A Truly Hybrid Approach to Substructuring Problems Using Mixed Assembly and Implicit Solving Strategies 0 5 101520253035404550 0 20 40 60 80 100 120 Frequency [Hz] Relative error | ΔY / Y| [%] Error on Y 77 AB − Magnitude Standard LM FBS Mixed approach 0 5 101520253035404550 0 0.5 1 1.5 2 2.5 3 3.5 Error on Y 77 AB − Phase Frequency [Hz] Phase error ∠(Y + ΔY) − ∠ Y [rad] Fig. 14: Standard deviations found from sensitivity analysis 0 5 101520253035404550 −120 −100 −80 −60 −40 −20 0 Y 77 AB with confidence interval − LM FBS Magnitude [dB] Frequency [Hz] Average FRF CI 0 5 101520253035404550 −120 −100 −80 −60 −40 −20 0 Y 77 AB with confidence interval − Alternative FBS Magnitude [dB] Frequency [Hz] Fig. 15: Assembled FRFs obtained with LM FBS (left) and alternative FBS (right) and their confidence intervals 5 Conclusions & Recommendations In this paper we presented a first attempt to develop an alternative method for experimental frequency based substructuring. Since the traditional FBS methods suffer from large sensitivity to measurement errors due to the inversion of the interface flexibility FRFs, the aim of this work was to derive an “inverse free” methodology for FBS. To this end, a framework for the assembly of substructure FRF models was presented in section 2. This framework allowed to derive a “mixed assembly” procedure, in which dynamic stiffness and receptance FRFs can be directly coupled without inversion. However, in order to actually solve the mixed assembled models in an inverse free manner, additional procedures were needed. Section 3 therefore presented an algorithm known as “sign counting” which allows to locate the eigenfrequencies of the assembled structure. Based on this information the assembled mode shapes could be extracted and, if needed, the assembled receptance FRFs reconstructed. A case study on a simple academic system in section 4 was used to illustrate the methodology. This showed that the method indeed is able to identify the modes and frequencies of the assembled system accurately. However, due to a lack of information on damping and difficulties with scaling the mode shapes, reconstructing the assembled receptance FRFs is somewhat more tricky. Nonetheless, reasonably accurate results could still be obtained. For the method to be truly promising, we needed to asses its sensitivity to measurement errors. Using Monte Carlo simulations with randomly distributed errors on the subsystem FRFs, the sensitivity of the method could be quantified and compared to classic FBS methods. The results indicated that the alternative FBS method seems indeed to be less sensitive to errors in the subsystem descriptions. In contrast to the classic FBS methods the proposed method seems most sensitive around anti-resonances, but seems to suffer less from spurious modes around the subsystem resonances. 345
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