  We develop a theory to address the non-equilibrium dynamics of phonons in a
one-dimensional trapped ion system. We elaborate our earlier results obtained
in Phys. Rev. Lett. {\bf 111}, 170406 (2013) to chart out the mechanism of
dynamics-induced cooling and entanglement generation between phonons in these
systems when subjected to a linear ramp protocol inducing site-specific tuning
of on-site interactions between the phonons. We further extend these studies to
non-linear ramps and periodic drive protocols and identify the optimal ramp
protocol for minimal cooling and entanglement generation time. We qualitatively
address the effect of noise arising out of fluctuation of the intensity of the
laser used to generate entanglement and provide a detailed discussion of a
realistic experimental setup which may serve as a test bed for our theory.
