Resonances of a nonlinear SDOF system with time-delay in linear feedback control

The primary and subharmonic resonances of a nonlinear single-degree-of-freedom (SDOF) system under feedback control with a time delay have been studied by means of an asymptotic perturbation technique. Both external (forcing) and parametric excitations have been included. By means of the averaging method and multiple scales method, two slow-flow equations for the amplitude and phase of the primary and subharmonic resonances and all other parameters are obtained, respectively. The steady state solutions (fixed points) for the original system are investigated. The stability of the fixed points is examined by using the variational method. The effect of the feedback gains, time-delay, the coefficient of cubic term, the coefficients of external and parametric excitations on the steady state responses are investigated and the results are presented as plots of the steady state response amplitude versus the detuning parameter. The results obtained by the two methods are in excellent agreement. There exist saddle node bifurcations for the case of primary resonance and the solutions lose stability for the case of resonance subharmonic.