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2020 Vol.11, Issue 4 Preview Page

General Article

Approximate analytical solution for free axial vibration of a cracked cantilever beam
Hillal Ayas1†
,
Mokhtar Touati2
,
Mohamed Chabaat3
1Associate Professor, Department of Civil Engineering and Hydraulics, University of Jijel, B.P. 98 Ouled Aissa Jijel, Algeria.
2Associate Professor, Built Environmental Research Lab., Dept. of Structures and Materials, Civil Engineering Faculty, University of Sciences and Technology Houari Boumediene, B. P. 32 El Alia, Bab Ezzouar, Algiers, Algeria
3Professor, Built Environmental Research Lab., Dept. of Structures and Materials, Civil Engineering Faculty, University of Sciences and Technology Houari Boumediene, B. P. 32 El Alia, Bab Ezzouar, Algiers, Algeria
*hillal.ayas@gmail.com
31 December 2020. pp. 209-221
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Abstract

In this paper, free vibration analysis of a cracked cantilever bar by an Approximate Analytical approach is investigated. The crack is modeled by an equivalent axial spring with stiffness Kx according to Castigliano’s theorem. The effect of different crack depth ratio on the longitudinal frequencies and mode shapes in the cracked bar has been analyzed. It is shown that an increase in the crack depth ratio produces a decrease in the fundamental longitudinal natural frequency of a cracked bar. On the other hand, approximate analytical results indicate that the variation of a normalized longitudinal natural frequency with respect to the crack depth ratio give comparable results with those obtained from the numerical solution of an analytical equation of a cracked beam.

