27 A Complex Power Approach to Characterise Joints in Experimental Dynamic Substructuring 283 A fAi uA b B uB b gA b gB b A fAi uA b B uB b A fAi uA b B uB b Cc Kc a b c Fig. 27.1 Assembly of two substructures A and B using one node at each interface. (a) Assembly of two substructures A and B. u are displacements, g internal forces and f applied forces. (b) Assembly using a rigid connection, i.e. no relative displacement at the interface. (c) Assembly using a compliant connection, i.e. relative displacement at the interface. The interfaces are connected by a linear spring and dashpot in parallel Starting with a recap of the DS framework, a rigid connection is considered first, i.e. no relative displacement between substructures at the interface is allowed. This is followed by the introduction of the compliant interface model. In both cases, an expression in terms of system matrices as well as the system’s receptance is considered. 27.2.1 Rigid Connection Traditional substructuring techniques couple substructures as if they are rigidly connected, i.e. no relative displacement of the interface nodes between the substructures is allowed, see Fig. 27.1b. This results in two conditions that need to be satisfied when coupling substructures: 1. Compatibility: this condition states that the Degrees of Freedom (DoFs) associated with coinciding interface nodes of the respective substructures are equal. The compatibility condition can be expressed by BuD0 (27.2) where as Bis a signed boolean matrix operating on the interface DoFs. This expression states that any pair of matching interface DoFs uA b anduB b have the same displacement, i.e. uA b uB b D0. 2. Equilibrium: this condition requires force equilibrium between the interface DoFs. The equilibrium condition can be expressed by LTg D0 (27.3) where Lis a boolean matrix localising the interface DoFs from the global set (for more details on constructingBand L, the reader is referred to [6]). This expression states that the sum of a matching pair of interface DoFs gA b and gB b should be equal to zero, i.e. gA b CgB b D0. The coupled system can be obtained by using either a primal or a dual formulation approach, as discussed in [6]. When the primal formulation is used, a unique set of interface DoFs is defined while the interface forces are eliminated as unknowns. In the dual formulation however, the full set of DoFs is retained. Additionally, the dual assembled system is obtained by satisfying the interface equilibrium a priori. For an interface model which allows for relative displacement between the two substructures, the dual formulation is therefore the only feasible formulation. Consequently, the equilibrium condition is implemented by defining g D BT (27.4) with being a Lagrange Multiplier representing the interface force intensity. It can be verified that Eq. (27.4) automatically satisfies the equilibrium condition [6].
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