Abstarct: The main purpose of this thesis is to increase the accuracy of the numerical finite element method and some of its elements for investigating the free vibration analysis of non-uniform rotating and non-rotating Euler-Bernoulli beams, and verifying the results by means of differential transform method as a semi-analytical technique. To this end, Petrov-Galerkin, Galerkin and Collocation methods, as the three well-known approaches of weighted-residual method, are firstly utilized for carrying out the free vibration analysis of non-rotating tapered general restrained beams. Then the results obtained by considering the mentioned methods are compared by the closed-form solution and differential transform method. The free vibration analysis of rotating tapered beams is subsequently investigated. It should be mentioned that, the governing differential equation of non-rotating Euler-Bernoulli beams is a particular form (when rotational speed is assumed zero) of the differential equation of rotating Euler-Bernoulli beams. Moreover, in order to enhance the accuracy of the finite element method shape functions, the governing differential equation of the rotating beams is used as an additional constraint. The results achieved by considering the new shape functions are also verified utilizing the conventional finite element method and differential transform method. Afterwards, the free vibration analysis of non-uniform rotating Euler-Bernoulli beams with exponentially varying cross-section is carried out by using differential transform and finite element methods. Finally, trigonometric functions are employed in order to derive more accurate finite element method shape functions. It is obviously shown that the results obtained by using the new trigonometric shape functions are significantly efficient and accurate.