be useful in the derivation of the fixed-interface normal modes and the interface constraint modes, where the bn boundary coordinates are assembled in the set bu and the in free (interior) coordinates are gathered in the set iu : bb bi bb bi b b b ib ii ib ii i i M M u K K u f M M u K K u 0 ª º ª º ª º ª º ª º + = « » « » « » « » « » ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ . (2) Restraining all boundary DOF and solving the eigenvalue problem ( ) 2 ii p ii p K M 0 −ω Φ = , (3) one obtains the fixed-interface normal modes, which can be stacked in the modal base in Φ . Upon normalization of the fixed-interface normal modes with respect to the mass matrix, the generalized orthogonality of the modes can be expressed in the following compact form: ( ) T T 2 in ii in ii in ii in nn p M I , K diag Φ Φ = Φ Φ =Λ = ω , (4) where ii I is ( i i n n× ) the identity matrix and nn Λ is the diagonal matrix of squared linear natural frequencies of the fixed-interface normal modes. According to the partitioning of Eq.(2), the fixed interface modal matrix nΦ for all coordinates of the FE model takes on the form bn n in 0ª º Φ =« » Φ¬ ¼ . (5) The interface constraint modes are the static deformations of the interior coordinates by imposing a unit displacement at one physical coordinate of the set bu , while the remaining DOF of the same set are restrained. This procedure is applied consecutively to all coordinates of the set bu . That is, bb bi bb bb ib ii ib ib K K I R K K 0 ª ºª º ª º = « »« » « » Ψ ¬ ¼¬ ¼ ¬ ¼ , (6) where bb R is the corresponding external force matrix and bb I is the ( b b n n× ) identity matrix. Consequently, the constraint-mode matrix cΨ is given by bb bb c 1 ib ii ib I I K K− ª º ª º Ψ = =« » « » ª º Ψ −« » ¬ ¼ ¬ ¼ ¬ ¼ . (7) 3.2 Craig-Bampton method The displacement-based CMS methods are based on a Rayleigh-Ritz transformation of the form u q =Φ , (8) where the solution is approximated in a reduced subspace by representing the physical coordinates, u, in terms of component generalized coordinates q. The matrix Φ is called the reduction base, Ritzvector base, dynamic component mode superset, or simply the transformation matrix. The columns of the transformation matrix consist of preselected component modes of the following type: rigid body modes, normal modes of the undamped system (global vibration modes), attachment modes, 39
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