Large torsional oscillations in a suspension bridge: Multiple periodic solutions to a nonlinear wave equation

We consider a forced nonlinear wave equation on a bounded domain which, under certain physical assumptions, models the torsional oscillation of the main span of a suspension bridge. We use Leray-Schauder degree theory to prove that, under small periodic external forcing, the undamped equation has multiple periodic solutions. To establish this multiplicity theorem, we prove an abstract degree theoretic result that can be used to prove multiplicity of solutions for more general operators and nonlinearities. 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 computed solutions are consistent with the behavior observed at Tacoma Narrows on the day of its collapse.