  We present an analysis of the dynamics of a nanomechanical resonator coupled
to a superconducting single electron transistor (SSET) in the vicinity of the
Josephson quasiparticle (JQP) and double Josephson quasiparticle (DJQP)
resonances. For weak coupling and wide separation of dynamical timescales, we
find that for either superconducting resonance the dynamics of the resonator is
given by a Fokker-Planck equation, i.e., the SSET behaves effectively as an
equilibrium heat bath, characterised by an effective temperature, which also
damps the resonator and renormalizes its frequency. Depending on the gate and
drain-source voltage bias points with respect to the superconducting resonance,
the SSET can also give rise to an instability in the mechanical resonator
marked by negative damping and temperature within the appropriate Fokker-Planck
equation. Furthermore, sufficiently close to a resonance, we find that the
Fokker-Planck description breaks down. We also point out that there is a close
analogy between coupling a nanomechanical resonator to a SSET in the vicinity
of the JQP resonance and Doppler cooling of atoms by means of lasers.
