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Titrate:
SDLD313 - Système mass-arises to 2 ddl (damping hysteretic)
Date:
23/06/03
Author (S):
O. Key NICOLAS
:
V2.01.313-A Page:
1/10

Organization (S): EDF-R & D/AMA

Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
Document: V2.01.313

SDLD313 - Système masses spring with 2 DDL with
damping hysteretic

Summary:

This one-way problem consists in carrying out a harmonic analysis of a mechanical structure
composed of a whole of mass-springs with damping hysteretic and subjected to an excitation
sinusoidal. This test of mechanics of the structures corresponds to a dynamic analysis of a discrete model
having a linear behavior. It includes/understands three modelings.

Via modeling A, one tests the discrete elements in translation (mass, arises), them
options AMOR_HYST of AFFE_CARA_ELEM.
Via modeling B, one tests the elements of beam (POU_D_T), options AMOR_HYST of
DEFI_MATERIAU,
Via modeling C, one tests calculation modal (MODE_ITER_SIMULT) complex.

On the first two modelings, one tests the definition of a force of specific excitation harmonic and
the operator of harmonic calculation of answer (DYNA_LINE_HARM [U4.54.02]). In addition, several are tested
operators of postprocessing: RECU_FONCTION [U4.62.03], TEST_FONCTION [U4.72.02], RECU_CHAMP
[U4.62.01].

Results obtained for the first two modelings (field of displacement for different
frequencies of excitation) are in concord with the results of guide VPCS. Results obtained for
the third modeling are in concord with the semi-analytical results.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A

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Titrate:
SDLD313 - Système mass-arises to 2 ddl (damping hysteretic)
Date:
23/06/03
Author (S):
O. Key NICOLAS
:
V2.01.313-A Page:
2/10

1
Problem of reference

1.1 Geometry
We consider the system represented by the diagram below:


k1 (1+j * 1)
k2 (1+j * 2)

m
m2

1

X1
X2
WITH B C

Specific masses:
m & m
1
2

Stiffnesses of connection:
k1 & k2

Damping hysteretic:
1 & 2

1.2
Properties of material

Comes out from linear elastic translation
K1 = 28000
NR/m

K2 =
28000 NR/m
Specific mass
M1 =
10 kg

M2 =
5 kg
Damping hysteretic
1 =
0.1

2 =
0.0

1.3
Boundary conditions and loadings

Boundary conditions:
Points A, B, C embedded out of DY and DZ
Embedded points a: (DX = 0).

Loading: Force concentrated sinusoidal of variable frequency at the point C

Not C

Fx = 0
F sint = 2 F 0 Hz

F
Hz

21.0543
4

0
F = constant =
NR

100

1.4 Conditions
initial

Without object for the study of the permanent harmonic mode.
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A

Code_Aster ®
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Titrate:
SDLD313 - Système mass-arises to 2 ddl (damping hysteretic)
Date:
23/06/03
Author (S):
O. Key NICOLAS
:
V2.01.313-A Page:
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2
Reference solution

2.1
Method of calculation

The system of differential equations of the second command coupled is form:

M u&+ Ku = F
0 0 0
1+.
0 1 J
- 1 - 0 1
. J
0




with M = 0 10 0
and
K = 28000-1-.
0 1 J
2 + 0 1
. J
-
1




0 0 5

0
- 1
1


The solution with a harmonic excitation F = F0 E J T j2 = - 1
(
) is form U =u0 ejt, it
who leads to: (
2
K - M) U = F
0
0

This system is solved for all.

2.2
Sizes and results of reference

Displacement according to X of the point C for certain frequencies.
Eigen frequencies and damping reduced.

2.3
Uncertainties on the solution

Semi-analytical solution.

