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Titrate:
SDLS109 - Fréquences clean of a thick cylindrical ring
Date:
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J.M. PROIX, Key S. CAILLAUD
:
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Organization (S): EDF-R & D/AMA

Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
Document: V2.03.109

SDLS109 - Fréquences clean of a ring
cylindrical thick

Summary:

This test is inspired by a vibratory study carried out on collector VVP of the N4 sections. This collector is
thick and a maximum report/ratio thickness presents on average radius of 0,13. This value, being able to be typical
of an industrial structure, is slightly higher than the limiting value of usually recognized validity
for the plates and hulls. In this study, the modeling of the collector in hulls is then evaluated by
comparison with a voluminal model on a ring.

This test makes it possible to evaluate the algorithm of search for eigenvalues MODE_ITER_SIMULT [U4.52.03] with
operators of rigidity and mass corresponding to following modelings:

1) plates of the type DKQ (finite element MEDKQU4) and DSQ (finite element MEDSQU4),
2) plates of the type DKT (finite element MEDKTR3) and DST (finite element MEDSTR3) with an ear grid
and in star grid,
3) three-dimensional hulls of COQUE_3D type (finite elements MEC3QU9H and MEC3TR7H),
4) sections of hull in plane constraints of type COQUE_C_PLAN (finite element METCSE3),
5) telegraphic elements with kinematics of beam and modes of Fourier TUYAU (finite element METUSEG3) and
TUYAU_6M (finite element MET6SEG3).

The results obtained are compared with the solution resulting from a voluminal modeling of the ring (element
finished MECA_HEXA8) revealing the modes of Fourier of command 2 (ovalization) and 3 (trifoliate) like 2
modes except plan. The variations on the frequencies of the modes of ovalization and trifoliate are close to:

·
0.4% in DSQ and COQUE_3D in QUAD9,
·
0.7% in COQUE_3D in TRIA7, COQUE_C_PLAN, TUYAU and TUYAU_6M,
·
1% for the DKQ, DKT and DST.
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:
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1
Problem of reference

1.1 Geometry

Rm
L
T


It is of a cylindrical ring, average radius Rm = 0.369 m, about thickness T = 0.048 m and of
length L = 0.05 Mr.

1.2
Properties of material

The material is homogeneous, isotropic, elastic linear. The elastic coefficients are:
E = 185.000 MPa and = 0.3.
The density is constant and is worth: = 7800 kg.m3.

1.3
Boundary conditions and loadings

The structure is free in space.

1.4
Order of magnitude of the Eigen frequencies

The required clean modes correspond to the modes of Fourier of command 2 and 3 of the ring.
frequencies of a ring can be estimated starting from an analytical model of curved beam
of Euler [bib1]. For a mode of Fourier of command N, the frequency is worth:

N (N2 -)
1
E I
F
y
=
N
2

R2
2
m
m (N +)
1

Lt3
where: I =
=
y
and m
Lt
12

For the modes of ovalization (N = 2) and trifoliate (N = 3), the corresponding frequencies are worth
respectively 211.65 Hz and 598.64 Hz. The search for clean modes is carried out on the tape
200 - 800 Hz in order to collect these 2 modes of Fourier.

1.5 Reference
bibliographical

[1]
Blevins R.D., Formulas for natural frequency and shape mode, N.Y.: Van Nostrand Reynhold
Company, 1979, 492 p.
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2
Modeling of reference

2.1
Characteristics of the modeling of reference

Voluminal elements 3D


The discretized geometry is represented above. The elements 3D are voluminal with 8 nodes of
type HEXA8. The number of nodes on the circumference is 600, on thickness 9 and length 9.

2.2
Characteristics of the grid

A number of nodes: 48600
A number of meshs and type: 38400 HEXA8

2.3 Functionalities
tested

Commands
AFFE_MODELE
AFFE
MODELISATION = “3D”
CALC_MATR_ELEM OPTION “RIGI_MECA”
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ = (200., 800.)
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3
Results of the modeling of reference

3.1
Values of reference tested

Frequencies of the clean modes of ovalization, trifoliate and except plan.

Mode
Eigen frequencies (Hz)
ovalization 210.55
210.55
trifoliate 587.92
587.92
except plan
205.89
205.89
588.88
588.88

3.2 Remarks

The axisymmetric problem has double modes in the plan and except plan.
The modes except plan have the following deformations:

205.89 Hz:
588.88 Hz:



3.3 Uncertainties

Uncertainty results from the analysis of convergence of the grid where Eigen frequencies with
grid of reference 600x8x8 are compared with those of the grids 500x7x9.

