Plates are common structural elements of most engineering structures,
including aerospace,
automotive, and civil engineering structures. The study of plates from
theoretical perspective
as well as experimental viewpoint is fundamental to understanding of the
behavior of such structures.
The dynamic characteristics of plates, such as natural vibrations,
transient responses for the external forces and so on, are especially
of importance in actual environments.
In this paper, we conside the envelope surface created by the
vibrations of a square plate on a weakly nonliner elastic foundation
and analyze the stability of the uniform solution of the governing
equation for the envelope surface.
We derive the two-dimensional equation that governs the spatial and
temporal evolution of the envelope surface on cubic nonlinear elastic
foundation.
The fact that the governing equation becomes the quintic nonlinear
Schrödinger equation is shown.
Also we obtain the stability condition of the uniform solution of the
quintic nonlinear Schrödinger equation.