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Code_Aster
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Version
5.0
Titrate:
SDND101 Lâcher of a system masses spring with shock
Date:
30/08/01
Author (S):
Fe WAECKEL
,
G. JACQUART
Key
:
V5.01.101-B
Page:
1/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
Organization (S):
EDF/RNE/AMV















Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
V5.01.101 document



SDND101 - To release of a system masses spring
with shock




Summary

This problem corresponds to a transitory analysis by modal recombination of a nonlinear discrete system
with a degree of freedom. Non-linearity consists of a contact with shock on a rigid level. The mass is launched
with a nonnull initial speed against the obstacle. The initial play between the material point and the obstacle is null. It
problem makes it possible to test the postprocessing of the forces of impact: velocity impact, duration of shock…
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Code_Aster
®
Version
5.0
Titrate:
SDND101 Lâcher of a system masses spring with shock
Date:
30/08/01
Author (S):
Fe WAECKEL
,
G. JACQUART
Key
:
V5.01.101-B
Page:
2/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
1
Problem of reference
1.1 Geometry
K
m
Uo
Kchoc
.

1.2
Properties of materials
The system consists of a mass m and a spring of stiffness K. The stop of shock has a stiffness
equalize in Kchoc.
Mass
m = 100 kg
Stiffness
K = 10
4
NR/m
Normal rigidity of shock
Kchoc = 10
6
NR/m

1.3 Conditions
initial
The system is initially in position at rest (
U
0
= 0) and have an initial speed
&
U
0
> 0. One
will choose for the application an initial speed
&
U
0
= 1 m/s.
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Code_Aster
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Version
5.0
Titrate:
SDND101 Lâcher of a system masses spring with shock
Date:
30/08/01
Author (S):
Fe WAECKEL
,
G. JACQUART
Key
:
V5.01.101-B
Page:
3/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
2
Reference solution
2.1
Method of calculation used for the reference solution
During the phase of impact, the system is solution of the differential equation:
m the U.K. the U.K.U
U
U
U
C
.&&
.
&
&.
+
+
=
=
=
+
0
0
0
0
0
with
and
X
+
indicate the positive value of
X
.
The analytical solution of this problem is:
()
U U
T
C
C
= & sin
0
where
C
C
K
K
m
=
+
.
Speed is cancelled for
T
U
C
&=
=
0
2