2.4 References
bibliographical

[1]
J. PIRANDA: Note of use of the software of modal analysis MODAN - Version 0.2 (1990).
Laboratory of Mécanique Appliquée - Université de Franche County - Besancon (France).
Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A

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Titrate:
SDLD313 - Système mass-arises to 2 ddl (damping hysteretic)
Date:
23/06/03
Author (S):
O. Key NICOLAS
:
V2.01.313-A Page:
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3 Modeling
With

3.1
Characteristics of modeling

Discrete element of rigidity in translation

y
X
WITH B
C


Characteristics of the elements
DISCRET:
with nodal masses
M_T_D_N

and matrices of rigidity
K_T_D_L

Limiting conditions:
in all the nodes
DDL_IMPO:
(TOUT:“YES” DY: 0. , DZ: 0. )
with the node end A

(GROUP_NO: WITH DX: 0. )

Names of the nodes:
Not A = N1
With = N1
Not B = N2
B = N2
Not C = N3
C = N3

3.2
Characteristics of the grid

A number of nodes: 3
A number of meshs and types: 2 SEG2

3.3 Functionalities
tested

Commands
DISCRETE AFFE_CARA_ELEM GROUP_MA “K_T_D_L'

GROUP_MA
AMOR_HYST
“A_T_D_L'
“MECHANICAL” AFFE_MODELE VERY “DIS_T'

GROUP_NO
“DIS_T'
AFFE_CHAR_MECA DDL_IMPO GROUP_NO


FORCE_NODALE
NOEUD
DEFI_LIST_REEL BEGINNING



INTERVALLE
RECU_FONCTION LIST_FREQ



Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A

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Titrate:
SDLD313 - Système mass-arises to 2 ddl (damping hysteretic)
Date:
23/06/03
Author (S):
O. Key NICOLAS
:
V2.01.313-A Page:
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3.4
Sizes tested and results

Parts real and imaginary of component DX of the displacement of the point C.

Frequency Reference
Aster %
Difference
0.00
7.1075E-03
7.1074964639321E-03
1.08E-04
­ 3.5360E-04
­ 3.5360678925035E-04
3.36870E+00 9.388216E-03
9.3882649899583E-03
5.31E-04
­ 7.31196E-04
­ 7.3120610001073E-04
6.48480E+00 ­ 5.0269E-03
­ 5.0349198344062E-03
0.012
­ 7.07103E-02
­ 7.0708581052416E-02
8.00060E+00 ­ 9.54931E-03
­ 9.5490053525137E-03
0.003
­ 2.2154E-03
­ 2.2153458282190E-03
1.18746E+01 ­ 4.23259E-05
­ 4.2266734408325E-05
0.016
­ 3.57193E-04
­ 3.5719325443817E-04
1.34747E+01 2.35524E-03
2.3552527130123E-03
5.34E-04
­ 5.01765E-04
­ 5.0176685846530E-04
1.55802E+01 ­ 1.6395374E-02
­ 1.6420641488151E-02
0.039
­ 6.871471E-02
­ 6.8704047854161E-02
2.10543E+01 ­ 1.88977E-03
­ 1.8897660707219E-03
2.08E-04
­ 5.53314E-06
­ 5.5328629109043E-06

3.5 Remarks

Contents of the file results:

Values of the displacement of component DX of the point C for all the frequencies of 0 with
2.10543E+01Hz by step of 3.3687.

Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A

Code_Aster ®
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Titrate:
SDLD313 - Système mass-arises to 2 ddl (damping hysteretic)
Date:
23/06/03
Author (S):
O. Key NICOLAS
:
V2.01.313-A Page:
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4 Modeling
B

4.1
Characteristics of modeling

Continuous element of beam type in traction

y
X
WITH B
C


Characteristics of the elements
DISCRET: masses
nodal
M_T_D_N
POUTRE:
matrices of rigidity
POU_D_T

Limiting conditions:
in all the nodes
DDL_IMPO:
(TOUT:“YES” DY: 0. , DZ: 0. )
with the node end A

(GROUP_NO: WITH DX: 0. )

Names of the nodes:
Not A = N1
With = N1
Not B = N2
B = N2
Not C = N3
C = N3

4.2
Characteristics of the grid

A number of nodes: 3
A number of meshs and types: 2 SEG2

4.3 Functionalities
tested

Commands



DISCRETE AFFE_CARA_ELEM
GROUP_MA “M_T_D_L'

DEFI_MATERIEU ELAS

AMOR_HYST

AFFE_MODELE ALL
“MECHANICAL” “POU_D_T'
GROUP_NO
“DIS_T'
AFFE_CHAR_MECA DDL_IMPO
GROUP_NO


FORCE_NODALE
NOEUD
CALC_MATR_ELEM RIGI_MECA_HYST


DEFI_LIST_REEL BEGINNING



INTERVALLE
RECU_FONCTION LIST_FREQ

Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A

Code_Aster ®
Version
7.0
Titrate:
SDLD313 - Système mass-arises to 2 ddl (damping hysteretic)
Date:
23/06/03
Author (S):
O. Key NICOLAS
:
V2.01.313-A Page:
7/10

4.4
Sizes tested and results

Parts real and imaginary of component DX of the displacement of the point C.

