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Numerical Investigation on Vibration and Stability of Cutting Fluid Delivery Viscoelastic Conduits

  • H. S. Sunil KumarEmail author
  • R. B. Anand
  • D. L. Prabhakara
Research Article - Mechanical Engineering
  • 2 Downloads

Abstract

A fluid-conveying cantilever pipe is likely to lose stability by flutter when the fluid is conveyed at certain critical velocity. In the present work, in order to avoid instability and reduce the possibility of unbounded vibrations, parametric studies and numerical investigations were carried out to fine-tune the fluid-conveying cantilever pipe by using a sliding mass and a sliding spring. To elucidate the flow mechanism, mathematical and classical formulations have been implemented using Hamilton’s principles and the numerical experimentation has been carried out using finite element method. Parametric studies on the critical velocity of fluid have been carried out in which various parameters such as the position and stiffness of the spring and position of the sliding mass were considered. The results revealed that when the discrete spring was provided in the first half of the conduit from the support, there was a significant improvement in the flutter velocity and providing only lumped mass with or without spring would not enhance the critical flutter velocity.

Keywords

Flutter Instability Cantilever pipe Critical velocity Eigenvalues 

List of symbols

[C]

Nonsymmetric damping matrix

[k]

Nonsymmetric stiffness matrix

[M]

Symmetric mass matrix

[\(\lambda \)]

Transformation matrix

C

Damping coefficient

d

Nodal displacement

DOF

Degree of freedom

E

Young’s modulus

\({E}^{*}\)

Viscous resistance coefficient

I

Area moment of inertia

\({i}_{\mathrm{m}}\)

Forward node of mass

Im ()

Imaginary part of ()

\({i}_{\mathrm{s}}\)

Forward node of spring

k

Spring stiffness

L

Total length of pipe

\({L}_{\mathrm{m}}\)

Position of concentrated mass (measured from the support)

\({L}_{\mathrm{s}}\)

Position of spring measured from the support

M

External sliding mass

\({m}_{\mathrm{f}}\)

Mass of incompressible fluid per unit length

\({m}_{\mathrm{p} }\)

Mass of pipe per unit length

N

Shape function

R

No. of elements

Re ()

Real part of ()

t

Time

T

Total kinetic energy

\(T_{\mathrm{F}}\)

Total kinetic energy of the fluid

\(T_{\mathrm{F1}}\)

Kinetic energy equivalent to axial component of velocity of fluid

\(T_{\mathrm{F2}}\)

Kinetic energy equivalent to lateral velocity of pipe which carries fluid

\(T_{\mathrm{F3}}\)

Kinetic energy equivalent to lateral component of velocity

\(T_{\mathrm{M} }\)

Kinetic energy of the lumped mass

\(T_{\mathrm{p}}\)

kinetic energy of entire pipe

u

Nondimensional velocity of fluid

U

Elastic potential energy of the pipe

\(U_{\mathrm{cr}}\)

Nondimensional critical velocity

\(U_{\mathrm{s}}\)

Strained energy stored in the spring

v

Fluid velocity

W

Work done by fluid force

\({W}_{\mathrm{c} }\)

Work done by conservative component of the fluid force

X

Amplitude of x(t)

\(\alpha \)

Nondimensional stiffness of the discrete spring

\(\beta \)

Mass of the fluid to mass of fluid + mass of pipe ratio

\(\gamma \)

Structural damping ratio

\(\delta {W}_{\mathrm{id}}\)

Virtual work done due to structural damping

\(\delta {W}_{\mathrm{nc}}\)

Virtual work done by nonconservative component of the fluid force

\(\eta \)

Nondimensional position of the mass

\(\lambda \)

Eigen value

\(\xi \)

Nondimensional position of the spring

\(\tau \)

Period of oscillation, time

\(\psi \)

Nondimensional ratio of concentrated mass to the mass of the pipe + fluid

\(\omega \)

