HHT-based nonlinear signal processing method for parametric and non-parametric identification of dynamical systems

Abstract This paper presents a time–frequency signal processing method based on Hilbert–Huang transform (HHT) and a sliding-window fitting (SWF) technique for parametric and non-parametric identification of nonlinear dynamical systems. The SWF method is developed to reveal the limitations of conventional signal processing methods and to perform further decomposition of signals. Similar to the short-time Fourier transform and wavelet transform, the SWF uses windowed regular harmonics and function orthogonality to extract time-localized regular and/or distorted harmonics. On the other hand, HHT uses the apparent time scales revealed by the signal's local maxima and minima to sequentially sift components of different time scales, starting from high- to low-frequency ones. Because HHT does not use pre-determined basis functions and function orthogonality for component extraction, it provides more accurate time-varying amplitudes and frequencies of extracted components for accurate estimation of system characteristics and nonlinearities. Methods are developed to reduce the end effect caused by Gibbs’ phenomenon and other mathematical and numerical problems of HHT analysis. For parametric identification of a nonlinear one-degree-of-freedom system, the method processes one free damped transient response and one steady-state response and uses amplitude-dependent dynamic characteristics derived from perturbation analysis to determine the type and order of nonlinearity and system parameters. For non-parametric identification, the method uses the maximum displacement states to determine the displacement–stiffness curve and the maximum velocity states to determine the velocity-damping curve. Moreover, the SWF method and a synchronous detection method are used for further decomposition of components extracted by HHT to improve the accuracy of parametric and non-parametric estimations. Numerical simulations of several nonlinear systems show that the proposed method can provide accurate parametric and non-parametric identifications of different nonlinear dynamical systems.

[1]  Charles R. Farrar,et al.  Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: A literature review , 1996 .

[2]  Douglas E. Adams,et al.  Health monitoring of structural materials and components : methods with applications , 2007 .

[3]  Michael Feldman,et al.  Non-linear system vibration analysis using Hilbert transform--I. Free vibration analysis method 'Freevib' , 1994 .

[4]  Alan V. Oppenheim,et al.  Discrete-Time Signal Pro-cessing , 1989 .

[5]  Earl H. Dowell,et al.  Studies in nonlinear aeroelasticity , 1988 .

[6]  N. Huang,et al.  A new view of nonlinear water waves: the Hilbert spectrum , 1999 .

[7]  S. S. Shen,et al.  A confidence limit for the empirical mode decomposition and Hilbert spectral analysis , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[8]  Robert T. Ratay.  Structural Condition Assessment , 2005 .

[9]  P. D. McFadden,et al.  A review of time-frequency methods for structural vibration analysis , 2003 .

[10]  Nii O. Attoh-Okine,et al.  The Hilbert-Huang Transform in Engineering , 2005 .

[11]  P. F. Pai,et al.  Non-Linear Structural Dynamics Characterization using a Scanning Laser Vibrometer , 2003 .

[12]  P. Pai.  Nonlinear vibration characterization by signal decomposition , 2007 .

[13]  E. Bedrosian.  A Product Theorem for Hilbert Transforms , 1963 .

[14]  Hoon Sohn,et al.  A review of structural health monitoring literature 1996-2001 , 2002 .

[15]  Ali H. Nayfeh,et al.  Energy Transfer From High-to Low-Frequency Modes in a Flexible Structure via Modulation , 1994 .

[16]  David J. Ewins,et al.  Modal Testing: Theory, Practice, And Application , 2000 .

[17]  Michael Feldman,et al.  Non-linear system vibration analysis using Hilbert transform--II. Forced vibration analysis method 'Forcevib' , 1994 .

[18]  P. Frank Pai,et al.  Highly Flexible Structures : Modeling, Computation, and Experimentation , 2007 .

[19]  Maher N. Bismarck-Nasr.  Finite Elements in Aeroelasticity of Plates and Shells , 1996 .

[20]  N. Huang,et al.  The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[21]  Gábor Stépán,et al.  Nonlinear Dynamics of High-Speed Milling—Analyses, Numerics, and Experiments , 2005 .

[22]  E. Dowell.  Panel flutter - A review of the aeroelastic stability of plates and shells , 1970 .

[23]  N. Huang,et al.  A study of the characteristics of white noise using the empirical mode decomposition method , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[24]  H. Saunders,et al.  Structural Integrity Monitoring , 1985 .

[25]  Victor Giurgiutiu,et al.  Piezoelectric Wafer Embedded Active Sensors for Aging Aircraft Structural Health Monitoring , 2002 .

[26]  G. Tomlinson,et al.  Nonlinearity in Structural Dynamics: Detection, Identification and Modelling , 2000 .

[27]  L. S. Qu,et al.  General Interpolated Fast Fourier Transform: A New Tool for Diagnosing Large Rotating Machinery , 2005 .

[28]  Balakumar Balachandran,et al.  Dynamics and stability of milling process , 2001 .

[29]  A. Nayfeh.  Introduction To Perturbation Techniques , 1981 .

[30]  Chuh Mei,et al.  Review of Nonlinear Panel Flutter at Supersonic and Hypersonic Speeds , 1999 .

[31]  A. Nuttall,et al.  On the quadrature approximation to the Hilbert transform of modulated signals , 1966 .

[32]  Christopher H. M. Jenkins,et al.  Gossamer spacecraft : membrane and inflatable structures technology for space applications , 2001 .

[33]  Lu Huang,et al.  Dynamics- and Laser-Based Boundary Effect Evaluation Method for Damage Inspection of One- and Two-Dimensional Structures , 2006 .

[34]  A. Nayfeh,et al.  Linear and Nonlinear Structural Mechanics , 2002 .

[35]  E. O. Brigham,et al.  The Fast Fourier Transform , 1967, IEEE Transactions on Systems, Man, and Cybernetics.

[36]  Truong Q. Nguyen,et al.  Wavelets and filter banks , 1996 .

[37]  Anthony N. Palazotto,et al.  Detection and identification of nonlinearities by amplitude and frequency modulation analysis , 2008 .

[38]  F. Chang,et al.  Detection and monitoring of hidden fatigue crack growth using a built-in piezoelectric sensor/actuator network: I. Diagnostics , 2004 .

[39]  Patrick Flandrin,et al.  Time-Frequency/Time-Scale Analysis , 1998 .

[40]  S. Hahn.  Hilbert Transforms in Signal Processing , 1996 .

[41]  E. Dowell,et al.  Chaotic Vibrations: An Introduction for Applied Scientists and Engineers , 1988 .

[42]  K. Worden,et al.  Past, present and future of nonlinear system identification in structural dynamics , 2006 .

[43]  D. Zimmerman,et al.  Structural damage detection using a minimum rank update theory , 1994 .