Analysis of a nonlinear fish-bone model for suspension bridges with rigid hangers in the presence of flow effects

Abstract: We consider a dynamic system of nonlinear partial differential equations modeling the motions of a suspension bridge. This fish-bone model captures the flexural displacements of the bridge deck's mid-line, and each chordal filament's rotation angle from the centerline. These two dynamics are strongly coupled through the effect of cable-hanger, appearing through a sublinear function. Additionally, a structural nonlinearity of Woinowsky-Krieger type is included, allowing for large displacements. Well-posedness of weak solutions is shown and long-time dynamics are studied. In particular, to force the dynamics, we invoke a non-conservative potential flow approximation which, although greatly simplified from the full multi-physics fluid-structure interaction, provides a driver for non-trivial end behaviors. We describe the conditions under which the dynamics are uniformly stable, as well as demonstrate the existence of a compact global attractor under all nonlinear and non-conservative effects. To do so, we invoke the theory of quasi-stability, first explicitly constructing an absorbing ball via stability estimates and, subsequently, demonstrating a stabilizability estimate on trajectory differences applied to the aforesaid absorbing ball. Finally, numerical simulations are performed to examine the possible end behaviors of the dynamics.