Numerical computation of nonlinear normal modes in mechanical engineering

Algorithms; Industrial structures; Low dimensional; Nonlinear normal modes; Numerical computations; Strongly nonlinear; Weakly nonlinear systems; Computational methods

Abstract :

[en] This paper reviews the recent advances in computational methods for nonlinear normal modes (NNMs). Different algorithms for the computation of undamped and damped NNMs are presented, and their respective advantages and limitations are discussed. The methods are illustrated using various applications ranging from low-dimensional weakly nonlinear systems to strongly nonlinear industrial structures. © 2015 Elsevier Ltd.

Disciplines :

Aerospace & aeronautics engineering

Author, co-author :

Renson, Ludovic

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

Cochelin, B.; LMA, Centrale Marseille, Aix-Marseille Université, CNRS UPR 7051, Marseille Cedex, France

Language :

English

Title :

Numerical computation of nonlinear normal modes in mechanical engineering

Publication date :

2016

Journal title :

Journal of Sound and Vibration

ISSN :

0022-460X

eISSN :

1095-8568

Publisher :

Academic Press

Volume :

364

Pages :

177-206

Peer reviewed :

Peer Reviewed verified by ORBi

Available on ORBi :

since 11 November 2017

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116 (8 by ULiège)

Number of downloads

8 (3 by ULiège)

