Next, we take the results obtained from the case with highest frequency resolution and try to synthesize the receptance FRFs of the assembled system. The difficulty with doing so is that no damping values are obtained from the procedure outlined in the previous section. Hence, the modal damping values are estimated at the average of the modal damping values of the true system. One of the resulting FRFs (a driving point FRF at DoF 7) is shown in figure 13. It can be seen that the synthesized assembled FRF closely resembles the true FRF of systemAB, apart from some small discrepancies around resonances and anti-resonances. As explained, these are due to the incorrect damping values and some inaccuracies in the scaling of the mode shapes. 0 5 101520253035404550 −110 −100 −90 −80 −70 −60 −50 −40 −30 −20 Frequency [Hz] Magnitude [dB] Y 77 AB − Magnitude plot 0 5 101520253035404550 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 Frequency [Hz] Phase [rad] Y 77 AB − Phase plot Synthesized True Synthesized True Fig. 13: Synthesized vs. true assembled FRF 4.2 Sensitivity of Assembled FRFs Now that it was shown in the previous subsection that the proposed method is indeed capable of predicting the assembled mode shapes and FRFs, the question that remains is whether it is more robust than the conventional FBS methods when random measurement errors are present in the subsystem models. To quantify the sensitivity of the approach, a sensitivity analysis was performed in the form of Monte Carlo simulations. To that end, the receptance FRF matrix of component B, which was imagined to originate from measurements, was perturbed with a normally distributed random error with a standard deviation of ±2% on the magnitude and ±2 deg on the phase of the original FRF. This perturbed receptance matrix was then used in the assembly and identification procedures outlined in sections 2 and 3. This was done for a thousand different perturbations and subsequently the mean and standard deviation of the assembled FRFs were calculated to obtain the sensitivity of the approach. The same perturbed FRF matrices of Bwere also used in the LM FBS method, in order to compare sensitivities. Note that the dynamic stiffness matrix of substructure Awas not perturbed, as this component was assumed to be the analytical component in the assembly. The results of the sensitivity analysis are shown in figures 14 and 15 for the same assembled FRF considered above, the driving point FRF at DoF 7. From figure 14 it can be seen that the proposed alternative FBS method seems to perform slightly better than the LM FBS method, since the maximum standard deviation is lower. From a more detailed inspection of this figure, we see that the two methods behave quite differently. Where the LM FBS method (and in fact all classic FBS methods [28]) have the largest sensitivity around the eigenfrequencies of the subsystems, the alternative FBS method seems to suffer from this problem to a lesser extent. In fact, it seems to be most sensitive around the global anti-resonances. The effect of the sensitivities is shown in figure 15, where the average FRF is plotted in black and the 95% confidence interval is shown as a gray band around the FRF. One can see that the alternative FBS method indeed shows the largest uncertainty around its anti-resonances, which is also reflected by the fact that the magnitude at these frequencies is not synthesized entirely correctly (see for instance the first anti-resonance). However, where the LM FBS shows a big sensitivity around 16 Hz, possibly leading to spurious peaks in the assembled FRF, the alternative FBS method shows much smaller uncertainty. S.N. Voormeeren, P.L.C. van der Valk and D.J. Rixen 344
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