Skip to main content
SpringerLink
Log in

Nonlinear dynamics of rectangular plates: investigation of modal interaction in free and forced vibrations

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Nonlinear vibrations of thin rectangular plates are considered, using the von kármán equations in order to take into account the effect of geometric nonlinearities. Solutions are derived through an expansion over the linear eigenmodes of the system for both the transverse displacement and the Airy stress function, resulting in a series of coupled oscillators with cubic nonlinearities, where the coupling coefficients are functions of the linear eigenmodes. A general strategy for the calculation of these coefficients is outlined, and the particular case of a simply supported plate with movable edges is thoroughly investigated. To this extent, a numerical method based on a new series expansion is derived to compute the Airy stress function modes, for which an analytical solution is not available. It is shown that this strategy allows the calculation of the nonlinear coupling coefficients with arbitrary precision, and several numerical examples are provided. Symmetry properties are derived to speed up the calculation process and to allow a substantial reduction in memory requirements. Finally, analysis by continuation allows an investigation of the nonlinear dynamics of the first two modes both in the free and forced regimes. Modal interactions through internal resonances are highlighted, and their activation in the forced case is discussed, allowing to compare the nonlinear normal modes (NNMs) of the undamped system with the observable periodic orbits of the forced and damped structure.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amabili M.: Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments. Comput. Struct. 82, 2587–2605 (2004)

    Article  Google Scholar 

  2. Amabili M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  3. Anlas G., Elbeyli O.: Nonlinear vibrations of a simply supported rectangular metallic plate subjected to transverse harmonic excitation in the presence of a one-to-one internal resonance. Nonlinear Dyn. 30, 1–28 (2002)

    Article  MATH  Google Scholar 

  4. Awrejcewicz J., Krysko V.A., Krysko A.V.: Spatio-temporal chaos and solitons exhibited by von Kármán model. Int. J. Bifurc Chaos. 12, 1465–1513 (2002)

    Article  Google Scholar 

  5. Bilbao S.: A family of conservative finite difference schemes for the dynamical von Kármán plate equations. Numer. Methods Partial Differ. Equ. 24, 193–216 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bilbao S.: Numerical Sound Synthesis Finite Difference Schemes and Simulation in Musical Acoustics. Wiley, New York (2009)

    MATH  Google Scholar 

  7. Bilbao S.: Percussion synthesis based on models of nonlinear shell vibration. IEEE Trans. Audio Speech Lang. Process. 18, 872–880 (2010)

    Article  Google Scholar 

  8. Blanc F., Touzé C., Mercier J.-F., Ege K., Bonnet Ben-Dhia A.-S.: On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems. Mech. Syst. Signal Process. 36, 520–539 (2013)

    Article  Google Scholar 

  9. Boudaoud A., Cadot O., Odille B., Touzé C.: Observation of wave turbulence in vibrating plates. Phys. Rev. Lett. 100, 234504 (2008)

    Article  Google Scholar 

  10. Boumediene F., Duigou L., Boutyour E.H., Miloudi A., Cadou J.M.: Nonlinear forced vibration of damped plates by an asymptotic numerical method. Comput. Struct. 87, 1508–1515 (2009)

    Article  Google Scholar 

  11. Chaigne A., Lambourg C.: Time-domain simulation of damped impacted plates. I. Theory and experiments. J. Acoust. Soc. Am. 109, 1422–1432 (2001)

    Article  Google Scholar 

  12. Chaigne A., Touzé C., Thomas O.: Nonlinear vibrations and chaos in gongs and cymbals. Acoust. Sci. Technol. 26, 403–409 (2005)

    Article  Google Scholar 

  13. Chang S.I., Bajaj A.K., Krousgrill C.M.: Nonlinear oscillations of a fluttering plate. AIAA J. 4, 1267–1275 (1966)

    Article  Google Scholar 

  14. Chang S.I., Bajaj A.K., Krousgrill C.M.: Non-linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance. Nonlinear Dyn. 4, 433–460 (1993)

    Article  Google Scholar 

  15. Chen W.Q., Ding H.J.: On free vibration of a functionally graded piezoelectric rectangular plate. Acta Mechanica 153, 207–216 (2002)

