The axisymmetric vibrational characteristics of an elastic circular rod of finite length with stress‐free boundaries are analytically determined, and compared to approximate solutions. The solution approach consists of choosing a series of functions which term by term satisfy the governing equations and the boundary conditions on the shear stress. The boundary conditions on the axial and radial stress are satisfied by an orthogonalization procedure. This method yields an infinite eigenvalue matrix, the coefficients of which are transcendental functions of the frequency. It is found that the procedure converges and that sufficient accuracy is achieved, with truncation to a 20×20 eigenvalue matrix. Comparison is made of these results with the Pochhammer‐Chree solutions which do not permit satisfaction of the boundary conditions on the ends of the rod. This comparison shows many similarities, but also some dissimilarities. Comparisons are also made with the lowest mode of the free‐plate solution and the plane stress solution for very short rods, and with the one‐dimensional rod solution for long rods.