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A partially hinged, partially free rectangular plate is considered, with theaim to address the possible unstable end behaviors of a suspension bridgesubject to wind. This leads to a nonlinear plate evolution equation with anonlocal stretching active in the span-wise direction. The wind-flow in thechord-wise direction is modeled through a piston-theoretic approximation, whichprovides both weak (frictional) dissipation and non-conservative forces. Thelong-time behavior of solutions is analyzed from various points of view.Compact global attractors, as well as fractal exponential attractors, areconstructed using the recent quasi-stability theory. The non-conservativenature of the dynamics requires the direct construction of a uniformlyabsorbing ball, and this relies on the superlinearity of the stretching. Forsome parameter ranges, the non-triviality of the attractor is shown through thespectral analysis of the stationary linearized (non self-adjoint) equation andthe existence of multiple unimodal solutions is shown. Several stabilityresults, obtained through energy estimates under various smallness conditionsand/or assumptions on the equilibrium set, are also provided. Finally, theexistence of a finite set of determining modes for the dynamics isdemonstrated, justifying the usual modal truncation in engineering for thestudy of the qualitative behavior of suspension bridge dynamics.

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