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Titrate:
Calculation of matrix of mass added on modal basis
Date:
02/10/95
Author (S):
G. ROUSSEAU
Key:
R4.07.03-A
Page:
1/12
Organization (S): EDF/EP/AMV
Handbook of Référence
R4.07 booklet: Fluid coupling structure
Document: R4.07.03
Calculation of matrix of added mass
on modal basis
Summary:
This document has an aspect of the fluid coupling/structure: when a vibrating structure is
immersed in a fluid which one supposes at rest, incompressible and nonviscous, it feels forces of
pressure whose resultant is proportional to the acceleration of the structure in the fluid: the coefficient of
proportionality is homogeneous with a mass: it is called added mass. One specifies here the means of estimating one
stamp of mass added for one (or of) structure (S) to several degrees of freedom on the modal basis of
(of) structure (S) in the vacuum.
Handbook of Référence
R4.07 booklet: Fluid coupling structure
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Code_Aster ®
Version
3
Titrate:
Calculation of matrix of mass added on modal basis
Date:
02/10/95
Author (S):
G. ROUSSEAU
Key:
R4.07.03-A
Page:
2/12
Contents
1 Notations ................................................................................................................................................ 3
2 Introduction ............................................................................................................................................ 3
3 Recalls of the equations of the problem ..................................................................................................... 4
3.1 Equations in the fluid .................................................................................................................. 4
3.2 Equations in the structures ......................................................................................................... 5
3.3 Equations of the coupled problem - Mise in obviousness of the matrix of added mass ..................... 6
3.4 Some definitions ........................................................................................................................ 7
3.4.1 Definition 1 ............................................................................................................................. 7
3.4.2 Definition 2 ............................................................................................................................. 7
3.4.3 Definition 3 ............................................................................................................................. 7
3.5 Properties of the matrix of added mass ..................................................................................... 7
3.5.1 Theorem 1: the matrix of added mass is symmetrical ................................................... 7
3.5.2 Theorem 2: the matrix of added mass is definite positive ............................................. 8
3.5.3 Theorem 3 ............................................................................................................................ 8
3.5.4 Other properties .................................................................................................................... 9
4 Implementation numerical .................................................................................................................... 9
4.1 Resolution of the equation of Laplace by finite elements of volume ................................................. 9
4.2 Calculation of the coefficients of the matrix of mass added on modal basis .................................... 10
5 Implementation in Code_Aster .................................................................................................... 11
5.1 Thermal analogy ........................................................................................................................ 11
5.2 Implementation practices ................................................................................................................ 12
6 Bibliography ........................................................................................................................................ 12
Handbook of Référence
R4.07 booklet: Fluid coupling structure
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Code_Aster ®
Version
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Titrate:
Calculation of matrix of mass added on modal basis
Date:
02/10/95
Author (S):
G. ROUSSEAU
Key:
R4.07.03-A
Page:
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1 Notations
p
:
fluctuating pressure in the fluid,
!
:
contour of the structure indexed by!
“
X
:
S
the field of displacements in the structure
!
!,
F, S: density of the fluid, the structure,
X
:
I!
clean mode of command I of the structure! in air
has, a:
generalized co-ordinates, speeds, accelerations
I
I
! “!
relating to mode I of the structure
has "
! in air
I
“!
:
the tensor of the constraints in the structure
:
the fluid vector of flow
H
:
the matrix of rigidity of the fluid
v
:
the field fluid speeds
N
:
the interior normal of the fluid.
2 Introduction
Many industrial components are in contact with fluid environments, which more is often in
flow. These surrounding fluid environments disturb the vibratory characteristics of the structures,
in particular their modal characteristics. This action of the fluid on the structure results in
effects of fluid coupling/structure.
One supposes the incompressible, perfect fluid environment here surrounding and at rest. One will show that then,
a structure which vibrates with a small amplitude in this fluid modifies the field of pressure in
fluid at rest, and thus feels a compressive force, proportional to its acceleration.
proportionality factor is a mass. It describes the inertial effect of the fluid on the structure: it is
why one names this mass masses added fluid on the structure.
