We consider a second order system of two ODE's which arises as a single mode Galerkin projection of the so-called fish-bone [2] model of suspension bridges. The two unknown represent flexural and torsional modes of vibration of the deck of the bridge. 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 establish a condition depending on a set of 3 parameters under which the flexural motions are unstable provided the energy is sufficiently large.

We study the asymptotics for the lengths LN (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 → 0, the instability tongues have the same behavior that occurs in the case of the Mathieu equation: LN (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.

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.

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We study a single mode for the Kirchhoff string vibrating in space.\ud In 3D a single mode is generally almost periodic in contrast to the 2D periodic case. In order to show a complete geometrical description of a single mode we prove some monotonicity properties of the almost periods of the solution, with respect to the mechanical energy and the momentum. As a consequence of these properties, we observe that a planar single mode in 3D is always unstable, while it is known that a single mode in 2D is stable (under a suitable denition of stability), if the energy is small