in which the symbol ⊗ represents the Kronecker product between two matrices [25], given by A⊗B= A11B A11B A1mB A21B A22B An1B AnmB ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ (19) where A is an n×m matrix. Using these results, the expression for mass uncertainty in Eq. (16) can be rewritten as vec ΔMCBT i ( ) = R CBT i ⊗R CBT i ⎡⎣ ⎤⎦vec Δmi ( ) (20) in which RCBT i = MCB i φ CB i , vec ΔMCBT i ( ) is a n CB i ( )2 ×1 vector, Δpm i is n s i ( )2 ×1. The uncertainty matrices Δmi and ΔMCBT i are both symmetric, so only the lower triangular terms must be included in the propagation analysis. The vech X( ) operator [25] can be used to extract the lower triangular terms of a symmetric n×n matrix and stack them column-wise in a n n+1 ( ) / 2 dimensional column vector. The elimination matrix, Sn [26], can be used to relate the vec X( ) and vech X( ) operators as vech X( ) =Snvec X( ) (21) where Sn is a n n+1 ( ) / 2 ⎡⎣ ⎤⎦ ×n 2 full row rank matrix with a single 1.0 in each row. The inverse elimination matrix, or duplication matrix, can be formed such that vec X( ) =Sn −1vech X( ) (22) in which Sn −1 is an n2 × n n+1 ( ) / 2 ⎡⎣ ⎤⎦ full column rank matrix, also with a single 1.0 in each row, and S nSn −1 = I n n+1 ( )/2 . Applying these definitions to Eq. (20) produces vech ΔMCBT i ( ) =S nCB i R CBT i ⊗R CBT i ⎡⎣ ⎤⎦ S nCB i ( ) −1 vech Δmi ( ) (23) or using the simplifying notation, ΔpMCBT i =vech ΔMCBT i ( ) , Δp m i =vech Δmi ( ) , and Ri =S nCB i R CBT i ⊗R CBT i ⎡⎣ ⎤⎦ S nCB i ( ) −1 , Eq. (23) becomes ΔpMCBT i = Ri Δp m i (24) where matrix Ri has dimension n CB i n CB i +1 ( ) / 2 ⎡⎣ ⎤⎦ × n s i n s i +1 ( ) / 2 ⎡⎣ ⎤⎦ . The corresponding equation relating substructure modal stiffness and CB stiffness uncertainties is given by ΔpK CBT i = Ri Δp k i (25) Variance in free substructure modal mass can be related to variance in the CB substructure mass using linear covariance propagation [12]. Taking the expectation of the outer product of Eq. (25) with itself gives E ΔpMCBT i Δp MCBT iT ( ) =C ΔMCBT i = Ri E Δp m i Δp m iT ( ) RiT = RiC Δm i RiT (26) in which CΔMCBT i and C Δm i are the covariance matrices for uncertainty in CB substructure mass and free interface substructure modal mass, respectively. The analogous stiffness equation is given by 140
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