.
The force of shock is then maximum and is worth
F
K U T
K U
C
U
C
C
max
&
(
)
&
=
=
=0
0
.
By construction, the duration of the shock is worth
T
T
shock
U
=
=
2
0
&
.
The system returns to the position U = 0 with speed -
&
U
0
.
In the field U < the 0 system has as an equation
m the U.K.U
.&&
.
+
= 0
with for initial conditions
U
1
0
=
and
&
&.
U
U
1
0
= -
Its solution is
(
)
U
U
T
K
m
= -
=
& sin.
0
0
0
0
where
.
Speed is cancelled for:
=
=
T
U & 0
0
2
.
By construction, the time of coasting flight is worth: Tvol =
2
0
=
T
u&
.
The system is thus periodic with alternatively a phase of time of shock of Tchoc duration where
the system describes an arch of sine in the field of U > 0 and one phase of coasting flight of duration
Tvol where the system describes an arch of sine in the field of U < 0.
The impulse with each impact is worth:
I
K U T dt
K U
mU
K
K
C
T
C
C
C
shock
=
=
=
+
()
&
&
0
0
2
0
2
2
1
.
2.2
Results of reference
The results taken for reference are the values of the moments of maximum force, the value of force
maximum, duration of the time of shock, the value of the impulse and touch-down speed as well as
impact elementary for the first two oscillations of the system numbers.
2.3
Uncertainty on the solution
Analytical solution.
2.4 References
bibliographical
[1]
G.JACQUART: Postprocessing of calculations of core and interns ITEM under stress
seismic - HP-61/95/074/A.
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Code_Aster
®
Version
5.0
Titrate:
SDND101 Lâcher of a system masses spring with shock
Date:
30/08/01
Author (S):
Fe WAECKEL
,
G. JACQUART
Key
:
V5.01.101-B
Page:
4/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
3 Modeling
With
3.1
Characteristics of modeling
The system mass-arises is modelized by an element of the type POI1 to node NO1. It is fixed with
to move according to axis X. Node NO1 is positioned out of O = (0. 0. 0.).
An obstacle of the type
PLAN_Z
(two parallel plans separated by a play) is used to simulate them
possible shocks of the system mass-arises against a rigid plan. One chooses to take axis OY for
normal in the plan of shock, is
NORM_OBST
: (0., 1., 0.). Not to be obstructed by the rebound of
the oscillator on the symmetrical level, one very pushes back the aforementioned far (cf [Figure. 3.1-a]). One thus chooses
to locate the origin of the obstacle in
ORIG_OBS: (- 1. 0. 0.)
.
Zloc
Yloc
ORIG_OBS
(- 1, 0, 0)
NO1
U0 (0,0,0)
X
K
m
PLAY
Appear 3.1-a: modelized Geometry
It remains to define the parameter
PLAY
who gives the half-spacing between the plans in contact. One
wish here a play real no one, from where
PLAY
: 1. If one wishes a real play of J, it is necessary, in the case of figure
presented, to impose
PLAY
: 1+ J.
Temporal integration is carried out with the algorithm of Euler and a pitch of time of 5.10
- 4
S. All them
no calculation are filed. It is considered that reduced damping
I
for the whole of the modes
calculated is null.
3.2
Characteristics of the mesh
The mesh consists of a node and a mesh of the type
POI1
.
3.3 Functionalities
tested
Controls
Keys Doc. V5
“MECHANICAL” AFFE_MODELE
“DIST_T'
[U4.41.01]
DISCRETE AFFE_CARA_ELEM
M_T_D_N
[U4.42.01]
K_T_D_N
AFFE_CHAR_MECA DDL_IMPO
[U4.44.01]
MODE_ITER_INV OPTION
NEAR
[U4.52.04]
AFFE_CHAM_NO SIZE
“DEPL_R”
[U4.44.11]
PROJ_VECT_BASE VECT_ASSE
[U4.63.13]
PROJ_MATR_BASE MATR_ASSE
[U4.63.12]
STANDARD DEFI_OBSTACLE
“PLAN_Z”
[U4.44.21]
DYNA_TRAN_MODAL ETAT_INIT
VITE_INIT_GENE
[U4.53.21]
SHOCK
POST_DYNA_MODA_T RESU_GENE
[U4.84.02]
SHOCK
OPTION
“IMPACT”
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Code_Aster
®
Version
5.0
Titrate:
SDND101 Lâcher of a system masses spring with shock
Date:
30/08/01
Author (S):
Fe WAECKEL
,
G. JACQUART
Key
:
V5.01.101-B
Page:
5/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
4
Results of modeling A
4.1 Values
tested
For the first two shocks, one compares with the analytical values the computed values of the moment
where the impact occurs, of the maximum force of shock, the time of shock, the impulse and speed
of impact. One also tests the value of the absolute extremum force of impact.
First shock:
Time (S)
Reference
Aster %
difference
INST 1,5630E02
1,55000E02
­ 0,832
F_MAX 9,9500E+03
9,95269E+03 0,027
T_CHOC 3,1260E02 3,15000E02 0,768
IMPULSE 1,9805E+02 1,98093E+02
0,022
V_IMPACT ­ 1.
­ 1,00031E+00
0,031

Second shock:
Time (S)
Reference
Aster %
difference
INST 3,6100E01
3,61000E01
0
F_MAX 9,9500E+03
9,95478E+03
0,048
T_CHOC 3,1260E02
3,15000E02
0,768
IMPULSE 1,9805E+02
1,98093E+02
0,022
V_IMPACT ­ 1,0000E+00
­ 1,00031E+00
0,031
Time (S)
Reference
Aster %
difference
F_MAX_ABS 9,95E+03
9,95478E+03
0,048

4.2 Parameters
of execution
Version: STA 5.02
Machine: SGI Origin 2000
Time CPU to use: 2,2 seconds
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Code_Aster
®
Version
5.0
Titrate:
SDND101 Lâcher of a system masses spring with shock
Date:
30/08/01
Author (S):
Fe WAECKEL
,
G. JACQUART
Key
:
V5.01.101-B
Page:
6/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
5
Summary of the results
One notes, on the whole of the sizes, a very good agreement with the produced analytical solution.
The sizes the least best represented are the duration of shock and the moment of shock (to better than 1%
however). This problem is not related on the precision of calculation but to the only fact that a pitch of time
of integration of 5.10
- 4
S.A. be selected what over durations as short as 0,03 S produces already one
temporal inaccuracy of 1,66%. To supplement this synthesis, one could carry out a test of
convergence by decreasing the pitch of calculation.