On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models

  • Arthur GivoisEmail author
  • Aurélien Grolet
  • Olivier Thomas
  • Jean-François Deü
Original paper


This paper presents a general methodology to compute nonlinear frequency responses of flat structures subjected to large amplitude transverse vibrations, within a finite element context. A reduced-order model (ROM) is obtained by an expansion onto the eigenmode basis of the associated linearized problem, including transverse and in-plane modes. The coefficients of the nonlinear terms of the ROM are computed thanks to a non-intrusive method, using any existing nonlinear finite element code. The direct comparison to analytical models of beams and plates proves that a lot of coefficients can be neglected and that the in-plane motion can be condensed to the transverse motion, thus giving generic rules to simplify the ROM. Then, a continuation technique, based on an asymptotic numerical method and the harmonic balance method, is used to compute the frequency response in free (nonlinear mode computation) or harmonically forced vibrations. The whole procedure is tested on a straight beam, a clamped circular plate and a free perforated plate for which some nonlinear modes are computed, including internal resonances. The convergence with harmonic numbers and oscillators is investigated. It shows that keeping a few of them is sufficient in a range of displacements corresponding to the order of the structure’s thickness, with a complexity of the simulated nonlinear phenomena that increase very fast with the number of harmonics and oscillators.


Geometric nonlinearities Non-intrusive stiffness evaluation procedure Reduced-order finite element model Continuation method 



The French Ministry of Research is warmly thanked for the financial support of this study, through the Ph.D. Grant of the first author.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest


  1. 1.
    Chen, C., Zanette, D.H., Czaplewsk, D.A., Shaw, S., López, D.: Direct observation of coherent energy transfer in nonlinear micromechanical oscillators. Nat. Commun. 8, 15523 (2017)Google Scholar
  2. 2.
    Thomas, O., Mathieu, F., Mansfield, W., Huang, C., Trolier-McKinstry, S., Nicu, L.: Efficient parametric amplification in MEMS with integrated piezoelectric actuation and sensing capabilities. Appl. Phys. Lett. 102(16), 163504 (2013). Google Scholar
  3. 3.
    Dezest, D., Thomas, O., Mathieu, F., Mazenq, L., Soyer, C., Costecalde, J., Remiens, D., Deü, J.-F., Nicu, L.: Wafer-scale fabrication of self-actuated piezoelectric nanoelectromechanical resonators based on lead zirconate titanate (PZT). J. Micromech. Microengineering 25(3), 035002 (2015)Google Scholar
  4. 4.
    Quinn, D.D., Triplett, A.L., Vakakis, A.F., Bergman, L.A.: Energy harvesting from impulsive loads using intentional essential nonlinearities. J. Vibr. Acoust. 133(1), 011004 (2011)Google Scholar
  5. 5.
    Ducceschi, M., Touzé, C.: Modal approach for nonlinear vibrations of damped impacted plates: application to sound synthesis of gongs and cymbals. J. Sound Vibr. 344, 313–331 (2015)Google Scholar
  6. 6.
    Monteil, M., Thomas, O., Touzé, C.: Identification of mode couplings in nonlinear vibrations of the steelpan. Appl. Acoust. 89, 1–15 (2015). Google Scholar
  7. 7.
    Grolet, A., Thouverez, F.: Free and forced vibration analysis of a nonlinear system with cyclic symmetry: application to a simplified model. J. Sound Vibr. 331(12), 2911–2928 (2012)Google Scholar
  8. 8.
    Renson, L., Noël, J.P., Kerschen, G.: Complex dynamics of a nonlinear aerospace structure: numerical continuation and normal modes. Nonlinear Dyn. 79(2), 1293–1309 (2015)Google Scholar
  9. 9.
    Noor, A.K.: Recent advances in reduction methods for nonlinear problems. Comput. Struct. 13(1–3), 31–44 (1981)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Slaats, P.M.A., Jongh, J.D., Sauren, A.A.H.J.: Model reduction tools for nonlinear structural dynamics. Comput. Struct. 54(6), 1155–1171 (1995)zbMATHGoogle Scholar
  11. 11.
    von Kármán, T.: Festigkeitsprobleme im Maschinenbau. Encykl. Math. Wiss. 4(4), 311–385 (1910)zbMATHGoogle Scholar
  12. 12.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, London (2008)zbMATHGoogle Scholar
  13. 13.
