Skip to main content
SpringerLink
Log in

Tutorial on Nonlinear System Identification

  • Conference paper
  • First Online:
Nonlinear Dynamics, Volume 1

  • 1696 Accesses

Abstract

Because nonlinearity is now a frequent occurrence in real-life applications, the practitioner should understand the resulting dynamical phenomena and account for them in the design process. This tutorial focuses on nonlinear system identification, which extracts relevant information about nonlinearity directly from experimental measurements. Specifically, the identification process is a progression through three steps, namely detection, characterization and parameter estimation. The tutorial presents these steps in detail and illustrates them using real aerospace structures.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Van Der Auweraer, H.: Testing in the age of virtual prototyping. In: Proceedings of International Conference on Structural Dynamics Modelling, Funchal (2002)

    Google Scholar 

  2. Ljung, L.: System Identification - Theory for the User. Prentice-Hall, Englewood Cliffs (1987)

    MATH  Google Scholar 

  3. Soderstrom, T., Stoica, P.: System Identification. Prentice-Hall, Englewood Cliffs (1989)

    MATH  Google Scholar 

  4. Ibrahim, S.R., Mikulcik, E.C.: A time domain modal vibration test technique. Shock Vib. Bull. 43, 21–37 (1973)

    Google Scholar 

  5. Juang, J.S., Pappa, R.S.: An eigensystem realization algorithm for modal parameter identification and model reduction. AIAA J. Guid. Control Dyn. 12, 620–627 (1985)

    Article  MATH  Google Scholar 

  6. Van Overschee, P., De Moor, B.: Subspace Identification for Linear Systems: Theory, Implementation, Applications. Kluwer Academic Publishers, Dordrecht (1996)

    Book  MATH  Google Scholar 

  7. Peeters, B., Van Der Auweraer, H., Guillaume, P.: The PolyMAX frequency domain method: a new standard for modal parameter estimation. Shock Vib. 11, 395–409 (2004)

    Article  Google Scholar 

  8. Allemang, R.J., Brown, D.L.: A unified matrix polynomial approach to modal identification. J. Sound Vib. 211, 301–322 (1998)

    Article  MATH  Google Scholar 

  9. Allemang, R.J., Phillips, A.W.: The unified matrix polynomial approach to understanding modal parameter estimation: an update. In: Proceedings of the International Seminar on Modal Analysis (ISMA), Leuven (2004)

    Google Scholar 

  10. Amabili, M., Paidoussis, M.: Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Appl. Mech. Rev. 56, 349–381 (2003)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  12. White, S.W., Kim, S.K., Bajaj, A.K., Davies, P., Showers, D.K., Liedtke, P.E.: Experimental techniques and identification of nonlinear and viscoelastic properties of flexible polyurethane foam. Nonlinear Dyn. 22, 281–313 (2000)

    Article  MATH  Google Scholar 

  13. Schultze, J.F., Hemez, F.M., Doebling, S.W., Sohn, H.: Application of non-linear system model updating using feature extraction and parameter effect analysis. Shock Vib. 8, 325–337 (2001)

    Article  Google Scholar 

  14. Singh, R., Davies, P., Bajaj, A.K.: Identification of nonlinear and viscoelastic properties of flexible polyurethane foam. Nonlinear Dyn. 34, 319–346 (2003)

    Article  MATH  Google Scholar 

  15. Richards, C.M., Singh, R.: Characterization of rubber isolator nonlinearities in the context of single- and multi-degree-of-freedom experimental systems. J. Sound Vib. 247, 807–834 (2001)

    Article  Google Scholar 

  16. Caughey, T.K., Vijayaraghavan, A.: Free and forced oscillations of a dynamic system with linear hysteretic damping. Int. J. Non-Linear Mech. 5, 533–555 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tomlinson, G.R., Hibbert, J.H.: Identification of the dynamic characteristics of a structure with Coulomb friction. J. Sound Vib. 64, 233–242 (1979)

