Skip to main content
SpringerLink
Log in

The Variational Formulation of a Rod in Torsional Vibration for Crack Identification

  • Chapter
  • First Online:
Analytical Methods in Rotor Dynamics

Part of the book series: Mechanisms and Machine Science ((Mechan. Machine Science,volume 9))

  • 3441 Accesses

Abstract

In Chap. 10 the Hu-Washizu-Barr variational formulation is used for the development of the differential equation and boundary conditions for a cracked rod. Based on the general variational principle and independent assumptions about displacement, momentum, strain and stress fields of the cracked rod with one or more pairs of transverse symmetrically disposed open edge cracks along its length, the equations of motion in torsional vibration were derived. Crack is introduced as a stress disturbance function, and stress field is determined by fracture mechanics methods. Strain energy density theory has been used for an accurate evaluation of the stress disturbance function. The strain energy density criterion is based on local density of the energy field in the crack tip region, and no special assumptions on the direction in which the energy released by the separating crack surfaces is required.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Sih, G.C.: Multiple hierarchical scale-dependency on physical mechanisms of material damage: Macromechanical, microstructural and nanochemical, particle and continuum aspects of mesomechanics. In: Sih, G.C., Nait-Abdelaziz, M., Vu-Khanh, T. (eds.) Mesomechanics 2007, ISTE Ltd, London (2007)

    Google Scholar 

  2. Donaldson, B.K.: Analysis of Aircraft Structures an Introduction. McGraw-Hill, New York (1993)

    Google Scholar 

  3. Sih, G.C., Loeber, J.E.: Torsional vibration of an elastic solid containing a penny-shaped crack. J. Acoust. Soc. Am. 44(5), 1237–1245 (1968)

    Article  MATH  Google Scholar 

  4. Loeber, J.E., Sih, G.C.: Torsional wave scattering about a penny-shaped crack on a bimaterial interface. In: Sih, G.C. (eds.) Dynamic Crack Propagation, pp. 513–28. Noordhoff, Leyden (1973)

    Google Scholar 

  5. Wauer, J.: On the dynamics of cracked rotors: A literature survey. Appl. Mech. Rev. 43(1), 13–17 (1990)

    Google Scholar 

  6. Gasch, R.: A survey of the dynamic behavior of a simple rotating shaft with a transverse crack. J. Sound Vibr. 160, 313–332 (1993)

    Article  MATH  Google Scholar 

  7. Dimarogonas, A.D.: Vibration for engineers, 2nd edn. Prentice-Hall, Upper Saddle River (1996)

    Google Scholar 

  8. Dimarogonas, A.D.: Vibration of cracked structures: A state of the art review. Eng. Fract. Mech. 55(5), 831–857 (1996)

    Article  Google Scholar 

  9. Dimarogonas, A.D.: Dynamic response of cracked rotors. General Electric Co., Schenectady, New York (Internal Report) (1970)

    Google Scholar 

  10. Dimarogonas, A.D.: Dynamics of cracked shafts. General Electric Co., Schenectady, New York (Internal Report) (1971)

    Google Scholar 

  11. Pafelias, T.: Dynamic behaviour of a cracked rotor. General Electric Co., technical information series, no. DF-74-LS-79 (1974)

    Google Scholar 

  12. General Electric Co.: A methodology for predicting torsional fatigue life of turbine generator shafts using crack initiation plus propagation, EL-4333 research project 1531-1, Final report (1985)

    Google Scholar 

  13. Edwards, S., Lees, A.W., Friswell, M.I.: Fault diagnosis of rotating machinery. Shock Vibr. Digest Shock Vib. Dig. 30(1), 4–13 (1998)

    Article  Google Scholar 

  14. Meng, G., Hahn, E.J.: Dynamic response of a cracked rotor with some comments on crack detection ASME. J. Eng. Gas Turbines Power 119, 447–455 (1997)

    Article  Google Scholar 

  15. Sekhar, A.S., Prabhu, B.S.: Condition monitoring of cracked rotors through transient response. Mech. Mach. Theor. 33(8), 1167–1175 (1988)

    Article  Google Scholar 

  16. Bicego, V., Lucon, E., Rinaldi, C., Crudeli, R.: Failure analysis of a generator rotor with a deep crack detected during operation: Fractographic and fracture mechanics approach. Nucl. Eng. Des. 188, 173–183 (1999)

    Article  Google Scholar 

  17. Irretier, H.: Mathematical foundations of experimental modal analysis in rotor dynamics. Mech. Syst. Sign. Proces. 13(2), 183–191 (1999)

    Article  Google Scholar 

  18. He, Y., Guo, D., Chu, F.: Using genetic algorithms and finite element methods to detect shaft crack for rotor-bearing system. Math. Comput. Simul. 57, 95–108 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gounaris, G.D., Papadopoulos, C.A.: Crack identification in rotating shafts by coupled response measurements. Eng. Fract. Mech. 69, 339–352 (2002)

    Article  Google Scholar 

  20. Keiner, H., Gadala, M.S.: Comparison of different modelling techniques to simulate the vibration of a cracked rotor. J. Sound Vibr. 254(5), 1012–1024 (2002)

