Direct decoupling of substructures using primal and dual formulation it is pointed out that DoFs used to enforce equilibrium need not to be the same as DoFs used to enforce compatibility: this gives rise to the so called non collocated approach, as opposite to the traditional approach in which such DoFs are the same, which is called collocated. In this paper, direct decoupling techniques are considered. Specifically, primal formulation for decoupling is developed and compared with dual formulation: it is shown that both formulation provide the same result when the number of DoFs used to enforce compatibility is equal to the number of DoFs used to enforce equilibrium, i.e. in the collocated approach and in some special case of non collocated approach. On the contrary, when the number of DoFs used to enforce compatibility is different from the number of DoFs used to enforce equilibrium, the two approaches provide different results. The techniques are applied using simulated data from a torsional system describing a two-speed transmission. 2 Direct decoupling techniques The coupled structural systemAB(NAB DoFs) is assumed to be made by an unknown subsystemA (NA DoFs) and a residual subsystemB (NB DoFs) joined through a number of couplings (see fig. 1). The residual subsystem (B) can be made by one or more substructures. The degrees of freedom (DoFs) of the coupled system can be partitioned into internal DoFs (not belonging to the couplings) of subsystemA(a), internal DoFs of subsystemB(b), and coupling DoFs (c). COUPLED SYSTEM INTERNAL DOFS UNKNOWN SUBSYSTEM COUPLING DOFS RESIDUAL SUBSYSTEM INTERNAL DOFS FRFS AT COUPLING DOFS + FRFS AT SOME INTERNAL DOFS → NOISE + IDENTIFICATION ERRORS PHYSICAL (FE) MODEL A B Fig. 1 Scheme of the decoupling problem It is required to find the FRF of the unknown substructure A starting from the FRF of the coupled systemAB. The subsystemA can be extracted from the coupled systemABby cancelling the dynamic effect of the residual subsystemB. This 49
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