Nonlinear Resonance Response of Suspended Cables Under Multi-Frequency Excitations and Time-Delayed Feedback

This study investigates the nonlinear resonance responses of suspended cables subjected to multi-frequency excitations and time-delayed feedback. Two specific combinations and simultaneous resonances are selected for detailed examination. Initially, utilizing Hamilton’s variational principle, a nonlinear vibration control model of suspended cables under multi-frequency excitations and longitudinal time-delayed velocity feedback is developed, and the Galerkin method is employed to obtain the discrete model. Subsequently, focusing solely on single-mode discretization, analytical solutions for the two simultaneous resonances are derived using the method of multiple scales. The frequency response equations are derived, and the stability analysis is presented for two simultaneous resonance cases. The results demonstrate that suspended cables exhibit complex nonlinearity under multi-frequency excitations. Multiple solutions under multi-frequency excitation can be distinguished through the frequency–response and the detuning-phase curves. By adjusting the control gain and time delay, the resonance range, response amplitude, and phase of suspended cables can be modified.