Stability and Bifurcation Analysis of the Transverse Vibrations of a Cantilever Pipe Conveying Pulsating Two Phase Flow

A.S.  Adegoke; T.A.  Fashanu; A.A.  Oyediran

A.S. Adegoke Department of Mechanical Engineering, University of Lagos, Lagos, Nigeria
T.A. Fashanu Department of Systems Engineering, University of Lagos, Nigeria
A.A. Oyediran Department of Mechanical Engineering, University of Lagos, Lagos, Nigeria

Keywords: Cantilever pipe, Transverse vibration, Pulsating two-phase flow, Stability, Bifurcation

Abstract

This work presents analytical and numerical analysis of the stability and bifurcation of a cantilevered pipe conveying pulsating two phase flow. Multiple scale perturbation technique is used to obtain the stationary trivial and nontrivial solutions of its response amplitudes. Clearly, the system exhibits both stable and unstable solutions depending on the detuning of the frequencies. The fixed point of the non trivial solutions of the pipe’s dynamics is a saddle node bifurcation. On the other hand, trajectories of the trivial solutions present subcritical pitchfork and supercritical pitchfork bifurcations at the critical points. Numerical simulations are observed to be in agreement with analytical results. Also, a study on the effect of void fraction on the bifurcation points shows that at post-critical mixture velocity, a defining void fraction exists when there is a transition between the subcritical pitchfork bifurcation point and supercritical pitchfork bifurcation.

References

Gregory, R.W. and Païdoussis, M.P. (1966), “Unstable Oscillation of Tubular Cantilevers Conveying Fluid. I. Theory”, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences Vol. 293, pp.512–527.
Païdoussis, M.P. and Issid, N.T. (1974), “Dynamic Stability of Pipes Conveying Fluid”, Journal of Sound and Vibration, Vol. 33(3), pp. 267–294.
Shilling, R. and Lou, Y. K. (1980), “An Experimental Study on the Dynamic Response of a Vertical Cantilever Pipe Conveying Fluid” Journal of Energy Resource Technology, Vol. 102(3), pp. 129–135.
Semler, C., Li, G.X. and Païdoussis, M.P. (1994), “The Nonlinear Equations of Motion of Pipes Conveying Fluid” Journal of Sound and Vibration, Vol. 169, pp. 577–599.
Ghayesh, M.H., Païdoussis, M.P. and Amabili, M. (2013), “Nonlinear Dynamics of Cantilevered Extensible Pipes Conveying Fluid” Journal of Sound and Vibration, Vol. 332, pp. 6405–6418.
Monette, C. and Pettigrew, M.J. (2004), “Fluidelastic Instability of Flexible Tubes Subjected to Two –phase Internal Flow” Journal of Fluids and Structures, Vol. 19, pp. 943–956.
Adegoke, A. S. and Oyediran, A. A. (2018), “Natural Frequencies, Modes and Critical Velocities of Top Tensioned Cantilever Pipes Conveying Pressurized Steady Two-phase Flow Under Thermal Loading” Research on Engineering Structures and Materials, Vol. pp. 4 297-323.
Adegoke, A. S. and Oyediran, A. A. (2017), “The Analysis of Nonlinear Vibrations of Top-Tensioned Cantilever Pipes Conveying Pressurized Steady Two-Phase Flow under Thermal Loading”. Mathematical and Computational Applications, Vol. 22: 44.
Adegoke, A. S., Fashanu, T.A. and Oyediran, A. A. (2019), “Nonlinear Coupled Axial and Transverse Vibrations of a Cantilevered Pipe Conveying Pulsating Two Phase Flow”. First International Nonlinear Dynamics Conference Book of Abstract ISBN 978-88-944229-0-0 pp. 421-422
Wang, L., Yang, Y., Li, Y. and Wang, Y. (2018), “Dynamic Behaviours of Horizontal gas-Liquid Pipes Subjected to Hydrodynamic Slug Flow: Modelling and experiments” International Journal of Pressure Vessels and Pipping, Vol. 161, pp. 50-57.
An, C., and Su, J. (2015), “Dynamic Behavior of Pipes Conveying Gas-Liquid Phase Flow” Nuclear Engineering and Design, Vol. 292, pp. 204-212.
Woldesemayat, M.A. and Ghajar, A.J. (2007), “Comparison of Void Fraction Correlations for Different Flow Patterns in Horizontal and Upward Inclined Pipes” International Journal of Multiphase Flow, Vol.33, pp. 347–370.
Nayfeh, A.H. (2004), “Perturbation Methods” Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, ISBN 9780471399179.
Nayfeh, A.H. and Mook, D.T. (1995), “Nonlinear Oscillations” John Wiley and sons, Inc. ISBN 0471121428.
Thomsen, J.J. (2003), “Vibrations and Stability” Springer, ISBN 3540401407.
Nayfeh, A.H. and Balachandran, B. (2004), “Applied Nonlinear Dynamics” Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, ISBN 9780471593485.