Torsional vibrations of a shaft with non-uniform cross section

A solution is presented for the equation governing the propagation of torsional waves in shafts with variable cross sections. The solution permits one to treat several types of profiles, and in particular to compute the natural frequencies of an axisymmetrical shaft with various kinds of end conditions. Computed frequencies for three kinds of end conditions are compared with those of an equivalent cylindrical shaft, particularly with respect to the dependence on the geometrical characteristics.