2.2 Propagation of Uncertainty into Craig-Bampton Substructure The CB substructure representation is well suited as a building block for model validation of substructured systems. The ith substructure representation is generated using the coordinate transformation ui = uo i ua i ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ = φo i ψi 0 I ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ qi ua i ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ =TCB i u CB i (13) in which ua i represents the displacement of the substructure interface, and u o i is the displacement of the interior of the substructure. This representation is characterized by a combination of fixed interface substructure mode shapes, φo i , and a set of static shapes, Ψi = ψiT I ⎡ ⎣ ⎤ ⎦ T , called constraint modes, in which ψi =− K oo ( ) −1 Kao . The nCB i ×n CB i substructure mass and stiffness matrices in the CB space are then given by MCB i =T CB iT MiT CB i = I Mqa i Maq i MS i ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ KCB i =T CB iT KiT CB i = Ωo i 0 0 KS i ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ (14) where Ωo i is the matrix of fixed-interface eigenvalues for the ith substrcture, Mqa i is the mass coupling between fixed modal degrees of freedom and the physical interface degrees of freedom, and MS i and K S i represent the substructure mass and stiffness matrices statically reduced to the interface, respectively. If all of the fixed-interface modes are retained, then the transformation into the CB substructure representation is exact. However, a significant reduction in model size can be achieved by truncating the number of fixed-interface modes based on frequency. The uncertainty in ith substructure modal mass can be related to the uncertainty in the CB mass matrix using Δmi = φ CB iT ΔMCB i φ CB i (15) where φCB i are the n s i mass normalized nominal free substructure modes of interest in CB coordinates. Remember that the uncertainty in the substructure modal mass is obtained from a substructure test-analysis correlation process. The corresponding mode sets are sometimes called target modes. Pre- and post-multiplying each side of Eq, (15) by MCB i φ CB i and its transpose, respectively, gives MCB i φ CB i Δmi φ CB iT MCB i = P T iT ΔMCB i P T i = ΔMCBT i (16) in which the matrix PT i = φ CB i φ CB iT MCB i is an oblique projector [24] onto the column space spanned by the nominal substructure target modes being considered in the correlation analysis. Therefore, ΔMCBT i is the uncertainty in the CB mass matrix due to the substructure target modes. Note that PT i φ CB i = φ CB i φ CB iT MCB i φ CB i = φ CB i , therefore Eq. (15) can be rewritten as Δmi = φ CB iT P T iT ΔMCB i P T i φ CB i = φ CB iT ΔMCBT i φ CB i (17) This indicates that Δmi only contains information concerning the uncertainty from the target modes. The same can be said for the substructure modal stiffness uncertainty Δki . Therefore, the process of recovering both the CB mass and stiffness uncertainty matrices using Eq. (16) does not result in any additional loss of information. In order to proceed, the uncertainty matrices in Eq. (16) must be transformed to vectors using the vec X( ) operator, in which the columns in matrix X are stacked column-wise. It can be shown that for compatible matrices A, X, and B vec AXB ( ) = BT ⊗A ( )vec X( ) (18) 139
RkJQdWJsaXNoZXIy MTMzNzEzMQ==