Abstract
In this paper, the phenomenon of natural frequencies shifting due to the nonlinear stiffness effects from membrane stress is studied using a nonlinear reduced order model based on backbone curves. The structure chosen for study in this paper is a rectangular plate with a pinned constraint along all edges. To analytically explore the frequency varying phenomenon, a four nonlinear-mode based reduced-order model that contains both single-mode and coupled-mode nonlinear terms is derived. The process of deriving the reduced order model is based on a normal form transformation, combined with a Galerkin type decomposition of the governing partial differential equation of the plate. This allows a low number of ordinary differential equations to be obtained, which in turn can be used to derive backbone curves that relate directly to the nonlinear normal modes (NNMs). The frequency shifting is then investigated relative to the backbone curves. Modal interactions, caused by nonlinear terms are shown to cause the frequency shifts. In the final part of the paper, an attempt is made to quantify the frequency shifting due to different nonlinear effects.
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References
Chia, C.-Y.: Nonlinear Analysis of Plates. McGraw-Hill, New York (1980)
Mignolet, M.P., Przekop, A., Rizzi, S.A., Spottswood, S.M.: A review of indirect/non-intrusive reduced order modeling of nonlinear geometric structures. J. Sound Vib. 332 (10), 2437–2460 (2013)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)
Nash, M.: Nonlinear structural dynamics by finite element model synthesis. PhD thesis, Imperial College London, University of London (1978)
Shi, Y., Mei, C.: A finite element time domain modal formulation for large amplitude free vibrations of beams and plates. J. Sound Vib. 193 (2), 453–464 (1996)
Wagg, D., Neild, S.: Nonlinear Vibration with Control: For Flexible and Adaptive Structures. Springer, Berlin (2014)
Muravyov, A.A., Rizzi, S.A.: Determination of nonlinear stiffness with application to random vibration of geometrically nonlinear structures. Commun. Strateg. 81 (15), 1513–1523 (2003)
McEwan, M.I., Wright, J.R., Cooper, J.E., Leung, A.Y.T.: A finite element/modal technique for nonlinear plate and stiffened panel response prediction. In: Proceedings of the 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit Technical Papers, pp. 3061–3070 (2001)
McEwan, M.I., Wright, J.R., Cooper, J.E., Leung, A.Y.T.: A combined modal/finite element analysis technique for the dynamic response of a non-linear beam to harmonic excitation. J. Sound Vib. 243 (4), 601–624 (2001)
Hollkamp, J.J., Gordon, R.W.: Reduced-order models for nonlinear response prediction: implicit condensation and expansion. J. Sound Vib. 318 (4), 1139–1153 (2008)
Lewandowski, R.: On beams membranes and plates vibration backbone curves in cases of internal resonance. Meccanica 31 (3), 323–346 (1996)
Touzé, C., Thomas, O., Chaigne, A.: Asymmetric non-linear forced vibrations of free-edge circular plates. Part 1: theory. J. Sound Vib. 258 (4), 649–676 (2002)
Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, Cambridge (2008)
Pierre, C., Jiang, D., Shaw, S.: Nonlinear normal modes and their application in structural dynamics. Math. Probl. Eng. 2006, 1–15 (2006)
Touzé, C., Amabili, M.: Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modelling of harmonically forced structures. J. Sound Vib. 298 (4), 958–981 (2006)
Kerschen, G., Peeters, M., Golinval, J.-C., Vakakis, A.F.: Nonlinear normal modes, Part I: a useful framework for the structural dynamicist. Mech. Syst. Signal Process. 23 (1), 170–194 (2009)
Liu, X., Cammarano, A., Wagg, D.J., Neild, S.A.: A study of the modal interaction amongst three nonlinear normal modes using a backbone curve approach. In: Nonlinear Dynamics, vol. 1, pp. 131–139. Springer, Berlin (2016)
Liu, X., Cammarano, A., Wagg, D.J., Neild, S.A., Barthorpe, R.J.: Nonlinear modal interaction analysis for a three degree-of-freedom system with cubic nonlinearities. In: Nonlinear Dynamics, vol. 1, pp. 123–131. Springer, Berlin (2016)
Liu, X., Cammarano, A., Wagg, D.J., Neild, S.A., Barthorpe, R.J.: N-1 modal interactions of a three-degree-of-freedom system with cubic elastic nonlinearities. Nonlinear Dyn. 83 (1–2), 497–511 (2016)
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Liu, X., Wagg, D.J., Neild, S.A. (2017). An Explanation for Why Natural Frequencies Shifting in Structures with Membrane Stresses, Using Backbone Curve Models. In: Kerschen, G. (eds) Nonlinear Dynamics, Volume 1. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-54404-5_2
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DOI: https://doi.org/10.1007/978-3-319-54404-5_2
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