Advertisement

Nonlinear Dynamics

, Volume 98, Issue 3, pp 2097–2114 | Cite as

Geometrically exact equation of motion for large-amplitude oscillation of cantilevered pipe conveying fluid

  • Wei Chen
  • Huliang Dai
  • Qingqing Jia
  • Lin WangEmail author
Original paper
  • 107 Downloads

Abstract

Theoretical modeling and dynamic analysis of cantilevered pipes conveying fluid are presented with particular attention to geometric nonlinearities in the case of large-amplitude oscillations. To derive a new version of nonlinear equation of motion, the rotation angle of the centerline of the pipe is utilized as the generalized coordinate to describe the motion of the pipe. By using variational operations on energies of the pipe system with respect to either lateral displacement or rotation angle of the centerline, two kinds of new equations of motion of the cantilever are derived first based on Hamilton’s principle. It is interesting that these two governing equations are geometrically exact, different-looking but essentially equivalent. With the aid of Taylor expansion, one of the newly developed equations of motion can be degenerated into previous Taylor-expansion-based governing equation expressed in the form of lateral displacement. Then, the proposed new equation of motion is linearized to determine the stability of the cantilevered pipe system. Finally, nonlinear analyses are conducted based on the current geometrically exact model. It is shown that the cantilevered pipe would undergo limit-cycle oscillation after flutter instability is induced by the internal fluid flow. As expected, quantitative agreement between geometrically exact model and Taylor-expansion-based model can be achieved when the oscillation amplitude of the pipe is relatively small. However, remarkable difference between the results of oscillation amplitudes predicted using these two models would occur for large-amplitude oscillations. The main reason is that in the Taylor-expansion-based model, high-order geometric nonlinearities have been neglected when applying the Taylor expansion, thus yielding some deviation when large-amplitude oscillations are generated. Consequently, the proposed new geometrically exact equation of motion is more reliable for large-amplitude oscillations of cantilevered pipes conveying fluid.

Keywords

Cantilevered pipe conveying fluid Geometric nonlinearity Large-amplitude oscillation Critical flow velocity Stability 

Notes

Acknowledgements

The authors would like to gratefully acknowledge the financial support of the National Natural Science Foundation of China (Nos. 11622216, 11672115) to this work.

Compliance with ethical standards

Conflict of interest

The authors have no conflict of interest.

Ethical standard

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards.

Human and animal rights

This article does not contain any studies with animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.

