2 Prediction of the Coupled Impedance from Frequency Response Data 23 The structures were excited in the all x, y, z and accelerations were measured in the same directions at point 1 and point 2. The measure of rotations for a simple structure like this is very difficult and becomes impossible for more realistic structures, thus moments and rotational responses were disregarded. All the cross combination of input force and output acceleration were measured. Therefore the size of the FRF matrices is 6 6. The experimental data were processed and H1 method was used as the FRF estimator [12]. Figure 2.4 shows the comparison between the modulus of the measured accelerance FRF function Ge of the coupled structure at point and those obtained applying Eq. (2.10) GeRS, viz. coupling the measured FRF matrices of the source GeS and the receiver GeR. Only the component along the z-axis is showed as an example but the same results apply to all other directions. As expected, this first sets of experimental results shows that the coupling model gives good prediction at low frequency. At higher frequency results become less and less reliable. Differences can be due to several factors such as a lack in modelling the connections between the source and the receiver, uncertainties in measuring, e.g. force applied in z direction is not perfectly aligned along the z axis and so on. However, it is expected that the main errors have been introduced neglecting moments and rotational responses. This is investigated in the next section. 2.4 Numerical Model The aim of this section is validating numerically the coupling technique and investigating the influence of the moments and rotational displacements in the FRF coupling model. The structure depicted in Fig. 2.2 is discretised using BEAM elements with 6DOFs per node, three translations and three rotations. The receiver is discretised using 324 elements while the source using 40 elements. Source and receiver are solved separately to obtain their FRF matrices, GnS and GnR. These are coupled using Eq. (2.8) to obtain GnRS. The FRF matrix, Gn, obtained from an FE model of the coupled structure, is also evaluated to model the real coupled structured and compare the results. Figure 2.5 shows the comparison between the direct frequency responses function GnRS and Gn at point 1. It can be seen that the curves are in perfect agreement and thus coupling approach theoretically predicts the dynamics accurately when all the forcing and moments and displacements and rotations are considered. The experimental measure of the rotational displacements can be so difficult that in many practical cases only translation forces and responses are considered. In order to simulate numerically this realistic situation, the FRF matrices of the source and the receiver are evaluated only for the translational degrees of freedom. Figure 2.6 shows the comparison between FRF for the coupled structure Gn and those obtained coupling the FRF matrices of the source and the receiver GnRS when only the translational degrees of 0 500 1000 1500 2000 2500 3000 10-3 10-2 10-1 100 101 102 Frequency (Hz) |a1/F1|(m/s 2/N) Gn Gn RS Fig. 2.5 Numerical case study. Comparison between numerical FRF for the coupled structure Gn and those obtained coupling the FRF matrices of the source and the receiver GnRS using Eq. (2.12)
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