Advertisement

Meccanica

, Volume 49, Issue 8, pp 1901–1916 | Cite as

An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems

  • L. Renson
  • G. Deliége
  • G. Kerschen
Nonlinear Dynamics and Control of Composites for Smart Engi design

Abstract

This paper addresses the numerical computation of nonlinear normal modes defined as two-dimensional invariant manifolds in phase space. A novel finite-element-based algorithm, combining the streamline upwind Petrov–Galerkin method with mesh moving and domain prediction–correction techniques, is proposed to solve the manifold-governing partial differential equations. It is first validated using conservative examples through the comparison with a reference solution given by numerical continuation. The algorithm is then demonstrated on nonconservative examples.

Keywords

Nonlinear normal modes Invariant manifolds Nonconservative systems Modal analysis Finite element method 

Notes

Acknowledgements

The author L. Renson would like to acknowledge the Belgian National Fund for Scientific Research (FRIA fellowship) for its financial support.

References

  1. 1.
    Rosenberg RM (1966) On nonlinear vibrations of systems with many degrees of freedom. Adv Appl Mech 9:155–242CrossRefGoogle Scholar
  2. 2.
    Vakakis AF, Manevitch LI, Mlkhlin YV, Pilipchuk VN, Zevin AA (2008) Normal modes and localization in nonlinear systems. Wiley-VCH Verlag GmbHGoogle Scholar
  3. 3.
    Shaw SW, Pierre C (1993) Normal modes for non-linear vibratory systems. J Sound Vib 164:40CrossRefMathSciNetGoogle Scholar
  4. 4.
    Vakakis AF (1997) Non-linear normal modes (NNMs) and their applications in vibration theory: an overview. Mech Syst Signal Process 11(1):3–22ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    Jezequel L, Lamarque CH (1991) Analysis of non-linear dynamical systems by the normal form theory. J Sound Vib 149(3):429–459ADSCrossRefGoogle Scholar
  6. 6.
    Lacarbonara W, Camillacci R (2004) Nonlinear normal modes of structural systems via asymptotic approach. Int J Solids Struct 41(20):5565–5594CrossRefMATHGoogle Scholar
  7. 7.
    Gendelman OV (2004) Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment. Nonlinear Dyn 37(2):115–128CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Lenci S, Rega G (2007) Dimension reduction of homoclinic orbits of buckled beams via the non-linear normal modes technique. Int J Non-linear Mech 42(3):515–528CrossRefMATHGoogle Scholar
  9. 9.
    Warminski J (2010) Nonlinear normal modes of a self-excited system driven by parametric and external excitations. Nonlinear Dyn 61(4):677–689CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Lee YS, Kerschen G, Vakakis AF, Panagopoulos P, Bergman L, McFarland DM (2005) Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment. Physica D 204(1-2):41–69ADSCrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Kerschen G, Peeters M, Golinval JC, Vakakis AF (2009) Nonlinear normal modes, part I: A useful framework for the structural dynamicist. Mech Syst Signal Process 23(1):170–194ADSCrossRefGoogle Scholar
  12. 12.
    Arquier R, Bellizzi S, Bouc R, Cochelin B (2006) Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes. Comput Struct 84(24-25):1565–1576CrossRefMathSciNetGoogle Scholar
  13. 13.
    Peeters M, Viguié R, Sérandour G, Kerschen G, Golinval JC (2009) Nonlinear normal modes, part II: Toward a practical computation using numerical continuation techniques. Mech Syst Signal Process 23(1):195–216ADSCrossRefGoogle Scholar
  14. 14.
    Kerschen G, Peeters M, Golinval JC, Stephan C (2013) Nonlinear normal modes of a full-scale aircraft. AIAA Journal Aircr 50:1409–1419CrossRefGoogle Scholar
  15. 15.
    Touzé C, Amabili M (2006) Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. J Sound Vib 298(4-5):958–981ADSCrossRefGoogle Scholar
  16. 16.
    Boivin N, Pierre C, Shaw SW (1995) Nonlinear normal modes, invariance, and modal dynamics approximations of nonlinear systems. Nonlinear Dyn 8:32MathSciNetGoogle Scholar
  17. 17.
    Chen S-L, Shaw SW (1996) Normal modes for piecewise linear vibratory systems. Nonlinear Dyn 10:135–164CrossRefMathSciNetGoogle Scholar
  18. 18.
    Boivin N, Pierre C, Shaw SW (1996) Non-linear modal analysis of the forced response of structural systems. AIAA J 31:22Google Scholar
  19. 19.
