Code_Aster ®
Version
7.2
Titrate:
FDLV110 - Calcul of mass added on modes obtained

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.110-A Page:
1/8

Organization (S): EDF-R & D/AMA, EDF-DPN/UTO
Handbook of Validation
V8.01 booklet: Fluid
Document: V8.01.110

FDLV110 - Calcul 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). This one 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 grid
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).
Handbook of Validation
V8.01 booklet: Fluid
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
FDLV110 - Calcul of mass added on modes obtained

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.110-A Page:
2/8

1
Problem of reference

1.1 Geometry
L
surface fluid
k2
Z
IH
R
y
F
X
Re
springs
fluid
k1
roll medium
external cylinder


5
L=50m; Rf=1m; Ri= m; R
3
e=3m; k1=109 N.m1; k2=0.5 107 N.m1; fluide=1000 kg.m3;
s=7800 kg.m3; thickness of the hull: 50 cm.

1.2
Properties of materials

Fluid: density 1 = 1000 kg.m3; 2 = 750 kg.m3.

Structure: = 7800 kg/m3; E = 2.1 1011 Pa; = 0.3 (steel).
S

1.3
Boundary conditions and loadings

The external cylinder on the one hand is connected to a fixed frame via the four springs of unit stiffness k1,
connected in addition to the cylinder medium by four springs of unit stiffness k2. The two structures are
rigid in this reference solution.

Handbook of Validation
V8.01 booklet: Fluid
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
FDLV110 - Calcul of mass added on modes obtained

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.110-A Page:
3/8

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
k2 = 107 N.m1. These springs are embedded with the interface with the external cylinder which constitutes
second substructure (interface of the type CRAIG-BAMPTON).

IH
k2


Mass cylinder 1: m1 = 2.041 106 kg

The cylinder being rigid, its movement can be modelled by a system mass-arises with a degree
of freedom:

m1


Achacune of its ends, the cylinder is connected to two springs in parallel: equivalent stiffness
of each one = 2k2 is k'

1
2k
1
4k
The Eigen frequency is worth then:
2
F =
=
, that is to say: F = 0 705
.
Hz
2
1
m
2
1
m

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 frame:

Re
k1
K


Mass cylinder 2: m2=3,674 106 kg
Handbook of Validation
V8.01 booklet: Fluid
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
FDLV110 - Calcul of mass added on modes obtained

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.110-A Page:
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Equivalent stiffness of a fastener of this cylinder by the system of springs in series k1 and k2 being worth
9,9.106 N.m1 (four fasteners of the same type connect in parallel the cylinder to an embedding),
Eigen frequency is given by:

1
4k2 1
K (k2 + 1
K)
F =
, that is to say: F =.
0.522Hz
2
1
m

N.B.: the third cylinder (interior cylinder) was not modelled 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:

m2
m1
k3=4k1
k4=4k2


The Eigen frequencies of this system are given by the exact formula [bib2]:

1
K
K
K
K
K
K 2
K K
F =
3 + 4 + 4 ± 3 + 4 + 4 - 4 3 4
,
I
32
m
m
m

2
2
1
m
m
m
2
2
1
m m
2
1
2

that is to say

F = 0.497

and

Hz
F = 5.
Hz
263
1
2
.

The two clean modes admit, for numerical value:

X


X
3
1

-
.

1
1
- 9 10

=
and
=


.
X
- 3

5. 10
X
2

1

1
2 2

2.2
Calculation of the matrix of added mass

Fluid potentials

Beginning again [bib1], it is established that:


2
R + 2
2
2
R

2
2
()
1
2 I
F
R +
E
I
R
= R
+
I
R
E
R
1
;
I
(- 3
5.e)
2
+


×

R
2
2
2
2
E
2
2

R - R
R - R
R -
I
F
E
I
E
I
R
and


2
R + 2
2
2
R

2
2
(2)

=
I
F
E
R
I
R
I
R
E
R
1
(- - 3
9.e)
2
+
2
+




× I
R
+
;R
2
2
2
2nd
2
2

R - R
R - R
R -
I
F
E
I
E
I
R
Handbook of Validation
V8.01 booklet: Fluid
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
FDLV110 - Calcul of mass added on modes obtained

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.110-A Page:
5/8

The shape of the matrix of added mass, in this configuration, is:

11
12
M
M
M =
has
has
has

21
22
M has
M has

With:


2
2

2
2
R + R

11
2 I
F
R +
I
E
R
M
=
has
L
R
+
I
= 753
,
1
.106 kg,
2
2
2
2


R - R
R -
I
F
E
I
R



2
2
R + R
22
2
I
E

M
=
has
L
R

E
=,
2.676 10
. 6 kg,


2
2


R -
E
I
R



2
2

2
2
R + R

2
2
-
I
F
R + R

I
E
-
R + R
12



I
E

M
=
has
L
5
(10
.
3)
2
× R
+
- 9
(10
.
3)
2
× R

I
E
= 15318
-
kg.
2
2
2
2
2
2


R - R
R -
I
F
E
I
R
R -
E
I
R



The coefficient of inertial coupling
12
M A is regarded as negligible in front of the coefficients
of added car-mass
11
M has and 22
M A. Les 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 - Bibliographie and establishment in
Code_Aster - HP-61/95/064
[2]
BLEVINS R.D.: Formulated for Natural frequency and shape mode, ED. Krieger
Handbook of Validation
V8.01 booklet: Fluid
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
FDLV110 - Calcul of mass added on modes obtained

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.110-A Page:
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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

Commands


CALC_MATR_AJOU OPTION
“NOEUD_DOUBLE”


“MASS_AJOU”



Handbook of Validation
V8.01 booklet: Fluid
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
FDLV110 - Calcul of mass added on modes obtained

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.110-A Page:
7/8

4
Results of modeling A

4.1 Values
tested


Frequencies analytical
Calculated frequencies
Variation (%)
in air (Hz)
in air by Aster (Hz)





First mode in air
0.497 0.496
0.2
Second mode in air
5.263 5.147
0.2





Mass added theoretical Masse added calculated
Variation (%)
(kg)
(kg)



M11
1.753 106 1.741
106 0.6
M22
2.675 106 2.567
106 4.0





Frequencies analytical
Water frequencies
Variation (%)
modes out of water (Hz) calculated by Aster (Hz)



First mode out of water
0.365
0.364
0.2
Second mode out of water
4.004 4.061
1.0

Handbook of Validation
V8.01 booklet: Fluid
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
FDLV110 - Calcul of mass added on modes obtained

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.110-A Page:
8/8

5
Summary of the results

The calculation of mass added on modes estimated by under-structuring is satisfactory. This has
license to validate option “NOEUD_DOUBLE” of command “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.

Handbook of Validation
V8.01 booklet: Fluid
HT-66/04/005/A

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