Molecular Dynamics Simulations (MD) at Constant-Temperature or Constant-total Energy for the conformational Global-Minimum (GM) of elastin tetrapeptide Boc-Gly-Leu-Gly-Gly-NMe have been performed. The thermalization problem concerning the initial state of Constant-Temperature MD has been solved developing two effective strategies. In the first one, the run starts from the room-temperature state reached by Molecular Dynamics Simulated Annealing (SA). In the second one, one starts from the annealed-state at low-temperature and performs a long constant-low-temperature run until the initial conformer is perfectly equilibrated. Then, the low-temperature equilibrated-state is used as initial state for MD at room-temperature. heuristic criteria on order to define the onset of steady-state have been established monitoring the hystories of collective parameters (e.g., the total energy, temperature, end-to-end distance, etc.) and their amplitude fluctuations. Moreover, the equilibrium between the system and the heat bath is verified analyzing the total linear momentum conservation by the time evolution of center mass velocity. The slow drift of total energy during Constant-total Energy MD has been corrected using a loose coupling between the system and the heat bath. Moreover, we have verified that the roto-translational motions do not affect significantly the properties of molecular vibrations. The librations of peptide unit inside the type II beta-turn [Gly1]C = 0 ... HN[Gly4], previously detected, were confirmed. Large -Gly-Gly- chain motions were identified and modeled as fluctuations occurring between the tetratepeptide GM and the saddle-point corresponding to the transition state of the conversion toward the extended-chain conformation. All these peptide motions could contribute to the elasticity mechanism of elastin.