Mode Shape Analysis of Multiple Cracked Functionally Graded Timoshenko Beams

Abstract

The present paper addresses free vibration of multiple cracked Timoshenko beams made of Functionally Graded Material (FGM). Cracks are modeled by rotational spring of stiffness calculated from the crack depth and material properties vary according to the power law throughout the beam thickness. Governing equations for free vibration of the beam are formulated with taking into account actual position of the neutral plane. The obtained frequency equation and mode shapes are used for analysis of the beam mode shapes in dependence on the material and crack parameters. Numerical results validate usefulness of the proposed herein theory and show that mode shapes are good indication for detecting multiple cracks in Timoshenko FGM beams.

Free and forced vibration analysis of multiple cracked FGM multi span continuous beams using dynamic stiffness method

“…• Likely to the natural frequencies, mode shapes of cracked continuous FGM beam has critical points where crack makes no change in certain mode shape like a simple span beam (Lien, Đuc, et al, 2017b). For instance, crack at position of 0.2m from the left end of the second span does not change the first mode shape or crack at position 1.05m and 1.12m make no change in second mode shape (Fig.…”

“…The frequency equation in the form of third-order determinant that significantly simplifies calculating natural frequencies for FGM Euler-Bernoulli beam containing an arbitrary number of open edge cracks was established by Aydin (Aydin, 2013). The natural frequencies and mode shapes of multiple cracked FGM Timoshenko beam were investigated in (Lien, Duc, & Khiem, 2017a), (Lien, Đuc, & Khiem, 2017b) using the rotational spring model of cracks and actual position of neutral plane (Eltaher, Alshorbagy, & Mahmoud, 2013). The obtained results show that, natural frequencies calculated with respect to neutral axis are over estimated than those calculated at the centroidal axis.…”

Free vibration analysis of cracked functionally graded non-uniform beams

“…The use of FG materials eliminates interlaminar stress concentration and due to its unique properties provides strength and toughness of the structure [2]. A large number of studies related to vibration characteristics of intact and cracked FG uniform beams are available [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Furthermore the studies have been extended in mechanical analysis of small-sized structures [23][24][25][26][27][28][29][30][31].…”

Crack identification in multiple cracked beams made of functionally graded material by using stationary wavelet transform of mode shapes

“…As the FEM is formulated on the base of frequency independent polynomial shape function, it could not be used to capture all necessary high frequencies and mode shapes of interest. An alternative approach called DSM fulfilled the gap of FEM by using frequency-dependent shape functions that are c 2019 Vietnam Academy of Science and Technology found as exact solution of vibration problem in the frequency domain [17][18][19][20][21]. Although exact solutions of the vibration problem are not easily constructed for complete structures, but they, if were available, enable to study exact response of the beam in arbitrary frequency range.…”