  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.
