Static deflection and large amplitude vibrations of a rubber plate are analyzed. Both the geometrical and physical (material) nonlinearities are taken into account. The properties of the plate hyperelastic material are described by the Neo-Hookean law. A method for building a local model, which approximates the plate behavior around a deformed configuration, is proposed. This local model takes the form of a system of ordinary differential equations with quadratic and cubic nonlinearities. The results obtained with the help of this local model are compared to the solution of the exact model, and are found to be accurate. The difference between the model retaining both physical and geometrical non-inearities and a model with only geometrical nonlinearities is also analyzed. It is found that influence of physically induced nonlinearities at moderate strains is significant.