Vol.:(0123456789) 1 3 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:488 https://doi.org/10.1007/s40430-020-02558-1 TECHNICAL PAPER Damped vibration analysis of cracked Timoshenko beams with restrained end conditions Yasar Pala 1  · Semih Beycimen 1  · Caglar Kahya 1 Received: 21 May 2019 / Accepted: 3 August 2020 / Published online: 24 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 investi- gated. Frequencies in complex forms have been obtained for both cracked Euler–Bernoulli and Timoshenko beams. Depend- ing upon the crack-depth and crack-location, frequencies have been tabulated in each case. The results have also been com- pared 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 efect of the crack on the beam vibration for various boundary conditions. Mahmoud [1] developed an approach to determine the strain den- sity factor using the equivalent load in a simply supported undamped Euler–Bernoulli beam containing a single-sided and double-sided open cracks under the infuence of the moving load. Modal analysis was performed to obtain the natural frequencies of a pre-stressed fxed-fxed beam and studied the efects of crack depth by Masoud et al. [2]. Reis and Pala [3] and Pala and Reis [4] investigated the efects of the inertial, centripetal, and Coriolis forces on the dynamic response of a cracked cantilever and cracked simply sup- ported 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 dou- ble edges. Hasan [6] used a perturbation method to fnd the crack efects on the eigen frequencies of an Euler–Bernoulli 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 Kraw- czuk 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 efects 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 eigenval- ues. Free and forced vibration analyses of a cracked can- tilever Euler–Bernoulli beam were studied by Orhan [11]. This study showed that free vibration analysis is more useful to fnd the crack location while forced vibration analysis is more suitable to determine crack depth. Inverse method on the free vibration of a Timoshenko beam was conducted to fnd the crack properties such as crack location and crack size [12, 13]. The vibration of a simply supported beam con- taining an open crack was solved by Lin [12], either directly or via inverse method by using the transfer matrix method. Rizos et al. [14] studied an inverse problem which intended to determine the location of the crack. Liang et al. [15] also discussed a similar problem with the fnite element method. Gounaris and Papadoppoulos [16] presented a method for the determination of crack location and crack depth. Apart from these studies, Thalapil et al. [17] investigated the crack which is parallel to the longitudinal direction of the beam. Technical Editor: José Roberto de França Arruda. * Semih Beycimen semihbeycimen@uludag.edu.tr Yasar Pala mypala@uludag.edu.tr Caglar Kahya ckahya@uludag.edu.tr 1 Department of Mechanical Engineering, Faculty of Engineering, Bursa Uludag University, Bursa, Turkey