Direct decoupling of substructures using primal and dual formulation Fig. 22 Rotational mobility at the coupling DoF of subsystem A: true (—), computed from fitted perturbed FRF (∗∗∗) with compatibility at DoFs θ3 and θ5 and equilibrium at DoFs θ4 and θ5 0 20 40 60 80 100 120 140 160 180 200 10−5 100 105 Frequency [Hz] Magnitude [rad s−1/(N m)] 0 20 40 60 80 100 120 140 160 180 200 −4 −2 0 2 4 Frequency [Hz] Phase [rad] Fig. 23 Rotational mobility at the coupling DoF of subsystem A: true (—), computed from fitted perturbed FRF (∗∗∗) with compatibility at DoFs θ4 and θ5 and equilibrium at DoFs θ3 and θ5 0 20 40 60 80 100 120 140 160 180 200 10−5 100 105 Frequency [Hz] Magnitude [rad s−1/(N m)] 0 20 40 60 80 100 120 140 160 180 200 −4 −2 0 2 4 Frequency [Hz] Phase [rad] 5 Summary and discussion The paper considers the decoupling problem, i.e. the identification of the dynamic behaviour of a structural subsystem, starting from the known dynamic behaviour of the assembled system, and from information about the residual subsystem, i.e. the remaining part of the structural system. Direct decoupling techniques are considered, that consist in adding to the assembled system a fictitious subsystem, which is the negative of the residual subsystem. Starting from the 3-field formulation (dynamic balance, compatibility and equilibrium), a primal approach for decoupling is developed and compared with a dual approach. Compatibility and equilibrium can be required either at coupling DoFs only (standard interface), or at additional internal DoFs of the residual subsystem (extended interface): for the dual approach, the choice of interface DoFs determines a set of frequencies at which the decoupling problem is ill conditioned. Apparently, when using an extended interface, the problem is singular at all frequencies, although this singularity is easily removed by smart inversion techniques. To avoid this problem, the non collocated approach, in which DoFs used to enforce equilibrium are not the same as DoFs used to enforce compatibility, can be considered: in this case the number of DoFs used to enforce compatibility must be less than the 73
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