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Scopus citations^®

145

Scopus citations^®
without self-citations

130

OpenCitations

101

OpenAlex citations

154

Bibliography

D.J. Ewins Modal Testing: Theory, Practice and Application 2nd edition 2000 Research Studies Press LTD Baldock, Hertfordshire, England
M. Degener, Ground vibration testing for validation of large aircraft structural dynamics, Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, Manchester, UK, 1995.
J.R. Ahlquist, J.M. Carreño, H. Climent, R. de Diego, J. de Alba, Assessment of nonlinear structural response in A400M GVT, Proceedings of the International Modal Analysis Conference, Jacksonville, FL, USA, 2010.
J.P. Noël, L. Renson, G. Kerschen, B. Peeters, S. Manzato, J. Debille, Nonlinear dynamic analysis of an F-16 aircraft using GVT data, Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, Bristol, UK, 2013.
N. Boechler, G. Theocharis, and C. Daraio Bifurcation-based acoustic switching and rectification Nature Materials 10 9 2011 665 668 10.1038/nmat3072
D. Antonio, D.H. Zanette, and D. Lopez Frequency stabilization in nonlinear micromechanical oscillators Nature Communications 3 2012 806 10.1038/ncomms1813
C.M. Donahue, P.W.J. Anzel, L. Bonanomi, T.A. Keller, and C. Daraio Experimental realization of a nonlinear acoustic lens with a tunable focus Applied Physics Letters 104 1 2014
A.F. Vakakis, O.V. Gendelman, L.A. Bergman, D.M. McFarland, G. Kerschen, Y. S. Lee, Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, Solid Mechanics and Its Applications, Vol. 156, Springer, The Netherlands, 2009.
M.A. Karami D.J. Inman Powering pacemakers from heartbeat vibrations using linear and nonlinear energy harvesters Applied Physics Letters 100 (4) 2012
J.P. Noël, L. Renson, and G. Kerschen Complex dynamics of a nonlinear aerospace structure: experimental identification and modal interactions Journal of Sound and Vibration 333 12 2014 2588 2607
A.F. Vakakis, L.I. Manevitch, Y.V. Mlkhlin, V.N. Pilipchuk, and A.A. Zevin Normal Modes and Localization in Nonlinear Systems 2008 Wiley-VCH Verlag GmbH Weinheim, Germany
B. Peeters, A. Carrella, J. Lau, M. Gatto, G. Coppotelli, Advanced shaker excitation signals for aerospace testing, Proceedings of the International Modal Analysis Conference (IMAC), Jacksonville, USA, 2011.
L. Jezequel, and C.H. Lamarque Analysis of non-linear dynamical systems by the normal form theory Journal of Sound and Vibration 149 3 1991 429 459
A.H. Nayfeh, and S.A. Nayfeh Nonlinear normal modes of a continuous system with quadratic nonlinearities Journal of Vibration and Acoustics 117 2 1995 199 205
O.V. Gendelman Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment Nonlinear Dynamics 37 2 2004 115 128
S. Lenci, and G. Rega Dimension reduction of homoclinic orbits of buckled beams via the non-linear normal modes technique International Journal of Non-Linear Mechanics 42 3 2007 515 528
E.L. Allgower, and K. Georg Introduction to Numerical Continuation Methods, Classics in Applied Mathematics 2003 SIAM Philadelphia, PA, USA
E.J. Doedel Lecture notes on numerical analysis of nonlinear equations B. Krauskopf, H.M. Osinga, J. Galán-Vioque, Numerical Continuation Methods for Dynamical Systems 2007 Springer The Netherlands
R. Seydel, Practical Bifurcation and Stability Analysis, Interdisciplinary Applied Mathematics, Vol. 5, Springer, New York, 2010.
W.J.F. Govaerts Numerical Methods for Bifurcations of Dynamical Equilibria 2000 Society for Industrial and Applied Mathematics Philadelphia, PA, USA
H. Dankowicz, and F.F. Schilder Recipes for Continuation, Computational Science and Engineering Vol. 11 2013 SIAM Philadelphia, PA, USA
G. Kerschen, M. Peeters, J.C. Golinval, and A.F. Vakakis Nonlinear normal modes, part I: a useful framework for the structural dynamicist Mechanical Systems and Signal Processing 23 1 2009 170 194
Y.V. Mikhlin, and K.V. Avramov Nonlinears normal modes for vibrating mechanical systems. Review of theoretical developments Applied Mechanics Reviews 63 6 2010 060802 060821
