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Code_Aster
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Titrate:
Beam élasto - acoustic
Date:
04/07/96
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.03-A
Page:
1/10
Manual of Reference
R4.02 booklet: Accoustics
HP-51/96/038/A
Organization (S):
EDF/EP/AMV
Manual of Reference
R4.02 booklet: Accoustics
Document: R4.02.03
Beam élasto - acoustic
Summary:
One presents an element of coupling élasto - acoustic right which applies to an element of structure of the type
beam of Timoshenko. This element makes it possible to carry out, into vibroacoustic, the modal analysis of a piping
straight line containing of the fluid under pressure (water, vapor…). One can also carry out calculations of answer to
fluid sources (flow masses, volume flow rate, pressure) by modal recombination. Boundary conditions
applicable to the nodes of these elements are of Dirichlet type: displacement, pressure or potential can y
to be imposed.
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Code_Aster
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Titrate:
Beam élasto - acoustic
Date:
04/07/96
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.03-A
Page:
2/10
Manual of Reference
R4.02 booklet: Accoustics
HP-51/96/038/A
Contents
1 Notations ................................................................................................................................................ 3
2 Introduction ............................................................................................................................................ 3
3 the model of beam élasto-accoustics ................................................................................................. 4
3.1 Assumptions ...................................................................................................................................... 4
3.2 Functional calculus of the coupled problem ................................................................................................... 4
3.2.1 Contribution of piping ................................................................................................... 4
3.2.2 Contribution of the fluid ............................................................................................................. 5
3.2.2.1 Term corresponding to the contribution of
~
p
............................................................ 6
3.2.2.2 Term corresponding to the contribution of
p
............................................................ 6
3.2.3 Term of coupling ................................................................................................................ 6
3.2.3.1 Parallel section ........................................................................................................ 6
3.2.3.2 Melts of pipe ............................................................................................................ 6
3.2.4 Functional calculus of the system coupled in the case of pipings .............................................. 7
3.3 Discretization by finite elements ...................................................................................................... 7
3.4 Establishment in Code_Aster .................................................................................................... 8
4 Bibliography ........................................................................................................................................ 10
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Code_Aster
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Titrate:
Beam élasto - acoustic
Date:
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Key:
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R4.02 booklet: Accoustics
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1 Notations
P
:
instantaneous total pressure in a point of the fluid
p
:
pressure realized on a cross-section
~
p
:
fluctuating pressure
U
:
displacement of the structure
:
potential of displacements of the fluid
0
:
density of the acoustic fluid
S
:
density of the structure
, F
:
pulsation, frequency
C
:
speed of sound in the fluid
, K
:
wavelength, numbers wave



:
tensor of the stresses of the structure



:
tensor of the structural deformations
FD
:
element of volume
dA
:
element of surface
:
surface interaction between piping and the fluid
S
F
:
cross-section of the fluid
S
S
:
cross-section of piping
2 Introduction
In order to be able to carry out calculations of dynamic response of structures filled of fluid to
fluids, of the elements of coupling fluid-structure 3D were developed in Code_Aster
(cf [bib2]).
These voluminal elements have the advantage of allowing a fine description of the structure in
particular places like, for example, connection enters a main piping and one
pricking of instrumentation. On the other hand, their systematic use for the analysis of ramified networks
and complexes would lead to costs of modeling (realization of mesh) and of calculation prohibitory.
For this reason, and in order to facilitate simplified studies of dynamic behavior of
pipings, one developed an element of beam right élasto-accoustics allowing to realize, with
lower cost of calculation, and calculation labor of overall behavior of the parts
straight lines of pipings in low frequency.
One finds hereafter a presentation of the finite elements of pipings of the type beam élasto-accoustics.
The vibratory behavior of the networks of pipings is conditioned by the flow of the fluid which
traverses.
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Code_Aster
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Titrate:
Beam élasto - acoustic
Date:
04/07/96
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.03-A
Page:
4/10
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R4.02 booklet: Accoustics
HP-51/96/038/A
3
The model of beam élasto-accoustics
3.1 Assumptions
One studies low frequency the vibrations of a piping elastic, linear homogeneous and isotropic
coupled to a compressible fluid.
The effects due to viscosity and the flow of the fluid are neglected.
Pipings are lengthened bodies. Indeed, their transverse dimensions are much lower than
their length: D << L, and the thicknesses are such as one can neglect the modes of swelling and
of ovalization of the pipe. One can use a model of beam.
Low frequency the acoustic wavelengths associated the studied problems are large
compared to transverse and small dimensions compared to the longitudinal dimension of the circuit:
.L/C > 1 and
.D/C << 1. Compressibility acts indeed mainly on displacements
longitudinal. Transversely, it is considered that the fluid moves like a solid
indeformable, i.e. it acts like an added mass. Pressure in a cross-section of
pipe being then constant, it is said that the acoustic wave is plane.
3.2
Functional calculus of the coupled problem
One can write the variational formulation of the problem of the pipings filled with fluid from
and behavior equilibrium equations of the fluid and the pipe as well as boundary conditions. With
to leave the general functional calculus of the three-dimensional coupled problem ([bib1], [bib2]), one can write
functional calculus applied to the particular case of the beams.
The variational formulation of the 3D problem amounts minimizing the functional calculus:
(
)
()
()
[
]
(
)
}
F,
:
grad
U p
FD
P
C
P
C
FD
dA
S
S
F
=
-


