The dynamics of a thin elastic helical rod with circular cross section in relaxed state, i.e., the dynamics of a rod with intrinsic curvature and twisting, was discussed. Based on the Kirchhoff's kinetic analogy, the dynamical equations of the elastic rod were expressed by the Euler's angles. The inertial effects of the linear and angular accelerations of the cross section were considered. The stability in spatial and time domain of a helical rod with circular cross section in relaxed state was discussed in the sense of first approximation. We proved that the stability conditions were satisfied in the spatial domain, and in the time domain when the wave number was larger than 1. The propagation of the elastic wave of bending/twisting deformation was discussed, and the relationship between the propagation speed and the wave number was obtained.