interface modes obtained from the statically condensed problem [4]. In this paper we propose a similar method, but now suitable for the Dual Craig-Bampton method. In section 2 we shortly recall the Dual Craig-Bamtpon method and in section 3 we propose an alternative, but equivalent, formulation of that substructuring method. In section 4 the concept of interface modes for the dual interface problem is introduced in order to reduce the interface problem. The method is then tested on two problems in sections 5 and 6. In section 7 we go deeper in the discussion of modes with interface incompatibility and interface modes. We conclude with some general remarks and an outlook on further research in section 8. 2 The Dual Craig-Bampton in a nuttshell The Dual Craig-Bampton method was proposed in [12] as a Dual version of the classical Craig-Bampton substructuring technique [3]. Here we shortly recall the method and introduce the notations used in this paper. Let us consider the linear dynamic equations for a structure Ma¨ua +Kaua =fa(t) (1) where Ma, Ka, ua and fa are respectively the global (assembled) mass and stiffness matrix, and the associated displacements and applied forces. Assume that the problem is decomposed in Ns non-overlapping substructures, the dynamic equilibrium equations can be locally expressed by ⎧⎪ ⎪⎨ ⎪⎪⎩ M(s) ¨u(s) +K(s)u(s) + b(s)T λ 0 =f(s) s =1,...Ns Ns ∑s=1 b(s)u(s) b =0 (2) where the superscript (s) indicates that the quantity belongs to substructure s, and where b(s) is a signed Boolean matrix such that the second equation in (2) represents the equality of displacements on the interface (i.e. the interface compatibility condition). ub is the restriction of the displacement degrees of freedom (dofs) to the boundary (interface) and λ represent the Lagrange multipliers associated to the interface compatibility condition such that b(s)T λ are the interface internal forces. Let us also use the block diagonal notation M=⎡ ⎢⎣ M(1) 0 . . . 0 M(Ns) ⎤ ⎥⎦ K= ⎡ ⎢⎣ K(1) 0 . . . 0 K(Ns) ⎤ ⎥⎦ (3) Daniel J. Rixen 312
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