  We propose two measurement-based schemes to cool a nonlinear mechanical
resonator down to energies close to that of its ground state. The protocols
rely on projective measurements of a spin degree of freedom, which interacts
with the resonator through a Jaynes-Cummings interaction. We show the
performance of these cooling schemes, that can be either concatenated -- i.e.
built by repeating a sequence of dynamical evolutions followed by projective
measurements -- or single-shot. We characterize the performance of both cooling
schemes with numerical simulations, and pinpoint the effects of decoherence and
noise mechanisms. Due to the ubiquity and experimental relevance of the
Jaynes-Cummings model, we argue that our results can be applied in a variety of
experimental setups.