K.V. Avramov, and Y.V. Mikhlin Review of applications of nonlinear normal modes for vibrating mechanical systems Applied Mechanics Reviews 65 2 2013 020801
M. Aluko, and H.-C. Chang PEFLOQ: an algorithm for the bifurcational analysis of periodic solutions of autonomous systems Computers & Chemical Engineering 8 6 1984 355 365
M. Holodniok, and M. Kubicek DERPER - an algorithm for the continuation of periodic solutions in ordinary differential equations Journal of Computational Physics 55 2 1984 254 267
C. Franke, and C. Führer Collocation methods for the investigation of periodic motions of constrained multibody systems Multibody System Dynamics 5 2 2001 133 158
E.J. Doedel, R.C. Paffenroth, H.B. Keller, D.J. Dichmann, J. Galán-Vioque, and A. Vanderbauwhede Computation of periodic solutions of conservative systems with application to the 3-body problem International Journal of Bifurcation and Chaos 13 06 2003 1353 1381
F.J. Muñoz Almaraz, E. Freire, J. Galán, E. Doedel, and A. Vanderbauwhede Continuation of periodic orbits in conservative and hamiltonian systems Physica D: Nonlinear Phenomena 181 1-2 2003 1 38
M. Dellnitz, and A. Hohmann A subdivision algorithm for the computation of unstable manifolds and global attractors Numerische Mathematik 75 3 1997 293 317
B. Krauskopf, H.M. Osinga, E.J. Doedel, M.E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, and O. Junge A survey of methods for computing (un)stable manifolds of vector fields International Journal of Bifurcation and Chaos 15 3 2005 763 791
M.E. Henderson Computing invariant manifolds by integrating fat trajectories SIAM Journal on Applied Dynamical Systems 4 4 2005 832 882
J. Guckenheimer, and A. Vladimirsky A fast method for approximating invariant manifolds SIAM Journal on Applied Dynamical Systems 3 3 2004 232 260
A.M. Lyapunov The General Problem of the Stability of Motion 1947 Princeton University Press Princeton, NJ, USA
A. Kelley On the Liapounov subcenter manifold Journal of Mathematical Analysis and Applications 18 3 1967 472 478
R.M. Rosenberg Normal modes of nonlinear dual-mode systems Journal of Applied Mechanics 27 2 1960 263 268
R.M. Rosenberg The normal modes of nonlinear n-degree-of-freedom systems Journal of Applied Mechanics 29 1 1962 7 14
R.M. Rosenberg On nonlinear vibrations of systems with many degrees of freedom Advances in Applied Mechanics 9 1966 155 242
Y.S. Lee, G. Kerschen, A.F. Vakakis, P. Panagopoulos, L. Bergman, and D.M. McFarland Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment Physica D: Nonlinear Phenomena 204 1-2 2005 41 69
S.W. Shaw, and C. Pierre Non-linear normal modes and invariant manifolds Journal of Sound and Vibration 150 1 1991 170 173
S.W. Shaw, and C. Pierre Normal modes for non-linear vibratory systems Journal of Sound and Vibration 164 1993 40
H.A. Ardeh, M.S. Allen, Instantaneous center manifolds and nonlinear normal modes, Proceedings of the ASME 2012 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Chicago, 2012.
H.A. Ardeh, M.S. Allen, Investigating cases of jump phenomenon in a nonlinear oscillatory system, Proceedings of the International Modal Analysis Conference (IMAC), California, USA, 2013.
M. Peeters, R. Viguié, G. Sérandour, G. Kerschen, and J.C. Golinval Nonlinear normal modes, part II: toward a practical computation using numerical continuation techniques Mechanical Systems and Signal Processing 23 1 2009 195 216
R. Arquier, S. Bellizzi, R. Bouc, and B. Cochelin Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes Computers & Structures 84 24-25 2006 1565 1576
J.C. Simo, and N. Tarnow The discrete energy-momentum method, conserving algorithms for nonlinear elastodynamics Zeitschrift für Angewandte Mathematik und Physik 43 5 1992 757 792
J.C. Slater A numerical method for determining nonlinear normal modes Nonlinear Dynamics 10 1 1996 19 30
A.H. Nayfeh, and B. Balachandran Applied Nonlinear Dynamics, Analytical, Computational, and Experimental Methods 2007 Wiley-VCH Verlag GmbH Weinheim, Germany