When several structures are in contact of the same fluid, when one of the structures is put at
to vibrate, not only it feels the inertia of the fluid, but it modifies the field of pressure around
interfaces with the fluid of all the other structures. The efforts that each one feels are
proportional to the acceleration of the vibrating structure: there still proportionality factors
called added masses of coupling are masses.
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Titrate:
Calculation of matrix of mass added on modal basis
Date:
02/10/95
Author (S):
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Key:
R4.07.03-A
Page:
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3
Recalls of the equations of the problem
3.1
Equations in the fluid
It is supposed that K vibrating structures are immersed in a true fluid (nonviscous),
incompressible and at rest. One neglects the effect of gravity. One can thus write the equations
of Euler associated with the fluid at rest:
N
y or X 2
!
Z
N
X or x1
`
· conservation of the mass:
F +div (v) =
F
0
éq 3.1-1
T
· conservation of the momentum:
v
1
+ (v · )
v +
p
= 0
éq 3.1-2
T
F
Because of incompressibility of the fluid, the equation [éq 3.1-1] becomes:
div v = 0
éq 3.1-3
In the volume of the fluid, one neglects the convection induced by the movement of low amplitude of
the structure. The equation [éq 3.1-2] thus becomes:
v
1
+
p
= 0
éq 3.1-4
T
F
v
While deriving [éq 3.1-3] compared to time and by deferring the expression of according to
T
pressure in this equation, one obtains:
div p = 0
that is to say:
p = 0 in
who is the equation of Laplace in a fluid at rest.
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Titrate:
Calculation of matrix of mass added on modal basis
Date:
02/10/95
Author (S):
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Key:
R4.07.03-A
Page:
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With the fluid interface/structure, one can write that the normal acceleration of the wall of the structure is
equalize with the normal acceleration of the fluid (continuity of the normal accelerations - condition
of impermeability of the structure). One uses here following convention for the normal: it is about
normal external with the structure, directed structure towards the fluid.
v
· N = X " · N
T
S!
With the equation [éq 3.1-4], one obtains:
v
p· N = - F
·N = - X " ·N
T
F S!
That is to say: p
(
) = - “X ·
N on
N
F S
!
!
!, fluid interface/structure of the structure indexed by!.
In short, the fluid problem consists in solving an equation of Laplace with boundary condition
of type von Neumann:
p = 0 in
p
(
) = - F “xs ·N on 1, 1 =
N 1
$!
!=1, K
éq 3.1-5
#
p
(
) = 0 out of 2, 2 = -
N2
1
3.2
Equations in the structures
Let us consider K structures elastic divings in a fluid environment. The equation of their movement in
presence of fluid is written:
! index of structure! {, 0 #, K}, M “X + K X = 0 in, volume of the structure
!
!
!
!
S
!
!
!, N = -
pn on!, contour of the structure!
M! is the matrix of mass of the structure, K! its matrix of rigidity. The boundary condition on
contour of the structures translates the continuity of the normal constraint to the fluid interface/structure (it
tensor of the fluid constraints being tiny room to its nondeviatoric part, fluid being perfect). In
integrating on the contour of each structure this normal constraint, a force F is obtained!
resultant of the structure/compressive forces of the fluid to the fluid interface. This force is the integral of
field of pressure on contour!% of each structure:
! index of structure! {, 0 #, K}, F = -
!
pnd
!
The field of pressure checks the problem [éq 3.1-5].
Handbook of Référence
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Titrate:
Calculation of matrix of mass added on modal basis
Date:
02/10/95
Author (S):
G. ROUSSEAU
Key:
R4.07.03-A
Page:
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3.3 Equations of the coupled problem - Mise in obviousness of the matrix of
mass added
Ultimately, the fluid coupled problem/structure is written:
p = 0 in
! {
p
0, K
#},
= -
“
X
N on
S
N
F
·
!
!
!
! {0, K
#}, M “X
+ K X
0 in
=
S
! !
! !
!
! {0, K
#}, F = - pn
D
!
on!
!
éq 3.3-1
One will show from now on that the effort that feel the immersed structures is proportional to their
acceleration. A good means of showing that is to place itself in the modal base of the structures
in the vacuum. One can thus break up acceleration on this basis (which is in fact the meeting of
modal bases of each structure). As follows:
xS (R, T) = has (T
I
) Xi (R)
!
