eesti teaduste
akadeemia kirjastus
Estonian Journal of Engineering
Exploring vibrations of cracked beams by the Haar wavelet method; pp. 58–75
PDF | doi: 10.3176/eng.2012.1.05

Ülo Lepik

Free and forced vibrations of cracked Euler–Bernoulli beams are analysed by the Haar wavelet method. Stiffness reduction for open V-shaped edge cracks is calculated. To demonstrate the efficiency of the proposed method, three problems about bending and vibration of cantilever beams are solved. The obtained results are compared with available data of other authors.


  1. Dimarogonas, A. D. Vibration of cracked structures: a state of the art review. Eng. Fract. Mech., 1996, 55, 831–857.

  2. Doebling, S. M., Farrar, C. R. and Prime, M. B. A summary review of vibration-based damage identification methods. Shock Vibr. Digest, 1998, 30, 91–105.

  3. Shifrin, E. I. and Ruotolo, R. Natural frequencies of a beam with an arbitrary number of cracks. J. Sound Vibr., 1999, 222, 409–423.

  4. Ruotolo, R. and Surace, C. Natural frequencies of a bar with multiple cracks. J. Sound Vibr., 2004, 272, 301–316.

  5. Lin, H.-P. Direct and inverse methods on free vibration analysis of simply supported beams with a crack. Eng. Struct., 2004, 26, 427–436.

  6. Wang, J. and Qiao, P. Vibration of beams with arbitrary discontinuities and boundary condition. J. Sound Vibr., 2007, 308, 12–27.

  7. Lee, J. Identification of multiple cracks in a beam using vibration amplitudes. J. Sound Vibr., 2009, 326, 205–212.

  8. Skrinar, M. Elastic beam finite element with an arbitrary number of transverse cracks. Finite Element Anal. Design, 2009, 45, 181–189.

  9. Orhan, S. Analysis of free and forced vibration of a cracked cantilever beam. NDT E Int., 2007, 40, 443–450.

10. Christides, S. and Barr, A. D. S. One dimensional theory of cracked Bernoulli–Euler beams. Int. J. Mech. Sci., 1984, 26, 639–648.

11. Sinha, J. K., Friswell, M. I. and Edwards, S. Simplified models for the location of cracks in beam structures using measured vibration data. J. Sound Vibr., 2002, 251, 13–38.

12. Carneiro, C. H. S. and Inman, D. J. Comments on the free vibrations of beams with a single edge crack. J. Sound Vibr., 2001, 244, 729–737.

13. Pugno, N. and Surace, C. Evaluation of the non-linear dynamic response to harmonic excitation of a beam with several breathing cracks. J. Sound Vibr., 2009, 235, 473–489.

14. Bovsunovski, A. P. and Matveev, V. V. Analytical approach to the determination of dynamic characteristics of a beam with a closing crack. J. Sound Vibr., 2000, 235, 415–434.

15. Chondros, T. G., Dimarogonas, A. D. and Yao, J. Vibration of a beam with a breathening crack. J. Sound Vibr., 2001, 239, 57–67.

16. Caddemi, S., Calio, I. and Marietta, M. The non-linear dynamic response of the Euler-Bernoulli beams with an arbitrary number of switching cracks. Int. J. Non-Linear Mech., 2010, 45, 714–726.

17. Biondi, B. and Caddemi, S. Closed form solutions of Euler–Bernoulli beam with singularities. Int. J. Solids Struct., 2005, 42, 3027–3044.

18. Koplow, M. A., Bhattacharyya, A. and Mann, P. M. Closed form solutions for the dynamic response of Euler–Bernoulli beams with step changes in cross-sections. J. Sound Vibr., 2006, 295, 214–225.

19. Wang, J. and Qiao, P. Vibration of beams with arbitrary discontinuities and boundary conditions. J. Sound Vibr., 2007, 308, 17–27.

20. Caddemi, S. and Calio, I. Exact closed form solution for the vibration modes of the Euler–Bernoull beam with multiple open cracks. J. Sound Vibr., 2000, 327, 749–762.

21. Caddemi, S. and Calio, I. Exact solution of the multi-cracked Euler–Bernoulli column. Int. J. Solids Struct., 2008, 45, 1332–1353.

22. Gentile, A. and Messina, A. On the continuous wavelet transforms applied to discrete vibrational data for detecting open cracks in damaged beams. Int. J. Solids Struct., 2003, 40, 295–315.

23. Zhang, W., Wang, Z. and Ma, H. Crack identification in stepped cantilever beam combining wavelet analysis with transform matrix. Acta Mech. Solida Sinica, 2009, 22, 360–368.

24. Douka, E., Loutridis, S. and Trochidis, A. Crack identification in beams using wavelet analysis. Int. J. Solids Struct., 2003, 40, 3557–3569.

25. Zhong, S. and Ojadiji, S. O. Crack detection in simply supported beams without baseline modal parameters by stationary wavelet transform. Mech. Syst. Signal Process., 2007, 21, 1853–1884.

26. Quek, S.-T., Wang, Q., Zhang, L. and Ang, K. Sensitivity analysis of crack detection in beams by wavelet technique. Int. J. Mech. Sci., 2001, 43, 2899–2910.

27. Kim, B. H., Park, T. and Voyladis, G. Z. Damage estimation in beam-like structures using the multi-resolution analysis. Int. J. Solids Struct., 2006, 43, 4238–4257.

28. Lepik, Ü. Haar wavelet method for solving higher order differential equations. Int. J. Math. Comput., 2008, 1, 84–94.

29. Lepik, Ü. Numerical solution of evolution equations by the Haar wavelet method. Appl. Math. Comput., 2007, 185, 695–704.

30. Lepik, Ü. Haar wavelet method for solving stiff differential equations. Math. Modelling and Analysis, 2009, 10, 1–17.

31. Cerri, M. M. and Vestroni, F. Identification of damage due to open cracks by change of measured frequencies. In Proc. 16th IMETA Congress of Theoretical and Applied Mechanics. Ferrari, 2003.

32. Bilello, E. Theoretical and Experimental Investigation of Damaged Beam Under Moving System. PhD Thesis. Universita degli Studi di Palermo. Palermo, Italy, 2001.

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