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Meccanica

pp 1–12 | Cite as

Stability analysis of pipes conveying fluid with fractional viscoelastic model

  • M. Javadi
  • M. A. NoorianEmail author
  • S. Irani
Article
  • 45 Downloads

Abstract

Divergence and flutter instabilities of pipes conveying fluid with fractional viscoelastic model has been investigated in the present work. Attention is concentrated on the boundaries of the stability. Based on the Euler–Bernoulli beam theory for structural dynamics, viscoelastic fractional model for damping and, plug flow model for fluid flow, equation of motion has been derived. The effects of gravity, and distributed follower forces are also considered. By transferring the equation of motion to the Laplace domain and using the Galerkin method, the characteristic equations are obtained. By solving the eigenvalue problem, frequencies and dampings of the system have been obtained versus flow velocity. Some numerical test cases have been studied with viscoelastic fractional model and the effect of the fractional derivative order and the retardation time is investigated for various boundary conditions.

Keywords

Pipes conveying fluid Fractional viscoelastic model Stability analysis Galerkin method 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest

References

  1. 1.
    Bourrières F (1939) Sur un phénomène d’oscillation auto-entretenue en mécanique des fluides réels. E. Blondel La RougeryGoogle Scholar
  2. 2.
    Dodds HL, Runyan HL (1965) Effect of high-velocity fluid flow on the bending vibrations and static divergence of a simply supported pipe. National Aeronautics and Space Administration 2870Google Scholar
  3. 3.
    Païdoussis MP, Li GX (1993) Pipes conveying fluid: a model dynamical problem. J Fluids Struct 7(2):137–204CrossRefGoogle Scholar
  4. 4.
    Païdoussis MP (1998) Fluid–structure interactions slender structures and axial flow, vol 1. Academic Press, LondonGoogle Scholar
  5. 5.
    Amabili M, Pellicano F, Païdoussis MP (1999) Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid. Part I: stability. J Sound Vib 225(4):655–700ADSCrossRefGoogle Scholar
  6. 6.
    Lakis A, Laveau A (1991) Non-linear dynamic analysis of anisotropic cylindrical shells containing a flowing fluid. Int J Solids Struct 28(9):1079–1094CrossRefzbMATHGoogle Scholar
  7. 7.
    Firouz-Abadi RD, Noorian MA, Haddadpour H (2010) A fluid–structure interaction model for stability analysis of shells conveying fluid. J Fluids Struct 26(5):747–763CrossRefGoogle Scholar
  8. 8.
    Weaver DS, Unny TE (2010) On the dynamic stability of fluid-conveying pipes. J Appl Mech 40(1):747–763Google Scholar
  9. 9.
    Caputo M, Mainardi F (1971) Linear models of dissipation in anelastic solids. La Riv del Nuovo Cimento (1971–1977) 1(2):161–198ADSCrossRefGoogle Scholar
  10. 10.
    Caputo M, Mainardi F (1971) A new dissipation model based on memory mechanism. Pure Appl Geophys 91(1):134–147ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Oldham K, Spanier J (1974) The fractional calculus theory and applications of differentiation and integration to arbitrary order, vol 111. Elsevier, AmsterdamzbMATHGoogle Scholar
  12. 12.
    Bagley RL, Torvik J (1983) Fractional calculus-a different approach to the analysis of viscoelastically damped structures. AIAA J 21(5):741–748ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Bagley RL, Torvik J (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27(3):201–210ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Permoon MR, Haddadpour H, Javadi M (2018) Nonlinear vibration of fractional viscoelastic plate: primary, subharmonic, and superharmonic response. Int J Non-linear Mech 99:154–164CrossRefGoogle Scholar
  15. 15.
    Asgari M, Permoon MR, Haddadpour H (2017) Stability analysis of a fractional viscoelastic plate strip in supersonic flow under axial loading. Meccanica 52(7):1495–1502MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yang T, Fang B (2013) Asymptotic analysis of an axially viscoelastic string constituted by a fractional differentiation law. Int J Non-Linear Mech 49:170–174CrossRefGoogle Scholar
  17. 17.
    Di Paola M, Heuer R, Pirrotta A (2013) Fractional visco-elastic Euler–Bernoulli beam. Int J Solids Struct 50(22–23):3505–3510CrossRefGoogle Scholar
  18. 18.
    Yang T, Fang B (2012) Stability in parametric resonance of an axially moving beam constituted by fractional order material. Arch Appl Mech 82(12):1763–1770CrossRefzbMATHGoogle Scholar
  19. 19.
    Rossikhin Y, Shitikova MV (2010) Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl Mech Rev 63(1):010801ADSCrossRefGoogle Scholar
  20. 20.
    Rossikhin Y, Shitikova MV (2012) On fallacies in the decision between the caputo and Riemann–Liouville fractional derivatives for the analysis of the dynamic response of a nonlinear viscoelastic oscillator. Mech Res Commun 45:22–27CrossRefGoogle Scholar
  21. 21.
    Agrawal OP (2004) Analytical solution for stochastic response of a fractionally damped beam. J Vib Acoust 126(4):561–566CrossRefGoogle Scholar
  22. 22.
    Di Lorenzo S, Di Paola M, Pinnola FP, Pirrotta A (2014) Stochastic response of fractionally damped beams. Probab Eng Mech 35:37–43CrossRefGoogle Scholar
  23. 23.
    Spanos PD, Malara G (2014) Nonlinear random vibrations of beams with fractional derivative elements. J Eng Mech 140(9):04014069CrossRefGoogle Scholar
  24. 24.
    Liaskos KB, Pantelous AA, Kougioumtzoglou IA, Meimaris AT (2018) Implicit analytic solutions for the linear stochastic partial differential beam equation with fractional derivative terms. Syst Control Lett 121:38–49MathSciNetCrossRefGoogle Scholar
  25. 25.
    Drozdov AD (1997) Stability of a viscoelastic pipe filled with a moving fluid. ZAMM-J Appl Math Mech/Z Angew Math Mech 77(9):689–700MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yin Y, Zhu K (1973) Oscillating flow of a viscoelastic fluid in a pipe with the fractional maxwell model. Appl Math Comput 173(1):48–52MathSciNetGoogle Scholar
  27. 27.
    Wang L (2012) Flutter instability of supported pipes conveying fluid subjected to distributed follower forces. Acta Mech Solida Sin 25(1):46–52CrossRefGoogle Scholar
  28. 28.
    Deng J, Liu Y, Zhang Z, Liu W (2017) Dynamic behaviors of multi-span viscoelastic functionally graded material pipe conveying fluid. Proc Inst Mech Eng Part C: J Mech Eng Sci 231(17):3181–3192CrossRefGoogle Scholar
  29. 29.
    Deng J, Liu Y, Zhang Z, Liu W (2017) Stability analysis of multi-span viscoelastic functionally graded material pipes conveying fluid using a hybrid method. Eur J Mech-A/Solids 65:257–270MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zhou XQ, Yu DY, Shao XY, Wang CY, Zhangand S (2017) Dynamics characteristic of steady fluid conveying in the periodical partially viscoelastic composite pipeline. Compos Part B: Eng 111:387–408CrossRefGoogle Scholar
  31. 31.
    Tang Y, Zhen Y, Fang B (2018) Nonlinear vibration analysis of a fractional dynamic model for the viscoelastic pipe conveying fluid. Appl Math Model 56:123–136MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of Aerospace EngineeringK.N. Toosi University of TechnologyTehranIran

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