  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.
