90 S. W. B. Klaassen and D. J. Rixen YA YB YJ YAJB uA b −uB b =Δu Fig. 10.2 Component A is connected to component B via a mass-less joint represented by YJ. Due to the weakened connection the compatibility condition states that uA b −uB b = u The coupled admittance matrix YAB is then: YAB =Y−YBT BYBT −1 BY (10.10) Equation (10.10) is a single-line equation of LM-FBS to couple models. Note that, although only two models were coupled in the presented example, the equation holds for multiple components. 10.2.1.1 Weakening in the Interface: Adding Joint Dynamics Note that in the previous part the LM-FBS method is derived with strict compatibility and equilibrium between the components, and thus a rigid connection. In order to add a linear flexible joint one of two things can be done: Either the joint is added as a separate substructure into Eq. (10.10), which as explained before, is done easily. Or a joint is added as a relaxation of the compatibility condition between components Aand B; this method is extensively described in [6] but will be shortly repeated here. In Fig. 10.2 a flexible joint YJ is added between the interfaces of component Aand B. Because of the joint, a gap can occur between the boundary DoF of the two components altering the compatibility condition from Eq. (10.5): uA b −u B b = u → Bu = u (10.11) This gap is a response to the boundary forces λwhich act on the joint: u=YJ λ (10.12) Substituting these relations into Eq. (10.4) results in: Bu=BY f −BTλ = u =YJ λ (10.13) which again, is solved for λ: λ= BYBT +YJ −1 BYf (10.14) Similar to above, the weakly-coupled admittance matrix YAJB is found to be: YAJB =Y−YBT BYBT +YJ −1 BY (10.15)
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