Code_Aster
®
Version
4.0
Titrate:
SDLL15 Beam hurled, embed-free
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.015-B
Page:
1/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Organization (S):
EDF/IMA/MN
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.015 document
SDLL15 - Beam 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 Mechanics of the 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.
Code_Aster
®
Version
4.0
Titrate:
SDLL15 Beam hurled, embed-free
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.015-B
Page:
2/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
1
Problem of reference
1.1 Geometry
L
With
B
Z, W
y, v
X, U
C
y
C
Co-ordinates of the points (in m):
With
B
C
X
0.
10.
10.
y
0.
0.
y
C
Z
0.
0.
0.
length of the beam: AB = L = 10 m
specific mass out of C: m
C
= 1000 kg
Tubular section:
external diameter
of = 0.350 m
internal diameter
di = 0.320 m
surface
To = 1.57865 10
2
m
2
inertia
I
y
= I
Z
= 2.21899 10
4
m
4
polar inertia
IP = 4.43798 10
4
m
4
2 studied cases:
1) teststemyç = 0.
2) teststemyç = 1. m
1.2
Material properties
E = 2.1 10
11
AP
= 7800 kg/m
3
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.
Code_Aster
®
Version
4.0
Titrate:
SDLL15 Beam hurled, embed-free
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.015-B
Page:
3/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
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),
·
bending 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:
F
1
=
1
2
3rd I
Z
L
3
m
C
+
0.24 M
(
)
with M = total mass of the beam.
When the mass is offset, the modes of bending (X, Z) and of torsion are coupled, as well as
modes of bending (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
v
W
U
v
W
Z
y
Z
X
y
X
C
C
C
B
B
B
C
C
C
C
C
C
X
y
Z
B
B
B
=
+
-
-
+
+
-
0
0
0
U
C
=
U
B
-
Z
B
for this test
v
C
=
v
B
W
C
=
W
B
+
X
B
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:
F
1
analytical solution
other frequencies ± 1%
Problem 2:
± 1%
2.4 References
bibliographical
[1]
Working group Analyzes Dynamic. Commission of Validation of the Software packages of Calculation of
Structures. French company of Mécaniens. (1988)
Code_Aster
®
Version
4.0
Titrate:
SDLL15 Beam hurled, embed-free
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.015-B
Page:
4/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
3 Modeling
With
3.1
Characteristics of modeling
Element of beam
POU_D_E
and discrete element
DIS_TR
With
B
y
X
C
Cutting: beam AB: 20 meshs SEG2.
Limiting conditions:
with the node end A
DDL_IMPO: (NODE: WITH DX: 0., DY: 0., DZ: 0., DRX: 0., DRY: 0., DRZ: 0.)
Nodal mass out of B with an eccentricity
E
y
= 0.
Case 1
E
y
= 1.
Case 2
Names of the nodes:
Points
With = N100
B = N200
3.2
Characteristics of the mesh
A number of nodes:
21
A number of meshs and types:
20 SEG2
3.3 Functionalities
tested
Controls
Keys
AFFE_CARA_ELEM
BEAM
“CIRCLE”
ALL
[U4.24.01]
DISCRETE
“M_TR_D_N'
AFFE_CHAR_MECA
DDL_IMPO
NODE
[U4.25.01]
AFFE_MATERIAU
ALL
[U4.23.02]
AFFE_MODELE
“MECHANICAL”
“POU_D_E”
ALL
[U4.22.01]
“DIS_TR”
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_SIMULT
METHOD
“TRIA_DIAG”
[U4.52.01]
CALC_FREQ
OPTION
“PLUS_PETITE”
NMAX_FREQ
: 10 cases 1
: 8 cases 2
Code_Aster
®
Version
4.0
Titrate:
SDLL15 Beam hurled, embed-free
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.015-B
Page:
5/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
4
Results of modeling A
4.1 Values
tested
Case
Nature of the mode
clean
Frequency
Reference
Hz
Aster
% difference
bending 1,2
1.65
1.6554
0.33
bending 3,4
16.07
16.0712
0.
CASE 1
bending 5,6
50.02
50.0240
0.
traction 1
76.47
76.4727
0.
teststemyç = 0.
torsion 1
80.47
80.4688
0.
bending 7,8
103.20
103.20444
0.
fz + to 1
1.636
1.6363
0.
fy + tr 2
1.642
1.6416
0.
CASE 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
X
B
0.03
3.039 10
2
1.321
1
W
C
/W
B
1.030
1.030
0.
2
U
C
/v
B
0.148
0.148
0.
3
U
C
/v
B
2.882
2.880
0.07
4
W
C
/W
B
0.922
0.923
0.108
5
X
B
1.922
1.92268
0.036
with:
F
Z
+ to = bending X, Z + torsion
F
y
+ tr = bending X, y + traction
4.2 Remarks
Calculations carried out by:
MODE_ITER_SIMULT
METHOD: “TRI_DIAG”
OPTION: “PLUS_PETITE” NMAX_FREQ:
10 Cases 1
8 Cases 2
In the test, one cannot check the values of the reports/ratios
U
C
v
B
for modes 2 and 3 (except
manually). With regard to the values of
W
C
W
B
, the technique is as follows: if one imposes
W
B
=
1
(control
NORM_MODE
), one has then
W
C
W
B
=
1
+
X
B
and one can make checks on
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
Overall dimension memory:
8 megawords
Time CPU To use:
7.2 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLL15 Beam hurled, embed-free
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.015-B
Page:
6/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
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.