Code_Aster
®
Version
7.2
Titrate:
FDLV110 - Calculation of mass added on modes obtained
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.110-A
Page:
1/8
Manual of Validation
V8.01 booklet: Fluid
HT-66/04/005/A
Organization (S):
EDF-R & D/AMA, EDF-DPN/UTO
Manual of Validation
V8.01 booklet: Fluid
Document: V8.01.110
FDLV110 - Calculation of mass added on modes
obtained by under-structuring
Summary:
This test of the field of the modal analysis and the interaction fluid-structure implements the calculation of mass
added on a structure made up of three concentric cylinders separated by two rings from fluid (water)
which one supposes the behavior governed by the potential theory (fluid true, incompressible at rest).
model is three-dimensional for water. The structure is represented by elements of the mean hull type
in modeling A (the structure is rigid in the reference solution). The aforementioned is characterized by two
clean modes evaluated by dynamic under-structuring, with interface of the type CRAIG-BAMPTON.
The interest of the test lies in the use of the functionality `
NOEUD_DOUBLE
`of the operator `
CALC_MATR_AJOU
`.
This functionality makes it possible to calculate the effects of added mass of a structure represented by a mesh
surface (without thickness) which is bathed in a fluid. The fluids chosen in this case-test are of
densities different on both sides from the intermediate cylinder (water at different temperatures).
Code_Aster
®
Version
7.2
Titrate:
FDLV110 - Calculation of mass added on modes obtained
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.110-A
Page:
2/8
Manual of Validation
V8.01 booklet: Fluid
HT-66/04/005/A
1
Problem of reference
1.1 Geometry
Z
y
X
fluid
L
surface fluid
roll medium
springs
external cylinder
R
F
R
I
R
E
K
1
K
2
L=50m; R
F
=1m; R
I
=
5
3
m; R
E
=3m; K
1
=10
9
N.m
1
; K
2
=0.5 10
7
N.m
1
;
fluid
=1000 kg.m
3
;
S
=7800 kg.m
3
; thickness of the hull: 50 cm.
1.2
Properties of materials
Fluid: density
1
= 1000 kg.m
3
;
2
= 750 kg.m
3
.
Structure:
S
= 7800 kg/m
3
;
E
= 2.1 10
11
AP;
= 0.3 (steel).
1.3
Boundary conditions and loadings
The external cylinder on the one hand is connected to a fixed jig via the four springs of unit stiffness K
1
,
connected in addition to the cylinder medium by four springs of unit stiffness K
2
. The two structures are
rigid in this reference solution.
Code_Aster
®
Version
7.2
Titrate:
FDLV110 - Calculation of mass added on modes obtained
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.110-A
Page:
3/8
Manual of Validation
V8.01 booklet: Fluid
HT-66/04/005/A
2
Reference solution
One calculates the clean modes of the system after having checked those of each substructure. One
evaluate then the mass added on the modes in air.
2.1
Decomposition in substructures
First substructure: roll intermediate
The first substructure is consisted of the intermediate cylinder and four springs of stiffness
K
2
= 10
7
N.m
1
. These springs are embedded with the interface with the external cylinder which constitutes
second substructure (interface of the type CRAIG-BAMPTON).
R
I
K
2
Mass cylinder 1: m
1
= 2.041 10
6
kg
The cylinder being rigid, its movement can be modelized by a system mass-arises with a degree
of freedom:
m
1
Achacune of its ends, the cylinder is connected to two springs in parallel: equivalent stiffness
of each one is K
'
= 2k
2
The Eigen frequency is worth then:
1
2
1
4
2
1
2
2
1
m
K
m
K
F
=
=
, that is to say:
=
F
Hz
0 705
.
Second substructure: roll external
The second substructure is the external cylinder connected on the one hand to the interface by the same ones
springs, in addition with a fixed jig:
R
E
K
1
K
Mass cylinder 2: m
2
=3,674 10
6
kg
Code_Aster
®
Version
7.2
Titrate:
FDLV110 - Calculation of mass added on modes obtained
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.110-A
Page:
4/8
Manual of Validation
V8.01 booklet: Fluid
HT-66/04/005/A
Equivalent stiffness of a fastener of this cylinder by the system of springs in series K
1
and K
2
being worth
9,9 10
6
N.m
1
(four fasteners of the same type connect in parallel the cylinder to an embedding),
Eigen frequency is given by:
(
)
1
1
2
1
2
4
2
1
m
K
K
K
K
F
+
=
, that is to say:
Hz
.
F
522
0
=
N.B.: the third cylinder (interior cylinder) was not modelized in our case because it agint of one
fixed cylinder. It thus constitutes a fixed wall of the fluid field.
Modes in air of the structure supplements (intermediate cylinder and external cylinder)
It is a system with two degrees of freedom:
m
2
m
1
K
3
=4k
1
K
4
=4k
2
The Eigen frequencies of this system are given by the exact formula [bib2]:
F
K
m
K
m
K
m
K
m
K
m
K
m
K K
m m
I
=
+
+
±
+
+
-
1
2
4
32
3
2
4
2
4
1
3
2
4
2
4
1
2
3 4
1
2
,
that is to say
Hz
and
Hz
263
5
497
0
2
1
.
