Craig R., and Bampton M. Coupling of substructures for dynamic analysis. AIAA Journal 6 (1968) 1313-1319
Ewins D.J. Modal Testing: Theory, Practice and Application. second ed. (2000), Research Studies Press Ltd., Hertfordshire
Friswell M.I., and Mottershead J.E. Finite Element Model Updating in Structural Dynamics (1995), Kluwer Academic Publishers, London
S.W. Doebling, C.R. Farrar, M.B. Prime, D.W. Shevitz, Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review, Los Alamos National Laboratory Report LA-13070-MS, 1996.
Kerschen G., Worden K., Vakakis A.F., and Golinval J.C. Past, present and future of nonlinear system identification in structural dynamics. Mechanical Systems and Signal Processing 20 (2006) 505-592
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 29 (1962) 7-14
Rosenberg R.M. On nonlinear vibrations of systems with many degrees of freedom. Advances in Applied Mechanics 9 (1966) 155-242
Rand R. Nonlinear normal modes in two-degree-of-freedom systems. Journal of Applied Mechanics 38 (1971) 561
Rand R. A higher-order approximation for nonlinear normal modes in two-degree-of-freedom systems. International Journal of Non-Linear Mechanics 6 (1971) 545-547
Rand R. A direct method for nonlinear normal modes. International Journal of Non-Linear Mechanics 9 (1974) 363-368
Manevitch L.I., and Mikhlin Y.V. On periodic solutions close to rectilinear normal vibration modes. Prikladnaya Matematika I Mekhanika 36 (1972) 1051-1058
A.F. Vakakis, Analysis and identification of linear and nonlinear normal modes in vibrating systems, Ph.D. Dissertation, California Institute of Technology, 1990.
Caughey T.K., Vakakis A.F., and Sivo J.M. Analytical study of similar normal modes and their bifurcations in a class of strongly nonlinear systems. International Journal of Non-Linear Mechanics 25 (1990) 521-533
Vakakis A.F. Non-similar normal oscillations in a strongly non-linear discrete system. Journal of Sound and Vibration 159 (1992) 341-361
King M.E., and Vakakis A.F. An energy-based formulation for computing nonlinear normal-modes in undamped continuous systems. Journal of Vibration and Acoustics 116 (1994) 332-340
Vakakis A.F., Manevitch L.I., Mikhlin Y.V., Pilipchuk V.N., and Zevin A.A. Normal Modes and Localization in Nonlinear Systems (1996), Wiley, New York
Vakakis A.F. Non-linear normal modes and their applications in vibration theory: an overview. Mechanical Systems and Signal Processing 11 (1997) 3-22
Shaw S.W., and Pierre C. Non-linear normal modes and invariant manifolds. Journal of Sound and Vibration 150 (1991) 170-173
S.W. Shaw, C. Pierre, On nonlinear normal modes, ASME Winter Annual Meeting, 1992.
Shaw S.W., and Pierre C. Normal modes for non-linear vibratory systems. Journal of Sound and Vibration 164 (1993) 85-124
Shaw S.W., and Pierre C. Normal modes of vibration for non-linear continuous systems. Journal of Sound and Vibration 169 (1994) 319-347
Jézéquel L., and Lamarque C.H. Analysis of nonlinear dynamic systems by the normal form theory. Journal of Sound and Vibration 149 (1991) 429-459
Nayfeh A.H., and Nayfeh S.A. Nonlinear normal modes of a continuous system with quadratic nonlinearities. Journal of Vibration and Acoustics 117 (1995) 199-205
Mikhlin Y.V. Normal vibrations of a general class of conservative oscillators. Nonlinear Dynamics 11 (1996) 1-15
Qaisi M.I. Non-linear normal modes of a continuous system. Journal of Sound and Vibration 209 (1998) 561-569
Agnes G.S., and Inman D.J. Performance of nonlinear vibration absorbers for multi-degrees-of-freedom systems using nonlinear normal modes. Nonlinear Dynamics 25 (2001) 275-292
