Code_Aster
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Version
4.0
Titrate:
SDLL04 Beam hurled on two supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.004-C
Page:
1/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Organization (S):
EFD/IMA/MN
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.004 document
SDLL04 - Beam hurled on two supports,
coupled to a system mass-arises
Summary:
This plane problem consists in seeking the frequencies of vibration of a mechanical structure made up of one
beam embed-slide and of a mass connected to the beam by a spring. Stiffness of the spring and mass
depend on a variable parameter, which will make it possible to highlight the displacement of the frequencies
clean for a small disturbance of the model. This test of Mechanics of the Structures corresponds to an analysis
dynamics of a linear model having a linear behavior. It includes/understands only one modeling.
This problem makes it possible to test the element of beam of Timoshenko in bending, the calculation of the Eigen frequencies
by the method of the iterations opposite and the method of Lanczos, the discrete elastic connection between one
specific mass and a node of a beam.
The results obtained are in concord with the results given in guide VPCS. It well is observed
unfolding of the Eigen frequencies induced by the disturbance of the initial model (beam hurled on two
supports).
Code_Aster
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Titrate:
SDLL04 Beam hurled on two supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.004-C
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/4
B
D
X, U
y, v
With
L
C
Length: L = 10
has
=
AD
B
=
DB
(
)
m
E
=
WITH L
=
780
Kg
K
E
=
4
m
E
=
780
4
NR/m
Cross-section:
surface
A=1.10
2
M
2
moment of inertia
Iz = 3.9 10
6
m
4
3 cases studied:
= 0.
= 0.001
= 0.01
Co-ordinates of the points (meters):
With
B
C
D
X
0.
10.
2.5
2.5
y
0.
0.
qcq_0
0.
1.2
Material properties
E
= 2.10
11
AP
= 7.800. kg/m
3
1.3
Boundary conditions and loadings
Not a:
U = v = 0.
Not b:
v = 0.
Not C:
U = 0.
= 0. vertical slide
1.4 Conditions
initial
Without object for the modal analysis.
Code_Aster
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Titrate:
SDLL04 Beam hurled on two supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.004-C
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 SDLL04/89 of the guide VPCS which presents
method of calculation in the following way:
The equation with the own pulsations of the complete system is written:
() ()
()
() ()
()
(
)
R L
R has
R B
R L
R has
R B
R L
I
I
I
I
I
I
I
I
C
C
sin
sin
sin
HS
HS
HS
/
-
=
-
2
2
2
2
with:
=
m
E
WITH L
R
I
4
=
I
2
With
E I
C
=
K
E
m
E
has
+
B
=
L
In secondary absence of system,
K
E
, m
E
=
0
, one finds well the Eigen frequencies of the beam
hurled on two supports.
F
I
L
I.E.(internal excitation)
With
I
I
=
=
2
2
2
2
1
2
When the secondary system is granted exactly on the first mode of this beam, them
new Eigen frequencies of the system can be obtained by the approximate formulas:
(
)
F
m
M
F
F
F
F
E
1 2
1
1
1
3
2
1 0 5
1 0 5
,
*
*
.
.
= ±
= ±
with
M
1
modal mass of the beam without secondary system for a normalized clean mode with 1
at point D.
2.2
Results of reference
The first two Eigen frequencies for
= 0.
The first three Eigen frequencies for
= 0.001 and
= 0.01.
2.3
Uncertainty on the solution
< 4
% for the first modes if the system is granted to the first mode.
2.4 References
bibliographical
[1]
NOUR-OMID, SACKMAN, KIUREGHIAN. Modal characterization off equipment continous
structure system. Newspaper off Sound and Vibration, V.88 n°4, p. 459, 472 (1983).
Code_Aster
®
Version
4.0
Titrate:
SDLL04 Beam hurled on two supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.004-C
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
One uses right beams of Timoshenko
POU_D_T
and of the discrete elements
DIS_T
.
y
With
C
D
X
B
Cutting:
AD: 5 meshs SEG2
DB: 15 meshs SEG2
CD: 1 mesh SEG2
Modeling:
POU_D_T
for all the meshs of beam AB
DIS_T
for the mesh CD and the point C
For all the structure
DZ = DRX = DRY = 0
Limiting conditions:
in all the nodes of
beam AB:
with the nodes
ends:
out of C:
DDL_IMPO: (GROUP_NO: NPOUTRE DZ:0., DRX:0, DRY:0.)
(GROUP_NO: WITH DX: 0., DY: 0. ) (GROUP_NO: B DY: 0. )
(GROUP_NO: C DX: 0., DZ: 0. )
Names of the nodes:
Not A = N1
Not B = N21
Not C = N22
Not D = N6
3.2
Characteristics of the mesh
A number of nodes:
22
A number of meshs and types:
21 meshs SEG2
1 mesh P0I1
3.3 Functionalities
tested
Controls
Keys
AFFE_CARA_ELEM
BEAM
“GENERAL”
ALL
[U4.24.01]
GROUP_MA
DISCRETE
“K_T_D_L'
“M_T_D_N'
AFFE_CHAR_MECA
DDL_IMPO
ALL
[U4.25.01]
GROUP_NO
AFFE_MATERIAU
GROUP_MA
[U4.23.02]
AFFE_MODELE
“MECHANICAL”
“POU_D_T'
GROUP_MA
[U4.22.01]
“DIS_T'
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_SIMULT
METHOD
“TRI_DIAG”
[U4.52.02]
CALC_FREQ
OPTION
“PLUS_PETITE”
NMAX_FREQ
MODE_ITER_INV
CALC_FREQ
OPTION
“NEAR”
[U5.52.01]
FREQ
Code_Aster
®
Version
4.0
Titrate:
SDLL04 Beam hurled on two supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.004-C
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
Frequency (Hz)
Command of the clean mode
Reference
Aster
% difference
0.
bending 1
bending 2
1.5707
6.2831
1.5707
6.2812
0.
0.03
0.001
1 bending
2 bending
3 bending 2
1.5460
1.5958
6.2336
1.5460
1.5957
6.2817
0.
0.
0.77
0.01
1 bending
2 bending
3 bending 2
1.4937
1.6506
6.2874
1.4936
1.6506
6.2854
0.
0.
0.03
4.2 Remarks
For
= 0, one carried out:
MODE_ITER_SIMULT METHOD: “TRI_DIAG”
OPTION: “PLUS_PETITE”
NMAX_FREQ: 2
For
= 0.001, one carried out:
MODE_ITER_INV
OPTION: “NEAR”
LIST_FREQ: (1.5, 1.6, 6.5)
For
= 0.01, one carried out:
MODE_ITER_INV
OPTION: “ADJUSTS”
LIST_FREQ: (1. , 7.)
Contents of the file results:
Case 1: the first 2 Eigen frequencies, clean vectors and modal parameters.
Case 2: the first 3 Eigen frequencies and modal parameters.
Case 3: the first 3 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:
8 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLL04 Beam hurled on two supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.004-C
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 unfolding of the Eigen frequencies induced by the disturbance of the initial model is perfectly
represented.