EXACT-SOLUTIONS FOR THE LONGITUDINAL VIBRATION OF NONUNIFORM RODS

The objective of this paper is to present exact analytical solutions f or the longitudinal vibration of rods with non-uniform cross-section. Using appropriate transformations, the equation of motion of axial vib ration of a rod with varying cross-section is reduced to analytically solvable standard differential equations whose form depends upon the s pecific area variation. Solutions are obtained for a rod with a polyno mial area variation and for a sinusoidal rod. The solutions are obtain ed in terms of special functions such as Bessel and Neumann as well as trignometric functions. Simple formulas to predict the natural freque ncies of non-uniform rods with various end conditions are presented. T he natural frequencies of non-uniform rods for these end conditions ar e calculated, and their dependence on taper is discussed. The governin g equation for the problem is the same as that of wave propagation thr ough ducts with non-uniform cross-sections. Therefore solutions presen ted here can be used to investigate such problems. (C) 1997 Academic P ress Limited.