    Chu, H.-N., Herrmann, G.: Influence of large amplitudes on free flexural vibrations of rectangular elastic plates. J. Appl. Mech 23, 532–540 (1956)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Thomas, O., Bilbao, S.: Geometrically non-linear flexural vibrations of plates: in-plane boundary conditions and some symmetry properties. J. Sound Vibr. 315(3), 569–590 (2008)Google Scholar
  15. 15.
    Woinowski-Krieger, S.: The effect of axial force on the vibration of hinged bars. J. Appl. Mech. 17(2), 35–36 (1950)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Eisley, J.G.: Nonlinear vibration of beams and rectangular plates. Z. Angew. Math. Phys. (ZAMP) 15(2), 167–175 (1964)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ho, C.H., Scott, R.A., Eisley, J.G.: Non-planar, non-linear oscillations of a beam. part II: Free motions. J. Sound Vibr. 47(3), 333–339 (1976)zbMATHGoogle Scholar
  18. 18.
    Sridhar, S., Mook, D.T., Nayfeh, A.H.: Non-linear resonances in the forced responses of plates, part II: asymmetric responses of circular plates. J. Sound Vibr. 59(2), 159–170 (1975)zbMATHGoogle Scholar
  19. 19.
    Touzé, C., Thomas, O., Chaigne, A.: Asymmetric non-linear forced vibrations of free-edge circular plates, part I: theory. J. Sound Vibr. 258(4), 649–676 (2002). Google Scholar
  20. 20.
    Ducceschi, M., Touzé, C., Bilbao, S., Webb, C.J.: Nonlinear dynamics of rectangular plates: investigation of modal interaction in free and forced vibrations. Acta Mech. 22(1), 213–232 (2014)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Capiez-Lernout, E., Soize, C., Mignolet, M.P.: Computational stochastic statics of an uncertain curved structure with geometrical nonlinearity in three-dimensional elasticity. Computational Mechanics 49(1), 87–97 (2012)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Touzé, C., Vidrascu, M., Chapelle, D.: Direct finite element computation of non-linear modal coupling coefficients for reduced-order shell models. Comput. Mech. 54(2), 567–580 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ribeiro, P., Petyt, M.: Non-linear vibration of beams with internal resonance by the hierarchical finite element method. J. Sound Vibr. 224(4), 591–624 (1999)Google Scholar
  24. 24.
    Stoykov, S., Ribeiro, P.: Periodic geometrically nonlinear free vibrations of circular plates. J. Sound Vibr. 315, 536–555 (2008)Google Scholar
  25. 25.
    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 Vibr. 243(4), 601–624 (2001)Google Scholar
  26. 26.
    Muravyov, A.A., Rizzi, S.A.: Determination of nonlinear stiffness with application to random vibration of geometrically nonlinear structures. Comput. Struct. 81(15), 1513–1523 (2003)Google Scholar
  27. 27.
    Mignolet, M.P., Przekop, A., Rizzi, S., Spottswood, S.: A review of indirect/non-intrusive reduced-order modeling of nonlinear geometric structures. J. Sound Vibr. 332(10), 2437–2460 (2013)Google Scholar
  28. 28.
    Perez, R., Wang, X.Q., Mignolet, M.P.: Nonintrusive structural dynamic reduced-order modeling for large deformations: enhancements for complex structures. J. Comput. Nonlinear Dyn. 9(3), 031008 (2014)Google Scholar
  29. 29.
    Murthy, R., Wang, X.Q., Perez, R., Mignolet, M.P., Richter, L.A.: Uncertainty-based experimental validation of nonlinear reduced-order models. J. Sound Vibr. 331(5), 1097–1114 (2012)Google Scholar
  30. 30.
    Claeys, M., Sinou, J.-J., Lambelin, J.-P., Alcoverro, B.: Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 19, 4196–4212 (2014)Google Scholar
  31. 31.
    O’Hara, P., Hollkamp, J.J.: Modeling vibratory damage with reduced-order models and the generalized finite element method. J. Sound Vibr. 333(24), 6637–6650 (2014)Google Scholar
  32. 32.
