Skip to main content
SpringerLink
Log in

Dynamic characteristics of multilayered beams with viscoelastic layers described by the fractional Zener model

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

This paper concerns the dynamic analysis of composite beams containing elastic and viscoelastic (VE) layers. A method for determination of the dynamic characteristics of multilayered beams with VE layers (i.e., natural frequencies, non-dimensional damping ratios and modes of vibration) is presented. The Euler–Bernoulli beam theory and the Timoshenko theory are used to describe the elastic and VE layers, respectively. To describe the mechanical properties of a VE layer, the four-parameter rheological model with fractional derivative is applied. A number of particular rheological models are particular cases of this general model. The virtual work principle and the finite element method together with the Laplace transform are used to derive the equation of motion in the frequency domain. The dynamic characteristics of a beam with VE layers are obtained as the solution to a properly defined nonlinear eigenvalue problem. The continuation method is adopted for solving the nonlinear eigenproblem. Several conclusions concerning the accuracy of the method and variability of results are presented on the basis of numerical studies.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

  1. Kerwin, E.M.J.: Damping of flexural waves by a constrained viscoelastic layer. J. Acoust. Soc. Am. 31, 952–962 (1959)

    Article  Google Scholar 

  2. Mead, D.J., Markus, S.: The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions. J. Sound Vib. 10, 163–75 (1969)

    Article  MATH  Google Scholar 

  3. Di Taranto, R.A.: Theory of vibratory bending of elastic and viscoelastic, layered, finite-length beams. J. Appl. Mech. 32, 881–886 (1965)

    Article  Google Scholar 

  4. Backstrom, D., Nilsson, A.C.: Modeling the vibration of sandwich beams using frequency-dependent parameters. J. Sound Vib. 300, 589–611 (2007)

    Article  Google Scholar 

  5. Alvelid, M.: Sixth order differential equation for sandwich beam deflection including transverse shear. Compos. Struct. 102, 29–37 (2013)

    Article  Google Scholar 

  6. Zapfe, J.A., Lesieutre, G.A.: A discrete layer beam finite element for the dynamic analysis of composite sandwich beams with integral damping layers. Comput. Struct. 70, 647–666 (1999)

    Article  MATH  Google Scholar 

  7. Sainsbury, M.G., Zhang, Q.J.: The Galerkin element method, applied to the vibration of damped sandwich beams. Comput. Struct. 71, 239–256 (1999)

    Article  Google Scholar 

  8. Banerjee, J.R., Sobey, A.J.: Dynamic stiffness formulation and free vibration analysis of a three-layered sandwich beam. Int. J. Solids Struct. 42, 2181–2197 (2005)

    Article  MATH  Google Scholar 

  9. Zhang, J., Zheng, G.T.: The Biot model and its application in viscoelastic composite structures. J. Vib. Acoust. 129, 533–540 (2007)

    Article  Google Scholar 

  10. Plagianakos, T.S., Saravanos, D.A.: High-order layerwise finite element for the damped free-vibration response of thick composite and sandwich composite plates. Int. J. Numer. Methods Eng. 77, 1593–1626 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Galucio, A.C., Deu, J.F., Ohayon, R.: Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Comput. Mech. 33, 282–291 (2004)

    Article  MATH  Google Scholar 

  12. Lin, F., Rao, M.D.: Vibration analysis of a multiple-layered viscoelastic structure using the Biot damping model. AIAA J. 48, 624–634 (2010)

    Article  Google Scholar 

  13. Won, S.G., Bae, S.H., Cho, J.R., Bae, S.R., Jeong, W.B.: Three-layered damped element for forced vibration analysis of symmetric sandwich structures with a viscoelastic core. Finite Elem. Anal. Des. 68, 39–51 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, Y., Inman, D.J.: Finite element analysis and experimental study on dynamic properties of a composite beam with viscoelastic damping. J. Sound Vib. 332, 6177–6191 (2013)

    Article  Google Scholar 

  15. Araujo, A.L., Mota Soares, C.M., Mota Soares, C.A., Herskovits, J.: Optimal design and parameter estimation of frequency dependent viscoelastic laminated sandwich composite plates. Compos. Struct. 92, 2321–2327 (2010)

    Article  MATH  Google Scholar 

  16. Ghinet, S., Atalla, N.: Modeling thick composite laminate and sandwich structure with linear viscoelastic damping. Comput. Struct. 89, 1547–1561 (2011)

    Article  Google Scholar 

  17. Johnson, C., Keinholz, D., Rogers, L.: Finite element prediction of damping in structures with constrained viscoelastic layers. AIAA J. 20, 1284–1290 (1982)

    Article  Google Scholar 

  18. Hamed, E., Rabinovitch, O.: Modeling and dynamics of sandwich beams with a viscoelastic soft core. AIAA J. 47, 2194–2211 (2009)