S.L. Lau, and Y.K. Cheung Amplitude incremental variational principle for nonlinear vibration of elastic systems ASME Journal of Applied Mechanics 28 1981 959 964
T.M. Cameron, and J.H. Griffin An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems Journal of Applied Mechanics 56 1 1989 149 154
B. Cochelin, and C. Vergez A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions Journal of Sound and Vibration 324 1-2 2009 243 262
A. Grolet, and F. Thouverez On a new harmonic selection technique for harmonic balance method Mechanical Systems and Signal Processing 30 2012 43 60
G. Von Groll, and D.J. Ewins The harmonic balance method with arc-length continuation in rotor/stator contact problems Journal of Sound and Vibration 241 2 2001 223 233
A. Lazarus, and O. Thomas A harmonic-based method for computing the stability of periodic solutions of dynamical systems Comptes Rendus Mécanique 338 9 2010 510 517
H.B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, in: P.H. Robinowitz (Ed.), Application of Bifurcation Theory: Proceedings of an Advanced Seminar, 1977.
E. Doedel Nonlinear numerics Journal of Franklin Institute 334B 1997 1049 1073
P. Ribeiro, and M. Petyt Non-linear free vibration of isotropic plates with internal resonance International Journal of Non-Linear Mechanics 35 2 2000 263 278
B. Cochelin A path-following technique via an asymptotic-numerical method Computers & Structures 53 5 1994 1181 1192
B. Cochelin, N. Damil, and M. Potier-Ferry Asymptotic-numerical methods and Pade approximants for non-linear elastic structures International Journal for Numerical Methods in Engineering 37 7 1994 1187 1213
B. Cochelin, N. Damil, and M. Potier-Ferry Méthode Asymptotique Numérique 2007 Hermès - Lavoisier Paris, France (in French)
P. Vannucci, B. Cochelin, N. Damil, and M. Potier-Ferry An asymptotic-numerical method to compute bifurcating branches International Journal for Numerical Methods in Engineering 41 8 1998 1365 1389
S. Baguet, and B. Cochelin On the behaviour of the anm continuation in the presence of bifurcations Communications in Numerical Methods in Engineering 19 6 2003 459 471
B. Cochelin, and M. Medale Power series analysis as a major breakthrough to improve the efficiency of asymptotic numerical method in the vicinity of bifurcations Journal of Computational Physics 236 2013 594 607
R.J. Kuether, L. Renson, T. Detroux, C. Grappasonni, G. Kerschen, and M.S. Allen Nonlinear normal modes, modal interactions and isolated resonance curves Journal of Sound and Vibration 351 2015 299 310
S. Tsakirtzis, P.N. Panagopoulos, G. Kerschen, O. Gendelman, A.F. Vakakis, and L.A. Bergman Complex dynamics and targeted energy transfer in linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments Nonlinear Dynamics 48 3 2006 285 318
G. Kerschen, J.J. Kowtko, D.M. McFarland, L.A. Bergman, and A.F. Vakakis Theoretical and experimental study of multimodal targeted energy transfer in a system of coupled oscillators Nonlinear Dynamics 47 1-3 2006 285 309
F. Georgiades, M. Peeters, G. Kerschen, J.C. Golinval, and M. Ruzzene Modal analysis of a nonlinear periodic structure with cyclic symmetry AIAA Journal 47 4 2009 1014 1025
K.R. Jayaprakash, Y. Starosvetsky, A.F. Vakakis, M. Peeters, and G. Kerschen Nonlinear normal modes and band zones in granular chains with no pre-compression Nonlinear Dynamics 63 3 2011 359 385
G. Kerschen, M. Peeters, J.C. Golinval, and C. Stéphan Nonlinear modal analysis of a full-scale aircraft Journal of Aircraft 50 5 2013 1409 1419
L. Renson, J.P. Noël, and G. Kerschen Complex dynamics of a nonlinear aerospace structure: numerical continuation and normal modes Nonlinear Dynamics 79 2 2015 1293 1309
M. Peeters, G. Kerschen, and J.C. Golinval Dynamic testing of nonlinear vibrating structures using nonlinear normal modes Journal of Sound and Vibration 330 3 2011 486 509
R.J. Kuether, M.S. Allen, Computing nonlinear normal modes using numerical continuation and force appropriation, Proceedings of the ASME 2012 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Chicago, 2012.
