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Titrate:
SDLL15 Poutre hurled, embed-free
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.015-B Page:
1/6
Organization (S): EDF/IMA/MN
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.015 document
SDLL15 - Poutre hurled, embed-free,
with mass or offset inertia
Summary:
This three-dimensional problem consists in calculating the frequencies and the modes of vibration of a structure
mechanics made up of a right beam slim, embed-free, with tubular section and of a mass
offset attached at the loose lead of the beam. This test of Mécanique of Structures corresponds to one
analyze dynamic of a linear model having a linear behavior. It comprises only one modeling.
This problem makes it possible to test the element of beam of Euler Bernouilli, the model of specific mass and calculation
modal by the method of Lanczos.
The results obtained are in concord with those of guide VPCS. Two calculations carried out (eccentricity of
the specific mass null or different from zero) make it possible to highlight the coupling of different
modes when the specific mass is offset.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
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Titrate:
SDLL15 Poutre hurled, embed-free
Date:
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B. QUINNEZ
Key:
V2.02.015-B Page:
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1
Problem of reference
1.1 Geometry
Z, W
y, v
C
With
teststemyç
B
L
X, U
Co-ordinates of the points (in m):
With
B
C
X
0.
10.
10.
y
0.
0.
teststemyç
Z
0.
0.
0.
length of the beam: AB = L = 10 m
specific mass out of C: mc = 1000 kg
Tubular section:
external diameter
of = 0.350 m
internal diameter
di = 0.320 m
surface
With = 1.57865 102 m2
inertia
Iy = Iz = 2.21899 104 m4
polar inertia
IP = 4.43798 104 m4
2 studied cases:
1) teststemyç = 0.
2) teststemyç = 1. m
1.2
Material properties
E = 2.1 1011 Pa
= 7800 kg/m3
1.3
Boundary conditions and loadings
Not A embedded: (U = v = W = 0, = = =
X
y
Z
0).
1.4 Conditions
initial
Without object for the modal analysis.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
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Titrate:
SDLL15 Poutre hurled, embed-free
Date:
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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 SDLL15/89 of the guide VPCS which presents
method of calculation in the following way:
The problem with not offset mass leads to uncoupled modes:
· traction and compression (effect of the mass alone),
· torsion (effect of inertia around neutral fiber),
· inflection in plans X, y and X, Z (effect of the mass).
The various Eigen frequencies are given with a model by finite elements of beam
of Euler (slim beam).
For the first mode with an unbalance, a method of Rayleigh gives the formula
approached:
3rd I
F = 1
Z
1
2
L3 m
(+
)
C
0.24 M
with M = total mass of the beam.
When the mass is offset, the modes of inflection (X, Z) and of torsion are coupled, as well as
modes of inflection (X, y) and of traction and compression.
For the clean mode, the components at the point B make it possible to calculate the components in the center
of gravity of the mass (point C) by:
U
U
0
Z
- y
C
B
C
C
xB
v
= v
+ - Z
0
+ X
C
B
C
C
y
B
W
W + y - X
0
C
B
C
C
Z
B
U = U -
C
B
zB
for this test
v = v
C
B
W = W +
C
B
xB
2.2
Results of reference
Case 1: the first 10 clean modes.
Case 2: the first 8 clean modes.
2.3
Uncertainty on the solution
Problem 1:
f1 analytical solution
other frequencies ± 1%
Problem 2:
± 1%
2.4 References
bibliographical
[1]
Working group Analyze Dynamique. Commission of Validation of Progiciels de Calcul of
Structures. French company of Mécaniens. (1988)
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
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Titrate:
SDLL15 Poutre hurled, embed-free
Date:
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3 Modeling
With
3.1
Characteristics of modeling
Element of beam POU_D_E and discrete element DIS_TR
y
C
With
B
X
Cutting: beam AB: 20 meshs SEG2.
Limiting conditions:
with the node end A
DDL_IMPO: (NOEUD: WITH DX: 0., DY: 0., DZ: 0., DRX: 0., DRY: 0., DRZ: 0.)
Nodal mass out of B with an eccentricity
ey= 0.
Case 1
ey= 1.
Case 2
Names of the nodes:
Points
With = N100
B = N200
3.2
Characteristics of the grid
A number of nodes:
21
A number of meshs and types:
20 SEG2
3.3 Functionalities
tested
Commands
Keys
AFFE_CARA_ELEM
POUTRE
“CERCLE”
TOUT
[U4.24.01]
DISCRET
“M_TR_D_N'
AFFE_CHAR_MECA
DDL_IMPO
NOEUD
[U4.25.01]
AFFE_MATERIAU
TOUT
[U4.23.02]
AFFE_MODELE
“MECANIQUE”
“POU_D_E”
TOUT
[U4.22.01]
“DIS_TR”
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_SIMULT
METHODE
“TRIA_DIAG”
[U4.52.01]
CALC_FREQ
OPTION
“PLUS_PETITE”
NMAX_FREQ
: 10 cases 1
: 8 cases 2
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
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Titrate:
SDLL15 Poutre hurled, embed-free
Date:
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4
Results of modeling A
4.1 Values
tested
Case
Nature of the mode
Frequency Hz
% difference
clean
Reference
Aster
inflection 1,2
1.65
1.6554
0.33
inflection 3,4
16.07
16.0712
0.
CAS 1
inflection 5,6
50.02
50.0240
0.
traction 1
76.47
76.4727
0.
teststemyç = 0.
torsion 1
80.47
80.4688
0.
inflection 7,8
103.20
103.20444
0.
fz + to 1
1.636
1.6363
0.
fy + tr 2
1.642
1.6416
0.
CAS 2
fy + tr 3
13.46
13.4551
0.
fz + to 4
13.59
13.5919
0.
teststemyç = 1.
fz + to 5
28.90
28.8972
0.
fy + tr 6
31.96
31.9594
0.
fz + to 7
61.61
61.6091
0.
fy + tr 8
63.93
63.9289
0.
Mode
0.03
3.039 102
1.321
X B
1
WC/wB
1.030
1.030
0.
2
CPU/vB
0.148
0.148
0.
3
CPU/vB
2.882
2.880
0.07
4
WC/wB
0.922
0.923
0.108
5
1.922
1.92268
0.036
X B
with:
fz + to = inflection X, Z + torsion
fy + tr = inflection X, y + traction
4.2 Remarks
Calculations carried out by:
MODE_ITER_SIMULT
METHODE: “TRI_DIAG”
OPTION: “PLUS_PETITE” NMAX_FREQ:
10 Cases 1
8 Cases 2
U
In the test, one cannot check the values of the reports/ratios C for modes 2 and 3 (except
vB
W
manually). With regard to the values of
C, the technique is as follows: if one imposes
wB
W
W =
C =
B
1 (command NORM_MODE), one has then
1 + X and one can make checks on
W
B
B
values of X.
B
Contents of the file results:
Case 1: the first 11 Eigen frequencies, clean vectors and modal parameters.
Case 2: the first 9 Eigen frequencies, clean vectors and modal parameters.
4.3 Parameters
of execution
Version: 3.02.21
Machine: CRAY C90
System:
UNICOS 8.0
Obstruction memory:
8 megawords
Time CPU To use:
7.2 seconds
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:
SDLL15 Poutre hurled, embed-free
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.015-B Page:
6/6
5
Summary of the results
The modeling of unbalance gives exact results for the 8 frequencies of reference.
The precision of the clean modes is about 0.1% until mode 4.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A