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.

提出了一种在不同边界条件下，存在裂缝的情况下分析梁动力特性的新颖分析方法。应用季莫申科梁理论并引入辅助函数，使用哈密顿方法导出运动方程。固有频率是通过应用Euler-Bernoulli方法获得的，并通过自由振动中Euler-Bernoulli梁控制方程的相应辅助函数得出。为了证明所提出方法的有效性，提出了典型结果，并将其与文献中提供的一些结果进行了比较。考虑了不同的边界条件，并计算并比较了固有频率。结果表明获得了非常好的结果。这种方法对于研究季莫申科梁的振动问题非常有效。所提出的方法的新颖之处在于，尽管两种理论的辅助函数都不同，但是在两种情况下，动力学问题都可以追溯到对承受轴向载荷的Euler–Bernoulli梁的研究。