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Version
5.0
Titrate:
SDLL02 - Poutre hurled, embed-free, folded up on it even
Date:
19/08/02
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.02.002-B Page:
1/10

Organization (S): EDF/AMA, DeltaCAD

Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.002 document

SDLL02 - Poutre hurled, embed-free,
folded up on it even

Summary:

This two-dimensional problem consists in seeking the frequencies and the modes of vibration of a structure
mechanics, made up of a hurled, embedded beam free and folded up on itself.

The problem arising does not have physical significance. It on the other hand makes it possible to validate the search of
Eigen frequencies of inflection multiples and the search of the modes double in a subspace of command 2.

In this test, one carries out three different modelings:

· in the first modeling, the boundary conditions are imposed using parameters of
Lagrange (command AFFE_CHAR_MECA) and the clean values and vectors are calculated by
method of Lanczos (command MODE_ITER_SIMULT, method: “TRI_DIAG”),
· in the second modeling, the boundary conditions are imposed by removing degrees
of freedom in the matrices of mass and stiffness (command AFFE_CHAR_CINE) and the values and
clean vectors are calculated by the method of Bathe and Wilson (command MODE_ITER_SIMULT,
method: “JACOBI”),
· in the third modeling, one checks the behavior of modeling COQUE_C_PLAN in
dynamics. The eigenvalues and the clean modes are calculated with the command
MODE_ITER_SIMULT and with the method of SORENSEN.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SDLL02 - Poutre hurled, embed-free, folded up on it even
Date:
19/08/02
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.02.002-B Page:
2/10

1
Problem of reference

1.1 Geometry

y
y, v
y
With
B
H

C
X
Z
X, U
B
L


The geometrical characteristics of the beam constituting the mechanical model are as follows:

Length: L = 0.5 m

Rectangular cross-section:

Height:
H = 0.005 m
Width:
B = 0.050 m
Surface:
With = 2.5 10­4 m2
Moment of inertia:
Iz = 5.208 10­10 m4

The co-ordinates (in meters) of the points characteristic of the whole of the beams are:

WITH B
C
X 0.
0.5
0.
y 0.
0. 0.

1.2
Material properties

The properties of material constituting the beam are:

E = 2.1 1011 Pa

= 0.3

= 7.800. kg/m3

1.3
Boundary conditions and loadings

The boundary condition which characterizes this problem is the embedding of point A and is written:

U = v = 0., = 0.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SDLL02 - Poutre hurled, embed-free, folded up on it even
Date:
19/08/02
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.02.002-B Page:
3/10

2
Reference solution

2.1
Method of calculation used for the reference solution

The reference solution is that given in card SDLL02/89 of the guide VPCS which presents
method of calculation in the following way:

By the method of stiffness dynamic, one shows that the folded up beam admits frequencies
double, solution of:
cos
() = 0

I = 2
(I - 1) 2
2

E I
F
I
Z
I = 1
I =1, 2,…
2 L2
With

For a rectangular section, one obtains:

E
F
(
) 2
I = 2 I - 1 R
I = 1, 2,…
8 L2
12

This formulation neglects the deformations of shearing action and inertia of rotation (beam of Euler-
Bernoulli).

For the clean modes, the forms are given in guide VPCS. They are normalized with 1 or ­ 1 with
not greater amplitude. There are results only for modes 1, 2, 3, 4, 7 and 8. By
example, the forms of the first two clean modes are as follows:

1
1
0.707
­ 0.707
mode 1
F
mode 2
1 = 11.76 Hz


2.2
Results of reference

The results of reference are the first eight Eigen frequencies and displacements of the points
B, C for the clean modes 1, 2, 3, 4, 7 and 8. In Code_Aster, the modes are normalized to 1 with
not greater amplitude (command NORM_MODE). To be able to make comparisons with
the results of reference, the latter were corrected (multiplication by ­ 1 if necessary).

2.3
Uncertainty on the solution

There is no uncertainty on the solution because it is analytical.

2.4 References
bibliographical

[1]
PIRANDA J.: Run and Travaux Dirigés de Vibrations of Structures - Option Mécanique -
École Nationale Supérieure of Mécanique and Micromécanique - Laboratoire de Mécanique
Applied - Besancon (France (1983).)
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SDLL02 - Poutre hurled, embed-free, folded up on it even
Date:
19/08/02
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.02.002-B Page:
4/10

3 Modeling
With

3.1
Characteristics of modeling

y
B
With
X
C


The beam in 20 meshs SEG2 was cut out (10 for part AB and 10 for part BC).

The modeling used for the beams is that of Euler Bernoulli (POU_D_E).

Two-dimensional solutions are sought. One can thus block for all the nodes it
displacement DZ and rotations DRX and DRY.

The end of the beam (not A) is embedded from where in this point:

DX = DY = 0. DRZ = 0.

3.2
Characteristics of the grid

The grid contains 21 nodes and 20 meshs of the type SEG2.

