Abstract
The objective of this chapter is to introduce nonlinear normal modes (NNMs) to structural dynamicists who are not acquainted with them. Specifically, this chapter describes how the concept of modes can be extended to the nonlinear case. It also describes, in simple terms, the fundamental properties of NNMs, including frequency-energy dependence, harmonics, bifurcation, and stability.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Craig R, Bampton M (1968) Coupling of substructures for dynamic analysis. AIAA J 6:1313–1319
Ewins DJ (2000) Modal testing: theory, practice and application, 2nd edn. Research Studies Press LTD, Hertfordshire
Friswell MI, Mottershead JE (1995) Finite element model updating in structural dynamics. Kluwer Academic Publishers, London
Doebling SW, Farrar CR, Prime MB, Shevitz DW (1996) 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
Kerschen G, Worden K, Vakakis AF, Golinval JC (2006) Past, present and future of nonlinear system identification in structural dynamics. Mech Syst Signal Process 20:505–592
Rosenberg RM (1960) Normal modes of nonlinear dual-mode systems. J Appl Mech 27: 263–268
Rosenberg RM (1962) The normal modes of nonlinear n-degree-of-freedom systems. J Appl Mech 29:7–14
Rosenberg RM (1966) On nonlinear vibrations of systems with many degrees of freedom. Adv Appl Mech 9:155–242
Rand R (1971) Nonlinear normal modes in two-degree-of-freedom systems. Journal of Applied Mechanics 38:561.
Rand R (1971) A higher-order approximation for nonlinear normal modes in two-degree-of-freedom systems. Int J Nonlinear Mech 6:545–547
Rand R (1974) A direct method for nonlinear normal modes. Int J Nonlinear Mech 9:363–368
Manevitch LI, Mikhlin YV (1972) On periodic solutions close to rectilinear normal vibration modes. PMM 36:1051–1058
Vakakis AF (1990) Analysis and identification of linear and nonlinear normal modes in vibrating systems. Ph.D. Dissertation, California Institute of Technology
Caughey TK, Vakakis AF, Sivo JM (1990) Analytical study of similar normal modes and their bifurcations in a class of strongly nonlinear systems. Int J Nonlinear Mech 25:521–533
Vakakis AF (1992) Non-similar normal oscillations in a strongly non-linear discrete system. J Sound Vib 159:341–361
King ME, Vakakis AF (1994) An energy-based formulation for computing nonlinear normal-modes in undamped continuous systems. J Sound Vib 116:332–340
Vakakis AF, Manevitch LI, Mikhlin YV, Pilipchuk VN, Zevin AA (1996) Normal modes and localization in nonlinear systems. John Wiley & Sons, New York
Vakakis AF (1997) Non-linear normal modes and their applications in vibration theory: an overview. Mech Syst Signal Process 11:3–22
Shaw SW, Pierre C (1991) Non-linear normal modes and invariant manifolds. J Sound Vib 150:170–173
Shaw SW, Pierre C (1992) On nonlinear normal modes. ASME Winter Annual Meeting
Shaw SW, Pierre C (1993) Normal modes for non-linear vibratory systems. J Sound Vib 164:85–124
Shaw SW, Pierre C (1994) Normal modes of vibration for non-linear continuous systems. J Sound Vib 169:319–347
Jezequel L, Lamarque CH (1991) Analysis of nonlinear dynamic systems by the normal form theory. J Sound Vib 149:429–459
Yabuno H, Nayfeh AH (2001) Nonlinear normal modes of a parametrically excited cantilever beam. Nonlinear Dyn 25:65–77
Xie WC, Lee HP, Lim SP (2003) Nonlinear dynamic analysis of MEMS switches by nonlinear modal analysis. Nonlinear Dyn 31:243–256
Mazzilli CEN, Soares MES, Baracho Neto GP (2004) Non-linear normal modes of a simply supported beam: continuous system and finite-element models. Comput Struct 82: 2683–2691
Gendelman OV (2004) Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment. Nonlinear Dyn 37:115–128
Lacarbonara W, Camillacci R (2004) Nonlinear normal modes of structural systems via asymptotic approach. Int J Solids Struct 41:5565–5594
Lee YS, Kerschen G, Vakakis AF, Panagopoulos PN, Bergman LA, McFarland DM (2005) Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment. Physica D 204:41–69
Dick AJ, Balachandran B, Mote CD (2006) Nonlinear vibration modes in micro-resonator arrays. In: Smart Structures and Materials 2006: Modeling, Signal Processing, and Control, Proceedings of the SPIE, vol 6166, pp 206–217
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 298:958–981
Srinil N, Rega G (2007) Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II: Internal resonance activation, reduced-order models and nonlinear normal modes. Nonlinear Dyn 48:253–274
Lenci S, Rega G (2007) Dimension reduction of homoclinic orbits of buckled beams via the non-linear normal modes technique. Int J Nonlinear Mech 42:515–528
Lacarbonara W, Paolone A, Vestroni F (2007) Non-linear modal properties of non-shallow cables. Int J Nonlinear Mech 42:542–554
Cochelin B, Vergez C (2009) A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions. J Sound Vib 324:243–262
Laxalde D, Thouverez F, Complex non-linear modal analysis for mechanical systems: application to turbomachinery bladings with friction interfaces. J Sound Vib 322 (2009), 1009–1025.
