Abstract The natural frequencies and mode shapes of in-plane vibration of the structures described in the title are determined. Both radial sliding and fixed supports are symmetrically placed. The ring responsible is expressed in terms of a complete set of periodic functions. An infinite Rayleigh-Ritz method with an extended functional (i.e., Lagrangian multipliers are used) is applied. Due to the symmetry of the supported ring, the problem dissociates into four independent solutions corresponding to different vibration modes, usually named “symmetric” and “antisymmetric” ones. The methodology is based upon the use of theoretically infinite series. However, a small number of terms is sufficient to attain accurate results from the engineering viewpoint.