F
.
F
=
=
.
The two clean modes admit, for numerical value:
X
X
X
X
1
2 1
3
1
2 2
3
1
5 10
9 10
1
=
= -
-
-
.
.
and
.
2.2
Calculation of the matrix of added mass
Fluid potentials
Beginning again
[bib1], it is established that:
()
()
()
(
)
-
+
-
+
+
-
+
×
-
=
-
+
×
-
+
+
-
+
=
-
-
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
2
1
2
2
2
2
2
3
2
2
2
2
2
2
2
2
2
1
1
;
.
9
.
5
;
I
E
E
I
E
I
E
I
E
F
I
F
I
I
I
E
E
I
E
I
E
I
E
F
I
F
I
I
R
R
R
R
R
R
R
R
R
R
R
R
R
R
E
R
R
R
R
R
E
R
R
R
R
R
R
R
R
R
and
Code_Aster
®
Version
7.2
Titrate:
FDLV110 - Calculation of mass added on modes obtained
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.110-A
Page:
5/8
Manual of Validation
V8.01 booklet: Fluid
HT-66/04/005/A
The shape of the matrix of added mass, in this configuration, is:
=
22
21
12
11
has
has
has
has
has
M
M
M
M
M
With:
.
,
,
kg
kg
kg
15318
)
10
.
9
(
)
10
.
5
(
10
.
676
,
2
10
.
753
,
1
2
2
2
2
2
3
2
2
2
2
2
2
2
2
2
3
12
6
2
2
2
2
2
22
6
2
2
2
2
2
2
2
2
2
11
-
=
-
+
×
-
-
+
+
-
+
×
=
=
-
+
=
=
-
+
+
-
+
=
-
-
I
E
E
I
E
I
E
E
I
F
I
F
I
I
has
I
E
E
I
E
has
I
E
E
I
F
I
F
I
I
has
R
R
R
R
R
R
R
R
R
R
R
R
R
R
L
M
R
R
R
R
R
L
M
R
R
R
R
R
R
R
R
R
L
M
The coefficient of inertial coupling
12
has
M
is regarded as negligible in front of the coefficients
of added car-mass
11
has
M
and
22
has
M
. The Eigen frequencies of the system do not depend, in first
approximation, that of these two last coefficients.
2.3
Results of reference
Analytical result.
2.4 References
bibliographical
[1]
ROUSSEAU G., LUU H.T.: Mass, damping and stiffness added for a structure
vibrating placed in a potential flow - Bibliography and establishment in
Code_Aster - HP-61/95/064
[2]
BLEVINS R.D.: Formulated for Natural frequency and shape mode, ED. Krieger
Code_Aster
®
Version
7.2
Titrate:
FDLV110 - Calculation of mass added on modes obtained
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.110-A
Page:
6/8
Manual of Validation
V8.01 booklet: Fluid
HT-66/04/005/A
3 Modeling
With
3.1
Characteristics of modeling
For the system 3D on which one calculates the added coefficients:
Roll:
2400 meshs QUAD4
elements of hulls MEDKQU4
12 meshs SEG2
elements springs MECA_DIS_T_L
Fluid:
3600 meshs QUAD4
thermal elements THER_FACE4
on cylindrical surfaces
7200 meshs HEXA8
thermal elements THER_HEXA8
in fluid annular volume
3.2 Functionalities
tested
Controls
CALC_MATR_AJOU OPTION
“NOEUD_DOUBLE”
“MASS_AJOU”
Code_Aster
®
Version
7.2
Titrate:
FDLV110 - Calculation of mass added on modes obtained
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.110-A
Page:
7/8
Manual of Validation
V8.01 booklet: Fluid
HT-66/04/005/A
4
Results of modeling A
4.1 Values
tested
Frequencies analytical
in air (Hz)
Calculated frequencies
in air by Aster (Hz)
Variation (%)
First mode in air
0.497 0.496
0.2
Second mode in air
5.263 5.147
0.2
Mass added theoretical
(kg)
Mass added calculated
(kg)
Variation (%)
M
11
1.753 10
6
1.741
10
6
0.6
M
22
2.675 10
6
2.567
10
6
4.0
Frequencies analytical
modes out of water (Hz)
Water frequencies
calculated by Aster (Hz)
Variation (%)
First mode out of water
0.365
0.364
0.2
Second mode out of water
4.004 4.061
1.0
Code_Aster
®
Version
7.2
Titrate:
FDLV110 - Calculation of mass added on modes obtained
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.110-A
Page:
8/8
Manual of Validation
V8.01 booklet: Fluid
HT-66/04/005/A
5
Summary of the results
The calculation of mass added on modes estimated by under-structuring is satisfactory. This has
license to validate the option `
NOEUD_DOUBLE
'of the control `
CALC_MATR_AJOU
'. The variation observed on
the second coefficient of added mass is explained by the discretization of the second cylinder.
a many elements are a little insufficient to calculate in an exact way the integral of the field of
pressure on the structure.