Soil Effect on the Finite Simply Supported Plate Vibration

We are interested in the vibration prediction for a finite flexural plate lying on a semi-infinite soil, whose surface is free, except under the plate. Both the plate equation and the Navier equations are solved, using their bidimensional spatial Fourier Transforms. A cousin problem is the one of the acoustic radiation of the unbaffled plate, a one velocity problem. In this soil problem, two velocities are taken into account, the soil shear and dilatation velocities, considered as a visco-elastic homogeneous medium. Finally, expanding the plate displacement on its modes, linear systems in plate displacement amplitude are solved. As for the unbaffled acoustic radiation problem, equivalent vibratory radiation impedances set is proposed, totally new, describing the modal coupling between the plate modes and the soil. It is shown, contrary to the acoustic one velocity problem that the sign of the imaginary part of the complex vibratory radiation terms is negative at very low frequency, and positive above, meaning that the soil adds stiffness to the plate at low frequency and mass above. The soil effect on the plate vibration is of first importance, highly decreasing the plate vibration by more than 30 dB even for thick concrete plates.