Frequencies of Circular Plates Weakened Along an Internal Concentric Circle and Elastically Restrained Edge Against Translation

The present study deals with the derivation of an exact solution for the problem of obtaining the natural frequencies of the vibration of circular plates weakened along an internal concentric circle due to the presence of a radial crack and elastically restrained along the outer edge of the plate against translation. The frequencies of the circular plates are computed for varying values of the elastic translational restraint, the radius of the radial crack, and the extent of the weakening duly simulated by considering the radial crack as a radial elastic rotational restraint on the plate. The results for the first six modes of the plate vibrations are computed. The effects of the elastic edge restraint, the radius of the weakened circle, and the extent of the weakening represented by an elastic rotational restraint on the vibration behavior of thin circular plates are studied in detail. The internal weakening due to a crack resulted in decreasing the fundamental frequency of the plate. The exact method of solution and the results presented in this paper are expected to be of specific use in analyzing the effect of a radial crack on the fundamental natural frequency of the circular plate in the presence of a translational restraint existing along the outer edge of the plate. These exact solutions can be used to check the numerical or approximate results.