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SDLD325 - Transitory dynamic Réponse of a system mass-arises
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:
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Organization (S): EDF-R & D/AMA, IRCN
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
V2.01.325 document
SDLD325 - Transitory dynamic Réponse of one
system mass-arises deadened to 2 ddl


Summary:

This problem consists in analyzing the dynamic response of a system made up of a unit of
mass-spring-damping devices with 2 ddl from which the stiffnesses of the springs are very different under excitation from
crenel type in 1 ddl.

Via this problem, one tests the sensitivity of diagrams of integration on physical space
(DYNA_LINE_TRAN [U4.53.02]) or modal space (DYNA_TRAN_MODAL [U4.53.21]) with respect to the report/ratio of
rigidities.

The results in displacement and speed are compared with an average of results coming from codes
industrialists and of a method of integration numerical of type - Newmark improved.

Handbook of Validation
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HT-66/04/005/A

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Titrate:
SDLD325 - Transitory dynamic Réponse of a system mass-arises
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1
Problem of reference

1.1 Geometry

y, v
K
C
K
B
1
2
With
m
m
F
X, U
0 (T)
C
C

T (S)
0
1


1.2
Material properties

Stiffnesses of connection: K = 28.103 N.m1

2 cases:

·
k1 = K/10, k2 = 10k
·
k1 = 10k, k2 = K/10

Specific mass: m = 10 kg

One-way viscous damping: C = 50 kg.s1

1.3
Boundary conditions and loadings

Embedded end A.


(T) = 1 if 0 T S

1
Force applied at end b: F (T) = F0 (T) with
and F0 = 5N.



(T) = 0 if not

1.4 Conditions
initial


The system is at rest with T = 0: (
U)
0 = 0 and
()
0 = 0.
dt
Handbook of Validation
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Code_Aster ®
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Titrate:
SDLD325 - Transitory dynamic Réponse of a system mass-arises
Date:
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Author (S):
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:
V2.01.325-A Page:
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2
Reference solution

2.1
Method of calculation used for the reference solution

The search of the transitory response of this problem to damping nonproportional can be
carried out by numerical integration in real space:

[M] {U &} + [C] {U &} + [K] {U} = {F
N
N
N
}.

For that, the answer was calculated with two industrial codes:

·
PERMAS: Diagram of integration of Newmark (= 0,25 and = 0,5) T = 10­4s;
·
ABAQUS: Diagram of integration of Hilbert-Hugues-Taylor [bib1] (= ­ 0,05) T = 10­4s;

and method of integration of - Newmark improved [bib2]:

[M] [C] [K]

+
+

+
+


+
+

{
F
F
F
2 M
K
U
N2
N 1
N
n+2}
{
} {
} {}
[] []
=
+
-

{U
n+}
t2
2t
3
3


t2
1
3

[M] [C] [K]
+ -
+
-

{U
N}
t2
2t
3

where N, n+1, n+2 respectively indicate the calculations carried out at times tn, tn+1 = tn+t and tn+2 = tn+2t
where T is the increment of appointed time.

To start, one takes:

·
u0 and U 1 = U
-
0 - T
u&0
·
F
2F
F
- =
-
1
0
1

The step of adopted time is T = 10­5s.

2.2
Results of reference

Displacement and speed of the point end B.

Displacement of the point B for k2/k1=100
3,20E-03
2,40E-03
1,60E-03
(m) 8,00E-04
U B
0,00E+00
0,00
0,50
1,00
1,50
2,00
2,50
3,00
- 8,00E-04
- 1,60E-03
time (S)

Handbook of Validation
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SDLD325 - Transitory dynamic Réponse of a system mass-arises
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Displacement of the point B for k2/k1=0,01
3,00E-03
2,50E-03
2,00E-03
1,50E-03
1,00E-03
(m)
U B 5,00E-04
0,00E+00
0,00
0,50
1,00
1,50
2,00
2,50
- 5,00E-04
- 1,00E-03
- 1,50E-03
time (S)


2.3
Uncertainty on the solution

Average of numerical solutions.

