9 On Dynamic Substructuring of Systems with Localised Nonlinearities 107 9.2.1 Craig-Bampton Reduction The Craig-Bampton is one of the most prominent and long standing techniques in CMS and is already implemented in numerous commercial finite element softwares [17]. In the Craig-Bampton method each sub-structure can be reduced individually and then usually the reduced sub-structures are coupled together using a primal assembly. However, in this case only the linear system will be reduced whilst the coupling procedure will be carried out within the integration algorithm. The key aspects of the reduction are that the degrees of freedom (DOFs) of each substructure are partitioned into internal DOFs, xi, and external DOFs, xb, i.e., those which lie on the interface between substructures. Mii Mib Mbi Mbb ¨xi ¨xb + Kii Kib Kbi Kbb xi xb =0 (9.1) The internal DOFs, which in most cases are the majority, can be significantly reduced using a modal decomposition from the eigenvalue problem in equation 9.2. Given that for structural dynamics the lowest eigenmodes are dominant, the majority of these modes can be discarded, with only the lower modes, r, retained. These are known as the fixed interface normal modes. M−1 ii Kii = Σ T, = r d (9.2) In addition to the fixed interface normal modes, constraint modes, , are also used to characterise the boundary DOFs and the static response. Constraint modes correspond to the static deformation shape due to a unit displacement applied at a boundary DOF. As such there exist as many, as there exist boundary DOFs. These are calculated as follows. =−K−1 ii Kib (9.3) Using both the fixed interface and constraint modes, the original high dimensional coordinate set can be approximated on a significantly lower coordinate set, with the reduction of DOFs dependant upon the number of fixed interface modes that are discarded. The reduced coordinate set consists of the generalised internal coordinates q and the retained boundary coordinates xb. xi xb ≈ r 0 I q xb =CB q xb (9.4) The mass and stiffness matrices of the sub-structures can then be reduced by projection onto these reduced coordinate sets. ˆK=CBTKCB ˆM=CBTMCB (9.5) In the Craig-Bampton method the reduction basis consists of the generalised internal coordinates, whilst the boundary coordinates are retained as physical; this simplifies the coupling procedure. 9.2.2 Integration and Coupling The coupled integration scheme used herein is inspired by that presented in [16], this comprises a technique designed for hybrid simulation in which each of the substructures is solved as a free solution over the time interval with coupling enforced at the end of the time interval using a method similar to the FETI method [6]. The partitioned design of this integration method allows it to be applied in a hybrid test in that, the 2 systems are solved in parallel with coupling only enforced at the end of each time step. This means that in the context of a hybrid test, displacements can be enforced on the physical system, with the resulting restoring force being measured and being used to solve the free problem in the physical system and finally coupling being enforced at the end of the time step. The coupled equations of motion for the numerical and physical systems to be integrated are shown in equations 9.6 and 9.7. These present the typical equations of motion of a nonlinear mechanical system in state space form. However, these
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