Vibration of a Rectangular Plate Carrying a Massive Machine with Elastic Supports

In engineering practice, a massive machine may be placed on a plate supported by beams considered as elastic boundary conditions. The vibration of the plate due to the periodic excitation of the massive machine will cause noises or damages to the building in which the machine is housed. An analytical approach for the vibration analysis of a rectangular plate carrying a massive machine with uniform elastic supports is presented. The machine is simplified as a distributed mass. The transverse plate displacement is determined by the superposition of a two-dimensional (2D) Fourier cosine series and several supplementary functions. All the unknown Fourier coefficients are calculated directly from the Rayleigh–Ritz formulation. To validate the present approach, several numerical examples with classical boundary conditions are presented. The results reveal good agreement between the analytical results and those based on the finite element analysis (ANSYS). The effects of the plate size, location of the machine, and support stiffness on the modal, and transient response of the plate are investigated. From the results it is found that the transient displacement amplitude of the plate decreases almost linearly as the thickness increases, it increases nonlinearly along with the increase in the support stiffness, and that the optimal position for deploying the transformer is the center of the plate.