Closed-form solutions for the free longitudinal vibration of inhomogeneous rods

This paper aims at presenting a family of exact solutions for the longitudinal vibration of variable area rods. Area variations that give solutions in terms of the confluent hypergeometric function are being sought for and the governing differential equation is appropriately reduced to the confluent hypergeometric differential equation, using a generic transformation. The eigenfrequencies of rods with certain area variations, subjected to classical boundary conditions, are obtained and the parametric space of the solutions obtained is studied. These solutions are also highly useful in other topics of study such as torsional vibration of rods and wave propagation in ducts with variable cross-sectional areas, since the governing differential equations are very similar.