Abstract
In this paper, nonlinear resonances in a coupled shaker-beam-top mass system are investigated both numerically and experimentally. The imperfect, vertical beam carries the top mass and is axially excited by the shaker at its base. The weight of the top mass is below the beam’s static buckling load. A semi-analytical model is derived for the coupled system. In this model, Taylor-series approximations are used for the inextensibility constraint and the curvature of the beam. The steady-state behavior of the model is studied using numerical tools. In the model with a single beam mode, parametric and direct resonances are found, which affect the dynamic stability of the structure. The model with two beam modes not only shows an additional second harmonic resonance, but also reveals some extra small resonances in the low-frequency range, some of which can be identified as combination resonances. The experimental steady-state response is obtained by performing a (stepped) frequency sweep-up and sweep-down with respect to the harmonic input voltage of the amplifier-shaker combination. A good correspondence between the numerical and experimental steady-state responses is obtained.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Nayfeh, A., Pai, P.: Linear and Nonlinear Structural Mechanics. Wiley-VCH, Weinheim (2004)
Virgin, L.: Vibration of Axially Loaded Structures. Cambridge University Press, New York (2007)
Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, Cambridge (2008)
Zavodney, L., Nayfeh, A.: The non-linear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment. Int. J. Non-Linear Mech. 24(2), 105–125 (1989)
Yabuno, H., Ide, Y., Aoshima, N.: Nonlinear analysis of a parametrically excited cantilever beam (effect of the tip mass on stationary response). JSME Int. J. Ser. B Fluids Therm. Eng. 41(3), 555–562 (1998)
Son, I.-S., Uchiyama, Y., Lacarbonara, W., Yabuno, H.: Simply supported elastic beams under parametric excitation. Nonlinear Dyn. 53, 129–138 (2008)
Winterflood, J., Barber, T., Blair, D.: Using Euler buckling springs for vibration isolation. Class. Quantum Gravity 19(7), 1639–1645 (2002)
Virgin, L., Davis, R.: Vibration isolation using buckled struts. J. Sound Vib. 260, 965–973 (2003)
Mallon, N., Fey, R., Nijmeijer, H.: Dynamic stability of a base-excited thin beam with top mass. In: Proc. of the 2006 ASME IMECE, Nov. 5-10, Paper 13148, pp. 1–10, Chicago, IL (2006)
Mettler, E.: Dynamic buckling. In: Flügge, W. (ed.) Handbook of Engineering Mechanics. McGraw-Hill, New York (1962)
Gibson, C.: Elementary Geometry of Differentiable Curves: An Undergraduate Introduction. Cambridge University Press, Cambridge (2001)
Anderson, T., Nayfeh, A., Balachandran, B.: Experimental verification of the importance of the nonlinear curvature in the response of a cantilever beam. J. Vib. Acoust. 118(1), 21–27 (1996)
Yabuno, H., Okhuma, M., Lacarbonara, W.: An experimental investigation of the parametric resonance in a buckled beam. In: Proceedings of the ASME DETC’03, pp. 2565–2574, Chicago, IL (2003)
Xu, X., Pavlovskaia, E., Wiercigroch, M., Romeo, F., Lenci, S.: Dynamic interactions between parametric pendulum and electro-dynamical shaker. Z. Angew. Math. Mech. 87(2), 172–186 (2007)
McConnell, K.: Vibration Testing, Theory and Practice. Wiley, New York (1995)
Mallon, N.: Dynamic stability of thin-walled structures: a semi-analytical and experimental approach. Ph.D. thesis, Eindhoven University of Technology (2008)
Preumont, A.: Mechatronics, Dynamics of Electromechanical and Piezoelectric Systems. Springer, Berlin (2006)
Atluri, S.: Nonlinear vibrations of a hinged beam including nonlinear inertia effects. J. Appl. Mech. 40(1), 121–126 (1973)
Noijen, S., Mallon, N., Fey, R., Nijmeijer, H., Zhang, G.: Periodic excitation of a buckled beam using a higher order semi-analytic approach. Nonlinear Dyn. 50(1–2), 325–339 (2007)
Kraaij, C.: A semi-analytical buckling approach: modeling and validation. Tech. Rep. DCT 2008.095, Eindhoven University of Technology (2008)
Verbeek, G., de Kraker, A., van Campen, D.: Nonlinear parametric identification using periodic equilibrium states. Nonlinear Dyn. 7, 499–515 (1995)
Doedel, E., Paffenroth, R., Champneys, A., Fairgrieve, T., Kuznetsov, Y., Oldeman, B., Sandstede, B., Wang, X.: AUTO97: Continuation and bifurcation software for ordinary differential equations (with HOMCONT), Technical Report, Concordia University (1998)
Thomsen, J.: Vibrations and Stability; Advanced Theory, Analysis, and Tools, 2nd edn. Springer, Berlin (2003)
Zaretzky, C., da Silva, M.C.: Experimental investigation of non-linear modal coupling in the response of cantilever beams. J. Sound Vib. 174(2), 145–167 (1994)
Ribeiro, P., Carneiro, R.: Experimental detection of modal interactions in the non-linear vibration of a hinged-hinged beam. J. Sound Vib. 277(4–5), 943–954 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by the Dutch Technology Foundation STW, Applied Science Division of NWO and the Technology Programme of the Ministry of Economic Affairs (STW project EWO.5792).
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Fey, R.H.B., Mallon, N.J., Kraaij, C.S. et al. Nonlinear resonances in an axially excited beam carrying a top mass: simulations and experiments. Nonlinear Dyn 66, 285–302 (2011). https://doi.org/10.1007/s11071-011-9959-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-9959-8