Journal of Vibration Engineering & Technologies

, Volume 6, Issue 6, pp 483–494 | Cite as

Free Vibration Analysis of Functionally Graded Shaft System with a Surface Crack

  • Debabrata Gayen
  • Debabrata ChakrabortyEmail author
  • Rajiv Tiwari
Original Paper



Free vibration analysis of functionally graded (FG) non-spinning simply supported shaft having a transverse surface crack has been carried out. Material properties of the FG shaft are assumed to be graded radially following the power law of material gradation.


Local flexibility coefficients as a function of depth of crack are computed with the help of Paris’ equation along with Castigliano’s theorem. Finite element (FE) formulation has been done using Timoshenko beam elements with two nodes and having four degrees of freedom per node. Based on the formulation, FE analysis has been performed to understand the effects of shaft’s slenderness, crack’s depth and location, gradient parameter and thermal gradient on the free vibration response of the cracked FG shaft.


FE analysis show that for a cracked FG shaft, both gradation parameter and crack parameters have significant influence on the free vibration response.


Gradation parameter could be suitably chosen in the design of FG shafts so as to minimize the increase in flexibility due to appearance of cracks.


Functionally graded material Material gradient index Flexibility coefficient Cracked shaft Temperature effects 


  1. 1.
    Niino M, Hirai T, Watanabe R (1987) The functionally gradient materials. J Jpn Soc Compos Mater 13:254–264CrossRefGoogle Scholar
  2. 2.
    Koizumi M (1993) The concept of FGM. Ceram Trans 34:3–10Google Scholar
  3. 3.
    Reddy JN, Chin CD (1998) Thermoelastical analysis of functionally graded cylinders and plates. J Therm Stress 21(6):593–626CrossRefGoogle Scholar
  4. 4.
    Shen H-S (2009) Functionally graded materials nonlinear analysis of plates and shells. CRC, Taylor & Francis, Boca RatonCrossRefGoogle Scholar
  5. 5.
    Lanhe W (2004) Thermal buckling of a simply supported moderately thick rectangular FGM plate. Compos Struct 64(2):211–218CrossRefGoogle Scholar
  6. 6.
    Azadi M (2011) Free and forced vibration analysis of FG beam considering temperature dependency of material properties. J Mech Sci Technol 25(1):69–80CrossRefGoogle Scholar
  7. 7.
    Piovan MT, Sampaio R (2009) A study on the dynamics of rotating beams with functionally graded properties. J Sound Vib 327(1–2):134–143CrossRefGoogle Scholar
  8. 8.
    Kona M, Ray K (2010) Parametric instability and control of functionally graded beams. J Vib Eng Technol 9(1):105–118Google Scholar
  9. 9.
    Mohanty SC, Dash RR, Rout T (2014) Parametric instability of functionally graded Timoshenko beam in high temperature environment. J Vib Eng Technol 2(3):205–228Google Scholar
  10. 10.
    Gayen D, Roy T (2014) Finite element based vibration analysis of functionally graded spinning shaft system. J Mech Eng Sci Part C 228(18):3306–3321CrossRefGoogle Scholar
  11. 11.
    Dimarogonas AD (1996) Vibration of cracked structures: a state of the art review. Eng Fract Mech 55(5):831–857CrossRefGoogle Scholar
  12. 12.
    Papadopoulos CA, Dimarogonas AD (1987) Coupled longitudinal and bending vibrations of a rotating shaft with an open crack. J Sound Vib 117(1):81–93CrossRefGoogle Scholar
  13. 13.
    Papadopoulos CA (2004) Some comments on the calculations of the local flexibility of cracked shafts. J Sound Vib 278:1205–1211CrossRefGoogle Scholar
  14. 14.
    Sekhar AS, Prabhu BS (1992) Crack detection and vibration characteristics of cracked shafts. J Sound Vib 157(2):375–381CrossRefGoogle Scholar
  15. 15.
    Sinou JJ, Lees AW (2005) The influence of cracks in rotating shafts. J Sound Vib 285(4–5):1015–1037CrossRefGoogle Scholar
  16. 16.
    Darpe AK, Chawla A, Gupta K (2002) Analysis of the response of a cracked Jeffcott rotor to axial excitation. J Sound Vib 249(3):429–445CrossRefGoogle Scholar
  17. 17.
    Yang J, Chen Y (2008) Free vibration and buckling analyses of functionally graded beams with edge cracks. Compos Struct 83(1):48–60CrossRefGoogle Scholar
  18. 18.
    Ke LL, Yang J, Kitipornchai S, Xiang Y (2009) Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials. Mech Adv Mater Struct 16(6):488–502CrossRefGoogle Scholar
  19. 19.
    Ferezqi HZ, Tahani M, Toussi HE (2010) Analytical approach to free vibrations of cracked Timoshenko beams made of functionally graded materials. Mech Adv Mater Struct 17(5):353–365CrossRefGoogle Scholar
  20. 20.
    Wei D, Liu Y, Xiang Z (2012) An analytical method for free vibration analysis of functionally graded beams with edge cracks. J Sound Vib 331(7):1686–1700CrossRefGoogle Scholar
  21. 21.
    Aydin K (2013) Free vibration of functionally graded beams with arbitrary number of surface cracks. Eur J Mech A Solids 42:112–124MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tada H, Paris PC, Irwin GR (1973) The stress analysis of cracks handbook. Del Research Corporation, HellertownGoogle Scholar
  23. 23.
    Touloukian YS (1967) Thermophysical properties of high temperature solid materials. McMillan, New YorkGoogle Scholar

Copyright information

© Krishtel eMaging Solutions Private Limited 2018

Authors and Affiliations

  • Debabrata Gayen
    • 1
  • Debabrata Chakraborty
    • 1
    Email author
  • Rajiv Tiwari
    • 1
  1. 1.Mechanical Engineering DepartmentIndian Institute of Technology GuwahatiGuwahatiIndia

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