u=⎡ ⎢⎣ u(1) .. . u(Ns) ⎤ ⎥⎦ f = ⎡ ⎢⎣ f(1) .. . f(Ns) ⎤ ⎥⎦ B= B(1) ··· B(Ns) B(s) = b(s) 0 so that the dynamic equilibrium of a substructured system can be written as M0 0 0 ¨u λ + KBT B 0 u λ = f 0 (4) The solution of the dynamic equilibrium for a substructure (second set of equations in (4) ) can be found in the form u(s) =u (s) stat + n(s)−m(s) ∑r=1 θ(s) r η (s) r (5) where n(s) is the dimension of the local problem, θ(s) r are the free interface modes number of substructure s solution of K(s) −ω2 r M(s) θ(s) r =0 (6) and where the quasi-static solutionu(s) stat is givenby u(s) stat =−K(s)+B(s)T λ + m(s) ∑i=1 R(s) i α (s) i (7) In this last equation K(s)+ is a pseudo-inverse in case the substructure are floating, hencewhen m(s) local rigid body modes R(s) i exist (see e.g. [7]). If the substructure has no rigid body modes when disconnected from its neighbors, K(s)+ =K(s)− 1 is simply the inverse of K. Defining the projector P(s) =I−M(s)R(s)R(s) T such that R(s) T P(s) =0 , P(s)M(s)R(s) =0 (8) we can compute a generalized inverse where the rigid body modes have been filtered out, namely G(s) =P(s) T K(s)+P(s) (9) so that the quasi-static solution can also be written as u(s) stat =−G(s)B(s)T λ + m(s) ∑i=1 R(s) i α (s) i (10) Interface Reduction in the Dual Craig-Bampton method 313
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