REDUCED MODELS In order to determine a dynamic reduced model, an undamped linear multi-dof dynamic model of the whole system is considered. In the present paper dissipative terms, such as viscous damping or hysteretic effects, are neglected because the aim is preliminarily to focus on mechanical systems lightly damped or with proportional damping. The equations of motion of the n generalized coordinates vector t x are described, in time domain, through the following well-known matrix expression: t t t f Mx K x (4) where M and K are respectively the real and positive-definite symmetric mass matrix and the real and positivesemidefinite symmetric stiffness matrix obtained by a standard FE code; the system may be subject to a generalized force set vector t f . The n dofs x of the FE system may be generically partitioned in master mx and slave sx dofs. Each condensation criterion expresses the dofs x of the system as a function at least of the master dofs mx and of a set of possible auxiliary non-physical dofs, as modal coordinates q: » ¼ º « ¬ ª » ¼ º « ¬ ª q x T x x x m s m (5) where T is the transformation matrix. Thus, Equation (5) can be rewritten as: » ¼ º « ¬ ª » ¼ º « ¬ ª » ¼ º « ¬ ª » ¼ º « ¬ ª » ¼ º « ¬ ª 0 f x x K K K K x x M M M M m s m ss sm ms mm s m ss sm ms mm (6) where the mass and stiffness matrices of the reduced model, having m < n dofs, are related to the transformation matrix T as: M T MT T r and K T KT T r . (7) In the present paper, the applied reduction technique is the well-known Serep (System Equivalent Reduction Expansion Process) methodology [14], originally developed as an experimental modal technique able to identify equivalent dynamic systems. The Serep technique is a low computational cost methodology because it only needs an ordinary modal analysis of the single sub-structure or an experimental modal identification. It permits flexible boundary conditions in substructuring problems to be taken into account, but it is also affected by a small numerical stability, due to the master nodes chosen for the reduction procedure. It is chosen in order to evaluate the performance of MoGeSeC in nodes selection because it tries to reduce the ill-conditioning of matrices. In Serep approach [14] the relationship between master and slave dofs of the model is derived from the modal superposition: q ĭ ĭ x x » ¼ º « ¬ ª » ¼ º « ¬ ª s m s m (8) where the eigenvector matrix ĭ is split into master (subscript m) and slave (subscript s) dofs and q is the modal coordinates vector. 285
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