We consider the forced sine-Gordon equation on a bounded domain, which models the torsional oscillation of the main span of a suspension bridge. We use Leray-Schauder degree theory to prove that, under small external forcing, the undamped equation has multiple periodic solutions. Using physical constants from the engineers' reports of the collapse of the Tacoma Narrows Bridge, we solve the damped equation numerically and observe that multiple periodic solutions exist and that whether the span oscillates with small or large amplitude depends only on its initial displacement and velocity. Moreover, we observe that the qualitative properties of our solutions are consistent with the behavior observed at Tacoma Narrows on the day of its collapse.
We also consider a nonlinear system which models the coupled vertical-torsional motion of the main span. Again, we prove the existence of multiple periodic solutions to the undamped system via a Leray-Schauder degree theory argument. We solve the damped system numerically and replicate the phenomena that were observed at Tacoma Narrows on the day of its collapse.