The response of a nonlinear oscillator is characterized by its instantaneous amplitude (IA) and instantaneous frequency (IF) features, which can be significantly affected by the physical properties of the system. Accordingly, the system properties could be inferred from the IA and IF of its response if both instantaneous features can be identified accurately. To fulfill such an idea, a nonlinear system parameter identification method is proposed in this paper with the aid of polynomial chirplet transform (PCT), which has been proved a powerful tool for processing nonstationary signals. First, the PCT is used to extract the instantaneous characteristics, i.e., IA and IF, from nonlinear system responses. Second, instantaneous modal parameters estimation was adopted to extract backbone and damping curves, which characterize the inherent nonlinearities of the system. Third, the physical property parameters of the system were estimated through fitting the identified average nonlinear characteristic curves. Finally, the proposed nonlinear identification method is experimentally validated through comparing with two Hilbert transform (HT) based methods.
Issue Section:
Research Papers
References
1.
Von Karman
, T.
, 1940
, “The Engineer Grapples With Nonlinear Problem
,” Bull. Am. Math. Soc.
, 46
(8
), pp. 615
–683
.2.
Amabili
, M.
, and Paidoussis
, M.
, 2003
, “Review of Studies on Geometrically Nonlinear Vibrations and Dynamics of Circular Cylindrical Shells and Panels, With and Without Fluid-Structure Interaction
,” ASME Appl. Mech. Rev.
, 56
(4
), pp. 349
–381
.3.
Schmidt
, G.
, and Tondl
, A.
, 1986
, Non-Linear Vibration
, Cambridge University Press
, Cambridge, UK
.4.
Nayfeh
, A. H.
, and Mook
, D. T.
, 1979
, Nonlinear Oscillations
, Wiley-Interscience
, New York
.5.
Verhulst
, F.
, 1999
, Nonlinear Differential Equations and Dynamical Systems
, 2nd, ed., Springer
, Berlin
.6.
Nayfeh
, A. H.
, and Pai
, P. F.
, 2004
, Linear and Nonlinear Structural Mechanics
, Wiley-Interscience
, New York
.7.
Babitsky
, V. I.
, and Krupenin
, V. L.
, 2001
, Vibrations of Strongly Nonlinear Discontinuous Systems
, Springer
, Berlin
.8.
Worden
, K.
, and Tomlinson
, G. R.
, 2002
, Nonlinearity in Structural Dynamics: Detection, Identification and Modeling
, CRC Press
, Boca Raton, FL
.9.
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, Report No. LA-13070-MS
.10.
Cabrera
, D.
, Sancho
, F.
, Sanchez
, R. V.
, Zurita
, G.
, Cerrada
, M.
, Li
, C.
, and Vasquez
, R. E.
, 2015
, “Fault Diagnosis of Spur Gearbox Based on Random Forest and Wavelet Packet Decomposition
,” Front. Mech. Eng.
, 10
(3
), pp. 277
–286
.11.
Falco
, M.
, Liu
, M.
, Nguyen
, S. H.
, and Chelidze
, D.
, 2014
, “Nonlinear System Identification and Modeling of a New Fatigue Testing Rig Based on Inertial Forces
,” ASME J. Vib. Acoust.
, 136
(4
), p. 041001.12.
Kerschen
, G.
, Worden
, K.
, and Vakakis
, A. F.
, 2006
, “Past, Present and Future of Nonlinear System Identification in Structural Dynamics
,” Mech. Syst. Signal Process.
, 20
(3
), pp. 505
–592
.13.
Socha
, L.
, and Pawleta
, M.
, 2001
, “Are Statistical Linearization and Standard Equivalent Linearization the Same Methods in the Analysis of Stochastic Dynamic Systems?
,” J. Sound Vib.
, 248
(2
), pp. 387
–394
.14.
Iwan
, W. D.
, and Mason
, A. B.
, 1980
, “Equivalent Linearization for Systems Subjected to Non-Stationary Random Excitation
,” Int. J. Non-Linear Mech.
, 15
(2
), pp. 71
–82
.15.
Miles
, H. J.
, 1995
, “Identification of Weakly Non-Linear Systems Using Equivalent Linearization
,” J. Sound Vib.
, 185
(3
), pp. 473
–481
.16.
Soize
, C.
, 1994
, “Stochastic Linearization Method With Random Parameters and Power Spectral Density Calculation
,” 6th International Conference on Structural Safety and Reliability
, Innsbruck, Austria, Aug. 9–13, pp. 217
–222
.17.
Billings
, S. A.
, and Tsang
, K. M.
, 1989
, “Spectral Analysis for Nonlinear Systems, Part I: Parametric Non-Linear Spectral Analysis
,” Mech. Syst. Signal Process.
, 3
(4
), pp. 319
–339
.18.
