Direct decoupling of substructures using primal and dual formulation 4.2 Results First of all, the case of standard interface (rotational mobility only at the coupling DoF θ5) is considered. In Fig. 5, the true drive point rotational mobility at the coupling DoF of subsystemAis compared with the corresponding FRF computed using the dual formulation, Eq. (24), starting from the fitted perturbed FRF of the coupled system. The same result is obtained using the primal formulation, Eq. (14). Fig. 5 Rotational mobility at the coupling DoF of subsystem A: true (—), computed from fitted perturbed FRF (∗∗∗) without additional internal DoFs 0 20 40 60 80 100 120 140 160 180 200 10−5 100 105 Frequency [Hz] Magnitude [rad s−1/(N m)] 0 20 40 60 80 100 120 140 160 180 200 −4 −2 0 2 4 Frequency [Hz] Phase [rad] As discussed in previous papers [2, 3], the predicted rotational mobility of subsystemAis badly identified at frequencies around 30, 70 and 150 Hz. This depends on ill conditioning due to unmeasured internal DoFs, as explained in section 2.3. In fact, the coupled systemABand the residual subsystemB, with the ”measured” coupling DoF θ5 grounded, share three resonance frequencies, namely fn1 =28.94 Hz, fn2 =72.23Hz and fn3 =151.5 Hz. Around these frequencies, [ ˆHAB] −[ ˆHB] is ill-conditioned and noise is greatly amplified. A way to circumvent this problem is to use an extended interface, i.e. to assume that the FRF matrix of the coupled system is known not only at the coupling DoF but also at a subset of the three internal DoFs ( θ1, θ3, θ4) of the residual subsystem B. The predicted rotational mobility of the unknown subsystemA, obtained using collocated approach with all the internal DoFs, is shown in Fig. 6: in this case, the residual subsystem is fully grounded and no ill-conditioned frequencies appear. The predicted rotational mobility of the unknown subsystemA, obtained using non collocated approach with compatibility at all DoFs and equilibrium at the coupling DoF θ5, is shown in Fig. 7 for primal formulation and in Fig. 8 for dual formulation. As discussed in section 2.3.2, ill conditioned frequencies are the same as for standard interface but they affect the predicted FRFs to a lower extent if compared with the standard interface. 65
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