The projection basis follows a similar reasoning. However, it deserves more attention because the ingredients are different: the constraint modes are not the direct transposes of the constraint modes in the right basis due to the unsymmetric nature of the component matrices. The left basis might be written following the left eigenvector problem and writing that basis in a Craig-Bampton like scheme, TT cb,L = I −KbiK−1 ii 0 φPi,L T +φP b,L TKbiK−1 ii . (30) To facilitate a numerically robust implementation, the modified mode vectors in equations (29) and (30) are orthogonalized with respect to the mass matrix. 3.2 Pseudo-vectors with unique global generalized dof Use of global pseudo-vectors instead of the fixed interface modes has another advantage: since these modes are global to the system one can choose to associate them to a unique global degree of freedom on the system level. This operation might be summarized with a few matrix transformations. To exemplify the operations, we are going to consider a system that is divided into two components. Starting from the modal expansion where one set of generalized set of dofs are kept, the transformation equation might be written as ⎡ ⎣ qb qi,1 qi,2 ⎤ ⎦ =⎡ ⎣ I φP b −K−1 ii,1Kib,1 φPi,1 −K−1 ii,2Kib,2 φPi,2 ⎤ ⎦ ˜qb ηP , (31) where qb represents the common boundary dofs and qi,1 and qi,2 represents the internal dofs of component 1 and 2, respectively. To assemble the components, a transformation step is necessary using the first row of equation (31), namely, ˜qb =qb −φP b η. Use of this relation results in ⎡ ⎣ qb qi,1 qi,2 ⎤ ⎦ =⎡ ⎢⎢ ⎣ I 0 −K−1 ii,1Kib,1 φPi,1 +K−1 ii,1Kib,1 φP b −K−1 ii,2Kib,2 φPi,2 +K−1 ii,2Kib,2 φP b ⎤ ⎥⎥ ⎦ qb ηP , (32) indicating that component contributions of the amplitudes of the global modes can be assembled similarly to the interface degrees of freedom. This has an advantage in comparison to the conventional Craig-Bampton method, where the fixed interface modes are local to a component and thus generate separate degrees of freedom at the global level. With the just outlined scheme, the reduced system matrices are reduced one more level with respect to the conventional Craig-Bampton approach. Umut Tabak and Daniel J. Rix 272
RkJQdWJsaXNoZXIy MTMzNzEzMQ==