Sharp corner displacement functions have been well used in the past to accelerate the numerical solutions of two-dimensional free vibration problems, such as plates, to obtain accurate frequencies and mode shapes. The present analysis derives such functions for three-dimensional (3D) bodies of revolution where a sharp boundary discontinuity is present (e.g., a stepped shaft, or a circumferential V notch), undergoing arbitrary modes of deformation. The 3D equations of equilibrium in terms of displacement components, expressed in cylindrical coordinates, are transformed to a new coordinate system having its origin at the vertex of the corner. An asymptotic analysis in the vicinity of the sharp corner reduces the equations to a set of coupled, ordinary differential equations with variable coefficients. By a suitable transformation of variables the equations are simplified to a set of equations with constant coefficients. These are solved, the boundary conditions along the intersecting corner faces are applied, and the resulting eigenvalue problems are solved for the characteristic equations and corner functions.