  We study theoretically dynamics in a Josephson junction coupled to a
mechanical resonator looking at the signatures of the resonance in d.c.
electrical response of the junction. Such a system can be realized
experimentally as a suspended ultra-clean carbon nanotube brought in contact
with two superconducting leads. A nearby gate electrode can be used to tune the
junction parameters and to excite mechanical motion. We augment theoretical
estimations with the values of setup parameters measured in the samples
fabricated.
  We show that charging effects in the junction give rise to a mechanical force
that depends on the superconducting phase difference. The force can excite the
resonant mode provided the superconducting current in the junction has
oscillating components with a frequency matching the resonant frequency of the
mechanical resonator. We develop a model that encompasses the coupling of
electrical and mechanical dynamics. We compute the mechanical response (the
effect of mechanical motion) in the regime of phase bias and d.c. voltage bias.
We thoroughly investigate the regime of combined a.c. and d.c. bias where
Shapiro steps are developed and reveal several distinct regimes characteristic
for this effect. Our results can be immediately applied in the context of
experimental detection of the mechanical motion in realistic superconducting
nano-mechanical devices.