References

  1. 1.
    Kumar, K.A., Sugunamma, V., Sandeep, N., Reddy, J.V.R.: Numerical examination of MHD nonlinear radiative slip motion of non-newtonian fluid across a stretching sheet in the presence of a porous medium. Heat Transf. Res. 50(12), 1163–1181 (2019)CrossRefGoogle Scholar
  2. 2.
    Kumar, K.A., Sugunamma, V., Sandeep, N.: Numerical exploration of MHD radiative micropolar liquid flow driven by stretching sheet with primary slip: a comparative study. J. Non-Equilib. Thermodyn. 44(2), 101–122 (2019)CrossRefGoogle Scholar
  3. 3.
    Kumar, K.A., Reddy, J.V.R., Sugunamma, V., Sandeep, N.: MHD flow of chemically reacting Williamson fluid over a curved/flat surface with variable heat source/sink. Int. J. Fluid Mech. Res. (2019) (Forthcoming Article)Google Scholar
  4. 4.
    Kumar, K.A., Reddy, J.V.R., Sugunamma, V., Sandeep, N.: Simultaneous solutions for MHD flow of Williamson fluid over a curved sheet with nonuniform heat source/sink. Heat Transf. Res. 50(6), 581–603 (2019)CrossRefGoogle Scholar
  5. 5.
    Kumar, K.A., Sugunamma, V., Sandeep, N.: Impact of non-linear radiation on MHD non-aligned stagnation point flow of micropolar fluid over a convective surface. J. Non-Equilib. Thermodyn. 43(4), 327–345 (2018)CrossRefGoogle Scholar
  6. 6.
    Herrmann, G., Nemat-Nasser, S.: Instability modes of cantilevered bars induced by fluid flow through attached pipes. Int. J. Solids Struct. 3, 39–52 (1967)CrossRefGoogle Scholar
  7. 7.
    Païdoussis, M.P.: Dynamics of tubular cantilevers conveying fluid. J. Mech. Eng. Sci. 12, 85–103 (1970)CrossRefGoogle Scholar
  8. 8.
    Benjamin, T.B.: Dynamics of a system of articulated pipes conveying fluid. II. Experiments. Proc. R. Soc. Lond. A 261, 487–499 (1961)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gregory, R.W., Païdoussis, M.P.: Unstable oscillation of tubular cantilevers conveying fluid. II. Experiments. Proc. R. Soc. Lond. A 293, 528–542 (1966)CrossRefGoogle Scholar
  10. 10.
    Benjamin, T.B.: Dynamics of a system of articulated pipes conveying fluid. I. Theory. Proc. R. Soc. Lond. A 261, 457–486 (1961)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gregory, R.W., Païdoussis, M.P.: Unstable oscillation of tubular cantilevers conveying fluid. I. Theory. Proc. R. Soc. Lond. A 293, 512–527 (1966)CrossRefGoogle Scholar
  12. 12.
    Zhang, Y.L., Chen, L.Q.: External and internal resonances of the pipe conveying fluid in the supercritical regime. J. Sound Vib. 332(9), 2318–2337 (2013)CrossRefGoogle Scholar
  13. 13.
    Ding, H., Ji, J.C., Chen, L.Q.: Nonlinear vibration isolation for fluid-conveying pipes using quasi-zero stiffness characteristics. Mech. Syst. Signal. Pr. 121, 675–688 (2019)CrossRefGoogle Scholar
  14. 14.
    Zhang, Y.L., Chen, L.Q.: Internal resonance of pipes conveying fluid in the supercritical regime. Nonlinear Dyn. 67(2), 1505–1514 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Païdoussis, M.P., Issid, N.T.: Dynamic stability of pipes conveying fluid. J. Sound Vib. 33, 267–294 (1974)CrossRefGoogle Scholar
  16. 16.
    Païdoussis, M.P., Issid, N.T.: Experiments on parametric resonance of pipes containing pulsatile flow. J. Appl. Mech. 43, 198–202 (1976)CrossRefGoogle Scholar
  17. 17.
    Païdoussis, M.P., Sundararajan, C.: Parametric and combination resonances of a pipe conveying pulsating fluid. J. Appl. Mech. 42, 780–784 (1975)CrossRefGoogle Scholar
  18. 18.
    Wang, Y., Wang, L., Ni, Q., et al.: Non-planar responses of cantilevered pipes conveying fluid with intermediate motion constraints. Nonlinear Dyn. 93(2), 505–524 (2018)CrossRefGoogle Scholar
  19. 19.
    Rong, B., Lu, K., Rui, X.T., et al.: Nonlinear dynamics analysis of pipe conveying fluid by Riccati absolute nodal coordinate transfer matrix method. Nonlinear Dyn. 92(2), 699–708 (2018)CrossRefGoogle Scholar
