Analytical and Numerical Analysis of Time-Delayed NDF Control for Nonlinear Rotating Pendulums with Stiffness Modulation Under Simultaneous Resonance Conditions

In this paper, the nonlinear dynamics and vibration suppression of a harmonically excited rotating pendulum are studied by a novel bidirectional coupled second-order auxiliary negative derivative feedback (NDF) controller. The governing equations are obtained by Lagrangian mechanics, extended to the case of quintic nonlinearities, for which the variable torsional stiffness is the nonlinear restoring torque. A unique feature of the proposed control configuration is the addition of two separate time-delay parameters in the bidirectional coupling, which allows us to investigate the effects of delayed interactions between the primary system and the auxiliary controller. The resulting coupled nonlinear system is analytically investigated by the averaging method under the simultaneous resonance conditions and the steady-state response and stability characteristics are obtained by the eigenvalue analysis. The analytical predictions are validated with direct numerical simulations by the fourth-order Runge–Kutta method. It is shown that the analytical predictions are effective at vibration suppression and the analytical and numerical responses are in good agreement. Furthermore, a detailed parametric study is presented to investigate the effects of the dual time delays, control gains, controller damping, and excitation amplitude on the nonlinear response and stability. The proposed framework thus integrates a bidirectional coupled second-order auxiliary NDF controller, dual time-delay effects, quintic nonlinear dynamics and simultaneous-resonance analysis into a unified vibration-control strategy.