Analytical Solutions for The Torsional Vibrations of Variable Cross-Section Rods

Abstract

The objective of this paper is to present exact analytical solutions for the torsional vibration of rods with nonuniform cross-section. Using appropriate transformations the equation of motion of torsional vibration of a rod with varying cross-section is reduced to analytically solvable standard differential equations whose form depends upon the specific area variation. Solutions are obtained for a rod with for a polynomial area variation. The solutions are obtained in terms of special functions such as Bessel and Neumann functions. Simple formulas to predict the natural frequencies of non-uniform rods with various end conditions are presented. The natural frequencies of variable cross-section rods for these end conditions are calculated and their dependence on taper is discussed