  The motion of a forced vibro-impacting inclined energy harvester is
investigated in parameter regimes with asymmetry in the number of impacts on
the bottom and top of the device. This motion occurs beyond a grazing
bifurcation, at which alternating top and bottom impacts are supplemented by a
zero velocity impact with the bottom of the device. For periodic forcing, we
obtain semi-analytical expressions for the asymmetric periodic motion with a
ratio of 2:1 for the impacts on the device bottom and top, respectively. These
expressions are derived via a set of nonlinear maps between different pairs of
impacts, combined with impact conditions that provide jump discontinuities in
the velocity. Bifurcation diagrams for the analytical solutions are
complemented by a linear stability analysis around the 2:1 asymmetric periodic
solutions, and are validated numerically. For smaller incline angles, a second
grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For
larger incline angles, period doubling bifurcations precede this bifurcation.
The converted electrical energy per impact is reduced for the asymmetric
motions, and therefore less desirable under this metric.