The points characteristic of the grid are as follows:

Not A = A
Not B = B
Not C = C

3.3 Functionalities
tested

The functionalities tested are summarized in this table:

Commands



AFFE_CARA_ELEM
POUTRE
“RECTANGLE”
TOUT

AFFE_CHAR_MECA
DDL_IMPO
TOUT

NOEUD

AFFE_MATERIAU
TOUT

AFFE_MODELE
“MECANIQUE”
“POU_D_E”
TOUT

RECU_CHAMP
“NUMERO_ORDRE”
“DEPL”

DEFI_MATERIAU
ELAS

MODE_ITER_SIMULT
METHODE
“TRI_DIAG”

CALC_FREQ
OPTION
“PLUS_PETITE”

NORM_MODE
“TRAN”


One insists on commands MODE_ITER_SIMULT (method of Lanczos) for the calculation of the modes
clean and AFFE_CHAR_MECA for the imposition of the boundary conditions.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SDLL02 - Poutre hurled, embed-free, folded up on it even
Date:
19/08/02
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.02.002-B Page:
5/10

4
Results of modeling A

4.1 Values
tested

For the frequencies of vibration of the structure, there are the following results:

Identification Reference Aster %
difference
Frequency 1
11.76
11.7642
0.04
Frequency 2
11.76
11.7642
0.04
Frequency 3
105.88
105.8811
0.00
Frequency 4
105.88
105.8812
0.00
Frequency 5
294.10
294.1780
0.03
Frequency 6
294.10
294.1806
0.03
Frequency 7
576.44
576.9802
0.09
Frequency 8
576.44
577.0079
0.10

For the modes of vibration of the structure, there are the following results:

Identification Points
Size Reference
Aster %
difference
Frequency 1
B
DY
­ 0.707
­ 0.69845
­ 1.2
C
DY
1.
1.
Frequency 2
B
DY
0.707
0.72615
2.7
C
DY
1.
1.
Frequency 3
B
DY
0.707
0.70711
0.01
C
DY
1.
1.
Frequency 4
B
DY
­ 0.370
­ 0.37015
0.04
C
DY
0.523
0.52347
0.09
Frequency 7
B
DY
0.707
0.70711
0.02
C
DY
1.
1.
Frequency 8
B
DY
­ 0.388
­ 0.38847
0.12
C
DY
0.549
0.54937
0.07

4.2 Remarks

For the Eigen frequencies, the results obtained are correct. It is the same for the results
obtained concerning the clean modes. The tolerance is lower than 2% for the whole of the modes
except for the mode 2 where the tolerance lies between 2 and 3%.

4.3 Parameters
of execution

Version: NEW 3.03.09
Machine: CRAY C90



Obstruction memory:
8 MW
Time CPU To use:
5 seconds
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SDLL02 - Poutre hurled, embed-free, folded up on it even
Date:
19/08/02
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.02.002-B Page:
6/10

5 Modeling
B

5.1
Characteristics of modeling

y
B
With
X
C


The beam in 20 meshs SEG2 was cut out (10 for part AB and 10 for part BC).

The modeling used for the beams is that of Euler Bernouilli (POU_D_E).

Two-dimensional solutions are sought. One can thus block for all the nodes it
displacement DZ and rotations DRX and DRY.

The end of the beam (not A) is embedded from where in this point:

DX = DY = 0. DRZ = 0.

5.2
Characteristics of the grid

The grid contains 21 nodes and 20 meshs of the type SEG2.

The points characteristic of the grid are as follows:

Not A = A
Not B = B
Not C = C

5.3 Functionalities
tested

The functionalities tested are summarized in this table:

Commands



AFFE_CARA_ELEM
POUTRE
“RECTANGLE”
TOUT

AFFE_CHAR_CINE
MECA_IMPO
TOUT

NOEUD

AFFE_MATERIAU
TOUT

AFFE_MODELE
“MECANIQUE”
“POU_D_E”
TOUT

RECU_CHAMP
“NUMERO_ORDRE”
“DEPL”

DEFI_MATERIAU
ELAS

MODE_ITER_SIMULT
METHODE
“JACOBI”

CALC_FREQ
OPTION
“PLUS_PETITE”

NORM_MODE
“TRAN”


One insists on commands MODE_ITER_SIMULT (method of Bathe and Wilson) for the calculation of
clean modes and AFFE_CHAR_CINE for the imposition of the boundary conditions.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SDLL02 - Poutre hurled, embed-free, folded up on it even
Date:
19/08/02
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.02.002-B Page:
7/10

6
Results of modeling B

6.1 Values
tested

For the frequencies of vibration of the structure, there are the following results:

Identification Reference Aster %
difference
Frequency 1
11.76
11.7642
0.04
Frequency 2
11.76
11.7642
0.04
Frequency 3
105.88
105.8811
0.00
Frequency 4
105.88
105.8812
0.00
Frequency 5
294.10
294.1780
0.03
Frequency 6
294.10
294.1806
0.03
Frequency 7
576.44
576.9802
0.09
Frequency 8
576.44
577.0079
0.1

For the modes of vibration of the structure, there are the following results:

