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SDLD21 - System mass-arises to 8 ddl with viscous shock absorber
Date:
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Author (S):
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Key:
V2.01.021-C
Page:
1/10
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
Organization (S):
EDF-R & D/AMA














Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
V2.01.021 document



SDLD21 - System mass-arises to 8 ddl
with viscous shock absorber




Summary:

This one-way problem consists in carrying out a harmonic analysis of a mechanical structure
composed of a whole of mass-springs with shock absorbers viscous 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 this problem, one tests the discrete elements in translation (mass, arises, shock absorber),
the definition of a force of specific excitation harmonic, the operator of calculation modal (
MODE_ITER_SIMULT
[U4.52.03]) into quadratic and the operator of harmonic calculation of answer (
DYNA_LINE_HARM
[U4.54.02]).
In addition, several operators of postprocessing are tested:
RECU_FONCTION
[U4.62.03],
TEST_FONCTION
[U4.72.02],
RECU_CHAMP
[U4.62.01].

Results obtained (field of displacement, speed and acceleration for various frequencies of excitation)
are in concord with the results of guide VPCS.

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Code_Aster
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Titrate:
SDLD21 - System mass-arises to 8 ddl with viscous shock absorber
Date:
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Author (S):
O. NICOLAS
Key:
V2.01.021-C
Page:
2/10
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
1
Problem of reference
1.1 Geometry
X, U
P1
U1
With
C
m
C
K
K
m
P2
U2
m
C
K
P3
U3
m
C
K
P8
B
U8

Specific masses:
m
P1
= m
P2
= m
P3
= ...... = m
P8
= m
Stiffnesses of connection:
K
AP1
= K
P1P2
= K
P2P3
= ...... = K
P8B
= K
Viscous damping:
C
AP1
= C
P1P2
= C
P2P3
= ...... = C
P8B
= C

1.2
Material properties
Comes out from linear elastic translation
K =
10
5
NR/m
Specific mass
m =
10 kg
One-way viscous damping
C =
50 NR/(m/s)

1.3
Boundary conditions and loadings
Boundary conditions:
Embedded points A and b: (U = 0).
Loading: Force concentrated sinusoidal of variable frequency at the point P
4
Not P
4
F
X
4
=
F
0
sin
T
=
2
F
5 Hz
F
40 Hz
F
0
=
constant
=
1 NR
Other points P
I
F
X
I
=
0

1.4 Conditions
initial
Without object for the study of the permanent harmonic mode.
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Titrate:
SDLD21 - System mass-arises to 8 ddl with viscous shock absorber
Date:
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Key:
V2.01.021-C
Page:
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Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The system of differential equations of the second command coupled is form:














-
-
-
-
-
-
+
=














-
-
-
-
-
-
=










=
=
+
+
2
1
1
.
.
.
.
.
.
2
1
1
2
1
1
2
5
10
2
1
1
.
.
.
.
.
.
2
1
1
2
1
1
2
50
10
10
.
10
10
with
K
C
M
F
U
K
U
C
U
M
&
&&
The solution
with a harmonic excitation
F
=
F
0
E
J
T
J
2
= -
1
(
)
is form
U
=
U
0
E
J
T
, it
who leads to:
K
-
M
2
+
J
C
(
)
U
0
=
F
0
This system can be solved for all
, either directly, or by using the modal transformation with
to leave the real clean modes obtained by the associated conservative system
K
-
M
2
(
)
=
0
.
It admits N solutions clean (8 in this case)
i2
and associated vectors
I
gathered in
spectral matrix
=
i2
[]
and the modal matrix
=
I
[]
.
The modal transformation consists in writing:
U
0
=
Q
what leads to:
-
2
I
+
J
[
]
Q
=
T
F
0
I
is the identity,
here
is diagonal
=
II
[]
because damping is proportional
C
=
K
(
)
.
The answer is written:
U
0
=
I T
I
i2
-
2
+
J
II
I
=
1
N
F
0
One obtains the exact solution by taking all the clean modes.
One deduces some:
&
&&
U
J
U
U
U
0
0
0
2
0
=
= -
and
2.2
Results of reference
Displacement according to X of the point P
4
for certain frequencies.
2.3
Uncertainty on the solution
Semi-analytical solution.
2.4 Reference
bibliographical
[1]
J. PIRANDA: Note of use of the software of modal analysis MODAN - Version 0.2 (1990).
Laboratory of Mechanics Applied - University of Frank County - Besancon (France).
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Code_Aster
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Version
7.0
Titrate:
SDLD21 - System mass-arises to 8 ddl with viscous shock absorber
Date:
23/06/03
Author (S):
O. NICOLAS
Key:
V2.01.021-C
Page:
4/10
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
3 Modeling
With
3.1
Characteristics of modeling
Discrete element of rigidity in translation
With
B
P
1
y
X
P
2
P
3
P
4
P
5
P
6
P
7
P
8
Characteristics of the elements
DISCRETE
:
with nodal masses
M_T_D_N
and matrices of rigidity
K_T_D_L
and matrices of damping
A_T_D_L
Limiting conditions:
in all the nodes
DDL_IMPO:
(ALL:“YES” DY: 0. , DZ: 0. )
with the nodes ends
(GROUP_NO: AB DX: 0. )
Names of the nodes:
Not A = N1
P
1
= N2
Not B = N10
P
2
= N3
.............
P
8
= N9

