102 M. Scheel and A.T. Johansson To improve the constraints, more modes could be added to the transmission simulator mode shape matrix. For experimental coupling, the condition number of this matrix is believed to be crucial. In order to lower the condition number, more sensors have to be included along with more modes. However, this approach is limited by the number of sensors available. It is known that the bolts introduce nonlinearities to the interface [21]. These might become more apparent by measuring close to the joints. On the other hand, mounting the sensors on the brackets enables excitation and measurement in the same direction due to the plain surface. Other possible sources of errors include the measurement setup. It was found that stinger resonance has no impact on the frequency range of interest. The pronounced differences in the measurements above 400 Hz could be due to resonances in the suspension. However, the modes in this frequency range are dominated by an in-plane motion which may be particularly sensitive to the input direction. A slight misalignment of the stinger could then cause large deviations. Similar issues were reported in the work of Gibanica et al. [11]. In total eleven sensors, listed in Sect. 9.3.2, turned out to be defective and where removed from the system identification procedure. Repeating the measurement to achieve a full set of measured points could possibly improve the substructuring results. Liljerehn and Abrahamsson [4] found that the peak height of the identified FRF has a large influence on the substructuring results. The identified models in this work are believed to replicate the measurements well enough, as indicated by the results. However, the absolute value of the error for the first mode in the receptance FRF (used for coupling) is larger than for the other modes due to larger displacements. This might be an explanation for the rather high error in the substructuring results for the first mode. The high damping errors could be a sign of an inappropriate damping model. The main source of dissipation is likely to be in the joints, which is probably unsatisfactorily represented by modal damping [10]. 9.6 Concluding Remarks In this paper, the transmission simulator technique is transferred to state-space synthesis of substructures. The interface is coupled with MCFS similar to FBS, and the method is then verified with measurements of the Ampair A600 wind turbine. Three blades attached to the hub were measured, and coupled with two negative FE models of the hub to arrive at a system with one hub and three blades. Physical properties of the identified models like passivity and adherence to Netwon’s second law were checked and enforced if necessary. The substructuring results correlate well with the measured three-bladed hub. The overall behavior is captured but the damping is overestimated. Moreover, substructuring predicted the system’s behavior better than the nominal FE model. The transmission simulator was successfully applied to state-space substructuring, and it was shown that CMS and statespace coupling yield the same results. The results may improve with better measurements since they are believed to be a great source of errors, yet the transmission simulator with least squares constraints can level out such kinds of measurement errors. More work should be done to find a way to judge the constraint equations with state-space models. References 1. de Klerk, D., Rixen, D.J., Voormeeren, S.N.: General Framework for dynamic substructuring: history, review and classification of techniques. AIAAJ. 46(5), 1169–1181 (2008) 2. Su, T.-J., Juang, J.-N.: Substructure system identification and synthesis. J. Guid. Control. Dyn. 17(5), 1087–1095 (1994) 3. Sjövall, P., Abrahamsson, T.: Component system identification and state-space model synthesis. Mech. Syst. Signal Process. 21(7), 2697–2714 (2007) 4. Liljerehn, A., Abrahamsson, T.: Experimental-analytical substructure model sensitivity analysis for cutting machine chatter prediction. In: Proceedings of the 30th International Modal Analysis Conference, Jacksonville, FL (2012) 5. Liljerehn, A., Abrahamsson, T.: Dynamic sub-structuring with passive state-space components. In: 26th International Conference on Noise and Vibration Engineering, ISMA, Leuven (2014) 6. Craig, R.R., Kurdila, A.J.: Fundamentals of Structural Dynamics. Wiley, Hoboken, NJ (2006) 7. Allen, M.S., Mayes, R.L., Bergman, E.J.: Experimental modal substructuring to couple and uncouple substructures with flexible fixtures and multi-point connections. J. Sound Vib. 329(23), 4891–4906 (2010) 8. Mayes, R., Hunter, P.S., Simmermacher, T.W.: Combining lightly damped experimental substructures with analytical substructures. In: Proceedings of the 25th International Modal Analysis Conference, Orlando, FL (2008) 9. Rohe, D.P., Mayes, R.L.: Coupling of a bladed hub to the tower of the ampair 600 wind turbine using the transmission simulator method. In: Proceedings of the 31st International Modal Analysis Conference, Garden Grove, CA, pp. 193–206 (2013)
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