The Frequency of Torsional Vibration of a Tapered Beam

A solution for the equation of torsional vibration of tapered beams has been found in terms of Bessel functions for beams satisfying the following conditions: (a) the cross sections along the span are similar in shape; and (b) the torsional stiffness of a section can be expressed as a power of a linear function of distance along the span. The method of applying the analysis to actual cases has been described. Charts are given from which numerical values can be immediately obtained for most cases of practical importance. The theoretical values of the frequency ratio have been experimentally checked on five beams having different amounts of taper.