dynamic stability of periodically stiffened pipes conveying fluid dr. osama j. aldraihem dept. of...
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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.
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