A number of elements
Eigen frequencies (Hz)
Circumference Thickness
Length
Ovalization Trifoliate
100
3
3
232.03
648.62
200
3
3
225.20
628.74
400
5
8
221.02
616.34
500
7
9
210.64
588.18
600
8
8
210.55
587.92

Uncertainty is 0.05% on the frequency.

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4 Modeling
With

4.1
Characteristics of modeling

Plates DKQ and DSQ.



The discretized geometry is represented above. Elements DKQ and DSQ are facets
plane with four nodes of the type QUAD4. The number of nodes on the circumference is 100 and on
length 5.

4.2
Characteristics of the grid

A number of nodes: 500
A number of meshs and type: 400 QUAD4

4.3 Functionalities
tested

Commands
AFFE_MODELE
AFFE
MODELISATION = “DKT” or “DST”
AFFE_CARA_ELEM
COQUE
EPAIS = 0.048
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ = (200., 800.)
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5
Results of modeling A

5.1 Values
tested

(Frequencies as Hertz)

Reference mode
Aster
DKQ %
Aster difference
DSQ %
difference
ovalization 210.55 211.48
0.44
209.57
­ 0.46
210.55
211.48
0.44
209.57
­ 0.46
trifoliate 587.92
598.23
1.75
586.30
­ 0.27
587.92
598.23
1.75
586.30
­ 0.27
except plan
205.89
234.70
13.99


205.89
234.70
13.99


588.88
646.34
9.75
533.02
­ 9.49
588.88
646.34
9.75
533.02
­ 9.49

5.2 Remarks

Modelings in plates DKQ and DSQ do not make it possible to represent the modes correctly
except plan. One can think that that is due to the number of meshs over the too low length (4 with the place
from 8 in the modeling of reference). The first frequency except plan in plates DSQ must be
lower than 200 Hz.

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6 Modeling
B

Plates DKT and DST - ear grid



The geometry discretized on the average radius Rm = 0.369 m is represented above. Elements
DKT and DST are plane facets with three nodes of the type TRIA3 laid out out of ears. The number of
nodes on the circumference is 100 and over length 5.

6.1
Characteristics of the grid

A number of nodes: 500
A number of meshs and type: 800 TRIA3

6.2 Functionalities
tested

Commands
AFFE_MODELE
AFFE
MODELISATION = “DKT” or “DST”
AFFE_CARA_ELEM
COQUE
EPAIS = 0.048
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ = (200., 800.)
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7
Results of modeling B

7.1 Values
tested

(Frequencies as Hertz)

Reference mode
Aster
DKT %
Aster difference
DST %
difference
ovalization 210.55 211.54
0.47
203.69
­ 3.25
210.55
211.54
0.47
203.69
­ 3.25
trifoliate 587.92
598.64
1.82
568.70
­ 3.27
587.92
598.64
1.82
568.70
­ 3.27
except plan
205.89
254.89
23.80
202.38
­ 1.70
205.89
254.89
23.80
202.38
­ 1.70
588.88
707.53
20.14
617.18
4.81
588.88
707.53
20.14
617.18
4.81

7.2 Remarks

Modeling in plates DKT does not make it possible to represent the modes except plan correctly.
The errors on the frequencies in DST plates are relatively important.

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8 Modeling
C

Plates DKT and DST - star grid



The geometry discretized on the average radius Rm = 0.369 m is represented above. Elements
DKT and DST are plane facets with three nodes of the type TRIA3 laid out out of stars. The number
nodes on the circumference is 100 and over length 5.

8.1
Characteristics of the grid

A number of nodes: 500
A number of meshs and type: 800 TRIA3

8.2 Functionalities
tested

Commands
AFFE_MODELE
AFFE
MODELISATION = “DKT” or “DST”
AFFE_CARA_ELEM
COQUE
EPAIS = 0.048
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ = (200., 800.)