Nondimensional natural frequency

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References

  1. 1.
    Ezugwu, E.O.; Bonney, J.; Yamane, Y.: An overview of the machinability of aeroengine alloys. J. Mater. Process. Technol. 134, 233–253 (2003)CrossRefGoogle Scholar
  2. 2.
    Machado, A.R.; Wallbank, J.; Pashby, I.R.; Ezugwu, E.O.: Tool performance and chip control when machining Ti6A14V and inconel 901 using high pressure coolant supply. Mach. Sci. Technol. 2, 1–12 (1998)CrossRefGoogle Scholar
  3. 3.
    De Lacalle, L.N.L.; Pérez-Bilbatua, J.; Sánchez, J.A.; Llorente, J.I.; Gutiérrez, A.; Albóniga, J.: Using high pressure coolant in the drilling and turning of low machinability alloys. Int. J. Adv. Manuf. Technol. 16, 85–91 (2000)CrossRefGoogle Scholar
  4. 4.
    Hong, H.; Riga, A.T.; Cahoon, J.M.; Scott, C.G.: Machinability of steels and titanium alloys under lubrication. Wear 162–164, 34–39 (1993)CrossRefGoogle Scholar
  5. 5.
    Wang, Z.Y.; Rajurkar, K.P.; Fan, J.: Turning: Ti–6Al–4V alloy with cryogenic cooling. Trans. N. Am. Manuf. Res. Inst. SME 1, 3–8 (1996)Google Scholar
  6. 6.
    Hong, S.Y.; Ding, Y.: Cooling approaches and cutting temperatures in cryogenic machining of Ti–6Al–4V. Int. J. Mach. Tools Manuf. 41, 1417–1437 (2001)CrossRefGoogle Scholar
  7. 7.
    Hong, S.Y.; Markus, I.; Jeong, W.C.: New cooling approach and tool life improvement in cryogenic machining of titanium alloy Ti–6Al–4V. Int. J. Mach. Tools Manuf. 41, 2245–2260 (2001)CrossRefGoogle Scholar
  8. 8.
    Timoshenko, S.P.; Gere, J.M.: Theory of Elastic Stability. McGraw-Hill, New York (1961)Google Scholar
  9. 9.
    Paıdoussis, M.P.; Li, G.X.: Pipes conveying fluid: A model dynamical problem. J. Fluids Struct. 7, 137–204 (1993)CrossRefGoogle Scholar
  10. 10.
    Herrmann, G.: Dynamics and stability of mechanical system with follower forces. NASA Contractor Report CR-1782 National Aeronautical and Space Administration (1971)Google Scholar
  11. 11.
    Koiter, W.T.: Unrealistic follower forces. J. Sound Vib. 194, 636 (1996)CrossRefGoogle Scholar
  12. 12.
    Sugiyama, Y.; Langthjem, M.A.; Ryu, B.J.: Realistic follower forces. J. Sound Vib. 225, 779–782 (1999)CrossRefGoogle Scholar
  13. 13.
    Bolotin, V.V.: Dynamic instabilities in mechanics of structures. Appl. Mech. Rev. 52, R1–R9 (1999)CrossRefGoogle Scholar
  14. 14.
    Sugiyama, Y.; Sekiya, T.: Survey of the experimental studies of instability of elastic systems subjected to non-conservative forces. J. Jpn. Soc. Aeronaut. Space Sci. 19, 61–68 (1971)Google Scholar
  15. 15.
    Kuiper, G.L.; Metrikine, A.V.: Dynamic stability of a submerged, free-hanging riser conveying fluid. J. Sound Vib. 280, 1051–1065 (2005)CrossRefGoogle Scholar
  16. 16.
    D. Royalance.: Lecture note on engineering viscoelasticity. Department of Materials Science and Engineering, MIT, Cambridge. Available through I’Net (2001)Google Scholar
  17. 17.
    Benjamin, T.B.: Dynamics of a system of articulated pipes conveying fluid. I. Experiments. Proc. R. Soc. A Math. Phys. Eng. Sci. 261, 457–486 (1961)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Benjamin, T.B.: Dynamics of a system of articulated pipes conveying fluid. II. Experiments. Proc. R. Soc. A Math. Phys. Eng. Sci. 261, 487–499 (1961)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gregory, R.W.; Paidoussis, M.P.: Unstable oscillation of tubular cantilevers conveying fluid. I-Theory. Experiments. Proc. R. Soc. A Math. Phys. Eng. Sci. 293, 512–527 (1966)zbMATHGoogle Scholar
  20. 20.
    Gregory, R.W.; Paidoussis, M.P.: Unstable oscillation of tubular cantilevers conveying fluid. II. Experiments. Proc. R. Soc. A Math. Phys. Eng. Sci. 293, 528–542 (1966)Google Scholar
  21. 21.
    Sallstrom, J.H.: Fluid-conveying beams in transverse vibrations. Doctoral dissertation, Division of Solid Mechanics, Chalmers University of Technology (1992)Google Scholar
  22. 22.
    Paidoussis, M.P.: Fluid-Structure Interactions: Slender Structures and Axial Flow. Academic Press, London (1998)Google Scholar
  23. 23.
    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. Non-Linear Mech. 33, 15–32 (1998)CrossRefzbMATHGoogle Scholar
