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Version
4.0
Titrate:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A Page:
1/6
Organization (S): EDF/IMA/MN
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.009 document
SDLL09 - Vibration of a slim beam
of variable rectangular section (embed-free)
Summary:
This plane problem consists in seeking the frequencies of vibration of a free fixed beam with section
rectangular variable. This test comprises only one modeling.
The variation of section of the beam is either homothetic, or nonhomothetic. Characteristics of
section of the beam are given according to meshs' in two different ways:
· section and inertias,
· height and width.
This problem thus makes it possible to thus test the element of beam with variable section for a prismatic structure
that the calculation of the frequencies of vibration by iterations opposite. In addition, in the operator
AFFE_CARA_ELEM [U4.24.01], one tests the remanence of certain key words.
The results obtained are in concord with those given in guide VPCS.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
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Version
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Titrate:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A Page:
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1
Problem of reference
1.1 Geometry
y, v
L
y
y
bo
b1
B
H
H
O
h1
H
With
1
Z
O
X, U
Z
B
Z, W
1
bo
Rectangular sections
Length of the beam:
L = 1 m
Rectangular section:
Initial cross-section
Final cross-section
Case 1
Case 2
height:
Ho = 0.04 m
= 0.04 m
h1 = 0.01 m
width:
bo = 0.04 m
= 0.05 m
b1 = 0.01 m
surface:
Ao = 1.6 103 m2
= 2.103 m2
A1 = 1.104 m2
inertia:
lzo = 2.1333 107 m4
= 2.6667 107 m4
Iz1 = 8.3333 1010 m4
Co-ordinates of the points (m):
With
B
X
0.
1.
y
0.
0.
Z
0.
0.
1.2
Material properties
E = 2.1011 Pa
= 7.800 kg/m3
1.3
Boundary conditions and loadings
Not a: embedded U = v = 0 = 0.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
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Version
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Titrate:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A Page:
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2
Reference solution
2.1
Method of calculation used for the reference solution
The reference solution is that given in card SDLL09/89 of the guide VPCS which presents
method of calculation in the following way:
Exact calculation by numerical integration of the differential equation of the inflection of the beams (Théorie
of Euler-Bernouilli).
2
2
v
E
I
Z
x2
2
= -
v
With
x2
t2
where Iz and A vary with the X-coordinate.
One obtains:
E
F
=
1
() h1
I
,
2 I
L2
12
with:
= h0 = 4
h1
= b0 = 4 or 5
b1
1
2
3
4
5
= 4
23.289
73.9
165.23
299.7
478.1
= 5
24.308
75.56
167.21
301.9
480.4
2.2
Results of reference
the first 5 clean modes of inflection.
2.3
Uncertainty on the solution
Semi-analytical solution.
2.4 References
bibliographical
H.H. MABIE, C.B. ROGERS, Transverse vibrations off double-tapered cantilever beams - Journal off the
Acoustical Society off America, n° 51, p. 1771-1774 (1972).
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
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Version
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Titrate:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A Page:
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3 Modeling
With
3.1
Characteristics of modeling
Modeling: Elements of beam POU_D_E
y
B
With
X
Cutting:
beam AB: 30 meshs SEG2 of section variable
15 meshs in “Générale section”
15 meshs in “Rectangulaire section”
Limiting conditions:
in all the nodes
DDL_IMPO: (TOUT:“YES” DZ: 0., DRX: 0., DRY: 0. )
at end A
(Node: WITH DX: 0., DY: 0., DRZ: 0. )
Names of the nodes:
Not A = N100
Not B = N200
3.2
Characteristics of the grid
Grid:
A number of nodes: 31
A number of meshs and types: 30 SEG2
3.3 Functionalities
tested
Commands
Keys
AFFE_CARA_ELEM
POUTRE
“GENERALE”
MAILLE
[U4.24.01]
AFFE_CARA_ELEM
POUTRE
“RECTANGLE”
MAILLE
[U4.24.01]
AFFE_CHAR_MECA
DDL_IMPO
[U4.25.01]
AFFE_MODELE
“MECANIQUE”
“POU_D_E”
[U4.22.01]
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_INV
OPTION
“AJUSTE”
[U4.52.01]
Test of the remanence of a key word (SECTION and CARA).
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
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Titrate:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A Page:
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4
Results of modeling A
4.1 Values
tested
Identification
Reference
Aster
% difference
frequency in HZ
Case 1 Ho/h1 = 4 bo/b1 = 4
homothetic
inflection 1
54.18
54.1354
0.08
inflection 2
171.94
171.7122
0.13
inflection 3
384.40
383.8764
0.14
inflection 4
697.24
696.1877
0.15
inflection 5
1112.28
1110.4727
0.16
Cases 2 Ho/h1 = 4 bo/b1 = 5
nonhomothetic
inflection 1
56.55
56.4984
0.09
inflection 2
175.19
175.5306
+0.19
inflection 3
389.01
388.3426
0.17
inflection 4
702.36
700.9879
0.20
inflection 5
1117.63
1115.4275
0.20
4.2 Remarks
Calculations by
MODE_ITER_INV (…
CALC_FREQ: (FREQ: (53. , 1150.))) ;
4.3 Parameters
of execution
Version: NEW3.03.15
Machine: CRAY C90
Obstruction memory:
8 MW
Time CPU To use:
7.72 seconds
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A Page:
6/6
5
Summary of the results
Good establishment of the element of non-prismatic beam with a fine grid.
A coarser modeling would be sufficient.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A