F. Wang, A.K. Bajaj, and K. Kamiya Nonlinear normal modes and their bifurcations for an inertially coupled nonlinear conservative system Nonlinear Dynamics 42 2005 33
F. Wang, and A. Bajaj Nonlinear normal modes in multi-mode models of an inertially coupled elastic structure Nonlinear Dynamics 47 1 2007 25 47
T. Detroux, L. Renson, G. Kerschen, The harmonic balance method for advanced analysis and design of nonlinear mechanical system, Proceedings of the International Modal Analysis Conference, Orlando, USA 2014.
B. Zhou, F. Thouverez, and D. Lenoir Essentially nonlinear piezoelectric shunt circuits applied to mistuned bladed disks Journal of Sound and Vibration 333 9 2014 2520 2542
E. Sarrouy, A. Grolet, and F. Thouverez Global and bifurcation analysis of a structure with cyclic symmetry International Journal of Non-Linear Mechanics 46 5 2011 727 737
A. Grolet, and F. Thouverez Free and forced vibration analysis of a nonlinear system with cyclic symmetry: application to a simplified model Journal of Sound and Vibration 331 12 2012 2911 2928
J. Shao, and B. Cochelin Theoretical and numerical study of targeted energy transfer inside an acoustic cavity by a non-linear membrane absorber International Journal of Non-Linear Mechanics 64 2014 85 92
R. Lewandowski Computational formulation for periodic vibration of geometrically nonlinear structures - part 2: numerical strategy and examples International Journal of Solids and Structures 34 15 1997 1949 1964
P. Ribeiro, and M. Petyt Nonlinear vibration of plates by the hierarchical finite element and continuation methods International Journal of Mechanical Sciences 41 4-5 1999 437 459
S. Stoykov, and P. Ribeiro Periodic geometrically nonlinear free vibrations of circular plates Journal of Sound and Vibration 315 3 2008 536 555
S. Stoykov, and P. Ribeiro Nonlinear free vibrations of beams in space due to internal resonance Journal of Sound and Vibration 330 18-19 2011 4574 4595
R. Lewandowski Non-linear free vibrations of beams by the finite element and continuation methods Journal of Sound and Vibration 170 5 1994 577 593
S. Karkar, Numerical Methods for Nonlinear Dynamical Systems. Application to Self-Sustained Oscillations in Musical Instruments, PhD Thesis, Laboratoire de Mécanique et dacoustique, 2012.
S. Karkar, C. Vergez, B. Cochelin, Numerical tools for musical instruments acoustics: analysing nonlinear physical models using continuation of periodic solutions, Proceedings of Acoustics 2012 Nantes, France, 2012.
E.H. Moussi, S. Bellizzi, B. Cochelin, and I. Nistor Nonlinear normal modes of a two degrees-of-freedom piecewise linear system Mechanical Systems and Signal Processing 64-65 2015 266 281
MHI, Root Cause Analysis Report for Tube Wear Identified in the Unit 2 and Unit 3 Steam Generators of San Onofre Nuclear Generating Station, NRC, 2013.
S. Karkar, B. Cochelin, and C. Vergez A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems Journal of Sound and Vibration 333 12 2014 2554 2567
E.J. Doedel, R.C. Paffenroth, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B.E. Oldeman, B. Sandstede, X.J. Wang, Auto2000: Continuation and Bifurcation Software for Ordinary Differential Equations. Available via (http://cmvl.cs.concordia.ca/), 2000.
A. Dhooge, W. Govaerts, and Y.A. Kuznetsov Matcont: a matlab package for numerical bifurcation analysis of odes ACM Transactions on Mathematical Software 29 2 2003 141 164
G. Sérandour, M. Peeters, G. Kerschen, J.C. Golinval, Computation of nonlinear normal modes, part II: numerical continuation in auto, Proceedings of the ENOC, Saint Petersburg, Russia, 2008.
M. Ducceschi, C. Touzé, S. Bilbao, C.J. Webb, Nonlinear dynamics of rectangular plates: investigation of modal interaction in free and forced vibrations. Acta Mechanica (2013) 1-20.
NNMcont - A Matlab Package for the Computation of Nonlinear Normal Modes (http://www.ltas-vis.ulg.ac.be/cmsms/index.php?page=nnm#package), 2014.
M. Jerschl, D. Süß, K. Willner, Numerical continuation methods for the concept of non-linear normal modes, Proceedings of the International Modal Analysis Conference (IMAC), Orlando, USA, 2014.