!
!
i=1
By deferring this expression in the second equation of the system [éq 3.3-1], one is brought to
to seek the field of pressure in the form:
p =
(T) p has
“I”!
I (R)
!
=,
1, K i=1,
! #
#
By deferring in the problem [éq 3.3-1] these expressions, one has to solve in the fluid as much
problems of Laplace whom one chose of modes for each structure. This results in:
p
I =
0 in
!
p
! {1, #, K}, I {1, #,}
,
!
I
= -
X
N on
N
F
I ·
!
!
!
[semi (A) + K (A) = (F) in
! ] “!
[!I]!
I!
!
The “matrices” of mass and rigidity written in these bases are diagonal.
Each component of the effort of pressure resulting projected on modal basis is written:
K
I {1,
,
#
}
! {1, K
#}, (F
! ) = - has "
p X ·
N NR D
I
jk
jk
I
J
!
K =1 j=1
!
One can then write the vector of the effort generalized of pressure on a structure immersed under
matric form:
(F) = - [m] “with m = p X has ·N
I
I jk
jk
I jk
jk I D
!
!
!
!
!
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Titrate:
Calculation of matrix of mass added on modal basis
Date:
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Here! is fixed: the matrix [semi jk
! ] is called matrix of mass added of the fluid on the structure of
contour!. When one considers the modal base of the whole of K structures, one generalizes
notation of the matrix of added mass [semi jk
! ] on modal basis in the vacuum! varying from 1 to K.
This matrix is in general not diagonal.
3.4 Some
definitions
3.4.1 Definition
1
When! = K (even structure) and I = J (even command of mode), the coefficient semi I!! is the car-mass
added mode I of the structure!. It is about additional inertia due to the fluid moved by
mode of command I of the structure, taking into account the geometrical containments induced in the fluid by
presence of the other presumedly fixed structures.
3.4.2 Definition
2
When! = K (even structure) and I J (different commands of mode), the coefficient semi J!! is the mass
added coupling between the modes of command I and J of the structure!. In air, these terms of mass
extradiagonaux are null, because the modes are orthogonal between them. Taking into account the expression
general of the coefficient semi jk
! , modes I and J can be coupled in mass, because the field of
pressure p J! created by the mode J of the structure! is not necessarily orthogonal with the mode of command
I of this same structure. It is enough that this structure is immersed in an environment
not comprising geometrical symmetry so that this coefficient is nonnull. In an environment
symmetrical, on the other hand, the orthogonality of the field of pressure with the mode is observed.
3.4.3 Definition
3
When! K (different structures) and I J (different commands of mode), the coefficient semi jk
! is
mass added coupling between the modes of command I and J respectively of the structures! and K. It
coefficient translates the inertial effort which makes undergo the structure K vibrating on its mode of command J to the structure
! vibrating on its mode I.
3.5
Properties of the matrix of added mass
3.5.1 Theorem 1: the matrix of added mass is symmetrical
To simplify the demonstration, we will consider a single structure immersed in a fluid
perfect, incompressible and nonviscous. We break up the movement of the structure on its basis
modal (truncated with N modes), but the result can be just as easily shown in “physical” base
(i.e the base of the nodal functions of interpolation). Lastly, the result spreads with the case of K
structures immersed in the same fluid.
One must show that:
m = p X · N D = m = p X · N D
ij
I J
ji
J I
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Titrate:
Calculation of matrix of mass added on modal basis
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·
pi (respectively pj) represents the field of pressure created in the fluid and to the interface
with the structure by the mode of command I (respectively of command J) of the structure,
·
X J (respectively Xi) respectively represents the modal deformation of the mode of command J (
of command I).
However:
p
0 in
fluid volume
= 0 in fluid volume
I =
pj
p
and
p
I
J
= - F Xi ·N on
= - F X J ·N on
N
N
From where, by using the formula of Green with a normal directed of the structure towards the fluid and
harmonicity of pi and p J:
1
p
m = p X · N D = -
p
J D
ij
I J
I
N
F
1
= -
(p p
D -
p
· p
D
I J
I
J)
F
& '
(
)
(
0
1
= -
(p p
D -
p
· p
D
J I
J I)
F
& '
(
)
(
0
1
p
= -
p
I D = p X ·N D
J
N
J
I
F
= mji
C.Q.F.D.
