Abstract: In a two-degree-of-freedom system with delayed nonlineardynamical vibration absorber, the method of multiple scales is employed tostudy its performance of vibration suppression. The amplitude-delay responsecurves of the primary system are obtained. The results show that for thedelayed nonlinear dynamical vibration absorber, the gain and the delay maybe tuned to control the vibration of the primary system. With the delayvarying for a fixed gain, it is easy to see that the vibration can besuppressed at some values of the delay under the conditions of the system isstable. The ``maximum vibration suppression point'' may be predicted atwhich the amplitude of the primary system is the minimum for a fixed gain.For different values of the gain, the amplitudes of the primary system aredifferent at their ``maximum vibration suppression point''. Correspondingly,the optimum values of the gain and the delay may be obtained to suppress theamplitude of the primary system to a minimum. The results of this paper showthat the vibration of the primary system can be suppressed about 90\% whenthe optimum values of the gain and the delay are obtained by comparing withthe nonlinear dynamical vibration absorber. All the analytical predictionsof this paper are in good agreement with the numerical simulation.