A Truly Hybrid Approach to Substructuring Problems Using Mixed Assembly and Implicit Solving Strategies 2 A General Framework for Substructure Assembly This section addresses an important aspect of dynamic substructuring, namely the assemblyof component models to obtain the structural dynamic model of the total system. Traditionally one uses in FBS either so-called “primal” or “dual” approaches, as outlined in [14]. However, in order to derive the “inverse free” FBSmethod we need a third option: “mixed assembly”. In this section, we will present a general framework for substructure assembly which allows to derive this mixed assemblymethod. In general, structural dynamic component models in the frequency domain can be expressed either in terms of stiffness or flexibility. The former can be the case for dynamic stiffness FRF matrices (possibly condensed) originating fromfinite element (FE) models, while the latter is the case for experimentally obtained receptance models (where a force is applied and dynamic responses are measured) or models synthesized fromanalytical mode shapes.1 Hence, the substructure DoF vector either contains displacement DoF or force DoF. Given the different representations of the substructure models, three assembly cases can be distinguished: 1. Assembly of displacements to displacements: “stiffness assembly”. 2. Assembly of forces to forces: “flexibility assembly”. 3. Assembly of displacements to forces: “mixed assembly”. These three cases are illustrated in figure 1 and will be treated in detail in the subsequent sections, where we will use a so called three field variational formulation to derive the required assembly procedures. Such an approach is needed to tackle the mixed assembly problem; for the more straightforward cases of dynamic stiffness and receptance assembly a two field formulation is already sufficient. 2 1 /w1D K / wdD K `wdD K ` w1D K ` 2 1 /w1D K / wdD K ` 1 / wdD K ` 2 /w1D K wAD w{D wsD .w1D .wdD Vw1D VwdD .wdD ` wdD K Vw1D Fig. 1: Three different assembly cases 2.1 Dynamic Stiffness Assembly In this section we treat the assembly of components expressed in terms of dynamic stiffness. We start fromthe linear, discrete and possibly condensed dynamic stiffness formof the equations of motion of a substructure s connected to other substructures: Z(s)(ω)u(s)(ω)=f(s)(ω)+g(s)(ω) (1) Here Z(s) denotes the substructure’s dynamic stiffness matrix, u(s) the vector of displacement degrees of freedom, f(s) the external excitation vector andg(s) the vector of connection forces felt fromconnected substructures. Note that in the subsequent discussion we will omit the explicit frequency dependence for clarity. We partition the subsystemDoF and the associated matrices in an internal i and boundary b part and write the DoF vector and vector of connection forces as: u(s) = u(s) i u (s) b , g(s) = 0 g (s) b 1 Note that a FE model can often easily be used to obtain a receptance FRF matrix of the structure by simple performing a model analysis and synthesizing the FRFs. However, when the model contains frequency dependent behavior, for example fromviscoelastic effects, the identification of mode shapes is impossible and one either has to work directly with the dynamic stiffness model or invert it at every frequency to obtain the receptance matrix. 331
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