3 An alternative formulation of the Dual Craig-Bampton As recalled in the previous section, the Dual Craig-Bampton as proposed earlier in [12] uses the residual attachment modes. This leads to the reduced matrices (19, 18) where the coupling between attachment modes and internal modes appears in the reduced stiffness matrix ˜K. In this paper dealing with interface reduction, we will see that it might be more appropriate to use an other equivalent form of the Dual Craig Bampton. To this end we use attachment modes G(s) (9) and not the residual ones G(s) res (14). Thus the approximation is now written as u λ = ⎡ ⎢⎢⎢ ⎢⎣ R(1) Θ(1) 0 −G1B(1) T . . . . . . .. . 0 R(Ns) Θ(Ns) −GNsB(Ns) T 0 ··· 0 I ⎤ ⎥⎥⎥ ⎥⎦ ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣ α(1) η(1) .. . α(Ns) η(Ns) λ ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎦ (22) = R Θ −GBT 0 0 I ⎡⎣ α η λ ⎤ ⎦ (23) = ¯Tdual ⎡ ⎣ α η λ ⎤ ⎦ (24) When only a small number of vibration modes are considered (typically the lowest modes in Θ), this representation is an approximation yielding a reduction basis. Using this reduction basis to transform the system we find the following reduced problem ¯M⎡ ⎣ ¨α ¨η ¨λ ⎤ ⎦ +¯K⎡ ⎣ α η λ ⎤ ⎦ = ¯TT dual f 0 (25) where the reduced matrices are now ¯M= ¯TT dual M0 0 0 ¯Tdual (26) =⎡ ⎣ RTMR 0 −RTMGBT 0 ΘTMΘ −ΘTMGBT −BGMR−BGMΘ BGMGBT ⎤ ⎦ =⎡ ⎣ I 0 0 0 I −Ω−2ΘTBT 0 −BΘΩ−2 MI ⎤ ⎦ ¯K= ¯TT dual KBT B 0 ¯Tdual (27) Daniel J. Rixen 316
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