L. Azrar, B. Cochelin, N. Damil, and M. Potier-Ferry An asymptotic-numerical method to compute the postbuckling behaviour of elastic plates and shells International Journal for Numerical Methods in Engineering 36 8 1993 1251 1277
J.M. Cadou, M. Potier-Ferry, B. Cochelin, and N. Damil Anm for stationary Navier-Stokes equations and with Petrov-Galerkin formulation International Journal for Numerical Methods in Engineering 50 4 2001 825 845
J.M. Cadou, M. Potier-Ferry, and B. Cochelin A numerical method for the computation of bifurcation points in fluid mechanics European Journal of Mechanics - B/Fluids 25 2 2006 234 254
Manlab - An Interactive Path-Following and Bifurcation Analysis Software (http://manlab.lma.cnrs-mrs.fr/), 2012.
I. Charpentier On higher-order differentiation in nonlinear mechanics Optimization Methods and Software 27 2 2011 221 232
T. D Burton Numerical calculation of nonlinear normal modes in structural systems Nonlinear Dynamics 49 3 2007 425 441
D. Laxalde, and M. Legrand Nonlinear modal analysis of mechanical systems with frictionless contact interfaces Computational Mechanics 47 4 2011 469 478
C. Touzé, and M. Amabili 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
F. Blanc, C. Touzé, J.F. Mercier, K. Ege, and A.S. Bonnet Ben-Dhia On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems Mechanical Systems and Signal Processing 36 2 2013 520 539
S. Bellizzi, and R. Bouc A new formulation for the existence and calculation of nonlinear normal modes Journal of Sound and Vibration 287 3 2005 545 569
L. Renson, G. Deliége, and G. Kerschen An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems Meccanica 49 8 2014 1901 1916
J. Carr Applications of Centre Manifold Theory, Applied Mathematical Sciences Vol. 35 1981 Springer-Verlag New York, USA
E. Pesheck, C. Pierre, and S.W. Shaw A new Galerkin-based approach for accurate non-linear normal modes through invariant manifolds Journal of Sound and Vibration 249 5 2002 971 993
E. Pesheck, Reduced Order Modeling of Nonlinear Structural Systems Using Nonlinear Normal Modes and Invariant Manifolds. PhD Thesis, University of Michigan, 2000.
J.-L. Lions Optimal Control of Systems Governed by Partial Differential Equations 1971 Springer-Verlag Berlin, Heidelberg, Germany
A. Ern, and J.-L. Guermond Theory and Practice of Finite Elements, Applied Mathematical Sciences Vol. 159 2004 Springer New York
A.N. Brooks, and T.J.R. Hughes Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations Computer Methods in Applied Mechanics and Engineering 32 1-3 1982 199 259
L. Renson, Nonlinear Modal Analysis of Conservative and Nonconservative Aerospace Structures, PhD Thesis, Aerospace and Mechanical Engineering, Liège, 2014.
S. Bellizzi, and R. Bouc An amplitude-phase formulation for nonlinear modes and limit cycles through invariant manifolds Journal of Sound and Vibration 300 3-5 2007 896 915
D. Noreland, S. Bellizzi, C. Vergez, and R. Bouc Nonlinear modes of clarinet-like musical instruments Journal of Sound and Vibration 324 3-5 2009 983 1002
E.J. Doedel, B. Krauskopf, and H.M. Osinga Global bifurcations of the Lorenz manifold Nonlinearity 19 2006
L. Renson, C. Touzé, G. Kerschen, Computation of damped nonlinear normal modes with internal resonances: a boundary value approach, Proceedings of the ENOC, Vienna, Austria, 2014.
B. Krauskopf, and H.M. Osinga Computing invariant manifolds via the continuation of orbit segments B. Krauskopf, H.M. Osinga, J. Galán-Vioque, Numerical Continuation Methods for Dynamical Systems 2007 Springer The Netherlands
M. Elmegard, Mathematical Modeling and Dimension Reduction in Dynamical Systems, PhD Thesis, 2014.
S. Wiggins Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Applied Mathematical Sciences Vol. 105 1994 Springer-Verlag New York, USA
D. Laxalde, and F. Thouverez Complex non-linear modal analysis for mechanical systems: application to turbomachinery bladings with friction interfaces Journal of Sound and Vibration 322 4-5 2009 1009 1025