    Article  Google Scholar 

  18. Sherif, H.A., Abu Omar, T.M.: Mechanism of energy dissipation in mechanical system with dry friction. Tribol. Int. 37, 235–244 (2004)

    Article  Google Scholar 

  19. Al-Bender, F., Symens, W., Swevers, J., Van Brussel, H.: Theoretical analysis of the dynamic behavior of hysteresis elements in mechanical systems. Int. J. Non-Linear Mech. 39, 1721–1735 (2004)

    Article  MATH  Google Scholar 

  20. Babitsky, V.I., Krupenin, V.L.: Vibrations of Strongly Nonlinear Discontinuous Systems. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  21. Rhee, S.H., Tsang, P.H.S., Yen, S.W.: Friction-induced noise and vibrations of disc brakes. Wear 133, 39–45 (1989)

    Article  Google Scholar 

  22. Von Karman, T.: The engineer grapples with nonlinear problems. Bull. Am. Math. Soc. 46, 615–683 (1940)

    Article  MATH  Google Scholar 

  23. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley-Interscience, New York (1979)

    MATH  Google Scholar 

  24. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Addison-Wesley, Reading (1994)

    Google Scholar 

  25. Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems, 2nd edn. Springer, Berlin (1999)

    MATH  Google Scholar 

  26. Rand, R.: Lecture Notes on Nonlinear Vibrations, Cornell (2003). Notes freely available at http://tam.cornell.edu/Rand.html.

  27. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. Springer, New York (1983)

    MATH  Google Scholar 

  28. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)

    Book  MATH  Google Scholar 

  29. Caughey, T.K.: Equivalent linearisation techniques. J. Acoust. Soc. Am. 35, 1706–1711 (1963)

    Article  MathSciNet  Google Scholar 

  30. Iwan, W.D.: A generalization of the concept of equivalent linearization. Int. J. Non-Linear Mech. 8, 279–287 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rosenberg, R.M.: The normal modes of nonlinear n-degree-of-freedom systems. J. Appl. Mech. 29, 7–14 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rosenberg, R.M.: On nonlinear vibrations of systems with many degrees of freedom. Adv. Appl. Mech. 9, 155–242 (1966)

    Article  Google Scholar 

  33. Rand, R.: A direct method for nonlinear normal modes. Int. J. Non-Linear Mech. 9, 363–368 (1974)

    Article  MATH  Google Scholar 

  34. Shaw, S.W., Pierre, C.: Normal modes for non-linear vibratory systems. J. Sound Vib. 164, 85–124 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  35. Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., Zevin, A.A.: Normal Modes and Localization in Nonlinear Systems. Wiley, New York (1996)

    Book  MATH  Google Scholar 

  36. Vakakis, A.F.: Non-linear normal modes and their applications in vibration theory: an overview. Mech. Syst. Signal Process. 11, 3–22 (1997)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  38. O’Malley, R.E.: Singular Perturbation Methods for Ordinary Differential Equations. Springer, New York (1991)

    Book  MATH  Google Scholar 

  39. Kevorkian, J., Cole, J.D.: Multiple Scales and Singular Perturbation Methods. Springer, New York (1996)

    Book  MATH  Google Scholar 

  40. Chan, H.S.Y., Chung, K.W., Xu, Z.: A perturbation-incremental method for strongly non-linear oscillators. Int. J. Non-Linear Mech. 31, 59–72 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  41. Chen, S.H., Cheung, Y.K.: A modified Lindstedt-Poincaré method for a strongly nonlinear two degree-of-freedom system. J. Sound Vib. 193, 751–762 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  42. Pilipchuk, V.N.: The calculation of strongly nonlinear systems close to vibration-impact systems. Prikl. Mat. Mech. (PMM) 49, 572–578 (1985)

    Google Scholar 

  43. Manevitch, L.I.: Complex representation of dynamics of coupled oscillators. In: Mathematical Models of Nonlinear Excitations, Transfer Dynamics and Control in Condensed Systems. Kluwer Academic/Plenum Publishers, New York (1999)