Yabuno H., and Nayfeh A.H. Nonlinear normal modes of a parametrically excited cantilever beam. Nonlinear Dynamics 25 (2001) 65-77
Xie W.C., Lee H.P., and Lim S.P. Nonlinear dynamic analysis of MEMS switches by nonlinear modal analysis. Nonlinear Dynamics 31 (2003) 243-256
Mazzilli C.E.N., Soares M.E.S., and Baracho Neto G.P. Non-linear normal modes of a simply supported beam: continuous system and finite-element models. Computer and Structures 82 (2004) 2683-2691
Gendelman O.V. Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment. Nonlinear Dynamics 37 (2004) 115-128
Lacarbonara W., and Camillacci R. Nonlinear normal modes of structural systems via asymptotic approach. International Journal of Solids and Structures 41 (2004) 5565-5594
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
Sinha S.C., Redkar S., and Butcher E.A. Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds. Journal of Sound and Vibration 284 (2005) 985-1002
Serra Villa C.V., Sinou J.J., and Thouverez F. The invariant manifold approach applied to nonlinear dynamics of a rotor-bearing system. European Journal of Mechanics A/Solids 24 (2005) 676-689
Musienko A.I., and Manevitch L.I. Short wavelength dynamics of the system of nonlinear oscillators coupled by stretched weightless beam. Chaos, Solitons and Fractals 26 (2005) 107-116
Dick A.J., Balachandran B., and Mote C.D. Nonlinear vibration modes in micro-resonator arrays. Smart Structures and Materials 2006: Modeling, Signal Processing, and Control, Proceedings of the SPIE vol. 6166 (2006) 206-217
Pak C.H. On the coupling of non-linear normal modes. International Journal of Non-Linear Mechanics 41 (2006) 716-725
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 (2006) 958-981
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 on Applied Mathematics 66 (2006) 648-679
Bellizzi S., and Bouc R. An amplitude-phase formulation for nonlinear modes and limit cycles through invariant manifolds. Journal of Sound and Vibration 300 (2007) 896-915
Srinil N., and Rega G. Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II: internal resonance activation, reduced-order models and nonlinear normal modes. Nonlinear Dynamics 48 (2007) 253-274
Lenci S., and Rega G. Dimension reduction of homoclinic orbits of buckled beams via the non-linear normal modes technique. International Journal of Non-Linear Mechanics 42 (2007) 515-528
Lacarbonara W., Paolone A., and Vestroni F. Non-linear modal properties of non-shallow cables. International Journal of Non-Linear Mechanics 42 (2007) 542-554
Nayfeh A.H. Nonlinear Interactions: Analytical, Computational and Experimental Methods (2000), Wiley-Interscience, New York
Pierre C., Jiang D., and Shaw S.W. Nonlinear normal modes and their application in structural dynamics. Mathematical Problems in Engineering 10847 (2006) 1-15
Slater J.C. A numerical method for determining nonlinear normal modes. Nonlinear Dynamics 10 (1996) 19-30
E. Pesheck, Reduced-order modeling of nonlinear structural systems using nonlinear normal modes and invariant manifolds, Ph.D. Thesis, University of Michigan, Ann Arbor, 2000.
Jiang D., Pierre C., and Shaw S.W. Large-amplitude non-linear normal modes of piecewise linear systems. Journal of Sound and Vibration 272 (2004) 869-891
Arquier R., Bellizzi S., Bouc R., and Cochelin B. Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes. Computers & Structures 84 (2006) 1565-1576
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
Burton T.D. Numerical calculation of nonlinear normal modes in structural systems. Nonlinear Dynamics 49 (2007) 425-441
