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Free and forced vibration modelling of a delaminated beam structure using a Green’s function method

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Abstract

This work proposes an analytical modelling method for free and forced vibration analyses of a delaminated beam structure based on the Green’s function method, which considers the connection among sub-beams to generate a general model of a beam with different delaminations. This analytical modelling allows the investigation on the dynamic response of a delaminated beam under frequency-varying excitations. Two models have been developed, the ‘free mode’ and ‘constrained mode’ models, for simulating the dynamic response of a delaminated beam with its solution obtained from utilizing the Green’s functions on multiple segments of the beam structure. The accuracy of the proposed models is verified by comparisons of results from the finite element models and the previous works that utilized the classical beam theory, demonstrating consistent dynamic characteristics of a delaminated beam structure. It is found that the delamination location has dominant influences on the deformation of the beam and its natural frequencies, compared to the delamination size and depth, with the higher frequency excitation generally has more influences on beam deformation compared to the low frequency excitation for a particular delamination configuration. The results demonstrated the effectiveness of the proposed modelling method for free and forced vibration analyses of a beam with delamination.

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References

  1. Tracy, J.J., Pardoen, G.C.: Effect of delamination on the fiexural stiffness of composite laminates. Thin-Walled Struct. 6, 371–383 (1988)

    Article  Google Scholar 

  2. Grady J.E., Meyn E.H.: Vibration testing of impact-damaged composite lainates. NASA Tech. Memorandum, 1989: p. 1–7.

  3. Saravanos, D.A., Hopkins, D.A.: Effects of delaminations on the damped dynamic characteristics of composite laminates: mechanics and experiments. NASA Tech. Memorandum, 1995: p. 1–34.

  4. Wang, J.T.S., Liu, Y.Y., Gibby, J.A.: Vibrations of split beams. J. Sound Vib. 84(4), 491–502 (1982)

    Article  MATH  Google Scholar 

  5. Mujumdar, P.M., Suryanarayan, S.: Flexural vibrations of beams with delaminations. J. Sound Vib. 125(3), 441–461 (1988)

    Article  MATH  Google Scholar 

  6. Luo, H., Hanagud, S.: Dynamics of delaminated beams. Int. J. Solids Struct. 37(10), 1501–1519 (2000)

    Article  MATH  Google Scholar 

  7. Torabi, K., Shariati-Nia, M., Heidari-Rarani, M.: Experimental and theoretical investigation on transverse vibration of delaminated cross-ply composite beams. Int. J. Mech. Sci. 115–116, 1–11 (2016)

    Article  Google Scholar 

  8. Della, C.N., Shu, D.: Vibration of delaminated composite laminates: a review. Appl. Mech. Rev. 60(1), 1–20 (2007)

    Article  Google Scholar 

  9. Jafari-Talookolaei, R.A., Abedi, M., Hajianmaleki, M.: Vibration characteristics of generally laminated composite curved beams with single through-the-width delamination. Compos. Struct. 138, 172–183 (2016)

    Article  Google Scholar 

  10. Shu, D.: Vibration of sandwich beams with double delaminations. Compos. Sci. Technol. 54(1), 101–109 (1995)

    Article  Google Scholar 

  11. Shen, M.H. Grady, J.: Free vibrations of delaminated beams. 1991: 3017–3025.

  12. Della, C.N., Shu, D.: Vibration of beams with double delaminations. J. Sound Vib. 282(3–5), 919–935 (2005)

    Article  Google Scholar 

  13. Della, C.N., Shu, D.: Vibration of delaminated multilayer beams. Compos. B Eng. 37(2–3), 227–236 (2005)

    Article  Google Scholar 

  14. Shams, S., Torabi, A.R., Narab, M.F., Amiri Atashgah, M.A.: Free vibration analysis of a laminated beam using dynamic stiffness matrix method considering delamination. Thin-Walled Struct. 166, 1–19 (2021)

    Article  Google Scholar 

  15. Szekrényes, A., Máté, P., Hauck, B.: On the dynamic stability of delaminated composite beams under free vibration. Acta Mech. 233(4), 1485–1512 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kargarnovin, M.H., Ahmadian, M.T., Jafari-Talookolaei, R.A.: Analytical solution for the dynamic analysis of a delaminated composite beam traversed by a moving constant force. J. Vib. Control 19(10), 1524–1537 (2012)

    Article  MathSciNet  Google Scholar 

  17. Ju, F., Lee, H.P., Lee, K.H.: Dynamic response of delaminated composite beams with intermittent contact in delaminated segments. Compos. Eng. 4, 1211–1224 (1994)

    Article  Google Scholar 

  18. Pölöskei, T., Szekrényes, A.: Dynamic stability analysis of delaminated composite beams in frequency domain using a unified beam theory with higher order displacement continuity. Compos. Struct. 272, 1–14 (2021)

