Archive of Applied Mechanics

, Volume 82, Issue 10–11, pp 1461–1479 | Cite as

Nonlinear system identification of the dynamics of a vibro-impact beam: numerical results

  • Mehmet Kurt
  • Heng Chen
  • Young S. LeeEmail author
  • D. Michael McFarland
  • Lawrence A. Bergman
  • Alexander F. Vakakis
Special Issue


We study the dynamics of a cantilever beam with two rigid stops of certain clearances by performing nonlinear system identification (NSI) based on the correspondence between analytical and empirical slow-flow dynamics. The NSI method in this work can proceed in two directions: One for the numerical data obtained from a reduced-order model by means of the assumed-mode method, and the other for the experimental data measured at the same positions as in numerical simulations. This paper focuses on the analysis of the numerical data, providing qualitative comparison with some experimental results; the latter task will be discussed in detail in a companion paper. First, we perform empirical mode decomposition (EMD) on the acceleration responses measured at ten, almost evenly-spaced, spanwise positions along the beam leading to sets of intrinsic modal oscillators governing the vibro-impact dynamics at different time scales. In particular, the EMD analysis can separate any nonsmooth effects caused by vibro-impacts of the beam and the rigid stops from the smooth (elastodynamic) response, so that nonlinear modal interactions caused by vibro-impacts can be explored only with the remaining smooth components. Then, we establish nonlinear interaction models (NIMs) for the respective intrinsic modal oscillators, where the NIMs invoke slowly-varying forcing amplitudes that can be computed from empirical slow-flows. By comparing the spatio-temporal variations of the nonlinear modal interactions for the vibro-impact beam and those of the underlying linear model (i.e., the beam with no rigid constraints), we demonstrate that vibro-impacts significantly influence the lower-frequency modes, introducing spatial modal distortions, whereas the higher frequency modes tend to retain their linear dynamics between impacts. We introduce a linear correlation coefficient as a measure for studying the linear dependency between the slowly-varying complex forcing amplitudes for the linear and vibro-impact beams and demonstrate that only a set of lower-frequency modes are strongly influenced by vibro-impacts, capturing most of the essential nonlinear dynamics. These results demonstrate the efficacy of the proposed approach to analyze strongly nonlinear measured time series and provide physical insight for strong nonlinear dynamical interactions.


Nonlinear system identification Empirical mode decomposition Vibro-impact beam Intrinsic mode oscillation Nonlinear interaction model 