E. Doedel, AUTO, Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, 2007.
Seydel R. Practical Bifurcation and Stability Analysis, from Equilibrium to Chaos. second ed. (1994), Springer, Berlin
Argoul P., and Le T.P. Continuous wavelet transform for modal identification using free decay response. Journal of Sound and Vibration 277 (2004) 73-100
Garibaldi L., Ruzzene M., Fasana A., and Piombo B. Identification of non-linear damping mechanisms using the wavelet transform. Mecanique Industrielle et Materiaux 51 (1998) 92-94
Staszewski W.J. Identification of non-linear systems using multi-scale ridges and skeletons of the wavelet transform. Journal of Sound and Vibration 214 (1998) 639-658
Nayfeh A.H., and Nayfeh S.A. On nonlinear modes of continuous systems. Journal of Vibration and Acoustics 116 (1994) 129-136
Nayfeh A.H., Chin C.M., and Nayfeh S.A. On nonlinear normal modes of systems with internal resonance. Journal of Vibration and Acoustics 118 (1996) 340-345
Lacarbonara W., Rega G., and Nayfeh A.H. Resonant non-linear normal modes, part I: analytical treatment for structural one-dimensional systems. International Journal of Non-Linear Mechanics 38 (2003) 851-872
Lacarbonara W., and Rega G. Resonant non-linear normal modes, part II: activation/orthogonality conditions for shallow structural systems. International Journal of Non-Linear Mechanics 38 (2003) 873-887
Guckenheimer J., and Holmes P. Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (1983), Springer, New York
Nayfeh A.H., and Mook D.T. Nonlinear Oscillations (1979), Wiley-Interscience, New York
Vakakis A.F. Inducing passive nonlinear energy sinks in vibrating systems. Journal of Vibration and Acoustics 123 (2001) 324-332
Lee Y.S., Vakakis A.F., Bergman L.A., McFarland D.M., and Kerschen G. Suppression of aeroelastic instability by means of broadband passive TET: part I, theory. AIAA Journal 45 (2007) 693-711
Boivin N., Pierre C., and Shaw S.W. Non-linear modal analysis of structural systems featuring internal resonances. Journal of Sound and Vibration 182 (1995) 336-341
Jiang D., Pierre C., and Shaw S.W. The construction of non-linear normal modes for systems with internal resonance. International Journal of Non-linear Mechanics 40 (2005) 729-746
King M.E., and Vakakis A.F. An energy-based approach to computing resonant nonlinear normal modes. Journal of Applied Mechanics 63 (1995) 810-819
Oueini S., Chin C.M., and Nayfeh A.H. Dynamics of a cubic nonlinear vibration absorber. Nonlinear Dynamics 20 (1999) 283-295
Lewandowski R. Computational formulation for periodic vibration of geometrically nonlinear structures, part I: theoretical background. International Journal of Solids and Structures 34 (1997) 1925-1947
Lewandowski R. Computational formulation for periodic vibration of geometrically nonlinear structures, part II: numerical strategy and examples. International Journal of Solids and Structures 34 (1997) 1949-1964
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
Pak C.H. Synge's concept of stability applied to non-linear normal modes. International Journal of Non-Linear Mechanics 41 (2006) 657-664
Recktenwald G., and Rand R. Stability of strongly nonlinear normal modes. Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1128-1132
Boivin N., Pierre C., and Shaw S.W. Non-linear normal modes, invariance, and modal dynamics approximations of non-linear systems. Nonlinear Dynamics 8 (1995) 315-346
Mazzilli C.E.N., and Baracho Neto O.G.P. Evaluation of non-linear normal modes for finite-element models. Computer and Structures 80 (2002) 957-965
Li X., Ji J.C., and Hansen C.H. Non-linear normal modes and their bifurcation of a two DOF system with quadratic and cubic non-linearity. International Journal of Non-linear Mechanics 41 (2006) 1028-1038
Nayfeh A.H. On direct methods for constructing nonlinear normal modes of continuous systems. Journal of Vibration and Control 1 (1995) 389-430
Touzé C., Thomas O., and Huberdeau A. Asymptotic non-linear normal modes for large-amplitude vibrations of continuous structures. Computers & Structures 82 (2004) 2671-2682
Touzé C., Thomas O., and Chaigne A. Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. Journal of Sound and Vibration 273 (2004) 77-101
Lewandowski R. Application of the Ritz method to the analysis of non-linear free vibrations of beams. Journal of Sound and Vibration 114 (1987) 91-101
Szemplinska-Stupnicka W. Non-linear normal modes and the generalized Ritz method in the problems of vibrations of non-linear elastic continuous systems. International Journal of Non-linear Mechanics 18 (1983) 149-165
Szemplinska-Stupnicka W. The Behavior of Nonlinear Vibrating Systems-Volume II: Advanced Concepts and Application to Multi-degree-of-freedom Systems (1990), Kluwer Academic Publishers, Dordrecht
Benamar R., Bennouna M.M.K., and White R.G. The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures, part I: simply supported and clamped-clamped beams. Journal of Sound and Vibration 149 (1991) 179-195
Burton T.D., and Hamdan M.N. On the calculation of non-linear normal modes in continuous systems. Journal of Sound and Vibration 197 (1996) 117-130
Pak C.H. Nonlinear Normal Mode Dynamics for Two-degree-of-freedom Systems (1999), Inha University Press, Seoul
F. Pérignon, Vibration forcées de structures minces, élastiques, non-linéaires, Ph.D. Thesis, Université de la Méditerrané, Marseille, 2004.
