Abstract
Nonlinear Normal Modes (NNMs) have a clear conceptual relation to the classical linear normal modes (LNMs), yet they offer a solid theoretical framework for interpreting a wide class of non-linear dynamical phenomena with no linear counterpart. The main difficulty associated with NNMs is that their calculation for large-scale models is expensive, particularly for distributed nonlinearities. Repeated direct time integrations need to be carried out together with extensive sensitivity analysis to reproduce the frequency-energy dependence of the modes of interest.In the present paper, NNMs are computed from a reduced model obtained using a quadratic transformation comprising LNMs and Modal Derivatives (MDs). Previous studies have shown that MDs can capture the essential dynamics of geometrically nonlinear structures and can greatly reduce the computational cost of time integration.A direct comparison with the NNMs computed from another standard reduction technique highlights the capability of the proposed reduction method to capture the essential nonlinear phenomena. The methodology is demonstrated using simple examples with 2 and 4 degrees of freedom.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ahlquist JR, Carreño JM, Climent H, de Diego R, de Alba J (2010) Assessment of nonlinear structural response in A400M GVT. In: Proceedings of the international modal analysis conference, Jacksonville, 2010
Noël JP, Renson L, Kerschen G, Peeters B, Manzato S, Debille J (2013) Nonlinear dynamic analysis of an F-16 aircraft using GVT data. In: Proceedings of the international forum on aeroelasticity and structural dynamics, Bristol, 2013
Renson L, Noël JP, Kerschen G (2015) Complex dynamics of a nonlinear aerospace structure: numerical continuation and normal modes. Nonlinear Dyn. doi:10.1007/s11071-014-1743-0
Rosenberg RM (1960) Normal modes of nonlinear dual-mode systems. J Appl Mech 27(2):263–268
Rosenberg RM (1966) On nonlinear vibrations of systems with many degrees of freedom. Adv Appl Mech 9:155–242
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
Vakakis AF, Manevitch LI, Mlkhlin YV, Pilipchuk VN, Zevin AA (2008) Normal modes and localization in nonlinear systems. Wiley-VCH Verlag GmbH, Weinheim
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
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
Kerschen G, Peeters M, Golinval JC, Stéphan C (2013) Nonlinear modal analysis of a full-scale aircraft. J Aircr 50(5):1409–1419
Idelsohn SR, Cardona A (1985) A reduction method for nonlinear structural dynamic analysis. Comput Methods Appl Mech Eng 49(3):253–279
Tiso P (2011) Optimal second order reduction basis selection for nonlinear transient analysis. In: Modal analysis topics, vol 3. Springer, New York, pp 27–39
Wenneker F, Tiso P (2014) A substructuring method for geometrically nonlinear structures. In:Â Dynamics of coupled structures, vol 1. Springer, Heidelberg
Rutzmoser JB, Rixen DJ, Tiso P (2014) Model order reduction using an adaptive basis for geometrically nonlinear structural dynamics. In: Conference on noise and vibration engineering, Leuven, 20–25 Sept 2014
Acknowledgement
The author L. Renson is a Marie-Curie COFUND Postdoctoral Fellow of the University of Liège, co-funded by the European Union
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 The Society for Experimental Mechanics, Inc.
About this paper
Cite this paper
Sombroek, C., Renson, L., Tiso, P., Kerschen, G. (2016). Bridging the Gap Between Nonlinear Normal Modes and Modal Derivatives. In: Kerschen, G. (eds) Nonlinear Dynamics, Volume 1. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-15221-9_32
Download citation
DOI: https://doi.org/10.1007/978-3-319-15221-9_32
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15220-2
Online ISBN: 978-3-319-15221-9
eBook Packages: EngineeringEngineering (R0)