Abstarct: In this thesis, the analysis of linear and nonlinear vibrations of thin and thick-walled cylindrical shells with constant thickness made of functionally graded porous viscoelastic, porous viscoelastic, viscoelastic, functionally graded (porous) and elastic materials is presented. The cylinder is under time-varying internal dynamic pressure. The first and second type Zener model are used to model the viscoelastic behavior of the cylinder, and the property of porosity functionality is modeled by considering the power and cosine functions distribution through thickness. In deriving the motion equations, the displacement field is estimated by using the first-order shear deformation theory which taking into account the effect of transverse normal strain. The constitutive relations are according to the linear Biot theory of porous materials and the strain-displacement relations are the von karman nonlinear equations. The shell motion equations and the boundary conditions are determined using the Hamilton principle. The motion equations consist of a set of coupled nonlinear partial differential equations with constant coefficients. These equations are solved by the multiple scales method of the perturbation theory. The cases studies in this analysis are: determination of the linear and nonlinear natural frequencies and mode shapes, dependency of the frequency to the amplitude, investigating the effect of the geometric parameters, porous material heterogeneity and viscoelastic material properties on the frequency and calculating the response in terms of the time and space. Also the results are compared with a numerical solution. The results show that the frequency of porous viscoelastic cylindrical shells is less than the viscoelastic and it is higher than the porous case. Under the same conditions, the porous viscoelastic and viscoelastic cylindrical shells have the softening behavior but the elastic cylindrical shells and porous have the hardening behavior. In the porous viscoelastic cylindrical shells, the natural frequency decreases and the amplitude increases with increasing the porosity index.