2020/07/03 by Denis Bonheure, Filippo Gazzola, Bonheure, Denis +5 · 1 citation
Engineering · #35B41 #35G31 #35Q74 #70J10 #74H40 #74K20 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Fluid Dynamics and Thin Films #Fluid Dynamics and Vibration Analysis #Vibration and Dynamic Analysis
paper · pdf · doi:10.48550/arxiv.2007.01801
openalex publication_date 2020/07/03 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
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.