Empirical mode decomposition


Frequency-energy plot


Fourier transform


Hilbert transform


Intrinsic mode function


Intrinsic modal oscillator


Nonlinear interaction model


Nonlinear system identification


Proper orthogonal decomposition


Reduced-order model




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ewins D.J.: Modal Testing: Theory and Practice. Research Studies Press, UK (1990)Google Scholar
  2. 2.
    Brandon J.A.: Some insights into the dynamics of defective structures. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 212, 441–454 (1998)CrossRefGoogle Scholar
  3. 3.
    Kerschen G., Golinval J.-C., Vakakis A.F., Bergman L.A.: The method of proper orthogonal decomposition for order reduction of mechanical systems: an overview. Nonlinear Dyn. 41, 147–170 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Kerschen G., Worden K., Vakakis A.F., Golinval J.-C.: Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 20, 505–592 (2005)CrossRefGoogle Scholar
  5. 5.
    Feeny B.F., Kappagantu R.: On the physcal interpretation of proper orthogonal modes in vibrations. J. Sound Vib. 211, 607–616 (1998)CrossRefGoogle Scholar
  6. 6.
    Kerschen G., Golinval J.C.: Physical interpretation of the proper orthogonal modes using the singular value decomposition. J. Sound Vib. 249, 849–865 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bellizzi S., Sampaio R.: POMs analysis of randomly vibrating systems obtained from Karhunen-Loève expansion. J. Sound Vib. 297, 774–793 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Allison T.C., Miller A.K., Inman D.J.: A deconvolution-based approache to structural dynamics system identification and response prediction. J. Vib. Acoust. 130, 031010 (2008)CrossRefGoogle Scholar
  9. 9.
    Chelidze D., Zhou W.: Smooth orthogonal decomposition-based vibration mode identification. J. Sound Vib. 292, 461–473 (2006)CrossRefGoogle Scholar
  10. 10.
    Silva W.: Identification of nonlinear aeroelastic systems based on the Volterra theory: progress and opportunities. Nonlinear Dyn. 39, 25–62 (2005)zbMATHCrossRefGoogle Scholar
  11. 11.
    Li L.M., Billings S.A.: Analysis of nonlinear oscillators using Volterra series in the frequency domain. J. Sound Vib. 330, 337–355 (2011)CrossRefGoogle Scholar
  12. 12.
    Mariani S., Ghisi A.: Unscented Kalman filtering for nonlinear structural dynamics. Nonlinear Dyn. 49, 131–150 (2007)zbMATHCrossRefGoogle Scholar
  13. 13.
    Masri S., Caughey T.: A nonparametric identification techanique for nonlinear dynamic systems. J. Appl. Mech. 46, 433–441 (1979)zbMATHCrossRefGoogle Scholar
  14. 14.
    Leontaritis, I.J., Billings, S.A.: Input–output parametric models for nonlinear systems. Part I. Deterministic nonlinear systems; Part II. Stochastic nonlinear systems. Int. J. Control 41, 303–328; 329–344 (1985)Google Scholar
  15. 15.
    Thothadri M., Casas R.A., Moon F.C., D’Andrea R., Johnson C.R. Jr: Nonlinear system identification of multi-degree-of-freedom systems. Nonlinear Dyn. 32, 307–322 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Feldman, M.: Non-linear system vibration analysis using Hilbert transform–I. Free vibration analysis method ‘FREEVIB’; II. Forced vibration analysis method ‘FORCEVIB’. Mech. Syst. Signal Proces. 8, 119–127; 309–318 (1994)Google Scholar
  17. 17.
    Feldman M.: Time-varying vibration decomposition and analysis based on the Hilbert transform. J. Sound Vib. 295, 518–530 (2006)zbMATHCrossRefGoogle Scholar
  18. 18.
    Ma X., Azeez M.F.A., Vakakis A.F.: Non-linear normal modes and non-parametric system identification of non-linear oscillators. Mech. Syst. Signal Process. 14, 37–48 (2000)CrossRefGoogle Scholar
  19. 19.
    Georgiou I.: Advanced proper orthogonal decomposition tools: using reduced order models to identify normal modes of vibration and slow invariant manifolds in the dynamics of planar nonlinear rods. Nonlinear Dyn. 41, 69–110 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Galvanetto U., Surace C., Tassotti A.: Structural damage detection based on proper orthogonal decomposition: experimental verification. AIAA J. 46, 1624–1630 (2008)CrossRefGoogle Scholar
  21. 21.
    Cusumano J.P., Bae B.-Y.: Period-infinity periodic motions, chaos, and spatial coherence in a 10 degree of freedom impact oscillator. Chaos, Solitons Fractals 3, 515–535 (1993)zbMATHCrossRefGoogle Scholar
  22. 22.
    Cusumano J.P., Sharkady M.T., Kimble B.W.: Experimental measurements of dimensionality and spatial coherence in the dynamics of a flexible-bea impact oscillator. Philos. Trans. R. Soc. Ser. A 347, 421–438 (1994)CrossRefGoogle Scholar
  23. 23.
