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Variational Formulation of Consistent: Continuous Cracked Structural Members

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Analytical Methods in Rotor Dynamics

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

Abstract

In Chap. 9, modeling and formulation of the governing dynamic equations for cracked Euler-Bernoulli beams in flexural vibration are studied. The results of three independent evaluations of the lowest natural frequency of lateral vibrations of beams with single-edge cracks and various end conditions are investigated: continuous cracked beam vibration theory, lumped crack flexibility model vibration analysis, a finite element method, and experimental results. For the case of torsional vibration of a shaft with a peripheral crack, the Hu-Washizu-Barr variational formulation is adopted for obtaining the differential equation of motion, with plausible assumptions about displacements, momentum, strain and stress fields, along with the associated boundary conditions. For the experimental procedure crack propagation and formation of stationary cracks is achieved by a vibration technique. Continuous cracked beam theory agrees better with experimental results than lumped crack theory.

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References

  1. Irwin, G.R.: Analysis of stresses and strains near the end of a crack traversing a plate. J. Appl. Mech. 24, 361–364 (1957)

    Google Scholar 

  2. Irwin, G.R.: “Fracture”, in Handbuch der Physik, 6th edn, pp. 551–590. Springer, Heidelberg (1958)

    Google Scholar 

  3. Dimarogonas, A.D.: Vibration for Engineers, 2nd edn. Prentice Hall, Upper Saddle River (1996)

    Google Scholar 

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

    Article  Google Scholar 

  5. Chondros, T.: Dynamic Response of Cracked Beams, M.Sc. Thesis, University of Patras, Greece (1977)

    Google Scholar 

  6. Chondros, T.G., Dimarogonas, A.D.: Identification of cracks in circular plates welded at the contour, American society of mechanical engineers design engineering technical conference, St. Louis, Paper 79-DET-106, (1979)

    Google Scholar 

  7. Chondros, T.G., Dimarogonas, A.D.: Identification of cracks in welded joints of complex structures. J. Sound Vib. 69, 531–538 (1980)

    Google Scholar 

  8. Dimarogonas, A.D., Massouros, G.: Torsional vibration of a shaft with a circumferential crack. Eng. Fract. Mech. 15, 439–444 (1980)

    Article  Google Scholar 

  9. Gudmundson, P.: The dynamic behaviour of slender structures with cross-sectional cracks. J. Mech. Phys Solids 31, 329–345 (1983)

    Article  MATH  Google Scholar 

  10. Yuen, M.M.: A numerical study of the Eigen parameters of a damaged Cantilever. J. Sound Vib. 103(3), 301–310 (1985)

    Article  Google Scholar 

  11. Cuntze, R., Hajek, M.: Eigenfrequenzen eines angerissenen Kragträgers, Ingenieur-Archiv, vol. 55, 237–241. Springer, Heidelberg (1985)

    Google Scholar 

  12. Chondros, T.G., Dimarogonas, A.D.: Dynamic sensitivity of structures to cracks. J Vib. Acoust. Stress Reliab Des. 111, 251–256 (1989)

    Google Scholar 

  13. Actis, R.I., Dimarogonas, A. D.: Non-linear effects due to closing cracks in vibrating beams 12th ASME Conference on Mechanical Engineering, Vibration and Noise, Montreal, 17–20 Sept 1989

    Google Scholar 

  14. Araujo Gomes, A.J.M., Montalvao da Silva.: Theoretical and experimental data on crack depth effects in the dynamic behaviour of free–free beams, J. M. international modal analytical conference IMAC, 9, Union College, Schenectady, 274–283 1991

    Google Scholar 

  15. Narkis, Y.: Identification of crack location in vibrating simply supported beams. J. Sound Vib. 172, 549–558 (1994)

    Article  MATH  Google Scholar 

  16. Krawczuk, M., Ostachowicz, W.M.: Transverse natural vibrations of a cracked beam loaded with a constant axial force. J. Vib. Acoust. Trans. ASME 115(4), 524–533 (1995)

    Article  Google Scholar 

  17. Boltezar, Strancar, M.B., Kuhelj, A.: Identification of transverse crack location in flexural vibrations of free–free-beams. J. Sound Vib. 211(5), 729–734 (1998)