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9
Results of modeling C

9.1 Values
tested

(Frequencies as Hertz)

Reference mode
Aster
DKT %
Aster difference
DST %
difference
ovalization 210.55 211.54
0.47
208.20
­ 1.11
210.55
211.54
0.47
208.20
­ 1.11
trifoliate 587.92
598.58
1.81
581.00
­ 1.18
587.92
598.58
1.81
581.00
­ 1.18
except plan
205.89
284.38
38.12
225.23
9.39
205.89
284.38
38.12
225.23
9.39
588.88
797.24
35.38
690.73
17.29
588.88
797.24
35.38
690.73
17.29

9.2 Remarks

Modelings in plates DKT and DST do not make it possible to represent the modes correctly
except plan.

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10 Modeling
D

10.1 Characteristics of modeling

COQUE_3D grid in QUAD9.


The geometry discretized on the average radius Rm = 0.369 m is represented above. Elements
COQUE_3D is meshs with 9 nodes of the type QUAD9 making it possible to take into account the radius of
curvature of the ring. The nodes mediums are of course the average circumference. The number of
nodes on the circumference is 40 and over length 5.

10.2 Characteristics of the grid

A number of nodes: 200
A number of meshs and type: 40 QUAD9

10.3 Functionalities
tested

Commands
CREA_MAILLAGE
MODI_MAILLAGE
OPTION = “QUAD8_9”
AFFE_MODELE
AFFE
MODELISATION = “COQUE_3D”
AFFE_CARA_ELEM
COQUE
A_CIS = 0.833333
COEF_RIGI_DRZ = 1.E-5
EPAIS = 0.048
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ = (200., 800.)
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11
Results of modeling D

11.1 Values
tested

(Frequencies as Hertz)

Reference mode
Code_Aster %
difference
ovalization 210.55
209.91
­ 0.30
210.55
209.91
­ 0.30
trifoliate 587.92
586.51
­ 0.24
587.92
586.51
­ 0.24
except plan
205.89
205.14
­ 0.36
205.89
205.14
­ 0.36
588.88
587.55
­ 0.23
588.88
587.55
­ 0.23

11.2 Remarks

All the frequencies are correctly estimated.

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12 Modeling
E

12.1 Characteristics of modeling

COQUE_3D grid in TRIA7.


The geometry discretized on the average radius Rm = 0.369 m is represented above. Elements
COQUE_3D is meshs with 7 nodes of the type TRIA7 making it possible to take into account the radius of
curvature of the ring. The nodes mediums are of course the average circumference. The number of
nodes on the circumference is 40 and over length 5.

12.2 Characteristics of the grid

A number of nodes: 280
A number of meshs and type: 160 TRIA7

12.3 Functionalities
tested

Commands
CREA_MAILLAGE
MODI_MAILLAGE
OPTION = “TRIA6_7”
AFFE_MODELE
AFFE
MODELISATION = “COQUE_3D”
AFFE_CARA_ELEM
COQUE
A_CIS = 0.833333
COEF_RIGI_DRZ = 1.E-5
EPAIS = 0.048
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ = (200., 800.)
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13
Results of modeling E

13.1 Values
tested

(Frequencies as Hertz)

Reference mode
Code_Aster %
difference
ovalization 210.55 211.19
0.30
210.55
211.19
0.30
trifoliate 587.92
590.98
0.52
587.92
590.98
0.52
except plan
205.89
205.81
­ 0.04
205.89
205.81
­ 0.04
588.88
595.38
1.10
588.88
595.38
1.10

13.2 Remarks

The frequencies are less better estimated that with elements QUAD9.

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14 Modeling
F

14.1 Characteristics of modeling

COQUE_C_PLAN.



The geometry discretized on the average radius Rm = 0.369 m is represented above. Elements
COQUE_C_PLAN are meshs with 3 nodes of the type SEG3 making it possible to take into account the radius
of curvature of the ring. The nodes mediums are of course the average circumference. The number of
nodes on the circumference is 100.

14.2 Characteristics of the grid

A number of nodes: 100
A number of meshs and type: 50 SEG3

14.3 Functionalities
tested

Commands
AFFE_MODELE
AFFE
MODELING = “COQUE_C_PLAN”
AFFE_CARA_ELEM
COQUE
A_CIS = 0.833333
MODI_METRIQUE = “NOT”
EPAIS = 0.048
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ = (200., 800.)
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15
Results of modeling F

15.1 Values
tested

(Frequencies as Hertz)

Reference mode
Code_Aster %
difference
ovalization 210.55
209.14
­ 0.67
210.55
209.14
­ 0.67
trifoliate 587.92
583.28
­ 0.79
587.92
583.28
­ 0.79
except plan
205.89


205.89
588.88
588.88

15.2 Remarks

Modeling in plane constraints does not make it possible to reveal the modes except plan of
the ring.

