An Instability Result for Suspension Bridges

We consider a class of second order systems of two ODEs which arise as single mode Galerkin projections of the so-called fish-bone [BG] model of suspension bridges. The two unknowns represent flexural and torsional modes of vibration of the deck of the bridge. The nonlinear elastic responses F of the cables are supposed to be generalizations of the slackening regime. In the first part, under the assumption of sub-linear growth for F we establish a condition depending on a set of 3 parameters under which the flexural motions are unstable provided the energy is sufficiently large. In the last part of the paper we numerically investigate the effect of slackening for different model functions, either sub-linear or super-linear. Finally, we examine the different types of bifurcations that give rise to instability of flexural modes.