    Boivin N, Pierre C, Shaw SW (1995) Non-linear modal analysis of structural systems featuring internal resonances. J Sound Vib 182:6CrossRefGoogle Scholar
  20. 20.
    Pesheck E, Pierre C, Shaw SW (2002) A new Galerkin-based approach for accurate non-linear normal modes through invariant manifolds. J Sound Vib 249(5):971–993ADSCrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Pesheck E (2000) Reduced order modeling of nonlinear structural systems using nonlinear normal modes and invariant manifolds. PhD thesis, University of MichiganGoogle Scholar
  22. 22.
    Jiang D, Pierre C, Shaw SW (2004) Large-amplitude non-linear normal modes of piecewise linear systems. J Sound Vib 272(3–5):869–891ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    Jiang D, Pierre C, Shaw SW (2005) Nonlinear normal modes for vibratory systems under harmonic excitation. J Sound Vib 288(4–5):791–812ADSCrossRefGoogle Scholar
  24. 24.
    Pesheck E, Boivin N, Pierre C, Shaw SW (2001) Nonlinear modal analysis of structural systems using multi-mode invariant manifolds. Nonlinear Dyn 25:183–205CrossRefMATHGoogle Scholar
  25. 25.
    Jiang D, Pierre C, Shaw SW (2005) The construction of non-linear normal modes for systems with internal resonance. Int J Non-linear Mech 40(5):729–746CrossRefMATHGoogle Scholar
  26. 26.
    Legrand M, Jiang D, Pierre C, Shaw SW (2004) Nonlinear normal modes of a rotating shaft based on the invariant manifold method. Int J Rotat Mach 10(4):319–335CrossRefGoogle Scholar
  27. 27.
    Blanc F, Touzé C, Mercier JF, Ege K, Bonnet Ben-Dhia AS (2013) On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems. Mech Syst Signal Process 36(2):520–539ADSCrossRefGoogle Scholar
  28. 28.
    Bellizzi S, Bouc R (2005) A new formulation for the existence and calculation of nonlinear normal modes. J Sound Vib 287(3):545–569ADSCrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Bellizzi S, Bouc R (2007) An amplitude–phase formulation for nonlinear modes and limit cycles through invariant manifolds. J Sound Vib 300(3–5):896–915ADSCrossRefGoogle Scholar
  30. 30.
    Laxalde D, Thouverez F (2009) Complex non-linear modal analysis for mechanical systems: application to turbomachinery bladings with friction interfaces. J Sound Vib 322(4–5):1009–1025ADSCrossRefGoogle Scholar
  31. 31.
    Larsson S, Thomée V (2003) Partial differential equations with numerical methods, vol 45. Springer, Berlin HeidelbergMATHGoogle Scholar
  32. 32.
    Renardy M, Rogers R (2004) An introduction to partial differential equations, vol 13. Springer, Berlin HeidelbergGoogle Scholar
  33. 33.
    Stein K, Tezduyar TE, Benney R (2004) Automatic mesh update with the solid-extension mesh moving technique. Comput Methods Appl Mech Eng 193(21–22):2019–2032ADSCrossRefMATHGoogle Scholar
  34. 34.
    Xu Z, Accorsi M (2004) Finite element mesh update methods for fluid-structure interaction simulations. Finite Elem Anal Des 40(9–10):1259–1269CrossRefGoogle Scholar
  35. 35.
    Ern A, Guermond J-L (2004) Theory and practice of finite elements, vol 159 of Applied Mathematical Sciences. Springer, New YorkGoogle Scholar
  36. 36.
    Donea J, Huerta A (2005) Steady transport problems. Finite element methods for flow problems. Wiley, New YorkGoogle Scholar
  37. 37.
    Brooks AN, Hughes TJR (1982) Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 32(1–3):199–259ADSCrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Hughes TJR, Tezduyar TE (1984) Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Comput Methods Appl Mech Eng 45(1–3):217–284ADSCrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Touzé C, Thomas O, Huberdeau A (2004) Asymptotic non-linear normal modes for large-amplitude vibrations of continuous structures. Comput Struct 82(31–32):2671–2682CrossRefGoogle Scholar
  40. 40.
    Johnson C, Nävert U, Pitkäranta J (1984) Finite element methods for linear hyperbolic problems. Comput Methods Appl Mech Eng 45(1–3):285–312ADSCrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Seydel R (2010) Practical bifurcation and stability analysis, volume 5 of Interdisciplinary Applied Mathematics. Springer, New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Space Structures and Systems Laboratory, Aerospace and Mechanical Engineering DepartmentUniversity of LiègeLiègeBelgium
  2. 2.Computational Nonlinear Mechanics, Aerospace and Mechanical Engineering DepartmentUniversity of LiègeLiègeBelgium

Personalised recommendations