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

TL;DR

This paper develops a smoothed piecewise-linear model of a forced impacting beam, analyzes its bifurcation structure, and compares the results with experimental data, revealing how smoothing affects bifurcations.

Contribution

It introduces a smoothed model derived from first principles and investigates how smoothing influences bifurcations, aligning theoretical results with experiments.

Findings

Smoothing can induce bifurcations in the impact oscillator.

Bifurcation structures depend on the localization of the transition.

Model predictions agree well with experimental measurements.

Abstract

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 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.