3.5.2 Theorem 2: the matrix of added mass is definite positive
One returns to the reference [bib1] for the complete demonstration.
3.5.3 Theorem
3
Let us suppose that one has K structures having properties of linear elasticity identical and who are
immersed in the same fluid. Moreover, these structures admit two degrees of freedom of
displacement in the plan Oxy (cf diagram). Each one of these structures admits the same spectrum
F
F
1,
, N,
*
* of Eigen frequencies in the vacuum.
For any Eigen frequency fn, there exists 2K Eigen frequencies {1,
# 2K} of the coupled system
fluid/structure checking I {1,
2
, K},
#
F
I
N
One returns to the reference [bib1] for the complete demonstration.
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Titrate:
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3.5.4 Others
properties
· the coefficients of added car-mass are always positive
One always supposes that one has only one structure immersed in a true fluid, incompressible and with
rest. The demonstration spreads without difficulty with K immersed structures.
One must show that:
I index of mode {1,
#}
N, m = p X · N D
II
I
I
0
However:
1
p
m = p X ·N D = -
p
I D
II
I I
I
N
F
1
= -
(p p
D -
p
· p
D
I I
I I)
F
& '
(
)
(
0
1
=
((p
) 2D
I
F
0
· let us suppose that one has K structures immersed in the same fluid. It is supposed that they have one
only degree of freedom of translation according to OX. Then the sum of all the coefficients of mass
added this matrix gives the car-mass added on the whole of K structures moving
very of the same sinusoidal rectilinear motion.
One returns to the reference [bib2] for the complete demonstration.
4
Implementation numerical
4.1
Resolution of the equation of Laplace by finite elements of volume
Let us take again the fluid problem of Laplace with boundary condition of the type von Neumann:
p = 0 in
p
(
) = - F “xs ·N on 1, 1 =
N 1
$!
!=1, K
#
p
(
) = 0 out of 2, 2 = -
N2
1
Let us write a variational formulation of this problem:
v p
D =
0
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Titrate:
Calculation of matrix of mass added on modal basis
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Author (S):
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Key:
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By using the formula of Green with a normal which one supposes directed of the structure towards the fluid
(thus interior with fluid volume) and by posing = 1 2:
p
v · p D + v D =
0
N
That is to say:
v · p D =
v X " D
F
éq 4.1-1
N
1
One considers a partition of volume in a finished number of elements. On this discretization of
field, one can write the approximate shape of the hydrodynamic field of pressure:
NR
p = NR (R) p
I
I
i=1
Ni represents the nodal functions of interpolation definite on the elements: they are worth 1 with the node
n°i, and 0 on all the others.
Then, by taking as function-tests v successively the nodal functions of interpolation, one
a system of NR equations obtains while deferring in [éq 4.1-1]:
NR
J = 1, NR;
p
#
NR (R) · NR (R) D =
NR X D
I
I
J
F
J “N”
i=1
1
what can be written in the form:
HP = with vector of components J = F NR jx " N "
D
éq 4.1-2
with
H stamps coefficients ij
H = Ni·
NR J D
In any rigor, this system is singular. It admits an infinity of solutions differing from a constant. It
is thus necessary to impose a pressure (boundary condition of the Dirichlet type) in a point of the fluid for raising
indetermination on the solution.
These precautions taken, by reversing the system [éq 4.1-2], one obtains the field of pressure in all
the volume of fluid, including with the fluid interface/structure, where it interests us obviously.
4.2 Calculation of the coefficients of the matrix of mass added on basis
modal
It is necessary to estimate the value of the integral numerically:
m
= p X · N D
I jk
jk I
!
!
éq 4.2-1
!
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Titrate:
Calculation of matrix of mass added on modal basis
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starting from a field with the nodes of pressure represented by a vector column noted Pjk and of a field
with the nodes of displacement corresponding to a modal deformation of structure in air and represented
by the vector column Xi! . However, on the fluid interface/structure, the field of pressure approximate pjk due to
the discretization of the interface in NR elements of edge can be written:
NR
p = Nm (R) p
jk
jkm
m=1
while the field of “modal” displacement is written on this same discretization:
NR
I
X = N (R) I
!
