Direct decoupling of substructures using primal and dual formulation Since the full set of internal DoFs can be difficult to measure in practice, results obtained by adding only one internal DoF are considered. The resonant frequencies of the residual subsystemB, with the measured DoFs grounded (the coupling DoF and one additional internal DoF), are shown in Table 3 versus the added internal DoF. Table3 Resonant frequencies of the residual subsystemBwith extended interface DoFs grounded (the coupling DoF θ5 and one additional internal DoF) AddedDoF fn1[Hz] fn2[Hz] θ1 69.4 151.5 θ3 37.3 143.4 θ4 31.5 87.15 The predicted rotational mobility of the unknown subsystemA obtained using collocated approach with the additional internal DoF θ1 is shown in Fig. 9: in this case, troubles arise at frequencies close to 70 and 150 Hz, as shown in Table 3, around which the inversion of an ill conditioned matrix is performed. The predicted rotational mobility of the unknown subsystemAobtained using non collocated approach with compatibility at DoFs θ1 and θ5 and equilibrium at the coupling DoF θ5 is shown in Fig. 10 for primal formulation and in Fig. 11 for dual formulation. Ill conditioned frequencies are the same as for standard interface. Fig. 12 shows the predicted FRF of subsystemAusing collocated approach with the additional internal DoF θ3: in this case, troubles could arise at frequencies close to 37 and 143 Hz as shown in Table 3, but they do not appear. Using non collocated approach with compatibility at DoFs θ3 and θ5 and equilibrium at the coupling DoF θ5, results are shown in Fig. 13 for primal formulation and in Fig. 14 for dual formulation. Again, ill conditioned frequencies are the same as for standard interface. Fig. 15 shows the predicted FRF of subsystemAusing collocated approach with the additional internal DoF θ4: in this case, troubles could arise at frequencies close to 31 and 87 Hz as shown in Table 3, but only very slight effects can be noticed. Using non collocated approach with compatibility at DoFs θ4 and θ5 and equilibrium at the coupling DoF θ5, results are shown in Fig. 16 for primal formulation and in Fig. 17 for dual formulation. Again, ill conditioned frequencies are the same as for standard interface. Generally, collocated approach seems to provide the best results, because in non collocated approach, with equilibrium enforced only at coupling DoFs, ill conditioning occurs at the same frequencies that are troublesome with standard interface. 67
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