Dynamic Stability of Periodically Stiffened Pipes Conveying Fluid Dr. Osama J. Aldraihem Dept. of Mechanical Engineering King Saud University, Saudi Arabia INES 2003

 Motivation and Objectives  Previous Works  Modeling  Stability Analysis and Dynamic Response  Results and Discussions  Conclusions  Work in Progress/ Future Research Presentation Outline

Motivation Engineering Examples: Trans-Arabian pipeline “TAPLINE”

Motivation Heat exchanger tubesCoriolis mass flow meter

 Objectives  To present a general model for periodically stiffened pipes  To evaluate the stability of stiffened pipes  To investigate the stability for clamped-free periodic pipes of various design parameters  Pipe Construction Pipes Conveying Fluid

 Housner [1952] was the first to investigate the dynamic stability of uniform pipes supported at both ends and conveying fluids.  Benjamin [1961] was the first to correctly derive the Hamilton’s principle of continuous flexible pipes.  Païdoussis [1997] has presented a comprehensive survey of the dynamics and stability of slender structures subjected to moving fluid.  Maalawi and Ziada [2002] is focused on the static instability of stepped pipes conveying fluid.  Aldraihem and Baz [2002] studied the dynamic stability of stepped beams under the action of moving loads. Previous Works

 Main assumptions: (1) the pipe is symmetric and obeys the Euler-Bernoulli theory; (2) the fluid is incompressible and of mass m f per unit length; (3) the pipe’s cells are identical and made of isotropic materials. Modeling

 An approach that accounts for the out-release energy of a flowing fluid in a pipe should be used.  The approach is essentially the Hamilton’s principle with some modification to encompass the fluid out- flow energy(was first devised by Benjamin [1961] and then elaborated by McIver [1973]). Traditional Hamilton’s Principle New terms Formulation

 Kinetic Energy  Strain Energy  Work by Non-Conservative Forces Pipe System Energies

with boundary conditions pairs At x = 0 W = 0 or W’ = 0 or At x = L W = 0 or W’ = 0 or Inertia Force Flexural Restoring Force Internal Damping Force Coriolis Effect term Centrifugal Force Gravitational Force Distributed-Parameter Model

 Buckling of Column COMPARING TERMS  Pipe Conveying Fluid Source of Instability in Pipe Conveying Fluid

 Using a one-dimensional beam element, yields  Cast in a first order form Where Finite Element Model

 The stability of the pipe system in the neighborhood of the equilibrium depends upon the eigenvalues of the matrix [A].  If the real parts of the eigenvalues are negative, the pipe is asymptotically stable;  If at least one of the eigenvalues has a positive real part, the pipe is unstable;  If at least one of the eigenvalues has no real part, the pipe is marginally stable. Stability Analysis

 The pipe response is obtained by where Dynamic Response

Results and Discussions Material Properties Aluminum: E = 76 GPa  = 2840 kg/m 3  Control Parameters: m f,  A, EI, U and L Mass ratio Speed ratio Using Dimensionless quantities:

Results and Discussions Geometrical Properties Cantilever pipes Inner diameter: D i = 14 mm Outer diameter: D o = 16 mm Length: L = mm Fixed at the left end (x = 0) Free at the other end (x = L) Pipes are exposed to flowing fluids traveling at constant speed U form the fixed end toward the free end

Performance of Periodic Pipes

Effect of Cell Length Ratio Ls/Lu on Stability

Effect of Step Factor f on the Stability

Conclusions  Pipes stability is predicted by FEM that accounts for periodic cells and the interaction between the flowing fluid and pipe vibration.  The effect of the number of cells, cell length ratio and step factor on the stability characteristics are examined.  Results demonstrated that periodically stiffened pipes exhibit significantly improved stability characteristics.  The stability characteristics of stiffened pipes with four and more cells are comparable.  The effect of the cell length ratio on the stability appears to be important for large values of mass ratio.  Increasing the step factor enlarges the stable region of the pipe.

Work in Progress/ Future Work  Work in Progress : Dynamic analyses of pipes with periodic rings made of piezoelectric and viscoelastic materials.  Future Work: The present numerical results will be verified experimentally.