+








-
1
2
2
2
2
0
2
0
2
2
0
2






-
-
-
-
U
U
U
u.n
2
with:
S
, the field of the structure
F
, the field of the fluid
, the fluid surface of interaction - structure.
3.2.1 Contribution of piping
The model of beam used is that of Timoshenko with deformations of shearing action and inertia of
rotation of the cross section. It corresponds to modeling
POU_D_T
it takes again calculations
elementary. One does not take into account the effects of ovalization [bib3].
The terms associated with piping in the variational formulation are written then:
() ()
[
]






U
U
S
U
S ds
L
S
:
-
2 2
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Code_Aster
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Titrate:
Beam élasto - acoustic
Date:
04/07/96
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Key:
R4.02.03-A
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R4.02 booklet: Accoustics
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with:
L
, the average fiber of piping and
S
S
, the section of piping to the X-coordinate
S
(cf [Figure 3.2.1-a]).
M (S)
S (S)
S (s+ds)
F
D
L
M (S)
F
F
S
S
N
N
K
J
I
J
Appear 3.2.1-a: geometry of piping
3.2.2 Contribution of the fluid
In this paragraph, one is interested only in the fluid part of the functional calculus, i.e., term
of coupling put aside, at the end which is written in 3D:
(
)
P
C
P
C
FD
F
2
0
2
0
2
0
2
2
2
-
-
-
-
-








grad
éq 3.2.2-1
It is supposed that the pressure breaks up into two terms:
()
(
) ()
()
(
)
P
,
p,
~p
,
M S T
S T
M S T
=
+
where
p
is the value realized on a cross-section of the pressure:
()
()
()
()
(
)
p S T
S
M S T DM
S
,
S
P
,
F
S
F
=
1
and
~
p
is a term of fluctuating pressure which corresponds to the contribution of the transverse modes.
According to the assumptions of the paragraph [§1],
p
check the equation 1-D of Helmholtz and
~
p
the equation of
Laplace (incompressible). The integral [éq 3.2.2-1] thus breaks up into two terms.
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Titrate:
Beam élasto - acoustic
Date:
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Key:
R4.02.03-A
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R4.02 booklet: Accoustics
HP-51/96/038/A
3.2.2.1 Term corresponding to the contribution of
~
p
In the movements perpendicular to the axis of the pipe, one considers that the fluid intervenes
only by its added mass [bib4], the term related to
~
p
is thus a term of inertia:
()
0
2
U
T
2
L
F
S ds
U
T
being transverse components of the vector displacement of the structure and
S
F
the section of
fluid with the X-coordinate
S
.
3.2.2.2 Term corresponding to the contribution of
p
P
C
P
C
S
S ds
L
F
2
0
2
2
0
2
2
0
2
-
-
-
-
-










