Long-time dynamics of a hinged-free plate driven by a nonconservative force

Abstract

A partially hinged, partially free rectangular plate is considered, with the aim of addressing the possible unstable end behaviors of a suspension bridge subject to wind. This leads to a nonlinear plate evolution equation with a nonlocal stretching active in the spanwise direction. The wind-flow in the chordwise direction is modeled through a piston-theoretic approximation, which provides both weak (frictional) dissipation and nonconservative forces. The long-time behavior of solutions is analyzed from various points of view. Compact global attractors, as well as fractal exponential attractors, are constructed using the recent quasi-stability theory. The nonconservative nature of the dynamics requires the direct construction of a uniformly absorbing ball, and this relies on the superlinearity of the stretching. For some parameter ranges, the nontriviality of the attractor is shown through the spectral analysis of the stationary linearized (non-self-adjoint) equation and the existence of multiple unimodal solutions is shown. Several stability results, obtained through energy estimates under various smallness conditions and/or assumptions on the equilibrium set, are also provided. Finally, the existence of a finite set of determining modes for the dynamics is demonstrated, justifying the usual modal truncation in engineering for the study of the qualitative behavior of suspension bridge dynamics.