<p>This paper presents a novel viscoelastic foundation model for the first time, for the damped vibration analysis of functionally graded sandwich beams. The proposed viscoelastic foundation is established by two layers: the visco-Winkler layer and the visco-Pasternak layer. Each layer involves two coefficients, including the coefficient of elasticity and the coefficient of viscosity. Both hard- and soft-core functionally graded sandwich beams are considered in detail. The governing equations of motion are generated using Reddy’s third-order shear deformation theory and Hamilton’s principle. Navier’s technique is used to achieve complex eigenvalues and eigenvectors of the damped vibration of the beams. The correctness and efficiency of the current algorithm are authenticated through a comparison study; then the proposed algorithm and calculation program are employed to explore the damped vibration characteristics of the sandwich beams. A detailed numerical analysis is carried out to illustrate the effects of several coefficients on the damped vibration characteristics of the sandwich beams. The outcomes of this study showed that the influences of the damping coefficients are substantial on the damped vibration behaviors of the sandwich beams.</p>