Effects of delayed feedback control on nonlinear vibration absorber system

The delayed feedback control is applied to suppress the vibration of vertical displacement in a two-degree-of-freedom nonlinear system with external excitation. Effects of both positive and negative feedback control are observed when the primary resonance and 1:1 internal resonance occur in the system simultaneously. The method of multiple scales is employed to obtain analytical solutions of the system under consideration. With the time delay varying for a fixed gain, it is seen that the vibration can be suppressed at some values of the delay. These values form a called “vibration suppression region”, where one can easily observe the “maximum vibration suppression point”. At this point, the vertical vibration could be suppressed about 65% for the positive feedback and 86% for the negative one by comparing with the nonlinear vibration absorber without delayed feedback control. However, the vibration suppression is invalid at some values of the delay. Effects of the gain on the vibration suppression are also investigated. The dynamical behaviors became complex with the gain increasing at some values of the delay, but the vibration can be suppressed more efficiently at other values. It implies that one can always choose a suitable delay and gain to achieve a vibration suppression. All analytical predictions of this paper are in good agreement with the numerical simulation.