3.2
Characteristics of the mesh
A number of nodes: 10
A number of meshs and types: 9 SEG2

3.3 Functionalities
tested
Controls
DISCRETE AFFE_CARA_ELEM GROUP_MA
GROUP_MA
GROUP_MA
“K_T_D_L'
“A_T_D_L'
“M_T_D_N'
AFFE_MODELE ALL
GROUP_NO
“MECHANICAL” “DIS_T'
“DIS_T'
AFFE_CHAR_MECA DDL_IMPO
FORCE_NODALE
GROUP_NO
NODE
DYNA_LINE_HARM MATR_AMOR
DEFI_LIST_REEL BEGINNING
INTERVAL
RECU_FONCTION LIST_FREQ
TEST_FONCTION
TEST_RESU
IMPR_RESU
LIRE_RESU
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Titrate:
SDLD21 - System mass-arises to 8 ddl with viscous shock absorber
Date:
23/06/03
Author (S):
O. NICOLAS
Key:
V2.01.021-C
Page:
5/10
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
3.4
Results of modeling A
Parts real and imaginary of the component
DX
displacement of the point P
4
.
Frequency Reference
Aster %
Difference
5.00
1.0237
E4
­ 8.5187
E6
1.02369
E4
­ 8.51874
E6
0.0004
5.50
4.5066
E4
­ 7.7914
E4
4.50662
E4
­ 7.79143
E4
0.0004
6.00
­ 9.4101
E5
­ 1.0585
E5
­ 9.41096
E5
­ 1.05851
E5
0.0002
10.00
8.4143
E7
­ 1.0335
E6
8.41427
E7
­ 1.03346
E6
0.0024
15.00
1.2656
E5
­ 5.6652
E6
1.26556
E5
­ 5.66517
E6
0.0032
20.00
2.9784
E6
­ 6.6970
E6
2.97844
E6
­ 6.69700
E6
0.0003
25.00
­ 1.2536
E6
­ 5.2703
E6
­ 1.25362
E6
­ 5.27033
E6
0.0008
30.00
­ 2.0904
E6
­ 5.4821
E6
­ 2.09042
E6
­ 5.48215
E6
0.0009
35.00
­ 4.5447
E6
­ 1.1190
E6
­ 4.54473
E6
­ 1.11903
E6
0.0011
39.50
­ 2.6895
E6
­ 3.0505
E7
­ 2.68949
E6
­ 3.05048
E7
0.0003