    Ehrhardt, D.A., Allen, M.S., Beberniss, T.J., Neild, S.A.: Finite element model calibration of a nonlinear perforated plate. J. Sound Vibr. 392, 280–294 (2017)Google Scholar
  33. 33.
    Kim, K., Radu, A.G., Wang, X.Q., Mignolet, M.P.: Nonlinear reduced-order modeling of isotropic and functionally graded plates. Int. J. Non-linear Mech. 49, 100–110 (2013)Google Scholar
  34. 34.
    Lazarus, A., Thomas, O., Deü, J.-F.: Finite elements reduced order models for nonlinear vibrations of piezoelectric layered beams with applications to NEMS. Finite Elem. Anal. Des. 49(1), 35–51 (2012)MathSciNetGoogle Scholar
  35. 35.
    Hollkamp, J., Gordon, R.: Reduced-order models for nonlinear response prediction: implicit condensation and expansion. J. Sound Vibr. 318(4), 1139–1153 (2008)Google Scholar
  36. 36.
    Kim, K., Khanna, V., Wang, X., Mignolet, M.: Nonlinear reduced order modeling of flat cantilevered structures. In: Proceedings of the 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, p. 2492 (2009)Google Scholar
  37. 37.
    Nash, M.: Nonlinear structural dynamics by finite element modal synthesis, Ph.D. thesis, Imperial College—University of London (1978)Google Scholar
  38. 38.
    Rizzi, S.A., Przekop, A.: System identification-guided basis selection for reduced-order nonlinear response analysis. J. Sound Vibr. 315(3), 467–485 (2008)Google Scholar
  39. 39.
    Przekop, A., Guo, X., Rizzi, S.A.: Alternative modal basis selection procedures for reduced-order nonlinear random response simulation. J. Sound Vibr. 331(17), 4005–4024 (2012)Google Scholar
  40. 40.
    Kuether, R.J., Deaner, B., Hollkamp, J.J., Allen, M.S.: Evaluation of geometrically nonlinear reduced-order models with nonlinear normal modes. AIAA J. 53(11), 3273–3285 (2015)Google Scholar
  41. 41.
    Idelsohn, S.R., Cardona, A.: A reduction method for nonlinear structural dynamic analysis. Comput. Methods Appl. Mech. Eng. 49(3), 253–279 (1985)zbMATHGoogle Scholar
  42. 42.
    Sombroek, C.S.M., Tiso, P., Renson, L., Kerschen, G.: Numerical computation of nonlinear normal modes in a modal derivative subspace. Comput. Struct. 195, 34–36 (2018)Google Scholar
  43. 43.
    Jain, S., Tiso, P., Rutzmoser, J.B., Rixen, D.J.: A quadratic manifold for model order reduction of nonlinear structural dynamics. Comput. Struct. 188, 80–94 (2017)Google Scholar
  44. 44.
    Rutzmoser, J.B., Rixen, D.J., Tiso, P., Jain, S.: Generalization of quadratic manifolds for reduced order modeling of nonlinear structural dynamics. Comput. Struct. 192, 196–209 (2017)Google Scholar
  45. 45.
    Boumediene, F., Miloudi, A., Cadou, J., Duigou, L., Boutyour, E.: Nonlinear forced vibration of damped plates by an asymptotic numerical method. Comput. Struct. 87(23–24), 1508–1515 (2009)zbMATHGoogle Scholar
  46. 46.
    Boumediene, F., Duigou, L., Boutyour, E., Miloudi, A., Cadou, J.: Nonlinear forced vibration of damped plates coupling asymptotic numerical method and reduction models. Comput. Mech. 47(4), 359–377 (2011)zbMATHGoogle Scholar
  47. 47.
    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)Google Scholar
  48. 48.
    Lamarque, C.-H., Touzé, C., Thomas, O.: An upper bound for validity limits of asymptotic analytical approaches based on normal form theory. Nonlinear Dyn. 70(3), 1931–1949 (2012)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Renson, L., Kerschen, G., Cochelin, B.: Numerical computation of nonlinear normal modes in mechanical engineering. J. Sound Vibr. 364, 177–206 (2016)Google Scholar
  50. 50.
