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

TL;DR

This paper investigates the long-term behavior of a nonlinear, partially hinged-free rectangular plate model of a suspension bridge under wind forces, revealing the existence of attractors, stability conditions, and modal truncation justifications.

Contribution

It develops a comprehensive analysis of the nonlinear plate dynamics with non-conservative forces, constructing attractors and stability results using quasi-stability theory and spectral analysis.

Findings

Existence of global and fractal exponential attractors.

Identification of parameter ranges with non-trivial attractors.

Demonstration of finite determining modes justifying modal truncation.

Abstract

A partially hinged, partially free rectangular plate is considered, with the aim to address 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 span-wise direction. The wind-flow in the chord-wise direction is modeled through a piston-theoretic approximation, which provides both weak (frictional) dissipation and non-conservative 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 non-conservative 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 non-triviality 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.