Parts real and imaginary of the component
DX
speed of the point P
4
.
Frequency Reference
Aster %
Difference
5.00
2.6762
E4
3.2160
E3
2.6762
E4
3.21603
E3
0.000
5.50
2.6925
E2
1.5574
E2
2.69252
E2
1.55737
E2
0.001
6.00
3.9904
E4
­ 3.5475
E3
3.99052
E4
­ 3.54752
E3
0.000
10.00
6.4937
E5
5.2869
E5
6.49347
E5
5.28685
E5
0.002
15.00
5.3393
E4
1.1928
E3
5.33929
E4
1.19276
E3
0.003
20.00
8.4157
E4
3.7428
E4
8.41570
E4
3.74282
E4
0.001
25.00
8.2786
E4
­ 1.9691
E4
8.27862
E4
­ 1.96919
E4
0.001
30.00
1.0333
E3
­ 3.9403
E4
1.03334
E3
­ 3.94035
E4
0.001
35.00
2.4608
E4
­ 9.9943
E4
2.46089
E4
­ 9.99439
E4
0.001
39.50
7.5709
E5
­ 6.6749
E4
7.57086
E5
­ 6.67494
E4
0.000
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Code_Aster
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Titrate:
SDLD21 - System mass-arises to 8 ddl with viscous shock absorber
Date:
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Author (S):
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Key:
V2.01.021-C
Page:
6/10
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
Parts real and imaginary of the component
DX
acceleration of the point P
4
.
Frequency Reference
Aster %
Difference
5.00
­ 1.0103
E1
8.4076
E3
­ 1.01035
E1
8.40766
E3
0.000
5.50
­ 5.3819
E1
9.3047
E1
­ 5.38190
E1
9.30470
E1
0.000
6.00
1.3374
E1
1.5044
E2
1.33738
E1
1.50439
E2
0.000
10.00
­ 3.3218
E3
4.0801
E3
­ 3.32182
E3
4.07996
E3
0.002
15.00
­ 1.1242
E1
5.0322
E2
­ 1.12415
E1
5.03217
E2
0.003
20.00
­ 4.7033
E2
1.0575
E1
­ 4.70337
E2
1.05755
E1
0.001
25.00
3.0931
E2
1.3004
E1
3.09320
E2
1.30040
E1
0.001
30.00
7.4273
E2
1.9478
E1
7.42739
E2
1.94780
E1
0.001
35.00
2.1979
E1
5.4116
E2
2.19788
E1
5.41178
E2
0.001
39.50
1.6566
E1
1.8789
E2
1.65662
E1
1.87898
E2
0.000

3.5 Remarks
Contents of the file results:
Values of the displacement of the component
DX
point P
4
for all the frequencies from 5 to 40 Hz
by pitch of 0.5 (Case initial test of VPCS).
Values the speed and the acceleration of the component
DX
point P
4
for some
frequencies of vibration.
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Titrate:
SDLD21 - System mass-arises to 8 ddl with viscous shock absorber
Date:
23/06/03
Author (S):
O. NICOLAS
Key:
V2.01.021-C
Page:
7/10
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
4 Modeling
B
4.1
Characteristics of modeling
Discrete element of rigidity in translation
With
B
P
1
y
X
P
2
P
3
P
4
P
5
P
6
P
7
P
8
Characteristics of the elements
DISCRETE
:
with nodal masses
M_T_D_N
and matrices of rigidity
K_T_D_L
and matrices of damping
A_T_D_L
Limiting conditions:
in all the nodes
DDL_IMPO:
(ALL:“YES” DY: 0. , DZ: 0. )
with the nodes ends
(GROUP_NO: AB DX: 0. )
Names of the nodes:
Not A = N1
P
1
= N2
Not B = N10
P
2
= N3
.............
P
8
= N9

4.2
Characteristics of the mesh
A number of nodes: 10
A number of meshs and types: 9 SEG2

4.3 Functionalities
tested
Controls
DISCRETE AFFE_CARA_ELEM GROUP_MA
GROUP_MA
GROUP_MA
“K_T_D_L'
“A_T_D_L'
“M_T_D_N'
AFFE_MODELE ALL
GROUP_NO
“MECHANICAL” “DIS_T'
“DIS_T'
AFFE_CHAR_MECA DDL_IMPO
FORCE_NODALE
GROUP_NO
NODE
MODE_ITER_SIMULT
MACRO_PROJ_BASE
DYNA_LINE_HARM MATR_AMOR
REST_BASE_PHY
DEFI_LIST_REEL BEGINNING
INTERVAL
RECU_FONCTION LIST_FREQ
TEST_FONCTION
TEST_RESU
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Titrate:
SDLD21 - System mass-arises to 8 ddl with viscous shock absorber
Date:
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Author (S):
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Key:
V2.01.021-C
Page:
8/10
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
4.4
Results of modeling B
Parts real and imaginary of the component
DX
displacement of the point P
4
.
Frequency Reference
Aster %
Difference
5.00
1.0237
E4
­ 8.5187
E6
1.02369
E4
­ 8.51874
E6
0.0004
5.50
4.5066
E4
­ 7.7914
E4
4.50662
E4
­ 7.79143
E4
0.0004
6.00
­ 9.4101
E5
­ 1.0585
E5
­ 9.41096
E5
­ 1.05851
E5
0.0002
10.00
8.4143
E7
­ 1.0335
E6
8.41427
E7
­ 1.03346
E6
0.0024
15.00
1.2656
E5
­ 5.6652
E6
1.26556
E5
­ 5.66517
E6
0.0032
20.00
2.9784
E6
­ 6.6970
E6
2.97844
E6
­ 6.69700
E6
0.0003
25.00
­ 1.2536
E6
­ 5.2703
E6
­ 1.25362
E6
­ 5.27033
E6
0.0008
30.00
­ 2.0904
E6
­ 5.4821
E6
­ 2.09042
E6
­ 5.48215
E6
0.0009
35.00
­ 4.5447
E6
­ 1.1190
E6
­ 4.54473
E6
­ 1.11903
E6
0.0011
39.50
­ 2.6895
E6
­ 3.0505
E7
­ 2.68949
E6
­ 3.05048
E7
0.0003

