( ) ( ) ( ) ( ) ( ) ( ) T T cb cbj cb cb cb cb cb ext k cb j T T T j jj j j j cbj jj ˆ q K K M 0 q ˆ f f q ,q ˆ qˆ ˆ 0 M q ˆ ˆ ˆ ˆ K K ª º ª º ª º Φ Φ ª º ª º ª º « » « » « » + = + « » « » « » « » « » « » « »« » Ψ Ψ ¬ ¼ ¬ ¼¬ ¼ « » « » « » ¬ ¼ ¬ ¼ ¬ ¼ (21) is obtained, where the reduced mass and stiffness matrices are ( ) ( ) ( ) T T T FE FE FE cbj cb j jj j j jj j j ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ K K , K K , M M = Φ Ψ = Ψ Ψ = Ψ Ψ . (22) Moreover, the mass-orthogonal local Ritz-vectors are normalized, in order to satisfy jj Mˆ I = and ( ) 2 jj jj j ˆ ˆ ˆ K diag =Λ = ω . It should be pointed out that the matrices cb M and cb K , as well as the generalized coordinates cb q of Eq.(21), are the same as those of Eq.(13). The original Craig-Bampton mode base is more or less accurate for a predefined frequency range. The latter frequency range is determined by the considered fixed-interface normal modes, and an upper frequency limit, uω , may be declared. Due to the fact that for the computation of the massorthogonal local Ritz-vectors the content of the Craig-Bampton component modes is totally removed, the eigenfrequencies of the local mass-orthogonal Ritz-vectors are above uω . Consequently, the dynamics of the mass-orthogonalized local Ritz-vectors can be neglected, while the quasistatic influence on the local displacement field can not. It should be mentioned that the frequencies of the mass-orthogonal local Ritz-vectors are - from a physical point of view - just numerical quantities, but regarding numerical time integration, they specify whether the inertia effects have to be taken into account or not. In case of weak stiffness coupling between the DOF of the global vibration modes and the local Ritz- vectors, it can be moreover assumed that ( )T jj j cbj cb jj j ˆ ˆ ˆ ˆ ˆ M q K q K q + , (23) where a ‘weak stiffness coupling’ means that ( )T cbj cb jj j ˆ ˆ ˆ K q K q . (24) A mathematical proof of the range of Eq.(24) is left to a future contribution. Applying the assumption of Eq.(23) to Eq.(21) leads to a reduced equation of motion in the form of ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 T 1 T T cb cb cb cbj jj cbj cb cb cbj jj j ext k cb j 1 T T j jj j ext k cb j cbj cb ˆ ˆ ˆ ˆ ˆ ˆ ˆ M q K K K K q K K f f q ,q ˆ ˆ ˆ ˆ ˆ 0 q K f f q ,q K q − − − ª º ª º ª º + − = Φ − Ψ + « » « »¬ ¼ ¬ ¼ ¬ ¼ ª º ª º = − Ψ + − « » ¬ ¼ ¬ ¼ , (25) where the differential equations due to the local Ritz-vector DOF are replaced by a set of (nonlinear) algebraic constrain equations. A comparison of the equation of motion, Eq.(21), and its modified form, Eq.(25), shows that the dimension of the system of differential equations, which needs to be time integrated, is much smaller for the latter formulation. The penalty which needs to be paid is an additional set of algebraic equations. 5 Numerical Example In this section a generic dynamical example will be presented in order to illustrate the proposed algorithm and to demonstrate its advantages. 42
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