    Article  Google Scholar 

  21. Kalkat, M., Yildirim, S., Uzmay, I.: Rotor dynamics analysis of rotating machine systems using artificial neural networks. Int. J. Rotating Mach. 9, 255–262 (2003)

    Google Scholar 

  22. Yang, B., Suh, C.S.: Interpretation of crack-induced rotor non-linear response using instantaneous frequency. Mech. Syst. Signal Proces. 18, 491–513 (2004)

    Article  Google Scholar 

  23. Sekhar, A.S.: Model-based identification of two cracks in a rotor system. Mech. Syst. Sig. Proces. 18, 977–983 (2004)

    Article  Google Scholar 

  24. Sekhar, A.S.: Detection and monitoring of crack in a coast-down rotor supported on fluid film bearings. Tribol. Int. 37, 279–287 (2004)

    Article  Google Scholar 

  25. Seibold, S., Weinert, K.: A time domain method for the localization of cracks in rotors. Eur. J. Mech. A/Solids 21, 793–810 (2002)

    Article  MathSciNet  Google Scholar 

  26. Andrieux, S., Vare, C.: A 3D cracked beam model with unilateral contact. Application to rotors. J. Sound Vibr. 194(1), 67–82 (1996)

    Article  Google Scholar 

  27. Chondros, T.G.: The continuous crack flexibility method for crack identification. Fatigue Fract. Eng. Mater. Struct. 24, 643–650 (2001)

    Article  Google Scholar 

  28. Chondros, T.G.: Variational formulation of a rod under torsional vibration for crack identification. Fatigue Fract. Eng. Mater. Struct. 44(1), 95–104 (2005)

    Google Scholar 

  29. Chondros, T.G., Labeas, G.: Torsional vibration of a cracked rod by variational formulation and numerical analysis. J. Sound Vibr. 301(3–5), 994–1006 (2007)

    Article  Google Scholar 

  30. Christides, S., Barr, A.D.S.: One-dimensional theory of cracked Bernoulli-Euler beams. Int. J. Mech. Sci. 26(11/12), 639–648 (1984)

    Article  Google Scholar 

  31. Barr, A.D.S.: An extension of the Hu-Washizu variational principle in linear elasticity for dynamic problems. J. Appl. Mech. Trans. ASME 33(2), 465 (1966)

    Article  MathSciNet  Google Scholar 

  32. Hu, H.C.: On some variational principles in the theory of elasticity and plasticity. Sci. Sin. 4, 33–55 (1955)

    MATH  Google Scholar 

  33. Tada, H., Paris, P., Sih., G.C.: The stress analysis of cracks handbook. Del Research Corporation, Hellertown, Pennsylvania (1973, 1985)

    Google Scholar 

  34. Sneddon, I.N.: The distribution of stress in the neighborhood of a crack in an elastic solid. Proc. Roy. Soc. Lond. A, 187 (1946)

    Google Scholar 

  35. Sih, G.C.: Some basic problems in fracture mechanics and new concepts. Eng. Fract. Mech. 5, 365–377 (1973)

    Article  Google Scholar 

  36. Sih, G.C., Mcdonald, B.: Fracture mechanics applied to engineering problems, strain energy density fracture criterion. Eng. Fract. Mech. 6, 493–507 (1974)

    Article  Google Scholar 

  37. Ismail, A.E., Ariffin, A.K., Abdullah, S., Ghazali, M.J., Daud, R.: Mode III stress intensity factors of surface crack in round bars. Adv. Mater. Res. 214, 92–96 (2011)

    Google Scholar 

  38. Sih, G.C.: Mechanics of fracture initiation and propagation. Kluver, Boston (1991)

    Book  Google Scholar 

  39. Love, A.E.H., The mathematical theory of elasticity, 4th edn. Cambridge University Press, Cambridge (1952)

    Google Scholar 

  40. Dimarogonas, A.D., Massouros, G.: Torsional vibration of a shaft with a circumferential crack. Eng. Fract. Mech. 15(3–4), 439–444 (1981)

    Article  Google Scholar 

  41. Wauer, J.: Modelling and formulation of equation of motion for cracked rotating shafts. Int. J. Sol. Str. 26(4), 901–914 (1990)

    Article  MATH  Google Scholar 

  42. Ansys, Inc. ANSYS ver. 7.1 (2003)

    Google Scholar 

  43. Chondros, T.G., Dimarogonas, A.D.: Influence of cracks on the dynamic characteristics of structures. J. Vibr. Acoust. Stress Reliab. Des. 111, 251–256 (1989)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Van Godewijckstraat, 3311 GX, Dordrecht, Netherlands

    Andrew D. Dimarogonas

  2. Mechanical Engineering and Aeronautics, University of Patras, Ostfold Fylke, University Campus, Rio, 26500, Patras, Greece

    Stefanos A. Paipetis & Thomas G. Chondros

Authors
  1. Andrew D. Dimarogonas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefanos A. Paipetis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas G. Chondros
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas G. Chondros .

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Dimarogonas, A.D., Paipetis, S.A., Chondros, T.G. (2013). The Variational Formulation of a Rod in Torsional Vibration for Crack Identification. In: Analytical Methods in Rotor Dynamics. Mechanisms and Machine Science, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5905-3_10

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-5905-3_10

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-5904-6

  • Online ISBN: 978-94-007-5905-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation