The constant needs of the industry impel the engineering community in seeking of new concepts and new strategies in order to improve the structural response of structures as well as to enhance the endurance of materials. This is particularly true in the case of rotating blades that are subjected to severe environmental conditions such as high temperatures as well as mechanical conditions such as high rotating accelerations, centrifugal forces, geometric stiffening, among others. Creation and discovery of advanced composite materials containing functionally graded materials is a solution for this problem. Utilizing functionally graded materials is effective because of high mechanical and thermal resistance and also is not difficult to delimitate. An important characteristic in the dynamic analysis of functionally graded beams is computational of their natural frequencies and mode shapes. This is important as the functionally graded beams structures often operate in complex environment conditions with high temperature and are frequently exposed to a variety of dynamic excitations. The characteristics of rotating flexible structures differ significantly from those of non-rotating flexible structures. Centrifugal inertia force due to rotational motion causes the variation of bending stiffness, which naturally results in the variations of natural frequencies and mode shapes. Moreover, the stiffness property of functionally graded structures can be easily modulated through changing power-law exponent of functionally graded materials. Power-law exponent dictates the material variation profile through the predetermined direction of the beam. The finite element method is one of the most powerful numerical procedures for solving complex problems in various branches of engineering and physics. Nowadays, some advanced elements have been introduced, among them is element with 8 degrees of freedom. In normal element (i.e. element with 4 degrees of freedom), a beam element is modeled using two nodes at the end and every node has two degrees of freedom. Therefore, a large number of elements are needed for modeling of the beam and to achieve an acceptable accuracy. By using element with 8 degrees of freedom, some derivational terms are added to the nodes of element and displacement function obtains extra degrees of freedom. Thus, the same accuracy can be achieved by using much less number of elements. This results in rapid convergence. In this thesis, the free vibration analysis of rotating and non-rotating beams made of functionally graded materials for purely out-of-plane motion (flap) and purely in-plane motion (lag) is investigated. Numerical finite element method with two type element 4 and 8 degrees of freedom is applied to find the natural frequencies and mode shapes of functionally graded beams in the bending mode of vibration by taking into account the taper ratio and the rotation simultaneously. In this study, as well as, a nonlinear model is developed appealing to a nonlinear strain–displacement relation. Numerical Newark’s method is employed to solve the nonlinear model and to obtain a numerical approximation of the motion equations. The element mass, damping and stiffness matrices are derived and the effects of hub radius, slenderness ratio, rotation, taper ratio, material constituents and shear deformation on the beam natural frequencies are investigated. Keywords: Natural Frequencies, Functionally Graded Materials, Tapered Beam, Finite Element Method, Numerical Newark’s Method, Flap, Lag