    Article  MATH  Google Scholar 

  16. Chia C.Y.: Nonlinear Analysis of Plates. Mc Graw Hill, New York (1980)

    Google Scholar 

  17. Chu, H.N., Herrmann, G.: Influence of large amplitudes on free flexural vibrations of rectangular elastic plates. J. Appl. Mech. 23 (1956)

  18. Doaré O., Michelin S.: Piezoelectric coupling in energy-harvesting fluttering flexible plates: linear stability analysis and conversion efficiency. J. Fluids Struct. 27, 1357–1375 (2011)

    Article  Google Scholar 

  19. Doedel, E., Paffenroth, R.C., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Oldeman, B.E., Sandstede, B., Wang, X.: Auto2000: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont). Technical report, Concordia University, Canada (2002)

  20. Düring G., Josserand C., Rica S.: Weak turbulence for a vibrating plate: can one hear a Kolmogorov spectrum?. Phys. Rev. Lett. 97, 025503 (2006)

    Article  Google Scholar 

  21. Fu Y.M., Chia C.Y.: Nonlinear bending and vibration of symmetrically laminated orthotropic elliptical plate with simply supported edge. Acta Mech. 74, 155–170 (1988)

    Article  MATH  Google Scholar 

  22. Gao Y., Xu B., Huh H.: Electromagneto-thermo-mechanical behaviors of conductive circular plate subject to time-dependent magnetic fields. Acta Mech. 210, 99–116 (2010)

    Article  MATH  Google Scholar 

  23. Géradin M., Rixen D.: Mechanical Vibrations. Wiley, New York (1997)

    Google Scholar 

  24. Golinval, J.C., Stephan, C., Lubrina, P., Peeters, M., Kerschen, G.: Nonlinear normal modes of a full-scale aircraft. In: 29th International Modal Analysis Conference, Jacksonville, USA (2011)

  25. Gordnier R.E., Visbal M.R.: Development of a three-dimensional viscous aeroelastic solver for nonlinear panel flutter. J. Fluids Struct. 16, 497–527 (2002)

    Article  Google Scholar 

  26. Hagedorn P., DasGupta A.: Vibrations and Waves in Continuous Mechanical Systems. Wiley, Chichester (2007)

    Book  MATH  Google Scholar 

  27. 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, 170–194 (2009)

    Article  Google Scholar 

  28. Kung G.C., Pao Y.-H.: Nonlinear flexural vibrations of a clamped circular plate. J. Appl. Mech. 39, 1050–1054 (1972)

    Article  Google Scholar 

  29. Legge K.A., Fletcher N.H.: Nonlinearity, chaos, and the sound of shallow gongs. J. Acoust. Soc. Am. 86, 2439–2443 (1989)

    Article  Google Scholar 

  30. Leissa A.: Vibration of Plates. Acoustical Society of America, New York (1993)

    Google Scholar 

  31. Li W.L.: Vibration analysis of rectangular plates with general elastic support. J. Sound Vib. 273, 619–635 (2003)

    Article  Google Scholar 

  32. Luo A.C.J, Huang J.: Analytical solutions for asymmetric periodic motions to chaos in a hardening Duffing oscillator. Nonlinear Dyn. 72, 417–438 (2013)

    Article  MathSciNet  Google Scholar 

  33. Meenen J., Altenbach H.: A consistent deduction of von Kármán-type plate theories from three-dimensional nonlinear continuum mechanics. Acta Mech. 147, 1–17 (2001)

    Article  MATH  Google Scholar 

  34. Mordant N.: Are there waves in elastic wave turbulence?. Phys. Rev. Lett. 100, 234505 (2008)

    Article  Google Scholar 

  35. Mordant N.: Fourier analysis of wave turbulence in a thin elastic plate. Eur. Phys. J. B 76, 537–545 (2010)

    Article  MATH  Google Scholar 

  36. Moussa M.O., Moumni Z., Doaré O., Touzé C., Zaki W.: Non-linear dynamic thermomechanical behaviour of shape memory alloys. J. Intell. Mater. Syst. Struct. 23, 1593–1611 (2012)