4.5 Remarks
Contents of the file results:
Values of the displacement of the component
DX
point P
4
for all the frequencies from 5 to 40 Hz
by pitch of 0.5 (Case initial test of VPCS).
Values the speed and the acceleration of the component
DX
point P
4
for some
frequencies of vibration.
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Code_Aster
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Version
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Titrate:
SDLD21 - System mass-arises to 8 ddl with viscous shock absorber
Date:
23/06/03
Author (S):
O. NICOLAS
Key:
V2.01.021-C
Page:
9/10
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
5 Modeling
C
5.1
Characteristics of modeling
Discrete element of rigidity in translation
With
B
P
1
y
X
P
2
P
3
P
4
P
5
P
6
P
7
P
8
Characteristics of the elements
DISCRETE
:
with nodal masses
M_T_D_N
and matrices of rigidity
K_T_D_L
and matrices of damping
A_T_D_L
Limiting conditions:
in all the nodes
DDL_IMPO:
(ALL:“YES” DY: 0. , DZ: 0. )
with the nodes ends
(GROUP_NO: AB DX: 0. )
Names of the nodes:
Not A = N1
P
1
= N2
Not B = N10
P
2
= N3
.............
P
8
= N9

5.2
Characteristics of the mesh
A number of nodes: 10
A number of meshs and types: 9 SEG2

5.3 Functionalities
tested
Controls
DISCRETE AFFE_CARA_ELEM GROUP_MA
GROUP_MA
GROUP_MA
“K_T_D_L'
“A_T_D_L'
“M_T_D_N'
AFFE_MODELE ALL
GROUP_NO
“MECHANICAL” “DIS_T'
“DIS_T'
AFFE_CHAR_MECA DDL_IMPO
FORCE_NODALE
GROUP_NO
NODE
MODE_ITER_SIMULT MATR_AMOR
DYNA_LINE_HARM AMOR_REDUIT
DEFI_LIST_REEL BEGINNING
INTERVAL
RECU_FONCTION LIST_FREQ
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Titrate:
SDLD21 - System mass-arises to 8 ddl with viscous shock absorber
Date:
23/06/03
Author (S):
O. NICOLAS
Key:
V2.01.021-C
Page:
10/10
Manual of Validation
V2.01 booklet: Linear dynamics of the discrete systems
HT-66/03/008/A
5.4
Results of modeling C
Eigen frequencies of the structure for the sequence numbers from 1 to 5.
Sequence number
Reference
Aster %
Difference
1
5.5271 5.5271848238694
0.002
2
10.8868 1.088524727521
­ 0.014
3
15.9155 1.5910519939851
­ 0.031
4
20.4606 20.449995091940
­ 0.052
5
24.384 24.366059022201
­ 0.074

Damping reduce structure for the sequence numbers from 1 to 5.
Sequence number
Reference
Aster %
Difference
1
0.00868241 8.6824088833463D-03
­ 1.29E-05
2
0.017101 1.7101007166284D-02
4.19E-05
3
0.025 2.5000000000002D-02
9.19E-12
4
0.0321394 3.2139380484326D-02
­ 6.07E-05
5
0.0383022 3.8302222155950D-02
5.78E-05


6
Summary of the results
The results obtained are excellent, which is normal for a direct integration.