Mode Interactions of an Inclined Cable Excited Close to Its Second Natural Frequency

In a cable-stayed bridge the bridge deck is supported by a large number of inclined cables of varying lengths and inclinations, which are suspended from a bridge tower. Vertical excitation of the bridge deck, for example, due to traffic loading, is influencing the cables. This may result in quite complex dynamics of those cables, whose basic vibration modes are near resonances with the excitation. These dynamics are nonlinear due to gravitational sag and tension variations in the cable. We consider here a single inclined cable of length L, whose lower support is periodically excited in the vertical direction with a frequency that is close to the second natural frequency ω2 of the cable. This system is described mathematically by a system of six ordinary differential equations for the contributions of three basic modes of vibration: the second in-plane mode (of vertical motion) and the first and second out-of-plane modes (of horizontal motion). The participating basic modes and the parameters are chosen to represent a scaled-down cable experiment that was designed to represent a full-scale cable of a realistic cable-stayed bridge. More specifically, we present a comprehensive bifurcation study of the different possible coupled-mode responses of the cable, which are distinguished by which of the two out-of-plane modes participate in the dynamics apart from the directly excited in-plane mode. These coupled modes are equilibria of the ODE model, and they are continued numerically with the package AUTO in the dimensionless amplitude ∆/L and the frequency ratio Ω/ω2 of the excitation. Each type of coupled-mode response, of which there are four, is computed and represented as a surface of equilibria in (∆/L,Ω/ω2, ||N ||)-space, where ||N || is a suitable norm. We also compute curves of bifurcations, along which the stability of the equilibria changes and different surfaces meet. Overall, we obtain a geometric picture of the two-parameter bifurcation diagram of the coupled-mode response of the cable. Chosen cross sections, for fixed ∆/L and for fixed Ω/ω2, illucidate its intricate properties.