Free and force vibration analysis of cracked Euler-Bernoulli beam with spectral finite element method

In this article, a spectral finite element (SFE) formulation and its solution are described for free and force vibrations of cracked Euler-Bernoulli beam. The formulation based on SFE algorithm includes deriving partial differential equations of motion, spectral displacement field, dynamic shape functions, and dynamic stiffness matrix. Frequency-domain dynamic shape functions are derived from an exact solution of governing wave equations. The cracked beam with an open crack is modeled as two segments connected by a massless rotational spring at the crack position and frequency-domain dynamic stiffness matrix for cracked Euler-Bernoulli beam is extracted. By considering free vibration of the cracked beam, its natural frequencies are derived for different boundary conditions. In the SFE model, It is possible to represent the whole length of beam only by two spectral elements, while it may not be possible to do that in finite element (FE) model, for reaching the same order of accuracy. The accuracy of results obtained from SFE formulation is compared with that of either FE method or analytical formulations. The SFE results display remarkable superiority with respect to those of FE, for reducing the number of elements as well as increasing numerical accuracy.