1 Department of Applied Mathematics and Computer Science, Technical University of Denmark2 Mathematics, Department of Applied Mathematics and Computer Science, Technical University of Denmark3 Department of Mechanical Engineering, Technical University of Denmark4 Solid Mechanics, Department of Mechanical Engineering, Technical University of Denmark5 Dynamical Systems, Department of Applied Mathematics and Computer Science, Technical University of Denmark6 University of Auckland7 Dynamical systems, Department of Mathematics, Technical University of Denmark
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 bifurca tions 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 twoparameter continuation of periodic orbits in the driving frequency and/or forcing amplitude. The results are in good agreement with experimental measurements.
Nonlinear Dynamics, 2014, Vol 77
Vibro-impacting beam; Piecewise-linear; Piecewise-smooth; Smoothing; Cantilever beam; Single-degree-of-freedom model