The Cauchy function and characteristic series were applied to solve the boundary value problem of free transverse vibrations of vertically mounted, elastically supported tapered cantilever beams. A concentrated mass was attached at the same distance from the base. The beams were subjected to universal axial loads - conservative and follow wing tangential forces - and distributed loads along the cantilever length. The general form of characteristic equation was obtained taking into account the shape of the tapered cantilever, elastic foundation and nonhomogeneous material. Bernstein-Kieropian double estimators of natural frequency were calculated based on free coefficients of the characteristic series. Good agreement was obtained between the calculated natural frequency results and the exact values available in the literature.

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