The in-plane wave motion is analytically examined to address the stationary deflection, natural frequency splitting, and mode contamination of the rotationally ring-shaped periodic structures (RRPS). The governing equation is developed by the Hamilton's principle where the structure is modeled as a thin ring with equally-spaced particles, and the centrifugal effect is included. The free responses are captured by the perturbation method and determined as closed-form expressions. The results imply that the response of stationary RRPS is characterized as standing wave, and the natural frequencies can split when the wave number n and particle number N satisfying 2n/N = int. Also the splitting behavior is determined by the relative angle between the particle and wave antinode. The coefficients of the mode contamination are also obtained. For rotating RRPS, the invariant deflections due to the centrifugal force are estimated at different rotating speeds. It is found that, for certain waves satisfying 2n/N = int, the natural frequency exceeds that of the corresponding smooth ring at the critical speed, and furthermore, the critical speed of the backward traveling wave is lower than that of the forward one. The contamination coefficients of the two kinds of waves are also obtained and they have different magnitudes. All results verify that the splitting and contamination can be determined by the relationship among the mode order, wave number, particle number, and relative position between the particle and antinode. Numerical examples and comparisons with the existing results in the literature are presented.