The study of partially embedded piles is an important topic in civil engineering due to its use in several engineering fields. The aim of the present work is to derive an exact solution for the natural frequencies and vibration modes of these piles and, based on these results, to analyze the nonlinear behavior of the column-foundation system. In order to study the free vibration problem, the energy functional of the system is obtained, allowing the derivation of the partial differential equations of motion for the buried and unburied parts of the pile. The possible solutions of these equations are then obtained as a function of a set of non-dimensional parameters characteristic of the problem. The consideration of the geometric nonlinearity of the beam leads to a set of nonlinear differential equations. The solution of these equations is obtained by employing the Ritz method to discretize the column in space and the methods of Galerkin- Urabe and Harmonic Balance to derive an approximate solution for the resulting equations of motion. Based on these results, the influence of the geometrical and physical parameters of the column-foundation system on the free and forced non-linear vibrations of the pile is analyzed.