Abstract The governing differential equations for out-of-plane vibrations of curved non-uniform beams of constant radius are derived. Two physical parameters are introduced to simplify the analysis. The explicit relations between the flexural displacement, its first three order derivatives and the torsional displacement are derived. With these explicit relations, the two coupled governing characteristic differential equations can be decoupled and reduced to a sixth order ordinary differential equation with variable coefficients in the torsional displacement. It is shown that if the material and geometric properties of the beam are in arbitrary polynomial forms of spatial variable, then exact solutions for the out-of-plane vibrations of the beam can be obtained. The derived explicit relations can also be used to reduce the difficulty in experimental measurements. Finally, the influence of taper ratio, center angle and arc length on the first two natural frequencies of the beams is illustrated.