Dynamic Stability of Periodically Stiffened Pipes

Conveying Fluid

Dr. Osama J. AldraihemDept. 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 tubes Coriolis 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 mf per unit

length; (3) the pipe’s cells are identicaland made ofisotropic 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]).

0}{0

22

1

dtwwUwUmdxwwUmWVT LLLf

Lt

t

fnc

Traditional Hamilton’s Principle

New terms

Formulation

Kinetic Energy

dxcUUwwUwmwATL

f 0

222 222

1

Strain Energy

L

f dxgmAwEIU0

2 )(22

1

Work by Non-Conservative Forces

L

nc dxwwIW0

Pipe System Energies

with boundary conditions pairs

wUwUmwEIwI f 0 wEIwI

0 wEIwI 0 wEIwI

At x = 0 W = 0 or

W’ = 0 or

At x = L W = 0 or W’ = 0 or

gmAwUmwUmwIwEIwmA ffff )(2)( 2

InertiaForce

FlexuralRestoringForce

InternalDampingForce

CoriolisEffect term

CentrifugalForce

GravitationalForce

Distributed-Parameter Model

Buckling of Column

0 wPwEIwm

COMPARINGTERMS

02)( 2 wUmwUmwEIwmA fff

Pipe Conveying Fluid

Source of Instability in Pipe Conveying Fluid

Using a one-dimensional beam element, yields

eqeqeqeq fKCM

Cast in a first order form

ZAZ ][

Z

][][][][

][]0[][ 11

eqeqeqeq CMKM

IA

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

t

tAtA

o

dFeZetZ )]([][ )0()(

where

}{][

}0{1

eqeq fMF

Dynamic Response

Results and Discussions

Material Properties

Aluminum: E = 76 GPa = 2840 kg/m3

Control Parameters: mf, A, EI, U and L

Am

m

f

f

Mass ratio EI

mULu fSpeed ratio

Using Dimensionless quantities:

Results and Discussions

Geometrical Properties

Cantilever pipes Inner diameter: Di = 14 mm

Outer diameter: Do = 16 mm

Length: L = 983.3 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

0.0 0.2 0.4 0.6 0.8 1.04

8

12

16

20

24

Ls/Lu = 1/2f = 1 .25

U niform

2 C ells

3 C ells

4 C ells

8 C ells

16 C ells

u

S ta b le

F lu tte r

f

Effect of Cell Length Ratio Ls/Lu on Stability

0.0 0.2 0.4 0.6 0.8 1.04

8

12

16

20

24

Four C ell P ipef = 1.25

Uniform

Ls/Lu = 1/3

Ls/Lu = 1/4

Ls/Lu = 1/5

Curve 7

S ta b le

F lu tte r

Effect of Step Factor f on the Stability

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

Four C ell P ipeLs/Lu = 1/2

f =1 (U niform )

f = 1.1

f = 1.25

f = 1.5

f = 2

S tab le

F lu tte r

Conclusions Pipes stability is predicted by FEM that accounts for periodiccells 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 fourand 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.