Nonlinear Dynamics

, Volume 77, Issue 3, pp 951–966 | Cite as

Bifurcation analysis of a smoothed model of a forced impacting beam and comparison with an experiment

  • M. Elmegård
  • B. Krauskopf
  • H. M. Osinga
  • J. Starke
  • J. J. Thomsen
Original Paper


A piecewise-linear model with a single degree of freedom is derived from first principles for a driven vertical cantilever beam with a localized mass and symmetric stops. The aim is to show that this model constitutes a considerable step toward developing a vibro-impact model that is able to make qualitative and quantitative predictions of the observed dynamics. The resulting piecewise-linear dynamical system is smoothed by a switching function (nonlinear homotopy). For the chosen smoothing function, it is shown that the smoothing can induce bifurcations in certain parameter regimes. These induced bifurcations disappear when the transition of the switching is sufficiently and increasingly localized as the impact becomes harder. The bifurcation structure of the impact oscillator response is investigated via the one- and two-parameter continuation of periodic orbits in the driving frequency and/or forcing amplitude. The results are in good agreement with experimental measurements.


Vibro-impacting beam Piecewise-linear Piecewise-smooth Smoothing  Cantilever beam Single-degree-of-freedom model 



ME thanks the Department of Mathematics at The University of Auckland for its kind hospitality and the Idella Fondation for financial support. JS and JJT acknowledge funding from the Danish Research Council FTP under the project number 09-065890/FTP. The authors thank Emil Bureau, Frank Schilder and Ilmar Santos for fruitful discussions and for providing the data from [10].


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • M. Elmegård
    • 1
  • B. Krauskopf
    • 2
  • H. M. Osinga
    • 2
  • J. Starke
    • 1
  • J. J. Thomsen
    • 3
  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKongens LyngbyDenmark
  2. 2.Department of MathematicsUniversity of AucklandAucklandNew Zealand
  3. 3.Department of Mechanical EngineeringTechnical University of DenmarkKongens LyngbyDenmark

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