The dynamic behavior of long pipelines due to the wide applications in power plants, refineries, petrochemical plants, oil rigs and offshore structures are allocated to a special place. One of the most common defects in structures is crack that can have dangerous effect on the structural behavior which may cause it to damage. Fatigue cracks often exist in structural members that are subjected to repeated loading. The crack models analyses fall largely into two categories: (1) open crack models and (2) open and closed or breathing crack models. Most researchers have used open crack models in their studies and have claimed that the change in natural frequency might be a parameter used to detect the presence of crack .However, the assumption that crack is always open in vibration is not realistic because compressive loads may close the crack. In open crack model it is assumed that the crack in the vibration of the beam always remains open. Several studies have been conducted over the last two decades shows that fatigue crack in the vibration of the structure can be opened and closed. It causes stiffness change when system vibration, the most important effect is influenced of the harmonic components in the dynamic response of the system. Furthermore, the breathing crack due to decrease in the natural frequency is lower than the one predicted by open crack model. This means that using open crack model show that fatigue crack depth is less than the actual one. In this thesis, free vibrations of breathing crack pipe is investigated by using perturbation methods. For this purpose, the transverse vibrations of the simply supported beam and pipe in the first mode is modeled as a single degree of freedom (SDOF) system by using equivalent mass, stiffness, and damping. In this model, due to the opened and closed form of the crack during vibration, it is assumed that the equivalent stiffness of the system is varying as a harmonic function. The spring stiffness of the beam is obtained analytically and for the pipe is compared with the experimental values. Then, the nonlinear vibration equations are solved by using the Lindstedts-Poincare perturbation method and for validation, the results are compared with the results obtained by the Runge-Kutta fourth-order method. Moreover, the effects of crack depth and location on the natural frequency and the displacement of the beam and pipe are studied. Super harmonics component is observed in the system response which is due to the nonlinearity of the system. The increasing of the non-linear effects which are due to the change in the crack parameters causes some changes to the super harmonics amplitudes. By investigating of the results it is concluded that damping value is not a constant amount and it increases with increasing the crack depth and position. It is proved that the open crack model shows lower amounts of crack depth in comparison with the actual predict. Keywords: Crack pipe, Breathing crack, Perturbation method, Natural frequency.