MODAL GEOMETRICAL SELECTION CRITERION The basic idea of MoGeSeC is to consider both the geometrical and the modal features of the body. In fact the nodes selection has to consider both the disposition of nodes in the body volume and the modal shapes that have to be modelled or evaluated. If only geometrical or modal information are used, some important dynamic features could be underestimated. For example, considering just geometrical information, a torsional behaviour could be erased if the nodes are selected on a axis parallel to the rotational axis. In the same way, using only modal information, some areas of the body could be not represented in the model. Starting from this consideration, MoGeSeC selects the additional master dofs evaluating the maximum value of a weight function vector W, estimated for each physical node of the FE model or of all the possible experimental locations: T g m W W W diag (1) where the geometrical vector gW and the modal vector mW are evaluated for each physical node p by means of expressions: > @ ¦ ¦ ) ) ) » » ¼ º « « ¬ ª 3 1 1 2 , 2 , 2 , , 1 2 2 2 , 2 1 max 1 1 1 max 1 r j k j p z j p y j p x m m p r i k i p i p i p g g p W x x y y z z W W W (2) and the geometrical grid coordinates of the candidate p are p p p x y z , , , while the geometrical grid coordinates of the r previously selected master nodes are i i ix y z , , ; the corresponding modal displacements of the candidate p for the j normalized modal shape are j p z j p y j p x , , , , , ) ) ) . Exponents k1 and k2 are weight parameters that allow flexibility to be increased. Both geometrical and modal terms are normalized to unit values. It is possible to stress the following milestones of MoGeSeC methodology: x actual physical nodes and the three coordinate displacements of the model are taken into account; x as Matta and Kammer suggest in [8], [15], only translational generalized displacements are considered, because rotations are negligible; x the reduced dynamic model should be able to replicate the modal properties of master nodes, such as representative model nodes or boundary connection nodes, and should be expanded by adding additional nodes without iterative procedures and by maintaining the previous ones; x the aim of adding a new node is to improve numerical stability of matrices: this is done selecting the node with the furthest possible distance from the other master nodes and selection is also weighted with a high modal participation; x the analytical expression for the geometrical term is based on a generalized gravitational force form; the sum of distances between the new candidates and the previously master nodes are evaluated through the power of parameter k1 and the repeated choice of the same node is prevented; x analogously, the sum of modal displacements of candidates is weighted through the power of parameter k2; if k2 is set equal to 1, the square norms sum of modal shapes is adopted; x parameters k1 and k2 are arbitrarily set through applying the selection criterion to a wide number of case study models; they are initially set respectively to 2 and 1 and the parameter investigation for the optimal choice is discussed in some previous papers [19], [20]. In Figure 1 it may be evinced the local minimum for the function surface in the case of the first 150 modes of an exhaust pipeline; similar behaviour can be pointed out for the other considered models. For each model an optimal choice of k1 and k2 parameters can be done in the concurrence of surface minimum. 283
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