Nonlinear hybrid-mode resonant forced oscillations of sagged inclined cables at avoidances

We investigate non-linear forced oscillations of sagged inclined cables under planar 1:1 internal resonance at avoidance. To account for frequency avoidance phenomena and associated hybrid modes actually distinguishing inclined cables from horizontal cables, asymmetric inclined static configurations are considered. Emphasis is placed on highlighting nearly tuned 1:1 resonant interactions involving coupled hybrid modes. The inclined cable is subjected to a uniformly distributed vertical harmonic excitation at primary resonance of a high-frequency mode. Approximate non-linear partial-differential equations of motion, capturing overall displacement coupling and dynamic extensibility effect, are analytically solved based on a multi-mode discretization and a second-order multiple scales approach. Bifurcation analyses of both equilibrium and dynamic solutions are carried out via a continuation technique, highlighting the influence of system parameters on internally resonant forced dynamics of avoidance cables. Direct numerical integrations of modulation equations are also performed to validate the continuation prediction and characterize non-linear coupled dynamics in post-bifurcation states. Depending on the elasto-geometric (cable sag and inclination) and control parameters, and on assigned initial conditions, the hybrid modal interactions undergo several kinds of bifurcations and non-linear phenomena, along with meaningful transition from periodic to quasi-periodic and chaotic responses. Moreover, corresponding spatio-temporal distributions of cable non-linear dynamic displacement and tension are manifested.