Unpredictable Behavior of a Partially Damped System of PDEs Modeling Suspension Bridges

We consider a nonlinear nonlo cal coupled system of beam-wave equations governing the dynamics of a fish-bone plate modeling the behavior of suspension bridges. Due to the physics of the problem, the beam equation is both damped and forced, whereas the wave equation is ``isolated"". For the single beam equation, which is dissipative, we show that, in the presence of a stationary force, the unique steady solution is globally stable. We then prove the counterpart of such a result in the periodic setting for small forces only, since multiple periodic solutions may instead arise in general. These results do not hold for the coupled beam-wave system, for which we prove that, even in the unforced case, the unique stationary solution is unstable and the \omega -limit set contains infinitely many periodic solutions. We also discuss more stability and multiplicity issues, including sufficient conditions of torsional stability and explicit examples of multiple periodic solutions. The main employed techniques rely on careful estimates of suitable modified energy functionals, fixed point arguments, and stability criteria for the Hill equation. Overall, our results show how hard it is to forecast the general behavior of bridges, and the striking differences between the sole beam equation and the coupled system might represent the cause of some not fully understood paradoxes and erroneous conclusions.