Billings
, S. A.
, and Tsang
, K. M.
, 1989
, “Spectral Analysis for Nonlinear Systems, Part II: Interpretation of Nonlinear Frequency Response Functions
,” Mech. Syst. Signal Process.
, 3
(4
), pp. 341
–359
.19.
Storer
, D. M.
, and Tomlinson
, G. R.
, 1993
, “Recent Developments in the Measurements and Interpretation of Higher Order Functions From Non-Linear Structures
,” Mech. Syst. Signal Process.
, 7
(2
), pp. 173
–189
.20.
Schetzen
, M.
, 1980
, The Volterra and Wiener Theories of Nonlinear Systems
, Wiley
, New York
.21.
Scussel
, O.
, and da Silva
, S.
, 2016
, “Output-Only Identification of Nonlinear System Via Volterra Series
,” ASME J. Vib. Acoust.
, 138
(4
), p. 041012
.22.
Riche
, R. L.
, Gualandris
, D.
, Thomas
, J. J.
, and Hemez
, F.
, 2001
, “Neural Identification of Non-Linear Dynamic Structures
,” J. Sound Vib.
, 248
(2
), pp. 247
–265
.23.
Chassiakos
, A. G.
, and Masri
, S. F.
, 1996
, “Modelling Unknown Structural Systems Through the Use of Neural Networks
,” Earthquake Eng. Struct. Dyn.
, 25
(2
), pp. 117
–128
.24.
Cinar
, A.
, 1995
, “Nonlinear Time Series Models for Multivariable Dynamic Processes
,” Chemom. Intell. Lab. Syst.
, 30
(1
), pp. 147
–158
.25.
Zou
, Y.
, Tong
, L.
, and Steven
, G. P.
, 2000
, “Vibration-Based Model-Dependent Damage (Delamination) Identification and Health Monitoring for Composite Structures-a Review
,” J. Sound Vib.
, 230
(2
), pp. 357
–378
.26.
Kerschen
, G.
, Vakakis
, A. F.
, Lee
, Y. S.
, Mcfarland
, D. M.
, and Bergman
, L. A.
, 2008
, “Toward a Fundamental Understanding of the Hilbert–Huang Transform in Nonlinear Structural Dynamics
,” J. Vib. Control
, 14
(1–2
), pp. 77
–105
.27.
Sun
, Y.
, Zhuang
, C.
, and Xiong
, Z.
, 2015
, “Transform Operator Pair Assisted Hilbert-Huang Transform for Signals With Instantaneous Frequency Intersections
,” ASME J. Vib. Acoust.
, 137
(6
), p. 061016
.28.
Feldman
, M.
, and Braun
, S.
, 1995
, “Identification of Non-Linear System Parameters Via the Instantaneous Frequency: Application of the Hilbert Transform and Wigner–Ville Technique
,” 13th International Modal Analysis Conference
, Nashville, TN, Feb. 13–16, pp. 637
–642
.29.
Franco
, H.
, and Pauletti
, R. M. O.
, 1997
, “Analysis of Nonlinear Oscillations by Gabor Spectrograms
,” Nonlinear Dyn.
, 12
(3
), pp. 215
–236
.30.
Bellizzi
, S.
, Guillemain
, P.
, and Kronland-Martinet
, R.
, 2001
, “Identification of Coupled Non-Linear Modes From Free Vibration Using Time–Frequency Representations
,” J. Sound Vib.
, 243
(2
), pp. 191
–213
.31.
Argoul
, P.
, and Le
, T. P.
, 2003
, “Instantaneous Indicators of Structural Behaviour Based on the Continuous Cauchy Wavelet Analysis
,” Mech. Syst. Signal Process.
, 17
(1
), pp. 243
–250
.32.
Staszewski
, W. J.
, 1998
, “Identification of Non-Linear Systems Using Multi-Scale Ridges and Skeletons of the Wavelet Transform
,” J. Sound Vib.
, 214
(4
), pp. 639
–658
.33.
Pailwal
, D.
, Choudhur
, A.
, and Govandhan
, T.
, 2014
, “Identification of Faults Through Wavelet Transform Vis-à-Vis Fast Fourier Transform of Noisy Vibration Signals Emanated From Defective Rolling Element Bearings
,” Front. Mech. Eng.
, 9
(2
), pp. 130
–141
.34.
Pai
, P. F.
, 2013
, “Time–Frequency Analysis for Parametric and Non-Parametric Identification of Nonlinear Dynamical Systems
,” Mech. Syst. Signal Process.
, 36
(2
), pp. 332
–353
.35.
Feldman
, M.
, 1994
, “Non-linear System Vibration Analysis Using Hilbert Transform—I. Free Vibration Analysis Method ‘FREEVIB’
,” Mech. Syst. Signal Process.