    Ritto T.G., Buezas F.S., Sampaio R.: A new measure of efficiency for model reduction: application to a vibroimpact system. J. Sound Vib. 330, 1977–1984 (2011)CrossRefGoogle Scholar
  24. 24.
    Azeez M.F.A., Vakakis A.F.: Proper orthogonal decomposition (POD) of a class of vibroimpact oscillations. J. Sound Vib. 240, 859–889 (2001)CrossRefGoogle Scholar
  25. 25.
    Lee Y.S., Vakakis A.F., McFarland D.M., Bergman L.A.: A global-local approach to system identification: a review. Struct. Control Health Monit. 17, 742–760 (2010)CrossRefGoogle Scholar
  26. 26.
    Huang N., Shen Z., Long S., Wu M., Shih H., Zheng Q., Yen N.-C, Tung C., Liu H.: The empirical mode decompostion and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 454, 903–995 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Lee Y.S., Tsakirtzis S., Vakakis A.F., Bergman L.A., McFarland D.M.: Physics-based foundation for empirical mode decomposition. AIAA J. 47, 2938–2963 (2009)CrossRefGoogle Scholar
  28. 28.
    Lee Y.S., Tsakirtzis S., Vakakis A.F., McFarland D.M., Bergman L.A.: A time-domain nonlinear system identification method based on multiscale dynamic partitions. Meccanica 46, 625–649 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lee Y.S., Vakakis A.F., McFarland D.M., Bergman L.A.: Nonlinear system identification of the dynamics of aeroelastic instability suppression based on targeted energy transfers. Aeronaut. J. 114, 61–82 (2010)Google Scholar
  30. 30.
    Tsakirtzis S., Lee Y.S., Vakakis A.F., Bergman L.A., McFarland D.M.: Modeling of nonlinear modal interactions in the transient dynamics of an elastic rod with an essentially nonlinear attachment. Commun. Nonlinear Sci. Numer. Simul. 15, 2617–2633 (2010)zbMATHCrossRefGoogle Scholar
  31. 31.
    Dawes J.H.P.: Review: the emergence of a coherent structure for coherent structures: localized states in nonlinear systems. Philos. Trans. R. Soc. Ser. A 368, 3519–3534 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Chati M., Rand R., Mukherjee S.: Modal analysis of a cracked beam. J. Sound Vib. 207, 249–270 (1997)zbMATHCrossRefGoogle Scholar
  33. 33.
    Chen H.G., Yan Y.J., Jiang J.S.: Vibration-based damage detection in composite wingbox structures by HHT. Mech. Syst. Signal Process. 21, 307–321 (2007)CrossRefGoogle Scholar
  34. 34.
    Mane, M.: Experiments in Vibro-Impact Beam Dynamics and a System Exhibiting a Landau-Zener Quantum Effect. MS Thesis (unpublished), Univeristy of Illinois at Urbana-Champaign (2010)Google Scholar
  35. 35.
    Blevins R.D.: Formulas for Natural Frequency and Mode Shape. Krieger, New York (1995)Google Scholar
  36. 36.
    Manevitch L.: The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables. Nonlinear Dyn. 25, 95–109 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Lochak P., Meunier C.: Multiphase Averaging for Classical Systems: With Applications to Adiabatic Theorems. Springer, New York (1988)zbMATHCrossRefGoogle Scholar
  38. 38.
    Lee Y.S., Nucera F., Vakakis A.F., McFarland D.M., Bergman L.A.: Periodic orbits and damped transitions of vibro-impact dynamics. Phys. D 238, 1868–1896 (2009)zbMATHCrossRefGoogle Scholar
  39. 39.
    Nordmark A.B.: Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators. Nonlinearity 14, 1517–1542 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Deléchelle E., Lemoine J., Niang O.: Empirical mode decomposition: an analytical approach for sifting process. IEEE Signal Process. Lett. 12, 764–767 (2005)CrossRefGoogle Scholar
  41. 41.
    Lee, Y.S., Chen, H., Vakakis, A.F., McFarland, D.M., Bergman, L.A.: Nonlinear system identification of vibro-impact nonsmooth dynamical systems (AIAA-2011-2067). In: 52nd AIAA Structures, Structural Dynamics and Materials Conference, Denver, Colorado, 4–7 April 2011 (2011)Google Scholar
  42. 42.
    Gibbons J.D.: Nonparametric Statistical Inference. 2nd edn. M. Dekker, New York (1985)zbMATHGoogle Scholar
  43. 43.
    Chen, H., Kurt, M., Lee, Y.S., McFarland, D.M., Bergman, L.A., and Vakakis, A.F.: System identification of a vibro-impact beam with a view toward structural health monitoring. Exp. Mech. (submitted)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Mehmet Kurt
    • 1
  • Heng Chen
    • 2
  • Young S. Lee
    • 2
    Email author
  • D. Michael McFarland
    • 3
  • Lawrence A. Bergman
    • 3
  • Alexander F. Vakakis
    • 4
  1. 1.Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Mechanical and Aerospace EngineeringNew Mexico State UniversityLas CrucesUSA
  3. 3.Department of Aerospace EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  4. 4.Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations