Abstract
Free vibration of beams with an open crack at a clamped end, based on the exact, small-strain and linear two-dimensional (2-D) elasticity theory, is studied. The crack is described by a very narrow gap which is considered to have free boundaries. In the analysis, the cracked beam with mixed boundary conditions is divided from the tip of crack along the beam length into two sub-beams with pure boundary conditions. The 2-D displacement field in each sub-beam is individually developed by the Ritz method. The admissible functions of the displacements in each sub-beam are taken as the Chebyshev polynomials multiplied by suitable boundary characteristic functions which satisfy the geometric boundary conditions of the sub-beam. The total eigenvalue equation of the cracked beams can be established by using the displacement continuity conditions at the interface of two sub-beams to integrate the sub-eigenvalue equations. The method is called as the sub-domain Chebyshev–Ritz method. Three kinds of boundary conditions (clamped–simply supported, clamped–clamped and cantilevered) are considered in numerical examples. The convergence studies show that the first eight natural frequencies of the cracked beams have an accuracy of at least three significant figures. Compared with finite-element solutions and available results in literature, the present solutions exhibit excellent agreement. The effects of size parameters such as the height–span ratio and crack depth on the first several natural frequencies and mode shapes of cracked beams are performed in detail.
Similar content being viewed by others
References
Ashok D, Chandrupatle A, Tirupathi R (2002) Introduction to finite elements in engineering. Pearson Schweiz Ag 44(2):69
Aydin K (2013) Influence of crack and slenderness ratio on the eigenfrequencies of Euler-Bernoulli and Timoshenko beams. Mech Adv Mater Struct 20(5):339–352
Barad KH, Sharma DS, Vyas V (2013) Crack detection in cantilever beam by frequency based method. Procedia Eng 51:770–775
Behzad M, Ebrahimi A, Meghdari A (2008) A new continuous model for flexural vibration analysis of a cracked beam. Pol Marit Res 15(2):32–39
Biswal AR, Behera RK, Roy T (2014) Vibration analysis of a Timoshenko beam with transverse open crack by finite element method. Appl Mech Mater 592–594:2102–2106
Chondros TG (1977) Dynamic response of cracked beams. M.Sc. Thesis, University of Patras, Greece
Chondros TG, Dimarogonas AD (1980) Identification of crack in welded joints of complex structure. J Sound Vib 69(4):531–538
Chondros TG, Dimarogonas AD, Yao J (1998) A continuous cracked beam vibration theory. J Sound Vib 215(1):17–34
Christides S, Barr ADS (1984) One-dimensional theory of cracked Bernoulli-Euler beams. Int J Mech Sci 26(11):639–648
Dimarogonas AD (1976) Vibration engineering. West Publisher
Dimarogonas AD (1996) Vibration of cracked structures: a state of the art review. Eng Fract Mech 55(5):831–857
Dimarogonas AD, Paipetis SA, Chondros TG (1983) Analytical methods in rotor dynamics. Applied Science Publishers
Ebrahimi A, Behzad M, Meghdari A (2010) A continuous vibration theory for beams with a vertical edge crack. Sci Iran 17(3):194–204
Ebrahimi A, Heydari M, Behzad M (2014) A continuous vibration theory for rotors with an open edge crack. J Sound Vib 333(15):3522–3535
Fang JS, Zhou D (2016) Free vibration analysis of rotating axially functionally graded tapered Timoshenko beams. Int J Struct Stab Dyn 16(05):197–220
Gomes HM, Almeida FJFD (2014) An analytical dynamic model for single-cracked beams including bending, axial stiffness, rotational inertia, shear deformation and coupling effects. Appl Math Model 38(3):938–948
Ju FD, Mimovich ME (1988) Experimental diagnosis of fracture damage in structures by the modal frequency method. J Vib Acoust 110(4):456
Khaji N, Shafiei M, Jalalpour M (2009) Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions. Int J Mech Sci 51(9–10):667–681
Khassetarash A, Hassannejad R (2016) Energy dissipation caused by fatigue crack in beam-like cracked structures. J Sound Vib 363:247–257
Labib A, Kennedy D, Featherston C (2014) Free vibration analysis of beams and frames with multiple cracks for damage detection. J Sound Vib 333(20):4991–5003
Lele SP, Maiti SK (2002) Modelling of transverse vibration of short beams for crack detection and measurement of crack extension. J Sound Vib 257(3):559–583
Liu J, Shao YM, Zhu WD (2016) Free vibration analysis of a cantilever beam with a slant edge crack. J Mech Eng Sci 231(5):203–210
Loya JA, Rubio L, Fernández-Sáez J (2006) Natural frequencies for bending vibrations of Timoshenko cracked beams. J Sound Vib 290(3):640–653
Nahvi H, Jabbari M (2005) Crack detection in beams using experimental modal data and finite element model. Int J Mech Sci 47(10):1477–1497
Panteliou SD, Chondros TG, Argyrakis VC, Dimarogonas AD (2001) Dam** factor as an indicator of crack severity. J Sound Vib 241(2):235–245
Papadopoulos CA, Dimarogonas AD (1987a) Coupling of bending and torsional vibration of a cracked Timoshenko shaft. Ing Archiv 57(4):257–266
Papadopoulos CA, Dimarogonas AD (1987b) Coupled longitudinal and bending vibrations of a rotating shaft with an open crack. J Sound Vib 117(1):81–93
Petroski HJ (1981) Simple static and dynamic models for the cracked elastic beam. Int J Fract 17(4):71–76
Rubio L, Fernández-Sáez J, Morassi A (2016) Identification of an open crack in a beam with variable profile by two resonant frequencies. J Vib Control 24(5):839–859
Swamidas ASJ, Seshadri R, Yang X (2004) Identification of cracking in beam structures using Timoshenko and Euler formulations. J Eng Mech 130(11):1297–1308
Thomson WJ (1949) Vibration of slender bars with discontinuities in stiffness. J Appl Mech 16:203–207
Timoshenko SP, Goodier JC (1970) Theory of elasticity. McGraw-Hill, New York
Yokoyama T, Chen MC (1998) Vibration analysis of edge-cracked beams using a line-spring model. Eng Fract Mech 59(3):403–409
Zhao JL, Zhou D, Zhang JD, Hu CB (2020) Free vibration characteristics of multi-cracked beam based on Chebyshev–Ritz method. J ZheJiang Univ (engineering Science) 54(4):778–786
Zhou D, Lo SH (2015) Three-dimensional free vibration analysis of doubly-curved shells. J Vib Control 21(12):2306–2324
Zhou D, Au FTK, Cheung YK, Lo SH (2006) Effect of built-in edges on 3-D vibrational characteristics of thick circular plates. Int J Solids Struct 43(7–8):1960–1978
Acknowledgements
This work is financially supported by the National Natural Science Foundation of China (Grant Nos. 52108149; 51778285; 51878347) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20180151).
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Liu, D., Jiang, J., Han, H. et al. Free vibration analysis of beams with an open crack at clamped end by the sub-domain Chebyshev–Ritz method. Iran J Sci Technol Trans Civ Eng 47, 415–430 (2023). https://doi.org/10.1007/s40996-022-01021-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40996-022-01021-6