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

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## 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.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.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.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.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.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.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.Païdoussis, M.P.: Dynamics of tubular cantilevers conveying fluid. J. Mech. Eng. Sci.
**12**, 85–103 (1970)CrossRefGoogle Scholar - 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.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.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.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.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.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.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.Païdoussis, M.P., Issid, N.T.: Dynamic stability of pipes conveying fluid. J. Sound Vib.
**33**, 267–294 (1974)CrossRefGoogle Scholar - 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.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.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.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.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.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.Païdoussis, M.P.: Fluid-Structure Interactions: Slender Structures and Axial Flow. Academic Press, London (1998)Google Scholar
- 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.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.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.Bajaj, A.K.: Bifurcation to periodic solutions in rotationally symmetric discrete mechanical systems. Ph.D. Thesis, University of Minnesota (1981)Google Scholar
- 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.Holmes, P.J.: Pipes supported at both ends cannot flutter. J. Appl. Mech.
**45**, 619–622 (1978)CrossRefGoogle Scholar - 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.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.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.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.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.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.Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (2005)zbMATHGoogle Scholar
- 36.Lacarbonara, W.: Nonlinear Structural Mechanics: Theory, Dynamical Phenomena and Modeling. Springer, New York (2013)CrossRefGoogle Scholar
- 37.Farokhi, H., Ghayesh, M.H.: Extremely large oscillations of cantilevers subject to motion constraints. J. Appl. Mech.
**86**, 031001 (2019)CrossRefGoogle Scholar - 38.Stoker, J.J.: Nonlinear Elasticity. Gordon and Breach Science Publishers, New York (1968)zbMATHGoogle Scholar