Nonlinear normal modes, Part 2: Toward a pratical computation using numerical continuation techniques

Peeters, Maxime; Viguié, Régis; Sérandour, Guillaume; Kerschen, Gaëtan; Golinval, Jean-Claude

doi:10.1016/j.ymssp.2008.04.003

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Nonlinear normal modes, Part 2: Toward a pratical computation using numerical continuation techniques

Peeters, Maxime; Viguié, Régis; Sérandour, Guillaume et al.

2009 • In Mechanical Systems and Signal Processing, 23, p. 195-216

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https://hdl.handle.net/2268/18253

DOI
10.1016/j.ymssp.2008.04.003

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Keywords :

Nonlinear dynamics; Nonlinear normalmodes; Modal analysis; Shooting method; Continuation methodsofperiodicmotions

Abstract :

[en] The conceptofnonlinearnormalmodes(NNMs)isdiscussedinthepresentpaperandits companion,PartI.OnereasonofthestilllimiteduseofNNMsinstructuraldynamicsis that theircomputationisoftenregardedasimpractical.However,whenresortingto numericalalgorithms,weshowthattheNNMcomputationispossiblewithlimited implementationeffort,whichpavesthewaytoapracticalmethodfordeterminingthe NNMsofnonlinearmechanicalsystems.Theproposedalgorithmreliesontwomain techniques,namelyashootingprocedureandamethodforthecontinuationofNNM motions. Thealgorithmisdemonstratedusingfourdifferentmechanicalsystems,a weakly andastronglynonlineartwo-degree-of-freedomsystem,asimplifieddiscrete modelofanonlinearbladeddiskandanonlinearcantileverbeamdiscretizedbythefinite elementmethod.

Disciplines :

Mechanical engineering
Mathematics
Computer science
Engineering, computing & technology: Multidisciplinary, general & others

Author, co-author :

Peeters, Maxime ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > LTAS - Vibrations et identification des structures

Viguié, Régis ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > LTAS - Vibrations et identification des structures

Sérandour, Guillaume ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > LTAS - Vibrations et identification des structures

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

Golinval, Jean-Claude ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > LTAS - Vibrations et identification des structures

Language :

English

Title :

Nonlinear normal modes, Part 2: Toward a pratical computation using numerical continuation techniques

Publication date :

2009

Journal title :

Mechanical Systems and Signal Processing

ISSN :

0888-3270

eISSN :

1096-1216

Publisher :

Academic Press, London, United Kingdom

Volume :

Pages :

195-216

Peer reviewed :

Peer Reviewed verified by ORBi

Available on ORBi :

