On analytical solutions of vibrations of rods with variable cross sections

Analytical solutions of several rods whose cross-sections vary in the axial direction are considered. The analysis in this paper uses two transformations that exist in the literature to help transform the equation of motion of the rod into a form similar to that of the one-dimensional Schroedinger equation where the shape of the cross-section of the rod is governed by a potential function that satisfies a second order differential equation. By solving this second order differential equation, it is possible to obtain a class of shapes that have a common form of solution that are determined from the well known solution of the Schroedinger equation. There are very few analytical solutions available in the literature especially when the area of cross-section varies. In this paper, it is shown that the set of available solutions for variable cross-sectional rods are particular cases of the general analysis. In addition, analytical solutions to several new variable cross-sections are considered. This paper also considers an existing transformation that transforms the Sturm-Liouville equation to a particular form of the equation of motion under consideration. So, tracing the analysis backwards, for any second order differential equation for which existence of solutions are guaranteed, and that can be reduced in the above manner, it is possible to determine the form of the cross-section of the rod. In this paper, several shapes of the cross-section are investigated when the equation of motion is described by special functions, the Legendre, Hermite and Laugurre functions. The analysis given here is a general one and is applicable when the equation of motion is described by other special functions. © 1983.