Clelia Marchionna | Searcharxiv

arXiv2022

Transfer of energy from flexural to torsional modes for the Fish-bone Suspension bridge

We consider a conservative coupled oscillators systems 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 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 an our previous article 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.

math.CA↗

arXiv2019

On the instability tongues of the Hill equation coupled with a conservative nonlinear oscillator

We study the asymptotics for the lengths $L_N(q)$ of the instability tongues of Hill equations that arise as iso-energetic linearization of two coupled oscillators around a single-mode periodic orbit. We show that for small energies, i.e. $q\rightarrow 0$, the instability tongues have the same behavior that occurs in the case of the Mathieu equation: $L_N(q) = O(q^N)$. The result follows from a theorem which fully characterizes the class of Hill equations with the same asymptotic behavior. In addition, in some significant cases we characterize the shape of the instability tongues for small energies. Motivation of the paper stems from recent mathematical works on the theory of suspension bridges.

math.CA↗