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16 Modeling
G

16.1 Characteristics of modeling

TUYAU and TUYAU_6M.

AXE in SEG3


The geometry discretized on the axis of the ring is represented above. Elements TUYAU and
TUYAU_6M are meshs with 3 nodes of the type SEG3 representing the axis of the ring. The number of
nodes on the axis is 5.

16.2 Characteristics of the grid

A number of nodes: 5
A number of meshs and type: 2 SEG3

16.3 Functionalities
tested

Commands
AFFE_MODELE
AFFE
MODELISATION = “TUYAU” or “TUYAU_6M'
AFFE_CARA_ELEM
POUTRE
SECTION = “CIRCLE”
MODI_METRIQUE = “NOT”
CARA = (“R” “EP”)
VALE = (0.395, 0.048)
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ = (200., 800.)

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17
Results of modeling G

17.1 Values
tested

(Frequencies as Hertz)

Reference mode
Aster
TUYAU %
Aster difference
TUYAU_6M %
difference
ovalization 210.55 209.02
­ 0.72
209.02
­ 0.72
210.55
209.02
­ 0.72
209.02
­ 0.72
trifoliate 587.92
591.00
0.52
591.00
0.52
587.92
591.00
0.52
591.00
0.52
except plan
205.89
259.74
26.15
259.74
26.15
205.89
259.74
26.15
259.74
26.15
588.88
649.57
10.31
649.57
10.31
588.88
649.57
10.31
649.57
10.31

17.2 Remarks

Modelings in TUYAU (limited by construction to 3 modes of Fourier) and TUYAU_6M
do not allow to represent the modes except plan correctly. On the other hand, they provide them
same results on the outline views, close to the reference.

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18 Modeling
H

18.1 Characteristics of modeling

Modeling SHB8



The discretized geometry is represented above. Elements SHB8 are pressed on meshs
HEXA8. The number of elements on the circumference is 100 and over length 5.

18.2 Characteristics of the grid

A number of nodes: 1000
A number of meshs and type: 400 HEXA8, 400 QUAD4

18.3 Functionalities
tested

Commands
AFFE_MODELE
AFFE
MODELISATION = “SHB8”
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_SIMULT
CALC_FREQ
BANDAGE FREQ = (200., 800.)
CREA_MAILLAGE COQU_VOLU

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19
Results of modeling H

19.1 Values
tested

(Frequencies as Hertz)

Reference mode
Aster
DKQ %
difference
ovalization 210.55 210.71
0.08
210.55
210.71
0.08
trifoliate 587.92
590.84
0.5
587.92
590.84
0.5
except plan
205.89
208.05
1.05
205.89
208.05
1.05
588.88
595.3
1.09
588.88
595.3
1.09

20
Summary of the results

·
Even if the results obtained are honourable, models DKQ, DSQ, DKT, DST,
COQUE_C_PLAN, TUYAU and TUYAU_6M do not make it possible to estimate the modes except plan of
the ring. Only the model COQUE_3D with a grid in QUAD9 provides a good estimate
frequential of these modes with an error lower than 0.4%.
·
Models DSQ and COQUE_3D in QUAD9 give an error close to 0.3% for the modes
of ovalization and trifoliate. For the models COQUE_3D in TRIA7, COQUE_C_PLAN, TUYAU and
TUYAU_6M, this error border 0.7% and are higher than 1% for the DKQ, DKT and DST.
·
The best results of the DSQ compared to the DKQ confirm than the effect of shearing
transverse is not negligible in the hulls notable thickness.
·
On a developable geometry like that of the cylinder studied here, finite elements
quadrangular provide better results which triangular finite elements. DST,
tested on an ear grid and a symmetrical star grid, are much less
goods that DSQ. The same remark is also valid for the COQUE_3D in TRIA7 by
report/ratio with the COQUE_3D in QUAD9.
·
The performances of the COQUE_3D are good quantitatively and in calculating times.
richness of the interpolation, the taking into account of the curvature of the cylinder in the elements and
metric correction carried out in the thickness of the hull seem to explain these good results.
·
Elements SHB8 make it possible to about obtain all the modes with a maximum change
1%.

Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HT-66/04/005/A

Outline document