    Article  Google Scholar 

  37. Murphy K.D., Virgin L.N., Rizzi S.A.: Characterizing the dynamic response of a thermally loaded, acoustically excited plate. J. Sound Vib. 196, 635–658 (1996)

    Article  Google Scholar 

  38. Nayfeh A.H.: Nonlinear Oscillations. Wiley, New York (1995)

    Book  Google Scholar 

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

    Book  MATH  Google Scholar 

  40. Parlitz U., Lauterborn W.: Superstructure in the bifurcation set of the Duffing equation. Phys. Lett. A 107, 351–355 (1985)

    Article  MathSciNet  Google Scholar 

  41. Peeters M., Viguié R., Sérandour G., Kerschen G., Golinval J.-C.: Nonlinear normal modes, part II: toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23, 195–216 (2009)

    Article  Google Scholar 

  42. Ribeiro P.: Nonlinear vibrations of simply-supported plates by the p-version finite element method. Finite Elem. Anal. Des. 41, 911–924 (2005)

    Article  Google Scholar 

  43. Ribeiro P., Petyt M.: Geometrical non-linear, steady-state, forced, periodic vibration of plate, part I: model and convergence study. J. Sound Vib. 226, 955–983 (1999)

    Article  Google Scholar 

  44. Ribeiro P., Petyt M.: Geometrical non-linear, steady-state, forced, periodic vibration of plate, part II: stability study and analysis of multimodal response. J. Sound Vib. 226, 985–1010 (1999)

    Article  Google Scholar 

  45. Sathyamoorthy M.: Nonlinear vibrations of plates: an update of recent research developments. Appl. Mech. Rev. 49, S55–S62 (1996)

    Article  Google Scholar 

  46. Thomas O., Bilbao S.: Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties. J. Sound Vib. 315, 569–590 (2008)

    Article  Google Scholar 

  47. Thomas O., Touzé C., Chaigne A.: Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance. Int. J. Solids Struct. 42, 3339–3373 (2005)

    Article  MATH  Google Scholar 

  48. Thomsen J.J.: Vibrations and Stability. Springer, Berlin (2003)

    Book  Google Scholar 

  49. Touzé C., Bilbao S., Cadot O.: Transition scenario to turbulence in thin vibrating plates. J. Sound Vib. 331, 412–433 (2011)

    Article  Google Scholar 

  50. Touzé C., Thomas O., Amabili M.: Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates. Int. J. Non-Linear Mech. 46, 234–246 (2011)

    Article  Google Scholar 

  51. Touzé C., Thomas O., Chaigne A.: Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. J. Sound Vib. 273, 77–101 (2004)

    Article  Google Scholar 

  52. Touzé C., Thomas O., Huberdeau A.: Asymptotic non-linear normal modes for large-amplitude vibrations of continuous structures. Comput. struct. 82, 2671–2682 (2004)

    Article  Google Scholar 

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

    Article  Google Scholar 

  54. von Kármán T.: Festigkeitsprobleme im Maschinenbau. Encyklopädie der Mathematischen Wissenschaften 4, 311–385 (1910)

    Google Scholar 

  55. Yamaki N.: Influence of large amplitudes on flexural vibrations of elastic plates. Zeitschrift für Angewandte Mathematik und Mechanik 41, 501–510 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  56. Yang X.L., Sethna P.R.: Local and global bifurcations in parametrically excited vibrations of nearly square plates. Int. J. Non-Linear Mech. 26, 199–220 (1991)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Unité de Mecanique, ENSTA-ParisTech, 828 Boulevard des Maréchaux, Palaiseau, France

    Michele Ducceschi & Cyril Touzé

  2. James Clerk Maxwell Building, University of Edinburgh, Edinburgh, Scotland

    Stefan Bilbao & Craig J. Webb

Authors
  1. Michele Ducceschi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cyril Touzé
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan Bilbao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Craig J. Webb
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cyril Touzé.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ducceschi, M., Touzé, C., Bilbao, S. et al. Nonlinear dynamics of rectangular plates: investigation of modal interaction in free and forced vibrations. Acta Mech 225, 213–232 (2014). https://doi.org/10.1007/s00707-013-0931-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-013-0931-1

Keywords

Search

Navigation