Time-Frequency Method for Nonlinear System Identification and Damage Detection

Abstract

This paper presents a method for extracting system nonlinearities and time-localized transient response to impulsive loading by processing stationary/transient responses using the Hilbert—Huang transform (HHT) and a sliding-window fitting (SWF) technique. Time-dependent dynamic characteristics of nonlinear systems are derived using perturbation analysis. The SWF is introduced mainly to show the mathematical implications of HHT and the differences between HHT and the discrete Fourier transform. Similar to the wavelet transform the SWF uses windowed predetermined regular harmonics and function orthogonality to extract local harmonic components. It simultaneously decomposes a signal into just a few regular/distorted harmonics, and the obtained time-varying amplitudes and frequencies of the harmonics can reveal system nonlinearities. On the other hand the HHT uses the apparent time scales revealed by the signal's local maxima and minima and cubic splines of the extrema to sequentially sift components of different time scales, starting from high-frequency to low-frequency ones. Because HHT does not use predetermined basis functions and function orthogonality for component extraction, components are extracted without distortion and hence their time-varying amplitudes and frequencies can be accurately computed using the Hilbert transform to reveal system characteristics and nonlinearities. Moreover, because the first component extracted from HHT contains all discontinuities of the original signal, its time-varying frequency and amplitude are excellent indicators for pinpointing the time instants of impulsive loads. However, the discontinuity-induced Gibbs' phenomenon makes HHT analysis inaccurate around the two data ends. On the other hand, the SWF analysis suffers less from Gibbs' phenomenon at the two data ends, but it cannot extract accurate time-varying frequencies and amplitudes because the use of predetermined basis functions and function orthogonality in the sliding-window fitting process distorts the extracted components. Numerical and experimental results show that the proposed method with the use of HHT can provide accurate extraction of intrawave amplitude and phase modulations, distorted harmonic response under a single-frequency harmonic excitation, softening and hardening effects, different orders of nonlinearity, interwave amplitude and phase modulations, multiple-mode vibrations caused by internal/ external resonances, and time instants of impact loading on a structure. These are key phenomena for performing dynamics-based system identification and damage detection.

Keywords

Hilbert-Huang transform nonlinear signal processing nonlinearity identification structural damage detection Gibbs' phenomenon

Get full access to this article

View all access options for this article.

References

Pai, P.F. (2007). Highly Flexible Structures: Modeling, Computation and Experimentation, Reston , VA: AIAA .

Doebling, S.W. , Farrar, C.R. , Prime, M.B. and Shevitz, D.W. (1996). Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature Review, Los Alamos National Laboratory .

Sohn, H. , Farrar, C.R. , Hemez, F.M. , Shunk, D.D. , Stinemates, D.W. and Nadler, B.R. (2003). A review of structural health monitoring literature: 1996-2001, Report No. LA-13976-MS, Los Alamos National Laboratory .

Humar, J. , Bagchi, A. and Xu, H. (2006). Performance of vibration-based techniques for the identification of structural damage , Structural Health Monitoring, 5(3), 215-241.

Pai, P.F. and Huang, L. (2006). Dynamics- and laser-based boundary effect evaluation method for damage inspection of one- and two-dimensional structures . Journal of Nondestructive Evaluation, 25(2), 83-105.

Nayfeh, A.H. and Mook, D.T. (1979). Nonlinear Oscillations, NY Wiley .

Nayfeh, A.H. and Pai, P.F. (2004). Linear and Nonlinear Structural Mechanics, New York: Wiley-Interscience.

Nayfeh, A.H. (1981). Introduction to Perturbation Techniques, New York: Wiley-Interscience.

Worden, K. and Tomlinson, G.R. (2001). Nonlinearity in Structural Dynamics: Detection, Identification and Modelling, Bristol and Philadelphia: Institute of Physics Publishing.

10.

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 emperical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis . In Proc. R. Soc. Lond. A, Vol. 454, pp. 903-995.

11.

Huang, N.E. , Shen, Z. and Long, S.R. (1999). A new view of nonlinear water waves: the hilbert spectrum . Annu. Rev. Fluid Mechanics , 31, 417-457.

12.

Huang, N.E. , Wu, M.C. , Long, S.R. , Shen, S.S.P. , Qu, W. , Gloersen, P. and Fan, K.L. (2003). A confidence limit for the empirical mode decomposition and Hilbert spectral analysis. Proc . R. Soc. Lond. A, Vol. 459, pp. 2317-2345.

13.

Wu, Z. and Huang, N.E. (2004). A study of the characteristics of white noise using the empirical mode decomposition method. Proc . R. Soc. Lond. A, Vol. 460, pp. 1597-1611.

14.

Huang, N.E. and Attoh-Okine, N.O. (eds) (2005). The Hilbert-Huang Transform in Engineering, Boca Raton, FL: CRC Press.

15.

Brigham, E.O. (1974). The Fast Fourier Transform, Englewood Cliffs, New Jersey: Prentice-Hall.

16.

Flandrin, P. (1999). Time-Frequency/Time-Scale Analysis, Academic Press, New York.

17.

Wylie, C.R. (1975). Advanced Engineering Mathematics, New York: McGraw-Hill.

18.

Zygmund, A. (1955). Trigonometrical Series, New York: Dover Publications.

19.

Strang, G. and Nguyen, T. (1997). Wavelets and Filter Banks, Wellesley, MA: Wellesley-Cambridge Press.

20.

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.

21.

Feldman, M. (1994). Nonlinear system vibration analysis using the Hilbert transform-II. Forced vibration analysis method `FORCEVIB' . Mechanical Systems and Signal Processing, 8, 309-318.

22.

Pai, P.F. and Nayfeh, A.H. (1990). Nonlinear nonplanar oscillations of a cantilever beam under lateral base excitations . Int. J. Nonlinear Mechanics, 25, 455-474.

23.

Beard, S. and Chang, F.K. (1997). Active damage detection in filament wound composite tubes using built-in sensors and actuators . Journal of Intelligent Material Systems and Structures, 8(10), 891-898.

24.

Flandrin, P. , Rilling, G. and Goncalves, P. (2004). Emperical mode decomposition as a filterbank . IEEE Signal Proc. Lett., 11(2), 112-114.

25.

Yang, J.N. , Huang, H. and Lin, S. (2006). Sequential nonlinear least-square estimation for damage identification of structures . Int. J. Nonlinear Mechanics , 41(1), 124-140.

26.

Yang, J.N. , Lei, Y. , Pan, S. and Huang, N. (2003). Identification of linear structures based on Hilbert-Huang transform. II: complex modes . J. Earthquake Eng. Struct. Dyn. , 32(10), 1533-1554.

27.

Zhang, R. , Ma, S. and Hartzell, S. (2003). Signatures of seismic source in EMD-based characterization of the 1994 Northridge, California, earthquake recordings . Bulletin of Seismological Society of America, 93, 501-518.

28.

Hahn, S.L. (1996). Hilbert Transforms in Signal Processing, Boston: Artech House.