Skip to main content
SpringerLink
Log in

Nonlinear vibrational response of a single edge cracked beam

  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

The nonlinear vibrational response of a breathing cracked beam was investigated. The study was done by using a new crack stiffness model to examine some of the nonlinear behaviors of a cantilever beam with a breathing crack. The quadratic polynomial stiffness equation of the cracked beam was derived based on the hypothesis that the breathing process of a crack depends on the vibration magnitude. The Galerkin method combined with the stiffness equation was used to simplify the cracked beam into a Single-degree-of-freedom (SDOF) lumped system with nonlinear terms. The multi scale method was adopted to analyze the nonlinear amplitude frequency response of the beam. The applicability of the stiffness model was discussed and parameter sensitivity studies on the dynamic response were carried out by the SDOF model for a cantilever beam. Results indicate that the new stiffness model provides an efficient tool to study the vibrational nonlinearities introuduced by the breathing crack. Therefore, it might be used to develop a nonlinear identification method of a crack in a beam.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. W. Lee, Crack identification method for tapered cantilever pipe-type beam using natural frequencies, International J. of Steel Structures, 16 (2) (2016) 467–476.

    Article  Google Scholar 

  2. M. J. Mungla, D. S. Sharma and R. R. Trivedi, Identification of a crack in clamped-clamped beam using frequencybased method and genetic algorithm, Procedia Engineering, 144 (2016) 1426–1434.

    Article  Google Scholar 

  3. N. T. Khiem and L. K. Toan, A novel method for crack detection in beam-like structures by measurements of natural frequencies, J. of Sound and Vibration, 333 (18) (2014) 4084–4103.

    Article  Google Scholar 

  4. Y. Narkis, Identification of crack location in vibrating simply supported beams, J. of Sound and Vibration, 172 (4) (1994) 549–558.

    Article  MATH  Google Scholar 

  5. N. T. Khiem and H. T. Tran, A procedure for multiple crack identification in beam-like structures from vibration mode, J. of Vibration and Control, 20 (9) (2014) 1417–1427.

    Article  MathSciNet  Google Scholar 

  6. R. O. Curadelli, J. D. Riera, D. Ambrosini and M. G. Amani, Damage detection by means of structural damping identification, Engineering Structures, 30 (12) (2008) 3497–3504.

    Article  Google Scholar 

  7. E. Douka, G. Bamnios and A. Trochidis, A method for determining the location and depth of cracks in doublecracked beams, Applied Acoustics, 65 (10) (2004) 997–1008.

    Article  Google Scholar 

  8. Z. A. Jassim, N. N. Ali, F. Mustapha and N. A. A. Jalil, A review on the vibration analysis for a damage occurrence of a cantilever beam, Engineering Failure Analysis, 31 (2013) 442–461.

    Article  Google Scholar 

  9. M. Krawczuk and W. M. Ostachowicz, Modelling and vibration analysis of a cantilever composite beam with a transverse open crack, J. of Sound and Vibration, 183 (1) (1995) 69–89.

    Article  MATH  Google Scholar 

  10. C. A. Papadopoulos and A. D. Dimarogonas, Coupled longitudinal and bending vibrations of a rotating shaft with an open crack, J. of Sound and Vibration, 117 (1) (1987) 81–93.

    Article  Google Scholar 

  11. T. G. Chondros and A. D. Dimarogonas, Vibration of a cracked cantilever beam, J. of Vibration and Acoustics, 120 (3) (1998) 742–746.

    Article  MATH  Google Scholar 

  12. M. Heydari, A. Ebrahimi and M. Behzad, Continuous model for flexural vibration analysis of Timoshenko beams with a vertical edge crack, Archive of Applied Mechanics, 85 (5) (2015) 601–615.

    Article  Google Scholar 

  13. G. Bamnios and A. Trochides, Dynamic behaviour of a cracked cantilever beam, Applied Acoustics, 45 (2) (1995) 97–112.

    Article  Google Scholar 

  14. P. Gudmundson, The dynamic behaviour of slender structures with cross-sectional cracks, J. of the Mechanics and Physics of Solids, 31 (4) (1983) 329–345.

    Article  MATH  Google Scholar 

  15. S. Orhan, Analysis of free and forced vibration of a cracked cantilever beam, NDT&E International, 40 (6) (2007) 443–450.

    Article  Google Scholar 

  16. W. M. Ostachowicz and M. Krawczuk, Analysis of the effect of cracks on the natural frequencies of a cantilever beam, J. of Sound and Vibration, 150 (2) (1991) 191–201.

    Article  Google Scholar 

  17. M. Balci and O. Gundogdu, Estimation of physical properties of laminated composites via the method of inverse vibration problem, J. of Mechanical Science and Technology, 31 (1) (2017) 29–36.

    Article  Google Scholar 

  18. T. G. Chondros, A. D. Dimarogonas and J. Yao, Vibration of a beam with a breathing crack, J. of Sound and Vibration, 239 (1) (2001) 57–67.

    Article  Google Scholar 

  19. M. Kisa and J. Brandon, The effects of closure of cracks on the dynamics of a cracked cantilever beam, J. of Sound and Vibration, 238 (1) (2000) 1–18.

    Article  Google Scholar 

  20. G. I. Giannopoulos, S. K. Georgantzinos and N. K. Anifantis, Coupled vibration response of a shaft with a breathing crack, J. of Sound and Vibration, 336 (3) (2015) 191–206.

    Article  Google Scholar 

  21. M. Chati, R. Rand and S. Mukherjee, Modal analysis of a cracked beam, J. of Sound and Vibration, 207 (2) (1997) 249–270.

