The present study deals with an exact analysis of free transverse vibrations of annular plates having small core and sliding inner edge and the outer edge being elastically restrained based on classical plate theory. This study focuses mainly on the influence of variations in the elastic restraint parameters on the fundamental frequencies of plate vibration. The natural frequencies for the first six modes of annular plate vibrations are computed for different materials and varying values of the radius parameter and these natural frequencies may correspond to either axisymmetric and/or non-axisymmetric modes of plate vibration. The extensive data of values of fundamental frequency parameter presented in this paper is believed to be of use in the design of acoustic underwater transducers, ocean and naval structures, compressor and pump elements, offshore platforms. These results may serve as bench mark values for researchers to validate their results obtained using approximate numerical methods. © 2016 Elsevier Ltd. All rights reserved.