where ( ) ( ) ( ) ( ) T T T T FE FE FE FE cbj cb j cbj cb j jj j j jj j j M M , K K , M M , K K =Φ Ψ =Φ Ψ=ΨΨ=ΨΨ . (14) The combined reduced system, Eq.(13), is usually not decoupled. For problems with nonlinear local effects, it is useful to separate the force vector f of Eq.(1) in an external non-state dependent force vector ext f and a state dependent force vector kf : ext k f f f = + (15) The external force vector ext f is the vector of nodal forces due to applied external forces and does not include forces, which result form local effects of the system (e.g. contact forces). The latter forces are assembled in the force vector kf . Regarding local contact effects, the forces of the vector kf are formulated as a nonlinear function of the state variable vector, namely, ( ) k k f f u = . (16) Thus, in terms of a component modal model the vector kf depends on the generalized coordinates cb q and j q . 4.1 Mass-orthogonal local Ritz-vectors Due to the fact that the extension of the Craig-Bampton component modes with JIM leads to a reduced system which is not decoupled, see Eq.(13), it is not possible to neglect the inertia effects of the joint interface mode group. To achieve a mass decoupled reduced system, the JIM have to be modified so that they are mass-orthogonal to the component modes of the Craig-Bampton mode base. To get mass-orthogonal local Ritz-vectors, j ˆΨ , it is necessary to remove the content of the component modes, which are assembled to the transformation matrix cb Φ , from the local Ritz-vectors j Ψ . This can be formally written as j j cb ˆΨ =Ψ −Φ κ, subject to ( )T FE cb j cbj M ˆ M 0 Φ Ψ = = , (17) where κ is a scaling matrix. Premultiplying the first part of Eq.(17) with ( )T FE cb M Φ yields ( ) ( ) ( ) 1 T T FE FE cb cb cb j M M − κ= Φ Φ Φ Ψ . (18) Finally, Eqs.(17) and (18) may be combined to give: ( ) ( ) ( ) 1 T T FE FE j j cb cb cb cb j ˆ M M − Ψ =Ψ −Φ Φ Φ Φ Ψ , (19) where the modified matrix j ˆΨ contains mass-orthogonal local Ritz-vectors. The transformation matrix thus obtained can be written as cb j ˆ ˆ u q ª º = Φ Ψ ¬ ¼ . (20) Using the latter reduction rule, a reduced system in the form 41
RkJQdWJsaXNoZXIy MTMzNzEzMQ==