  24. 24.
    Ryu, S.U.; Sugiyama, Y.; Ryu, B.J.: Eigenvalue branches and modes for flutter of cantilevered pipes conveying fluid. Comput. Struct. 80, 1231–1241 (2002)CrossRefGoogle Scholar
  25. 25.
    Fernández, M.Á.; Le Tallec, P.: Linear stability analysis in fluid–structure interaction with transpiration. Part II: numerical analysis and applications. Comput. Methods Appl. Mech. Eng. 192, 4837–4873 (2003)CrossRefzbMATHGoogle Scholar
  26. 26.
    Sarkar, A.; Païdoussis, M.P.: A compact limit-cycle oscillation model of a cantilever conveying fluid. J. Fluids Struct. 17, 525–539 (2003)CrossRefGoogle Scholar
  27. 27.
    Wang, X.: Instability analysis of some fluid-structure interaction problems. Comput. Fluids. 32, 121–138 (2003)CrossRefzbMATHGoogle Scholar
  28. 28.
    Sarkar, A.; Paidoussis, M.P.: A cantilever conveying fluid: coherent modes versus beam modes. Int. J. Non-Linear. Mech. 39, 467–481 (2004)CrossRefzbMATHGoogle Scholar
  29. 29.
    Lin, Y.H.; Huang, R.C.; Chu, C.L.: Optimal modal vibration suppression of a fluid-conveying pipe with a divergent mode. J. Sound Vib. 271, 577–597 (2004)CrossRefGoogle Scholar
  30. 30.
    Païdoussis, M.P.; Sarkar, A.; Semler, C.: A horizontal fluid-conveying cantilever: spatial coherent structures, beam modes and jumps in stability diagram. J. Sound Vib. 280, 141–157 (2005)CrossRefGoogle Scholar
  31. 31.
    Zou, G.P.; Cheraghi, N.; Taheri, F.: Fluid-induced vibration of composite natural gas pipelines. Int. J. Solids Struct. 42, 1253–1268 (2005)CrossRefGoogle Scholar
  32. 32.
    Païdoussis, M.P.; Semler, C.; Wadham-Gagnon, M.: A reappraisal of why aspirating pipes do not flutter at infinitesimal flow. J. Fluids Struct. 20, 147–156 (2005)CrossRefGoogle Scholar
  33. 33.
    Païdoussis, M.P.; Semler, C.; Wadham-Gagnon, M.; Saaid, S.: Dynamics of cantilevered pipes conveying fluid. Part 2: dynamics of the system with intermediate spring support. J. Fluids Struct. 23, 569–587 (2007)CrossRefGoogle Scholar
  34. 34.
    Païdoussis, M.P.: The canonical problem of the fluid-conveying pipe and radiation of the knowledge gained to other dynamics problems across applied mechanics. J. Sound Vib. 310, 462–492 (2008)CrossRefGoogle Scholar
  35. 35.
    Païdoussis, M.P.; Luu, T.P.; Prabhakar, S.: Dynamics of a long tubular cantilever conveying fluid downwards, which then flows upwards around the cantilever as a confined annular flow. J. Fluids Struct. 24, 111–128 (2008)CrossRefGoogle Scholar
  36. 36.
    Wang, L.; Ni, Q.: A reappraisal of the computational modelling of carbon nanotubes conveying viscous fluid. Mech. Res. Commun. 36, 833–837 (2009)CrossRefzbMATHGoogle Scholar
  37. 37.
    Hellum, A.; Mukherjee, R.; Hull, A.J.: Flutter instability of a fluid-conveying fluid-immersed pipe affixed to a rigid body. J. Fluids Struct. 27, 1086–1096 (2011)CrossRefGoogle Scholar
  38. 38.
    Rinaldi, S.; Païdoussis, M.P.: Theory and experiments on the dynamics of a free-clamped cylinder in confined axial air-flow. J. Fluids Struct. 28, 167–179 (2012)CrossRefGoogle Scholar
  39. 39.
    Giacobbi, D.B.; Rinaldi, S.; Semler, C.; Païdoussis, M.P.: The dynamics of a cantilevered pipe aspirating fluid studied by experimental, numerical and analytical methods. J. Fluids Struct. 30, 73–96 (2012)CrossRefGoogle Scholar
  40. 40.
    Zhang, T.; Ouyang, H.; Zhang, Y.O.; Lv, B.L.: Nonlinear dynamics of straight fluid-conveying pipes with general boundary conditions and additional springs and masses. Appl. Math. Model 40(17–18), 7880–7900 (2016)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Bahaadini, R.; Saidi, A.R.; Hosseini, M.: On dynamics of nanotubes conveying nanoflow. Int. J. Eng. Sci. 123, 181–196 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Paz, M.: Structural Dynamics: Theory and Computation, 2nd edn. Springer, Berlin (2012)Google Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringNational Institute of TechnologyTiruchirappalliIndia
  2. 2.Sahyadri College of Engineering and ManagementMangaloreIndia

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