Comprehensive Study of Free Vibration of Rectangular Mindlin’s Plates with Rotationally Constrained Edges Using Dynamic Timoshenko Trial Functions

A comprehensive theoretical study of the free vibration of rotationally restrained rectangular uniform isotropic Mindlin’s plate is presented. The plate mode shape is assumed to be a weighted combination of the product of the Timoshenko beam functions in the either direction, which are previously generated for rotationally constrained boundary conditions. The effect of the uniformly distributed rotational spring constant (modelling the edge) participates in the potential energy of the plate. The Rayleigh-Ritz method has been used to generate the natural frequencies and plate mode shapes for various intermediate boundary conditions, asymptoting to those of the plates with all possible (six) classical boundary conditions. Plates with various thickness ratios have been studied to converge the results to the corresponding Kirchhoff’s frequencies. The eigenvectors from the eigenvalue problem have been scrutinized to establish the beam-wise modal participation from either direction into the final plate mode shape. The square Mindlin’s plate mode shapes have been generated to establish the various types of frequencies; which have been innovatively named and categorized as the (i) single frequencies, (ii) repeated frequencies (identical twins) and (iii) non-repeated frequencies (fraternal twins). Plates with different rectangular aspect ratios have been also analysed to show the deviation in the frequencies and mode shapes from the square plate. Also, their asymptotic behaviour to the corresponding Timoshenko beam at extreme aspect ratios has been established.