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The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pastemak foundations are investigated by the interpolating matrix method.The research is executed in view of a three-parameter foundation which includes the effects of the Winkler coefficient,Pastak coefficient and damping coefficient of the elastic medium.The goving equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pastak foundations are transformed into ordinary differential equations with variable coefficients in light of the bending rotation angle and transverse displacement.All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time,and a good convergence and accuracy of the proposed method is verified through two numerical examples.The influences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with different taper ratios are analyzed.A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.