Dynamic analogy between Timoshenko and Euler-Bernoulli beams

In this paper, a novel analytically method for analyzing the dynamic behavior of beams, under different boundary conditions and in presence of cracks, is proposed. Applying the Timoshenko beam theory and introducing the auxiliary functions, the equation of motion is derived using the Hamiltonian approach. The natural frequencies are obtained by applying the Euler-Bernoulli method and are derived by the corresponding auxiliary functions of the governing equation of the Euler-Bernoulli beam in free vibration. In order to demonstrate the efficiency of the proposed approach, typical results are presented and compared with some results available in the literature. Different boundary conditions were considered, and natural frequencies were calculated and compared. It is shown that very good results are obtained. This approach is very effective for the study of the vibration problem of Timoshenko beams. The novelty of the proposed approach is that although the auxiliary functions are different for the two theories, in both cases the dynamic problem is traced to the study of an Euler-Bernoulli beam subjected to an axial load.