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Topics in Nonlinear Dynamics, Volume 3 pp 287–299Cite as

Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam

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Abstract

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. 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 vibroimpact 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 in between impacts.

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References

  1. Ewins DJ (1990) Modal testing: theory and practice. Research Studies Press, UK

    Google Scholar 

  2. Brandon JA (1998) Some insights into the dynamics of defective structures. Proc Inst Mech Eng Part C: J Mech Eng Sci 212(6):441–454

    Article  Google Scholar 

  3. Kerschen G, Golinval J-C, Vakakis AF, Bergman LA (2005) The method of proper orthogonal decomposition for order reduction of mechanical systems: an overview. Nonlinear Dyn 41(1–3):147–170

    Article  MathSciNet  MATH  Google Scholar 

  4. Kerschen G, Worden K, Vakakis AF, Golinval J-C (2005) Past, present and future of nonlinear system identification in structural dynamics. Mech Syst Signal Process 20(3):505–592

    Article  Google Scholar 

  5. Feeny BF, Kappagantu R (1998) On the physcal interpretation of proper orthogonal modes in vibrations. J Sound Vib 211:607–616

    Article  Google Scholar 

  6. Kerschen G, Golinval JC (2002) Physical interpretation of the proper orthogonal modes using the singular value decomposition. J Sound Vib 249:849–865

    Article  MathSciNet  MATH  Google Scholar 

  7. Bellizzi S, Sampaio R (2006) POMs analysis of randomly vibrating systems obtained from Karhunen-Loeve expansion. J Sound Vib 297:774–793

    Article  MathSciNet  MATH  Google Scholar 

  8. Allison TC, Miller AK, Inman DJ (2008) A deconvolution-based approach to structural dynamics system identification and response prediction. J Vib Acoust 130:031010

    Article  Google Scholar 

  9. Chelidze D, Zhou W (2006) Smooth orthogonal decomposition-based vibration mode identification. J Sound Vib 292:461–473

    Article  Google Scholar 

  10. Silva W (2005) Identification of nonlinear aeroelastic systems based on the Volterra theory: progress and opportunities. Nonlinear Dyn 39:25–62

    Article  MATH  Google Scholar 

  11. Li LM, Billings SA (2011) Analysis of nonlinear oscillators using Volterra series in the frequency domain. J Sound Vib 330:337–355

    Article  Google Scholar 

  12. Mariani S, Ghisi A (2007) Unscented Kalman filtering for nonlinear structural dynamics. Nonlinear Dyn 49:131–150

    Article  MATH  Google Scholar 

  13. Masri S, Caughey T (1979) A nonparametric identification techanique for nonlinear dynamic systems. Trans ASME J Appl Mech 46:433–441

    Article  MATH  Google Scholar 

  14. Leontaritis IJ, Billings SA (1985) Input–output parametric models for nonlinear systems. Part I. Deterministic nonlinear systems; part II. Stochastic nonlinear systems. Int J Control 41:303–344

    Article  MathSciNet  MATH  Google Scholar 

  15. Thothadri M, Casas RA, Moon FC, D’Andrea R, Johnson CR (2003) Nonlinear system identification of multi-degree-of-freedom systems. Nonlinear Dyn 32:307–322

    Article  MathSciNet  MATH  Google Scholar 

  16. Feldman M (1994) Non-linear system vibration analysis using Hilbert transform-I. Free vibration analysis method ‘FREEVIB’; and II. Forced vibration analysis method ‘FORCEVIB’. Mech Syst Signal Process 8(2):119–127, and 8(3); 309–318

    Article  Google Scholar 

  17. Feldman M (2006) Time-varying vibration decomposition and analysis based on the Hilbert transform. J Sound Vib 295:518–530

    Article  MATH  Google Scholar 

  18. Ma X, Azeez MFA, Vakakis AF (2000) Non-linear normal modes and non-parametric system identification of non-linear oscillators. Mech Syst Signal Process 14:37–48

    Article  Google Scholar 

  19. Georgiou I (2005) 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

    Article  MathSciNet  MATH  Google Scholar 

  20. Galvanetto U, Surace C, Tassotti A (2008) Structural damage detection based on proper orthogonal decomposition: experimental verification. AIAA J 46:1624–1630

    Article  Google Scholar 

  21. Cusumano JP, Bae B-Y (1993) Period-infinity periodic motions, chaos, and spatial coherence in a 10 degree of freedom impact oscillator. Chaos, Soliton Fract 3:515–535

    Article  MATH  Google Scholar 

  22. Cusumano JP, Sharkady MT, Kimble BW (1994) Experimental measurements of dimensionality and spatial coherence in the dynamics of a flexible-beam impact oscillator. Philos Trans R Soc Ser A 347:421–438

    Article  Google Scholar 

  23. Ritto TG, Buezas FS, Sampaio R (2011) A new measure of efficiency for model reduction: application to a vibroimpact system. J Sound Vib 330:1977–1984

    Article  Google Scholar 

  24. Azeez MFA, Vakakis AF (2001) Proper orthogonal decomposition (POD) of a class of vibroimpact oscillations. J Sound Vib 240:859–889

