Analysis of the nonlinear axial vibrations of a cantilevered pipe conveying pulsating two-phase flow

Document Type : Research Paper


1 Department of Mechanical Engineering, University of Lagos, Nigeria

2 Department of Systems Engineering, University of Lagos, Nigeria


The parametric resonance of the axial vibrations of a cantilever pipe conveying harmonically perturbed two-phase flow is investigated using the method of multiple scale perturbation. The nonlinear coupled and uncoupled planar dynamics of the pipe are examined for a scenario when the axial vibration is parametrically excited by the pulsating frequencies of the two phases conveyed by the pipe. Away from the internal resonance condition, the stability regions are determined analytically. The stability boundaries are found to reduce as the void fraction is increasing. With the amplitude of the harmonic velocity fluctuations of the phases taken as the control parameters, the presence of internal resonance condition results in the occurrence of both axial and transverse resonance peaks due to the transfer of energy between the planar directions. However, an increase in the void fraction is observed to reduce the amplitude of oscillations due to the increase in mass content in the pipe and which further dampens the motions of the pipe.


[1]       Ibrahim R.A., 2010, Mechanics of Pipes Conveying Fluids-Part 1, Fundamental Studies, Journal of Pressure Vessel Technology 113.
[2]       Ginsberg J.H., 1973, The dynamic stability of a pipe conveying a pulsatile flow, International Journal of Engineering Science 11: 1013-1024.
[3]       Chen S.S., 1971, Dynamic Stability of a Tube Conveying Fluid, ASCE Journal of Engineering Mechanics 97: 1469-1485.
[4]       Païdoussis M.P.  and Issid N.T., 1974, Dynamic stability of pipes conveying fluid” Journal of Sound and Vibration 33: 267–294.
[5]       Paidoussis M.P., Sundararajan C., 1975, Parametric and combination resonances of a pipe conveying pulsating fluid’’, Journal of Applied Mechanics 42: 780-784.
[6]       Nayfeh A.H.  and Mook D.T., 1995, Nonlinear Oscillations, John Wiley and sons, Inc. ISBN 0471121428.
[7]       Semler C., Païdoussis M.P., 1996, Nonlinear Analysis of the Parametric Resonance of a Planar Fluid-Conveying Cantilevered Pipe, Journal of Fluids and Structures 10: 787–825.
[8]       Namachchivaya N.S., Tien W.M., 1989, Bifurcation Behaviour of Nonlinear Pipes Conveying Pulsating Flow”, Journal of Fluids and Structures 3: 609–629.
[9]       Panda L.N. and. Kar R.C., 2008, Nonlinear Dynamics of a Pipe Conveying Pulsating Fluid with Combination, Principal Parametric and Internal Resonances, Journal of Sound and Vibration 309: 375-406.
[10]    Mohammadi M. and Rastgoo A., 2018, Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium, Journal of Mechanics of Advanced Materials and Structures,
[11]    Asemi S.R., Mohammadi M. and Farajpour A., 2014, A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory, Latin American Journal of Solids and Structures 11 (9): 1515–1540.
[12]    Mohammadi M. and Rastgoo A., 2019, Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core, Structural Engineering and Mechanics 69 (2): 131–143.
[13]    Danesh M., Farajpour A. and Mohammadi M., 2011, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications 39 (1): 23–27
[14]    Adegoke A. S.  and Oyediran A. A., 2007, The Analysis of Nonlinear Vibrations of Top-Tensioned Cantilever Pipes Conveying Pressurized Steady Two-Phase Flow under Thermal Loading, Mathematical and Computational Applications 22.
[15]    Ghayesh M.H., Païdoussis M.P.  and Amabili M., 2013, Nonlinear Dynamics of Cantilevered Extensible Pipes Conveying Fluid, Journal of Sound and Vibration 332: 6405–6418.
[16]    Woldesemayat M.A., Ghajar A.J., 2007, Comparison of void fraction correlations for different flow patterns in horizontal and upward inclined pipes, International Journal of Multiphase Flow 33: 347–370.
[17]    Oz H.R., 2001, Nonlinear Vibrations and Stability Analysis of Tensioned Pipes Conveying Fluid with Variable Velocity, International Journal of Nonlinear Mechanics 36: 1031-1039.
[18]    Nayfeh A.H., 2004, Perturbation Methods, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, ISBN 9780471399179.
[19]    Thomsen J.J., 2003, Vibrations and Stability, Springer, ISBN 3540401407.
Volume 51, Issue 2
December 2020
Pages 311-322
  • Receive Date: 09 January 2020
  • Revise Date: 22 January 2020
  • Accept Date: 24 January 2020