316 M. Brehm and A. Deraemaeker 30.4 Conclusions This paper contributes with a novel time-domain approach to the problem of uncertainty propagation and quantification in virtual testing with respect to Fourier transformed responses of a dynamical system with a deterministic structural system description, but random excitations. Under the assumption of a multivariate normal distribution of excitations in time domain, the statistics of the Fourier transformed responses have been derived analytically. In addition, the novel approach considered typical signal processing techniques, such as linear combinations and windowing. To verify the novel approach, it was compared to a sample-based approach using a Latin hypercube sampling scheme. The evaluation of the results related to a three-degree-of-freedom system showed that the novel approach was 30 times faster and more precise. For an increasing length of the time series, it has been shown that the results obtained by the novel approach converged to the results derived by the approach based on the frequency domain estimators. This verified additionally the correctness of the proposed novel approach. Further research is needed to implement the effect of measurement errors into the novel approach to increase its applicability as a realistic virtual testing technique. Moreover, the application of the novel approach is foreseen for the design of damage indicators applied in vibration-based structural health monitoring systems. Acknowledgements The research presented in this article was carried out within the postdoctoral project “Dynamic Strain Sensing for SHM” funded through an incentive grant for scientific research by the Belgian funding organization F.R.S.-FNRS to which the authors like to express their gratitude for financial support. References 1. Bathe KJ (1996) Finite element procedures. Prentice Hall, Englewood Cliffs 2. Brehm M, Massart TJ, Deraemaeker A (2012) Application of an updated notched beam model using an implicit gradient cracking approach for the purpose of damage detection based on modal strains. In: Proceedings of international conference on noise and vibration engineering (ISMA), Leuven, Belgium, 17–19 September 2012 3. Brehm M, Zabel V, Bucher C (2013) Optimal reference sensor positions using output-only vibration test data. Mech Syst Signal Process 41(1–2):196–225 4. Deraemaeker A, Reynders E, De Roeck G, Kullaa J (2008) Vibration based structural health monitoring using output-only measurements under changing environment. Mech Syst Signal Process 22(1):34–56 5. Mao Z, Todd M (2013) Statistical modeling of frequency response function estimation for uncertainty quantification. Mech Syst Signal Process 38(2):333–345 6. Natke HG (1989) Baudynamik, 3rd edn. B.G. Teubner Stuttgart, Germany 7. Natke HG (1992) Einführung in die Theorie und Praxis der Zeitreihen- und Modalanalyse, 3rd edn. Vieweg & Sohn, Braunschweig/Wiesbaden, Germany 8. Oppenheim AV, Schafer RW, Buck JR (1999) Discrete-time signal processing, 2nd edn. Prentice-Hall, Englewood Cliffs 9. Preumont A (1982) Frequency domain analysis of time integration operators. Earthquake Eng Struct Dyn 10:691–697 10. Stein M (1987) Large sample properties of simulations using Latin hypercube sampling. Technometrics 29(2):143–151 11. Tondreau G (2013) Damage localization in civil engineering structures using dynamic strain measurements. Ph.D. thesis, Université libre de Bruxelles, Belgium 12. Tondreau G, Deraemaeker A (2013) Local modal filters for automated data-based damage localization using ambient vibrations. Mech Syst Signal Process 39(1–2):162–180 13. Zhang F (2011) Matrix theory: basic results and techniques. Springer, Berlin
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