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An Explanation for Why Natural Frequencies Shifting in Structures with Membrane Stresses, Using Backbone Curve Models

Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)

Abstract

In this paper, the phenomenon of natural frequencies shifting due to the nonlinear stiffness effects from membrane stress is studied using a nonlinear reduced order model based on backbone curves. The structure chosen for study in this paper is a rectangular plate with a pinned constraint along all edges. To analytically explore the frequency varying phenomenon, a four nonlinear-mode based reduced-order model that contains both single-mode and coupled-mode nonlinear terms is derived. The process of deriving the reduced order model is based on a normal form transformation, combined with a Galerkin type decomposition of the governing partial differential equation of the plate. This allows a low number of ordinary differential equations to be obtained, which in turn can be used to derive backbone curves that relate directly to the nonlinear normal modes (NNMs). The frequency shifting is then investigated relative to the backbone curves. Modal interactions, caused by nonlinear terms are shown to cause the frequency shifts. In the final part of the paper, an attempt is made to quantify the frequency shifting due to different nonlinear effects.

Keywords

Nonlinear reduced order model Backbone curves Nonlinear modal interaction Second-order normal form method Thin plate 

References

  1. 1.
    Chia, C.-Y.: Nonlinear Analysis of Plates. McGraw-Hill, New York (1980)Google Scholar
  2. 2.
    Mignolet, M.P., Przekop, A., Rizzi, S.A., Spottswood, S.M.: A review of indirect/non-intrusive reduced order modeling of nonlinear geometric structures. J. Sound Vib. 332 (10), 2437–2460 (2013)CrossRefGoogle Scholar
  3. 3.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)MATHGoogle Scholar
  4. 4.
    Nash, M.: Nonlinear structural dynamics by finite element model synthesis. PhD thesis, Imperial College London, University of London (1978)Google Scholar
  5. 5.
    Shi, Y., Mei, C.: A finite element time domain modal formulation for large amplitude free vibrations of beams and plates. J. Sound Vib. 193 (2), 453–464 (1996)CrossRefMATHGoogle Scholar
  6. 6.
    Wagg, D., Neild, S.: Nonlinear Vibration with Control: For Flexible and Adaptive Structures. Springer, Berlin (2014)MATHGoogle Scholar
  7. 7.
    Muravyov, A.A., Rizzi, S.A.: Determination of nonlinear stiffness with application to random vibration of geometrically nonlinear structures. Commun. Strateg. 81 (15), 1513–1523 (2003)Google Scholar
  8. 8.
    McEwan, M.I., Wright, J.R., Cooper, J.E., Leung, A.Y.T.: A finite element/modal technique for nonlinear plate and stiffened panel response prediction. In: Proceedings of the 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit Technical Papers, pp. 3061–3070 (2001)Google Scholar
  9. 9.
    McEwan, M.I., Wright, J.R., Cooper, J.E., Leung, A.Y.T.: A combined modal/finite element analysis technique for the dynamic response of a non-linear beam to harmonic excitation. J. Sound Vib. 243 (4), 601–624 (2001)CrossRefGoogle Scholar
  10. 10.
    Hollkamp, J.J., Gordon, R.W.: Reduced-order models for nonlinear response prediction: implicit condensation and expansion. J. Sound Vib. 318 (4), 1139–1153 (2008)CrossRefGoogle Scholar
  11. 11.
    Lewandowski, R.: On beams membranes and plates vibration backbone curves in cases of internal resonance. Meccanica 31 (3), 323–346 (1996)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Touzé, C., Thomas, O., Chaigne, A.: Asymmetric non-linear forced vibrations of free-edge circular plates. Part 1: theory. J. Sound Vib. 258 (4), 649–676 (2002)Google Scholar
  13. 13.
    Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, Cambridge (2008)CrossRefMATHGoogle Scholar
  14. 14.
    Pierre, C., Jiang, D., Shaw, S.: Nonlinear normal modes and their application in structural dynamics. Math. Probl. Eng. 2006, 1–15 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Touzé, C., Amabili, M.: Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modelling of harmonically forced structures. J. Sound Vib. 298 (4), 958–981 (2006)CrossRefGoogle Scholar
  16. 16.
    Kerschen, G., Peeters, M., Golinval, J.-C., Vakakis, A.F.: Nonlinear normal modes, Part I: a useful framework for the structural dynamicist. Mech. Syst. Signal Process. 23 (1), 170–194 (2009)MathSciNetGoogle Scholar
  17. 17.
    Liu, X., Cammarano, A., Wagg, D.J., Neild, S.A.: A study of the modal interaction amongst three nonlinear normal modes using a backbone curve approach. In: Nonlinear Dynamics, vol. 1, pp. 131–139. Springer, Berlin (2016)Google Scholar
  18. 18.
    Liu, X., Cammarano, A., Wagg, D.J., Neild, S.A., Barthorpe, R.J.: Nonlinear modal interaction analysis for a three degree-of-freedom system with cubic nonlinearities. In: Nonlinear Dynamics, vol. 1, pp. 123–131. Springer, Berlin (2016)Google Scholar
  19. 19.
    Liu, X., Cammarano, A., Wagg, D.J., Neild, S.A., Barthorpe, R.J.: N-1 modal interactions of a three-degree-of-freedom system with cubic elastic nonlinearities. Nonlinear Dyn. 83 (1–2), 497–511 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Society for Experimental Mechanics, Inc. 2017

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of SheffieldSheffieldUK
  2. 2.Department of Mechanical EngineeringUniversity of BristolBristolUK

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