Internal resonance between in-plane and out-of-plane modes of vibration of inclined cables subjected to vertical support excitation

Inclined cables are important structural elements of cable-stayed bridges. When the bridge deck oscillates, large amplitude cable vibrations can arise in various modes as a result of the low cable damping, parametric excitation or non-linear modal coupling. The resulting vibrations are undesirable and potentially damaging to the long-term performance of the bridge. The phenomena can be modelled considering internal resonances between in-plane and out-of-plane modes of vibration of the cable. % Here they have been studied using a four-mode model that represents the response of an inclined cable vertically excited at the lower end (i.e. from the deck) with a frequency that is close to the natural frequency of the second cable mode in each plane and twice the frequency of the first mode in each plane. The modal equations of the model are investigated using the software package AUTO for the numerical continuation of solutions of a system of ODEs. This allows the identification of the important solution branches in the cable model responsible for unwanted vibration behaviour. Here they have been studied using a four-mode model that represents the response of an inclined cable vertically excited at the lower end (i.e. from the deck) with a frequency that is close to the natural frequency of the second cable mode in each plane and twice the frequency of the first mode in each plane. The modal equations of the model are investigated using the software package AUTO for the numerical continuation of solutions of a system of ODEs. This allows the identification of the important solution branches in the cable model responsible for unwanted vibration behaviour. The result of our analysis is that we identify amplitudes of excitation above which modes other than the directly excited mode (the second in-plane mode) start contributing to the response of the cable. In addition, we show that the response amplitudes in these additional modes is of similar magnitude to the amplitudes in the directly excited mode, which could be considered an issue in the design of cable-stayed bridges. In summary, by using a numerical continuation technique we predict when the response of the cable will change from a single in-plane mode to coupled responses in two or more modes, in-plane or in both planes, and the modal amplitudes involved in these coupled responses.