Keywords
crack
free vibration
axial spring
longitudinal natural frequency
castigliano’s theorem
References
1
M. Chabaat, S. Djouder, and M. Touati, Semi Empirical Stress Analysis of a Brittle Material in a Vicinity of a Stress Concentrator. Applied Mechanics and Materials, Trans Tech Publications, 2005, pp. 243-252. 10.4028/www.scientific.net/AMM.3-4.243
2
M. Chabaat and H. Ayas, Complex variable Green's functions for crack-microcracks interactions. Key Engineering Materials, Trans Tech Publications, 2011. pp. 123-128. 10.4028/www.scientific.net/KEM.465.123
3
A. Morassi, Crack-induced changes in eigenparameters of beam structures. Journal of Engineering Mechanics. 119(9) (1993), pp. 1798-1803. 10.1061/(ASCE)0733-9399(1993)119:9(1798)
4
A.D. Dimarogonas, Vibration of cracked structures: a state of the art review. Engineering fracture mechanics. 55(5) (1996), pp. 831-857. 10.1016/0013-7944(94)00175-8
5
P. Cacciola, N. Impollonia, and G. Muscolino, Crack detection and location in a damaged beam vibrating under white noise. Computers & structures. 81(18-19) (2003), pp. 1773-1782. 10.1016/S0045-7949(03)00201-3
6
X.F. Yang, A.S.J. Swamidas, and R. Seshadri, Crack identification in vibrating beams using the energy method. Journal of sound and vibration. 244(2) (2001), pp. 339-357. 10.1006/jsvi.2000.3498
7
V.V. Matveev and A.P. Bovsunovsky, Vibration-based diagnostics of fatigue damage of beam-like structures. Journal of Sound and Vibration. 249(1) (2002), pp. 23-40. 10.1006/jsvi.2001.3816
8
T.G. Chondros, A.D. Dimarogonas, and J. Yao, A consistent cracked bar vibration theory. Journal of Sound and Vibration. 200(3) (1997), pp. 303-313. 10.1006/jsvi.1996.0718
9
K.R. Collins, R.H. Plaut, and J. Wauer, Free and forced longitudinal vibrations of a cantilevered bar with a crack. Journal of vibration and acoustics, 114(2) (1992), pp. 171-177. 10.1115/1.2930246
10
F. Vestroni and D. Capecchi, Damage detection in beam structures based on frequency measurements. Journal of Engineering Mechanics. 126(7) (2000), pp. 761-768. 10.1061/(ASCE)0733-9399(2000)126:7(761)
11
T.G. Chondros, A.D. Dimarogonas, and J. Yao, Vibration of a beam with a breathing crack. Journal of Sound and vibration. 239(1) (2001), pp. 57-67. 10.1006/jsvi.2000.3156
12
T.G. Chondros, A.D. Dimarogonas, and J. Yao, A continuous cracked beam vibration theory. Journal of sound and vibration. 215(1) (1998), pp. 17-34. 10.1006/jsvi.1998.1640
13
A.D. Dimarogonas and C.A. Papadopoulos, Vibration of cracked shafts in bending. Journal of sound and vibration. 91(4) (1983), pp. 583-593. 10.1016/0022-460X(83)90834-9
14
R.D. Adams, P. Cawley, C.J. Pye, and B.J. Stone, A vibration technique for non-destructively assessing the integrity of structures. Journal of Mechanical Engineering Science. 20(2) (1978), pp. 93-100. 10.1243/JMES_JOUR_1978_020_016_02
15
W.T. Springer, K.L. Lawrence, and T.J. Lawley, Damage assessment based on the structural frequency-response function. Experimental Mechanics, 28(1) (1988), pp. 34-37. 10.1007/BF02328993
16
J.A. Loya, L. Rubio, and J. Fernández-Sáez, Natural frequencies for bending vibrations of Timoshenko cracked beams. J. of Sound and Vibration, 290(3-5) (2006), pp. 640-653. 10.1016/j.jsv.2005.04.005
17
F.B. Sayyad and B. Kumar, Approximate analytical method for damage detection in beam. International Journal of Damage Mechanics. 21(7) (2012), pp. 1064-1075. 10.1177/1056789511429142
18
F.B. Sayyad, B. Kumar, and S.A. Khan, Approximate analytical method for damage detection in free-free beam by measurement of axial vibrations. International Journal of Damage Mechanics, 22(1) (2013), pp. 133-142. 10.1177/1056789512440897
19
H. Ayas, M. Chabaat, and L. Amara, Dynamic analysis of a cracked bar by the method of characteristics. International Journal of Structural Integrity. 10(4) (2019), pp. 438-453. 10.1108/IJSI-01-2018-0001
20
J.F. Doyle, Wave Propagation in Structures. 1997, Springer, New York, NY.
21
P.F. Rizos, N. Aspragathos, and A.D. Dimarogonas, Identification of crack location and magnitude in a cantilever beam from the vibration modes. Journal of sound and vibration. 138(3) (1990), pp. 381-388. 10.1016/0022-460X(90)90593-O
22
W.M. Ostachowicz and M. Krawczuk, Analysis of the effect of cracks on the natural frequencies of a cantilever beam. Journal of sound and vibration. 150(2) (1991), pp. 191-201. 10.1016/0022-460X(91)90615-Q
23
R. Ruotolo and C. Surace, Natural frequencies of a bar with multiple cracks. Journal of Sound and Vibration. 272(1-2) (2004), pp. 301-316. 10.1016/S0022-460X(03)00761-2
24
H. Tada, P. Paris, and G. Irwin, The analysis of cracks handbook. 2000, New York: ASME Press, vol. 2, pp. 1.
25
G.C. Sih, Some basic problems in fracture mechanics and new concepts. Engineering fracture mechanics. 5(2) (1973), pp. 365-377. 10.1016/0013-7944(73)90027-1
26
S.S. Rao, Vibration of continuous systems. 2007, New York : Wiley.
Information
  • Publisher :Sustainable Building Research Center (ERC) Innovative Durable Building and Infrastructure Research Center
  • Publisher(Ko) :건설구조물 내구성혁신 연구센터
  • Journal Title :International Journal of Sustainable Building Technology and Urban Development
  • Volume : 11
  • No :4
  • Pages :209-221
  • Received Date : 2019-12-15
  • Accepted Date : 2020-12-11
  • DOI :https://doi.org/10.22712/susb.20200016
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