Periodic Solutions and Torsional Instability in a Nonlinear Nonlocal Plate Equation

A thin and narrow rectangular plate having the two short edges hinged and the two long edges free is considered. A nonlinear nonlocal evolution equation describing the deformation of the plate is introduced: well-posedness and existence of periodic solutions are proved. The natural phase space is a particular second order Sobolev space that can be orthogonally split into two subspaces containing, respectively, the longitudinal and the torsional movements of the plate. Sufficient conditions for the stability of periodic solutions and of solutions having only a longitudinal component are given. A stability analysis of the so-called prevailing mode is also performed. Some numerical experiments show that instabilities may occur. This plate can be seen as a simplified and qualitative model for the deck of a suspension bridge, which does not take into account the complex interactions between all the components of a real bridge.