  20. 20.
    Liu, Z.Y., Wang, L., Sun, X.P.: Nonlinear forced vibration of cantilevered pipes conveying fluid. Acta Mech. Solida Sin. 31(1), 32–50 (2018)CrossRefGoogle Scholar
  21. 21.
    Tang, Y., Yang, T., Fang, B.: Fractional dynamics of fluid-conveying pipes made of polymer-like materials. Acta Mech. Solida Sin. 31(2), 243–258 (2018)CrossRefGoogle Scholar
  22. 22.
    Païdoussis, M.P.: Fluid-Structure Interactions: Slender Structures and Axial Flow. Academic Press, London (1998)Google Scholar
  23. 23.
    Holmes, P.J.: Bifurcations to divergence and flutter in flow-induced oscillations: a finite-dimensional analysis. J. Sound Vib. 53, 471–503 (1977)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Cheg, E., Dowell, E.H.: A theoretical analysis of nonlinear effects on the flutter and divergence of a tube conveying fluid. Flow-Induced Vibrations, pp. 65–81. Wiley, New York (1979)Google Scholar
  25. 25.
    Lundgren, T.S., Sethna, P.R., Bajaj, A.K.: Stability boundaries for flow induced motions of tubes with an inclined terminal nozzle. J. Sound Vib. 64, 553–571 (1979)CrossRefGoogle Scholar
  26. 26.
    Bajaj, A.K.: Bifurcation to periodic solutions in rotationally symmetric discrete mechanical systems. Ph.D. Thesis, University of Minnesota (1981)Google Scholar
  27. 27.
    Rousselet, J., Herrmann, G.: Dynamic behaviour of continuous cantilevered pipes conveying fluid near critical velocities. J. Appl. Mech. 48, 943–947 (1981)CrossRefGoogle Scholar
  28. 28.
    Holmes, P.J.: Pipes supported at both ends cannot flutter. J. Appl. Mech. 45, 619–622 (1978)CrossRefGoogle Scholar
  29. 29.
    Semler, C., Li, G.X., Païdoussis, M.P.: The nonlinear equations of motion of pipes conveying fluid. J. Sound Vib. 169, 577–599 (1994)CrossRefGoogle Scholar
  30. 30.
    Païdoussis, M.P., Semler, C.: Nonlinear dynamics of a fluid-conveying cantilevered pipe with an intermediate spring support. J. Fluids Struct. 7(3), 269–298 (1993)CrossRefGoogle Scholar
  31. 31.
    Païdoussis, M.P., Moon, F.C.: Nonlinear and chaotic fluidelastic vibrations of a flexible pipe conveying fluid. J. Fluids Struct. 2, 567–591 (1988)CrossRefGoogle Scholar
  32. 32.
    Païdoussis, M.P., Semler, C.: Non-linear dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end. Int. J. Nonlinear Mech. 33(1), 15–32 (1998)CrossRefGoogle Scholar
  33. 33.
    Tang, D.M., Dowell, D.H.: Chaotic oscillations of a cantilevered pipe conveying fluid. J. Fluids Struct. 2(3), 263–283 (1998)CrossRefGoogle Scholar
  34. 34.
    Païdoussis, M.P., Semler, C.: Nonlinear and chaotic oscillations of a constrained cantilevered pipe conveying fluid: a full nonlinear analysis. Nonlinear Dyn. 4, 655–670 (1993)CrossRefGoogle Scholar
  35. 35.
    Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (2005)zbMATHGoogle Scholar
  36. 36.
    Lacarbonara, W.: Nonlinear Structural Mechanics: Theory, Dynamical Phenomena and Modeling. Springer, New York (2013)CrossRefGoogle Scholar
  37. 37.
    Farokhi, H., Ghayesh, M.H.: Extremely large oscillations of cantilevers subject to motion constraints. J. Appl. Mech. 86, 031001 (2019)CrossRefGoogle Scholar
  38. 38.
    Stoker, J.J.: Nonlinear Elasticity. Gordon and Breach Science Publishers, New York (1968)zbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Wei Chen
    • 1
    • 2
  • Huliang Dai
    • 1
    • 2
  • Qingqing Jia
    • 1
    • 2
  • Lin Wang
    • 1
    • 2
    Email author
  1. 1.Department of MechanicsHuazhong University of Science and TechnologyWuhanChina
  2. 2.Hubei Key Laboratory for Engineering Structural Analysis and Safety AssessmentWuhanChina

Personalised recommendations