    Article  Google Scholar 

  18. Ceri, N., Vestroni, F.: Detection of damage in beams subjected to diffuse cracking. J. Sound Vib. 234(2), 259–276 (2000)

    Article  Google Scholar 

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

    Article  Google Scholar 

  20. Mcadams, D.A., Tumer, I.Y.: Toward intelligent fault detection in turbine blades: variational vibration models of damaged pinned–pinned beams. J. Vib. Acoust. 127, 467–474 (2005)

    Article  Google Scholar 

  21. Chondros, T.G., Dimarogonas, A.D.: A consistent cracked bar vibration theory. J. Sound Vib. 200, 303–313 (1997)

    Article  Google Scholar 

  22. Chondros, T.G., Dimarogonas, A.D.: Vibration of a cracked cantilever beam. J. Vib. Acoust. 120, 742–746 (1998)

    Article  Google Scholar 

  23. Chondros, T.G., Dimarogonas, A.D., Yao, J.: A continuous cracked beam vibration theory. J. Sound Vib. 215(1), 17–34 (1998)

    Article  MATH  Google Scholar 

  24. Chondros, T.G., Dimarogonas, A.D., Yao, J.: Longitudinal vibration of a bar with a breathing crack. Eng. Fract. Mech. J. 61, 503–518 (1998)

    Article  Google Scholar 

  25. Chondros, T.G., Dimarogonas, A.D., Yao, J.: Longitudinal vibration of a continuous cracked bar. Eng. Fract. Mech. J. 61, 593–606 (1998)

    Article  Google Scholar 

  26. Chondros, T.G.: Vibration of a beam with a breathing crack. J. Sound Vib. 239(1), 57–67 (2001)

    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 Vib. 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. Sinica 4, 33–55 (1955)

    MATH  Google Scholar 

  33. Shen, M.H.H., Pierre, C.: Natural modes of Bernoulli-Euler beams with symmetric cracks. J. Sound Vib. 138(1), 115–134 (1990)

    Article  Google Scholar 

  34. Shen, M.H.H., Pierre, C.: Free vibrations of beams with a single-edge crack. J. Sound Vib. 170(2), 237–259 (1994)

    Article  MATH  Google Scholar 

  35. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil. Mag. 41, 744–746 (1921)

    Google Scholar 

  36. Tada, H., Paris, P., Sih, G.C.: The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown (1973, 1985)

    Google Scholar 

  37. Benthem, J.P., Koiter, W.T.: Asymptotic approximations to crack problems. In: Sih, G.C. (ed.) Methods of Analysis and Solutions of Crack Problems, Noordhoff, Leyden (1973)

    Google Scholar 

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

    Article  Google Scholar 

  39. Sih, G.C., Mcdonald, B.: Fracture mechanics applied to engineering problems. Strain Energ Density Fract. Criterion Eng. Fract. Mech. 6, 493–507 (1974)

    Google Scholar 

  40. Sih, G.C., Barthelemy, B.M.: Mixed mode fatigue crack growth predictions. Eng. Fract. Mech. 13, 439–451 (1980)

    Article  Google Scholar 

  41. Timoshenko, S., Goodier, J.N.: Theory of Elasticity, McGraw-Hill, Inc., New York (1953)

    Google Scholar 

  42. Sokolnikoff, I.S.: Mathematical Theory of Elasticity, 2nd edn. McGraw-Hill, New York (1956)

    MATH  Google Scholar 

  43. Szabo, I.: Höhere Technische Mechanik. Springer, Heidelberg (1964)

    Book  MATH  Google Scholar 

  44. ANSYS, Inc., ANSYS ver. 7.1 (2003)

    Google Scholar 

  45. Love A.E.H.: The Mathematical Theory of Elasticity 4th edn. Cambridge University Press, Cambridge (1952)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mechanical Engineering Department, Washington University, St. Louis, Missouri, USA

    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
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  2. Stefanos A. Paipetis
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  3. Thomas G. Chondros
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Correspondence to Thomas G. Chondros .

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Dimarogonas, A.D., Paipetis, S.A., Chondros, T.G. (2013). Variational Formulation of Consistent: Continuous Cracked Structural Members. In: Analytical Methods in Rotor Dynamics. Mechanisms and Machine Science, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5905-3_9

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  • DOI: https://doi.org/10.1007/978-94-007-5905-3_9

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