    Article  Google Scholar 

  19. Palfalvi, A.: A comparison of finite element formulations for dynamics of viscoelastic beams. Finite Elem. Anal. Des. 44, 814–818 (2008)

    Article  Google Scholar 

  20. Johnson, A.R., Tessler, A., Dambach, M.: Dynamics of thick viscoelastic beams. J. Eng. Mater. Technol. 119, 273–278 (1997)

    Article  Google Scholar 

  21. Bagley, R.L., Torvik, P.J.: Fractional calculus: a different approach to the analysis of viscoelastically damped structures. AIAA J. 21, 741–748 (1983)

    Article  MATH  Google Scholar 

  22. Lima, A.M.G., Guaraldo-Neto, B., Sales, T.P., Rade, D.A.: A time-domain modeling of systems containing viscoelastic materials and shape memory alloys as applied to the problem of vibration attenuation. Eng. Struct. 68, 85–95 (2014)

    Article  Google Scholar 

  23. Park, S.W.: Analytical modeling of viscoelastic dampers for structural and vibration control. Int. J. Solids Struct. 38, 8065–8092 (2001)

    Article  MATH  Google Scholar 

  24. Lewandowski, R., Bartkowiak, A., Maciejewski, H.: Dynamic analysis of frames with viscoelastic dampers: a comparison of dampers models. Struct. Eng. Mech. 41, 113–137 (2012)

    Article  Google Scholar 

  25. Lewandowski, R., Pawlak, Z.: Dynamic analysis of frames with viscoelastic dampers modelled by rheological models with fractional derivatives. J. Sound Vib. 330, 923–936 (2011)

    Article  Google Scholar 

  26. Pawlak, Z., Lewandowski, R.: The continuation method for the eigenvalue problem of structures with viscoelastic dampers. Comput. Struct. 125, 53–61 (2013)

    Article  Google Scholar 

  27. Singh, M.P., Chang, T.S., Nandana, H.: Algorithms for seismic analysis of MDOF systems with fractional derivatives. Eng. Struct. 33, 2371–2381 (2011)

    Article  Google Scholar 

  28. Schmidt, A., Gaul, L.: Finite element formulation of viscoelastic constitutive equations using fractional time derivatives. Nonlinear Dyn. 29, 37–55 (2002)

    Article  MATH  Google Scholar 

  29. Cortés, F., Elejabarrieta, M.J.: Homogenised finite element for transient dynamic analysis of unconstrained layer damping beams involving fractional derivative models. Comput. Mech. 40, 313–324 (2007)

    Article  MATH  Google Scholar 

  30. Enelund, M., Lesieutre, G.A.: Time domain modeling of damping using anelastic displacement fields and fractional calculus. Int. J. Solids Struct. 36, 4447–4472 (1999)

    Article  MATH  Google Scholar 

  31. Sorrentino, S., Fasana, A.: Finite element analysis of vibrating linear systems with fractional derivative viscoelastic models. J. Sound Vib. 299, 839–853 (2007)

    Article  Google Scholar 

  32. Di Paola, M., Pirrotta, A., Valenza, A.: Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results. Mech. Mater. 43, 799–806 (2011)

    Article  Google Scholar 

  33. Di Paola, M., Fiore, V., Piannola, F.P., Valanza, A.: On the influence of the initial ramp for a correct definition of the parameters of fractional viscoelastic materials. Mech. Mater. 69, 63–70 (2014)

    Article  Google Scholar 

  34. Podlubny, I.: Fractional Differential Equation. Academic Press, New York (1999)

    MATH  Google Scholar 

  35. Rao, D.K.: Frequency and loss factors of sandwich beams under various boundary conditions. J. Mech. Eng. Sci. 20, 271–282 (1978)

    Article  Google Scholar 

  36. Adhikari, S.: Structural Dynamic Analysis with Generalized Damping Models: Analysis. ISTE, London, Wiley, New York (2014)

    MATH  Google Scholar 

  37. Daya, E.M., Azrar, L., Potier-Ferry, M.: An amplitude equation for the non-linear vibration of viscoelastically damped sandwich beams. J. Sound Vib. 271, 789–813 (2004)

    Article  Google Scholar 

Download references

Acknowledgments

This study is supported by the National Science Centre, Poland, as part of Project No. 2013/09/B/ST8/01733, carried out in the years 2014–2016. This support is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Poznan University of Technology, ul. Piotrowo 5, 60-965, Poznan, Poland

    Roman Lewandowski & Marcin Baum

Authors
  1. Roman Lewandowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcin Baum
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roman Lewandowski.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lewandowski, R., Baum, M. Dynamic characteristics of multilayered beams with viscoelastic layers described by the fractional Zener model. Arch Appl Mech 85, 1793–1814 (2015). https://doi.org/10.1007/s00419-015-1019-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-015-1019-2

Keywords

Search

Navigation