2.4 References
bibliographical

[1]
H. Mr. HILBERT, T.J.R HUGUES and R.L. TAYLOR “Improved numerical dissipation for time
integration algorithms in structural dynamics “Earthquake Engineering and Structural
Dynamics, Vol.5, 1977, pp. 283-292
[2]
Structural N.M. NEWMARK “A method off computation for dynamics” Proceeding ASCE
J.Eng. Mech. DIV E-3, July 1959, pp. 67-94

Handbook of Validation
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SDLD325 - Transitory dynamic Réponse of a system mass-arises
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3 Modeling
With

3.1
Characteristics of modeling

Discrete elements of rigidity, damping and mass.

y
B
C
A.
.
.
X
N1
N2
N3


Characteristics of the elements:

DISCRET: nodal mass
M_T_D_N
rigidity
linear
K_T_D_L (kN1N2 = K/10, kN2N3 = 10k)
damping
linear
A_T_D_L

Boundary conditions: with the node N1 DDL_IMPO DX = DY = DZ = 0.

Names of the nodes: WITH = N1, C = N2, B = N3.

Methods of calculation:

·
Integration on physical space with Newmark (= 0,25, = 0,5)
No time T = 10­3s

·
Integration on the modal basis supplements with Euler
No time T = 10­3s then modal recombination

·
Integration on the modal basis supplements with T adaptive
No initial time T = 10­3s then modal recombination

Duration of observation: 3 S.

3.2
Characteristics of the grid

A number of nodes: 3

A number of meshs and type: 2 meshs SEG2

3.3
Functionalities tested

Commands



DISCRETE AFFE_CARA_ELEM
NET “K_T_D_L'

MAILLE
“A_T_D_L'

NOEUD
“M_T_D_N'

MODE_ITER_SIMULT
OPTION: “CENTER”


DYNA_LINE_TRAN NEWMARK

MATR_AMOR

DYNA_TRAN_MODAL EULER

AMOR_GENE

ADAPT



REST_BASE_PHYS



Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A

Code_Aster ®
Version
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Titrate:
SDLD325 - Transitory dynamic Réponse of a system mass-arises
Date:
16/02/04
Author (S):
E. BOYERE, T. QUESNEL Key
:
V2.01.325-A Page:
6/10


4
Results of modeling A

4.1 Values
tested

·
Displacement (m) of the point B


Displacement displacement
Displacement
Time Reference
Aster Difference
Aster Difference
Aster Difference
(S)
NEWMARK
(%)
EULER
(%)
ADAPT
(%)
0,27 3,0927 E-3 3,09263 E-3
- 0,002
3,09254 E-3
- 0,005
3,09278 E-3
0,003
0,53 8,7953 E-4 8,79902 E-4
0,042
8,79515 E-4
- 0,002
8,79583 E-4
0,006
0,80 2,4669 E-3 2,46677 E-3
- 0,005
2,46666 E-4
- 0,010
2,46688 E-4
- 0,001
1,25 - 1,0980 E-3 - 1,09829 E-3
0,026
- 1,09248 E-4
- 0,502
- 1,09844 E-4
0,040
1,51 7,8754 E-4 7,87625 E-4
0,011
7,82702 E-4
- 0,614
7,87760 E-4
0,028
1,78 - 5,6508 E-4 - 5,65131 E-4
0,009
- 5,61709 E-4
- 0,597
- 5,65265 E-4
0,033
2,05 4,0502 E-4 4,05155 E-4
0,033
4,02581 E-4
- 0,602
4,05168 E-4
0,037
2,31 - 2,9012 E-4 - 2,90070 E-4
- 0,017
- 2,88252 E-4
- 0,644
- 2,90192 E-4
0,025
2,58 2,0831 E-4 2,08323 E-4
0,006
2,06960 E-4
- 0,648
2,08376 E-4
0,032
2,85 - 1,4943 E-4 - 1,49462 E-4
0,022
- 1,48425 E-4
- 0,672
- 1,49477 E-4
0,032