3.2.3 Term of coupling
3.2.3.1 Section
current
According to the references [bib4] and [bib5], one shows that the term of coupling C:
C
dA
R
S ds
dS
ds ds
F
L
F
L
= -
=
-
+


0
0
0
u.n
U. J
u.i
Equilibrium equations of the structure and the equation of propagation plane waves (Helmholtz)
in the fluid are thus coupled with the level of the bent parts and the right parts where there is one
change of section of piping. In the case of a pipe with constant cross-section:
R
dS
ds
C
F
=
=
and
thus
0
0
There is thus no coupling between the movements of beam of the structure and displacements
longitudinal of the fluid in the right parts of the circuit. In this case, the fluid is characterized
only by its added mass related to transverse displacements.
3.2.3.2 Melts of pipe
Sf
In the case of a bottom of pipe, one notes
T
S
F
= +
, the total surface of interaction enters the fluid and
piping.
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Titrate:
Beam élasto - acoustic
Date:
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Key:
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R4.02 booklet: Accoustics
HP-51/96/038/A
In the case of a right piping with closed constant section, the term of coupling
C
is worth then:
C
dA
dA
S
S
F
F
= -
= -
+
0
0
u.n
u.n
It is the basic effect.
This term is added at the end of coupling of a parallel section. Thus, a free node which carries out one
condition of null flow through the section [bib6] carries out an acoustic basic condition. Indeed, one
incidental plane wave is completely considered on the bottom: the acoustic pressure in the conduit obeys
the equation of Helmoltz with normal gradient of pressure no one (fluid displacements and solid being null).

2
2
2
0
0
p
p
(
p)
X
K
X
S
F
+
=
=




One seeks the solution in the form:
p
cos (
)
cos (
)
=
-
+
+
With
T kx
B
T kx
i.e. in the form of a linear combination of an incidental plane acoustic wave and one
considered wave.
The condition of null gradient on the bottom, checked for every moment, imposes:
WITH B
=
The considered wave is thus “equal” to the incidental wave (coefficient of reflection equal to the unit).
3.2.4 Functional calculus of the system coupled in the case of pipings
In the particular case that we treat of a not bent piping, with constant section,
functional calculus of the coupled problem is thus written in the following form:
(
)
()
()
()
[
]
}
F,
:
U p
S ds
S U
S
ds
P
C
P
C
S ds
dA
S
S S
F
L
L
F
S
L
F
=
-
+


+
-














-
1
2
2
2
2
0
2
0
2
0
2
0
2
2
2
0






-
-
-
-



U
U
U
u.n
T
2
3.3
Discretization by finite elements
The solution
(
)
U p
,
sought the functional calculus minimizes
F
. The approximation by finite elements of
complete problem leads then to the symmetrical system:
K
K
U
MR. M
M
M
M
M
H
U
F
F
fl
T
fl
T
0
0
0
0
0
0
0
0
0
0
0
0
2
2
2
2
0
.
.
C
p
C
C
p










-
+
















=






K
and
M
being respectively matrices of rigidity and mass of the structure,
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Code_Aster
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Titrate:
Beam élasto - acoustic
Date:
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Author (S):
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Key:
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R4.02 booklet: Accoustics
HP-51/96/038/A
K
F
,
M
fl
,
H
being fluid matrices, respectively obtained starting from the quadratic forms:
p S ds
p S ds
S
S ds
F
F
L
L
F
L
2
2
,
,





M
F
being the fluid matrix obtained starting from the quadratic form:
()
p S
ds
F
L 0
U
T
2
M



being the matrix of coupling obtained starting from the bilinear form:
p
dA
S
F
0
U N
.
While discretizing linearly
p
and
, one thus has:
p p L X
L
p xL
L X
L
X
L
1
=
- +
=
- +
2
1
2
and
,
L
being the longor of the element considered.
In this case, the elementary matrix of stiffness of the fluid is written:
{
}
K
F
F
1
2
1
2
S L
3
p p
/
/
p
p
=