    Book  Google Scholar 

  44. Qaisi, M.I., Kilani, A.W.: A power-series solution for a strongly non-linear two-degree-of-freedom system. J. Sound Vib. 233, 489–494 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  45. Rhoads, J.F., Shaw, S.W., Turner, K.L., Baskaran, R.: Tunable MEMS filters that exploit parametric resonance. J. Vib. Acoust. (2005, in press)

    Google Scholar 

  46. Vakakis, A.F., Gendelman, O.: Energy pumping in nonlinear mechanical oscillators: part II — resonance capture. J. Appl. Mech. 68, 42–48 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  47. Vakakis, A.F., McFarland, D.M., Bergman, L.A., Manevitch, L.I., Gendelman, O.: Isolated resonance captures and resonance capture cascades leading to single- or multi-mode passive energy pumping in damped coupled oscillators. J. Vib. Acoust. 126, 235–244 (2004)

    Article  Google Scholar 

  48. Kerschen, G., Lee, Y.S., Vakakis, A.F., McFarland, D.M., Bergman, L.A.: Irreversible passive energy transfer in coupled oscillators with essential nonlinearity. SIAM J. Appl. Math. (2005, in press)

    Google Scholar 

  49. Nichols, J.M., Nichols, C.J., Todd, M.D., Seaver, M., Trickey, S.T., Virgin, L.N.: Use of data-driven phase space models in assessing the strength of a bolted connection in a composite beam. Smart Mater. Struct. 13, 241–250 (2004)

    Article  Google Scholar 

  50. Epureanu, B.I., Hashmi, A.: Parameter reconstruction based on sensitivity vector fields. J. Vib. Acoust. (2005, submitted)

    Google Scholar 

  51. Adams, D.E., Allemang, R.J.: Survey of nonlinear detection and identification techniques for experimental vibrations structural dynamic model through feedback. In: Proceedings of the International Seminar on Modal Analysis (ISMA), Leuven, pp. 269–281 (1998)

    Google Scholar 

  52. Worden, K.: Nonlinearity in structural dynamics: the last ten years. In: Proceedings of the European COST F3 Conference on System Identification and Structural Health Monitoring, Madrid, pp. 29–52 (2000)

    Google Scholar 

  53. Duffing, G.: Erzwungene Schwingungen bei Veranderlicher Eigenfrequenz (Forced Oscillations in the Presence of Variable Eigenfrequencies). Vieweg, Braunschweig (1918)

    MATH  Google Scholar 

  54. Roache, P.J.: Verification and Validation in Computational Science and Engineering. Hermosa Publications, Albuquerque (1998)

    Google Scholar 

  55. Doebling, S.: Structural dynamics model validation: pushing the envelope. In: Proceedings of International Conference on Structural Dynamics Modelling, Funchal (2002)

    Google Scholar 

  56. Schlesinger, S., Crosbie, R.E., Gagne, R.E., Innis, G.S., Lalwani, C.S., Loch, J., Sylvester, R.J., Wright, R.D., Kheir, N., Bartos, D.: Terminology for model credibility. Simulation 32, 103–104 (1979)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, Space Structures and Systems Laboratory (S3L), Structural Dynamics Research Group, University of Liège, Liège, Belgium

    G. Kerschen

Authors
  1. G. Kerschen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Kerschen .

Editor information

Editors and Affiliations

  1. Liege, Belgium

    Gaetan Kerschen

Rights and permissions

Reprints and permissions

Copyright information

© 2016 The Society for Experimental Mechanics, Inc.

About this paper

Cite this paper

Kerschen, G. (2016). Tutorial on Nonlinear System Identification. In: Kerschen, G. (eds) Nonlinear Dynamics, Volume 1. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-29739-2_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-29739-2_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-29738-5

  • Online ISBN: 978-3-319-29739-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation