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Version
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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

Organization (S): EDF/RNE/AMV

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

SDND101 - Lâcher 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…
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A

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

1
Problem of reference

1.1 Geometry

K
m
Kchoc
.Uo


1.2
Properties of materials

The system consists of a mass m and a spring of stiffness K. The thrust of shock has a stiffness
equalize in Kchoc.

Mass
m = 100 kg
Stiffness
K = 104 NR/m
Normal rigidity of shock
Kchoc = 106 NR/m

1.3 Conditions
initial

The system is initially in position at rest (U0 = 0) and has an initial speed &
U0 > 0. One
will choose for the application an initial speed &
U0 = 1 m/s.

Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A

Code_Aster ®
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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

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:

Mr. U & + k.u + K U + = 0 with U = 0 and U
0
& = U
C
0
&.
0
X + indicates the positive value of X.

The analytical solution of this problem is:


U&
K + K
U =
0
(
sin T
C
c)
=

where C
.
C
m

Speed is cancelled for tu&= =
0
2.
C
U&
The force of shock is then maximum and is worth F
= K U (T
) = K
0
max
C
u&=0
C.
C
By construction, the duration of the shock is worth T
= 2t
shock
u&=0.

The system returns to the position U = 0 with speed - &
U0.

In the field U < the 0 system has as an equation Mr. U & + k.u = 0 with for initial conditions
u1 = 0 and &u1 = - &
U.
0
U&
K
0

Its solution is U = -
if (
N .t
0
) where =
0

.
m
0

Speed is cancelled for: T
=
u&=0
2.
0
By construction, the time of coasting flight is worth: Tvol = 2t u&=0.

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.

Tchoc
U&
2mU
0
&
The impulse with each impact is worth: I =
K U (T) dt = 2K

0
C
C
=
.
2

K
0
C
1 +

Kc

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 impact 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 REP under stress
seismic - HP-61/95/074/A.
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A

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

3 Modeling
With

3.1
Characteristics of modeling

The system mass-arises is modelled 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 pushes back very this one far (cf [Figure. 3.1-a]). One thus chooses
to locate the origin of the obstacle in ORIG_OBS: (- 1. 0. 0.).

Yloc
JEU
K
m
Zloc
X
ORIG_OBS
NO1
(- 1, 0, 0)
U0 (0,0,0)
Appear 3.1-a: Géométrie modelled

It remains to define the parameter JEU which gives the half-spacing between the plans in contact. One
wish here a play real no one, from where JEU: 1. If one wishes a real play of J, it is necessary, in the case of figure
presented, to impose JEU: 1+ J.

Temporal integration is carried out with the algorithm of Euler and a step of times of 5.10-4 S. Tous them
no calculation are filed. It is considered that damping reduces I for the whole of the modes
calculated is null.

3.2
Characteristics of the grid

The grid consists of a node and a mesh of the type POI1.

3.3 Functionalities
tested

Commands



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
PROCHE

[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]
CHOC



POST_DYNA_MODA_T RESU_GENE


[U4.84.02]
CHOC
OPTION
“IMPACT”

Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A

Code_Aster ®
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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

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
IMPULSION 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
IMPULSION 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
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A

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

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 step 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 step of calculation.
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A

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