    Article  Google Scholar 

  19. Pölöskei, T., Szekrényes, A.: Dynamic stability analysis of reduced delaminated planar beam structures using extended Craig-Bampton method. Appl. Math. Model. 102, 153–169 (2022)

    Article  MathSciNet  Google Scholar 

  20. Damanpack, A.R., Bodaghi, M.: A new sandwich element for modeling of partially delaminated sandwich beam structures. Compos. Struct. 256, 1–30 (2021)

    Article  Google Scholar 

  21. Kargarnovin, M.H., Ahmadian, M.T., Jafari-Talookolaei, R.-A.: Dynamics of a delaminated timoshenko beam subjected to a moving oscillatory mass. Mech. Based Des. Struct. Mach. 40(2), 218–240 (2012)

    Article  Google Scholar 

  22. Kargarnovin, M.H., Ahmadian, M.T., Jafari-Talookolaeia, R.A.: Forced vibration of delaminated Timoshenko beams subjected to a moving load. Sci. Eng. Compos. Mater. 19(2), 145–157 (2012)

    Article  Google Scholar 

  23. Abu-Hilal, M.: Forced vibration of Euler-Bernoulli beams by means of dynamic Green functions. J. Sound Vib. 267(2), 191–207 (2003)

    Article  MATH  Google Scholar 

  24. Kukla, S., Zamojska, I.: Frequency analysis of axially loaded stepped beams by Green’s function method. J. Sound Vib. 300(3–5), 1034–1041 (2007)

    Article  Google Scholar 

  25. Zhao, X., Zhao, Y.R., Gao, X.Z., Li, X.Y., Li, Y.H.: Green׳s functions for the forced vibrations of cracked Euler-Bernoulli beams. Mech. Syst. Signal Process. 68–69, 155–175 (2016)

    Article  Google Scholar 

  26. Ghannadiasl, A., Ajirlou, S.K.: Forced vibration of multi-span cracked Euler-Bernoulli beams using dynamic Green function formulation. Appl. Acoust. 148, 484–494 (2019)

    Article  Google Scholar 

  27. Chen, B., Zhao, X., Li, Y.H., Guo, Y.: Forced vibration analysis of multi-cracked Timoshenko beam with the inclusion of damping by virtue of Green’s functions. Appl. Acoust. 155, 477–491 (2019)

    Article  Google Scholar 

  28. Albassam, B.A.: Vibration control of a flexible beam structure utilizing dynamic Green’s function. J. King Saud Univ. Eng. Sci. 33, 186–200 (2021)

    Google Scholar 

  29. Lee, J.: Free vibration analysis of delaminated composite beams. Comput. Struct. 74(2), 121–129 (2000)

    Article  Google Scholar 

  30. Li, X.Y., Zhao, X., Li, Y.H.: Green’s functions of the forced vibration of Timoshenko beams with damping effect. J. Sound Vib. 333(6), 1781–1795 (2014)

    Article  Google Scholar 

  31. Ghannadiasl, A., Mofid, M.: Dynamic green function for response of timoshenko beam with arbitrary boundary conditions. Mech. Based Des. Struct. Mach. 42(1), 97–110 (2013)

    Article  Google Scholar 

  32. Liu, Y., Shu, D.W.: Free vibration analysis of rotating Timoshenko beams with multiple delaminations. Compos. B Eng. 44(1), 733–739 (2013)

    Article  Google Scholar 

  33. Kargarnovin, M.H., Jafari-Talookolaei, R.A., Ahmadian, M.T.: Vibration analysis of delaminated Timoshenko beams under the motion of a constant amplitude point force traveling with uniform velocity. Int. J. Mech. Sci. 70, 39–49 (2013)

    Article  Google Scholar 

  34. Kargarnovin, M.H., Ahmadian, M.T., Jafari-Talookolaei, R.A., Abedi, M.: Semi-analytical solution for the free vibration analysis of generally laminated composite Timoshenko beams with single delamination. Compos. B Eng. 45(1), 587–600 (2013)

    Article  Google Scholar 

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Acknowledgements

The authors acknowledge the support received from Ningbo Science and Technology Bureau—Ningbo Natural Science Foundation Programme  (Project code 202003N4183), China.

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Authors and Affiliations

  1. Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham Ningbo China, Ningbo, China

    Xuan Li & Dunant Halim

  2. Nottingham Ningbo China Beacons of Excellence Research and Innovation Institute, University of Nottingham Ningbo China, Ningbo, China

    Dunant Halim

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  1. Xuan Li
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Li, X., Halim, D. Free and forced vibration modelling of a delaminated beam structure using a Green’s function method. Acta Mech 234, 2889–2906 (2023). https://doi.org/10.1007/s00707-023-03527-0

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  • DOI: https://doi.org/10.1007/s00707-023-03527-0

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