·
Speed (Mr. s1) of the point B

Speed Speed Speed
Time Reference
Aster Difference
Aster Difference
Aster Difference
(S)
NEWMARK
(%)
EULER
(%)
ADAPT
(%)
0,11 1,8347 E-2 1,82400 E-2
- 0,583
1,84067 E-2
0,326
1,83510 E-2
0,022
0,39 - 1,3140 E-2 - 1,31120 E-2
- 0,213
1,31472 E-2
0,055
- 1,31407 E-2
0,006
0,66 9,3509 E-3 9,34550 E-3
- 0,058
9,36556 E-3
0,157
9,35335 E-2
0,026
0,93 - 6,7080 E-3 - 6,71303 E-3
0,075
- 6,70399 E-3
- 0,060
- 6,70788 E-3
- 0,002
1,11 - 1,5863 E-2 - 1,57872 E-2
- 0,478
- 1,57871 E-2
- 0,478
- 1,58789 E-2
0,100
1,37 1,1157 E-2 1,12034 E-2
0,416
1,10701 E-2
- 0,779
1,11521 E-2
- 0,044
1,64 - 7,9838 E-3 - 7,97210 E-3
- 0,147
- 7,94957 E-3
- 0,429
- 7,98789 E-3
0,051
1,90 5,7108 E-3 5,71217 E-3
0,024
5,67244 E-3
- 0,672
5,71139 E-3
0,010
2,17 - 4,0998 E-3 - 4,09898 E-3
- 0,020
- 4,07584 E-3
- 0,584
- 4,10120 E-3
0,034
2,44 2,9405 E-3 2,94126 E-3
0,026
2,92375 E-3
- 0,576
2,94154 E-3
0,035
2,71 - 2,1073 E-3 - 2,10817 E-3
0,041
- 2,09494 E-3
- 0,586
- 2,10808 E-3
0,037
2,97 1,5105 E-3 1,51036 E-3
- 0,009
1,50087 E-3
- 0,638
1,51084 E-3
0,022

4.2 Remarks

The results are tested on the level of the respective peaks of displacement and speed where values
are most significant.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A

Code_Aster ®
Version
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Titrate:
SDLD325 - Transitory dynamic Réponse of a system mass-arises
Date:
16/02/04
Author (S):
E. BOYERE, T. QUESNEL Key
:
V2.01.325-A Page:
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5 Modeling
B

5.1
Characteristics of modeling

Discrete elements of rigidity, damping and mass.

y
B
C
A.
.
.
X
N1
N2
N3


Characteristics of the elements:

DISCRET: nodal mass
M_T_D_N
rigidity
linear
K_T_D_L (kN1N2 = 10k, kN2N3 = K/10)
damping
linear
A_T_D_L

Boundary conditions: with the node N1 DDL_IMPO DX=DY=DZ=0.

Names of the nodes: WITH = N1, C = N2, B = N3.

Methods of calculation:

·
Integration on physical space with Newmark (= 0,25, = 0,5)
No time T = 10­3s

·
Integration on the modal basis supplements with Euler
No time T = 10­3s then modal recombination

·
Integration on the modal basis supplements with T adaptive
No initial time T = 10­3s then modal recombination

Duration of observation: 2,5 S.

5.2
Characteristics of the grid

A number of nodes: 3

A number of meshs and type: 2 meshs SEG2

5.3 Functionalities
tested

Commands



DISCRETE AFFE_CARA_ELEM
NET “K_T_D_L'

MAILLE
“A_T_D_L'

NOEUD
“M_T_D_N'

MODE_ITER_SIMULT
OPTION: “CENTER”


DYNA_LINE_TRAN NEWMARK

MATR_AMOR

DYNA_TRAN_MODAL EULER

AMOR_GENE

ADAPT



REST_BASE_PHYS



Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A

Code_Aster ®
Version
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Titrate:
SDLD325 - Transitory dynamic Réponse of a system mass-arises
Date:
16/02/04
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:
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6
Results of modeling B

