Hilbert transform; Nonlinear systems; component; model identification method; multi-degree-of-freedom
Abstract :
[en] The Hilbert transform is one of the most successful approaches to tracking the varying nature of vibration of a large class of nonlinear systems, thanks to the extraction of backbone curves from experimental data. Because signals with multiple frequency components do not admit a well-behaved Hilbert transform, this transform is inherently limited to the analysis of single-degree-of-freedom systems; this shortcoming is potentially overcome by the Hilbert-Huang transform (HHT). In this study, the joint application of the complexification-averaging method and the HHT enables us to develop a new technique, the slow-flow model identification method. Through an experimental application, we demonstrate that the proposed method is adequate for characterizing and identifying multi-degreeof-freedom nonlinear systems.
Worden, K. and Tomlinson, G.R., 2001, Nonlinearity in Structural Dynamics: Detection, Identification and Modelling, Institute of Physics Publishing, Bristol, Philadelphia.
Kerschen, G., Worden, K., Vakakis, A.F. and Golinval, J.C., 2006, .Past, present and future of nonlinear system identification in structural dynamics,. Mechanical Systems and Signal Processing 20, 505-592.
Masri, S.F. and Caughey, T.K., 1979, .A nonparametric identification technique for nonlinear dynamic problems,. Journal of Applied Mechanics 46, 433-44.
Feldman, M., 1994, .Nonlinear system vibration analysis using the Hilbert transform - I. Free vibration analysis method FREEVIB,. Mechanical Systems and Signal Processing 8, 119-127.
Feldman, M., 1997, .Non-linear free vibration identification via the Hilbert transform,. Journal of Sound and Vibration 208, 475-489.
Garibaldi, L., Ruzzene, M., Fasana, A. and Piombo, B., 1998, .Identification of non-linear damping mechanisms using the wavelet transform,. Mecanique Industrielle et Materiaux 51, 92-94.
Bellizzi, S., Gullemain, P. and Kronland-Martinet, R., 2001, .Identification of coupled non-linear modes from free vibration using time-frequency representation,. Journal of Sound and Vibration 243, 191-213.
Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.C., Tung, C.C. and Liu, H.H., 1998, .The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,. Proceedings of the Royal Society of London, Series A. Mathematical, Physical and Engineering Sciences 454, 903-995.
Kakurin, A.M and Orlovsky, I.I., 2005, .Hilbert-Huang transform in MHD plasma diagnostics,. Plasma Physics Reports 31, 1054-1063.
Huang, N.E., Wu M.L., Qu W.D., Long, S.R. and Shen, S.S.P., 2003, .Applications of Hilbert-Huang transform to nonstationary financial time series analysis,. Applied Stochastic Models in Business and Industry 19, 245-268.
Huang, N.E.,Wu M.L., QuW.D., Long, S.R. and Shen, S.S.P., Editors, 2003, Hilbert-Huang Transform and Its Applications, World Scientific, Singapore.
Yang, J.N., Lei, Y., Lin, S. and Huang, N., 2004, .Hilbert-Huang based approach for structural damage detection., Journal of Engineering Mechanics 130, 85-95.
Yu, D., Cheng, J. and Yang, Y., 2005, .Application of EMD method and Hilbert spectrum to the fault diagnosis of roller bearings,. Mechanical Systems and Signal Processing 19, 259270.
Brenner, M. and Prazenica, C., 2005, .Aeroelastic flight data analysis with the Hilbert-Huang algorithm,. AIAA Atmospheric Flight Mechanics Conference and Exhibit, San Francisco, 2005-5917.
Pai, F.P. and Hu, J., 2006, .Nonlinear vibration characterization by signal decomposition,. Proceedings of the 24th International Modal Analysis Conference (IMAC), St. Louis.
Yang, J.N., Lei, Y., Pan, S.W. and Huang, N., 2003, .System identification of linear structures based on Hilbert-Huang spectral analysis; Part 1: Normal modes,. Earthquake Engineering and Structural Dynamics 32, 1443-1467.
Yang, J.N., Lei, Y., Pan, S.W. and Huang, N., 2003, .System identification of linear structures based on Hilbert-Huang spectral analysis; Part 2: Complex modes,. Earthquake Engineering and Structural Dynamics 32, 1533-1554.
Kerschen, G., Vakakis, A.F., Lee, Y.S., McFarland, D.M. and Bergman, L.A., 2006, .Toward a fundamental understanding of the Hilbert-Huang Transform in nonlinear structural dynamics,. Proceedings of the 24th International Modal Analysis Conference (IMAC), St-Louis.
Sanders, J.A. and Verhulst, F., 1985, Averaging Methods in Nonlinear Dynamics Systems, Springer-Verlag, New York.
Manevitch, L.I., Complex representation of dynamics of coupled oscillators in Mathematical Models of Nonlinear Excitations, Transfer Dynamics and Control in Condensed Systems, New York, Kluwer Academic/Plenum Publishers, pp. 269-300, 1999.
Lochak, P. and Meunier, C., 1988, Multiphase Averaging for Classical Systems with Applications to Adiabatic Theorems, Springer Verlag, Berlin and New York.
Kerschen, G., Vakakis, A.F., Lee, Y.S., McFarland, D.M. and Bergman, L.A., Toward a Fundamental Understanding of the Hilbert-Huang Transform in Nonlinear Dynamics, Journal of Vibration and Control, in press.
Kerschen, G., Lee, Y.S., Vakakis, A.F., McFarland, D.M. and Bergman, L.A., 2006, .Irreversible passive energy transfer in coupled oscillators with essential nonlinearity,. SIAM Journal on Applied Mathematics 66, 648-679.
Rilling, G., Flandrin, P. and Goncalves, P., 2003, .On empirical mode decomposition and its algorithms,. IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing NSIP-03, Grado.
McFarland, D.M., Bergman, L.A. and Vakakis, A.F., 2005, .Experimental study of non-linear energy pumping occurring at a single fast frequency,. International Journal of Non-linear Mechanics 40, 891-899.
