Acta Mechanica Solida Sinica

, Volume 23, Issue 5, pp 459–470 | Cite as

Free Vibration Analysis of Simply Supported Beam with Breathing Crack using Perturbation Method

Article

Abstract

In this paper, a new analytical method for vibration analysis of a cracked simply supported beam is investigated. By considering a nonlinear model for the fatigue crack, the governing equation of motion of the cracked beam is solved using perturbation method. The solution of the governing equation reveals the superharmonics of the fundamental frequency due to the nonlinear effects in the dynamic response of the cracked beam. Furthermore, considering such a solution, an explicit expression is also derived for the system damping changes due to the changes in the crack parameters, geometric dimensions and mechanical properties of the cracked beam. The results show that an increase in the crack severity and approaching the crack location to the middle of the beam increase the system damping. In order to validate the results, changes in the fundamental frequency ratios against the fatigue crack severities are compared with those of experimental results available in the literature. Also, a comparison is made between the free response of the cracked beam with a given crack depth and location obtained by the proposed analytical solution and that of the numerical method. The results of the proposed method agree with the experimental and numerical results.

Key words

free vibration cracked beam breathing crack damping factor perturbation method 

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References

  1. [1]
    Chondros, T.G., The continuous crack flexibility method for crack identification. Fatigue & Fracture of Engineering Materials & Structures, 2001, 24: 643–650.CrossRefGoogle Scholar
  2. [2]
    Loya, J.A. and Ferna’ndez-Sa’ez, L.R.J., Natural frequencies for bending vibrations of Timoshenko cracked beams. Journal of Sound and Vibration, 2006, 290: 640–653.CrossRefGoogle Scholar
  3. [3]
    Faverjon, B. and Sinou, J., Identification of an open crack in a beam using an a posteriori error estimator of the frequency response functions with noisy measurements. European Journal of Mechanics A/Solids, 2009, 28: 75–85.CrossRefGoogle Scholar
  4. [4]
    Lam, H.F., Ng, C.T. and Veidt, M., Experimental characterization of multiple cracks in a cantilever beam utilizing transient vibration data following a probabilistic approach. Journal of Sound and Vibration, 2007, 305: 34–49.CrossRefGoogle Scholar
  5. [5]
    Ke, L.L., Yang, J., Kitipornchai, S. and Xiang, Y., Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials. Mechanics of Advanced Materials and Structures, 2009, 16: 488–502.CrossRefGoogle Scholar
  6. [6]
    Friswell, M.I. and Penny, J.E.T., A simple nonlinear model of a cracked beam. In: Proceedings 10th international modal analysis conference, San Diego, CA, 1992: 516–521.Google Scholar
  7. [7]
    Kisa, M. and Brandon, J., The effects of closure of cracks on the dynamics of a cracked cantilever beam. Journal of Sound and Vibration, 2000, 238(1): 1–18.CrossRefGoogle Scholar
  8. [8]
    Rytter, A., Vibrational Based Inspection of Civil Engineering Structures. PhD Thesis, University of Aalborg, 1993.Google Scholar
  9. [9]
    Abraham, O.N.L. and Brandon, J.A., Modeling of the opening and closure of a crack. Journal of Vibration and Acoustics, 1995, 117: 370–377.CrossRefGoogle Scholar
  10. [10]
    Cheng, S.M., Wu, X.J. and Wallace, W., Vibrational response of a beam with a breathing crack. Journal of Sound and Vibration, 1999, 225(1): 201–208.CrossRefGoogle Scholar
  11. [11]
    Ariaei, A., Ziaei-Rad, S. and Ghayour, M., Vibration analysis of beams with open and breathing cracks subjected to moving masses. Journal of Sound and Vibration, 2009, 326: 709–724.CrossRefGoogle Scholar
  12. [12]
    Douka, E. and Hadjileontiadis, J.L., Time-frequency analysis of the free vibration response of a beam with a breathing crack. NDT&E International, 2005, 38: 3–10.CrossRefGoogle Scholar
  13. [13]
    Loutridis, S., Douka, E. and Hadjileontiadis, J.L., Forced vibration behavior and crack detection of cracked beams using instantaneous frequency. NDT&E International, 2005, 38: 411–419.CrossRefGoogle Scholar
  14. [14]
    Zhang, W. and Testa, R.B., Closure effects on fatigue crack detection. Journal of Engineering Mechanics, 1999, 125: 1125–1132.CrossRefGoogle Scholar
  15. [15]
    Bovsunovsky, A.P. and Surace, C., Considerations regarding superharmonic vibrations of a cracked beam and the variation in the damping caused by the presence of the crack. Journal of Sound and Vibration, 2005, 288: 865–886.CrossRefGoogle Scholar
  16. [16]
    Curadelli, R.O., Riera, J.D., Ambrosini, D. and Amani, M.G., Damage detection by means of structural damping identification. Engineering Structures, 2008, 30: 3497–3504.CrossRefGoogle Scholar
  17. [17]
    Dimarogonas, A.D., Vibration of cracked structures: a state of the art review. Engineering Fracture Mechanics, 1996, 55: 831–857.CrossRefGoogle Scholar
  18. [18]
    Chondros, T.G. and Dimarogonas, A.D., A continuous cracked beam vibration theory. Journal of Sound and Vibration, 1998, 215(1): 17–34.CrossRefGoogle Scholar
  19. [19]
    Dimarogonas, A.D. and Paipetis, S.A., Analytical Methods in Rotor Dynamics. London: Elsevier Applied Science, 1986.MATHGoogle Scholar
  20. [20]
    Meirovitch, L., Analytical Methods in Vibrations. The Macmillan Company, 1967.Google Scholar
  21. [21]
    Clough, R.W. and Penzien, J., Dynamics of Structures. New York: McGraw-Hill, Inc, 1975.MATHGoogle Scholar
  22. [22]
    Chondros, T.G., Dimarogonas, A.D. and Yao, J., Vibration of a beam with a breathing crack. Journal of Sound and Vibration, 2001, 239(1): 57–67.CrossRefGoogle Scholar
  23. [23]
    Meirovitch, L., Elements of Vibration Analysis. New York: McGraw-Hill, 1986.MATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2010

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of TabrizTabrizIran

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