Utilizing time-delays to quench the nonlinear vibrations of a two-degree-of-freedom system

Although, time-delays are considered an undesirable phenomenon in the active control system, this article shows how to utilize time-delays to mitigate the oscillations of a two-degree-of-freedom nonlinear model simulating a horizontally supported Jeffcott-rotor system. The multiple scales method is conducted to obtain a second-order asymptotic solution to the system governing equations. The slow-flow modulating equations of both the amplitudes and phases are extracted. The conditions that make the time-delayed controller works as a damper or exciter are clarified. The effects of the control gains and time-delays on the system stability are investigated. The analyses illustrated that both negative and positive displacement feedback control can mitigate the system vibrations to an excellent level if the time-delays are chosen within their optimal values. A method for selecting the optimal values of the time-delays is included. Then, the acquired analytical results are approved by solving the system equations numerically. The analytical and numerical results confirmed that the vibration peak has been reduced by about 80% without time-delays, and by 95% with the optimal values of the time-delays. Finally, a comparison with recently published articles concerning time-delayed feedback control is included.