Axisymmetric forced vibrations of an elastic free circular plate on a tensionless two parameter foundation

The static and dynamic responses of a circular elastic plate on a two-parameter tensionless foundation are investigated by assuming that the external load is rotationally symmetric and the plate experiences axially symmetric displacements. In the solution procedure, the vertical displacement of the foundation is determined by the corresponding governing equation, whereas the vertical displacement of the plate is expressed in series in terms of the mode shapes of the completely free circular plate. For the case of complete contact, the corresponding governing equation of the plate incorporated with the edge reaction from the foundation is satisfied through the Galerkin's approximation technique. The contact radius is obtained by requiring that the surface of the foundation satisfies the corresponding continuity equations. It is shown that the problem displays a highly nonlinear character due to the lift-off of the plate from the foundation and the numerical treatment of the governing equation is accomplished by adopting iterative processes in terms of the contact radius. The governing equation of the problem is solved numerically for the static and dynamic cases and the results are presented in figures to demonstrate the nonlinear behavior of the plate for various values of the parameters of the problem comparatively.