Abstarct: In this research It's been investigated nonlinear vibration of a Euler-Bernoulli beam is connected to a nonlinear energy sink, once bonded to the ground and again in a free mode. in the present work, which consists of analytical and numerical sectors, the nonlinear sink of the energy sink in reducing the vibration of two different mechanical systems has been investigated. in general, the nonlinear energy sink is capable of extracting energy from the initial system under external loads in one way, passive, and in a wide range of excitation frequencies. the governing equations of the system using euler-Bernoulli theory are extracted and the mathematical model of the structure has been demonstrated using Newton method. then, we use the galerkin method for discretization of motion equations. in the end, the dynamic analysis of the system dynamics and the use of the concepts of nonlinear normal modes have been investigated. in this study, the original system parameters are assumed to change with time. investigation is do in the absorbing parameters by employing nonlinear energy well with the nonlinear stiffness of the third order and optimal absorbent parameters are calculated to reduce beam vibrations and increase the effective frequency range of vibration reduction. in all studies, the analysis of dynamic type is in steady flow. to investigate the convergence of beam response, stability analysis, robustness of the beam against variation of the amplitude of the excitation force, obtaining the maximum efficiency of the nonlinear sink of energy with parameter variations and the nonlinear dynamic analysis of the system are discussed in this thesis.