, 8
(2
), pp. 119
–127
.36.
Feldman
, M.
, 1994
, “Non-Linear System Vibration Analysis Using Hilbert Transform—II. Forced Vibration Analysis Method ‘FORCEVIB’
,” Mech. Syst. Signal Process.
, 8
(3
), pp. 309
–318
.37.
Feldman
, M.
, 2007
, “Considering High Harmonics for Identification of Non-Linear Systems by Hilbert Transform
,” Mech. Syst. Signal Process.
, 21
(2
), pp. 943
–958
.38.
Feldman
, M.
, 2006
, “Time-Varying Vibration Decomposition and Analysis Based on the Hilbert Transform
,” J. Sound Vib.
, 295
(3–5
), pp. 518
–530
.39.
Feldman
, M.
, 1997
, “Non-Linear Free Vibration Identification Via the Hilbert Transform
,” J. Sound Vib.
, 208
(3
), pp. 475
–489
.40.
Feldman
, M.
, 2012
, “Nonparametric Identification of Asymmetric Nonlinear Vibration Systems With the Hilbert Transform
,” J. Sound Vib.
, 331
(14
), pp. 3386
–3396
.41.
Peng
, Z. K.
, Meng
, G.
, and Chu
, F. L.
, 2011
, “Polynomial Chirplet Transform With Application to Instantaneous Frequency Estimation
,” IEEE Trans. Instrum. Meas.
, 60
(9
), pp. 3222
–3229
.42.
Yang
, Y.
, Zhang
, W.
, Peng
, Z.
, and Meng
, G.
, 2013
, “Multicomponent Signal Analysis Based on Polynomial Chirplet Transform
,” IEEE Trans. Ind. Electron.
, 60
(9
), pp. 3948
–3956
.43.
Wang
, L. L.
, Zhang
, J. H.
, Wang
, C.
, and Hu
, S. Y.
, 2003
, “Time–Frequency Analysis of Nonlinear Systems: The Skeleton Linear Model and the Skeleton Curves
,” ASME J. Vib. Acoust.
, 125
(2
), pp. 170
–177
.44.
Weaver
, W.
, Timoshenko
, S. P.
, and Young
, D. H.
, 1990
, Vibration Problems in Engineering
, 5th ed., Wiley
, New York
.45.
Feldman
, M.
, 2011
, Hilbert Transform Application in Mechanical Vibration
, Wiley
, Chichester, UK
.46.
Deng
, Y.
, Peng
, Z. K.
, Yang
, Y.
, Zhang
, W. M.
, and Meng
, G.
, 2013
, “Identification of Nonlinear Vibration Systems Based on Parametric TFA
,” Chin. J. Theor. Appl. Mech.
, 45
(6
), pp. 992
–996
(in Chinese).47.
Boashash
, B.
, 1992
, “Estimating and Interpreting the Instantaneous Frequency of a Signal. I. Fundamentals
,” Proc. IEEE
, 80
(4
), pp. 520
–538
.48.
Boashash
, B.
, 1992
, “Estimating and Interpreting the Instantaneous Frequency of a Signal. II. Algorithms and applications
,” Proc. IEEE
, 80
(4
), pp. 540
–568
.49.
Kwok
, H. K. C.
, and Jones
, D. L.
, 1995
, “Instantaneous Frequency Estimation Using an Adaptive Short-Time Fourier Transform
,” Twenty-Ninth Asilomar Conference on Signals, Systems and Computers
, Pacific Grove, CA, Oct. 30–Nov. 1, pp. 543
–546
.50.
Shui
, P. L.
, Bao
, Z.
, and Su
, H. T.
, 2008
, “Nonparametric Detection of FM Signals Using Time–Frequency Ridge Energy
,” IEEE Trans. Signal Process.
, 56
(5
), pp. 1749
–1760
.51.
Li
, Y. Y.
, Huang
, X. Q.
, and Mao
, W. X.
, 2005
, “Effect of the Cubic Nonlinear Factors for Displacement and Velocity on Amplitude Frequency Characteristics of Dry Friction System for Metal Rubber
,” J. Mech. Strength
, 27
(4
), pp. 436
–439
(in Chinese).52.
Ling
, R. J.
, Weng
, J. S.
, and Jin
, Z. L.
, 2009
, “Non-Linear Finite Element Analysis on Stiffness and Hysteresis Characteristic of Leaf Spring
,” J. Chongqing Inst. Technol. (Nat. Sci)
, 23
(1
), pp. 19
–23
(in Chinese).53.
Li
, S. H.
, and Yang
, S. P.
, 2006
, “Research Status of Hysteretic Nonlinear Models
,” J. Dyn.
Control, 4
(2
), pp. 8
–15
(in Chinese).Copyright © 2016 by ASME
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