  We use a DEM simulation and analytical considerations to study the dynamics
of a self-energised object, modelled as a disc, oscillating horizontally within
a two-dimesional bed of denser and smaller particles. We find that, for given
material parameters, the immersed object (IO) may rise, sink or not change
depth, depending on the oscillation amplitude and frequency, as well as on the
initial depth. With time, the IO settles at a specific depth that depends on
the oscillation parameters. We construct a phase diagram of this behaviour in
the oscillation frequency and velocity amplitude variable space. We explain the
observed rich behaviour by two competing effects: climbing on particles, which
fill voids opening under the disc, and sinking due to bed fluidisation. We
present a cavity model that allows us to derive analytically general results,
which agree very well with the observations and explain quantitatively the
phase diagram. Our specific analytical results are the following. (i)
Derivation of a critical frequency, $f_c$, above which the IO cannot float up
against gravity. We show that this frequency depends only on the gravitational
acceleration and the IO size. (ii) Derivation of a minimal amplitude,
$A_{min}$, below which the IO cannot rise even if the frequency is below $f_c$.
We show that this amplitude also depends only on the gravitational acceleration
and the IO size. (iii) Derivation of a critical value, $g_c$, of the IO's
acceleration amplitude, below which the IO cannot sink. We show that the value
of $g_c$ depends on the characteristics of both the IO and the granular bed,as
well as on the initial IO's depth.
