Vibration suppression of suspended cables with three-to-one internal resonances via time-delay feedback

Based on the time-delay feedback control, the vibration suppression of suspended cables with three-to-one internal resonances are investigated. Initially, the nonlinear differential equation of motion for a suspended cable under time-delay feedback control is considered, and a discrete model is derived using the Galerkin method. Subsequently, the method of multiple scales is employed to perturbatively solve the discrete time-delay differential equation, determining the modulation equations around the first primary resonance. Steady-state and periodic solutions of the modulation equations are detected numerically. Numerical results indicate that the internal resonance enhances the nonlinear dynamical complexity of the controlled suspended cable. It is observed that the time delay and control gain affect the controlled system: in particular, an increase in control gain leads to a reduction in response amplitude. By adjusting the time delay and control gain, the critical excitation can be altered, an aspect that could be very useful from a practical point of view. This research sheds light on the intricate dynamics of suspended cable and provides a theoretical foundation for designing more effective control strategies in engineering applications.