D. Laxalde, L. Salles, L. Blanc, F. Thouverez, Non-linear modal analysis for bladed disks with friction contact interfaces, Proceedings of GT2008, ASME Turbo Expo 2008: Power for Land, Sea and Air, Berlin, Germany 2008.
M. Krack, L. Panning-von Scheidt, and J. Wallaschek A method for nonlinear modal analysis and synthesis: application to harmonically forced and self-excited mechanical systems Journal of Sound and Vibration 332 25 2013 6798 6814
D. Jiang, C. Pierre, and S.W. Shaw Large-amplitude non-linear normal modes of piecewise linear systems Journal of Sound and Vibration 272 3-5 2004 869 891
E. Pesheck, C. Pierre, and S.W. Shaw Accurate reduced-order models for a simple rotor blade model using nonlinear normal modes Mathematical and Computer Modelling 33 10-11 2001 1085 1097
M. Legrand, D. Jiang, C. Pierre, and S.W. Shaw Nonlinear normal modes of a rotating shaft based on the invariant manifold method International Journal of Rotating Machinery 10 4 2004 319 335
D. Jiang, C. Pierre, and S.W. Shaw Nonlinear normal modes for vibratory systems under harmonic excitation Journal of Sound and Vibration 288 4-5 2005 791 812
N. Boivin, C. Pierre, and S.W. Shaw Non-linear modal analysis of structural systems featuring internal resonances Journal of Sound and Vibration 182 1995 6
E. Pesheck, N. Boivin, C. Pierre, and S.W. Shaw Nonlinear modal analysis of structural systems using multi-mode invariant manifolds Nonlinear Dynamics 25 2001
D. Jiang, C. Pierre, and S.W. Shaw The construction of non-linear normal modes for systems with internal resonance International Journal of Non-Linear Mechanics 40 5 2005 729 746
M. Krack, L. Panning-von Scheidt, and J. Wallaschek On the computation of the slow dynamics of nonlinear modes of mechanical systems Mechanical Systems and Signal Processing 42 1-2 2014 71 87
M. Krack, L. Panning-von Scheidt, and J. Wallaschek A high-order harmonic balance method for systems with distinct states Journal of Sound and Vibration 332 21 2013 5476 5488
R.J. Kuether, and M.S. Allen A numerical approach to directly compute nonlinear normal modes of geometrically nonlinear finite element models Mechanical Systems and Signal Processing 46 1 2014 1 15
LOCA - Library of Continuation Algorithms (http://www.cs.sandia.gov/LOCA/), 2014.
K. Lust, and D. Roose An adaptive Newton-Picard algorithm with subspace iteration for computing periodic solutions SIAM Journal on Scientific Computing 19 4 1998 1188 1209
D. Bindel, M. Friedman, W. Govaerts, J. Hughes, and Y.A. Kuznetsov Numerical computation of bifurcations in large equilibrium systems in Matlab Journal of Computational and Applied Mathematics 261 2014 232 248
F. Wenneker, P. Tiso, A substructuring method for geometrically nonlinear structures, Proceedings of the International Modal Analysis Conference (IMAC), Orlando, Florida, USA, 2014.
G.I. Cirillo, A. Mauroy, L. Renson, G. Kerschen, R. Sepulchre, Global parametrization of the invariant manifold defining nonlinear normal modes using the Koopman operator, Proceedings of the ASME 2015 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Boston, 2015.
M. Peeters, G. Kerschen, and J.C. Golinval Modal testing of nonlinear vibrating structures based on nonlinear normal modes: experimental demonstration Mechanical Systems and Signal Processing 25 4 2011 1227 1247
J.P. Noël, L. Renson, C. Grappasonni, G. Kerschen, Identification of nonlinear normal modes of engineering structures under broadband forcing, Mechanical Systems and Signal Processing (2015), in press, http://dx.doi.org/10.1016/j.ymssp.2015.04.016.
D.A.W. Barton, and S.G. Burrow Numerical continuation in a physical experiment: investigation of a nonlinear energy harvester Journal of Computational and Nonlinear Dynamics 6 1 2010
D.A.W. Barton, B.P. Mann, and S.G. Burrow Control-based continuation for investigating nonlinear experiments Journal of Vibration and Control 18 4 2012 509 520
E. Bureau, F. Schilder, I. Ferreira Santos, J.J. Thomsen, and J. Starke Experimental bifurcation analysis of an impact oscillator - tuning a non-invasive control scheme Journal of Sound and Vibration 332 22 2013 5883 5897