We consider an elastic helical medium whose tensor stiffness twirls uniformly along the helix axis. We are interested in analyzing the band structure when the whole material is externally forced to rotate around the helix axis to a fixed constant frequency. Departing from a general dynamic description of the elastic phenomena, we establish a set of equations for the displacement vector and the stress tensor. These equations allow us to calculate the band structure parametrized by the externally imposed rotating frequency. We find that the band structure strongly depends on the rotation frequency, and we show that backward and forward modes propagate differently, particularly for the longitudinal and right-polarized modes.