Haterbouch M., and Benamar R. Geometrically nonlinear free vibrations of simply supported isotropic thin circular plates. Journal of Sound and Vibration 280 (2005) 903-924
Bellizzi S., and Bouc R. A new formulation for the existence and calculation of nonlinear normal modes. Journal of Sound and Vibration 287 (2005) 545-569
Keller H.B. Numerical Solution of Two-point Boundary Value Problems (1976), Society of Industrial and Applied Mathematics, Philadelphia
Nayfeh A.H., and Balachandran B. Applied Nonlinear Dynamics: Analytical, Computational and Experimental Method (1995), Wiley, New York
Kerschen G., Kowtko J.J., McFarland D.M., Bergman L.A., and Vakakis A.F. Theoretical and experimental study of multimodal targeted energy transfer in a system of coupled oscillators. Nonlinear Dynamics 47 (2007) 285-309
Wang F.X., Bajaj A.K., and Kamiya K. Nonlinear normal modes and their bifurcations for an inertially coupled nonlinear conservative system. Nonlinear Dynamics 42 (2005) 233-265
Cochelin B., Damil N., and Potier-Ferry M. Asymptotic numerical methods and Padé approximants for nonlinear elastic structures. International Journal of Numerical Methods in Engineering 37 (1994) 1187-1213
R. Arquier, Une méthode de calcul des modes de vibrations non-linéaires de structures, Ph.D. Thesis, Université de la Méditerrané, Marseille, 2007.
Pesheck E., Pierre C., and Shaw S.W. A new Galerkin-based approach for accurate non-linear normal modes through invariant manifolds. Journal of Sound and Vibration 249 (2002) 971-993
Soares M.E.S., and Mazzilli C.E.N. Nonlinear normal modes of planar frames discretised by the finite element method. Computers and Structures 77 (2000) 485-493
Thouverez F. Presentation of the ECL benchmark. Mechanical Systems and Signal Processing 17 (2003) 195-202
Mallat S. A Wavelet Tour of Signal Processing. second ed. (1999), Academic Press, London
Göge D., Böswald M., Füllekrug U., and Lubrina P. Ground vibration testing of large aircraft-state-of-the-art and future perspectives. Proceedings of the International Modal Analysis Conference. Orlando (2007)
C. Touzé, M. Amabili, O. Thomas, Reduced-order models for large-amplitude vibrations of shells including in-plane inertia, Computer Methods in Applied Mechanics and Engineering, 2008, submitted for publication.
Pesheck E., Pierre C., and Shaw S.W. Accurate reduced-order models for a simple rotor blade model using nonlinear normal modes. Mathematical and Computer Modelling 33 (2001) 1085-1097
Apiwattanalunggarn P., Shaw S., and Pierre C. Component mode synthesis using nonlinear normal modes. Nonlinear Dynamics 41 (2005) 17-46
Butcher E.A., and Lu R. Order reduction of structural dynamic systems with static piecewise linear nonlinearities. Nonlinear Dynamics 49 (2007) 375-399
Amabili M., and Touzé C. Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: comparison of POD and asymptotic nonlinear normal modes methods. Journal of Fluids and Structures 23 (2007) 885-903
Castanier M.P., and Pierre C. Modeling and analysis of mistuned bladed disk vibration: status and emerging directions. Journal of Propulsion and Power 22 (2006) 384-396
Vakakis A.F., and Cetinkaya C. Mode localization in a class of multidegree-of-freedom nonlinear-systems with cyclic symmetry. SIAM Journal on Applied Mathematics 53 (1993) 265-282
Aubrecht J., and Vakakis A.F. Localized and non-localized nonlinear normal modes in a multi-span beam with geometric nonlinearities. Journal of Vibration and Acoustics 118 (1996) 533-542
Wang Y., McFarland D.M., Vakakis A.F., and Bergman L.A. Seismic base isolation by nonlinear mode localization. Archive of Applied Mechanics 74 (2005) 387-414
A.F. Vakakis, O. Gendelman, L.A. Bergman, D.M. McFarland, G. Kerschen, Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, Springer, in press.
