Code_Aster ®
Version
4.0
Titrate:
Simple SDNL100 Pendule in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B Page:
1/6
Organization (S): EDF/IMA/MN
Handbook of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
Document: V5.02.100
SDNL100 - Simple Pendule 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 operator DYNA_NON_LINE [U4.32.02].
Handbook of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A
Code_Aster ®
Version
4.0
Titrate:
Simple SDNL100 Pendule in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B Page:
2/6
1
Problem of reference
1.1 Geometry
Z
X
P
O
·
·
G
1
()
A rigid pendulum COp length 1 and center of gravity G oscillates around the point O.
The angular position of the pendulum is located 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): 1.108 NR
1.3
Boundary conditions and loadings
The pendulum is articulated at the point fixes O. Sous l' action de 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.
= +
= 0
2
Handbook of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A
Code_Aster ®
Version
4.0
Titrate:
Simple SDNL100 Pendule in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B Page:
3/6
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 fixes O, 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]:
2n
L
T = 2
1+
2
has
0
sin
G
N
2
n=1
with
2 N - 1
has =
N
2 N
2.2
Results of reference
For L = 0.5 m, G = 9.81 m/s2 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 times the calculated sum.
2.4 References
bibliographical
[1]
J. HAAG, “Les vibratory movements”, P.U.F. (1952).
Handbook of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A
Code_Aster ®
Version
4.0
Titrate:
Simple SDNL100 Pendule in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B Page:
4/6
3 Modeling
With
3.1
Characteristics of modeling
The pendulum is modelled 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 step of times equal to
T/40.
3.2 Functionalities
tested
Order
DYNA_NON_LINE for great displacements.
3.3
Characteristics of the grid
A number of nodes:
2
A number of meshs and types:
1 mesh SEG2
Handbook of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A
Code_Aster ®
Version
4.0
Titrate:
Simple SDNL100 Pendule in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B Page:
5/6
4
Results of modeling A
4.1 Values
tested
Moment
Size
Reference
Aster
% difference
Tolerance
T/4 0.4186
DXp
1.
0.97518
2.48
rel 2.5
DZ
1.
0.99969
0.03
rel 0.05
p
T/2 0.8372
DXp
2.
2.00000
0.0
rel 0.01
DZ
0.
6.29E4
-
ABS 0.0007
p
3T/4 1.2558
DXp
1.
1.07453
7.45
rel 7.5
DZ
1.
0.99722
0.28
rel 0.3
p
T 1.6744
DXp
0.
6.50E7
-
ABS 1.E-6
DZ
0.
1.40E3
-
ABS 1.5E-3
p
4.2 Remarks
· Temporal integration is done by the method of NEWMARK (rule of the trapezoid),
· With each step 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
Obstruction memory:
8 megawords
Time CPU To use:
55.6 seconds
Handbook of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A
Code_Aster ®
Version
4.0
Titrate:
Simple SDNL100 Pendule in great oscillation
Date:
01/12/98
Author (S):
J.M. PROIX, P. MASSIN
Key:
V5.02.100-B Page:
6/6
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.
Handbook of Validation
V5.02 booklet: Nonlinear dynamics of the linear structures
HI-75/98/040 - Ind A