As we know, when systems with mass and elasticity are exposed to an oscillating force with a frequency close to its natural frequency, will experience large amplitude vibration, which can cause static failures or accelerating fatigue phenomenon. Therefore, the natural frequencies of such structures and factors affecting them and the force acting these structures and their origin and their frequency must be determined and structural design done in such a way that the natural frequencies of these structural are enough away from frequence of forces applied to them to avoid resonance or large amplitude vibration. it will be more important if we are faced with a long and thin structures. on of the structures that are subject to resonance or large amplitude vibration are corrugated circular cylindrical tubes with a diameter greater than 36 inches. these structures are widely used in water transfer, and wastewater collection and transfer system from the cities and the transfer rainwater from the wide roads(culverts). The origin of loads acting on these structures can be due to the Car crossing the road, machinery working at ground level, earth tremor and fluid flow passing through them. If the frequency of the forces acting on these structures are close to the natural frequency of the structure, they will experience large amplitude vibration and resonance which will cause static failures or accelerating fatigue phenomenon. Therefore, the natural frequencies of the structure and the factors effective them should be identified and considered in their design In this thesis, linear and nonlinear free vibration analysis of corrugated circular cylindrical shells with sinusoidal corrugation have been performed. The corrugated circular cylindrical shell is made equivalent with an uncorrugated, orthotropic circular cylindrical shell. The undamped linear free vibration analysis has been studied using Donnell's shallow shell theory and natural frequencies were obtained. More over, natural frequencies were obtained using Abaqus software. The accuracy of the proposed method was determined by comparing the results of this study with the results obtained from Abaqus software. The effect of the geometric shell characteristics, i.e., radius, length and thickness of shell and depth and period of the corrugation on the natural frequencies are investigated. Non-linear free vibration analysys of corrugated circular cylindrical shells are examined. Non-linearities due to large-amplitude shell motion are considered by using the non-linear Donnell's shallow shell theory. A modal expansion is used for the normal displacement that satisfies the boundary conditions for the normal displacement exactly, but the boundary conditions for the in-plane displacements are approximately satisfied by an averaging technique. The system is discretizd by Galerkin's method, and is investigated by using a model involving three degrees of freedom. Galerkin technique is used to reduce the problem to a system of three coupled non-linear ordinary differential equations for the modal amplitudes. These non-linear differential equations are solved for arbitrary initial conditions by using the multiple-time-scale technique. Comparison of results with numerical solution showed the accuracy of multiple-time-scale results. natural frequencies of the modal amplitudes are obtained and internal resonance is investigated. Keywords : Corrugated shell, Nonlinear vibration, Multiple-time-scale technique, Perturbation method.