since 10 August 2009

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497

Scopus citations^®
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435

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334

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488

Bibliography

E. Pesheck, Reduced-order modelling of nonlinear structural systems using nonlinear normal modes and invariant manifolds, Ph.D. Thesis, University of Michigan, Ann Arbor, 2000.
Keller H.B. Numerical Methods in Bifurcation Problems (1987), Springer, Berlin
E. Doedel, Auto, software for continuation and bifurcation problems in ordinary differential equations 〈http://indy.cs.concordia.ca/auto/〉.
Seydel R. Practical bifurcation and stability analysis. From Equilibrium to Chaos. 2nd ed. (1994), Springer, Berlin
Nayfeh A.H., and Balachandran B. Applied Nonlinear Dynamics, Analytical, Computational, and Experimental Methods (1995), Wiley-Interscience, Chichester
Padmanabhan C., and Singh R. Analysis of periodically excited nonlinear-systems by a parametric continuation technique. Journal of Sound and Vibration 184 1 (1995) 35-58
Sundararajan P., and Noah S.T. An algorithm for response and stability of large order non-linear systems-application to rotor systems. Journal of Sound and Vibration 4 1 (1998) 695-723
Ribeiro P. Non-linear forced vibrations of thin/thick beams and plates by the finite element and shooting methods. Computers and Structures 82 17-19 (2004) 1413-1423
Govaerts W., Kuznetsov Y.A., and Dhooge A. Numerical continuation of bifurcations of limit cycles in MATLAB. SIAM Journal on Scientific Computing 27 1 (2005) 231-252
Touzé C., and Amabili M. Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298 4-5 (2006) 958-981
C. Touzé, M. Amabili, O. Thomas, Reduced-order models for large-amplitude vibrations of shells including in-plane inertia, in: Proceedings of the EUROMECH Colloquium on Geometrically Nonlinear Vibrations, 2007.
Munoz-Almaraz F.J., Freire E., Galán J., Doedel E., and Vanderbauwhede A. Continuation of periodic orbits in conservative and hamiltonian systems. Physica D 181 (2003) 1-38
Doedel E.J., Paffenroth R.C., Keller H.B., Dickmann D.J., Galan Vioque J., and Vanderbauwhede A. Computation of periodic solutions of conservative systems with application to the 3-body problem. International Journal of Bifurcations and Chaos 13 6 (2003) 1-29
Slater J.C. A numerical method for determining nonlinear normal modes. Nonlinear Dynamics 10 1 (1996) 19-30
Lee Y.S., Kerschen G., Vakakis A.F., Panagopoulos P.N., Bergman L.A., and McFarland D.M. Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment. Physica D 204 (2005) 41-69
Wang F.X., and Bajaj A.K. Nonlinear normal modes in multi-mode models of an inertially coupled elastic structure. Nonlinear Dynamics 47 (2007) 25-47
Arquier R., Bellizzi S., Bouc R., and Cochelin B. Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes. Computers and Structures 84 24-25 (2006) 1565-1576
Leine R.I., and Nijmeijer H. Dynamics and Bifurcations of Non-smooth Mechanical Systems (2004), Springer, Berlin
Rosenberg R.M. Normal modes of nonlinear dual-mode systems. Journal of Applied Mechanics 27 (1960) 263-268
Rosenberg R.M. The normal modes of nonlinear n-degree-of-freedom systems. Journal of Applied Mechanics 30 1 (1962) 7-14
Rosenberg R.M. On nonlinear vibrations of systems with many degrees of freedom. Advances in Applied Mechanics 242 9 (1966) 155-242
Shaw S.W., and Pierre C. Nonlinear normal modes and invariant manifolds. Journal of Sound and Vibration 150 1 (1991) 170-173
S.W. Shaw, C. Pierre, On nonlinear normal modes, in: ASME Winter Annual Meeting, 1992.
Shaw S.W., and Pierre C. Normal modes for nonlinear vibratory systems. Journal of Sound and Vibration 164 1 (1993) 85-124
Shaw S.W., and Pierre C. Normal modes of vibration for nonlinear continuous systems. Journal of Sound and Vibration 169 3 (1994) 319-347
Kerschen G., Lee Y.S., Vakakis A.F., McFarland D.M., and Bergman L.A. Irreversible passive energy transfer in coupled oscillators with essential nonlinearity. SIAM Journal of Applied Mathematics 66 2 (2006) 648-679
A.F. Vakakis, L.I. Manevitch, Y.V. Mikhlin, V.N. Pilipchuk, A.A. Zevin, Normal Modes and Localization in Nonlinear Systems, Wiley, New York, 1996.
Vakakis A.F., and Rand R.H. Normal modes and global dynamics of a 2-degree-of-freedom nonlinear-system; part I: low energies. International Journal of Non-Linear Mechanics 27 (1992) 861-874
Vakakis A.F., and Rand R.H. Normal modes and global dynamics of a 2-degree-of-freedom nonlinear-system; part II: high energies. International Journal of Non-Linear Mechanics 27 (1992) 875-888
Parker T.S., and Chua L.O. Practical Numerical Algorithms for Chaotic Systems (1989), Springer, New York
Brüls O., and Eberhard P. Sensitivity analysis for dynamic mechanical systems with finite rotations. International Journal for Numerical Methods in Engineering 1 (2006) 1-29
N. Mihajlović, Torsional and lateral vibrations in flexible rotor systems with friction, Ph.D. Thesis, Eindhoven University of Technology, 2005.
Allgower E.L., and Georg K. Introduction to Numerical Continuation Methods, Classics in Applied Mathematics (2003), SIAM, Philadelphia, PA
Géradin M., and Rixen D. Mechanical Vibrations, Theory and Application to Structural Dynamics (1994), Wiley, New York
Gendelman O.V., Manevitch L.I., Vakakis A.F., and McCloskey R. Energy pumping in nonlinear mechanical oscillators: part I-dynamics of the underlying hamiltonian systems. Journal of Applied Mechanics 68 (2001) 34-41
Vakakis A.F., and Gendelman O.V. Energy pumping in nonlinear mechanical oscillators: part II-resonance capture. Journal of Sound and Vibration 68 (2001) 42-48
Thouverez F. Presentation of the ECL benchmark. Mechanical Systems and Signal Processing 17 (2003) 195-202