Transfer of Energy from Flexural to Torsional Modes for the Fish-Bone Suspension Bridge Model

We consider a conservative coupled oscillators system which arises as a simplified model of the interaction of flexural and torsional modes of vibration along the deck of the so-called fish-bone (Berchio and Gazzola in Nonlinear Anal 121:54-72, 2015) model of suspension bridges. The elastic response of the cables is supposed to be asymptotically linear under traction, and asymptotically constant when compressed (a generalization of the slackening regime). We show that for vibrations of sufficiently large amplitude, transfer of energy from flexural modes to torsional modes may occur provided a certain condition on the parameters is satisfied. The main result is a non-trivial extension of a theorem in Marchionna and Panizzi (Nonlinear Anal 140:12-28, 2016) to the case when the frequencies of the normal modes are no more supposed to be the same. Several numerical computations of instability diagrams for various slackening models respecting our assumptions are presented.