A modal solution for finite length rods with non-uniform area

Abstract

This talk derives a modal solution to the displacement field of a finite length rod whose area is varying with respect to its length. This method facilitates a solution to any problem where the area and derivative of the area can be represented as analytical functions. The problem begins by writing the longitudinal displacement of the non-uniform area rod as a series of indexed coefficients multiplied by the eigenfunctions of the uniform area rod. This series solution is inserted into the non-uniform area rod equation, multiplied by a single p-indexed eigenfunction and integrated over the interval of the rod. The resultant expressions can be written as a set of linear algebraic equations and this yields a solution to the displacement of the system. Five example problems are included: the first one has a non-uniform area that corresponds with a known analytical solution, the second has an area that can be represented by a Fourier series, and the third and fourth have areas that do not have a known analytical solution and the fifth is a generic second order non-constant coefficient differential equation. Four of these problems are verified with other methods. Convergence of the series solution is discussed.