Frequency Reference
Aster %
Difference
0.00
7.1075E-03
7.1074964639321E-03
1.08E-04
­ 3.5360E-04
­ 3.5360678925035E-04
3.36870E+00 9.388216E-03
9.3882649899583E-03
5.31E-04
­ 7.31196E-04
­ 7.3120610001073E-04
6.48480E+00 ­ 5.0269E-03
­ 5.0349198344064E-03
0.012
­ 7.07103E-02
­ 7.0708581052416E-02
8.00060E+00 ­ 9.54931E-03
­ 9.5490053525137E-03
0.003
­ 2.2154E-03
­ 2.2153458282190E-03
1.18746E+01 ­ 4.23259E-05
­ 4.2266734408325E-05
0.016
­ 3.57193E-04
­ 3.5719325443817E-04
1.34747E+01 2.35524E-03
2.3552527130123E-03
5.34E-04
­ 5.01765E-04
­ 5.0176685846530E-04
1.55802E+01 ­ 1.6395374E-02
­ 1.6420641488152E-02
0.039
­ 6.871471E-02
­ 6.8704047854161E-02
2.10543E+01 ­ 1.88977E-03
­ 1.8897660707219E-03
2.08E-04
­ 5.53314E-06
­ 5.5328629109043E-06

4.5 Remarks

Contents of the file results:

Values of the displacement of component DX of the point C for all the frequencies of 0 with
2.10543E+01Hz by step of 3.3687.

Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A

Code_Aster ®
Version
7.0
Titrate:
SDLD313 - Système mass-arises to 2 ddl (damping hysteretic)
Date:
23/06/03
Author (S):
O. Key NICOLAS
:
V2.01.313-A Page:
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5 Modeling
C

5.1
Characteristics of modeling

Discrete element of rigidity in translation

y
X
WITH B
C


Characteristics of the elements
DISCRET:
with nodal masses
M_T_D_N

and matrices of rigidity
K_T_D_L

Limiting conditions:
in all the nodes
DDL_IMPO:
(TOUT:“YES” DY: 0. , DZ: 0. )
with the node end A

(GROUP_NO: WITH DX: 0. )

Names of the nodes:
Not A = N1
With = N1
Not B = N2
B = N2
Not C = N3
C = N3

5.2
Characteristics of the grid

A number of nodes: 3
A number of meshs and types: 2 SEG2

5.3 Functionalities
tested

Commands
DISCRETE AFFE_CARA_ELEM GROUP_MA “K_T_D_L'

GROUP_MA
AMOR_HYST
“A_T_D_L'
“MECHANICAL” AFFE_MODELE VERY “DIS_T'

GROUP_NO
“DIS_T'
AFFE_CHAR_MECA DDL_IMPO GROUP_NO


FORCE_NODALE
NOEUD
MODE_ITER_SIMULT SORENSEN
“COMPLEXE”

DEFI_LIST_REEL BEGINNING



INTERVALLE
RECU_FONCTION LIST_FREQ



Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A

Code_Aster ®
Version
7.0
Titrate:
SDLD313 - Système mass-arises to 2 ddl (damping hysteretic)
Date:
23/06/03
Author (S):
O. Key NICOLAS
:
V2.01.313-A Page:
9/10

5.4
Sizes tested and results

Eigen frequencies and reduced depreciation.

Eigen frequencies:

Sequence number
Reference
Aster %
Difference
1
6.4537 6.44568
­ 0.124
2 15.5806 1.55612
­ 0.124

Reduced depreciation:

Sequence number
Reference
Aster %
Difference
1
0.05 0.05
­ 1.39E-14
2 0.05 0.05
2.78E-14

Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A

Code_Aster ®
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7.0
Titrate:
SDLD313 - Système mass-arises to 2 ddl (damping hysteretic)
Date:
23/06/03
Author (S):
O. Key NICOLAS
:
V2.01.313-A Page:
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6
Summary of the results

The results obtained are excellent.

Handbook of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A

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