Publications on the vibration of a Euler–Bernoulli beam carrying thin discs at the ends are available. In this paper the thin discs are replaced with rigid bodies, whose axial width is included in the analysis. The centre of mass of the bodies is assumed to be on the beam axis but away from the beam end. Among the boundary conditions considered are: the classical clamped, pinned, sliding of free; the lateral translation and rotation of an unrestrained rigid body; the translation and/or rotation of a restrained rigid body, and other special cases. The frequency equations and the first three frequency parameters are tabulated for several sets of the system parameters and selected combinations of 10 boundary conditions