The low-frequency dynamics (<20 meV) of pure and sodium-doped trans polyacetylene are investigated using a combination of incoherent neutron scattering spectroscopy and molecular dynamics simulations. The simulations are performed using a molecular mechanics potential function and including explicitly the three-dimensional crystal environments of the molecules. Both the experiments and the simulations indicate that doping results in a marked change in the vibrational density of states of the polyene chains in the direction perpendicular to the chain axes, a broad minimum appearing at similar to 16 meV. This spectral region is dominated by intramolecular torsional displacements. The results also suggest that the mean-square displacements of the polyacetylene atoms become more isotropic on doping. The contributions of various rigid-body motions to the simulation-derived mean-square displacements and vibrations are described.