In-plane flexural vibrations of nonuniform circular rings embedded in Winkler-type foundations

Two approximate analytical solutions are obtained for the problem being considered by means of the Ritz method in the case of nonhomogeneous Winkler-type embedment of the ring. One solution makes use of polynomial coordinate functions that contain an exponential optimization parameter ‘‘γ,’’ which allows for minimization of the eigenvalues, while the other solution consists in approximating the ring response by means of sinusoidal functions. An independent numerical solution is also obtained using the finite element method.