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

Renson, Ludovic; Deliège, Geoffrey; Kerschen, Gaëtan

doi:10.1007/s11012-014-9875-3

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Article (Scientific journals)

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

Renson, Ludovic; Deliège, Geoffrey; Kerschen, Gaëtan

2014 • In Meccanica, 49 (8), p. 1901-1916

Peer reviewed

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https://hdl.handle.net/2268/162025

DOI
10.1007/s11012-014-9875-3

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Keywords :

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

Abstract :

[en] 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.

Disciplines :

Aerospace & aeronautics engineering

Author, co-author :

Renson, Ludovic ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire de structures et systèmes spatiaux

Deliège, Geoffrey ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > LTAS-Mécanique numérique non linéaire

Kerschen, Gaëtan ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire de structures et systèmes spatiaux

Language :

English

Title :

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

Publication date :

2014

Journal title :

Meccanica

ISSN :

0025-6455

eISSN :

1572-9648

Publisher :

Kluwer Academic Publishers, Netherlands

Special issue title :

Nonlinear Dynamics and Control of Composites for Smart Engineering Design

Volume :

Issue :

Pages :

1901-1916

Peer reviewed :

Peer reviewed

Funders :

FRIA - Fonds pour la Formation à la Recherche dans l'Industrie et dans l'Agriculture

Available on ORBi :

since 24 January 2014

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Bibliography

Rosenberg RM (1966) On nonlinear vibrations of systems with many degrees of freedom. Adv Appl Mech 9:155-242
Vakakis AF, Manevitch LI, Mlkhlin YV, Pilipchuk VN, Zevin AA (2008) Normal modes and localization in nonlinear systems. Wiley-VCH Verlag GmbH
Shaw SW, Pierre C (1993) Normal modes for non-linear vibratory systems. J Sound Vib 164:40
Vakakis AF (1997) Non-linear normal modes (NNMs) and their applications in vibration theory: an overview. Mech Syst Signal Process 11(1):3-22
Jezequel L, Lamarque CH (1991) Analysis of non-linear dynamical systems by the normal form theory. J Sound Vib 149(3):429-459
Lacarbonara W, Camillacci R (2004) Nonlinear normal modes of structural systems via asymptotic approach. Int J Solids Struct 41(20):5565-5594
Gendelman OV (2004) Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment. Nonlinear Dyn 37(2):115-128
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):515-528
Warminski J (2010) Nonlinear normal modes of a self-excited system driven by parametric and external excitations. Nonlinear Dyn 61(4):677-689
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):41-69
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):170-194
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):1565-1576
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):195-216
Kerschen G, Peeters M, Golinval JC, Stephan C (2013) Nonlinear normal modes of a full-scale aircraft. AIAA Journal Aircr 50:1409-1419
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):958-981
Boivin N, Pierre C, Shaw SW (1995) Nonlinear normal modes, invariance, and modal dynamics approximations of nonlinear systems. Nonlinear Dyn 8:32
Chen S-L, Shaw SW (1996) Normal modes for piecewise linear vibratory systems. Nonlinear Dyn 10:135-164
Boivin N, Pierre C, Shaw SW (1996) Non-linear modal analysis of the forced response of structural systems. AIAA J 31:22
Boivin N, Pierre C, Shaw SW (1995) Non-linear modal analysis of structural systems featuring internal resonances. J Sound Vib 182:6
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):971-993
Pesheck E (2000) Reduced order modeling of nonlinear structural systems using nonlinear normal modes and invariant manifolds. PhD thesis, University of Michigan
Jiang D, Pierre C, Shaw SW (2004) Large-amplitude non-linear normal modes of piecewise linear systems. J Sound Vib 272(3-5):869-891
Jiang D, Pierre C, Shaw SW (2005) Nonlinear normal modes for vibratory systems under harmonic excitation. J Sound Vib 288(4-5):791-812
Pesheck E, Boivin N, Pierre C, Shaw SW (2001) Nonlinear modal analysis of structural systems using multi-mode invariant manifolds. Nonlinear Dyn 25:183-205
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):729-746
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):319-335
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):520-539
Bellizzi S, Bouc R (2005) A new formulation for the existence and calculation of nonlinear normal modes. J Sound Vib 287(3):545-569
Bellizzi S, Bouc R (2007) An amplitude-phase formulation for nonlinear modes and limit cycles through invariant manifolds. J Sound Vib 300(3-5):896-915
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):1009-1025
Larsson S, Thomée V (2003) Partial differential equations with numerical methods, vol 45. Springer, Berlin Heidelberg
Renardy M, Rogers R (2004) An introduction to partial differential equations, vol 13. Springer, Berlin Heidelberg
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):2019-2032
Xu Z, Accorsi M (2004) Finite element mesh update methods for fluid-structure interaction simulations. Finite Elem Anal Des 40(9-10):1259-1269
Ern A, Guermond J-L (2004) Theory and practice of finite elements, vol 159 of Applied Mathematical Sciences. Springer, New York
Donea J, Huerta A (2005) Steady transport problems. Finite element methods for flow problems. Wiley, New York
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):199-259
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):217-284
Touzé C, Thomas O, Huberdeau A (2004) Asymptotic non-linear normal modes for large-amplitude vibrations of continuous structures. Comput Struct 82(31-32):2671-2682
Johnson C, Nävert U, Pitkäranta J (1984) Finite element methods for linear hyperbolic problems. Comput Methods Appl Mech Eng 45(1-3):285-312
Seydel R (2010) Practical bifurcation and stability analysis, volume 5 of Interdisciplinary Applied Mathematics. Springer, New York