Introducing a New Crack Disturbance Function for Transverse Vibration Analysis of a Cracked Beam

علوم کاربردی و محاسباتی در مکانیک

Article 2, Volume 25, Issue 2, October 2014, Pages 19-32   

Document Type: Original Article

DOI: 10.22067/fum-mech.v25i2.17086

Authors

mousa rezaee^* ; saeed lotfan; vahid arab maleki

Abstract

In this paper, in order to analyze the transverse vibration of a uniform Bernoulli-Euler beam containing one single edge crack, a new continuous model is proposed for the cracked section. To this end, by using the fracture mechanics, the crack is modeled as a continuous disturbance in the stress and strain fields. By applying the Hamilton’s principle, the equation of motion and the corresponding boundary conditions of the system are derived. The resulting equation is solved by the Galerkin method, and the natural frequencies and mode shapes are obtained. In order to consider the opening and closing effects of the crack, the stiffness at the crack location is modeled by a bilinear function. The results show that the changes in vibration frequencies for a breathing crack are smaller than ones caused by an open crack. The results have been validated by the experimental and theoretical data reported in the previous studies. There is a good agreement between the results obtained through the proposed method and those obtained from the reported experimental data. This agreement shows that present model is more accurate than the previous ones, and it can predict the vibration behavior of beams more precisely.

Keywords

Vibration analysis; Cracked beam; Hamilton’s principle; Disturbance function; Bilinear stiffness model

References

1. Dimarogonas, A.D., "Vibration of cracked structures: a state of the art review", Engineering Fracture Mechanics, Vol. 55, pp. 831-857, (1996).

2. Doebling, S.W., Farrar, C.R. and Prime, M.B., "A summary review of vibration-based damage identification methods", The Shock and Vibration Digest. Vol. 30, pp. 91-105, (1998).

3. Ming-Hung Hsu, "Vibration analysis of edge-cracked beam on elastic foundation with axial loading using the differential quadrature method", Comput. Methods Appl. Mech. Engrg. Vol. 194, pp. 1–17, (2005).

4. Wauer, J., "On the dynamics of cracked rotors: a literature survey", Applied Mechanics Reviews, Vol. 43, pp. 13-17, (1990).

5. Wauer, J., "Modelling and formulation of equations of motion for cracked rotating shafts", International Journal of Solids and Structures, Vol. 26, pp. 901-914, (1990).

6. Hai-Ping Lin, "Direct and inverse methods on free vibration analysis of simply supported beams with a crack", Engineering Structures, Vol. 26, pp. 427-436, (2004).

7. Sorrenito, S., Marchesiello, S. and Piombo, B.A.D., "A new analytical technique for vibration analysis of non-proportionally damped beams", Journal of Sound and Vibration, Vol. 265,

pp. 765-782, (2003).

8. Carneiro, S.H.S. and Inman, D.J., "Comments on the free vibration of beams with a single-edge crack", Journal of Sound and Vibration, Vol. 244, pp. 729-737, (2001).

9. Zheng, T. and Ji, T., "An approximate method for determining the static deflection and natural frequency of a cracked beam", Journal of Sound and Vibration, Vol. 331, pp. 2654-2670, (2012).

10. Chondros, T.G. and Dimarogonas, A.D., "Identification of cracks in welded joints of complex structures", Journal of Sound and Vibration, Vol. 69, pp. 531-538, (1980).

11. Chondros, T.G., Dimarogonas, A.D. and Yao, J., "A consistent bar vibration theory", Journal of Sound and Vibration, Vol. 200, pp. 303-313, (1997).

12. Christides, S. and Barr, A.D.S., "One-dimensional theory of cracked Bernoulli-Euler beams", International Journal of Mechanical Sciences, Vol. 26, pp. 639-648, (1984).

13. Christides, S. and Barr, A.D.S., "Torsional vibration of cracked beams of noncircular cross-section", International Journal of Mechanical Sciences, Vol. 28, pp. 473-490, (1986).

14. Shen, M.H.H. and Pierre, C., "Free vibrations of beams with a single-edge crack", Journal of Sound and Vibration, Vol. 170, pp. 237-259, (1994).

15. Gudmundson, P., "The dynamic behavior of slender structures with cross-sectional cracks", Journal of Mechanics and Physics of Solids, Vol. 31, pp. 329-345, (1983).

16. Friswell, M.I. and Penny, J.E.T., "A simple nonlinear model of a cracked beam", Proceeding of 10th International Modal Analysis Conference, SEM society for Experimental Mechanics INC, pp.

516-521, (1992).

17. Gomes, H.M. and Almeida, F.J.F., "An analytical dynamic model for single-cracked beams including bending, axial stiffness, rotational inertia, shear deformation and coupling effects", Applied Mathematical Modeling, Vol. 38, pp. 938-948, (2014).

18. رضائی، موسی و عرب‌ملکی، وحید، «ارائه‌ی مدل غیرخطی جدید برای بررسی رفتار ارتعاشات عرضی تیر ترک‌دار با ترک خستگی»، نشریه علوم کاربردی و محاسباتی در مکانیک، سال بیست و دوم، شماره دو، (1390).

19. Tada, H., Paris, P.C. and Irvin, G.R., "The stress analysis of cracks handbook", Hellertown, Pennsylvania: Del Research Crop, (1985).

20. Paris, P.C. and Sih, G.C., "Stress analysis of cracks", Fracture toughness testing, ASME STP381, pp. 30-82, (1965).

21. Gdoutos, E.E., "Fracture mechanics: an introduction", Vol. 123, Springer, (2005).

22. Rao, S.S., "Vibration of continuous systems", John Wiley & Sons, University of Miami, (1976).

23. Douka, E. and Hadjileontiadis, L.J., "Time-frequency analysis of the free vibration response of a beam with a breathing crack", NDT&E International, Vol. 38, pp. 3-10, (2005).

24. Chondros, T.G., Dimarogonas, A.D. and Yao, J., "Vibration of a beam with a breathing crack", Journal of Sound and Vibration, Vol. 239, pp. 57-67, (2001).

25. Han-Ik Yoon, In-Soo Son and Sung-Jin Ahn, "Free vibration analysis of Euler-Bernoulli beam with double cracks", Journal of Mechanical Science and Technology, Vol. 21, pp. 476-485, (2007).

26. Rezaee, M. and Hassannejad, R., "Free vibration analysis of simply supported beam with breathing crack using perturbation method", Acta Mechanica Solida Sinica, Vol. 23, No. 5, (2010).