    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
  51. 51.
    Sombroek, C.S.M., Renson, L., Tiso, P., Kerschen, G.: Bridging the gap between nonlinear normal modes and modal derivatives. Nonlinear Dyn. 1, 349–361 (2016)Google Scholar
  52. 52.
    Kuether, R.J., Allen, M.S.: A numerical approach to directly compute nonlinear normal modes of geometrically nonlinear finite element models. Mech. Syst. Signal Process. 46(1), 1–15 (2014)Google Scholar
  53. 53.
    Ciarlet, P.G.: A justification of the von-Kármán equations. Arch. Rat. Mech. Analysis 73, 349–389 (1980)zbMATHGoogle Scholar
  54. 54.
    Millet, O., Hamdouni, A., Cimetière, A.: A classification of thin plate models by asymptotic expansion of non-linear three-dimensional equilibrium equations. Int. J. Non-linear Mech. 36(1), 165–186 (2001)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Lacarbonara, W., Yabuno, H.: Refined models of elastic beams undergoing large in-plane motions: theory and experiment. Int. J. Solids Struct. 43, 5066–5084 (2006)zbMATHGoogle Scholar
  56. 56.
    Thomas, O., Sénéchal, A., Deü, J.-F.: Hardening/softening behaviour and reduced order modelling of nonlinear vibrations of rotating cantilever beams. Nonlinear Dyn. 86(2), 1293–1318 (2016)Google Scholar
  57. 57.
    Bilbao, S., Thomas, O., Touzé, C., Ducceschi, M.: Conservative numerical methods for the full von Kármán plate equations 31(6), 1948–1970 (2015).
  58. 58.
    Cottanceau, E., Thomas, O., Véron, P., Alochet, M., Deligny, R.: A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables. Finite Elem. Anal. Des. 139, 14–34 (2017)MathSciNetGoogle Scholar
  59. 59.
    Chang, Y., Wang, X., Capiez-Lernout, E., Mignolet, M., Soize, C.: Reduced order modelling for the nonlinear geometric response of some curved structures. In: International Forum on Aeroelasticity and Structural Dynamics, IFASD 2011, AAAF-AIAA, Paper–IFASD (2011)Google Scholar
  60. 60.
    Grolet, A.: Dynamique non-linéaire des structures mécaniques: application aux systèmes à symétrie cyclique. Ecully, Ecole centrale de Lyon (2013). Ph.D. thesisGoogle Scholar
  61. 61.
  62. 62.
    Cochelin, B., Vergez, C.: A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions. J. Sound Vibr. 324(1), 243–262 (2009)Google Scholar
  63. 63.
    Cochelin, B., Damil, N., Potier-Ferry, M.: Méthode asymptotique numérique. Hermes Lavoissier, Paris (2007)zbMATHGoogle Scholar
  64. 64.
    Karkar, S., Cochelin, B., Vergez, C.: A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities. J. Sound Vibr. 332(4), 968–977 (2013)Google Scholar
  65. 65.
    Munoz-Almaraz, F., 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)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Manevitch, A., Manevitch, L.: Free oscillations in conservative and dissipative symmetric cubic two-degree-of-freedom systems with closed natural frequencies. Meccanica 38, 335–348 (2003)MathSciNetzbMATHGoogle Scholar
  67. 67.
    King, M., Vakakis, A.: Mode localization in a system of coupled flexible beams with geometric nonlinearities. Z. Angew. Math. Mech. 75(2), 127–139 (1995)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Grolet, A., Thouverez, F.: Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases. Mech. Syst. Signal Process. 52–53, 529–547 (2015)Google Scholar
  69. 69.
    Park, C.I.: Frequency equation for the in-plane vibration of a clamped circular plate. J. Sound Vibr. 313(1), 325–333 (2008)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Laboratoire d’Ingénierie des Systèmes Physiques et Numériques (LISPEN EA 7515)Arts et Métiers ParisTechLilleFrance
  2. 2.Laboratoire de Mécanique des Structures et des Systèmes Couplés (LMSSC EA 3196)Conservatoire National des Arts et MétiersParisFrance

Personalised recommendations