1 Experimental-Analytical Dynamic Substructuring of Ampair Testbed: A State-Space Approach 3 number of input nodes and applying Maxwell-Betti’s reciprocity principle (see [5]). The passivity criterion states that the power supplied to the system is non-negative and can be zero only for components without damping. Models derived from first principles satisfy the passivity criterion implicitly, but this is not necessarily the case for experimentally identified models. The passivity problem for state space models is commonly known, for details see for example [14,30]. The passivity of a state space model is linked to the positive real (PR) lemma. In this paper, an engineering solution to the passivity optimization problem defined in [14] based on averaging of the BandCmatrices has been used. 1.2.2 Substructuring Substructuring is the process of coupling two structures together by enforcing two conditions at their common interface; compatibility and force equilibrium. This paper will treat only what de Klerk et al. [1] refers to as the primal formulation, which implies that the displacements are defined and interface forces are eliminated. For brevity, the explicit time dependency is dropped from the equations in the following. For (analytically derived) internal models as in Eq. (1.4), enforcing compatibility and force equilibrium amounts to an assembly procedure parallel to that used for Finite Element Models, see further de Klerk et al. [1], such that synthesis of two systems 1 and 2 can be described by: 8 ˆˆ< ˆˆ: K1 0 0 K2 q1 q2 C V1 0 0 V2 Pq1 Pq2 C M1 0 0 M2 Rq1 Rq2 Df Cg Eq D0 LTg D0 (1.5) where Ecan be thought of as a signed boolean matrix enforcing compatibility,1 g are the interface forces, and the matrix L is the nullspace of E. For details, refer to [1]. When coupling two structures, which are described by m1 and m2 degrees of freedom (DOFs), the assembled structure will consist of m1 Cm2 mc coordinates, where mc is the number of coupling constraints. The de Klerk et al. paper [1] generalizes the formulation using the two matrices EandLto include also models using generalized coordinates and outer models (Eq. (1.2)), i.e. FRF coupling. In a following subsection, the state-space method [14] will be put into the same format. 1.2.3 State-Space Coupling The state-space coupling used here is described in the paper by Sjövall and Abrahamsson [14]. At its core is the coupling formof Eq. (1.6), where the i:th first-order system to be coupled is rewritten to second-order form at the interface in order for the compatibility and force equilibrium conditions to be applicable 8 ˆˆˆˆ< ˆˆˆˆ: 2 4 Ryc Pyc Pxb 3 5D 2 4 Ai vv Ai vd Ai vb I 0 0 0 Ai bd Ai bb 3 5 2 4 Pyc yc xb 3 5C 2 4 Bi vv Bi vb 0 0 0 Bi bb 3 5 uc ud yc yd D 0 I 0 Ci bv Ci bd Ci bb (1.6) Here, the subscript c refers to coupling degrees of freedom, bother degrees of freedom, vvelocity outputs andd displacement outputs. Note that for state-space coupling there exist twice as many states as DOFs (a product of rewriting a second-order differential equation on first-order form) and thus if n D 2m, for two structures with n1 and n2 states and mc coupling coordinates the coupled system will containn1 Cn2 2mc states. 1This matrix is denotedBin de Klerk et al. The notation cannot be adopted here as theBmatrix is strongly associated to the state-space formulation in Eq. (1.3).
RkJQdWJsaXNoZXIy MTMzNzEzMQ==