Longitudinal vibrations of arbitrary non-uniform rods

Free and steady-state forced longitudinal vibrations of non-uniform rods are investigated by an iteration method, which results in a series solution. The series obtained are convergent and linearly independent. Its convergence is verified by convergence tests, its linear independence confirmed by the nonzero value of the corresponding Wronski determinant. Then, the solution obtained is an exact one reducible to a classical solution for the case of uniform rods. In order to verify the method, two examples are presented as an application of the proposed method. The results obtained are equivalent to the method in literature. In contrast to the proposed method capable of dealing with arbitrary non-uniform rods in principle, the method in literature is confined to work on special cases.