Skip to main content
SpringerLink
Log in

Simplifying Transformations for Nonlinear Systems: Part I, An Optimisation-Based Variant of Normal Form Analysis

  • Conference paper
Nonlinear Dynamics, Volume 1

Abstract

This paper introduces the idea of a ‘simplifying transformation’ for nonlinear structural dynamic systems. The idea simply stated; is to bring under one heading, those transformations which ‘simplify’ structural dynamic systems or responses in some sense. The equations of motion may be cast in a simpler form or decoupled (and in this sense, nonlinear modal analysis is encompassed) or the responses may be modified in order to isolate and remove certain components. It is the latter sense of simplification which is considered in this paper. One can regard normal form analysis in a way as the removal of superharmonic content from nonlinear system response. In the current paper, this problem is cast in an optimisation form and the differential evolution algorithm is used.

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. Kerschen G, Golinval J-C, Vakakis AF, Bergman LA (2005) The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn 41(1–3):147–169

    Article  MathSciNet  MATH  Google Scholar 

  2. Vakakis AF (1997) Non-linear normal modes (nnms) and their applications in vibration theory: an overview. Mech Syst Signal Process 11(1):3–22

    Article  MathSciNet  Google Scholar 

  3. Worden K, Tomlinson GR (2000) Nonlinearity in structural dynamics: detection, identification and modelling. CRC Press, Boca Raton

    Book  Google Scholar 

  4. Worden K, Tomlinson GR (2001) Nonlinearity in experimental modal analysis. Philos Trans R Soc Lond Ser A Math Phys Eng Sci 359(1778):113–130

    Google Scholar 

  5. Worden K, Green PL (2014) A machine learning approach to nonlinear modal analysis. In: Catbas FN (ed) Dynamics of civil structures, vol 4. Springer, New York, pp 521–528

    Chapter  Google Scholar 

  6. Rosenberg RM (1962) The normal modes of nonlinear n-degree-of-freedom systems. J Appl Mech 29(1):7–14

    Article  MATH  Google Scholar 

  7. Shaw SW, Pierre C (1993) Normal modes for non-linear vibratory systems. J Sound Vib 164(1):85–124

    Article  MathSciNet  Google Scholar 

  8. Neild SA, Wagg DJ (2011) Applying the method of normal forms to second-order nonlinear vibration problems. Proc R Soc A Math Phys Eng Sci 467(2128):1141–1163

    Article  MathSciNet  MATH  Google Scholar 

  9. Poncelet F, Kerschen G, Golinval J-C, Verhelst D (2007) Output-only modal analysis using blind source separation techniques. Mech Syst Signal Process 21(6):2335–2358

    Article  Google Scholar 

  10. Dervilis N, Wagg DJ, Green PL, Worden K (2014) Nonlinear modal analysis using pattern recognition. In: Proceedings of ISMA2014, pp 3017–3027

    Google Scholar 

  11. Worden K, Manson G, Sohn H, Farrar CR (2005) Extreme value statistics from differential evolution for damage detection. In: Proceedings of the 23rd international modal analysis conference (IMAC XXIII), Orlando, pp 2009–3

    Google Scholar 

  12. Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 11(4):341–359

    Article  MathSciNet  MATH  Google Scholar 

  13. Qin AK, Suganthan PN (2005) Self-adaptive differential evolution algorithm for numerical optimization. In: The 2005 IEEE congress on evolutionary computation, vol 2. IEEE, pp 1785–1791

    Google Scholar 

  14. Cross E (2012) On structural health monitoring in changing environmental and operational conditions. Ph.D. thesis

    Google Scholar 

  15. Welch P (1967) The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans Audio Electroacoust 15(2):70–73

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The support of the UK Engineering and Physical Sciences Research Council (EPSRC) through grant reference number EP/J016942/1 and EP/K003836/2 is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK

    N. Dervilis, K. Worden & D. J. Wagg

  2. Department of Mechanical Engineering, Queens Building, University of Bristol, Bristol, BS8 1TR, UK

    S. A. Neild

Authors
  1. N. Dervilis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Worden
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. J. Wagg
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. A. Neild
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Dervilis .

Editor information

Editors and Affiliations

  1. Space Structures and Systems Laboratory, University of Liège, Liège, Belgium

    Gaëtan Kerschen

Rights and permissions

Reprints and permissions

Copyright information

© 2016 The Society for Experimental Mechanics, Inc.

About this paper

Cite this paper

Dervilis, N., Worden, K., Wagg, D.J., Neild, S.A. (2016). Simplifying Transformations for Nonlinear Systems: Part I, An Optimisation-Based Variant of Normal Form Analysis. In: Kerschen, G. (eds) Nonlinear Dynamics, Volume 1. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-15221-9_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-15221-9_28

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15220-2

  • Online ISBN: 978-3-319-15221-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation