Damped vibration analysis of cracked Timoshenko beams with restrained end conditions
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(2020) 42:488
TECHNICAL PAPER
Damped vibration analysis of cracked Timoshenko beams with restrained end conditions Yasar Pala1 · Semih Beycimen1 · Caglar Kahya1 Received: 21 May 2019 / Accepted: 3 August 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020
Abstract Damped vibration of a cracked Timoshenko beam with ends supported with damper, linear and rotational springs is investigated. Frequencies in complex forms have been obtained for both cracked Euler–Bernoulli and Timoshenko beams. Depending upon the crack-depth and crack-location, frequencies have been tabulated in each case. The results have also been compared in terms of the ratio of the beam depth to the beam length. Modal shapes for various conditions have also been plotted. Keywords Damped cracked · Timoshenko · Euler–Bernoulli · Vibration · Mode shapes
1 Introduction Several authors have studied the effect of the crack on the beam vibration for various boundary conditions. Mahmoud [1] developed an approach to determine the strain density factor using the equivalent load in a simply supported undamped Euler–Bernoulli beam containing a single-sided and double-sided open cracks under the influence of the moving load. Modal analysis was performed to obtain the natural frequencies of a pre-stressed fixed-fixed beam and studied the effects of crack depth by Masoud et al. [2]. Reis and Pala [3] and Pala and Reis [4] investigated the effects of the inertial, centripetal, and Coriolis forces on the dynamic response of a cracked cantilever and cracked simply supported beams, respectively. Chondros et al. [5] developed a theory for modeling the lateral vibration of a cracked Euler–Bernoulli beam with open cracks on a single or double edges. Hasan [6] used a perturbation method to find the crack effects on the eigen frequencies of an Euler–Bernoulli Technical Editor: José Roberto de França Arruda. * Semih Beycimen [email protected] Yasar Pala [email protected] Caglar Kahya [email protected] 1
Department of Mechanical Engineering, Faculty of Engineering, Bursa Uludag University, Bursa, Turkey
beam on an elastic foundation. The presence of a crack can be modeled by means of rotational spring. Lele and Maiti [7] applied this method to Euler–Bernoulli beam while Krawczuk et al. [8] applied it for the Timoshenko beam. Loya et al. [9] modelled the crack as two massless springs, such as an extensional and a rotational springs, to involve the effects of the transmission of both shear force and bending moment. The vibration of the beam was also investigated for the case of multiple cracks. Lin et al. [10] examined the free vibration of the beam containing multiple cracks using semi-analytical and semi-digital hybrid method. In that study, the crack was considered as a rotational spring only. Transfer matrix method was used to calculate the eigenvalues. Free and forced vibration analyses of a cracked cantilever Euler–Bernoulli beam were studied by Orhan [11]. This study showed that free vibration analysis is more useful to f
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