Derivation of explicit relationships in the determination of the natural frequency of Euler-Bernoulli cracked beams on elastic foundation using Rayleigh method

In this paper, an approximate solution based on the Rayleigh’s method is sought to analyze the vibration behavior of Euler-Bernoulli cracked beam resting on an elastic foundation. The modeling of the elastic foundation is implemented using the Winkler elastic spring theory and the stiffness factor of the elastic spring is specified corresponding to material characteristics of the elastic foundation. The Dirac’s delta function is used to apply the crack opening mode in the equation of the Rayleigh in which the factor of this function can be identified in terms of the stiffness factor of an equivalent rotational spring by considering material and geometric parameters of the crack. In the present analysis, explicit relationships are originally established to obtain the natural frequency in three boundary conditions of simply supported-simply supported, clamped-free and clamped-clamped. In this method, the natural frequency of the first mode is determined as the ratio of the maximum enriched potential energy and the maximum kinetic energy. Based on these relationships, the effects of the crack depth, the crack position and the elastic foundation on the response of natural frequency of the beam are investigated. The results of the analysis demonstrate that increasing the crack depth decreases the natural frequency of the beam containing the crack; while the elastic foundation increases the natural frequency of the cracked beam. The comparison of the results of proposed relationships with those of full modeling of the structure in Abaqus software shows a great accuracy of the present analysis.