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


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.


Flutter Instability Cantilever pipe Critical velocity Eigenvalues 

List of symbols


Nonsymmetric damping matrix


Nonsymmetric stiffness matrix


Symmetric mass matrix

[\(\lambda \)]

Transformation matrix


Damping coefficient


Nodal displacement


Degree of freedom


Young’s modulus


Viscous resistance coefficient


Area moment of inertia


Forward node of mass

Im ()

Imaginary part of ()


Forward node of spring


Spring stiffness


Total length of pipe


Position of concentrated mass (measured from the support)


Position of spring measured from the support


External sliding mass


Mass of incompressible fluid per unit length

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

Mass of pipe per unit length


Shape function


No. of elements

Re ()

Real part of ()




Total kinetic energy


Total kinetic energy of the fluid


Kinetic energy equivalent to axial component of velocity of fluid


Kinetic energy equivalent to lateral velocity of pipe which carries fluid


Kinetic energy equivalent to lateral component of velocity

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

Kinetic energy of the lumped mass


kinetic energy of entire pipe


Nondimensional velocity of fluid


Elastic potential energy of the pipe


Nondimensional critical velocity


Strained energy stored in the spring


Fluid velocity


Work done by fluid force

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

Work done by conservative component of the fluid force


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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© 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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