6.1 Values
tested

·
Displacement (m) of the point B


Displacement displacement
Displacement
Time Reference
Aster Difference
Aster Difference
Aster Difference
(S)
NEWMARK
(%)
EULER
(%)
ADAPT
(%)
0,19 2,9334 E-3 2,93325 E-3
- 0,005
2,93308 E-3
- 0,011
2,93355 E-3
0,005
0,38 1,0959 E-3 1,09605 E-3
0,014
1,09625 E-3
0,032
1,09573 E-3
- 0,015
0,57 2,2468 E-3 2,24664 E-3
- 0,007
2,24647 E-3
- 0,015
2,24690 E-3
0,005
0,76 1,5260 E-3 1,52615 E-3
0,010
1,52627 E-3
0,017
1,52595 E-3
- 0,003
0,95 1,9773 E-3 1,97725 E-3
- 0,002
1,97718 E-3
- 0,006
1,97739 E-3
0,005
1,19 - 1,2107 E-3 - 1,21113 E-3
0,036
- 1,20839 E-3
- 0,191
- 1,21142 E-3
0,060
1,38 7,5880 E-4 7,59030 E-4
0,030
7,56994 E-4
- 0,238
7,59422 E-4
0,082
1,57 - 4,7553 E-4 - 4,75637 E-4
0,023
- 4,74180 E-4
- 0,284
- 4,75974 E-4
0,093
1,76 2,9796 E-4 2,98011 E-4
0,017
2,97002 E-4
- 0,322
2,98273 E-4
0,105
1,95 - 1,8668 E-4 - 1,86695 E-4
0,008
- 1,86012 E-4
- 0,358
- 1,86890 E-4
0,113
2,14 1,1694 E-4 1,16943 E-4
0,002
1,16489 E-4
- 0,385
1,17076 E-4
0,116
2,33 - 7,3246 E-5 - 7,32415 E-5
- 0,006
- 7,29453 E-5
- 0,411
- 7,33309 E-5
0,116

·
Speed (Mr. s1) of the point B

Speed Speed Speed
Time Reference
Aster Difference
Aster Difference
Aster Difference
(S)
NEWMARK
(%)
EULER
(%)
ADAPT
(%)
0,09 2,4261 E-2 2,42719 E-2
0,045
2,42772 E-2
0,067
2,42563 E-2
- 0,019
0,28 - 1,5210 E-2 - 1,52159 E-2
0,039
- 1,52111 E-2
0,007
- 1,52087 E-2
- 0,009
0,47 9,5332 E-3 9,53598 E-3
0,029
9,52994 E-3
- 0,034
9,53446 E-3
0,013
0,66 - 5,9745 E-3 - 5,97590 E-3
0,023
- 5,97018 E-3
- 0,072
- 5,97614 E-3
0,028
0,85 3,7438 E-3 3,74438 E-3
0,015
3,73979 E-3
- 0,107
3,74519 E-3
0,037
1,08 - 2,6037 E-2 - 2,60274 E-2
- 0,037
- 2,59908 E-2
- 0,177
- 2,60402 E-2
0,012
1,27 1,6302 E-2 1,62945 E-2
- 0,046
1,62664 E-2
- 0,218
1,63040 E-2
0,013
1,46 - 1,0204 E-2 - 1,01990 E-2
- 0,049
- 1,01797 E-2
- 0,238
- 1,02065 E-2
0,024
1,66 6,3887 E-3 6,39331 E-3
0,072
6,37778 E-3
- 0,171
6,39477 E-3
0,095
1,85 - 4,0059 E-3 - 4,00851 E-3
0,065
3,99659 E-3
- 0,232
- 4,01048 E-3
0,114
2,04 2,5114 E-3 2,51292 E-3
0,061
2,50425 E-3
- 0,285
2,51465 E-3
0,130
2,23 - 1,5743 E-3 - 1,57516 E-3
0,055
- 1,56902 E-3
- 0,355
- 1,57652 E-3
0,141
2,42 9,8676 E-4 9,87206 E-4
0,045
9,82986 E-4
- 0,382
9,88220 E-4
0,148

6.2 Remarks

The results are tested on the level of the respective peaks of displacement and speed where values
are most significant.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A

Code_Aster ®
Version
6.4
Titrate:
SDLD325 - Transitory dynamic Réponse of a system mass-arises
Date:
16/02/04
Author (S):
E. BOYERE, T. QUESNEL Key
:
V2.01.325-A Page:
9/10


7
Summary of the results

For two modelings, the results are precise with an error lower than 1%.

Integration on modal basis with a diagram with adaptive step gives the best results for one
restricted calculating time.

For information, here various times CPU User used for the resolution of modelings A and
B.

CPU To use
DYNA_LINE_TRAN DYNA_TRAN_MODAL DYNA_TRAN_MODAL
(dryness) (NEWMARK) (EULER) (Adaptatif)
Modeling A
69,82
0,50 *
0,81 *
Modeling B
57,29
0,42 *
0,50 *

(*): MODE_ITER_SIMULT = 0,23 S: calculating time of the modal base to add.

Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/04/005/A

Code_Aster ®
Version
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Titrate:
SDLD325 - Transitory dynamic Réponse of a system mass-arises
Date:
16/02/04
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E. BOYERE, T. QUESNEL Key
:
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