Vertical vibration of rigid bodies with rectangular bases on elastic media

Abstract

Abstract Dual double integral equations completely representing the mixed boundary‐value problem of the vertical vibration of rigid bodies with rectangular bases on semi‐infinite elastic media are exactly formulated for the first time. An exact solution of these equations—even for the static case—is at present formidable, but an approximate solution is given by reducing the pair of dual double integral equations to another pair which shows an approximate relation between the three‐dimensional and axi‐symmetric systems. This new method is used to generate response curves for Poisson's ratio 0.25 and side ratios 1, 2, 3, 4 and 5, from known results for circular bases, for the dynamic problem. Results obtained by using this technique are in good agreement with experiments in Reference 3 where expanded rubber is used as the elastic medium. The present theory which is applicable for any Poisson's ratio also shows some measure of agreement with the theory of Elorduy et al. 4 which is limited, by its numerical nature, to a Poisson's ratio of 0.25.