Abstract: The homogeneous torsional vibration as a special case of the vibrationof a helical thin elastic rod is discussed in this paper. A rod withnoncircular cross section and a rod with circular cross section andintrinsic curvature are under investigation. In the process of homogeneoustorsional vibration all cross sections of the rod have a same torsionalangle, and the configuration of centerline of the rod is not affected by thevibration. It is shown that the torsional vibration is caused by theasymmetry of the cross section and the intrinsic curvature of the rod. Thetorsional vibration of the rod is analogous to the motion of a simplependulum, and there exists an exact solution for the dynamical equation. Thesimilar vibration of an annular rod is a special case when the pitch angleof the helix is equal to zero.