Exact solutions for the longitudinal vibration of non-uniform rods

Abstract

The objective of this paper is to present exact analytical solutions for the longitudinal vibration of rods with non-uniform cross-section. Using appropriate transformation, the equation of motion of axial 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 a polynomial area vibration and for a sinusoidal rod. The solutions are obtained in terms of special functions such as Bessel and Neumann as well as trigonometric functions. Simple formulas to predict the natural frequencies of non-uniform rods with various end conditions are presented. The natural frequencies of non-uniform rods for these end conditions are calculated, and their dependence on taper is discussed. The governing equation for the problem is the same as that of wave propagation through ducts with non-uniform cross-sections. Therefore solutions presented here can be used to investigate such problems.