Torsional Vibrations and Design Charts for Hollow Shafts with Variable Cross Section 891149
Shafts with variable cross section are encountered in many applications such as engine crankshafts, ship and aircraft propellers, turbomachinery, and bearing housings. The present study investigates torsional vibration characteristics of elastic homogeneous nonprismatic hollow shafts. Integrating the wave-propagation equation, the spatial angle of twist is expressed as a linear combination of Bessel functions. The natural frequencies of vibration are obtained from the zeros of the Bessel functions that result upon satisfying the appropriate end conditions. Three different types of end conditions are considered: both ends fixed, both ends free, and one free-one fixed end. A design chart has been developed to determine the natural frequency of a nonprismatic hollow shaft upon specifying its geometric profile, end dimensions, and end conditions. By comparing the natural frequencies of a nonprismatic shaft and its equivalent prismatic hollow shaft, it was found that the error associated with neglecting the variation in the shaft cross section across its axis is pronounced for the case of one end free-one end fixed. The error becomes a minimum when the ratio between the inner radius and outer radius at x=0 is a minimum.