1
1 2
1 2
1
The elementary matrix of coupling is written:
{
}
M
=
-




0
1 0
0
1
S
U
U
F
1
2
1
2
The various matrices of fluid mass elementary are written:
{
}
{
}
M
H
fl
F
1
2
1
2
F
1
2
1
2
S L
/
/
p
p
S
=




= -
-
-




3
1
1 2
1 2
1
1
1
1
1

L
3.4
Establishment in Code_Aster
On the principles which we have just described, a vibroacoustic element of beam, Timoshenko
for the piping part, right to constant section, was established in Code_Aster. It belongs to
modeling
“FLUI_STRU”
phenomenon
“MECHANICAL”
.
This element has 8 ddl by node: displacements and rotations of piping, pressure and it
potential of displacement of the fluid (cf [Figure 3.4-a]). The formulation is written for displacements
buildings in the local reference mark with the element made up of neutral fiber (axis X) and of the main axes
of inertia (axis Y, axis Z) of the section. Two scalars
p
and
(pressure and potential of
fluid displacements) are invariants by change of reference mark.
On each node of this element, one can impose boundary conditions of the Dirichlet type in
pressure, potential of fluid displacements and displacements (translation or rotation).
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Titrate:
Beam élasto - acoustic
Date:
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R4.02 booklet: Accoustics
HP-51/96/038/A
p,
X
Y
Z
Z
Y
p,
Uz
Uy
Ux
Uy
Uz
Ux
Ury
Urx
Urz
Ury
Urx
Urz
Appear 3.4-a: element of beam filled with fluid
This element to date makes it possible only to calculate the clean modes of a right piping filled of
fluid and to make harmonic calculation of answer. Effects of curvature or abrupt widening
of section are not taken into account for the moment, but these fluid-structure effects, lorqu' one has
business with not very dense fluids like the vapor of a piping of intake, do not seem to have
a determining importance on the calculation of the first modes: the correct mechanical representation
elbow (coefficient of flexibility) seems sufficient to calculate these frequencies [bib7].
In modal analysis, one can quote the case of a right piping filled with fluid with loose lead:
Structure
Fluid
Appear 3.4-b: beam filled with fluid embedded - free
the Eigen frequency of the mode of traction and compression of this fluid coupled system/structure is
data by the relation:
tg (
)
L
C
S
S
E
C
S
F
=
0 2
One indicates by:
E
: Young modulus of solid material
S
S
: section of the solid
S
F
: section of the fluid
It is supposed here that the speed of speed of sound in the fluid is equal to the speed of sound in
solid
C
E
S
S
=
[bib7].
The transitory calculation of response for this type of finite element
(
)
U,
p
is not yet available in
Code_Aster.
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Code_Aster
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Titrate:
Beam élasto - acoustic
Date:
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Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.03-A
Page:
10/10
Manual of Reference
R4.02 booklet: Accoustics
HP-51/96/038/A
4 Bibliography
[1]
H.J.P. MORAND; R. OHAYON Interaction fluid-structures Series Search in
Mathematics Applied n° 23 Masson Edition, July 1992
[2]
F. WAECKEL Analyzes modal into vibroacoustic in ASTER. Note intern DER
HP-61/91.160
[3]
R. OHAYON Variational analysis off has slender fluid-structure system: the elasto-acoustic
beam. With new
symmetric formulation. International newspaper for numerical methods in
engineering, vol22, 637-647 (1986)
[4]
R.J. GIBERT Vibrations of the structures. Interactions with the fluids. Collection of Management
studies and Search of E.D.F. Eyrolles Edition, September 1988
[5]
S. FRIKHA Analyzes experimental dynamic stresses applied to a portion of
structure modélisable by the theory of the beams. Thesis of doctorate in Mechanical Engineering,
ENSAM, February 1992
[6]
LI-LIN Identification of the acoustic sources induced by the singularities of a circuit
hydraulics. Thesis of doctor engineer, University Paris 6, April 1988
[7]
F. WAECKEL, C. DUVAL Notes principle and of use of the pipes
implemented in Code_Aster Intern DER HP-61/92.138 Notes