Nonlinear Dynamics

, Volume 79, Issue 2, pp 1293–1309 | Cite as

Complex dynamics of a nonlinear aerospace structure: numerical continuation and normal modes

  • L. Renson
  • J. P. Noël
  • G. Kerschen
Original Paper


This paper investigates the dynamics of a real-life aerospace structure possessing a strongly nonlinear component with multiple mechanical stops. A full-scale finite element model is built for gaining additional insight into the nonlinear dynamics that was observed experimentally, but also for uncovering additional nonlinear phenomena, such as quasiperiodic regimes of motion. Forced/unforced, damped/undamped numerical simulations are carried out using advanced techniques and theoretical concepts such as numerical continuation and nonlinear normal modes.


Aerospace structure Piecewise-linear nonlinearities  Numerical continuation Nonlinear normal modes Modal interactions 



This paper was prepared in the framework of the European Space Agency (ESA) Technology Research Programme study “Advancement of Mechanical Verification Methods for Nonlinear Spacecraft Structures (NOLISS)” (ESA contract No. 21359/08/NL/SFe). The authors also thank LMS-SAMTECH for providing access to the SAMCEF finite element software. The authors L. Renson and J. P. Noël are Research Fellows (FRIA fellowship) of the Fonds de la Recherche Scientifique—FNRS, which is gratefully acknowledged.


