Abstarct: Nonlinear Energy Sink (NES) is one of the most efficient passive vibration absorbers that consists of a mass, attach to a nonlinear spring. In this study, two separate systems will be examined in Strongly Modulated Response (SMR) and cascade response. For the first system, the effect of the NES on a system that consists of a Linear Oscillator (LO) that has a baxse stimulation is studied. The second NES system that studied analytically, consists of a LO with a baxse stimulation and an NES that is connected to ground with a nonlinear spring called Grounded NES (GNES). For cascade response, after obtaining the equations of motion, the harmonic balance method applied to analytically solve the equations. Then, a slow invariant manifold is obtained and Its changes are examined with respect to the change of parameters. For SMR, the system is studied in the vicinity of 1:1 resonance. For this purpose, after obtaining the equations of motion, some variables were applied such as Manevitch complex variable method (CX-A) and then, multiple scale method is applied to solve the equations analytically and SIM exxpression is obtained. At last, the energy dissipation ratio of the NES and the percentage of the instantaneous total energy stored in the NES are obtained and compared with the amplitude-time diagram. The results show that if the effect of the nonlinearity becomes less influential, the occurrence of energy pumping will be less probable. Also, when the stimulation amplitude becomes smaller, the value of the energy dissipation ratio of the NES increase and the time to complete the process will be reduced.