to the coupled system displacement vector uSYS using the transformation uG = uCB 1 uCB 2 uCB nsub ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎭ ⎪ ⎪ = T1 T2 Tnsub ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ uQ uA ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ =TuSYS (28) where uQ = q 1T q2T qnsubT { } T is a partition of the coupled system displacement containing all of the component modal degrees of freedom from the substructures, and uA is the partition containing all the non-redundant substructure interface degrees of freedom. For example, in the case of two CB substructures uG = q1 ua 1 q2 ua 2 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ = I 0 0 0 0 I 0 I 0 0 0 I ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ q1 q2 ua 1 ⎧ ⎨ ⎪⎪ ⎩ ⎪ ⎪ ⎫ ⎬ ⎪⎪ ⎭ ⎪ ⎪ = T 1 T2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ uSYS (29) in which T1 = I 0 0 0 0 I ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ T2 = 0 I 0 0 0 I ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Uncertainty in the uncoupled CB substructure mass matrices can then be related to the uncertainty in the system coupled mass matrix using the expression ΔMSYS =T T ΔMGT = T 1T T2T TnsubT ⎡ ⎣ ⎤ ⎦ ΔMCBT 1 0 0 0 ΔMCBT 2 0 0 0 0 0 ΔMCBT nsub ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ T1 T2 Tnsub ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ (30) Transforming to system modal coordinates then gives the uncertainty in the system modal mass 141
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