X!N
n=1
Thus, by deferring these two expressions in the integral [éq 4.2-1], one obtains:
NR
NR
NR
semi jk (Nm (R) p) [N (R) Xi X ·N + N (R) X ·N D
!
jk
!
X
I y
!
y]
m
N
N
m=1
n=1
n=1
!
NR
NR
NR
NR
m
p
(Nm (R) N (R) N D) X +
p
(Nm (R) N (R) N D) X
I jk
!
jk
X
I X
!
jk
y
I y
m
N
m
! N
m=1 n=1
m=1 n=1
!
!
One supposes in the demonstration that the problem is two-dimensional.
This can be put in the shape of a scalar product, utilizing a product stamps vector:
m
T
T
= P A X + P A X
with A stamps coefficients NR NR N D
I jk
jk
X I X
jk
y I y
X
I J X
!
!
!
!
and A stamps coefficients NR NR N D
y
I J y
!
5
Implementation in Code_Aster
5.1 Analogy
thermics
To solve the problem of Laplace in pressure, a thermal analogy is used: it is about
in hover to solve the equation of heat with a thermal material of conductivity equalizes with
the unit. As follows:
p = 0 in
div
(gradT) = 0 in T = 0 if
=
1
p
T
(
) = - F “xs · N in
(
) = N
in
N
N
T represents the temperature in the medium, it plays the part of the pressure in the fluid environment. N is
the normal heat flow to the wall, it plays the part of the term - F “xs · N which is comparable to the variation
in the course of the time of the flow of mass (fluid) to the wall of the structure. This quantity - F “xs · N is in
homogeneous effect with a mass divided by a surface and a time squared.
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Titrate:
Calculation of matrix of mass added on modal basis
Date:
02/10/95
Author (S):
G. ROUSSEAU
Key:
R4.07.03-A
Page:
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5.2
Implementation practical
A new operator CALC_MASS_AJOU was developed to take into account the inertial coupling
(added mass) between structures bathed in the same true, incompressible fluid and with
rest. The fluid is described by equivalent thermal characteristics (operator
DEFI_MATERIAU [U4.23.01]) and the part of the grid representing are affected by elements
thermics (operator AFFE_MODELE [U4.22.01]).
The operator uses five obligatory key words:
· the key word: MODELE_FLUIDE: it is on this model that one solves the problem of Laplace with
boundary conditions of Von Neumann (or its thermal problem are equivalent),
· the key word: MODE_MECA (or CHAM_NO, or MODELE_GENE): this key word makes it possible to calculate them
boundary conditions of the flow type to the wall of the structure,
· the key word: MODELE_INTERFACE: it is on this model which includes/understands all the elements
thermics of edge of the fluid interface/structure which one calculates the scalar product mentioned
in the paragraph [§4.2],
· the key word: CHAM_MATER: it is about fluid material (described by characteristics
thermics equivalent),
· the key word: CHARGE: it is a thermal load (temperature imposed in a node
unspecified of the fluid grid) which corresponds to the boundary condition of Dirichlet for raising
the singularity of the problem of Laplace (see [§4.1]).
One thus obtains a matrix of generalized added mass. This matrix having a profile line of
full sky but (operator NUME_DDL_GENE [U4.55.07]) can be summoned with the matrix of mass
generalized of the structure by using operator COMB_MATR_ASSE [U4.53.01]. This allows
to calculate the coupled modes fluid/structure of the immersed structures (“wet” modes) (operator
MODE_ITER_SIMULT or MODE_ITER_INV [U4.52.02], [U4.52.01]).
6 Bibliography
[1]
C. CONCA, J. PLANCHARD, B. THOMAS, Mr. VANNINATHAN: “Mathematical Problems
in fluid coupling/structure " _ EYROLLES (1994).
[2]
F. BEAUD, G. ROUSSEAU: “Validation inter-software of the calculation of mass added with
Code_Aster and code CALIFE ", HT-32/95/004/A
Handbook of Référence
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