Time-delay dynamics of the MR damper–cable system with one-to-one internal resonances

In this study, we investigate the nonlinear resonant response of a magnetorheological (MR) damper–stay cable system with time delay, and the one-to-one internal resonance is considered. Based on Hamilton’s principle, the motion equations of the MR damper–cable system are derived, and the Galerkin method is applied to obtain the discrete model. Then, the method of multiple scales is applied to determine the modulation equations and the second-order solution of the nonlinear response of the MR damper–cable system. Following, the equilibrium solution and dynamic solution of the modulation equations are examined via the Newton–Raphson method and shooting method. The results show that the equilibrium solution may undergo Hopf bifurcation, resulting in the periodic solution. Moreover, the effects of the time delay and the inclination angle on the resonant response of the MR damper–cable system are investigated as well as those of the damper position. The numerical results show that the time delay increases the amplitudes of in-plane and out-of-plane modes and results in the more remarkable hardening behavior and relatively poor mitigation performance of the MR damper. However, the large time delay may suppress the complex chaotic modulation motion of the MR damper–cable system.