Code_Aster
®
Version
4.0
Titrate:
Simple SDNL100 Pendulum in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B
Page:
1/6
Manual of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A
Organization (S):
EDF/IMA/MN
Manual of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
Document: V5.02.100
SDNL100 - Simple pendulum in great oscillation
Summary:
The object of this test is to calculate the movement of a heavy bar articulated at a point fixes by one of its
ends, free elsewhere and oscillating with great amplitude in a vertical plane.
Interest: to test the element of cable with two nodes - which is in fact an element of bar - in dynamics and sound
operation in the operator
DYNA_NON_LINE
[U4.32.02].
Code_Aster
®
Version
4.0
Titrate:
Simple SDNL100 Pendulum in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B
Page:
2/6
Manual of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A
1
Problem of reference
1.1 Geometry
()
G
Z
X
O
1
P
·
·
A rigid pendulum COp length 1 and center of gravity G oscillates around the point O.
The angular position of the pendulum is identified by:
= -
1.2
Material properties
Linear density of the pendulum: 1. kg/m
Axial rigidity (produced Young modulus by the surface of the cross-section): 1.10
8
NR
1.3
Boundary conditions and loadings
The pendulum is articulated at the point fixes O. Under the action of gravity, its end P oscillates on
half-circle (
) of center O and 1. There is no friction.
1.4 Conditions
initial
The pendulum is released without speed of horizontal position COp.
= +
=
2
0
Code_Aster
®
Version
4.0
Titrate:
Simple SDNL100 Pendulum in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B
Page:
3/6
Manual of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A
2
Reference solution
2.1
Method of calculation used for the reference solution
The period
T
of a mobile pendulum without friction around the point O fixes
,
whose mass is
concentrated in the center of gravity G (OG = L) and whose maximum angular amplitude is
0
is given
by the series [bib1]:
T
G
has
has
N
N
N
N
N
N
=
+
=
-
=
2
1
2
2
1
2
2
0
2
1
L
with
sin
2.2
Results of reference
For L = 0.5 m, G = 9.81 m/s
2
and
0
=
/2, one finds:
T
= 1.6744 S
2.3
Uncertainty on the solution
One summoned the terms of the series until N = 12 inclusively, the last term taken into account being
lower than 10
- 5
time the calculated sum.
2.4 References
bibliographical
[1]
J. HAAG, “movements vibratory”, P.U.F. (1952).
Code_Aster
®
Version
4.0
Titrate:
Simple SDNL100 Pendulum in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B
Page:
4/6
Manual of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A
3 Modeling
With
3.1
Characteristics of modeling
The pendulum is modelized by an element of cable with 2 nodes, identical to an element of bar of
constant section.
Discretizations:
·
space: an element of cable MECABL2
·
temporal: analyze movement over one period supplements T per pitch of times equal to
T/40.
3.2 Functionalities
tested
Order
DYNA_NON_LINE
for great displacements.
3.3
Characteristics of the mesh
A number of nodes:
2
A number of meshs and types:
1 mesh SEG2
Code_Aster
®
Version
4.0
Titrate:
Simple SDNL100 Pendulum in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B
Page:
5/6
Manual of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A
4
Results of modeling A
4.1 Values
tested
Moment
Size
Reference
Aster
% difference
Tolerance
T/4 0.4186
DX
p
DZ
p
1.
1.
0.97518
0.99969
2.48
0.03
rel 2.5
rel 0.05
T/2 0.8372
DX
p
DZ
p
2.
0.
2.00000
6.29E4
0.0
-
rel 0.01
ABS 0.0007
3T/4 1.2558
DX
p
DZ
p
1.
1.
1.07453
0.99722
7.45
0.28
rel 7.5
rel 0.3
T 1.6744
DX
p
DZ
p
0.
0.
6.50E7
1.40E3
-
-
ABS 1.E-6
ABS 1.5E-3
4.2 Remarks
·
Temporal integration is done by the method of NEWMARK (rule of the trapezoid),
·
With each pitch of time, convergence is reached in less than 8 iterations.
4.3 Parameters
of execution
Version: 3.06.11
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
55.6 seconds
Code_Aster
®
Version
4.0
Titrate:
Simple SDNL100 Pendulum in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B
Page:
6/6
Manual of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A
5
Summary of the results
One sees on this case-test that temporal integration by the “rule of the trapezoid” of Newmark does not modify
that very slightly the frequency and does not bring parasitic damping, since at the end of one
period one returns to very little close with the initial position.