82 W. D’Ambrogio and A. Fregolent 8.2.3 Prediction of the FRF of the Unknown Subsystem The FRF of the unknown subsystemUcan be obtained by back-substituting in Eq. (8.1 ), and by isolating u at the left hand side: uD H HBE T BCHBE T CBCH f (8.17) which is in the formu DHf, so that the FRF of the unknown subsystemUis: HU DH HBE T BCHBE T CBCH (8.18) With the dual assembly, the rows and the columns of HU corresponding to compatibility and equilibrium DoFs appear twice. Furthermore, when using an extended or mixed interface, HU contains some meaningless rows and columns: those corresponding to the internal DoFs of the residual substructure R. Obviously, only meaningful and independent entries are retained. 8.3 Test Bed The assembled system is an aluminium tree structure (Fig. 8.2). The residual substructure Ris a cantilever column with two staggered short arms and the unknown substructure Uis a horizontal beam. The horizontal beam is bolted to the top of the column, involving both translational and rotational DoFs. The geometrical dimensions are shown in Table 8.1. The cross section is 40 mm 8 mm for all beams, with the short side along the z-direction. The experimental FRFs of the assembled system RU up to 2000 Hz are obtained by applying impact excitation and measuring the resulting accelerations along z-direction at seven locations (3, 6, 9, 10, 11, 13, 20), as shown in Fig. 8.3. For Fig. 8.2 Sketch of the test structure Table 8.1 Geometrical dimensions (mm) a b c d e l 540 420 60 100 240 600
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