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Modelling of nonlinear modal interactions in the transient dynamics of an elastic rod with an essentially nonlinear attachment. (English) Zbl 1222.74028

Summary: We perform system identification and modelling of the strongly nonlinear modal interactions in a system composed of a linear elastic rod with an essentially nonlinear attachment at its end. Our method is based on slow/fast decomposition of the transient dynamics of the system, combined with empirical mode decomposition (EMD) and Hilbert transforms. The derived reduced order models (ROMs) are in the form of sets of uncoupled linear oscillators (termed intrinsic modal oscillators–IMOs), each corresponding to a basic frequency of the dynamical interaction and forced by transient excitations that represent the nonlinear modal interactions between the rod and the attachment at each of these basic frequencies. A main advantage of our proposed technique is that it is nonparametric and multi-scale, so it is applicable to a broad range of linear as well as nonlinear dynamical systems. Moreover, it is computationally tractable and conceptually meaningful, and it leads to reduced order models of rather simple form that fully capture the basic strongly nonlinear resonant interactions between the subsystems of the problem.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] Cheng, J.; Yu, D.; Yang, Y., Research on the intrinsic mode function (IMF) criterion in EMD method, Mech Syst Signal Proc, 20, 817-824 (2006)
[2] Cheng, J.; Yu, D.; Yang, Y., Application of support vector regression machines to the processing of end effects of Hilbert-Huang transform, Mech Syst Signal Proc, 21, 3, 1197-1211 (2007)
[3] Deering, R.; Kaiser, J. F., The use of a masking signal to improve empirical mode decomposition, Proc IEEE Int Conf Acoust Speech Signal Process (ICASSP ’05), 4, 485-488 (2005), Philadelpia, March
[4] Ewins, D. J., Modal testing: theory, practice and application (2000), Research Studies Press Ltd.: Research Studies Press Ltd. Hertfordshire, UK
[5] Georgiades F. Nonlinear localization and targeted energy transfer phenomena in vibrating systems with smooth and non-smooth stiffness nonlinearities, PhD thesis. Dept of applied mathematical and physical sciences. Athens, Greece: National Technical University of Athens; 2006.; Georgiades F. Nonlinear localization and targeted energy transfer phenomena in vibrating systems with smooth and non-smooth stiffness nonlinearities, PhD thesis. Dept of applied mathematical and physical sciences. Athens, Greece: National Technical University of Athens; 2006.
[6] Georgiades, F.; Vakakis, A. F.; Kerschen, G., Broadband passive targeted energy transfer from a linear dispersive rod to a lightweight essentially nonlinear end attachment, Int J Nonl Mech, 42, 773-788 (2007)
[7] Huang, N.; Shen, Z.; Long, S.; Wu, M.; Shih, H.; Zheng, Q., The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis, Proc Royal Soc London, Series A Math Phys Sc, 454, 903-995 (1998) · Zbl 0945.62093
[8] Huang, N.; Shen, Z.; Long, S., A new view of nonlinear water waves: the Hilbert spectrum, Rev Fluid Mech, 31, 417-457 (1999)
[9] Huang, Y.-P.; Li, X. Y.; Zhang, R.-B., A research on local mean in empirical mode decomposition, (Computational science - ICCS 2007. Computational science - ICCS 2007, Lecture notes in computer science, vol. 4489 (2007), Springer Verlag: Springer Verlag Berlin and New York), 125-128
[10] Junsheng, C.; Dejie, Y.; Yu, Y., The application of energy operator demodulation approach based on EMD in machinery fault diagnosis, Mech Syst Signal Proc, 21, 2, 668-677 (2007)
[11] Kercshen, G.; Worder, K.; Vakakis, A. F.; Golinval, J.-C., Past, present and future of nonlinear system identification in structural dynamics, Mech Syst Signal Proc, 20, 505-592 (2006)
[12] Kopsinis, Y.; McLaughlin, S., Improved EMD using doubly-iterative sifting and higher order spline interpolation, EURASIP J Adv Signal Proc, 8, 2 (2008), [Article no. 45] · Zbl 1184.94110
[13] Kourdis, P.; Vakakis, A. F., Some results on the dynamics of the linear parametric oscillator with general time-varying frequency, Applied Math Comp, 183, 1235-1248 (2006) · Zbl 1113.70018
[14] Lee YS, Tsakirtzis S, Vakakis AF, Bergman L, McFarland DM. Physics-based foundation for empirical mode decomposition: correspondence between intrinsic mode functions and slow flows. AIAA J, 2009, inpress.; Lee YS, Tsakirtzis S, Vakakis AF, Bergman L, McFarland DM. Physics-based foundation for empirical mode decomposition: correspondence between intrinsic mode functions and slow flows. AIAA J, 2009, inpress.
[15] Lee YS, Tsakirtzis S, Vakakis AF, Bergman L, McFarland DM. A Time-domain nonparametric nonlinear system identification method based on dynamic slow-fast partitions. Meccanica, 2009, submitted for publication.; Lee YS, Tsakirtzis S, Vakakis AF, Bergman L, McFarland DM. A Time-domain nonparametric nonlinear system identification method based on dynamic slow-fast partitions. Meccanica, 2009, submitted for publication.
[16] Nayfeh, A. H.; Mook, D., Nonlinear oscillations (1979), Wiley Interscience: Wiley Interscience New York
[17] Pai, P. F., Nonlinear vibration characterization by signal decomposition, J Sound Vib, 307, 3-5, 527-544 (2007)
[18] Panagopoulos, P. N.; Georgiades, F.; Tsakirtzis, S.; Vakakis, A. F.; Bergman, L. A., Multi-scaled analysis of the damped dynamics of an elastic continuum with an essentially nonlinear end attachment, Int J Solids Str, 44, 6256-6278 (2007) · Zbl 1178.74083
[19] Sharpley RC, Vatchev V. Analysis of the intrinsic mode functions, IMI technical reports 12. Dept. of Mathematics, University of Southern California; 2004.; Sharpley RC, Vatchev V. Analysis of the intrinsic mode functions, IMI technical reports 12. Dept. of Mathematics, University of Southern California; 2004.
[20] Tsakirtzis S. Passive targeted energy transfers from elastic continua to essentially nonlinear attachments for suppressing dynamical disturbances, PhD thesis. Dept. of applied mathematical and physical sciences. Athens, Greece: National Technical University of Athens; 2006.; Tsakirtzis S. Passive targeted energy transfers from elastic continua to essentially nonlinear attachments for suppressing dynamical disturbances, PhD thesis. Dept. of applied mathematical and physical sciences. Athens, Greece: National Technical University of Athens; 2006.
[21] Vakakis, A. F.; Gendelman, O.; Bergman, L. A.; McFarland, D. M.; Kerschen, G.; Lee, Y. S., Passive nonlinear targeted energy transfer in mechanical and structural systems: I and II (2008), Springer Verlag: Springer Verlag Berlin and New York
[22] Wu, Z.; Huang, N. E., Ensemble empirical mode decomposition: a noise-assisted data analysis method, Adv Adapt Data Analysis, 1, 1, 1-41 (2009)
[23] Yang, J. N.; Lei, Y.; Pan, S.; Huang, N., System identification of linear structures based on Hilbert-Huang spectral analysis, part 1: normal modes, Earthquake Eng Str Dyn, 32, 9, 1443-1467 (2003)
[24] Yu, D.; Cheng, J.; Yang, Y., Application of EMD method and hilbert spectrum to the fault diagnosis of roller bearings, Mech Syst Signal Proc, 19, 259-270 (2005)
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