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Suspended cables are commonly used structures in many civil and industrial applications. In cold regions, where ice accretions can break the axial-symmetry of the cable cross-section, the occurrence of the galloping phenomenon is often observed, and cables experience high amplitude/low frequency oscillations, even for moderate and uniform wind [1]. The phenomenon can be explained as a Hopf bifurcation happening at a certain critical wind velocity, and limit cycles occurring in post-critical conditions. In [2], a mechanical model of cable was proposed, where the wind was assumed to: (a) modify the equilibrium configuration initially occupied by the cable under self-weight, through the mean component; (b) induce the Hopf bifurcation from the modified equilibrium through the dynamic component. There it was observed that, besides the wind velocity which represents the main bifurcation parameter, a second parameter, namely the initial orientation of the ice accretion, was crucial to determine whether the critical mode was of symmetrical or anti-symmetrical shape. Furthermore, a third parameter, namely the initial sag-to-span ratio, was able to change the critical frequency of the symmetric mode.

It mean that there exist a critical value of the three parameters (wind mean velocity, initial orientation of the ice accretion and sag-to-span ratio) at which a 1:1 resonant double Hopf bifurcation occurs, i.e. where the symmetric and anti-symmetric modes have both the same critical wind velocity and frequency.

Here, the critical conditions for the occurrence of the mentioned codimension-three bifurcation are sought and the post-critical analysis is carried out, paying special attention to the transitions from non-resonant to resonant double Hopf bifurcation.

Acknowledgements

This work is partially funded by the European Union - Next Generation EU, Mission 4 Component 2 Investment 1.1, in the framework of the project PRIN 2022 PNRR, “P2022ZT5X5 - Smart Under-Ground Infra-Structures for Secure Communities and Post-Disaster Emergency Response: Eco-Friendly Seismic Protection Solutions” (CUP: E53D2301762 0001, University of L'Aquila).

References

[1] Den Hartog J., Mechanical Vibrations. Dover Publications, New York (1985).

[2] Zulli D., Piccardo G., Luongo A. “On the nonlinear effects of the mean wind force on the galloping onset in shallow cables”. Nonlinear Dynamics, 103, pages 3127–3148 (2021)