constrained modes, inertia-relief modes and local Ritz-vectors. Other types of assumed modes e.g. Krylov vectors [14] or nonlinear normal modes [12] may also be employed as component modes. The transformation matrix of the fixed-interface Craig-Bampton method, which approximates the solution in a reduced subspace, consists of two parts: the interface constrained modes, which have been defined by Eq.(7), and the first km fixed-interface normal modes of Eq.(5), which are assembled to the matrix kΦ . Typically, only those km normal modes are retained and assembled to the matrix kΦ , which are associated with an eigenfrequency below the maximum frequency of interest. That is, [ ] cb c k Φ = Ψ Φ . (9) It is worth noting that the Craig-Bampton transformation preserves all physical coordinates of set bu in the reduced subsystem. Eq.(8) and Eq.(9) together with the equation of motion form the reduced equation of motion or the so called component modal model. The reduced equation of motion in generalized coordinates is ( )T cb cb cb cb cb M q K q f + = Φ , (10) where the reduced mass and stiffness matrices, cb M and cb K , are defined as: ( ) ( ) T T FE FE cb cb cb cb cb cb M M , K K = Φ Φ = Φ Φ (11) 4 On a proper extension of the Craig-Bampton reduction base by local Ritz-vectors In this section we shortly review and modify the fixed-interface CMS method developed by Witteveen and Irschik [15], where the Craig-Bampton method is extended by Ritz-vectors needed for nonlinear local effects. This type of Ritz-vectors will be further denoted as ‘local Ritz-vectors’. In the subsection 4.1 the local Ritz-vectors will be modified in order to get an efficient formulation of the reduced equation of motion. The typical Craig-Bampton CMS [3] does preserve the nodal DOF at interfaces. The interface is defined as those DOF on which external forces may act. According to the partitioning of Eq.(2), the interface DOF have to be interpreted as boundary DOF. Furthermore, each interface DOF leads to an additional constraint mode in the final mode base. For an accurate and local application of contact and friction laws in the contact region, the nodal DOF of the involved contact surfaces need to be considered as interface DOF. This leads to an inefficient number of modes. In [13], [14] and [16] local Ritz-vectors, so called Joint Interface Modes (JIM), have been introduced. The latter modes together with the component modes of the Craig-Bampton method form a suitable Rayleigh-Ritz coordinate transformation for problems including local contact and friction effects. For the computation of the JIM, Newton’s third law (principle of equivalence of forces) across the contact region is explicitly accounted for. This leads to a remarkable small number of JIM, which have to be considered in order to approximate the solution of the contact problem. The considered j m JIM are assembled in the matrix j Ψ . The extension of the classical Craig-Bampton transformation [3] with the matrix j Ψ leads to a Rayleigh-Ritz coordinate transformation in the form of cb j u q ª º = Φ Ψ ¬ ¼ . (12) Substituting Eq.(12) into Eq.(1) and premultiplying the resulting equation by T cb j ª º Φ Ψ ¬ ¼ gives ( ) ( ) ( ) ( ) T cb cbj cb cbj cb cb cb T T T j j cbj jj cbj jj j M M q K K q f q q M M K K ª º ª º ª º Φ ª º ª º « » « » « » + = « » « » « » « » « » Ψ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ , (13) 40
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