Abstract A simple and unified approach is presented for the vibration analysis of a generally supported beam. The flexural displacement of the beam is sought as the linear combination of a Fourier series and an auxiliary polynomial function. The polynomial function is introduced to take all the relevant discontinuities with the original displacement and its derivatives at the boundaries and the Fourier series now simply represents a residual or conditioned displacement that has at least three continuous derivatives. As a result, not only is it always possible to expand the displacement in a Fourier series for beams with any boundary conditions, but also the solution converges at a much faster speed. The reliability and robustness of the proposed technique are demonstrated through numerical examples.