Closed-Form Solution for the Natural Frequencies of Low-Speed Cracked Euler–Bernoulli Rotating Beams

Abstract

In this study, two closed-form solutions for determining the first two natural frequencies of the flapwise bending vibration of a cracked Euler–Bernoulli beam at low rotational speed have been developed. To solve the governing differential equations of motion, the Frobenius method of solution in power series has been used. The crack has been modeled using two undamaged parts of the beam connected by a rotational spring. From the previous results, two novel polynomial expressions have been developed to obtain the first two natural frequencies as a function of angular velocity, slenderness ratio, cube radius and crack characteristics (depth and location). These expressions have been formulated using multiple regression techniques. To the knowledge of the authors, there is no similar expressions in the literature, which calculate, in a simple way, the first two natural frequencies based on beam features and crack parameters, without the need to know or solve the differential equations of motion governing the beam. In summary, the derived natural frequency expressions provide an extremely simple, practical, and accurate instrument for studying the dynamic behavior of rotating cracked Euler–Bernoulli beams at low angular speed, especially useful, in the future, to establish small-scale wind turbines’ maintenance planes.