Non-linear time delay saturation controller for reduction of a non-linear vibrating system via 1:4 internal resonance

In this paper, active control with time delay was implemented for eliminating the vibration of the non-linear system excited by external and parametric forcing amplitude in the presence of 1:4 internal resonances. Multiple time scale method is applied to determine first-order approximations of the controlled system. The stability of the system is investigated analytically using Lyapunov first method and numerically using frequency, force-response curves and phase-plane method for the considered resonance case. Effects of different parameters on the steady state response of the controlled system are investigated. Variation of some parameters leads to the bending of the frequency, force-response curves and hence to the jump phenomenon occurrence.