Walter D’Ambrogio and Annalisa Fregolent number of DoFs used to enforce equilibrium. However, it is shown that, when equilibrium is enforced at coupling DoFs only, the use of non collocated approach gives rise to an interface flexibility matrix that is ill conditioned at the same frequencies as for standard interface. Furthermore, it is shown that primal formulation and dual 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 greater than the number of DoFs used to enforce equilibrium, the two approaches provide different results: in fact, using the primal approach, compatibility is enforced exactly whilst equilibrium of constraint forces is only approximate; the opposite occurs using the dual approach, i.e. compatibility is only approximate whilst equilibrium of constraint forces is enforced exactly. The techniques are applied using simulated data from a torsional system describing a two-speed transmission. Results agree with theory. Acknowledgements This research is supported by grants from University of Rome La Sapienza. Appendix: Format and properties of dynamic stiffness matrices To obtain the condensed dynamic stiffness matrices of the coupled structure ABand of the residual substructure B, master DoFs M(interface DoFs to be measured and to be reatined after condensation) and slave DoFs S (unmeasured and to be eliminated after condensation) must be defined. According to which option is selected for interface DoFs, master DoFs and slave DoFs are: a) When standard interface is considered: • M≡c, that is interface DoFs are just the coupling DoFs; • for the residual subsystemB, S ≡b, that is the slave DoFs are all its internal DoFs; • for the coupled systemAB, S≡a∪b, that is the slave DoFs are all its internal DoFs. b) When extended interface is considered, also some internal DoFs i of the residual substructure are included among master DoFs: • M≡c∪i, as stated previously; • for the residual subsystemB, S≡b\i ≡u, that is the slave DoFs are given by the set difference between its internal DoFs and the internal DoFs included in the interface; • for the coupled systemAB, S≡a∪u. b1) A special case occurs when i ≡b. In this case, u is an empty set. 74
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