    Article  Google Scholar 

  25. Lee YS, Vakakis AF, McFarland DM, Bergman LA (2010) A global–local approach to system identification: a review. Struct Control Health Monitor 17:742–760

    Article  Google Scholar 

  26. Huang N, Shen Z, Long S, Wu M, Shih H, Zheng Q, Yen N-C, Tung C, Liu H (1998) The empirical mode decompostion and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc Roy Soc London, Ser A Math Phys Sci 454:903–995

    Article  MathSciNet  MATH  Google Scholar 

  27. Lee YS, Tsakirtzis S, Vakakis AF, Bergman LA, McFarland DM (2009) Physics-based foundation for empirical mode decomposition: correspondence between intrinsic mode functions and slow flows. AIAA J 47(12):2938–2963

    Article  Google Scholar 

  28. Lee YS, Tsakirtzis S, Vakakis AF, McFarland DM, Bergman LA (2011) A time-domain nonlinear system identification method based on multiscale dynamic partitions. Meccanica 46:625–649

    Article  MathSciNet  Google Scholar 

  29. Lee YS, Vakakis AF, McFarland DM, Bergman LA (2010) Time-domain nonlinear system identification of the dynamics of aeroelastic instability suppression based on targeted energy transfers. The Aeronaut J 114:1151

    Google Scholar 

  30. Tsakirtzis S, Lee YS, Vakakis AF, Bergman LA, McFarland DM (2010) Modeling of nonlinear modal interactions in the transient dynamics of an elastic rod with an essentially nonlinear attachment. Commun Nonlinear Sci Numer Simul 15(9):2617–2633

    Article  MATH  Google Scholar 

  31. Dawes JHP (2010) Review: the emergence of a coherent structure for coherent structures: localized states in nonlinear systems. Philos Trans Roy Soc Ser A 368:3519–3534

    Article  MathSciNet  Google Scholar 

  32. Chati M, Rand R, Mukherjee R (1997) Modal analysis of a cracked beam. J Sound Vib 207:249–270

    Article  MATH  Google Scholar 

  33. Chen HG, Yan YJ, Jiang JS (2007) Vibration-based damage detection in composite wingbox structures by HHT. Mech Syst Signal Process 21:307–321

    Article  Google Scholar 

  34. Mane M (2010) Experiments in vibro-impact beam dynamics and a system exhibiting a Landau-Zener quantum effect. MS thesis (unpublished), Univeristy of Illinois at Urbana-Champaign

    Google Scholar 

  35. Blevins RD (1995) Formulas for natural frequency and mode shape. Krieger, Malabar

    Google Scholar 

  36. Lee YS, Nucera F, Vakakis AF, McFarland DM, Bergman LA (2009) Periodic orbits and damped transitions of vibro-impact dynamics. Physica D 238:1868–1896

    Article  MATH  Google Scholar 

  37. Nordmark AB (2001) Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators. Nonlinearity 14:1517–1542

    Article  MathSciNet  MATH  Google Scholar 

  38. Delechelle E, Lemoine J, Niang O (2005) Empirical mode decomposition: an analytical approach for sifting process. IEEE Signal Process Lett 12:764–767

    Article  Google Scholar 

  39. Lee YS, Chen H, Vakakis AF, McFarland DM, Bergman LA (2011) Nonlinear system identification of vibro-impact nonsmooth dynamical systems (AIAA-2011-2067). Fifty-second AIAA structures, structural dynamics and materials conference, Denver, 4–7 Apr 2011

    Google Scholar 

  40. Gibbons JD (1985) Nonparametric statistical inference, 2nd edn. Marcel Dekker, New York

    MATH  Google Scholar 

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Acknowledgments

This material is based upon work supported by the National Science Foundation under Grants Number CMMI-0927995 and CMMI-0928062.

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, New Mexico State University, 1040 S. Horseshoe St., Las Cruces, NM, 88003, USA

    H. Chen & Y. S. Lee

  2. Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 S. Wright St., Urbana, IL, 61801, USA

    M. Kurt & A. F. Vakakis

  3. Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green St., Urbana, IL, 61801, USA

    D. M. McFarland & L. A. Bergman

Authors
  1. H. Chen
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  2. M. Kurt
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  3. Y. S. Lee
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  4. D. M. McFarland
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  5. L. A. Bergman
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  6. A. F. Vakakis
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Corresponding author

Correspondence to Y. S. Lee .

Editor information

Editors and Affiliations

  1. Purdue University, West Lafayette, 47908, USA

    D. Adams

  2. University of Liege, Liege, Belgium

    G. Kerschen

  3. LMS International, Interleuvenlaan 68, Leuven, 3001, Belgium

    A. Carrella

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© 2012 The Society for Experimental Mechanics, Inc. 2012

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Chen, H., Kurt, M., Lee, Y.S., McFarland, D.M., Bergman, L.A., Vakakis, A.F. (2012). Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam. In: Adams, D., Kerschen, G., Carrella, A. (eds) Topics in Nonlinear Dynamics, Volume 3. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-2416-1_23

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  • DOI: https://doi.org/10.1007/978-1-4614-2416-1_23

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  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-2415-4

  • Online ISBN: 978-1-4614-2416-1

  • eBook Packages: EngineeringEngineering (R0)

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