Assuming now that the solution in the substructure is represented in an approximate way by using only the first n(s) θ internal modes, one writes u(s) −G(s)B(s) T λ +R(s)α(s) +Θ(s)η(s) (11) where Θ(s) is a matrix containing in its column the first n(s) θ internal modes of the substructure. Instead of using the so-called attachment modes G(s) representing the static response to interface forces, it is common to use the residual attachment modes (see for instance the MacNeal and the Rubin methods [11, 14]). To find the residual flexibility, we observe that the flexibility has as spectral expansion G(s) = n(s)−m(s) ∑r=1 θ(s) r θ (s)T r ω(s)2 r (12) so that the dynamic response of a substructure can be equivalently approximated by u(s) =−G (s) resB(s)T λ +R(s)α(s) +Θ(s)η(s) (13) where G(s) res = n(s)−m(s) ∑ r=n (s) θ +1 θ(s) r θ (s)T r ω(s)2 r =G(s) − n(s) θ ∑r=1 θ(s) r θ (s)T r ω(s)2 r (14) Using the block diagonal notations we can write u=−GresB Tλ+Rα+Θη (15) where Gres =⎡ ⎢⎢ ⎣ G(1) res 0 . . . 0 G(Ns) res ⎤ ⎥⎥ ⎦ R=⎡ ⎢⎣ R(1) 0 . . . 0 R(Ns) ⎤ ⎥⎦ Θ= ⎡ ⎢⎣ Θ(1) 0 . . . 0 Θ(Ns) ⎤ ⎥⎦ α=⎡ ⎢⎣ α(1) .. . α(Ns) ⎤ ⎥⎦ η = ⎡ ⎢⎣ η(1) .. . η(Ns) ⎤ ⎥⎦ The approximation (13) can be written in matrix form as Daniel J. Rixen 314
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