=⎡ ⎣ 0 0 RTBT 0 ΘTKΘ −ΘTKGBT +ΘTBT BR −BGKΘ+BΘ BGKGBT −2BGBT ⎤ ⎦ =⎡ ⎣ 0 0 RTBT 0 Ω2 0 BR 0 −FI ⎤ ⎦ where FI = Ns ∑s=1 B(s)G(s)B(s)T =BGBT (28) MI = Ns ∑s=1 B(s)G(s)M(s)G(s)B(s)T =BGMGBT (29) Let us summarize both formulations: ⎡ ⎣ I 0 0 0 I 0 0 0 Mres ⎤ ⎦ ⎡ ⎣ ¨α ¨η ¨λ ⎤ ⎦ + ⎡ ⎣ 0 0 RTBT 0 Ω2 ΘTBT BR BΘ −Fres ⎤ ⎦ ⎡ ⎣ α η λ ⎤ ⎦ =TT dual f 0 (17) ⎡ ⎣ I 0 0 0 I −Ω−2ΘTBT 0 −BΘΩ−2 MI ⎤ ⎦ ⎡ ⎣ ¨α ¨η ¨λ ⎤ ⎦ +⎡ ⎣ 0 0 RTBT 0 Ω2 0 BR 0 −FI ⎤ ⎦ ⎡ ⎣ α η λ ⎤ ⎦ =¯TT dual f 0 (25) Remarks 1. Let us note that although the reduced matrices are different, the approximations are based on identical subspaces and hence the results obtained with the formulations (17) and (25) are identical. It is however noteworthy that when the residual attachment modes are used, the coupling between the attachment modes on the vibration modes Θ appears in the reduced stiffness, see (17), whereas when the non-residualized attachment modes are used, the coupling appears in the mass matrix, see (25). 2. In the reduced stiffness matrix (27) we observe a coupling between the attachment modes and the rigid body modes. This can be explained by the fact that the rigid body modes must be considered as part of the static solutions as explained before, so that the block 0 RTBT BR FI must be seen as the dually assembled statically condensed flexibility, similar to the statically condensed (Guyan) stiffness matrix in the classical Craig-Bampton method. It then becomes clear that the form (25) is the true counterpart of the classical Craig-Bampton. Interface Reduction in the Dual Craig-Bampton method 317
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