3. Let us stress again that exactly like in the original formulation(17) of the Dual Craig-Bampton, the last set of equations in (25) represent a weakened compatibility condition. This can be seen from the fact that this last set of equation originates from −BG I Substructure equilibrium equations Interface compatibility Thus the weak compatibility condition imposed in the reduced method can be interpreted as follows. The local equilibrium is only satisfied weakly do to the approximation of the local dofs, resulting in a force error resf on the substructure. If that force error would be applied to the substructure, static displacements resu =Gresf would result. Then the weak compatibility states that the compatibility error on the interface can be as big as the incompatibility of resu on the interface. 4. For the original formulation (17) it can easily be verified that setting Mres = 0 would enforce exact interface compatibility and in that case the Dual CraigBampton is equivalent to the MacNeal method [12]. For the new form (25) such an equivalence can not be found. However one can verify that if all modes are included in Θ (in which case the method is a mere transformation but not a reduction) the exact compatibility is retrieved: the last set of equations in (25) can be written as B −ΘΩ−2 ¨η+GMGBT ¨λ+Rα−GBTλ =−BGf WhenΘcontains all modes, G=ΘΩ−2ΘT and we can write B −ΘΩ−2 ¨η+ΘΩ−4ΘTBT ¨λ+Rα−GBTλ =−BΘΩ−2ΘTf or BΘΩ−2 −¨η+Ω−2ΘTBT ¨λ+ΘTf +B Rα−GBTλ =0 Taking in account the modal equilibrium equations corresponding to the first set of equations in (25), BΘΩ−2 Ω2η +B Rα−GBTλ =Bu=0 concluding the verification that when all modes are included, exact compatibility is satisfied. 5. The fact that a weak compatibility is allowed in the Dual Craig-Bampton implies that the infinite eigenvalues related to the Lagrange multipliers λ in the non-reduced problem (2) are no becoming finite. Since those eigenvalues are related to the compatibility constrained it can be showed that they are negative. In practice those negative eigensolutions will appear only in the higher spectrum is the reduction space is rich enough. Also using so-called model truncation Daniel J. Rixen 318
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