  We propose a very accurate computational scheme for the dynamics of a
classical oscillator coupled to a molecular junction driven by a finite bias,
including the finite mass effect. We focus on two minimal models for the
molecular junction: Anderson-Holstein (AH) and two-site Su-Schrieffer-Heeger
(SSH) models. As concerns the oscillator dynamics, we are able to recover a
Langevin equation confirming what found by other authors with different
approaches and assessing that quantum effects come from the electronic
subsystem only. Solving numerically the stochastic equation, we study the
position and velocity distribution probabilities of the oscillator and the
electronic transport properties at arbitrary values of electron-oscillator
interaction, gate and bias voltages. The range of validity of the adiabatic
approximation is established in a systematic way by analyzing the behaviour of
the kinetic energy of the oscillator. Due to the dynamical fluctuations, at
intermediate bias voltages, the velocity distributions deviate from a gaussian
shape and the average kinetic energy shows a non monotonic behaviour. In this
same regime of parameters, the dynamical effects favour the conduction far from
electronic resonances where small currents are observed in the infinite mass
approximation. These effects are enhanced in the two-site SSH model due to the
presence of the intermolecular hopping t. Remarkably, for sufficiently large
hopping with respect to tunneling on the molecule, small interaction strengths
and at intermediate bias (non gaussian regime), we point out a correspondence
between the minima of the kinetic energy and the maxima of the dynamical
conductance.