Identification Points
Size Reference
Aster %
difference
Frequency 1
B
DY
­ 0.707
­ 0.69658
­ 1.47
C
DY
1.
1.
Frequency 2
B
DY
0.707
0.73038
3.31
C
DY
1.
1.
Frequency 3
B
DY
0.707
0.70711
0.02
C
DY
1.
1.
Frequency 4
B
DY
­ 0.370
­ 0.37014
0.04
C
DY
0.523
0.52347
0.09
Frequency 7
B
DY
0.707
0.70711
0.02
C
DY
1.
1.
Frequency 8
B
DY
­ 0.388
­ 0.38846
0.12
C
DY
0.549
0.54937
0.07

6.2 Remarks

For the Eigen frequencies, the results obtained are correct. It is the same for the results
obtained concerning the clean modes. The tolerance is lower than 2% for the whole of the modes
except for the mode 2 where the tolerance lies between 3 and 4%.

6.3 Parameters
of execution

Version: NEW 3.03.09
Machine: CRAY C90



Obstruction memory:
8 MW
Time CPU To use:
5 seconds
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SDLL02 - Poutre hurled, embed-free, folded up on it even
Date:
19/08/02
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.02.002-B Page:
8/10

7 Modeling
C

7.1
Characteristics of modeling

y
B
With
X
C


The beam in 20 meshs SEG3 was cut out (10 for part AB and 10 for part BC).

Modeling used is COQUE_C_PLAN.

The end of the beam (not A) is embedded from where in this point:

DX = DY = DRZ = 0.

7.2
Characteristics of the grid

The grid contains 41 nodes and 20 meshs of the type SEG3.

The points characteristic of the grid are as follows:

Not A = A
Not B = B
Not C = C

7.3 Functionalities
tested

The functionalities tested are summarized in this table:

Commands



AFFE_CARA_ELEM
COQUE
EPAIS
AFFE_MODELE
“MODELISATION”
“COQUE_C_PLAN”
MODE_ITER_SIMULT
METHODE
“SORENSEN”
CALC_FREQ
OPTION
“PLUS_PETITE”
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SDLL02 - Poutre hurled, embed-free, folded up on it even
Date:
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Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.02.002-B Page:
9/10

8
Results of modeling C

8.1 Values
tested

For the frequencies of vibration of the structure, there are the following results:

Identification Reference Aster %
difference
Frequency 1
11.76
11.767
0.063
Frequency 2
11.76
11.804
0.377
Frequency 3
105.88
106.533
0.616
Frequency 4
105.88
107.556
1.583
Frequency 5
294.10
299.685
1.899
Frequency 6
294.10
304.643
3.585
Frequency 7
576.44
599.025
3.918
Frequency 8
576.44
613.506
6.430

For the modes of vibration of the structure, there are the following results:

Identification Points
Size Reference
Aster %
difference
Frequency 1
B
DY
­ 0.707
- 0.707
0.015
C
DY
1.
1.
0.
Frequency 2
B
DY
0.707
0.707
0.015
C
DY
1.
1.
0.
Frequency 3
B
DY
0.707
0.707
0.015
C
DY
1.
1.
0.
Frequency 4
B
DY
­ 0.370
- 0.373
0.713
C
DY
0.523
0.527
0.763
Frequency 7
B
DY
0.707
0.707
0.015
C
DY
1.
1.
0.
Frequency 8
B
DY
­ 0.388
- 0.403
3.990
C
DY
0.549
0.571
3.936

8.2 Remarks

In this case-test, where the results are independent of the Young modulus, it is not necessary of
to modify the Young modulus retained for modeling, as in the case of the static analysis
linear, to take account of the real width of the beam.

8.3 Parameters
of execution

Version: NEW 5.04.17
Machine: SGI-Origin2000 R12000


Obstruction memory: 16 megabytes
Time CPU To use:
2.49 seconds
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SDLL02 - Poutre hurled, embed-free, folded up on it even
Date:
19/08/02
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.02.002-B Page:
10/10

9
Summary of the results

· Modelings A and B of the Poutre type:
The problem is dealt with with very good precision on the first eight frequencies (tolerance
< 0.1%) for two modelings tested. The components of modes 3, 4, 7 and 8 are
also obtained with a good precision of about 0.1%. The precision on mode 1 is of
the command of 1% for the method of Lanczos and 0.5% for the method of Bathe and Wilson. In it
who relates to mode 2, the precision degrades himself: it is about 2.7% for the method of
Lanczos and about 3.3% for the method of Bathe and Wilson. Complementary tests
(use of the method of Lanczos by imposing boundary conditions by the command
AFFE_CHAR_CINE) make it possible to think that these differences come from the method of
seek eigenvalues used.

· Modeling COQUE_C_PLAN
The precision on the results is good for the first three frequencies, the error is of the command
from 0.6%. It is degraded as the frequency increases, the error passes from 4th
frequency with 8th of 1.5% to 6.4%. More the frequency is high plus the difference between the frequencies
double is important. The error on the modes is satisfactory for the first 7 modes
(<0.7%), it is higher for 8th (<4%). A finer grid should allow best
to represent the modal deformations associated with the high frequencies.

Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/02/001/A

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