A Modal Solution for Finite Length Rods with Non-Uniform Area

This paper derives a modal solution to the displacement field of a finite length rod whose area varies with respect to its length. This new 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 and multiplied by a single p-indexed eigenfunction. This equation is then integrated over the interval of the rod. Although the resultant expressions are not orthogonal, they can be written as a set of linear algebraic equations which can be solved to yield the unknown coefficients. Once these are known, the displacement of the system can be calculated. 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, 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. It is shown that this new model is almost two orders of magnitude faster than corresponding finite element analysis.