Kuether RJ, Allen MS (2014) A numerical approach to directly compute nonlinear normal modes of geometrically nonlinear finite element models. Mech Syst Signal Process 46:1–15
Hill TL, Cammarano A, Neild SA, Wagg DJ (2015) Interpreting the forced responses of a two-degree-of-freedom nonlinear oscillator using backbone curves. J Sound Vib 349: 276–288
Renson L, A. Gonzalez-Buelga, Barton DAW, Neild SA, Robust identification of backbone curves using control-based continuation. Journal of Sound and Vibration 367 (2016), 145–158.
Hill TL, Cammarano A, Neild SA, Barton DAW (2017) Identifying the significance of nonlinear normal modes. Proc R Soc A 473:20160789
Scheel M, Peter S, Leine RI, Krack M (2018) A phase resonance approach for modal testing of structures with nonlinear dissipation. J Sound Vib 435:56–73
Ribeiro P, Thomas O (2017) Nonlinear modes of vibration and internal resonances in nonlocal beams. J Comput Nonlinear Dyn 12:031017
Song M, Renson L, Noel JP, Moaveni B, Kerschen G (2018) Bayesian model updating of nonlinear systems using nonlinear normal modes. Struct Control Health Monit 25:e2258
Haller G, Ponsioen S (2016) Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction. Nonlinear Dyn 86:1493–1534
Krack M (2015) Nonlinear modal analysis of nonconservative systems: extension of the periodic motion concept. Comput Struct 154:59–71
Peeters M, Viguié R, Sérandour G, Kerschen G, Golinval JC (2009) Nonlinear normal modes, Part II: toward a practical computation using numerical continuation. Mech Syst Signal Process 23:170–194
Nayfeh AH (2000) Nonlinear interactions: analytical, computational and experimental methods. Wiley-Interscience, New York
SS. Oueini, Chin CM, Nayfeh AH (1999) Dynamics of a cubic nonlinear vibration absorber. Nonlinear Dyn 20:283–295
Noel JP, Renson L, Kerschen G (2014) Complex dynamics of a nonlinear aerospace structure: experimental identification and modal interactions. J Sound Vib 333:2588–2607
Renson L, Noel JP, Kerschen G (2015) Complex dynamics of a nonlinear aerospace structure: numerical continuation and normal modes. Nonlinear Dyn 79:1293–1309
Claeys M, Sinou JJ, Lambelin JP, Todeschini R (2016) Modal interactions due to friction in the nonlinear vibration response of the “Harmony” test structure: experiments and simulations. J Sound Vib 376:131–148
Kerschen G, Peeters M, Golinval JC, Stephan C (2013) Nonlinear modal analysis of a full-scale aircraft. AIAA J Aircr 50:1409–1419
Pak CH (2006) Synge’s concept of stability applied to non-linear normal modes. Int J Nonlinear Mech 41:657–664
Recktenwald G, Rand R (2007) Stability of strongly nonlinear normal modes. Commun Nonlinear Sci Numer Simul 12:1128–1132
King ME, Vakakis AF (1995) An energy-based approach to computing resonant nonlinear normal modes. J Appl Mech 63:810–819
Boivin N, Pierre C, Shaw SW (1995) Non-linear modal analysis of structural systems featuring internal resonances. J Sound Vib 182:336–341
Slater JC (1996) A numerical method for determining nonlinear normal modes. Nonlinear Dyn 10:19–30
Cochelin B, Damil N, Potier-Ferry M (1994) Asymptotic numerical methods and Padé approximants for nonlinear elastic structures. Int J Numer Methods Eng 37:1187–1213
Pérignon F (2004) Vibration forcées de structures minces, élastiques, non-linéaires. Ph.D. Thesis, Université de la Méditerrané, Marseille
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 84:1565–1576
Thouverez F (2003) Presentation of the ECL benchmark. Mech Syst Signal Process 17: 195–202
Pesheck E (2000) Reduced-order modeling of nonlinear structural systems using nonlinear normal modes and invariant manifolds. Ph.D. Thesis, University of Michigan, Ann Arbor
Touzé C, Thomas O, Chaigne A (2004) Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. J Sound Vib 273:77–101
C. Touzé, Amabili M, Thomas O (2007) Reduced-order models for large-amplitude vibrations of shells including in-plane inertia. In: Proceedings of the EUROMECH Colloquium on Geometrically Nonlinear Vibrations, Porto
Peeters M, Kerschen G, Golinval JC (2011) Modal testing of nonlinear vibrating structures based on nonlinear normal modes: experimental demonstration. Mech Syst Signal Process 25:1227–1247
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Society for Experimental Mechanics
About this entry
Cite this entry
Kerschen, G., Vakakis, A.F. (2022). Modal Analysis of Nonlinear Mechanical Systems. In: Allemang, R., Avitabile, P. (eds) Handbook of Experimental Structural Dynamics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4547-0_35
Download citation
DOI: https://doi.org/10.1007/978-1-4614-4547-0_35
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4546-3
Online ISBN: 978-1-4614-4547-0
eBook Packages: EngineeringReference Module Computer Science and Engineering