  In this paper, we theoretically propose and investigate a feasible
experimental scheme for realizing the dynamical Casimir effect (DCE) of phonons
in an optomechanical setup formed by a ground-state precooled mechanical
oscillator (MO) inside a Fabry-P{\'e}rot cavity, which is driven by an
amplitude-modulated classical laser field in the dispersive (far-detuned)
regime. The time modulation of the driving field leads to the parametric
amplification of the mechanical vacuum fluctuations of the MO, which results in
the generation of Casimir phonons over time scales longer than the cavity
lifetime. We show that the generated phonons exhibit quadrature squeezing,
bunching effect, and super-Poissonian statistics which are controllable by the
externally modulated laser pump. In particular, we find that the scheme allows
for a perfect squeezing transfer from one mechanical quadrature to another when
the laser frequency is varied from red detuning to blue detuning. Moreover, by
analyzing the effect of the thermal noise of the MO environment, we find that
there exists a critical temperature above which there is no phonon quadrature
squeezing to occur. We also show that in the presence of time modulation of the
driving laser the linewidth narrowing of the displacement spectrum of the MO
can be considered as a signature of the generation of Casimir phonons.