    Article  MATH  Google Scholar 

  22. R. Ruotolo, C. Surace, P. Crespo and D. Storer, Harmonic analysis of the vibrations of a cantilevered beam with a closing crack, Computers & Structures, 61 (6) (1996) 1057–1074.

    Article  MATH  Google Scholar 

  23. A. P. Bovsunovsky and V. V. Matveev, Analytical approach to the determination of dynamic characteristics of a beam with a closing crack, J. of Sound and Vibration, 235 (3) (2000) 415–434.

    Article  Google Scholar 

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

    Article  Google Scholar 

  25. N. Pugno, C. Surace and R. Ruotolo, Evaluation of the non-linear dynamic response to harmonic excitation of a beam with several breathing cracks, J. of Sound and Vibration, 235 (5) (2000) 749–762.

    Article  Google Scholar 

  26. M. Rezaee and R. Hassannejad, Free vibration analysis of simply supported beam with breathing crack using perturbation method, Acta Mechanica Solida Sinica, 23 (5) (2010) 459–470.

    Article  Google Scholar 

  27. O. Giannini, P. Casini and F. Vestroni, Nonlinear harmonic identification of breathing in beams, Computers and Structures, 129 (2013) 166–177.

    Article  Google Scholar 

  28. A. D. Dimarogonas, Vibration of cracked structures: A state of the art review, Engineering Fracture Mechanics, 55 (5) (1996) 831–857.

    Article  Google Scholar 

  29. C. Surace, R. Ruotolo and D. Storer, Detecting nonlinear behaviour using the Volterra series to assess damage in beam-like structures, J. of Theoretical and Applied Mechanics, 49 (3) (2011) 905–926.

    Google Scholar 

  30. Y. C. Chu and M. H. Shen, Analysis of forced bilinear oscillators and the application to cracked beam dynamics, AIAA J., 30 (10) (1992) 2512–2519.

    Article  MATH  Google Scholar 

  31. D. Younesian, S. R. Marjani and E. Esmailzadeh, Nonlinear vibration analysis of harmonically excited cracked beams on viscoelastic foundations, Nonlinear Dynamics, 71 (1-2) (2013) 109–120.

    Article  MathSciNet  Google Scholar 

  32. A. Chatterjee, Nonlinear dynamics and damage assessment of a cantilever beam with breathing edge crack, J. of Vibration and Acoustics, 133 (5) (2011) 1–6.

    Article  Google Scholar 

  33. S. L. Tsyfansky and V. I. Beresnevich, Detection of fatigue cracks in flexible geometrically non-linear bars by vibration monitoring, J. of Sound and Vibration, 213 (1) (1998) 159–168.

    Article  Google Scholar 

  34. M. H. H. Shen and Y. C. Chu, Vibrations of beams with a fatigue crack, Computers & Structures, 45 (1) (1992) 79–93.

    Article  Google Scholar 

  35. H. Ma, J. Zeng, Z. Lang and Y. Guo, Analysis of the dynamic characteristics of a slant-cracked cantilever beam, Mechanical Systems and Signal Processing, 75 (15) (2016) 261–279.

    Article  Google Scholar 

  36. F. Semperlotti, K. W. Wang and E. C. Smith, Localization of a breathing crack using super-harmonic signals due to system nonlinearity, AIAA J., 47 (9) (2009) 2076–2086.

    Article  Google Scholar 

  37. U. Andreaus, P. Casini and F. Vestroni, Non-linear dynamics of a cracked cantilever beam under harmonic excitation, International J. of Non-linear Mechanics, 42 (3) (2007) 566–575.

    Article  Google Scholar 

  38. U. Andreaus and P. Baragatti, Cracked beam identification by numerically analysing the nonlinear behaviour of the harmonically forced response, J. of Sound and Vibration, 330 (4) (2011) 721–742.

    Article  Google Scholar 

  39. A. D. Dimarogonas, S. A. Paipetis and T. G. Chondros, Analytical methods in rotor dynamics, Springer Dordrecht Heidelberg (2013).

    Google Scholar 

  40. M. Lalanne, P. Berthier and J. D. Hagopian, Mechanical vibrations for engineers, Wiley, New York (1983).

    MATH  Google Scholar 

  41. R. Mulyukov, S. Mikhailov, R. Zaripova and D. Salimonenko, Damping properties of 18Cr-10Ni stainless steel with submicrocrystalline structure, Materials Research Bulletin, 31 (6) (1996) 639–645.

    Article  Google Scholar 

  42. S. D. Panteliou, T. G. Chondros, V. C. Argyrakis and A. D. Dimarogonas, Damping factor as an indicator of crack severity, J. of Sound and Vibration, 241 (2) (2011) 235–245.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Aeronautic Manufacturing Engineering, Nanchang Hangkong University, Nanchang, JX, 330063, China

    Wenguang Liu

  2. Department of Aerospace Engineering and Mechanics, The University of Alabama, Tuscaloosa, AL, 35487, USA

    Mark E. Barkey

Authors
  1. Wenguang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark E. Barkey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenguang Liu.

Additional information

Recommended by Associate Editor Junhong Park

Wenguang Liu received his Ph.D. from Nanjing University of Aeronautics and Astronautics, China. He is currently working as Associate Professor in Nanchang Hangkong University, China.

Mark E. Barkey received his Ph.D. from the University of Illinois at Urbana- Champaign, USA. He is currently working as Professor at the University of Alabama, USA.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, W., Barkey, M.E. Nonlinear vibrational response of a single edge cracked beam. J Mech Sci Technol 31, 5231–5243 (2017). https://doi.org/10.1007/s12206-017-1016-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-017-1016-1

Keywords

Search

Navigation