19 Methods for Component Mode Synthesis Model Generation for Uncertainty Quantification 179 Œ‰ D ŒKnn 1 ŒKna ; (19.1) where a represents interface degrees of freedom and n the interior degrees of freedom. 19.2.4 Craig-Bampton Transformation Matrix The Craig-Bampton transformation matrix, Wis assembled using the component and constraint modes. This matrix transforms the generalized degrees of freedom used to synthesize the reduced order model back into the original system’s degrees of freedom. This takes the form un ua D ˆ‰ 0 I „ƒ‚… W pk ua ; (19.2) where pk is the vector of modal coordinates, and un and ua are displacements in generalized and physical coordinates, respectively. 19.2.5 Reduced Stiffness and Mass Matrices Once the Craig-Bampton transformation matrix is assembled, it is used to calculate the reduced mass and stiffness matrices for the component. These matrices are then used to assemble the reduced order model back into the main system and include both interface as well as modal degrees of freedom. They are calculated as h OMCBi DŒW T ŒM ŒW (19.3) hOKCBi DŒW T ŒK ŒW : (19.4) 19.3 Craig-Bampton Generation for UQ Studies As was mentioned in Sect. 19.1, there is significant computational cost associated with calculating the fixed-interface modes, which is outlined in Sect. 19.2.2. It is also noted that for extremely large component models, there can be significant cost associated with inverting the stiffness matrix in Eq. (19.1). However, for this study it is assumed that the component model is a small enough size such that the fixed-interface mode calculation dominates. For a UQ study involving hundreds or thousands of parameter sets, re-calculating the fixed-interface modes for each set would put too much computational burden on the analysis. Presented in the next two subsections are methods which perturb the Craig-Bampton model to account for uncertain parameter sets without recalculating the fixed-interface modes. 19.3.1 REMAP Technique The first technique presented is further separated into two sub techniques, REMAP1 and REMAP2. Both of these technique use the nominal fixed-interface modes when calculating the Craig Bampton transformation matrix. The assumption is that the perturbed state, due to the uncertain parameter set, is small enough that major changes to the fixed-interface modal matrix do not occur. For REMAP1, both the constraint modes, ‰, as well as the component mass and stiffness matrices are updated to reflect the UQ parameters. The Craig Bampton transformation matrix and reduced mass and stiffness matrices are shown in Eqs. (19.5), (19.7) and (19.8), respectively. REMAP2 uses all nominal entries in the Craig Bampton transformation matrix,
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