Abstract
A structure undergoing large amplitude deformations can exhibit nonlinear behavior which is not predicted by traditional linear theories. Structures with some initial curvature offer an additional complication due to buckling and snap through phenomena, and can exhibit softening, hardening and, internal resonance. As a structure transitions into a region of nonlinear response, a structure’s nonlinear normal modes (NNMs) can provide insight into the forced responses of the nonlinear system. Mono-harmonic excitations can often be used to experimentally isolate a dynamic response in the neighborhood of a single NNM. This is accomplished with an extension of the modal indicator function and force appropriation to ensure the dynamic response of the structure is on the desired NNM. This work explores these methods using two structures: a nominally-flat beam and a curved axi-symmetric plate. Single-point force appropriation is used by manually tuning the excitation frequency and amplitude until the mode indicator function is satisfied for the fundamental harmonic. The results show a reasonable estimate of the NNM backbone, the occurrence of internal resonance, and couplings between the underlying linear modes along the backbone.
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References
Ewins DJ (2000) Modal analysis theory, practice, and application, 2nd edn. Research Studies Press LTD., Baldock, Hertfordshire, England
Allemang RJ, Brown DL (1998) A unified matrix polynomial approach to modal identification. J Sound Vib 211:301–322
Lewis RC, Wrisley DL (1950) A system for the excitation of pure natural modes of complex structure. J Aeronaut Sci (Institute of the Aeronautical Sciences) 17:705–722, 1950/11/01
Fraeijs de Veubeke B (1956) A variational approach to pure mode excitation based on characteristic phase lag theory. North Atlantic Treaty Organization, Advisory Group for Aerospace Research and Development, Paris
Wright JR, Cooper JE, Desforges MJ (1999) Normal mode force appropriation – theory and application. Mech Syst Signal Process 13:217–240
Rosenberg RM (1960) Normal modes of nonlinear dual-mode systems. J Appl Mech 27:263–268
Vakakis AF, Manevitch LI, Mikhlin YV, Pilipchuk VM, Zeven AA (1996) Normal modes and localization in nonlinear systems. Wiley, New York
Kerschen G, Peeters M, Golinval JC, Vakakis AF (2009) Nonlinear normal modes, part I: a useful framework for the structural dynamicist. Mech Syst Signal Process 23:170–194
Kerschen G, Worden K, Vakakis AF, Golinval J-C (2006) Past, present and future of nonlinear system identification in structural dynamics. Mech Syst Signal Process 20:505–592
Worden K, Tomlinson GR (2001) Nonlinearity in structural dynamics: detection, identification, and modeling. Institute of Physics, Bristol/Philadelphia
Atkins PA, Wright JR, Worden K (2000) An extension of force appropriation to the identification of non-linear multi-degree of freedom systems. J Sound Vib 237:23–43
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
Peeters M, Kerschen G, Golinval JC (2011) Dynamic testing of nonlinear vibrating structures using nonlinear normal modes. J Sound Vib 330:486–509
Peeters M, Kerschen G, Golinval JC, Stéphan C, Lubrina P (2011) Nonlinear normal modes of a full-scale aircraft. In: Proulx T (ed) Modal analysis topics, vol 3. Springer, New York, pp 223–242
Ehrhardt D, Harris R, Allen M (2014) Numerical and experimental determination of nonlinear normal modes of a circular perforated plate. International modal analysis conference XXXII, Orlando, pp 239–251
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
Kuether RJ, Allen MS (2012) Computing nonlinear normal modes using numerical continuation and force appropriation, presented at the 24th conference on mechanical vibration and noise
Gordon RW, Hollkamp JJ (2011) Reduced-order models for acoustic response prediction. Air Force Research Laboratory, AFRL-RB-WP-TR-2011-3040, Dayton
Hollkamp JJ, Gordon RW (2008) Reduced-order models for nonlinear response prediction: implicit condensation and expansion. J Sound Vib 318:1139–1153
Virgin LN (2007) Vibration of axially loaded structures. Cambridge University Press, Cambridge, UK
Virgin LN (2000) Introduction to experimental nonlinear dynamics: a case study in mechanical vibration. Cambridge University Press, Cambridge, UK
Gordon RW, Hollkamp JJ, Spottswood SM (2003) Non-linear response of a clamped-clamped beam to random base excitation, presented at the VIII international conference on recent advances in structural dynamics, Southampton
Ehrhardt D, Yang S, Beberniss T, Allen M (2014) Mode shape comparison using continuous-scan laser Doppler vibrometry and high speed 3D digital image correlation. In: International modal analysis conference XXXII, Orlando, pp 321–331
Kuether RJ, Deaner B, Allen MS, Hollkamp JJ (2014) Evaluation of geometrically nonlinear reduced order models with nonlinear normal modes. AIAA J, (Submitted)
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© 2016 The Society for Experimental Mechanics, Inc.
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Ehrhardt, D.A., Allen, M.S., Beberniss, T.J. (2016). Measurement of Nonlinear Normal Modes Using Mono-harmonic Force Appropriation: Experimental Investigation. 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_22
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DOI: https://doi.org/10.1007/978-3-319-15221-9_22
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