  1. 1.
    Oueini, S.S., Nayfeh, A.H.: Analysis and application of a nonlinear vibration absorber. J. Vib. Control 6(7), 999–1016 (2000)CrossRefGoogle Scholar
  2. 2.
    Vakakis, A.F., Gendelman, O., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems. Series: Solid Mechanics and Its Applications, vol. 156. Springer, Berlin (2009)Google Scholar
  3. 3.
    Gourdon, E., Alexander, N.A., Taylor, C.A., Lamarque, C.H., Pernot, S.: Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results. J. Sound Vib. 300(3–5), 522–551 (2007)CrossRefGoogle Scholar
  4. 4.
    Carrella, A., Ewins, D.J.: Identifying and quantifying structural nonlinearities in engineering applications from measured frequency response functions. Mech. Syst. Signal Process. 25(3), 1011–1027 (2011)CrossRefGoogle Scholar
  5. 5.
    Ahlquist, J.R., Carreno, J.M., Climent, H., Diego, R., Alba, J.: Assessment of nonlinear structural response in A400M GVT. In: Proceedings of the International Modal Analysis Conference, Jacksonville, USA (2011)Google Scholar
  6. 6.
    Fuellekrug, U., Goege, D.: Identification of weak non-linearities within complex aerospace structures. Aerosp. Sci. Technol. 23(1), 53–62 (2012)CrossRefGoogle Scholar
  7. 7.
    Platten, M.F., Wright, J.R., Cooper, J.E., Dimitriadis, G.: Identification of a nonlinear wing structure using an extended modal model. J. Aircr. 46(1), 1614–1626 (2009)CrossRefGoogle Scholar
  8. 8.
    Noël, J.P., Renson, L., Kerschen, G., Peeters, B., Manzato, S., Debille, J.: Nonlinear dynamic analysis of an F-16 aircraft using GVT data. In: Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, Bristol, UK (2013)Google Scholar
  9. 9.
    Carney, K.S., Yunis, I., Smith, K., Peng, C.Y.: Nonlinear dynamic behavior in the Cassini spacecraft modal survey. In: Proceedings of the International Modal Analysis Conference, Orlando, USA (1997)Google Scholar
  10. 10.
    Link, M., Boeswald, M., Laborde, S., Weiland, M., Calvi, A.: Non-linear experimental modal analysis and application to satellite vibration test data. In: Proceedings of the COMPDYN Conference, Corfu, Greece (2011)Google Scholar
  11. 11.
    Petrov, E.P., Ewins, D.J.: Advanced modeling of underplatform dampers for analysis of bladed disk vibration. J. Turbomach. 129, 143–150 (2007)CrossRefGoogle Scholar
  12. 12.
    Laxalde, D., Thouverez, F., Sinou, J.J., Lombard, J.P.: Qualitative analysis of forced response of blisks with friction ring dampers. Eur. J. Mech. A/Solids 26, 676–687 (2007)zbMATHCrossRefGoogle Scholar
  13. 13.
    Wei, F., Liang, L., Zheng, G.T.: Parametric study for dynamics of spacecraft with local nonlinearities. AIAA J. 48, 1700–1707 (2010)CrossRefGoogle Scholar
  14. 14.
    Knowles, J.A.C., Krauskopf, B., Lowenberg, M.H.: Numerical continuation applied to landing gear mechanism analysis. J. Aircr. 48, 1254–1262 (2011)CrossRefGoogle Scholar
  15. 15.
    Noël, J.P., Renson, L., Kerschen, G.: Complex dynamics of a nonlinear aerospace structure: experimental identification and modal interactions. J. Sound Vib. 333(12), 2588–2607 (2014)CrossRefGoogle Scholar
  16. 16.
    Muñoz-Almaraz, F.J., Freire, E., Galán, J., Doedel, E., Vanderbauwhede, A.: Continuation of periodic orbits in conservative and hamiltonian systems. Phys. D Nonlinear Phenom. 181(1–2), 1–38 (2003)zbMATHCrossRefGoogle Scholar
  17. 17.
    Dankowicz, H., Schilder, F.: Recipes for Continuation. Computational Science and Engineering, vol. 11, p. 579. SIAM (2013). doi: 10.1137/1.9781611972573
  18. 18.
    Vakakis, A.F., Manevitch, L.I., Mlkhlin, Y.V., Pilipchuk, V.N., Zevin, A.A.: Normal Modes and Localization in Nonlinear Systems. Wiley-VCH Verlag GmbH, Weinheim (2008)Google Scholar
  19. 19.
    Shaw, S.W., Pierre, C.: Normal modes for non-linear vibratory systems. J. Sound Vib. 164, 40 (1993)MathSciNetGoogle Scholar
  20. 20.
    Russell, A.G.: Thick skin, faceted, CFRP, monocoque tube structure for smallsats. In: European Conference on Spacecraft Structures, Materials and Mechanical Testing, Noordwijk, The Netherlands (2000)Google Scholar
  21. 21.
    Camarasa, P., Kiryenko, S.: Shock attenuation system for spacecraft and adaptor (SASSA). In: European Conference on Spacecraft Structures, Materials and Mechanical Testing, Toulouse, France (2009)Google Scholar
  22. 22.
    Bampton, M.C.C., Craig Jr, R.R.: Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968)zbMATHCrossRefGoogle Scholar
  23. 23.
    Masri, S.F., Caughey, T.K.: A nonparametric identification technique for nonlinear dynamic problems. J. Appl. Mech. 46(2), 433–447 (1979)zbMATHCrossRefGoogle Scholar
  24. 24.
    Renson, L., Kerschen, G., Newerla, A.: Nonlinear modal analysis of the SmallSat spacecraft. In: Proceedings of the International Modal Analysis Conference, Orlando, USA (2012)Google Scholar
  25. 25.
    Renson, L., Noël, J.-P., Kerschen, G., Newerla, A.: Nonlinear modal analysis of the SmallSat spacecraft. In: Proceedings of the European Conference on Spacecraft Structures, Materials and Environmental Testing, Noordwijk, The Netherlands (2012)Google Scholar
  26. 26.
    Seydel, R.: Practical Bifurcation and Stability Analysis, Interdisciplinary Applied Mathematics, vol. 5. Springer, New York (2010)CrossRefGoogle Scholar
  27. 27.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, vol. 112. Springer, New York (2004)CrossRefGoogle Scholar
  28. 28.
    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)CrossRefGoogle Scholar
  29. 29.
    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(1), 195–216 (2009)Google Scholar
  30. 30.
    Kerschen, G., Peeters, M., Golinval, J.C., Stéphan, C.: Nonlinear modal analysis of a full-scale aircraft, J. Aircr. 50(5), 1409–1419 (2013)Google Scholar
  31. 31.
    Lee, Y.S., Kerschen, G., Vakakis, A.F., Panagopoulos, P., Bergman, L., McFarland, D.M.: Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment. Physica D 204(1–2), 41–69 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Lee, Y.S., Nucera, F., Vakakis, A.F., McFarland, D.M., Bergman, L.A.: Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments. Physica D 238(18), 1868–1896 (2009)Google Scholar
  33. 33.
    Vakakis, A.F.: Non-linear normal modes (NNMs) and their applications in vibration theory: an overview. Mech. Syst. Signal Process. 11(1), 3–22 (1997)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Georgiades, F., Peeters, M., Kerschen, G., Golinval, J.C., Ruzzene, M.: Modal analysis of a nonlinear periodic structure with cyclic symmetry. AIAA J. 47(4), 1014–1025 (2009)CrossRefGoogle Scholar
  35. 35.
    Laxalde, D., Thouverez, F.: Complex non-linear modal analysis for mechanical systems: application to turbomachinery bladings with friction interfaces. J. Sound Vib. 322(4–5), 1009–1025 (2009)CrossRefGoogle Scholar
  36. 36.
    Renson, L., Deliége, G., Kerschen, G.: An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems. Meccanica (2014)Google 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

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